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Fairness Provision in the IEEE 802.11e Infrastructure Basic Service Set † Feyza Keceli, Inanc Inan, and Ender Ayanoglu Center for Pervasive Communications and Computing Department of Electrical Engineering and Computer Science The Henry Samueli School of Engineering University of California, Irvine, 92697-2625 Email: {fkeceli, iinan, ayanoglu}@uci.edu Abstract Most of the deployed IEEE 802.11e Wireless Local Area Networks (WLANs) use infrastructure Basic Service Set (BSS) in which an Access Point (AP) serves as a gateway between wired and wireless domains. We present the unfairness problem between the uplink and the downlink flows of any Access Category (AC) in the 802.11e Enhanced Distributed Channel Access (EDCA) when the default settings of the EDCA parameters are used. We propose a simple analytical model to calculate the EDCA parameter settings that achieve weighted fair resource allocation for all uplink and downlink flows. We also propose a simple model-assisted measurement-based dynamic EDCA parameter adaptation algorithm. Moreover, our dynamic solution addresses the differences in the transport layer and the Medium Access Control (MAC) layer interactions of User Datagram Protocol (UDP) and Transmission Control Protocol (TCP). We show that proposed Contention Window (CW) and Transmit Opportunity (TXOP) limit adaptation at the AP provides fair UDP and TCP access between uplink and downlink flows of the same AC while preserving prioritization among ACs. I. INTRODUCTION IEEE 802.11 Wireless Local Area Network (WLAN) is built around a Basic Service Set (BSS) [1]. While a number of stations may gather to form an independent BSS with no connectivity to the wired network, the common deployment is the infrastructure BSS which includes an Access Point (AP). The AP provides the connection to the wired network. The IEEE 802.11 standard [1] defines Distributed Coordination Function (DCF) as a contention based Medium Access Control (MAC) mechanism. The 802.11e standard [2] updates the MAC layer of the † This work is supported by the Center for Pervasive Communications and Computing, and by National Science Foundation under Grant No. 0434928. Any opinions, findings, and conclusions or recommendations expressed in this material are those of authors and do not necessarily reflect the view of the National Science Foundation. http://arxiv.org/abs/0704.1842v2 former 802.11 standard for Quality-of-Service (QoS) provision. In particular, the Enhanced Distributed Channel Access (EDCA) function of 802.11e is an enhancement of the DCF. The EDCA scheme (similarly to DCF) uses Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) and slotted Binary Exponential Backoff (BEB) mechanism as the basic access method. The major enhancement to support QoS is that EDCA differentiates packets using different priorities and maps them to specific Access Categories (ACs) that use separate queues at a station. Each ACi within a station (0 ≤ i ≤ 3) contends for the channel independently of the others. Levels of services are provided through different assignments of the AC-specific EDCA parameters; Contention Window (CW) sizes, Arbitration Interframe Space (AIFS) values, and Transmit Opportunity (TXOP) limits. The DCF and the EDCA are defined such that each station in a BSS uses the same contention parameter set. Therefore, fair access can be achieved in the MAC layer for all the contending stations in terms of the average number of granted transmissions, over a sufficiently long interval. However, this does not translate into achieving fair share of bandwidth between uplink and downlink flows in the 802.11e infrastructure BSS. An AC of the AP which serves all downlink flows has the same access priority with the same AC of the stations that serve uplink flows. Therefore, an approximately equal number of accesses that an uplink AC may get is shared among all downlink flows in the same AC of the AP. This leads to the uplink/downlink unfairness problem in the WLAN where each individual downlink flow gets comparably lower bandwidth than each individual uplink flow gets at high load. This phenomenon will be described further in Section II-A. We deal with weighted fair channel access between the uplink and the downlink flows of the same AC in the IEEE 802.11e infrastructure BSS. Using a simple analytical approach, we calculate the EDCA parameter settings that achieve a given utilization ratio between the uplink and the downlink transmissions. Comparing with simulation results, we noticed that sticking only with analytical results that are based on ideal condition assumptions may result in inaccuracies in a real WLAN scenario. Therefore, we also propose a simple model-assisted measurement-based dynamic EDCA parameter adaptation algorithm that provides weighted fair resource allocation in an arbitrary scenario. Most of the data traffic in the Internet is carried by Transmission Control Protocol (TCP), while most of the real-time applications use User Datagram Protocol (UDP). UDP employs one-way unreliable communication. On the other hand, TCP defines reliable bi-directional communication where the forward link data rate depends on the rate of received Acknowledgment (ACK) packets in the backward link. Another key contribution of this study is that our solution considers the effects of this difference on the design of the weighted fairness support algorithm. II. BACKGROUND In this section, we first present the uplink/downlink unfairness problem in the IEEE 802.l1(e) WLAN at high traffic load. Next, we provide a brief review of the literature on this subject. A. Problem Definition In the 802.11e WLAN, at high load, a bandwidth asymmetry exists between contending upload and download flows which use the same AC. This is due to the fact that the MAC layer contention parameters are all equal for the AP and the stations. If n stations and an AP are always contending for the access to the wireless channel using the same AC, each host ends up having approximately 1/(n+ 1) share of the total transmissions over a long time interval. This results in n/(n + 1) of the transmissions to be in the uplink, while only 1/(n + 1) of the transmissions belonging to the downlink flows. This is the WLAN uplink/downlink unfairness problem stated previously. The uneven bandwidth share results in downlink flows experiencing significantly lower throughput and larger delay. The congestion at the AP may result in considerable packet loss depending on the size of interface buffers. The results may even be more catastrophic in the case of TCP flows. The TCP receiver returns TCP ACK packets to the TCP transmitter in order to confirm the successful reception of data packets. In the case of multiple uplink and downlink flows in the WLAN, returning TCP ACKs of upstream TCP data are queued at the AP together with the downstream TCP. When the bandwidth asymmetry in the forward and reverse path builds up the queue in the AP, the dropped packets impair the TCP flow and congestion control mechanisms which assume equal transmission rate both in the forward and reverse path [3]. TCP’s timeout mechanism initiates a retransmission of a data packet if it has not been acknowledged during a timeout duration. However, any received TCP ACK can cumulatively acknowledge all the data packets sent before the data packet for which the ACK is intended to. When the packet loss is severe in the AP buffer, downstream flows will experience frequent timeouts resulting in significantly low throughput. On the other hand, due to the cumulative property of TCP ACK mechanism, upstream flows with high congestion windows will not experience such frequent timeouts. In this case, it is a low probability that many consecutive TCP ACK losses occur for the same flow. Conversely, flows with low congestion window (fewer packets currently on flight) may experience frequent timeouts and decrease their congestion windows even more. Therefore, a number of upstream flows may starve in terms of throughput while others enjoy a high throughput. This results in unfairness between the TCP upstream flows on top of the unfairness between the uplink and the downlink. Fig. 3 shows the average throughput of individual flows for a scenario of 10 uplink UDP, 10 downlink UDP, 10 uplink TCP and 10 downlink TCP connections in an ns-2 simulation [4],[5]. Each connection is initiated by a separate station. All stations employ 54 Mbps data rate at the physical layer. The packet size is 1500 bytes for all flows. UDP flows are mapped to an AC with CWmin = 31 and CWmax = 511. TCP flows use an AC with CWmin = 63 and CWmax = 1023. For both ACs, AIFSN values are set to 2 and TXOP limits are 0. Other simulation parameters are as stated in Section IV. The results illustrate the throughput unfairness of the uplink and the downlink flows. The throughput unfairness between uplink TCP connections is also significant. Moreover, data packet losses at the AP buffer have almost shut down all downlink TCP connections. B. Related Work There are two groups of studies in the literature related to this work. The first group works within the constraints of the default 802.11 contention parameters. In [6], the effect of the AP buffer size in the wireless channel bandwidth allocation for TCP is studied. The proposed solution of [6] is to manipulate advertised receiver windows of the TCP packets at the AP. Uplink/downlink fairness problem is studied in [7] using per-flow queueing. A simplified approach is proposed in [8] where two separate queues for TCP data and ACKs are used. In our previous work, we proposed using congestion control and filtering techniques at the MAC layer to solve the TCP uplink unfairness problem [9]. Two queue management strategies are proposed in [10] to improve TCP fairness. A rate-limiter approach is used in [11] which requires available instantaneous WLAN bandwidth estimation in both directions. The second group proposes changes at the MAC layer access parameters to achieve improved fairness. Our work also falls into this category. AIFS and CW differentiation is proposed for improved fairness and channel utilization in [12]. A simulation-based analysis is carried out for a specific scenario consisting of TCP and audio flows both in the uplink and the downlink. An experimental study is carried out in [13] to decide on CW and TXOP values of the AP and the stations for a scenario with TCP uplink and downlink flows. Both solutions propose that individual uplink and downlink streams use separate ACs. No guidelines are provided on how to decide on the EDCA parameters that achieve fair resource allocation for an arbitrary scenario. Also, the interaction of TCP flow and congestion control mechanisms with the MAC is not addressed. In [14], it is proposed that the AP accesses the channel in Point Interframe Space (PIFS) completion without any backoff when the interface queue size goes over a threshold. The use of TXOP is evaluated in [15] for temporal fairness provisioning among stations employing different data rates. Achieving weighted fairness between uplink and downlink in DCF is studied through mean backoff distribution adjustment in [16]. A mechanism that dynamically tunes CW and TXOP values in order to prevent delay asymmetry of realtime UDP flows is proposed in [17]. An adaptive priority control mechanism is employed in [18] to balance the uplink and downlink delay of VoIP traffic. III. WEIGHTED FAIR ACCESS BETWEEN UPLINK AND DOWNLINK FLOWS In this section, we first describe the simple analytical model we propose in order to find the AIFS, CWmin, and TXOP settings of the ACs that provide weighted fairness between uplink and downlink flows. Next, we propose a parameter adaptation algorithm which dynamically updates the analytically calculated CW and TXOP values of the AP regarding simple network measurements. As we will describe in Section III-D, our dynamic solution also addresses the effects of the slow-start phase of TCP. Every beacon interval, the AP announces the values of the AC-specific EDCA parameters to the stations. The stations overwrite their EDCA parameter settings with the new values if any change is detected. Due to the specific design of the EDCA Parameter Set element in the beacon packet, the stations can only employ CW values that are integer powers of 2, i.e., the AP encodes the corresponding 4-bit fields of CWmin and CWmax in an exponent form. A key point which the studies in the literature have missed is that the CW settings of the ACs at the AP are not restricted to the powers of 2. The ACs at the AP may use any value and this value does not have to be equal to what is announced via beacons. A. Analytical Model Fair access between uplink and downlink flows using the same AC can be provided by assigning different EDCA parameters for the AP and the stations. This results in two Traffic Classes (TCs) using the same AC. While uplink flows constitute the first TC, downlink flows constitute the second TC. In the analysis, we will treat the case with one AC (thus 2 TCs), since we address the weighted fairness problem between the uplink and downlink flows that are mapped to the same AC. Moreover, we only formulate the situation when there is only one TC per station, therefore no internal collisions can occur. Note that, this does not cause any loss of generality, since the analysis can be extended for larger number of ACs or TCs as in [19], and larger number of ACs per station as in [20],[21]. Our analysis considers the fact that the difference in AIFS creates the so-called contention zones as shown in Fig. 1 [19],[22],[23],[21]. First, we calculate the average collision probability of each TC according to the long term occupancy of AIFS and backoff slots in saturation. The average collision probability of a TC is a function of transmission probabilities of all TCs. Next, we formulate the average transmission probability for each TC, which is a function of average collision probability of the same TC. This results in a set of nonlinear equations which can be solved numerically. We define pci,x as the probability that TCi experiences a collision given that it has observed the medium idle for AIFSx and transmits in the current slot (note AIFSx ≥ AIFSi should hold). For notational sim- plicity, let uplink flows belong to TC0 and downlink flows belong to TC1. Let di = AIFSNi−AIFSNmin where AIFSNmin = min(AIFSN0, AIFSN1) and AIFSi = SIFS + AIFSNi · Tslot. Following the slot homogeneity assumption of [24], assume that each TCi transmits with constant probability, τi. Also, let the total number of TCi in the BSS be Ni (note that N1 = 1). Then, pci,x = 1− i′:di′≤dx (1− τi′) (1− τi) . (1) We use the Markov chain shown in Fig. 2 to find the long term occupancy of contention zones. Each state represents the nth backoff slot after completion of the AIFSmin idle interval following a transmission period. The Markov analysis uses the fact that a backoff slot is reached if no transmission occurs in the previous slot. Moreover, the number of states is limited by the maximum idle time between two successive transmissions which is Wmin = min(CWi,max) for a saturated scenario. The probability that at least one transmission occurs in a backoff slot in contention zone x is ptrx = 1− i′:di′≤dx (1− τi′) Ni′ . (2) The long term occupancy of the backoff slots b′n in Fig. 2 can be obtained from the steady-state solution. Then, the average collision probability pci is found by weighing zone specific collision probabilities pci,x according to the long term occupancy of contention zones (thus backoff slots) pci = ∑Wmin n=di+1 pci,xb ∑Wmin n=di+1 where x = max y | dy = max (dz | dz ≤ n) which shows x is assigned the highest index value within a set of TCs that have AIFSN smaller than equal to n+ AIFSNmin. Given pci , we can calculate the expected number of backoff slots Ei[tbo] that TCi waits before attempting a transmission. Let Wi,k = 2 min(k,mi)(CWi,min + 1)− 1 be the CW size of TCi at backoff stage k where CWi,max = 2 mi(CWi,min + 1)− 1, 0 ≤ mi < ri. Note that, when the retry limit ri is reached, any packet is discarded. Ei[tbo] = (prici) pk−1ci (1− pci) 1− prici pk−1ci (1− pci) . (4) Then as also shown in [23], the transmission probability of TCi can be calculated as Ei[tbo] + 1 . (5) The nonlinear system of equations (1)-(5) can be solved numerically to calculate average collision and transmission probabilities of each TCi for an arbitrary setting of EDCA parameters. We provide the validation of the proposed analytical model in [19]. B. Weighted Fairness between Uplink and Downlink Flows Let γi be the probability that the transmitted packet belongs to an arbitrary user from TCi given that the transmission is successful. Also, let psi,n be the probability that a successfully transmitted packet at backoff slot n belongs to ACi. Then, n=di+1 psi,n psj,n , (6) psi,n = (1− τi) i′:di′≤n−1 (1− τi′) Ni′ , if n ≥ di + 1 0, if n < di + 1. Let U denote the utilization ratio between the downlink and the uplink transmissions of an AC. Let NTXOP,i denote the maximum number of packets that can fit in one TXOP of TCi. Then, for our running example with one AC, γ1 ·NTXOP,1 γ0 ·NTXOP,0 . (8) 1) Implementation of the Numerical Solution: Without loss of generality, the EDCA parameters of the stations, AIFS0, CWmin,0, and NTXOP,0, are fixed at predetermined values . Then, the EDCA parameters of the TC at the AP, AIFS1, CWmin,1, and NTXOP,1, that achieve a required utilization ratio Ur can be calculated numerically as follows. 1) We assume AIFS differentiation is only used for the prioritization between the ACs not the TCs (thus AIFS0 = AIFS1). 2) When AIFS0 = AIFS1, after some algebra on (6)-(8), τ1 · (1− τ0) ·NTXOP,1 τ0 · (1− τ1) ·NTXOP,0 . (9) Therefore, τ1 can be written in terms of τ0, NTXOP,0, NTXOP,1, and Ur. A numerical solution for τ0 and τ1 for given Ur and a fixed value of NTXOP,1 (initially, NTXOP,1 = 1) is obtained using (1)-(5). 3) CWmin,1 can be calculated as follows (the formula below is obtained using (8) and (9) in [25, Section IV-A]), CWmin,i = 2− τi (1− prici)(1− 2pci)(1− pci) (1− pci) 2(1− (2pci) mi+1) + 2mip ci (1− 2pci)(1− pci)(1− p ri−mi−1 4) A simple controller block checks whether the prioritization among ACs are maintained or not for the new configuration. This block ensures that CWmin of a low priority AC (at the AP or a station) is not smaller than CWmin of a higher priority AC. Therefore, if analytically calculated CWmin,1 value does not satisfy the controller block requirements, NTXOP,1 is doubled and the algorithm returns to step 3. The larger NTXOP,1 is, the larger CWmin,1 will be. 5) If the calculated CWmin,1 is not an integer, it is rounded to the closest integer value. A few remarks on the implementation are as follows. • A numerical solution also exists when AIFS0 and AIFS1 are not equal, but the implementation differs since (9) does not hold. In such a case, AIFS1 is also assigned an initial value as NTXOP,1 and the nonlinear system of equations (1)-(8) is solved numerically. According to the controller block requirements on CWmin,1, the procedure may be repeated for updated values of AIFS1 and NTXOP,1. • As previously mentioned, our formulation is valid for the situation when there is only one TC per station (including the AP). As an approximation, we assume that (1)-(8) still holds when there are multiple TCs at the AP. Indeed as only a few collisions are avoided when the internal collision procedure is run at the AP [26], the solution of (1)-(8) will be very close to an extension that exactly formulates the virtual collisions at the AP. In this case, if the AIFS values of TCs within an AC remains equal, it can be shown that (9) still holds for the TCs of the same AC. Therefore, we use the implementation procedure previously stated for scenarios when larger number of ACs exist as long as there is one AC (or TC) per station and multiple TCs at the AP. 2) Proposed BEB Algorithm for non-integer CW values: As specified in [2], the initial value of CW is set to the AC-specific CWmin. At each unsuccessful transmission, the value of CW is doubled until the maximum AC-specific CWmax limit is reached. The value of CW is reset to the AC-specific CWmin if the transmission is successful, or the retry limit is reached thus the packet is dropped. The proposed analytical calculation for weighted fairness may decide a non-integer value of CWmin,1 thus W1,k, k < r1. The simplest approach is rounding to the closest integer and employing the rounded value in the BEB. Instead, we also propose the AP to choose integer W ′1,k values from a probability distribution that satis- fies E[W ′1,k] = W1,k. For example, it is straightforward to show a simple discrete probability distribution such as Pr(W ′1,k = ⌊W1,k⌋) = ⌈W1,k⌉ −W1,k and Pr(W 1,k = ⌈W1,k⌉) = W1,k − ⌊W1,k⌋ holds. According to the proposed algorithm, the EDCA function at the AP decides on the interval (0,W ′1,k) to select the backoff value regarding the given simple discrete probability distribution. Fig. 4 shows the downlink/uplink access ratio for increasing number of uplink and downlink flows. We assume equal AIFSN = 2 for all the stations and the AP, and analytically calculate CWmin,1 that achieves downlink/uplink access ratio of Ur = 1 when CWmin,0 = 127, NTXOP,0 = 1, and NTXOP,1 is varied from 1 to 4. The performance of rounding the analytically calculated CW values is compared with the performance of the proposed BEB algorithm that uses the stated discrete probability distribution function. As the results imply, the proposed BEB algorithm maintains perfect weighted fairness while rounding the analytically calculated value may result in slight inaccuracies in terms of utilization ratio. As the number of uplink stations increase, CWmin,1 that achieves Ur = 1 decreases. As Fig. 4 shows the effect of rounding is much more noticeable when CWmin,1 is small. The effect of rounding becomes negligible as NTXOP,1 (thus CWmin,1) is increased. C. Dynamic Parameter Adaptation The IEEE 802.11 infrastructure BSS exhibits some non-ideal conditions which most of the analytical models ignore to maintain simplicity. For example, • Accurate information on the instantaneous number of active flows may not always be available to the AP [27]. • If a station and the AP collide, the station’s transmission results in failure since the destination (the AP) is not in listen mode. However, there is some probability that the transmission of the AP results in success as a consequence of the capture effect depending on the spatial distribution and the power levels of the stations [28]. Such non-ideal conditions make finding the optimum EDCA setting analytically hard for any scenario. This also limits the use of proposed BEB algorithm for non-integer CW values. We propose a simple model-assisted measurement-based dynamic algorithm to adapt the analytically calculated CWmin values for such scenarios. The AP carries out the dynamic adaptation for each AC every β beacon intervals which is called an adaptation interval in the sequel. If it is detected as a new flow starting transmission or as an old flow becoming inactive at the last adaptation interval, the algorithm decides on new good EDCA parameters using the proposed analytical model which results in weighted fair resource allocation for the estimated number of uplink and downlink flows in ideal conditions. Otherwise, fine tuning on the CW and the TXOP values of the AC at the AP is carried out to make measured U as close as to Ur. We use a simple algorithm to estimate the number of active flows. More advanced approaches [27] can also be used. The AP counts the number of unique source and destination MAC addresses observed from incoming frames to estimate the number of uplink and downlink flows respectively. Let nu and nd denote the number of uplink and downlink flows labeled as active. If the AP receives a packet with the corresponding MAC address not on its list, it adds the new MAC address to the list and increments nu or nd. If the AP does not receive any packet with the corresponding MAC address during the last adaptation interval, it deletes the MAC address from the list and decrements nu or nd. Then, we define the required utilization ratio as . (11) If Ur has been changed during the last adaptation interval, EDCA parameters are analytically calculated for U = Ur and the fine tuning phase is skipped. Otherwise, solely fine tuning on CWmin is performed as follows. Every adaptation interval, the AP measures the number of successful uplink and downlink transmissions, ntu and ntd respectively where ntd/ntu is the measured U of the last adaptation interval. < (1 − α) · Ur, then CWmin,1 is decremented (where 0 ≤ α ≤ 1). Similarly, if > (1 + α) · Ur, then CWmin,1 is incremented. Otherwise, no action is taken. Note that using steps equal in value to 1 in the CWmin adaptation is sufficient since the analytical calculation will provide a good initial guess. D. TCP-MAC Interactions TCP defines a reliable bi-directional communication where the forward link data rate depends on the rate of the received ACK packets in the backward link. This behavior of TCP constitutes the main difference between TCP and UDP access in the WLAN. The key observation is that, if we assume there are no packet losses in TCP connections (infinitely large interface buffers at the AP and the stations), the TCP access is fair irrespective of the EDCA parameter selection (which is not the case for UDP). This is due to the fact that the slow link limits the throughput for all TCP flows. However, when the buffer size at the AP (bottleneck) is limited, significant unfairness and low channel utilization is experienced as previously shown in Fig. 3. Therefore, for fair resource allocation and high channel utilization, packet losses at the AP buffer should be minimized. We configure our adaptation algorithm considering the TCP dynamics to achieve this objective. None of the work in the literature on IEEE 802.11 MAC upload/download fairness considered the asymmetry in the forward and backward link packet rate during the slow-start phase of the TCP con- nections. During the slow-start phase, the packet rate in the forward link is twice the packet rate in the backward link. When the congestion avoidance phase is entered, the forward and the backward link packet rates become equal. When this asymmetry during slow-start is neglected, the download traffic is penalized with longer queueing delays. Depending on the buffer availability, significant packet loss may even occur during the slow-start. These may considerably affect the short-term fairness and the channel utilization. Our solution is simple yet effective. Considering each TCP data and ACK streams of each connection as individual active flows, the parameter adaptation algorithm of Section III-C is used. Since TCP is fair irrespective of the EDCA parameter selection as long as there are no packet losses, fine tuning on CWmin is always skipped. Therefore, the AP does not have to measure ntu and ntd . On the other hand, fine tuning is carried out on TXOP assignments to overcome increased rate of downlink TCP data flows during slow-start. Since the forward to backward link packet rate ratio is 2 during the slow-start, the analytically calculated TXOP duration is multiplied by 2. Our approach is adapting the duration of the TXOP depending on the number of packets buffered at the interface queue. If the number of packets goes over a threshold value th, doubled TXOPs are enabled until the number goes below the threshold again. Assigning best-effort data flows a non-zero TXOP or a small CWmin may not be a favorable approach when multimedia flows coexist in the WLAN. The controller block located at the AP should check whether the QoS for admitted realtime flows is preserved or not in the WLAN with the CWmin and TXOP values calculated for uplink/downlink fairness. IV. NUMERICAL AND SIMULATION RESULTS We carried out simulations in ns-2 [4] in order to evaluate the performance of the proposed weighted fairness adaptation algorithm. For the simulations, we employ the IEEE 802.11e EDCA MAC simulation module for ns-2.28 [5]. We consider a network topology where each wireless station initiates a connection with a wired station where the WLAN traffic is relayed to the wired network through the AP. The stations are uniformly distributed on a circle and the AP is located at the center. The power thresholds are set so that every station can hear the other’s transmission. The data connections use either UDP or TCP NewReno. The UDP traffic uses a Constant Bit Rate (CBR) application. The TCP traffic uses a File Transfer Protocol (FTP) agent which models bulk data transfer. The default TCP NewReno parameters in ns-2 are used. The UDP traffic is mapped to a higher priority AC than the TCP traffic. All the stations are assumed to have 802.11g PHY using 54 Mbps and 6 Mbps as the data and basic rate respectively [29]. The packet size is 1500 bytes for all flows. The buffer size at the stations and the AP is set to 200 packets. We found β = 5, α = 0.5, and th = 50 packets to be appropriate through extensive simulations. Fig. 5 shows the average throughput of individual flows for a scenario of 10 uplink UDP, 10 downlink UDP, 10 uplink TCP and 10 downlink TCP connections (same scenario as in Fig. 3). At the stations, UDP flows are mapped to an AC with CWmin = 31 and CWmax = 511. TCP flows use an AC with CWmin = 63 and CWmax = 1023. For both ACs, AIFSN values are set to 2 and TXOP limits are 0. Unless otherwise stated, all data connections of the stations in other experiments use these ACs (thus these EDCA parameters). At the AP, we run the proposed algorithm designed for weighted fairness support in the downlink and uplink. Since the number of downlink and uplink flows are equal for both ACs, we define the downlink/uplink utilization requirement as Ur = 1. The analytical model decides on the CW and the TXOP that achieves Ur = 1. Fine tuning on CW is carried out for the fairness of UDP flows. The TXOP is adaptively doubled according to the proposed algorithm for TCP flows. The results illustrate that U = 1 is perfectly achieved in terms of throughput for both UDP and TCP flows. We have tested the proposed algorithm for a range of network conditions. a) Experiment 1: In the first set of experiments, we generate an equal number of TCP and UDP flows both in the uplink and downlink. Each flow starts at the same time and the simulation duration is 100 seconds. The wired link delay (denoted as Round Trip Time (RTT) in the titles of the figures) is equal for all flows (30 ms). Fig. 6 shows the total throughput of TCP and UDP flows in each direction for the proposed algorithm. The results for the default 802.11e EDCA are also included for comparison. As the results depict, U = 1 is perfectly achieved in terms of average throughput for the proposed algorithm, while the default scheduler cannot maintain fair access. Fig. 7 shows the total throughput of TCP and UDP flows as well as the total system throughput for the proposed algorithm and the default case. The proposed algorithm can maintain more efficient channel utilization than the default EDCA while providing fair access. In Fig. 8, we present the performance in terms of fairness between individual TCP or UDP flows in the same direction for the proposed algorithm and the default EDCA. The performance metric we use is the widely used fairness index [30]. The fairness index, f , is defined as follows: if there are n concurrent connections and the throughput achieved by connection i is equal to xi, 1 ≤ i ≤ n, then i=1 xi) i=1 x . (12) As the results imply, the proposed algorithm also provides fair access between UDP and TCP flows of the same direction. However, the default EDCA results in unfair resource allocation even between the TCP flows of the same direction. As we have described in Section II-A, the unfairness is more significant between TCP uplink flows. Although no unfair behavior is expected between UDP flows in the same direction, we have included these results in Fig. 8 for the sake of completeness. b) Experiment 2: We have repeated the simulation set of experiment 1 when the wired link delay is varied for TCP flows. The wired link delay of the first TCP connection is set to 24 ms and each newly generated TCP connection is assigned 4 ms larger wired link delay than the previous one. Therefore, the second TCP connection has 28 ms wired link delay, the third one has 32 ms wired link delay and so on. This holds for both uplink and downlink connections. UDP wired link delay is constant for each connection. Fig. 9 shows the average throughput of each TCP and UDP flow in each direction. Fig. 10 shows the total throughput of TCP and UDP flows as well as the total system throughput. Fig. 11 shows the performance in terms of fairness between individual TCP or UDP flows in the same direction. As the results show, the performance of the proposed algorithm in terms of fair resource allocation is independent of the duration of the wired link delay. High channel utilization and perfect fairness is maintained. On the other hand, as the comparison of Fig. 8 and Fig. 11 imply, the performance of default EDCA depends on the duration of the wired link delay. In the case of varying wired link delays, the unfairness between individual TCP flows both in the downlink and uplink is even worse. c) Experiment 3: In the third set of experiments, we also generate an equal number of TCP and UDP flows both in the uplink and downlink. In this scenario, each uplink or downlink flow starts at different times and the simulation duration is 300 seconds. The wired link delay is equal for all flows. The first downlink UDP connection starts at t = 5 s. The first uplink UDP connection starts at t = 10 s. The first uplink TCP connection starts at t = 7 s. The first downlink TCP connection starts at t = 12 s. Then, a new flow of the same type arrives every 10 s. No other flow arrives after 200 s. Fig. 12 and Fig. 13 show the instantaneous UDP and TCP throughput of individual uplink and downlink flows respectively for default EDCA. The unfairness between uplink and downlink for both UDP and TCP and the unfairness between individual TCP flows both in the uplink and downlink are evident. Fig. 14 and Fig. 15 show the instantaneous UDP and TCP throughput of individual uplink and downlink flows respectively when the proposed algorithm is enabled. As the results imply, the proposed algorithm adaptively updates EDCA parameters and always maintains instantaneous Ur (as calculated in (11). d) Experiment 4: We have repeated the simulation set of experiment 3 when the wired link delay is varied for TCP flows. We set different wired link delays using the way as previously stated. Fig. 16 and Fig. 17 show the instantaneous UDP and TCP throughput of individual uplink and downlink flows respec- tively for default EDCA. The unfairness between individual TCP flows both in the uplink and downlink are more pronounced when compared with the equal wired link delay scenario. Fig. 18 and Fig. 19 show the instantaneous UDP and TCP throughput of individual uplink and downlink flows respectively for the proposed algorithm. Since the proposed algorithm adaptively updates EDCA parameters, it maintains fair resource allocation. The downlink flows does not starve in terms of throughput. e) Experiment 5: We have repeated the simulation set of experiment 3 when half of the TCP flows model short flows. The flow generation times follow the rules of experiment 3. The simulation duration is 450 s. No other flow arrives after 300 s. The short and long TCP flows are alternatively initiated both in the downlink and uplink. The short TCP flows consist of 31 packets and leave the system after all the data is transferred. Fig. 20 shows the total transmission duration for individual short TCP flows for the proposed algorithm and the default EDCA. Note that flow indices from 1 to 15 represent uplink TCP flows while flow indices from 16 to 30 represent downlink TCP flows. The file transfers with short durations can be completed in a considerably shorter time when the proposed algorithm is used. At high load, short flows experience significantly long delays and connection timeouts when default constant EDCA parameter selection is used. f) Experiment 6: We have repeated the simulation set of experiment 5 when the wired link delay is varied for TCP flows. We set different wired link delays using the way as previously stated. Fig. 21 shows the total transmission duration for individual short TCP flows for the proposed algorithm and the default EDCA. The comparison of Fig. 20 and Fig. 21 reveals that the proposed algorithm performance in terms of short TCP flow completion time is independent of varying wired link delays among the flows. g) Experiment 7: In another set of experiments, we consider three types of traffic sources; audio, video, and data. The audio traffic model implements a Voice-over-IP (VoIP) application as a Constant Bit Rate (CBR) traffic profile at 24 kbps. The constant audio packet size is 60 bytes. Although not presented here, similar results and discussion hold when the silence suppression scheme is used and the audio traffic exhibits on-off traffic characteristics. For the video source models, we have used traces of real H.263 video streams [31]. The mean and maximum video payload size is 2419 bytes and 3112 bytes respectively. The mean video data rate is 255 kbps. The audio flows are mapped to an AC with CWmin = 7 and CWmax = 15. The video flows use an AC with CWmin = 15 and CWmax = 31. For both ACs, AIFSN values are set to 2 and TXOP limits are 0. Fig. 22 shows the average throughput of uplink and downlink data flows when there are 5 voice and 5 video flows both in the uplink and downlink (a total of 20 flows with QoS requirements). Similarly, Fig. 23 shows the average throughput of uplink and downlink data flows when there are 10 voice and 10 video flows both in the uplink and downlink. We also compare the results with the proposed algorithm of [13]. As the results of default EDCA and [13] imply, sticking with constant EDCA parameters for any number of flows does not result in fair access no matter which EDCA parameter setting is used. On the other hand, the proposed adaptive algorithm effectively manages fair resource allocation for any number of stations. Note that we have not included the average throughput of the flows with QoS requirements in Fig. 22 and Fig. 23, since all audio and video flows get necessary bandwidth to serve offered load with zero packet loss rate. Fig. 24 compares the average delay of each QoS flow in each direction for default EDCA and the proposed algorithm when there are a total of 20 flows with QoS requirements. Similarly, Fig. 25 compares the average delay of each QoS flow in each direction for default EDCA and the proposed algorithm when there are a total of 40 flows with QoS requirements. As the results show, the QoS flows experience slightly larger delays when the proposed algorithm is used (due to smaller CW and larger TXOP assignment for data flows). On the other hand, the delay increase is well within the limits of QoS requirements. Moreover, fair resource allocation for data flows is provided. V. CONCLUSIONS We have proposed a model-assisted measurement-based dynamic EDCA parameter adaptation algorithm that achieves a predetermined utilization ratio between uplink and downlink flows of the same AC while keeping the prioritization among ACs. The key contribution is that depending on simple network measures, the proposed algorithm dynamically adapts the EDCA parameters calculated via a proposed analytical model. Another key insight is that the proposed algorithm differentiates the way of adaptation between UDP and TCP flows regarding their characteristics. The proposed algorithm is fully compliant with the 802.11e standard. We propose AP to use any CW value, not necessarily exponents of 2. Our observation is that the 802.11e standard does not restrict the CW settings of the ACs at the AP to be the powers of 2, while the CW setting of the ACs at the STA should be powers of 2 due to the definition of specific fields in the beacon packet. Our approach provides the AP the freedom of satisfying any required utilization ratio through fine tuning on CW settings. Via simulations, it is shown that fair resource allocation between uplink and downlink flows of an AC can be maintained in a wide-range of scenarios when the proposed model-assisted measurement-based dynamic EDCA parameter adaptation algorithm is used. 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Kulapala, “Network Performance Evaluation Using Frame Size and Quality Traces of Single-Layer and Two-Layer Video: A Tutorial,” IEEE Communications Surveys and Tutorials, vol. 6, no. 2, pp. 58–78, Third Quarter 2004. [Online]. Available: http://www.eas.asu.edu/trace http://www.eas.asu.edu/trace Transmission/ Collision period AIFSN AIFSN No Tx Zone 1 Zone 0 TC1 in Backoff TC0 in Backoff Fig. 1. EDCA backoff after busy medium. 1 d0 Wmin tr 1-p tr 1-p Fig. 2. Transition through backoff slots in different contention zones for the example given in Fig.1. 0 5 10 15 20 25 30 35 40 Flow index Fig. 3. Total throughput of 10 uplink UDP (indices 1-10), 10 downlink UDP (indices 11-20), 10 uplink TCP (indices 21-30) and 10 downlink TCP (indices 31-40 flows) when the AP and the stations use equal EDCA parameters. 0 5 10 15 20 25 Number of active uplink and downlink flows non−integer round, N TXOP,1 round, N TXOP,1 round, N TXOP,1 Fig. 4. The downlink/uplink access ratio for increasing number of uplink and downlink flows. 0 5 10 15 20 25 30 35 40 45 Flow index Fig. 5. Total throughput of 10 uplink UDP (indices 1-10), 10 downlink UDP (indices 11-20), 10 uplink TCP (indices 21-30) and 10 downlink TCP (indices 31-40 flows) when the AP uses the proposed adaptation algorithm to achieve Ur = 1. 5 10 15 20 25 Number of TCP/UDP flows at each direction Equal RTT for each TCP flow Default − UDP uplink Default − UDP downlink Default − TCP uplink Default − TCP downlink Proposed − UDP uplink Proposed − UDP downlink Proposed − TCP uplink Proposed − TCP downlink Fig. 6. The total throughput of TCP and UDP flows in each direction (experiment 1). 5 10 15 20 25 Number of TCP/UDP flows at each direction Equal RTT for each TCP flow Default − UDP Default − TCP Default − Total Proposed − UDP Proposed − TCP Proposed − Total Fig. 7. The total throughput of TCP and UDP flows as well as the total system throughput (experiment 1). 5 10 15 20 25 Number of TCP/UDP flows at each direction Equal RTT for each TCP flow Default − UDP uplink Default − UDP downlink Default − TCP uplink Default − TCP downlink Proposed − UDP uplink Proposed − UDP downlink Proposed − TCP uplink Proposed − TCP downlink Fig. 8. Fairness index of individual TCP or UDP flows in the same direction (experiment 1). 5 10 15 20 25 Number of TCP/UDP flows at each direction Different RTT for each TCP flow Default − UDP uplink Default − UDP downlink Default − TCP uplink Default − TCP downlink Proposed − UDP uplink Proposed − UDP downlink Proposed − TCP uplink Proposed − TCP downlink Fig. 9. The total throughput of TCP and UDP flows in each direction (experiment 2). 5 10 15 20 25 Number of TCP/UDP flows at each direction Different RTT for each TCP flow Default − UDP Default − TCP Default − Total Proposed − UDP Proposed − TCP Proposed − Total Fig. 10. The total throughput of TCP and UDP flows as well as the total system throughput (experiment 2). 5 10 15 20 25 Number of TCP/UDP flows at each direction Different RTT for each TCP flow Default − UDP uplink Default − UDP downlink Default − TCP uplink Default − TCP downlink Proposed − UDP uplink Proposed − UDP downlink Proposed − TCP uplink Proposed − TCP downlink Fig. 11. Fairness index of individual TCP or UDP flows in the same direction (experiment 2). 0 50 100 150 200 250 300 Time (s) UDP: Equal RTT for each TCP flow, Default Uplink Downlink Fig. 12. The instantaneous UDP throughput of individual uplink and downlink flows for default EDCA (experiment 3). 0 50 100 150 200 250 300 Time (s) TCP: Equal RTT for each TCP flow, Default Uplink Downlink Fig. 13. The instantaneous TCP throughput of individual uplink and downlink flows for default EDCA (experiment 3). 0 50 100 150 200 250 300 Time (s) UDP: Equal RTT for each TCP flow, Proposed Uplink Downlink Fig. 14. The instantaneous UDP throughput of individual uplink and downlink flows for the proposed algorithm (experiment 3). 0 50 100 150 200 250 300 Time (s) TCP: Equal RTT for each TCP flow, Proposed Uplink Downlink Fig. 15. The instantaneous TCP throughput of individual uplink and downlink flows for the proposed algorithm (experiment 3). 0 50 100 150 200 250 300 Time (s) UDP: Different RTT for each TCP flow, Default Uplink Downlink Fig. 16. The instantaneous UDP throughput of individual uplink and downlink flows for default EDCA (experiment 4). 0 50 100 150 200 250 300 Time (s) UDP: Different RTT for each TCP flow, Proposed Uplink Downlink Fig. 17. The instantaneous TCP throughput of individual uplink and downlink flows for default EDCA (experiment 4). 0 50 100 150 200 250 300 Time (s) TCP: Different RTT for each TCP flow, Default Uplink Downlink Fig. 18. The instantaneous UDP throughput of individual uplink and downlink flows for the proposed algorithm (experiment 4). 0 50 100 150 200 250 300 Time (s) TCP: Different RTT for each TCP flow, Proposed Uplink Downlink Fig. 19. The instantaneous TCP throughput of individual uplink and downlink flows for the proposed algorithm (experiment 4). 0 5 10 15 20 25 30 Flow index Equal RTT for each TCP flow Default Proposed Fig. 20. The total transmission duration for individual short TCP flows(experiment 5). 0 5 10 15 20 25 30 Flow index Different RTT for each TCP flow Default Proposed Fig. 21. The total transmission duration for individual short TCP flows (experiment 6). 5 10 15 20 25 Number of TCP flows at each direction Equal RTT for each TCP flow, number of QoS Flows: 4*5 Default − TCP uplink Default − TCP downlink Proposed − TCP uplink Proposed − TCP downlink [13] − TCP uplink [13] − TCP downlink Fig. 22. The average throughput of uplink and downlink data flows when there are 5 voice and 5 video flows both in the uplink and downlink (experiment 7). 5 10 15 20 25 Number of TCP flows at each direction Equal RTT for each TCP flow, number of QoS Flows: 4*10 Default − TCP uplink Default − TCP downlink Proposed − TCP uplink Proposed − TCP downlink [13] − TCP uplink [13] − TCP downlink Fig. 23. The average throughput of uplink and downlink data flows when there are 10 voice and 10 video flows both in the uplink and downlink (experiment 7). 5 10 15 20 25 Number of TCP flows at each direction Equal RTT for each TCP flow , number of QoS Flows: 4*5 Default − VoIP uplink Default − VoIP downlink Default − Video uplink Default − Video downlink Proposed − VoIP uplink Proposed − VoIP downlink Proposed − Video uplink Proposed − Video downlink Fig. 24. The average delay of each QoS flow in each direction when there are a total of 20 flows with QoS requirements (experiment 7). 5 10 15 20 25 Number of TCP flows at each direction Equal RTT for each TCP flow , number of QoS Flows: 4*10 Default − VoIP uplink Default − VoIP downlink Default − Video uplink Default − Video downlink Proposed − VoIP uplink Proposed − VoIP downlink Proposed − Video uplink Proposed − Video downlink Fig. 25. The average delay of each QoS flow in each direction when there are a total of 40 flows with QoS requirements (experiment 7). Introduction Background Problem Definition Related Work Weighted Fair Access between Uplink and Downlink Flows Analytical Model Weighted Fairness between Uplink and Downlink Flows Implementation of the Numerical Solution Proposed BEB Algorithm for non-integer CW values Dynamic Parameter Adaptation TCP-MAC Interactions Numerical and Simulation Results Conclusions References

Most of the deployed IEEE 802.11e Wireless Local Area Networks (WLANs) use infrastructure Basic Service Set (BSS) in which an Access Point (AP) serves as a gateway between wired and wireless domains. We present the unfairness problem between the uplink and the downlink flows of any Access Category (AC) in the 802.11e Enhanced Distributed Channel Access (EDCA) when the default settings of the EDCA parameters are used. We propose a simple analytical model to calculate the EDCA parameter settings that achieve weighted fair resource allocation for all uplink and downlink flows. We also propose a simple model-assisted measurement-based dynamic EDCA parameter adaptation algorithm. Moreover, our dynamic solution addresses the differences in the transport layer and the Medium Access Control (MAC) layer interactions of User Datagram Protocol (UDP) and Transmission Control Protocol (TCP). We show that proposed Contention Window (CW) and Transmit Opportunity (TXOP) limit adaptation at the AP provides fair UDP and TCP access between uplink and downlink flows of the same AC while preserving prioritization among ACs.

Introduction Background Problem Definition Related Work Weighted Fair Access between Uplink and Downlink Flows Analytical Model Weighted Fairness between Uplink and Downlink Flows Implementation of the Numerical Solution Proposed BEB Algorithm for non-integer CW values Dynamic Parameter Adaptation TCP-MAC Interactions Numerical and Simulation Results Conclusions References

Radiation from Kinetic Poynting Flux Acceleration Edison Liang1 and Koichi Noguchi1 ABSTRACT We derive analytic formulas for the power output and critical frequency of radiation by electrons accelerated by relativistic kinetic Poynting flux, and validate these results with Particle- In-Cell plasma simulations. We find that the in-situ radiation power output and critical frequency are much below those predicted by the classical synchrotron formulae. We discuss potential astrophysical applications of these results. Subject Headings: Acceleration of particles– Radiation mechanisms: non-thermal - Gamma- rays:bursts Online Material: color figures 1. INTRODUCTION In popular paradigms of radiation from blazars, pulsar wind nebulae (PWN), gamma-ray bursters (GRB) and other gamma-ray sources, relativistic outflow energy (hydrodynamic or electromagnetic) from the central compact object (black hole or neutron star) is first converted into relativistic nonthermal electrons via some collisionless dissipation mechanisms (e.g. shocks, Dermer 2003, Meszaros 2002, Lyubarski 2005). These nonthermal electrons are then hypothesized to radiate synchrotron-like radiation, including small-pitch-angle synchrotron (Epstein and Petrosian 1973, Lloyd and Petrosian 2000), or “jitter” radiation if the magnetic field is too chaotic (e.g. due to Weibel instability, Weibel 1958, Medvedev 2000, Medvedev et al 2005). In addition, inverse Comptonization of the synchrotron photons (SSC) or external soft photons (EC) may account for the high-energy (e.g. MeV-TeV) gamma-rays (Dermer et al 2000, 2003). Most popular astrophysical models invoke the classical synchrotron formulas (Rybicki and Lightman 1979). However, two outstanding questions remain unsolved: (a) exactly how is the outflow energy converted into nonthermal electron energy via collisionless shocks (CS, Hoshino et al 1992, Gallant et al 1992, Silva et al 2003, Nishikawa et al 2003, Spitkovski 2006), or electromagnetic Poynting flux (PF, Smolsky & Usov 2000, Lyutikov & Blackman 2002, Van Putten & Levinson 2003, Lyutikov and Blanford 2003)? (b) do the accelerated electrons always radiate synchrotron radiation, since the synchrotron models do not agree with observations in many cases (Fenimore 2002, Dermer & Chang 1999, Preece et al 2000)? In this paper we present concrete examples of acceleration mechanisms whose radiation process is drastically different from classical synchrotron radiation. Over the past few years we have used sophisticated Particle-in-Cell (PIC) codes for relativistic collisionless plasmas (Langdon and Lasinksi 1976, Birdsall & Langdon 1991, Langdon 1992) to study nonthermal electron acceleration and radiation processes (Liang et al 2003, Liang & Nishimura 2004, Nishimura et al 2003, Liang & Noguchi 2005, 2006). A unique feature of our PIC simulations is that the power radiated in-situ by each superparticle (=numerical representation of a charged particle) can be computed simultaneously as the superparticles are accelerated by the Lorentz force (Noguchi et al 2005, Liang and Noguchi 2005, 2006). This approach provides a fully self-consistent treatment of the intrinsic radiation output during the acceleration process. In this paper we focus on the radiation of plasmas accelerated directly by intense electromagnetic pulses or Poynting flux (PF), and derive analytic formulas for this radiation from first-principles. Section 2 reviews the basic physics of PF acceleration. Section 3 briefly summarizes the key result of the numerical radiation power. In Section 4 we derive the critical frequency of PF radiation. In Section 5 we derive an analytic formula for the radiation power output. In Section 6 we speculate on the astrophysics scenarios of PF acceleration. In Section 7 we apply the analytic formulas to a sample PF model of long GRBs. Section 8 is devoted to discussion and summary. A common misconception about PIC simulations is that such simulations are too small in physical scale (measured in units of plasma skin depths and electron gyroradii) to be relevant to macroscopic astrophysical phenomena. However, unlike MHD simulations, the purpose of PIC simulations is not to try to reproduce macroscopic phenomena, but to discover and quantify microphysical laws governing particle energization, radiation mechanisms, wave-particle interaction and dissipation processes, which operate at the level of plasma skin depths and gyroradii. Once discovered via numerical simulations, such physical laws should be rederived analytically from first principles. These validated laws are then applicable to macroscopic phenomena irrespective of the space and time scales. This is the approach we will adopt in this paper. 2. ACCELERATION BY KINETIC POYNTING FLUX In this paper we are interested in relativistic collisionless plasmas whose Coulomb mean free paths are much larger than the relevant plasma scale sizes (see Sec.7 for sample numbers). The interaction of such plasmas with intense EM fields lies outside the classical MHD regime and can only be treated correctly with kinetic theory. Here we define “Poynting flux” as a directed plasma outflow dominated and driven kinetically by transverse electromagnetic (EM) fields with Ωe/ωpe=B/(4πnm)1/2 >1, (Ωe = eB/m =electron gyrofrequency, ωpe =(4πne2/m)1/2 = electron plasma frequency, m=electron mass, n=imbedded electron density, c = 1 throughout this paper except in Sec.7), in the absence of a flow-aligned longitudinal magnetic field. Hence we will not consider diffusive particle acceleration by scattering with classical Alfven and whistler waves (see discussions in Sec.8) in a background magnetic field, or electrostatic acceleration by longitudinal plasma (Langmuir) waves (Boyd and Sanderson 1969). Instead we focus on particle acceleration by the ponderomotive (JxB) force. Astrophysical examples of such kinetic electromagnetic outflows include the equatorial stripe wind of pulsars and magnetars (Lyubarsky 2005, Skjaeraasen et al 2005), and the front end of a low-density magnetic tower jet driven by magnetized accretion disks around black holes (Koide et al 2004). There are two opposite situations in which a kinetic PF can efficiently transfer its EM energy into the nonthermal kinetic energy of a plasma: pushing or pulling the plasma with the JxB force (Fig.1). The common “pushing” mechanism involves an intense EM pulse impinging on an overdense plasma (ωpe>2π/λ, λ=characteristic wavelength of the EM field) from the outside (Fig.1a). This phenomenon is well known from the interaction of intense radio waves with the ionosphere, or from laser-plasma interactions (Kruer et al 1975). The intense EM field induces a ponderomotive (jxB) force near the critical surface (where ωpe=2π/λ, Fig.1a), which accelerates the surface electrons inward to Lorentz factors characterized by the local dimensionless vector potential ao (=eBλ/2πme, Wilks et al 1992). When ao>>1, the EM field “snowplows” all upstream electrons. The reflection front moves forward relativistically so that the EM pulse suffers little reflection (Kruer et al 1975). In extreme cases (e.g. in an e+e- plasma) the bulk of the EM energy is transferred to relativistic particles instead of being reflected. However, this type of “pushing” PF acceleration mainly produces quasi-Maxwellian “superthermal” electron spectra with kT ~ ao (Fig.2a, Wilks et al 1992), instead of the power-law spectra (Preece et al 2000) commonly observed in astrophysics. We call this “pushing” acceleration by an intense EM pulse “leading Poynting or ponderomotive acceleration” or LPA (Fig.1a). PIC simulations and analytic theory (Harteman and Kerman 1996) suggest that the maximum Lorentz factor achievable by LPA is limited to max(Ωe2/ωpe2, ao2/2) due to energy and momentum conservation, since all upstream electrons must share the PF momentum and energy. However, because the snowplow is moving at almost light speed and the electrons are highly collisionless, in most cases no forward shock (in the conventional sense) is observed to form in the upstream plasma. Astrophysically, LPA is potentially relevant to the interaction of pulsar or magnetar winds with a dense environment in the kinetic limit. In contrast to the LPA, the “trailing Poynting or ponderomotive acceleration” or TPA, occurs when an intense EM pulse pulls, instead of pushes, an overdense plasma (Fig.1b, TPA replaces the acronym DRPA used in our early publications, Liang et al 2003, Liang & Nishimura 2004). Consider for example a situation in which a strongly magnetized, overdense collisionless plasma with B/(4πnm)1/2 >1 suddenly expands due to force imbalance. The expansion disrupts the sustaining current, so that 4πJ < Curl B. In the absence of an external EMF regenerating the current, the excess displacement current (∂E/∂t) then generates a transverse EM pulse, which tries to escape from the embedding plasma (Fig.1b). As the EM pulse emerges from the plasma, it “pulls” out the surface electrons via the jxB force, where j is the self-induced polarization current (Boyd and Sanderson 1969). When the jxB force is ultra-intense, the accelerated electrons can stay comoving with the group velocity of the EM pulse which is < c due to plasma loading, and the acceleration can be sustained. Unlike the LPA case in which the maximum Lorentz factor is limited by momentum conservation, TPA transfers the PF energy and momentum to a decreasing number of fast electrons, as slower electrons continuously fall behind the pulse (Liang & Nishimura 2004). We find that the maximum Lorentz factor achieved by TPA is unlimited until radiation damping or dephasing (e.g. due to wave-front curvature) become important. PIC simulations show that TPA always accelerates the high-energy electrons into a simple power law independent of the initial conditions or the pulse width (Fig.2b). The TPA mechanism is exceedingly robust and efficient, typically converting > 50% of the EM energy into accelerated particle energy over a short distance(see Sec.7). Physically both LPA and TPA are caused by a relativistic E x B drift in which the transverse EM field comoves with the particle drift velocity. The key difference is that in LPA the plasma load snowplowed by the EM pulse increases or stays constant with time, thereby limiting the Lorentz factor, whereas the TPA Lorentz factor increases indefinitely due to decreasing plasma loading (Liang et al 2003, Liang and Nishimura 2004). Density-wise, LPA involves collisionless compression of the plasma, while TPA involves rarefaction of the plasma. In contrast to shocks, in which bulk flow energy is converted into EM energy, either via compression of upstream fields or via Weibel (1959) and other instabilities, LPA and TPA converts EM energy into accelerated particle energy. We emphasize that both LPA and TPA are strictly kinetic phenomena with no analog in the MHD limit. The detailed physics of LPA and TPA has been reviewed extensively elsewhere (Liang and Noguchi 2005, 2006), so they will not be repeated here. TPA may be relevant to radiation in astrophysics at two different levels: global and local. Globally, EM pulses with large-scale ordered fields may be generated due to reconnection of a magnetic tower jet or magnetar wind (Koide et al 2004), or from the merger of strongly magnetized neutron stars into a black hole. For example, TPA can take place when the front end of a disconnected magnetic tower jet propagates down the steep density gradient of the collapsar envelope and turns into an unconfined kinetic EM pulse (see Sec.7). Similarly, when a millisecond magnetar or a merging strongly magnetized neutron star binary collapses to form a black hole, part of its energy may be emitted in the form of an intense EM pulse. However, TPA may also occur at the local level even in the absence of large scale ordered EM fields. For example, relativistic EM turbulence generated by shocks and shear layers may dissipate via the TPA mechanism as the nonlinear waves propagate into low density regions with Ωe/ωpe>1. In this case sustained comoving particle acceleration can last locally until dephasing occurs. We emphasize that the only piece of physics invoked in the radiation calculation below is that the particle is accelerated locally by a nearly comoving transverse EM wave. No assumption about the global geometry, topology or size scale of the EM field is required. Hence the radiation formulas derived in this paper have more general validity and much broader applications than the small scale PIC simulations performed so far based on the simplistic LPA and TPA scenarios (Liang and Noguchi 2005, 2006). 3. NUMERICAL RADIATION POWER OUTPUT In this paper we focus on the radiation output of electrons (and positrons) accelerated by a comoving kinetic Poynting flux. Numerically, we compute the radiation power output by incorporating the relativistic dipole formula (Rybicki & Lightman 1979) into our PIC code: Prad = 2e2 (F||2 + γ2F+2)/3m2 (1) where γ = Lorentz factor, F|| = force component along velocity v, and F+ = force component orthogonal to v. We compute the total power loss of each superparticle in the PIC simulation, by interpolating the field data from the cell boundaries to the instantaneous superparticle position, so that F and v refer to the same time and space point. We have carefully calibrated this numerical procedure with known analytic results. Fig.3 compares the numerical radiation output for an isotropic thermal plasma in a static uniform B field with that computed using the analytic synchrotron formula (Rybicki and Lightman 1979). Their excellent agreement, especially for the high-energy electrons, validates our numerical algorithm. However, PIC simulation cannot be used to compute the radiation spectrum numerically because the PIC simulation time step (typically = 0.25 gyroperiod) is too large to accommodate the high frequencies. The upper panels of Figs.4 & 5 illustrate the evolution of the Prad distribution of superparticles in sample LPA and TPA runs. In both cases a plane, linearly polarized EM pulse accelerates the same slab of overdense e+e- plasma, one from the outside and one from the inside. While the energies of the pairs increase monotonically due to the acceleration, the power radiated by the electrons initially rises to a maximum, but then declines monotonically. We find that in both cases Prad << the classical synchrotron power Psyn = 2e4B2p+2/3m2 (mp+=momentum orthogonal to B). This suppression of radiative power can be understood as follows. As particles are accelerated to higher and higher γ, v aligns increasingly with the Lorentz force F. So the F+ term in Eq.(1) decreases relative to the F|| term. However, Psyn comes only from the F+ term (Rybicki & Lightman 1979). Hence Psyn >> Prad for high γ particles. In Sec.5 we will rigorously derive a general analytic formula for Prad. Here we will only state that, for electrons almost comoving with the EM pulse, Prad can be approximated by: Panalytic = Psyn sin4α (2) where α is the angle between p+ and the Poynting vector k (Fig.6). Fig.7 compares the numerical Prad with Panalytic for the runs of Figures 4 & 5. It shows good correlation for the high-γ particles. Since sinα <<1 for high-γ particles (c.f. Fig.8), Eq.(2) explains why Prad << Psyn. The rise and decline of Prad in Figs.4 & 5 are caused by the competition between increasing γ (and Psyn) and decreasing α. 4. RADIATION CRITICAL FREQUENCY A prominent feature of GRB and blazar spectra is the presence of a low energy spectral break Epk (hundreds of keV for classical GRBs, radio-IR for blazars). This spectral break is an indicator of the overall spectral hardness, and is usually interpreted as the critical frequency of synchrotron radiation ωcrsyn ~ 1.5Ω eγop+o (Rybicki & Lightman 1979) by electrons with low energy cutoff γo. This interpretation of the spectral break, together with some assumptions about energy equipartition, is often used to constrain the Lorentz factor and magnetic field of the source. However, as we show below, for radiation emitted by TPA and LPA electrons, the asymptotic critical frequency ωcr is << ωcrsyn. To derive the formula for ωcr, we follow the approach of Landau and Lifshitz (1980): ωcr is determined by the time (measured in detector frame) it takes the radiation beam of opening angle 1/γ to sweep past the detector due to the curvature of the particle trajectory. For electrons comoving or almost comoving with the PF, the parallel momentum px (x is the direction along k, Fig.6) increases monotonically while pz (momentum along E) asymptotes to a constant (Liang & Nishimura 2004, note that py along B is constant to first order). Hence the change in the radiation beam direction due to bending of particle trajectory is dominated by the change in px: Δθ ~ pzΔpx/px2. From the Lorentz force equation we have dγ/dt=eEzpz/mγ. Hence the time in the laboratory frame for the radiation beam to change by an angle Δθ~2/γ is Δt=2γ2m/(eEzpz2) where we have used the approximation γ~px(>>pz, py). This translates into a duration in the detector frame Δtob=Δt/2γ2=m/eEzpz2. Thus the critical frequency (Rybicki and Lightman 1979): ωcr=1.5/Δtob=1.5eEzpz2/m=1.5Ωepz2=1.5Ωe p+2sin2α ∼ ωcrsyn sin2α (3). Since sinα<<1 at high γ (Fig.8), ωcr << ωcrsyn In Sec.6 we will discuss the major implications of this result for modeling GRB and blazar data. . In the lower panels of Figs.4 & 5 we plot the evolution of ωcr of the same LPA and TPA runs. These snapshots highlight the evolution of the spectral hardness. Again we see that ωcr first rises to a maximum before declining monotonically due to the competition between increasing γ and decreasing α. However the decline of Prad is more rapid than ωcr due to the extra factors of sinα. 5. RADIATION POWER FORMULA FOR KINETIC PF ACCELERATION In this section we derive a general analytic approximation for the power radiated by an electron accelerated locally by a comoving kinetic PF. While the following derivation assumes linearly polarized plane waves for simplicity, the results should be valid in general 3D geometry as long as the wave front curvature and transverse gradients are << 1/(acceleration distance) (distance for e-folding increase in γ). We emphasize that this radiation formula should be applicable to any acceleration by transverse EM fields almost comoving with the local ExB drift velocity. Hence its potential applications in astrophysics are much broader than the specific LPA or TPA scenarios discussed above. For particle motion in a linearly polarized plane wave with (E,B) = (Ez, By) (Fig.6), we have Fx=-evzBy; Fy=0; Fz=e(Ez+vxBy). Here x is the direction of Poynting vector k. After a little algebra we find: F||=eEzvz/v; F+2=e2By2[sin2α(v2-vw2)+(vx-vw)2] (4) where vw=-Ez/By is the local profile speed of the EM field (vw<1 due to plasma loading) and sinα=vz/v. Substituting Eq.(4) into Eq.(1) we obtain: Panalytic = 2e4By2[sin2α(γ2-1)(1-vw2)+γ2(vx-vw)2]/3m2 (5). Hence the power radiated by a PF-accelerated electron depends in general on two key parameters: the local EM field profile speed vw and the angle α between velocity v and Poynting vector k. Eq.(5) simplifies in various special limits: A. Comoving particles (vx=vw): In this case Eq.(5) simplifies to Panalytic = 2e4By2(pz2+py2)sin2α/3m2 when γ>>1. In all of our runs, pz>>py at late times. So this reduces to Panalytic = 2e4By2p+2sin4α/3m2 which is Eq.(2). As Fig.7 shows, Eq.(2) is a good approximation for electrons comoving with vw. However, Eq.(2) is not a good approximation for electrons out of phase with vw (note that electrons can have vx > vw or vx < vw). Liang and Nishimura (2004) suggested that vw corresponds roughly to the peak γ of the particle distribution function f(γ). B. Vacuum pulse limit (vw=1): In the limit vw=1, the PF propagates as a vacuum EM pulse, Eq.(5) becomes for γ >>1: Panalytic =2e4By2γ2 (1-vx)2/3m2~e4By2γ2sin4α/6m2 (6) Since p+ ~ γ, Eq.(6) has the same form as Eq.(2) but is a factor of 4 less in magnitude. It defines the lower limit to the radiative power loss of a PF accelerated electron since in reality vw<1. C. Slightly subluminal PF (1-vw=ε<<1) For most astrophysics applications, the PF will be slightly subluminal. We can simplify Eq.(5) by Taylor expanding 1-vw=ε<<1 to lowest order. Eq.(5) then reduces to Panalytic ~ 2e4By2γ2(ε + sin2α/2)2/3m2 (7). This formula fully explains the physical origin of the result Prad << Psyn. In relativistic PF acceleration, both ε and sinα are <<1. Fig.9 shows an example for which the best-fit ε = 0.03. Eq.(7) shows that Panalytic behaves differently depending on whether ε >> or << sin2α/2. In the former case Panalytic depends only on the EM field profile speed vw and not on α: Panalytic ~2e4By2γ2ε2/3m2 (8) In the second case we regain Eq.(6) which depends only on α and not on vw. Therefore when we model astrophysical data using these formulas, we obtain different physical information depending on the ratio ε:sin2α/2, which depends on the PF initial condition, such as To, Ωe/ωpe .etc. Eqs.(2), (6) & (8), which contain only 3 unknowns: (By, γ, α) or (By, γ, ε), are much easier to use for modeling astrophysical data than Eq.(5) or Eq.(7), which contain 4 unknowns. In practice, ε fluctuates rapidly in both time and space so its extraction from astrophysical data would be more difficult, whereas PIC simulation results suggest that sinα has a narrower range for high-γ particles (Fig.8). So the above analytic approximation is most useful for data modeling in the regime ε << sin2α/2, which seems to be the case for GRB’s (cf. Sec.7) and may also be the case for blazars. We emphasize that the above analytic radiation formulas are derived from first principles independent of any PIC simulation results. Hence their validity is completely independent of any numerical simulation size scale. In fact we have carefully validated these analytic formulas using simulations spanning a dynamic range of >105 (i.e. PF pulse widths ranging from 102 to107 gyroradii). One may argue that the above result comes about only because we work in the lab frame, and that the classical synchrotron formula must apply if we transform to a (primed) Lorentz frame in which E’ = 0 and B’ is static. This is indeed true, and one can rederive the above formulas using appropriate Lorentz transformations of the classical synchrotron power formula (Rybicki and Lightman 1979). However, finding the Lorentz frame with E’ = 0 is impractical, since E/B varies rapidly in both space and time due to modulation by self-generated currents and current instabilities (Liang and Nishimura 2004). There is no single Lorentz transformation that can lead to E’ = 0 for any meaningful fraction of the EM pulse. So in practice it is more convenient to work in a global lab frame, measure E, B and p in this frame and use the above analytic formulas. Such lab-frame quantities are more relevant to astrophysical observations anyway. 6. IMPLICATIONS FOR ASTROPHYSICAL PF MODELS Using PIC simulations and analytic methods, we have demonstrated above that electrons accelerated by Poynting flux which comoves with the local ExB drift velocity, radiate at a rate (in the lab-frame) much below the classical synchrotron power, and the critical frequency of their radiation spectrum is also much below the classical synchrotron critical frequency. These results have major implications for the interpretation of astrophysical data from GRBs and blazars, if their energy supply is coming from Poynting flux. One scenario would be that the classical synchrotron model applies only to the radiation zone, which is separate from the particle acceleration zone, and the synchrotron model (B, γ) values refer only to the radiation zone but not the acceleration zone, which likely has higher B and γ. An alternative scenario is that the observed radiation is intrinsic to the acceleration process. Then we have to use Eq.(5) or its various limits (Eq.(2)-(8)) to model the astrophysics data. This will lead to much higher (B, γ) values for the source than in conventional synchrotron models since ε and α are <<1. Clearly the overall energetics and parameters of the two models will be very different and such differences may be testable. As an example of the application of the results of Secs. 4 & 5, we will consider a simplistic model of long GRBs in Sec.7. Another important consequence of the suppression of synchrotron radiation in PF scenarios is that inverse Comptonization of external soft photons (EC) may dominate even when the (lab-frame) magnetic energy density greatly exceeds the external soft photon energy density (Rybicki & Lightman 1979), and the conventional SSC + EC model of blazars (Dermer & Boettcher 2002) may need to be revamped. These scenarios will be reconsidered in future papers. 7. APPLICATION TO A PF MODEL OF LONG GRB’s Currently there is no universally accepted model of GRB energization and radiation. Two popular paradigms are hydrodynamic versus electromagnetic outflows from a central engine (e.g. a newly formed black hole), dissipating at a distance of 1014-15 cm via nonthermal electrons and gamma-rays (Meszaros 2002, Piran 2000). If GRBs are indeed the manifestations of intense PF outflow, a dissipation mechanism such as TPA is attractive due to its high energy conversion efficiency and power-law spectra (cf. Sec.2). Liang and Nishimura (2004) also pointed out several tantalizing similarities between the observable properties of TPA in e+e- plasmas and GRB phenomenology. To demonstrate the utility of the results of Secs.4&5, here we apply the analytic radiation formulas to a simplistic “toy” model of classical long GRBs, assuming that the PF contains only e+e- pairs with no ions (e-ion models will be considered in subsequent papers). We will derive the value of the spectral break energy Epk from these formulas. The underlying astrophysical framework is that some central engine activity lasting 10’s of seconds launches an intense EM pulse of width ~ 1012 cm and energy ~1051 ergs, loaded with only low-density e+e- plasma so that Ωe/ωpe >1. This intense EM pulse initially propagates through the collapsar envelope as a non-dissipative subluminal MHD pulse as long as the ambient density is high enough so that the formal Alfven speed vA=B/(4πρp)1/2 < c (ρp= ambient proton mass density). But the pulse will eventually reach a point where the envelope ion density is so low that vA > c, and the MHD pulse turns into a freely-expanding kinetic EM pulse. This triggers the TPA dissipation and rapid conversion of EM energy into e+e- kinetic energy. We have performed simulations of relativistic magnetosonic pulses propagating down steep density gradients. The preliminary results seem to support the above picture. For long GRBs it is useful to scale the burst parameters with the following benchmark values (Fishman & Meegan 1998, Preece et al 2000): total energy E51=Etot/1051erg, burst duration T30=T90/30sec, prompt-γ emission distance R14=R/1014cm. We assume that the EM pulse is a quasi-spherical shell with thickness ΔR=cT90=1012cmT30 (in this section we write out c explicitly) and solid angle Ω4π=Ω/4π. To simplify the model we assume that the shell is uniform with mean field B and mean lepton (e- + e+) density n. All physical quantities are measured in the “lab- frame”, which we assume to be the rest frame of the GRB central engine or host galaxy. In reality the field, density and momentum profiles are highly structured due to current instabilities (Liang and Nishimura 2004), and the following parameters refer primarily to those leptons at the peak of the momentum distribution function. All our simulations with pair plasmas suggest that at late times, particle energy Eparticle ≥ 0.6Etot, EM energy (= 2EB)≤ 0.4Etot (Fig.10, see also Liang et al 2003). Let N = total number of leptons (e+ + e-) in the pulse and Γ = average Lorentz factor of the lepton distribution = <γf(γ)>. (Γ ~ the group veocity Lorentz factor of the EM pulse Γw=(1-vw2)-1/2, Liang and Nishimura 2004). We thus have dimensionally in cgs units: NΓmc2 ~ 6x1050 E51 (9) B2ΔRR2Ω~16πx1050 E51 (10) Eq.(10) gives: B ~2x105 G (R14-1 Ω4π-1/2 E511/2T30-1/2) (11) Next we estimate Γ for the radiation epoch by invoking the condition: radiative cooling rate = particle acceleration rate. Liang and Nishimura (2004) derived from the Lorentz force equation the particle acceleration rate dΓ/dt = fΩe/Γ, where f is a fudge parameter of O(1) that depends on the initial conditions. We emphasize that this formula depends solely on the comoving nature of the EM pulse and is independent of other global assumption of the TPA. Using Eq.(7) and assuming ε << sinα (and check for consistency below), we obtain: f ecB/Γ = e4By2Γ2sin4α/6m2c3 (12). The peak value of the late-time sinα distribution for high-γ electrons (Fig.8) lies in the range 0.01 to 0.2 in the simulations performed so far. Hence we scale sinα with 0.1 below: α.1=sinα/0.1. Solving Eq.(12) for Γ we obtain: Γ ~1.2x105 (f 1/3 R141/3 Ω4π1/6 E51-1/6T301/6α.1-4/3) (13) Hence ε~1/Γ << sinα and our assumption is justified. Note that this Γ is measured in the lab- frame so it is actually smaller than the composite Lorentz factor of internal shock models (with a bulk Lorentz factor of 102 times internal Lorentz factors of 103-104). Using this in Eq.(9) we find: N~ 6x1051 (f -1/3 R14-1/3 Ω4π-1/6 E517/6T301/6 α.1 4/3) (14) From Eqs.(3), (11) and (13) we obtain the value of the spectral break energy, taken as the critical frequency corresponding to Γ: Epk = hωcr/2π ~ 490 keV(f 2/3 R14-1/3 Ω4π-1/6 E511/6T30-1/6 α.1 -2/3) (15) This value agrees with the observed spectral breaks of typical long GRBs: Epk ~ 250 keV (1+z) ~ 500 keV for z ~1 (Preece et al 2000) in the host-Galaxy frame. Note that Epk in Eq.(15) depends only weakly on the various uncertainty factors. Eq.(14) gives the mean lepton density: n = N/(ΩΔRR2)~ 5x1010(f -1/3 R14-7/3 Ω4π-1/6 E517/6T30-7/6 α.14/3) (16) and the frequency ratio: Ωe/ωpe ~ 250 (f 1/6 R141/6 Ω4π-5/12 E51-1/12T301/12 α.1-2/3) >>1 (17) which justifies our EM-domination assumption. At this density the pairs are completely collisionless (Coulomb mean free path > 1020cm). We note that the local acceleration (=cooling) time of an individual electron with the above B, Γ and sinα values is very short: tcool =tcool ~10-2 sec. However, this should not conflict with the 30 second observed GRB duration. The radiation duration of the shell is determined by the conversion time of overall EM energy into particle energy, which is proportional to the light crossing time of the shell thickness ΔR/c, since the EM pulse takes at least that long to emerge and energize the embedded plasma (Fig.10). Moreover, radiation emitted by the front and back of the plasma arrive at the detector with a time delay of ΔR/c. These two effects combine to make the GRB duration measured by the detector ~ ΔR/c = 30 sec, irrespective of the short acceleration/cooling time of individual leptons. An analogy with internal shock models is in order here. Even though the internal shock thickness and particle acceleration length are << 1012 cm, the GRB radiation duration is governed by the shock crossing time of the colliding shells, which have thicknesses of ~ 1012 cm. We note that 1012cm corresponds to ~1014 gyroradii. Even the acceleration/cooling distance of ~108cm equals 1010 gyroradii. Both scales are much larger than the largest PIC simulation we have performed (~107 gyroradii). Hence it is tempting to question the applicability of our results to the GRB regimes. But we emphasize that the only physics used to derive the radiation rate Eq.(7) and the acceleration rate of Liang and Nishimura (2004) is the comoving assumption of the local EM wave with the local ExB drift speed. This assumption is completely independent of the global geometry, structure and size scale of the fields and plasmas, which affect only the duratiion and longevity of the acceleration process. In addition, we have validated the analytic radiation and acceleration rates with simulations spanning over 4 decades in physical size (103 to107 gyroradii). This gives us added confidence in their general validity. However one puzzle remains. What makes the EM pulse dissipation to occur at R~1014 cm from the central engine, two orders of magnitude larger than the EM pulse width and six orders of magnitudes larger than the lepton acceleration/cooling length? We speculate that it may be the environment which determines this dissipation distance. Here we venture a somewhat speculative but plausible scenario that gives rise to such a large dissipation distance. In reality, the TPA action takes place not at a sharp boundary but in an external density gradient whose scale height is much larger than the acceleration length. In such cases we believe that the Liang-Nishimura (2004) acceleration rate and the radiation cooling rate of Sec.5 are applicable only when the ambient ion mass density drops below the internal pair mass density. Otherwise the EM expansion and particle acceleration would be inhibited by the external ion inertia (cf. discussion at the beginning of Sec.7). In the collapsar model the GRB progenitor is likely surrounded by a Wolf-Rayet wind whose mass density ~ A 5x1011 r-2 g.cm-1 (Chevalier and Li 2000) where the parameter A depends on the mass loss rate. Hence the PF “breakout” or dissipation distance, using the pair density of Eq.(16), becomes rbreakout ~ A1/21014 cm. In other words, the TPA action is inhibited until the PF reaches an ambient ion mass density of ≤ 5x10-17 g.cm-3, and this can only occur at a distance ≥1014 cm in a Wolf-Rayet wind. Fig.11 illustrates the relevant length scales of this scenario. 8. DISCUSSION AND SUMMARY Using PIC simulations and analytic theory, we have shown in this paper that when electrons are accelerated by a comoving Poynting flux with Ωe/ωpe >1, the in-situ radiation power output and critical frequency are much lower than those given by the classical synchrotron formulas. This is because the most energetic electrons have their momentum closely aligned with the local Poynting vector or ExB drift direction. We apply our analytic formulas for the radiation power output and critical frequency to a simple PF model of classical long GRBs, and find that the predicted spectral break energy lies in the range of observed data. Besides the LPA and TPA mechanisms which involve ponderomotive acceleration of overdense plasmas, there are many other Poynting flux scenarios that may result in nonthermal particle acceleration. For example, electron acceleration by comoving longitudinal wakefields generated by PF in an underdense plasma (similar to laser accelerators in the laboratory, Tajima and Dawson 1979) may occur in special astrophysical situations. We have also not considered Poynting flux propagating along flow-aligned guide fields such as Alfven and whistler waves. Preliminary PIC simulation results suggest that linear Alfven waves (ΔB << Bo where Bo is the longitudinal guide field) cannot accelerate nonthermal particles efficiently via the ponderomotive force, since the net E x B drift direction is misaligned from the (strong) guide field. On the other hand, if the Alfven wave is highly nonlinear (ΔB > Bo), it behaves like transverse EM waves. Then the TPA results may apply to first order. 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Figure Captions Fig.1 Sample PIC simulation outputs illustrating the two different mechanisms of kinetic Poynting flux acceleration of an e+e- plasma slab: (a) in LPA, an intense EM pulse incident on an overdense plasma interface induces a ponderomotive force (JxB is along Poynting vector k) that snowplows relativistically all upstream electrons which must share the Poynting flux momentum, thus limiting their Lorentz factor; (b) in TPA, an intense EM pulse escaping from an overdense plasma induces a ponderomotive force which pulls out the surface electrons relativistically. Only the fastest electrons can keep up with the EM pulse, so that the plasma loading of the EM pulse decreases with time. TPA leads to the sustained comoving acceleration of a decreasing number of fast electrons, with no limit to their Lorentz factor. In all figures of this paper, x is expressed in units of 3c/ωpe. Fig.2 (a) Electron energy spectrum accelerated by LPA resembles a superthermal quasi- Maxwellian distribution; (b) electron energy spectra accelerated by TPA for different initial PF thicknesses (103 and 104 c/ωpe) both show a robust power-law of index ~ -3 to -4. The low- energy spectral breaks correspond roughly to the Lorentz factor of the EM pulse group velocity . Fig.3 Calibration of the numerical radiation power Prad computed from the PIC simulation (Eq.(1)) against the analytic synchrotron formula Psyn for a 5 MeV thermal plasma in a static uniform B field shows excellent agreement. The scatter at low energies is due to small errors from interpolating the field values to the particle position. In all figures of this paper, Prad, Psyn and Panalytic are expressed in units of 2e2Ωe2/2700. Fig.4 Upper panel: Snapshots of Prad distribution (dots) and By profile (solid, in units of Bo/15) vs. x for an e+e- plasma slab initially located at x=180 with n=16ncr, thickness = 12c/ωpe and snowplowed by a vacuum EM pulse with Ωe/ωpe=10 from left to right. Prad of the accelerated electrons reaches a maximum at ~2 light crossing times after the EM pulse hits the plasma surface, followed by rapid monotonic decay. This behavior is caused by the decrease in angle α competing with the increase in electron energy. Lower panel: Snapshots of the critical frequency (Eq.3) distribution show that the evolution of the spectral hardness of radiation follows that of Prad. ωcr is expressed in units of10Ωe. Fig.5 Upper panel: Snapshots of Prad distribution (dots) and By profile (solid, in units of Bo/150) vs. x for a e+e- plasma slab accelerated by TPA with initial plasma temperature kTo=0.005m, thickness Lo=12c/ωpe, Ωe/ωpe=10 and initially located at x=180. Prad of the accelerated electrons reaches a maximum at ~5 light crossing times after the emergence of the EM pulse, followed by monotonic decay which is slower than in the LPA case. Lower panel: Snapshots of the critical frequency (Eq.3) distribution shows that the evolution of the spectral hardness of radiation follows that of Prad. ωcr is expressed in units of10Ωe. Note that ωcr of Fig.5 is a factor of 10 larger than that of Fig.4. The radiation at t=0 is thermal cyclotron radiation due to the finite initial temperature. But it has very low ωcr Fig.6 Diagram showing the angle α between the Poynting vector k and p+, the momentum component orthogonal to B. Fig.7 Scatter plot of Prad compared with Panalytic of Eq.(2) for the runs of (a) Fig.4 and (b) Fig.5. At these times most of the high–γ particles are comoving with the EM pulse. Fig.8 Scatter plot of the distribution of Lorentz factor γ vs. sinα=pz/γ for a Ωe/ωpe=10 e+e- slab accelerated by TPA shows that the highest–γ particles have their sin α distribution peaking in the range ~ 0.01- 0.2. Fig.9 Scatter plot of Prad compared with Panalytic of Eq.(7) for an Ωe/ωpe=10 e+e- slab accelerated by TPA, At this time the best correlation for high-γ electrons is obtained when ε = 0.03. Fig.10 Decay curves of EM energy for an Ωe/ωpe=10, kTo=5MeV e+e- slab TPA’s with different initial thicknesses: (A) Lo=10800c/ωpe; (B) Lo=90c/ωpe; (C) Lo=12c/ωpe. This confirms that the conversion time of EM energy into particle energy is directly proportional to the light crossing time Lo/c. Fig.11 Diagram illustrating the different size scales in the “breakout” of a PF from a Wolf-Rayet wind model of long GRBs. The wavy arrow denotes the (lab-frame) PF thickness (ΔR=1012cm) along the observer line of sight. The PF breakout distance (~1014 cm) is determined by the radius at which the wind mass density drops below the PF pair mass density (~5x10-17 g.cm-3). Despite the short acceleration/cooling length (~3x108cm) of individual leptons accelerated by the PF, the detector-measured GRB duration at infinity is ~ΔR/c=30 sec due to the transit time of the PF crossing rbreakout and the light path difference between the front and back of the PF (upper-right space-time diagram). Plasma Plasma Entering Exiting EM pulse Fig.1 Fig.2 Fig.3 Fig.6 Fig.7 Panalytic sin α Fig.8

We derive analytic formulas for the power output and critical frequency of radiation by electrons accelerated by relativistic kinetic Poynting flux, and validate these results with Particle-In-Cell plasma simulations. We find that the in-situ radiation power output and critical frequency are much below those predicted by the classical synchrotron formulae. We discuss potential astrophysical applications of these results.

Radiation from Kinetic Poynting Flux Acceleration Edison Liang1 and Koichi Noguchi1 ABSTRACT We derive analytic formulas for the power output and critical frequency of radiation by electrons accelerated by relativistic kinetic Poynting flux, and validate these results with Particle- In-Cell plasma simulations. We find that the in-situ radiation power output and critical frequency are much below those predicted by the classical synchrotron formulae. We discuss potential astrophysical applications of these results. Subject Headings: Acceleration of particles– Radiation mechanisms: non-thermal - Gamma- rays:bursts Online Material: color figures 1. INTRODUCTION In popular paradigms of radiation from blazars, pulsar wind nebulae (PWN), gamma-ray bursters (GRB) and other gamma-ray sources, relativistic outflow energy (hydrodynamic or electromagnetic) from the central compact object (black hole or neutron star) is first converted into relativistic nonthermal electrons via some collisionless dissipation mechanisms (e.g. shocks, Dermer 2003, Meszaros 2002, Lyubarski 2005). These nonthermal electrons are then hypothesized to radiate synchrotron-like radiation, including small-pitch-angle synchrotron (Epstein and Petrosian 1973, Lloyd and Petrosian 2000), or “jitter” radiation if the magnetic field is too chaotic (e.g. due to Weibel instability, Weibel 1958, Medvedev 2000, Medvedev et al 2005). In addition, inverse Comptonization of the synchrotron photons (SSC) or external soft photons (EC) may account for the high-energy (e.g. MeV-TeV) gamma-rays (Dermer et al 2000, 2003). Most popular astrophysical models invoke the classical synchrotron formulas (Rybicki and Lightman 1979). However, two outstanding questions remain unsolved: (a) exactly how is the outflow energy converted into nonthermal electron energy via collisionless shocks (CS, Hoshino et al 1992, Gallant et al 1992, Silva et al 2003, Nishikawa et al 2003, Spitkovski 2006), or electromagnetic Poynting flux (PF, Smolsky & Usov 2000, Lyutikov & Blackman 2002, Van Putten & Levinson 2003, Lyutikov and Blanford 2003)? (b) do the accelerated electrons always radiate synchrotron radiation, since the synchrotron models do not agree with observations in many cases (Fenimore 2002, Dermer & Chang 1999, Preece et al 2000)? In this paper we present concrete examples of acceleration mechanisms whose radiation process is drastically different from classical synchrotron radiation. Over the past few years we have used sophisticated Particle-in-Cell (PIC) codes for relativistic collisionless plasmas (Langdon and Lasinksi 1976, Birdsall & Langdon 1991, Langdon 1992) to study nonthermal electron acceleration and radiation processes (Liang et al 2003, Liang & Nishimura 2004, Nishimura et al 2003, Liang & Noguchi 2005, 2006). A unique feature of our PIC simulations is that the power radiated in-situ by each superparticle (=numerical representation of a charged particle) can be computed simultaneously as the superparticles are accelerated by the Lorentz force (Noguchi et al 2005, Liang and Noguchi 2005, 2006). This approach provides a fully self-consistent treatment of the intrinsic radiation output during the acceleration process. In this paper we focus on the radiation of plasmas accelerated directly by intense electromagnetic pulses or Poynting flux (PF), and derive analytic formulas for this radiation from first-principles. Section 2 reviews the basic physics of PF acceleration. Section 3 briefly summarizes the key result of the numerical radiation power. In Section 4 we derive the critical frequency of PF radiation. In Section 5 we derive an analytic formula for the radiation power output. In Section 6 we speculate on the astrophysics scenarios of PF acceleration. In Section 7 we apply the analytic formulas to a sample PF model of long GRBs. Section 8 is devoted to discussion and summary. A common misconception about PIC simulations is that such simulations are too small in physical scale (measured in units of plasma skin depths and electron gyroradii) to be relevant to macroscopic astrophysical phenomena. However, unlike MHD simulations, the purpose of PIC simulations is not to try to reproduce macroscopic phenomena, but to discover and quantify microphysical laws governing particle energization, radiation mechanisms, wave-particle interaction and dissipation processes, which operate at the level of plasma skin depths and gyroradii. Once discovered via numerical simulations, such physical laws should be rederived analytically from first principles. These validated laws are then applicable to macroscopic phenomena irrespective of the space and time scales. This is the approach we will adopt in this paper. 2. ACCELERATION BY KINETIC POYNTING FLUX In this paper we are interested in relativistic collisionless plasmas whose Coulomb mean free paths are much larger than the relevant plasma scale sizes (see Sec.7 for sample numbers). The interaction of such plasmas with intense EM fields lies outside the classical MHD regime and can only be treated correctly with kinetic theory. Here we define “Poynting flux” as a directed plasma outflow dominated and driven kinetically by transverse electromagnetic (EM) fields with Ωe/ωpe=B/(4πnm)1/2 >1, (Ωe = eB/m =electron gyrofrequency, ωpe =(4πne2/m)1/2 = electron plasma frequency, m=electron mass, n=imbedded electron density, c = 1 throughout this paper except in Sec.7), in the absence of a flow-aligned longitudinal magnetic field. Hence we will not consider diffusive particle acceleration by scattering with classical Alfven and whistler waves (see discussions in Sec.8) in a background magnetic field, or electrostatic acceleration by longitudinal plasma (Langmuir) waves (Boyd and Sanderson 1969). Instead we focus on particle acceleration by the ponderomotive (JxB) force. Astrophysical examples of such kinetic electromagnetic outflows include the equatorial stripe wind of pulsars and magnetars (Lyubarsky 2005, Skjaeraasen et al 2005), and the front end of a low-density magnetic tower jet driven by magnetized accretion disks around black holes (Koide et al 2004). There are two opposite situations in which a kinetic PF can efficiently transfer its EM energy into the nonthermal kinetic energy of a plasma: pushing or pulling the plasma with the JxB force (Fig.1). The common “pushing” mechanism involves an intense EM pulse impinging on an overdense plasma (ωpe>2π/λ, λ=characteristic wavelength of the EM field) from the outside (Fig.1a). This phenomenon is well known from the interaction of intense radio waves with the ionosphere, or from laser-plasma interactions (Kruer et al 1975). The intense EM field induces a ponderomotive (jxB) force near the critical surface (where ωpe=2π/λ, Fig.1a), which accelerates the surface electrons inward to Lorentz factors characterized by the local dimensionless vector potential ao (=eBλ/2πme, Wilks et al 1992). When ao>>1, the EM field “snowplows” all upstream electrons. The reflection front moves forward relativistically so that the EM pulse suffers little reflection (Kruer et al 1975). In extreme cases (e.g. in an e+e- plasma) the bulk of the EM energy is transferred to relativistic particles instead of being reflected. However, this type of “pushing” PF acceleration mainly produces quasi-Maxwellian “superthermal” electron spectra with kT ~ ao (Fig.2a, Wilks et al 1992), instead of the power-law spectra (Preece et al 2000) commonly observed in astrophysics. We call this “pushing” acceleration by an intense EM pulse “leading Poynting or ponderomotive acceleration” or LPA (Fig.1a). PIC simulations and analytic theory (Harteman and Kerman 1996) suggest that the maximum Lorentz factor achievable by LPA is limited to max(Ωe2/ωpe2, ao2/2) due to energy and momentum conservation, since all upstream electrons must share the PF momentum and energy. However, because the snowplow is moving at almost light speed and the electrons are highly collisionless, in most cases no forward shock (in the conventional sense) is observed to form in the upstream plasma. Astrophysically, LPA is potentially relevant to the interaction of pulsar or magnetar winds with a dense environment in the kinetic limit. In contrast to the LPA, the “trailing Poynting or ponderomotive acceleration” or TPA, occurs when an intense EM pulse pulls, instead of pushes, an overdense plasma (Fig.1b, TPA replaces the acronym DRPA used in our early publications, Liang et al 2003, Liang & Nishimura 2004). Consider for example a situation in which a strongly magnetized, overdense collisionless plasma with B/(4πnm)1/2 >1 suddenly expands due to force imbalance. The expansion disrupts the sustaining current, so that 4πJ < Curl B. In the absence of an external EMF regenerating the current, the excess displacement current (∂E/∂t) then generates a transverse EM pulse, which tries to escape from the embedding plasma (Fig.1b). As the EM pulse emerges from the plasma, it “pulls” out the surface electrons via the jxB force, where j is the self-induced polarization current (Boyd and Sanderson 1969). When the jxB force is ultra-intense, the accelerated electrons can stay comoving with the group velocity of the EM pulse which is < c due to plasma loading, and the acceleration can be sustained. Unlike the LPA case in which the maximum Lorentz factor is limited by momentum conservation, TPA transfers the PF energy and momentum to a decreasing number of fast electrons, as slower electrons continuously fall behind the pulse (Liang & Nishimura 2004). We find that the maximum Lorentz factor achieved by TPA is unlimited until radiation damping or dephasing (e.g. due to wave-front curvature) become important. PIC simulations show that TPA always accelerates the high-energy electrons into a simple power law independent of the initial conditions or the pulse width (Fig.2b). The TPA mechanism is exceedingly robust and efficient, typically converting > 50% of the EM energy into accelerated particle energy over a short distance(see Sec.7). Physically both LPA and TPA are caused by a relativistic E x B drift in which the transverse EM field comoves with the particle drift velocity. The key difference is that in LPA the plasma load snowplowed by the EM pulse increases or stays constant with time, thereby limiting the Lorentz factor, whereas the TPA Lorentz factor increases indefinitely due to decreasing plasma loading (Liang et al 2003, Liang and Nishimura 2004). Density-wise, LPA involves collisionless compression of the plasma, while TPA involves rarefaction of the plasma. In contrast to shocks, in which bulk flow energy is converted into EM energy, either via compression of upstream fields or via Weibel (1959) and other instabilities, LPA and TPA converts EM energy into accelerated particle energy. We emphasize that both LPA and TPA are strictly kinetic phenomena with no analog in the MHD limit. The detailed physics of LPA and TPA has been reviewed extensively elsewhere (Liang and Noguchi 2005, 2006), so they will not be repeated here. TPA may be relevant to radiation in astrophysics at two different levels: global and local. Globally, EM pulses with large-scale ordered fields may be generated due to reconnection of a magnetic tower jet or magnetar wind (Koide et al 2004), or from the merger of strongly magnetized neutron stars into a black hole. For example, TPA can take place when the front end of a disconnected magnetic tower jet propagates down the steep density gradient of the collapsar envelope and turns into an unconfined kinetic EM pulse (see Sec.7). Similarly, when a millisecond magnetar or a merging strongly magnetized neutron star binary collapses to form a black hole, part of its energy may be emitted in the form of an intense EM pulse. However, TPA may also occur at the local level even in the absence of large scale ordered EM fields. For example, relativistic EM turbulence generated by shocks and shear layers may dissipate via the TPA mechanism as the nonlinear waves propagate into low density regions with Ωe/ωpe>1. In this case sustained comoving particle acceleration can last locally until dephasing occurs. We emphasize that the only piece of physics invoked in the radiation calculation below is that the particle is accelerated locally by a nearly comoving transverse EM wave. No assumption about the global geometry, topology or size scale of the EM field is required. Hence the radiation formulas derived in this paper have more general validity and much broader applications than the small scale PIC simulations performed so far based on the simplistic LPA and TPA scenarios (Liang and Noguchi 2005, 2006). 3. NUMERICAL RADIATION POWER OUTPUT In this paper we focus on the radiation output of electrons (and positrons) accelerated by a comoving kinetic Poynting flux. Numerically, we compute the radiation power output by incorporating the relativistic dipole formula (Rybicki & Lightman 1979) into our PIC code: Prad = 2e2 (F||2 + γ2F+2)/3m2 (1) where γ = Lorentz factor, F|| = force component along velocity v, and F+ = force component orthogonal to v. We compute the total power loss of each superparticle in the PIC simulation, by interpolating the field data from the cell boundaries to the instantaneous superparticle position, so that F and v refer to the same time and space point. We have carefully calibrated this numerical procedure with known analytic results. Fig.3 compares the numerical radiation output for an isotropic thermal plasma in a static uniform B field with that computed using the analytic synchrotron formula (Rybicki and Lightman 1979). Their excellent agreement, especially for the high-energy electrons, validates our numerical algorithm. However, PIC simulation cannot be used to compute the radiation spectrum numerically because the PIC simulation time step (typically = 0.25 gyroperiod) is too large to accommodate the high frequencies. The upper panels of Figs.4 & 5 illustrate the evolution of the Prad distribution of superparticles in sample LPA and TPA runs. In both cases a plane, linearly polarized EM pulse accelerates the same slab of overdense e+e- plasma, one from the outside and one from the inside. While the energies of the pairs increase monotonically due to the acceleration, the power radiated by the electrons initially rises to a maximum, but then declines monotonically. We find that in both cases Prad << the classical synchrotron power Psyn = 2e4B2p+2/3m2 (mp+=momentum orthogonal to B). This suppression of radiative power can be understood as follows. As particles are accelerated to higher and higher γ, v aligns increasingly with the Lorentz force F. So the F+ term in Eq.(1) decreases relative to the F|| term. However, Psyn comes only from the F+ term (Rybicki & Lightman 1979). Hence Psyn >> Prad for high γ particles. In Sec.5 we will rigorously derive a general analytic formula for Prad. Here we will only state that, for electrons almost comoving with the EM pulse, Prad can be approximated by: Panalytic = Psyn sin4α (2) where α is the angle between p+ and the Poynting vector k (Fig.6). Fig.7 compares the numerical Prad with Panalytic for the runs of Figures 4 & 5. It shows good correlation for the high-γ particles. Since sinα <<1 for high-γ particles (c.f. Fig.8), Eq.(2) explains why Prad << Psyn. The rise and decline of Prad in Figs.4 & 5 are caused by the competition between increasing γ (and Psyn) and decreasing α. 4. RADIATION CRITICAL FREQUENCY A prominent feature of GRB and blazar spectra is the presence of a low energy spectral break Epk (hundreds of keV for classical GRBs, radio-IR for blazars). This spectral break is an indicator of the overall spectral hardness, and is usually interpreted as the critical frequency of synchrotron radiation ωcrsyn ~ 1.5Ω eγop+o (Rybicki & Lightman 1979) by electrons with low energy cutoff γo. This interpretation of the spectral break, together with some assumptions about energy equipartition, is often used to constrain the Lorentz factor and magnetic field of the source. However, as we show below, for radiation emitted by TPA and LPA electrons, the asymptotic critical frequency ωcr is << ωcrsyn. To derive the formula for ωcr, we follow the approach of Landau and Lifshitz (1980): ωcr is determined by the time (measured in detector frame) it takes the radiation beam of opening angle 1/γ to sweep past the detector due to the curvature of the particle trajectory. For electrons comoving or almost comoving with the PF, the parallel momentum px (x is the direction along k, Fig.6) increases monotonically while pz (momentum along E) asymptotes to a constant (Liang & Nishimura 2004, note that py along B is constant to first order). Hence the change in the radiation beam direction due to bending of particle trajectory is dominated by the change in px: Δθ ~ pzΔpx/px2. From the Lorentz force equation we have dγ/dt=eEzpz/mγ. Hence the time in the laboratory frame for the radiation beam to change by an angle Δθ~2/γ is Δt=2γ2m/(eEzpz2) where we have used the approximation γ~px(>>pz, py). This translates into a duration in the detector frame Δtob=Δt/2γ2=m/eEzpz2. Thus the critical frequency (Rybicki and Lightman 1979): ωcr=1.5/Δtob=1.5eEzpz2/m=1.5Ωepz2=1.5Ωe p+2sin2α ∼ ωcrsyn sin2α (3). Since sinα<<1 at high γ (Fig.8), ωcr << ωcrsyn In Sec.6 we will discuss the major implications of this result for modeling GRB and blazar data. . In the lower panels of Figs.4 & 5 we plot the evolution of ωcr of the same LPA and TPA runs. These snapshots highlight the evolution of the spectral hardness. Again we see that ωcr first rises to a maximum before declining monotonically due to the competition between increasing γ and decreasing α. However the decline of Prad is more rapid than ωcr due to the extra factors of sinα. 5. RADIATION POWER FORMULA FOR KINETIC PF ACCELERATION In this section we derive a general analytic approximation for the power radiated by an electron accelerated locally by a comoving kinetic PF. While the following derivation assumes linearly polarized plane waves for simplicity, the results should be valid in general 3D geometry as long as the wave front curvature and transverse gradients are << 1/(acceleration distance) (distance for e-folding increase in γ). We emphasize that this radiation formula should be applicable to any acceleration by transverse EM fields almost comoving with the local ExB drift velocity. Hence its potential applications in astrophysics are much broader than the specific LPA or TPA scenarios discussed above. For particle motion in a linearly polarized plane wave with (E,B) = (Ez, By) (Fig.6), we have Fx=-evzBy; Fy=0; Fz=e(Ez+vxBy). Here x is the direction of Poynting vector k. After a little algebra we find: F||=eEzvz/v; F+2=e2By2[sin2α(v2-vw2)+(vx-vw)2] (4) where vw=-Ez/By is the local profile speed of the EM field (vw<1 due to plasma loading) and sinα=vz/v. Substituting Eq.(4) into Eq.(1) we obtain: Panalytic = 2e4By2[sin2α(γ2-1)(1-vw2)+γ2(vx-vw)2]/3m2 (5). Hence the power radiated by a PF-accelerated electron depends in general on two key parameters: the local EM field profile speed vw and the angle α between velocity v and Poynting vector k. Eq.(5) simplifies in various special limits: A. Comoving particles (vx=vw): In this case Eq.(5) simplifies to Panalytic = 2e4By2(pz2+py2)sin2α/3m2 when γ>>1. In all of our runs, pz>>py at late times. So this reduces to Panalytic = 2e4By2p+2sin4α/3m2 which is Eq.(2). As Fig.7 shows, Eq.(2) is a good approximation for electrons comoving with vw. However, Eq.(2) is not a good approximation for electrons out of phase with vw (note that electrons can have vx > vw or vx < vw). Liang and Nishimura (2004) suggested that vw corresponds roughly to the peak γ of the particle distribution function f(γ). B. Vacuum pulse limit (vw=1): In the limit vw=1, the PF propagates as a vacuum EM pulse, Eq.(5) becomes for γ >>1: Panalytic =2e4By2γ2 (1-vx)2/3m2~e4By2γ2sin4α/6m2 (6) Since p+ ~ γ, Eq.(6) has the same form as Eq.(2) but is a factor of 4 less in magnitude. It defines the lower limit to the radiative power loss of a PF accelerated electron since in reality vw<1. C. Slightly subluminal PF (1-vw=ε<<1) For most astrophysics applications, the PF will be slightly subluminal. We can simplify Eq.(5) by Taylor expanding 1-vw=ε<<1 to lowest order. Eq.(5) then reduces to Panalytic ~ 2e4By2γ2(ε + sin2α/2)2/3m2 (7). This formula fully explains the physical origin of the result Prad << Psyn. In relativistic PF acceleration, both ε and sinα are <<1. Fig.9 shows an example for which the best-fit ε = 0.03. Eq.(7) shows that Panalytic behaves differently depending on whether ε >> or << sin2α/2. In the former case Panalytic depends only on the EM field profile speed vw and not on α: Panalytic ~2e4By2γ2ε2/3m2 (8) In the second case we regain Eq.(6) which depends only on α and not on vw. Therefore when we model astrophysical data using these formulas, we obtain different physical information depending on the ratio ε:sin2α/2, which depends on the PF initial condition, such as To, Ωe/ωpe .etc. Eqs.(2), (6) & (8), which contain only 3 unknowns: (By, γ, α) or (By, γ, ε), are much easier to use for modeling astrophysical data than Eq.(5) or Eq.(7), which contain 4 unknowns. In practice, ε fluctuates rapidly in both time and space so its extraction from astrophysical data would be more difficult, whereas PIC simulation results suggest that sinα has a narrower range for high-γ particles (Fig.8). So the above analytic approximation is most useful for data modeling in the regime ε << sin2α/2, which seems to be the case for GRB’s (cf. Sec.7) and may also be the case for blazars. We emphasize that the above analytic radiation formulas are derived from first principles independent of any PIC simulation results. Hence their validity is completely independent of any numerical simulation size scale. In fact we have carefully validated these analytic formulas using simulations spanning a dynamic range of >105 (i.e. PF pulse widths ranging from 102 to107 gyroradii). One may argue that the above result comes about only because we work in the lab frame, and that the classical synchrotron formula must apply if we transform to a (primed) Lorentz frame in which E’ = 0 and B’ is static. This is indeed true, and one can rederive the above formulas using appropriate Lorentz transformations of the classical synchrotron power formula (Rybicki and Lightman 1979). However, finding the Lorentz frame with E’ = 0 is impractical, since E/B varies rapidly in both space and time due to modulation by self-generated currents and current instabilities (Liang and Nishimura 2004). There is no single Lorentz transformation that can lead to E’ = 0 for any meaningful fraction of the EM pulse. So in practice it is more convenient to work in a global lab frame, measure E, B and p in this frame and use the above analytic formulas. Such lab-frame quantities are more relevant to astrophysical observations anyway. 6. IMPLICATIONS FOR ASTROPHYSICAL PF MODELS Using PIC simulations and analytic methods, we have demonstrated above that electrons accelerated by Poynting flux which comoves with the local ExB drift velocity, radiate at a rate (in the lab-frame) much below the classical synchrotron power, and the critical frequency of their radiation spectrum is also much below the classical synchrotron critical frequency. These results have major implications for the interpretation of astrophysical data from GRBs and blazars, if their energy supply is coming from Poynting flux. One scenario would be that the classical synchrotron model applies only to the radiation zone, which is separate from the particle acceleration zone, and the synchrotron model (B, γ) values refer only to the radiation zone but not the acceleration zone, which likely has higher B and γ. An alternative scenario is that the observed radiation is intrinsic to the acceleration process. Then we have to use Eq.(5) or its various limits (Eq.(2)-(8)) to model the astrophysics data. This will lead to much higher (B, γ) values for the source than in conventional synchrotron models since ε and α are <<1. Clearly the overall energetics and parameters of the two models will be very different and such differences may be testable. As an example of the application of the results of Secs. 4 & 5, we will consider a simplistic model of long GRBs in Sec.7. Another important consequence of the suppression of synchrotron radiation in PF scenarios is that inverse Comptonization of external soft photons (EC) may dominate even when the (lab-frame) magnetic energy density greatly exceeds the external soft photon energy density (Rybicki & Lightman 1979), and the conventional SSC + EC model of blazars (Dermer & Boettcher 2002) may need to be revamped. These scenarios will be reconsidered in future papers. 7. APPLICATION TO A PF MODEL OF LONG GRB’s Currently there is no universally accepted model of GRB energization and radiation. Two popular paradigms are hydrodynamic versus electromagnetic outflows from a central engine (e.g. a newly formed black hole), dissipating at a distance of 1014-15 cm via nonthermal electrons and gamma-rays (Meszaros 2002, Piran 2000). If GRBs are indeed the manifestations of intense PF outflow, a dissipation mechanism such as TPA is attractive due to its high energy conversion efficiency and power-law spectra (cf. Sec.2). Liang and Nishimura (2004) also pointed out several tantalizing similarities between the observable properties of TPA in e+e- plasmas and GRB phenomenology. To demonstrate the utility of the results of Secs.4&5, here we apply the analytic radiation formulas to a simplistic “toy” model of classical long GRBs, assuming that the PF contains only e+e- pairs with no ions (e-ion models will be considered in subsequent papers). We will derive the value of the spectral break energy Epk from these formulas. The underlying astrophysical framework is that some central engine activity lasting 10’s of seconds launches an intense EM pulse of width ~ 1012 cm and energy ~1051 ergs, loaded with only low-density e+e- plasma so that Ωe/ωpe >1. This intense EM pulse initially propagates through the collapsar envelope as a non-dissipative subluminal MHD pulse as long as the ambient density is high enough so that the formal Alfven speed vA=B/(4πρp)1/2 < c (ρp= ambient proton mass density). But the pulse will eventually reach a point where the envelope ion density is so low that vA > c, and the MHD pulse turns into a freely-expanding kinetic EM pulse. This triggers the TPA dissipation and rapid conversion of EM energy into e+e- kinetic energy. We have performed simulations of relativistic magnetosonic pulses propagating down steep density gradients. The preliminary results seem to support the above picture. For long GRBs it is useful to scale the burst parameters with the following benchmark values (Fishman & Meegan 1998, Preece et al 2000): total energy E51=Etot/1051erg, burst duration T30=T90/30sec, prompt-γ emission distance R14=R/1014cm. We assume that the EM pulse is a quasi-spherical shell with thickness ΔR=cT90=1012cmT30 (in this section we write out c explicitly) and solid angle Ω4π=Ω/4π. To simplify the model we assume that the shell is uniform with mean field B and mean lepton (e- + e+) density n. All physical quantities are measured in the “lab- frame”, which we assume to be the rest frame of the GRB central engine or host galaxy. In reality the field, density and momentum profiles are highly structured due to current instabilities (Liang and Nishimura 2004), and the following parameters refer primarily to those leptons at the peak of the momentum distribution function. All our simulations with pair plasmas suggest that at late times, particle energy Eparticle ≥ 0.6Etot, EM energy (= 2EB)≤ 0.4Etot (Fig.10, see also Liang et al 2003). Let N = total number of leptons (e+ + e-) in the pulse and Γ = average Lorentz factor of the lepton distribution = <γf(γ)>. (Γ ~ the group veocity Lorentz factor of the EM pulse Γw=(1-vw2)-1/2, Liang and Nishimura 2004). We thus have dimensionally in cgs units: NΓmc2 ~ 6x1050 E51 (9) B2ΔRR2Ω~16πx1050 E51 (10) Eq.(10) gives: B ~2x105 G (R14-1 Ω4π-1/2 E511/2T30-1/2) (11) Next we estimate Γ for the radiation epoch by invoking the condition: radiative cooling rate = particle acceleration rate. Liang and Nishimura (2004) derived from the Lorentz force equation the particle acceleration rate dΓ/dt = fΩe/Γ, where f is a fudge parameter of O(1) that depends on the initial conditions. We emphasize that this formula depends solely on the comoving nature of the EM pulse and is independent of other global assumption of the TPA. Using Eq.(7) and assuming ε << sinα (and check for consistency below), we obtain: f ecB/Γ = e4By2Γ2sin4α/6m2c3 (12). The peak value of the late-time sinα distribution for high-γ electrons (Fig.8) lies in the range 0.01 to 0.2 in the simulations performed so far. Hence we scale sinα with 0.1 below: α.1=sinα/0.1. Solving Eq.(12) for Γ we obtain: Γ ~1.2x105 (f 1/3 R141/3 Ω4π1/6 E51-1/6T301/6α.1-4/3) (13) Hence ε~1/Γ << sinα and our assumption is justified. Note that this Γ is measured in the lab- frame so it is actually smaller than the composite Lorentz factor of internal shock models (with a bulk Lorentz factor of 102 times internal Lorentz factors of 103-104). Using this in Eq.(9) we find: N~ 6x1051 (f -1/3 R14-1/3 Ω4π-1/6 E517/6T301/6 α.1 4/3) (14) From Eqs.(3), (11) and (13) we obtain the value of the spectral break energy, taken as the critical frequency corresponding to Γ: Epk = hωcr/2π ~ 490 keV(f 2/3 R14-1/3 Ω4π-1/6 E511/6T30-1/6 α.1 -2/3) (15) This value agrees with the observed spectral breaks of typical long GRBs: Epk ~ 250 keV (1+z) ~ 500 keV for z ~1 (Preece et al 2000) in the host-Galaxy frame. Note that Epk in Eq.(15) depends only weakly on the various uncertainty factors. Eq.(14) gives the mean lepton density: n = N/(ΩΔRR2)~ 5x1010(f -1/3 R14-7/3 Ω4π-1/6 E517/6T30-7/6 α.14/3) (16) and the frequency ratio: Ωe/ωpe ~ 250 (f 1/6 R141/6 Ω4π-5/12 E51-1/12T301/12 α.1-2/3) >>1 (17) which justifies our EM-domination assumption. At this density the pairs are completely collisionless (Coulomb mean free path > 1020cm). We note that the local acceleration (=cooling) time of an individual electron with the above B, Γ and sinα values is very short: tcool =tcool ~10-2 sec. However, this should not conflict with the 30 second observed GRB duration. The radiation duration of the shell is determined by the conversion time of overall EM energy into particle energy, which is proportional to the light crossing time of the shell thickness ΔR/c, since the EM pulse takes at least that long to emerge and energize the embedded plasma (Fig.10). Moreover, radiation emitted by the front and back of the plasma arrive at the detector with a time delay of ΔR/c. These two effects combine to make the GRB duration measured by the detector ~ ΔR/c = 30 sec, irrespective of the short acceleration/cooling time of individual leptons. An analogy with internal shock models is in order here. Even though the internal shock thickness and particle acceleration length are << 1012 cm, the GRB radiation duration is governed by the shock crossing time of the colliding shells, which have thicknesses of ~ 1012 cm. We note that 1012cm corresponds to ~1014 gyroradii. Even the acceleration/cooling distance of ~108cm equals 1010 gyroradii. Both scales are much larger than the largest PIC simulation we have performed (~107 gyroradii). Hence it is tempting to question the applicability of our results to the GRB regimes. But we emphasize that the only physics used to derive the radiation rate Eq.(7) and the acceleration rate of Liang and Nishimura (2004) is the comoving assumption of the local EM wave with the local ExB drift speed. This assumption is completely independent of the global geometry, structure and size scale of the fields and plasmas, which affect only the duratiion and longevity of the acceleration process. In addition, we have validated the analytic radiation and acceleration rates with simulations spanning over 4 decades in physical size (103 to107 gyroradii). This gives us added confidence in their general validity. However one puzzle remains. What makes the EM pulse dissipation to occur at R~1014 cm from the central engine, two orders of magnitude larger than the EM pulse width and six orders of magnitudes larger than the lepton acceleration/cooling length? We speculate that it may be the environment which determines this dissipation distance. Here we venture a somewhat speculative but plausible scenario that gives rise to such a large dissipation distance. In reality, the TPA action takes place not at a sharp boundary but in an external density gradient whose scale height is much larger than the acceleration length. In such cases we believe that the Liang-Nishimura (2004) acceleration rate and the radiation cooling rate of Sec.5 are applicable only when the ambient ion mass density drops below the internal pair mass density. Otherwise the EM expansion and particle acceleration would be inhibited by the external ion inertia (cf. discussion at the beginning of Sec.7). In the collapsar model the GRB progenitor is likely surrounded by a Wolf-Rayet wind whose mass density ~ A 5x1011 r-2 g.cm-1 (Chevalier and Li 2000) where the parameter A depends on the mass loss rate. Hence the PF “breakout” or dissipation distance, using the pair density of Eq.(16), becomes rbreakout ~ A1/21014 cm. In other words, the TPA action is inhibited until the PF reaches an ambient ion mass density of ≤ 5x10-17 g.cm-3, and this can only occur at a distance ≥1014 cm in a Wolf-Rayet wind. Fig.11 illustrates the relevant length scales of this scenario. 8. DISCUSSION AND SUMMARY Using PIC simulations and analytic theory, we have shown in this paper that when electrons are accelerated by a comoving Poynting flux with Ωe/ωpe >1, the in-situ radiation power output and critical frequency are much lower than those given by the classical synchrotron formulas. This is because the most energetic electrons have their momentum closely aligned with the local Poynting vector or ExB drift direction. We apply our analytic formulas for the radiation power output and critical frequency to a simple PF model of classical long GRBs, and find that the predicted spectral break energy lies in the range of observed data. Besides the LPA and TPA mechanisms which involve ponderomotive acceleration of overdense plasmas, there are many other Poynting flux scenarios that may result in nonthermal particle acceleration. For example, electron acceleration by comoving longitudinal wakefields generated by PF in an underdense plasma (similar to laser accelerators in the laboratory, Tajima and Dawson 1979) may occur in special astrophysical situations. We have also not considered Poynting flux propagating along flow-aligned guide fields such as Alfven and whistler waves. Preliminary PIC simulation results suggest that linear Alfven waves (ΔB << Bo where Bo is the longitudinal guide field) cannot accelerate nonthermal particles efficiently via the ponderomotive force, since the net E x B drift direction is misaligned from the (strong) guide field. On the other hand, if the Alfven wave is highly nonlinear (ΔB > Bo), it behaves like transverse EM waves. Then the TPA results may apply to first order. 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Figure Captions Fig.1 Sample PIC simulation outputs illustrating the two different mechanisms of kinetic Poynting flux acceleration of an e+e- plasma slab: (a) in LPA, an intense EM pulse incident on an overdense plasma interface induces a ponderomotive force (JxB is along Poynting vector k) that snowplows relativistically all upstream electrons which must share the Poynting flux momentum, thus limiting their Lorentz factor; (b) in TPA, an intense EM pulse escaping from an overdense plasma induces a ponderomotive force which pulls out the surface electrons relativistically. Only the fastest electrons can keep up with the EM pulse, so that the plasma loading of the EM pulse decreases with time. TPA leads to the sustained comoving acceleration of a decreasing number of fast electrons, with no limit to their Lorentz factor. In all figures of this paper, x is expressed in units of 3c/ωpe. Fig.2 (a) Electron energy spectrum accelerated by LPA resembles a superthermal quasi- Maxwellian distribution; (b) electron energy spectra accelerated by TPA for different initial PF thicknesses (103 and 104 c/ωpe) both show a robust power-law of index ~ -3 to -4. The low- energy spectral breaks correspond roughly to the Lorentz factor of the EM pulse group velocity . Fig.3 Calibration of the numerical radiation power Prad computed from the PIC simulation (Eq.(1)) against the analytic synchrotron formula Psyn for a 5 MeV thermal plasma in a static uniform B field shows excellent agreement. The scatter at low energies is due to small errors from interpolating the field values to the particle position. In all figures of this paper, Prad, Psyn and Panalytic are expressed in units of 2e2Ωe2/2700. Fig.4 Upper panel: Snapshots of Prad distribution (dots) and By profile (solid, in units of Bo/15) vs. x for an e+e- plasma slab initially located at x=180 with n=16ncr, thickness = 12c/ωpe and snowplowed by a vacuum EM pulse with Ωe/ωpe=10 from left to right. Prad of the accelerated electrons reaches a maximum at ~2 light crossing times after the EM pulse hits the plasma surface, followed by rapid monotonic decay. This behavior is caused by the decrease in angle α competing with the increase in electron energy. Lower panel: Snapshots of the critical frequency (Eq.3) distribution show that the evolution of the spectral hardness of radiation follows that of Prad. ωcr is expressed in units of10Ωe. Fig.5 Upper panel: Snapshots of Prad distribution (dots) and By profile (solid, in units of Bo/150) vs. x for a e+e- plasma slab accelerated by TPA with initial plasma temperature kTo=0.005m, thickness Lo=12c/ωpe, Ωe/ωpe=10 and initially located at x=180. Prad of the accelerated electrons reaches a maximum at ~5 light crossing times after the emergence of the EM pulse, followed by monotonic decay which is slower than in the LPA case. Lower panel: Snapshots of the critical frequency (Eq.3) distribution shows that the evolution of the spectral hardness of radiation follows that of Prad. ωcr is expressed in units of10Ωe. Note that ωcr of Fig.5 is a factor of 10 larger than that of Fig.4. The radiation at t=0 is thermal cyclotron radiation due to the finite initial temperature. But it has very low ωcr Fig.6 Diagram showing the angle α between the Poynting vector k and p+, the momentum component orthogonal to B. Fig.7 Scatter plot of Prad compared with Panalytic of Eq.(2) for the runs of (a) Fig.4 and (b) Fig.5. At these times most of the high–γ particles are comoving with the EM pulse. Fig.8 Scatter plot of the distribution of Lorentz factor γ vs. sinα=pz/γ for a Ωe/ωpe=10 e+e- slab accelerated by TPA shows that the highest–γ particles have their sin α distribution peaking in the range ~ 0.01- 0.2. Fig.9 Scatter plot of Prad compared with Panalytic of Eq.(7) for an Ωe/ωpe=10 e+e- slab accelerated by TPA, At this time the best correlation for high-γ electrons is obtained when ε = 0.03. Fig.10 Decay curves of EM energy for an Ωe/ωpe=10, kTo=5MeV e+e- slab TPA’s with different initial thicknesses: (A) Lo=10800c/ωpe; (B) Lo=90c/ωpe; (C) Lo=12c/ωpe. This confirms that the conversion time of EM energy into particle energy is directly proportional to the light crossing time Lo/c. Fig.11 Diagram illustrating the different size scales in the “breakout” of a PF from a Wolf-Rayet wind model of long GRBs. The wavy arrow denotes the (lab-frame) PF thickness (ΔR=1012cm) along the observer line of sight. The PF breakout distance (~1014 cm) is determined by the radius at which the wind mass density drops below the PF pair mass density (~5x10-17 g.cm-3). Despite the short acceleration/cooling length (~3x108cm) of individual leptons accelerated by the PF, the detector-measured GRB duration at infinity is ~ΔR/c=30 sec due to the transit time of the PF crossing rbreakout and the light path difference between the front and back of the PF (upper-right space-time diagram). Plasma Plasma Entering Exiting EM pulse Fig.1 Fig.2 Fig.3 Fig.6 Fig.7 Panalytic sin α Fig.8

A New Algebraic Structure of Finite Quantum Systems and the Modified Bessel Functions Kazuyuki FUJII ∗ Department of Mathematical Sciences Yokohama City University Yokohama, 236-0027 Japan Abstract In this paper we present a new algebraic structure (a super hyperbolic system in our terminology) for finite quantum systems, which is a generalization of the usual one in the two–level system. It fits into the so–called generalized Pauli matrices, so they play an important role in the theory. Some deep relation to the modified Bessel functions of integer order is pointed out. By taking a skillful limit finite quantum systems become quantum mechanics on the circle developed by Ohnuki and Kitakado. ∗E-mail address : fujii@yokohama-cu.ac.jp http://arxiv.org/abs/0704.1844v2 Quantum Computation is usually based on two–level system of atoms (qubit theory). In the realistic construction of quantum logic gates we must solve some Schrödinger equa- tions. Then the Pauli matrices {σ1, σ3} is essentially used and not only the periodic functions {cos(x), sin(x)} but also the hyperbolic functions {cosh(x), sinh(x)} play an important role. On the other hand, they are deeply related to the modified Bessel functions of integer order {In(x) | n ∈ Z}. The functions are in general given by the generating function. Atom has usually many (finite or infinite) energy levels. However, to treat infinitely many ones at the same time is not realistic, so we treat an atom with finite (for example n) energy levels. We call this a finite quantum system and for this system the so–called generalized Pauli matrices {Σ1,Σ3} play a crucial role, see for example [1], [2] and [3]. In this system we have a natural question on what functions corresponding to the hy- perbolic functions are. In the paper we present such a system {c0(x), c1(x), · · · , cn−1(x)} (a super hyperbolic system in our terminology) as a “natural” generalization of {cosh(x), sinh(x)}. Moreover, we define a generating matrix based on the generalized Pauli matrices as a “natural” generalization of the generating function and obtain interesting results by taking some traces. Lastly, we want to take a limit of finite quantum systems, which is of course impossible. However, there is a bypass. That is, by taking a skillful limit finite quantum systems become quantum mechanics on the circle developed by Ohnuki and Kitakado [4]. Through this paper we have a clear and unified picture of quantum systems. First of all we make some mathematical preliminaries on the 2–level system. Let {σ1, σ2, σ3} be Pauli matrices and 12 the unit matrix :  0 1  , σ2 =  0 −i  , σ3 =  1 0  , 12 =  1 0  . (1) List the well–known properties of σ1 and σ3 : σ21 = σ 3 = 12, σ 1 = σ1, σ 3 = σ3, σ3σ1 = −σ1σ3 = eπiσ1σ3. (2) Let W be the Walsh–Hadamard matrix  1 1  = W−1 , (3) then we can diagonalize σ1 as σ1 = Wσ3W −1 by making use of W . The modified Bessel functions of integer order {Ik(x) | k ∈ Z} are given by the generating function (w+ 1 Ik(x)w k. (4) Now let us list some (well–known) important properties (see for example [5]) : 1 = I0(x) + 2 (−1)kI2k(x), ex = I0(x) + 2 Ik(x), e −x = I0(x) + 2 (−1)kIk(x) cosh(x) = I0(x) + 2 I2k(x), sinh(x) = 2 I2k−1(x). In the following we set c0(x) ≡ cosh(x) = (2k)! , c1(x) ≡ sinh(x) = x2k+1 (2k + 1)! for simplicity. The fundamental equation c20(x)− c21(x) = 1 (6) is interpreted as a simple relation Sσ3S = σ3 ⇐⇒ σ3Sσ3S = 12 ⇐⇒ (σ3S)2 = 12 for S defined by  c0(x) c1(x) c1(x) c0(x)  = c0(x)12 + c1(x)σ1 = exσ1 . (7) Next we would like to extend the 2–level system to general n–level one. To make our purpose clearer we treat the 3–level case in detail. Let σ be exp(2πi ), then we have σ3 = 1, σ̄ = σ2, 1 + σ + σ2 = 0. (8) Let Σ1 and Σ3 be generators of generalized Pauli matrices in the case of n = 3, namely , Σ3 = . (9) Then it is easy to see Σ31 = Σ 3 = 13, Σ 1 = Σ 3 = Σ 3, Σ3Σ1 = σΣ1Σ3. (10) Now we can show that Σ1 can be diagonalized by making use of the matrix 1 1 1 1 σ2 σ 1 σ σ2 ∈ U(3) (11) Σ1 = WΣ3W † = WΣ3W −1. (12) In fact 1 1 1 1 σ2 σ 1 σ σ2 1 1 1 1 σ σ2 1 σ2 σ 0 0 3 3 0 0 0 3 0 = Σ1, where we have used the relations in (8). From (5) we set c0(x) = (3k)! , c1(x) = x3k+1 (3k + 1)! , c2(x) = x3k+2 (3k + 2)! . (13) Then it is easy to check c0(x) = ex + eσx + eσ , c1(x) = ex + σ2eσx + σeσ , c2(x) = ex + σeσx + σ2eσ by use of σ in (8) or reversely ex = c0(x)+c1(x)+c2(x), e σx = c0(x)+σc1(x)+σ 2c2(x), e σ2x = c0(x)+σ 2c1(x)+σc2(x). Now, our question is as follows : What is the fundamental equation that {c0(x), c1(x), c2(x)} satisfy ? The answer is given by the equation (c0(x) + c1(x) + c2(x))(c0(x) + σc1(x) + σ 2c2(x))(c0(x) + σ 2c1(x) + σc2(x)) = exeσxeσ 2x = e(1+σ+σ 2)x = e0 = 1. By expanding the left-hand side and using the relations (8) we obtain c30(x) + c 1(x) + c 2(x)− 3c0(x)c1(x)c2(x) = 1. (15) Next let us consider the addition formulas. By expanding eσxeσy = eσ(x+y) ⇐= eσt = c0(t) + σc1(t) + σ2c2(t) we have c0(x)c0(y) + c1(x)c2(y) + c2(x)c1(y) = c0(x+ y), c0(x)c1(y) + c1(x)c0(y) + c2(x)c2(y) = c1(x+ y), c0(x)c2(y) + c1(x)c1(y) + c2(x)c0(y) = c2(x+ y). (16) From here let us give a unified approach by use of the generalized Pauli matrices {Σ1,Σ3} above. We consider the matrix exΣ1 = c0(x)13 + c1(x)Σ1 + c2(x)Σ c0(x) c2(x) c1(x) c1(x) c0(x) c2(x) c2(x) c1(x) c0(x) . (17) Then by Σ1 = WΣ3W † in (12) c0(x) c2(x) c1(x) c1(x) c0(x) c2(x) c2(x) c1(x) c0(x) = exΣ1 = W exΣ3W † = W so taking the determinant leads to ∣∣∣∣∣∣∣∣∣ c0(x) c2(x) c1(x) c1(x) c0(x) c2(x) c2(x) c1(x) c0(x) ∣∣∣∣∣∣∣∣∣ = e(1+σ+σ 2)x = 1. Namely, we recovered (15). On the other hand, by use of (11) it is straightforward to show W † = 1 1 1 1 σ2 σ 1 σ σ2 1 1 1 1 σ σ2 1 σ2 σ ex + eσx + eσ ex + σ2eσx + σeσ ex + σeσx + σ2eσ so we recovered (14). The matrix form is very convenient. Moreover, we can give new relations. For that we consider the simple equation exΣ1eyΣ 1 = exΣ1+yΣ 1 . (18) The left hand side is exΣ1eyΣ 1 = (c0(x)13 + c1(x)Σ1 + c2(x)Σ 1)(c0(y)13 + c1(y)Σ 1 + c2(y)Σ1) = (c0(x)c0(y) + c1(x)c1(y) + c2(x)c2(y))13 + (c0(x)c2(y) + c1(x)c0(y) + c2(x)c1(y))Σ1 + (c0(x)c1(y) + c1(x)c2(y) + c2(x)c0(y))Σ because Σ 1 = Σ 1. The right hand side is exΣ1+yΣ 1 = eW (xΣ3+yΣ )W † = W exΣ3+yΣ 3W † = W exσ+yσ ex+y + exσ+yσ + exσ ex+y + σ2exσ+yσ + σexσ ex+y + σexσ+yσ + σ2exσ so we obtain c0(x)c0(y) + c1(x)c1(y) + c2(x)c2(y) = ex+y + exσ+yσ + exσ c0(x)c2(y) + c1(x)c0(y) + c2(x)c1(y) = ex+y + σ2exσ+yσ + σexσ c0(x)c1(y) + c1(x)c2(y) + c2(x)c0(y) = ex+y + σexσ+yσ + σ2exσ . (19) Next, let us consider the matrix exΣ1+yΣ 1 . If we set y = 1/x, then the matrix exΣ1+(1/x)Σ 1 is similar to (4) the generating function of modified Bessel functions of integer order. Therefore from (4) it is reasonable to consider (wΣ1+ 1 1) = e (wΣ1+ 1 1 ) = Ik(x)w kΣk1 . (20) In the following we call this the generating matrix of modified Bessel functions of integer order. Let us look for some typical properties. The result is (wΣ1+ 1 (w+ 1 ) + e (wσ+ 1 σ2) + e (wσ2+ 1 I3k(x)w (wΣ1+ 1 (w+ 1 ) + σe (wσ+ 1 σ2) + σ2e (wσ2+ 1 I3k−1(x)w 3k−1, (wΣ1+ 1 1)Σ21 (w+ 1 ) + σ2e (wσ+ 1 σ2) + σe (wσ2+ 1 I3k−2(x)w where σ−1 = σ2 and σ−2 = σ. A comment is in order. In the case of n = 2 the generating matrix is (wσ1+ 1 1) = e (w+ 1 )σ1 = cosh 12 + sinh because σ1 is hermitian, so the situation becomes much easier. From the lesson for the case of n = 3, let us set up the general case. Let {Σ1,Σ3} be generalized Pauli matrices  . . . . . .  , Σ3 =  . . .  where σ is a primitive element σ = exp(2πi ) which satisfies σn = 1, σ̄ = σn−1, 1 + σ + · · ·+ σn−1 = 0. (23) Then it is easy to see Σn1 = Σ 3 = 1n, Σ 1 = Σ 1 , Σ 3 = Σ 3 , Σ3Σ1 = σΣ1Σ3. (24) If we define a Vandermonde matrix W based on σ as  1 1 1 · · · 1 1 1 σn−1 σ2(n−1) · · · σ(n−2)(n−1) σ(n−1)2 1 σn−2 σ2(n−2) · · · σ(n−2)2 σ(n−1)(n−2) 1 σ2 σ4 · · · σ2(n−2) σ2(n−1) 1 σ σ2 · · · σn−2 σn−1  , (25) then it is not difficult to see Σ1 = WΣ3W † = WΣ3W −1. (26) That is, Σ1 can be diagonalized by making use of W . We set cj(x) = xkn+j (kn+ j)! for 0 ≤ j ≤ n− 1. It is of course cj(x) and easy to see exΣ1 = c0(x)1n + c1(x)Σ1 + c2(x)Σ 1 + · · ·+ cn−2(x)Σn−21 + cn−1(x)Σn−11  c0(x) cn−1(x) · · · c2(x) c1(x) c1(x) c0(x) cn−1(x) · · · c2(x) c1(x) c0(x) cn−1(x) . . . . . . . . . cn−2(x) · · · c1(x) c0(x) cn−1(x) cn−1(x) cn−2(x) · · · c2(x) c1(x) c0(x)  . (28) Let us look for the fundamental equation that {c0(x), c1(x), · · · , cn−2(x), cn−1(x)} sat- isfy. By use of (26) exΣ1 = exWΣ3W = W exΣ3W † we have ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ c0(x) cn−1(x) · · · c2(x) c1(x) c1(x) c0(x) cn−1(x) · · · c2(x) c1(x) c0(x) cn−1(x) . . . . . . . . . cn−2(x) · · · c1(x) c0(x) cn−1(x) cn−1(x) cn−2(x) · · · c2(x) c1(x) c0(x) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ . . . ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ = ex(1+σ+σ 2+···+σn−2+σn−1) = e0 = 1 (29) because W is unitary (|W | = 1). For example n = 2 c20(x)− c21(x) = 1 (⇐= (6)) n = 3 c30(x) + c 1(x) + c 2(x)− 3c0(x)c1(x)c2(x) = 1 (⇐= (15)) n = 4 c40(x)− c41(x) + c42(x)− c43(x)− 2c20(x)c22(x) + 2c21(x)c23(x) −4c20(x)c1(x)c3(x) + 4c0(x)c21(x)c2(x)− 4c1(x)c22(x)c3(x) + 4c0(x)c2(x)c23(x) = 1. We call {c0(x), c1(x), · · · , cn−1(x)} the super hyperbolic system. The addition formulas are given by the simple equation exΣ1eyΣ1 = e(x+y)Σ1 and become cj(x+ y) = k+l=j (mod n) ck(x)cl(y) for 0 ≤ j ≤ n− 1. (31) More explicitly, cj(x+y) = c0(x)cj(y)+c1(x)cj−1(y)+· · ·+cj(x)c0(y)+cj+1(x)cn−1(y)+· · ·+cn−1(x)cj+1(y). The new relations are given by the simple equation exΣ1eyΣ 1 = exΣ1+yΣ and become ck(x)cn−j+k(y) + ck(x)ck−j(y) = σk(n−j)exσ k+yσn−k (32) for 0 ≤ j ≤ n− 1. The generating matrix of modified Bessel functions of integer order is given by (wΣ1+ 1 Ik(x)w kΣk1 (33) and from this we have (wΣ1+ 1 (wσl+ 1 σ−l) = Ink−j(x)w nk−j (34) for 0 ≤ j ≤ n− 1. The result in the case of j = 0 is known in [6] and [7]. We want to take a (formal) limit n −→ ∞. That is, what is Σ1 −→ ?, Σ3 −→ ? It is of course impossible to take a limit with this form. For that let us make a small change. We set n = 2N + 1 and Σ̃1 = Σ1, Σ̃3 =  . . . . . .  where σ = exp( 2πi ). Here we rewrite Σ̃3 as Σ̃3 = exp( G̃) where  . . . . . .  . (36) The commutator [G̃, Σ̃1] becomes [G̃, Σ̃1] =  0 −2N . . . . . .   . . . . . .  (mod 2N + 1). That is, we have the relation [G̃, Σ̃1] = Σ̃1 (mod 2N + 1). (37) In this stage, it may be better to write the (finite dimensional) Hilbert space as 2N+1 = VectC{|−N〉, · · · , |−1〉, |0〉, |1〉, · · · , |N〉} because G̃|n〉 = n|n〉. Now, if we take a formal limit N −→ ∞ then we have the fundamental relation [G,W ] = W (38) where  . . . . . .  , W =  . . . . . . 0 . . . . . .  where the notations {G,W} in [4] were used. Note that G is a hermitian operator and W a unitary operator on the Hilbert space L2(Z) = cn|n〉 | |cn|2 < ∞ ; W |n〉 = |n+ 1〉, G|n〉 = n|n〉. (40) The relation (38) is just the fundamental one in quantum mechanice on the circle devel- oped by Ohnuki and Kitakado [4]. A comment is in order. There is some freedom on the choice of G. That is, if we choose G like G −→ G+ α1, 0 ≤ α < 1 the relation (38) still holds. α is interpreted as a kind of abelian gauge induced in quantum mechanice on the circle. Therefore it may be better to write the generators {Gα ≡ G + α1,W} in place of {G,W} in [4]. We don’t repeat the contents, so see [4] and its references. Readers may find many interesting problems from the paper. For example, we can consider the generating operator (wW+ 1 W †). We leave some calculations to readers. In this paper we developed the super hyperbolic structure for (all) finite quantum systems, and defined the generating matrix for the modified Bessel functions of integer order and obtained some interesting results. We also gave a connection to quantum mechanics on the circle by Ohnuki and Kitakado by taking a skillful limit. Our motivation is to apply the development in the paper to qudit theory based on finite quantum systems, which will be reported in another paper. Acknowledgment. The author wishes to thank K. Funahashi for helpful comments and suggestions. References [1] K. Fujii : Exchange Gate on the Qudit Space and Fock Space, J. Opt. B : Quantum Semiclass. Opt, 5(2003), S613, quant-ph/0207002. http://arxiv.org/abs/quant-ph/0207002 [2] K. Fujii : How to Treat an N-Level System : A Proposal, quant-ph/0302050. [3] K. Fujii, K. Funahashi and T. Kobayashi : Jarlskog’s Parametrization of Uni- tary Matrices and Qudit Theory, Int. J. Geom. Meth. Mod. Phys. 3(2006), 269, quant-ph/0508006. [4] Y. Ohnuki and S. Kitakado : Fundamental Algebra for Quantum Mechanics on SD and Gauge Potentials, J. Math. Phys. 34(1993), 2827. [5] E. T. Whittaker and G. N. Watson : A Course of MODERN ANALYSIS, 1990, Cambridge University Press. [6] V. Barsan and S. Cojocaru : Bessel functions of integer order in terms of hyperbolic functions, math-ph/0703010. [7] S. Cojocaru : Green’s function of a finite chain and the discrete Fourier transform, Int. J. Mod. Phys. 20(2006), 593, arXiv : 0704.2898 (math-ph). http://arxiv.org/abs/quant-ph/0302050 http://arxiv.org/abs/quant-ph/0508006 http://arxiv.org/abs/math-ph/0703010

In this paper we present a new algebraic structure (a super hyperbolic system in our terminology) for finite quantum systems, which is a generalization of the usual one in the two-level system. It fits into the so-called generalized Pauli matrices, so they play an important role in the theory. Some deep relation to the modified Bessel functions of integer order is pointed out. By taking a skillful limit finite quantum systems become quantum mechanics on the circle developed by Ohnuki and Kitakado.

A New Algebraic Structure of Finite Quantum Systems and the Modified Bessel Functions Kazuyuki FUJII ∗ Department of Mathematical Sciences Yokohama City University Yokohama, 236-0027 Japan Abstract In this paper we present a new algebraic structure (a super hyperbolic system in our terminology) for finite quantum systems, which is a generalization of the usual one in the two–level system. It fits into the so–called generalized Pauli matrices, so they play an important role in the theory. Some deep relation to the modified Bessel functions of integer order is pointed out. By taking a skillful limit finite quantum systems become quantum mechanics on the circle developed by Ohnuki and Kitakado. ∗E-mail address : fujii@yokohama-cu.ac.jp http://arxiv.org/abs/0704.1844v2 Quantum Computation is usually based on two–level system of atoms (qubit theory). In the realistic construction of quantum logic gates we must solve some Schrödinger equa- tions. Then the Pauli matrices {σ1, σ3} is essentially used and not only the periodic functions {cos(x), sin(x)} but also the hyperbolic functions {cosh(x), sinh(x)} play an important role. On the other hand, they are deeply related to the modified Bessel functions of integer order {In(x) | n ∈ Z}. The functions are in general given by the generating function. Atom has usually many (finite or infinite) energy levels. However, to treat infinitely many ones at the same time is not realistic, so we treat an atom with finite (for example n) energy levels. We call this a finite quantum system and for this system the so–called generalized Pauli matrices {Σ1,Σ3} play a crucial role, see for example [1], [2] and [3]. In this system we have a natural question on what functions corresponding to the hy- perbolic functions are. In the paper we present such a system {c0(x), c1(x), · · · , cn−1(x)} (a super hyperbolic system in our terminology) as a “natural” generalization of {cosh(x), sinh(x)}. Moreover, we define a generating matrix based on the generalized Pauli matrices as a “natural” generalization of the generating function and obtain interesting results by taking some traces. Lastly, we want to take a limit of finite quantum systems, which is of course impossible. However, there is a bypass. That is, by taking a skillful limit finite quantum systems become quantum mechanics on the circle developed by Ohnuki and Kitakado [4]. Through this paper we have a clear and unified picture of quantum systems. First of all we make some mathematical preliminaries on the 2–level system. Let {σ1, σ2, σ3} be Pauli matrices and 12 the unit matrix :  0 1  , σ2 =  0 −i  , σ3 =  1 0  , 12 =  1 0  . (1) List the well–known properties of σ1 and σ3 : σ21 = σ 3 = 12, σ 1 = σ1, σ 3 = σ3, σ3σ1 = −σ1σ3 = eπiσ1σ3. (2) Let W be the Walsh–Hadamard matrix  1 1  = W−1 , (3) then we can diagonalize σ1 as σ1 = Wσ3W −1 by making use of W . The modified Bessel functions of integer order {Ik(x) | k ∈ Z} are given by the generating function (w+ 1 Ik(x)w k. (4) Now let us list some (well–known) important properties (see for example [5]) : 1 = I0(x) + 2 (−1)kI2k(x), ex = I0(x) + 2 Ik(x), e −x = I0(x) + 2 (−1)kIk(x) cosh(x) = I0(x) + 2 I2k(x), sinh(x) = 2 I2k−1(x). In the following we set c0(x) ≡ cosh(x) = (2k)! , c1(x) ≡ sinh(x) = x2k+1 (2k + 1)! for simplicity. The fundamental equation c20(x)− c21(x) = 1 (6) is interpreted as a simple relation Sσ3S = σ3 ⇐⇒ σ3Sσ3S = 12 ⇐⇒ (σ3S)2 = 12 for S defined by  c0(x) c1(x) c1(x) c0(x)  = c0(x)12 + c1(x)σ1 = exσ1 . (7) Next we would like to extend the 2–level system to general n–level one. To make our purpose clearer we treat the 3–level case in detail. Let σ be exp(2πi ), then we have σ3 = 1, σ̄ = σ2, 1 + σ + σ2 = 0. (8) Let Σ1 and Σ3 be generators of generalized Pauli matrices in the case of n = 3, namely , Σ3 = . (9) Then it is easy to see Σ31 = Σ 3 = 13, Σ 1 = Σ 3 = Σ 3, Σ3Σ1 = σΣ1Σ3. (10) Now we can show that Σ1 can be diagonalized by making use of the matrix 1 1 1 1 σ2 σ 1 σ σ2 ∈ U(3) (11) Σ1 = WΣ3W † = WΣ3W −1. (12) In fact 1 1 1 1 σ2 σ 1 σ σ2 1 1 1 1 σ σ2 1 σ2 σ 0 0 3 3 0 0 0 3 0 = Σ1, where we have used the relations in (8). From (5) we set c0(x) = (3k)! , c1(x) = x3k+1 (3k + 1)! , c2(x) = x3k+2 (3k + 2)! . (13) Then it is easy to check c0(x) = ex + eσx + eσ , c1(x) = ex + σ2eσx + σeσ , c2(x) = ex + σeσx + σ2eσ by use of σ in (8) or reversely ex = c0(x)+c1(x)+c2(x), e σx = c0(x)+σc1(x)+σ 2c2(x), e σ2x = c0(x)+σ 2c1(x)+σc2(x). Now, our question is as follows : What is the fundamental equation that {c0(x), c1(x), c2(x)} satisfy ? The answer is given by the equation (c0(x) + c1(x) + c2(x))(c0(x) + σc1(x) + σ 2c2(x))(c0(x) + σ 2c1(x) + σc2(x)) = exeσxeσ 2x = e(1+σ+σ 2)x = e0 = 1. By expanding the left-hand side and using the relations (8) we obtain c30(x) + c 1(x) + c 2(x)− 3c0(x)c1(x)c2(x) = 1. (15) Next let us consider the addition formulas. By expanding eσxeσy = eσ(x+y) ⇐= eσt = c0(t) + σc1(t) + σ2c2(t) we have c0(x)c0(y) + c1(x)c2(y) + c2(x)c1(y) = c0(x+ y), c0(x)c1(y) + c1(x)c0(y) + c2(x)c2(y) = c1(x+ y), c0(x)c2(y) + c1(x)c1(y) + c2(x)c0(y) = c2(x+ y). (16) From here let us give a unified approach by use of the generalized Pauli matrices {Σ1,Σ3} above. We consider the matrix exΣ1 = c0(x)13 + c1(x)Σ1 + c2(x)Σ c0(x) c2(x) c1(x) c1(x) c0(x) c2(x) c2(x) c1(x) c0(x) . (17) Then by Σ1 = WΣ3W † in (12) c0(x) c2(x) c1(x) c1(x) c0(x) c2(x) c2(x) c1(x) c0(x) = exΣ1 = W exΣ3W † = W so taking the determinant leads to ∣∣∣∣∣∣∣∣∣ c0(x) c2(x) c1(x) c1(x) c0(x) c2(x) c2(x) c1(x) c0(x) ∣∣∣∣∣∣∣∣∣ = e(1+σ+σ 2)x = 1. Namely, we recovered (15). On the other hand, by use of (11) it is straightforward to show W † = 1 1 1 1 σ2 σ 1 σ σ2 1 1 1 1 σ σ2 1 σ2 σ ex + eσx + eσ ex + σ2eσx + σeσ ex + σeσx + σ2eσ so we recovered (14). The matrix form is very convenient. Moreover, we can give new relations. For that we consider the simple equation exΣ1eyΣ 1 = exΣ1+yΣ 1 . (18) The left hand side is exΣ1eyΣ 1 = (c0(x)13 + c1(x)Σ1 + c2(x)Σ 1)(c0(y)13 + c1(y)Σ 1 + c2(y)Σ1) = (c0(x)c0(y) + c1(x)c1(y) + c2(x)c2(y))13 + (c0(x)c2(y) + c1(x)c0(y) + c2(x)c1(y))Σ1 + (c0(x)c1(y) + c1(x)c2(y) + c2(x)c0(y))Σ because Σ 1 = Σ 1. The right hand side is exΣ1+yΣ 1 = eW (xΣ3+yΣ )W † = W exΣ3+yΣ 3W † = W exσ+yσ ex+y + exσ+yσ + exσ ex+y + σ2exσ+yσ + σexσ ex+y + σexσ+yσ + σ2exσ so we obtain c0(x)c0(y) + c1(x)c1(y) + c2(x)c2(y) = ex+y + exσ+yσ + exσ c0(x)c2(y) + c1(x)c0(y) + c2(x)c1(y) = ex+y + σ2exσ+yσ + σexσ c0(x)c1(y) + c1(x)c2(y) + c2(x)c0(y) = ex+y + σexσ+yσ + σ2exσ . (19) Next, let us consider the matrix exΣ1+yΣ 1 . If we set y = 1/x, then the matrix exΣ1+(1/x)Σ 1 is similar to (4) the generating function of modified Bessel functions of integer order. Therefore from (4) it is reasonable to consider (wΣ1+ 1 1) = e (wΣ1+ 1 1 ) = Ik(x)w kΣk1 . (20) In the following we call this the generating matrix of modified Bessel functions of integer order. Let us look for some typical properties. The result is (wΣ1+ 1 (w+ 1 ) + e (wσ+ 1 σ2) + e (wσ2+ 1 I3k(x)w (wΣ1+ 1 (w+ 1 ) + σe (wσ+ 1 σ2) + σ2e (wσ2+ 1 I3k−1(x)w 3k−1, (wΣ1+ 1 1)Σ21 (w+ 1 ) + σ2e (wσ+ 1 σ2) + σe (wσ2+ 1 I3k−2(x)w where σ−1 = σ2 and σ−2 = σ. A comment is in order. In the case of n = 2 the generating matrix is (wσ1+ 1 1) = e (w+ 1 )σ1 = cosh 12 + sinh because σ1 is hermitian, so the situation becomes much easier. From the lesson for the case of n = 3, let us set up the general case. Let {Σ1,Σ3} be generalized Pauli matrices  . . . . . .  , Σ3 =  . . .  where σ is a primitive element σ = exp(2πi ) which satisfies σn = 1, σ̄ = σn−1, 1 + σ + · · ·+ σn−1 = 0. (23) Then it is easy to see Σn1 = Σ 3 = 1n, Σ 1 = Σ 1 , Σ 3 = Σ 3 , Σ3Σ1 = σΣ1Σ3. (24) If we define a Vandermonde matrix W based on σ as  1 1 1 · · · 1 1 1 σn−1 σ2(n−1) · · · σ(n−2)(n−1) σ(n−1)2 1 σn−2 σ2(n−2) · · · σ(n−2)2 σ(n−1)(n−2) 1 σ2 σ4 · · · σ2(n−2) σ2(n−1) 1 σ σ2 · · · σn−2 σn−1  , (25) then it is not difficult to see Σ1 = WΣ3W † = WΣ3W −1. (26) That is, Σ1 can be diagonalized by making use of W . We set cj(x) = xkn+j (kn+ j)! for 0 ≤ j ≤ n− 1. It is of course cj(x) and easy to see exΣ1 = c0(x)1n + c1(x)Σ1 + c2(x)Σ 1 + · · ·+ cn−2(x)Σn−21 + cn−1(x)Σn−11  c0(x) cn−1(x) · · · c2(x) c1(x) c1(x) c0(x) cn−1(x) · · · c2(x) c1(x) c0(x) cn−1(x) . . . . . . . . . cn−2(x) · · · c1(x) c0(x) cn−1(x) cn−1(x) cn−2(x) · · · c2(x) c1(x) c0(x)  . (28) Let us look for the fundamental equation that {c0(x), c1(x), · · · , cn−2(x), cn−1(x)} sat- isfy. By use of (26) exΣ1 = exWΣ3W = W exΣ3W † we have ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ c0(x) cn−1(x) · · · c2(x) c1(x) c1(x) c0(x) cn−1(x) · · · c2(x) c1(x) c0(x) cn−1(x) . . . . . . . . . cn−2(x) · · · c1(x) c0(x) cn−1(x) cn−1(x) cn−2(x) · · · c2(x) c1(x) c0(x) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ . . . ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ = ex(1+σ+σ 2+···+σn−2+σn−1) = e0 = 1 (29) because W is unitary (|W | = 1). For example n = 2 c20(x)− c21(x) = 1 (⇐= (6)) n = 3 c30(x) + c 1(x) + c 2(x)− 3c0(x)c1(x)c2(x) = 1 (⇐= (15)) n = 4 c40(x)− c41(x) + c42(x)− c43(x)− 2c20(x)c22(x) + 2c21(x)c23(x) −4c20(x)c1(x)c3(x) + 4c0(x)c21(x)c2(x)− 4c1(x)c22(x)c3(x) + 4c0(x)c2(x)c23(x) = 1. We call {c0(x), c1(x), · · · , cn−1(x)} the super hyperbolic system. The addition formulas are given by the simple equation exΣ1eyΣ1 = e(x+y)Σ1 and become cj(x+ y) = k+l=j (mod n) ck(x)cl(y) for 0 ≤ j ≤ n− 1. (31) More explicitly, cj(x+y) = c0(x)cj(y)+c1(x)cj−1(y)+· · ·+cj(x)c0(y)+cj+1(x)cn−1(y)+· · ·+cn−1(x)cj+1(y). The new relations are given by the simple equation exΣ1eyΣ 1 = exΣ1+yΣ and become ck(x)cn−j+k(y) + ck(x)ck−j(y) = σk(n−j)exσ k+yσn−k (32) for 0 ≤ j ≤ n− 1. The generating matrix of modified Bessel functions of integer order is given by (wΣ1+ 1 Ik(x)w kΣk1 (33) and from this we have (wΣ1+ 1 (wσl+ 1 σ−l) = Ink−j(x)w nk−j (34) for 0 ≤ j ≤ n− 1. The result in the case of j = 0 is known in [6] and [7]. We want to take a (formal) limit n −→ ∞. That is, what is Σ1 −→ ?, Σ3 −→ ? It is of course impossible to take a limit with this form. For that let us make a small change. We set n = 2N + 1 and Σ̃1 = Σ1, Σ̃3 =  . . . . . .  where σ = exp( 2πi ). Here we rewrite Σ̃3 as Σ̃3 = exp( G̃) where  . . . . . .  . (36) The commutator [G̃, Σ̃1] becomes [G̃, Σ̃1] =  0 −2N . . . . . .   . . . . . .  (mod 2N + 1). That is, we have the relation [G̃, Σ̃1] = Σ̃1 (mod 2N + 1). (37) In this stage, it may be better to write the (finite dimensional) Hilbert space as 2N+1 = VectC{|−N〉, · · · , |−1〉, |0〉, |1〉, · · · , |N〉} because G̃|n〉 = n|n〉. Now, if we take a formal limit N −→ ∞ then we have the fundamental relation [G,W ] = W (38) where  . . . . . .  , W =  . . . . . . 0 . . . . . .  where the notations {G,W} in [4] were used. Note that G is a hermitian operator and W a unitary operator on the Hilbert space L2(Z) = cn|n〉 | |cn|2 < ∞ ; W |n〉 = |n+ 1〉, G|n〉 = n|n〉. (40) The relation (38) is just the fundamental one in quantum mechanice on the circle devel- oped by Ohnuki and Kitakado [4]. A comment is in order. There is some freedom on the choice of G. That is, if we choose G like G −→ G+ α1, 0 ≤ α < 1 the relation (38) still holds. α is interpreted as a kind of abelian gauge induced in quantum mechanice on the circle. Therefore it may be better to write the generators {Gα ≡ G + α1,W} in place of {G,W} in [4]. We don’t repeat the contents, so see [4] and its references. Readers may find many interesting problems from the paper. For example, we can consider the generating operator (wW+ 1 W †). We leave some calculations to readers. In this paper we developed the super hyperbolic structure for (all) finite quantum systems, and defined the generating matrix for the modified Bessel functions of integer order and obtained some interesting results. We also gave a connection to quantum mechanics on the circle by Ohnuki and Kitakado by taking a skillful limit. Our motivation is to apply the development in the paper to qudit theory based on finite quantum systems, which will be reported in another paper. Acknowledgment. The author wishes to thank K. Funahashi for helpful comments and suggestions. References [1] K. Fujii : Exchange Gate on the Qudit Space and Fock Space, J. Opt. B : Quantum Semiclass. Opt, 5(2003), S613, quant-ph/0207002. http://arxiv.org/abs/quant-ph/0207002 [2] K. Fujii : How to Treat an N-Level System : A Proposal, quant-ph/0302050. [3] K. Fujii, K. Funahashi and T. Kobayashi : Jarlskog’s Parametrization of Uni- tary Matrices and Qudit Theory, Int. J. Geom. Meth. Mod. Phys. 3(2006), 269, quant-ph/0508006. [4] Y. Ohnuki and S. Kitakado : Fundamental Algebra for Quantum Mechanics on SD and Gauge Potentials, J. Math. Phys. 34(1993), 2827. [5] E. T. Whittaker and G. N. Watson : A Course of MODERN ANALYSIS, 1990, Cambridge University Press. [6] V. Barsan and S. Cojocaru : Bessel functions of integer order in terms of hyperbolic functions, math-ph/0703010. [7] S. Cojocaru : Green’s function of a finite chain and the discrete Fourier transform, Int. J. Mod. Phys. 20(2006), 593, arXiv : 0704.2898 (math-ph). http://arxiv.org/abs/quant-ph/0302050 http://arxiv.org/abs/quant-ph/0508006 http://arxiv.org/abs/math-ph/0703010

CALT 68-2643 UCSD/PTH 07-04 The Lee-Wick Standard Model Benjamı́n Grinstein,1, ∗ Donal O’Connell,2, † and Mark B. Wise2, ‡ 1Department of Physics, University of California at San Diego, La Jolla, CA 92093 2California Institute of Technology, Pasadena, CA 91125 (Dated: October 22, 2018) Abstract We construct a modification of the standard model which stabilizes the Higgs mass against quadratically divergent radiative corrections, using ideas originally discussed by Lee and Wick in the context of a finite theory of quantum electrodynamics. The Lagrangian includes new higher derivative operators. We show that the higher derivative terms can be eliminated by introducing a set of auxiliary fields; this allows for convenient computation and makes the physical interpre- tation more transparent. Although the theory is unitary, it does not satisfy the usual analyticity conditions. ∗bgrinstein@ucsd.edu †donal@theory.caltech.edu ‡wise@theory.caltech.edu http://arxiv.org/abs/0704.1845v3 mailto:bgrinstein@ucsd.edu mailto:donal@theory.caltech.edu mailto:wise@theory.caltech.edu I. INTRODUCTION The extreme fine-tuning needed to keep the Higgs mass small compared to the Planck scale (i.e., the hierarchy puzzle) has motivated many extensions of the minimal standard model. All of these contain new physics, beyond that in the minimal standard model, which might be observed at the Large Hadron Collider (LHC). The most widely explored of these extensions is low energy supersymmetry. In this paper we introduce another extension of the standard model that solves the hierarchy puzzle. Our approach builds on the work of Lee and Wick [1, 2] who studied the possibility that the regulator propagator in Pauli-Villars corresponds to a physical degree of freedom. Quantum electrodynamics with a photon propagator that includes the regulator term is a higher derivative version of QED. The higher derivative propagator contains two poles, one corresponding to the massless photon, and the other corresponding to a massive Lee- Wick-photon (LW-photon). A problem with this approach is that the residue of the massive LW-photon pole has the wrong sign. Lee and Wick argued that one can make physical sense of such a theory. There is no problem with unitarity since the massive LW-photon is not in the spectrum; it decays through its couplings to ordinary fermions. However, the wrong sign residue moves the poles in the photon two point function that are associated with this massive resonance from the second sheet to the physical sheet, introducing time dependence that grows exponentially. Lee and Wick and Cutkosky et al. [3] propose a modification of the usual integration contour in Feynman diagrams that removes this growth and preserves unitarity of the S matrix1. This was further discussed in [5, 6]. The theory of QED that Lee and Wick studied is finite. In this paper we propose to extend their idea to the standard model, removing the quadratic divergence associated with the Higgs mass, and thus solving the hierarchy problem. In the LW-standard model, every field in the minimal standard model has a higher derivative kinetic term that introduces a corresponding massive LW-resonance. These masses are additional free parameters in the theory and must be high enough to evade current experimental constraints. For the non- Abelian gauge bosons the higher derivative kinetic term has, because of gauge invariance, new higher derivative interactions. Hence the resulting theory is not finite; however, we 1 The consistency of this approach is controversial [4]. argue that it does not give rise to a quadratic divergence in the Higgs mass, and so solves the hierarchy puzzle. A power counting argument and some explicit one loop calculations are given to demonstrate this. For explicit calculations, and to make the physics clearer, it is useful to remove the higher derivative terms in the Lagrangian density by introducing auxiliary LW-fields that, when integrated out, reproduce the higher derivative terms in the action. The LW-standard model2 has a new parameter for each standard model field, which cor- responds physically to the tree-level mass of its LW-partner resonance. Explicit calculations can be performed in this theory at any order in perturbation theory, and the experimental consequences for physics at the LHC, and elsewhere, can be studied. The nonperturbative formulation of Lee-Wick theories has been studied in [7, 8]. Lee-Wick theories are unusual; however, even if one does not take the particular model we present as the correct theory of nature at the TeV scale our work does suggest that a further examination of higher deriva- tive theories is warranted. Some previous work on field theories with non-local actions that contain terms with an infinite number of derivatives can be found in Ref. [9]. II. A TOY MODEL To illustrate the physics of Lee-Wick theory [1, 2, 8] in a simple setting, we consider in this section a theory of one self-interacting scalar field, φ̂, with a higher derivative term. The Lagrangian density is Lhd = ∂µφ̂∂ (∂2φ̂)2 − m2φ̂2 − gφ̂3, (1) so the propagator of φ̂ in momentum space is given by D̂(p) = p2 − p4/M2 −m2 . (2) For M ≫ m, this propagator has poles at p2 ≃ m2 and also at p2 ≃ M2. Thus, the propagator describes more than one degree of freedom. We can make these new degrees of freedom manifest in the Lagrangian density in a simple way. First, let us introduce an auxiliary scalar field φ̃, so that we can write the theory as ∂µφ̂∂ m2φ̂2 − φ̃∂2φ̂+ M2φ̃2 − gφ̂3. (3) 2 LW extension of the standard model would be more precise. Since L is quadratic in φ̃, the equations of motion of φ̃ are exact at the quantum level. Removing φ̃ from L with their equations of motion reproduces Lhd in Eq. (1). Next, we define φ = φ̂+ φ̃. In terms of this variable, the Lagrangian in Eq. (3) becomes, after integrating by parts, ∂µφ̃∂ M2φ̃2 − m2(φ− φ̃)2 − g(φ− φ̃)3. (4) In this form, it is clear that there are two kinds of scalar field: a normal scalar field φ and a new field φ̃, which we will refer to as an LW-field. The sign of the quadratic Lagrangian of the LW-field is opposite to the usual sign so one may worry about stability of the theory, even at the classical level. We will return to this point. If we neglect the mass m for simplicity, the propagator of φ̃ is given by D̃(p) = p2 −M2 . (5) The LW-field is associated with a non-positive definite norm on the Hilbert space, as indi- cated by the unusual sign of its propagator. Consequently, if this state were to be stable, unitarity of the S matrix would be violated. However, as emphasized by Lee and Wick, uni- tarity is preserved provided that φ̃ decays. This occurs in the theory described by Eq. (4) because φ̃ is heavy and can decay to two φ-particles. In the presence of the mass m, there is mixing between the scalar field φ and the LW- scalar φ̃. We can diagonalize this mixing without spoiling the diagonal form of the derivative terms by performing a symplectic rotation on the fields: cosh θ sinh θ sinh θ cosh θ  . (6) This transformation diagonalizes the Lagrangian if tanh 2θ = −2m2/M2 1− 2m2/M2 . (7) A solution for the angle θ exists provided M > 2m. The Lagrangian (4) describing the system becomes ′∂µφ′ − m′2φ′2 − ′∂µφ̃′ + M ′2φ̃′2 − (cosh θ − sinh θ)3g(φ′ − φ̃′)3, (8) wherem′ andM ′ are the masses of the diagonalized fields. Notice the form of the interaction; we can define g′ = (cosh θ − sinh θ)3g and then drop the primes to obtain a convenient Lagrangian for computation.3 Introducing the LW-fields makes the physics of the theory clear. There are two fields; the heavy LW-scalar decays to the lighter scalar. At loop level, the presence of the heavier scalar improves the convergence of loop graphs at high energy consistent with our expectations from the higher derivative form of the theory. We can use the familiar technology of perturbative quantum field theory (appropriately modified [3]) to compute quantum corrections to the physics. It is worth pausing for a moment to consider loop corrections to the two point function of the LW-field. Using the one loop self energy, the full propagator for the LW-scalar is given, near p2 = M2, by D̃(p) = p2 −M2 p2 −M2 (−iΣ(p2)) −i p2 −M2 + · · · p2 −M2 + Σ(p2) . (9) Note that, unlike for ordinary scalars, there is a plus sign in front of the self energy Σ(p2) in the denominator. This sign is significant; for example, if one defines the width in the usual way (i.e., near the pole the propagator has denominator p2 −M2 + iMΓ) then, from a one loop computation of the self energy Σ, the width of the LW-field is (for Im p2 > 0) Γ = − . (10) This width differs in sign from widths of the usual particles we encounter. With this result in hand, we can demonstrate how unitarity of the theory is maintained in an explicit example. Consider φφ scattering in this theory. From unitarity, the imaginary part of the forward scattering amplitude, M, must be a positive quantity. Near p2 = M2, the scattering is dominated by the φ̃ pole and therefore the imaginary part of the amplitude is given for Im p2 > 0 by ImM = −g2 (p2 −M2)2 +M2Γ2 . (11) The unusual sign of the propagator is compensated by the unusual sign of the decay width. As another consequence of this sign, the poles associated with these LW-particles occur on the physical sheet of the analytic continuation of the S matrix, in violation of the usual rules 3 In the following, we will always assume that M ≫ m so that g′ ≃ g. of S matrix theory. These signs are also associated with exponential growth of disturbances, which is related to the stability concerns alluded to earlier. Lee and Wick, and Cutkowsky et al argued that one can nevertheless make sense of these theories by modifying the usual contour prescription for momentum integrals. The Feynman iǫ prescription can be thought of as a deformation of the contour such that the poles on the real axis are appropriately above or below the contour. The Lee-Wick prescription is equivalent to imposing the boundary condition that there are no outgoing exponentially growing modes. It is well known that such future boundary conditions cause violations of causality. In the Lee-Wick theory the acausal effects occur only on microscopic scales, and show up as a peculiar time ordering of events; for example, the decay products of a Lee-Wick particle appear at times before the Lee-Wick particle itself is created. It is believed that this theory does not produce violations of causality, or any paradoxes, on a macroscopic scale [6]. III. THE HIERARCHY PROBLEM AND LEE-WICK THEORY In this section, we consider a scalar in the fundamental representation interacting with gauge bosons. We find the Lagrange density for the LW version of such a theory and show by power counting appropriate to the higher derivative version of the theory that the scalar mass is free of quadratic divergences. We then show by an explicit one loop calculation that the ordinary scalar and the massive LW-fields do not receive a quadratically divergent contribution to their pole masses. A. Gauge Fields The higher derivative Lagrangian in the gauge sector is Lhd = − tr F̂µνF̂ D̂µF̂µν D̂λF̂λ , (12) where F̂µν = ∂µÂν−∂νµ− ig[µ, Âν ], and µ = ÂAµTA with TA the generators of the gauge group G in the fundamental representation. We can now eliminate the higher derivative term by introducing auxiliary massive gauge bosons Ã. Each gauge boson is described by a Lagrangian L = − tr F̂µνF̂ µν −M2A tr õõ + 2 tr F̂µνD̂µÃν , (13) where D̂µÃν = ∂µÃν − ig[µ, Ãν ]. To diagonalize the kinetic terms, we introduce shifted fields defined by µ = Aµ + õ. (14) The Lagrangian becomes L = −1 trFµνF DµÃν −Dνõ DµÃν −Dνõ − ig tr õ, Ãν g2 tr õ, Ãν õ, Ãν − 4ig tr õ, Ãν DµÃν −M2A tr õà . (15) Note that for a U(1) gauge boson all the commutators vanish, there are no traces and an extra overall factor of 1/2. To perform perturbative calculations, we must introduce a gauge fixing term. We could introduce such a term in the higher derivative Lagrangian, Eq. (12), in terms of the La- grangian involving A and Ã, Eq. (15), or even in the Lagrangian with mixed kinetic terms for  and Ã, Eq. (13). As is usual in gauge theories, all of these choices will yield the same results for physical quantities, but they may differ for unphysical quantities. Different gauge choices can differ on how divergent unphysical quantities are. Therefore, we will only compute physical pole masses below. In these computations, we introduce a covariant gauge fixing term for the gauge bosons, AAµ , in the two field description of the theory given in Eq. (15). In this choice of gauge, the propagator for the gauge bosons is given by DABµν (p) = δ ηµν − (1− ξ) , (16) while the propagator for the LW-gauge field is D̃ABµν (p) = δ AB −i p2 −M2A ηµν − . (17) B. Scalar Matter Let us move on to consider scalar matter transforming in the fundamental representation of the gauge group. In ordinary field theory, such a scalar field has a quadratic divergence in its pole mass. The higher derivative Lagrangian is given in terms of the scalar field φ̂ by Lhd = D̂µφ̂ D̂µφ̂ D̂µD̂ D̂νD̂ − V (φ̂). (18) We eliminate the higher derivative term by introducing an LW-scalar multiplet φ̃. Then the Lagrangian is given in terms of the two fields φ̂ and φ̃ by D̂µφ̂ D̂µφ̂ +M2φφ̃ D̂µφ̂ D̂µφ̃ D̂µφ̃ D̂µφ̂ − V (φ̂), (19) where the covariant derivative is D̂µ = ∂µ + ig A. (20) For simplicity we take the ordinary scalar to have no potential at tree level, V (φ̂) = 0. It is not hard to include a potential for φ̂ in the analysis, and to show that the potential does not change our results. We diagonalized the pure gauge sector by shifting the gauge fields; in terms of the shifted gauge fields the hatted covariant derivative is D̂µ = Dµ + igà A, (21) where Dµ = ∂µ+ igA A is the usual covariant derivative. To diagonalize the scalar kinetic terms, we again shift the field φ̂ = φ− φ̃. (22) The scalar Lagrangian becomes L = (Dµφ)†Dµφ− (Dµφ̃)†Dµφ̃+M2φφ̃†φ̃+ ig(Dµφ)†ÃAµTAφ+ g2φ†ÃAµTAÃBµTBφ −igφ†ÃAµTADµφ− ig(Dµφ̃)†ÃAµTAφ̃+ igφ̃†ÃAµTADµφ̃− g2φ̃†ÃAµTAÃBµTBφ̃. (23) C. Power Counting Having defined the higher derivative and LW forms of the theory, we present a power counting argument for the higher derivative version of the theory which indicates that the only physical divergences in the theory are logarithmic. Since the power counting argument depends on the behaviour of Feynman graphs at high energies, we only need to consider the terms in the Lagrangian which are most important at high energies. For the perturbative power counting argument in the higher derivative theory, it is nec- essary to fix the gauge. We choose to add a covariant gauge fixing term −(∂µÂAµ)2/2ξ to the Lagrange density and introduce Faddeev-Popov ghosts that couple to the gauge bosons in the usual way. Then the propagator for the gauge field is D̂ABµν (p) = δ AB −i p2 − p4/M2A ηµν − (1− ξ) − ξ pµpν . (24) We work in ξ = 0 gauge. Note that ξ = 0 corresponds to Landau gauge and that the gauge boson propagator scales as p−4 at high energy. The propagator for the scalar in the fundamental representation is D̂ab(p) = δab p2 − p4/M2 . (25) At large momenta the scalar propagator scales as p−4 while the Faddeev-Popov ghost prop- agator scales as p−2, as usual. There are three kinds of vertices: those where only gauge bosons interact, vertices where gauge bosons interact with two scalars, and vertices where two ghosts interact with one gauge boson. A vertex where n vectors interact (with no scalars) scales as p6−n while a vertex with two scalars and n vectors scales as p4−n. The vertex between two ghosts and one gauge field scales as one power of p, as usual. Consider an arbitrary Feynman graph with L loops, I ′ internal vector lines, I internal scalar lines, Ig internal ghost lines, and with V n or Vn vertices with n vectors and zero or two scalar particles, respectively. We also suppose there are Vg ghost vertices. Then the superficial degree of divergence, d, is d = 4L− 4I ′ − 4I − 2Ig + V ′n(6− n) + Vn(4− n) + Vg. (26) We can simplify this expression using some identities. First, the number of loops is related to the total number of propagators and vertices by L = I + I ′ + Ig − (V ′n + Vn)− Vg + 1, (27) while the total number of lines entering or leaving the vertices is related to the number of propagators and external lines by (nV ′n + (n + 2)Vn) + 3Vg = 2(I + I ′ + Ig) + E + E ′ + Eg, (28) where E is the number of external scalars, E ′ is the number of external vectors, and Eg is the number of external ghosts. Finally, because the Lagrangian is quadratic in the number of scalars and ghosts, the number of scalar lines and ghost lines is separately conserved. Thus, Vn = 2I + E, 2Vg = 2Ig + Eg. (29) With these identities in hand, we may express the superficial degree of divergence as d = 6− 2L− E − E ′ − 2Eg. (30) Gauge invariance removes the potential quadratic divergence in the gauge boson two point function. Scalar mass renormalizations have E = 2, so that d = 4 − 2L. Consequently, the only possible quadratic divergence in the scalar mass is at one loop. However, gauge invariance also removes the divergence in the scalar mass renormalization, because two of the derivatives must act on the external legs. To see this, note that the interaction involves φ†D4φ ∼ φ†(∂2 + ∂ ·A + A · ∂ + A2)2φ. (31) Since we are working in Lorentz gauge, ∂ ·A = 0. We may ignore the A2 term compared to the A · ∂ term, as it is less divergent. Thus the most divergent terms in the interaction are φ†A · ∂∂2φ or φ†∂2A · ∂φ, where the φ acted on by the derivatives is an internal line. But by integration by parts and use of the gauge condition, we see that, at one loop, we can always take one of the derivatives to act on the external scalar. Thus the theory at hand is at most logarithmically divergent.4 D. One loop Pole Mass The power counting argument above was presented in the higher derivative version of the theory. As a check of the formalism we show, in the LW version of the theory, that the shift in the pole masses of the ordinary scalar, the LW-scalar and the LW-gauge boson do not receive quadratically divergent contributions at one loop. It is important to compute a physical quantity since it is for these that the higher derivative theory and the theory with LW-fields give equivalent results5. We perform the computations in Feynman gauge, using 4 It may seem that adding operators with more than four derivatives could yield a finite theory, but that is not the case. These theories are still logarithmically divergent. 5 We have fixed different gauges in our discussion of the power counting argument in the higher derivative theory and our explicit computations in the LW version of the theory. Consequently, we can only expect agreement between these theories for physical quantities. FIG. 1: One loop mass renormalization of the normal scalar field. The curly line is a gauge field while the zigzag line is the LW-gauge field. The dashed line represents the scalar field. the propagators in Eqs. (16) and (17), and regulate our diagrams where necessary using dimensional regularization with dimension n. 1. The normal scalar At one loop, there are four graphs contributing to the scalar mass, as shown in Figure 1. We find −iΣa(0) = g2C2(N) (2π)n (32a) −iΣb(0) = −g2C2(N) (2π)n k2 −M2A (32b) −iΣc(0) = −g2C2(N) (2π)n (32c) −iΣd(0) = −g2C2(N) (2π)n . (32d) We see that the quartic and quadratic divergences in this expressions cancel in the sum, so that the mass is only logarithmically divergent. FIG. 2: One loop mass renormalization of the LW-scalar field. The dotted line represents the LW-scalar field while the other propagators are as in Figure 1. 2. The LW-scalar At one loop the shift in the pole mass is determined by the self energy Σ(p2) evaluated at p2 = M2φ. The Feynman graphs are shown in Figure 2. We find −iΣa(M2φ) = −g2C2(N) (2π)n (33a) −iΣb(M2φ) = g2C2(N) (2π)n k2 −M2 (33b) −iΣc(M2φ) = g2C2(N) (2π)n k2 − 2p · k 4M2φ − 4p · k k2(k2 − 2p · k) (33c) −iΣd(M2φ) = g2C2(N) (2π)n 4M2φ − 2p · k (k2 −M2A)(k2 − 2p · k) . (33d) Once again, the quartic and quadratic divergence cancel in the sum of the graphs, so that there is only a logarithmic divergence in the mass of the LW-scalar. 3. The LW-vector For the LW-vectors the self energy tensor has the form ΣABµν (p 2) = δAB Σ(p2)ηµν + Σ ′(p2)pµpν . (34) The shift in the pole mass is determined by Σ(M2A). The relevant graphs are shown in Figure 3. They are very divergent. There are individual terms in Figure 3(c) that diverge FIG. 3: One loop mass renormalization of the LW-vector field. The propagators are as in Figure 1. as the sixth power of a momentum cutoff. However these cancel. There is also a quartic divergence in diagrams (b), (c) and (d) that cancels between them. To check that the quadratic divergence cancels we regulate the diagrams with dimensional regularization. In n dimensions, a quadratic divergence manifests itself as a pole at n = 2. Hence, we set n = 2− ǫ, expand about ǫ = 0 and extract the 1/ǫ part of Σ(M2A). We find that −iΣa(M2A) = C2(G) (35a) −iΣb(M2A) = C2(G) (35b) −iΣc(M2A) = C2(G) (35c) −iΣd(M2A) = C2(G) . (35d) As expected, the 1/ǫ pole cancels in the sum. Finally, we note that there are quadratic divergences in ΣABµν (p 2). Only the gauge invariant physical quantity Σ(M2A) must be free of quadratic divergences. IV. LEE-WICK STANDARD MODEL LAGRANGIAN Now that we have understood why the radiative correction to the Higgs mass cancels in these higher derivative theories, we move on to discuss the Lagrangian which describes the standard model extended to include a Lee-Wick partner for each particle. The gauge sector is as before. A. The Higgs Sector A higher derivative Lee-Wick Higgs sector was considered previously in [7]. We take the higher derivative Lagrangian for the Higgs doublet Ĥ to be Lhd = D̂µĤ D̂µĤ D̂µD̂ D̂νD̂ − V (Ĥ), (36) where the covariant derivative is given by D̂µ = ∂µ + ig A + ig2Ŵ a + ig1B̂µY, (37) while the potential is V (Ĥ) = Ĥ†Ĥ − . (38) We can then eliminate the higher derivative term by introducing an LW-Higgs doublet H̃. As before, we then diagonalize the Lagrangian by introducing the shifted field Ĥ = H − H̃. To diagonalize the gauge field Lagrangian, we introduced Lee-Wick gauge bosons Ã, B̃, and W̃ as well as the usual gauge fields A, B and W . In terms of these fields the covariant derivative is D̂µ = Dµ + igà A + ig2W̃ a + ig1B̃µY, (39) where Dµ = ∂µ + igA A + ig2W a + ig1BµY (40) is the usual standard model covariant derivative. We introduce the notation õ = gà A + g2W̃ a + g1B̃µY (41) for the LW-gauge bosons. The Lee-Wick form of the Higgs Lagrangian is then L = (DµH)†DµH − DµH̃ +M2HH̃ †H̃ − V (H, H̃) + i (DµH)† õH − iH†ÃµDµH +H†ÃµÃµH − i µH̃ + iH̃†ÃµD µH̃ − H̃†ÃµÃµH̃, (42) where V is given by the expression V (H, H̃) = V (H − H̃) H†H − v H†H − v H̃†H̃ − λ H†H − v H̃†H +H†H̃ H̃†H̃ H̃†H̃ H̃†H̃ . (43) In unitary gauge, we write  , H̃ = h̃+iP̃√  . (44) With this choice, the mass Lagrangian for the Higgs scalar, its partner, the charged LW- Higgs and pseudoscalar LW-Higgs fields is Lmass = − v2(h− h̃)2 + M h̃h̃+ P̃ P̃ + 2h̃+h̃− . (45) There is mixing between the usual Higgs scalar and its partner; this mixing can be treated perturbatively. It is possible to diagonalize the mass matrices of these particles via a sym- plectic rotation, which preserves the diagonal form of the kinetic terms. The Higgs vacuum expectation value induces masses for the gauge bosons. First, we focus on the mass Lagrangian for the LW-gauge bosons. In terms of the SU(2) and U(1) LW-gauge fields, the Lagrangian is Lmass = W̃ aµW̃ g1g2v W̃ 3µB̃ B̃µB̃ B̃µB̃ W̃ aµW̃ aµ. (46) There is mixing between the W̃ 3 and B̃ LW-gauge fields. We can diagonalize this Lagrangian by writing cosφ sinφ − sin φ cosφ  , (47) where the mixing angle is given by tan 2φ = g1g2v M21 −M22 + (g22 − g21) . (48) We expect that M1,2 lie in the TeV range, so that φ is a small angle. There is also mixing between the gauge fields and the LW-gauge fields. We will treat this mixing perturbatively. The Lagrangian describing this mixing is Lmix = M2W W+µ W̃ −µ + W̃+µ W +M2ZZµ cos θW W̃ 3µ − sin θW B̃µ , (49) where θW is the Weinberg angle andMW , MZ are the usual tree level standard model masses for the W and Z gauge bosons. One consequence of the mixing is that there is a tree level correction to the electroweak ρ parameter ∆ρ = ρ− 1 = −sin 2 θWM . (50) The current experimental constraint on this parameter is |∆ρ| . 10−3 [10] which leads to M1 & 1TeV. B. Fermion Kinetic Terms For simplicity, we discuss explicitly the case of a single left-handed quark doublet QL. It is straightforward to generalize this work to the other representations, and to include generation indices. The higher derivative theory is Lhd = Q̂LiD̂/ Q̂L + Q̂LiD̂/D̂/D̂/ Q̂L. (51) Naive power counting of the possible divergences in this higher derivative theory shows that there are potential quadratic divergences in one loop graphs containing two external gauge bosons and a fermionic loop. However, gauge invariance forces these graphs to be propor- tional to two powers of the external momentum so that the graphs are only logarithmically divergent. In this case, this cancellation is most easily understood in the LW description of the theory, which we now construct. We eliminate the higher derivative term by introducing LW-quark doublets Q̃L, Q̃ R which form a real representation of the gauge groups. The Lagrangian in this formulation becomes L = Q̂LiD̂/ Q̂L +MQ Q̃LQ̃ R + Q̃ + Q̃LiD̂/ Q̂L + Q̂LiD̂/ Q̃L − Q̃′RiD̂/ Q̃′R. (52) Eliminating the LW-fermions with their equations of motion Q̃′R = − Q̂L, Q̃L = D̂/D̂/ Q̂L, (53) FIG. 4: One loop graphs involving fermions which are potentially quadratically divergent. The solid lines represent fermion propagators while the curly and zigzag lines represent gauge bosons and LW-gauge bosons, respectively. reproduces the higher derivative Lagrangian, Eq. (51). To diagonalize the kinetic terms, we introduce the shift Q̂L = QL − Q̃L, and the La- grangian becomes L = QLiD/QL − Q̃LiD/ Q̃L − Q̃′RiD/ Q̃′R +MQ Q̃LQ̃ R + Q̃ −QLγµÃµQL + Q̃LγµÃµQ̃L + Q̃′RγµÃ µQ̃′R. (54) Note that Q̃L and Q̃ R combine into a single Dirac spinor of mass MQ. Now let us return to the issue of potential quadratic divergences in the theory. Inspec- tion of the Lagrangian, Eq. (54), shows that the only one loop graphs involving fermionic loops are the graphs of Figure 4. Figure 4a is a one loop correction to the gauge boson propagator, and consequently is proportional to p2, where p is the momentum flowing into the graph. Thus, the graph is logarithmically divergent, as is well known. Figure 4b is a one loop correction to the LW-gauge boson propagator. One might think that this graph could introduce a quadratic divergence of the LW-gauge boson mass. However, the vertices between the fermions and the gauge bosons are equal to the vertices between the fermions and the LW-gauge bosons, as can be seen in Eq. (54). Thus, Figure 4b is logarithmically divergent. Higher loop graphs in the theory are at most logarithmically divergent by power counting. C. Fermion Yukawa Interactions To simplify the discussion in this section, we will neglect neutrino masses. In the higher derivative formulation, the fermion Yukawas are LY = giju ûiRĤǫQ̂ L − g d̂iRĤ L − g L + h.c., (55) where repeated flavor indices are summed. In the formulation of the theory in which there are no higher derivatives, and in which the kinetic terms are diagonal, this becomes LY = giju (uiR − ũiR)(H − H̃)ǫ(Q L − Q̃ L)− g (diR − d̃iR)(H† − H̃†)(QjL − Q̃ −gije (eiR − ẽiR)(H† − H̃†)(L L − L̃ L) + h.c.. (56) The presence of the LW-fields in this equation improves the degree of convergence at one loop. For example, consider a one loop correction to the Higgs two point function coming from the first term of Eq. (56). Various degrees of freedom can propagate in the loop: the uR and QL quarks, and also the ũR and Q̃L LW-quarks. The presence of the LW-quarks cancels the quadratic divergence in the loop with only the quarks. The sum of these four graphs reproduces the result one would find by computing the corresponding correction in the higher derivative formulation of the theory, Eq. (55). To simplify the flavor structure of the theory, we adopt the principle of minimal flavor violation [11]. This forces all LW-fermions in the same representation of the gauge group have the same mass. Now the Yukawas can be diagonalized in the standard fashion. For notational brevity, we choose to use the same symbol for the weak and mass eigenstates. In terms of the mass eigenstate fields6, miu(u R − ũiR)(H − H̃)ǫ(QiL − Q̃iL)−mid(diR − d̃iR)(H† − H̃†)(QiL − Q̃iL) −mie(eiR − ẽiR)(H† − H̃†)(LiL − L̃iL) + h.c. , (57) where  , Q̃L = V d̃L  , Q̃′R = V d̃′R  . (58) Here V is the usual CKM matrix. The LW-fermions decay via the Yukawa interactions; for example, ν̃e → e−h̃+ → e−tb̄. LW-gauge bosons can decay to pairs of ordinary fermions. All the heavy LW-particles decay in this theory, so the only sources of missing energy in collider experiments are the usual standard model neutrinos. 6 They are mass eigenstate fields when mixing between the normal and LW-fields is neglected. This mixing can be treated as a perturbation. V. CONCLUSIONS In this paper we have developed an extension of the minimal standard model that is free of quadratic divergences. It is based on the work of Lee and Wick who constructed a finite version of QED by associating the regulator propagator in Pauli-Villars with a physical degree of freedom. Our model is a higher derivative theory and as such contains propagators with wrong sign residues about the new poles. Lee and Wick, and Cutkosky et al. provide a prescription for handling this issue. The LW-particles associated with these new poles are not in the spectrum, but instead decay to ordinary degrees of freedom. Their resummed propagators do not satisfy the usual analyticity properties since the poles are on the physical sheet. Lee and Wick (see also Cutkosky et al.) propose deforming integration contours in Feynman diagrams so that there is no catastrophic exponential growth as time increases. This amounts to a future boundary condition and so LW-theories violate the usual causal conditions. While the Lee Wick interpretation is peculiar it seems to be consistent, at least in perturbation theory, and predictions for physical observables can be made order by order in perturbation theory. Since the extension of the standard model presented here is free of quadratic divergences it solves the hierarchy problem. Our theory contains one new parameter, the mass of the LW- partner, for each field. We reduced the number of parameters by imposing minimal flavor violation to simplify the flavor structure of the theory. To make the physical interpreta- tion clearer and the calculations easier we introduced auxiliary LW-fields. The Lagrangian written in terms of these fields does not contain any higher derivative terms. When the LW-fields are integrated out the higher derivative theory is recovered. This paper focused on the the structure of the Lagrange density for the Lee-Wick exten- sion of the standard model. We constructed the Lagrange density, examined the divergence structure and showed how to introduce auxiliary fields to clarify the physical interpretation. For the future, a more extensive discussion of the phenomenology of the theory including its implications for LHC physics is appropriate. Acknowledgments DOC would like to thank Stephen Adler for a helpful discussion and for pointing out a useful reference. The work of BG was supported in part by the US Department of Energy under contract DE-FG03-97ER40546, while the work of DOC and MBW was supported in part by the US Department of Energy under contract DE-FG03-92ER40701. [1] T. D. Lee and G. C. Wick, Nucl. Phys. B 9, 209 (1969). [2] T. D. Lee and G. C. Wick, Phys. Rev. D 2, 1033 (1970). [3] R. E. Cutkosky, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, Nucl. Phys. B 12, 281 (1969). [4] N. Nakanishi, Phys. Rev. D 3, 811 (1971). [5] T. D. Lee and G. C. Wick, Phys. Rev. D 3, 1046 (1971). [6] S. Coleman, In “Erice 1969, Ettore Majorana School On Subnuclear Phenomena”, New York, 282 (1970). [7] K. Jansen, J. Kuti and C. Liu, Phys. Lett. B 309, 119 (1993) [arXiv:hep-lat/9305003]; K. Jansen, J. Kuti and C. Liu, Phys. Lett. B 309, 127 (1993) [arXiv:hep-lat/9305004]. [8] D. G. Boulware and D. J. Gross, Nucl. Phys. B 233, 1 (1984). [9] D. Evens, J. W. Moffat, G. Kleppe and R. P. Woodard, Phys. Rev. D 43, 499 (1991); J. W. Moffat, arXiv:hep-ph/0003171. [10] W. M. Yao et al. [Particle Data Group], J. Phys. G 33, 1 (2006). [11] R. S. Chivukula and H. Georgi, Phys. Lett. B 188, 99 (1987); L. J. Hall and L. Randall, Phys. Rev. Lett. 65, 2939 (1990); G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B 645, 155 (2002) [arXiv:hep-ph/0207036]. http://arxiv.org/abs/hep-lat/9305003 http://arxiv.org/abs/hep-lat/9305004 http://arxiv.org/abs/hep-ph/0003171 http://arxiv.org/abs/hep-ph/0207036 Introduction A Toy Model The Hierarchy Problem and Lee-Wick Theory Gauge Fields Scalar Matter Power Counting One loop Pole Mass The normal scalar The LW-scalar The LW-vector Lee-Wick Standard Model Lagrangian The Higgs Sector Fermion Kinetic Terms Fermion Yukawa Interactions Conclusions Acknowledgments References

We construct a modification of the standard model which stabilizes the Higgs mass against quadratically divergent radiative corrections, using ideas originally discussed by Lee and Wick in the context of a finite theory of quantum electrodynamics. The Lagrangian includes new higher derivative operators. We show that the higher derivative terms can be eliminated by introducing a set of auxiliary fields; this allows for convenient computation and makes the physical interpretation more transparent. Although the theory is unitary, it does not satisfy the usual analyticity conditions.

Introduction A Toy Model The Hierarchy Problem and Lee-Wick Theory Gauge Fields Scalar Matter Power Counting One loop Pole Mass The normal scalar The LW-scalar The LW-vector Lee-Wick Standard Model Lagrangian The Higgs Sector Fermion Kinetic Terms Fermion Yukawa Interactions Conclusions Acknowledgments References

SPECHT MODULES AND KAZHDAN–LUSZTIG CELLS IN TYPE Bn MEINOLF GECK, LACRIMIOARA IANCU AND CHRISTOS PALLIKAROS Abstract. Dipper, James and Murphy generalized the classical Specht module theory to Hecke algebras of type Bn. On the other hand, for any choice of a monomial order on the parameters in type Bn, we obtain corresponding Kazhdan–Lusztig cell modules. In this paper, we show that the Specht modules are naturally isomorphic to the Kazhdan–Lusztig cell modules if we choose the dominance order on the parameters, as in the “asymptotic case” studied by Bonnafé and the second named author. We also give examples which show that such an isomorphism does not exist for other choices of monomial orders. 1. Introduction LetHn be the generic Iwahori–Hecke algebra of type An−1 or Bn. For any partition or bipartition λ of n, we have a corresponding Specht module S̃λ, as defined by Dipper–James [5] (in type An−1) and Dipper–James–Murphy [6] (in type Bn). On the other hand, we have the cell modules arising from the theory of Kazhdan–Lusztig cells; see Lusztig [13], [14]. Now McDonough–Pallikaros [15] showed that, in type An−1, the Specht modules and Kazhdan–Lusztig cell modules are naturally isomorphic. The main purpose of this paper is to prove an analogous result for type Bn. Note that, contrary to the situation in type An−1, there are many different types of Kazhdan–Lusztig cell modules in type Bn, depending on the choice of a monomial order on the two parameters in type Bn. We will show that it is precisely the “asymptotic case” studied in [3] which yields an isomorphism with the Specht modules of Dipper–James–Murphy. In Theorem 3.6, we show the existence of a canonical isomorphism between a Specht module and a Kazhdan–Lusztig left cell module in the “asymptotic case” (where both of them are labelled by the appropriate bipartition of n). Both the Specht modules and the Kazhdan–Lusztig cells have certain standard bases. We show that, for a suitable ordering of these bases, the matrix of the canonical isomorphism is triangular with 1 on the diagonal. Our proof essentially relies on the combinatorial description [3] of the left cells in the “asymptotic case”. This allows us to determine explicitly (in terms of reduced expressions of elements) certain distinguished left cells for every bipartition of n; see Proposition 2.6. In Section 4, we give examples which show that the Specht modules are not isomorphic to Kazhdan–Lusztig cell modules for choices of the monomial order which are different from the “asymptotic case”. 2. Kazhdan–Lusztig bases and cells In this section, we recall the basic definitions concerning Kazhdan–Lusztig bases and cells, fol- lowing Lusztig [13], [14]. We also recall some of the main results of [3], [4], [8] concerning the “asymptotic case” in type Bn. This will allow us, see Proposition 2.6, to describe explicit reduced expressions for the elements in certain distinguished left cells in type Bn. Date: June, 2007. 2000 Mathematics Subject Classification. Primary 20C08; Secondary 20G40. http://arxiv.org/abs/0704.1846v2 2 Geck, Iancu and Pallikaros 2.A. Basic definitions. In [14], an Iwahori–Hecke algebra with possibly unequal parameters is defined with respect to an integer-valued weight function on W . Following a suggestion of Bonnafé [4], we can slightly modify Lusztig’s definition so as to include the more general setting in [13] as well. Let Γ be an abelian group (written additively) and let A = Z[Γ] be the free abelian group with basis {εγ | γ ∈ Γ}. There is a well-defined ring structure on A such that εγεγ = εγ+γ all γ, γ′ ∈ Γ. (Hence, if Γ = Z, then A is nothing but the ring of Laurent polynomials in an indeterminate ε.) We write 1 = ε0 ∈ A. Given a ∈ A we denote by aγ the coefficient of ε γ , so that γ∈Γ aγε γ . We say that a function L : W → Γ is a weight function if L(ww′) = L(w) + L(w′) whenever we have ℓ(ww′) = ℓ(w) + ℓ(w′) where ℓ : W → N is the usual length function. (We denote N = {0, 1, 2, . . .}.) Let H = H(W,S,L) be the generic Iwahori–Hecke algebra over A with parameters {vs | s ∈ S} where vs := ε L(s) for s ∈ S. The algebra H is free over A with basis {Tw | w ∈W}, and the multiplication is given by the rule TsTw = Tsw if ℓ(sw) > ℓ(w), Tsw + (vs − v s )Tw if ℓ(sw) < ℓ(w), where s ∈ S and w ∈W . Now assume that there is a total order 6 on Γ compatible with the group structure. (In the setting of [14], Γ = Z with the natural order.) The following definitions will depend on the choice of this total order. We denote by A>0 the set of Z-linear combinations of elements ε γ where γ > 0. Similarly, we define A>0, A60 and A<0. We assume throughout that L(s) > 0 for all s ∈ S. Having fixed a total order on Γ, we have a corresponding Kazhdan–Lusztig basis {Cw | w ∈W} of H. The element Cw is self-dual with respect to a certain ring involution of H, and we have Cw = Tw + P ∗y,w Ty ∈ H, where < denotes the Bruhat–Chevalley order on W and P ∗y,w ∈ A<0 for all y < w in W ; see [13, §6]. (In the framework of [14], the polynomials P ∗y,w are denoted py,w and the basis elements Cw are denoted cw.) Given x, y ∈W , we write CxCy = hx,y,z Cz where hx,y,z ∈ A. We have the following more explicit multiplication rules (see [13, §6]): for w ∈ W and s ∈ S, we TsCw = Csw − v s Cw + M sz,w Cz if sw > w, vsCw if sw < w, where the elements M sz,w ∈ A are determined as in [13, §3]. We recall the definition of the left cells of W and the corresponding left cell representations of H (see [13] or [14]). Note again that these depend on the choice of a total order on Γ. We write z ←L y if there exists some s ∈ S such that hs,y,z 6= 0, that is, Cz occurs in CsCy (when expressed in the C-basis). Let 6L be the pre-order relation on W generated by ←L, that is, we have z 6L y if there exist elements z = z0, z1, . . . , zk = y such that zi−1 ←L zi for 1 6 i 6 k. The equivalence relation associated with 6L will be denoted by ∼L and the corresponding equivalence classes are called the left cells of W . Specht modules and Kazhdan–Lusztig cells 3 Similarly, we can define a pre-order 6R by considering multiplication by Cs on the right in the defining relation. The equivalence relation associated with 6R will be denoted by ∼R and the corresponding equivalence classes are called the right cells of W . We have x 6R y ⇔ x This follows by using the anti-automorphism ♭ : H → H given by T ♭w = Tw−1 ; we have C w = Cw−1 ; see [14, 5.6]. Thus, any statement concerning the left pre-order relation 6L has an equivalent version for the right pre-order relation 6R, via ♭. Finally, we define a pre-order 6LR by the condition that x 6LR y if there exists a sequence x = x0, x1, . . . , xk = y such that, for each i ∈ {1, . . . , k}, we have xi−1 6L xi or xi−1 6R xi. The equivalence relation associated with 6LR will be denoted by ∼LR and the corresponding equivalence classes are called the two-sided cells of W . Each left cell C gives rise to a representation of H. This is constructed as follows (see [13, §7]). IC = 〈Cy (y ∈W ) | y 6L w for some w ∈ C〉A, ÎC = 〈Cy (y ∈W ) | y 6L w for some w ∈ C and y 6∈ C〉A. These are left ideals in H. Hence [C]A = IC/ÎC is a left H-module; it is free over A with basis {ew | w ∈ C} where ew denotes the class of Cw modulo ÎC. Explicitly, the action of H on [C]A is given by Cw.ex = hw,x,y ey for all x ∈ C and w ∈W. 2.B. The “asymptotic case” in type Bn. Now let Γ = Z2; then A = Z[Γ] is nothing but the ring of Laurent polynomials in two independent indeterminates V = ε(1,0) and v = ε(0,1). Let W = Wn be the Coxeter group of type Bn (n > 2), with generators, relations and weight function L : Wn → Γ given by the following diagram: ✐ ✐ ✐ · · · ✐ where a, b ∈ Γ. Let Hn be the corresponding generic two-parameter Iwahori–Hecke algebra over A = Z[Γ], where we set V := vt = ε b and v := vs1 = · · · = vsn−1 = ε (Note that any Hecke algebra of type Bn can be obtained from Hn by “specialisation”; see also Remark 3.8 below.) In order to obtain Kazhdan–Lusztig cells and the corresponding cell modules, we have to specify a total order 6 on Γ. Note that there are infinitely many such total orders: For example, we have all the weighted lexicographic orders, given by (i, j) < (i′, j′) if and only if xi+ yj < xi′ + yj′ or xi+ yj = xi′ + yj′ and i < i′, where x, y are fixed positive real numbers. Here, we shall take for 6 the lexicographic order on Γ such that (i, j) < (i′, j′) ⇐⇒ i < i′ or i = i′ and j < j′. This is the set-up originally considered by Bonnafé–Iancu [3]; it is called the “asymptotic case” in type Bn. We shall need some notation from [3]. Given w ∈Wn, we denote by ℓt(w) the number of occurrences of the generator t in a reduced expression for w, and call this the “t-length” of w. 4 Geck, Iancu and Pallikaros The parabolic subgroup Sn = 〈s1, . . . , sn−1〉 is naturally isomorphic to the symmetric group on {1, . . . , n}, where si corresponds to the basic transposition (i, i + 1). For 1 6 l 6 n − 1, we set Σl,n−l := {s1, . . . , sn−1} \ {sl}. For l = 0 or l = n, we also set Σ0,n = Σn,0 = {s1, . . . , sn−1}. Then we have the Young subgroup Sl,n−l = 〈Σl,n−l〉 = S{1,...,l} ×S{l+1,...,n}. Let Yl,n−l be the set of distinguished left coset representatives of Sl,n−l in Sn. We have the parabolic subalgebra Hl,n−l = 〈Tσ | σ ∈ Sl,n−l〉A ⊆ Hn. We denote by 6L,l the Kazhdan–Lusztig (left) pre-order relation on Sl,n−l and by ∼L,l the cor- responding equivalence relation. The symbols 6R,l, 6LR,l, ∼R,l and ∼LR,l have a similar meaning. Furthermore, as in [3, §4], we set a0 = 1 and al := t(s1t)(s2s1t) · · · (sl−1sl−2 · · · s1t) for l > 0. Then, by [3, Prop. 4.4], the set Yl,n−lal is precisely the set of distinguished left coset represen- tatives of Sn in Wn whose t-length equals l. Furthermore, every element w ∈ Wn has a unique decomposition w = awalσwb w where l = ℓt(w), σw ∈ Sl,n−l and aw, bw ∈ Yl,n−l; see [3, 4.6]. We call this the Clifford normal form of w. Theorem 2.1 (Bonnafé–Iancu [3, §7]). Assume that we are in the “asymptotic case” defined above. Let x, y ∈Wn. Then x ∼L y if and only if l := ℓt(x) = ℓt(y), bx = by and σx ∼L,l σy. Example 2.2. Let l ∈ {0, . . . , n} and C be a left cell of Sl,n−l. Since this group is a direct product, we can write C = C1 · C2 where C1 is a left cell in S{1,...,l} and C2 is a left cell in S{l+1,...,n}. Now Theorem 2.1 implies that (a) Yl,n−l al C is a left cell of Wn (in the “asymptotic case”). Now recall from [3, 4.1] that al = wlσl where wl is the longest element of the parabolic subgroup Wl = 〈t, s1, . . . , sl−1〉 (of type Bl) and σl is the longest element of S{1,...,l}. Since wl is central in Wl and conjugation with σl preserves the left cells of S{1,...,l}, we conclude that alC1al is a left cell of S{1,...,l}, too. Furthermore, al commutes with all elements of S{l+1,...,n} and so alCal is a left cell of Sl,n−l. Applying (a) now yields that (b) Yl,n−l C al is a left cell of Wn (in the “asymptotic case”). This example will be useful in the proof of Proposition 2.6 below. 2.C. Bitableaux. Let Λn be the set of all bipartitions of n. We write such a bipartition in the form λ = (λ1|λ2) where λ1 and λ2 are partitions such that |λ1| + |λ2| = n. For λ ∈ Λn, let T(λ) be the set of all standard λ-bitableaux. (Whenever we speak of bitableaux, it is understood that the filling is by the numbers 1, . . . , n.) The generalized Robinson–Schensted correspondence of [3] is a bijection T(λ)× T(λ), w 7→ (P (w), Q(w)). Thus, to each element w ∈Wn, we associate a pair of λ-bitableux for some λ ∈ Λn; in this case, we also write w λ and say that w is type λ. The following result provides an explicit combinatorial description of the left, right and two-sided cells in the “asymptotic case” in type Bn. Theorem 2.3. Assume we are in the “asymptotic case” defined in §2.B. Let x, y ∈Wn. Specht modules and Kazhdan–Lusztig cells 5 (a) (Bonnafé–Iancu [3, §7]) We have x ∼L y if and only if Q(x) = Q(y). Furthermore, x ∼R y if and only if P (x) = P (y). (b) (Bonnafé [4, §3 ]) We have x ∼LR y if and only if all of P (x), P (y), Q(x) and Q(y) have the same shape. Now let C be a left cell of Wn. We shall say that C is of type λ ∈ Λn if the bitableaux Q(x) (where x ∈ C) have shape λ. Theorem 2.4 (Geck [8, Theorem 6.3]). Let C and C′ be left cells of Wn (in the “asymptotic case”) which have the same type λ ∈ Λn. Then the left cell modules [C]A and [C ′]A are canonically isomorphic. In fact, there is a bijection C↔ C′ which induces an Hn-module isomorphism [C]A [C′]A. The above results show that, in order to study the left cell modules of Hn, it is sufficient to exhibit one particular left cell of type λ, for each given λ ∈ Λn. For this purpose, we shall need some further combinatorial notions from Dipper–James–Murphy [6, §3]. So let us fix a bipartition λ = (λ1|λ2) ∈ Λn, where l = |λ1| and 0 6 l 6 n. Let t λ be the “canonical” standard bitableau of shape λ defined in [6, p. 508]. Thus, tλ is a pair consisting of the “canonical” λ1-tableau t λ1 (obtaining by filling the rows in order from left to right by the numbers 1, . . . , l) and the “canonical” λ2-tableau t ′λ2 (obtained by filling the rows in order from left to right by the numbers l + 1, . . . , n). The symmetric group Sn acts (on the left) on bitableaux by permuting the entries. If t is any bitableau of shape λ, denote by d(t) the unique element of Sn which sends t λ to t. Thus, we have d(t).tλ = t for any λ-bitableau t. Now let Tr(λ) denote the set of all row-standard λ-bitableaux. Y λ := {d(t) | t ∈ Tr(λ)} is the set of distinguished left coset representatives of the parabolic subgroup Sλ in Sn; see [6, p. 509]. Applying this to the bipartition ((l), (n − l)), we find that Yl,n−l = Y ((l),(n−l)). Now we also define Trl (λ) to be the set of all t = (t1|t2) ∈ T r(λ) where t1 is filled by the numbers 1, . . . , l and t2 is filled by the numbers l + 1, . . . , n. Then, by the same argument as above, Y λl := {d(t) | t ∈ T l (λ)} is the set of distinguished left coset representatives of the parabolic subgroup Sλ inside Sl,n−l. Hence, considering the chain of parabolic subgroups Sλ ⊆ Sl,n−l ⊆ Sn, we obtain a decomposition Y λ = Yl,n−l · Y where ℓ(yd(t)) = ℓ(y) + ℓ(d(t)) for all y ∈ Yl,n−l and t ∈ T l (λ). Now we have the following purely combinatorial result. Lemma 2.5. In the above setting, let s ∈ Tr(λ), t ∈ Trl (λ) and y ∈ Yl,n−l be such that d(s) = y d(t). Then s is a standard bitableau if and only if t is a standard bitableau. Proof. We have s = d(s).tλ = (yd(t)).tλ = y.(d(t).tλ) = y.t. The permutation y ∈ Yl,n−l has the property that y(i) < y(i + 1) for 1 6 i < l and y(i) < y(i + 1) for l 6 i < n. Now it is an easy combinatorial exercise to see that s is standard if and only if t is standard; we omit further details. � 6 Geck, Iancu and Pallikaros Proposition 2.6. Let λ = (λ1|λ2) ∈ Λn and l = |λ1|. Let σλ ∈ Sλ be the longest element and Cλ be the left cell (with respect to the “asymptotic case”) containing σλal ∈Wn. Then Cλ has type (λ∗2|λ1) and we have Cλ = {d(t)σλ al | t ∈ T(λ)}, where ℓ(d(t)σλal) = ℓ(d(t)) + ℓ(σλal) for all t ∈ T(λ). Proof. By relation (♠) in the proof of [10, Prop. 5.4], the element alσλ has type (λ 2|λ1). Now since σλal = (alσλ) −1 it follows that σλal also has type (λ 2|λ1). Now, by [15, Lemma 3.3] (extended to the direct product of two symmetric groups), the set C := {d(t)σλ | t ∈ Tl(λ)} is the left cell of Sl,n−l containing σλ, where Tl(λ) is the set of all standard λ-bitableaux in T l (λ). Hence, by Example 2.2(b), we have Cλ = {y d(t)σλ al | y ∈ Yl,n−l, t ∈ Tl(λ)}. Furthermore, ℓ(y d(t)σλ al) = ℓ(y d(t)) + ℓ(σλ al). Now it remains to use Lemma 2.5. � Remark 2.7. In the above setting, it is not difficult to prove the following related result. Let x ∈Wn and l := ℓt(x). Then we have: x 6L σλal ⇐⇒ x = d(s)σλal where s is a row-standard λ-bitableaux. This follows from the properties of the Clifford normal form of the elements in Wn established in [3, §7] and the refinement obtained in [8, Theorem 5.11]. As we do not need this result in this paper, we omit further details. 3. Specht modules We keep the setup of the previous section, where we consider the Iwahori–Hecke algebra Hn of type Bn, defined over a polynomial ring A = Z[V ±1, v±1] in two independent indeterminates. We now consider the Specht modules defined by Dipper–James–Murphy [6]. The definition is based on the construction of a new basis of Hn, which is of the form {xst = Td(s) xλ Td(t)−1 | λ ∈ Λn and s, t ∈ T(λ)} where the element xλ is defined in [6, 4.1]; note that the definition of xλ does not rely on the choice of a total order on Γ. (An explicit description of xλ will be given in Lemma 3.2 below.) Let Nλ ⊆ Hn be the A-submodule spanned by all xst where s and t are standard µ-bitableaux such that λ E µ. Here, E denotes the dominance order on bipartitions, which is defined as follows; see Dipper–James–Murphy [6, §3]: Let λ = (λ1|λ2) and µ = (µ1|µ2) be bipartitions of n, with parts λ1 = (λ 1 > λ 1 > · · · > 0), λ2 = (λ 2 > λ 2 > · · · > 0), µ1 = (µ 1 > µ 1 > · · · > 0), µ2 = (µ 2 > µ 2 > · · · > 0). Then λ E µ if 1 (∀j) and |λ1|+ 2 6 |µ1|+ 2 (∀j). By [6, Cor. 4.13], Nλ is a two-sided ideal of Hn. Since the basis elements Tw (w ∈Wn) are invertible in Hn, we conclude that µ∈Λn;λEµ HnxµHn. Specht modules and Kazhdan–Lusztig cells 7 Similarly, we have the two-sided ideal N̂λ spanned by all xst where s and t are standard µ-bitableaux such that λ ⊳ µ (that is, λ E µ but λ 6= µ). Definition 3.1 (Dipper–James–Murphy [6, Def. 4.19]). Let λ ∈ Λn. The corresponding Specht module is defined by S̃λ := Mλ/(Mλ ∩ N̂λ) where Mλ = Hnxλ. By [6, Theorem 4.20], S̃λ is free over A, with standard basis {xs | s ∈ T(λ)} where xs denotes the class modulo Mλ ∩ N̂λ of the element x Our task will be to identify these Specht modules with certain Kazhdan–Lusztig left cells modules. For this purpose, assume from now on that we have chosen a total order on Γ such that we are in the “asymptotic case” defined in §2.B. Our first result, which is based on Bonnafé [4], identifies xλ in terms of the corresponding Kazhdan–Lusztig basis of Hn. Lemma 3.2. Let λ = (λ1|λ2) ∈ Λn and l = |λ1|. Then V lvl(l−1)−ℓ(σλ) xλ = Tσl Calσλ = CσλalTσl , where the elements σl, al and σλ are defined in §2. Proof. In [6, 4.1], the element xλ is defined as the product of three commuting factors u l , xλ1 , xλ2 . Bonnafé’s formula [4, Prop. 2.5] shows that V lvl(l−1) u+l = CalTσl = TσlCal . Furthermore, by Lusztig [14, Cor. 12.2], we have xλ1xλ2 = v ℓ(σλ)Cσλ . Finally, by [4, Prop. 2.3], we have CalCσλ = Calσλ and CσλCal = Cσλal . This yields the desired formulas. � Corollary 3.3. Let λ ∈ Λn. Then M λ = HnCalσλ = HnCσλalTσl . Proof. Clear by Lemma 3.2; just note v, V and Tσl are invertible in Hn. � Proposition 3.4. Let λ ∈ Λn. Then we have Nλ = 〈Cy (y ∈Wn) | y (ν1|ν2) where (λ1|λ2) E (ν2|ν 1 )〉A(a) ⊇ 〈Cy (y ∈Wn) | y 6LR alσλ〉A, N̂λ = 〈Cy (y ∈Wn) | Cy ∈ N λ and y 6∼LR alσλ〉A.(b) Proof. (a) The equality is proved in [10, Theorem 1.5]. Now let y ∈ Wn be such that y 6LR alσλ. Assume that y (µ1|µ2). Then Proposition 2.6 and [10, Prop. 5.4] show that (µ1|µ 2) E (λ or, equivalently, (λ1|λ2) E (µ2|µ 1). Thus, we have Cy ∈ N λ, as required. (b) Since N̂λ is the sum of all Nµ where µ ∈ Λn and λ ⊳ µ, the equality in (a) also implies that N̂λ = 〈Cy (y ∈Wn) | y (ν1|ν2) where (λ1|λ2) ⊳ (ν2|ν 1)〉A. So (b) follows from the description of the two-sided cells in Theorem 2.3(b). � Now we are ready to construct a canonical homomorphism from a Specht module to a certain Kazhdan–Lusztig cell module. Lemma 3.5. Let λ = (λ1|λ2) ∈ Λn and l = |λ1|. Let Cλ be the left cell of Wn containing σλal (with respect to the “asymptotic case”); see Proposition 2.6. Then there is a unique Hn-module homomorphism ϕλ : S̃ λ → [Cλ]A which sends the class of xλ ∈M λ in S̃λ to the class of Cσλal ∈ Iλ in [Cλ]A. 8 Geck, Iancu and Pallikaros Proof. Recall that [Cλ]A = Iλ/Îλ, where Iλ = 〈Cy | y ∈Wn such that y 6L σλal〉A, Îλ = 〈Cy | y ∈Wn such that y 6L σλal and y 6∈ Cλ〉A. We define ζλ := V −(l−1) vℓ(σλ)−l(l−1) T−1σl ∈ Hn. (Note that any element of the T -basis is invertible in Hn.) Then the map ρλ : Hn →Hn, h 7→ h ζλ, (that is, right multiplication by ζλ) is a left Hn-module isomorphism. By Lemma 3.2, Corollary 3.3 and the definition of 6L, we have ρλ(xλ) = Cσλal and ρλ(M λ) = HnCσλal ⊆ Iλ. Now, by Proposition 3.4, we certainly have Iλ ∩ N̂ λ ⊆ Îλ and so λ ∩ N̂λ) ⊆ HnCσλal ∩ N̂ λ ⊆ Iλ ∩ N̂ λ ⊆ Îλ. Hence, recalling also that S̃λ = Mλ/Mλ∩ N̂λ, we obtain a well-defined Hn-module homomorphism ϕλ : S̃ λ → [Cλ]A, m+ (M λ ∩ N̂λ) 7→ mζλ + Îλ, having the desired properties. The unicity of ϕλ is clear since S̃ λ is generated, as an Hn-module, by the class of xλ. � Next, we would like to obtain more detailed information about the matrix of ϕλ : S̃ λ → [Cλ]A with respect to the standard bases of the two modules. The aim will be to show that this matrix is triangular with 1 on the diagonal; in particular, this will show that ϕλ is an isomorphism. Recall that the Specht module S̃λ has a standard basis {xs | s ∈ T(λ)}; see Definition 3.1. On the other hand, by the definition of cell modules and Proposition 2.6, [Cλ]A has a standard basis {ed(s)σλal | s ∈ T(λ)} where ed(s)σλal denotes the class modulo Îλ of the element Cd(s)σλal ∈ Iλ. So, for any t ∈ T(λ), we write ϕλ(xt) = s∈T(λ) gs,t ed(s)σλal where gs,t ∈ A. Thus, Gλ := s,t∈T(λ) is the matrix of ϕλ with respect to the standard bases of S̃ λ and [Cλ]A, respectively. Now we can state the main result of this paper. Theorem 3.6. The map ϕλ : S̃ λ → [Cλ]A constructed in Lemma 3.5 is an isomorphism. More precisely, the following hold. For any s, t ∈ T(λ), we have gt,t = 1 for all t ∈ T(λ), gs,t = 0 unless d(s) 6 d(t), gs,t ∈ v Z[v−1] if s 6= t; here, 6 denotes the Bruhat–Chevalley order. Thus, the matrix Gλ has an upper unitriangular shape for a suitable ordering of the set T(λ). Proof. We begin with the following computation inside the parabolic subgroup Sn ⊆ Wn. Let t ∈ Tr(λ). By the multiplication rules for the Kazhdan–Lusztig basis, Td(t)Cσλ equals Cd(t)σλ plus a Z[v, v−1]-linear combination of terms Cx where x ∈ Sn, x 6L,n σλ and x < d(t)σλ. Now, the condition x 6L,n σλ implies that x can be written as x = d(s)σλ for some s ∈ T r(λ) (see, for Specht modules and Kazhdan–Lusztig cells 9 example, [15, 2.9]). Then the condition x = d(s)σλ < d(t)σλ implies that d(s) < d(t) (see [14, 9.10(f)]). Thus, we obtain (∗) Td(t) Cσλ = s∈Tr(λ) s,tCd(s)σλ for any t ∈ T r(λ), where g′ s,t ∈ Z[v, v −1] for all s, t ∈ Tr(λ); furthermore, g′ t,t = 1 and g s,t = 0 unless d(s) 6 d(t) and d(s)σλ 6L,n d(t)σλ. To pass from Sn to Wn, we use the following argument. First note that al is a distinguished right coset representative of Sn in Wn. By [4, Prop. 2.3], we have Cσal = CσCal for any σ ∈ Sn. Hence, multiplying (∗) on the right by Cal , we obtain Td(t) Cσλal = s∈Tr(λ) s,tCd(s)σλal for any t ∈ T r(λ). Now assume that t ∈ T(λ). Let s ∈ Tr(λ) be such that g′ s,t 6= 0. Then d(s)σλ 6L,n d(t)σλ and so d(s)σλal 6L d(t)σλal; see [14, Prop. 9.11]. Hence, using Proposition 2.6, we find that Td(t) Cσλal ≡ s∈T(λ) s,tCd(s)σλal mod Îλ. Passing to the quotient Iλ → [Cλ]A = Iλ/Îλ, we obtain Td(t).eσλal = s∈T(λ) s,t ed(s) σλal for any t ∈ T(λ). Now note that ϕλ(xt) = ϕλ(Td(t).x̄λ) = Td(t).ϕλ(x̄λ) = Td(t).eσλal , where x̄λ denotes the class of xλ in S̃λ. Thus, we see that g′ s,t = gs,t for all s, t ∈ T(λ). Consequently, the coefficients gs,t have the property that gt,t = 1 and gs,t = 0 unless d(s) 6 d(t). Hence, for a suitable ordering of the rows and columns, the matrix Gλ is unitriangular and ϕλ is an isomorphism. It remains to prove that gs,t ∈ v Z[v−1] for s 6= t. We will actually show that g′ s,t ∈ v Z[v−1] for all s, t ∈ Tr(λ) such that s 6= t. This is seen as follows. We can invert the equations (∗) and obtain Cd(t)σλ = s∈Tr(λ) g̃s,tTd(s) Cσλ for any t ∈ T r(λ), where the g̃s,t’s are the entries of the inverse of the matrix s,t∈Tr(λ) . A comparison with [7, Prop. 3.3] shows that g̃s,t = p d(s)σλ,d(t)σλ ∈ v−1Z[v−1] if s 6= t. Hence we also have g′ s,t ∈ v Z[v−1] for s 6= t. � Remark 3.7. Let λ ∈ Λn and C be any left cell such that C and Cλ are contained in the same two-sided cell. Then, by Theorem 2.4, [C]A and [Cλ]A are canonically isomorphic as Hn-modules. Hence, in combination with Theorem 3.6, we conclude that S̃λ ∼= [C]A. Thus, any left cell module of Hn is canonically isomorphic to a Specht module. Remark 3.8. The above results also hold for specialized algebras. More precisely, let R be any commutative ring (with 1) and fix two invertible elements Q, q ∈ R which admit square roots Q1/2 and q1/2 in R. Then we have a unique ring homomorphism θ : A→ R such that θ(V ) = Q1/2 and θ(v) = q1/2. We can extend scalars from A to R and set Hn,R = R⊗A Hn, S̃ R = R⊗A S̃ λ, [C]R = R⊗A [C]A, 10 Geck, Iancu and Pallikaros for any λ ∈ Λn and any left cell C of Wn. Then S̃ R precisely is the Specht module of the algebra of type Bn with parameters Q, q, as considered by Dipper–James–Murphy [6]. By Theorem 3.6 and Remark 3.7, we have an induced canonical isomorphism ϕ̃R : S̃ −→ [C]R whenever C is in the same two-sided cell as Cλ. 4. Counterexample Recall that Hn is defined over the ring of Laurent polynomials A = Z[V ±1, v±1] in two indepen- dent indeterminates. In the previous sections, we considered the Kazhdan–Lusztig cell modules of Hn with respect to the “asymptotic case” [3], that is, assuming that the group of monomials {V ivj | i, j ∈ Z} is endowed with the pure lexicographic order where V ivj < vi if and only if i < i′ or i = i′ and j < j′. But there are many other monomial orders, each giving rise to a Kazhdan–Lusztig basis of Hn and corresponding cell modules. The aim of this section is to show that, in general, the Dipper–James–Murphy Specht modules S̃λ cannot be identified with cell modules for these other choices of a monomial order. We do this in two ways: (1) by a concrete example in type B3 and (2) by a general argument involving non-semisimple specialisations of Hn. 4.A. An example in type B3. Let n = 3; then W3 = 〈t, s1, s2〉. Let λ = ((1), (2)) ∈ Λ3 and consider the corresponding Specht module S̃λ. By Theorem 3.6, it is isomorphic to [Cλ]A, where Cλ is a left cell with respect to the “asymptotic case”. We have l = 1 and σλal = s2t. Using Proposition 2.6, we find that Cλ = {s2t, s1s2t, s2s1s2t}. The corresponding left cell representation ρλ : H3 →M3(A) is given by Tt 7→ V V v−1V −1v V v−2 + V −1v2 0 −V −1 0 0 0 −V −1 Ts1 7→ −v−1 0 0 1 v 0 0 0 v  , Ts2 7→ v 1 0 0 −v−1 0 0 1 v Now let us choose a different monomial order on {V ivj | i, j ∈ Z}, namely, the weighted lexico- graphic order where V ivj < V i ′ def ⇐⇒ i+ j < i′ + j′ or i+ j = i′ + j′ and i < i′. (In particular, we have v < V < v2.) By an explicit computation, one can show that, in this case, the Kazhdan–Lusztig cell modules are all irreducible over K (the field of fractions of A), in accordance with [2, Conjecture A+]. (We are in the case r = 1 of that conjecture.) Furthermore, there are precisely three left cells C1,C2,C3 such that [Ci]K ∼= S̃ K ; they are given as follows: C1 = {s1s2s1, s1ts1s2s1, ts1s2s1}, C2 = {s1s2s1t, s1ts1s2s1t, ts1s2s1t}, C3 = {s1s2s1ts1, s1ts1s2s1ts1, ts1s2s1ts1}. Specht modules and Kazhdan–Lusztig cells 11 The corresponding left cell representations are all identical and given by: Tt 7→ −V −1 0 0 0 −V −1 0 1 V v−1 + V −1v V Ts1 7→ v 0 0 0 v 1 0 0 −v−1  , Ts2 7→ v V v−2 + V −1v2 0 0 −v−1 0 0 1 v Denote this representation by ρ : H3 → M3(A). Now one checks that Pρλ(Ts) = ρ(Ts)P for s ∈ {t, s1, s2} where 0 0 V v−2 + V −1v2 0 1 0 1 0 0 Thus, P defines a non-trivial module homomorphism between ρλ and ρ. Since these representations are irreducible over K, the matrix P is uniquely determined up to scalar multiples. But we see that there is no scalar λ ∈ K such that λP ∈ M3(A) and det(λP ) ∈ A ×. Hence, S̃λ will not be isomorphic to any Kazhdan–Lusztig cell module with respect to the above weighted lexicographic order. 4.B. General cell modules. Let k be a field and fix an element ξ ∈ k×. Let a, b ∈ Z>0 and consider the specialisation A→ k such that V 7→ ξb and v 7→ ξa. Let Hn,k = k ⊗A Hn be the corresponding specialized algebra. As in Remark 3.8, we also have corresponding Specht modules S̃λk for Hn,k. Now, for each λ ∈ Λn, there is a certain Hn,k-invariant bilinear form φλ : S̃ k × S̃ k → k; see [6, §5]. Let rad(φλ) be the radical of that form and set Dλ = S̃λk/rad(φλ). Then D λ is either 0 or an absolutely irreducible Hn,k-module; furthermore, we have Irr(Hn,k) = {D µ | µ ∈ Λ♣} where Λ♣ = {λ ∈ Λ | Dλ 6= 0}; see Dipper–James–Murphy [6, Theorem 6.6]. The conjecture in [6, 8.13] about an explicit combi- natorial description of Λ♣ has recently been proved by Ariki–Jacon [1]. Now consider the Kazhdan–Lusztig basis {Cw} of Hn,k with respect to the weight function L : Wn → Z such that L(t) = b and L(si) = a for all i. Assume that Lusztig’s conjectures (P1)– (P15) in [14, 14.2] on Hecke algebras with unequal parameters hold. (This is the case, for example, in the “equal parameter case” where a = b; see [14, Chap. 15].) Using these properties, it is shown in [9] that Hn,k has a natural “cellular structure” in the sense of Graham–Lehrer [12]. The elements of the “cellular basis” are certain linear combinations of the basis elements {Cw}. Then, by the general theory of “cellular algebras”, for any λ ∈ Λn, we have a “cell module” Wk(λ) for Hn,k and this cell module is naturally equipped with an Hn,k-invariant bilinear form gλ : Wk(λ)×Wk(λ)→ k. Let rad(gλ) be the radical of that form and set L λ = Wk(λ)/rad(gλ). Then, again, L λ is either 0 or an absolutely irreducible Hn,k-module; furthermore, we have Irr(Hn,k) = {L µ | µ ∈ Λ♠} where Λ♠ = {λ ∈ Λ | Lλ 6= 0}; see Graham–Lehrer [12, §3] and [9, Example 4.4]. Given these two settings, it is natural to ask if S̃λk ∼= Wk(λ) and, subsequently, if Λ ♣ = Λ♠ ? In the case where Hn,k is semisimple, it is shown in [9, Example 4.4] that S̃ ∼= Wk(λ); furthermore, by the general theory of “cellular algebras” [12] and the results in [6], we have Λ♣ = Λ♠ = Λ in 12 Geck, Iancu and Pallikaros this case. However, if Hn,k is not semisimple, then the answer to these questions is negative, as can be seen from the fact that Λ♣ 6= Λ♠ in general; see [11] and the references there. By [11, Theorem 2.8] it is true, however, that Λ♣ = Λ♠ if b > (n − 1)a > 0 which corresponds precisely to the “asymptotic case” discussed in this paper. Indeed, by [8, Corollary 6.3], the basis {Cw} of Hn,k is cellular under this assumption on a, b, and by Theorem 3.6, we have Wk(λ) ∼= S̃ Acknowledgements. The final form of this paper grew out of several discussions which the three authors could hold thanks to the hospitality of various institutions. Part of this work was done while all three authors enjoyed the hospitality of the Bernoulli Center at the EPFL Lausanne (Switzerland) in 2005. C.P. thanks the University of Aberdeen (Scotland) for an invitation in November 2006; M.G. and L.I. would like to thank the University of Cyprus at Nicosia for an invitation in March 2007. References [1] S. Ariki and N. Jacon, Dipper–James–Murphy’s conjecture for Hecke algebras of type B, preprint; available at math.RT/0703447. [2] C. Bonnafé, M. Geck, L. Iancu and T. Lam, On domino insertion and Kazhdan–Lusztig cells in type Bn, preprint; available at math.RT/0609279. [3] C. Bonnafé and L. Iancu, Left cells in type Bn with unequal parameters, Represent. Theory 7 (2003), 587–609. [4] C. Bonnafé, Two-sided cells in type B in the asymptotic case, J. Algebra 304 (2006), 216–236. [5] R. Dipper and G. D. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. 52 (1986), 20–52. [6] R. Dipper, G. D. James and G. E. Murphy, Hecke algebras of type Bn at roots of unity, Proc. London Math. Soc. 70 (1995), 505–528. [7] M. Geck, On the induction of Kazhdan–Lusztig cells, Bull. London Math. Soc. 35 (2003), 608–614. [8] M. Geck, Relative Kazhdan–Lusztig cells, Represent. Theory 10 (2006), 481–524. [9] M. Geck, Hecke algebras of finite type are cellular, Invent. Math. (2007), to appear. [10] M. Geck and L. Iancu, Lusztig’s a-function in type Bn in the asymptotic case. Special issue celebrating the 60th birthday of George Lusztig, Nagoya J. Math. 182 (2006), 199–240. [11] M. Geck and N. Jacon, Canonical basic sets in type B. Special issue in honour of Gordon Douglas James, J. Algebra 306 (2006), 104–127. [12] J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), 1–34. [13] G. Lusztig, Left cells in Weyl groups, Lie Group Representations, I (R. L. R. Herb and J. Rosenberg, eds.), Lecture Notes in Math., vol. 1024, Springer-Verlag, 1983, pp. 99–111. [14] G. Lusztig, Hecke algebras with unequal parameters, CRM Monographs Ser. 18, Amer. Math. Soc., Provi- dence, RI, 2003. [15] T. P. McDonough and C. A. Pallikaros, On relations between the classical and the Kazhdan–Lusztig representations of symmetric groups and associated Hecke algebras, J. Pure and Applied Algebra 203 (2005), 133–144. M.G. and L.I.: Department of Mathematical Sciences, King’s College, Aberdeen University, Ab- erdeen AB24 3UE, Scotland, U.K. E-mail address: m.geck@maths.abdn.ac.uk E-mail address: l.iancu@maths.abdn.ac.uk C.P.: Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus E-mail address: pallikar@ucy.ac.cy http://arxiv.org/abs/math/0703447 http://arxiv.org/abs/math/0609279 1. Introduction 2. Kazhdan–Lusztig bases and cells 2.A. Basic definitions 2.B. The ``asymptotic case'' in type Bn 2.C. Bitableaux 3. Specht modules 4. Counterexample 4.A. An example in type B3 4.B. General cell modules References

Dipper, James and Murphy generalized the classical Specht module theory to Hecke algebras of type $B_n$. On the other hand, for any choice of a monomial order on the parameters in type $B_n$, we obtain corresponding Kazhdan--Lusztig cell modules. In this paper, we show that the Specht modules are naturally equivalent to the Kazhdan--Lusztig cell modules {\em if} we choose the dominance order on the parameters, as in the ``asymptotic case'' studied by Bonnaf\'e and the second named author. We also give examples which show that such an equivalence does not hold for other choices of monomial orders.

Introduction LetHn be the generic Iwahori–Hecke algebra of type An−1 or Bn. For any partition or bipartition λ of n, we have a corresponding Specht module S̃λ, as defined by Dipper–James [5] (in type An−1) and Dipper–James–Murphy [6] (in type Bn). On the other hand, we have the cell modules arising from the theory of Kazhdan–Lusztig cells; see Lusztig [13], [14]. Now McDonough–Pallikaros [15] showed that, in type An−1, the Specht modules and Kazhdan–Lusztig cell modules are naturally isomorphic. The main purpose of this paper is to prove an analogous result for type Bn. Note that, contrary to the situation in type An−1, there are many different types of Kazhdan–Lusztig cell modules in type Bn, depending on the choice of a monomial order on the two parameters in type Bn. We will show that it is precisely the “asymptotic case” studied in [3] which yields an isomorphism with the Specht modules of Dipper–James–Murphy. In Theorem 3.6, we show the existence of a canonical isomorphism between a Specht module and a Kazhdan–Lusztig left cell module in the “asymptotic case” (where both of them are labelled by the appropriate bipartition of n). Both the Specht modules and the Kazhdan–Lusztig cells have certain standard bases. We show that, for a suitable ordering of these bases, the matrix of the canonical isomorphism is triangular with 1 on the diagonal. Our proof essentially relies on the combinatorial description [3] of the left cells in the “asymptotic case”. This allows us to determine explicitly (in terms of reduced expressions of elements) certain distinguished left cells for every bipartition of n; see Proposition 2.6. In Section 4, we give examples which show that the Specht modules are not isomorphic to Kazhdan–Lusztig cell modules for choices of the monomial order which are different from the “asymptotic case”. 2. Kazhdan–Lusztig bases and cells In this section, we recall the basic definitions concerning Kazhdan–Lusztig bases and cells, fol- lowing Lusztig [13], [14]. We also recall some of the main results of [3], [4], [8] concerning the “asymptotic case” in type Bn. This will allow us, see Proposition 2.6, to describe explicit reduced expressions for the elements in certain distinguished left cells in type Bn. Date: June, 2007. 2000 Mathematics Subject Classification. Primary 20C08; Secondary 20G40. http://arxiv.org/abs/0704.1846v2 2 Geck, Iancu and Pallikaros 2.A. Basic definitions. In [14], an Iwahori–Hecke algebra with possibly unequal parameters is defined with respect to an integer-valued weight function on W . Following a suggestion of Bonnafé [4], we can slightly modify Lusztig’s definition so as to include the more general setting in [13] as well. Let Γ be an abelian group (written additively) and let A = Z[Γ] be the free abelian group with basis {εγ | γ ∈ Γ}. There is a well-defined ring structure on A such that εγεγ = εγ+γ all γ, γ′ ∈ Γ. (Hence, if Γ = Z, then A is nothing but the ring of Laurent polynomials in an indeterminate ε.) We write 1 = ε0 ∈ A. Given a ∈ A we denote by aγ the coefficient of ε γ , so that γ∈Γ aγε γ . We say that a function L : W → Γ is a weight function if L(ww′) = L(w) + L(w′) whenever we have ℓ(ww′) = ℓ(w) + ℓ(w′) where ℓ : W → N is the usual length function. (We denote N = {0, 1, 2, . . .}.) Let H = H(W,S,L) be the generic Iwahori–Hecke algebra over A with parameters {vs | s ∈ S} where vs := ε L(s) for s ∈ S. The algebra H is free over A with basis {Tw | w ∈W}, and the multiplication is given by the rule TsTw = Tsw if ℓ(sw) > ℓ(w), Tsw + (vs − v s )Tw if ℓ(sw) < ℓ(w), where s ∈ S and w ∈W . Now assume that there is a total order 6 on Γ compatible with the group structure. (In the setting of [14], Γ = Z with the natural order.) The following definitions will depend on the choice of this total order. We denote by A>0 the set of Z-linear combinations of elements ε γ where γ > 0. Similarly, we define A>0, A60 and A<0. We assume throughout that L(s) > 0 for all s ∈ S. Having fixed a total order on Γ, we have a corresponding Kazhdan–Lusztig basis {Cw | w ∈W} of H. The element Cw is self-dual with respect to a certain ring involution of H, and we have Cw = Tw + P ∗y,w Ty ∈ H, where < denotes the Bruhat–Chevalley order on W and P ∗y,w ∈ A<0 for all y < w in W ; see [13, §6]. (In the framework of [14], the polynomials P ∗y,w are denoted py,w and the basis elements Cw are denoted cw.) Given x, y ∈W , we write CxCy = hx,y,z Cz where hx,y,z ∈ A. We have the following more explicit multiplication rules (see [13, §6]): for w ∈ W and s ∈ S, we TsCw = Csw − v s Cw + M sz,w Cz if sw > w, vsCw if sw < w, where the elements M sz,w ∈ A are determined as in [13, §3]. We recall the definition of the left cells of W and the corresponding left cell representations of H (see [13] or [14]). Note again that these depend on the choice of a total order on Γ. We write z ←L y if there exists some s ∈ S such that hs,y,z 6= 0, that is, Cz occurs in CsCy (when expressed in the C-basis). Let 6L be the pre-order relation on W generated by ←L, that is, we have z 6L y if there exist elements z = z0, z1, . . . , zk = y such that zi−1 ←L zi for 1 6 i 6 k. The equivalence relation associated with 6L will be denoted by ∼L and the corresponding equivalence classes are called the left cells of W . Specht modules and Kazhdan–Lusztig cells 3 Similarly, we can define a pre-order 6R by considering multiplication by Cs on the right in the defining relation. The equivalence relation associated with 6R will be denoted by ∼R and the corresponding equivalence classes are called the right cells of W . We have x 6R y ⇔ x This follows by using the anti-automorphism ♭ : H → H given by T ♭w = Tw−1 ; we have C w = Cw−1 ; see [14, 5.6]. Thus, any statement concerning the left pre-order relation 6L has an equivalent version for the right pre-order relation 6R, via ♭. Finally, we define a pre-order 6LR by the condition that x 6LR y if there exists a sequence x = x0, x1, . . . , xk = y such that, for each i ∈ {1, . . . , k}, we have xi−1 6L xi or xi−1 6R xi. The equivalence relation associated with 6LR will be denoted by ∼LR and the corresponding equivalence classes are called the two-sided cells of W . Each left cell C gives rise to a representation of H. This is constructed as follows (see [13, §7]). IC = 〈Cy (y ∈W ) | y 6L w for some w ∈ C〉A, ÎC = 〈Cy (y ∈W ) | y 6L w for some w ∈ C and y 6∈ C〉A. These are left ideals in H. Hence [C]A = IC/ÎC is a left H-module; it is free over A with basis {ew | w ∈ C} where ew denotes the class of Cw modulo ÎC. Explicitly, the action of H on [C]A is given by Cw.ex = hw,x,y ey for all x ∈ C and w ∈W. 2.B. The “asymptotic case” in type Bn. Now let Γ = Z2; then A = Z[Γ] is nothing but the ring of Laurent polynomials in two independent indeterminates V = ε(1,0) and v = ε(0,1). Let W = Wn be the Coxeter group of type Bn (n > 2), with generators, relations and weight function L : Wn → Γ given by the following diagram: ✐ ✐ ✐ · · · ✐ where a, b ∈ Γ. Let Hn be the corresponding generic two-parameter Iwahori–Hecke algebra over A = Z[Γ], where we set V := vt = ε b and v := vs1 = · · · = vsn−1 = ε (Note that any Hecke algebra of type Bn can be obtained from Hn by “specialisation”; see also Remark 3.8 below.) In order to obtain Kazhdan–Lusztig cells and the corresponding cell modules, we have to specify a total order 6 on Γ. Note that there are infinitely many such total orders: For example, we have all the weighted lexicographic orders, given by (i, j) < (i′, j′) if and only if xi+ yj < xi′ + yj′ or xi+ yj = xi′ + yj′ and i < i′, where x, y are fixed positive real numbers. Here, we shall take for 6 the lexicographic order on Γ such that (i, j) < (i′, j′) ⇐⇒ i < i′ or i = i′ and j < j′. This is the set-up originally considered by Bonnafé–Iancu [3]; it is called the “asymptotic case” in type Bn. We shall need some notation from [3]. Given w ∈Wn, we denote by ℓt(w) the number of occurrences of the generator t in a reduced expression for w, and call this the “t-length” of w. 4 Geck, Iancu and Pallikaros The parabolic subgroup Sn = 〈s1, . . . , sn−1〉 is naturally isomorphic to the symmetric group on {1, . . . , n}, where si corresponds to the basic transposition (i, i + 1). For 1 6 l 6 n − 1, we set Σl,n−l := {s1, . . . , sn−1} \ {sl}. For l = 0 or l = n, we also set Σ0,n = Σn,0 = {s1, . . . , sn−1}. Then we have the Young subgroup Sl,n−l = 〈Σl,n−l〉 = S{1,...,l} ×S{l+1,...,n}. Let Yl,n−l be the set of distinguished left coset representatives of Sl,n−l in Sn. We have the parabolic subalgebra Hl,n−l = 〈Tσ | σ ∈ Sl,n−l〉A ⊆ Hn. We denote by 6L,l the Kazhdan–Lusztig (left) pre-order relation on Sl,n−l and by ∼L,l the cor- responding equivalence relation. The symbols 6R,l, 6LR,l, ∼R,l and ∼LR,l have a similar meaning. Furthermore, as in [3, §4], we set a0 = 1 and al := t(s1t)(s2s1t) · · · (sl−1sl−2 · · · s1t) for l > 0. Then, by [3, Prop. 4.4], the set Yl,n−lal is precisely the set of distinguished left coset represen- tatives of Sn in Wn whose t-length equals l. Furthermore, every element w ∈ Wn has a unique decomposition w = awalσwb w where l = ℓt(w), σw ∈ Sl,n−l and aw, bw ∈ Yl,n−l; see [3, 4.6]. We call this the Clifford normal form of w. Theorem 2.1 (Bonnafé–Iancu [3, §7]). Assume that we are in the “asymptotic case” defined above. Let x, y ∈Wn. Then x ∼L y if and only if l := ℓt(x) = ℓt(y), bx = by and σx ∼L,l σy. Example 2.2. Let l ∈ {0, . . . , n} and C be a left cell of Sl,n−l. Since this group is a direct product, we can write C = C1 · C2 where C1 is a left cell in S{1,...,l} and C2 is a left cell in S{l+1,...,n}. Now Theorem 2.1 implies that (a) Yl,n−l al C is a left cell of Wn (in the “asymptotic case”). Now recall from [3, 4.1] that al = wlσl where wl is the longest element of the parabolic subgroup Wl = 〈t, s1, . . . , sl−1〉 (of type Bl) and σl is the longest element of S{1,...,l}. Since wl is central in Wl and conjugation with σl preserves the left cells of S{1,...,l}, we conclude that alC1al is a left cell of S{1,...,l}, too. Furthermore, al commutes with all elements of S{l+1,...,n} and so alCal is a left cell of Sl,n−l. Applying (a) now yields that (b) Yl,n−l C al is a left cell of Wn (in the “asymptotic case”). This example will be useful in the proof of Proposition 2.6 below. 2.C. Bitableaux. Let Λn be the set of all bipartitions of n. We write such a bipartition in the form λ = (λ1|λ2) where λ1 and λ2 are partitions such that |λ1| + |λ2| = n. For λ ∈ Λn, let T(λ) be the set of all standard λ-bitableaux. (Whenever we speak of bitableaux, it is understood that the filling is by the numbers 1, . . . , n.) The generalized Robinson–Schensted correspondence of [3] is a bijection T(λ)× T(λ), w 7→ (P (w), Q(w)). Thus, to each element w ∈Wn, we associate a pair of λ-bitableux for some λ ∈ Λn; in this case, we also write w λ and say that w is type λ. The following result provides an explicit combinatorial description of the left, right and two-sided cells in the “asymptotic case” in type Bn. Theorem 2.3. Assume we are in the “asymptotic case” defined in §2.B. Let x, y ∈Wn. Specht modules and Kazhdan–Lusztig cells 5 (a) (Bonnafé–Iancu [3, §7]) We have x ∼L y if and only if Q(x) = Q(y). Furthermore, x ∼R y if and only if P (x) = P (y). (b) (Bonnafé [4, §3 ]) We have x ∼LR y if and only if all of P (x), P (y), Q(x) and Q(y) have the same shape. Now let C be a left cell of Wn. We shall say that C is of type λ ∈ Λn if the bitableaux Q(x) (where x ∈ C) have shape λ. Theorem 2.4 (Geck [8, Theorem 6.3]). Let C and C′ be left cells of Wn (in the “asymptotic case”) which have the same type λ ∈ Λn. Then the left cell modules [C]A and [C ′]A are canonically isomorphic. In fact, there is a bijection C↔ C′ which induces an Hn-module isomorphism [C]A [C′]A. The above results show that, in order to study the left cell modules of Hn, it is sufficient to exhibit one particular left cell of type λ, for each given λ ∈ Λn. For this purpose, we shall need some further combinatorial notions from Dipper–James–Murphy [6, §3]. So let us fix a bipartition λ = (λ1|λ2) ∈ Λn, where l = |λ1| and 0 6 l 6 n. Let t λ be the “canonical” standard bitableau of shape λ defined in [6, p. 508]. Thus, tλ is a pair consisting of the “canonical” λ1-tableau t λ1 (obtaining by filling the rows in order from left to right by the numbers 1, . . . , l) and the “canonical” λ2-tableau t ′λ2 (obtained by filling the rows in order from left to right by the numbers l + 1, . . . , n). The symmetric group Sn acts (on the left) on bitableaux by permuting the entries. If t is any bitableau of shape λ, denote by d(t) the unique element of Sn which sends t λ to t. Thus, we have d(t).tλ = t for any λ-bitableau t. Now let Tr(λ) denote the set of all row-standard λ-bitableaux. Y λ := {d(t) | t ∈ Tr(λ)} is the set of distinguished left coset representatives of the parabolic subgroup Sλ in Sn; see [6, p. 509]. Applying this to the bipartition ((l), (n − l)), we find that Yl,n−l = Y ((l),(n−l)). Now we also define Trl (λ) to be the set of all t = (t1|t2) ∈ T r(λ) where t1 is filled by the numbers 1, . . . , l and t2 is filled by the numbers l + 1, . . . , n. Then, by the same argument as above, Y λl := {d(t) | t ∈ T l (λ)} is the set of distinguished left coset representatives of the parabolic subgroup Sλ inside Sl,n−l. Hence, considering the chain of parabolic subgroups Sλ ⊆ Sl,n−l ⊆ Sn, we obtain a decomposition Y λ = Yl,n−l · Y where ℓ(yd(t)) = ℓ(y) + ℓ(d(t)) for all y ∈ Yl,n−l and t ∈ T l (λ). Now we have the following purely combinatorial result. Lemma 2.5. In the above setting, let s ∈ Tr(λ), t ∈ Trl (λ) and y ∈ Yl,n−l be such that d(s) = y d(t). Then s is a standard bitableau if and only if t is a standard bitableau. Proof. We have s = d(s).tλ = (yd(t)).tλ = y.(d(t).tλ) = y.t. The permutation y ∈ Yl,n−l has the property that y(i) < y(i + 1) for 1 6 i < l and y(i) < y(i + 1) for l 6 i < n. Now it is an easy combinatorial exercise to see that s is standard if and only if t is standard; we omit further details. � 6 Geck, Iancu and Pallikaros Proposition 2.6. Let λ = (λ1|λ2) ∈ Λn and l = |λ1|. Let σλ ∈ Sλ be the longest element and Cλ be the left cell (with respect to the “asymptotic case”) containing σλal ∈Wn. Then Cλ has type (λ∗2|λ1) and we have Cλ = {d(t)σλ al | t ∈ T(λ)}, where ℓ(d(t)σλal) = ℓ(d(t)) + ℓ(σλal) for all t ∈ T(λ). Proof. By relation (♠) in the proof of [10, Prop. 5.4], the element alσλ has type (λ 2|λ1). Now since σλal = (alσλ) −1 it follows that σλal also has type (λ 2|λ1). Now, by [15, Lemma 3.3] (extended to the direct product of two symmetric groups), the set C := {d(t)σλ | t ∈ Tl(λ)} is the left cell of Sl,n−l containing σλ, where Tl(λ) is the set of all standard λ-bitableaux in T l (λ). Hence, by Example 2.2(b), we have Cλ = {y d(t)σλ al | y ∈ Yl,n−l, t ∈ Tl(λ)}. Furthermore, ℓ(y d(t)σλ al) = ℓ(y d(t)) + ℓ(σλ al). Now it remains to use Lemma 2.5. � Remark 2.7. In the above setting, it is not difficult to prove the following related result. Let x ∈Wn and l := ℓt(x). Then we have: x 6L σλal ⇐⇒ x = d(s)σλal where s is a row-standard λ-bitableaux. This follows from the properties of the Clifford normal form of the elements in Wn established in [3, §7] and the refinement obtained in [8, Theorem 5.11]. As we do not need this result in this paper, we omit further details. 3. Specht modules We keep the setup of the previous section, where we consider the Iwahori–Hecke algebra Hn of type Bn, defined over a polynomial ring A = Z[V ±1, v±1] in two independent indeterminates. We now consider the Specht modules defined by Dipper–James–Murphy [6]. The definition is based on the construction of a new basis of Hn, which is of the form {xst = Td(s) xλ Td(t)−1 | λ ∈ Λn and s, t ∈ T(λ)} where the element xλ is defined in [6, 4.1]; note that the definition of xλ does not rely on the choice of a total order on Γ. (An explicit description of xλ will be given in Lemma 3.2 below.) Let Nλ ⊆ Hn be the A-submodule spanned by all xst where s and t are standard µ-bitableaux such that λ E µ. Here, E denotes the dominance order on bipartitions, which is defined as follows; see Dipper–James–Murphy [6, §3]: Let λ = (λ1|λ2) and µ = (µ1|µ2) be bipartitions of n, with parts λ1 = (λ 1 > λ 1 > · · · > 0), λ2 = (λ 2 > λ 2 > · · · > 0), µ1 = (µ 1 > µ 1 > · · · > 0), µ2 = (µ 2 > µ 2 > · · · > 0). Then λ E µ if 1 (∀j) and |λ1|+ 2 6 |µ1|+ 2 (∀j). By [6, Cor. 4.13], Nλ is a two-sided ideal of Hn. Since the basis elements Tw (w ∈Wn) are invertible in Hn, we conclude that µ∈Λn;λEµ HnxµHn. Specht modules and Kazhdan–Lusztig cells 7 Similarly, we have the two-sided ideal N̂λ spanned by all xst where s and t are standard µ-bitableaux such that λ ⊳ µ (that is, λ E µ but λ 6= µ). Definition 3.1 (Dipper–James–Murphy [6, Def. 4.19]). Let λ ∈ Λn. The corresponding Specht module is defined by S̃λ := Mλ/(Mλ ∩ N̂λ) where Mλ = Hnxλ. By [6, Theorem 4.20], S̃λ is free over A, with standard basis {xs | s ∈ T(λ)} where xs denotes the class modulo Mλ ∩ N̂λ of the element x Our task will be to identify these Specht modules with certain Kazhdan–Lusztig left cells modules. For this purpose, assume from now on that we have chosen a total order on Γ such that we are in the “asymptotic case” defined in §2.B. Our first result, which is based on Bonnafé [4], identifies xλ in terms of the corresponding Kazhdan–Lusztig basis of Hn. Lemma 3.2. Let λ = (λ1|λ2) ∈ Λn and l = |λ1|. Then V lvl(l−1)−ℓ(σλ) xλ = Tσl Calσλ = CσλalTσl , where the elements σl, al and σλ are defined in §2. Proof. In [6, 4.1], the element xλ is defined as the product of three commuting factors u l , xλ1 , xλ2 . Bonnafé’s formula [4, Prop. 2.5] shows that V lvl(l−1) u+l = CalTσl = TσlCal . Furthermore, by Lusztig [14, Cor. 12.2], we have xλ1xλ2 = v ℓ(σλ)Cσλ . Finally, by [4, Prop. 2.3], we have CalCσλ = Calσλ and CσλCal = Cσλal . This yields the desired formulas. � Corollary 3.3. Let λ ∈ Λn. Then M λ = HnCalσλ = HnCσλalTσl . Proof. Clear by Lemma 3.2; just note v, V and Tσl are invertible in Hn. � Proposition 3.4. Let λ ∈ Λn. Then we have Nλ = 〈Cy (y ∈Wn) | y (ν1|ν2) where (λ1|λ2) E (ν2|ν 1 )〉A(a) ⊇ 〈Cy (y ∈Wn) | y 6LR alσλ〉A, N̂λ = 〈Cy (y ∈Wn) | Cy ∈ N λ and y 6∼LR alσλ〉A.(b) Proof. (a) The equality is proved in [10, Theorem 1.5]. Now let y ∈ Wn be such that y 6LR alσλ. Assume that y (µ1|µ2). Then Proposition 2.6 and [10, Prop. 5.4] show that (µ1|µ 2) E (λ or, equivalently, (λ1|λ2) E (µ2|µ 1). Thus, we have Cy ∈ N λ, as required. (b) Since N̂λ is the sum of all Nµ where µ ∈ Λn and λ ⊳ µ, the equality in (a) also implies that N̂λ = 〈Cy (y ∈Wn) | y (ν1|ν2) where (λ1|λ2) ⊳ (ν2|ν 1)〉A. So (b) follows from the description of the two-sided cells in Theorem 2.3(b). � Now we are ready to construct a canonical homomorphism from a Specht module to a certain Kazhdan–Lusztig cell module. Lemma 3.5. Let λ = (λ1|λ2) ∈ Λn and l = |λ1|. Let Cλ be the left cell of Wn containing σλal (with respect to the “asymptotic case”); see Proposition 2.6. Then there is a unique Hn-module homomorphism ϕλ : S̃ λ → [Cλ]A which sends the class of xλ ∈M λ in S̃λ to the class of Cσλal ∈ Iλ in [Cλ]A. 8 Geck, Iancu and Pallikaros Proof. Recall that [Cλ]A = Iλ/Îλ, where Iλ = 〈Cy | y ∈Wn such that y 6L σλal〉A, Îλ = 〈Cy | y ∈Wn such that y 6L σλal and y 6∈ Cλ〉A. We define ζλ := V −(l−1) vℓ(σλ)−l(l−1) T−1σl ∈ Hn. (Note that any element of the T -basis is invertible in Hn.) Then the map ρλ : Hn →Hn, h 7→ h ζλ, (that is, right multiplication by ζλ) is a left Hn-module isomorphism. By Lemma 3.2, Corollary 3.3 and the definition of 6L, we have ρλ(xλ) = Cσλal and ρλ(M λ) = HnCσλal ⊆ Iλ. Now, by Proposition 3.4, we certainly have Iλ ∩ N̂ λ ⊆ Îλ and so λ ∩ N̂λ) ⊆ HnCσλal ∩ N̂ λ ⊆ Iλ ∩ N̂ λ ⊆ Îλ. Hence, recalling also that S̃λ = Mλ/Mλ∩ N̂λ, we obtain a well-defined Hn-module homomorphism ϕλ : S̃ λ → [Cλ]A, m+ (M λ ∩ N̂λ) 7→ mζλ + Îλ, having the desired properties. The unicity of ϕλ is clear since S̃ λ is generated, as an Hn-module, by the class of xλ. � Next, we would like to obtain more detailed information about the matrix of ϕλ : S̃ λ → [Cλ]A with respect to the standard bases of the two modules. The aim will be to show that this matrix is triangular with 1 on the diagonal; in particular, this will show that ϕλ is an isomorphism. Recall that the Specht module S̃λ has a standard basis {xs | s ∈ T(λ)}; see Definition 3.1. On the other hand, by the definition of cell modules and Proposition 2.6, [Cλ]A has a standard basis {ed(s)σλal | s ∈ T(λ)} where ed(s)σλal denotes the class modulo Îλ of the element Cd(s)σλal ∈ Iλ. So, for any t ∈ T(λ), we write ϕλ(xt) = s∈T(λ) gs,t ed(s)σλal where gs,t ∈ A. Thus, Gλ := s,t∈T(λ) is the matrix of ϕλ with respect to the standard bases of S̃ λ and [Cλ]A, respectively. Now we can state the main result of this paper. Theorem 3.6. The map ϕλ : S̃ λ → [Cλ]A constructed in Lemma 3.5 is an isomorphism. More precisely, the following hold. For any s, t ∈ T(λ), we have gt,t = 1 for all t ∈ T(λ), gs,t = 0 unless d(s) 6 d(t), gs,t ∈ v Z[v−1] if s 6= t; here, 6 denotes the Bruhat–Chevalley order. Thus, the matrix Gλ has an upper unitriangular shape for a suitable ordering of the set T(λ). Proof. We begin with the following computation inside the parabolic subgroup Sn ⊆ Wn. Let t ∈ Tr(λ). By the multiplication rules for the Kazhdan–Lusztig basis, Td(t)Cσλ equals Cd(t)σλ plus a Z[v, v−1]-linear combination of terms Cx where x ∈ Sn, x 6L,n σλ and x < d(t)σλ. Now, the condition x 6L,n σλ implies that x can be written as x = d(s)σλ for some s ∈ T r(λ) (see, for Specht modules and Kazhdan–Lusztig cells 9 example, [15, 2.9]). Then the condition x = d(s)σλ < d(t)σλ implies that d(s) < d(t) (see [14, 9.10(f)]). Thus, we obtain (∗) Td(t) Cσλ = s∈Tr(λ) s,tCd(s)σλ for any t ∈ T r(λ), where g′ s,t ∈ Z[v, v −1] for all s, t ∈ Tr(λ); furthermore, g′ t,t = 1 and g s,t = 0 unless d(s) 6 d(t) and d(s)σλ 6L,n d(t)σλ. To pass from Sn to Wn, we use the following argument. First note that al is a distinguished right coset representative of Sn in Wn. By [4, Prop. 2.3], we have Cσal = CσCal for any σ ∈ Sn. Hence, multiplying (∗) on the right by Cal , we obtain Td(t) Cσλal = s∈Tr(λ) s,tCd(s)σλal for any t ∈ T r(λ). Now assume that t ∈ T(λ). Let s ∈ Tr(λ) be such that g′ s,t 6= 0. Then d(s)σλ 6L,n d(t)σλ and so d(s)σλal 6L d(t)σλal; see [14, Prop. 9.11]. Hence, using Proposition 2.6, we find that Td(t) Cσλal ≡ s∈T(λ) s,tCd(s)σλal mod Îλ. Passing to the quotient Iλ → [Cλ]A = Iλ/Îλ, we obtain Td(t).eσλal = s∈T(λ) s,t ed(s) σλal for any t ∈ T(λ). Now note that ϕλ(xt) = ϕλ(Td(t).x̄λ) = Td(t).ϕλ(x̄λ) = Td(t).eσλal , where x̄λ denotes the class of xλ in S̃λ. Thus, we see that g′ s,t = gs,t for all s, t ∈ T(λ). Consequently, the coefficients gs,t have the property that gt,t = 1 and gs,t = 0 unless d(s) 6 d(t). Hence, for a suitable ordering of the rows and columns, the matrix Gλ is unitriangular and ϕλ is an isomorphism. It remains to prove that gs,t ∈ v Z[v−1] for s 6= t. We will actually show that g′ s,t ∈ v Z[v−1] for all s, t ∈ Tr(λ) such that s 6= t. This is seen as follows. We can invert the equations (∗) and obtain Cd(t)σλ = s∈Tr(λ) g̃s,tTd(s) Cσλ for any t ∈ T r(λ), where the g̃s,t’s are the entries of the inverse of the matrix s,t∈Tr(λ) . A comparison with [7, Prop. 3.3] shows that g̃s,t = p d(s)σλ,d(t)σλ ∈ v−1Z[v−1] if s 6= t. Hence we also have g′ s,t ∈ v Z[v−1] for s 6= t. � Remark 3.7. Let λ ∈ Λn and C be any left cell such that C and Cλ are contained in the same two-sided cell. Then, by Theorem 2.4, [C]A and [Cλ]A are canonically isomorphic as Hn-modules. Hence, in combination with Theorem 3.6, we conclude that S̃λ ∼= [C]A. Thus, any left cell module of Hn is canonically isomorphic to a Specht module. Remark 3.8. The above results also hold for specialized algebras. More precisely, let R be any commutative ring (with 1) and fix two invertible elements Q, q ∈ R which admit square roots Q1/2 and q1/2 in R. Then we have a unique ring homomorphism θ : A→ R such that θ(V ) = Q1/2 and θ(v) = q1/2. We can extend scalars from A to R and set Hn,R = R⊗A Hn, S̃ R = R⊗A S̃ λ, [C]R = R⊗A [C]A, 10 Geck, Iancu and Pallikaros for any λ ∈ Λn and any left cell C of Wn. Then S̃ R precisely is the Specht module of the algebra of type Bn with parameters Q, q, as considered by Dipper–James–Murphy [6]. By Theorem 3.6 and Remark 3.7, we have an induced canonical isomorphism ϕ̃R : S̃ −→ [C]R whenever C is in the same two-sided cell as Cλ. 4. Counterexample Recall that Hn is defined over the ring of Laurent polynomials A = Z[V ±1, v±1] in two indepen- dent indeterminates. In the previous sections, we considered the Kazhdan–Lusztig cell modules of Hn with respect to the “asymptotic case” [3], that is, assuming that the group of monomials {V ivj | i, j ∈ Z} is endowed with the pure lexicographic order where V ivj < vi if and only if i < i′ or i = i′ and j < j′. But there are many other monomial orders, each giving rise to a Kazhdan–Lusztig basis of Hn and corresponding cell modules. The aim of this section is to show that, in general, the Dipper–James–Murphy Specht modules S̃λ cannot be identified with cell modules for these other choices of a monomial order. We do this in two ways: (1) by a concrete example in type B3 and (2) by a general argument involving non-semisimple specialisations of Hn. 4.A. An example in type B3. Let n = 3; then W3 = 〈t, s1, s2〉. Let λ = ((1), (2)) ∈ Λ3 and consider the corresponding Specht module S̃λ. By Theorem 3.6, it is isomorphic to [Cλ]A, where Cλ is a left cell with respect to the “asymptotic case”. We have l = 1 and σλal = s2t. Using Proposition 2.6, we find that Cλ = {s2t, s1s2t, s2s1s2t}. The corresponding left cell representation ρλ : H3 →M3(A) is given by Tt 7→ V V v−1V −1v V v−2 + V −1v2 0 −V −1 0 0 0 −V −1 Ts1 7→ −v−1 0 0 1 v 0 0 0 v  , Ts2 7→ v 1 0 0 −v−1 0 0 1 v Now let us choose a different monomial order on {V ivj | i, j ∈ Z}, namely, the weighted lexico- graphic order where V ivj < V i ′ def ⇐⇒ i+ j < i′ + j′ or i+ j = i′ + j′ and i < i′. (In particular, we have v < V < v2.) By an explicit computation, one can show that, in this case, the Kazhdan–Lusztig cell modules are all irreducible over K (the field of fractions of A), in accordance with [2, Conjecture A+]. (We are in the case r = 1 of that conjecture.) Furthermore, there are precisely three left cells C1,C2,C3 such that [Ci]K ∼= S̃ K ; they are given as follows: C1 = {s1s2s1, s1ts1s2s1, ts1s2s1}, C2 = {s1s2s1t, s1ts1s2s1t, ts1s2s1t}, C3 = {s1s2s1ts1, s1ts1s2s1ts1, ts1s2s1ts1}. Specht modules and Kazhdan–Lusztig cells 11 The corresponding left cell representations are all identical and given by: Tt 7→ −V −1 0 0 0 −V −1 0 1 V v−1 + V −1v V Ts1 7→ v 0 0 0 v 1 0 0 −v−1  , Ts2 7→ v V v−2 + V −1v2 0 0 −v−1 0 0 1 v Denote this representation by ρ : H3 → M3(A). Now one checks that Pρλ(Ts) = ρ(Ts)P for s ∈ {t, s1, s2} where 0 0 V v−2 + V −1v2 0 1 0 1 0 0 Thus, P defines a non-trivial module homomorphism between ρλ and ρ. Since these representations are irreducible over K, the matrix P is uniquely determined up to scalar multiples. But we see that there is no scalar λ ∈ K such that λP ∈ M3(A) and det(λP ) ∈ A ×. Hence, S̃λ will not be isomorphic to any Kazhdan–Lusztig cell module with respect to the above weighted lexicographic order. 4.B. General cell modules. Let k be a field and fix an element ξ ∈ k×. Let a, b ∈ Z>0 and consider the specialisation A→ k such that V 7→ ξb and v 7→ ξa. Let Hn,k = k ⊗A Hn be the corresponding specialized algebra. As in Remark 3.8, we also have corresponding Specht modules S̃λk for Hn,k. Now, for each λ ∈ Λn, there is a certain Hn,k-invariant bilinear form φλ : S̃ k × S̃ k → k; see [6, §5]. Let rad(φλ) be the radical of that form and set Dλ = S̃λk/rad(φλ). Then D λ is either 0 or an absolutely irreducible Hn,k-module; furthermore, we have Irr(Hn,k) = {D µ | µ ∈ Λ♣} where Λ♣ = {λ ∈ Λ | Dλ 6= 0}; see Dipper–James–Murphy [6, Theorem 6.6]. The conjecture in [6, 8.13] about an explicit combi- natorial description of Λ♣ has recently been proved by Ariki–Jacon [1]. Now consider the Kazhdan–Lusztig basis {Cw} of Hn,k with respect to the weight function L : Wn → Z such that L(t) = b and L(si) = a for all i. Assume that Lusztig’s conjectures (P1)– (P15) in [14, 14.2] on Hecke algebras with unequal parameters hold. (This is the case, for example, in the “equal parameter case” where a = b; see [14, Chap. 15].) Using these properties, it is shown in [9] that Hn,k has a natural “cellular structure” in the sense of Graham–Lehrer [12]. The elements of the “cellular basis” are certain linear combinations of the basis elements {Cw}. Then, by the general theory of “cellular algebras”, for any λ ∈ Λn, we have a “cell module” Wk(λ) for Hn,k and this cell module is naturally equipped with an Hn,k-invariant bilinear form gλ : Wk(λ)×Wk(λ)→ k. Let rad(gλ) be the radical of that form and set L λ = Wk(λ)/rad(gλ). Then, again, L λ is either 0 or an absolutely irreducible Hn,k-module; furthermore, we have Irr(Hn,k) = {L µ | µ ∈ Λ♠} where Λ♠ = {λ ∈ Λ | Lλ 6= 0}; see Graham–Lehrer [12, §3] and [9, Example 4.4]. Given these two settings, it is natural to ask if S̃λk ∼= Wk(λ) and, subsequently, if Λ ♣ = Λ♠ ? In the case where Hn,k is semisimple, it is shown in [9, Example 4.4] that S̃ ∼= Wk(λ); furthermore, by the general theory of “cellular algebras” [12] and the results in [6], we have Λ♣ = Λ♠ = Λ in 12 Geck, Iancu and Pallikaros this case. However, if Hn,k is not semisimple, then the answer to these questions is negative, as can be seen from the fact that Λ♣ 6= Λ♠ in general; see [11] and the references there. By [11, Theorem 2.8] it is true, however, that Λ♣ = Λ♠ if b > (n − 1)a > 0 which corresponds precisely to the “asymptotic case” discussed in this paper. Indeed, by [8, Corollary 6.3], the basis {Cw} of Hn,k is cellular under this assumption on a, b, and by Theorem 3.6, we have Wk(λ) ∼= S̃ Acknowledgements. The final form of this paper grew out of several discussions which the three authors could hold thanks to the hospitality of various institutions. Part of this work was done while all three authors enjoyed the hospitality of the Bernoulli Center at the EPFL Lausanne (Switzerland) in 2005. C.P. thanks the University of Aberdeen (Scotland) for an invitation in November 2006; M.G. and L.I. would like to thank the University of Cyprus at Nicosia for an invitation in March 2007. References [1] S. Ariki and N. Jacon, Dipper–James–Murphy’s conjecture for Hecke algebras of type B, preprint; available at math.RT/0703447. [2] C. Bonnafé, M. Geck, L. Iancu and T. Lam, On domino insertion and Kazhdan–Lusztig cells in type Bn, preprint; available at math.RT/0609279. [3] C. Bonnafé and L. Iancu, Left cells in type Bn with unequal parameters, Represent. Theory 7 (2003), 587–609. [4] C. Bonnafé, Two-sided cells in type B in the asymptotic case, J. Algebra 304 (2006), 216–236. [5] R. Dipper and G. D. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. 52 (1986), 20–52. [6] R. Dipper, G. D. James and G. E. Murphy, Hecke algebras of type Bn at roots of unity, Proc. London Math. Soc. 70 (1995), 505–528. [7] M. Geck, On the induction of Kazhdan–Lusztig cells, Bull. London Math. Soc. 35 (2003), 608–614. [8] M. Geck, Relative Kazhdan–Lusztig cells, Represent. Theory 10 (2006), 481–524. [9] M. Geck, Hecke algebras of finite type are cellular, Invent. Math. (2007), to appear. [10] M. Geck and L. Iancu, Lusztig’s a-function in type Bn in the asymptotic case. Special issue celebrating the 60th birthday of George Lusztig, Nagoya J. Math. 182 (2006), 199–240. [11] M. Geck and N. Jacon, Canonical basic sets in type B. Special issue in honour of Gordon Douglas James, J. Algebra 306 (2006), 104–127. [12] J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), 1–34. [13] G. Lusztig, Left cells in Weyl groups, Lie Group Representations, I (R. L. R. Herb and J. Rosenberg, eds.), Lecture Notes in Math., vol. 1024, Springer-Verlag, 1983, pp. 99–111. [14] G. Lusztig, Hecke algebras with unequal parameters, CRM Monographs Ser. 18, Amer. Math. Soc., Provi- dence, RI, 2003. [15] T. P. McDonough and C. A. Pallikaros, On relations between the classical and the Kazhdan–Lusztig representations of symmetric groups and associated Hecke algebras, J. Pure and Applied Algebra 203 (2005), 133–144. M.G. and L.I.: Department of Mathematical Sciences, King’s College, Aberdeen University, Ab- erdeen AB24 3UE, Scotland, U.K. E-mail address: m.geck@maths.abdn.ac.uk E-mail address: l.iancu@maths.abdn.ac.uk C.P.: Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus E-mail address: pallikar@ucy.ac.cy http://arxiv.org/abs/math/0703447 http://arxiv.org/abs/math/0609279 1. Introduction 2. Kazhdan–Lusztig bases and cells 2.A. Basic definitions 2.B. The ``asymptotic case'' in type Bn 2.C. Bitableaux 3. Specht modules 4. Counterexample 4.A. An example in type B3 4.B. General cell modules References

Growing Directed Networks: Stationary in-degree probability for arbitrary out-degree Daniel Fraiman Departamento de Matemática y Ciencias, Universidad de San Andrés, Buenos Aires, Argentina. We compute the stationary in-degree probability, Pin(k), for a growing network model with di- rected edges and arbitrary out-degree probability. In particular, under preferential linking, we find that if the nodes have a light tail (finite variance) out-degree distribution, then the corresponding in-degree one behaves as k−3. Moreover, for an out-degree distribution with a scale invariant tail, Pout(k) ∼ k−α, the corresponding in-degree distribution has exactly the same asymptotic behavior only if 2 < α < 3 (infinite variance). Similar results are obtained when attractiveness is included. We also present some results on descriptive statistics measures such as the correlation between the number of in-going links, Din, and outgoing links, Dout, and the conditional expectation of Din given Dout, and we calculate these measures for the WWW network. Finally, we present an applica- tion to the scientific publications network. The results presented here can explain the tail behavior of in/out-degree distribution observed in many real networks. PACS numbers: 05.65.+b, 89.75.Kd, 87.23.Ge, 02.50.Cw I. INTRODUCTION Barabási and Albert [1] discovered that several net- works in nature have a strange topological characteris- tic: they have a scale-free [2, 3, 4] degree distribution, P (k) ∼ k−α, where the degree of a vertex is defined as the total number of its connections. Nowadays, this em- pirical behavior is confirmed in a great number of com- pletely different empirical networks, from biological net- works to e-mail networks, including scientific publication networks. In [1] they also proposed a model (B-A model) for explaining this behavior. The model can be formu- lated as follows: 1) start with a network with N nodes, connected by j edges in an arbitrary way, and 2) at each time step a new node, with m edges, appears, and each of edges connects to the existing nodes according to some probability law, π. The probability that a new edge at- taches to a node with degree k, πk, was defined [1] as proportional to the degree of the node. In particular, they showed that with this attachment law, , (1) where Nk is the number of nodes with degree k, the sta- tionary degree distribution has a power law tail, P (k) ∼ k−3. In [5] they computed the stationary degree prob- ability (not only the tail behavior) or limit degree dis- tribution for a model similar to the B-A one, but for a generalization of the preferential linking attachment law. They introduced a new parameter, the attractiveness, A (in their case A ≥ 0), and defined the attachment law as: (A+ k)Nkin∑ (A+ j)N jin , (2) where Nkin is the number of nodes with in-degree equal k. They found in this case that P (k) ∼ k−(2+A/m), be- ing more flexible for comparing to empirical networks. Typically, degree distribution of real networks satisfy, P (k) ∼ k−α with 2 ≤ α ≤ 3. But the B-A model and similar ones [5], no matter which is the attachment law, have a mayor drawback, the number (m) of edges that arise from new nodes is a fixed number. In almost all real networks, the new nodes do not have the same number of edges. On the other hand, the number of edges of a ran- dom selected new node (from a real network) is a random variable. So, in order to be more realistic, we will study the behavior of the B-A model when new nodes with a random number of edges appear, but in the more general context of directed growing networks. In this context new questions arises. Directed networks are characterized by the fact that the edges are directed (arrows), each node has edges that point at it, and others that born in it. The in-degree of a node is defined as the number of incoming edges, and the out-degree as the number of its outgoing edges. The most studied directed growing networks have been the WWW network [7, 8, 12], and the scientific publications network [6]. In the first one, each node represents a web page and the hyper-links (references to other web pages) represents the directed edges or links. In the second one, each paper is a node, and its references the directed links. In this last case, the in-degree distribution represents the distribution of citations for a random selected paper, and the out-degree distribution represents the number of ref- erences of a random selected paper. Empirical directed growing networks follow in general one of two possible behaviors. In the first case they have an out-degree expo- nential distribution, Pout(k) ∼ ak (0 < a < 1), or an out- degree distribution taking finitely many values, associ- ated with an in-degree one distribution with a power law tail Pin(k) ∼ k−α where typically α ≈ 3. In the second case the out-degree distribution satisfies Pout(k) ∼ k−β , and is associated with Pin(k) ∼ k−α with α ≈ β. Exam- ples, such as biological, WWW, or communication net- works, can be found in [2, 3, 4, 9]. In this paper, we address the question of why the em- pirical growing directed networks show this strange gen- eral behavior for the tail of the in/out degree distribu- tions. We study a particular growing network model (a generalization of [1] to be precise), obtaining the sta- tionary joint in-out degree distribution, Pin,out(j, k), and some of its derivatives, such as the marginal distribution, Pin(k), the covariance, and the conditional expectation of the number of in-links given the number of out-links. In particular, studying in detail Pin(k), we prove (for the model presented here) that it is expected to observe the in/out tail behavior reported for real networks [2, 3, 4]. Finally we present an application to the most “pure” (ex- tremely few double arrows) growing directed network: the scientific publication network. In this application, we show the relevance of having an expression for the limit in-degree distribution (Pin(k)) for an arbitrary out- degree one (Pout(k)). II. GROWING DIRECTED NETWORK MODEL Before describing the model, it is important to remark that real directed growing networks have in general a con- siderable asymmetry between the in-links and out-links of a node. For example, nobody will care much about how many references (out-links) an own paper has, but people are interested in the number of cites (in-links) that their own paper has. That is why we are going to treat the out-links from a new node and the in-links in a completely different way. In particular, a node can receive (with positive probability), a connection from a new node at any moment, but typically a node can not change who their pointers (the set of nodes it is point- ing to) are. This is very clear in the scientific publica- tions network. In this network the in-degree distribution has been extensively study [6, 8], whereas the out-degree distribution has been poorly reported [10, 11]. Neverthe- less, in the case of the WWW network, the outgoing links (hyper-links) can change at any moment and new hyper- links can be aggregated or old hyper-links can be redi- rected. In [7, 8] they proposed some models for describing this network taking into account the characteristics men- tioned above. However these models do not consider that the new nodes have a particular out-degree distribution, i.e. the models are constructed under the hypothesis that new nodes have a fixed number of out-links. The mayor problem of both models is that the nodes (webpages) do not have a controlled number of out-links, they can have a huge number of them which does not seem realistic. Our strategy for modeling these networks is completely different to the ones proposed in [7, 8], for us, most of the variability in the number of out-links is explained when the node appear, defined as “intrinsic” variability, and not as a product of updating nodes. We think that in many real networks the updating of nodes can give a small correction compared with the “intrinsic” variabil- ity. This assumption is at the core of our model. In a real network the “intrinsic” variability is given by differ- ent reasons that are hard to know (why does a randomly selected scientific paper has a number of references with some particular distribution?), but typically the problem of trying to understand it is not a mayor question. Now, we describe the growing network model: 1) ini- tially the network consists of N nodes connected in a given arbitrary way, 2) at each time step, say time step n + 1, a node with Dout outgoing-edges appear, where Dout is a random variable ( P (Dout = k) = 1), and 3) each new directed edge points out to an existing node with some probability law πn+1 (uniform, preferential linking, etc.). Fig. 1 shows an scheme of the model. If FIG. 1: Scheme of the growing network model. In each temporal step a new node (shown in black) with Dout out- links appear; these links point towards existing nodes. Dout is not a fixed number, on the contrary is a random vari- able. The degree vector at time 0, and 1 is: ~N0 = (1, 4, 0, 0, 1, 0, 0, 0, ..., 0, ...), ~N1 = (1, 4, 1, 0, 0, 1, 0, 0, ..., 0, ...). πn+1 is an arbitrary function that depends on the de- gree vector at time n, ~Nn = (N1n, N n, ..., N n , ...) and/or ~Nin,n ( ~Nout,n), then the growing network model, de- scribed above is a Markov chain taking values in NN0 or N0 ×NN 0 with transition probabilities given by πn+1. In this work (under the Markovian hypothesis), we show an easy way to compute stationary (in/out) degree probabil- ities for arbitrary πn+1. An important part of this article is devoted to the study of the model under the law: πkn+1 := (A+ k)Nkn∑ (A+ j)N jn , (3) and in Section 2.4 we show some results under different π’s. The law of eq. 3 corresponds to preferential linking on degree with attractiveness. This probability is well defined for values of A greater or equal to -B, where B = min {k : P (Dout = k) > 0}. (4) For this attachment law, the model is in fact an exten- sion of the Albert-Barabási model, although in this case Dout is a random variable with an arbitrary distribution, P (Dout = k) with k ∈ N, and the edges are directed. The limit (stationary) in-degree distribution and the limit de- gree distribution have not been reported, even for simple cases as Dout taking values 1 and 2, with probabilities p1 and 1−p1 respectively. Moreover, even in the undirected case, it is not known if in general the limit degree distri- bution (P (k)) satisfies a superposition principle (linear combination). A. Stationary Probabilities The number of out-links does not depend on time (see Appendix A for additional details), therefore, the limit out-degree distribution satisfies Pout(k) ≡ P (Dout = k). Note that the out-degree distribution is defined a priori (in accordance with the specific network), imposing in this way the asymmetry mentioned before between the in and out links. We are interested in obtaining the limit degree distribution, P (k), and the limit in-degree one, Pin(k). In order to compute this last probability distri- bution, we first compute the stationary joint degree and out-degree distribution, Pdeg,out(j, k). If the network is distributed according to the stationary probability, then the probability that a randomly chosen node has k out- links and j total links, ~D = (D,Dout) = (j, k), is given Pdeg,out(j, k) = P ( ~D = (j, k)) = lim deg,out,n∑ j,k∈N,No deg,out,n where Nh,ideg,out,n is the number of nodes with h total links from which i are out-links at time n. The last equality holds by the Law of Large Numbers. Clearly, the joint in- out degree can be computed from this last one, Pin,out(j− k, k) = Pdeg,out(j, k), and also the in-degree and degree probability taking marginal distributions. deg,out,n+1 depends on: 1) N deg,out,n, and 2) the tran- sition probabilities, πdeg,out,n+1. As it is usual for Markov chains, we associate to the transition probabilities of this chain some random variables that we now describe. In the first place, there is the out-degree, Dout, of the new node. Secondly, we consider at each time n + 1 a se- quence of independent and identical distributed bivari- ate random vectors {~Zi}, taking value (j, k), j, k ∈ N, with probability πj,kdeg,out,n+1, which depends on the state of the chain at time n. This way, the growing network dynamics can be written as: deg,out,n+1 = N deg,out,n + ∆ n ∀j ≥ k ∈ N (5) where ∆j,kn = Dout∑ δ~Zi=(j−1,k) − δ~Zi=(j,k) for j > k δDout=j − Dout∑ δ~Zi=(j,j) for j = k The random vector ~Zi indicates to which type of node the i link (of the new node) is pointing to. For example, if ~Z1 = (3, 2), a new link is pointing to an existing node with 2 out-links and 1 in-link (or 3 total links). Clearly, in order to have a good representation of the growing network process, the probability law of Zi must be equal to πj,kdeg,out,n+1, as we impose. Equations 5 and 6 can be read in the following way: if at time n + 1 a new node with Dout = m out-links is aggregated, then N deg,out,n+1 grows by one, and m components of the degree vector undergo a “shift”. As the network continues to grow, the goal is to find whether there exists a limit distribution for the in-out degree. For very large values of n, given a randomly selected node, what is the probability that this one has j links, of which k are out-links, Pdeg,out(j, k)?. The following property shows a way of computing Pdeg,out(j, k) which has interest on itself. Property: Pdeg,out(j, k) is the solution of: Pdeg,out(j, k) = 〈∆j,kn /Θn〉 ∀j ≥ k ∈ N, (7) where Θn is the event that imposes that the empirical dis- tribution at time n is equal to the stationary distribution, i.e. Θn = { deg,out,nP l,m∈N deg,out,n = Pdeg,out(h, i) ∀h, i ∈ N}. The preceding property says that if the process at time n is distributed according to the stationary probability, Pdeg,out, it will remain there in expectation. This tech- nique for finding stationary probabilities seems much eas- ier (see Appendix B) than previous approaches [1, 5, 18]. Using the property mentioned above and eq. 6, it is easy to see that the stationary joint deg-out distribution, Pdeg,out, satisfies: Pdeg,out(j, k) = π j−1,k deg,out〈Dout〉 − π deg,out〈Dout〉 Pdeg,out(j, j) = Pout(j)− π deg,out〈Dout〉 for j > k ∈ N, where 〈Dout〉 = kPout(k). These two equations contain all the information about the limit joint in-out degree distribution, being a crucial result in this paper. It is important to note that since we have conditioned on the fact that at time n the process is distributed according to the stationary probability, the link attachment probability does not depend on time. Now, πj,kdeg,out denotes the stationary probability that a new link (from a new node) point to an existing node with j − k in-degree links and k out-degree links. Under preferential linking on degree with attractiveness (eq. 3), the stationary attachment law remains: deg,out = 〈D〉+A Pdeg,out(j, k). (9) where 〈D〉 = kPdeg(k). The marginal distribution of eq. 9, πk = deg,out, is the stationary version of π presented in eq. 3. Replacing eq. 9 in eq. 8, and using 〈D〉 = 2〈Dout〉 (for each new node with k out-links, the total degree increases by 2k) we obtain: Pdeg,out(j, k) = Ψ(j +A, 3 + δ) Ψ(k +A, 2 + δ) Pout(k), (10) where Ψ(a, b) ≡ Γ[a]Γ[b] Γ[a+b] ta−1(1−t)b−1dt (Beta func- tion), and δ = A/〈Dout〉. From eq. 10, taking marginal distributions is trivial to obtain: Pin,out(j, k) = Ψ(j + k +A, 3 + δ) Ψ(k +A, 2 + δ) Pout(k) (a) P (k) = Ψ(k +A, 3 + δ) Pout(j) Ψ(j +A, 2 + δ) Pin(k) = Pout(j) Ψ(j + k +A, 3 + δ) Ψ(j +A, 2 + δ) Eq. 11 shows the joint stationary in-out degree probabil- ity, the degree distribution and the in-degree distribution. In the stationary regime (for the probability) the propor- tion of nodes with j in-links and k out-links (eq. 11 (a)), depends on the attractiveness, and on the out-degree dis- tribution through two quantities: 〈Dout〉 and Pout(k). The same happens for P (k) and Pin(k). Eq. 11 (b) shows the stationary degree probability for arbitrary out-degree distribution (see Appendix B for a simpler derivation). Note that just by replacing Pout(k) by δk=m (this means a non-random Dout and equal to m) we obtain the known result [5] for undirected networks. Eq. 11 (c) constitutes one of the main results of the paper. Replacing Pout(k) by the empirical value, we can check whether the model is adequate for the network under study. Moreover, it is possible to see that a superposition principle does not hold, either for P (k), Pin(k), or Pin,out(k, j). They can- not be written as P (k) = Pout(j)Qj(k), where Qj(k) is the stationary probability for a fixed number j of out- links. The superposition principle will be valid for the three limit distributions only when the attractiveness vanishes (preferential linking). In this way, the prefer- ential linking generalization (the inclusion of attractive- ness) introduced in [5] has the advantage of enlarging the power exponent values of the degree distribution, with the drawback of loosing a superposition principle. If we allow the appearance of new nodes with zero out-links (P (Dout = k) = Pout(k) with k ∈ No), then the results presented in equations 11 (b) and (c), still hold after switching the initial index in the summation from 1 to 0 and taking k ∈ No = N ∪ {0}. In this last case, the attractiveness must be greater o equal zero (see eq. 4). B. Descriptive Statistics Before trying to describe a real network by a model, some first checks are recommendable. One typical mea- sure that has been extensively used is the clustering coef- ficient, that is a measure of how connected the neighbors of a node are. We are going to discuss much simpler de- scriptive measures that also serve as tools for looking for the “best” model. Therefore, it is important to have ana- lytical devices for comparing with real data in the search of a good model. 1. Covariance and conditional expectation A measure of dependence between the in-degree and the out-degree can give an idea of which is the attach- ment law that better describes the empirical data. The covariance between Dout and Din, Cov(Din, Dout) = 〈DinDout〉 − 〈Din〉〈Dout〉 is an adequate statistical mea- sure for this purpose. For example, in the case where the law of attachment is preferential linking on in-degree (eq. 2) this measure is obviously zero. For the case stud- ied in detail here, preferential linking on degree (eq. 3), it is straightforward to see that the covariance between Dout and Din in the particular case A = 0, satisfies the following equation: Cov(Din, Dout) = Cov(D,Dout) = V ar(Dout) (12) where V ar(Dout) = Cov(Dout, Dout). The covariance is always positive or zero (for non random Dout), as it is expected for this type of attachment law. Eq. 12 in- stead can be written in terms of the correlation, r = Cov(Din,Dout)√ V ar(Din)V ar(Dout) , in the following way: V ar(Dout) V ar(Din) . (13) It is surprising that the correlation satisfy this simple re- lation between the standard deviations, r is the ratio be- tween σout ( V ar(Dout)) and σin ( V ar(Din)). Since the correlation coefficient is always less or equal 1, we obtain the following inequality: V ar(Dout) ≤ V ar(Din). (14) Although it is very easy for real network to estimate the variance of the number of out and in links, and also the covariance (or correlation) between the in and out-degree, these measures are not typically reported (see Appendix C for results on the WWW network). On the other hand, the first right term of the covari- ance always satisfies: 〈DinDout〉 = k〈Din/Dout = k〉Pout(k), (15) where 〈Din/Dout = k〉 is the conditional expectation of the number of in-links given that the node has k out- links. From equations 12 and 15 it is very easy to see that: 〈Din/Dout〉 = 〈D/Dout〉 = Dout. (16) The relationship between 〈Din/Dout〉 and Dout can be a second check to make before modeling. For a real network this can be done in the following way, choose all the nodes that have a number Dout of outgoing links, and take the mean of the number of in-links over this set of nodes. If the conditional mean is equal to Dout for all values of Dout, then this is an indication that the model can be adequate. For non null attractiveness it is hard to obtain an- alytical results, nevertheless, we compute numerically 〈Din/Dout〉 for different values of Dout and attractive- ness. From eq. 11 (a) and the definition of conditional expectation, it is easy to obtain: 〈Din/Dout〉 = Ψ(j +Dout +A, 3 + δ) Ψ(Dout +A, 2 + δ) . (17) Fig. 2 (a) shows the numerical results of 〈Din/Dout〉 based on eq. 17. For any value of the attractiveness and 〈Dout〉, the conditional expectation follows a linear rela- tion with Dout: 〈Din/Dout〉 = f(A, 〈Dout〉)Dout + g(A, 〈Dout〉). (18) The slope, f(A, 〈Dout〉), and the intercept, g(A, 〈Dout〉), of this straight line satisfies: f(A, 〈Dout〉) = 0 g(A, 〈Dout〉) = 〈Dout〉, as it is shown in Fig. 2 (b) and (c). For positive values of attractiveness the slope is smaller than one, going to zero as the attractiveness goes to infinity. In the case A → ∞, Din and Dout are independent (always with the same expectation). Finally, for negative values of A the slope is greater than one. Studying the empirical relationship between 〈Din/Dout〉 and Dout can give some insight on the model. Moreover, if this relationship is linear, from Fig. 2 (b) and (c), it is possible to have a first estimation of the attractiveness. In Appendix C we show the statistical measures presented here for the WWW network. It is important to note that equations 12 (which in- cludes 13, 14), and 18 (which include 16) holds for any out-degree distribution (Pout(k)). These results do not depend on the details (shape) of the out-degree distri- bution. Nevertheless, there exist some measures that do not share this nice property. For example, the condi- tional number of out-links given the number of in-links, 〈Dout/Din〉, depends explicitly on Pout(k), as can be seen FIG. 2: (a) Conditional expectation of in-degree given the out-degree. Each straight line correspond to a different value of attractiveness (specified in the graph). (b) Slope and (c) Intercept of the type of straight lines shown in (a) as a func- tion of the attractiveness for two different values of < Dout >. in the following equation: 〈Dout/Din = k〉 = Ψ(k+j+A,3+δ) Ψ(j+A,2+δ) Pout(j) Ψ(h+k+A,3+δ) Ψ(h+A,2+δ) Pout(h) . (20) Next, we present another measure useful for model se- lection. 2. Relationship between the distribution tails Now, we study the relationship between the tails of the in-degree and the out-degree distributions. In the case A = 0, if the out-degree distribution has finite expectation (〈Dout〉 < ∞) and a scale invariant tail, Pout(k) ∼ k−(2+β), it is not difficult (from eq. 11 (b)) to see that the limit degree distribution and the in-degree distribution have the following tail behavior: P (k) ∼ Pin(k) ∼ k−(2+β) 0 < β < 1 log(k)k−3 β = 1 k−3 β > 1 Eq. 21 constitute our second main result: if the out- degree distribution has finite variance and a scale invari- ant tail, Pout(k) ∼ k−(2+β), then the limit in-degree dis- tribution has also a scale invariant tail, Pin(k) ∼ k−α. Moreover, for 0 < β < 1, α is equal to the out-degree ex- ponent. This last result can explain why in so many real networks the in and out power exponents are so similar, taking values in a range from 2 to 3. In the case β > 1, α = 3, regardless of the value of β. For the frontier case (finite/infinite variance) of β = 1, the limit distribution decays at a slower rate than k−3. Precisely, it decays as Pin(k) ∼ log(k)k−3. In the general case of prefer- ential linking with attractiveness for Pout(k) ∼ k−(2+β), the regimes are similar to the non-attractiveness case. In this case the only difference is that there is now a sepa- ratrix curve between them, as it is shown in Fig. 3. The behavior is regulated by δ ≡ A/Eo and β. For δ > 1 + β the limit out degree Pin(k) ∼ k−(2+β), and in this case the (in) degree distribution has exactly the same tail as the out-degree, even for large β. For δ < 1 + β, Pin(k) behaves as k−(3+δ). Finally on the separatrix curve, δ = 1 + β, the behavior is given by log(k)k−(3+δ). Note that δ (A/〈Dout〉) can not be smaller than -1, since 〈Dout〉 must be (see eq. 4) greater than -A. For out-degree distributions with exponential tails, as a geometric, Poisson, or finite range distributions, the in- degree distribution satisfies that Pin(k) ∼ k−(3+δ), even for negatives values of δ. In [11] they show that for the PRL citation network the out-degree distribution has an exponential decay, and the in-degree one has a power law tail with α near 3, just as described before for the null attractiveness case. We remark the following: a) if the model is adequate for describing a real growing network, and this network has an out-degree distribution with ex- ponential tail, and a scale invariant in-degree distribution with a power between 2 and 3, then attractiveness param- eter must be negative, and b) if the empirical in-degree distribution has a scale invariant tail with a power less than 2, then the model presented here is not adequate for describing this network. Keeping in mind the last point, the new estimations [12] of the in-degree power exponent of the WWW network, would rule out the model for de- scribing this particular network. FIG. 3: Stationary in-degree probability tail under pref- erential linking with attractiveness for an out-degree with Pout(k) ∼ 1k2+β as a function of δ = and β. The hor- izontal axis corresponds to preferential linking (A = 0). In the separatrix curve, δ = β − 1, Pin(k) ∼ log(k)k3+δ = log(k) C. Application: scientific publications network The scientific publications network has two advantages that define it as the most “pure”: 1) extremely few dou- ble arrows, and 2) all the variability in the number of out-links is “intrinsic”. These two features guarantee that our model (see Fig. 1) is adequate for describing the scientific network. Nevertheless, it is not clear which is the attachment law (π) such that we can obtain a good mimic of the growing network process. Fig. 4 shows the citation distribution for all scientific publications published in 1981 from the ISI dataset cited between 1981 and 1997 (see [6]). Clearly, this distribu- tion represent the in-degree one (see Appendix D). Un- fortunately the out-degree distribution (Pout(k)), i.e. the number of references that has a randomly selected paper, has not been reported. This makes impossible to test the growing model by a plug-in approach (see eq. 11 (c)). Nevertheless, we take the following strategy: we suppose a geometric out-degree distribution Pout(k) = p(1 − p)k with k ∈ No, a preferential linking on degree attach- ment law (eq. 9 with A = 0), and finally we estimate p. Probably the empirical out-degree distribution (Pout(k)) does not fall in any family of parametric distributions. However, a well estimated in-degree distribution will be a positive result, since the in-degree distribution is ob- tained as a result of a theoretical computation based on the out-degree distribution. In order to estimate p, we first compute the average number of citations in the ISI network (〈cites〉 = 8.573) and impose the condition that FIG. 4: Citation distribution for all papers published in 1981 (from the ISI) cited between 1981 and 1997. The theoret- ical citation (in-degree) curves are calculated by eq. 11 (c) assuming that A=0, and the out-degree distribution is geo- metric, Pout(k) = p(1 − p)k for k ∈ No. The dashed line correspond to p = 0.104 (T = 0.115), and the solid one to p = 0.0817 (T = 0.023) but with Pout(0) = 0.3 and Pout(k) = 0.7622781p(1 − p)k for k ∈ N . Inset: Difference between the empirical cumulative distribution and the theo- retical cumulative distribution. Data from [15]. 〈cites〉 = 〈references〉 = kPout(k) = 8.573 we ob- tain p = 1/(9.573). The dashed line in Fig. 4 corre- spond to this case. If we estimate separately the case k = 0, and assume that the out-degree distribution is such that Pout(0) = a, Pout(k) = cp(1 − p)k for k ∈ N with c = (1−a)/(1−p), we obtain p = (1−a)/8.573 after taking the mean value condition. Curiously, for a = 0.3 (p = 0.0817) the theoretical in-degree probability (solid line) is extremely similar to the empirical one in all the range of the distribution, which can not be achieved with an oversimplified model where Pout(k) = δk=m. This is not the only Pout(k) that fits perfectly well, hence we do not assert that the estimated Pout(k) must be similar to the real cites distribution. Moreover, the estimated Pout(k) does not seem very adequate, since under this probability distribution 30% of all scientific publications do not contain any reference (yet, note that in [10] it was reported that 10% of all publications do not contain any reference). In order to have a better notion of the goodness of fitness we compute the Kolmogorov statistic, T = max |G(k)| = max |F bPin(k)− FP theoin (k)| (22) where FP (k) is the cumulative distribution, FP (k) = P (j), P theoin correspond to the theoretical in-degree distribution showed in eq. 11 (c) assuming a particular Pout(k), and P̂in correspond to the empirical citation dis- tribution. One advantage of the proposed estimator in eq. 22 is that it is possible to test whether the model (in- cluding the attachment law) is adequate for describing the real network. In our application, the null hypothesis is Ho: the real growing network has an underlying link attachment law that is preferential on degree. For the simplest case where T compares an empirical distribution with a theoretical one, but without estimating parame- ters, the null hypothesis will be rejected (at a 0.05 level of significance) only if T > 0.0015. In the case shown with solid line T = 0.023, and for the case where Pout(k) is geometric (dashed line) T = 0.115. Clearly, T is a good measure for ranking models (or model selection). The in- set of Fig. 4 shows the function G(k) for both out-degree distributions proposed, for the geometric (dashed line) case the maximum distance between the cumulative dis- tributions (see eq. 22) occurs at k = 0, and for the other case (solid line) at k = 10. As we mentioned at the beginning of this section, the model is adequate for the scientific publication network, but the attachment law is completely unknown. We have proposed one, preferential linking on degree, but we do not have the possibility to corroborate it. This is one of the reasons why we are going to study the model under different attachment laws. The only weak argument in fa- vor of the law given by eq. 3, is that review papers, that have a huge number of references, are typically highly cited compared with regular articles that have a small number of references. In this way, the correlation be- tween Din and Dout will be positive, which is a virtue of the law defined in eq. 3. D. Different attachment laws Clearly, it may happen that for a real network the in- formal checks (covariance, variance and conditional ex- pectation) discussed before might be not consistent with the observables of the model. In this case, three things may be happening: 1) the link attachment law is not ade- quate, 2) the model is not correct, or 3) both before. The first point is related to the mechanism of linking: pref- erential, uniform, non linear preferential, or may have some age dependency as described in [16, 17]. The sec- ond point correspond to the growing mechanism, that can be seen as the core of the model. For example, up- dating of nodes, or a very high proportion of double links can be present, that are not considered in the model. In this section we discuss only the alternative where the at- tachment law is different from the one proposed in eq. 3 (preferential linking on degree), but the core of the model remains true. 1. Preferential linking on in-degree In [5] they studied a model where the attachment law depends on the in-degree and on the attractiveness. The proposed law was the following: πkin = (A+ k)Nkin∑ (A+ j)N jin , (23) where Nkin is the number of nodes with in-degree equal k. In principle, this can be a good law for the scientific publications network. The joint attachment law in this case is given by: deg,out = j − k +A 〈Dout〉+A Pdeg,out(j, k), (24) where we have used that 〈Din〉 = kPin(k) = 〈Dout〉. Replacing eq. 24 in eq. 8, it is very easy to compute the stationary probabilities: Pin(k) = Ψ(k +A, 2 + δ) Ψ(A, 1 + δ) P (k) = Ψ(A, 1 + δ) Pout(j)Ψ(k − j +A, 2 + δ) (b) Pin,out(j, k) = Pdeg,out(j + k, k) = Pin(j)Pout(k) (c) where k, j ∈ No. This case is specially easy to solve because, for a randomly selected node, the num- ber of out-links (Dout) and the number of in-links (Din) are independent random variables (Pin,out(k, j) = Pin(k)Pout(j)). This mean: r = 0 (a) 〈Din/Dout = k〉 = 〈Dout〉 (b) 〈Dout/Din = k〉 = 〈Dout〉 (c). One big difference between the previous attachment law (eq. 3) and this one (eq. 23) is that Pin(k) depends only on the mean number of out-links (〈Dout〉) by δ (δ = A/〈Dout〉), and not on the shape of the out-degree distribution (Pout(k)). For A > 0 and k >> 1, Pin(k) behaves as k−(2+δ) no matter which is Pout(k) (only de- pends on 〈Dout〉). Therefore, under the attachment law given by eq. 23, the tail of the out-degree distribution does not give any information about the tail of in-degree distribution, contrary to what happens for the law of eq. 3. In addition, for this new attachment law the cor- relation between Din and Dout is zero (eq. 26 (a)), and the conditional expectation of Din (Dout) given Dout = k (Din = k) does not depend on k (eq. 26 (b) and (c)). Note that πkin in eq. 23 is well defined only for posi- tive or zero values of attractiveness. But, only strictly positive values of A are interesting, since for A = 0 we get that the stationary probability is Pin(k) = δk=0. This last result is easy is to understand: new nodes appear but they can not be pointed by other nodes (A = 0), and in this way the network will be formed by almost all nodes with zero in-links and only a few (given by the initial condition of the network) with many in-links. Clearly, in the limit n→∞ the proportion of nodes with k in-links goes to a delta function (δk=0). 2. Uniform attachment law It is thus clear that even when preferential linking is an accepted mechanism of link attachment, it is neces- sary to study [18, 19] alternative types. For the uniform attachment law on degree: deg,out = Pdeg,out(n, k) by means of the same technology (replacing πn,kdeg,out in eq. 8) we obtain: P (k) = 1 + 〈Dout〉 Pout(j)( 〈Dout〉 1 + 〈Dout〉 Pin(k) = 1 + 〈Dout〉 〈Dout〉 1 + 〈Dout〉 Note that, Pin(k) depends only on 〈Dout〉 (and not on Pout(k)), and decays exponentially fast. For an out-degree with Pout(k) ∼ k−(2+β), P (k) behaves as k−(2+β)f(k)−1, where f(k) is an increasing function of k that grows more slowly than log(k). It is important to remark that for empirical (finite) networks, the f(k)−1 term will be very difficult to discriminate (f(k) grows at a rate slower than log(log(k))). This behavior may be hard to “separate” from P (k) ∼ k−(2+β), but the in- degree distribution will sort out any possible confusion about the link attachment law. III. CONCLUSIONS For the model presented here, we showed a simple way to compute the stationary probabilities. This model was constructed in order to take into account the main fea- tures of real directed growing networks with the prop- erty that almost all the variability in the number of out- links is “intrinsic” (see Section 2). From the station- ary Property, we showed how to compute the stationary joint in-out degree distribution for an arbitrary out de- gree distribution, and arbitrary link attachment law (π). We studied three different π’s, paying special attention to the preferential linking on degree with attractiveness mechanism (πkn = (A+k)NknP (A+j)N ). Once obtained the joint probability, we compute: (1) Pin(k) as a function of Pout(k). (2) The correlation between Din and Dout. (3) The conditional expectation of Din(Dout) given Dout(Din). From Pin(k) we studied the relationship between the distribution tails, giving a possible explanation for the in/out degree tail relationship reported for many real net- works. The statistical measures mentioned in (2) and (3) were studied for the WWW network, obtaining a good agreement with some of the analytical results presented in this paper. Nevertheless, we cannot say that the model is appropriate to describe this network (an important part of the variability would be not “intrinsic”). Finally, we showed an application to the scientific pub- lications network. In this network: (a) New publications continuously [21] appear (grow- ing network) and do not disappear. (b) The structure is rigid. Published papers cannot change their references, only new papers can change the number of citations of already published works. (c) The publication that is forthcoming has a non pre- dictable number of references, Dout (random vari- able) (d) Even knowing Dout, the cited papers by the forth- coming publication are unpredictable (there is a law of attachment, π). The model we proposed considers the four points men- tioned above. The main difference with other mod- els, is that the number of out-links (references) of a new node (paper) is treated now as a random variable. Therefore, if the distribution of the number of references (Pout(k)) is known, an important part ((a),(b) and (c)) of the scientific network will be well described by the model. But, the distribution of the number of references of the forthcoming publication (out-degree distribution) has not been reported. In addition, the attachment law ((d)) of the scientific publication network is completely unknown, and difficult to estimate it. Thus, we proposed a simple out degree distribution (geometric) and an at- tachment law of preferential linking on degree (we also consider preferential linking on in-degree and uniform at- tachment). With these two assumptions, we found a very good fit. This application also served to discuss how to compare various models. In this matter, we proposed a measure (eq. 22) frequently used in statistics to compare two distributions. From a modeling point of view, we see our results as a further step from which more complex models may be built in order to be closer to reality. The model can be seen as the skeleton to construct more sophisticated models. For example, it does not seem difficult to in- corporate in the model double links (a mixed out-links distribution) in order to be closer to the metabolic net- work, or some updates in the nodes to mimic the WWW network. Other important issue to explore is what hap- pens when Pout(k) depends on time in a simple para- metric way. This last point is related with accelerating networks [20]. We thank A. Calabrese, A. Cuevas, M. O’Connell, and G. Solovey for critical reading of the manuscript, I. Ar- mendáriz, and P. Ferrari for useful discussions, and A.L. Barabási and S. Redner for their generosity in sharing network data. APPENDIX A: COMMENTS ON THE MODEL Being rigorous, the model as it was presented in Sec- tion 2 is not well defined. Yet, as we discuss in this appendix, this is not a serious problem (all the results presented before hold). The difficulty is that Pout(k) is any probability distribution. In particular, it includes the ones that take infinitely values (such as geometric, or any one with exponential or power law tails). The prob- lem can be stated as follows: if a new node, for example has 1000 links and the network has 100 nodes, what do we must do with the remaining 900 links?. We describe below the correct form of the model (that can be implemented): (1) Initially the network consists of n nodes connected in a given arbitrary way. (2) At each time step starting from n+1, say time step m, a node with D̃mout outgoing-edges appear. D̃ is a random variable with law Qmout(k) (Q out(k) ≡ P (D̃mout = k), and P (D̃mout = k) = 1). (3) Each new directed edge points out to an existing node with some probability law πm (uniform, pref- erential linking, etc.). The distribution of the number of out-links from a new node at time m (the networks has m−1 nodes) is defined by the following equation: Qmout(k) = P (Dout = k/Dout < m). (A1) Qmout(k) is the conditional distribution of Dout given Dout ≤ m − 1. From definition A1 is very easy to see that Qmout(k) converge to Pout(k), Qmout(k) = Pout(k), (A2) as the network grows, where Pout(k) is the distribution defined a priori (see Section 2). From this last conver- gence we can see that the model with this correction (we have only changed Pout(k) by Qmout(k)) has exactly the same asymptotic behavior that was obtained for the model presented in Section 2. Therefore, all the results presented in this paper also hold for the corrected model. The general conclusion would be:“small effects disappear at ∞”. See, for instance Section 2.4.1 were we discuss why for A=0, Pin(k) converges to δk=0. APPENDIX B: A CLOSED EQUATION FOR P (k) If we were only interested on the stationary degree dis- tribution (P (k)), the computation is much easier than the one presented in Section 2.1, since there is a closed equa- tion for P (k). The growing network dynamics is given Nkn+1 = N n + ∆ n (a) ∆kn = δDout=k + Dout∑ δYi=k−1 − δYi=k (b) where {Yi}1≤k≤n is a sequence of independent and identi- cal distributed random variables, taking value k (k ∈ N) with probability πkn+1. Property: ~P ≡ (P (1), P (2), . . . , P (k), . . . ) is the so- lution of: 〈∆kn/ = ~P 〉 = P (k) ∀k ∈ N. (B2) Replacing ∆kn by eq. B1 (b) in eq. B2, we get: 〈δDout=k + Dout∑ δYi=k−1 − δYi=k/ = ~P 〉 = P (k). From this last equation it is trivial to obtain that the stationary degree probability satisfies: P (k) = Pout(k) + (π k−1 − πk)〈Dout〉 (B4) where πk is the stationary probability that a new link is attached to a node with degree j. Under preferential link- ing on degree linking with attractiveness, the stationary attachment law, πk, remains equal to (k+A)P (k)〈D〉+A . Replac- ing πk in eq. B4, and using 〈D〉 = 2〈Dout〉, it is easy to conclude that the limit degree distribution (P (k)) is given by eq. B5. P (k) = Ψ(k +A, 3 + δ) Pout(j) Ψ(j +A, 2 + δ) . (B5) APPENDIX C: WWW NETWORK As we have mentioned in the Section 2.2.1, it is difficult to find articles on networks that report the simple de- scriptive measures (covariance, variance and conditional expectation) for nodes discussed here. However, a de- tailed statistical analysis of the topological properties of four different WWW networks have been reported re- cently [12]. In [12] the covariance and the variance of the number of out-going links (Dout) and in-going links (Din) are reported, which we give in Table 1. The first thing Cov(Din, Dout) V ar(Dout) V ar(Din) WBGC01 155.682 171.61 40080.04 WGUK02 524.244 750.76 20534.89 WBGC03 348.486 870.25 54980742 WGIT04 3478.75 4502.41 776866 TABLE I: Descriptive statistical measures for 4 WWW net- works. Data from [12]. that can be noted is that for all the domains studied V ar(Dout) < V ar(Din), consistent with eq. 14. More- over, Cov(Din, Dout) and V ar(Dout) have similar values (consistent with eq. 12), the relative differences seems large only for WBGC03. In order to compare in a bet- ter way these last two quantities, Table 2 shows r and V ar(Dout) V ar(Din) for the same data. We can see that WBGC01 and WGIT04 have very similar values of r and R (see eq. 13). In order to study the relationship between WBGC01 0.0594 0.0654 WGUK02 0.1335 0.1912 WBGC03 0.0016 0.004 WGIT04 0.0588 0.0761 TABLE II: Correlation (r) and R for 4 WWW networks. Data computed from Table 1. 〈Din/Dout〉 and Dout is necessary to have the complete data. At this point, we analyze the WWW data obtained from [13] presented in [14]. We built up a database with the information of the number of out-links and in-links ((Dout, Din)) for each of the 325729 nodes. In order to have a good estimation of the conditional expectation, we first restrict the study to the values of Dout such that there exist at least 500 nodes. Fig. 5 (a) shows the rela- tionship between Dout and the conditional mean of Din (〈Din/Dout〉) given Dout. Interestingly, there is a strong relationship between both. For values of the Dout smaller than 20 there is a clear linear relationship between them. A robust regression (least median of squares) estimation between 〈Din/Dout〉 and Dout gives a slope of 0.523 and an intercept of 1.739. In the case Dout is greater than 20 it seems that 〈Din/Dout〉 grows faster than linear, but it is not clear if this effect is real (based on Fig. 5 (b)). The graph presented in Fig. 5 (b) is similar to the one in (a), but now we study the values of Dout such that there exist at least 30 nodes. A plot of two different representations of the joint in-out distribution is given in Fig 5 (c) and (d), to have an idea of the shape of the joint law, while (e) shows a scatter plot on a larger grid. Besides, the in- degree variance (V ar(Din) = 1346.85) is greater than the out-degree one (V ar(Dout) = 461.25), consistent with eq. 14. Fig 5 (f) shows the conditional standard devia- tion of Din given Dout, σin/out = V ar(Din/Dout). Un- like the conditional expectation, the conditional variance does not seem to have any relationship with Dout. In [14] the authors showed the empirical out de- gree (Pout(k)) and in degree (Pin(k)) distributions (see Fig. 6), and reported a power exponent of 2.45 for out- degree and of 2.1 for the in-degree [22]. This is the first empirical evidence that the model presented here can not describe in a good way the WWW network, in the model the power law exponents are equal. The second evidence is that r and R are not similar, r = 0.2244 and R = 0.5852. APPENDIX D: COMMENT ON THE SCIENTIFIC PUBLICATION NETWORK In the scientific publication network it is implicit that we are under the hypothesis that the citation distribu- tion for all papers published in 1981 can be treated as the stationary in-degree distribution of a growing net- work model. But, why can be treated in this way only studying the papers of a particular year (1981)?. This is just because: if the total scientific network has arrived (today in 2007) to a proportion of papers with k citations that do not change with time (stationary), then the arti- cles published in 1981 are a sample of this distribution. [1] A.L. Barabási, and R. Albert, Science 286, 509 (1999). [2] R. Albert, A.L. Barabási, Rev. Mod. Phys. 74, 47 (2002). [3] S.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51, 1079 (2002). [4] H. Jeong, B. Tombor, R. Albert, Z. N./ Oltvai, and A.L. Barabási, Nature 407, 651 (2000). [5] S. N. Dorogovtsev, J. F. F. Mendes, A. N. Samukhin, Phys. Rev. Lett. 85, 4633 (2000). [6] S. Redner, Eur. Phys. J. B 4, 131 (1998). [7] P. L. Krapivsky, G. J. Rodgers, and S. Redner, Phys. Rev. Lett. 86, 5401 (2001). [8] B. Tadić, Physica A 293, 273 (2001). [9] M.E.J. Newman, SIAM Review 45, 167 (2003). [10] D.J. Price, Science 149, 510 (1965). [11] R. Lambiotte, http://www.lambiotte.be/talks/vienna2006.pdf (2006). [12] M. A. Serrano, A. Maguitman, M Bogua, S. Fortunato and A. Vespignani, ACM Trans. Web, 1, No.2. Article 10 (2007). [13] http://www.nd.edu/∼networks/resources.htm [14] R. Albert, H. Jeong and A.L. Barabási, Nature, 401, 130 (1999). [15] http://physics.bu.edu/∼redner/projects/citation/isi.html [16] S. N. Dorogovtsev, J. F. F. Mendes, Phys. Rev. E, 62, 1842 (2000). [17] K.B. Hajra, P. Sen, Physica A, 346, 44 (2005). [18] P. L. Krapivsky, S. Redner, F. Leyvraz, Phys. Rev. Lett., 85, 4629 (2000). [19] P. L. Krapivsky, S. Redner, Phys. Rev. E, 63, 66123 (2001). [20] S.N. Dorogovtsev and J.F.F. Mendes, Handbook of Graphs and Networks: From the Genome to the Inter- net (Wiley-VCH, Berlin, 2002), 318. [21] Probably a non-homogeneous Poisson process provides a good description of the arrival of new publications. But as we are interested in asymptotic distributions, which are independent (except in the pathological cases where explosions might occur) of the arrival process, it is suffi- cient to study the time step process, where in each step a new publication is aggregatted. [22] Our estimations of the exponents have some differences from the ones in [14], but the difference between the in and out exponents is still appreciable. http://www.lambiotte.be/talks/vienna2006.pdf http://www.nd.edu/~networks/resources.htm http://physics.bu.edu/~redner/projects/citation/isi.html FIG. 5: Conditional mean of Din given Dout, when for each value of Dout there exist at least: (a) 500, and (b) 30 nodes. Data presented as a confidence interval of 95%. (c) and (d) Different representations of the joint in-out density of the links in a node. (e) Scatter plot of Din as a function of Dout. (f) Conditional standard deviation of Din given Dout, σin�out. FIG. 6: Pout(k) and Pin(k) as a function of k+1. This graph was presented in [14]. Introduction Growing Directed Network Model Stationary Probabilities Descriptive Statistics Covariance and conditional expectation Relationship between the distribution tails Application: scientific publications network Different attachment laws Preferential linking on in-degree Uniform attachment law Conclusions Comments on the model A closed equation for P(k) WWW network Comment on the scientific publication network References

We compute the stationary in-degree probability, $P_{in}(k)$, for a growing network model with directed edges and arbitrary out-degree probability. In particular, under preferential linking, we find that if the nodes have a light tail (finite variance) out-degree distribution, then the corresponding in-degree one behaves as $k^{-3}$. Moreover, for an out-degree distribution with a scale invariant tail, $P_{out}(k)\sim k^{-\alpha}$, the corresponding in-degree distribution has exactly the same asymptotic behavior only if $2<\alpha<3$ (infinite variance). Similar results are obtained when attractiveness is included. We also present some results on descriptive statistics measures %descriptive statistics such as the correlation between the number of in-going links, $D_{in}$, and outgoing links, $D_{out}$, and the conditional expectation of $D_{in}$ given $D_{out}$, and we calculate these measures for the WWW network. Finally, we present an application to the scientific publications network. The results presented here can explain the tail behavior of in/out-degree distribution observed in many real networks.

Introduction Growing Directed Network Model Stationary Probabilities Descriptive Statistics Covariance and conditional expectation Relationship between the distribution tails Application: scientific publications network Different attachment laws Preferential linking on in-degree Uniform attachment law Conclusions Comments on the model A closed equation for P(k) WWW network Comment on the scientific publication network References

UV stable, Lorentz-violating dark energy with transient phantom era Maxim Libanov and Valery Rubakov Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect, 7a, Moscow, 117312, Russia Eleftherios Papantonopoulos Department of Physics, National Technical University of Athens, Zografou Campus GR 157 73, Athens, Greece M. Sami Centre for Theoretical Physics, Jamia Millia, New Delhi-110025, India Shinji Tsujikawa Department of Physics, Gunma National College of Technology, Gunma 371-8530, Japan Phantom fields with negative kinetic energy are often plagued by the vacuum quantum instability in the ultraviolet region. We present a Lorentz-violating dark energy model free from this problem and show that the crossing of the cosmological constant boundary w = −1 to the phantom equation of state is realized before reaching a de Sitter attractor. Another interesting feature is a peculiar time- dependence of the effective Newton’s constant; the magnitude of this effect is naturally small but may be close to experimental limits. We also derive momentum scales of instabilities at which tachyons or ghosts appear in the infrared region around the present Hubble scale and clarify the conditions under which tachyonic instabilities do not spoil homogeneity of the present/future Universe. I. INTRODUCTION The compilations of various observational data show that the Universe has entered the stage of an accelerated expansion around the redshift z ∼ 1 [1, 2, 3, 4, 5, 6]. The equation of state (EOS) parameter w of Dark Energy (DE) responsible for the acceleration of the Universe has been constrained to be close to w = −1. However, the phantom EOS (w < −1) is still allowed by observations and even favored by some analyses of the data [7]. It is also possible that the EOS of DE crossed the cosmological constant boundary (w = −1) in relatively near past [8]. The presence of the phantom corresponds to the violation of weak energy condition, the property which is generally difficult to accommodate within the framework of field theory. The simplest model which realizes the phantom EOS is provided by a minimally coupled scalar field with a negative kinetic term [9, 10] (see also Refs. [11, 12]). The negative kinetic energy is generally problematic because it leads to a quantum instability of the vacuum in the ultraviolet (UV) region [10, 13, 14, 15, 16, 17]: the vacuum is unstable against the catastrophic particle production of ghosts and normal (positive energy) fields. There have been a number of attempts to realize the phantom EOS without having the pathological behaviour in the UV region. One example is scalar-tensor gravity in which a scalar field φ with a positive kinetic term is coupled to Ricci scalar R [18, 19]. This coupling leads to the modification of gravitational constant, but it was shown in Ref. [20] that there are some parameter regions in which a phantom effective EOS is achieved without violating local gravity constraints in the present Universe. Another example is provided by the so-called modified gravity, including f(R) gravity models [21] and the Gauss- Bonnet (GB) models [22]. In f(R) models it is possible to obtain a strongly phantom effective EOS, but in that case the preceding matter epoch is practically absent [23]. For GB DE models it was shown in Ref. [24] that the crossing of the cosmological constant boundary, w = −1, is possible, but local gravity experiments place rather strong constraints on the effective GB energy fraction [25]. In addition, tensor perturbations are typically plagued by instabilities in the UV region if the GB term is responsible for the accelerated expansion of the Universe [26]. Thus, it is generally not so easy to construct viable modified gravity models that realize the phantom effective EOS without violating cosmological and local gravity constraints. The third example is the Dvali-Gabadadze-Porrati (DGP) braneworld model [27] and its extension [28] with a GB term in the bulk, which allow for the possibility to have w < −1 [29, 30]. However, it was shown in Ref. [31] that the DGP model contains a ghost mode, which casts doubts on the viability of the self-accelerating solution. While the above models more or less correspond to the modification of gravity, it was recently shown that in the Einstein gravity in a Lorentz-violating background the phantom EOS can be achieved without any inconsistency in the UV region [32, 33]. In particular, in the model of Ref. [33] Lorentz invariance is broken in the presence of a vector field Bµ which has two-derivative kinetic terms similar to those given in Ref. [34]. The effect of the Lorentz violation http://arxiv.org/abs/0704.1848v2 is quantified by a parameter Ξ ≡ BµBµ/M2, where M is an UV cut-off scale. In analogy to Ref. [35] the vector field also has one-derivative coupling ǫ∂µΦB µ with a scalar field Φ, where ǫ is a small parameter that characterizes an IR scale. In the UV region, where the spatial momentum p is much larger than ǫ, ghosts, tachyons and super-luminal modes are not present. Meanwhile tachyons or ghosts can appear in the IR region p <∼ ǫ. This is not problematic provided that ǫ is close to the present Hubble scale. In this paper we apply this Lorentz-violating model to dark energy and study the cosmological dynamics in detail in the presence of mass terms in the potential, V = 1 m2Φ2 − 1 M2X2 (where X2 = BµB µ). We show that the model has a de Sitter attractor responsible for the late-time acceleration. At early times DE naturally has normal EOS with w > −1, while the phantom EOS can be realized between the matter-dominated era and the final de Sitter epoch. We clarify the conditions under which the cosmological constant boundary crossing to the phantom region occurs. Interestingly, in a range of parameters this crossing takes place at the epoch when Ωm ∼ ΩDE thus making the crossing potentially observable. Another interesting feature of our model is the time-dependence of the effective Newton’s constant. It is naturally weak, but may well be comparable with current experimental limits. Moreover, the effective Newton’s constant G∗(t) has a peculiar behaviour correlated with the deviation of w from −1. We also derive momentum scales of instabilities of perturbations, first in Minkowski spacetime. This is the extension of the work [33] that mainly focused on the case of massless scalar (m = 0). We show that in the UV region (p ≫ ǫ) the model does not have any unhealthy states such as ghosts, tachyons or super-luminal modes. In the IR region (p <∼ ǫ) tachyons or ghosts appear, depending on the momentum. Finally, we study the evolution of perturbations in the cosmological background and estimate the amplitude of perturbations amplified by the tachyonic instability around the scale of the present Hubble radius. The perturbations remain to be smaller than the background fields under certain restriction on the model parameters. This paper is organized as follows. In Sec. II we present our Lorentz-violating model and derive basic equations describing spatially flat Friedmann–Robertson–Walker cosmology in the presence of DE, radiation and non-relativistic matter. In Sec. III the cosmological dynamics is discussed in detail analytically and numerically with an emphasis on the occurrence of a phantom phase before reaching a de Sitter attractor. The time-dependence of the effective gravitational constant is also considered. In Sec. IV we study the Minkowski spectrum of field perturbations and clarify the properties of tachyons and ghosts in the IR region. We then discuss the tachyonic amplification of field perturbations around the present Hubble scale in the cosmological background. We summarize our results in Sec. V. Appendix A contains the derivation of the effective “Newtonian gravitational constant” in our model. In Appendix B we derive the fixed points of the system by rewriting the equations in autonomous form. We analyse the stability of the fixed points and show analytically that the cosmological evolution proceeds from radiation-dominated stage through matter-dominated stage to the final de Sitter regime. II. LORENTZ-VIOLATING MODEL We study a 4-dimensional Lorentz-violating model whose Lagrangian density includes a vector field Bµ and a scalar field Φ: L = −1 α(Ξ)gνλDµBνD µBλ + β(Ξ)DµBνD µΦ + ǫ∂µΦB µ − V (B,Φ) , (1) where Ξ = BµB µ/M2 with M being an UV cut-off scale of the effective theory. The dimensionless parameters α and β are the functions of Ξ, and ǫ is a free positive parameter that characterizes an IR scale. The first two terms in (1) are familiar in two-derivative theory [34], whereas the one-derivative term ǫ∂µΦB µ is introduced following the approach of Ref. [35]. We study dynamics of flat Friedmann-Robertson-Walker (FRW) Universe ds2 = N 2(t)dt2 − a2(t)dx2 , (2) where N (t) is a Lapse function and a(t) is a scale factor. In the case of spatially homogeneous fields with Bi = 0 (i = 1, 2, 3), the Lagrangian (1) reads −gL = γ 2 − 3α 2 + ǫa3φ̇X − a3NV (X,φ) , (3) where X = B0/N , φ is the homogeneous part of the field Φ and γ(X) = M2β(X)− α(X) . (4) Hereafter we study the case in which the following condition holds α > γ > 0 . This is required to avoid a superluminal propagation in Minkowski spacetime [33], as we will see later. Throughout this paper we assume that α and γ are of order unity. For fixed X , the second term in the Lagrangian (3) has precisely the form of the Einstein–Hilbert action specified to the flat FRW metric. Hence, it leads to the change of the “cosmological” effective Planck mass [33] m2pl,cosm = m pl + 4παX 2 . (5) Another effective Planck mass mpl,Newton determines the strength of gravitational interactions at distances much shorter than the cosmological scale; in general, these two effective Planck masses are different [19, 35, 36]. We show in Appendix A that the “Newtonian” Planck mass in our model is given by m2pl,Newton = m pl − 4παX2 . (6) Both effective Planck masses depend on time via X = X(t). Since the time-dependent terms in (5) and (6) differ by sign only, it will be sufficient to study one of these effective masses. In what follows we concentrate on the “Newtonian” mass (6) for definiteness. In this paper we focus on the case in which the potential V takes a separable form: V = W (φ) + U(X) . (7) We take into account the contributions of non-relativistic matter and radiation whose energy densities ρm and ρr, respectively, satisfy ρ̇m + 3Hρm = 0 , (8) ρ̇r + 4Hρr = 0 . (9) The energy density of the fields is derived by taking the derivative with respect to N of the action S = ρ = − 1 Ẋ2 − 3α H2X2 + φ̇2 + V . (10) We set N = 1 for the rest of this paper. The Friedmann equation is given by γẊ2 − 3α H2X2 + φ̇2 +W (φ) + U(X) + ρm + ρr , (11) where κ2 = 8π/m2pl. The equations of motion for the homogeneous fields φ and χ are Ẍ + 3HẊ γ,XẊ 2 − 3 2X2 − 3αH2X + ǫφ̇ = U,X , (12) −(φ̈+ 3Hφ̇)− ǫ(Ẋ + 3HX) = W,φ , (13) where γ,X = dγ/dX , etc. Taking the time-derivative of Eq. (11) and using Eqs. (12) and (13), we obtain Ḣ = −κ ρ+ p+ ρm + where ρ+ p = ǫφ̇X + αḢX2 + 2αHXẊ + γẊ2 + φ̇2 + α,XHX 2Ẋ . (14) In what follows we assume for simplicity that α and γ are constants, i.e., α,X = γ,X = 0. Following Ref. [33] we consider the simplest potential for the fields, W (φ) = m2φ2 , U(X) = −1 M2X2 , (15) which allows for a possibility to realize a phantom phase. III. DYNAMICS OF DARK ENERGY One way to analyse the cosmological dynamics in our model is to make use of the autonomous equations, the techniques widely used in the context of dark energy studies [6, 38, 39]. This approach is presented in Appendix B, where we analytically confirm that our model can lead to the sequence of radiation, matter and accelerated epochs. Also, in Appendix B we derive the conditions under which the de Sitter solution given below is an attractor. Here we first present a simpler analysis based on the slow-roll approximation. Then we give numerical solutions to eqs. (8), (9), (11), (12), (13), exhibiting transient phantom behaviour, and study their dependence on various parameters of our model, including the initial values of the fields. A. Final and initial stages One immediate point to note is that in the absence of radiation and matter, the system of equations (11), (12), (13) has a de Sitter solution, H = const, for which φ and X are also independent of time, provided that . (16) Indeed, for constant H,φ and X eqs. (11), (12), (13) reduce to a simple algebraic system H2X2 − M 3αH2 = M2 , −3ǫHX = m2φ . (17) Once the inequality (16) is satisfied, this system has a solution Mmplǫ√ 3ǫ2/m2 − 2α XA = − 3ǫ2/m2 − 2α . (18) We will see in what follows, and elaborate in Appendix B, that in a range of parameters this solution is an attractor which corresponds to the de Sitter phase in asymptotic future (hence the notation). In order to use this for dark energy we require that the mass scale M is of the order of the present Hubble parameter H0. Then the Newtonian effective Planck mass, Eq. (6), is given by m2pl,Newton = m 3ǫ2/m2 − 2α . (19) In order that the change of the Planck mass be small, we impose the condition αm . (20) It is worth noting that under this condition, the contribution of the field φ in the energy density dominates in the de Sitter regime, φ2A ≫ X2A = X2A . (21) Thus, as the system approaches the de Sitter attractor, the total energy density in the Universe becomes determined by the scalar field energy density. Another point to note is that at early times (at the radiation-dominated epoch already), when the Hubble parameter is large enough, the term (−3αH2X) in Eq. (12) drives the field X to zero, the relevant time being of the order of the Hubble time. Soon after that the field φ obeys the usual scalar field equation in the expanding Universe, so the Hubble friction freezes this field out. Thus, the initial data for the interesting part of the DE evolution are Xi = 0 , φi = const . (22) The value of φi is a free parameter of the cosmological evolution in our model. Since at early times the field X is close to zero, its effect on the evolution of the field φ is negligible. The field φ slowly rolls down its potential, and its energy density dominates over that of X . Therefore, EOS for DE at early times is normal, w > −1, with w being close to −1. We refer to this regime as quintessence stage. As we will see below, in a range of parameters, the system eventually crosses the cosmological constant boundary w = −1 and passes through a transient phantom phase before reaching the de Sitter asymptotics (18). B. Slow roll phantom regime The approach to the de Sitter solution (18) occurs in the slow roll regime. To see how this happens, we truncate Eqs. (12) and (13) to ǫφ̇− 3αH2X = U,X , (23) −3ǫHX = W,φ . (24) This truncation is legitimate provided that in addition to the usual slow-roll conditions φ̈ ≪ Hφ̇ and Ẍ ≪ HẊ, the following conditions are satisfied: φ̇ ≪ ǫX , (25) ǫφ̇X ≪ V , (26) Ẋ ≪ HX . (27) [When writing inequalities, we always mean the absolute values of the quantities.] Note that we do not impose the condition ǫφ̇ ≫ 3αH2X unlike in Ref. [33], since the term 3αH2X is not necessarily negligible relative to the term U,X in Eq. (23). From Eq. (23) we obtain X = − ǫφ̇ , (28) where ξ ≡ 1− 3αH . (29) Note that ξ may be considered as a measure of the deviation from the de Sitter regime (18). Substituting Eq. (28) into Eq. (24) we get the following equation 3Hφ̇ = ξW̃,φ , (30) where W̃ (φ) ≡ m Equation (30) shows that the field φ rolls up the potential W (φ) for ξ > 0, i.e., for . (31) This is the region in which the phantom equation of state (w < −1) is realized; indeed, Eq. (14) gives ρ+ p ≈ ǫφ̇X = ξXU,X ≡ −ξM2X2. Another way to understand the phantom behaviour is to notice that when the system approaches the de Sitter regime, the field φ dominates the energy density, see Eq. (21), so the energy density increases as the field φ rolls up. Let us find out whether the slow roll conditions (25), (26) and (27) are indeed satisfied. Making use of Eq. (28) we obtain that the condition (25) is equivalent to ǫ2 ≫ ξM2 , (32) while using Eqs. (28) and (30) we rewrite the condition (26) as ǫ2 ≫ ξm . (33) The second inequality ensures also the validity of the relation (27); this can be seen by taking the time derivative of Eq. (24). The latter two inequalities are automatically valid at small ξ, that is near the de Sitter solution (18). We conclude that the approach to the de Sitter solution indeed occurs in the slow roll regime, and that the phantom phase is indeed realised provided that the relation (31) holds. Our analysis implies also that the de Sitter solution (18) is an attractor: the Hubble parameter slowly increases towards its de Sitter value, ξ decreases, and the dynamics gets frozen as ξ → 0. Since the field φ dominates the energy density at the phantom slow roll stage, the condition (31) takes a simple φ < φA = MmPl√ , (34) where we made use of (20). The latter relation translates into the range of initial conditions which eventually lead to the transient phantom behaviour, φi <∼ φA . (35) Indeed, during the radiation- and matter-dominated stages the field φ remains almost constant, and at the quintessence stage it also does not roll down much. Recalling again that φ dominates the energy density, we rewrite the inequality (33) as εs ≪ 1, where ξ . (36) The parameter εs may be viewed as the slow roll parameter for the field φ. Indeed, one observes that = εs , (37) which, together with Eq. (30), justifies this interpretation. From Eqs. (10), (14), (28) and (30) it follows that during the slow roll phantom stage, the EOM parameter of DE is given by w = −1− εs . (38) Hence, the appreciable deviation from w = −1 occurs when εs is not much smaller than unity, i.e., when φi is appreciably smaller than φA. In the next section we confirm these expectations by numerical analysis, and also show explicitly that in a range of parameters, the cosmological evolution proceeds from radiation-dominated to matter-dominated epoch, and then to the slow roll phantom stage, before finally ending up in the de Sitter regime (18). C. Numerical solutions In our numerical analysis we choose initial conditions Ẋ = φ̇ = X = 0 with nonzero values of φ, ρm and ρr. This choice corresponds to the initial data (22). We have also tried many other initial conditions and found that the results are not sensitive to the initial values of X , Ẋ and φ̇, in accord with the discussion in the end of Sec. III A. In Fig. 1 we plot the cosmological evolution for the model parameters α = 1, γ = 1/2, ǫ/m = 3, M/m = 1 and the initial value φi = 0.5φA. We find it convenient to present the plots in terms of the variable N = ln a . (39) Figure 1 clearly shows that the sequence of radiation, matter and de Sitter epochs can be achieved together with the w = −1 crossing. The DE EOS parameter w is nearly a constant, w ≃ −1, during the radiation and matter epochs because the fields are almost frozen. At the transition era from matter domination to DE domination, w begins to grow because the kinetic energies of the fields become important; this is the quintessence phase. However, the system soon enters the phantom phase during which the field φ rolls up the potential. Hence the equation of state w crosses the cosmological constant boundary w = −1 and reaches a minimum value wmin < −1. The solution finally approaches the de Sitter regime (18) from the phantom side. Of some interest is also the EOS parameter of the entire system, weff = . (40) It is seen from Fig. 1, this parameter also becomes smaller than −1 soon after w = −1 crossing. - 1 . 5 - 1 . 0 - 0 . 5 0 0 . 0 0 . 5 0 1 . 0 1 . 5 0 5 1 0 1 5 FIG. 1: Cosmological evolution for the model parameters α = 1, γ = 1/2, ǫ/m = 3 and M/m = 1. We choose initial conditions X = Ẋ = φ̇ = 0, φi = 0.5φA, and Ωr,i = 0.99, Ωm,i = 0.01. Shown is the evolution of ΩDE, Ωm, Ωr , w and weff as functions of N ≡ ln a. Note that the present epoch corresponds to Ωm = 0.3 and ΩDE = 0.7, which is denoted by a vertical line. After the cosmological constant boundary crossing, the DE EOS parameter w reaches a minimum wmin = −1.19 and then increases towards the de Sitter value w = −1 from the phantom side. The pattern shown in Fig. 1 is generic in our model, provided that its parameters and initial data obey m/ǫ ≪ 1, M/ǫ ≪ 1 and φi < φA (in fact, the inequalities here need not be strong). The strengths of the effects depend, of course, on the values of these parameters. In particular, the minimum value wmin is related to the slow-roll parameter εs, in accord with Eq. (38). If the field φ evolves very slowly, one has εs ≪ 1, so wmin is close to −1. On the contrary, the appreciable deviation from w = −1 occurs if εs is not very much smaller than unity. FIG. 2: The minimum value of the EOS parameter w of DE, its value w0 at the present epoch (Ωm = 0.3, ΩDE = 0.7) and its maximum value as functions of m/ǫ for M/ǫ = 1/30 and the initial value φi = 0.5φA. Since the field φ is practically frozen during the radiation- and matter-dominated epochs, and evolves rather slowly later on, the dependence of the cosmological evolution on the parameters of the model and on the initial value φi can be understood, at qualitative level, by inspecting Eq. (36). For the qualitative discussion of the evolution well before the asymptotic de Sitter regime sets in, the parameter ξ in (36) may be set equal to 1, while the value of φ may be set equal to its initial value φi. Then Eq. (36) implies that with other parameters and φi fixed, for smaller m/ǫ one gets smaller absolute value of εs at its minimum, leading to the value of wmin closer to −1. This is shown in Fig. 2. From Fig. 1 it is clear, however, that the minimum of w occurs after the present epoch (ΩDE ≃ 0.7 and Ωm ≃ 0.3). Again, this is a rather generic feature of our model. Therefore, instead of wmin, more interesting quantities are the present value w0 of the DE EOS parameter and also its maximum value before the cosmological constant boundary crossing. These quantities are also given in Fig. 2. Overall, the behaviour shown in Fig. 1 is more pronounced at larger m/ǫ, once other parameters of solutions are kept fixed. According to Eq. (36), the initial value of the field φ is also important to determine the amplitude of εs and hence w: a smaller φi results in a stronger deviation of wmin from −1. The present value w0 also becomes more negative (stronger deviating from −1), while the cosmological constant boundary crossing occurs earlier. From the numerical analysis we find that at relatively large values of m/ǫ the increase of M has the opposite effect, while at smaller m/ǫ the effects due to the variation of M are small. These properties are illustrated in Figs. 3 and 4. As discussed above, the phantom phase occurs only if the initial value of φ obeys (35), otherwise the de Sitter attractor is approached from the quintessence side, w > −1. This is illustrated in Fig. 5. - 2 . 0 - 1 . 8 - 1 . 6 - 1 . 4 - 1 . 2 - 1 . 0 - 0 . 8 0 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 m/e=1/3, M/e=1/3 m/e=1/3, M/e=1/30 m/e=1/10, M/e=1/3 m/e=1/10, M/e=1/30 FIG. 3: The dependence of the present value w0 (i.e., w at Ωm = 0.3, ΩDE = 0.7) on the initial value of φ, for different sets of the model parameters. - 1 . 0 - 0 . 5 0 0 . 0 0 . 5 0 1 . 0 1 . 5 2 . 0 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 m/e=1/3, M/e=1/3 m/e=1/3, M/e=1/30 m/e=1/10, M/e=1/3 m/e=1/10, M/e=1/30 FIG. 4: The dependence of the redshift (from the present epoch, Ωm = 0.3, ΩDE = 0.7) of the cosmological constant boundary crossing, w = −1, on the initial value of φ. - 1 . 5 - 1 . 0 - 0 . 5 0 0 . 0 0 . 5 0 1 . 0 1 . 5 0 5 1 0 1 5 2 0 FIG. 5: Cosmological evolution for the model parameters α = 1, γ = 1/2, ǫ/m = 3 and M/m = 0.1. We choose initial conditions X = Ẋ = φ̇ = 0, φi = 3φA, and Ωr,i = 0.99, Ωm,i = 0.01. The present epoch corresponds to ΩDE = 0.7 and Ωm = 0.3, which is denoted by a vertical line. In this case the cosmological constant boundary crossing is not realized because φ remains always larger than φA. At the end of this section we discuss the variation of the effective gravitational constant. According to Eq. (6), the effective Newton’s constant that determines the interaction deep inside the horizon scale is given by G∗ = G(1 − 4πGαX2)−1 . (41) Its variation in time is conveniently expressed in terms of the following quantity d lnG∗ d ln a ≡ Ġ∗ 8παXẊ H(m2pl − 4παX2) . (42) The typical experimental and observational constraints on the variation of G∗ in the present Universe are given by |Ġ∗/G∗| <∼ 10−12 yr−1 [37], which translates into the condition Ġ∗/G∗ <∼ 10 −2H0 . (43) - 0 . 0 1 0 0 . 0 0 . 0 1 0 0 . 0 2 0 0 . 0 3 0 0 . 0 4 0 0 5 1 0 1 5 2 0 ( a ) ( b ) FIG. 6: The evolution of the variation of the effective gravitational constant d lnG∗/d ln a as function of N = ln a and redshift z. The cases (a) and (b) correspond to the model parameters and initial conditions given of Figs. 1 and 5, respectively. The black points represent the values at the present epoch (Ωm ≃ 0.3). FIG. 7: The maximum value of d lnG∗/d ln a as a function of m/ǫ. Other parameters and initial conditions are the same as in Fig. 2. In Fig. 6 we plot the evolution of the quantity d lnG∗/d lna for the model parameters and initial conditions given in Figs. 1 and 5. At the present epoch (Ωm ≃ 0.3) we obtain the values Ġ∗/G∗ = 3.5× 10−2H−10 and 2.4× 10−2H 0 for these two cases, respectively. Comparing Fig. 6 with Figs. 1 and 5 one observes that the variation of the gravitational constant is correlated in time with the deviation of w from −1. This is clear from Eq. (41) too: the gravitational constant varies when the field X changes in time, while the latter occurs during the transition from the matter dominated stage to the final de Sitter attractor. It is precisely at this transition stage that w substantially deviates from −1. Figure 7 shows the maximum value of d lnG∗/d lna as a function of m/ǫ. Again, the variation of the effective Newton’s constant is more pronounced at larger m/ǫ. This means that it is correlated with the amplitude of the deviation of w from −1. The dependence of d lnG∗/d lna on M and on the initial value of φ is rather weak. It is worth pointing out that as long as the deviation of the EOS from w = −1 is not so significant, the models satisfy the constraint (43), and also that our model suggests that the variation of G∗ is close to the present upper bound on Ġ∗/G∗. IV. MOMENTUM SCALES OF INSTABILITIES In this section the momentum scales of instabilities are present in our model. We first study dispersion relations in the Minkowski space-time and then proceed to those in the FRW space-time. We wish to clarify the conditions under which a tachyon or a superluminal mode appears by considering dispersion relations. We also evaluate the energy of the modes to find out a ghost state. A. Minkowski spectrum Let us consider the perturbations for the fields, B0 = X + b0 , Bi = bi , Φ = φ+ ϕ . (44) The quadratic Lagrangian for perturbations, following from the general expression (1), is Lb0,bi,ϕ = ∂µb0∂ µb0 + ∂µbi∂ µbi + µϕ+ ǫ∂0ϕb0 − ǫ∂iϕbi 2 , (45) where m20 = U,XX , m 1 = − , m2ϕ = W,φφ . (46) For our model (15) one has −m20 = m21 = M2 and m2ϕ = m2. In what follows we concentrate on this case, and assume the following relations, see (20) and (32), ǫ ≫ M . (47) Varying the Lagrangian (45) with respect to bi, b0 and ϕ, we obtain the equations for the field perturbations. In order to find the spectrum of the system we write the solutions in the form b0 = b̃0e = b̃0e i(ωt−p·r), bi = b̃ie and ϕ = ϕ̃eipµx . The transverse mode of the vector field Bi has the dispersion relation ω20 = p . (48) The three scalar modes b̃i = (pi/p)b̃L, b̃0 and ϕ̃ satisfy the following equations ω2 − p2 − M b̃L + i pϕ̃ = 0 , (49) ω2 − p2 + M b̃0 − i ωϕ̃ = 0 , (50) ω2 − p2 −m2 ϕ̃− iǫωb̃0 − iǫpb̃L = 0 . (51) Expressing b̃L and b̃0 in terms of ϕ̃ from Eqs. (49), (50) and plugging them into Eq. (51), we find that the eigenfre- quencies corresponding to three mixed states satisfy (z −m2) z − M − ǫ2z = 0 , (52) where z ≡ ω2 − p2 . (53) The spectrum in the case m = 0 was studied in Ref. [33]. Our purpose here is to extend the analysis to the case of non-zero m. Denoting the solutions of Eq. (52) as z1, z2 and z3, we obtain the relation z1z2z3 = − . (54) Once the conditions (47) are satisfied, then one can show that if the relation z1 < z2 < z3 (55) holds at some momentum, then the inequality (55) is satisfied for all momenta. In the limits p → ∞ and p → 0, we obtain the following dispersion relations, respectively. • (A) UV limit (p → ∞) ω1 = p− +O(M2/p) , (56) ω2 = p+ 2p3ǫ2(α+ γ) +O(1/p5) , ω3 = p+ +O(M2/p) . We see that ω1 < ω2 < ω3 and z1 < 0, z2,3 > 0. In all three cases the group velocities ∂ωi/∂p are less than 1, so neither mode is superluminal at high three-momenta, provided that α > γ. The two-derivative terms in the Lagrangian (45) dominate in the UV limit, so there are neither ghosts nor tachyons in this limit. • (B) IR limit (p → 0) ω1 = − 1 +O(m2/ǫ2,M2/ǫ2) , (57) +m2 +O(M2) . We see again that ω1 < ω2 < ω3 and z1 < 0, z2,3 > 0. This means that using the property (55) we can identify the modes: the first one has the behaviour (56) and (57), and so on. It follows from Eq. (54) that zi never vanish. In fact the coefficients in Eq. (52) are regular at all momenta, so zi are regular as well. Therefore, zi never change signs and hence z1 < 0, z2,3 > 0 for all momenta. This means, in particular, that the second and third modes never become tachyonic. Let us discuss the dangerous mode with the dispersion relation ω = ω1(p) in some detail. The expression for the fields in each mode is b̃L,i = −iǫp γzi +M · Ci , b̃0,i = iǫω(αzi −M2) · Ci , ϕ̃i = (γzi +M 2)(αzi −M2) · Ci , (58) where Ci are the normalization factors. Setting ω 2 = 0 in Eq. (52), we obtain three critical momenta p21,2 = ǫ2 −M2 −m2 ± ǫ2 −M2  , (59) p23 = . (60) Under the conditions (47), the critical momenta p21,2 are approximately given by p21 ≃ ǫ2 −M2 −m2 , p22 ≃ , (61) so that p21 > p 3 > p 2 > 0. The tachyonic mode (ω 1 < 0) is present for 0 < p 2 < p22 and p 3 < p 2 < p21. In order to find whether there are ghosts we calculate the energy of the modes (58), Ei(p) = 2ω 2|Ci|2 αǫ2p2(γzi +M 2)2 + ǫ2(γp2 −M2)(αzi −M2)2 + (γzi +M2)2(αzi −M2)2 . (62) For the modes with ω2,3 we have E2,3(p) > 0. For the mode with ω1 the energy is equal to zero at p = p3 = M/ While ω21 > 0 for p 2 < p23 ≡ M2/γ, the energy E1(p) changes its sign at this momentum. Thus the mode with ω1 is a ghost for p2 < M2/γ. We summarize the properties of the dangerous mode as follows: • (i) p2 > (ǫ2 −M2)/α−m2: healthy • (ii) M2/γ < p2 < (ǫ2 −M2)/α−m2: tachyon • (iii) m2M2/ǫ2 < p2 < M2/γ: ghost, but not tachyon • (iv) 0 < p2 < m2M2/ǫ2: tachyon. Unlike the case m = 0 [33] the tachyon is present in the deep IR region (iv). To end up the discussion of the modes in Minkowski space-time, we give the expressions for the minimum values of ω2 in the tachyonic regions, • (ii): ω2min = − 4α(α+ γ) at p2 = γ + 2α γ + α . (63) • (iv): ω2min = − at p2 = 0 . Note that |ω2min| is relatively large in the region (ii), so this region is the most problematic. B. Evolution of perturbations in cosmological background Finally we discuss the evolution of field perturbations in the FRW background (2). In the cosmological context the physical momentum p is related to the comoving momentum k as p = k/a. Once the parameters of the model and initial data are such that the cosmological boundary crossing occurs, the present epoch (ΩDE ≃ 0.7) typically corresponds to the phantom region. From Fig. 1 one can see that the Hubble parameter does not change much during the transition from the phantom epoch to the final de Sitter era. Hence the present value of the Hubble parameter (H0) is of the same order as the value H = M/ 3α in the de Sitter asymptotics. This means that the value p3 = M/ is of the same order as H0 provided that γ and α are of order unity. The tachyon appears when the momentum p = k/a of the dangerous mode becomes smaller than (ǫ2 −M2)/α−m2 and temporally disappears when the mode crosses the value M/√γ. Hence this instability is present for the modes which are inside the Hubble radius and satisfy M/ γ < p < (ǫ2 −M2)/α−m2, but it is absent for the modes deep inside the Hubble radius, satisfying p > (ǫ2 −M2)/α−m2. After the Hubble radius crossing (k = aH), the tachyonic instability disappears in the momentum region m2M2/ǫ2 < p2 < M2/γ, but the tachyon appears again for p2 < m2M2/ǫ2. Note that the ghost existing at m2M2/ǫ2 < p2 < M2/γ is a not a problem because of its low energy [13, 40]. In what follows we discuss the evolution of field perturbations in the two tachyonic regimes. Before doing that it is instructive to study the high-momentum regime that sets the initial data for the tachyonic evolution. 1. p2 ≫ (ǫ2 −M2)/α−m2 We denote the overall amplitude of the dangerous mode as ϕ. Since the modes are deep inside the Hubble radius (k/a ≫ H) in the regime we discuss here, the field perturbation χ approximately satisfies χ+ k2χ ≃ 0 , (64) where χ = aϕ and η is conformal time defined by η = a−1dt. Taking the asymptotic Minkowski vacuum state, χ = e−ikη/ 2k, the squared amplitude of the field perturbation ϕ is given by [41] (2π)3 |ϕ|2 = . (65) Since the maximum momentum at which the tachyon appears is k/a ≃ ǫ/ α, one has the following estimate for the amplitude of the field perturbation at the beginning of the tachyonic instability, . (66) As usual, this amplitude characterizes the contribution of a logarithmic interval of momenta into 〈ϕ2(x)〉. 2. M2/γ < p2 < (ǫ2 −M2)/α−m2 This interval of momenta is dangerous, as the perturbations undergo the tachyonic amplification. Since the modes are still inside the Hubble radius, one can neglect the gravitational effects on the “frequency” ω when estimating the growth of field perturbations. By the time the tachyonic amplification ends up, the amplitude of field perturbations is estimated as ϕ ≃ ϕi exp |ω1|dt = ϕi exp . (67) Recall that p1 ≃ ǫ/ α and p3 = M/ γ. The largest value of |ω21 | is approximately given by (63). Substituting this value into Eq. (67) and recalling that the background changes slowly (H ≃ const), one finds that the amplitude of the field perturbation after exit from the tachyonic regime is of order ϕ ≃ ǫ α(γ + α) , (68) where we used Eq. (66). Recall now that H is of the same order as M/ 3α during the phantom phase. Hence the large ratio ǫ/M leads to a strong amplification of field perturbations. From (18), the homogeneous field φ at the phantom and de Sitter phase is estimated as φ ≃ . The requirement that the perturbation ϕ is smaller than the background field φ leads to the constraint α(γ + α) . (69) As an example, in the case m = H0 = 10 −42 GeV, M = 3αH0, α = 1 and γ = 1/2, we obtain the constraint ǫ/M <∼ 70. As long as α and γ are of order one, the ratio ǫ/M should not be too much larger than unity. 3. 0 < p2 < m2M2/ǫ2 After the Hubble radius crossing, the effect of the cosmic expansion can no longer be neglected when estimating the “frequencies” of the field perturbations. Since there are no tachyonic instabilities for m2M2/ǫ2 < p2 < M2/γ, we consider the evolution of perturbations in the region 0 < p2 < m2M2/ǫ2. In Ref. [33] the equations for the field perturbations were derived in the slow-rolling background under the condition p2 ≪ M2,m2. The equation for the perturbation χ = aϕ is approximately given by k2 − 1 − a2m χ = 0 . (70) Here we neglected the contribution of metric perturbations on the r.h.s. of this equation. Note that metric pertur- bations works as a back reaction effect after the field perturbation is sufficiently amplified. The growth rate of the perturbation χ is mainly determined by the terms in the parenthesis of Eq. (70) rather than the backreaction of metric perturbations. The last term corresponds to the tachyonic mass term, which already appeared in Minkowski spacetime [see Eq. (57)]. Since the term (d2a/dη2)/a is of order a2H2, one can estimate the ratio of the tachyonic mass relative to this gravitational term: δ ≡ a 2m2M2/ǫ2 (d2a/dη2)/a . (71) If we use the de Sitter value H = M/ 3α, this ratio is estimated as δ = 3αm2/ǫ2 ≪ 1. Hence the gravitational term (d2a/dη2)/a dominates over the tachyonic mass. In the de Sitter background with a = −1/(Hη) the approximate solutions to Eq. (70) can be obtained by setting χ = −(C/H)η−1+δ̃. One finds that δ̃ = −m2M2/(3H2ǫ2) for the growing solution, thereby giving ϕ = Cη 3H2ǫ2 ∝ aδ/3 . (72) In Ref. [33] it was shown that the physical temporal component of the vector field perturbations evolves as b0/a ∝ aδ/3, whereas the physical spatial component of the vector field decreases as Bi/a ∝ a−1+δ/3. The growth rate of ϕ and b0/a is small due to the condition δ ≪ 1. So, the second tachyonic instability is harmless for the past and present cosmological evolution. However, we notice that since the de Sitter solution is a late-time attractor, the perturbations ϕ and b0/a become larger than the homogeneous background fields in the distant future. At this stage we expect that the contribution of metric perturbations can be also important. V. CONCLUSIONS In this paper we have studied the dynamics of dark energy in a Lorentz-violating model with the action given in (1). The model involves a vector field Bµ and a scalar field Φ with mass terms M and m, respectively. The presence of the one-derivative term ǫ∂µΦB µ leads to an interesting dynamics at the IR scales larger than ǫ−1. The phantom equation of state can be realized without having ghosts, tachyons or superluminal modes in the UV region. We have taken into account the contributions of radiation and non-relativistic matter and studied the cosmological evolution of the system. Interestingly, there exists a de Sitter attractor solution that can be used for the late-time acceleration. The phantom regime is not an attractor, but we have found that in a range of parameters, the phantom stage occurs during the transition from the matter epoch to the final de Sitter attractor. As is seen, e.g., in Fig. 1 the equation of state parameter w of dark energy crosses the cosmological constant boundary towards the phantom region. We clarified the conditions under which the w = −1 crossing is realized together with the existence of the stable de Sitter solution. In the model studied in this paper, the effective Newton’s constant is time-dependent. We have found, however, that this dependence is typically mild, though for interesting values of parameters it is close to the experimental bounds. We have also considered the field perturbations in Minkowski spacetime and obtained the momentum scales of instabilities present in the IR region (p <∼ ǫ). We have found that either tachyons or ghosts appear for the spatial momenta p smaller than (ǫ2 −M2)/α−m2, while in the UV region there are no unhealthy modes. In the cosmo- logical context the presence of tachyons at the IR scales leads to the amplification of large-scale field perturbations whose wavelengths are roughly comparable to the present Hubble radius. There are two tachyonic regions of spatial momenta in this model: (a) one is sub-horizon and its momenta are characterized by M2/γ < p2 < (ǫ2−M2)/α−m2; (b) another is super-horizon and has 0 < p2 < m2M2/ǫ2. In the region (a) we derived the condition under which the perturbations always remain smaller than the homogenous fields, see Eq. (69). While the existence of the phantom phase requires that ǫ > M , the condition (69) shows that ǫ cannot be very much larger than M . Thus the allowed range of ǫ is constrained to be relatively narrow. In the tachyonic region (b) the growth of the perturbations is estimated as ϕ ∝ aαm2/ǫ2 . Since the growth rate is suppressed by the factor m2/ǫ2, this effect is negligible in the past and at present, though the inhomogeneities can start to dominate over the homogenous fields in the distant future. There are several issues yet to be understood. The presence of the tachyonic instability on sub-horizon scales may lead to the variation of the gravitational potential, which can be an additional source of the late-time integrated Saches-Wolfe effect on the CMB power spectrum. Another property of this model is the peculiar time-dependence of the effective Newton’s constant, which may result in interesting phenomenology. The model studied in this paper is likely to belong to a wider class of Lorentz-violating theories exhibiting the phantom behaviour (see Refs. [42] for a number of Lorentz-violating models). It would be interesting to understand how generic are the features we found in our particular model — late-time de Sitter attractor, transient phantom stage, time-dependent Newton’s constant, sub-horizon tachyons, super-horizon ghosts, etc. One more direction is to modify our model in such a way that it would be capable of describing inflationary epoch rather than the late-time acceleration. Since it is known that the spectra of scalar and tensor perturbations produced during the phantom inflationary phase are typically blue-tilted [43], this model may give rise to some distinct features in the CMB spectrum. ACKNOWLEDGEMENTS This work is supported by RFBR grant (M. L. and V.R., 05-02-17363-a), grant of the President of Russian Federation (M. L. and V.R., NS-7293.2006.2), INTAS grant (M. L., YSF 04-83-3015), grant of Dynasty Foundation awarded by the Scientific Board of ICFPM (M.L.), the European Union through the Marie Curie Research and Training Network UniverseNet (E. P., MRTN-CT-2006-035863) and JSPS (S. T., No. 30318802). Appendix A We are going to find the effective Newton’s constant that determines the strength of gravitational interactions at distances shorter than all scales present in our model, including ǫ−1, M−1, m−1 as well as the Hubble distance. To this end, we neglect the last two terms in the action (1), and also neglect the time dependence of the background fields φ and X . We also neglect the space-time curvature of the Universe, and therefore consider our model in Minkowski space-time. Let us impose the gauge h0i = 0, where hµν is the metric perturbation about the Minkowski background. Then the quadratic Lagrangian for perturbations of metric, vector and scalar fields is readily calculated, ḃi + X∂ih00 ∂ibj − Xḣij ḃ0 + Xḣ00 ∂ib0 + Xḣ00 ϕ̇2 − (∂iϕ)2 . (73) where X2 = B20 is the background value. Clearly the scalar field ϕ decouples in our approximation, so we will not consider it in what follows. By varying the quadratic action with respect to h00 and hij , one obtains (00)- and (ij)-components of the linearized energy-momentum tensor for perturbations (note that T µν = −2δS/δhµν). Specifying further to scalar perturbations with bi = ∂ibL and choosing conformal Newtonian gauge, h00 = 2Φ, hij = −2Ψδij, one obtains (we keep the standard notation for the Newtonian potential, even though the same notation was used for the original scalar field in the main text) T 00 = αX(X∆Φ−∆ḃL) + γX (b0 +XΦ) , T ij = αX∂i∂j ḃL − δijαX2Ψ̈ , (74) where ∆ = ∂i∂i and = ∂ 0 −∆. The field equations for b0 and bL in the absence of sources for these fields read (b0 +XΦ) = 0 , − bL +X(Φ̇− Ψ̇) = 0 . (75) Now, the longitudinal (proportional to ∂i∂j) part of the (ij)-component of the Einstein equations, in the absence of external anisotropic stresses, gives Φ + Ψ = 8πGαXḃL , (76) while the trace part and (00)-component are ∆(Φ + Ψ) = −4πGαX2Ψ̈− 4πGpext , (77) −∆Ψ = 4πG(αX2∆Φ− αX∆ ˙bL) + 4πGρext , (78) where ρext and pext are energy density and pressure of an external source. For time-independent, pressureless source it is consistent to take all perturbations independent of time and set bL = 0. Then one finds, as usual, Ψ = −Φ and obtains the following equation for the Newtonian potential, (1 − 4πGαX2)∆Φ = 4πGρext . (79) Thus, the effective Newton’s constant in the background field X is G∗ = G(1 − 4πGαX2)−1 . (80) This means that the effective Planck mass entering the Newton’s law is given by (6). Appendix B A. Autonomous equations Let us define the following dimensionless variables which are convenient for studying the dynamical system [6, 38]: , x2 = , x3 = , x4 = , x5 = , x6 = . (81) Then we obtain the following autonomous equations x′1 = −3x1 − x2x5 + 5 − x1 , (82) x′2 = −3x2 − x1x5 − 3 x4x5 − x3x5 − x2 , (83) x′3 = x2x5 − x3 , (84) x′4 = x1 , (85) x′5 = −x5 , (86) x′6 = −2x6 − x6 , (87) 1 + x21 + x 2 − x23 + x24(3α+ x25) + x26/3 + 2(ǫ/M)x2x4x5 + 4(α/ γ)x1x4 1 + 3αx24 where prime denotes the derivative with respect to N ≡ ln(a) . Equation (11) gives the constraint = 1− x21 − x22 − x23 + x24 3α+ x25 − x26 . (88) Note that the above equations are invariant under the simultaneous change of the signs of φ and X . Hence it is not restrictive to study the case of positive φ. Note also that we study the case of an expanding Universe with H > 0. B. Fixed points By setting x′i = 0 one formally finds the following six fixed points: • (A) de Sitter (i): (x1, x2, x3, x4, x5, x6) = 0, 0, ǫ 3ǫ2/m2−2α ,− 1√ 3(3ǫ2/m2−2α) 3α, 0 • (B) de Sitter (ii): (x1, x2, x3, x4, x5, x6) = (0, 0, const, 0, 0, 0) , • (C) matter: (x1, x2, x3, x4, x5, x6) = (0, 0, 0, 0, 0, 0) , • (D) radiation: (x1, x2, x3, x4, x5, x6) = (0, 0, 0, 0, 0, 1) , • (E1) kinetic point (i): (x1, x2, x3, x4, x5, x6) = (0, 1, 0, 0, 0, 0) , • (E2) kinetic point (ii): (x1, x2, x3, x4, x5, x6) = (0,−1, 0, 0, 0, 0) . The fixed point (A) is precisely the de Sitter solution (18) that we discussed in Sec. III A. We will comment on its stability shortly. The point (B) is also in some sense a de Sitter point. It exists even in the absence of the field X and satisfies the relation 3H2 = κ2W (φ). To reach the solution (B), the Hubble parameter needs to increase towards infinity (M/H → 0), and the field φ needs to diverge as well. The point (C) corresponds to matter-dominated era satisfying Ωm = 1 and weff = 0, whereas the point (D) describes radiation-dominated epoch with Ωr = 1 and weff = 1/3. The points (E1) and (E2) are kinetic solutions satisfying ΩDE = 1 and weff = 1. These solutions are used neither for dark energy nor for radiation/matter dominated epochs. A cosmologically viable trajectory starts from the radiation point (D), connects to the matter solution (C) and finally approaches the de Sitter point (A). [Note that the initial data (22) indeed correspond to x1, x2, x3, x4, x5 → 0 as t → 0.] To see that this sequence of events is indeed possible, let us study the stability of the fixed points against perturbations. Let us consider linear perturbations δxi. By perturbing Eqs. (82)-(86) we obtain δx′1 = −3− H − c1x1 δx1 + x5 − c2x1 δx2 − c3x1δx3 x25 − − c4x1 δx4 + x4x5 − c5x1 δx5 − c6x1δx6 , (89) δx′2 = − x5 + c1x2 δx1 − 3 + c2x2 + δx2 − x5 + c3x2 x5 + c4x2 δx4 − x1 + 3 x3 + c5x2 δx5 − c6x2δx6 , (90) δx′3 = −c1x3δx1 + x5 − c2x3 δx2 − c3x3δx3 − c4x3δx4 + x2 − c5x3 δx5 − c6x3δx6 , (91) δx′4 = δx1 , (92) δx′5 = −c1x5δx1 − c2x5δx2 − c3x5δx3 − c4x5δx4 − + c5x5 δx5 − c6x5δx6 , (93) δx′6 = −c1x6δx1 − c2x6δx2 − c3x6δx3 − c4x6δx4 − c5x6δx5 − + c6x6 δx6 , (94) where δ(H ′/H) = i=1 ciδxi with c1 = − 3x1 + 6(α/ 1 + 3αx24 x4 , c2 = − 3x2 + 3(ǫ/M)x4x5 1 + 3αx24 , c3 = 1 + 3αx24 c4 = − 3x4(3α+ x 5) + 3(ǫ/M)x2x5 + 6(α/ 1 + 3αx24 − 6αx4 1 + 3αx24 c5 = − 3x24x5 + 3(ǫ/M)x2x4 1 + 3αx24 , c6 = − 1 + 3αx24 . (95) The stability of fixed points can be analyzed by considering eigenvalues of the 6 × 6 matrix M for perturbations along the lines of Ref. [6, 38]. The stability of the de Sitter point (A) is important for having the late-time accelerated epoch. This depends upon the two ratios ǫ/m and M/m once the parameters α and γ are fixed. When α = 1 and γ = 1/2, for example, the parameter range of ǫ/m is determined by the ratio M/m. We find that the point (A) is a stable attractor if the following conditions hold: • (i) When M/m = 0.1, ǫ/m > 0.817, • (ii) When M/m = 1, ǫ/m > 1.35, • (iii) When M/m = 10, ǫ/m > 3.52, When ǫ ≫ m, the stability of the point (A) is ensured automatically unless the ratio M/m is too much larger than unity. In view of (20) the case ǫ ≫ m is of particular interest. For another de Sitter point (B), the eigenvalues are − 3,−3,−3 9− 12α , 0,−1/2 , (96) This means that this point is marginally stable. The zero eigenvalue comes from the perturbation equation for δx5. If H continues to increase toward the solution (B), this eigenvalue actually obtains a small negative value, as can be seen from Eq. (93). Thus in such a case the point (B) is stable. 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Rev. D 74, 063505 (2006); P. G. Ferreira, B. M. Gripaios, R. Saffari and T. G. Zlosnik, Phys. Rev. D 75, 044014 (2007). [43] M. Baldi, F. Finelli and S. Matarrese, Phys. Rev. D 72, 083504 (2005). http://arxiv.org/abs/hep-th/0307138 http://arxiv.org/abs/hep-th/0604153 Introduction Lorentz-violating model Dynamics of dark energy Final and initial stages Slow roll phantom regime Numerical solutions Momentum scales of instabilities Minkowski spectrum Evolution of perturbations in cosmological background p2 (2-M2)/-m2 M2/<p2<(2-M2)/-m2 0<p2<m2M2/2 Conclusions ACKNOWLEDGEMENTS Appendix A Appendix B Autonomous equations Fixed points References

Phantom fields with negative kinetic energy are often plagued by the vacuum quantum instability in the ultraviolet region. We present a Lorentz-violating dark energy model free from this problem and show that the crossing of the cosmological constant boundary w=-1 to the phantom equation of state is realized before reaching a de Sitter attractor. Another interesting feature is a peculiar time-dependence of the effective Newton's constant; the magnitude of this effect is naturally small but may be close to experimental limits. We also derive momentum scales of instabilities at which tachyons or ghosts appear in the infrared region around the present Hubble scale and clarify the conditions under which tachyonic instabilities do not spoil homogeneity of the present/future Universe.

Introduction Lorentz-violating model Dynamics of dark energy Final and initial stages Slow roll phantom regime Numerical solutions Momentum scales of instabilities Minkowski spectrum Evolution of perturbations in cosmological background p2 (2-M2)/-m2 M2/<p2<(2-M2)/-m2 0<p2<m2M2/2 Conclusions ACKNOWLEDGEMENTS Appendix A Appendix B Autonomous equations Fixed points References

Response of degree-correlated scale-free networks to stimuli Sheng-Jun Wang,1 An-Cai Wu,1 Zhi-Xi Wu,1 Xin-Jian Xu,2 and Ying-Hai Wang1∗ 1Institute of Theoretical Physics, Lanzhou University, Lanzhou Gansu 730000, China 2Departamento de Fı́sica da Universidade de Aveiro, 3810-193 Aveiro, Portugal (Dated: August 24, 2021) The response of degree-correlated scale-free attractor networks to stimuli is studied. We show that degree- correlated scale-free networks are robust to random stimuli as well as the uncorrelated scale-free networks, while assortative (disassortative) scale-free networks are more (less) sensitive to directed stimuli than uncorrelated networks. We find that the degree-correlation of scale-free networks makes the dynamics of attractor systems different from uncorrelated ones. The dynamics of correlated scale-free attractor networks result in the effects of degree correlation on the response to stimuli. PACS numbers: 89.75.Hc, 87.18.Sn, 05.50.+q, 05.40.-a Many complex systems have the ability to react to low lev- els of special stimuli, whereas, they can maintain their state when exposed to high levels of other irrelevant stimuli [1]. If we take the units of response as nodes and the interactions be- tween responding units as edges, the structure of some these systems can be described as complex networks. In neural net- works or social networks, for example, the nodes are individ- ual neurons or persons. It is an interesting problem that how one system have both the sensitivity to the right stimuli and robustness in the face of the wrong one. And the problem is also important for designing large artificial complex systems. The source of the ability of networked complex systems to in- corporate the two complementary attributes have been investi- gated using network models. It was shown that the power-law shape degree distributions of networks give rise to the sensi- tivity and robustness in a system [1]. The topology of real networks is also characterized by de- gree correlation [2, 3, 4]. In a network with degree correla- tion, there exist certain relationships between network nodes. The degree correlations are often named respectively as “as- sortative mixing”, i.e. a preference for high-degree nodes to attach to other high-degree nodes, while “ disassortative mixing” — high-degree nodes attach to low-degree ones [4]. It has been pointed out that the existence of degree correla- tions among nodes is an important property of real networks [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. The percolation [4] and disease spreading [15] on correlated networks have been stud- ied. And more effects of degree correlation on network struc- ture and function have attracted attention [16, 17, 18]. There- fore, the extension of previous results for uncorrelated net- work model about responding to stimuli is necessary. In this paper, we study the response of degree-correlated scale-free networks to stimuli following the work contributed by Bar-Yam and Epstein [1]. Numerical investigation reveals that the dynamical process of the evolution of attractor sys- tems on correlated scale-free networks is different from un- correlated networked systems. The special dynamics of cor- related attractor systems result in the different responding behavior from uncorrelated systems. The degree-correlated ∗Electronic address: yhwang@lzu.edu.cn scale-free network is robust in the face of wrong stimuli as uncorrelated networks. In assortative networks, the sensitiv- ity to right stimuli is enhanced. While in the disassortative networks the sensitivity to right stimuli is weaker than uncor- related networks. And, the relation between the sensitivity to stimuli and the degree of correlation is not monotonic. We consider the method for modelling the response of com- plex systems proposed in [1]. We use a model of attrac- tor networks [19, 20], where the node states si = ±1, i ∈ {1, · · · , N} are binary. The state of the system is the set of node states {si}. The dynamical equations of the attractor system are si(t+ 1) = sign( Jijsj(t)), (1) with symmetric influence matrix Jij . Using the Hebbian im- printing rule Jij = j , (2) we can set the states {sαi }α=1,··· ,n as the stable states of the network dynamics (attractor). cij is the entry of the symmet- ric adjacent matrix which is equal to 1 when node i connects to node j, and zero otherwise. An attractor is stable to pertur- bation and thus can represent a functional state of systems. In simulations, we randomly choose two functional states of the system {sαi }α=1,2, and the influence is Jij = α=1 cijs External stimuli are modelled by changing the signs of a spec- ified set of nodes. When the states of some nodes are flipped, the system either evolves back to its initial state or switches to other stable system states. The response of networked sys- tems is described as a process of switching between attractors. The size of the basin of attraction, the number of nodes whose states can be changed before the dynamics of the network fails to bring the system back to its original state, indicates the de- gree of stability of the system. We calculate the size of the basin of attraction in different cases of stimuli to reveal the sensitivity and robustness of the network model. Generally, degree-correlated networks can be generated from uncorrelated ones by means of reshuffling method pro- posed in [5]. Starting from a given network, at each step two http://arxiv.org/abs/0704.1849v1 edges of the network are chosen at random. The four nodes attached to the two edges are ordered with respect to their de- grees. Then with probability p, the edges are rewired in such a way that one edge connects the two nodes with the smaller degrees and the other connects the two nodes with the larger degrees; otherwise, the edges are randomly rewired. In the case when one or both of these new edges already existed in the network, the step is discarded and a pair of other edges is selected. A repeated application of the rewiring step leads to an assortative networks. For producing disassortative net- works, we modify the way for building new edges used in above reshuffling method as that the node of the largest degree connects to the nodes of the smallest degree and two other nodes are connected. It is worth noting that the algorithm does not change the degree distribution in the given network [5]. Before investigating the effect of the degree correlation on the response, we review the results on uncorrelated attrac- tor networks [1], where the system was characterized by the scale-free networks which have the power-law shape degree distribution P (k) ∼ k−γ . The size of the basin of attrac- tion for two kinds of stimuli, namely, the random stimuli (randomly chosen nodes are flipped) and the directed stim- uli (means flipping sequentially the nodes of greatest degree) were studied on scale-free attractor network systems. The re- lation between the size of the basin of attraction for random stimuli br and directed stimuli bm, which are all normalized by network size N , are derived: bm = b (γ−1)/(γ−2) r . (3) The derivation was based on a assumption that the response of attractor networks occurs if the sum of edges coming from stimulated nodes exceeds a threshold which is the same for both random and directed stimuli. For Barabási-Albert (BA) scale-free networks [21], the distribution exponent γ = 3 and thus bm = b r. So the scale-free networks are robust to random stimuli and sensitive to directed stimuli. Let us first calculate the average size of the basin of attrac- tion for random stimuli br and directed stimuli bm on degree- correlated BA networks. According to [1], we use the network size N = 1000 and average degree 〈k〉 = 20 in all simula- tions. Figure 1 shows the average size of attractor basin versus the degree of correlation which is quantified by the Pearson correlation coefficient r [4]. To compare with uncorrelated case, in Fig. 1 we also plot the predicted size of the attrac- tor basin for directed stimuli b′m which is calculated using the size of the attractor basin for random stimuli br following Eq. (3). Restricted by the reshuffling method, we can not gener- ate networks with strong degree correlation |r| → 1 [5]. In simulations, the region of the Pearson correlation coefficient r is about from −0.3 to 0.3. Although the region is small, it nearly covers all the values of the Pearson correlation coeffi- cient r of realistic complex networks shown in [4]. Therefore, we interest in systems with the Pearson correlation coefficient belonging to the region about from -0.3 to 0.3. In Fig. 1 we can see the effects of the degree correla- tion of scale-free networks on the size of the basion of at- traction. Comparing the size of attractor basin b m predicted using Eq.(3) (the curve with triangles) with the size obtained -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 FIG. 1: (Color online) The size of attractor basin of scale-free net- works as a function of Pearson correlation coefficient r in the case of directed (square) and random stimuli (circle). All networks have the same network size N = 1000 and average degree 〈k〉 = 20. Each curve is an average of 1000 realizations. The predicted curve of b′m calculated using Eq.(3) is shown as the curve with triangles. by computer simulations (the curve with squares), one can see that the relation between the size of attractor basin for random stimuli br and directed stimuli bm derived in uncorrelated case is not satisfied in correlated scale-free networks. When r ≈ 0 the numerical result of the attractor basin for directed stimuli bm is identical with the prediction of uncorrelated networks b′m.[22] For assortative case r > 0, the basin of attraction for directed stimuli is less than the value of uncorrelated net- work. This means that the assortative scale-free network is more sensitive to directed stimuli than uncorrelated scale-free networks. For disassortative case, the size of attractor basin undergoes a non-monotonic process with the variance of Pear- son correlation coefficient. The sensitivity of disassortative scale-free networks is weaker than uncorrelated systems. The size of the basin of attraction for random stimuli br decreases monotonically with the increase of r. And the slope is small. The robustness of scale-free networks to random stimuli re- tains when these networks are degree correlated. To understand the underlying mechanism of the effect of degree correlation on response, we analyze the dynamics of attractor networks. We assume that there are n functional states in an attractor system. Substitute of Eq. (2) into Eq. (1) gives si(t+ 1) = sign( j sj(t)) = sign( sαj sj(t)), (4) where Gi is the set of nodes adjacent to node i (the neighbors of node i). We use the functional state {s1i } as the original system state, and the stimulated system state is denoted as i }. Thus the first step of the evolution is like si(1) = sign(s sαj s j ). (5) The functional states {sαi }α=2,··· ,n are uncorrelated with the stimulated state {sβi }, since the functional states are chosen at random. Thus the second term in the bracket at the right side of Eq. (5) is approximately equal to 0, and this term can be taken as noise [19]. For an arbitrary node i, if much less than half nodes in Gi are flipped by the stimulus, then si(1) = s if much more than half nodes in Gi are flipped, si(1) = −s In general, the fraction of flipped nodes in Gi increases as stimuli are enhanced. Because of the influence of noise, when the fraction of flipped nodes in Gi is near but less than 0.5, the node i choose a state s1i or −s i at random. In the case of uncorrelated networks, for both random and directed stimuli, the fraction of flipped nodes in neighbors of each node is equal to the fraction f of edges coming from flipped nodes in a network. This property determines a criti- cal condition for uncorrelated systems responding to stimuli: near half edges in a network come from the stimulated nodes. We obtained the critical value of f on the system with two functional states by numerical simulation, which is fc = 0.46 for both random and directed stimuli. When stimuli are large enough to satisfy the critical condition, all nodes in uncorre- lated networks choose their states at random with the help of noise term. Then, the system state {si(1)} becomes a random state, and evolves to one of attractors randomly. The analysis of the above property gives an insight of the dynamics of un- correlated networks that the uncorrelated networks responds to both kinds of stimuli as a whole. Figure 2 shows numerical results of the critical fraction of edges attached to stimulated nodes versus the Pearson correla- tion coefficient of reshuffling scale-free networks. When net- works are degree-correlated, the difference between the crit- ical fraction fc for random stimuli and directed stimuli is re- markable. The result shows that the mentioned assumption used for deriving Eq.(3) in [1] is not appropriate for degree- correlated scale-free networks. In Fig. 2, one can note that the critical fraction fc for random stimuli varies slightly. Under random stimuli, for correlated scale-free networks, the frac- tion of flipped nodes in the neighbor of each node is approx- imately equal to the fraction f of edges coming from flipped nodes in a network. The dynamics of degree-correlated scale- free networks under random stimuli have the same character- istic as uncorrelated networks: the attractor systems respond to random stimuli as a whole. Under directed stimuli, the vari- ation of fc versus the Pearson correlation coefficient indicates that the dynamics of directed stimulated attractor networks are affected seriously by degree-correlation. Next we numerically investigate the dynamical process of the evolution of the attractor system in the case of directed stimuli, and reveal the underlying mechanism of the effect of degree correlation. To do this, we give a directed stimuli with size equal to 235 to a realization of the uncorrelated net- work. The stimulus is larger than the average attractor basin -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 f c FIG. 2: (Color online) The critical value of the number of edges attached to flipped nodes as a function of Pearson correlation coef- ficient in the case of directed (square) and random (circle) stimuli. Each curve is an average of 1000 realizations. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 FIG. 3: The number of flipped nodes in the process of the evolution of the uncorrelated system. Inset: the number of nodes whose state si(t) is the same as s for uncorrelated scale-free attractor systems given in Fig. 1 which is equal to 215(±12). In Fig. 3 the dynamical process of the evolution of the system is represented by the number of flipped nodes (Nf ). At the first step of the evolution, the num- ber of the flipped nodes is 488, which is near half of the net- work size. And then the system evolves to another imprinted functional state, as shown in the inset of Fig. 3. The evolu- tion shows that the uncorrelated scale-free networks response to directed stimuli as a whole, as the above analysis. For assortative networks, we give a directed stimulus with the size 170 to attractor systems. Although the size of the 0 10 20 30 40 50 60 0 10 20 30 40 50 FIG. 4: (Color online) The number of flipped nodes in the process of the evolution of two assortative systems with r = 0.13 (square) and r = 0.15 (circle). Inset: the number of nodes whose state si(t) is the same as s2i . stimuli is smaller than the mentioned average attractor basin of uncorrelated networks, the system responds to the stim- ulus with the process of the change of the system state, as shown in Fig. 4. We note that the number of flipped nodes increases gradually. In contrast with uncorrelated scale-free networks, the evolution shows that the assortative scale-free network system does not make response as a whole. In as- sortative scale-free networks, a group of nodes of large de- gree preferentially connect to the nodes of greatest degrees, i.e. stimulated nodes, and thus they are easier to get the con- dition for changing their states. So the set of flipped nodes can be extended by the assortative mixing. The assortative scale- free network system evolves as a hierarchical cascade [23] that progresses from higher to lower degree classes. Therefore the basin of attraction of assortative network system decreases and the system is more sensitive to directed stimuli. With the increase of Pearson correlation coefficient, the cluster coefficient of assortative networks are increased by the degree based reshuffling steps [5]. The cluster property also effects the dynamics of assortative scale-free networks. In Fig. 4 we show two numerical simulations with different types of dynamics. For one kind of dynamics (square), the stable sys- tem states are the functional states imprinted by Hebbian rule, as the uncorrelated networks. The upper curve (square) in the inset of Fig. 4 shows that a system evolves into the second functional state. For another kind of dynamics (circle), the stable system state at the end of evolution is not the imprinted functional state. The lower curve (circle) in the inset of Fig. 4 shows the discrepancy. In this kind of systems, cluster forms between stimulated nodes which have a high density of edges within them, with a lower density of edges between other groups of nodes. So these stimulated nodes hold their states on −s1i . Additionally, the state of some low-degree nodes which connect tightly to the cluster is also held. These nodes held by the cluster structure result in the difference between the sys- 0 5 10 15 20 0 5 10 15 20 FIG. 5: The number of flipped nodes in the process of disassortative system, r = −0.16. Inset: system with r = −0.30. The size of stimuli is 245. tem state and the imprinted functional state. There is a critical value rc, for the networks used in simulations rc = 0.32, be- low which two types of dynamics are possible (and larger the value of r is, more frequently the second type of dynamics oc- cur), while above which systems only responde to stimuli by the second type of dynamics. Because of the cluster property of assortative networks, too larger assortative mixing is not expected for response of networks. In the limit of r → 1, net- works disintegrate into isolated clusters, each of them consists of nodes with certain degree k. Directed stimuli cannot in- duce these systems to change their functional states, but only change few clusters and leave the other nodes on their initial states. For the disassortative system, we choose a reshuffling scale- free network realization with Pearson correlation coefficient r = −0.16 which has the lowest sensitivity to directed stim- uli as shown in Fig. 1. We give the disassortative network a directed stimulus with size 245 which is larger than the av- erage attractor basin of the uncorrelated scale-free networks. Fig. 5 shows the dynamical process of the evolution of the system. Although more than half of nodes flip their states at the first step, the system state is attracted into the origi- nal functional state. In disassortative networks, nodes with large degrees preferentially connect to the nodes with small ones. Under directed stimuli, the fraction of stimulated nodes in the neighbors of the nodes in middle degree class is less than the fraction of the edge coming from stimulated nodes. Thus, more nodes need to be stimulated than uncorrelated sys- tems for inducing the system into random state, and the basin of attraction of disassortative system extends. For larger disassortative mixing systems, the second im- printed functional state cannot be reached. The inset of Fig. 5 shows the dynamical process of evolution of a network real- ization with r = −0.30. The system is induced into stable os- cillation state, which is established by the interaction between large and small nodes. The system with large disassortative mixing property is easier to responde the directed stimuli by evolving into stable oscillation states. This structural prop- erty leads to the non-monotonic behavior of sensitivity versus Pearson correlation coefficient shown in Fig. 1. Additionally, it is notable that the too large disassortative degree correlation also destroys the ability of systems to responde directed stim- uli with imprinted functional states, as the too large assortative degree correlation. In summary, we have studied the effect of the degree corre- lation on the response of scale-free networks to stimuli. Cor- related scale-free networks retain the robustness to random stimuli. In the region of Pearson correlation coefficient in which we interest, assortative scale-free networks are more sensitive to directed stimuli than uncorrelated ones; and the sensitivity of scale-free networks are weaken when networks are disassortative. We found that the effects of degree corre- lation result from the properties of the dynamics of degree- correlated network systems. Uncorrelated networks responde to stimuli as a whole. While the degree correlation of a net- work destroys the identical critical condition of all nodes for the response to directed stimuli. Assortative scale-free net- works reduce the need on the size of directed stimuli to be responded via a cascade that progresses from higher to lower degree classes. The disassortative correlation extends the size of the basin of attraction by the nodes in middle degree class which has less stimulated neighbors and stay on initial state. But the response of too large assortative and disassortative scale-free networks is destroyed by the structure property, and imprinted functional states cannot be reached. Since many realistic complex networks have both scale-free and degree- correlated properties, the intuitive description of the dynamics might contribute to understanding of the attributes of realistic networks. This work was supported by the Fundamental Research Fund for Physics and Mathematics of Lanzhou Univer- sity under Grant No. Lzu05008. X.-J. 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The response of degree-correlated scale-free attractor networks to stimuli is studied. We show that degree-correlated scale-free networks are robust to random stimuli as well as the uncorrelated scale-free networks, while assortative (disassortative) scale-free networks are more (less) sensitive to directed stimuli than uncorrelated networks. We find that the degree-correlation of scale-free networks makes the dynamics of attractor systems different from uncorrelated ones. The dynamics of correlated scale-free attractor networks result in the effects of degree correlation on the response to stimuli.

Response of degree-correlated scale-free networks to stimuli Sheng-Jun Wang,1 An-Cai Wu,1 Zhi-Xi Wu,1 Xin-Jian Xu,2 and Ying-Hai Wang1∗ 1Institute of Theoretical Physics, Lanzhou University, Lanzhou Gansu 730000, China 2Departamento de Fı́sica da Universidade de Aveiro, 3810-193 Aveiro, Portugal (Dated: August 24, 2021) The response of degree-correlated scale-free attractor networks to stimuli is studied. We show that degree- correlated scale-free networks are robust to random stimuli as well as the uncorrelated scale-free networks, while assortative (disassortative) scale-free networks are more (less) sensitive to directed stimuli than uncorrelated networks. We find that the degree-correlation of scale-free networks makes the dynamics of attractor systems different from uncorrelated ones. The dynamics of correlated scale-free attractor networks result in the effects of degree correlation on the response to stimuli. PACS numbers: 89.75.Hc, 87.18.Sn, 05.50.+q, 05.40.-a Many complex systems have the ability to react to low lev- els of special stimuli, whereas, they can maintain their state when exposed to high levels of other irrelevant stimuli [1]. If we take the units of response as nodes and the interactions be- tween responding units as edges, the structure of some these systems can be described as complex networks. In neural net- works or social networks, for example, the nodes are individ- ual neurons or persons. It is an interesting problem that how one system have both the sensitivity to the right stimuli and robustness in the face of the wrong one. And the problem is also important for designing large artificial complex systems. The source of the ability of networked complex systems to in- corporate the two complementary attributes have been investi- gated using network models. It was shown that the power-law shape degree distributions of networks give rise to the sensi- tivity and robustness in a system [1]. The topology of real networks is also characterized by de- gree correlation [2, 3, 4]. In a network with degree correla- tion, there exist certain relationships between network nodes. The degree correlations are often named respectively as “as- sortative mixing”, i.e. a preference for high-degree nodes to attach to other high-degree nodes, while “ disassortative mixing” — high-degree nodes attach to low-degree ones [4]. It has been pointed out that the existence of degree correla- tions among nodes is an important property of real networks [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. The percolation [4] and disease spreading [15] on correlated networks have been stud- ied. And more effects of degree correlation on network struc- ture and function have attracted attention [16, 17, 18]. There- fore, the extension of previous results for uncorrelated net- work model about responding to stimuli is necessary. In this paper, we study the response of degree-correlated scale-free networks to stimuli following the work contributed by Bar-Yam and Epstein [1]. Numerical investigation reveals that the dynamical process of the evolution of attractor sys- tems on correlated scale-free networks is different from un- correlated networked systems. The special dynamics of cor- related attractor systems result in the different responding behavior from uncorrelated systems. The degree-correlated ∗Electronic address: yhwang@lzu.edu.cn scale-free network is robust in the face of wrong stimuli as uncorrelated networks. In assortative networks, the sensitiv- ity to right stimuli is enhanced. While in the disassortative networks the sensitivity to right stimuli is weaker than uncor- related networks. And, the relation between the sensitivity to stimuli and the degree of correlation is not monotonic. We consider the method for modelling the response of com- plex systems proposed in [1]. We use a model of attrac- tor networks [19, 20], where the node states si = ±1, i ∈ {1, · · · , N} are binary. The state of the system is the set of node states {si}. The dynamical equations of the attractor system are si(t+ 1) = sign( Jijsj(t)), (1) with symmetric influence matrix Jij . Using the Hebbian im- printing rule Jij = j , (2) we can set the states {sαi }α=1,··· ,n as the stable states of the network dynamics (attractor). cij is the entry of the symmet- ric adjacent matrix which is equal to 1 when node i connects to node j, and zero otherwise. An attractor is stable to pertur- bation and thus can represent a functional state of systems. In simulations, we randomly choose two functional states of the system {sαi }α=1,2, and the influence is Jij = α=1 cijs External stimuli are modelled by changing the signs of a spec- ified set of nodes. When the states of some nodes are flipped, the system either evolves back to its initial state or switches to other stable system states. The response of networked sys- tems is described as a process of switching between attractors. The size of the basin of attraction, the number of nodes whose states can be changed before the dynamics of the network fails to bring the system back to its original state, indicates the de- gree of stability of the system. We calculate the size of the basin of attraction in different cases of stimuli to reveal the sensitivity and robustness of the network model. Generally, degree-correlated networks can be generated from uncorrelated ones by means of reshuffling method pro- posed in [5]. Starting from a given network, at each step two http://arxiv.org/abs/0704.1849v1 edges of the network are chosen at random. The four nodes attached to the two edges are ordered with respect to their de- grees. Then with probability p, the edges are rewired in such a way that one edge connects the two nodes with the smaller degrees and the other connects the two nodes with the larger degrees; otherwise, the edges are randomly rewired. In the case when one or both of these new edges already existed in the network, the step is discarded and a pair of other edges is selected. A repeated application of the rewiring step leads to an assortative networks. For producing disassortative net- works, we modify the way for building new edges used in above reshuffling method as that the node of the largest degree connects to the nodes of the smallest degree and two other nodes are connected. It is worth noting that the algorithm does not change the degree distribution in the given network [5]. Before investigating the effect of the degree correlation on the response, we review the results on uncorrelated attrac- tor networks [1], where the system was characterized by the scale-free networks which have the power-law shape degree distribution P (k) ∼ k−γ . The size of the basin of attrac- tion for two kinds of stimuli, namely, the random stimuli (randomly chosen nodes are flipped) and the directed stim- uli (means flipping sequentially the nodes of greatest degree) were studied on scale-free attractor network systems. The re- lation between the size of the basin of attraction for random stimuli br and directed stimuli bm, which are all normalized by network size N , are derived: bm = b (γ−1)/(γ−2) r . (3) The derivation was based on a assumption that the response of attractor networks occurs if the sum of edges coming from stimulated nodes exceeds a threshold which is the same for both random and directed stimuli. For Barabási-Albert (BA) scale-free networks [21], the distribution exponent γ = 3 and thus bm = b r. So the scale-free networks are robust to random stimuli and sensitive to directed stimuli. Let us first calculate the average size of the basin of attrac- tion for random stimuli br and directed stimuli bm on degree- correlated BA networks. According to [1], we use the network size N = 1000 and average degree 〈k〉 = 20 in all simula- tions. Figure 1 shows the average size of attractor basin versus the degree of correlation which is quantified by the Pearson correlation coefficient r [4]. To compare with uncorrelated case, in Fig. 1 we also plot the predicted size of the attrac- tor basin for directed stimuli b′m which is calculated using the size of the attractor basin for random stimuli br following Eq. (3). Restricted by the reshuffling method, we can not gener- ate networks with strong degree correlation |r| → 1 [5]. In simulations, the region of the Pearson correlation coefficient r is about from −0.3 to 0.3. Although the region is small, it nearly covers all the values of the Pearson correlation coeffi- cient r of realistic complex networks shown in [4]. Therefore, we interest in systems with the Pearson correlation coefficient belonging to the region about from -0.3 to 0.3. In Fig. 1 we can see the effects of the degree correla- tion of scale-free networks on the size of the basion of at- traction. Comparing the size of attractor basin b m predicted using Eq.(3) (the curve with triangles) with the size obtained -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 FIG. 1: (Color online) The size of attractor basin of scale-free net- works as a function of Pearson correlation coefficient r in the case of directed (square) and random stimuli (circle). All networks have the same network size N = 1000 and average degree 〈k〉 = 20. Each curve is an average of 1000 realizations. The predicted curve of b′m calculated using Eq.(3) is shown as the curve with triangles. by computer simulations (the curve with squares), one can see that the relation between the size of attractor basin for random stimuli br and directed stimuli bm derived in uncorrelated case is not satisfied in correlated scale-free networks. When r ≈ 0 the numerical result of the attractor basin for directed stimuli bm is identical with the prediction of uncorrelated networks b′m.[22] For assortative case r > 0, the basin of attraction for directed stimuli is less than the value of uncorrelated net- work. This means that the assortative scale-free network is more sensitive to directed stimuli than uncorrelated scale-free networks. For disassortative case, the size of attractor basin undergoes a non-monotonic process with the variance of Pear- son correlation coefficient. The sensitivity of disassortative scale-free networks is weaker than uncorrelated systems. The size of the basin of attraction for random stimuli br decreases monotonically with the increase of r. And the slope is small. The robustness of scale-free networks to random stimuli re- tains when these networks are degree correlated. To understand the underlying mechanism of the effect of degree correlation on response, we analyze the dynamics of attractor networks. We assume that there are n functional states in an attractor system. Substitute of Eq. (2) into Eq. (1) gives si(t+ 1) = sign( j sj(t)) = sign( sαj sj(t)), (4) where Gi is the set of nodes adjacent to node i (the neighbors of node i). We use the functional state {s1i } as the original system state, and the stimulated system state is denoted as i }. Thus the first step of the evolution is like si(1) = sign(s sαj s j ). (5) The functional states {sαi }α=2,··· ,n are uncorrelated with the stimulated state {sβi }, since the functional states are chosen at random. Thus the second term in the bracket at the right side of Eq. (5) is approximately equal to 0, and this term can be taken as noise [19]. For an arbitrary node i, if much less than half nodes in Gi are flipped by the stimulus, then si(1) = s if much more than half nodes in Gi are flipped, si(1) = −s In general, the fraction of flipped nodes in Gi increases as stimuli are enhanced. Because of the influence of noise, when the fraction of flipped nodes in Gi is near but less than 0.5, the node i choose a state s1i or −s i at random. In the case of uncorrelated networks, for both random and directed stimuli, the fraction of flipped nodes in neighbors of each node is equal to the fraction f of edges coming from flipped nodes in a network. This property determines a criti- cal condition for uncorrelated systems responding to stimuli: near half edges in a network come from the stimulated nodes. We obtained the critical value of f on the system with two functional states by numerical simulation, which is fc = 0.46 for both random and directed stimuli. When stimuli are large enough to satisfy the critical condition, all nodes in uncorre- lated networks choose their states at random with the help of noise term. Then, the system state {si(1)} becomes a random state, and evolves to one of attractors randomly. The analysis of the above property gives an insight of the dynamics of un- correlated networks that the uncorrelated networks responds to both kinds of stimuli as a whole. Figure 2 shows numerical results of the critical fraction of edges attached to stimulated nodes versus the Pearson correla- tion coefficient of reshuffling scale-free networks. When net- works are degree-correlated, the difference between the crit- ical fraction fc for random stimuli and directed stimuli is re- markable. The result shows that the mentioned assumption used for deriving Eq.(3) in [1] is not appropriate for degree- correlated scale-free networks. In Fig. 2, one can note that the critical fraction fc for random stimuli varies slightly. Under random stimuli, for correlated scale-free networks, the frac- tion of flipped nodes in the neighbor of each node is approx- imately equal to the fraction f of edges coming from flipped nodes in a network. The dynamics of degree-correlated scale- free networks under random stimuli have the same character- istic as uncorrelated networks: the attractor systems respond to random stimuli as a whole. Under directed stimuli, the vari- ation of fc versus the Pearson correlation coefficient indicates that the dynamics of directed stimulated attractor networks are affected seriously by degree-correlation. Next we numerically investigate the dynamical process of the evolution of the attractor system in the case of directed stimuli, and reveal the underlying mechanism of the effect of degree correlation. To do this, we give a directed stimuli with size equal to 235 to a realization of the uncorrelated net- work. The stimulus is larger than the average attractor basin -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 f c FIG. 2: (Color online) The critical value of the number of edges attached to flipped nodes as a function of Pearson correlation coef- ficient in the case of directed (square) and random (circle) stimuli. Each curve is an average of 1000 realizations. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 FIG. 3: The number of flipped nodes in the process of the evolution of the uncorrelated system. Inset: the number of nodes whose state si(t) is the same as s for uncorrelated scale-free attractor systems given in Fig. 1 which is equal to 215(±12). In Fig. 3 the dynamical process of the evolution of the system is represented by the number of flipped nodes (Nf ). At the first step of the evolution, the num- ber of the flipped nodes is 488, which is near half of the net- work size. And then the system evolves to another imprinted functional state, as shown in the inset of Fig. 3. The evolu- tion shows that the uncorrelated scale-free networks response to directed stimuli as a whole, as the above analysis. For assortative networks, we give a directed stimulus with the size 170 to attractor systems. Although the size of the 0 10 20 30 40 50 60 0 10 20 30 40 50 FIG. 4: (Color online) The number of flipped nodes in the process of the evolution of two assortative systems with r = 0.13 (square) and r = 0.15 (circle). Inset: the number of nodes whose state si(t) is the same as s2i . stimuli is smaller than the mentioned average attractor basin of uncorrelated networks, the system responds to the stim- ulus with the process of the change of the system state, as shown in Fig. 4. We note that the number of flipped nodes increases gradually. In contrast with uncorrelated scale-free networks, the evolution shows that the assortative scale-free network system does not make response as a whole. In as- sortative scale-free networks, a group of nodes of large de- gree preferentially connect to the nodes of greatest degrees, i.e. stimulated nodes, and thus they are easier to get the con- dition for changing their states. So the set of flipped nodes can be extended by the assortative mixing. The assortative scale- free network system evolves as a hierarchical cascade [23] that progresses from higher to lower degree classes. Therefore the basin of attraction of assortative network system decreases and the system is more sensitive to directed stimuli. With the increase of Pearson correlation coefficient, the cluster coefficient of assortative networks are increased by the degree based reshuffling steps [5]. The cluster property also effects the dynamics of assortative scale-free networks. In Fig. 4 we show two numerical simulations with different types of dynamics. For one kind of dynamics (square), the stable sys- tem states are the functional states imprinted by Hebbian rule, as the uncorrelated networks. The upper curve (square) in the inset of Fig. 4 shows that a system evolves into the second functional state. For another kind of dynamics (circle), the stable system state at the end of evolution is not the imprinted functional state. The lower curve (circle) in the inset of Fig. 4 shows the discrepancy. In this kind of systems, cluster forms between stimulated nodes which have a high density of edges within them, with a lower density of edges between other groups of nodes. So these stimulated nodes hold their states on −s1i . Additionally, the state of some low-degree nodes which connect tightly to the cluster is also held. These nodes held by the cluster structure result in the difference between the sys- 0 5 10 15 20 0 5 10 15 20 FIG. 5: The number of flipped nodes in the process of disassortative system, r = −0.16. Inset: system with r = −0.30. The size of stimuli is 245. tem state and the imprinted functional state. There is a critical value rc, for the networks used in simulations rc = 0.32, be- low which two types of dynamics are possible (and larger the value of r is, more frequently the second type of dynamics oc- cur), while above which systems only responde to stimuli by the second type of dynamics. Because of the cluster property of assortative networks, too larger assortative mixing is not expected for response of networks. In the limit of r → 1, net- works disintegrate into isolated clusters, each of them consists of nodes with certain degree k. Directed stimuli cannot in- duce these systems to change their functional states, but only change few clusters and leave the other nodes on their initial states. For the disassortative system, we choose a reshuffling scale- free network realization with Pearson correlation coefficient r = −0.16 which has the lowest sensitivity to directed stim- uli as shown in Fig. 1. We give the disassortative network a directed stimulus with size 245 which is larger than the av- erage attractor basin of the uncorrelated scale-free networks. Fig. 5 shows the dynamical process of the evolution of the system. Although more than half of nodes flip their states at the first step, the system state is attracted into the origi- nal functional state. In disassortative networks, nodes with large degrees preferentially connect to the nodes with small ones. Under directed stimuli, the fraction of stimulated nodes in the neighbors of the nodes in middle degree class is less than the fraction of the edge coming from stimulated nodes. Thus, more nodes need to be stimulated than uncorrelated sys- tems for inducing the system into random state, and the basin of attraction of disassortative system extends. For larger disassortative mixing systems, the second im- printed functional state cannot be reached. The inset of Fig. 5 shows the dynamical process of evolution of a network real- ization with r = −0.30. The system is induced into stable os- cillation state, which is established by the interaction between large and small nodes. The system with large disassortative mixing property is easier to responde the directed stimuli by evolving into stable oscillation states. This structural prop- erty leads to the non-monotonic behavior of sensitivity versus Pearson correlation coefficient shown in Fig. 1. Additionally, it is notable that the too large disassortative degree correlation also destroys the ability of systems to responde directed stim- uli with imprinted functional states, as the too large assortative degree correlation. In summary, we have studied the effect of the degree corre- lation on the response of scale-free networks to stimuli. Cor- related scale-free networks retain the robustness to random stimuli. In the region of Pearson correlation coefficient in which we interest, assortative scale-free networks are more sensitive to directed stimuli than uncorrelated ones; and the sensitivity of scale-free networks are weaken when networks are disassortative. We found that the effects of degree corre- lation result from the properties of the dynamics of degree- correlated network systems. Uncorrelated networks responde to stimuli as a whole. While the degree correlation of a net- work destroys the identical critical condition of all nodes for the response to directed stimuli. Assortative scale-free net- works reduce the need on the size of directed stimuli to be responded via a cascade that progresses from higher to lower degree classes. The disassortative correlation extends the size of the basin of attraction by the nodes in middle degree class which has less stimulated neighbors and stay on initial state. But the response of too large assortative and disassortative scale-free networks is destroyed by the structure property, and imprinted functional states cannot be reached. Since many realistic complex networks have both scale-free and degree- correlated properties, the intuitive description of the dynamics might contribute to understanding of the attributes of realistic networks. This work was supported by the Fundamental Research Fund for Physics and Mathematics of Lanzhou Univer- sity under Grant No. Lzu05008. X.-J. 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Shock and Release Temperatures in Molybdenum Damian C. Swift∗ and Achim Seifter Group P-24, Physics Division, Los Alamos National Laboratory, MS E526, Los Alamos, New Mexico 87545, USA David B. Holtkamp and David A. Clark Group P-22, Physics Division, Los Alamos National Laboratory, MS D410, Los Alamos, New Mexico 87545, USA (Dated: March 12, 2007 – LA-UR-07-1660) Shock and release temperatures in Mo were calculated, taking account of heating from plastic flow predicted using the Steinberg-Guinan model. Plastic flow was calculated self-consistently with the shock jump conditions: this is necessary for a rigorous estimate of the locus of shock states accessible. The temperatures obtained were significantly higher than predicted assuming ideal hydrodynamic loading. The temperatures were compared with surface emission spectrometry measurements for Mo shocked to around 60GPa and then released into vacuum or into a LiF window. Shock loading was induced by the impact of a planar projectile, accelerated by high explosive or in a gas gun. Surface velocimetry showed an elastic wave at the start of release from the shocked state; the amplitude of the elastic wave matched the prediction to around 10%, indicating that the predicted flow stress in the shocked state was reasonable. The measured temperatures were consistent with the simulations, indicating that the fraction of plastic work converted to heat was in the range 70-100% for these loading conditions. PACS numbers: 62.50.+p, 62.20.Fe, 65.40.-b, 07.20.Ka Keywords: shock physics, shock temperature, plasticity, molybdenum I. INTRODUCTION The behavior of matter subjected to extreme condi- tions through dynamic loading is of interest from a direct physical standpoint, as dynamic loading is often the only practical way to induce extreme conditions, and because important engineering problems in hypervelocity impact and weaponry involve dynamic loading [1]. Temperature is a key parameter in understanding the properties of matter, as it governs the population of vibrational pro- cesses and excitations past energy barriers. Temperature is thus important for a physical understanding of many types of behavior and the associated models. The equa- tion of state (EOS) includes contributions from the ex- citation of atomic vibrations and electronic excitations. Plastic flow is mediated by the excitation of dislocations and twin boundaries past Peierls barriers. Phase changes depend on the thermodynamic state’s location in the phase diagram. The kinetics of phase changes are de- scribed by the nucleation and growth of the daughter phase in the matrix of the parent phase, requiring the excitation of atoms past barriers. Chemical reactions are governed by the excitation of atoms or electrons over barriers. Diffusion in condensed matter is the motion of atoms past the barriers formed by their neighbors. Con- ductivities include scattering contributions from thermal excitations. Temperature is notoriously difficult to measure dur- ∗Electronic address: dswift@lanl.gov ing dynamic loading, in particular for opaque materials [2]. Extreme states of matter are often hidden within a sheath of matter in a different state – this is the usual situation in shock loading experiments. Probes made of matter generally disrupt the state of interest, e.g. by pre- senting an impedance mismatch to compression waves. Much interesting physics in condensed matter occurs at compressions of a few tens of percent, where the heating may be in the range of a few hundred kelvin and the re- sulting thermal emissions are small. Most temperature measurements of shock-loaded systems have been made using photon emission spectroscopy, commonly called py- rometry [3, 4]. However, many materials of interest (e.g. metals) are opaque in the relevant region of the spectrum: infra-red through visible for shocks up to the terapascal regime. Emission from an opaque material comes from the surface, which cannot be maintained at the pressure of the initial shock for long enough to allow useful emis- sion spectra to be collected. A transparent window can be placed in contact with the sample to maintain an el- evated pressure, but the mismatch in shock impedance must be taken into account, along with the effect of heat conduction. Accurate temperature data have been ob- tained from transparent sample materials, where ther- mal radiation from the shocked state can escape from the sample [5, 6]. Neutron resonance spectroscopy (NRS) has been inves- tigated as a fundamentally different technique for mea- suring the temperature inside a dynamically-loaded spec- imen, which can be used on opaque materials [7]. Trial measurements of NRS temperatures were performed on shock-loadedMo, as a standard material for high pressure work; the shock temperature was found to lie significantly http://arxiv.org/abs/0704.1850v1 mailto:dswift@lanl.gov above the temperature predicted using reasonable EOS for Mo [7]. Measurements were also made using pyrome- try of the temperature of Mo which was shocked and then released into a LiF window or into vacuum; these exper- iments also yielded temperatures which lay significantly above EOS predictions [8]. Here we consider the effect of plastic flow on the Mo pyrometry experiments. Plastic flow was neglected in previous comparisons of predictions with the tempera- ture data. The contribution of plastic work to heating has been mentioned in studies of other metals [9] but has not been quantified in detail or consistently. The contri- bution to the total internal energy from plastic heating has been estimated in order to extract the scalar EOS from shock data [10], which involves a similar analysis of shock heating, though less general. II. CORRECTIONS AND SYSTEMATIC UNCERTAINTIES IN PYROMETRY Polycrystalline materials, such as the Mo for which the discrepancy in temperature measurements was re- ported, are heterogeneous in that they are composed of an aggregate of grains of different crystallographic orien- tation. The Mo samples were machined from material which had been prepared by pressing from powder, so there was a population of voids and there were impuri- ties concentrated along grain boundaries. On shock load- ing and subsequent release, different regions of the sam- ple would thus respond differently, producing a variation in local temperature. Given enough time, temperature variations disperse through thermal conduction, but this typically takes of order microseconds for grains tens of micrometers in size, which is long on the time scale of the experiments. We wish to compare pyrometry mea- surements of temperature in Mo with predictions using continuum models, so the temperature of interest is the mean, bulk value. Thermal radiation is described by the Stefan-Boltzmann relation, where the total power is pro- portional to the fourth power of temperature. Pyrometry measurements are often more accurate at shorter wave- lengths where the power varies with higher powers of tem- perature [11]. Thus unquantified temperature variations (spatial or temporal, within the respective resolutions of the detectors) lead to an overestimate of the mean tem- perature. Spatial variation of brightness temperature has been observed in shock-loaded Sn [12], which has a much lower flow stress than Mo. Pyrometry measurements from metals probe the sur- face temperature. The surface is prone to increased plastic work around surface features such as machining marks: when a rough surface in contact with material of lower impedance (or vacuum, in the extreme case) is shocked, flow may be exaggerated in pits and grooves. At sufficiently high shock pressures, localized jetting may occur. Any such localized increase in plastic flow will produce a higher local temperature, which may appear as a higher mean temperature as discussed above. For experiments in which the sample is observed through a transparent window to maintain an elevated pressure, there may be additional radiation from com- pression of any gas or glue in the gap between the sample and the window, or from the shocked window material it- self. These effects were considered and corrected for the Mo data [13]. After the shock passes from the sample into a window, the temperature of the shock state is gen- erally different to that in the sample, so heat conduction can alter the temperature of the sample surface. In multiple channel pyrometry measurements, thermal emission from the sample is recorded using a set of de- tectors responding to different ranges of photon wave- length. In the simplest case, a grey body spectrum can be fitted to the signals, and the mean emissivity and temperature deduced from the shape of the spectrum. In general, the emissivity of a material varies with state and wavelength, and also with surface roughness, which may change during dynamic loading. The emissivity may be measured directly, for example by ellipsometry. It is more common to assume that the multiple pyrometry channels over-sample the wavelength variation of the ra- diance and emissivity, at least over part of the spectrum, allowing both to be deduced. Again, these effects were taken into account for the Mo measurements [13]. Aside from gaps and glues between the sample and any window, it is common for components of the shocked target assembly to include sharp corners and low-density materials such as plastics, foams, and glues, as part of the engineering structure. As with glued windows, low density materials in general may be shock-loaded to a higher temperature than the sample. Sharp internal cor- ners, when shocked, may form jets with large amounts of plastic heating. Thermal emission from any of these sources may be present as a background against which the emission from the sample must be distinguished. The net effect of the complications associated with py- rometry measurements on metals is that it is possible to over-estimate the temperature. III. HEATING FROM PLASTIC WORK IN SHOCK AND RELEASE Shock compression involves the transit of a shock wave through each element of material. Subsequently, axial and lateral release and recompression waves reverberate through the sample until it ultimately comes to rest at zero pressure. In the simplest case, which many shock physics experiments are designed to achieve, the shock is steady and planar and the initial release is planar: the material is compressed and released uniaxially. In general, the shock and release may be converging or di- verging, but locally the compression and decompression induced is close to uniaxial. Specifically, the strains are not not isotropic. If a crystalline solid is subjected to non-isotropic strains then shear stresses must be induced, leading to plastic flow if the flow stress is exceeded. In continuum dynamics situations such as shock load- ing, simulations and analysis may be performed accu- rately by a scalar solution of the shock jump and isen- tropic expansion relations if the material is represented by an EOS, i.e. if the effects of elasticity and plastic flow are ignored. It is common practice to use spatially- resolving numerical simulations if the material is to be represented with any greater complexity. However, sim- ulations of shock waves are complicated by the need to ensure that the shock – discontinuous at the continuum level – is captured accurately without inducing numerical artifacts such as oscillations or excess heating. However, numerical methods have been developed to allow shock compression and ramp decompression to be simulated by a scalar solution for general material models including elasticity and plastic flow [14]. These numerical solutions were used to interpret the temperature measurements on Spatially-resolved simulations were also performed for comparison with velocity history measurements. These simulations used a Lagrangian representation of the shock experiments, integrated in time with a predictor- corrector numerical scheme employing artificial viscosity to stabilize the shock wave [15]. The material models were identical with those used in the scalar solution. The conservation equations for shock and release states in material dynamics are usually formulated in terms of compression and pressure. An order to take account of elastic-plastic effects, the equations were formulated in terms of the stress and strain tensors. Thus the Rankine- Hugoniot equations [16] for conservation across the shock were expressed in terms of the total stress normal to the shock, τn, rather than the pressure p: u2s = −v τn − τn0 v0 − v , (1) ∆up = −(τn − τn0)(v0 − v), (2) e = e0 − (τn + τn0)(v0 − v), (3) where v is specific volume (the reciprocal of the mass density ρ), e is specific internal energy, us is the speed of the shock wave with respect to the material, ∆up is the change in material speed normal to the shock wave (i.e., parallel to its direction of propagation), and sub- script 0 refers to the initial state. The specific internal energy was defined to exclude elastic strain energy, so the energy equation above included only the volumetric and plastic strain contributions to the volume change. The relation for adiabatic compression and release was expressed similarly: pdiv ~u : elastic (||σgrad~u|| − pdiv ~u) : plastic where τ is the stress tensor, σ the deviatoric stress σ ≡ τ − Tr τI = τ + pI, (5) grad~u the velocity gradient tensor, and div ~u its trace. For uniaxial compression, ||σgrad~u|| = σn∂~un/∂rn i.e. the product of the components in the direction normal to the wave, all others being zero. In the non-spatially- resolved calculations, the velocity gradient was simply the assumed or imposed strain rate. The state of the material was expressed in terms of ρ and e (allowing a mean pressure p(ρ, e) to be calculated from the EOS), a deviatoric elastic strain tensor ǫ (allow- ing the deviatoric stress contributions σ to be calculated), and a scalar equivalent plastic strain ǫ̃p, used to calculate work hardening. As discussed elsewhere [14], a hypere- lastic formulation using strain rather than a hypoelastic formulation using stress was preferred for consistency and accuracy in situations where shear strains are applied at different compressions. Thus the stress deviator σ was calculated from the instantaneous strain, σ = 2Gǫ, (6) where G(ρ, T ) is the shear modulus. Plastic flow was taken to occur using a von Mises yield surface [17]. De- formation was plastic rather than elastic if the scalar ef- fective shear stress fσ||σ2|| (7) exceeded the yield stress Y (ρ, e, ǫ̃p), in which case plastic strain for work hardening was accumulated at a rate ˙̃ǫp = ||ǫ̇ǫ||+ ||ǫǫ̇|| , (8) where ǫ̇ is the deviatoric part of the symmetric part of the velocity gradient, ǫ̇ ≡ Ė − Tr ĖI : Ė ≡ (U + UT ), U ≡ grad~u. If σ̃ < Y , the elastic deformation was simply ǫ̇. For uniaxial compression along the x-direction, the only non- zero component of U is [U ]100. If plastic flow occurs, then the material is always heated to some degree. Plastic flow occurs through the motion and generation of defects in the crystal lattice, such as dislocations. Usually in polycrystalline materials, defects accumulate during plastic deformation. Heating generally represents less than the total plastic work as some potential energy is absorbed in the structure of de- fects. The fraction of plastic work converted to heat fp is thought to be 0.85-0.95. It was assumed here to be 0.9. Thus the contribution of plastic work to heating was ėp = fp ||σė|| if σ̃ > Y and zero otherwise. When a metal is deformed, shear strains result in the accumulation of elastic energy until the flow stress is reached. Continued deformation results in plastic work. If the material work-hardens, the rate of plastic working increases. If thermal softening occurs, the rate of plastic working decreases but the stored elastic energy is also converted to plastic work. In an idealized material ex- hibiting a constant flow stress (elastic-perfectly plastic), arbitrarily large amounts of plastic work may be accu- mulated by large deformations – uniaxial as well as pure shear – beyond the flow stress. Ceramics may behave dif- ferently, the flow stress decreasing to a small fraction of its initial value as deformation continues beyond the elas- tic limit [9], presumably as interatomic bonds are broken and brittle damage occurs. Many transparent materials are ceramic; this reduction of the flow stress may explain why good agreement has been obtained between shock temperatures and predictions neglecting heating from the constitutive response [6]. Plastic flow is largely irreversible. If a sample of mate- rial is shock loaded and then released [24], the pressure reduces on release but further plastic work is done. Mo was represented by an empirical EOS fitted to shock compression data [18, 19, 20], with a deviatoric strength model developed and calibrated to data on the amplitude and shape of elastic waves running ahead of shocks [18]. The EOS was likely to be accurate to a few percent in temperature for the shock pressures of a few tens of gigapascals considered here. The Steinberg- Guinan strength model includes a prediction of the flow stress at elevated pressures. The flow stress, and hence the heating from plastic flow, was uncertain at the level of a few tens of percent. As discussed below, measurements of surface velocity provided an independent measure of the flow stress. Material models for continuum dynamics are often im- plemented in varying ways in different computer pro- grams. The results may depend on details such as the way in which numerical limits, e.g. on flow stress, are applied. In our simulations, the EOS was represented by an expression for pressure p in terms of mass density ρ and specific internal energy e. This is sufficient to allow the dynamical equations for the continuum to be inte- grated in time. Two different EOS were used, a tabular form from the ‘SESAME’ library [19], and an analytic form of the Grüneisen type, using the principal Hugoniot as the reference curve [18]: shock speed us in terms of particle speed up, us = c0 + s1up, (11) together with a relation for the Grüneisen parameter γ(ρ) = γ0 + b(ρ/ρ0 − 1). (12) The shear modulus G and flow stress Y followed the Steinberg-Guinan model [21], which includes explicit de- pendence on temperature T and accumulated plastic strain ǫp: G(p, T ) = G0 1 +Ap(ρ/ρ0) −1/3 −B(T − T0) Y (p, T ) = Y0f(ǫp)G(p, T )/G0 (14) f(ǫp) = min [(1 + βǫ̃p) n, Ymax/Y0] . (15) TABLE I: Grüneisen equation of state and Steinberg-Guinan strength parameters for Mo. equation of state strength ρ0 10.2 g/cm G0 125 GPa c0 5.143 km/s Y0 1.6 GPa s1 1.255 A 1.14 × 10 −2 GPa−1 γ0 1.59 B 1.52 × 10 −4 K−1 b 0.30 β 20 cp 2.43 × 10 −4 MJ/kg.K n 0.15 a 1.3 Ymax 2.8 GPa Source: [18] with unit conversions. Because of the scaling of flow stress by shear modulus, the maximum flow stress at high pressures was not lim- ited by the ‘maximum’ flow stress Ymax – this allowed the flow stress to be significantly greater than Ymax in the Mo impact experiments. The usual definition of the Steinberg-Guinan model [18] includes an explicit initial plastic strain from manufacture; we treated ǫp as a local material parameter in addition to ρ, e, and the elastic strain, and set ǫ̃p to a non-zero value in the initial condi- tions if required. The factors fǫ and fσ used in calculat- ing the scalar effective magnitudes of the corresponding tensors were chosen for consistency with the definitions of stress and strain used in deducing strength parameters for Mo from experiments: fǫ = fσ = 3/2. The SESAME EOS were defined as a pair of tables {p, e}(ρ, T ), so the p(ρ, e) relation was obtained by nu- merical inversion and the temperature was readily calcu- lated. Temperatures were calculated from the Grüneisen EOS with reference to a compression curve along which the temperature and specific internal energy were known, {Tr, er}(ρ), and using a specific heat capacity defined as a function of density cv(ρ) (constant in practice). The ref- erence curve chosen was the zero kelvin isotherm (‘cold curve’), Tr = 0K. This was calculated from the principal isentrope e(ρ)|s0 using the estimated density variation of Grüneisen parameter: er(ρ) = e(ρ)|s0 − T0cpe a(1−ρ0/ρ) )γ0−a . (16) The isentrope was calculated by numerical integration of the second law of thermodynamics, = −p(1/v, e). (17) Mechanical properties and temperatures calculated by ei- ther EOS gave the same result to o(1%), which consti- tutes good agreement for models in material dynamics. The Grüneisen EOS have slightly smoother loci, so the results presented below are from this EOS. Simulations were performed in units of millimeters, gigapascals, microseconds, kelvin, and Mg/m3=g/cm3. Parameters for Mo in these units are listed in Table I. projectile thermal radiation collecting lens; to spectrometer reflected laser light for Doppler velocimetryro window sample FIG. 1: Schematic of impact-induced shock experiments with surface temperature measurements. Aspect ratios are repre- sentative of the experiments discussed here. If the window is omitted, the experiment measures the free surface (zero normal stress) temperature. IV. PYROMETRY EXPERIMENTS ON MOLYBDENUM Pyrometry measurements of the temperature in shocked and released Mo have been made using two types of experiment. In both cases, the shock was induced by the impact of a flat projectile. The projectiles were ac- celerated using a high explosive launcher, as in the NRS experiments, and by a gas gun. The pyrometry measure- ment was performed at the surface opposite the impact, the shocked state releasing either to vacuum or into a LiF window to sustain an elevated pressure (Figs 1 and 2). In each case, the shock state was calculated using the published EOS and strength properties for the projectile and Mo target. In all cases, the impact conditions were calculated us- ing the scalar solution, and were repeated with and with- out strength in all components of the impact experiment. For experiments with a LiF window, the temperature in the window was also predicted; a high temperature would signal an increased possibility of thermal radiation from the window obscuring the emissions from the Mo sam- ple. Where an uncertainty in impact velocity was re- ported, the calculations were repeated for velocities at the extremes of uncertainty, giving an estimate of the uncertainty in pressure and temperature. Similar calculations were performed with and without strength in each component separately. The Steinberg- Guinan model is least appropriate for LiF, so this is the only component where it would be useful to make such additional comparisons. However, the effect on states releasing into LiF were dominated by the strength of the Mo, so the additional comparisons are omitted for clarity. An indication that the contribution of strength in the LiF is a small effect in the simulations is that the predicted shock temperature in LiF varied much less as a function of strength than did the temperature of any of the Mo states. Taking strength into account, on release into LiF, the 0 0.5 1 1.5 2 2.5 3 3.5 4 particle speed (km/s) Al-6061 projectile impact shock elastic release window release free surface release FIG. 2: Shock and release states induced in impact experi- ments with and without a window. Solid lines: shock Hugo- niots; dashed line: release adiabat. Example calculation for Al-6061 projectile traveling at 3.6 km/s, impacting stationary Mo target, releasing into LiF window or into vacuum. The initial states of the Mo and LiF are at the origin; the initial state of the projectile is at zero normal stress and 3.6 km/s. On impact, the shock states in the projectile and sample are at the elevated pressure intersection marked ‘impact shock.’ When the shocked state in the Mo releases into the LiF win- dow, the resulting state is the intersection marked ‘window release.’ When the shocked state in the Mo is released at a free surface, the resulting state is the zero normal stress state marked ‘free surface release.’ Release from the shocked state shows an inflexion when plastic flow occurs. normal stress in the Mo was lower than the in-plane stress because the elastic strain is a distension in the axial di- rection. For this reason, the calculations with strength have a lower normal stress: a result which may be coun- terintuitive. Various improvements could be made in future pyrom- etry measurements to reduce the temperature uncertain- ties. Some optimization could be performed by repeating experiments multiple times, adjusting detector gains and digitization ranges for best accuracy. However, the diffi- culty and cost of each projectile impact experiment can make multiple repeats impractical. A. Gas gun The projectile was Ta, 3mm thick, accelerated to 1.70 km/s using a two stage gas gun. The target was Mo, 5mm thick. Thermal emission was measured on re- lease into a LiF window, using a 7 channel pyrometer. The measured release temperature was 683± 41K. The shocked state in the Mo was calculated to be 58.7GPa and 645K, of which 51K was from plastic work. The state on release into LiF was thus calculated to be 24.8GPa and 614K, of which 82K was from plastic work. 200 300 400 500 600 700 800 temperature (K) Hugoniot release adiabat elastic LiF release solid: Steinberg-Guinan dashed: no strength FIG. 3: Temperature measurement from a shock of 59GPa, on release to 25GPa into LiF, compared with predictions based on the Steinberg-Guinan strength model and with strength neglected. The crosses on the release adiabats show where release pauses when a shock is transmitted into the LiF. When strength is included, the first portion of release is elastic; the elastic portion of the adiabat is marked; flow becomes plastic below the inflexion. The measured surface temperature was just 1.5 standard deviations above the temperature predicted using the Steinberg-Guinan strength model, and more than three standard deviations above the temperature predicted ig- noring material strength. (Table II and Fig. 3.) B. Forest Flyer The high explosive launcher used the Forest Flyer de- sign [22]. With this system, the projectile was slightly dished on impact, though this should not affect the py- rometry measurement significantly. The projectile was accelerating slightly, reverberating elastically from the acceleration process, and possibly had a porous region through its thickness as a result of tensile stresses dur- ing acceleration. The relatively strong reverberations in the projectile affect its effective speed on impact, and contributed to the uncertainty in material states. The projectile was Al-6061 alloy, 6mm thick, acceler- ated to 3.6 ± 0.1 km/s. The target was Mo, also 6mm thick. Six experiments were performed, four for release into a LiF window and two into vacuum. Surface emis- sion was measured with a 5 channel visible-near infrared pyrometer or a 4 channel near infrared pyrometer. The free surface temperature had a relatively large uncer- tainty, and the signals on release into LiF showed ev- idence of thermal emission from the LiF itself with a temperature of around 580K [13]. The measured release temperature was 762 ± 40K into LiF, and 568 ± 100K from the free surface. 200 300 400 500 600 700 800 900 temperature (K) Hugoniot release adiabat LiF release free surface solid: Steinberg-Guinan dashed: no strength FIG. 4: Temperature measurements from a shock of 64GPa, on release to 27GPa into LiF and to zero pressure, compared with predictions based on the Steinberg-Guinan strength model and with strength neglected. The release adiabat from the mean shock pressure is shown, along with adiabats re- flecting the uncertainty in shock pressure. The lines across the release adiabats show where release pauses when a shock is transmitted into the LiF. The shocked state in the Mo was calculated to be 63.9± 2.4GPa and 707±31K, of which 53±3K was from plastic work. The state on release into LiF was thus calculated to be 27.1 ± 1GPa and 670 ± 25K, of which 89 ± 1K was from plastic work. The state on release into vacuum was calculated to be 635± 23K, of which 126± 4K was from plastic work. The uncertainties are correlated: the smallest, mean, and largest of each go together. The surface temperature on release into LiF was 1.5-2.5 standard deviations of the temperature predicted using the Steinberg-Guinan strength model, and 3.5-4.5 stan- dard deviations from the temperature predicted with- out strength. The uncertainty in the free surface re- lease temperature was too large to discriminate between a purely hydrodynamic calculation (no strength) and the Steinberg-Guinan model – both lay within one standard deviation of the measurement. The predicted temper- ature of the LiF itself also matched the measurement to within the experimental uncertainties. (Table II and Fig. 4.) The velocity history of the surface of the sample was measured by laser Doppler velocimetry of the ‘VISAR’ type [23]. General features of the velocity history in- cluded a rapid acceleration when the shock reached the surface, a roughly constant peak velocity corresponding to the sustained pressure behind the shock, deceleration caused by the release wave from the rear of the projectile, and a slight re-acceleration as the sample was subjected to tensile stress causing spall type damage (Fig. 5). The epoch of peak velocity was not perfectly constant, but showed some acceleration. This was probably caused by the compression gradient in the projectile from the resid- 0 0.5 1 1.5 2 2.5 3 3.5 4 time after impact (µs) shock release FIG. 5: Surface velocity histories measured in Forest Flyer impact experiments with and without a LiF window. Separate lines are from different experiments. The upper two traces are from free surface release; the lower three are from release into a LiF window. ual accelerating pressure at impact, and any regions of porosity resulting from tensile damage as the projectile was accelerated by the relatively strong pressures induced by the detonating high explosive. The onset of release showed a clear elastic precursor (Fig. 6). The measured velocity histories were compared with spatially-resolved one-dimensional continuum dynamics simulations. The projectile was modeled as ideal, i.e. at uniform STP conditions and traveling at a constant 3.6 km/s with no reverberations. As a result, the peak ve- locity epoch was flatter than measured, but was in good agreement for amplitude and duration. Release into the LiF was also reproduced well overall. The shape of the elastic precursor to release was not reproduced perfectly using the Steinberg-Guinan strength model, but its am- plitude was reproduced to within around 10% and the time of arrival was in good agreement with the exper- iment. The difference in shape could be caused by in- adequacy in the Steinberg-Guinan model – for example, in the detailed work-hardening history in the shocked state – but is more likely to reflect density variations in the projectile as discussed above. The uniaxial strains greatly exceeded the elastic limit on release as well as on compression, so the plastic work should be dominated by the flow stress rather than the precise path before plastic flow occurred. Thus the agreement between calculated and observed amplitudes suggests that the plastic work should be correct to around 10%. Spallation did not affect the shock and release states of interest for the temperature measurements considered here. The simulations used a crude spall model of the minimum pressure type, with a minimum pressure of - 1.5GPa [18], meaning that the maximum tensile stress induced by the Mo as it was distended was 1.5GPa. 0 0.5 1 1.5 2 2.5 3 3.5 4 time after impact (µs) experiment experiment strength strength FIG. 6: Surface velocity history in Forest Flyer impact ex- periment with a free surface (upper traces) and a LiF window (lower traces). Each experimental records is compared with two continuum dynamics simulations, with (solid lines) and without strength (dashed lines) in the Mo. The elastic precur- sor to the release wave is evident where the experimental ve- locity histories deviate from the simulations without strength. No treatment of accumulating porosity was included, so the Mo as simulated continued to exert a tensile stress when in reality voids or cracks would open, reducing the stress. Thus the simulations of velocity history did not show a re-acceleration after the deceleration associated with the release wave. Tensile damage and spall can de- pend strongly on the strain rate and loading history. The simulated and observed release deceleration matched to within a few percent, suggesting that the published spall strength applies well to the loading history induced by these projectile impact experiments. V. CONCLUSIONS Shock and release temperatures were calculated self- consistently using the equation of state and a published constitutive model for Mo. Strength was calculated to make a significant difference to states in experiments ex- ploring pressures of tens of gigapascals. The high pres- sure flow stress predicted using the Steinberg-Guinan strength model matched the elastic release precursor ob- served using surface Doppler velocimetry, suggesting that the flow stress was correct to around 10%. The predicted temperatures were consistent with pyrometry measure- ments for shocks of around 60GPa, releasing into a LiF window or into vacuum. The LiF release temperatures were clearly more consistent with plastic work as pre- dicted using the Steinberg-Guinan model than with hy- drodynamic flow (no strength). The uncertainties in tem- perature were however too large to discriminate between strength models to better than several tens of percent in TABLE II: Shock and release states. no strength strength measured particle speed normal stress temperature particle speed normal stress temperature temperature (km/s) (GPa) (K) (km/s) (GPa) (K) (K) gas gun Mo shock 0.905 57.9 594 0.902 58.7 645 Mo release into LiF 1.374 25.4 532 1.337 24.8 614 683 ± 41 LiF shock 1.374 25.4 535 1.337 24.8 532 Forest Flyer Mo shock 0.97 ± 0.03 63.3 ± 2.4 654± 28 0.97± 0.03 63.9 ± 2.4 707± 31 Mo release into vacuum 1.95 ± 0.07 0 509± 19 1.91± 0.07 0 635± 23 566± 100 Mo release into LiF 1.48 ± 0.04 27.8 ± 1 581± 24 1.44± 0.04 27.1± 1 670± 25 762 ± 40 LiF shock 1.48 ± 0.04 27.8 ± 1 570± 17 1.44± 0.04 27.1± 1 566± 16 624± 100 flow stress. Heating from plastic work was calculated to be around 50K for shock pressures around 60GPa, 90K on sub- sequent release into LiF, and 125K on release at a free surface. The fraction of plastic work converted to heat was assumed to be 90% – the heating would have been about 10% greater if all the plastic work appeared as heat. Taking plastic flow into account, there was no dis- crepancy between predictions and measured release tem- peratures for Mo. This is a validation of the models of EOS and strength for Mo, and of the use of pyrometry to measure release temperatures in metals – though the pyrometry measurements obtained in these experiments were not precise enough to discriminate between models calibrated against similar mechanical data such as veloc- ity histories. The fraction of plastic work converted to heat was most likely close to 100%, though the uncer- tainty in the temperature measurements means that this figure cannot be justified statistically to better than a few tens of percent. Plastic flow makes a significant contribution to recon- ciling the temperature discrepancy observed in the neu- tron resonance spectrometry experiments on shocked Mo, although the complete explanation is more complicated and will be reported separately. Acknowledgments We would like to acknowledge the contribution of Carl Greeff for assistance and advice on equations of state and their uncertainties or certainties for Mo, of Ron Ra- bie, David Funk, Rob Hixson, and Chuck Forest for de- tailed information on the design and testing of the Forest Flyer loading system, and of Sheng-Nian Luo for gen- eral advice and comments on pyrometry and material dynamics. The gas gun experiments were performed by D.B. Holtkamp, P. Paulsen, P. Fiske, D. DeVore, J. Gar- cia, and L. Tabaka at Lawrence Livermore National Lab- oratory in 1999. The work was performed under the aus- pices of the U.S. Department of Energy under contracts W-7405-ENG-36 and DE-AC52-06NA25396. [1] M. Eremets, “High Pressure Experimental Methods” (Oxford University Press, 1996, New York). [2] S.-N. Luo and D.C. Swift, Physica B 388, 139 (2007). [3] S.B. Kormer, Sov. Phys. Usp. 21, 689700 (1965) – trans- lation. [4] M.B. Boslough and T.J. Ahrens, Rev. Sci. Instrum. 60, 3711-6 (1989). [5] S.-N. Luo, J.A. Akins, T.J. Ahrens, and P.D. Asimow, J. Geophys. Res. 109, B05205 (2004). [6] S.-N. Luo, D.C. Swift, R.N. Mulford, N.D. Drummond, and G.J. Ackland, J. Phys.: Cond. Matt., 16, 30, 5435 - 5442 (2004). [7] V.W. Yuan, J.D. Bowman, D.J. Funk, G.L. Morgan, R.L. Rabie, C.E. Ragan, J.P. Quintana, and H.L. Stacy, Phys. Rev. Lett. 94, 125504 (2005). [8] A. Seifter, K. Boborides, D.A. Clark, R.B. Corrow, D.B. Holtkamp, G.L. Morgan, J.R. Payton, P. Quintana, C.E. Ragan, P. Rodriguez, H.L. Stacey, W.S. Vogan, V.W. Yuan, and A.W. Obst, Temperature measurements of shock-loaded molybdenum (Los Alamos National Labo- ratory report LA-UR-04-2561), Proc. TEMPMEKO 2004 conference, held in Dubrovnik, Croatia, 22-27 June 2004. [9] S.A. Raikes and T.J. Ahrens, Geophys. J. of the Roy. Astron. Soc. 58, pp 717-747 (1979). [10] C.E. Morris and J.N. Fritz, J. Appl. Phys. 51, 2, pp 1244- 1246 (1980). [11] A. Seifter and A. Obst, About the proper wavelength for pyrometry on shock physics experiments, submitted to Int. J. of Thermophysics. [12] A. Seifter (Los Alamos National Laboratory), infra red camera measurements, in preparation. [13] A. Seifter (Los Alamos National Laboratory), detailed pyrometry analysis, in preparation. [14] D.C. Swift, Numerical solution of shock and ramp load- ing relations for general material properties, submitted, arXiv:cond-mat/0704.0008. [15] D. Benson, Computer Methods in Appl. Mechanics and Eng. 99, 235 (1992). [16] For a recent review and introduction, see e.g. M.R. Boslough and J.R. Asay, in J.R. Asay and M. Shahinpoor (Eds), “High-Pressure Shock Compres- sion of Solids” (Springer-Verlag, New York, 1992). [17] R. Hill, “The Mathematical Theory of Plasticity” (Clarendon Press, Oxford, 1950). [18] D.J. Steinberg, Equation of state and strength parame- ters for selected materials, Lawrence Livermore National Laboratory report UCRL-MA-106439 change 1 (1996). [19] K.S. Holian (Ed.), Los Alamos National Laboratory re- port LA-10160-MS (1984). [20] C. Greeff (Los Alamos National Laboratory), unpub- lished work and private communications. [21] D.J. Steinberg, S.G. Cochran, and M.W. Guinan, J. Appl. Phys. 51, 1498 (1980). [22] D.C. Swift, C.A. Forest, D.A. Clark, W.T. Buttler, M. Marr-Lyon, and P. Rightley, On High Explosive Launching of Projectiles for Shock Physics Experiments, submitted, arXiv:cond-mat/0702693. [23] L.M. Barker and R.E. Hollenbach, J. Appl. Phys. 43, 11, pp 4669-4675 (1972). [24] In purely hydrodynamic analyses, where the effect of plastic flow is ignored, release from a shocked state fol- lows an isentrope. This is no longer true when additional dissipative processes occur, such as plastic flow and vis- cosity, which lead to heating with an increase in entropy. The term ‘quasi-isentropic’ is sometimes used in this con- text, particularly for shockless compression; here we pre- fer to refer to the release loci as adiabats since this is a more specific term. http://arxiv.org/abs/cond-mat/0702693

Shock and release temperatures in Mo were calculated, taking account of heating from plastic flow predicted using the Steinberg-Guinan model. Plastic flow was calculated self-consistently with the shock jump conditions: this is necessary for a rigorous estimate of the locus of shock states accessible. The temperatures obtained were significantly higher than predicted assuming ideal hydrodynamic loading. The temperatures were compared with surface emission spectrometry measurements for Mo shocked to around 60GPa and then released into vacuum or into a LiF window. Shock loading was induced by the impact of a planar projectile, accelerated by high explosive or in a gas gun. Surface velocimetry showed an elastic wave at the start of release from the shocked state; the amplitude of the elastic wave matched the prediction to around 10%, indicating that the predicted flow stress in the shocked state was reasonable. The measured temperatures were consistent with the simulations, indicating that the fraction of plastic work converted to heat was in the range 70-100% for these loading conditions.

Shock and Release Temperatures in Molybdenum Damian C. Swift∗ and Achim Seifter Group P-24, Physics Division, Los Alamos National Laboratory, MS E526, Los Alamos, New Mexico 87545, USA David B. Holtkamp and David A. Clark Group P-22, Physics Division, Los Alamos National Laboratory, MS D410, Los Alamos, New Mexico 87545, USA (Dated: March 12, 2007 – LA-UR-07-1660) Shock and release temperatures in Mo were calculated, taking account of heating from plastic flow predicted using the Steinberg-Guinan model. Plastic flow was calculated self-consistently with the shock jump conditions: this is necessary for a rigorous estimate of the locus of shock states accessible. The temperatures obtained were significantly higher than predicted assuming ideal hydrodynamic loading. The temperatures were compared with surface emission spectrometry measurements for Mo shocked to around 60GPa and then released into vacuum or into a LiF window. Shock loading was induced by the impact of a planar projectile, accelerated by high explosive or in a gas gun. Surface velocimetry showed an elastic wave at the start of release from the shocked state; the amplitude of the elastic wave matched the prediction to around 10%, indicating that the predicted flow stress in the shocked state was reasonable. The measured temperatures were consistent with the simulations, indicating that the fraction of plastic work converted to heat was in the range 70-100% for these loading conditions. PACS numbers: 62.50.+p, 62.20.Fe, 65.40.-b, 07.20.Ka Keywords: shock physics, shock temperature, plasticity, molybdenum I. INTRODUCTION The behavior of matter subjected to extreme condi- tions through dynamic loading is of interest from a direct physical standpoint, as dynamic loading is often the only practical way to induce extreme conditions, and because important engineering problems in hypervelocity impact and weaponry involve dynamic loading [1]. Temperature is a key parameter in understanding the properties of matter, as it governs the population of vibrational pro- cesses and excitations past energy barriers. Temperature is thus important for a physical understanding of many types of behavior and the associated models. The equa- tion of state (EOS) includes contributions from the ex- citation of atomic vibrations and electronic excitations. Plastic flow is mediated by the excitation of dislocations and twin boundaries past Peierls barriers. Phase changes depend on the thermodynamic state’s location in the phase diagram. The kinetics of phase changes are de- scribed by the nucleation and growth of the daughter phase in the matrix of the parent phase, requiring the excitation of atoms past barriers. Chemical reactions are governed by the excitation of atoms or electrons over barriers. Diffusion in condensed matter is the motion of atoms past the barriers formed by their neighbors. Con- ductivities include scattering contributions from thermal excitations. Temperature is notoriously difficult to measure dur- ∗Electronic address: dswift@lanl.gov ing dynamic loading, in particular for opaque materials [2]. Extreme states of matter are often hidden within a sheath of matter in a different state – this is the usual situation in shock loading experiments. Probes made of matter generally disrupt the state of interest, e.g. by pre- senting an impedance mismatch to compression waves. Much interesting physics in condensed matter occurs at compressions of a few tens of percent, where the heating may be in the range of a few hundred kelvin and the re- sulting thermal emissions are small. Most temperature measurements of shock-loaded systems have been made using photon emission spectroscopy, commonly called py- rometry [3, 4]. However, many materials of interest (e.g. metals) are opaque in the relevant region of the spectrum: infra-red through visible for shocks up to the terapascal regime. Emission from an opaque material comes from the surface, which cannot be maintained at the pressure of the initial shock for long enough to allow useful emis- sion spectra to be collected. A transparent window can be placed in contact with the sample to maintain an el- evated pressure, but the mismatch in shock impedance must be taken into account, along with the effect of heat conduction. Accurate temperature data have been ob- tained from transparent sample materials, where ther- mal radiation from the shocked state can escape from the sample [5, 6]. Neutron resonance spectroscopy (NRS) has been inves- tigated as a fundamentally different technique for mea- suring the temperature inside a dynamically-loaded spec- imen, which can be used on opaque materials [7]. Trial measurements of NRS temperatures were performed on shock-loadedMo, as a standard material for high pressure work; the shock temperature was found to lie significantly http://arxiv.org/abs/0704.1850v1 mailto:dswift@lanl.gov above the temperature predicted using reasonable EOS for Mo [7]. Measurements were also made using pyrome- try of the temperature of Mo which was shocked and then released into a LiF window or into vacuum; these exper- iments also yielded temperatures which lay significantly above EOS predictions [8]. Here we consider the effect of plastic flow on the Mo pyrometry experiments. Plastic flow was neglected in previous comparisons of predictions with the tempera- ture data. The contribution of plastic work to heating has been mentioned in studies of other metals [9] but has not been quantified in detail or consistently. The contri- bution to the total internal energy from plastic heating has been estimated in order to extract the scalar EOS from shock data [10], which involves a similar analysis of shock heating, though less general. II. CORRECTIONS AND SYSTEMATIC UNCERTAINTIES IN PYROMETRY Polycrystalline materials, such as the Mo for which the discrepancy in temperature measurements was re- ported, are heterogeneous in that they are composed of an aggregate of grains of different crystallographic orien- tation. The Mo samples were machined from material which had been prepared by pressing from powder, so there was a population of voids and there were impuri- ties concentrated along grain boundaries. On shock load- ing and subsequent release, different regions of the sam- ple would thus respond differently, producing a variation in local temperature. Given enough time, temperature variations disperse through thermal conduction, but this typically takes of order microseconds for grains tens of micrometers in size, which is long on the time scale of the experiments. We wish to compare pyrometry mea- surements of temperature in Mo with predictions using continuum models, so the temperature of interest is the mean, bulk value. Thermal radiation is described by the Stefan-Boltzmann relation, where the total power is pro- portional to the fourth power of temperature. Pyrometry measurements are often more accurate at shorter wave- lengths where the power varies with higher powers of tem- perature [11]. Thus unquantified temperature variations (spatial or temporal, within the respective resolutions of the detectors) lead to an overestimate of the mean tem- perature. Spatial variation of brightness temperature has been observed in shock-loaded Sn [12], which has a much lower flow stress than Mo. Pyrometry measurements from metals probe the sur- face temperature. The surface is prone to increased plastic work around surface features such as machining marks: when a rough surface in contact with material of lower impedance (or vacuum, in the extreme case) is shocked, flow may be exaggerated in pits and grooves. At sufficiently high shock pressures, localized jetting may occur. Any such localized increase in plastic flow will produce a higher local temperature, which may appear as a higher mean temperature as discussed above. For experiments in which the sample is observed through a transparent window to maintain an elevated pressure, there may be additional radiation from com- pression of any gas or glue in the gap between the sample and the window, or from the shocked window material it- self. These effects were considered and corrected for the Mo data [13]. After the shock passes from the sample into a window, the temperature of the shock state is gen- erally different to that in the sample, so heat conduction can alter the temperature of the sample surface. In multiple channel pyrometry measurements, thermal emission from the sample is recorded using a set of de- tectors responding to different ranges of photon wave- length. In the simplest case, a grey body spectrum can be fitted to the signals, and the mean emissivity and temperature deduced from the shape of the spectrum. In general, the emissivity of a material varies with state and wavelength, and also with surface roughness, which may change during dynamic loading. The emissivity may be measured directly, for example by ellipsometry. It is more common to assume that the multiple pyrometry channels over-sample the wavelength variation of the ra- diance and emissivity, at least over part of the spectrum, allowing both to be deduced. Again, these effects were taken into account for the Mo measurements [13]. Aside from gaps and glues between the sample and any window, it is common for components of the shocked target assembly to include sharp corners and low-density materials such as plastics, foams, and glues, as part of the engineering structure. As with glued windows, low density materials in general may be shock-loaded to a higher temperature than the sample. Sharp internal cor- ners, when shocked, may form jets with large amounts of plastic heating. Thermal emission from any of these sources may be present as a background against which the emission from the sample must be distinguished. The net effect of the complications associated with py- rometry measurements on metals is that it is possible to over-estimate the temperature. III. HEATING FROM PLASTIC WORK IN SHOCK AND RELEASE Shock compression involves the transit of a shock wave through each element of material. Subsequently, axial and lateral release and recompression waves reverberate through the sample until it ultimately comes to rest at zero pressure. In the simplest case, which many shock physics experiments are designed to achieve, the shock is steady and planar and the initial release is planar: the material is compressed and released uniaxially. In general, the shock and release may be converging or di- verging, but locally the compression and decompression induced is close to uniaxial. Specifically, the strains are not not isotropic. If a crystalline solid is subjected to non-isotropic strains then shear stresses must be induced, leading to plastic flow if the flow stress is exceeded. In continuum dynamics situations such as shock load- ing, simulations and analysis may be performed accu- rately by a scalar solution of the shock jump and isen- tropic expansion relations if the material is represented by an EOS, i.e. if the effects of elasticity and plastic flow are ignored. It is common practice to use spatially- resolving numerical simulations if the material is to be represented with any greater complexity. However, sim- ulations of shock waves are complicated by the need to ensure that the shock – discontinuous at the continuum level – is captured accurately without inducing numerical artifacts such as oscillations or excess heating. However, numerical methods have been developed to allow shock compression and ramp decompression to be simulated by a scalar solution for general material models including elasticity and plastic flow [14]. These numerical solutions were used to interpret the temperature measurements on Spatially-resolved simulations were also performed for comparison with velocity history measurements. These simulations used a Lagrangian representation of the shock experiments, integrated in time with a predictor- corrector numerical scheme employing artificial viscosity to stabilize the shock wave [15]. The material models were identical with those used in the scalar solution. The conservation equations for shock and release states in material dynamics are usually formulated in terms of compression and pressure. An order to take account of elastic-plastic effects, the equations were formulated in terms of the stress and strain tensors. Thus the Rankine- Hugoniot equations [16] for conservation across the shock were expressed in terms of the total stress normal to the shock, τn, rather than the pressure p: u2s = −v τn − τn0 v0 − v , (1) ∆up = −(τn − τn0)(v0 − v), (2) e = e0 − (τn + τn0)(v0 − v), (3) where v is specific volume (the reciprocal of the mass density ρ), e is specific internal energy, us is the speed of the shock wave with respect to the material, ∆up is the change in material speed normal to the shock wave (i.e., parallel to its direction of propagation), and sub- script 0 refers to the initial state. The specific internal energy was defined to exclude elastic strain energy, so the energy equation above included only the volumetric and plastic strain contributions to the volume change. The relation for adiabatic compression and release was expressed similarly: pdiv ~u : elastic (||σgrad~u|| − pdiv ~u) : plastic where τ is the stress tensor, σ the deviatoric stress σ ≡ τ − Tr τI = τ + pI, (5) grad~u the velocity gradient tensor, and div ~u its trace. For uniaxial compression, ||σgrad~u|| = σn∂~un/∂rn i.e. the product of the components in the direction normal to the wave, all others being zero. In the non-spatially- resolved calculations, the velocity gradient was simply the assumed or imposed strain rate. The state of the material was expressed in terms of ρ and e (allowing a mean pressure p(ρ, e) to be calculated from the EOS), a deviatoric elastic strain tensor ǫ (allow- ing the deviatoric stress contributions σ to be calculated), and a scalar equivalent plastic strain ǫ̃p, used to calculate work hardening. As discussed elsewhere [14], a hypere- lastic formulation using strain rather than a hypoelastic formulation using stress was preferred for consistency and accuracy in situations where shear strains are applied at different compressions. Thus the stress deviator σ was calculated from the instantaneous strain, σ = 2Gǫ, (6) where G(ρ, T ) is the shear modulus. Plastic flow was taken to occur using a von Mises yield surface [17]. De- formation was plastic rather than elastic if the scalar ef- fective shear stress fσ||σ2|| (7) exceeded the yield stress Y (ρ, e, ǫ̃p), in which case plastic strain for work hardening was accumulated at a rate ˙̃ǫp = ||ǫ̇ǫ||+ ||ǫǫ̇|| , (8) where ǫ̇ is the deviatoric part of the symmetric part of the velocity gradient, ǫ̇ ≡ Ė − Tr ĖI : Ė ≡ (U + UT ), U ≡ grad~u. If σ̃ < Y , the elastic deformation was simply ǫ̇. For uniaxial compression along the x-direction, the only non- zero component of U is [U ]100. If plastic flow occurs, then the material is always heated to some degree. Plastic flow occurs through the motion and generation of defects in the crystal lattice, such as dislocations. Usually in polycrystalline materials, defects accumulate during plastic deformation. Heating generally represents less than the total plastic work as some potential energy is absorbed in the structure of de- fects. The fraction of plastic work converted to heat fp is thought to be 0.85-0.95. It was assumed here to be 0.9. Thus the contribution of plastic work to heating was ėp = fp ||σė|| if σ̃ > Y and zero otherwise. When a metal is deformed, shear strains result in the accumulation of elastic energy until the flow stress is reached. Continued deformation results in plastic work. If the material work-hardens, the rate of plastic working increases. If thermal softening occurs, the rate of plastic working decreases but the stored elastic energy is also converted to plastic work. In an idealized material ex- hibiting a constant flow stress (elastic-perfectly plastic), arbitrarily large amounts of plastic work may be accu- mulated by large deformations – uniaxial as well as pure shear – beyond the flow stress. Ceramics may behave dif- ferently, the flow stress decreasing to a small fraction of its initial value as deformation continues beyond the elas- tic limit [9], presumably as interatomic bonds are broken and brittle damage occurs. Many transparent materials are ceramic; this reduction of the flow stress may explain why good agreement has been obtained between shock temperatures and predictions neglecting heating from the constitutive response [6]. Plastic flow is largely irreversible. If a sample of mate- rial is shock loaded and then released [24], the pressure reduces on release but further plastic work is done. Mo was represented by an empirical EOS fitted to shock compression data [18, 19, 20], with a deviatoric strength model developed and calibrated to data on the amplitude and shape of elastic waves running ahead of shocks [18]. The EOS was likely to be accurate to a few percent in temperature for the shock pressures of a few tens of gigapascals considered here. The Steinberg- Guinan strength model includes a prediction of the flow stress at elevated pressures. The flow stress, and hence the heating from plastic flow, was uncertain at the level of a few tens of percent. As discussed below, measurements of surface velocity provided an independent measure of the flow stress. Material models for continuum dynamics are often im- plemented in varying ways in different computer pro- grams. The results may depend on details such as the way in which numerical limits, e.g. on flow stress, are applied. In our simulations, the EOS was represented by an expression for pressure p in terms of mass density ρ and specific internal energy e. This is sufficient to allow the dynamical equations for the continuum to be inte- grated in time. Two different EOS were used, a tabular form from the ‘SESAME’ library [19], and an analytic form of the Grüneisen type, using the principal Hugoniot as the reference curve [18]: shock speed us in terms of particle speed up, us = c0 + s1up, (11) together with a relation for the Grüneisen parameter γ(ρ) = γ0 + b(ρ/ρ0 − 1). (12) The shear modulus G and flow stress Y followed the Steinberg-Guinan model [21], which includes explicit de- pendence on temperature T and accumulated plastic strain ǫp: G(p, T ) = G0 1 +Ap(ρ/ρ0) −1/3 −B(T − T0) Y (p, T ) = Y0f(ǫp)G(p, T )/G0 (14) f(ǫp) = min [(1 + βǫ̃p) n, Ymax/Y0] . (15) TABLE I: Grüneisen equation of state and Steinberg-Guinan strength parameters for Mo. equation of state strength ρ0 10.2 g/cm G0 125 GPa c0 5.143 km/s Y0 1.6 GPa s1 1.255 A 1.14 × 10 −2 GPa−1 γ0 1.59 B 1.52 × 10 −4 K−1 b 0.30 β 20 cp 2.43 × 10 −4 MJ/kg.K n 0.15 a 1.3 Ymax 2.8 GPa Source: [18] with unit conversions. Because of the scaling of flow stress by shear modulus, the maximum flow stress at high pressures was not lim- ited by the ‘maximum’ flow stress Ymax – this allowed the flow stress to be significantly greater than Ymax in the Mo impact experiments. The usual definition of the Steinberg-Guinan model [18] includes an explicit initial plastic strain from manufacture; we treated ǫp as a local material parameter in addition to ρ, e, and the elastic strain, and set ǫ̃p to a non-zero value in the initial condi- tions if required. The factors fǫ and fσ used in calculat- ing the scalar effective magnitudes of the corresponding tensors were chosen for consistency with the definitions of stress and strain used in deducing strength parameters for Mo from experiments: fǫ = fσ = 3/2. The SESAME EOS were defined as a pair of tables {p, e}(ρ, T ), so the p(ρ, e) relation was obtained by nu- merical inversion and the temperature was readily calcu- lated. Temperatures were calculated from the Grüneisen EOS with reference to a compression curve along which the temperature and specific internal energy were known, {Tr, er}(ρ), and using a specific heat capacity defined as a function of density cv(ρ) (constant in practice). The ref- erence curve chosen was the zero kelvin isotherm (‘cold curve’), Tr = 0K. This was calculated from the principal isentrope e(ρ)|s0 using the estimated density variation of Grüneisen parameter: er(ρ) = e(ρ)|s0 − T0cpe a(1−ρ0/ρ) )γ0−a . (16) The isentrope was calculated by numerical integration of the second law of thermodynamics, = −p(1/v, e). (17) Mechanical properties and temperatures calculated by ei- ther EOS gave the same result to o(1%), which consti- tutes good agreement for models in material dynamics. The Grüneisen EOS have slightly smoother loci, so the results presented below are from this EOS. Simulations were performed in units of millimeters, gigapascals, microseconds, kelvin, and Mg/m3=g/cm3. Parameters for Mo in these units are listed in Table I. projectile thermal radiation collecting lens; to spectrometer reflected laser light for Doppler velocimetryro window sample FIG. 1: Schematic of impact-induced shock experiments with surface temperature measurements. Aspect ratios are repre- sentative of the experiments discussed here. If the window is omitted, the experiment measures the free surface (zero normal stress) temperature. IV. PYROMETRY EXPERIMENTS ON MOLYBDENUM Pyrometry measurements of the temperature in shocked and released Mo have been made using two types of experiment. In both cases, the shock was induced by the impact of a flat projectile. The projectiles were ac- celerated using a high explosive launcher, as in the NRS experiments, and by a gas gun. The pyrometry measure- ment was performed at the surface opposite the impact, the shocked state releasing either to vacuum or into a LiF window to sustain an elevated pressure (Figs 1 and 2). In each case, the shock state was calculated using the published EOS and strength properties for the projectile and Mo target. In all cases, the impact conditions were calculated us- ing the scalar solution, and were repeated with and with- out strength in all components of the impact experiment. For experiments with a LiF window, the temperature in the window was also predicted; a high temperature would signal an increased possibility of thermal radiation from the window obscuring the emissions from the Mo sam- ple. Where an uncertainty in impact velocity was re- ported, the calculations were repeated for velocities at the extremes of uncertainty, giving an estimate of the uncertainty in pressure and temperature. Similar calculations were performed with and without strength in each component separately. The Steinberg- Guinan model is least appropriate for LiF, so this is the only component where it would be useful to make such additional comparisons. However, the effect on states releasing into LiF were dominated by the strength of the Mo, so the additional comparisons are omitted for clarity. An indication that the contribution of strength in the LiF is a small effect in the simulations is that the predicted shock temperature in LiF varied much less as a function of strength than did the temperature of any of the Mo states. Taking strength into account, on release into LiF, the 0 0.5 1 1.5 2 2.5 3 3.5 4 particle speed (km/s) Al-6061 projectile impact shock elastic release window release free surface release FIG. 2: Shock and release states induced in impact experi- ments with and without a window. Solid lines: shock Hugo- niots; dashed line: release adiabat. Example calculation for Al-6061 projectile traveling at 3.6 km/s, impacting stationary Mo target, releasing into LiF window or into vacuum. The initial states of the Mo and LiF are at the origin; the initial state of the projectile is at zero normal stress and 3.6 km/s. On impact, the shock states in the projectile and sample are at the elevated pressure intersection marked ‘impact shock.’ When the shocked state in the Mo releases into the LiF win- dow, the resulting state is the intersection marked ‘window release.’ When the shocked state in the Mo is released at a free surface, the resulting state is the zero normal stress state marked ‘free surface release.’ Release from the shocked state shows an inflexion when plastic flow occurs. normal stress in the Mo was lower than the in-plane stress because the elastic strain is a distension in the axial di- rection. For this reason, the calculations with strength have a lower normal stress: a result which may be coun- terintuitive. Various improvements could be made in future pyrom- etry measurements to reduce the temperature uncertain- ties. Some optimization could be performed by repeating experiments multiple times, adjusting detector gains and digitization ranges for best accuracy. However, the diffi- culty and cost of each projectile impact experiment can make multiple repeats impractical. A. Gas gun The projectile was Ta, 3mm thick, accelerated to 1.70 km/s using a two stage gas gun. The target was Mo, 5mm thick. Thermal emission was measured on re- lease into a LiF window, using a 7 channel pyrometer. The measured release temperature was 683± 41K. The shocked state in the Mo was calculated to be 58.7GPa and 645K, of which 51K was from plastic work. The state on release into LiF was thus calculated to be 24.8GPa and 614K, of which 82K was from plastic work. 200 300 400 500 600 700 800 temperature (K) Hugoniot release adiabat elastic LiF release solid: Steinberg-Guinan dashed: no strength FIG. 3: Temperature measurement from a shock of 59GPa, on release to 25GPa into LiF, compared with predictions based on the Steinberg-Guinan strength model and with strength neglected. The crosses on the release adiabats show where release pauses when a shock is transmitted into the LiF. When strength is included, the first portion of release is elastic; the elastic portion of the adiabat is marked; flow becomes plastic below the inflexion. The measured surface temperature was just 1.5 standard deviations above the temperature predicted using the Steinberg-Guinan strength model, and more than three standard deviations above the temperature predicted ig- noring material strength. (Table II and Fig. 3.) B. Forest Flyer The high explosive launcher used the Forest Flyer de- sign [22]. With this system, the projectile was slightly dished on impact, though this should not affect the py- rometry measurement significantly. The projectile was accelerating slightly, reverberating elastically from the acceleration process, and possibly had a porous region through its thickness as a result of tensile stresses dur- ing acceleration. The relatively strong reverberations in the projectile affect its effective speed on impact, and contributed to the uncertainty in material states. The projectile was Al-6061 alloy, 6mm thick, acceler- ated to 3.6 ± 0.1 km/s. The target was Mo, also 6mm thick. Six experiments were performed, four for release into a LiF window and two into vacuum. Surface emis- sion was measured with a 5 channel visible-near infrared pyrometer or a 4 channel near infrared pyrometer. The free surface temperature had a relatively large uncer- tainty, and the signals on release into LiF showed ev- idence of thermal emission from the LiF itself with a temperature of around 580K [13]. The measured release temperature was 762 ± 40K into LiF, and 568 ± 100K from the free surface. 200 300 400 500 600 700 800 900 temperature (K) Hugoniot release adiabat LiF release free surface solid: Steinberg-Guinan dashed: no strength FIG. 4: Temperature measurements from a shock of 64GPa, on release to 27GPa into LiF and to zero pressure, compared with predictions based on the Steinberg-Guinan strength model and with strength neglected. The release adiabat from the mean shock pressure is shown, along with adiabats re- flecting the uncertainty in shock pressure. The lines across the release adiabats show where release pauses when a shock is transmitted into the LiF. The shocked state in the Mo was calculated to be 63.9± 2.4GPa and 707±31K, of which 53±3K was from plastic work. The state on release into LiF was thus calculated to be 27.1 ± 1GPa and 670 ± 25K, of which 89 ± 1K was from plastic work. The state on release into vacuum was calculated to be 635± 23K, of which 126± 4K was from plastic work. The uncertainties are correlated: the smallest, mean, and largest of each go together. The surface temperature on release into LiF was 1.5-2.5 standard deviations of the temperature predicted using the Steinberg-Guinan strength model, and 3.5-4.5 stan- dard deviations from the temperature predicted with- out strength. The uncertainty in the free surface re- lease temperature was too large to discriminate between a purely hydrodynamic calculation (no strength) and the Steinberg-Guinan model – both lay within one standard deviation of the measurement. The predicted temper- ature of the LiF itself also matched the measurement to within the experimental uncertainties. (Table II and Fig. 4.) The velocity history of the surface of the sample was measured by laser Doppler velocimetry of the ‘VISAR’ type [23]. General features of the velocity history in- cluded a rapid acceleration when the shock reached the surface, a roughly constant peak velocity corresponding to the sustained pressure behind the shock, deceleration caused by the release wave from the rear of the projectile, and a slight re-acceleration as the sample was subjected to tensile stress causing spall type damage (Fig. 5). The epoch of peak velocity was not perfectly constant, but showed some acceleration. This was probably caused by the compression gradient in the projectile from the resid- 0 0.5 1 1.5 2 2.5 3 3.5 4 time after impact (µs) shock release FIG. 5: Surface velocity histories measured in Forest Flyer impact experiments with and without a LiF window. Separate lines are from different experiments. The upper two traces are from free surface release; the lower three are from release into a LiF window. ual accelerating pressure at impact, and any regions of porosity resulting from tensile damage as the projectile was accelerated by the relatively strong pressures induced by the detonating high explosive. The onset of release showed a clear elastic precursor (Fig. 6). The measured velocity histories were compared with spatially-resolved one-dimensional continuum dynamics simulations. The projectile was modeled as ideal, i.e. at uniform STP conditions and traveling at a constant 3.6 km/s with no reverberations. As a result, the peak ve- locity epoch was flatter than measured, but was in good agreement for amplitude and duration. Release into the LiF was also reproduced well overall. The shape of the elastic precursor to release was not reproduced perfectly using the Steinberg-Guinan strength model, but its am- plitude was reproduced to within around 10% and the time of arrival was in good agreement with the exper- iment. The difference in shape could be caused by in- adequacy in the Steinberg-Guinan model – for example, in the detailed work-hardening history in the shocked state – but is more likely to reflect density variations in the projectile as discussed above. The uniaxial strains greatly exceeded the elastic limit on release as well as on compression, so the plastic work should be dominated by the flow stress rather than the precise path before plastic flow occurred. Thus the agreement between calculated and observed amplitudes suggests that the plastic work should be correct to around 10%. Spallation did not affect the shock and release states of interest for the temperature measurements considered here. The simulations used a crude spall model of the minimum pressure type, with a minimum pressure of - 1.5GPa [18], meaning that the maximum tensile stress induced by the Mo as it was distended was 1.5GPa. 0 0.5 1 1.5 2 2.5 3 3.5 4 time after impact (µs) experiment experiment strength strength FIG. 6: Surface velocity history in Forest Flyer impact ex- periment with a free surface (upper traces) and a LiF window (lower traces). Each experimental records is compared with two continuum dynamics simulations, with (solid lines) and without strength (dashed lines) in the Mo. The elastic precur- sor to the release wave is evident where the experimental ve- locity histories deviate from the simulations without strength. No treatment of accumulating porosity was included, so the Mo as simulated continued to exert a tensile stress when in reality voids or cracks would open, reducing the stress. Thus the simulations of velocity history did not show a re-acceleration after the deceleration associated with the release wave. Tensile damage and spall can de- pend strongly on the strain rate and loading history. The simulated and observed release deceleration matched to within a few percent, suggesting that the published spall strength applies well to the loading history induced by these projectile impact experiments. V. CONCLUSIONS Shock and release temperatures were calculated self- consistently using the equation of state and a published constitutive model for Mo. Strength was calculated to make a significant difference to states in experiments ex- ploring pressures of tens of gigapascals. The high pres- sure flow stress predicted using the Steinberg-Guinan strength model matched the elastic release precursor ob- served using surface Doppler velocimetry, suggesting that the flow stress was correct to around 10%. The predicted temperatures were consistent with pyrometry measure- ments for shocks of around 60GPa, releasing into a LiF window or into vacuum. The LiF release temperatures were clearly more consistent with plastic work as pre- dicted using the Steinberg-Guinan model than with hy- drodynamic flow (no strength). The uncertainties in tem- perature were however too large to discriminate between strength models to better than several tens of percent in TABLE II: Shock and release states. no strength strength measured particle speed normal stress temperature particle speed normal stress temperature temperature (km/s) (GPa) (K) (km/s) (GPa) (K) (K) gas gun Mo shock 0.905 57.9 594 0.902 58.7 645 Mo release into LiF 1.374 25.4 532 1.337 24.8 614 683 ± 41 LiF shock 1.374 25.4 535 1.337 24.8 532 Forest Flyer Mo shock 0.97 ± 0.03 63.3 ± 2.4 654± 28 0.97± 0.03 63.9 ± 2.4 707± 31 Mo release into vacuum 1.95 ± 0.07 0 509± 19 1.91± 0.07 0 635± 23 566± 100 Mo release into LiF 1.48 ± 0.04 27.8 ± 1 581± 24 1.44± 0.04 27.1± 1 670± 25 762 ± 40 LiF shock 1.48 ± 0.04 27.8 ± 1 570± 17 1.44± 0.04 27.1± 1 566± 16 624± 100 flow stress. Heating from plastic work was calculated to be around 50K for shock pressures around 60GPa, 90K on sub- sequent release into LiF, and 125K on release at a free surface. The fraction of plastic work converted to heat was assumed to be 90% – the heating would have been about 10% greater if all the plastic work appeared as heat. Taking plastic flow into account, there was no dis- crepancy between predictions and measured release tem- peratures for Mo. This is a validation of the models of EOS and strength for Mo, and of the use of pyrometry to measure release temperatures in metals – though the pyrometry measurements obtained in these experiments were not precise enough to discriminate between models calibrated against similar mechanical data such as veloc- ity histories. The fraction of plastic work converted to heat was most likely close to 100%, though the uncer- tainty in the temperature measurements means that this figure cannot be justified statistically to better than a few tens of percent. Plastic flow makes a significant contribution to recon- ciling the temperature discrepancy observed in the neu- tron resonance spectrometry experiments on shocked Mo, although the complete explanation is more complicated and will be reported separately. Acknowledgments We would like to acknowledge the contribution of Carl Greeff for assistance and advice on equations of state and their uncertainties or certainties for Mo, of Ron Ra- bie, David Funk, Rob Hixson, and Chuck Forest for de- tailed information on the design and testing of the Forest Flyer loading system, and of Sheng-Nian Luo for gen- eral advice and comments on pyrometry and material dynamics. The gas gun experiments were performed by D.B. Holtkamp, P. Paulsen, P. Fiske, D. DeVore, J. Gar- cia, and L. Tabaka at Lawrence Livermore National Lab- oratory in 1999. The work was performed under the aus- pices of the U.S. Department of Energy under contracts W-7405-ENG-36 and DE-AC52-06NA25396. [1] M. Eremets, “High Pressure Experimental Methods” (Oxford University Press, 1996, New York). [2] S.-N. Luo and D.C. Swift, Physica B 388, 139 (2007). [3] S.B. Kormer, Sov. Phys. Usp. 21, 689700 (1965) – trans- lation. [4] M.B. Boslough and T.J. Ahrens, Rev. Sci. Instrum. 60, 3711-6 (1989). [5] S.-N. Luo, J.A. Akins, T.J. Ahrens, and P.D. Asimow, J. Geophys. Res. 109, B05205 (2004). [6] S.-N. Luo, D.C. Swift, R.N. Mulford, N.D. Drummond, and G.J. Ackland, J. Phys.: Cond. Matt., 16, 30, 5435 - 5442 (2004). [7] V.W. Yuan, J.D. Bowman, D.J. Funk, G.L. Morgan, R.L. Rabie, C.E. Ragan, J.P. Quintana, and H.L. Stacy, Phys. Rev. Lett. 94, 125504 (2005). [8] A. Seifter, K. Boborides, D.A. Clark, R.B. Corrow, D.B. Holtkamp, G.L. Morgan, J.R. Payton, P. Quintana, C.E. Ragan, P. Rodriguez, H.L. Stacey, W.S. Vogan, V.W. Yuan, and A.W. Obst, Temperature measurements of shock-loaded molybdenum (Los Alamos National Labo- ratory report LA-UR-04-2561), Proc. TEMPMEKO 2004 conference, held in Dubrovnik, Croatia, 22-27 June 2004. [9] S.A. Raikes and T.J. Ahrens, Geophys. J. of the Roy. Astron. Soc. 58, pp 717-747 (1979). [10] C.E. Morris and J.N. Fritz, J. Appl. Phys. 51, 2, pp 1244- 1246 (1980). [11] A. Seifter and A. Obst, About the proper wavelength for pyrometry on shock physics experiments, submitted to Int. J. of Thermophysics. [12] A. Seifter (Los Alamos National Laboratory), infra red camera measurements, in preparation. [13] A. Seifter (Los Alamos National Laboratory), detailed pyrometry analysis, in preparation. [14] D.C. Swift, Numerical solution of shock and ramp load- ing relations for general material properties, submitted, arXiv:cond-mat/0704.0008. [15] D. Benson, Computer Methods in Appl. Mechanics and Eng. 99, 235 (1992). [16] For a recent review and introduction, see e.g. M.R. Boslough and J.R. Asay, in J.R. Asay and M. Shahinpoor (Eds), “High-Pressure Shock Compres- sion of Solids” (Springer-Verlag, New York, 1992). [17] R. Hill, “The Mathematical Theory of Plasticity” (Clarendon Press, Oxford, 1950). [18] D.J. Steinberg, Equation of state and strength parame- ters for selected materials, Lawrence Livermore National Laboratory report UCRL-MA-106439 change 1 (1996). [19] K.S. Holian (Ed.), Los Alamos National Laboratory re- port LA-10160-MS (1984). [20] C. Greeff (Los Alamos National Laboratory), unpub- lished work and private communications. [21] D.J. Steinberg, S.G. Cochran, and M.W. Guinan, J. Appl. Phys. 51, 1498 (1980). [22] D.C. Swift, C.A. Forest, D.A. Clark, W.T. Buttler, M. Marr-Lyon, and P. Rightley, On High Explosive Launching of Projectiles for Shock Physics Experiments, submitted, arXiv:cond-mat/0702693. [23] L.M. Barker and R.E. Hollenbach, J. Appl. Phys. 43, 11, pp 4669-4675 (1972). [24] In purely hydrodynamic analyses, where the effect of plastic flow is ignored, release from a shocked state fol- lows an isentrope. This is no longer true when additional dissipative processes occur, such as plastic flow and vis- cosity, which lead to heating with an increase in entropy. The term ‘quasi-isentropic’ is sometimes used in this con- text, particularly for shockless compression; here we pre- fer to refer to the release loci as adiabats since this is a more specific term. http://arxiv.org/abs/cond-mat/0702693

7 Spectrum of the Laplacian on Quaternionic Kähler Manifolds Shengli Kong, Peter Li∗and Detang Zhou† Abstract Let M4n be a complete quaternionic Kähler manifold with scalar curvature bounded below by −16n(n+2). We get a sharp estimate for the first eigenvalue λ1(M) of the Laplacian which is λ1(M) ≤ (2n+1) If the equality holds, then either M has only one end, or M is diffeo- morphic to R × N with N given by a compact manifold. Moreover, if M is of bounded curvature, M is covered by the quaterionic hy- perbolic space QHn and N is a compact quotient of the generalized Heisenberg group. When λ1(M) ≥ 8(n+2) , we also prove that M must have only one end with infinite volume. 0 Introduction Let Mn be a complete n-dimensional Riemannian manifold whose Ricci cur- vature bounded below by −(n − 1). It is well known from Cheng [Ch] that the first eigenvalue λ1(M) satisfies λ1(M) ≤ (n− 1)2 In [LW3], Li and Wang proved an analogous theorem for complete Käh- ler manifolds. They showed that if M2n is a complete Kähler manifold of complex dimension n with holomorphic bisectional curvature BKM bounded below by −1, then the first eigenvalue λ1(M) satisfies λ1(M) ≤ n ∗Research partially supported by NSF grant DMS-0503735 †Research partially supported by CAPES and CNPq of Brazil. http://arxiv.org/abs/0704.1851v1 Here BKM ≥ −1 means that Rīijj̄ ≥ −(1 + δij) for any unitary frame e1, . . . , en. In this paper, we prove the corresponding Laplacian comparison theorem for a quaterionic Kähler manifold M4n. As an application we get the sharp estimate λ1(M) for a complete quaterionic Kähler manifold M 4n with scalar curvature bounded below by −16n(n+ 2) as λ1(M) ≤ (2n + 1) It is an interesting question to ask what one can say about those manifolds when the above inequalities are realized as equalities. In works of Li and Wang [LW1] and [LW2], the authors obtained the following theorems. The first was a generalization of the theory of Witten-Yau [WY], Cai-Galloway [CG], and Wang [W] for conformally compact manifolds. The second was to answer the aforementioned question. Theorem 0.1. Let Mn be a complete Riemannian manifold of dimension n ≥ 3 with Ricci curvature bounded below by −(n − 1). If λ1(M) ≥ n − 2, then either (1) M has only one infinite volume end; or (2) M = R×N with warped product metric of the form ds2M = dt 2 + cosh2 t ds2N , where N is an (n − 1)-dimensional compact manifold of Ricci curvature bounded below by λ1(M). Theorem 0.2. Let Mn be a complete Riemannian manifold of dimension n ≥ 2 with Ricci curvature bounded below by −(n − 1). If λ1(M) ≥ (n−1)2 then either (1) M has no finite volume end; or (2) M = R×N with warped product metric of the form ds2M = dt 2 + e2t ds2N , where N is an (n − 1)-dimensional compact manifold of nonnegative Ricci curvature. In [LW3] and [LW5], Li and Wang also consider the Kähler case. They proved the following theorems. Theorem 0.3. Let Mn be a complete Kähler manifold of complex dimension n ≥ 1 with Ricci curvature bounded below by RicM ≥ −2(n + 1). If λ1(M) > , then M must have only one infinite volume end. Theorem 0.4. Let Mn be a complete Kähler manifold of complex dimension n ≥ 2 with holomorphic bisectional curvature bounded by BKM ≥ −1. If λ1(M) ≥ n 2, then either (1) M has only one end; or (2) M = R × N with N being a compact manifold. Moreover the metric on M is of the form ds2M = dt 2 + e4t ω22 + e ω2i , where {ω2, ω3, . . . , ω2n} are orthonormal coframe of N with Jdt = ω2. If M has bounded curvature, then we further conclude that M is covered by CHn and N is a compact quotient of the Heisenberg group. In [LW5], the authors pointed out that the assumption on the lower bound of λ1(M) in Theorem 0.3 is sharp, since one can construct M of the form M = Σ×N satisfying RicM ≥ −2(n + 1) (0.1) λ1(M) = n + 1 (0.2) with N being a compact Kähler manifold and Σ being a complete surface with at least two infinite volume ends. However, it is still an open question to characterized all those complete Kähler manifolds satisfying conditions ( 0.1) and ( 0.2). In sections 4 and 5, we will prove the following quaternionic Kähler ver- sions of the above theorems. Theorem 0.5. Let (M4n, g) be a complete quaternionic Kähler manifold with scalar curvature satisfying SM ≥ −16n(n + 2). If λ1(M) ≥ 8(n+2) , then M must have only one infinite volume end. Theorem 0.6. Let (M4n, g) be a complete quaternionic Kähler manifold with scalar curvature satisfying SM ≥ −16n(n + 2). If λ1(M) ≥ (2n+ 1) 2, then either (1) M has only one end, or (2) M is diffeomorphic to R×N where N is a compact manifold. Moreover, the metric is given by the form ds2M = dt 2 + e4t ω2p + e where {ω2, . . . , ω4n} are orthonormal coframes for N. If M is of bounded curvature then we further conclude that M is covered by the quaterionic hy- perbolic space QHn and N is a compact quotient of the generalized Heisenberg group. Remark 0.1. It is known that a horosphere in QHn is isometric to a certain generalized Heisenberg group with three-dimensional center and left-invariant Riemannian metric. Such generalized Heisenberg groups have compact quo- tients. For an explicit construction see for instance Example 2.6 in [G]. We don’t have an example to show that the bounded curvature condition in The- orem 0.6 is necessary. If such an example exists, its curvature should decay at exponentially in some directions. Perhaps it is interesting to restrict our attention to the special case when M4n = QHn/Γ is given by the quotient of the quaternionic hyperbolic space n with a discrete group of isometies Γ. In particular, it is instructional to compare with previous results by Corlette [C2] and Corlette-Iozzi [CI] where Lie group theoretic approach was used in understanding these manifolds. For example, in [CI], the authors proved a Patterson-Sullivan type formula for λ1(M) in terms of the Hausdorff dimension δ(Γ) of the limit set of Γ. More specifically, they proved that if Γ is geometrically finite, then for δ(Γ) ≥ 2n+1 one has λ1(M) = δ(Γ)((4n+ 2)− δ(Γ)). Hence in this case, the condition in Theorem 0.6 on λ1(M) = (2n + 1) equivalent to the condition δ(Γ) = 2n+ 1. In [C2] (Theorem 4.4), Corlette also pointed out that by a result of Kostant λ1(M) = 0 or λ1(M) ≥ 8n. On the other hand, it was also shown in [CI] that if Γ is geometrically finite and torsion free, then M = QHn/Γ must have at most one end with infinite volume. These two statements give an interesting comparison to Theorem 0.5 stated above. We would also like to point out to the interested readers that in [LW4] and [LW5] Li and Wang considered a more general class of manifolds satisfying a weighted Poincaré inequality. However, since quaternionic Kähler manifolds are automatically Einstein, the same type of questions are not interesting for this class of manifolds. Acknowledgement. This work was done when the third author was visiting the University of California, Irvine. He wishes to thank the institu- tion for its hospitality. He also would like to thank Professor J. Berndt for pointing out the paper of [G] to him. 1 Preliminaries on quaternionic Kähler man- ifolds In this section, we will recall basic properties of quaternionic Kähler manifolds that will be needed in the sequel. These properties were proved by Berger [B] and Ishihara [I] (also see [Be]). Let (Mn, g) be a Riemannian manifold, TM the tangent space of M and ∇ the Levi-Civita connection. The Riemannian curvature R : TM ⊗ TM ⊗ TM −→ TM is defined by R(X, Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z If {e1, · · · , en} is an orthonormal basis of TM , the components of curvature tensor is defined by Rijkl = 〈R(ei, ej)el, ek〉, the Ricci curvature is defined by RicM(X, Y ) = 〈R(X, ei)ei, Y 〉, and the scalar curvature is defined by i,j=1 〈R(ei, ej)ej , ei〉. Definition 1.1. A quaternionic Kähler manifold (M, g) is a Riemannian manifold with a rank 3 vector bundle V ⊂ End(TM) satisfying (a) In any coordinate neighborhood U ofM , there exists a local basis {I, J,K} of V such that I2 = J2 = K2 = −1 IJ = −JI = K JK = −KJ = I KI = −IK = J 〈IX, IY 〉 = 〈JX, JY 〉 = 〈KX,KY 〉 = 〈X, Y 〉 for all X, Y ∈ TM . (b) If φ ∈ Γ(V ), then ∇Xφ ∈ Γ(V ) for all X ∈ TM . Remark 1.1. It follows from (a) that dimM = 4n. A well known fact about 4n-dimensional Riemannian manifold is that it is quaternionic Kähler if and only if its restricted holonomy group is contained in Sp(n)Sp(1). The 4-dimensional Riemannian manifolds with holonomy Sp(1)Sp(1) are simply the oriented Riemanian manifolds, naturally we only consider those when n ≥ 2. Notice that in general I, J,K are not defined everywhere on M . For example, the canonical quaternionic projective space QP n admits no almost complex structure. On the other hand, the vector space generated by I, J,K is well defined at each point of M and this 3-dimensional subbundle V of End(TM) is in fact “globally parallel” under the Levi-Civita connection ∇ of g. A basic fact about the connection is the following lemma. Lemma 1.1. The condition (b) is equivalent to the following condition: ∇XI = c(X)J − b(X)K, ∇XJ = −c(X)I + a(X)K, ∇XK = b(X)I − a(X)J, where a, b, c are local 1-forms. Definition 1.2. Let (M, g) be a quaternionic Kähler manifold. We can define a 4-form by Ω = ω1 ∧ ω1 + ω2 ∧ ω2 + ω3 ∧ ω3, where ω1 = 〈·, I·〉, ω2 = 〈·, J ·〉, ω3 = 〈·, K·〉. Let {e1, Ie1, Je1, Ke1, · · · , en, Ien, Jen, Ken} be an orthonormal basis of TM and {θ1, Iθ1, Jθ1, Kθ1, · · · , θn, Iθn, Jθn, Kθn, } the dual basis. It follows θi ∧ Iθi + Jθi ∧Kθi θi ∧ Jθi +Kθi ∧ Iθi θi ∧Kθi + Iθi ∧ Jθi θi ∧ Iθi ∧ θj ∧ Iθj + θi ∧ Jθi ∧ θj ∧ Jθj + θi ∧Kθi ∧ θj ∧Kθj Jθi ∧Kθi ∧ Jθj ∧Kθj +Kθi ∧ Iθi ∧Kθj ∧ Iθj + Iθi ∧ Jθi ∧ Iθj ∧ Jθj θi ∧ Iθi ∧ Jθj ∧Kθj + θi ∧ Jθi ∧Kθj ∧ Iθj + θi ∧Kθi ∧ Iθj ∧ Jθj Lemma 1.2. The condition (b) is equivalent to the following condition: ∇Xω1 = c(X)ω2 − b(X)ω3, ∇Xω2 = −c(X)ω1 + a(X)ω3, ∇Xω3 = b(X)ω1 − a(X)ω2. where a, b, c are local 1-forms. Proof: It follows from the identities (∇Xω1)(Y, Z) = 〈Y, (∇XI)Z〉, (∇Xω2)(Y, Z) = 〈Y, (∇XJ)Z〉, (∇Xω3)(Y, Z) = 〈Y, (∇XK)Z〉. Using this lemma, we have that Theorem 1.1. The condition (b) is equivalent to that Ω is parallel, that is ∇XΩ = 0 for any X ∈ TM . In the following, we shall study the curvature of quaternionic Kähler manifold. First we have the following lemma. Lemma 1.3. If (M4n, g) is a quaternionic Kähler manifold, then [R(X, Y ), I] = γ(X, Y )J − β(X, Y )K, [R(X, Y ), J ] = −γ(X, Y )I + α(X, Y )K, [R(X, Y ), K] = β(X, Y )I − α(X, Y )J, where α, β and γ are local 2-forms given by α = da+ b ∧ c, β = db+ c ∧ a, γ = dc+ a ∧ b. Corollary 1.1. If (M4n, g) is a quarternionic Kähler manifold, then 〈R(X, Y )Z, IZ〉+ 〈R(X, Y )JZ,KZ〉 = α(X, Y ) |Z|2, 〈R(X, Y )Z, JZ〉+ 〈R(X, Y )KZ, IZ〉 = β(X, Y ) |Z|2, 〈R(X, Y )Z,KZ〉+ 〈R(X, Y )IZ, JZ〉 = γ(X, Y ) |Z|2. The following lemma is the key for quaternionic Kähler manifolds. Lemma 1.4. If (M4n, g) is a quaternionic Kähler manifold and n ≥ 2, then α(X, IY ) = β(X, JY ) = γ(X,KY ) = − n + 2 RicM(X, Y ). (1.1) As applications of the above lemma, one can show the following two main theorems on curvature of quaternionic Kähler manifolds. Theorem 1.2. If (M4n, g) is a quaternionic Kähler manifold and n ≥ 2, then (M4n, g) is Einstein, that is, there is a constant δ such that RicM(g) = 4(n+ 2)δg. Theorem 1.3. If (M4n, g) is a quaternionic Kähler manifold and n ≥ 2, (1) For any tangent vector X, the sectional curvature satisfies 〈R(X, IX)IX,X〉+ 〈R(X, JX)JX,X〉 +〈R(X,KX)KX,X〉 = 12δ |X|4. (2) For any tangent vector Y satisfying 〈Y,X〉 = 〈Y, IX〉 = 〈Y, JX〉 = 〈Y,KX〉 = 0, the sectional curvature satisfies 〈R(X, Y )Y,X〉+ 〈R(X, IY )IY,X〉+ 〈R(X, JY )JY,X〉+ 〈R(X,KY )KY,X〉 = 4δ |X|2 |Y |2, where 4(n+ 2)δ is the Einstein constant. Finally, we end this section with the following lemma. Lemma 1.5. Let γ : [a, b] → M be a geodesic with unit speed. If S = 16n(n + 2)δ, and XI(t), XJ(t), XK(t) are parallel vector fields along γ such that XI(a) = Iγ ′(a), XJ(a) = Jγ ′(a), XK(a) = Kγ ′(a), then K(γ′(t), XI(t)) +K(γ ′(t), XJ(t)) +K(γ ′(t), XK(t)) = 12δ, for all t and γ. Let Y be a tangent vector at γ(a) satisfying 〈γ′(a), Y 〉 = 0, 〈Iγ′(a), Y 〉 = 0, 〈Jγ′(a), Y 〉 = 0, and 〈Kγ′(a), Y 〉 = 0. If we denote the parallel vector fields Y (t), YI(t), YJ(t), and YK(t) along γ with initial data Y (a) = Y , YI(a) = IY, YJ(a) = JY , and YK(a) = KY , respectively, then K(γ′(t), Y (t)) +K(γ′(t), YI(t)) +K(γ ′(t), YJ(t)) +K(γ ′(t), YK(t)) = 4δ, for all t and γ. Proof. By the discussion above, we know the 3-dimensional vector space E(t) spanned by X(t), Y (t), Z(t) does not depend on the choice of I, J,K. Hence it is parallel under the Levi-Civita connection. We consider 〈R(·, γ′(t))γ′(t), ·〉 as a symmetric bilinear form on E(t). ThenK(γ′(t), X(t))+ K(γ′(t), Y (t)) + K(γ′(t), Z(t)) is its trace on E(t) which independent of the choice of orthonormal basis. By the computation above it is equal to 12δ. The same argument also applies to the second part of the lemma. � 2 Laplacian comparison theorem For a complete Riemannian manifold M and p ∈ M , let us denote the cut locus with respect to p by Cut(p). Theorem 2.1. Let (M4n, g) be a complete quaternionic Kähler manifold with scalar curvature SM ≥ 16n(n+2)δ and let r(x) be the distance function to a fixed point p ∈M . Then, for x /∈ Cut(p), ∆r(x) ≤ 6 coth 2r(x) + 4(n− 1) coth r(x) when δ = −1 (4n− 3)r−1(x) when δ = 0 6 cot 2r(x) + 4(n− 1) cot r(x) when δ = 1. (2.1) Proof. Let γ be the minimizing geodesic joining p to x. At x, we choose {e1, e2, · · · , en}, and two local almost complex structures I, J and K = IJ such that e1 = ∇r and {e1, Ie1, Je1, Ke1, e2, Ie2, Je2, Ke2, · · · , en, Ien, Jen, Ken} is an orthonormal frame. By parallel translating along γ we obtain an or- thonormal frame with e1 = ∇r. For convenience sake, we denote this frame by {ε1, ε2, · · · , ε4n}. Since |∇r| 2 = 1 on M\Cut(p), by taking covariant derivative of this equation, we have 0 = |∇r|2kl rikril + 2 ririkl, (2.2) for each k, l = 2, · · · , 4n. Since rikl = rkli + Rjkilrj, with Rijkl = 〈R(εi, εj)εl, εk〉, and r1 = 1, rj = 0, j = 2, · · · , 4n, we have rikril + rkl1 +R1k1l = 0. (2.3) In particular, if k = l, we have r2ik + rkk1 +K(ε1, εk) = 0, (2.4) where K(ε1, εk) = R1k1k is the sectional curvature of the 2-plane section spanned by ε1, εk. Using the inequality r2ik ≥ and setting f(t) = k=2 rkk, ( 2.4) implies that f ′(t) + f 2(t) + K(ε1, εk) ≤ 0. (2.5) By Lemma 1.5, we have f ′(t) + f 2(t) + 12δ ≤ 0. (2.6) Since a smooth Riemannian metric is locally Euclidean, then limt→0 tf(t) = 3. By a standard comparison argument for ordinary differential equations, we conclude that f(t) ≤ 6 cot 2t when δ = 1 3t−1 when δ = 0 6 coth 2t when δ = −1. (2.7) Similarly, using the inequality k=4i+1 r2ik ≥ k=4i+1 for 1 ≤ i ≤ n− 1, and setting hi(t) = ∑4i+4 k=4i+1 rkk, ( 2.4) implies that h′i(t) + h2i (t) + k=4i+1 K(ε1, εk) ≤ 0. (2.8) Together with Lemma 1.5 asserting that k=4i+1 K(ε1, εk) = 4δ, we have h′i(t) + h2i (t) + 4δ ≤ 0. (2.9) Hence, as before, we conclude that hi(t) ≤ 4 cot t when δ = 1 4t−1 when δ = 0 4 coth t when δ = −1. (2.10) The result follows from the equation ∆r(x) = f(r(x)) + i=1 hi(r(x)). � Remark 2.1. The estimate in Theorem 2.1 is sharp since the right hand sides are exactly the Laplacian of the distance functions of quaternionic hy- perbolic space QHn, quaternionic Euclidean space Qn and quaternionic pro- jective space QPn respectively. Remark 2.2. We actually proved the estimate for Hessian of the distance function. In particular, rkk ≤ 6 cot 2t when δ = 1 3t−1 when δ = 0 6 coth 2t when δ = −1. (2.11) Also for 1 ≤ i ≤ n− 1, we have k=4i+1 rkk ≤ 4 cot 2t when δ = 1 4t−1 when δ = 0 4 coth 2t when δ = −1. (2.12) Corollary 2.1. Let (M4n, g) be a complete quaternionic Kähler manifold with scalar curvature SM ≥ −16n(n + 2). Then for any point x ∈ M and r > 0, the area A(r) of the geodesic spheres centered at x satisfies A′(r) ≤ 6 coth 2r + 4(n− 1) coth r. (2.13) In particular, A(r) ≤ C(sinh 2r)3(sinh r)4(n−1) ≤ Ce(4n+2)r. Corollary 2.2. Let (M4n, g) be a complete quaternionic Kähler manifold with scalar curvature SM ≥ −16n(n + 2). Then for any point x ∈ M and 0 < r1 ≤ r2, the volume of the geodesic balls centered at x satisfies Vx(r2) Vx(r1) VQHn(r2) VQHn(r1) , (2.14) where VQHn(r) denotes the volume of the geodesic ball of radius r in QH n. In particular, λ1(M) ≤ (2n+ 1) Corollary 2.3. Let (M4n, g) be a complete quaternionic Kähler manifold with scalar curvature SM ≥ 16n(n+2) . Then it is compact, and the diameter d(M) ≤ π , which is the diameter of the model space QPn. Moreover, the volume of M is bounded by V (M) ≤ V (QPn), (2.15) where VQPn is the volume of QP 3 Quaternionic harmonicity In this section we will derive an over-determined system of harmonic func- tions with finite Dirichlet integral on a manifold with a parallel form. This result was first proved by Siu [S] for harmonic maps in his proof of the rigidity theorem for Kähler manifolds. Corlette [C1] gave a more systematic approach for harmonic map with finite energy from a finite-volume quaternionic hy- perbolic space or Cayley hyperbolic plane to a manifold with nonpositive curvature. In [L], the second author generalized Siu’s argument to harmonic functions with finite Dirichlet integral on a Kähler manifold. We will pro- vide an argument that generalizes Corlette’s argument to harmonic functions with finite Dirichlet integral on a complete manifold with a parallel form. We believe that it should be of independent interest. Theorem 3.1. Let M be a complete Riemannian manifold with a parallel p-form Ω. Assume that f is a harmonic function with its Dirichlet integral over geodesic balls centered at o of radius R satisfying the growth condition Bo(R) |∇f |2dv = o(R2) as R → ∞, then f satisfies d ∗ (df ∧ Ω) = 0. (3.1) Before we prove the theorem, let us first recall the following operators and some of the basic properties. For an oriented real vector space V with an inner product, we have the Hodge star operator ∗ : ∧pV → ∧n−pV. For any θ ∈ ∧1V and v ∈ V , we also have exterior multiplication and interior product operators ε(θ) : ∧pV → ∧p+1V, ℓ(v) : ∧pV → ∧p−1V. For θ ∈ ∧1V and v ∈ V is the dual of θ by the inner product, if ξ ∈ ∧pV we list the following identities among the operators: 1. ∗ ∗ ξ = (−1)p(n−p)ξ, 2. ∗ε(θ)ξ = (−1)pℓ(v) ∗ ξ, 3. ε(θ) ∗ ξ = (−1)p−1 ∗ ℓ(v)ξ, 4. ∗ε(θ) ∗ ξ = (−1)(p−1)(n−p)ℓ(v)ξ, 5. ℓ(v)ε(θ′)ξ + ε(θ)ℓ(v′)ξ = 0, where v⊥v′, 6. ℓ(v)ε(θ)ξ + ε(θ)ℓ(v)ξ = ξ. We are now ready to prove Theorem 3.1. Proof of Theorem 3.1. Let η : [0,+∞) → R be a smooth function satis- fying η′(t) ≤ 0, and η(t) = 1 when t ∈ [0, 1] 0 when t ∈ [2,+∞]. For R ≥ 1, we define the cut-off function φR(x) = η(r(x)/R), where r(x) is the distance function from a fixed point o ∈ M , then there is a positive constant C1 depending on η and C such that |∇φR(x)| ≤ C1R Since d2 = 0, then φ2R ∗ (df ∧ Ω) ∧ d ∗ (df ∧ ∗Ω) d(φ2R) ∧ ∗(df ∧ Ω) ∧ d ∗ (df ∧ ∗Ω) φ2R d ∗ (df ∧ Ω) ∧ d ∗ (df ∧ ∗Ω). (3.2) We claim that ∗ d ∗ (df ∧ Ω) = (−1)n−1d ∗ (df ∧ ∗Ω). (3.3) In fact, for any point x ∈ M , we can choose an orthonormal tangent basis i=1 in a neighborhood of x such that ∇eiej(x) = 0. Denote by {θ i}mi=1 the dual basis of {ei} i=1. Then for ω ∈ ∧ p(T ∗M) we have dω = ε(θi)∇eiω. Hence d ∗ (df ∧ ∗Ω) = d ∗ ε(df) ∗ Ω = (−1)(p−1)(m−p)d[ℓ(∇f)Ω] = (−1)(p−1)(m−p) ε(θi)∇ei(ℓ(∇f)Ω) = (−1)(p−1)(m−p) i,j=1 ε(θi)(∇ei∇ejf)(ℓ(ej)Ω) = (−1)(p−1)(m−p) i,j=1 fijε(θi)(ℓ(ej)Ω), where fij = ∇ei∇ejf and the facts Ω is parallel and ∇eiej(x) = 0 have been used. On the other hand, ∗d ∗ (df ∧ Ω) = ∗d ∗ ε(df)Ω ε(θi)∇ei(∗ε( fjθj)d[ℓ(∇f)Ω] i,j=1 fijε(θi) ∗ ε(θj)(Ω) = (−1)p(m−p−1) i,j=1 fijℓ(ei)ε(θj)Ω = (−1)p(m−p−1) fiiℓ(ei)ε(θi)Ω + i 6=j fijℓ(ei)ε(θj)Ω = (−1)p(m−p−1) fii[Ω− ε(θi)ℓ(ei)Ω]− i 6=j fijε(θj)ℓ(ei)Ω = (−1)p(m−p−1) i,j=1 fijε(θi)(ℓ(ej)Ω), (3.4) where we used fij = fji and i=1 fii = 0. So the claim is proved. By ( 3.2), we have φ2R |d ∗ (df ∧ Ω)| = (−1)m d(φ2R) ∧ ∗(df ∧ Ω) ∧ d ∗ (df ∧ ∗Ω) φ2R |d ∗ (df ∧ Ω)| |dφR| 2| ∗ (df ∧ Ω)|2dv (3.5) On the other hand, ( 3.3) and the fact that ω is bounded imply that there exists a constant C2 > 0, such that | ∗ (df ∧ Ω)| ≤ C2 |df | |d ∗ (df ∧ ∗Ω)| = |d ∗ (df ∧ Ω)|. Hence combining with ( 3.5) and using the definition of φR we conclude that Bo(R) |d ∗ (df ∧ Ω)|2dv ≤ C1R Bo(2R) |df |2dv. The assumption on the growth of the Dirichlet integral of f implies that the right hand side tends to zero as R → ∞. Therefore d ∗ (df ∧Ω) = 0, and the proof is complete. � Lemma 3.1. Let (M4n, g) be a quarternionic Kähler manifold and n ≥ 2. If f is a function on M satisfying d ∗ (df ∧ Ω) = 0 (3.6) for the 4-form Ω determined by the quaternionic Kähler structure, then f is quaternionic harmonic, namely, for any nonzero tangent vector X, fX,X + fIX,IX + fJX,JX + fKX,KX = 0 where fX,X = ∇df(X,X). Proof: Let A=1 = {e1, e2, · · · , en, Ie1, Ie2, · · · , Ien, Je1, Je2, · · · , Jen, Ke1, Ke2, · · · , Ken} be an orthonormal basis of TM and {ωA} the dual basis with e1 = . Since Ω is parallel, by ( 3.4) and ( 3.6), we have (∇eAdf) ∧ ℓ(eA)Ω A,B=1 feA,eB ωB ∧ ℓ(eA)Ω where we have used the fact that f is a harmonic function. Hence equation ( 3.6) implies A,B=1 feA,eB ωB ∧ ℓ(eA)Ω = 0 Comparing the coefficient of ωi∧Iωi∧Jωi∧Kωi on both sides by the explicit formula for Ω given before, we obtain that 6 (fei,ei + fIei,Iei + fJei,Jei + fKei,Kei) = 0 for all ei, (1 ≤ i ≤ n). So the proof is complete. � The following corollary is an immediate consequence of the lemma. Corollary 3.1. Let M4n be a complete quaternionic Kähler manifold. As- sume that f is a harmonic function with its Dirichlet integral satisfying the growth condition Bo(R) |∇f |2dv = o(R2) as R → ∞, then f must satisfy d ∗ (df ∧ Ω) = 0, (3.7) where Ω is the parallel 4-form determined by the quaternionic Kähler struc- ture. Moreover, f is quaternionic harmonic. 4 Uniqueness of infinite volume end Recall that for any complete manifold if λ1(M) > 0 then M must be non- parabolic. In particular,M must have at least one nonparabolic ends. It was also proved in [LW1] that under the assumption that λ1(M) > 0, an end is nonparabolic if and only if it has infinite volume. Let us assume that M has at least two nonparabolic ends, E1 and E2. A construction of Li-Tam [LT] asserts that one can construct a nonconstant bounded harmonic function with finite Dirichlet integral. The harmonic func- tion f can be obtained by taking a convergent subsequence of the harmonic functions fR, as R→ +∞, satisfying ∆fR = 0 on B(R), with boundary conditions fR = 1 on ∂B(R) ∩ E1 fR = 0 on ∂B(R) \ E1. It follows from the maximum principle that 0 ≤ fR ≤ 1, hence 0 ≤ f ≤ 1. We need the following estimates from [LW1](Lemma 1.1 and 1.2 in [LW1]), and [LW3](Lemma 5.1 in [LW3]). Lemma 4.1. Let M be a complete Riemannian manifold with λ1(M) > 0. Suppose M has at least two nonparabolic ends and E be an end of M . Then for the harmonic function f constructed above, it must satisfy the following growth conditions: 1. There exists a constant a such that f − a ∈ L2(E). Moreover, the function f − a must satisfy the decay estimate E(R+1)\E(R) (f − a)2 ≤ C exp(−2 λ1(E)R) for some constant C > 0 depending on f , λ1(E) and the dimension of 2. The Dirichlet integral of the function f must satisfy the decay estimate E(R+1)\E(R) |∇f |2 ≤ C exp(−2 λ1(E)R), exp(−2 λ1(E)r(x))|∇f | 2 ≤ CR for R sufficiently large. Lemma 4.2. Let M be a complete Riemannian manifold with at least two nonparabolic ends and λ1(M) > 0. Then for the harmonic function f con- structed above, for any t ∈ (inf f, sup f) and (a, b) ⊂ (inf f, sup f), L(a,b) |∇f |2 = (b− a) |∇f |, where l(t) = {x ∈M |f(x) = t}, L(a, b) = {x ∈M |a < f(x) < b}. Moreover, |∇f | = |∇f |. We are now ready to prove Theorem 0.5. Proof of Theorem 0.5: Suppose to the contrary that there exist two ends E1 and E2 with infinite volume. The assumption that λ1(M) > 0 implies that they are nonparabolic. By the construction above, there exists a harmonic function f with finite energy such that lim inf x→∞, x∈E1 f(x) = 1 lim inf x→∞, x∈E2 f(x) = 0. The Bochner formula implies that ∆|∇f |2 = RicM(∇f,∇f) + |∇ 2f |2. (4.1) We now choose an orthonormal basis {eA} A=1 satisfying {e1, e2, · · · , en, Ie1, Ie2, · · · , Ien, Je1, Je2, · · · , Jen, Ke1, Ke2, · · · , Ken} with e1 = |∇f | . Corollary 3.1 implies that f(in+1)(in+1) = 0. Therefore, applying the arithmetic-geometric means, we have |∇2f |2 = A,B=1 f 2AB ≥ f 211 + f 2(in+1)(in+1) + 2 f 21A ≥ f 211 + f(in+1)(in+1)) 2 + 2 f 21A |∇|∇f ||2, (4.2) hence combining with ( 4.1) we obtain ∆|∇f |2 ≥ −4(n + 2)|∇f |2 + |∇|∇f ||2. (4.3) If we write u = |∇f | 3 , then ∆u ≥ − 8(n+ 2) u. (4.4) We want to prove that the above inequality is actually an equality. The argument follows from that in [LW4] after making suitable modification to fit our situation. For any compactly supported smooth function φ on M , we 8(n+ 2) φu〈∇u,∇φ〉 − φ2|∇u|2 + λ1(M) (φu)2 φu〈∇u,∇φ〉 − φ2|∇u|2 + |∇(φu)|2 |∇φ|2u2. (4.5) Let us choose φ = ψχ to be the product of two compactly supported functions. For any ε ∈ (0, 1 ), we define χ(x) = 0 on L(0, σε) ∪ L(1− ε (log 2)−1(log f − log( ε )) on L( ε , ε) ∩ (M \E1) (log 2)−1(log(1− f)− log( ε )) on L(1− ε, 1− ε ) ∩ E1 1 otherwise. For R > 1 we define 1 on B(R− 1) R− r on B(R) \B(R− 1) 0 on M \B(R). Applying to the right hand side of ( 4.5), we obtain |∇φ|2u2 ≤ 2 |∇ψ|2χ2|∇f | 3 + 2 |∇χ|2ψ2|∇f | 3 . (4.6) Since RicM ≥ −4(n+2), then the local estimate of Cheng-Yau [CY] (see also [LW2]) implies that there exists a constant depending on n such that |∇f |(x) ≤ C|1− f(x)|. On E1, the first term of ( 4.6) satisfies |∇ψ|2χ2|∇f | |∇f |2 , (4.7) where Ω = E1 ∩ (B(R) \B(R − 1)) ∩ (L(1− ε, 1− ) ∪ L( ε , ε). Since 1 ≤ 4 (1− f)2 (1− f)2 ≤ 4Cε−2 exp(−2 λ1R), where in the last inequality we have used Lemma 4.1. Again by Lemma 4.1, from ( 4.7) we have |∇ψ|2χ2|∇f | 3 ≤ Cε− 3 exp(−2 λ1R). (4.8) For the second term of ( 4.6) we have |∇χ|2ψ2|∇f | ≤ (log 2)−2 L(1−ε,1− ε )∩E1∩B(R) |∇f | +2(1− f)−2 ≤ C(log 2)−2 L(1−ε,1− ε )∩E1∩B(R) |∇f |2(1− f)− Using the co-area formula and Lemma 4.2 we have L(1−ε,1− ε )∩E1∩B(R) |∇f |2(1− f)− ∫ 1− ε (1− t)− l(t)∩E1∩B(R) |∇f |dAdt |∇f |dA ∫ 1− ε (1− t)− = −3C[(1− t) |∇f |dA = 3Cε |∇f |dA. Combining the above inequality with ( 4.8) we have |∇φ|2u2 ≤ C(ε 3 exp(−2 λ1R) + ε 3 ). (4.9) A similar argument using f instead of 1 − f on the other end yields the estimate |∇φ|2u2 ≤ C(ε 3 exp(−2 λ1R) + ε Letting R → ∞ and ε→ 0, we have ∆u = − 8(n + 2) u (4.10) with λ1(M) = 8(n+2) , since f is nonconstant and u cannot be identically zero. Therefore all the inequalities used to prove ( 4.4) are equalities. Thus there exists a function µ, such that, (fAB) = , (4.11) where D1 and D2 are n× n matrices defined by · · · · · · Since f1α = 0 for α 6= 1 implies that |∇f | is constant along the level set of f . Moreover, regularity of the equation ( 4.10) implies that |∇f | can never be zero. Hence M must be diffeomorphic to R×N , where N is given by the level set of f . Also N must be compact since we assume that M has at least 2 ends. Fix a level set N0 of f , consider (−ε, ε)×N0 ⊂M . Note that {eA} is an orthonormal basis of TM such that e1 is the normal vector to N0 and {eα} are the tangent vectors of N0. We shall compute the sectional curvature K(e1, eα) = 〈R(e1, eα)eα, e1〉. We claim that ∇e1e1 = 0. Indeed it suffices to prove all integral curves η(t) of the vector field e1 = |∇f | emanating from N0 are geodesics. For any point η(t0), let γ be the geodesic realizing the distance between η(t0) and N0. Then γ is perpendicular to every level set Nt. So γ ′ is parallel to e1 along γ. This implies γ coincides with the integral curve of e1. Let (hαβ) with 2 ≤ α, β ≤ 4n be the second fundamental form of the level set of f . Then hαβ f1 = −fαβ , (4.12) ∇eαe1 = − hαβeβ. By the definition of curvature tensor, we have 〈R(e1, eα)e1, eα〉 = 〈∇e1∇eαe1 −∇eα∇e1e1 −∇[e1,eα]e1, eα〉 = 〈∇e1∇eαe1, eα〉 − 〈∇[e1,eα]e1, eα〉 = 〈∇e1∇eαe1, eα〉 − 〈∇∇e1eα−∇eαe1e1, eα〉 = 〈∇e1∇eαe1, eα〉 − 〈∇e1eα, eβ〉〈∇eβe1, eα〉 〈∇eαe1, eβ〉〈∇eβe1, eα〉 〈∇e1(hαβeβ), eα〉+ hαβ〈∇e1eα, eβ〉+ 〈(e1hαβ)eβ , eα〉 − hαβ〈∇e1eβ, eα〉 hαβ〈∇e1eα, eβ〉+ h = −e1hαα + 2 hαβ〈∇e1eα, eβ〉+ h Therefore K(e1, eα) = e1hαα − 2 hαβ〈∇e1eα, eβ〉 − h2αβ . (4.13) Since hαβ is diagonal, this implies that K(e1, eα) = e1hαα − h Combining with ( 4.11) and ( 4.12), we conclude that K(e1, e2) = K(e1, Ie2) = K(e1, Je2) = K(e1, Ke2) = 0 which implies M is Ricci flat by Theorem 1.3. This contradicts to the as- sumption that λ1 > 8(n+2) > 0. Therefore M must have only one end with infinite volume. � 5 Maximal first eigenvalue In this section, we will consider the case when λ1(M) is of maximal value. Proof of Theorem 0.6: According to Theorem 0.5, we know that M has exactly one nonparabolic end. Suppose thatM has more than one end. Then there must exist at least an end with finite volume. We divide the rest of the proof into several parts. The first part follows exactly as that in the proof of the corresponding theorem in the Kähler case (Theorem 3.1) in [LW5]. For completeness sake, we will give a quick outline of it. Part 1. Assume that E1 is such an end with finite volume given by M \ Bp(1). Then we can choose a ray η : [0,+∞) such that η(0) = p and η[1,+∞) ⊂ E1. The Busemann function corresponding to γ is defined by β(x) = lim [t− d(x, η(t))]. The Laplacian comparison theorem, Theorem 2.1, asserts that ∆β ≥ −2(2n+ 1) in the sense of distribution. We define the function f = exp((2n+ 1)β), and using the fact that |∇β| = 1 almost everywhere, we have ∆f = (2n+ 1) exp((2n+ 1)β)∆β + (2n+ 1)2 ≥ −(2n + 1)2f. Similar to the proof of above theorem, we conclude that for any compactly supported function φ, (∆f + (2n+ 1)2f)fφ2 f 2|∇φ|2. By choosing the function φ to be 1, on Bp(R); 2R−r(x) , on Bp(2R) \Bp(R); 0, on M \Bp(2R); we obtain f 2|∇φ|2 (Bp(2R)\Bp(R))∩E1 (Bp(R+i)\Bp(R+i−1))∩E1 (VE1(R + i)− VE1(R + i− 1)) exp(2(2n+ 1)(R+ i)) where VE1(R+ i) denotes the volume of the set E1∩Bp(R+ i). On the other hand, the volume estimate in Theorem 1.4 of [LW1] implies that VE1(∞)− VE1(R) ≤ C exp(−2(2n+ 1)R), hence VE1(R + i)− VE1(R + i− 1) = VE1(∞)− VE1(R + i− 1)− (VE1(∞)− VE1(R + i)) ≤ C exp(−2(2n+ 1)(R + i)). Therefore, we conclude that f 2|∇φ|2 ≤ Let us now denote E2 = M \ (Bp(1) ∪ E1) to be the other end of M . When x ∈ E2, following the argument in Theorem 3.1 of [LW4], we have β(x) ≤ −d(p, x) + 2. Therefore f 2|∇φ|2 ≤ (Bp(2R)\Bp(R))∩E2 (Bp(2R)\Bp(R))∩E2 exp(−2(2n+ 1)(r − 2)) Letting R → +∞, we conclude that ∆f + (2n+ 1)2f = 0, (5.1) and all inequalities used are indeed equalities and f is smooth by regularity of the equation ( 5.1). Moreover, |∇β| = 1, and ∆β = −2(2n+ 1). This implies that M must be diffeomorphic to R × N , where N is given by the level set of β. We choose an orthonormal basis {ei} i=1 as follows {e1, e2, · · · , en, Ie1, Ie2, · · · , Ien, Je1, Je2, · · · , Jen, Ke1, Ke2, · · · , Ken} with e1 = ∇β. Applying the Bochner formula to β, we get ∆|∇β|2 i,j=1 β2ij + RicM(∇β,∇β) + βi(∆β)i i,j=1 β2ij − 4(n+ 2). By the comparison theorem, we have, β(in+1)(in+1) = −6. Hence (βαβ) = where D1 and D2 are n× n matrices defined by · · · · · · Part 2. For a fix level set N0 of β, we consider (−ε, ε)×N0 ⊂M . Note that {ei} is an orthonormal basis of TM such that e1 is the normal vector to N0 and {eα}, for 2 ≤ α ≤ 4n, are the tangent vectors of N0. We shall compute the sectional curvature K(e1, eα) = 〈R(e1, eα)eα, e1〉. Since ∇e1e1 = 0 implies that the integral curves of e1 are geodesics. Let (hαγ) be the second fundamental form of the level set of ∇β. Then hαγ = 〈∇eαeγ, e1〉 = 〈∇eαeγ,∇β〉 = −βαγ ∇eαe1 = − hαγeγ. (5.2) By ( 4.13) in the proof of Theorem 0.5 we have 〈R(e1, eα)e1, eα〉 = −e1hαα + 2 hαγ〈∇e1eγ, eβ〉+ h2αγ . Since (hαγ) are constant and diagonal, then K(e1, eα) = −h In particular, we have K(e1, eα) = −4 when α = in+ 1, i = 1, 2, 3 −1 otherwise. On the other hand, we also have K(en+1, e2n+1) +K(en+1, e3n+1) = −12−K(e1, en+1) = −8 K(en+1, e2n+1) +K(e3n+1, e2n+1) = −8 K(e3n+1, e2n+1) +K(en+1, e3n+1) = −8, hence K(en+1, e2n+1) = K(en+1, e3n+1) = K(e2n+1, e3n+1) = −4. Since for α = 2, 3, · · · , n, K(Ie1, eα) = −〈R(Ie1, eα)Ie1, eα〉 = −〈IR(Ie1, eα)Ie1, Ieα〉 = 〈R(Ie1, eα)e1, Ieα〉 = 〈R(e1, Ieα)Ie1, eα〉 = K(e1, Ieα) = −1, and K(Je1, eα) = K(Ke1, eα) = −1, we have K(ein+1, eα) = −1, for all i = 0, 1, 2, 3 and α 6= 1, n+ 1, 2n+ 1, 3n+ 1. Let KN(eα, eγ) denote the sectional curvature of the level set with induced metric. By Gaussian equation, KN (eα, eγ)−K(eα, eγ) = hααhγγ, it is straightforward to obtain KN (en+1, e2n+1) = K N (en+1, e3n+1) = K N(e2n+1, e3n+1) = 0, KN(ein+1, eα) = 1, (5.3) for all i = 1, 2, 3 and α 6= 1, n+ 1, 2n+ 1, 3n+ 1. Part 3. There is a natural map ϕt between the level sets N0 and Nt given by the gradient flow of β. Since the integral curves are geodesics, dϕt(X) are Jacobi fields along corresponding curves. Let (N, g0) = N0 with the induced metric. We can consider ϕ as a flow on N . We claim that dϕt|V1 = e 2t id dϕt|V2 = e t id, where TN = V1 ⊕ V2, V1 = span{Ie1, Je1, Ke1} and V2 = V 1 . Indeed for any point q ∈ N0, denote e1(t) = ∇β(ϕ(t)) and {εα(t)} α=2 to be the parallel transport of the orthonormal base {eα} α=2 of N0 at q along ϕt(q). Since both V1 and V2 are ϕ-invariant, we have, in particular, 〈∇e1(t)εα, εγ〉 = 0, (5.4) when α ∈ {n+ 1, 2n+ 1, 3n+ 1}, and γ /∈ {n + 1, 2n+ 1, 3n+ 1}. Now we can compute R1α1γ . Then 〈R(e1, εα)e1, εγ〉 = 〈∇e1∇εαe1 −∇εα∇e1e1 −∇[e1,εα]e1, εγ〉 = 〈∇e1∇εγe1, εα〉 − 〈∇[e1,εα]e1, εγ〉 = 〈∇e1∇εαe1, εγ〉 − 〈∇∇e1εα−∇εαe1e1, εγ〉 = 〈∇e1∇εαe1, εγ〉 − 〈∇e1εα, ετ〉〈∇ετ e1, εγ〉 〈∇εαe1, ετ 〉〈∇ετe1, εγ〉 〈∇e1(hατετ ), εγ〉+ hγτ 〈∇e1εα, ετ 〉+ hατhτγ = −e1hαγ − hατ 〈∇e1ετ , εγ〉 hγτ 〈∇e1εα, ετ 〉+ hατhτγ. (5.5) We see that (hαγ) is diagonal and hαα = 2, when α = n + 1, 2n+ 1, 3n+ 1; 1, otherwise. Therefore, when α 6= γ, R1α1γ = −hαα〈∇e1εα, εγ〉+ hγγ〈∇e1εα, εγ〉 = (hγγ − hαα)〈∇e1εα, εγ〉. Since hαα = hγγ when α, γ ∈ {n+ 1, 2n+ 1, 3n+ 1} and α, γ /∈ {n+ 1, 2n+ 1, 3n+ 1}, using ( 5.4), we have R1α1γ = 0, for all α 6= γ. Define Jα(t) = e−2tεα, when α ∈ {n+ 1, 2n+ 1, 3n+ 1}; e−tεα, when α /∈ {n+ 1, 2n+ 1, 3n+ 1}. Since dϕt(eα)|t=0 = [e1, eα] = −∇eαe1, then we see that Jα satisfies the Jacobi equation and initial conditions Jα(0) = eα and J α(0) = eα = ∇ ∂ dϕt(eα)|t=0. By the uniqueness theorem for the Jacobi equations, we have dϕt(eα) = Jα. The claim is proved. Part 4. We have now a family of metrics on N written as ds2t = e ω2in+1 + e ω2in+α, and the metric of M can rewritten as ds2 = dt2 + e4t ω2p + e ω2α (5.6) where {ω2, ω3, ω4, . . . , ω4n} is the dual coframe to {e2, e3, e4, . . . , e4n} at N0. We also choose that Ie4s−3 = e4s−2, Je4s−3 = e4s−1, and Ke4s−3 = e4s for s = 1, . . . , n, with e1 = . In particular, the second fundamental form on Nt must be a diagonal matrix when written in terms of the basis {ei} i=2 with eigenvalues given by (〈∇eiej, e1〉) = 2I3 0 0 I4(n−1) , (5.7) where Ik denotes the k× k identity matrix. Also, the sectional curvatures of the sections containing e1 are given by K(e1, ep) = −4 for 2 ≤ p ≤ 4 K(e1, eα) = −1 for 5 ≤ α ≤ 4n. The Guass curvature equation also asserts that Rijkl = R̄ijkl + hlihkj − hkihlj, where R̄ijkl is the curvature tensor on Nt. In particular, Rijkl = R̄ijkl + δliδkj − δkiδlj if 5 ≤ i, j, k, l ≤ 4n R̄ijkl + 4δliδkj − 4δkiδlj if 2 ≤ i, j, k, l ≤ 4 R̄ijkl + 2 if 2 ≤ i = l ≤ 4 and 5 ≤ k = j ≤ 4n R̄ijkl + 2 if 2 ≤ k = j ≤ 4 and 5 ≤ i = l ≤ 4n R̄ijkl − 2 if 2 ≤ i = k ≤ 4 and 5 ≤ j = l ≤ 4n R̄ijkl − 2 if 2 ≤ j = l ≤ 4 and 5 ≤ i = k ≤ 4n R̄ijkl − 2 if 2 ≤ k = i ≤ 4 and 5 ≤ j = l ≤ 4n R̄ijkl otherwise. (5.8) We will now use ( 5.6) to compute the curvature tensor of M and hence N0. Using the orthonormal coframe η1 = ω1 = dt, ηp = e ηα = e for 2 ≤ p ≤ 4 and 5 ≤ α ≤ 4n, we obtain the first structural equations dη1 = 0, (5.9) dηp = 2e 2t ω1 ∧ ωp + e ωpq ∧ ωq + e ωpα ∧ ωα = −2ηp ∧ η1 + ωpq ∧ ηq + e ωpα ∧ ηα, (5.10) dηα = e t ω1 ∧ ωα + e ωαp ∧ ωp + e ωαβ ∧ ωβ = −ηα ∧ η1 + e ωαp ∧ ηp + ωαβ ∧ ηβ, (5.11) where ωij are the connection forms of N0. In the above and all subsequent computations, we will adopt the convention that 5 ≤ α, β ≤ 4n, 2 ≤ i, j ≤ 4n, 2 ≤ o, p, q, r ≤ 4, 2 ≤ s, t ≤ n, and 1 ≤ A,B ≤ 4n. Note that using the endomorphism I and the fact that ∇I = cJ − bK, we have ωij(X) = 〈∇̄Xej, ei〉 = 〈I∇Xej , Iei〉 = 〈∇XIej, Iei〉+ c(X) 〈Jej, Iei〉 − b(X) 〈kej , Iei〉 = 〈∇XIej, Iej〉+ c(X) 〈ej, Kei〉+ b(X)〈ej , Jei〉 for any tangent vector X to N0, where ∇̄ denotes the connection on N0. Hence we conclude that ωij = ωIiIj + c 〈ej, Kei〉+ b 〈ej, Jei〉, (5.12) where Ii denotes the index corresponding to Iei = eIi. Similarly, we have ωij = ωJiJj + c 〈ej, Kei〉+ a 〈ej, Iei〉, ωij = ωKiKj + b 〈ej, Jei〉+ a 〈ej , Iei〉. Together with ( 5.7), we conclude that ω2(4s−1)(e4s) = −1 = −ω2(4s)(e4s−1), ω2(4s−3)(e4s−2) = −1 = −ω2(4s−2)(e4s−3), for all 2 ≤ s ≤ n, and ω2α(eβ) = 0 otherwise. Similarly, ω2α(ep) = 〈∇epeα, e2〉 = −〈∇epIeα, e1〉 These identities imply that ω2(4s−3) = −ω4s−2, ω2(4s−2) = ω4s−3, ω2(4s−1) = −ω4s, ω2(4s) = ω4s−1. (5.13) A similar calculation using the endomorphisms J and K yield ω3(4s−3) = −ω4s−1, ω3(4s−2) = ω4s, ω3(4s−1) = ω4s−3, ω3(4s) = −ω4s−2, (5.14) ω4(4s−3) = −ω4s, ω4(4s−2) = −ω4s−1, ω4(4s−1) = ω4s−2, ω4(4s) = ω4s−3. (5.15) We claim that the connection forms are given by η1p = −ηp1 = 2ηp for 2 ≤ p ≤ 4, (5.16) η1α = −ηα1 = ηα for 5 ≤ α ≤ 4n, (5.17) ηpq = −ηqp = ωpq, (5.18) ηpα = −ηαp = e t ωpα, (5.19) η(4s)β = −ηβ(4s) ω(4s)β − (1−e −2t) η2 if β = 4s− 1 ω(4s)β + (1−e −2t) η3 if β = 4s− 2 ω(4s)β − (1−e −2t) η4 if β = 4s− 3 ω(4s)β if β 6= 4s− 1, 4s− 2, or 4s− 3, (5.20) η(4s−1)β = −ηβ(4s−1) ω(4s−1)β + (1−e −2t) η2 if β = 4s ω(4s−1)β − (1−e −2t) η4 if β = 4s− 2 ω(4s−1)β − (1−e −2t) η3 if β = 4s− 3 ω(4s−1)β if β 6= 4s, 4s− 2, or 4s− 3, (5.21) η(4s−2)β = −ηβ(4s−2) ω(4s−2)β − (1−e −2t) η3 if β = 4s ω(4s−2)β + (1−e −2t) η4 if β = 4s− 1 ω(4s−2)β − (1−e −2t) η2 if β = 4s− 3 ω(4s−2)β if β 6= 4s, 4s− 1, or 4s− 3, (5.22) η(4s−3)β = −ηβ(4s−3) ω(4s−3)β + (1−e −2t) η4 if β = 4s ω(4s−3)β + (1−e −2t) η3 if β = 4s− 1 ω(4s−3)β + (1−e −2t) η2 if β = 4s− 2 ω(4s−3)β if β 6= 4s, 4s− 1, or 4s− 2. (5.23) Indeed, if we substitute ( 5.16− 5.23) into the first structural equations dηA = ηA1 ∧ η1 + ηAq ∧ ηq + ηAβ ∧ ηβ we obtain ( 5.9), ( 5.10), and ( 5.11). To compute the curvature, we consider the second structural equations. In particular, dη1p − η1q ∧ ηqp − η1α ∧ ηαp = 2dηp − 2ηq ∧ ηqp − ηα ∧ ηαp = −4ηp ∧ η1 + ηα ∧ ηαp = −4ηp ∧ η1 + e t ωpα ∧ ηα. Hence using ( 5.13− 5.15), we have R1p1p = −4, R12(4s−1)(4s) =− 2 = −R12(4s)(4s−1), R12(4s−3)(4s−2) =− 2 = −R12(4s−2)(4s−3), R13(4s)(4s−2) =− 2 = −R13(4s−2)(4s), R13(4s−1)(4s−3) =2 = −R13(4s−3)(4s−1), R14(4s)(4s−3) =2 = −R14(4s−3)(4s), R14(4s−1)(4s−2) =2 = −R14(4s−2)(4s−1), R1pAB = 0, otherwise. Also, dη1α − η1q ∧ ηqα − η1β ∧ ηβα = dηα − 2ηq ∧ ηqα − ηβ ∧ ηβα = −ηα ∧ η1 + e t ωqα ∧ ηq, hence R1α1α = −1, R1(4s)(4s−1)2 =− 1 = −R1(4s−1)(4s)2, R1(4s)(4s−2)3 =1 = −R1(4s−2)(4s)3, R1(4s)(4s−3)4 =− 1 = −R1(4s−3)(4s)4, R1(4s−1)(4s−3)3 =− 1 = −R1(4s−3)(4s−1)3, R1(4s−1)(4s−2)4 =− 1 = −R1(4s−2)(4s−1)4, R1(4s−2)(4s−3)2 =− 1 = −R1(4s−3)(4s−2)2, R1αAB = 0 otherwise. Similarly, dηpq − ηp1 ∧ η1q − ηpr ∧ ηrq − ηpβ ∧ ηβq = dωpq + 4ηp ∧ ηq − ωpr ∧ ωrq − e 2tωpβ ∧ ωβq = Ω̄pq + (1− e 2t)ωpβ ∧ ωβq + 4ηp ∧ ηq, where Ω̄pq = R̄pqijωj ∧ ωi is the curvature form of N0. In particular, this implies that Rpqro = −4+e−4t R̄pqpq if r = p and o = q 4+e−4t R̄pqqp if r = q and o = p e−4t R̄pqro otherwise, (5.24) R23(4s)(4s−3) = e −2t R̄23(4s)(4s−3) − 2(e −2t − 1), R23(4s−1)(4s−2) = e −2t R̄23(4s−1)(4s−2) − 2(e −2t − 1), R24(4s)(4s−2) = e −2t R̄24(4s)(4s−2) − 2(e −2t − 1), R24(4s−1)(4s−3) = e −2t R̄24(4s−1)(4s−3) + 2(e −2t − 1), R34(4s)(4s−1) = e −2t R̄34(4s)(4s−1) − 2(e −2t − 1), R34(4s−2)(4s−3) = e −2t R̄34(4s−2)(4s−3) − 2(e −2t − 1), Rpqαβ = e −2t R̄pqαβ, otherwise. We now continue with our curvature computation and consider dηpα − ηp1 ∧ η1α − ηpq ∧ ηqα − ηpβ ∧ ηβα = d(et ωpα) + 2ηp ∧ ηα − ωpq ∧ e t ωqα − e t ωpβ ∧ ηβα = et η1 ∧ ωpα + etR̄pαijωj ∧ ωi + 2ηp ∧ ηα + e t ωpβ ∧ (ωβα − ηβα), where R̄pαij is the curvature tensor of N0. Using ( 5.13− 5.15) and ( 5.20− 5.23), we have R2(4s)ABηB ∧ ηA = η1 ∧ η(4s−1) + etR̄2(4s)ij ωj ∧ ωi − 2η(4s) ∧ η2 + (1− e −2t) η(4s) ∧ η2 + (1− e−2t) η(4s−3) ∧ η3 + (1− e −2t) η(4s−2) ∧ η4 = η1 ∧ η(4s−1) + etR̄2(4s)ij ωj ∧ ωi − (1 + e −2t) η(4s) ∧ η2 + (1− e−2t) η(4s−3) ∧ η3 + (1− e −2t) η(4s−2) ∧ η4. R2(4s−1)ABηB ∧ ηA = −η1 ∧ η(4s) + etR̄2(4s−1)ij ωj ∧ ωi − (1 + e −2t) η(4s−1) ∧ η2 + (1− e−2t)η(4s−2) ∧ η3 − (1− e −2t) η(4s−3) ∧ η4. R2(4s−2)ABηB ∧ ηA = η1 ∧ η4s−3 + etR̄2(4s−2)ij ωj ∧ ωi − (1 + e −2t) η(4s−2) ∧ η2 + (1− e−2t) η(4s−1) ∧ η3 − (1− e −2t) η(4s) ∧ η4. R2(4s−3)ABηB ∧ ηA = −η1 ∧ η4s−2 + etR̄2(4s−3)ij ωj ∧ ωi − (1 + e −2t) η(4s−3) ∧ η2 − (1− e−2t) η(4s) ∧ η3 + (1− e −2t) η(4s−1) ∧ η4. Similar formulas for the curvature tensors of the form R3αAB and R4αAB . Continuing with our computation of the second structural equations using ( 5.13− 5.15), we have dη(4s−1)(4s) − η(4s−1)1 ∧ η1(4s) − η(4s−1)q ∧ ηq(4s) − η(4s−1)β ∧ ηβ(4s) = dω(4s−1)(4s) + 2e −2t η1 ∧ η2 + (1− e −2t) dη2 + η(4s−1) ∧ η(4s) − e 2t ω(4s−1)q ∧ ωq(4s) − (ω(4s−1)(4s−2) − (1− e −2t) η4) ∧ (ω(4s−2)(4s) − (1− e −2t) η3) − (ω(4s−1)(4s−3) − (1− e −2t) η3) ∧ (ω(4s−3)(4s) + (1− e −2t) η4) R̄(4s−1)(4s)ij ωj ∧ ωi + (1− e 2t)ω(4s−1)q ∧ ωq(4s) + 2η1 ∧ η2 + (1− e−2t)ω2q ∧ ηq + e t(1− e−2t)ω2β ∧ ηβ − η(4s) ∧ η(4s−1) + (1− e−2t)ω(4s−1)(4s−2) ∧ η3 + (1− e −2t) η4 ∧ ω(4s−2)(4s) + (1− e−2t) η3 ∧ ω(4s−3)(4s) − (1− e −2t)ω(4s−1)(4s−3) ∧ η4 + 2(1− e−2t)2 η3 ∧ η4 R̄(4s−1)(4s)ij ωj ∧ ωi + (2− e −2t) η(4s−1) ∧ η(4s) − 2(1− e −2t) η(4s−3) ∧ η(4s−2) + 2η1 ∧ η2 + (1− e −2t)ω2q ∧ ηq + 2(1− e −2t) η(4r−3) ∧ η(4r−2) + 2(1− e−2t) η(4r−1) ∧ η(4r) + (1− e −2t) (ω(4s−1)(4s−2) − ω(4s−3)(4s)) ∧ η3 − (1− e−2t) (ω(4s−2)(4s) + ω(4s−1)(4s−3)) ∧ η4 + 2(1− e −2t)2 η3 ∧ η4. (5.25) Note that ( 5.22) asserts that (1− e−2t)ω2q ∧ ηq = (1− e −2t) (−ω14 ∧ η3 + c ∧ η3 + ω13 ∧ η4 − b ∧ η4) = (1− e−2t) (4e−2t η3 ∧ η4 + c ∧ η3 − bη4), (1− e−2t) (ω(4s−1)(4s−2) − ω(4s−3)(4s)) ∧ η3 = −(1− e−2t) c ∧ η3, −(1− e−2t) (ω(4s−2)(4s) + ω(4s−1)(4s−3)) ∧ η4 = (1− e−2t) b ∧ η4. Hence substituting into ( 5.25), we obtain R(4s−1)(4s)AB ηB ∧ ηA R̄(4s−1)(4s)ij ωj ∧ ωi + (2− e −2t) η(4s−1) ∧ η(4s) − 2(1− e −2t) η(4s−3) ∧ η(4s−2) + 2η1 ∧ η2 + 2(1− e−2t) η(4r−3) ∧ η(4r−2) + 2(1− e −2t) η(4r−1) ∧ η(4r) + 2(1− e −4t) η3 ∧ η4 = 2η1 ∧ η2 + R̄(4s−1)(4s)pq e −4t ηq ∧ ηp + 2(1− e −4t) η3 ∧ η4 + R̄(4s−1)(4s)pα e −3t ηα ∧ ηp R̄(4s−1)(4s)αβ e −2t ηβ ∧ ηα + (2− e −2t) η(4s−1) ∧ η(4s) − 2(1− e −2t) η(4s−3) ∧ η(4s−2) + 2(1− e−2t) η(4r−3) ∧ η(4r−2) + 2(1− e −2t) η(4r−1) ∧ η(4r). A similar computation yields the curvature tensor of the formR(4s−1)(4s−2)AB , R(4s−1)(4s−3)AB , R(4s−2)(4s−3)AB , R(4s−2)(4s)AB , and R(4s−3)(4s)AB . It remains to compute R(4s−3)(4r)AB ηB ∧ ηA = dη(4s−3)(4r) − η(4s−3)1 ∧ η1(4r) − η(4s−3)q ∧ ηq(4r) − η(4s−3)β ∧ ηβ(4r) = dω(4s−3)(4r) + η(4s−3) ∧ η(4r) − e 2t ω(4s−3)q ∧ ωq(4r) − η(4s−3)β ∧ ηβ(4r) R̄(4s−3)(4r)ij ωj ∧ ωi + (1− e 2t)ω(4s−3)q ∧ ωq(4r) − (1− e −2t) (η4 ∧ ω(4s)(4r) + η3 ∧ ω(4s−1)(4r))− (1− e −2t) (η2 ∧ ω4s−2)(4r) + ω(4s−3)(4r−1) ∧ η2 − ω(4s−3)(4r−2) ∧ η3 + ω(4s−3)(4r−3) ∧ η4) + η(4s−3) ∧ η(4r). (5.26) Using ( 5.12− 5.14), we can write ω(4s−3)q ∧ ωq(4r) = −η(4s− 2) ∧ η(4r−1) + η(4s−1) ∧ η(4r−2) − η(4s) ∧ η(4r−3)). Also using ( 5.12) asserts that ω(4s−3)(4r−1) = ω(4s−2)(4r), ω(4s−3)(4r−2) = −ω((4s−1)(4r) ω((4s−3)(4r−3) = ω((4s)(4r). Hence ( 5.26) becomes R(4s−3)(4r)AB ηB ∧ ηA R̄(4s−3)(4r)pq e −4t ηq ∧ ηp + R̄(4s−3)(4r)pβ e −3t ηβ ∧ ηp + R̄(4s−3)(4r)αβ e −2t ηβ ∧ ηα − (1− e2t) η(4s−2) ∧ η(4r−1) + (1− e −2t) η(4s−1) ∧ η(4r−2) − (1− e−2t) η(4s) ∧ η(4r−3) + η(4s−3) ∧ η4r. So we have determined all curvature tensors of M . Note that the quater- nionic curvatures satisfy K(e1, e2) +K(e1, e3) +K(e1, e4) = −12 K(e2, e1) +K(e2, e3) +K(e2, e4) = −12 + e −2t (KN(e2, e3) +K N (e2, e4)) K(e3, e1) +K(e3, e2) +K(e3, e4) = −12 + e −2t (KN(e3, e2) +K N (e3, e4)) K(e4, e1) +K(e4, e2) +K(e4, e3) = −12 + e −2t (K̄(e4, e2) +K N (e4, e3)). In particular, this implies that KN(e2, e3) = K N (e2, e4) = K N (e3, e4) = 0. Also, for 2 ≤ p ≤ 4, we have K(e1, e(4s−i)) = −4 K(ep, e(4s−i) = −4 + e −2t ( K̄(ep, e(4s−i))− 4) implying KN (ep, e(4s−i)) = 4. We also have K(e(4s), e(4s−1)) = −12 + e −2t ( KN(e(4s), e(4s−i)) + 9) implying KN (e(4s), e(4s−i)) = −9. Lastly, K(e(4s), e(4r−i)) = −4 + e KN(e(4s), e(4r−i)) implying KN (e(4s), e(4r−i)) = 0. The above computation determined the whole curvature tensor for M and N0. In particular, if M has bounded curvature, then from the formulas about the components of curvature tensors of M , all curvature components are determined as those of QHn. 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Shengli Kong Department of Mathematics http://arxiv.org/abs/math/0701865 University of California, Irvine Irvine, CA92697-3875, USA email: skong@math.uci.edu Peter Li Department of Mathematics University of California, Irvine Irvine, CA92697-3875, USA email:pli@math.uci.edu Detang Zhou Departamento de Geometria Insitituto de Matematica Universidade Federal Fluminense- UFF Centro, Niterói, RJ 24020-140, Brazil email: zhou@impa.br Preliminaries on quaternionic Kähler manifolds Laplacian comparison theorem Quaternionic harmonicity Uniqueness of infinite volume end Maximal first eigenvalue

Let $M^{4n}$ be a complete quaternionic K\"ahler manifold with scalar curvature bounded below by $-16n(n+2)$. We get a sharp estimate for the first eigenvalue $\lambda_1(M)$ of the Laplacian which is $\lambda_1(M)\le (2n+1)^2$. If the equality holds, then either $M$ has only one end, or $M$ is diffeomorphic to $\mathbb{R}\times N$ with N given by a compact manifold. Moreover, if $M$ is of bounded curvature, $M$ is covered by the quaterionic hyperbolic space $\mathbb{QH}^n$ and $N$ is a compact quotient of the generalized Heisenberg group. When $\lambda_1(M)\ge \frac{8(n+2)}3$, we also prove that $M$ must have only one end with infinite volume.

Introduction Let Mn be a complete n-dimensional Riemannian manifold whose Ricci cur- vature bounded below by −(n − 1). It is well known from Cheng [Ch] that the first eigenvalue λ1(M) satisfies λ1(M) ≤ (n− 1)2 In [LW3], Li and Wang proved an analogous theorem for complete Käh- ler manifolds. They showed that if M2n is a complete Kähler manifold of complex dimension n with holomorphic bisectional curvature BKM bounded below by −1, then the first eigenvalue λ1(M) satisfies λ1(M) ≤ n ∗Research partially supported by NSF grant DMS-0503735 †Research partially supported by CAPES and CNPq of Brazil. http://arxiv.org/abs/0704.1851v1 Here BKM ≥ −1 means that Rīijj̄ ≥ −(1 + δij) for any unitary frame e1, . . . , en. In this paper, we prove the corresponding Laplacian comparison theorem for a quaterionic Kähler manifold M4n. As an application we get the sharp estimate λ1(M) for a complete quaterionic Kähler manifold M 4n with scalar curvature bounded below by −16n(n+ 2) as λ1(M) ≤ (2n + 1) It is an interesting question to ask what one can say about those manifolds when the above inequalities are realized as equalities. In works of Li and Wang [LW1] and [LW2], the authors obtained the following theorems. The first was a generalization of the theory of Witten-Yau [WY], Cai-Galloway [CG], and Wang [W] for conformally compact manifolds. The second was to answer the aforementioned question. Theorem 0.1. Let Mn be a complete Riemannian manifold of dimension n ≥ 3 with Ricci curvature bounded below by −(n − 1). If λ1(M) ≥ n − 2, then either (1) M has only one infinite volume end; or (2) M = R×N with warped product metric of the form ds2M = dt 2 + cosh2 t ds2N , where N is an (n − 1)-dimensional compact manifold of Ricci curvature bounded below by λ1(M). Theorem 0.2. Let Mn be a complete Riemannian manifold of dimension n ≥ 2 with Ricci curvature bounded below by −(n − 1). If λ1(M) ≥ (n−1)2 then either (1) M has no finite volume end; or (2) M = R×N with warped product metric of the form ds2M = dt 2 + e2t ds2N , where N is an (n − 1)-dimensional compact manifold of nonnegative Ricci curvature. In [LW3] and [LW5], Li and Wang also consider the Kähler case. They proved the following theorems. Theorem 0.3. Let Mn be a complete Kähler manifold of complex dimension n ≥ 1 with Ricci curvature bounded below by RicM ≥ −2(n + 1). If λ1(M) > , then M must have only one infinite volume end. Theorem 0.4. Let Mn be a complete Kähler manifold of complex dimension n ≥ 2 with holomorphic bisectional curvature bounded by BKM ≥ −1. If λ1(M) ≥ n 2, then either (1) M has only one end; or (2) M = R × N with N being a compact manifold. Moreover the metric on M is of the form ds2M = dt 2 + e4t ω22 + e ω2i , where {ω2, ω3, . . . , ω2n} are orthonormal coframe of N with Jdt = ω2. If M has bounded curvature, then we further conclude that M is covered by CHn and N is a compact quotient of the Heisenberg group. In [LW5], the authors pointed out that the assumption on the lower bound of λ1(M) in Theorem 0.3 is sharp, since one can construct M of the form M = Σ×N satisfying RicM ≥ −2(n + 1) (0.1) λ1(M) = n + 1 (0.2) with N being a compact Kähler manifold and Σ being a complete surface with at least two infinite volume ends. However, it is still an open question to characterized all those complete Kähler manifolds satisfying conditions ( 0.1) and ( 0.2). In sections 4 and 5, we will prove the following quaternionic Kähler ver- sions of the above theorems. Theorem 0.5. Let (M4n, g) be a complete quaternionic Kähler manifold with scalar curvature satisfying SM ≥ −16n(n + 2). If λ1(M) ≥ 8(n+2) , then M must have only one infinite volume end. Theorem 0.6. Let (M4n, g) be a complete quaternionic Kähler manifold with scalar curvature satisfying SM ≥ −16n(n + 2). If λ1(M) ≥ (2n+ 1) 2, then either (1) M has only one end, or (2) M is diffeomorphic to R×N where N is a compact manifold. Moreover, the metric is given by the form ds2M = dt 2 + e4t ω2p + e where {ω2, . . . , ω4n} are orthonormal coframes for N. If M is of bounded curvature then we further conclude that M is covered by the quaterionic hy- perbolic space QHn and N is a compact quotient of the generalized Heisenberg group. Remark 0.1. It is known that a horosphere in QHn is isometric to a certain generalized Heisenberg group with three-dimensional center and left-invariant Riemannian metric. Such generalized Heisenberg groups have compact quo- tients. For an explicit construction see for instance Example 2.6 in [G]. We don’t have an example to show that the bounded curvature condition in The- orem 0.6 is necessary. If such an example exists, its curvature should decay at exponentially in some directions. Perhaps it is interesting to restrict our attention to the special case when M4n = QHn/Γ is given by the quotient of the quaternionic hyperbolic space n with a discrete group of isometies Γ. In particular, it is instructional to compare with previous results by Corlette [C2] and Corlette-Iozzi [CI] where Lie group theoretic approach was used in understanding these manifolds. For example, in [CI], the authors proved a Patterson-Sullivan type formula for λ1(M) in terms of the Hausdorff dimension δ(Γ) of the limit set of Γ. More specifically, they proved that if Γ is geometrically finite, then for δ(Γ) ≥ 2n+1 one has λ1(M) = δ(Γ)((4n+ 2)− δ(Γ)). Hence in this case, the condition in Theorem 0.6 on λ1(M) = (2n + 1) equivalent to the condition δ(Γ) = 2n+ 1. In [C2] (Theorem 4.4), Corlette also pointed out that by a result of Kostant λ1(M) = 0 or λ1(M) ≥ 8n. On the other hand, it was also shown in [CI] that if Γ is geometrically finite and torsion free, then M = QHn/Γ must have at most one end with infinite volume. These two statements give an interesting comparison to Theorem 0.5 stated above. We would also like to point out to the interested readers that in [LW4] and [LW5] Li and Wang considered a more general class of manifolds satisfying a weighted Poincaré inequality. However, since quaternionic Kähler manifolds are automatically Einstein, the same type of questions are not interesting for this class of manifolds. Acknowledgement. This work was done when the third author was visiting the University of California, Irvine. He wishes to thank the institu- tion for its hospitality. He also would like to thank Professor J. Berndt for pointing out the paper of [G] to him. 1 Preliminaries on quaternionic Kähler man- ifolds In this section, we will recall basic properties of quaternionic Kähler manifolds that will be needed in the sequel. These properties were proved by Berger [B] and Ishihara [I] (also see [Be]). Let (Mn, g) be a Riemannian manifold, TM the tangent space of M and ∇ the Levi-Civita connection. The Riemannian curvature R : TM ⊗ TM ⊗ TM −→ TM is defined by R(X, Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z If {e1, · · · , en} is an orthonormal basis of TM , the components of curvature tensor is defined by Rijkl = 〈R(ei, ej)el, ek〉, the Ricci curvature is defined by RicM(X, Y ) = 〈R(X, ei)ei, Y 〉, and the scalar curvature is defined by i,j=1 〈R(ei, ej)ej , ei〉. Definition 1.1. A quaternionic Kähler manifold (M, g) is a Riemannian manifold with a rank 3 vector bundle V ⊂ End(TM) satisfying (a) In any coordinate neighborhood U ofM , there exists a local basis {I, J,K} of V such that I2 = J2 = K2 = −1 IJ = −JI = K JK = −KJ = I KI = −IK = J 〈IX, IY 〉 = 〈JX, JY 〉 = 〈KX,KY 〉 = 〈X, Y 〉 for all X, Y ∈ TM . (b) If φ ∈ Γ(V ), then ∇Xφ ∈ Γ(V ) for all X ∈ TM . Remark 1.1. It follows from (a) that dimM = 4n. A well known fact about 4n-dimensional Riemannian manifold is that it is quaternionic Kähler if and only if its restricted holonomy group is contained in Sp(n)Sp(1). The 4-dimensional Riemannian manifolds with holonomy Sp(1)Sp(1) are simply the oriented Riemanian manifolds, naturally we only consider those when n ≥ 2. Notice that in general I, J,K are not defined everywhere on M . For example, the canonical quaternionic projective space QP n admits no almost complex structure. On the other hand, the vector space generated by I, J,K is well defined at each point of M and this 3-dimensional subbundle V of End(TM) is in fact “globally parallel” under the Levi-Civita connection ∇ of g. A basic fact about the connection is the following lemma. Lemma 1.1. The condition (b) is equivalent to the following condition: ∇XI = c(X)J − b(X)K, ∇XJ = −c(X)I + a(X)K, ∇XK = b(X)I − a(X)J, where a, b, c are local 1-forms. Definition 1.2. Let (M, g) be a quaternionic Kähler manifold. We can define a 4-form by Ω = ω1 ∧ ω1 + ω2 ∧ ω2 + ω3 ∧ ω3, where ω1 = 〈·, I·〉, ω2 = 〈·, J ·〉, ω3 = 〈·, K·〉. Let {e1, Ie1, Je1, Ke1, · · · , en, Ien, Jen, Ken} be an orthonormal basis of TM and {θ1, Iθ1, Jθ1, Kθ1, · · · , θn, Iθn, Jθn, Kθn, } the dual basis. It follows θi ∧ Iθi + Jθi ∧Kθi θi ∧ Jθi +Kθi ∧ Iθi θi ∧Kθi + Iθi ∧ Jθi θi ∧ Iθi ∧ θj ∧ Iθj + θi ∧ Jθi ∧ θj ∧ Jθj + θi ∧Kθi ∧ θj ∧Kθj Jθi ∧Kθi ∧ Jθj ∧Kθj +Kθi ∧ Iθi ∧Kθj ∧ Iθj + Iθi ∧ Jθi ∧ Iθj ∧ Jθj θi ∧ Iθi ∧ Jθj ∧Kθj + θi ∧ Jθi ∧Kθj ∧ Iθj + θi ∧Kθi ∧ Iθj ∧ Jθj Lemma 1.2. The condition (b) is equivalent to the following condition: ∇Xω1 = c(X)ω2 − b(X)ω3, ∇Xω2 = −c(X)ω1 + a(X)ω3, ∇Xω3 = b(X)ω1 − a(X)ω2. where a, b, c are local 1-forms. Proof: It follows from the identities (∇Xω1)(Y, Z) = 〈Y, (∇XI)Z〉, (∇Xω2)(Y, Z) = 〈Y, (∇XJ)Z〉, (∇Xω3)(Y, Z) = 〈Y, (∇XK)Z〉. Using this lemma, we have that Theorem 1.1. The condition (b) is equivalent to that Ω is parallel, that is ∇XΩ = 0 for any X ∈ TM . In the following, we shall study the curvature of quaternionic Kähler manifold. First we have the following lemma. Lemma 1.3. If (M4n, g) is a quaternionic Kähler manifold, then [R(X, Y ), I] = γ(X, Y )J − β(X, Y )K, [R(X, Y ), J ] = −γ(X, Y )I + α(X, Y )K, [R(X, Y ), K] = β(X, Y )I − α(X, Y )J, where α, β and γ are local 2-forms given by α = da+ b ∧ c, β = db+ c ∧ a, γ = dc+ a ∧ b. Corollary 1.1. If (M4n, g) is a quarternionic Kähler manifold, then 〈R(X, Y )Z, IZ〉+ 〈R(X, Y )JZ,KZ〉 = α(X, Y ) |Z|2, 〈R(X, Y )Z, JZ〉+ 〈R(X, Y )KZ, IZ〉 = β(X, Y ) |Z|2, 〈R(X, Y )Z,KZ〉+ 〈R(X, Y )IZ, JZ〉 = γ(X, Y ) |Z|2. The following lemma is the key for quaternionic Kähler manifolds. Lemma 1.4. If (M4n, g) is a quaternionic Kähler manifold and n ≥ 2, then α(X, IY ) = β(X, JY ) = γ(X,KY ) = − n + 2 RicM(X, Y ). (1.1) As applications of the above lemma, one can show the following two main theorems on curvature of quaternionic Kähler manifolds. Theorem 1.2. If (M4n, g) is a quaternionic Kähler manifold and n ≥ 2, then (M4n, g) is Einstein, that is, there is a constant δ such that RicM(g) = 4(n+ 2)δg. Theorem 1.3. If (M4n, g) is a quaternionic Kähler manifold and n ≥ 2, (1) For any tangent vector X, the sectional curvature satisfies 〈R(X, IX)IX,X〉+ 〈R(X, JX)JX,X〉 +〈R(X,KX)KX,X〉 = 12δ |X|4. (2) For any tangent vector Y satisfying 〈Y,X〉 = 〈Y, IX〉 = 〈Y, JX〉 = 〈Y,KX〉 = 0, the sectional curvature satisfies 〈R(X, Y )Y,X〉+ 〈R(X, IY )IY,X〉+ 〈R(X, JY )JY,X〉+ 〈R(X,KY )KY,X〉 = 4δ |X|2 |Y |2, where 4(n+ 2)δ is the Einstein constant. Finally, we end this section with the following lemma. Lemma 1.5. Let γ : [a, b] → M be a geodesic with unit speed. If S = 16n(n + 2)δ, and XI(t), XJ(t), XK(t) are parallel vector fields along γ such that XI(a) = Iγ ′(a), XJ(a) = Jγ ′(a), XK(a) = Kγ ′(a), then K(γ′(t), XI(t)) +K(γ ′(t), XJ(t)) +K(γ ′(t), XK(t)) = 12δ, for all t and γ. Let Y be a tangent vector at γ(a) satisfying 〈γ′(a), Y 〉 = 0, 〈Iγ′(a), Y 〉 = 0, 〈Jγ′(a), Y 〉 = 0, and 〈Kγ′(a), Y 〉 = 0. If we denote the parallel vector fields Y (t), YI(t), YJ(t), and YK(t) along γ with initial data Y (a) = Y , YI(a) = IY, YJ(a) = JY , and YK(a) = KY , respectively, then K(γ′(t), Y (t)) +K(γ′(t), YI(t)) +K(γ ′(t), YJ(t)) +K(γ ′(t), YK(t)) = 4δ, for all t and γ. Proof. By the discussion above, we know the 3-dimensional vector space E(t) spanned by X(t), Y (t), Z(t) does not depend on the choice of I, J,K. Hence it is parallel under the Levi-Civita connection. We consider 〈R(·, γ′(t))γ′(t), ·〉 as a symmetric bilinear form on E(t). ThenK(γ′(t), X(t))+ K(γ′(t), Y (t)) + K(γ′(t), Z(t)) is its trace on E(t) which independent of the choice of orthonormal basis. By the computation above it is equal to 12δ. The same argument also applies to the second part of the lemma. � 2 Laplacian comparison theorem For a complete Riemannian manifold M and p ∈ M , let us denote the cut locus with respect to p by Cut(p). Theorem 2.1. Let (M4n, g) be a complete quaternionic Kähler manifold with scalar curvature SM ≥ 16n(n+2)δ and let r(x) be the distance function to a fixed point p ∈M . Then, for x /∈ Cut(p), ∆r(x) ≤ 6 coth 2r(x) + 4(n− 1) coth r(x) when δ = −1 (4n− 3)r−1(x) when δ = 0 6 cot 2r(x) + 4(n− 1) cot r(x) when δ = 1. (2.1) Proof. Let γ be the minimizing geodesic joining p to x. At x, we choose {e1, e2, · · · , en}, and two local almost complex structures I, J and K = IJ such that e1 = ∇r and {e1, Ie1, Je1, Ke1, e2, Ie2, Je2, Ke2, · · · , en, Ien, Jen, Ken} is an orthonormal frame. By parallel translating along γ we obtain an or- thonormal frame with e1 = ∇r. For convenience sake, we denote this frame by {ε1, ε2, · · · , ε4n}. Since |∇r| 2 = 1 on M\Cut(p), by taking covariant derivative of this equation, we have 0 = |∇r|2kl rikril + 2 ririkl, (2.2) for each k, l = 2, · · · , 4n. Since rikl = rkli + Rjkilrj, with Rijkl = 〈R(εi, εj)εl, εk〉, and r1 = 1, rj = 0, j = 2, · · · , 4n, we have rikril + rkl1 +R1k1l = 0. (2.3) In particular, if k = l, we have r2ik + rkk1 +K(ε1, εk) = 0, (2.4) where K(ε1, εk) = R1k1k is the sectional curvature of the 2-plane section spanned by ε1, εk. Using the inequality r2ik ≥ and setting f(t) = k=2 rkk, ( 2.4) implies that f ′(t) + f 2(t) + K(ε1, εk) ≤ 0. (2.5) By Lemma 1.5, we have f ′(t) + f 2(t) + 12δ ≤ 0. (2.6) Since a smooth Riemannian metric is locally Euclidean, then limt→0 tf(t) = 3. By a standard comparison argument for ordinary differential equations, we conclude that f(t) ≤ 6 cot 2t when δ = 1 3t−1 when δ = 0 6 coth 2t when δ = −1. (2.7) Similarly, using the inequality k=4i+1 r2ik ≥ k=4i+1 for 1 ≤ i ≤ n− 1, and setting hi(t) = ∑4i+4 k=4i+1 rkk, ( 2.4) implies that h′i(t) + h2i (t) + k=4i+1 K(ε1, εk) ≤ 0. (2.8) Together with Lemma 1.5 asserting that k=4i+1 K(ε1, εk) = 4δ, we have h′i(t) + h2i (t) + 4δ ≤ 0. (2.9) Hence, as before, we conclude that hi(t) ≤ 4 cot t when δ = 1 4t−1 when δ = 0 4 coth t when δ = −1. (2.10) The result follows from the equation ∆r(x) = f(r(x)) + i=1 hi(r(x)). � Remark 2.1. The estimate in Theorem 2.1 is sharp since the right hand sides are exactly the Laplacian of the distance functions of quaternionic hy- perbolic space QHn, quaternionic Euclidean space Qn and quaternionic pro- jective space QPn respectively. Remark 2.2. We actually proved the estimate for Hessian of the distance function. In particular, rkk ≤ 6 cot 2t when δ = 1 3t−1 when δ = 0 6 coth 2t when δ = −1. (2.11) Also for 1 ≤ i ≤ n− 1, we have k=4i+1 rkk ≤ 4 cot 2t when δ = 1 4t−1 when δ = 0 4 coth 2t when δ = −1. (2.12) Corollary 2.1. Let (M4n, g) be a complete quaternionic Kähler manifold with scalar curvature SM ≥ −16n(n + 2). Then for any point x ∈ M and r > 0, the area A(r) of the geodesic spheres centered at x satisfies A′(r) ≤ 6 coth 2r + 4(n− 1) coth r. (2.13) In particular, A(r) ≤ C(sinh 2r)3(sinh r)4(n−1) ≤ Ce(4n+2)r. Corollary 2.2. Let (M4n, g) be a complete quaternionic Kähler manifold with scalar curvature SM ≥ −16n(n + 2). Then for any point x ∈ M and 0 < r1 ≤ r2, the volume of the geodesic balls centered at x satisfies Vx(r2) Vx(r1) VQHn(r2) VQHn(r1) , (2.14) where VQHn(r) denotes the volume of the geodesic ball of radius r in QH n. In particular, λ1(M) ≤ (2n+ 1) Corollary 2.3. Let (M4n, g) be a complete quaternionic Kähler manifold with scalar curvature SM ≥ 16n(n+2) . Then it is compact, and the diameter d(M) ≤ π , which is the diameter of the model space QPn. Moreover, the volume of M is bounded by V (M) ≤ V (QPn), (2.15) where VQPn is the volume of QP 3 Quaternionic harmonicity In this section we will derive an over-determined system of harmonic func- tions with finite Dirichlet integral on a manifold with a parallel form. This result was first proved by Siu [S] for harmonic maps in his proof of the rigidity theorem for Kähler manifolds. Corlette [C1] gave a more systematic approach for harmonic map with finite energy from a finite-volume quaternionic hy- perbolic space or Cayley hyperbolic plane to a manifold with nonpositive curvature. In [L], the second author generalized Siu’s argument to harmonic functions with finite Dirichlet integral on a Kähler manifold. We will pro- vide an argument that generalizes Corlette’s argument to harmonic functions with finite Dirichlet integral on a complete manifold with a parallel form. We believe that it should be of independent interest. Theorem 3.1. Let M be a complete Riemannian manifold with a parallel p-form Ω. Assume that f is a harmonic function with its Dirichlet integral over geodesic balls centered at o of radius R satisfying the growth condition Bo(R) |∇f |2dv = o(R2) as R → ∞, then f satisfies d ∗ (df ∧ Ω) = 0. (3.1) Before we prove the theorem, let us first recall the following operators and some of the basic properties. For an oriented real vector space V with an inner product, we have the Hodge star operator ∗ : ∧pV → ∧n−pV. For any θ ∈ ∧1V and v ∈ V , we also have exterior multiplication and interior product operators ε(θ) : ∧pV → ∧p+1V, ℓ(v) : ∧pV → ∧p−1V. For θ ∈ ∧1V and v ∈ V is the dual of θ by the inner product, if ξ ∈ ∧pV we list the following identities among the operators: 1. ∗ ∗ ξ = (−1)p(n−p)ξ, 2. ∗ε(θ)ξ = (−1)pℓ(v) ∗ ξ, 3. ε(θ) ∗ ξ = (−1)p−1 ∗ ℓ(v)ξ, 4. ∗ε(θ) ∗ ξ = (−1)(p−1)(n−p)ℓ(v)ξ, 5. ℓ(v)ε(θ′)ξ + ε(θ)ℓ(v′)ξ = 0, where v⊥v′, 6. ℓ(v)ε(θ)ξ + ε(θ)ℓ(v)ξ = ξ. We are now ready to prove Theorem 3.1. Proof of Theorem 3.1. Let η : [0,+∞) → R be a smooth function satis- fying η′(t) ≤ 0, and η(t) = 1 when t ∈ [0, 1] 0 when t ∈ [2,+∞]. For R ≥ 1, we define the cut-off function φR(x) = η(r(x)/R), where r(x) is the distance function from a fixed point o ∈ M , then there is a positive constant C1 depending on η and C such that |∇φR(x)| ≤ C1R Since d2 = 0, then φ2R ∗ (df ∧ Ω) ∧ d ∗ (df ∧ ∗Ω) d(φ2R) ∧ ∗(df ∧ Ω) ∧ d ∗ (df ∧ ∗Ω) φ2R d ∗ (df ∧ Ω) ∧ d ∗ (df ∧ ∗Ω). (3.2) We claim that ∗ d ∗ (df ∧ Ω) = (−1)n−1d ∗ (df ∧ ∗Ω). (3.3) In fact, for any point x ∈ M , we can choose an orthonormal tangent basis i=1 in a neighborhood of x such that ∇eiej(x) = 0. Denote by {θ i}mi=1 the dual basis of {ei} i=1. Then for ω ∈ ∧ p(T ∗M) we have dω = ε(θi)∇eiω. Hence d ∗ (df ∧ ∗Ω) = d ∗ ε(df) ∗ Ω = (−1)(p−1)(m−p)d[ℓ(∇f)Ω] = (−1)(p−1)(m−p) ε(θi)∇ei(ℓ(∇f)Ω) = (−1)(p−1)(m−p) i,j=1 ε(θi)(∇ei∇ejf)(ℓ(ej)Ω) = (−1)(p−1)(m−p) i,j=1 fijε(θi)(ℓ(ej)Ω), where fij = ∇ei∇ejf and the facts Ω is parallel and ∇eiej(x) = 0 have been used. On the other hand, ∗d ∗ (df ∧ Ω) = ∗d ∗ ε(df)Ω ε(θi)∇ei(∗ε( fjθj)d[ℓ(∇f)Ω] i,j=1 fijε(θi) ∗ ε(θj)(Ω) = (−1)p(m−p−1) i,j=1 fijℓ(ei)ε(θj)Ω = (−1)p(m−p−1) fiiℓ(ei)ε(θi)Ω + i 6=j fijℓ(ei)ε(θj)Ω = (−1)p(m−p−1) fii[Ω− ε(θi)ℓ(ei)Ω]− i 6=j fijε(θj)ℓ(ei)Ω = (−1)p(m−p−1) i,j=1 fijε(θi)(ℓ(ej)Ω), (3.4) where we used fij = fji and i=1 fii = 0. So the claim is proved. By ( 3.2), we have φ2R |d ∗ (df ∧ Ω)| = (−1)m d(φ2R) ∧ ∗(df ∧ Ω) ∧ d ∗ (df ∧ ∗Ω) φ2R |d ∗ (df ∧ Ω)| |dφR| 2| ∗ (df ∧ Ω)|2dv (3.5) On the other hand, ( 3.3) and the fact that ω is bounded imply that there exists a constant C2 > 0, such that | ∗ (df ∧ Ω)| ≤ C2 |df | |d ∗ (df ∧ ∗Ω)| = |d ∗ (df ∧ Ω)|. Hence combining with ( 3.5) and using the definition of φR we conclude that Bo(R) |d ∗ (df ∧ Ω)|2dv ≤ C1R Bo(2R) |df |2dv. The assumption on the growth of the Dirichlet integral of f implies that the right hand side tends to zero as R → ∞. Therefore d ∗ (df ∧Ω) = 0, and the proof is complete. � Lemma 3.1. Let (M4n, g) be a quarternionic Kähler manifold and n ≥ 2. If f is a function on M satisfying d ∗ (df ∧ Ω) = 0 (3.6) for the 4-form Ω determined by the quaternionic Kähler structure, then f is quaternionic harmonic, namely, for any nonzero tangent vector X, fX,X + fIX,IX + fJX,JX + fKX,KX = 0 where fX,X = ∇df(X,X). Proof: Let A=1 = {e1, e2, · · · , en, Ie1, Ie2, · · · , Ien, Je1, Je2, · · · , Jen, Ke1, Ke2, · · · , Ken} be an orthonormal basis of TM and {ωA} the dual basis with e1 = . Since Ω is parallel, by ( 3.4) and ( 3.6), we have (∇eAdf) ∧ ℓ(eA)Ω A,B=1 feA,eB ωB ∧ ℓ(eA)Ω where we have used the fact that f is a harmonic function. Hence equation ( 3.6) implies A,B=1 feA,eB ωB ∧ ℓ(eA)Ω = 0 Comparing the coefficient of ωi∧Iωi∧Jωi∧Kωi on both sides by the explicit formula for Ω given before, we obtain that 6 (fei,ei + fIei,Iei + fJei,Jei + fKei,Kei) = 0 for all ei, (1 ≤ i ≤ n). So the proof is complete. � The following corollary is an immediate consequence of the lemma. Corollary 3.1. Let M4n be a complete quaternionic Kähler manifold. As- sume that f is a harmonic function with its Dirichlet integral satisfying the growth condition Bo(R) |∇f |2dv = o(R2) as R → ∞, then f must satisfy d ∗ (df ∧ Ω) = 0, (3.7) where Ω is the parallel 4-form determined by the quaternionic Kähler struc- ture. Moreover, f is quaternionic harmonic. 4 Uniqueness of infinite volume end Recall that for any complete manifold if λ1(M) > 0 then M must be non- parabolic. In particular,M must have at least one nonparabolic ends. It was also proved in [LW1] that under the assumption that λ1(M) > 0, an end is nonparabolic if and only if it has infinite volume. Let us assume that M has at least two nonparabolic ends, E1 and E2. A construction of Li-Tam [LT] asserts that one can construct a nonconstant bounded harmonic function with finite Dirichlet integral. The harmonic func- tion f can be obtained by taking a convergent subsequence of the harmonic functions fR, as R→ +∞, satisfying ∆fR = 0 on B(R), with boundary conditions fR = 1 on ∂B(R) ∩ E1 fR = 0 on ∂B(R) \ E1. It follows from the maximum principle that 0 ≤ fR ≤ 1, hence 0 ≤ f ≤ 1. We need the following estimates from [LW1](Lemma 1.1 and 1.2 in [LW1]), and [LW3](Lemma 5.1 in [LW3]). Lemma 4.1. Let M be a complete Riemannian manifold with λ1(M) > 0. Suppose M has at least two nonparabolic ends and E be an end of M . Then for the harmonic function f constructed above, it must satisfy the following growth conditions: 1. There exists a constant a such that f − a ∈ L2(E). Moreover, the function f − a must satisfy the decay estimate E(R+1)\E(R) (f − a)2 ≤ C exp(−2 λ1(E)R) for some constant C > 0 depending on f , λ1(E) and the dimension of 2. The Dirichlet integral of the function f must satisfy the decay estimate E(R+1)\E(R) |∇f |2 ≤ C exp(−2 λ1(E)R), exp(−2 λ1(E)r(x))|∇f | 2 ≤ CR for R sufficiently large. Lemma 4.2. Let M be a complete Riemannian manifold with at least two nonparabolic ends and λ1(M) > 0. Then for the harmonic function f con- structed above, for any t ∈ (inf f, sup f) and (a, b) ⊂ (inf f, sup f), L(a,b) |∇f |2 = (b− a) |∇f |, where l(t) = {x ∈M |f(x) = t}, L(a, b) = {x ∈M |a < f(x) < b}. Moreover, |∇f | = |∇f |. We are now ready to prove Theorem 0.5. Proof of Theorem 0.5: Suppose to the contrary that there exist two ends E1 and E2 with infinite volume. The assumption that λ1(M) > 0 implies that they are nonparabolic. By the construction above, there exists a harmonic function f with finite energy such that lim inf x→∞, x∈E1 f(x) = 1 lim inf x→∞, x∈E2 f(x) = 0. The Bochner formula implies that ∆|∇f |2 = RicM(∇f,∇f) + |∇ 2f |2. (4.1) We now choose an orthonormal basis {eA} A=1 satisfying {e1, e2, · · · , en, Ie1, Ie2, · · · , Ien, Je1, Je2, · · · , Jen, Ke1, Ke2, · · · , Ken} with e1 = |∇f | . Corollary 3.1 implies that f(in+1)(in+1) = 0. Therefore, applying the arithmetic-geometric means, we have |∇2f |2 = A,B=1 f 2AB ≥ f 211 + f 2(in+1)(in+1) + 2 f 21A ≥ f 211 + f(in+1)(in+1)) 2 + 2 f 21A |∇|∇f ||2, (4.2) hence combining with ( 4.1) we obtain ∆|∇f |2 ≥ −4(n + 2)|∇f |2 + |∇|∇f ||2. (4.3) If we write u = |∇f | 3 , then ∆u ≥ − 8(n+ 2) u. (4.4) We want to prove that the above inequality is actually an equality. The argument follows from that in [LW4] after making suitable modification to fit our situation. For any compactly supported smooth function φ on M , we 8(n+ 2) φu〈∇u,∇φ〉 − φ2|∇u|2 + λ1(M) (φu)2 φu〈∇u,∇φ〉 − φ2|∇u|2 + |∇(φu)|2 |∇φ|2u2. (4.5) Let us choose φ = ψχ to be the product of two compactly supported functions. For any ε ∈ (0, 1 ), we define χ(x) = 0 on L(0, σε) ∪ L(1− ε (log 2)−1(log f − log( ε )) on L( ε , ε) ∩ (M \E1) (log 2)−1(log(1− f)− log( ε )) on L(1− ε, 1− ε ) ∩ E1 1 otherwise. For R > 1 we define 1 on B(R− 1) R− r on B(R) \B(R− 1) 0 on M \B(R). Applying to the right hand side of ( 4.5), we obtain |∇φ|2u2 ≤ 2 |∇ψ|2χ2|∇f | 3 + 2 |∇χ|2ψ2|∇f | 3 . (4.6) Since RicM ≥ −4(n+2), then the local estimate of Cheng-Yau [CY] (see also [LW2]) implies that there exists a constant depending on n such that |∇f |(x) ≤ C|1− f(x)|. On E1, the first term of ( 4.6) satisfies |∇ψ|2χ2|∇f | |∇f |2 , (4.7) where Ω = E1 ∩ (B(R) \B(R − 1)) ∩ (L(1− ε, 1− ) ∪ L( ε , ε). Since 1 ≤ 4 (1− f)2 (1− f)2 ≤ 4Cε−2 exp(−2 λ1R), where in the last inequality we have used Lemma 4.1. Again by Lemma 4.1, from ( 4.7) we have |∇ψ|2χ2|∇f | 3 ≤ Cε− 3 exp(−2 λ1R). (4.8) For the second term of ( 4.6) we have |∇χ|2ψ2|∇f | ≤ (log 2)−2 L(1−ε,1− ε )∩E1∩B(R) |∇f | +2(1− f)−2 ≤ C(log 2)−2 L(1−ε,1− ε )∩E1∩B(R) |∇f |2(1− f)− Using the co-area formula and Lemma 4.2 we have L(1−ε,1− ε )∩E1∩B(R) |∇f |2(1− f)− ∫ 1− ε (1− t)− l(t)∩E1∩B(R) |∇f |dAdt |∇f |dA ∫ 1− ε (1− t)− = −3C[(1− t) |∇f |dA = 3Cε |∇f |dA. Combining the above inequality with ( 4.8) we have |∇φ|2u2 ≤ C(ε 3 exp(−2 λ1R) + ε 3 ). (4.9) A similar argument using f instead of 1 − f on the other end yields the estimate |∇φ|2u2 ≤ C(ε 3 exp(−2 λ1R) + ε Letting R → ∞ and ε→ 0, we have ∆u = − 8(n + 2) u (4.10) with λ1(M) = 8(n+2) , since f is nonconstant and u cannot be identically zero. Therefore all the inequalities used to prove ( 4.4) are equalities. Thus there exists a function µ, such that, (fAB) = , (4.11) where D1 and D2 are n× n matrices defined by · · · · · · Since f1α = 0 for α 6= 1 implies that |∇f | is constant along the level set of f . Moreover, regularity of the equation ( 4.10) implies that |∇f | can never be zero. Hence M must be diffeomorphic to R×N , where N is given by the level set of f . Also N must be compact since we assume that M has at least 2 ends. Fix a level set N0 of f , consider (−ε, ε)×N0 ⊂M . Note that {eA} is an orthonormal basis of TM such that e1 is the normal vector to N0 and {eα} are the tangent vectors of N0. We shall compute the sectional curvature K(e1, eα) = 〈R(e1, eα)eα, e1〉. We claim that ∇e1e1 = 0. Indeed it suffices to prove all integral curves η(t) of the vector field e1 = |∇f | emanating from N0 are geodesics. For any point η(t0), let γ be the geodesic realizing the distance between η(t0) and N0. Then γ is perpendicular to every level set Nt. So γ ′ is parallel to e1 along γ. This implies γ coincides with the integral curve of e1. Let (hαβ) with 2 ≤ α, β ≤ 4n be the second fundamental form of the level set of f . Then hαβ f1 = −fαβ , (4.12) ∇eαe1 = − hαβeβ. By the definition of curvature tensor, we have 〈R(e1, eα)e1, eα〉 = 〈∇e1∇eαe1 −∇eα∇e1e1 −∇[e1,eα]e1, eα〉 = 〈∇e1∇eαe1, eα〉 − 〈∇[e1,eα]e1, eα〉 = 〈∇e1∇eαe1, eα〉 − 〈∇∇e1eα−∇eαe1e1, eα〉 = 〈∇e1∇eαe1, eα〉 − 〈∇e1eα, eβ〉〈∇eβe1, eα〉 〈∇eαe1, eβ〉〈∇eβe1, eα〉 〈∇e1(hαβeβ), eα〉+ hαβ〈∇e1eα, eβ〉+ 〈(e1hαβ)eβ , eα〉 − hαβ〈∇e1eβ, eα〉 hαβ〈∇e1eα, eβ〉+ h = −e1hαα + 2 hαβ〈∇e1eα, eβ〉+ h Therefore K(e1, eα) = e1hαα − 2 hαβ〈∇e1eα, eβ〉 − h2αβ . (4.13) Since hαβ is diagonal, this implies that K(e1, eα) = e1hαα − h Combining with ( 4.11) and ( 4.12), we conclude that K(e1, e2) = K(e1, Ie2) = K(e1, Je2) = K(e1, Ke2) = 0 which implies M is Ricci flat by Theorem 1.3. This contradicts to the as- sumption that λ1 > 8(n+2) > 0. Therefore M must have only one end with infinite volume. � 5 Maximal first eigenvalue In this section, we will consider the case when λ1(M) is of maximal value. Proof of Theorem 0.6: According to Theorem 0.5, we know that M has exactly one nonparabolic end. Suppose thatM has more than one end. Then there must exist at least an end with finite volume. We divide the rest of the proof into several parts. The first part follows exactly as that in the proof of the corresponding theorem in the Kähler case (Theorem 3.1) in [LW5]. For completeness sake, we will give a quick outline of it. Part 1. Assume that E1 is such an end with finite volume given by M \ Bp(1). Then we can choose a ray η : [0,+∞) such that η(0) = p and η[1,+∞) ⊂ E1. The Busemann function corresponding to γ is defined by β(x) = lim [t− d(x, η(t))]. The Laplacian comparison theorem, Theorem 2.1, asserts that ∆β ≥ −2(2n+ 1) in the sense of distribution. We define the function f = exp((2n+ 1)β), and using the fact that |∇β| = 1 almost everywhere, we have ∆f = (2n+ 1) exp((2n+ 1)β)∆β + (2n+ 1)2 ≥ −(2n + 1)2f. Similar to the proof of above theorem, we conclude that for any compactly supported function φ, (∆f + (2n+ 1)2f)fφ2 f 2|∇φ|2. By choosing the function φ to be 1, on Bp(R); 2R−r(x) , on Bp(2R) \Bp(R); 0, on M \Bp(2R); we obtain f 2|∇φ|2 (Bp(2R)\Bp(R))∩E1 (Bp(R+i)\Bp(R+i−1))∩E1 (VE1(R + i)− VE1(R + i− 1)) exp(2(2n+ 1)(R+ i)) where VE1(R+ i) denotes the volume of the set E1∩Bp(R+ i). On the other hand, the volume estimate in Theorem 1.4 of [LW1] implies that VE1(∞)− VE1(R) ≤ C exp(−2(2n+ 1)R), hence VE1(R + i)− VE1(R + i− 1) = VE1(∞)− VE1(R + i− 1)− (VE1(∞)− VE1(R + i)) ≤ C exp(−2(2n+ 1)(R + i)). Therefore, we conclude that f 2|∇φ|2 ≤ Let us now denote E2 = M \ (Bp(1) ∪ E1) to be the other end of M . When x ∈ E2, following the argument in Theorem 3.1 of [LW4], we have β(x) ≤ −d(p, x) + 2. Therefore f 2|∇φ|2 ≤ (Bp(2R)\Bp(R))∩E2 (Bp(2R)\Bp(R))∩E2 exp(−2(2n+ 1)(r − 2)) Letting R → +∞, we conclude that ∆f + (2n+ 1)2f = 0, (5.1) and all inequalities used are indeed equalities and f is smooth by regularity of the equation ( 5.1). Moreover, |∇β| = 1, and ∆β = −2(2n+ 1). This implies that M must be diffeomorphic to R × N , where N is given by the level set of β. We choose an orthonormal basis {ei} i=1 as follows {e1, e2, · · · , en, Ie1, Ie2, · · · , Ien, Je1, Je2, · · · , Jen, Ke1, Ke2, · · · , Ken} with e1 = ∇β. Applying the Bochner formula to β, we get ∆|∇β|2 i,j=1 β2ij + RicM(∇β,∇β) + βi(∆β)i i,j=1 β2ij − 4(n+ 2). By the comparison theorem, we have, β(in+1)(in+1) = −6. Hence (βαβ) = where D1 and D2 are n× n matrices defined by · · · · · · Part 2. For a fix level set N0 of β, we consider (−ε, ε)×N0 ⊂M . Note that {ei} is an orthonormal basis of TM such that e1 is the normal vector to N0 and {eα}, for 2 ≤ α ≤ 4n, are the tangent vectors of N0. We shall compute the sectional curvature K(e1, eα) = 〈R(e1, eα)eα, e1〉. Since ∇e1e1 = 0 implies that the integral curves of e1 are geodesics. Let (hαγ) be the second fundamental form of the level set of ∇β. Then hαγ = 〈∇eαeγ, e1〉 = 〈∇eαeγ,∇β〉 = −βαγ ∇eαe1 = − hαγeγ. (5.2) By ( 4.13) in the proof of Theorem 0.5 we have 〈R(e1, eα)e1, eα〉 = −e1hαα + 2 hαγ〈∇e1eγ, eβ〉+ h2αγ . Since (hαγ) are constant and diagonal, then K(e1, eα) = −h In particular, we have K(e1, eα) = −4 when α = in+ 1, i = 1, 2, 3 −1 otherwise. On the other hand, we also have K(en+1, e2n+1) +K(en+1, e3n+1) = −12−K(e1, en+1) = −8 K(en+1, e2n+1) +K(e3n+1, e2n+1) = −8 K(e3n+1, e2n+1) +K(en+1, e3n+1) = −8, hence K(en+1, e2n+1) = K(en+1, e3n+1) = K(e2n+1, e3n+1) = −4. Since for α = 2, 3, · · · , n, K(Ie1, eα) = −〈R(Ie1, eα)Ie1, eα〉 = −〈IR(Ie1, eα)Ie1, Ieα〉 = 〈R(Ie1, eα)e1, Ieα〉 = 〈R(e1, Ieα)Ie1, eα〉 = K(e1, Ieα) = −1, and K(Je1, eα) = K(Ke1, eα) = −1, we have K(ein+1, eα) = −1, for all i = 0, 1, 2, 3 and α 6= 1, n+ 1, 2n+ 1, 3n+ 1. Let KN(eα, eγ) denote the sectional curvature of the level set with induced metric. By Gaussian equation, KN (eα, eγ)−K(eα, eγ) = hααhγγ, it is straightforward to obtain KN (en+1, e2n+1) = K N (en+1, e3n+1) = K N(e2n+1, e3n+1) = 0, KN(ein+1, eα) = 1, (5.3) for all i = 1, 2, 3 and α 6= 1, n+ 1, 2n+ 1, 3n+ 1. Part 3. There is a natural map ϕt between the level sets N0 and Nt given by the gradient flow of β. Since the integral curves are geodesics, dϕt(X) are Jacobi fields along corresponding curves. Let (N, g0) = N0 with the induced metric. We can consider ϕ as a flow on N . We claim that dϕt|V1 = e 2t id dϕt|V2 = e t id, where TN = V1 ⊕ V2, V1 = span{Ie1, Je1, Ke1} and V2 = V 1 . Indeed for any point q ∈ N0, denote e1(t) = ∇β(ϕ(t)) and {εα(t)} α=2 to be the parallel transport of the orthonormal base {eα} α=2 of N0 at q along ϕt(q). Since both V1 and V2 are ϕ-invariant, we have, in particular, 〈∇e1(t)εα, εγ〉 = 0, (5.4) when α ∈ {n+ 1, 2n+ 1, 3n+ 1}, and γ /∈ {n + 1, 2n+ 1, 3n+ 1}. Now we can compute R1α1γ . Then 〈R(e1, εα)e1, εγ〉 = 〈∇e1∇εαe1 −∇εα∇e1e1 −∇[e1,εα]e1, εγ〉 = 〈∇e1∇εγe1, εα〉 − 〈∇[e1,εα]e1, εγ〉 = 〈∇e1∇εαe1, εγ〉 − 〈∇∇e1εα−∇εαe1e1, εγ〉 = 〈∇e1∇εαe1, εγ〉 − 〈∇e1εα, ετ〉〈∇ετ e1, εγ〉 〈∇εαe1, ετ 〉〈∇ετe1, εγ〉 〈∇e1(hατετ ), εγ〉+ hγτ 〈∇e1εα, ετ 〉+ hατhτγ = −e1hαγ − hατ 〈∇e1ετ , εγ〉 hγτ 〈∇e1εα, ετ 〉+ hατhτγ. (5.5) We see that (hαγ) is diagonal and hαα = 2, when α = n + 1, 2n+ 1, 3n+ 1; 1, otherwise. Therefore, when α 6= γ, R1α1γ = −hαα〈∇e1εα, εγ〉+ hγγ〈∇e1εα, εγ〉 = (hγγ − hαα)〈∇e1εα, εγ〉. Since hαα = hγγ when α, γ ∈ {n+ 1, 2n+ 1, 3n+ 1} and α, γ /∈ {n+ 1, 2n+ 1, 3n+ 1}, using ( 5.4), we have R1α1γ = 0, for all α 6= γ. Define Jα(t) = e−2tεα, when α ∈ {n+ 1, 2n+ 1, 3n+ 1}; e−tεα, when α /∈ {n+ 1, 2n+ 1, 3n+ 1}. Since dϕt(eα)|t=0 = [e1, eα] = −∇eαe1, then we see that Jα satisfies the Jacobi equation and initial conditions Jα(0) = eα and J α(0) = eα = ∇ ∂ dϕt(eα)|t=0. By the uniqueness theorem for the Jacobi equations, we have dϕt(eα) = Jα. The claim is proved. Part 4. We have now a family of metrics on N written as ds2t = e ω2in+1 + e ω2in+α, and the metric of M can rewritten as ds2 = dt2 + e4t ω2p + e ω2α (5.6) where {ω2, ω3, ω4, . . . , ω4n} is the dual coframe to {e2, e3, e4, . . . , e4n} at N0. We also choose that Ie4s−3 = e4s−2, Je4s−3 = e4s−1, and Ke4s−3 = e4s for s = 1, . . . , n, with e1 = . In particular, the second fundamental form on Nt must be a diagonal matrix when written in terms of the basis {ei} i=2 with eigenvalues given by (〈∇eiej, e1〉) = 2I3 0 0 I4(n−1) , (5.7) where Ik denotes the k× k identity matrix. Also, the sectional curvatures of the sections containing e1 are given by K(e1, ep) = −4 for 2 ≤ p ≤ 4 K(e1, eα) = −1 for 5 ≤ α ≤ 4n. The Guass curvature equation also asserts that Rijkl = R̄ijkl + hlihkj − hkihlj, where R̄ijkl is the curvature tensor on Nt. In particular, Rijkl = R̄ijkl + δliδkj − δkiδlj if 5 ≤ i, j, k, l ≤ 4n R̄ijkl + 4δliδkj − 4δkiδlj if 2 ≤ i, j, k, l ≤ 4 R̄ijkl + 2 if 2 ≤ i = l ≤ 4 and 5 ≤ k = j ≤ 4n R̄ijkl + 2 if 2 ≤ k = j ≤ 4 and 5 ≤ i = l ≤ 4n R̄ijkl − 2 if 2 ≤ i = k ≤ 4 and 5 ≤ j = l ≤ 4n R̄ijkl − 2 if 2 ≤ j = l ≤ 4 and 5 ≤ i = k ≤ 4n R̄ijkl − 2 if 2 ≤ k = i ≤ 4 and 5 ≤ j = l ≤ 4n R̄ijkl otherwise. (5.8) We will now use ( 5.6) to compute the curvature tensor of M and hence N0. Using the orthonormal coframe η1 = ω1 = dt, ηp = e ηα = e for 2 ≤ p ≤ 4 and 5 ≤ α ≤ 4n, we obtain the first structural equations dη1 = 0, (5.9) dηp = 2e 2t ω1 ∧ ωp + e ωpq ∧ ωq + e ωpα ∧ ωα = −2ηp ∧ η1 + ωpq ∧ ηq + e ωpα ∧ ηα, (5.10) dηα = e t ω1 ∧ ωα + e ωαp ∧ ωp + e ωαβ ∧ ωβ = −ηα ∧ η1 + e ωαp ∧ ηp + ωαβ ∧ ηβ, (5.11) where ωij are the connection forms of N0. In the above and all subsequent computations, we will adopt the convention that 5 ≤ α, β ≤ 4n, 2 ≤ i, j ≤ 4n, 2 ≤ o, p, q, r ≤ 4, 2 ≤ s, t ≤ n, and 1 ≤ A,B ≤ 4n. Note that using the endomorphism I and the fact that ∇I = cJ − bK, we have ωij(X) = 〈∇̄Xej, ei〉 = 〈I∇Xej , Iei〉 = 〈∇XIej, Iei〉+ c(X) 〈Jej, Iei〉 − b(X) 〈kej , Iei〉 = 〈∇XIej, Iej〉+ c(X) 〈ej, Kei〉+ b(X)〈ej , Jei〉 for any tangent vector X to N0, where ∇̄ denotes the connection on N0. Hence we conclude that ωij = ωIiIj + c 〈ej, Kei〉+ b 〈ej, Jei〉, (5.12) where Ii denotes the index corresponding to Iei = eIi. Similarly, we have ωij = ωJiJj + c 〈ej, Kei〉+ a 〈ej, Iei〉, ωij = ωKiKj + b 〈ej, Jei〉+ a 〈ej , Iei〉. Together with ( 5.7), we conclude that ω2(4s−1)(e4s) = −1 = −ω2(4s)(e4s−1), ω2(4s−3)(e4s−2) = −1 = −ω2(4s−2)(e4s−3), for all 2 ≤ s ≤ n, and ω2α(eβ) = 0 otherwise. Similarly, ω2α(ep) = 〈∇epeα, e2〉 = −〈∇epIeα, e1〉 These identities imply that ω2(4s−3) = −ω4s−2, ω2(4s−2) = ω4s−3, ω2(4s−1) = −ω4s, ω2(4s) = ω4s−1. (5.13) A similar calculation using the endomorphisms J and K yield ω3(4s−3) = −ω4s−1, ω3(4s−2) = ω4s, ω3(4s−1) = ω4s−3, ω3(4s) = −ω4s−2, (5.14) ω4(4s−3) = −ω4s, ω4(4s−2) = −ω4s−1, ω4(4s−1) = ω4s−2, ω4(4s) = ω4s−3. (5.15) We claim that the connection forms are given by η1p = −ηp1 = 2ηp for 2 ≤ p ≤ 4, (5.16) η1α = −ηα1 = ηα for 5 ≤ α ≤ 4n, (5.17) ηpq = −ηqp = ωpq, (5.18) ηpα = −ηαp = e t ωpα, (5.19) η(4s)β = −ηβ(4s) ω(4s)β − (1−e −2t) η2 if β = 4s− 1 ω(4s)β + (1−e −2t) η3 if β = 4s− 2 ω(4s)β − (1−e −2t) η4 if β = 4s− 3 ω(4s)β if β 6= 4s− 1, 4s− 2, or 4s− 3, (5.20) η(4s−1)β = −ηβ(4s−1) ω(4s−1)β + (1−e −2t) η2 if β = 4s ω(4s−1)β − (1−e −2t) η4 if β = 4s− 2 ω(4s−1)β − (1−e −2t) η3 if β = 4s− 3 ω(4s−1)β if β 6= 4s, 4s− 2, or 4s− 3, (5.21) η(4s−2)β = −ηβ(4s−2) ω(4s−2)β − (1−e −2t) η3 if β = 4s ω(4s−2)β + (1−e −2t) η4 if β = 4s− 1 ω(4s−2)β − (1−e −2t) η2 if β = 4s− 3 ω(4s−2)β if β 6= 4s, 4s− 1, or 4s− 3, (5.22) η(4s−3)β = −ηβ(4s−3) ω(4s−3)β + (1−e −2t) η4 if β = 4s ω(4s−3)β + (1−e −2t) η3 if β = 4s− 1 ω(4s−3)β + (1−e −2t) η2 if β = 4s− 2 ω(4s−3)β if β 6= 4s, 4s− 1, or 4s− 2. (5.23) Indeed, if we substitute ( 5.16− 5.23) into the first structural equations dηA = ηA1 ∧ η1 + ηAq ∧ ηq + ηAβ ∧ ηβ we obtain ( 5.9), ( 5.10), and ( 5.11). To compute the curvature, we consider the second structural equations. In particular, dη1p − η1q ∧ ηqp − η1α ∧ ηαp = 2dηp − 2ηq ∧ ηqp − ηα ∧ ηαp = −4ηp ∧ η1 + ηα ∧ ηαp = −4ηp ∧ η1 + e t ωpα ∧ ηα. Hence using ( 5.13− 5.15), we have R1p1p = −4, R12(4s−1)(4s) =− 2 = −R12(4s)(4s−1), R12(4s−3)(4s−2) =− 2 = −R12(4s−2)(4s−3), R13(4s)(4s−2) =− 2 = −R13(4s−2)(4s), R13(4s−1)(4s−3) =2 = −R13(4s−3)(4s−1), R14(4s)(4s−3) =2 = −R14(4s−3)(4s), R14(4s−1)(4s−2) =2 = −R14(4s−2)(4s−1), R1pAB = 0, otherwise. Also, dη1α − η1q ∧ ηqα − η1β ∧ ηβα = dηα − 2ηq ∧ ηqα − ηβ ∧ ηβα = −ηα ∧ η1 + e t ωqα ∧ ηq, hence R1α1α = −1, R1(4s)(4s−1)2 =− 1 = −R1(4s−1)(4s)2, R1(4s)(4s−2)3 =1 = −R1(4s−2)(4s)3, R1(4s)(4s−3)4 =− 1 = −R1(4s−3)(4s)4, R1(4s−1)(4s−3)3 =− 1 = −R1(4s−3)(4s−1)3, R1(4s−1)(4s−2)4 =− 1 = −R1(4s−2)(4s−1)4, R1(4s−2)(4s−3)2 =− 1 = −R1(4s−3)(4s−2)2, R1αAB = 0 otherwise. Similarly, dηpq − ηp1 ∧ η1q − ηpr ∧ ηrq − ηpβ ∧ ηβq = dωpq + 4ηp ∧ ηq − ωpr ∧ ωrq − e 2tωpβ ∧ ωβq = Ω̄pq + (1− e 2t)ωpβ ∧ ωβq + 4ηp ∧ ηq, where Ω̄pq = R̄pqijωj ∧ ωi is the curvature form of N0. In particular, this implies that Rpqro = −4+e−4t R̄pqpq if r = p and o = q 4+e−4t R̄pqqp if r = q and o = p e−4t R̄pqro otherwise, (5.24) R23(4s)(4s−3) = e −2t R̄23(4s)(4s−3) − 2(e −2t − 1), R23(4s−1)(4s−2) = e −2t R̄23(4s−1)(4s−2) − 2(e −2t − 1), R24(4s)(4s−2) = e −2t R̄24(4s)(4s−2) − 2(e −2t − 1), R24(4s−1)(4s−3) = e −2t R̄24(4s−1)(4s−3) + 2(e −2t − 1), R34(4s)(4s−1) = e −2t R̄34(4s)(4s−1) − 2(e −2t − 1), R34(4s−2)(4s−3) = e −2t R̄34(4s−2)(4s−3) − 2(e −2t − 1), Rpqαβ = e −2t R̄pqαβ, otherwise. We now continue with our curvature computation and consider dηpα − ηp1 ∧ η1α − ηpq ∧ ηqα − ηpβ ∧ ηβα = d(et ωpα) + 2ηp ∧ ηα − ωpq ∧ e t ωqα − e t ωpβ ∧ ηβα = et η1 ∧ ωpα + etR̄pαijωj ∧ ωi + 2ηp ∧ ηα + e t ωpβ ∧ (ωβα − ηβα), where R̄pαij is the curvature tensor of N0. Using ( 5.13− 5.15) and ( 5.20− 5.23), we have R2(4s)ABηB ∧ ηA = η1 ∧ η(4s−1) + etR̄2(4s)ij ωj ∧ ωi − 2η(4s) ∧ η2 + (1− e −2t) η(4s) ∧ η2 + (1− e−2t) η(4s−3) ∧ η3 + (1− e −2t) η(4s−2) ∧ η4 = η1 ∧ η(4s−1) + etR̄2(4s)ij ωj ∧ ωi − (1 + e −2t) η(4s) ∧ η2 + (1− e−2t) η(4s−3) ∧ η3 + (1− e −2t) η(4s−2) ∧ η4. R2(4s−1)ABηB ∧ ηA = −η1 ∧ η(4s) + etR̄2(4s−1)ij ωj ∧ ωi − (1 + e −2t) η(4s−1) ∧ η2 + (1− e−2t)η(4s−2) ∧ η3 − (1− e −2t) η(4s−3) ∧ η4. R2(4s−2)ABηB ∧ ηA = η1 ∧ η4s−3 + etR̄2(4s−2)ij ωj ∧ ωi − (1 + e −2t) η(4s−2) ∧ η2 + (1− e−2t) η(4s−1) ∧ η3 − (1− e −2t) η(4s) ∧ η4. R2(4s−3)ABηB ∧ ηA = −η1 ∧ η4s−2 + etR̄2(4s−3)ij ωj ∧ ωi − (1 + e −2t) η(4s−3) ∧ η2 − (1− e−2t) η(4s) ∧ η3 + (1− e −2t) η(4s−1) ∧ η4. Similar formulas for the curvature tensors of the form R3αAB and R4αAB . Continuing with our computation of the second structural equations using ( 5.13− 5.15), we have dη(4s−1)(4s) − η(4s−1)1 ∧ η1(4s) − η(4s−1)q ∧ ηq(4s) − η(4s−1)β ∧ ηβ(4s) = dω(4s−1)(4s) + 2e −2t η1 ∧ η2 + (1− e −2t) dη2 + η(4s−1) ∧ η(4s) − e 2t ω(4s−1)q ∧ ωq(4s) − (ω(4s−1)(4s−2) − (1− e −2t) η4) ∧ (ω(4s−2)(4s) − (1− e −2t) η3) − (ω(4s−1)(4s−3) − (1− e −2t) η3) ∧ (ω(4s−3)(4s) + (1− e −2t) η4) R̄(4s−1)(4s)ij ωj ∧ ωi + (1− e 2t)ω(4s−1)q ∧ ωq(4s) + 2η1 ∧ η2 + (1− e−2t)ω2q ∧ ηq + e t(1− e−2t)ω2β ∧ ηβ − η(4s) ∧ η(4s−1) + (1− e−2t)ω(4s−1)(4s−2) ∧ η3 + (1− e −2t) η4 ∧ ω(4s−2)(4s) + (1− e−2t) η3 ∧ ω(4s−3)(4s) − (1− e −2t)ω(4s−1)(4s−3) ∧ η4 + 2(1− e−2t)2 η3 ∧ η4 R̄(4s−1)(4s)ij ωj ∧ ωi + (2− e −2t) η(4s−1) ∧ η(4s) − 2(1− e −2t) η(4s−3) ∧ η(4s−2) + 2η1 ∧ η2 + (1− e −2t)ω2q ∧ ηq + 2(1− e −2t) η(4r−3) ∧ η(4r−2) + 2(1− e−2t) η(4r−1) ∧ η(4r) + (1− e −2t) (ω(4s−1)(4s−2) − ω(4s−3)(4s)) ∧ η3 − (1− e−2t) (ω(4s−2)(4s) + ω(4s−1)(4s−3)) ∧ η4 + 2(1− e −2t)2 η3 ∧ η4. (5.25) Note that ( 5.22) asserts that (1− e−2t)ω2q ∧ ηq = (1− e −2t) (−ω14 ∧ η3 + c ∧ η3 + ω13 ∧ η4 − b ∧ η4) = (1− e−2t) (4e−2t η3 ∧ η4 + c ∧ η3 − bη4), (1− e−2t) (ω(4s−1)(4s−2) − ω(4s−3)(4s)) ∧ η3 = −(1− e−2t) c ∧ η3, −(1− e−2t) (ω(4s−2)(4s) + ω(4s−1)(4s−3)) ∧ η4 = (1− e−2t) b ∧ η4. Hence substituting into ( 5.25), we obtain R(4s−1)(4s)AB ηB ∧ ηA R̄(4s−1)(4s)ij ωj ∧ ωi + (2− e −2t) η(4s−1) ∧ η(4s) − 2(1− e −2t) η(4s−3) ∧ η(4s−2) + 2η1 ∧ η2 + 2(1− e−2t) η(4r−3) ∧ η(4r−2) + 2(1− e −2t) η(4r−1) ∧ η(4r) + 2(1− e −4t) η3 ∧ η4 = 2η1 ∧ η2 + R̄(4s−1)(4s)pq e −4t ηq ∧ ηp + 2(1− e −4t) η3 ∧ η4 + R̄(4s−1)(4s)pα e −3t ηα ∧ ηp R̄(4s−1)(4s)αβ e −2t ηβ ∧ ηα + (2− e −2t) η(4s−1) ∧ η(4s) − 2(1− e −2t) η(4s−3) ∧ η(4s−2) + 2(1− e−2t) η(4r−3) ∧ η(4r−2) + 2(1− e −2t) η(4r−1) ∧ η(4r). A similar computation yields the curvature tensor of the formR(4s−1)(4s−2)AB , R(4s−1)(4s−3)AB , R(4s−2)(4s−3)AB , R(4s−2)(4s)AB , and R(4s−3)(4s)AB . It remains to compute R(4s−3)(4r)AB ηB ∧ ηA = dη(4s−3)(4r) − η(4s−3)1 ∧ η1(4r) − η(4s−3)q ∧ ηq(4r) − η(4s−3)β ∧ ηβ(4r) = dω(4s−3)(4r) + η(4s−3) ∧ η(4r) − e 2t ω(4s−3)q ∧ ωq(4r) − η(4s−3)β ∧ ηβ(4r) R̄(4s−3)(4r)ij ωj ∧ ωi + (1− e 2t)ω(4s−3)q ∧ ωq(4r) − (1− e −2t) (η4 ∧ ω(4s)(4r) + η3 ∧ ω(4s−1)(4r))− (1− e −2t) (η2 ∧ ω4s−2)(4r) + ω(4s−3)(4r−1) ∧ η2 − ω(4s−3)(4r−2) ∧ η3 + ω(4s−3)(4r−3) ∧ η4) + η(4s−3) ∧ η(4r). (5.26) Using ( 5.12− 5.14), we can write ω(4s−3)q ∧ ωq(4r) = −η(4s− 2) ∧ η(4r−1) + η(4s−1) ∧ η(4r−2) − η(4s) ∧ η(4r−3)). Also using ( 5.12) asserts that ω(4s−3)(4r−1) = ω(4s−2)(4r), ω(4s−3)(4r−2) = −ω((4s−1)(4r) ω((4s−3)(4r−3) = ω((4s)(4r). Hence ( 5.26) becomes R(4s−3)(4r)AB ηB ∧ ηA R̄(4s−3)(4r)pq e −4t ηq ∧ ηp + R̄(4s−3)(4r)pβ e −3t ηβ ∧ ηp + R̄(4s−3)(4r)αβ e −2t ηβ ∧ ηα − (1− e2t) η(4s−2) ∧ η(4r−1) + (1− e −2t) η(4s−1) ∧ η(4r−2) − (1− e−2t) η(4s) ∧ η(4r−3) + η(4s−3) ∧ η4r. So we have determined all curvature tensors of M . Note that the quater- nionic curvatures satisfy K(e1, e2) +K(e1, e3) +K(e1, e4) = −12 K(e2, e1) +K(e2, e3) +K(e2, e4) = −12 + e −2t (KN(e2, e3) +K N (e2, e4)) K(e3, e1) +K(e3, e2) +K(e3, e4) = −12 + e −2t (KN(e3, e2) +K N (e3, e4)) K(e4, e1) +K(e4, e2) +K(e4, e3) = −12 + e −2t (K̄(e4, e2) +K N (e4, e3)). In particular, this implies that KN(e2, e3) = K N (e2, e4) = K N (e3, e4) = 0. Also, for 2 ≤ p ≤ 4, we have K(e1, e(4s−i)) = −4 K(ep, e(4s−i) = −4 + e −2t ( K̄(ep, e(4s−i))− 4) implying KN (ep, e(4s−i)) = 4. We also have K(e(4s), e(4s−1)) = −12 + e −2t ( KN(e(4s), e(4s−i)) + 9) implying KN (e(4s), e(4s−i)) = −9. Lastly, K(e(4s), e(4r−i)) = −4 + e KN(e(4s), e(4r−i)) implying KN (e(4s), e(4r−i)) = 0. The above computation determined the whole curvature tensor for M and N0. In particular, if M has bounded curvature, then from the formulas about the components of curvature tensors of M , all curvature components are determined as those of QHn. 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Shengli Kong Department of Mathematics http://arxiv.org/abs/math/0701865 University of California, Irvine Irvine, CA92697-3875, USA email: skong@math.uci.edu Peter Li Department of Mathematics University of California, Irvine Irvine, CA92697-3875, USA email:pli@math.uci.edu Detang Zhou Departamento de Geometria Insitituto de Matematica Universidade Federal Fluminense- UFF Centro, Niterói, RJ 24020-140, Brazil email: zhou@impa.br Preliminaries on quaternionic Kähler manifolds Laplacian comparison theorem Quaternionic harmonicity Uniqueness of infinite volume end Maximal first eigenvalue

Microsoft Word - CurrentStressSSLb.doc Biased Structural Fluctuations due to Electron Wind Force O. Bondarchuk*, W.G. Cullen, M. Degawa and Ellen D. Williams Department of Physics University of Maryland at College Park College Park, MD 20742-4111 T. Bole and P.J. Rous Department of Physics University of Maryland Baltimore County 1000 Hilltop Circle Baltimore, MD 21250 Direct correlation between temporal structural fluctuations and electron wind force is demonstrated, for the first time, by STM imaging and analysis of atomically-resolved motion on a thin film surface under large applied current (105A/cm2). The magnitude of the momentum transfer between current carriers and atoms in the fluctuating structure is at least 5x to 15x (± one sigma range) larger than for freely diffusing adatoms. The corresponding changes in surface resistivity will contribute significant fluctuation signature to nanoscale electronic properties. PACS #s: 73.63.-b Electronic transport in nanoscale materials and structures 73.25.+i Surface conductivity and carrier phenomena 73.50.Td Noise processes and phenomena 68.37.Ef Scanning tunneling microscopy 68.55.-a Thin film structure and morphology * Present address: Chemical Physics Department, Fritz-Haber-Insitut der Max- Planck - Gesellschaft, Faradayweg 4-6, Berlin, 14195 Germany Introduction. Due to the size-scaling of fluctuations, the effects of statistical mechanics will be very different at the nanoscale than for macroscopic systems. The effects of nanoscale thermal fluctuations will impact molecular electronic and nanoelectronic contacts 1-3, device stability 4, 5, electromigration 6-14 and noise 15-17. In this work we quantify the relationship of thermal fluctuations with electrical transport by directly observing step fluctuations at the surface of a current-carrying metal film, as illustrated in Fig. 1. Carrier scattering causes a force due to momentum transfer, known as the electromigration wind force, and corresponding changes in the surface resistivity. By convention, this force is written in terms of an effective valence Z*, and the (macroscopic) applied electric field E: F=Z*eE 18-20. The momentum transfer force felt by atoms at surfaces depends on the local environment: atoms at step edges, near defects, or freely diffusing at the terrace experience different forces 21-23. These forces can cause substantial changes in surface morphology 24-27, due to mechanisms similar to those well known in electromigration-induced failure 6, 18-20. Despite its substantial impact upon the morphological evolution of materials, the electromigration wind force is extremely weak, and detecting its effects have required following changes in structure after long periods of current stressing. Here, we will describe direct observation of the effects of the electromigration force on a time scale of seconds by measuring the nanoscale fluctuations of atomic-layer steps 28 on the surface of a current-carrying metallic thin film. The fluctuations of a surface step are observed via a direct measurement of the position of one element of the step as a function of time, x(t). Near equilibrium, step fluctuations can be well-described using the continuum step model 28-31, which predicts that the time-correlation function grows as a power law for times less than the correlation time. For the system described here, steps on Ag(111), step motion is driven by step edge diffusion (SED) 30, 32, 33, for which the correlation function is: G t( ) " x(t) # x(0)[ ] . (1) Here x(t) is the position of the step perpendicular to the average step-edge orientation and the average is taken by repeated observations, ax=0.25 nm and ay=0.29 nm are the lattice constants perpendicular and parallel to the step edge. The time characteristic of thermal fluctuations of the step edge, , is determined by the step stiffness, ˜ " , and the hopping time for atomic motion along the step edge: 2$ 3 4( ) ˜ + a , (2) where Γ is the Gamma function, and the value of τh has been measured to decrease from ~ 3 µs to ~3 ns between 300K and 460K 33. Recently the step continuum model has been expanded to include the effect of an electromigration force acting perpendicular to a step that is fluctuating via SED 34. The correlation function deviates from the equilibrium result as: G(t) = a ± t " EM( )0 * 1+ e+2u EM( )du , ax2 1± 0.3487 1/ 2- (3) where the + and – signs correspond to a downhill and uphill direction of force respectively. The presence of the electromigration force gives rise to an additional time constant, , that represents the time when the time-correlation function begins to deviate significantly from its equilibrium behavior, T( ) ˜ # ax( ) ) ) " h , (4) Since the electromigration force is weak, T and << ˜ " a we anticipate that . The nature of the result indicated by Eq. 3 can be understood by analogy to the Bales-Zangwill `kinetic instability 35. In both cases, a diffusional bias perpendicular to the step edge results in spontaneous increased deviations from the equilibrium position when the bias favors diffusion in the step-downhill direction. For an up-hill bias, an anomalous straightening of the step edge occurs. In the remainder of this letter, we exploit this model. Measurement of the correlations of thermal step-edge fluctuations in the presence of current stressing yield the electromigration time constant (Eq. 3) and, as a result, the electromigration force felt by atoms diffusing along the step edge (Eq. 4). This represents a direct measurement of the effective valence of an atom at the surface of a current-carrying solid. The experimental methods for preparing atomically clean (111)-oriented silver films have been described previously 33, 36. The films used here were 100 to 200 nm thick and 1-2 mm wide, with micron-scale areas of flat (111)-oriented surface separated by deep pits, which covered about 50% of the film area at the smallest film thickness (100 nm). Atomic cleanliness was confirmed by atomic-resolution STM imaging. Imaging was performed using tunneling conditions of 0.6-0.8 nA and 1V, at a scan rate of ~9 pixels/ms, which are known not to perturb the measured step configurations 30, 32, 33. After completing the STM measurements on each sample, the sample temperature was measured using a thermocouple brought into direct contact with the film surface 33. The thermocouple values and the measured hopping time constant 33 were used to determine the sample temperatures. The temporal evolution of the step fluctuations was observed by repeated STM scans across the step boundaries as shown in Figure 2. The size of each image is 100 nm x 512 scans (56.6 ms/scan) and an electric current of 0.4 A (nominal current density 1x105 A/cm2) flows through the sample perpendicular to the step edges. The temperature of the current-stressed sample (due to Joule heating) was 370±30 K. By following the motion of the edges of the steps shown in Fig. 2, we simultaneously determine the spatial variations x(t) for up-hill and down-hill steps fluctuating at the same sample temperature under the same current density. The time correlation functions obtained from the measured x(t) are shown in Fig. 3. The magnitude of the correlation function grows more rapidly for the up-hill current than for the down-hill current, consistent with an electromigration force acting in the same direction as the electron flow in the sample (e.g. Z* < 0, an electron wind force). The data were fit to Eq. 3, with the thermal and electromigration time constants (τ4 and τEM respectively) as the only adjustable parameters. The results of these fits are shown as solid curves in Fig. 3, and demonstrate excellent agreement between the measured correlation functions and those predicted by the Langevin theory (eqn. 4). As can be seen in fig. 4, the two fits yield clear χ2 minima for τEM of 16s and 52s, respectively for the down-hill and up-hill steps. Measurements on two additional steps subject to up-hill current stressing at a higher nominal current density (Jnom = 4x105A/cm2) gave fitting results similar to those of Fig. 3, with time constants of 98s (325K) and 32s (350K). The uncertainties in the fit parameters were 15-40%. The correlation function for an unstressed sample measured at 325K is also shown in Fig. 3. There is no minimum in the chi-squared value for the fit as a function of electromigration time constant, with virtually no change in the goodness of fit occurring for values larger than 9000s (see fig. 4). This shows that data is well fit by a single parameter, τ4, as expected when no electromigration force is acting (i.e. #$). Similar results were consistently obtained for other steps measured without current stressing. The electromigration force is found using the measured values of τEM in Eqs. 1 and 4, given the step stiffness ˜ " . The stiffness is calculated 37, 38 using an effective kink energy of ε = 0.117 eV 39, 40. The four measurements of the electromigration time constant yield average values of the force per step-edge atom of -2.7x10-5 eV/nm for Jnom= 4x105A/cm2 (325-350K) and -9.7x10-6 eV/nm for Jnom= 1x105A/cm2 (370K). Thus, within the cumulative uncertainties in the measured forces of ±50%, the force increases in direct proportion to the current density. The measured values of the force can be used to determine the effective valence Z* if the local surface current density and the resistivity in the surface region are known. Since these local quantities cannot be measured directly, we first estimate them using the bulk current densities and the bulk resistivity of Ag, which is approximately 1.8x10-6Ω-cm at 325K and 2.2x10-6Ω-cm at 370K 41, 42. The resulting effective valence, obtained using the nominal current density, is Z* = -(4±2)x102. The magnitude is substantially larger than the predicted effective valence of an isolated Ag adatom on Ag(111), which is is Z*= "19 43. For atoms in a close- packed site along a step edge, with a perpendicular current direction, the direct force per step atom may be as much as 2x higher than the force on an adatom 21, 23, which would yield a predicted valence of Z* ~ -38, still much smaller magnitude than the measured value. A substantial systematic effect can be attributed to the film cross section, because as described earlier, at 100 nm film thickness there are vacancies in the film up to 50% of the surface area. Therefore the bulk current densities may be as much as 2x higher than the nominal values, and as a result the lower limit on the effective valence is Z* = -(2±1)x102. The remainder of the difference compared with the perpendicular force on a close-packed step-edge atom may arise from the highly kinked environment suggested in Fig. 1. Because diffusion is parallel to the step edge, only the component of the electromigration force tangential to the local step orientation will affect step-edge diffusion. The largest impact of the electromigration force will thus occur for the most highly kinked step regions. There have been no calculations of the electromigration force acting on low-symmetry kink sites. However, such sites have enhanced valence charge density 44, and also present anomalous barriers to step-edge diffusion 45. Such significant changes in local electronic structure may be reflected in significant changes in the scattering cross section (and thus Z* value). In addition, the geometric effect of the kink configuration is likely to enhance scattering via blocking 21, or constriction-induced enhancement of local current density 46 analogous to current crowding 47, 48. The forces measured above are related to equal and opposite forces on the charge carriers, which translate into changes in the surface resistivity 22, 43. This can be evaluated by treating the diffusing step-edge atoms as independent scattering sites of density nk = (LkLstep)-1, where Lk is the average distance between diffusing atoms along the step edge and Lstep is the average distance between steps. Then the change in the surface resistivity ρs due to the diffusing step-edge atoms is 43: , , , (5) where Lf is the film thickness, nAg = 58.5 nm-3 is the bulk carrier density for Ag, k is the wind force acting per atom (our measured value) and j are additional changes in force on the carriers due to the perturbation of atomic structure in the immediate vicinity of the step-edge atoms. Using the measured force per atom and the upper limit of the current density yields the component of the surface resistivity due to scattering at the diffusing step-edge atoms alone, ~ (3±1.5nm , where ρ0 is the bulk value of the Ag resistivity. The impact of diffusing step-edge atoms on surface resistivity will include the direct- scattering term measured here as well as the perturbed lattice terms ( k in Eq. 5). The latter term may contribute as much as 2/3 of the total resistivity change, thus it is reasonable to expect that scattering at the step-edge will contribute a total change in the surface resistivity $ (10 ± 5nm . As an example, for moderate step and kink densities (Lstep =10 nm and Lk =2 nm) and a very thin film (Lf = 10 nm), the change in surface resistivity due to scattering at the step-kink sites could be as large as 10% of the bulk resistivity. This effect will be significant in nanoelectronic devices carrying large current densities 49, 50. The present observation of biased temporal fluctuations under current stress is the first direct correlation of nanoscale structural fluctuations with the electromigration wind force. Because of systematic uncertainty in the actual current density, analysis of the results yields a lower limit for the magnitude of the electron wind force on the diffusing step-edge atoms. Taking the one-sigma limits on the statistical uncertainties, we find that the value is five to fifteen times larger than that on individual adatoms 43. Effects on resistivity, noise and electromigration susceptibility in metal nanoelectronic structures will thus be concomitantly higher than would have been expected 51. Acknowledgments This work was supported by the U.S. Department of Energy Award No. DOE-FG02-01ER45939. We also gratefully acknowledge support and SEF support from the NSF MRSEC under grant DMR 05-20471. Figure Captions Fig. 1: Schematic illustration of current flow perpendicular to average orientation of steps on the surface. Enhanced scattering from step sites at the surface is suggested by the arrows. The inset illustrates adatoms on the terraces and the kinked (thermally roughened) step edge. Fig. 2: STM data for a current stressed sample. Applied current was 0.4 A and nominal sample cross section 200 nm thick x 2 mm wide. Sample temperature = 380 K. Upper panel – repeated scans across the edges of three steps (pseudo image) (from left to right one downhill step and two uphill steps) show fluctuations of step position in time x(t). Lower panel – height vs. position across the image is consistent with steps a single layer (0.236 nm) high. Fig. 3: Time correlation functions, G(t) in units Å2, for the step fluctuations (x(t) data) extracted from repeated measurements as shown in Fig. 1 and described in the text. The data for the step- down current (open circles) is the average of 10 separate measurements, and for the step-up current (open squares) is the average is over 9 data sets. Fits to Eq. 4 are shown as the solid lines for each data set. Also shown is the measured correlation function (average of 19 separate measurements) for the unstressed sample (open triangles) fit to a single power law. Fig. 4 The reduced chi-squared plotted as a function of τEM, while holding the value of τ4 at its optimum value. Solid curve: step-down current. Dot-dashed curve: step-up current. Dashed curve: unstressed data. Note that, for the unstressed data, there is no clear minimum value of τEM, i.e. a fit to a single parameter (τ4) suffices to describe the data. Fig. 1: Schematic illustration of current flow perpendicular to average orientation of steps on the surface. Enhanced scattering from step sites at the surface is suggested by the arrows. The inset illustrates adatoms on the terraces and the kinked (thermally roughened) step edge. Fig.2: (Color on-line) STM data for a current stressed sample. Applied current was 0.4 A and nominal sample cross section 200 nm thick x 2 mm wide. Upper panel – repeated scans across the edges of three steps (pseudo image) (from left to right one downhill step and two uphill steps) show fluctuations of step position in time x(t). Lower panel – height vs. position across the steps of single layer height (0.236 nm). Fig. 3: (Color on-line) Time correlation functions, G(t) for the step fluctuations (x(t) data) extracted from repeated measurements. The data for the step-down current (open circles) is the average over 10 separate measurements, and for the step-up current (open squares) the average is over 9. Fits to Eq. 3 are shown as the solid lines for each data set. Also shown is the measured correlation function (average of 19 separate measurements) for an unstressed sample (open triangles) fit to a single power law. 0 20 40 60 80 100 Fig. 4 (color on-line)The reduced chi-squared plotted as a function of τEM, while holding the value of τ4 at its optimum value. Solid curve: step-down current. Dot-dashed curve: step-up current. Dashed curve: unstressed data. Note that, for the unstressed data, there is no clear minimum value of τEM that is consistent with the data, i.e. a fit to a single parameter (τ4) suffices to describe the data. References 1 V. V. Zhirnov and R. K. Cavin, Nature Materials 5, 11 (2006). 2 H. Basch, R. Cohen, and M. A. Ratner, Nano Letters 5, 1668 (2005). 3 C. Tao, , Nano Letters, in press (2007). 4 A. Bid, A. Bora, and A. K. Raychaudhuri, Phys. Rev. B 72, 113415 (1 (2005). 5 C.-H. Zhang, F. Kassubek, and C. A. Stafford, Phys. Rev. B 68, 165414 (1 (2003). 6 H. Yasunaga and A. Natori, Surface Sci. Rep. 15, 205 (1992). 7 P. J. Rous, T. L. Einstein, and E. D. Williams, Surface Sci. 315, L995 (1994). 8 M. R. Gungor and D. Maroudas, Surface Sci. 415, L1055 (1998). 9 O. Pierre-Louis and T. L. Einstein, Phys. Rev. B62, 13697 (2000). 10 P. Kuhn, et al., Phys. Rev. Lett. 94 (2005). 11 S. Heinze, N.-P. Wang, and J. Tersoff, Phys. Rev. Lett. 95, 186802 (2005). 12 B. Stahlmecke, et al., Appl. Phys. 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Direct correlation between temporal structural fluctuations and electron wind force is demonstrated, for the first time, by STM imaging and analysis of atomically-resolved motion on a thin film surface under large applied current (10e5 Amp/sqare cm). The magnitude of the momentum transfer between current carriers and atoms in the fluctuating structure is at least five to fifteen times (plus or minus one sigma range) larger than for freely diffusing adatoms. The corresponding changes in surface resistivity will contribute significant fluctuation signature to nanoscale electronic properties.

Introduction. Due to the size-scaling of fluctuations, the effects of statistical mechanics will be very different at the nanoscale than for macroscopic systems. The effects of nanoscale thermal fluctuations will impact molecular electronic and nanoelectronic contacts 1-3, device stability 4, 5, electromigration 6-14 and noise 15-17. In this work we quantify the relationship of thermal fluctuations with electrical transport by directly observing step fluctuations at the surface of a current-carrying metal film, as illustrated in Fig. 1. Carrier scattering causes a force due to momentum transfer, known as the electromigration wind force, and corresponding changes in the surface resistivity. By convention, this force is written in terms of an effective valence Z*, and the (macroscopic) applied electric field E: F=Z*eE 18-20. The momentum transfer force felt by atoms at surfaces depends on the local environment: atoms at step edges, near defects, or freely diffusing at the terrace experience different forces 21-23. These forces can cause substantial changes in surface morphology 24-27, due to mechanisms similar to those well known in electromigration-induced failure 6, 18-20. Despite its substantial impact upon the morphological evolution of materials, the electromigration wind force is extremely weak, and detecting its effects have required following changes in structure after long periods of current stressing. Here, we will describe direct observation of the effects of the electromigration force on a time scale of seconds by measuring the nanoscale fluctuations of atomic-layer steps 28 on the surface of a current-carrying metallic thin film. The fluctuations of a surface step are observed via a direct measurement of the position of one element of the step as a function of time, x(t). Near equilibrium, step fluctuations can be well-described using the continuum step model 28-31, which predicts that the time-correlation function grows as a power law for times less than the correlation time. For the system described here, steps on Ag(111), step motion is driven by step edge diffusion (SED) 30, 32, 33, for which the correlation function is: G t( ) " x(t) # x(0)[ ] . (1) Here x(t) is the position of the step perpendicular to the average step-edge orientation and the average is taken by repeated observations, ax=0.25 nm and ay=0.29 nm are the lattice constants perpendicular and parallel to the step edge. The time characteristic of thermal fluctuations of the step edge, , is determined by the step stiffness, ˜ " , and the hopping time for atomic motion along the step edge: 2$ 3 4( ) ˜ + a , (2) where Γ is the Gamma function, and the value of τh has been measured to decrease from ~ 3 µs to ~3 ns between 300K and 460K 33. Recently the step continuum model has been expanded to include the effect of an electromigration force acting perpendicular to a step that is fluctuating via SED 34. The correlation function deviates from the equilibrium result as: G(t) = a ± t " EM( )0 * 1+ e+2u EM( )du , ax2 1± 0.3487 1/ 2- (3) where the + and – signs correspond to a downhill and uphill direction of force respectively. The presence of the electromigration force gives rise to an additional time constant, , that represents the time when the time-correlation function begins to deviate significantly from its equilibrium behavior, T( ) ˜ # ax( ) ) ) " h , (4) Since the electromigration force is weak, T and << ˜ " a we anticipate that . The nature of the result indicated by Eq. 3 can be understood by analogy to the Bales-Zangwill `kinetic instability 35. In both cases, a diffusional bias perpendicular to the step edge results in spontaneous increased deviations from the equilibrium position when the bias favors diffusion in the step-downhill direction. For an up-hill bias, an anomalous straightening of the step edge occurs. In the remainder of this letter, we exploit this model. Measurement of the correlations of thermal step-edge fluctuations in the presence of current stressing yield the electromigration time constant (Eq. 3) and, as a result, the electromigration force felt by atoms diffusing along the step edge (Eq. 4). This represents a direct measurement of the effective valence of an atom at the surface of a current-carrying solid. The experimental methods for preparing atomically clean (111)-oriented silver films have been described previously 33, 36. The films used here were 100 to 200 nm thick and 1-2 mm wide, with micron-scale areas of flat (111)-oriented surface separated by deep pits, which covered about 50% of the film area at the smallest film thickness (100 nm). Atomic cleanliness was confirmed by atomic-resolution STM imaging. Imaging was performed using tunneling conditions of 0.6-0.8 nA and 1V, at a scan rate of ~9 pixels/ms, which are known not to perturb the measured step configurations 30, 32, 33. After completing the STM measurements on each sample, the sample temperature was measured using a thermocouple brought into direct contact with the film surface 33. The thermocouple values and the measured hopping time constant 33 were used to determine the sample temperatures. The temporal evolution of the step fluctuations was observed by repeated STM scans across the step boundaries as shown in Figure 2. The size of each image is 100 nm x 512 scans (56.6 ms/scan) and an electric current of 0.4 A (nominal current density 1x105 A/cm2) flows through the sample perpendicular to the step edges. The temperature of the current-stressed sample (due to Joule heating) was 370±30 K. By following the motion of the edges of the steps shown in Fig. 2, we simultaneously determine the spatial variations x(t) for up-hill and down-hill steps fluctuating at the same sample temperature under the same current density. The time correlation functions obtained from the measured x(t) are shown in Fig. 3. The magnitude of the correlation function grows more rapidly for the up-hill current than for the down-hill current, consistent with an electromigration force acting in the same direction as the electron flow in the sample (e.g. Z* < 0, an electron wind force). The data were fit to Eq. 3, with the thermal and electromigration time constants (τ4 and τEM respectively) as the only adjustable parameters. The results of these fits are shown as solid curves in Fig. 3, and demonstrate excellent agreement between the measured correlation functions and those predicted by the Langevin theory (eqn. 4). As can be seen in fig. 4, the two fits yield clear χ2 minima for τEM of 16s and 52s, respectively for the down-hill and up-hill steps. Measurements on two additional steps subject to up-hill current stressing at a higher nominal current density (Jnom = 4x105A/cm2) gave fitting results similar to those of Fig. 3, with time constants of 98s (325K) and 32s (350K). The uncertainties in the fit parameters were 15-40%. The correlation function for an unstressed sample measured at 325K is also shown in Fig. 3. There is no minimum in the chi-squared value for the fit as a function of electromigration time constant, with virtually no change in the goodness of fit occurring for values larger than 9000s (see fig. 4). This shows that data is well fit by a single parameter, τ4, as expected when no electromigration force is acting (i.e. #$). Similar results were consistently obtained for other steps measured without current stressing. The electromigration force is found using the measured values of τEM in Eqs. 1 and 4, given the step stiffness ˜ " . The stiffness is calculated 37, 38 using an effective kink energy of ε = 0.117 eV 39, 40. The four measurements of the electromigration time constant yield average values of the force per step-edge atom of -2.7x10-5 eV/nm for Jnom= 4x105A/cm2 (325-350K) and -9.7x10-6 eV/nm for Jnom= 1x105A/cm2 (370K). Thus, within the cumulative uncertainties in the measured forces of ±50%, the force increases in direct proportion to the current density. The measured values of the force can be used to determine the effective valence Z* if the local surface current density and the resistivity in the surface region are known. Since these local quantities cannot be measured directly, we first estimate them using the bulk current densities and the bulk resistivity of Ag, which is approximately 1.8x10-6Ω-cm at 325K and 2.2x10-6Ω-cm at 370K 41, 42. The resulting effective valence, obtained using the nominal current density, is Z* = -(4±2)x102. The magnitude is substantially larger than the predicted effective valence of an isolated Ag adatom on Ag(111), which is is Z*= "19 43. For atoms in a close- packed site along a step edge, with a perpendicular current direction, the direct force per step atom may be as much as 2x higher than the force on an adatom 21, 23, which would yield a predicted valence of Z* ~ -38, still much smaller magnitude than the measured value. A substantial systematic effect can be attributed to the film cross section, because as described earlier, at 100 nm film thickness there are vacancies in the film up to 50% of the surface area. Therefore the bulk current densities may be as much as 2x higher than the nominal values, and as a result the lower limit on the effective valence is Z* = -(2±1)x102. The remainder of the difference compared with the perpendicular force on a close-packed step-edge atom may arise from the highly kinked environment suggested in Fig. 1. Because diffusion is parallel to the step edge, only the component of the electromigration force tangential to the local step orientation will affect step-edge diffusion. The largest impact of the electromigration force will thus occur for the most highly kinked step regions. There have been no calculations of the electromigration force acting on low-symmetry kink sites. However, such sites have enhanced valence charge density 44, and also present anomalous barriers to step-edge diffusion 45. Such significant changes in local electronic structure may be reflected in significant changes in the scattering cross section (and thus Z* value). In addition, the geometric effect of the kink configuration is likely to enhance scattering via blocking 21, or constriction-induced enhancement of local current density 46 analogous to current crowding 47, 48. The forces measured above are related to equal and opposite forces on the charge carriers, which translate into changes in the surface resistivity 22, 43. This can be evaluated by treating the diffusing step-edge atoms as independent scattering sites of density nk = (LkLstep)-1, where Lk is the average distance between diffusing atoms along the step edge and Lstep is the average distance between steps. Then the change in the surface resistivity ρs due to the diffusing step-edge atoms is 43: , , , (5) where Lf is the film thickness, nAg = 58.5 nm-3 is the bulk carrier density for Ag, k is the wind force acting per atom (our measured value) and j are additional changes in force on the carriers due to the perturbation of atomic structure in the immediate vicinity of the step-edge atoms. Using the measured force per atom and the upper limit of the current density yields the component of the surface resistivity due to scattering at the diffusing step-edge atoms alone, ~ (3±1.5nm , where ρ0 is the bulk value of the Ag resistivity. The impact of diffusing step-edge atoms on surface resistivity will include the direct- scattering term measured here as well as the perturbed lattice terms ( k in Eq. 5). The latter term may contribute as much as 2/3 of the total resistivity change, thus it is reasonable to expect that scattering at the step-edge will contribute a total change in the surface resistivity $ (10 ± 5nm . As an example, for moderate step and kink densities (Lstep =10 nm and Lk =2 nm) and a very thin film (Lf = 10 nm), the change in surface resistivity due to scattering at the step-kink sites could be as large as 10% of the bulk resistivity. This effect will be significant in nanoelectronic devices carrying large current densities 49, 50. The present observation of biased temporal fluctuations under current stress is the first direct correlation of nanoscale structural fluctuations with the electromigration wind force. Because of systematic uncertainty in the actual current density, analysis of the results yields a lower limit for the magnitude of the electron wind force on the diffusing step-edge atoms. Taking the one-sigma limits on the statistical uncertainties, we find that the value is five to fifteen times larger than that on individual adatoms 43. Effects on resistivity, noise and electromigration susceptibility in metal nanoelectronic structures will thus be concomitantly higher than would have been expected 51. Acknowledgments This work was supported by the U.S. Department of Energy Award No. DOE-FG02-01ER45939. We also gratefully acknowledge support and SEF support from the NSF MRSEC under grant DMR 05-20471. Figure Captions Fig. 1: Schematic illustration of current flow perpendicular to average orientation of steps on the surface. Enhanced scattering from step sites at the surface is suggested by the arrows. The inset illustrates adatoms on the terraces and the kinked (thermally roughened) step edge. Fig. 2: STM data for a current stressed sample. Applied current was 0.4 A and nominal sample cross section 200 nm thick x 2 mm wide. Sample temperature = 380 K. Upper panel – repeated scans across the edges of three steps (pseudo image) (from left to right one downhill step and two uphill steps) show fluctuations of step position in time x(t). Lower panel – height vs. position across the image is consistent with steps a single layer (0.236 nm) high. Fig. 3: Time correlation functions, G(t) in units Å2, for the step fluctuations (x(t) data) extracted from repeated measurements as shown in Fig. 1 and described in the text. The data for the step- down current (open circles) is the average of 10 separate measurements, and for the step-up current (open squares) is the average is over 9 data sets. Fits to Eq. 4 are shown as the solid lines for each data set. Also shown is the measured correlation function (average of 19 separate measurements) for the unstressed sample (open triangles) fit to a single power law. Fig. 4 The reduced chi-squared plotted as a function of τEM, while holding the value of τ4 at its optimum value. Solid curve: step-down current. Dot-dashed curve: step-up current. Dashed curve: unstressed data. Note that, for the unstressed data, there is no clear minimum value of τEM, i.e. a fit to a single parameter (τ4) suffices to describe the data. Fig. 1: Schematic illustration of current flow perpendicular to average orientation of steps on the surface. Enhanced scattering from step sites at the surface is suggested by the arrows. The inset illustrates adatoms on the terraces and the kinked (thermally roughened) step edge. Fig.2: (Color on-line) STM data for a current stressed sample. Applied current was 0.4 A and nominal sample cross section 200 nm thick x 2 mm wide. Upper panel – repeated scans across the edges of three steps (pseudo image) (from left to right one downhill step and two uphill steps) show fluctuations of step position in time x(t). Lower panel – height vs. position across the steps of single layer height (0.236 nm). Fig. 3: (Color on-line) Time correlation functions, G(t) for the step fluctuations (x(t) data) extracted from repeated measurements. The data for the step-down current (open circles) is the average over 10 separate measurements, and for the step-up current (open squares) the average is over 9. Fits to Eq. 3 are shown as the solid lines for each data set. Also shown is the measured correlation function (average of 19 separate measurements) for an unstressed sample (open triangles) fit to a single power law. 0 20 40 60 80 100 Fig. 4 (color on-line)The reduced chi-squared plotted as a function of τEM, while holding the value of τ4 at its optimum value. Solid curve: step-down current. Dot-dashed curve: step-up current. Dashed curve: unstressed data. Note that, for the unstressed data, there is no clear minimum value of τEM that is consistent with the data, i.e. a fit to a single parameter (τ4) suffices to describe the data. References 1 V. V. Zhirnov and R. K. 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arXiv:0704.1853v1 [astro-ph] 14 Apr 2007 PASJ: Publ. Astron. Soc. Japan , 1–??, 〈publication date〉 c© 2019. Astronomical Society of Japan. Dissipation of Magnetic Flux in Primordial Star Formation: From Run-away Phase to Mass Accretion Phase Hideki Maki Department of Physics, Rikkyo University, Nishi-Ikebukuro, Tokyo 171-8501 hide-mk@jcom.home.ne.jp Hajime Susa Department of Physics, Konan University, Okamoto, Kobe 658-8501 susa@center.konan-u.ac.jp (Received 〈reception date〉; accepted 〈acception date〉) Abstract We investigate the dissipation of magnetic flux in primordial star-forming clouds throughout their collapse including the run-away collapse phase as well as the accretion phase. We solve the energy equation and the non-equilibrium chemical reactions in the collapsing gas, in order to obtain the radial distribution of the ionized fraction during the collapse. As a result, we find the ionized fraction is high enough for the magnetic field to couple with the gas throughout the evolution of the cloud. This result suggests that the jet formation from protostars as well as the activation of magneto-rotational instability in the accretion disk are enabled in the presence of the cosmological seed magnetic flux proposed by Langer, Puget & Aghanim (2003). Key words: accretion, accretion disks — diffusion — early universe — stars: formation — stars: magnetic fields 1. INTRODUCTION Typical mass or initial mass function of population III stars are fundamental parameters that have great impacts on subsequent structure formation of the universe. Those stars are expected to be as massive as 100−1000M⊙, which ionize/dissociate the surrounding media. As a result, star formation activities in the neighbourhood of the stars are highly regulated (e.g. Susa (2007) and the references are therein). There are two chief reasons that the population III stars are expected to be very massive. First one comes from numerical studies by several authors (Nakamura & Umemura 1999; Bromm, Coppi & Larson 1999; Abel, Bryan & Norman 2000; Nakamura & Umemura 2001; Bromm, Coppi & Larson 2002). Those studies reveal that the prestellar cores formed as fragments of primordial gas are as massive as ∼ 103M⊙ − 104M⊙. Second ground is the very high accretion rates of those stars, which is as high as 10−3− 10−2M⊙yr−1. These two facts are the direct consequence of relatively high temperature (∼ 1000K) of primordial gas because of inefficient cooling by H2. On the other hand, abundance pattern of the hyper metal-poor stars seems to be more consistent with that of “faint” supernovae as remnants of less massive (∼ 25M⊙) population III stars(Christlieb et al. 2004; Frebel et al. 2005; Iwamoto et al. 2005). Recent theoretical studies suggest the possibilities for the formation of such less massive population III stars. If the primordial gas was once ionized, enhanced fraction of H2 causes more efficient cooling and HD formation. Since HD molecules have lower excitation energy than that of H2, gas can be cooled below 100K. As a result, the mass of the fragments could be smaller than 100M⊙(O’Shea, Abel, Whalen & Norman 2005; Johnson & Bromm 2006). O’Shea & Norman (2007) also demonstrates that such low mass population III stars cloud be formed directly from the cosmological density fluctuations. Those new ideas are quite interesting and promising, however, we still do not know the actual mass of population III stars formed from ∼ 103M⊙ − 104M⊙ prestellar cores, which are commonly found in cosmological simulations. Since they are only 104− 106cm−3, we have to follow the subsequent evolution of the collapsing cloud. Further evolution of prestellar cores is first investigated by Omukai & Nishi (1998), and they found that the collapse proceeds in a run-away fashion and converges to Larson-Penston type similarity solution (Larson 1969; Penston 1969; Suto & Silk 1988) with polytropic index γ ≃ 1.09. They also find the mass accretion rate is very large compared to the present-day forming stars, although spherical symmetry is assumed in their radiation hydrodynamic simulations. Recently, Yoshida et al. (2006) have performed cosmological simulations in which the run-away collapsing core is traced up to a very dense regime (nH ≫ 1010cm−3), taking the Sobolev type line transfer approximations. They also find basically consistent results in 3D cosmological simulations with previous 1D results by Omukai & Nishi (1998). However, most of the mass of protostar is accumulated in the accretion phase, which has not been traced especially in http://arxiv.org/abs/0704.1853v1 2 Maki & Susa [Vol. , multi-dimensional simulations. Recent numerical simulations performed by Saigo, Matsumoto & Umemura (2005) show that disks would be formed at the center of collapsing primordial clouds, which might result in the disks surrounding the protostars, or binaries. Thus, the actual mass of a population III star should depends on the mechanism of the angular momentum transport in the accretion disk. There are a few possibilities of the angular momentum transport mechanism such as gravitational torque by the nonaxisymmetric structures in the accretion disk, the interaction among the fragments of the disk (Stone et al. 2000; Bodenheimer et al. 2000), and the turbulent viscosity triggered by Magneto-Rotational Instability (MRI) (Hawley & Balbus 1992; Sano, Inutsuka & Miyama 1998; Sano & Inutsuka 2001). As for MRI induced turbulence, the strength of magnetic field in the disk is the key quantity to activate the instability(Tan & Mckee 2004; Tan & Blackman 2004). It is also worth noting that recent MHD simulations by Machida et al. (2006) suggests the possibility of bipolar jets from proto-population III stars, which also could suppress the mass accretion onto the central core. The formation of jets also requires the presence of some level of magnetic field. Therefore, it is quite important to assess the magnetic field strength brought into the accretion disk from initial weak cosmological seed field (e.g. Langer, Puget & Aghanim (2003)). Maki & Susa (2004) investigated this issue by solving detailed chemical reaction rate equations coupled with energy equation in run-away collapsing core. They found that magnetic field is always frozen to the collapsing core in case the strength of initial magnetic field is smaller than 10−5(nH/10 3 cm−3)0.55 G. This is comparable to the maximal strength of magnetic field which allow the clouds to collapse. Therefore they conclude magnetic field is always frozen to the collapsing cloud in the run-away collapsing core. They also evaluate the minimal strength of magnetic field which is required to activate MRI assuming magnetic field is frozen to the gas not only in the run-away phase, but also in the accretion phase. This assumption is based upon the argument that the temperature of the accretion flow would rise faster than that of the core, because of the shock heating. As a result, the ionization degree is expected to be higher than those in run-away phase, which guarantee the gas to be frozen to the magnetic field lines. However, such heating is only important in the final phase of accretion, where the flow settle onto the accretion disk. In order to investigate the coupling of matter and magnetic field in the accretion phase, we need to calculate the actual thermal and chemical evolution of accretion flow. Aside from the issue on ionization degree, the ambipolar diffusion velocity is proportional to the square inverse of density (vambB ∝ ρ−2). As a result, the diffusion velocity increases rapidly in the mass accretion phase, since the density at a fixed radius decreases as the accretion proceeds. Thus, we need a detailed treatment to test the coupling between the gas and magnetic field in accretion phase. In this paper, we investigate the dissipation of magnetic field in collapsing primordial gas cloud from run-away phase to accretion phase, by solving detailed thermal and chemical rate equations. In the next section, we describe the formulations employed. In §3, results of our calculations are shown. The formation of jets and activation of MRI is discussed in §4. Final section is devoted to summary. 2. METHOD OF CALCULATIONS In order to evaluate the coupling of the gas and the magnetic field, we need to assess the amount of ions and electrons in collapsing primordial cloud. The collapse of the cloud is expected to proceed in run-away fashion (run-away phase) in the beginning, followed by the mass accretion phase after the formation of rotationally supported disk at the center. In this paper, we follow the chemical and thermal evolution of the materials to form star, throughout the two phases. We assume spherically symmetric collapse of the progenitor gas, although we expect the formation of rotationally supported disk in the very dense regime (nH ≫ 1010cm−3). It is also assumed that the magnetic field is so weak that the dynamics of the collapsing gas is not affected by the magnetic force. 2.1. Run-away Collapse Phase The run-away collapse phase is traced by solving 1-dimensional hydrodynamics. The equation of continuity is = 4πr2ρ, (1) whereas the equation of motion is =−4πr2 ∂p , (2) where m is the mass within radius r, ρ is the density of the cloud, u is the velocity and p is the pressure. Since the equation of state of collapsing primordial gas in run-away phase is known to be approximated by polytrope with γ = 1.09 (Omukai & Nishi 1998), we employ following equation: p=Kργ , γ = 1.09 (3) We solve above set of equations by spherically symmetric Lagrangian hydrodynamics code developed by ourselves, following the Piecewise Parabolic Method (PPM) described in Colella & Woodward (1984). No. ] Dissipation of Magnetic Flux in Primordial Star Formation 3 2.2. Mass Accretion Phase The dynamics of accretion phase could be approximated by simple free-fall, since the flow velocity is supersonic. Basic equations of free-falling accretion flow consists of the equation of continuity and the equation of motion: 4πρ(t0,r0)r 0dr0 = 4πρ(t,r)r 2(t)dr, (4) d2r (t) =−GM0 r2(t) , (5) where r, t and t0 represent the position of fluid element, time, and the time when the mass accretion starts, respectively. M0 denotes the mass within r0 at t= t0: 4πr′0ρ0dr 0. (6) The initial radius r0 is defined as r0 ≡ r(t0). Therefore, the the solution of above equation of motion is given as r = r(t; t0,r0). The velocity of the fluid element is given by u(t; t0,r0) =− 2E0 +2 , (7) where E0 is the total energy defined as E0 ≡ u2(t0)/2−GM0/r0. Using equation (4), the density at (t,r) is given as ρ(t,r) = (∂r/∂r0)t . (8) (∂r/∂r0)t is the partial differentiation by r0 keeping t fixed. An explicit expression of this term is given in appendix 1. In order to clarify the validity of the free-fall approximation, it is compared with the similarity solution with γ = 1.09 (Suto & Silk 1988) in accretion phase. The initial condition of the accretion flow is set as the final state of the run-away collapse with γ = 1.09. Figure 1 illustrates the density and velocity distributions of several snapshots. It is obvious that the free-fall approximation can describe the matter distribution in accretion phase very well for polytropic gas. In other words, the thermal evolution has little effects on the dynamics of the accretion flow. Thus, the dynamics can be approximated by free-fall formula, irrespective of the internal energy equation. Fig. 1. Time evolution of density/velocity profiles in accretion flow are shown. Dotted lines and marks represent the result with free-fall approximation, whereas the solid lines denote the results from the similarity solution in accretion phase with γ = 1.09. In the accretion flow, the effective polytropic index is basically unknown. Thus, we have to solve the following energy equation for each fluid element: 4 Maki & Susa [Vol. , =−p d −L(net). (9) In this expression, p denotes the pressure, which is related to the internal energy ε by the equation of state: p= (γad − 1)ρε, (10) where γad is the adiabatic exponent, L(net) denotes the net energy loss rate per unit mass. We take into account the rovibrational line cooling by H2(Lline), the continuum radiation from the gas (Lcont), and cooling and heating associated with chemical reactions (Ldiss, GH− , GH+ and G3body). L(net) is the sum of all these contributions. The cooling rate by the rovibrational transition of hydrogen molecules are assessed by the fitting formula given by Galli & Palla (1998) in case the lines are optically thin. In the optically thick regime, we employ the escape probability formalism, as described in equations (12)-(15) in Omukai (2000), except that we use actual velocity gradient in our calculations. The cooling rate owing to the continuum radiation is given by Lcont = 4σκgasT 4 (11) where σ denotes the Stefan-Boltzmann constant, κgas is the Planck mean opacity of gas (Lenzuni, Chernoff & Salpeter 1991), which includes bound-free absorption, free-free absorption, photodissociation, Rayleigh scattering, and collision- induced absorption. This formula is valid in case the accreting gas is optically thin for continuum radiation. We confirmed that the optical depth is smaller than unity throughout the accretion phase. Cooling due to the latent heat of H2 dissociation is given by Ldiss = 4.48 eV s−1 g−1, (12) where nH is the number density of hydrogen nuclei, and (dyH2/dt)− is the dissociation rate of H2. On the other hand, the gas is heated when a hydrogen molecule is formed. The heating rate per unit mass is given GH− = 1+ncr/nH eV s−1 g−1, (13) 1+ncr/nH eV s−1 g−1, (14) G3body = 3body 1+ncr/nH eV s−1 g−1, (15) where (dyH2/dt)H− , (dyH2/dt)H+ and (dyH2/dt)3body are the formation rates of H2 by H − process, H+2 process and three body reactions (Hollenbach & McKee 1979). ncr is the critical density defined as ncr = T 1/2 1.6yH exp +1.4yH exp 1.2× 104 T +1200 cm−3. (16) 2.3. Chemical Reactions Since the dissipation of the magnetic flux, as well as the temperature of the gas, strongly depends on the chemical abundances, we have to solve the non-equilibrium chemical reaction rate equations coupled with equation of motion and energy equation described in previous subsections 2.1 and 2.2. The evolution of the fraction of species i is followed by solving the equations, nHklmylym+ nHklmnylymyn, (i= 1,2,3, · · · ,24), (17) where yi ≡ ni/nH is the fraction of species i, klm [cm3 s−1] and klmn [cm6 s−1] are the reaction rate coefficients with respect to two-body processes and three-body processes, respectively. In our calculation, we include 24 species: e−, H+, H, H−, H2, H 2 , H 3 , D, D +, D−, HD, HD+, H2D +, He, He+, He++, HeH+, Li, Li+, Li++, Li3+, Li−, LiH, and LiH+. We employ the latest reaction rate coefficients in the following papers, Galli & Palla (1998), Omukai (2000), Stancil, Lepp & Dalgarno (1998), Flower (2002) and Lepp, Stancil & Dalgarno (2002). As for the radiative recombination, we use the rate coefficients based on Spitzer (1978). These reactions are the same as our previous No. ] Dissipation of Magnetic Flux in Primordial Star Formation 5 paper(Maki & Susa 2004). Here we stress the importance to include above rare elements such as Li, because the coupling of magnetic field with gas can be attained by very low fractional abundance of electrons and ions (Maki & Susa 2004). 2.4. Drift velocity of magnetic field Magnetic field is dissipated from star-forming gas via ohmic loss and ambipolar diffusion. We assess the drift velocity vBx of the field lines due to these two processes, which is compared to the accretion velocity of gas. We evaluate the drift velocity following the formulation by Nakano & Umebayashi (1986). There are two important quantities which characterize these diffusion processes. They are τν and ων which denote the viscous damping time of the relative velocity of charged particle ν to the neutral particles, and the cyclotron frequency of the charged particle ν, respectively. Then, τν is expressed as µνnnνnn〈σv〉νn , (18) where µνn is the reduced mass, nν , nn, and ρν are, the mean number density for the charged particle ν, the neutral particle n, and the mass density of charged particle ν, respectively. The averaged momentum-transfer rate coefficient for a particle ν colliding with a neutral particle is expressed by 〈σv〉νn . We use the empirical formulae for the momentum-transfer rate coefficients (Kamaya & Nishi 2000; Sano et al. 2000). According to Nakano & Umebayashi (1986), the drift velocity is given by vBx = (j ×B)x, (19) where A=A21 +A 2, (20) ρντνω 1+ τ2νω , (21) 1+ τ2νω , (22) B is the mean magnetic field in the primordial cloud, the suffix x means x direction component in local Cartesian coordinates where the z direction is taken as the direction of B. We replace (1/c)(j×B)x in equation (19) with the mean magnetic force B2/4πr, where B is the mean field strength in the cloud, r is the radius of the cloud at which we assess the drift velocity. 2.5. Initial Conditions We consider primordial star-forming gas clouds that formed in the mini-halos with ∼ 106 M⊙. We set uniform and spherical gas cloud with nH = 10 3 cm−3, T0 = 250 K, and M = 10 4 M⊙. Such clouds are commonly found 3D cosmological simulations (Bromm, Coppi & Larson 1999; Abel, Bryan & Norman 2000; Abel, Bryan & Norman 2002; Bromm, Coppi & Larson 2002; Yoshida et al. 2003; O’Shea & Norman 2006). We also use the cosmological abundance given by Galli & Palla (1998), as the initial fraction of the chemical compositions. The initial magnetic field strength is expected to be very weak, however, its magnitude is still under discussion. Thus, we regard the field strength as a free parameter of the calculations. 2.6. Switching from Run-away Phase to Accretion Phase The run-away phase and the subsequent accretion phase have to be treated in different formulation from each other as described in subsections 2.1 and 2.2. Thus, we have to switch the scheme from the method in run-away phase to that in accretion phase. In the light of physical arguments, two phases should be switched when the central run-away collapse is stopped due to thermal pressure or centrifugal force. In our calculation, we switch the scheme when the sonic point of the accreting flow go inside a certain small radius. We set this radius to be rsw = 700 R⊙, which is comparable to the disk size when the accreted mass is comparable to ∼ 1M⊙(Tan & Mckee 2004). It is also worth noting that the central density of the core at switching is ∼ 1016 cm−3. 3. RESULTS 3.1. Runaway Collapse Phase 3.1.1. The distribution of the chemical abundances Two snapshots of yi distribution during the run-away collapse phase are shown in Figure 2. The left panel shows the distribution at the time when the central density nH,c reaches 10 12 cm−3, whereas the right panel represents the 6 Maki & Susa [Vol. , Fig. 2. The distribution of the main species, e−, H+, H, H2, H , Li, and Li+ are plotted. The vertical axis denotes the fractional abundance yi of the above species, and the horizontal axis denotes the radius from the core center. Left panel represents the snapshot when the central density satisfies nH,c = 10 12 cm−3, whereas right panel shows the results at nH,c = 10 16 cm−3. Fig. 3. Drift velocity vBx as a function of the radius r and the field strength B at the central density . Contour map of vBx/u is shown on r − B plane for nH,c = 10 16 cm−3 Solid curves represent the constant loci along which log(vBx/u) equal to the values labeled on the curve. The dashed curve represents the critical field strength Bcr given by the equation (23). distribution at nH,c = 10 16 cm−3. The horizontal axis denotes the radius from the center of the primordial gas cloud r [cm], and the vertical axis is the fractional abundances yi of each species. Since the central density increases as the collapse proceeds, right panel represents the later epoch than that of the left. Roughly speaking, in both of the panels, fraction of electrons (ye) in the inner region are smaller than that in the outer part. These results could be simply understood since the recombination process in the inner dense region proceeds faster than in the outer envelope. We also find the opposite behavior in the inner most region of right panel, No. ] Dissipation of Magnetic Flux in Primordial Star Formation 7 at which ye increases as r decreases, because of the collisional ionization. It is also worth noting that Li becomes the main provider of electrons around r ∼ 1014cm at nH,c = 1016 cm−3. Because of the combined effects of these two (collisional ionization & presence of Li), ye never gets lower than 10 −11 as far as we consider nH,c < 10 16 cm−3 in run-away phase. 3.1.2. Drift velocities in run-away collapse phase In Figure 3, we show the ratio of the drift velocity vBx to the infall velocity u when nH,c = 10 16 cm−3 is satisfied. The vertical axis shows the magnetic field strength, whereas the horizontal axis shows the distance from the cloud center. The solid curves in Figure 3 show the contours of log10(vBx/u). The magnetic fields are dissipated in the region where log10(vBx/u)> 0, in contrast, the fields are frozen to the gas in the region where log10(vBx/u)< 0. The dashed curve in Figure 3 is the critical field strength Bcr that is defined by the equation GM(r)ρ(r) . (23) Note that since we are interested in the collapsing gas cloud, the magnetic force needs to be weaker than the gravi- tational force. Hence, our calculations are valid in the region where the field strength in the cloud satisfies B < Bcr. We find clearly from Figure 3 that the frozen-in condition vBx/u < 1 is almost always satisfied as long as B is less than the critical field strength Bcr. We also find the basically same results for other snapshots, i.e. the drift velocity is smaller than the infall velocity anytime and anywhere if B <Bcr. In addition, if B < Bcr is satisfied at some initial time and position (t0, r0), this condition also holds at some later epoch (t,r(t)). This statement is proved as follows: Combining the magnetic flux conservation equation 2πr0dr0B(t0,r0) = 2πr(t)drB(t,r(t)) (24) and the mass conservation law given in equation (4), we obtain B(t,r(t)) =B(t0,r0) ρ(t,r(t))r(t) ρ(t0,r0)r0 , (25) Using equations (23) and (25) we have B(t,r(t)) Bcr(t,r(t)) B(t0,r0) Bcr(t0,r0) ρ(t,r(t))r3 (t) ρ(t0,r0)r The second term in right side is always less than or equals to unity in the run-away collapsing cloud, since ρ ∝ r−2 in the envelope, and ρ ∝ r−3 in the core. Thus, if B(t0,r0)/Bcr(t0,r0) < 1 is satisfied, B is less than Bcr throughout the collapse. Remark that we assume the flux conservation as equation (25), which gives the maximal field strength. Therefore, above inequality (B <Bcr) also holds even if the magnetic flux is dissipated. In summary, it is concluded that the magnetic fields are always frozen to the whole cloud in the course of run-away collapse phase, if the magnetic force is much smaller than gravitational force at the beginning of the collapse. We also remark that the slight increase due to collisional ionization at nH,c =10 16 cm−3 implies that ye will become larger as the central density increases up to nH,c> 10 16 cm−3, since the temperature gets higher as the density increases for γ=1.09 polytropic gas. As a result, the magnetic field is expected to be frozen to the gas even for nH,c> 10 16 cm−3 in accretion phase. 3.2. Accretion Phase 3.2.1. Evolution of gas and chemical species in accretion phase The evolutionary sequences of number density nH, temperature T , velocity u, and electron fraction ye in the accretion phase are illustrated in Figure 4. Upper left panel illustrates the evolution of density profile. The solid curves labeled as 0-4 corresponds to the epoch at which the central accreted mass equals to 8.87× 10−2 M⊙,1.58 M⊙,12.7 M⊙,50.2 M⊙,100 M⊙, respectively. It is clear that the density for r → 0 decreases as the collapse proceeds, which is basically the same feature found in similarity solution of polytropic cloud in accretion phase. Because of the high accretion velocity at later snapshots (lower left), the gas is heated efficiently by adiabatic compression. As a result, temperature gets higher as the collapse proceeds, except at the beginning of the accretion phase, when the gas is cooled efficiently due to the enhanced fraction of H2 molecules. The ionization degree (electron fraction ye, lower right) also increases as the collapse proceeds (Figures.4 and 5), due to the higher temperature for later snapshots. As a result, ye never gets lower than 10 −11. We also find that Li is not so important as was in the final phase of run-away phase, since the hydrogen win back the position of chief provider of electrons in the accretion phase( Figure 5). 8 Maki & Susa [Vol. , Fig. 4. The evolutionary sequences of the primordial gas cloud in accretion phase are shown. Four panels (a)-(d) show the spatial distribution of following physical variables as functions of radius: (a) number density nH, (b) temperature T , (c) velocity u, and (d) electron fraction ye. Five time sequences (0-4) are plotted. Accreted central mass Mc(t) is used as a clock. Corresponding mass at the stages are: 0:Mc(t) = 8.87× 10 −2 M⊙, 1:Mc(t) = 1.58M⊙, 2:Mc(t) = 12.7M⊙, 3:Mc(t) = 50.2M⊙, 4:Mc(t) = 100M⊙. 3.2.2. Drift velocities in accretion phase As briefly discussed in section 1, the ambipolar diffusion velocity is proportional to the inverse square of gas density. Its dependence on various physical quantities are described as vambBx ∝ y−1e ρ−2B2r−1 (see equations (19)-(22)). On the other hand, the accretion velocity scales as u ∝ (Mc/r)1/2 for r → 0. Since we consider the accretion phase starting from the final phase of run-away collapse with γ = 1.09, the density profile and the central mass in accretion phase also depend on γ. According to Suto & Silk (1988), we have ρ(t,r) ∝ t(2−3γ)/2r−3/2 for r → 0 (27) Mc(t)∝ t4−3γ (28) Thus, the limiting behaviour of the infall velocity is described as u(t,r)∝ t(4−3γ)/2r−1/2 for r→ 0 (29) Combining above set of equations, we have the dependence of the ratio vambBx /u on t: vambBx ∝ y−1e B2t(9γ−8)/2r5/2 (30) This equation indicates that the ratio keeps growing in accretion phase for fixed r,B and ye, since 9γ− 8 is positive for γ = 1.09. In reality, however, ye increases rapidly as the accretion proceeds (see Figures.4 and 5), and it offsets the increase of vambBx /u. Figures 6 and 7 illustrate the contours of log10(vBx/u) on r−B plane for two epochs. Two figures correspond to the snapshots when the central accreted mass satisfies Mc = 12.7M⊙ and Mc = 100M⊙, respectively. The notations are same as Figure 3. It is clear that the drift velocity is always smaller than the accretion velocity in two snapshots, No. ] Dissipation of Magnetic Flux in Primordial Star Formation 9 Fig. 5. The radial distributions of the fractional abundances of the main species, e, H+, H, H2, H −, H+ , Li, and Li+ in the accretion phase. Two panels correspond to the snapshots at Mc(t) = 12.7 M⊙(left), and Mc(t) = 100 M⊙(right). Fig. 6. The same as Fig.3, except that the contours are plotted in the accretion phase characterized at Mc =12.7M⊙. as long as B <Bcr. In fact, we find that this is true all through the calculations. Besides, the equation (26) holds also in the accretion phase. Considering that the density profile in the accretion phase is ρ∝ r−1.5∼−2, B is less than Bcr throughout the collapse, if B(t0,r0)/Bcr(t0,r0) < 1, as discussed in §3.1.2. In other words, magnetic force is always negligible if it can be ignored in the beginning of the collapse. Thus, it is concluded that the dissipation of magnetic flux is negligible throughout the mass accretion phase, as well as the run-away collapse phase. 4. DISCUSSION We confirmed that the magnetic field is frozen to the star-forming primordial gas cloud even in the accretion phase. Here we discuss the possibility of jet formation and activation of MRI considering the magnetic field strength brought 10 Maki & Susa [Vol. , Fig. 7. The same as Fig.6, except Mc = 100M⊙. into the accretion disk surrounding the protostar. The magnetic field strength brought into the accretion disk is assessed under the frozen-in condition as follows: Bdisk =B0 rdisk . (31) Here B0 is the initial field strength, rdisk denotes the disk radius, whereas R describes the initial radius within which includes the total mass of the disk-star star system M∗disk: R= (3M∗disk/4πρ0) 1/3, (32) where ρ0 represents the initial density of the cloud. rdisk can be evaluated by equation (15) in Tan & Mckee (2004): rdisk ≃ 66.4AU M∗disk where fKep denotes the ratio of the rotation velocity to the Kepler velocity of accreting matter, which is found to be ∼ 0.5 in Abel, Bryan & Norman (2002). Thus, we have the magnetic field strength in the disk as follows: Bdisk ≃ 7.5× 10−10G 3.7× 10−16G 103cm−3 )−2/3 M∗disk )−40/21 Several possibilities to generate cosmological seed magnetic field have been proposed so far. Most of the mechanisms predict BIGM <∼ 10 −19G(Widrow 2002), except the magnetic field generated by radiation transfer effects of powerful ionizing sources such as quasars or first stars (Langer, Puget & Aghanim 2003). They suggests the possibility to generate coherent magnetic field with BIGM ∼ 10−11G. Since the magnetic field is frozen to the primordial gas at low densities (nH < 10 3cm−3), B0 at nH = 10 3cm−3 can be evaluated as B0 = 3.7× 10−16G 10−19G 103cm−3 , (35) Recently, 3-dimensional MHD simulations on primordial star formation have been performed by Machida et al. (2006), assuming ideal MHD condition is always satisfied in the collapsing gas. In fact, our present results guarantee this hypothesis. They found that the protostellar jet is driven in primordial environment if B0 >∼ 10 −9G at nH = 103cm−3. Comparing this condition with equation (35), it is concluded that 1) jets could be driven in first star forming clouds if the seed field is generated by the mechanism proposed by Langer, Puget & Aghanim (2003), 2) whereas the other mechanisms cannot generate the seed field enough to drive the jets. No. ] Dissipation of Magnetic Flux in Primordial Star Formation 11 MRI can be activated in the accretion disk in case the magnetic field in the disk is larger than a critical value (Tan & Blackman 2004): Bdisk >∼ 1.1× 10 M∗disk 10 M⊙ )1/4( 104 K )−3/4( )1/2( ρdisk 5× 10−10 g cm−3 )1/2( 600 R⊙ )−3/4 , (36) This threshold is assessed by the confrontation between the growth rate of MRI and the ohmic dissipation rate. Combining equations (34)- (36), we find that MRI is activated in case the seed field satisfy BIGM >∼ 1.5× 10 M∗disk )2.155 104 K )−3/4( )1/2( ρdisk 5× 10−10 g cm−3 )1/2( 600 R⊙ )−3/4 . (37) Therefore, MRI is driven only if the seed field generation mechanism by the transfer effects of ionizing radiation works. Based upon these arguments, we emphasize that the mechanism proposed by Langer, Puget & Aghanim (2003) should be scrutinized since their results still based upon the argument of order-estimation. 5. SUMMARY In this paper, we investigate the dissipation of magnetic flux in star-forming primordial gas cloud. We solve non- equilibrium chemical reaction equations, coupled with thermal and dynamical evolution of the collapsing cloud all through the run-away phase as well as the mass accretion phase. Thus, we obtain the detailed evolution of ionized fraction of the gas, which enables us to assess the coupling between gas and magnetic field. As a result, we find that the magnetic field is basically frozen to the gas anywhere in collapsing star-forming primordial clouds at any time. Based upon this result, we find the cosmological seed magnetic field generated by most of the mechanisms proposed so far is not sufficient to form jets as well as to activate MRI in the star-forming cloud. Only one mechanism proposed by Langer, Puget & Aghanim (2003) is able to create sufficient field strength. We thank Kazu Omukai for stimulating discussions. Noriaki Shibazaki and Ken Ohsuga are acknowledged for continuous encouragement. The analysis has been made with computational facilities at Rikkyo University. This work was supported in part by Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Grants-in-Aid, Specially Promoted Research 16002003 and Young Scientists (B) 17740110. Appendix 1. Density distribution of free-falling matter In this Appendix, we derive the explicit formula for which gives the density distribution of free-falling matter in equation (8). Equation of motion (5) is integrated as ṙ2 = −E0, E0 = u20, (E0 > 0), (A1) where u0 is the velocity of a fluid element at (t0,r0) where r0 denotes the position of the element at some initial time t0. A solution of this equation is given as r(α,α0) = (1− cosα), (A2) t(α,α0) = t0 − (2E0)3/2 [(α− sinα)− (α0 − sinα0)] , (A3) where α is the so-called development angle, and α0 is its value at t= t0. α0 also satisfies following relation (1− cosα0). 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We investigate the dissipation of magnetic flux in primordial star-forming clouds throughout their collapse including the run-away collapse phase as well as the accretion phase. We solve the energy equation and the non-equilibrium chemical reactions in the collapsing gas, in order to obtain the radial distribution of the ionized fraction during the collapse. As a result, we find the ionized fraction is high enough for the magnetic field to couple with the gas throughout the evolution of the cloud. This result suggests that the jet formation from protostars as well as the activation of magneto-rotational instability in the accretion disk are enabled in the presence of the cosmological seed magnetic flux proposed by Langer et al.(2003).

arXiv:0704.1853v1 [astro-ph] 14 Apr 2007 PASJ: Publ. Astron. Soc. Japan , 1–??, 〈publication date〉 c© 2019. Astronomical Society of Japan. Dissipation of Magnetic Flux in Primordial Star Formation: From Run-away Phase to Mass Accretion Phase Hideki Maki Department of Physics, Rikkyo University, Nishi-Ikebukuro, Tokyo 171-8501 hide-mk@jcom.home.ne.jp Hajime Susa Department of Physics, Konan University, Okamoto, Kobe 658-8501 susa@center.konan-u.ac.jp (Received 〈reception date〉; accepted 〈acception date〉) Abstract We investigate the dissipation of magnetic flux in primordial star-forming clouds throughout their collapse including the run-away collapse phase as well as the accretion phase. We solve the energy equation and the non-equilibrium chemical reactions in the collapsing gas, in order to obtain the radial distribution of the ionized fraction during the collapse. As a result, we find the ionized fraction is high enough for the magnetic field to couple with the gas throughout the evolution of the cloud. This result suggests that the jet formation from protostars as well as the activation of magneto-rotational instability in the accretion disk are enabled in the presence of the cosmological seed magnetic flux proposed by Langer, Puget & Aghanim (2003). Key words: accretion, accretion disks — diffusion — early universe — stars: formation — stars: magnetic fields 1. INTRODUCTION Typical mass or initial mass function of population III stars are fundamental parameters that have great impacts on subsequent structure formation of the universe. Those stars are expected to be as massive as 100−1000M⊙, which ionize/dissociate the surrounding media. As a result, star formation activities in the neighbourhood of the stars are highly regulated (e.g. Susa (2007) and the references are therein). There are two chief reasons that the population III stars are expected to be very massive. First one comes from numerical studies by several authors (Nakamura & Umemura 1999; Bromm, Coppi & Larson 1999; Abel, Bryan & Norman 2000; Nakamura & Umemura 2001; Bromm, Coppi & Larson 2002). Those studies reveal that the prestellar cores formed as fragments of primordial gas are as massive as ∼ 103M⊙ − 104M⊙. Second ground is the very high accretion rates of those stars, which is as high as 10−3− 10−2M⊙yr−1. These two facts are the direct consequence of relatively high temperature (∼ 1000K) of primordial gas because of inefficient cooling by H2. On the other hand, abundance pattern of the hyper metal-poor stars seems to be more consistent with that of “faint” supernovae as remnants of less massive (∼ 25M⊙) population III stars(Christlieb et al. 2004; Frebel et al. 2005; Iwamoto et al. 2005). Recent theoretical studies suggest the possibilities for the formation of such less massive population III stars. If the primordial gas was once ionized, enhanced fraction of H2 causes more efficient cooling and HD formation. Since HD molecules have lower excitation energy than that of H2, gas can be cooled below 100K. As a result, the mass of the fragments could be smaller than 100M⊙(O’Shea, Abel, Whalen & Norman 2005; Johnson & Bromm 2006). O’Shea & Norman (2007) also demonstrates that such low mass population III stars cloud be formed directly from the cosmological density fluctuations. Those new ideas are quite interesting and promising, however, we still do not know the actual mass of population III stars formed from ∼ 103M⊙ − 104M⊙ prestellar cores, which are commonly found in cosmological simulations. Since they are only 104− 106cm−3, we have to follow the subsequent evolution of the collapsing cloud. Further evolution of prestellar cores is first investigated by Omukai & Nishi (1998), and they found that the collapse proceeds in a run-away fashion and converges to Larson-Penston type similarity solution (Larson 1969; Penston 1969; Suto & Silk 1988) with polytropic index γ ≃ 1.09. They also find the mass accretion rate is very large compared to the present-day forming stars, although spherical symmetry is assumed in their radiation hydrodynamic simulations. Recently, Yoshida et al. (2006) have performed cosmological simulations in which the run-away collapsing core is traced up to a very dense regime (nH ≫ 1010cm−3), taking the Sobolev type line transfer approximations. They also find basically consistent results in 3D cosmological simulations with previous 1D results by Omukai & Nishi (1998). However, most of the mass of protostar is accumulated in the accretion phase, which has not been traced especially in http://arxiv.org/abs/0704.1853v1 2 Maki & Susa [Vol. , multi-dimensional simulations. Recent numerical simulations performed by Saigo, Matsumoto & Umemura (2005) show that disks would be formed at the center of collapsing primordial clouds, which might result in the disks surrounding the protostars, or binaries. Thus, the actual mass of a population III star should depends on the mechanism of the angular momentum transport in the accretion disk. There are a few possibilities of the angular momentum transport mechanism such as gravitational torque by the nonaxisymmetric structures in the accretion disk, the interaction among the fragments of the disk (Stone et al. 2000; Bodenheimer et al. 2000), and the turbulent viscosity triggered by Magneto-Rotational Instability (MRI) (Hawley & Balbus 1992; Sano, Inutsuka & Miyama 1998; Sano & Inutsuka 2001). As for MRI induced turbulence, the strength of magnetic field in the disk is the key quantity to activate the instability(Tan & Mckee 2004; Tan & Blackman 2004). It is also worth noting that recent MHD simulations by Machida et al. (2006) suggests the possibility of bipolar jets from proto-population III stars, which also could suppress the mass accretion onto the central core. The formation of jets also requires the presence of some level of magnetic field. Therefore, it is quite important to assess the magnetic field strength brought into the accretion disk from initial weak cosmological seed field (e.g. Langer, Puget & Aghanim (2003)). Maki & Susa (2004) investigated this issue by solving detailed chemical reaction rate equations coupled with energy equation in run-away collapsing core. They found that magnetic field is always frozen to the collapsing core in case the strength of initial magnetic field is smaller than 10−5(nH/10 3 cm−3)0.55 G. This is comparable to the maximal strength of magnetic field which allow the clouds to collapse. Therefore they conclude magnetic field is always frozen to the collapsing cloud in the run-away collapsing core. They also evaluate the minimal strength of magnetic field which is required to activate MRI assuming magnetic field is frozen to the gas not only in the run-away phase, but also in the accretion phase. This assumption is based upon the argument that the temperature of the accretion flow would rise faster than that of the core, because of the shock heating. As a result, the ionization degree is expected to be higher than those in run-away phase, which guarantee the gas to be frozen to the magnetic field lines. However, such heating is only important in the final phase of accretion, where the flow settle onto the accretion disk. In order to investigate the coupling of matter and magnetic field in the accretion phase, we need to calculate the actual thermal and chemical evolution of accretion flow. Aside from the issue on ionization degree, the ambipolar diffusion velocity is proportional to the square inverse of density (vambB ∝ ρ−2). As a result, the diffusion velocity increases rapidly in the mass accretion phase, since the density at a fixed radius decreases as the accretion proceeds. Thus, we need a detailed treatment to test the coupling between the gas and magnetic field in accretion phase. In this paper, we investigate the dissipation of magnetic field in collapsing primordial gas cloud from run-away phase to accretion phase, by solving detailed thermal and chemical rate equations. In the next section, we describe the formulations employed. In §3, results of our calculations are shown. The formation of jets and activation of MRI is discussed in §4. Final section is devoted to summary. 2. METHOD OF CALCULATIONS In order to evaluate the coupling of the gas and the magnetic field, we need to assess the amount of ions and electrons in collapsing primordial cloud. The collapse of the cloud is expected to proceed in run-away fashion (run-away phase) in the beginning, followed by the mass accretion phase after the formation of rotationally supported disk at the center. In this paper, we follow the chemical and thermal evolution of the materials to form star, throughout the two phases. We assume spherically symmetric collapse of the progenitor gas, although we expect the formation of rotationally supported disk in the very dense regime (nH ≫ 1010cm−3). It is also assumed that the magnetic field is so weak that the dynamics of the collapsing gas is not affected by the magnetic force. 2.1. Run-away Collapse Phase The run-away collapse phase is traced by solving 1-dimensional hydrodynamics. The equation of continuity is = 4πr2ρ, (1) whereas the equation of motion is =−4πr2 ∂p , (2) where m is the mass within radius r, ρ is the density of the cloud, u is the velocity and p is the pressure. Since the equation of state of collapsing primordial gas in run-away phase is known to be approximated by polytrope with γ = 1.09 (Omukai & Nishi 1998), we employ following equation: p=Kργ , γ = 1.09 (3) We solve above set of equations by spherically symmetric Lagrangian hydrodynamics code developed by ourselves, following the Piecewise Parabolic Method (PPM) described in Colella & Woodward (1984). No. ] Dissipation of Magnetic Flux in Primordial Star Formation 3 2.2. Mass Accretion Phase The dynamics of accretion phase could be approximated by simple free-fall, since the flow velocity is supersonic. Basic equations of free-falling accretion flow consists of the equation of continuity and the equation of motion: 4πρ(t0,r0)r 0dr0 = 4πρ(t,r)r 2(t)dr, (4) d2r (t) =−GM0 r2(t) , (5) where r, t and t0 represent the position of fluid element, time, and the time when the mass accretion starts, respectively. M0 denotes the mass within r0 at t= t0: 4πr′0ρ0dr 0. (6) The initial radius r0 is defined as r0 ≡ r(t0). Therefore, the the solution of above equation of motion is given as r = r(t; t0,r0). The velocity of the fluid element is given by u(t; t0,r0) =− 2E0 +2 , (7) where E0 is the total energy defined as E0 ≡ u2(t0)/2−GM0/r0. Using equation (4), the density at (t,r) is given as ρ(t,r) = (∂r/∂r0)t . (8) (∂r/∂r0)t is the partial differentiation by r0 keeping t fixed. An explicit expression of this term is given in appendix 1. In order to clarify the validity of the free-fall approximation, it is compared with the similarity solution with γ = 1.09 (Suto & Silk 1988) in accretion phase. The initial condition of the accretion flow is set as the final state of the run-away collapse with γ = 1.09. Figure 1 illustrates the density and velocity distributions of several snapshots. It is obvious that the free-fall approximation can describe the matter distribution in accretion phase very well for polytropic gas. In other words, the thermal evolution has little effects on the dynamics of the accretion flow. Thus, the dynamics can be approximated by free-fall formula, irrespective of the internal energy equation. Fig. 1. Time evolution of density/velocity profiles in accretion flow are shown. Dotted lines and marks represent the result with free-fall approximation, whereas the solid lines denote the results from the similarity solution in accretion phase with γ = 1.09. In the accretion flow, the effective polytropic index is basically unknown. Thus, we have to solve the following energy equation for each fluid element: 4 Maki & Susa [Vol. , =−p d −L(net). (9) In this expression, p denotes the pressure, which is related to the internal energy ε by the equation of state: p= (γad − 1)ρε, (10) where γad is the adiabatic exponent, L(net) denotes the net energy loss rate per unit mass. We take into account the rovibrational line cooling by H2(Lline), the continuum radiation from the gas (Lcont), and cooling and heating associated with chemical reactions (Ldiss, GH− , GH+ and G3body). L(net) is the sum of all these contributions. The cooling rate by the rovibrational transition of hydrogen molecules are assessed by the fitting formula given by Galli & Palla (1998) in case the lines are optically thin. In the optically thick regime, we employ the escape probability formalism, as described in equations (12)-(15) in Omukai (2000), except that we use actual velocity gradient in our calculations. The cooling rate owing to the continuum radiation is given by Lcont = 4σκgasT 4 (11) where σ denotes the Stefan-Boltzmann constant, κgas is the Planck mean opacity of gas (Lenzuni, Chernoff & Salpeter 1991), which includes bound-free absorption, free-free absorption, photodissociation, Rayleigh scattering, and collision- induced absorption. This formula is valid in case the accreting gas is optically thin for continuum radiation. We confirmed that the optical depth is smaller than unity throughout the accretion phase. Cooling due to the latent heat of H2 dissociation is given by Ldiss = 4.48 eV s−1 g−1, (12) where nH is the number density of hydrogen nuclei, and (dyH2/dt)− is the dissociation rate of H2. On the other hand, the gas is heated when a hydrogen molecule is formed. The heating rate per unit mass is given GH− = 1+ncr/nH eV s−1 g−1, (13) 1+ncr/nH eV s−1 g−1, (14) G3body = 3body 1+ncr/nH eV s−1 g−1, (15) where (dyH2/dt)H− , (dyH2/dt)H+ and (dyH2/dt)3body are the formation rates of H2 by H − process, H+2 process and three body reactions (Hollenbach & McKee 1979). ncr is the critical density defined as ncr = T 1/2 1.6yH exp +1.4yH exp 1.2× 104 T +1200 cm−3. (16) 2.3. Chemical Reactions Since the dissipation of the magnetic flux, as well as the temperature of the gas, strongly depends on the chemical abundances, we have to solve the non-equilibrium chemical reaction rate equations coupled with equation of motion and energy equation described in previous subsections 2.1 and 2.2. The evolution of the fraction of species i is followed by solving the equations, nHklmylym+ nHklmnylymyn, (i= 1,2,3, · · · ,24), (17) where yi ≡ ni/nH is the fraction of species i, klm [cm3 s−1] and klmn [cm6 s−1] are the reaction rate coefficients with respect to two-body processes and three-body processes, respectively. In our calculation, we include 24 species: e−, H+, H, H−, H2, H 2 , H 3 , D, D +, D−, HD, HD+, H2D +, He, He+, He++, HeH+, Li, Li+, Li++, Li3+, Li−, LiH, and LiH+. We employ the latest reaction rate coefficients in the following papers, Galli & Palla (1998), Omukai (2000), Stancil, Lepp & Dalgarno (1998), Flower (2002) and Lepp, Stancil & Dalgarno (2002). As for the radiative recombination, we use the rate coefficients based on Spitzer (1978). These reactions are the same as our previous No. ] Dissipation of Magnetic Flux in Primordial Star Formation 5 paper(Maki & Susa 2004). Here we stress the importance to include above rare elements such as Li, because the coupling of magnetic field with gas can be attained by very low fractional abundance of electrons and ions (Maki & Susa 2004). 2.4. Drift velocity of magnetic field Magnetic field is dissipated from star-forming gas via ohmic loss and ambipolar diffusion. We assess the drift velocity vBx of the field lines due to these two processes, which is compared to the accretion velocity of gas. We evaluate the drift velocity following the formulation by Nakano & Umebayashi (1986). There are two important quantities which characterize these diffusion processes. They are τν and ων which denote the viscous damping time of the relative velocity of charged particle ν to the neutral particles, and the cyclotron frequency of the charged particle ν, respectively. Then, τν is expressed as µνnnνnn〈σv〉νn , (18) where µνn is the reduced mass, nν , nn, and ρν are, the mean number density for the charged particle ν, the neutral particle n, and the mass density of charged particle ν, respectively. The averaged momentum-transfer rate coefficient for a particle ν colliding with a neutral particle is expressed by 〈σv〉νn . We use the empirical formulae for the momentum-transfer rate coefficients (Kamaya & Nishi 2000; Sano et al. 2000). According to Nakano & Umebayashi (1986), the drift velocity is given by vBx = (j ×B)x, (19) where A=A21 +A 2, (20) ρντνω 1+ τ2νω , (21) 1+ τ2νω , (22) B is the mean magnetic field in the primordial cloud, the suffix x means x direction component in local Cartesian coordinates where the z direction is taken as the direction of B. We replace (1/c)(j×B)x in equation (19) with the mean magnetic force B2/4πr, where B is the mean field strength in the cloud, r is the radius of the cloud at which we assess the drift velocity. 2.5. Initial Conditions We consider primordial star-forming gas clouds that formed in the mini-halos with ∼ 106 M⊙. We set uniform and spherical gas cloud with nH = 10 3 cm−3, T0 = 250 K, and M = 10 4 M⊙. Such clouds are commonly found 3D cosmological simulations (Bromm, Coppi & Larson 1999; Abel, Bryan & Norman 2000; Abel, Bryan & Norman 2002; Bromm, Coppi & Larson 2002; Yoshida et al. 2003; O’Shea & Norman 2006). We also use the cosmological abundance given by Galli & Palla (1998), as the initial fraction of the chemical compositions. The initial magnetic field strength is expected to be very weak, however, its magnitude is still under discussion. Thus, we regard the field strength as a free parameter of the calculations. 2.6. Switching from Run-away Phase to Accretion Phase The run-away phase and the subsequent accretion phase have to be treated in different formulation from each other as described in subsections 2.1 and 2.2. Thus, we have to switch the scheme from the method in run-away phase to that in accretion phase. In the light of physical arguments, two phases should be switched when the central run-away collapse is stopped due to thermal pressure or centrifugal force. In our calculation, we switch the scheme when the sonic point of the accreting flow go inside a certain small radius. We set this radius to be rsw = 700 R⊙, which is comparable to the disk size when the accreted mass is comparable to ∼ 1M⊙(Tan & Mckee 2004). It is also worth noting that the central density of the core at switching is ∼ 1016 cm−3. 3. RESULTS 3.1. Runaway Collapse Phase 3.1.1. The distribution of the chemical abundances Two snapshots of yi distribution during the run-away collapse phase are shown in Figure 2. The left panel shows the distribution at the time when the central density nH,c reaches 10 12 cm−3, whereas the right panel represents the 6 Maki & Susa [Vol. , Fig. 2. The distribution of the main species, e−, H+, H, H2, H , Li, and Li+ are plotted. The vertical axis denotes the fractional abundance yi of the above species, and the horizontal axis denotes the radius from the core center. Left panel represents the snapshot when the central density satisfies nH,c = 10 12 cm−3, whereas right panel shows the results at nH,c = 10 16 cm−3. Fig. 3. Drift velocity vBx as a function of the radius r and the field strength B at the central density . Contour map of vBx/u is shown on r − B plane for nH,c = 10 16 cm−3 Solid curves represent the constant loci along which log(vBx/u) equal to the values labeled on the curve. The dashed curve represents the critical field strength Bcr given by the equation (23). distribution at nH,c = 10 16 cm−3. The horizontal axis denotes the radius from the center of the primordial gas cloud r [cm], and the vertical axis is the fractional abundances yi of each species. Since the central density increases as the collapse proceeds, right panel represents the later epoch than that of the left. Roughly speaking, in both of the panels, fraction of electrons (ye) in the inner region are smaller than that in the outer part. These results could be simply understood since the recombination process in the inner dense region proceeds faster than in the outer envelope. We also find the opposite behavior in the inner most region of right panel, No. ] Dissipation of Magnetic Flux in Primordial Star Formation 7 at which ye increases as r decreases, because of the collisional ionization. It is also worth noting that Li becomes the main provider of electrons around r ∼ 1014cm at nH,c = 1016 cm−3. Because of the combined effects of these two (collisional ionization & presence of Li), ye never gets lower than 10 −11 as far as we consider nH,c < 10 16 cm−3 in run-away phase. 3.1.2. Drift velocities in run-away collapse phase In Figure 3, we show the ratio of the drift velocity vBx to the infall velocity u when nH,c = 10 16 cm−3 is satisfied. The vertical axis shows the magnetic field strength, whereas the horizontal axis shows the distance from the cloud center. The solid curves in Figure 3 show the contours of log10(vBx/u). The magnetic fields are dissipated in the region where log10(vBx/u)> 0, in contrast, the fields are frozen to the gas in the region where log10(vBx/u)< 0. The dashed curve in Figure 3 is the critical field strength Bcr that is defined by the equation GM(r)ρ(r) . (23) Note that since we are interested in the collapsing gas cloud, the magnetic force needs to be weaker than the gravi- tational force. Hence, our calculations are valid in the region where the field strength in the cloud satisfies B < Bcr. We find clearly from Figure 3 that the frozen-in condition vBx/u < 1 is almost always satisfied as long as B is less than the critical field strength Bcr. We also find the basically same results for other snapshots, i.e. the drift velocity is smaller than the infall velocity anytime and anywhere if B <Bcr. In addition, if B < Bcr is satisfied at some initial time and position (t0, r0), this condition also holds at some later epoch (t,r(t)). This statement is proved as follows: Combining the magnetic flux conservation equation 2πr0dr0B(t0,r0) = 2πr(t)drB(t,r(t)) (24) and the mass conservation law given in equation (4), we obtain B(t,r(t)) =B(t0,r0) ρ(t,r(t))r(t) ρ(t0,r0)r0 , (25) Using equations (23) and (25) we have B(t,r(t)) Bcr(t,r(t)) B(t0,r0) Bcr(t0,r0) ρ(t,r(t))r3 (t) ρ(t0,r0)r The second term in right side is always less than or equals to unity in the run-away collapsing cloud, since ρ ∝ r−2 in the envelope, and ρ ∝ r−3 in the core. Thus, if B(t0,r0)/Bcr(t0,r0) < 1 is satisfied, B is less than Bcr throughout the collapse. Remark that we assume the flux conservation as equation (25), which gives the maximal field strength. Therefore, above inequality (B <Bcr) also holds even if the magnetic flux is dissipated. In summary, it is concluded that the magnetic fields are always frozen to the whole cloud in the course of run-away collapse phase, if the magnetic force is much smaller than gravitational force at the beginning of the collapse. We also remark that the slight increase due to collisional ionization at nH,c =10 16 cm−3 implies that ye will become larger as the central density increases up to nH,c> 10 16 cm−3, since the temperature gets higher as the density increases for γ=1.09 polytropic gas. As a result, the magnetic field is expected to be frozen to the gas even for nH,c> 10 16 cm−3 in accretion phase. 3.2. Accretion Phase 3.2.1. Evolution of gas and chemical species in accretion phase The evolutionary sequences of number density nH, temperature T , velocity u, and electron fraction ye in the accretion phase are illustrated in Figure 4. Upper left panel illustrates the evolution of density profile. The solid curves labeled as 0-4 corresponds to the epoch at which the central accreted mass equals to 8.87× 10−2 M⊙,1.58 M⊙,12.7 M⊙,50.2 M⊙,100 M⊙, respectively. It is clear that the density for r → 0 decreases as the collapse proceeds, which is basically the same feature found in similarity solution of polytropic cloud in accretion phase. Because of the high accretion velocity at later snapshots (lower left), the gas is heated efficiently by adiabatic compression. As a result, temperature gets higher as the collapse proceeds, except at the beginning of the accretion phase, when the gas is cooled efficiently due to the enhanced fraction of H2 molecules. The ionization degree (electron fraction ye, lower right) also increases as the collapse proceeds (Figures.4 and 5), due to the higher temperature for later snapshots. As a result, ye never gets lower than 10 −11. We also find that Li is not so important as was in the final phase of run-away phase, since the hydrogen win back the position of chief provider of electrons in the accretion phase( Figure 5). 8 Maki & Susa [Vol. , Fig. 4. The evolutionary sequences of the primordial gas cloud in accretion phase are shown. Four panels (a)-(d) show the spatial distribution of following physical variables as functions of radius: (a) number density nH, (b) temperature T , (c) velocity u, and (d) electron fraction ye. Five time sequences (0-4) are plotted. Accreted central mass Mc(t) is used as a clock. Corresponding mass at the stages are: 0:Mc(t) = 8.87× 10 −2 M⊙, 1:Mc(t) = 1.58M⊙, 2:Mc(t) = 12.7M⊙, 3:Mc(t) = 50.2M⊙, 4:Mc(t) = 100M⊙. 3.2.2. Drift velocities in accretion phase As briefly discussed in section 1, the ambipolar diffusion velocity is proportional to the inverse square of gas density. Its dependence on various physical quantities are described as vambBx ∝ y−1e ρ−2B2r−1 (see equations (19)-(22)). On the other hand, the accretion velocity scales as u ∝ (Mc/r)1/2 for r → 0. Since we consider the accretion phase starting from the final phase of run-away collapse with γ = 1.09, the density profile and the central mass in accretion phase also depend on γ. According to Suto & Silk (1988), we have ρ(t,r) ∝ t(2−3γ)/2r−3/2 for r → 0 (27) Mc(t)∝ t4−3γ (28) Thus, the limiting behaviour of the infall velocity is described as u(t,r)∝ t(4−3γ)/2r−1/2 for r→ 0 (29) Combining above set of equations, we have the dependence of the ratio vambBx /u on t: vambBx ∝ y−1e B2t(9γ−8)/2r5/2 (30) This equation indicates that the ratio keeps growing in accretion phase for fixed r,B and ye, since 9γ− 8 is positive for γ = 1.09. In reality, however, ye increases rapidly as the accretion proceeds (see Figures.4 and 5), and it offsets the increase of vambBx /u. Figures 6 and 7 illustrate the contours of log10(vBx/u) on r−B plane for two epochs. Two figures correspond to the snapshots when the central accreted mass satisfies Mc = 12.7M⊙ and Mc = 100M⊙, respectively. The notations are same as Figure 3. It is clear that the drift velocity is always smaller than the accretion velocity in two snapshots, No. ] Dissipation of Magnetic Flux in Primordial Star Formation 9 Fig. 5. The radial distributions of the fractional abundances of the main species, e, H+, H, H2, H −, H+ , Li, and Li+ in the accretion phase. Two panels correspond to the snapshots at Mc(t) = 12.7 M⊙(left), and Mc(t) = 100 M⊙(right). Fig. 6. The same as Fig.3, except that the contours are plotted in the accretion phase characterized at Mc =12.7M⊙. as long as B <Bcr. In fact, we find that this is true all through the calculations. Besides, the equation (26) holds also in the accretion phase. Considering that the density profile in the accretion phase is ρ∝ r−1.5∼−2, B is less than Bcr throughout the collapse, if B(t0,r0)/Bcr(t0,r0) < 1, as discussed in §3.1.2. In other words, magnetic force is always negligible if it can be ignored in the beginning of the collapse. Thus, it is concluded that the dissipation of magnetic flux is negligible throughout the mass accretion phase, as well as the run-away collapse phase. 4. DISCUSSION We confirmed that the magnetic field is frozen to the star-forming primordial gas cloud even in the accretion phase. Here we discuss the possibility of jet formation and activation of MRI considering the magnetic field strength brought 10 Maki & Susa [Vol. , Fig. 7. The same as Fig.6, except Mc = 100M⊙. into the accretion disk surrounding the protostar. The magnetic field strength brought into the accretion disk is assessed under the frozen-in condition as follows: Bdisk =B0 rdisk . (31) Here B0 is the initial field strength, rdisk denotes the disk radius, whereas R describes the initial radius within which includes the total mass of the disk-star star system M∗disk: R= (3M∗disk/4πρ0) 1/3, (32) where ρ0 represents the initial density of the cloud. rdisk can be evaluated by equation (15) in Tan & Mckee (2004): rdisk ≃ 66.4AU M∗disk where fKep denotes the ratio of the rotation velocity to the Kepler velocity of accreting matter, which is found to be ∼ 0.5 in Abel, Bryan & Norman (2002). Thus, we have the magnetic field strength in the disk as follows: Bdisk ≃ 7.5× 10−10G 3.7× 10−16G 103cm−3 )−2/3 M∗disk )−40/21 Several possibilities to generate cosmological seed magnetic field have been proposed so far. Most of the mechanisms predict BIGM <∼ 10 −19G(Widrow 2002), except the magnetic field generated by radiation transfer effects of powerful ionizing sources such as quasars or first stars (Langer, Puget & Aghanim 2003). They suggests the possibility to generate coherent magnetic field with BIGM ∼ 10−11G. Since the magnetic field is frozen to the primordial gas at low densities (nH < 10 3cm−3), B0 at nH = 10 3cm−3 can be evaluated as B0 = 3.7× 10−16G 10−19G 103cm−3 , (35) Recently, 3-dimensional MHD simulations on primordial star formation have been performed by Machida et al. (2006), assuming ideal MHD condition is always satisfied in the collapsing gas. In fact, our present results guarantee this hypothesis. They found that the protostellar jet is driven in primordial environment if B0 >∼ 10 −9G at nH = 103cm−3. Comparing this condition with equation (35), it is concluded that 1) jets could be driven in first star forming clouds if the seed field is generated by the mechanism proposed by Langer, Puget & Aghanim (2003), 2) whereas the other mechanisms cannot generate the seed field enough to drive the jets. No. ] Dissipation of Magnetic Flux in Primordial Star Formation 11 MRI can be activated in the accretion disk in case the magnetic field in the disk is larger than a critical value (Tan & Blackman 2004): Bdisk >∼ 1.1× 10 M∗disk 10 M⊙ )1/4( 104 K )−3/4( )1/2( ρdisk 5× 10−10 g cm−3 )1/2( 600 R⊙ )−3/4 , (36) This threshold is assessed by the confrontation between the growth rate of MRI and the ohmic dissipation rate. Combining equations (34)- (36), we find that MRI is activated in case the seed field satisfy BIGM >∼ 1.5× 10 M∗disk )2.155 104 K )−3/4( )1/2( ρdisk 5× 10−10 g cm−3 )1/2( 600 R⊙ )−3/4 . (37) Therefore, MRI is driven only if the seed field generation mechanism by the transfer effects of ionizing radiation works. Based upon these arguments, we emphasize that the mechanism proposed by Langer, Puget & Aghanim (2003) should be scrutinized since their results still based upon the argument of order-estimation. 5. SUMMARY In this paper, we investigate the dissipation of magnetic flux in star-forming primordial gas cloud. We solve non- equilibrium chemical reaction equations, coupled with thermal and dynamical evolution of the collapsing cloud all through the run-away phase as well as the mass accretion phase. Thus, we obtain the detailed evolution of ionized fraction of the gas, which enables us to assess the coupling between gas and magnetic field. As a result, we find that the magnetic field is basically frozen to the gas anywhere in collapsing star-forming primordial clouds at any time. Based upon this result, we find the cosmological seed magnetic field generated by most of the mechanisms proposed so far is not sufficient to form jets as well as to activate MRI in the star-forming cloud. Only one mechanism proposed by Langer, Puget & Aghanim (2003) is able to create sufficient field strength. We thank Kazu Omukai for stimulating discussions. Noriaki Shibazaki and Ken Ohsuga are acknowledged for continuous encouragement. The analysis has been made with computational facilities at Rikkyo University. This work was supported in part by Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Grants-in-Aid, Specially Promoted Research 16002003 and Young Scientists (B) 17740110. Appendix 1. Density distribution of free-falling matter In this Appendix, we derive the explicit formula for which gives the density distribution of free-falling matter in equation (8). Equation of motion (5) is integrated as ṙ2 = −E0, E0 = u20, (E0 > 0), (A1) where u0 is the velocity of a fluid element at (t0,r0) where r0 denotes the position of the element at some initial time t0. A solution of this equation is given as r(α,α0) = (1− cosα), (A2) t(α,α0) = t0 − (2E0)3/2 [(α− sinα)− (α0 − sinα0)] , (A3) where α is the so-called development angle, and α0 is its value at t= t0. α0 also satisfies following relation (1− cosα0). 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teaching-for-transfer.dvi Teaching for transfer Sanjoy Mahajan Teaching and Learning Laboratory & Dept of Electrical Engineering and Computer Science MIT, Room 5-122 Cambridge, MA 02139 sanjoy@mit.edu 16 April 2007 Abstract. Students, after they leave our care, are called to solve the diverse problems offered by the world, so we should teach to increase transfer : the ability to apply fundamental principles to new problems and contexts. This ability is rare. The following pages are from a workshop for faculty on designing courses that pro- mote transfer. I discuss two design principles: to name the transferable ideas and to illustrate them with examples from diverse subjects. The discussion uses dimensional reasoning as the example of a valuable transferable idea, illustrating it with three diverse examples. Copyright 2007 Sanjoy Mahajan. This document is free software. You can modify and/or redistribute it under the terms of the GNU General Public License as published by the Free Software Foundation: either version 2 of the License or (at your option) any later version. The source code is available at arxiv.org/abs/0704.1854v1 from the ’Other formats’ link. Licensed under the GNU GPL v2 or later One theme to bind them all [I don’t give away the name of the technique yet, hence the coy section title, so that participants can discover it by discussion.] Pyramid volume [I ask the participants to discuss, in small groups, the flaws in the proposed volumes. Then we name the technique – dimensional analysis – for use in the next examples.] Here is a pyramid with a square base. It has height h and the side of the base is b. Comment on these proposed formulas for the volume: b. h3 + b2 + b c. b4 + h3 d. b4/h [To teach for transfer, people must practice transfer to other domains. So I ask the audience to find the worst flaw in the following argument, which comes from a domain far from physics or mathematics.] In many articles criticizing globalization (the kindler, gentler name for imperialism), you can read an argument like this one [from ‘Impunity for Multinationals’, www.globalpolicy.org/socecon/tncs/2002/0911impunity.htm, 11 Sept 2002]: In Nigeria, a relatively economically strong country, the GDP is $99 billion. The net worth of Exxon is $119 billion. ‘When multinationals have a net worth higher than the GDP of the country in which they operate, what kind of power relationship are we talking about?’ asks Laura Morosini. Find the most egregious fault in this argument. Tidal waves [And a hard example of dimensional analysis, to show that the wave speed is v = The speed of water waves in shallow water depends on the depth and on gravity, which provides the force that drives the waves. Find a formula that connects the speed v to the gravitational acceleration g and to the depth h. Tidal waves on the ocean are an example of shallow-water waves (!). How fast do they travel? Licensed under the GNU GPL v2 or later Why use diverse examples Here is a line of reasoning showing why diverse examples promote transfer. One example Concept Example A bare concept is difficult to grasp without examples. Examples help learners to understand the concept and, if chosen well, to separate the transferable concept from the illustrations. Suppose then that you explain a concept and illustrate it with an example. The concept and example merge in the learner’s mind, leaving him or her uncertain about the boundaries of the concept. Two examples Penum One remedy is to offer a second example. To the extent that the second example is similar to the first, the first concept plus example overlaps the second concept plus example. The over- lap includes a penumbra around the concept. The penumbra is smaller than when only one example is offered, and the two ex- amples delimit the boundaries of the concept more clearly than when only one example is offered. Progress! Diverse examples You can help the learner even more by taking the second (or third) example from a distant field: wherefore curriculum in- tegration. The penumbra shrinks and the concept stands out. ‘Only connect! That was the whole of her sermon. . .Live in fragments no longer.’ E. M. Forster, Howard’s End.

Students, after they leave our care, are called to solve the diverse problems of the world, so we should teach to increase transfer: the ability to apply fundamental principles to new problems and contexts. This ability is rare. The following pages are from a workshop for faculty on designing courses that promote transfer. I discuss two design principles: to name the transferable ideas and to illustrate them with examples from diverse subjects. The discussion uses dimensional reasoning as the example of a valuable transferable idea, illustrating it with three diverse examples.

teaching-for-transfer.dvi Teaching for transfer Sanjoy Mahajan Teaching and Learning Laboratory & Dept of Electrical Engineering and Computer Science MIT, Room 5-122 Cambridge, MA 02139 sanjoy@mit.edu 16 April 2007 Abstract. Students, after they leave our care, are called to solve the diverse problems offered by the world, so we should teach to increase transfer : the ability to apply fundamental principles to new problems and contexts. This ability is rare. The following pages are from a workshop for faculty on designing courses that pro- mote transfer. I discuss two design principles: to name the transferable ideas and to illustrate them with examples from diverse subjects. The discussion uses dimensional reasoning as the example of a valuable transferable idea, illustrating it with three diverse examples. Copyright 2007 Sanjoy Mahajan. This document is free software. You can modify and/or redistribute it under the terms of the GNU General Public License as published by the Free Software Foundation: either version 2 of the License or (at your option) any later version. The source code is available at arxiv.org/abs/0704.1854v1 from the ’Other formats’ link. Licensed under the GNU GPL v2 or later One theme to bind them all [I don’t give away the name of the technique yet, hence the coy section title, so that participants can discover it by discussion.] Pyramid volume [I ask the participants to discuss, in small groups, the flaws in the proposed volumes. Then we name the technique – dimensional analysis – for use in the next examples.] Here is a pyramid with a square base. It has height h and the side of the base is b. Comment on these proposed formulas for the volume: b. h3 + b2 + b c. b4 + h3 d. b4/h [To teach for transfer, people must practice transfer to other domains. So I ask the audience to find the worst flaw in the following argument, which comes from a domain far from physics or mathematics.] In many articles criticizing globalization (the kindler, gentler name for imperialism), you can read an argument like this one [from ‘Impunity for Multinationals’, www.globalpolicy.org/socecon/tncs/2002/0911impunity.htm, 11 Sept 2002]: In Nigeria, a relatively economically strong country, the GDP is $99 billion. The net worth of Exxon is $119 billion. ‘When multinationals have a net worth higher than the GDP of the country in which they operate, what kind of power relationship are we talking about?’ asks Laura Morosini. Find the most egregious fault in this argument. Tidal waves [And a hard example of dimensional analysis, to show that the wave speed is v = The speed of water waves in shallow water depends on the depth and on gravity, which provides the force that drives the waves. Find a formula that connects the speed v to the gravitational acceleration g and to the depth h. Tidal waves on the ocean are an example of shallow-water waves (!). How fast do they travel? Licensed under the GNU GPL v2 or later Why use diverse examples Here is a line of reasoning showing why diverse examples promote transfer. One example Concept Example A bare concept is difficult to grasp without examples. Examples help learners to understand the concept and, if chosen well, to separate the transferable concept from the illustrations. Suppose then that you explain a concept and illustrate it with an example. The concept and example merge in the learner’s mind, leaving him or her uncertain about the boundaries of the concept. Two examples Penum One remedy is to offer a second example. To the extent that the second example is similar to the first, the first concept plus example overlaps the second concept plus example. The over- lap includes a penumbra around the concept. The penumbra is smaller than when only one example is offered, and the two ex- amples delimit the boundaries of the concept more clearly than when only one example is offered. Progress! Diverse examples You can help the learner even more by taking the second (or third) example from a distant field: wherefore curriculum in- tegration. The penumbra shrinks and the concept stands out. ‘Only connect! That was the whole of her sermon. . .Live in fragments no longer.’ E. M. Forster, Howard’s End.

ACT-02-07, MIFP-07-11 Variations of the Hidden Sector in a Realistic Intersecting Brane Model Ching-Ming Chen,1 Tianjun Li,1, 2 V. E. Mayes,1 and Dimitri V. Nanopoulos1, 3, 4 1George P. and Cynthia W. Mitchell Institute for Fundamental Physics, Texas A&M University, College Station, TX 77843, USA 2 Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China 3Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, Woodlands, TX 77381, USA 4Academy of Athens, Division of Natural Sciences, 28 Panepistimiou Avenue, Athens 10679, Greece (Dated: November 1, 2018) Abstract Recently, we discussed the first example of a phenomenologically realistic intersecting D6-brane model. In this model, the gauge symmetry in the hidden sector is USp(2)1×USp(2)2×USp(2)3× USp(2)4. However, we find that the USp(2)1 × USp(2)2 gauge symmetry can be replaced by an U(2)12 gauge symmetry, and/or the USp(2)3 × USp(2)4 gauge symmetry can be replaced by an U(2)34 gauge symmetry since the USp(2) 2 stacks of D6-branes contribute to the same Ramond- Ramond tadpoles as those of the U(2) stacks. Thus, there are three non-equivalent variations of the hidden sector, and the corresponding gauge symmetries are U(2)12 × USp(2)3 × USp(2)4, U(2)34 × USp(2)1 × USp(2)2, and U(2)12 × U(2)34, respectively. Moreover, we study the hidden sector gauge symmetry breaking, discuss how to decouple the additional exotic particles, and briefly comment on the phenomenological consequences. PACS numbers: 11.10.Kk, 11.25.Mj, 11.25.-w, 12.60.Jv http://arxiv.org/abs/0704.1855v1 I. INTRODUCTION The goal of string phenomenology is to construct realistic standard-like string models with all moduli stabilized. In the early days, string model building was mainly concentrated on the weakly coupled heterotic string theory. After the second string revolution, consistent four- dimensional chiral models with non-Abelian gauge symmetry on Type II orientifolds were able to be constructed due to the advent of D-branes [1]. In particular, Type II orientifolds with intersecting D-branes, where the chiral fermions arise from the intersections of D-branes in the internal space [2] with T-dual description in terms of magnetized D-branes [3], have played an important role in string model building during the last few years. On Type IIA orientifolds with intersecting D6-branes, many non-supersymmetric three- family standard-like models and Grand Unified Theories (GUTs) were constructed [4, 5, 6]. Although these models were globally consistent, there generically existed uncancelled Neveu- Schwarz-Neveu-Schwarz (NSNS) tadpoles as well as the gauge hierarchy problem. To solve these two problems, semi-realistic supersymmetric standard-like models, Pati-Salam models, SU(5) models as well as flipped SU(5) models have been constructed in Type IIA theory on T6/(Z2 × Z2) [7, 8, 9, 10, 11, 12, 13, 14] and T6/(Z2 × Z′2) [15, 16] orientifolds with intersecting D6-branes, and some of their phenomenological consequences have been stud- ied [17, 18]. Moreover, the supersymmetric constructions in Type IIA theory on other orien- tifolds were also discussed [19]. There are two main constraints on supersymmetric D6-brane model building: RR tadpole cancellation conditions and four-dimensional N = 1 supersym- metric D6-brane configurations. Also, K-theory conditions provide minor constraints. In addition, to stabilize the closed-string moduli via supergravity fluxes, the flux models on Type II orientifolds have also been constructed [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. It is well known that there are two serious problems in almost all the supersymmetric D-brane models: no gauge coupling unification at the string scale, and the rank one problem in the Standard Model (SM) fermion Yukawa matrices. Although these problems can be solved in the flux models of Ref. [29] where the RR tadpole cancellation conditions are relaxed, these models are in the AdS vacua and the question of how to lift these AdS vacua to the Minkowski vacua or dS vacua correctly is still a big challenge. Recently, we found that there is one and only one intersecting D6-brane model on Type IIA T6/(Z2 × Z2) orientifold where the above problems can be solved [11, 29]. Moreover, this model may has a realistic low energy phenomenology [31]. Although its observable sector has unique phenomological properties, it is possible to have different stacks of the D6-branes in the hidden sector. In this paper, we discuss three non-equivalent variations of the hidden sector where the RR tadpoles are cancelled, the four-dimensional N = 1 supersymmetry is perserved, and the K-theory conditions are satisfied. These three variations seem to be the only possibilities. In the original model [11, 29], the gauge symmetry in the hidden sector is USp(2)1 × USp(2)2 × USp(2)3 × USp(2)4. Interestingly, we can replace the USp(2)1 × USp(2)2 gauge symmetry by an U(2)12 gauge symmetry, and/or the USp(2)3 × USp(2)4 gauge symmetry by an U(2)34 gauge symmetry since the contributions to the RR tadpoles from the USp(2) stacks of D6-branes are the same as those of the U(2) stacks. Thus, there are three non- equivalent variations, and the corresponding gauge symmetries in the hidden sector are U(2)12×USp(2)3×USp(2)4, U(2)34×USp(2)1×USp(2)2, and U(2)12×U(2)34, respectively. Moreover, we discuss the hidden sector gauge symmetry breaking, and consider how to decouple the additional exotic particles. Because the observable sector is the same, the discussions on phenomenological consequences, for example, the gauge coupling unification, supersymmetry breaking soft terms, low energy supersymmetric particle spectrum, dark matter density, and the SM fermion masses and mixings, are the same as those in Ref. [31, This paper is organized as follows. We briefly review the intersecting D6-brane model building on Type IIA T6/(Z2 × Z2) orientifold in Section II and the realistic intersecting D6-brane model in Section III. We study the three variations of the hidden sector in Section IV. Discussion and conclusions are given in Section V. II. INTERSECTING D6-BRANE MODEL BUILDING IN TYPE IIA THEORY ON T6/(Z2 × Z2) ORIENTIFOLD We briefly review the intersecting D6-brane model building in Type IIA theory on T6/(Z2 × Z2) orientifold [7, 8]. We consider T6 to be a six torus factorized as T6 = T2 × T2 ×T2 whose complex coordinates are zi, i = 1, 2, 3 for the i-th two torus, respec- tively. The θ and ω generators for the orbifold group Z2×Z2 act on the complex coordinates as following θ : (z1, z2, z3) → (−z1,−z2, z3) , ω : (z1, z2, z3) → (z1,−z2,−z3) . (1) We implement an orientifold projection ΩR, where Ω is the world-sheet parity, and R acts on the complex coordinates as R : (z1, z2, z3) → (z1, z2, z3) . (2) So, there are four kinds of orientifold 6-planes (O6-planes) for the actions of ΩR, ΩRθ, ΩRω, and ΩRθω, respectively. Also, we have two kinds of complex structures consistent with orientifold projection for a two torus – rectangular and tilted [32]. If we denote the TABLE I: General spectrum for intersecting D6-branes at generic angles, where Iaa′ = −23−k i=1(n a), and IaO6 = 2 3−k(−l1al2al3a + l1an2an3a + n1al2an3a + n1an2al3a). Moreover, M is the multiplicity, and aS and aA denote the symmetric and anti-symmetric representations of U(Na/2), respectively. Sector Representation aa U(Na/2) vector multiplet and 3 adjoint chiral multiplets ab+ ba M(Na ) = Iab = 2 i=1(n ab′ + b′a M(Na ) = Iab′ = −2−k i=1(n aa′ + a′a M(aS) = 12(Iaa′ − IaO6) ; M(aA) = 12(Iaa′ + IaO6) homology classes of the three cycles wrapped by the D6-brane stacks as nia[ai] +m a[bi] and nia[a i] +m a[bi] with [a i] = [ai] + [bi] for the rectangular and tilted tori respectively, we can label a generic one cycle by (nia, l a) in either case, where in terms of the wrapping numbers lia ≡ mia for a rectangular two torus and lia ≡ 2m̃ia = 2mia+nia for a tilted two torus. So, the homology three-cycles for stack a of Na D6-branes and its orientifold image a ′ take the form [Πa] = nia[ai] + 2 −βilia[bi] , [Πa′ ] = nia[ai]− 2 −βilia[bi] , (3) where βi = 0 if the i-th two torus is rectangular and βi = 1 if it is tilted. Also, we define k ≡ β1 + β2 + β3. For a stack of N D6-branes that do not lie on the top of any O6-plane, we obtain the U(N/2) gauge symmetry with three adjoint chiral superfields due to the orbifold projections. While for a stack of N D6-branes on the top of an O6-plane, we obtain the USp(N) gauge symmetry with three anti-symmetric chiral superfields. The bifundamental chiral superfields arise from the intersections of two different stacks of D6-branes or one stack of D6-branes and its ΩR image [7, 8]. In short, the general spectrum for intersecting D6-branes at generic angles, which is valid for both rectangular and tilted two tori, is given in Table I. Moreover, a model may contain additional non-chiral (vector-like) multiplet pairs from ab+ba, ab′+b′a, and aa′+ a′a sectors if two stacks of the corresponding D-branes are parallel and on the top of each other on one two torus. The multiplicity of the non-chiral multiplet pairs is given by the product of the intersection numbers on the other two two-tori. Before further discussions, let us define the products of wrapping numbers Aa ≡ −n1an2an3a, Ba ≡ n1al2al3a, Ca ≡ l1an2al3a, Da ≡ l1al2an3a, Ãa ≡ −l1al2al3a, B̃a ≡ l1an2an3a, C̃a ≡ n1al2an3a, D̃a ≡ n1an2al3a. TABLE II: Wrapping numbers of the four O6-planes. Orientifold Action O6-Plane (n1, l1)× (n2, l2)× (n3, l3) ΩR 1 (2β1 , 0)× (2β2 , 0) × (2β3 , 0) ΩRω 2 (2β1 , 0)× (0,−2β2)× (0, 2β3) ΩRθω 3 (0,−2β1)× (2β2 , 0)× (0, 2β3) ΩRθ 4 (0,−2β1)× (0, 2β2)× (2β3 , 0) The four-dimensional N = 1 supersymmetric models from Type IIA orientifolds with intersecting D6-branes are mainly constrained by the RR tadpole cancellation conditions and the four-dimensional N = 1 supersymmetric D6-brane configurations, and also constrained by the K-theory conditions: (1) RR Tadpole Cancellation Conditions The total RR charges of D6-branes and O6-planes must vanish since the RR field flux lines are conserved. And then we obtain the RR tadpole cancellation conditions as follows − 2kN (1) + NaAa = −2kN (2) + NaBa = −2kN (3) + NaCa = −2kN (4) + NaDa = −16, (5) where 2N (i) are the number of D6-branes wrapping along the i-th O6-plane which is defined in Table II. (2) Four-Dimensional N = 1 Supersymmetric D6-Brane Configurations The four-dimensional N = 1 supersymmetry can be preserved by the orientation projec- tion if and only if the rotation angle of any D6-brane with respect to the O6-plane is an element of SU(3) [2], or in other words, θ1+θ2+θ3 = 0 mod 2π, where θi is the angle between the D6-brane and the O6-plane in the i-th two torus. This supersymmetry conditions can be rewritten as [9] xAÃa + xBB̃a + xCC̃a + xDD̃a = 0, Aa/xA +Ba/xB + Ca/xC +Da/xD < 0, (6) where xA = λ, xB = λ2 β2+β3/χ2χ3, xC = λ2 β1+β3/χ1χ3, xD = λ2 β1+β2/χ1χ2, and χi = R i are the complex structure parameters. The positive parameter λ has been introduced to put all the variables A, B, C, and D on an equal footing. (3) K-theory Conditions The discrete D-brane RR charges classified by the Z2 K-theory groups in the presence of orientifolds, which are subtle and invisible by the ordinary homology [22, 33], should also be taken into account [21]. The K-theory conditions are 2−kÃa = 2−β1NaB̃a = 2−β2NaC̃a = 2−β3NaD̃a = 0 mod 4 . (7) III. THE REALISTIC INTERSECTING D6-BRANE MODEL There may be one and only one intersecting D6-brane model in Type IIA theory on T6/(Z2 × Z2) orientifold with a realistic phenomenology [11, 29, 31]. Let us briefly review it. We present the D6-brane configurations and intersection numbers in Table III, and its spectrum in Table IV. We put the a′, b, and c stacks of D6-branes on the top of each other on the third two torus, and then we have the additional vector-like particles from N = 2 subsectors. We have shown that the gauge symmetry in the observable sector can be broken down to the SM gauge symmetry via the Green-Schwarz mechanism, D6-brane splittings and su- persymmtry preserving Higgs mechanism. The gauge couplings for SU(4)C , SU(2)L and SU(2)R are unified at the string scale, and the additional exotic particles may be decoupled around the string scale. Also, we calculated the supersymmetry breaking soft terms, and the corresponding low energy supersymmetric particle spectrum that can be tested at the Large Hadron Collider (LHC). The observed dark matter density can also be generated. In addition, we can explain the SM quark masses and mixings, and the tau lepton mass. The neutrino masses and mixings may be generated via seesaw mechanism as well. Simi- lar to the GUTs [34], we have roughly the wrong fermion mass relation me/mµ ≃ md/ms, and the correct electron and muon masses can be generated via high-dimensional opera- stackN (n1,l1)(n2,l2)(n3,l3) A S b b ′ c c′ O61O62O63O64 a 8 ( 0,-1) ( 1, 1)( 1, 1) 0 0 3 0(3) -3 0(3) 1 -1 0 0 b 4 ( 3, 1) ( 1, 0)( 1,-1) -2 2 - - 0(6)0(1) 0 1 0 -3 c 4 ( 3,-1) ( 0, 1)( 1,-1) 2 -2 - - - - -1 0 3 0 O61 2 ( 1, 0) ( 1, 0)( 2, 0) - - - - - - - - - - O62 2 ( 1, 0) ( 0,-1)( 0, 2) - - - - - - - - - - O63 2 ( 0, -1)( 1, 0)( 0, 2) - - - - - - - - - - O64 2 ( 0, -1)( 0, 1)( 2, 0) - - - - - - - - - - TABLE III: The D6-brane configurations and intersection numbers on Type IIA T6/Z2 × Z2 orientifold. The gauge symmetry is [U(4)C × U(2)L × U(2)R]Observable × [USp(2)1 × USp(2)2 × USp(2)3 × USp(2)4]Hidden, the SM fermions and Higgs fields arise from the intersections on the first two torus, and the complex structure parameters are 2χ1 = 6χ2 = 3χ3 = 6. Also, the beta functions for all USp(2)i gauge symmetries are −3. tors [35]. Furthermore, all the USp(2)i gauge symmetries will become strong around the string scale [35]. IV. THREE VARIATIONS OF THE HIDDEN SECTOR In the realistic intersecting D6-brane model [11, 29], the observable sector is unique. Inter- estingly, we find three non-equivalent variations of the hidden sector where we can cancel the RR tadpoles, preserve the four-dimensional N = 1 supersymmetry, and satisfy the K-theory conditions. And it seems to us that there is no other variation. In the original model [11, 29], the gauge symmetry in the hidden sector is USp(2)1 × USp(2)2 × USp(2)3 × USp(2)4. We notice that the USp(2)1 × USp(2)2 gauge symmetry can be replaced by an U(2)12 gauge symmetry, and/or the USp(2)3 × USp(2)4 gauge symmetry by an U(2)34 gauge symmetry because the contributions to the RR tadpoles from the USp(2)2 stacks of D6-branes are the same as those of the U(2) stacks. Thus, there are three non-equivalent variations, and the corresponding gauge symmetries in the hidden sector are U(2)12 × USp(2)3 × USp(2)4, U(2)34×USp(2)1×USp(2)2, and U(2)12 ×U(2)34, respectively. Let us present them one by one in the following subsections. TABLE IV: The chiral and vector-like superfields, and their quantum numbers under the gauge symmetry SU(4)C × SU(2)L × SU(2)R × USp(2)1 × USp(2)2 × USp(2)3 × USp(2)4. Quantum Number Q4 Q2L Q2R Field ab 3× (4, 2, 1, 1, 1, 1, 1) 1 -1 0 FL(QL, LL) ac 3× (4, 1, 2, 1, 1, 1, 1) -1 0 1 FR(QR, LR) a1 1× (4, 1, 1, 2, 1, 1, 1) 1 0 0 Xa1 a2 1× (4, 1, 1, 1, 2, 1, 1) -1 0 0 Xa2 b2 1× (1, 2, 1, 1, 2, 1, 1) 0 1 0 Xb2 b4 3× (1, 2, 1, 1, 1, 1, 2) 0 -1 0 X i c1 1× (1, 1, 2, 2, 1, 1, 1) 0 0 -1 Xc1 c3 3× (1, 1, 2, 1, 1, 2, 1) 0 0 1 X i bS 2× (1, 3, 1, 1, 1, 1, 1) 0 2 0 T iL bA 2× (1, 1, 1, 1, 1, 1, 1) 0 -2 0 SiL cS 2× (1, 1, 3, 1, 1, 1, 1) 0 0 -2 T iR cA 2× (1, 1, 1, 1, 1, 1, 1) 0 0 2 SiR ab′ 3× (4, 2, 1, 1, 1, 1, 1) 1 1 0 3× (4, 2, 1, 1, 1, 1, 1) -1 -1 0 ac′ 3× (4, 1, 2, 1, 1, 1, 1) 1 1 Φi 3× (4, 1, 2, 1, 1, 1, 1) -1 0 -1 Φi bc 6× (1, 2, 2, 1, 1, 1, 1) 0 1 -1 Hi 6× (1, 2, 2, 1, 1, 1, 1) 0 -1 1 A. U(2)12 × USp(2)3 × USp(2)4 Hidden Sector In the first variation of the hidden sector, we replace the USp(2)1 × USp(2)2 gauge symmetry by an U(2)12 gauge symmetry. We present the D6-brane configurations and intersection numbers in Table V. Moreover, the particle spectrum has two parts: (1) the spectrum for old particles is given in Table IV by removing all the particles that are charged under USp(2)1 × USp(2)2; (2) the spectrum for the new particles is given in Table VI. The anomalies from the global U(1) of U(2)12 are cancelled by the Green-Schwarz mech- anism, and its gauge field obtains mass via the linear B ∧ F couplings. Then, the effective gauge symmetry is SU(2)12. The SU(2)12 gauge symmetry can be broken down to U(1)12 via D6-brane splitting. Interestingly, we do not have any additional chiral exotic particles that are charged under SU(4)C . The simple way to give masses to the extra exotic particles stackN (n1,l1)(n2,l2)(n3,l3) A S b b ′ c c′ d d′ O63O64 a 8 ( 0,-1) ( 1, 1) ( 1, 1) 0 0 3 0(3) -3 0(3) 0(2)0(1) 0 0 b 4 ( 3, 1) ( 1, 0) ( 1,-1) -2 2 - - 0(6)0(1) 1 0(1) 0 -3 c 4 ( 3,-1) ( 0, 1) ( 1,-1) 2 -2 - - - - -1 0(1) 3 0 d 4 ( 1, 0) ( 1,-1)( 1, 1) 0 0 - - - - - - -1 1 O63 2 ( 0, -1)( 1, 0) ( 0, 2) - - - - - - - - - - O64 2 ( 0, -1)( 0, 1) ( 2, 0) - - - - - - - - - - TABLE V: The D6-brane configurations and intersection numbers on Type IIA T6/Z2 ×Z2 orien- tifold. The complete gauge symmetry is [U(4)C × U(2)L × U(2)R]Observable × [U(2)12 ×USp(2)3 × USp(2)4]Hidden. TABLE VI: The new chiral superfields and their quantum numbers under the gauge symmetry SU(4)C × SU(2)L × SU(2)R × U(2)12 × USp(2)3 × USp(2)4. Representation Q4 Q2L Q2R Q12 Field bd 1× (1, 2, 1, 2, 1, 1) 0 1 0 -1 Xbd cd 1× (1, 1, 2, 2, 1, 1) 0 0 -1 1 Xcd d3 1× (1, 1, 1, 2, 2, 1) 0 0 0 -1 Xd3 d4 1× (1, 1, 1, 2, 1, 2) 0 0 0 1 Xd4 Xbd and Xcd is instanton effects [36, 37, 38, 39]. However, we do not have the suitable three- cycles wrapped by E2 instantons 1, and thus the instanton effects are not available. Similar results hold for the next two subsections. In addition, the USp(2)3 and USp(2)4 will become strong at about the string scale [35], and then we will have some composite particles in the U(2)12 anti-symmetric and symmetric representations, S d and T d from Xd3, and S d and T from Xd4, respectively. So we can break the U(1)12 by giving suitable string-scale vacuum expectation values (VEVs) to T d and T d, and we can give the string-scale VEVs to S d and S ′d. Note that we give the TeV-scale VEVs to S L and the string-scale VEVs to S R [31], we can give the GUT-scale masses to X ic3 and Xcd and the TeV-scale masses to the X b4 and 1 Note that the E2 branes must also wrap rigid cycles. Xbd via the high-dimensional operators [35]. Furthermore, if we could give the string-scale masses to the three U(2)12 adjoint chiral superfields and we do not break the SU(2)12 via D6-brane splitting, the SU(2)12 gauge symmetry will become strong around the string scale. Then we can have the singlet composite field S ′L in the U(2)L anti-symmetric representation with charge +2 under U(1)L from Xbd. And we can give the string-scale VEVs to S L and S while keeping the D-flatness of U(1)L. Therefore, we may also give the GUT-scale masses to the X ib4 and Xbd via the high-dimensional operators [35]. B. U(2)34 × USp(2)1 × USp(2)2 Hidden Sector stackN (n1,l1)(n2,l2)(n3,l3) A S b b ′ c c′ e e′ O61O62 a 8 ( 0,-1)( 1, 1) ( 1, 1) 0 0 3 0(3) -3 0(3) 0(2)0(0) 1 -1 b 4 ( 3, 1) ( 1, 0) ( 1,-1) -2 2 - - 0(6)0(1) 0(3) -3 0 1 c 4 ( 3,-1)( 0, 1) ( 1,-1) 2 -2 - - - - 0(3) 3 -1 0 e 4 ( 0, 1) (-1, 1) (-1, 1) 0 0 - - - - - - -1 1 O61 2 ( 1, 0) ( 1, 0) ( 2, 0) - - - - - - - - - - O62 2 ( 1, 0) ( 0,-1)( 0, 2) - - - - - - - - - - TABLE VII: The D6-brane configurations and intersection numbers on Type IIA T6/Z2 × Z2 orientifold. The gauge symmetry is [U(4)C × U(2)L × U(2)R]Observable × [U(2)34 × USp(2)1 × USp(2)2]Hidden. TABLE VIII: The new chiral superfields and their quantum numbers under the gauge symmetry SU(4)C × SU(2)L × SU(2)R × U(2)34 × USp(2)1 × USp(2)2. Representation Q4 Q2L Q2R Q34 Field be′ 3× (1, 2, 1, 2, 1, 1) 0 -1 0 -1 X i ce′ 3× (1, 1, 2, 2, 1, 1) 0 0 1 1 X i e1 1× (1, 1, 1, 2, 2, 1) 0 0 0 -1 Xe1 e2 1× (1, 1, 1, 2, 1, 2) 0 0 0 1 Xe2 In the second variation of the hidden sector, we replace the USp(2)3 × USp(2)4 gauge symmetry by an U(2)34 gauge symmetry. We present the D6-brane configurations and intersection numbers in Table VII. The particle spectrum also has two parts: (1) the spectrum for old particles is given in Table IV by removing all the particles that are charged under USp(2)3 × USp(2)4; (2) the spectrum for the new particles is given in Table VIII. Note that the wrapping numbers for the d stack of D6-branes are equivalent to those of the a stack by T duality and orientifold action, we can think that we have an U(6) gauge symmetry in the begining. Only the global U(1) of U(6) is anomalous U(1) symmetry, and its gauge field obtains mass via the linear B ∧ F couplings. After we put four D6-branes on the place with equivalent wrapping numbers (just like the D6-brane splittings), we break the SU(6) down to the SU(4)C × SU(2)34 × U(1)′ where the U(1)′ generator in SU(6) is TU(1)′ ≡ diag (1, 1, 1, 1,−2,−2) . (8) Thus, the left-handed and right-handed SM fermions have U(1)′ charges +1/2 6 and 6, respectively. In order to keep the gauge coupling unification, we have to break the U(1)′ so that it will not become part of the U(1)Y . In short, we have to break U(2)34 completely. Because the USp(2)1 and USp(2)2 will become strong at about the string scale [35], we will have some composite particles in the U(2)34 anti-symmetric and symmetric represen- tations, S e and T e from Xe1, and S e and T e from Xe2, respectively. So we can break the U(2)12 completely by giving suitable string-scale VEVs to S e, and T e. Moreover, we can have the singlet composite particle S ′L in the U(2)L anti-symmetric representation with charge +2 under U(1)L from Xb2. And then we can give the string-scale VEVs to S and S ′L while keeping the D-flatness of U(1)L. Note that S R also have string-scale VEVs, we may give the GUT-scale masses to Xb2, Xc1, X be′, and X ce′ via the high-dimensional operators [35]. Moreover, Xa1 and Xa2 may form the vector-like particles if we break the USp(2)1 and USp(2)2 down to the diagonal USp(2)D12 [31]. C. U(2)12 × U(2)34 Hidden Sector In the third variation of the hidden sector, we replace the USp(2)1 × USp(2)2 gauge symmetry by U(2)12, and replace the USp(2)3 × USp(2)4 gauge symmetry by U(2)34. We present the D6-brane configurations and intersection numbers in Table IX. The particle spectrum also has two parts: (1) the spectrum for old particles is given in Table IV by removing all the particles that are charged under USp(2)1×USp(2)2 ×USp(2)3 ×USp(2)4; (2) the spectrum for the new particles is given in Table X. stackN (n1,l1)(n2,l2)(n3,l3) A S b b ′ c c′ d d′ e e′ a 8 ( 0,-1)( 1, 1)( 1, 1) 0 0 3 0(3) -3 0(3) 0(2)0(1)0(2)0(0) b 4 ( 3, 1) ( 1, 0)( 1,-1) -2 2 - - 0(6)0(1) 1 0(1)0(3) -3 c 4 ( 3,-1)( 0, 1)( 1,-1) 2 -2 - - - - -1 0(1)0(3) 3 d 4 ( 1, 0) ( 1,-1)( 1, 1) 0 0 - - - - - - 0(1)0(2) e 4 ( 0, 1) (-1, 1)(-1, 1) 0 0 - - - - - - - - TABLE IX: The D6-brane configurations and intersection numbers on Type IIA T6/Z2×Z2 orien- tifold. The complete gauge symmetry is [U(4)C×U(2)L×U(2)R]Observable× [U(2)12×U(2)34]Hidden. TABLE X: The chiral and vector-like superfields, and their quantum numbers under the gauge symmetry SU(4)C × SU(2)L × SU(2)R × U(2)12 × U(2)34. Representation Q4 Q2L Q2R Q12 Q34 Field bd 1× (1, 2, 1, 2, 1) 0 1 0 -1 0 Xbd be′ 3× (1, 2, 1, 1, 2) 0 -1 0 0 -1 X i cd 1× (1, 1, 2, 2, 1) 0 0 -1 1 0 Xcd ce′ 3× (1, 1, 2, 1, 2) 0 0 1 0 1 X i de 1× (1, 1, 1, 2, 2) 0 0 0 1 -1 Xde 1× (1, 1, 1, 2, 2) 0 0 0 -1 1 Xde de′ 2× (1, 1, 1, 2, 2) 0 0 0 1 1 X i 2× (1, 1, 1, 2, 2) 0 0 0 -1 -1 Xi As discussed in above subsections, we can break the U(2)12 down to the U(1)12 gauge symmetry via Green-Schwarz mechanism and D6-brane splitting, and we have to break the U(2)34 gauge symmetry completely. In order to break the U(1)12 and U(2)34 gauge symmetries, we put the d and e stacks of D6-branes on the top of each other on the second two torus, and put the d and e′ stacks on the top of each other on the third two torus. Then, we have additional vector-like particles Xde, Xde, X de′, and X de′ , as given in Table X. And there exist the following Yukawa couplings W ⊃ yAijXbdX + yBijXcdX de′ , (9) where yAij and y ij are Yukawa couplings. If we give the diagonal string-scale VEVs to X and X de′ , we break the U(2)12×U(2)34 down to the diagonal U(2)D. Moreover, the Xbd and one linear combination of X ibe′, and the Xcd and one linear combination of X ce′ can have vector-like masses close to the string scale. Note that we can give the TeV-scale VEVs to SiL and the string-scale VEVs to SiR [31], we can give the GUT-scale masses to Xcd and the other two linear combinaions of X ice′, and the TeV-scale masses to Xbd and the other two linear combinations of X ibe′ via the high-dimensional operators [35]. Similar to the discussions in the above subsection A, if we can give the string-scale masses to the three U(2)12 adjoint chiral superfields and do not break the SU(2)12 gauge symmetry via D6-brane splitting, the SU(2)12 gauge symmetry will become strong around the string scale. Then we can have the singlet composite field S ′L in the U(2)L anti-symmetric representation with charge +2 under U(1)L from Xbd, and we can give the string-scale VEVs to S L and S L while keeping the D-flatness of U(1)L. Therefore, we may also give the GUT-scale masses to Xbd and the other two linear combinations of X ibe′ via the high-dimensional operators [35]. V. DISCUSSION AND CONCLUSIONS At present, there is only one known example of an intersecting D6-brane model with a realistic observable sector. Interestingly, there are three non-equivalent variations of the hidden sector in which the theoretical constraints on model building can be satisfied. There does not seem to be any other possible variation in the original model [11, 29], and the gauge symmetry in the hidden sector is USp(2)1 × USp(2)2 × USp(2)3 × USp(2)4. We noticed that the USp(2)1 × USp(2)2 gauge symmetry can be replaced by an U(2)12 gauge symmetry, and/or the USp(2)3 × USp(2)4 gauge symmetry can be replaced by an U(2)34 gauge symmetry because the USp(2)2 stacks of D6-branes contribute to the same RR tad- poles as those of the U(2) stacks. Thus, we obtained three non-equivalent variations, and the corresponding gauge symmetries in the hidden sector are U(2)12 × USp(2)3 × USp(2)4, U(2)34 ×USp(2)1 ×USp(2)2, and U(2)12 ×U(2)34, respectively. In addition, we studied the hidden sector gauge symmetry breaking, and discussed how to decouple the additional exotic particles. Because the observable sector is the same, the phenomenological discussions in the observable sector are the same as those in Ref. [31, 35]. Acknowledgments This research was supported in part by the Mitchell-Heep Chair in High Energy Physics (CMC), by the Cambridge-Mitchell Collaboration in Theoretical Cosmology (TL), and by the DOE grant DE-FG03-95-Er-40917 (DVN). 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[37] L. E. Ibanez and A. M. Uranga, JHEP 0703, 052 (2007). [38] M. Cvetic, R. Richter and T. Weigand, arXiv:hep-th/0703028. [39] L. E. Ibanez, A. N. Schellekens and A. M. Uranga, arXiv:0704.1079 [hep-th]. http://arxiv.org/abs/hep-th/0703280 http://arxiv.org/abs/hep-th/0609191 http://arxiv.org/abs/hep-th/0703028 http://arxiv.org/abs/0704.1079 Introduction Intersecting D6-Brane Model Building in Type IIA Theory on T6/(Z2Z2) Orientifold The Realistic Intersecting D6-Brane Model Three variations of the Hidden Sector U(2)12USp(2)3 USp(2)4 Hidden Sector U(2)34USp(2)1 USp(2)2 Hidden Sector U(2)12U(2)34 Hidden Sector Discussion and Conclusions Acknowledgments References

Recently, we discussed the first example of a phenomenologically realistic intersecting D6-brane model. In this model, the gauge symmetry in the hidden sector is USp(2)_1 x USp(2)_2 x USp(2)_3 x USp(2)_4. However, we find that the USp(2)_1 x USp(2)_2 gauge symmetry can be replaced by an U(2)_{12} gauge symmetry, and/or the USp(2)_3 x USp(2)_4 gauge symmetry can be replaced by an U(2)_{34} gauge symmetry since the USp(2)^2 stacks of D6-branes contribute to the same Ramond-Ramond tadpoles as those of the U(2) stacks. Thus, there are three non-equivalent variations of the hidden sector, and the corresponding gauge symmetries are U(2)_{12} x USp(2)_3 x USp(2)_4, U(2)_{34} x USp(2)_1 x USp(2)_2, and U(2)_{12} x U(2)_{34}, respectively. Moreover, we study the hidden sector gauge symmetry breaking, discuss how to decouple the additional exotic particles, and briefly comment on the phenomenological consequences.

Introduction Intersecting D6-Brane Model Building in Type IIA Theory on T6/(Z2Z2) Orientifold The Realistic Intersecting D6-Brane Model Three variations of the Hidden Sector U(2)12USp(2)3 USp(2)4 Hidden Sector U(2)34USp(2)1 USp(2)2 Hidden Sector U(2)12U(2)34 Hidden Sector Discussion and Conclusions Acknowledgments References

Axial Vector Tetraquark with Two s-quarks Yoshiko Kanada-En’yo1, Osamu Morimatsu2 and Tetsuo Nishikawa3 1 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502, Japan 2 Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization, Ibaraki 305-0801, Japan 3 Department of Physics, Tokyo Institute of Technology, 2-12-1, Oh-Okayama, Meguro, Tokyo 152-8551, Japan Possibility of an axial vector isoscalar tetraquark uds̄s̄ is discussed. If a f1 meson in the mass region 1.4 − 1.5 GeV consists of four quarks nsn̄s̄, the mass of the isoscalar uds̄s̄(ϑ+- meson) state with JP = 1+ is expected to be lower than that of the f1 meson. Within a flux-tube quark model, a possible resonant state of uds̄s̄(JP = 1+) is suggested to appear at ∼ 1.4 GeV with the width O(20 ∼ 50) MeV. We propose that the ϑ+-meson is the good candidate for the tetraquark search, which would be observed in the K+K+π− decay channel. The possibility of multiquark states has been discussed for a long time.1)–10) In particularly, the possible qqq̄q̄ states have been suggested in many theoretical efforts to understand light scalar mesons(for example, Refs.[1,3,11]). The 4q states were proposed in the description of f0(600) and f0(980), where the strong attraction between (qq)3̄ and (q̄q̄)3 play an important role. 1), 11) Here, (qq)3̄ and (q̄q̄)3 denote the color-anti-triplet quark pair and the color-triplet anti-quark pair, respectively. On the other hand, the KK molecule states were suggested to understand the properties of f0(980) and a0(980). 3) Even if the 4q components are dominant in a certain meson with the conventional flavor, it is difficult to find an direct evidence of the 4q components due to the mixing with the conventional qq̄ state via the annihilation of qq̄ pairs. Our main interest here is the possibility of “tetraquark” states which has the minimal 4-quark content. The recent observation of DsJ(2317) 12) and the reports of the pentaquark baryon Θ+(uudds̄),13) revived the motivation of the experimental and theoretical researches on multiquarks in hadron physics, though the existence of Θ+ yet to be well estab- lished. One of the striking characteristics of the Θ+ is its narrow width. For the theoretical interpretation of the narrow Θ+ state, the possibility of the spin-parity JP = 1/2+ and JP = 3/2− have been discussed by many groups.10), 14)–19) The unnatural spin and parity is a key of the suppressed width for the lowest decay channel. Now we turn to the discussion on the possibility of the tetraquarks. If we accept the interpretation of the pentaquark as the (ud)3̄(ud)3̄q̄ state based on the diquark picture by Jaffe and Wilczek, it is natural to expect that a tetraquark with the uds̄s̄ content may exist at the nearly same energy region by replacing a ud-diquark with s̄ quark. Firstly, one should consider the states with unnatural spin and parity, which cannot decay into two light hadrons (pseudo scalar mesons) in the S-wave channel. typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.1856v1 2 Letters Second, the exotic color configurations (qq)3̄(q̄q̄)3 would be essential to stabilize the exotic hadrons. Then, we propose a JP = 1+ uds̄s̄ state with the (qq)3̄(q̄q̄)3 configuration as the candidate of narrow tetraquark states. It should be stressed that two-body KK decays from any JP = 1+ uds̄s̄ state are forbidden because of the conservation of the total spin and parity. The lowest threshold energy of the allowed two-body decays is 1.39 GeV for the KK∗(895) channel. If the mass of the JP = 1+ uds̄s̄ state lies below(closely to) the KK∗, two-meson decay channels are (almost) closed, and hence, its width must be narrow. The tetraquark uds̄s̄ states were discussed in Ref.[1], and noted as E(KK)- mesons. In the MIT bag model,1) the theoretical mass for the isoscalar uds̄s̄(JP = 1+) state was predicted to be 1.65 GeV. Recently the isoscalar uds̄s̄ state with JP = 1− was suggested in analogy with the Θ+ by Burns et al.,20) and its mass was predicted to be ∼ 1.6 GeV. The isoscalar tetraquark uds̄s̄ in the flavor 1̄0 group was called ϑ+-meson in Ref.[20] in the association with the Θ baryon. In this paper, we investigate the ϑ+-meson(JP = 1+) with a constituent quark model. The theoretical method of the calculations is the same as that applied to the study of pentaquark and tetraquark in Refs.[17,21]. Namely, we apply the flux-tube quark model with antisymmetrized molecular dynamics(AMD)22) to 4q systems. Based on the picture of a flux-tube model, we ignore the coupling between the configurations (qq)3̄(q̄q̄)3 and (qq̄)1(qq̄)1, and solve the 4q dynamics with the variational method only in the model space within the exotic color configuration (qq)3̄(q̄q̄)3. By assuming that the f1 meson in the 1.4∼1.5 GeV mass region as the 4q state, we predict the mass and the width of the tetraquark ϑ+(JP = 1+). The method of AMD is a variational method. The adopted Hamiltonian is the same as that of previous works.17), 21) The Coulomb and color-magnetic terms of the OGE potential and the string potential are taken into account. The parameters in the Hamiltonian are chosen to reasonably reproduce the normal hadron spectra.21) In order to evaluate the ϑ+(1+) mass and the width, we adopt the observed data of the f1-mesons(f1(1420) and f1(1510)) in the 1.4∼1.5 GeV region as an input. The details of the formulations for fourquark systems are explained in Ref.[21]. As mentioned in,21) in order to predict the ϑ-meson mass, we need to deter- mine the mass shift parameter M0 in the string potential for fourquark (qq)3̄(q̄q̄)3 systems. Here, we use the f1-meson mass as the input to determine the mass shift. In the mass region 1∼1.6 GeV, three f1-mesons, f1(1285), f1(1420), and f1(1510) are known, though the f1(1510) is not well established. 23) In the P -wave qq̄ state, two f1-mesons are expected to appear in this energy region as the partners of the qq̄ nonet. It is considered that the lower one is dominated by the light-quark uū and dd̄ components(nn̄), and the major component of the higher one is the ss̄ state. In the general interpretation, the lowest f1-meson(f1(1285)) is regarded as the nn̄ state. However, there are two candidates (f1(1420) and f1(1510)) for the qq̄ partner of the f1(1285), and the assignment is not confirmed yet. In the constituent quark model calculation,24) the masses of the two 1++ qq̄ states in the P -wave qq̄ nonet are 1.24 and 1.48 GeV. The theoretical mass spectra of the 1++ qq̄ states seems to be consistent with the experimental ones if f1(1510) is assigned to be the partner of the Letters 3 f1(1285) in the flavor nonet. This is consistent with the assignment in Ref.[25]. On the other hand, an alternative interpretation of the f1(1285) and f1(1420) being the qq̄ partners is claimed in Refs.[23,26,27]. These interpretations lead to an indication that one of the f1(1420) or f1(1510) may be a non-qq̄ meson while the other and f1(1285) can be understood as the partners of the conventional P -wave qq̄ states. By ignoring the qq̄ annihilation, we calculate the mass of the JPC=1++ (us)3̄(ūs̄)3 state for the f1-meson with the present framework in the same way as for the tetraquark ϑ-meson. We adjust the mass shift parameter M0 by fitting the mass of the f1(1420) or the f1(1510). Then we get the ϑ +(1+) mass around 1.4 GeV. The obtained ϑ+(1+) wave function is dominated by the component with the spin-zero (ud)3̄ and the spin-one (s̄s̄)3 in the spatially symmetric configuration, (0s) This is consistent with the naive expectation in the diquark picture because the spin- zero (ud)3̄ gain the color-magnetic interaction while only the spin-one configuration is allowed in the spatially symmetric (s̄s̄)3 pair. As a result of the energy gain of the color magnetic interaction, the ϑ+(1+) mass is slightly lower than the fourquark f1 mass. Next we discuss the width of the ϑ+(1+) meson. As mentioned above, we suggest that the ϑ+(1+)-meson may appear in the energy region ∼1.4 GeV near the KK∗ threshold. The expected decay modes are KK∗ and KKπ. If the branching into KK∗ is small enough, the width should be narrow because the phase space for the three-body decays is suppressed in general. In order to discuss the stability of the ϑ+-meson, we consider only the two-hadron decay and give a rough estimation of the ϑ+ width assuming that the coupling of the fourquark f1 with the KK̄ ∗ and c.c. is the same as that of the ϑ+ with the KK∗. By considering the phase space for the two-body decays, we predict the width of ϑ+(1+) to be Γϑ = 20 ∼ 50 MeV. In summary, we discussed the possibility of the JP = 1+ state of the isoscalar tetraquark(S=+2), ϑ+-meson, with the uds̄s̄ content. If a f1 meson in the mass region 1.4−1.5 GeV consists of four quarks nsn̄s̄, the mass of the isoscalar uds̄s̄(ϑ+- meson) state with JP = 1+ is expected to be lower than that of the f1 meson. Within a flux-tube quark model, a possible resonant state of uds̄s̄(JP = 1+) is suggested to appear at ∼ 1.4 GeV with the width O(20 ∼ 50) MeV. We propose that the ϑ+-meson is the good candidate for the tetraquark search, which would be observed in the K+K+π− decay channel. Recently, the θ+(1−) and the θ+(0+) were predicted by Burns et al.20) and by Karlier and Lipkin,28) respectively. It should be pointed out that the allowed decay channels are different among these three predictions JP = 1+, 1−, and 0+ states of the ϑ+-mesons. The K+K+π− decay from the ϑ(JP = 1+) predicted in the present work is suitable for the experimental tetraquark search. As for the other candidates of the 4q states, it has been theoretically suggested that the scalar mesons like f0(600), f0(980) and a0(980) below 1 GeV might be interpreted as 4q states. We calculated the corresponding fourquarks with the present framework. Then, we found that the masses of fourquarks with the exotic color configurations (qq)3̄(q̄q̄)3 are much higher than these light scalar mesons. It indicates that these scalar mesons might be other than the dominant (qq)3̄(q̄q̄)3 state, but might be the hybrids of P -wave qq̄ and fourquark components with meson-meson 4 Letters tails in the outer region as argued in Ref.[27]. The authors would like to thank to H. Nemura and H. Hidaka for the valuable discussions. This work was supported by Japan Society for the Promotion of Science and Grants-in-Aid for Scientific Research of the Japan Ministry of Education, Science Sports, Culture, and Technology. The calculations of this work is supported by the computer system in YITP. 1) R. J. Jaffe, Phys. Rev.D15, 267 (1977); R. L. Jaffe, Phys. Rev.D15, 281 (1977). 2) R. J. Jaffe, Phys. Rev. Lett.38, 195 (1977). 3) J. Weinstein and N. Isgur, Phys. Rev.D27, 588 (1983); J. Weinstein and N. Isgur, Phys. Rev.D41, 2236 (1990). 4) F.E. Close, and H. J. Lipkin, Phys. Lett.B196, 245 (1987). 5) J. Carlson and V. R. Pandharipande, Phys. Rev.D43, 1652 (1991). 6) H. J. Lipkin, Nucl. Phys.A625, 207 (1997). 7) F. Stancu, Few Body Syst.Suppl.11, 33(1999). 8) T. Sakai, K. Shimizu, and K. Yazaki, Prog. Theor. Phys. Suppl. 137, 121 (2000). 9) By T. Barnes, F.E. Close, and H.J. Lipkin, Phys. Rev.D68, 054006 (2003). 10) M. Oka, Prog. Theor. Phys.112, 1 (2004), and references therein. 11) R. J. Jaffe and F. E. Low, Phys. Rev.D19, 2105 (1979). 12) BABAR Collaboration, B. Aubert et al., Phys. Rev. Lett.90, 242001 (2003). 13) LEPS collaboration, T. Nakano et al., Phys. Rev. Lett.91, 012002 (2003); LEPS collabo- ration, T. Nakano et al., Int. J. Mod. Phys. A20, 1543 (2005). 14) D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A359, 305 (1997). 15) R. Jaffe and F. Wilczek, Phys. Rev. Lett.91, 232003 (2003). 16) M. Karliner and H. J. Lipkin, Phys. Lett.B575, 249 (2003). 17) Y. Kanada-En’yo, M. Morimatsu and T. Nishikawa, hep-ph/0404144; Phys. Rev.C71, 045202 (2005); hep-ph/0410221 Proceedings of PENTAQUARK04, Spring8, Japan, 2004. 18) S. Takeuchi and K. Shimizu, Phys. Rev.C71, 062202 (2005). 19) T. Nishikawa, Y. Kanada-En’yo, O. Morimatsu and Y. Kondo, Phys. Rev.D71, 076004 (2005). 20) T. Burns, F. E. Close, and J. J. Dudek, Phys. Rev.D71, 014017 (2005). 21) Y. Kanada-En’yo, M. Morimatsu and T. Nishikawa, Phys. Rev.D71, 094005 (2005). 22) Y. Kanada-En’yo, H. Horiuchi and A. Ono, Phys. Rev. C 52, 628 (1995); Y. Kanada-En’yo and H. Horiuchi, Phys. Rev. C 52, 647 (1995). 23) S. Eidelman et al., Phys. Lett.B592, 1 (2004). 24) S. Godfrey and N. Isgur, Phys. Rev.D32, 189 (1985). 25) S. Godfrey and J. Napolitano, Rev. Mod. Phys.71, 1411 (1999). 26) F. E. Close and A. Kirk et al.Z. Phys. C76, 469 (1997). 27) F. E. Close and N. A. Törnqvist, J. Phys. G28 R249(2002). 28) M. Karliner and H. J. Lipkin, Phys. Lett.B612, 197 (2005). http://arxiv.org/abs/hep-ph/0404144 http://arxiv.org/abs/hep-ph/0410221

Possibility of an axial vector isoscalar tetraquark $ud\bar{s}\bar{s}$ is discussed. If a $f_1$ meson in the mass region $1.4-1.5$ GeV consists of four quarks $ns\bar{n}\bar{s}$, the mass of the isoscalar $ud\bar{s}\bar{s}$($\vartheta^+$-meson) state with $J^P=1^+$ is expected to be lower than that of the $f_1$ meson. Within a flux-tube quark model, a possible resonant state of $ud\bar{s}\bar{s}(J^{P}=1^{+})$ is suggested to appear at $\sim$ 1.4 GeV with the width ${\cal{O}}(20\sim 50)$ MeV. We propose that the $\vartheta^+$-meson is the good candidate for the tetraquark search, which would be observed in the $K^+K^+\pi^-$ decay channel.

Axial Vector Tetraquark with Two s-quarks Yoshiko Kanada-En’yo1, Osamu Morimatsu2 and Tetsuo Nishikawa3 1 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502, Japan 2 Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization, Ibaraki 305-0801, Japan 3 Department of Physics, Tokyo Institute of Technology, 2-12-1, Oh-Okayama, Meguro, Tokyo 152-8551, Japan Possibility of an axial vector isoscalar tetraquark uds̄s̄ is discussed. If a f1 meson in the mass region 1.4 − 1.5 GeV consists of four quarks nsn̄s̄, the mass of the isoscalar uds̄s̄(ϑ+- meson) state with JP = 1+ is expected to be lower than that of the f1 meson. Within a flux-tube quark model, a possible resonant state of uds̄s̄(JP = 1+) is suggested to appear at ∼ 1.4 GeV with the width O(20 ∼ 50) MeV. We propose that the ϑ+-meson is the good candidate for the tetraquark search, which would be observed in the K+K+π− decay channel. The possibility of multiquark states has been discussed for a long time.1)–10) In particularly, the possible qqq̄q̄ states have been suggested in many theoretical efforts to understand light scalar mesons(for example, Refs.[1,3,11]). The 4q states were proposed in the description of f0(600) and f0(980), where the strong attraction between (qq)3̄ and (q̄q̄)3 play an important role. 1), 11) Here, (qq)3̄ and (q̄q̄)3 denote the color-anti-triplet quark pair and the color-triplet anti-quark pair, respectively. On the other hand, the KK molecule states were suggested to understand the properties of f0(980) and a0(980). 3) Even if the 4q components are dominant in a certain meson with the conventional flavor, it is difficult to find an direct evidence of the 4q components due to the mixing with the conventional qq̄ state via the annihilation of qq̄ pairs. Our main interest here is the possibility of “tetraquark” states which has the minimal 4-quark content. The recent observation of DsJ(2317) 12) and the reports of the pentaquark baryon Θ+(uudds̄),13) revived the motivation of the experimental and theoretical researches on multiquarks in hadron physics, though the existence of Θ+ yet to be well estab- lished. One of the striking characteristics of the Θ+ is its narrow width. For the theoretical interpretation of the narrow Θ+ state, the possibility of the spin-parity JP = 1/2+ and JP = 3/2− have been discussed by many groups.10), 14)–19) The unnatural spin and parity is a key of the suppressed width for the lowest decay channel. Now we turn to the discussion on the possibility of the tetraquarks. If we accept the interpretation of the pentaquark as the (ud)3̄(ud)3̄q̄ state based on the diquark picture by Jaffe and Wilczek, it is natural to expect that a tetraquark with the uds̄s̄ content may exist at the nearly same energy region by replacing a ud-diquark with s̄ quark. Firstly, one should consider the states with unnatural spin and parity, which cannot decay into two light hadrons (pseudo scalar mesons) in the S-wave channel. typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.1856v1 2 Letters Second, the exotic color configurations (qq)3̄(q̄q̄)3 would be essential to stabilize the exotic hadrons. Then, we propose a JP = 1+ uds̄s̄ state with the (qq)3̄(q̄q̄)3 configuration as the candidate of narrow tetraquark states. It should be stressed that two-body KK decays from any JP = 1+ uds̄s̄ state are forbidden because of the conservation of the total spin and parity. The lowest threshold energy of the allowed two-body decays is 1.39 GeV for the KK∗(895) channel. If the mass of the JP = 1+ uds̄s̄ state lies below(closely to) the KK∗, two-meson decay channels are (almost) closed, and hence, its width must be narrow. The tetraquark uds̄s̄ states were discussed in Ref.[1], and noted as E(KK)- mesons. In the MIT bag model,1) the theoretical mass for the isoscalar uds̄s̄(JP = 1+) state was predicted to be 1.65 GeV. Recently the isoscalar uds̄s̄ state with JP = 1− was suggested in analogy with the Θ+ by Burns et al.,20) and its mass was predicted to be ∼ 1.6 GeV. The isoscalar tetraquark uds̄s̄ in the flavor 1̄0 group was called ϑ+-meson in Ref.[20] in the association with the Θ baryon. In this paper, we investigate the ϑ+-meson(JP = 1+) with a constituent quark model. The theoretical method of the calculations is the same as that applied to the study of pentaquark and tetraquark in Refs.[17,21]. Namely, we apply the flux-tube quark model with antisymmetrized molecular dynamics(AMD)22) to 4q systems. Based on the picture of a flux-tube model, we ignore the coupling between the configurations (qq)3̄(q̄q̄)3 and (qq̄)1(qq̄)1, and solve the 4q dynamics with the variational method only in the model space within the exotic color configuration (qq)3̄(q̄q̄)3. By assuming that the f1 meson in the 1.4∼1.5 GeV mass region as the 4q state, we predict the mass and the width of the tetraquark ϑ+(JP = 1+). The method of AMD is a variational method. The adopted Hamiltonian is the same as that of previous works.17), 21) The Coulomb and color-magnetic terms of the OGE potential and the string potential are taken into account. The parameters in the Hamiltonian are chosen to reasonably reproduce the normal hadron spectra.21) In order to evaluate the ϑ+(1+) mass and the width, we adopt the observed data of the f1-mesons(f1(1420) and f1(1510)) in the 1.4∼1.5 GeV region as an input. The details of the formulations for fourquark systems are explained in Ref.[21]. As mentioned in,21) in order to predict the ϑ-meson mass, we need to deter- mine the mass shift parameter M0 in the string potential for fourquark (qq)3̄(q̄q̄)3 systems. Here, we use the f1-meson mass as the input to determine the mass shift. In the mass region 1∼1.6 GeV, three f1-mesons, f1(1285), f1(1420), and f1(1510) are known, though the f1(1510) is not well established. 23) In the P -wave qq̄ state, two f1-mesons are expected to appear in this energy region as the partners of the qq̄ nonet. It is considered that the lower one is dominated by the light-quark uū and dd̄ components(nn̄), and the major component of the higher one is the ss̄ state. In the general interpretation, the lowest f1-meson(f1(1285)) is regarded as the nn̄ state. However, there are two candidates (f1(1420) and f1(1510)) for the qq̄ partner of the f1(1285), and the assignment is not confirmed yet. In the constituent quark model calculation,24) the masses of the two 1++ qq̄ states in the P -wave qq̄ nonet are 1.24 and 1.48 GeV. The theoretical mass spectra of the 1++ qq̄ states seems to be consistent with the experimental ones if f1(1510) is assigned to be the partner of the Letters 3 f1(1285) in the flavor nonet. This is consistent with the assignment in Ref.[25]. On the other hand, an alternative interpretation of the f1(1285) and f1(1420) being the qq̄ partners is claimed in Refs.[23,26,27]. These interpretations lead to an indication that one of the f1(1420) or f1(1510) may be a non-qq̄ meson while the other and f1(1285) can be understood as the partners of the conventional P -wave qq̄ states. By ignoring the qq̄ annihilation, we calculate the mass of the JPC=1++ (us)3̄(ūs̄)3 state for the f1-meson with the present framework in the same way as for the tetraquark ϑ-meson. We adjust the mass shift parameter M0 by fitting the mass of the f1(1420) or the f1(1510). Then we get the ϑ +(1+) mass around 1.4 GeV. The obtained ϑ+(1+) wave function is dominated by the component with the spin-zero (ud)3̄ and the spin-one (s̄s̄)3 in the spatially symmetric configuration, (0s) This is consistent with the naive expectation in the diquark picture because the spin- zero (ud)3̄ gain the color-magnetic interaction while only the spin-one configuration is allowed in the spatially symmetric (s̄s̄)3 pair. As a result of the energy gain of the color magnetic interaction, the ϑ+(1+) mass is slightly lower than the fourquark f1 mass. Next we discuss the width of the ϑ+(1+) meson. As mentioned above, we suggest that the ϑ+(1+)-meson may appear in the energy region ∼1.4 GeV near the KK∗ threshold. The expected decay modes are KK∗ and KKπ. If the branching into KK∗ is small enough, the width should be narrow because the phase space for the three-body decays is suppressed in general. In order to discuss the stability of the ϑ+-meson, we consider only the two-hadron decay and give a rough estimation of the ϑ+ width assuming that the coupling of the fourquark f1 with the KK̄ ∗ and c.c. is the same as that of the ϑ+ with the KK∗. By considering the phase space for the two-body decays, we predict the width of ϑ+(1+) to be Γϑ = 20 ∼ 50 MeV. In summary, we discussed the possibility of the JP = 1+ state of the isoscalar tetraquark(S=+2), ϑ+-meson, with the uds̄s̄ content. If a f1 meson in the mass region 1.4−1.5 GeV consists of four quarks nsn̄s̄, the mass of the isoscalar uds̄s̄(ϑ+- meson) state with JP = 1+ is expected to be lower than that of the f1 meson. Within a flux-tube quark model, a possible resonant state of uds̄s̄(JP = 1+) is suggested to appear at ∼ 1.4 GeV with the width O(20 ∼ 50) MeV. We propose that the ϑ+-meson is the good candidate for the tetraquark search, which would be observed in the K+K+π− decay channel. Recently, the θ+(1−) and the θ+(0+) were predicted by Burns et al.20) and by Karlier and Lipkin,28) respectively. It should be pointed out that the allowed decay channels are different among these three predictions JP = 1+, 1−, and 0+ states of the ϑ+-mesons. The K+K+π− decay from the ϑ(JP = 1+) predicted in the present work is suitable for the experimental tetraquark search. As for the other candidates of the 4q states, it has been theoretically suggested that the scalar mesons like f0(600), f0(980) and a0(980) below 1 GeV might be interpreted as 4q states. We calculated the corresponding fourquarks with the present framework. Then, we found that the masses of fourquarks with the exotic color configurations (qq)3̄(q̄q̄)3 are much higher than these light scalar mesons. It indicates that these scalar mesons might be other than the dominant (qq)3̄(q̄q̄)3 state, but might be the hybrids of P -wave qq̄ and fourquark components with meson-meson 4 Letters tails in the outer region as argued in Ref.[27]. The authors would like to thank to H. Nemura and H. Hidaka for the valuable discussions. This work was supported by Japan Society for the Promotion of Science and Grants-in-Aid for Scientific Research of the Japan Ministry of Education, Science Sports, Culture, and Technology. The calculations of this work is supported by the computer system in YITP. 1) R. J. Jaffe, Phys. Rev.D15, 267 (1977); R. L. Jaffe, Phys. Rev.D15, 281 (1977). 2) R. J. Jaffe, Phys. Rev. Lett.38, 195 (1977). 3) J. Weinstein and N. Isgur, Phys. Rev.D27, 588 (1983); J. Weinstein and N. Isgur, Phys. Rev.D41, 2236 (1990). 4) F.E. Close, and H. J. Lipkin, Phys. Lett.B196, 245 (1987). 5) J. Carlson and V. R. Pandharipande, Phys. Rev.D43, 1652 (1991). 6) H. J. Lipkin, Nucl. Phys.A625, 207 (1997). 7) F. Stancu, Few Body Syst.Suppl.11, 33(1999). 8) T. Sakai, K. Shimizu, and K. Yazaki, Prog. Theor. Phys. Suppl. 137, 121 (2000). 9) By T. Barnes, F.E. Close, and H.J. Lipkin, Phys. Rev.D68, 054006 (2003). 10) M. Oka, Prog. Theor. Phys.112, 1 (2004), and references therein. 11) R. J. Jaffe and F. E. Low, Phys. Rev.D19, 2105 (1979). 12) BABAR Collaboration, B. Aubert et al., Phys. Rev. Lett.90, 242001 (2003). 13) LEPS collaboration, T. Nakano et al., Phys. Rev. Lett.91, 012002 (2003); LEPS collabo- ration, T. Nakano et al., Int. J. Mod. Phys. A20, 1543 (2005). 14) D. Diakonov, V. Petrov and M. V. Polyakov, Z. Phys. A359, 305 (1997). 15) R. Jaffe and F. Wilczek, Phys. Rev. Lett.91, 232003 (2003). 16) M. Karliner and H. J. Lipkin, Phys. Lett.B575, 249 (2003). 17) Y. Kanada-En’yo, M. Morimatsu and T. Nishikawa, hep-ph/0404144; Phys. Rev.C71, 045202 (2005); hep-ph/0410221 Proceedings of PENTAQUARK04, Spring8, Japan, 2004. 18) S. Takeuchi and K. Shimizu, Phys. Rev.C71, 062202 (2005). 19) T. Nishikawa, Y. Kanada-En’yo, O. Morimatsu and Y. Kondo, Phys. Rev.D71, 076004 (2005). 20) T. Burns, F. E. Close, and J. J. Dudek, Phys. Rev.D71, 014017 (2005). 21) Y. Kanada-En’yo, M. Morimatsu and T. Nishikawa, Phys. Rev.D71, 094005 (2005). 22) Y. Kanada-En’yo, H. Horiuchi and A. Ono, Phys. Rev. C 52, 628 (1995); Y. Kanada-En’yo and H. Horiuchi, Phys. Rev. C 52, 647 (1995). 23) S. Eidelman et al., Phys. Lett.B592, 1 (2004). 24) S. Godfrey and N. Isgur, Phys. Rev.D32, 189 (1985). 25) S. Godfrey and J. Napolitano, Rev. Mod. Phys.71, 1411 (1999). 26) F. E. Close and A. Kirk et al.Z. Phys. C76, 469 (1997). 27) F. E. Close and N. A. Törnqvist, J. Phys. G28 R249(2002). 28) M. Karliner and H. J. Lipkin, Phys. Lett.B612, 197 (2005). http://arxiv.org/abs/hep-ph/0404144 http://arxiv.org/abs/hep-ph/0410221

What is the order of 2D polymer escape transition? Hsiao-Ping Hsu and Kurt Binder Institut für Physik, Johannes Gutenberg-Universität Mainz D-55099 Mainz, Staudinger Weg 7, Germany Leonid I. Klushin American University of Beirut, Department of Physics, Beirut, Lebanon Alexander M. Skvortsov Chemical-Pharmaceutical Academy, Prof. Popova 14, 197022 St. Petersburg, Russia. (Dated: August 11, 2021) An end-grafted flexible polymer chain in 3d space between two pistons undergoes an abrupt transition from a confined coil to a flower-like conformation when the number of monomers in the chain, N , reaches a critical value. In 2d geometry, excluded volume interactions between monomers of a chain confined inside a strip of finite length 2L transform the coil conformation into a linear string of blobs. However, the blob picture raises questions on the nature of this escape transition. To check the theoretical predictions based on the blob picture we study 2d single polymer chains with excluded volume interactions and with one end grafted in the middle of a strip of length 2L and width H by simulating self-avoiding walks on a square lattice with the pruned-enriched-Rosenbluth method (PERM). We estimate the free energy, the end-to-end distance, the number of imprisoned monomers, the order parameter, and its distribution. It is shown that in the thermodynamic limit of large N and L but finite L/N , there is a small but finite jump in several average characteristics, including the order parameter. We also present a theoretical description based on the Landau free energy approach, which is in good agreement with the simulation results. Both simulation results and the analytical theory indicate that the 2d escape transition is a weak first-order phase transition. I. INTRODUCTION A phenomenon that was called escape transition oc- curs upon progressive squeezing an end-grafted polymer chain between two pistons and has attracted great inter- est [1-16]. At weak deformation the chain is compressed uniformly into a relatively thick pan-cake conformation. Beyond certain critical compression, the chain configu- ration changes abruptly. One part of the chain forms a stem stretching from the grafting point to the piston edge, while the rest of the segments form a coiled crown outside the piston, thus escaping from the region under- neath the piston. An abrupt change from one state to another implies a first order transition. Various aspects of this problem were investigated: The escape transition of compressed polymer mushrooms in 3d space was inves- tigated thoroughly by scaling theory [1], numerical cal- culations [2, 3, 4], and computer modeling under good solvent [5], and theta solvent [6] conditions. The escape transition of star polymers was discussed in [7], and the escape transition of di-block-copolymers was considered in [8]. The influence of the curvature of the pistons was investigated in [9, 10], and the effect of adsorption be- tween the polymer chain and the surface of the piston was considered in [11]. A comparison between Monte Carlo simulations and experimental results by atomic force-electrochemical microscopy was recently presented in [12]. A rigorous analytical theory for the equilibrium and kinetic aspects of the escape transition for a Gaus- sian chain was constructed in [13, 14]. Metastability ef- fects, negative compressibility and the nonequivalence of the escape transition in two conjugate ensembles were analyzed for the same model in [15, 16]. The reason for studying the escape transition is that it gives the possibility to understand the phenomenon of a very unconventional phase transition. The concept of a phase transition requires taking a thermodynamic limit. For standard low-molecular weight systems as well as for macromolecular systems in condensed bulk matter finite size effects are usually negligible. In contrast to that, phase transitions at the level of a single macromolecule e.g., the coil-globule transition [17], or polymer adsorp- tion at an interface [18, 19] - do not have any analogies in the physics of low molecular mass systems. A sin- gle macromolecule always consists of a finite number of monomers N : computer modeling rarely deals with N larger than 104 so that finite-size effects in the single- molecule phase transitions are the rule rather than the exception. The situation is much more complicated in the case of the escape transition. It was shown [1] that the escape transition point (critical compression) depends on the relation between the chain length Na (a is the dis- tance between neighboring monomers) and the piston ra- dius L. Therefore, to analyze the escape transition in the thermodynamic limit, it is necessary to take both Na → ∞ and L → ∞ but Na/L = const. The physics of phase transitions is generally known to be strongly affected by the spatial dimensionality. For phase transitions at the level of a single macromolecule the spatial dimensionality is important because excluded volume effects are especially large for polymeric chains in two dimensions [20]. An ideal chain without excluded http://arxiv.org/abs/0704.1857v2 volume interactions retains a Gaussian coil conformation with the lateral size ∼ N1/2 even when it is confined in a 2d strip. There is a very pronounced difference between this state and the partially escaped state with a strongly stretched stem of size L ∼ Na. On the contrary, a 2d polymer chain with excluded volume interactions con- fined in a strip is already strongly elongated with the size ∼ Na . It is not clear whether the difference between the confined and the escaped states is large enough to result in a phase transition. To analyze the excluded volume effects of the escape transition, we study a flexible polymer chain containing N links of length a (N monomers) grafted in the middle of a strip of length 2L and width H , under good solvent conditions. Schematic drawings of a polymer chain in an imprisoned state and in an escaped state are shown in Fig. 1. First, we summarize the results of the escape transition for a 2d Gaussian ideal chain in Sec. II, and then we give the theoretical predictions based on the blob picture in Sec. III. In an experimental setup, the escape transition is driven by changing the piston separation H , while in a blob picture, the escape transition is studied by changing the chain length N or the strip length L at fixed H which is also the size of a blob. Comparisons between the escape transition behavior of the Gaussian chain model and that of the blob model in the thermo- dynamic limit are given in Sec. IV. In Sec. V we present our results from Monte Carlo (MC) simulations with the pruned-enriched Rosenbluth method (PERM) [21] In Sec. VI we provide a theoretical description based on the Landau free energy approach which is compared with MC results. A summary and discussions are given in Sec. VII. Detailed analyses of the variances of the im- prisoned monomers are given in the Appendix. II. ESCAPE FOR A 2D GAUSSIAN CHAIN For the escape transition of a Gaussian chain, a closed- form of the exact partition function was obtained earlier in [14]. The asymptotic form of the free energy F has two branches: = Fimp imprisoned state = Fesc escaped state . (1) Here d is the dimensionality of space and a factor of kBT is absorbed in the free energy throughout the paper. It was shown by direct numerical comparison that the sim- ple asymptotic expressions provide a very accurate de- scription. The two branches meet at the transition point which is given by , with d = 2 . (2) The average lateral forces acting on the grafting point are obtained by ∂F (N,L,H) n "blobs" crown FIG. 1: Schematic drawings of a flexible polymer chain of length N grafted in the middle of the strip of length 2L and width H , in a blob picture: (a) As the chain is imprisoned inside the strip, it forms a sequence of nb blobs. (b) As the chain length is larger than the maximum chain length N∗ of a chain in an imprisoned state, the chain partially escapes from the strip and forms an escaped state. A escaped state consists of a “stem” containing N∗ monomers and a “crown” containing N −N∗ monomers. which is related to the work required to pull a chain end into confined space by a unit distance. The average com- pression force fH = − ∂F (N,L,H) is related to the work of compression. It was shown earlier [14] that the average fraction of impris- oned monomers is given by the derivative Nimp = ∂F (N,L,H)/∂u of the free energy with respect to the effective confining potential u = (πa/2H) The average end-to-end distance per monomer is: N−1/2 imprisoned state +N−1/2 (1−Nimp/N)1/2 escaped state The Landau order parameter s for the escape transition was introduced in [14] as a stretching degree of the chain rN/N for the imprisoned state and as the stretching de- gree of the stem L/nimp for the escaped state. Note that the order parameter is defined for an instantaneous configuration with a given end-to-end distance rN and the number of imprisoned monomers nimp which may be quite different from the equilibrium average values R = 〈rN 〉 and Nimp = 〈nimp〉. The Landau free energy is defined as a function of the order parameter. The Lan- dau free energy for the imprisoned state is just the free energy of a coil as a function of its free end position, which has a standard parabolic form: Φimp(s) = N . (6) At s = simpeq = 0, the system is in equilibrium and the corresponding equilibrium free energy, i.e., the depth of the minimum, is Φimp(0) = (N/2d)(πa/H) 2. For the escaped state, Φesc(s) = . (7) The position of the minimum is at s = sesceq = πa/dH, and the corresponding depth is Φesc(s eq ) = πL/H . The binodal is determined by the condition that the two min- ima are equally deep, i.e. Φimp(s eq ) = Φesc(s eq ), which leads to the transition point described by Eq. (2). At the transition point, the average order parameter S =< s > jumps from simpeq to s eq . The Landau function allows one to analyze metastable states and to define the two lines where either one of the metastable minima vanishes [16]. Theoretical predictions for the escape transition of a Gaussian chain are summarized in Table I and shown in Fig 2. Detailed discussion and comparison with the prediction from the blob picture for 2d polymer chains are given in Sec. IV. III. BLOB PICTURE OF A 2D ESCAPE A free chain in d = 2, has an average end-to-end dis- tance given by (a is the distance between neighboring monomers) [17-20] RF = aN 3/4 (8) here the prefactors of order unity are neglected through- out. Based on the blob picture we have a cigar of blobs (nb blobs in total) in the confined situation. Thus the average end-to-end distance is R = nb(2rb) = nbH (9) where rb is the blob radius. Within a blob, self-avoiding walks (SAW) statistics holds, so if g monomers belong to a blob H = ag3/4 = 2rb , g = (H/a) 4/3 . (10) Since every monomer of a chain in an imprisoned state must be in a blob we furthermore have N = gnb = nb(H/a) 4/3 , nb = N(H/a) −4/3 . (11) This yields, together with Eq. (9), the formula for the end-to-end distance R/a = N(H/a)−1/3 (12) If H is of the same order as RF, Eq. (8), one finds R/a = N3/4, giving a smooth crossover to mushroom behavior, as expected. The free energy excess of the chain in an imprisoned state (in units of kBT ), relative to an unconfined mush- room, is simply the number of blobs, nb , Fimp = nb = N(H/a) −4/3 (13) We now define N = N∗ as the maximum chain length of an imprisoned chain. Then for N > N∗ the chain con- sists of a “stem” containing N∗ imprisoned monomers and an escaped “crown” comprising the rest N − N∗ monomers (Fig. 1b). Thus, we find N∗ from the con- dition that R becomes equal to L at the transition point, using Eq. (12) N∗ = (L/a)(H/a)1/3 . (14) Since the free energy of an unconfined mushroom is taken as zero reference point, the “crown” does not contribute to the excess free energy of the escaped chain, which is hence due to the stem only: Fesc = N ∗(H/a)−4/3 = L/H . (15) The average end-to-end distance of the escaped chain (in axial direction parallel to the confining boundaries) hence becomes Resc = L+ a(N −N∗)3/4 . (16) Equations (13) and (15) show that the free energy as a function ofN for fixed H and L, consists of two branches, i.e., Fimp for the imprisoned state (N < N ∗) and Fesc for the escaped state (N > N∗), meeting at N = N∗. The lateral and the compression forces are obtained from the free energy by using Eqs. (3) and (4). We use the same definition of the order parameter s as that for the Gaussian chain, so the average order parameter S S =< s >= R/Na imprisoned state L/N∗a = (H/a)−1/3 escaped state From Eqs. (12), (14), and (17), we find that the order parameter does not show any discontinuity at the tran- sition, but simply stays constant, i.e. S = (H/a)−1/3. Results of the theoretical predictions are listed in Table I and also shown in Fig. 2. IV. COMPARISON OF THE GAUSSIAN AND BLOB PICTURES In the thermodynamic limit N → ∞, L → ∞, L/N remains as a nontrivial variable along with H . In Fig. 2, theoretical predictions of the Gaussian chain model and of the blob model are shown by dotted and solid lines, respectively. The strip width, H , is fixed, and the ratio L/N is varied. The chain is in an imprisoned state if L is larger than the corresponding critical value, the Gaussian chain model or for the blob model, and in the escaped state for for L ]. All curves show only the scaling behavior disregarding nu- merical coefficients of order one, and the bond length a is taken as a unit length. In both models, the free energy per monomer is given by piecewise linear functions of L/N . For the escaped ~LN( ) f /NH −1/2 N N /N FIG. 2: Theoretical predictions for various averaged chain characteristics plotted against L/N at constant strip width H : (a) the free energy per monomer F/N , (b) the lateral force fL, (c) the compression force per monomer fH/N , (d) the fraction of imprisoned monomers Nimp/N , (e) the order parameter S, (f) the end-to-end distance per monomer R/N , . Gaussian model results are shown by dotted lines, blob model results - by solid lines. The chain is in an imprisoned state for L ], and in an escaped state for L ] for the Gaussian chain model [blob picture]. TABLE I: Theoretical predictions for the average values of the free energy per monomer F/N , the lateral force fL, the compression force per monomer fH/N , the fraction of imprisoned monomers Nimp/N , the order parameter S, and the reduced end-to-end distance R/Na based on the Gaussian chain model and the blob picture. Characteristics Imprisoned Escaped of the chain state state Gaussian Blob Gaussian Blob )2 ( a )4/3 π fL 0 0 )7/3 π 1 1 2 )1/3 L )1/3 π N−1/2 N−1/2 N−1/4 state, the slope of F/N vs. L/N is the same up to numer- ical coefficients of order 1, and scales as H−1. This slope has the meaning of the lateral force acting on the grafting point. It is proportional to the inverse size of a blob which is defined purely by confinement width H irrespective of whether excluded volume interactions are present or not. The free energy per monomer in the imprisoned state is independent of L/N , and scales as the inverse number of monomers in one blob, g−1 = (a/H) , where ν is the Flory exponent. The transition points in the two models may be quite far apart since they scale differently with H . In the setup where L/N is fixed and the piston sepa- ration is decreasing, the blob model predicts the transi- tion to happen at lower compression, as compared to the Gaussian chain. In both models the lateral force jumps from H−1 to zero at the respective transition point. As for compression forces, they are strongly affected by ex- cluded volume interactions (Fig. 2c). The difference in the plateau values for the imprisoned state reflects the lower compressibility of the self-avoiding walk as com- pared to the Gaussian chain. This corresponds directly to the plateau values of the free energy in Fig. 2a. In the Gaussian chain model, the compression force jumps at the transition point by a factor of 2, while the jump in the blob model is by a factor of 4/3. Figures 2a - 2c suggest that the behavior of both models is fundamentally the same and characteristic of a first-order phase transition. However, the next 3 graphs demonstrate qualitatively different predictions of the Gaussian and the blob models. For the Gaussian chain model, a transition from the confined state to the escaped state is accompanied by a jump in the average number of imprisoned monomers. At the transition point, one half of the total number of monomers are ejected outside to form a crown, see Fig. 2d. In contrast to that, the blob model predicts a smooth change without a jump. This directly affects the behavior of the average order parameter: while in the Gaussian chain model there is a pronounced jump, the blob model suggests that the order parameter does not change at all. The large constant value of S in the blob model is due to the cigar-shape conformation of the chain in an imprisoned state which is identical to the shape of the stem in the escaped state. The behavior of the average end-to-end distance is also qualitatively different for the two models. The plateau value for the imprisoned Gaussian chain, R/N = N−1/2, is characteristic of the ideal coil, while for the elongated cigar, this ratio is N -independent. The linear part of the curves corresponding to the escaped state simply repre- sents the dominant contribution of the stem to the overall end-to-end distance, R ≈ L irrespective of the model. At very low values of L/N corrections due to the crown size become large, as shown in Fig 2f. The size of the Gaus- sian chain demonstrates a jump at the transition point consistent with a strong conformation change accompa- nying the first-order transition. In the blob model, the chain size does not have any jump. It is clear that the predictions of the blob model pre- sented in Fig. 2 contain some internal contradictions. On the one hand, the picture of the two branches of the free energy meeting at some angle suggests a first order tran- sition. The jumps in the lateral and compression forces are a simple consequence of that. On the other hand, nothing dramatic happens to the chain conformation in the blob picture: the change from a completely confined state to a state with a small escaped tail is continuous, as evidenced in Figs 2e and 2f. The presence of discontinu- ity in the slope of the fraction of imprisoned monomers suggests that the transition should be classified as second order. A crucial question to ask is whether one can identify two distinct separate states with a bimodal distribution of some appropriate order parameter. This we address by employing a Monte-Carlo simulation of 2d self-avoiding chains undergoing the escape transition. V. MONTE CARLO SIMULATION Single polymer chains grafted in the middle of a strip of length 2L and width H are described by SAWs of N steps FIG. 3: Schematic drawing of a polymer chain growing as a self-avoiding walk inside a finite strip and grafted at (x = 0, y = 0). Monomers are allowed to sit on the lattice sites except for the lattice sites representing the walls {−L ≤ x ≤ L , y = 0} and {−L ≤ x ≤ L , y = H}. The first monomer is attached with a bond to the grafting site marked by a cross. Lengths are measured in units of the lattice spacing. 1000 0 30 60 90 120 150 N3/4/H H = 17 H = 33 H = 65 H = 129 1.944 NH-4/3 FIG. 4: (a) Free energy relative to a free chain, F (N,L,H) = − ln[(Z(N,L,H)/Z0(N)], plotted against N 3/4H−1 for L = 6400 and H = 17, 33, 65, and 129. The dashed curve is Fimp(N,L,H) = 1.944NH −4/3 and gives the best fit of the data. on a square lattice between two hard walls with distance H as shown in Fig. 3. Monomers are supposed to sit on lattice sites but they are forbidden to sit on the two walls, i.e. {−L ≤ x ≤ L , y = 0} and {−L ≤ x ≤ L , y = H}. For our simulations we employ PERM and PERMwith k- step Markovian anticipation as described in Ref. [21, 22]. PERM is a biased chain growth algorithm with popula- tion control. Polymer chains are built like random walks by adding one monomer at each step. Thus it has the advantage of estimating the partition sum and counting the imprisoned monomers directly. We simulate 2d SAWs starting at the grafting point of the strip of length L = 800, L = 1600, 3200, and 6400. The width of the strip is varied from H = 5 to H = 129. Depending on the chosen sizes of L andH , the total chain length is varied from 2500 to 50000 in order to cover the transition region. 1000 10 100 1000 L = 800 L = 1600 L = 3200 L = 6400 2.03 L/H FIG. 5: The log-log plot of the excess free energy of the escaped chain, Fesc(N,L,H), plotted against L/H for vari- ous values of L and H . The dashed line is Fesc(N,L,H) = 2.03L/H and gives the best fit of the data. 0.02 0.04 0.06 0.08 0 0.2 0.4 0.6 0.8 1 H = 17 H = 28 H = 33 H = 65 L = 800 L = 1600 L = 3200 L = 6400 2.03 L/(NH) FIG. 6: Free energy relative to a free chain divided by N , F (N,L,H)/N , plotted against L/N for various values of L and H . The dashed line extrapolated to zero is 2.03L/(NH). A. Free energy Let us first discuss the scaling behavior of the free en- ergy. The partition sum of a free SAW in infinite volume for N → ∞ scales as Z0(N) = µ γ−1 (18) with µ∞ being the critical fugacity per monomer, and with γ = 43/32 being a universal exponent [18]. As chains are still confined in a strip of width H , one should expect the scaling laws of the excess free energy including the crossover from the region of wide strips, RF << H , (RF ∼ Nν is the Flory radius with ν = 3/4 in d = 2), to the region of narrow strips, RF >> H , where chains are stretched. Since the length of the strip is finite, here we are more interested in another expected crossover behavior of the excess free energy from an im- prisoned and stretched chain state to an escaped state. Therefore, we plot the excess free energy F (N,L,H) = − ln(Z(N,L,H)/Z0(N)) against N3/4H−1 with the pre- cise estimate of µ∞ = 0.37905228 [23] in Fig. 4. The partition sum Z(N,L,H) is the total number of possible configurations of SAW of N steps partially confined in a strip of length 2L and of width H , which is estimated directly in the simulation. In Fig. 4, the sharp crossover behavior from the imprisoned states to the escaped states is indeed seen as N increases for a fixed value of H . The excess free energy of the escaped chain is independent of N , as predicted in a blob picture by Eqs. (13) and (15). The best fit of the free energy for imprisoned state is given by Fimp(N,L,H) ∼ 1.944(2)NH−4/3 . (19) It is in perfect agreement with the previous estimation in Ref. [22], where the fugacity per monomer scales as µH − µ∞ ∼ 0.737H−4/3 and hence the free energy of the chains of size N in the imprisoned state scales as Fimp ∼ 0.737µ∞ NH −4/3 ≈ 1.944NH−4/3. Values of the excess free energy of the escaped chain, Fesc(N,L,H) are determined by the horizontal curves shown in Fig. 4. Results for L = 800, 1600, 3200 and 6400 are shown in Fig. 5, where we obtain Fesc(N,L,H) = 2.03(3)L/H . (20) As H becomes comparable to L, i.e. L/H ∼ O(10), the data points deviate slightly from the straight line, indicating that there are further finite size corrections for small L/H . In order to compare with the theoreti- cal prediction shown in Fig. 2a, we plot F (N,L,H)/N against L/N for various values of L and H in Fig. 6. In the escaped regime, these straight lines extrapolated to L/N = 0 are indeed described by Eq. (20) very well. With conventional Monte Carlo simulations it is difficult to estimate the partition sum precisely. With PERM we do have very precise estimates of Z(N,L,H), and there- fore we can obtain the lateral and the compression forces by differentiating the estimated free energy, Eqs. (19) and (20), with respect to L and H respectively. B. Average characteristics of polymer chains The most straightforward way to understand how the conformations of the polymer chains change as they un- dergo the escape transition is to estimate the end-to-end distance, < x >. In Fig. 7, we plot the end-to-end dis- tance per monomer< x > /N versus L/N . As long as the polymer chains are imprisoned the curves are horizontal, i.e. the degree of chain stretching is constant, and < x > increases linearly with N as shown in Ref. [22]. As the width of the strip, H , decreases, the chains are stretched more. As L/N decreases, we see that there is a jump 0.15 0.45 0 0.15 0.3 0.45 0.6 0.75 0.9 H = 5 H = 9 H = 17 H = 33 L = 800 L = 1600 L = 3200 L = 6400 FIG. 7: Average end-to-end distance divided by N , < x > /N , plotted against L/N for various values of L and H . The dashed line is < x > /N = L/N . 0 0.1 0.2 0.3 0.4 0.5 H = 9 H = 17 H = 33 H = 64 H = 128 FIG. 8: Average fraction of the imprisoned monomers, < Nimp > /N , plotted against L/N for L = 800 and various values of H . The jump becomes more prominent as H de- creases. The ends of the extrapolated straight dashed lines to L/N = 0 all approach zero. in each curve to another branch where < x >= L This means that the chain stretching is abruptly increased so that the chain reaches the edge of the strip, indicating that the transition to a partially escaped conformation is first-order like. This is in contrast to the smooth behav- ior predicted by the blob picture and shown in Fig. 2f. A jump-wise change in the chain stretching suggests a similar change in the average number of imprisoned monomers, < Nimp >. In Fig. 8, we plot the fraction of imprisoned monomers < Nimp > /N versus L/N for L = 800. As long as the chain is imprisoned, < Nimp > /N = 1. With the increase in the number of monomers, N , each curve indeed develops a jump and the jump becomes more pronounced for smaller H . For a fixed L and N → ∞, < Nimp > /N → 0 in accord with both theoretical 0 0.15 0.3 0.45 0.6 0.75 0.9 H = 5 H = 9 H = 17 H = 33 L = 800 L = 1600 L = 3200 L = 6400 FIG. 9: Average order parameter S plotted against L/N for various values of L and H . models. In Fig. 9, we plot the average of the order parameter S, versus L/N . We see clear jumps between the two states. For a given value of H , the order parameter behaves as a step function. The data seem to suggest that the mag- nitude of the jump decreases with the size of the system, L. A detailed analysis of the distributions of s will be presented in the next section. We will see that the ap- parent decrease in the jump is due to poor sampling of the escaped state. C. Transition points For our simulations the transition points can be deter- mined by analyzing three quantities: (1) free energy, (2) variance of the number of imprisoned monomers Nimp, and (3) variance of the end-to-end distance x. Since at the transition point the free energy of the imprisoned state is equal to the free energy of the escaped state, Eqs. (19) and (20) give the following relation between L, N and H at the transition point: = 1.04(2)H1/3 . (21) This shows that a polymer chain of sizeN can be confined in a strip by tuning the length L or the width H of the strip. The abrupt change scenarios of < x > /N and < Nimp > /N shown in Fig. 7 and 8 indicate a phase tran- sition but it is difficult to locate precisely the transition point. It is clear that all the chain configurations can be di- vided into two subsets: imprisoned and escaped. Far from the transition point, only one subset is important in defining the average characteristics, but in the vicinity of the transition point both subsets contribute, as shown 0.37 0.375 0.38 0.385 0.39 0.395 FIG. 10: Average fraction of the imprisoned monomers < Nimp > /N (solid line) and two partial contributions, < Nimp >1 /N due to confined configurations, and < Nimp >2 /N due to escaped configurations (dotted lines) as functions of L/N near the transition point (L/N)tr = 0.381 for L = 3200 and H = 17. in Fig. 10. The average < Nimp > can be rewritten as < Nimp > = NimpWN (C1) + NimpWN (C2) WN (C1) + WN (C2) = < Nimp >1 + < Nimp >2 (22) where C1 (C2) denotes the imprisoned (escaped) con- figurations, WN (C1) (WN (C2)) are the total weights of the chain for obtaining the configuration C1 (C2), and < . . . >α denotes the partial contributions due to the imprisoned configurations (α = 1) or the escaped config- uration (α = 2). Similarly, < N2imp >=< N imp >1 + < N imp >2 . (23) Finally, the variance of Nimp, σ 2(Nimp) =< N imp > − < Nimp > 2 can be expressed as follows, σ2(Nimp) = σ 1(Nimp)+σ 2(Nimp)−2 < Nimp >1< Nimp >2 where σ21,2(Nimp) =< N imp >1,2 − < Nimp >21,2. It is shown in the Appendix that the variances of partial con- tributions to the total number of imprisoned monomers, σ21(Nimp) and σ 2(Nimp) are much better suited for locat- ing the transition point than the full variance σ2(Nimp) because of the very asymmetric behavior of the latter. We present σ21(Nimp)/N and σ 2(Nimp)/N as functions of L/N in Fig. 11 for various values of H and L. It is clear that for a fixed width H , the peaks become sharper as the length L of the strip increases. The escape transition points are identified with the positions of the peaks of σ21(Nimp) and σ 2(Nimp) as determined by a curve fitting. Values of the transition points, (L/N)tr,1 and (L/N)tr,2 are the same to the third digit for fixed values of L and H , so the transition point is taken as an average of them, i.e. (L/N)tr = ((L/N)tr,1 + (L/N)tr,2)/2. Results of (L/N)tr obtained by this method are listed in Table II and presented in Fig. 12. A more detailed discussion are given in the Appendix. A similar method was also used for determining the transition point from the variance of the end-to-end distance x. Results are also listed in Ta- ble IIand presented in Fig. 12. All the values of (N/L)tr are plotted in Fig. 12 against H1/3. The best fit gives = 1.025(35)H1/3 (25) which is consistent with the estimation of Eq. (21) within the error bar. VI. LANDAU THEORY: DISTRIBUTION OF THE ORDER PARAMETER A. Analytical theory The approach based on the Landau free energy is per- fectly suited for analyzing the relevant states in the es- cape problem, including the metastable states. In the Landau theory, all the configurations are first subdivided into subsets associated with a given value of the order parameter, s , and summation is performed separately within each subset. The full partition function can be obtained then by integrating over the order parameter: Z = exp(−F ) = ds exp(−Φ(s)) (26) where Φ(s) is the Landau free energy function, i.e. the non-equilibrium free energy taken as a function of the order parameter. In the vicinity of the first order tran- sition point, the Landau free energy is expected to have two minima (one stable and the other metastable). Our analysis will be based on finding the metastable minima and the associated thermodynamic characteristics. The proper choice of the order parameter is not always ob- vious, nor are there any standard recipes for making it. One criterion is quite clear: the average value of the or- der parameter should allow one to distinguish between two phases. For a first-order transition, the average or- der parameter changes jump-wise. We require that the properly chosen order parameter changes continuously as the system evolves from a metastable state, through the transition state at the top of the barrier, and eventu- ally falls into the equilibrium minimum. We have shown earlier [13] that these criteria are satisfied if the order parameter is defined as the chain stretching in the con- fined coil state, s = r/(Na) where r is the instantaneous end-to-end distance of the chain of N monomers, or as the stretching of the stem only in the flower conforma- tion: s = L/(na), where n is the number of monomers in the stem. In analogy to the Gaussian case, the Landau function consists of two branches that have to be intro- duced separately. As the chain is in an imprisoned state, (a) (b) 1500 3000 4500 6000 7500 9000 0.1 0.2 0.3 0.4 0.5 H=129 L = 800 L = 1600 L = 3200 L = 6400 1500 3000 4500 6000 7500 9000 0.1 0.2 0.3 0.4 0.5 H=129 L = 800 L = 1600 L = 3200 L = 6400 FIG. 11: Variances of the number of imprisoned monomers divided by N , (a) σ21(Nimp)/N , for the imprisoned state, and (b) σ22(Nimp)/N , for the escaped state, plotted against L/N . The height of peaks increases with L for a fixed value of H . TABLE II: Values of the transition points, (L/N)tr, determined from the analysis of the variances σ 1(Nimp) and σ 1(x) for the imprisoned states, and from the variances σ22(Nimp) and σ 2(x) for the escaped states. (L/N)tr,Nimp (L/N)tr,x H L = 800 1600 3200 6400 L = 800 1600 3200 6400 9 0.4790 0.4766 0.4703 0.4649 0.4790 0.4766 0.4723 0.4674 17 0.3831 0.3846 0.3810 0.3754 0.3829 0.3836 0.3810 0.3754 33 0.3027 0.3060 0.3071 0.3036 0.3024 0.3078 0.3070 0.3036 65 0.2363 0.2412 0.2434 0.2455 0.2357 0.2409 0.2433 0.2455 129 0.1816 0.1880 0.1916 0.1932 0.1803 0.1877 0.1914 0.1930 the Landau free energy is directly expressed in terms of the distribution of the end-to-end distance. There exists no closed-form formula for such a distribution of confined chains with excluded volume interactions. However, the distribution of the gyration radius for 3d chains confined in a tube was studied analytically and numerically in [24]. It was proposed that the free energy of a confined chain with a given gyration radius rg can be presented as a sum of two terms: F (rg) = N Acα +B , (27) where c is the segment volume concentration expressed as a function of the gyration radius and the confinement geometry, α and δ are linked to the space dimension d and the Flory exponent ν by α = (νd− 1)−1 and δ = (1− ν)−1. The first term describes the concentration effects in the des Cloizeaux [25] form, the second term is the Pincus [26] scaling form of the stretching free energy, and A and B are model-dependent numerical coefficients of order unity. Instead of rg, we use the same ansatz, Eq. (27), to describe the end-to-end distance distribution by taking c = Na2/rH , α = 2, and δ = 4. The free energy of the chain in an imprisoned state as a function of s is hence given by Φimp(s) = N , s ≤ L . (28) Since we prefer to keep the basic scaling formula of the Landau free energy in order to provide predictions in a simple analytical form, here we are not going to consider the further logarithmic correction terms as shown in [24]. In the thermodynamic limit, the average value of the order parameter for the imprisoned state, Simp, is found by locating the minimum of Φimp(s), i.e. dΦimp(s)/ds = 0 at s = simpeq , and hence Simp = s eq = (A/2B) 1/6(a/H)1/3 . (29) The minimum of the Landau free energy gives the free energy for the imprisoned state at equilibrium Fimp = Φimp(Simp) = 3B N . (30) 1.5 3 4.5 6 7.5 L = 800, Nimp L = 1600, Nimp L = 3200, Nimp L = 6400, Nimp L = 800, x L = 1600, x L = 3200, x L = 6400, x FIG. 12: Transition points (N/L)tr versus H . The dashed line is (N/L)tr = 1.025(35)H 1/3 and gives the best fit of the data. Compared with Eq. (13), this is indeed the correct scal- ing of the free energy. The end-to-end distance at equi- librium is found as RN = NaSimp = (A/2B) 1/6(H/a)−1/3Na , (31) which is consistent with the result of the blob model, Eq. (12). As the chain is in an escaped state, the formula of the free energy function is identical to Eq. (28), but corrected for the fact that only the n monomers that are part of the stem contribute: Φesc(s) = n s−3 +Bs3 , s ≥ L .(32) The average value of the order parameter in the escaped state, Sesc, is found by locating the minimum of Φesc(s) and is given by Sesc = s eq = (A/B) 1/6(a/H)1/3 (33) Thus, the free energy of the escaped chain at equilibrium Fesc = Φesc(Sesc) = 2(AB) The transition point is found from the condition that the two minima of the Landau free energy function are of equal depth. Using Eqs. (30) and (34) we get It is interesting to calculate the size of jumps implied by the Landau theory in the order parameter, the im- prisoned monomers and the end-to-end distance at the N /Nimpr (3a/2 )(A/B) (H/a) L/N S/Sesceq 5/3 1/6 −1/3 (3a/2 )(A/B) (H/a) L/N 5/3 1/6 −1/3 FIG. 13: Based on the Landau theory, the theoretical predic- tions of the average values of (a) the fraction of imprisoned monomers Nimp/N , and (b) the order parameter S, are plot- ted against L/N . The chain is in an imprisoned state for L/N > (3a/25/3)(A/B)1/6(a/H)1/3, and in an escaped state for L/N < (3a/25/3)(A/B)1/6(a/H)1/3. transition. Using Eqs. (29) and (33) we immediately get the reduced jump of the order parameter Sesc − Simp = 1− 2−1/6 ≈ 0.1091 , (36) which is independent of H and the coefficients A and B. For an imprisoned state, < Nimp >= N by defini- tion, while for the coexisting escaped state with the same choice of H , L, and N we have only < Nimp >= L/Sesc monomers. From Eqs. (33) and (35), we obtain the rela- tive reduction in the number of imprisoned monomers ∆Nimp N − L/Sesc ≈ 0.055 . (37) This number has a simple meaning of the fraction of the chain escaping out of the confinement at the transition point. It is much smaller than 1/2 in the Gaussian chain model, but non-zero in contrast to the blob model. Fi- nally, the reduced jump of the end-to-end distance is ob- tained by combining Eqs. (31) and (35) ≈ 0.0572 . (38) Eqs. (36)-(38) show that the sizes of jumps in S, < Nimp > /N and RN/L are universal quantities. Results for the average order parameter S and the average frac- tion of imprisoned monomers < Nimp > /N predicted by the Landau theory are shown in Fig.13. Comparing with the numerical results shown in Fig. 8 and Fig.9, we see that the Landau theory a good qualitative agreement. The predicted free energy of the chain at equilibrium, Eqs. (30) and (34) follow the same scaling behavior as obtained by the MC simulations shown in Eqs. (19) and (20). This allows us to identify the numerical values of the constants A and B for our model: A ≈ 1.057 and B ≈ 0.975. B. Numerical comparisons Here we focus on the results of the Landau free energy of polymer chains partially confined in a strip of width H = 28 and of length L = 800, 1600, 3200, and 6400. Since PERM gives the possibility to estimate directly the partition sum and the properly normalized histograms, the Landau free energy as a function of s, Φ(N,L,H, s), is given by Φ(N,L,H, s) = − ln P (N,L,H, s) Z0(N) where P (N,L,H, s) = walks δs,s′ is the histogram of s, and the partition sum of the partially confined chains can be written as Z(N,L,H) = P (N,L,H, s) (40) in accordance with Eq. (26). In Fig. 14, we plot four sets of results of the Landau free energy per monomer Φ(N,L,H, s)/N versus the order parameter s for L = 800, 1600, 3200 and 6400. Since the transition point is near H1/3, the histograms are obtained for N/L = 3.05, 3.10 and 3.15 for each set. The predicted analytical re- sults of ΦP (s) = Φimp(s) for the imprisoned state and ΦP (s) = Φesc(s) for the escaped state, given by Eqs. (28) and (32) are also shown for comparison. On the left-hand side of the branch points, due to the finite-size effect, we see that the excess free energy for the imprisoned state (the minimum of the curve) at s = s eq,L converges to the predicted value (the minimum of the curve ΦP (s)) of polymer chains confined in an infinite strip at s = simpeq slowly as L increases but s eq,L is slightly larger than s as L → ∞. The difference between those curves corre- sponding to the different ratio N/L is almost invisible for a fixed value of L as predicted by Eq. (28). On the right-hand side of the branch points, we see that only those curves for L = 800 finally develop a parabola-like behavior with fluctuations and they are more concave than those curves predicted by Eq. (32). It shows that PERM has difficulties to sample configurations in the es- caped regime as L increases and gives an explanation why we should not trust the size of those jumps that appear in Fig. 9 too much. However, one can easily overlook the existence of two minima in such a delicate situation. With PERM, at least we are able to give evidence for this two minimum picture of the first-order like transi- tion. We also see that additional finite-size correction terms should be taken into account for the theoretical predictions in Eqs. (28) and (32). 0.023 0.024 0.025 0.026 0.027 0.25 0.3 0.35 0.4 0.45 N/L=3.05 N/L=3.10 N/L=3.15 L= 800 L=1600 L=3200 L=6400 ΦP(s)/N FIG. 14: The Landau free energy divided by N , Φ(N,L,H, s)/N , plotted against s for various values of L and H = 28. The predicted Landau free energy functions, ΦP (s) = Φimp, Eq. (28), in the imprisoned regime and ΦP (s) = Φesc, Eq. (32), in the escaped regime are also plotted (dashed lines). Taking the results for L = 6400 as a reference, we plot the same data but shift all other curves by some constants, c0,L = −0.00235, −0.00109, −0.00044 for L = 800, 1600, and 3200 to make the three branch points for N/L = 3.05, 3.10, and 3.15 coincide with each other in Fig. 15. According to the prediction by Eq. (28), we should expect that the four curves for different values of L overlap with each other in the imprisoned regime. In fact, it is not the case but the difference between these curves decreases as L increases, and finally they will converge to one curve as L becomes very large. In the escaped regime, surprisingly, we see that those curves corresponding to different L all overlap with each other for a fixed ratio of N/L as predicted by Eq. (32). Although the lack of data for larger L, precludes very strong conclusions, we may assume that these curves all show the same behavior as the curve for L = 800, and do further analysis. In order to determine the transition point and extract an accurate value for the jump in the order parameter from simulations, we use two parabolic functions gimp(s) and gesc(s) to fit the numerical data in the imprisoned and escaped regimes, respectively: gimp(s) = a1,L(s− simpeq,L) 2 + c1,L (41) gesc(s) = a2(s− sesceq,L)2 + c2 + b2 where a1,L, c1,L, a2, c2, b2, s eq,L, and s eq,L are deter- mined by curve fitting, and results are shown in Table III and Fig. 15. From the condition of equal depth of minima gimp(s = s eq,L) = gesc(s = s eq,L), (43) 0.023 0.024 0.025 0.25 0.3 0.35 0.4 0.45 N/L=3.05 N/L=3.10 N/L=3.15 L = 6400 L = 3200 L = 1600 L = 800 FIG. 15: The Landau free energy divided by N , Φ(N,L,H, s)/N , plotted against s for various values of L and H = 28. The two minima of Φ(N,L,H, s)/N are determined by fitting g(s) = gimp(s), Eq. (41), in the imprisoned regime, and g(s) = gesc(s), Eq. (42), in the escaped regime, going through those lower points around the two minima, respec- tively. 0.06 0.08 0.12 0.14 0.16 0.18 0 0.0003 0.0006 0.0009 0.0012 0.0015 curve fitting estimation Eq. (36) FIG. 16: The reduced jump of the order parameter ∆S/Sesc plotted against L−1. we obtain the transition points (N/L)tr = 3.13(2), 3.10(2), 3.09(1), and 3.08(1) for L = 800, 1600, 3200, and 6400, respectively, which are in perfect agreement with the results given by free energy, Eq. (21), and the results given by the variance of the end-to-end distance and the imprisoned monomers, Eq. (25). The values for the reduced jump of the order parame- sesceq,L − s sesceq,L obtained by the curve fitting are plotted in Fig 16 against L−1 together with the direct estimates in the simulations TABLE III: Results of the coefficients a1,L, c1,L, a2, c2, b2, eq,L, and s eq,L for the curve fitting in Fig. 15. L a1,L c1,L s eq,L a2 c2 b2 s 800 0.6281 0.02277 0.2945 0.6518 0.0497 -0.0086 0.3510 1600 0.6336 0.02303 0.2986 3200 0.6675 0.02313 0.3007 6400 0.6872 0.02320 0.3017 and the prediction by the analytical theory, Eq. (36). We see that ∆S/Sesc decreases as L increase. As L → ∞, it remains finite and the value is slightly larger than the predicted value by the analytical theory. However, in view of the numerical uncertainties of our curve fitting we consider that the predictions of the analytical theory and the results by the MC simulations agree with each other quite well. VII. SUMMARY AND DISCUSSION In this paper we attack the problem of the 2d-escape transition by combining several approaches. We first compare two simple pictures of the transition predicted for Gaussian chains and by a blob model. This compar- ison is useful from a general pedagogical point of view since the two models are in a sense complimentary: each captures some essential features of the phenomenon while failing in some other aspects. Both models are attrac- tive because of their clarity, and although mathemati- cally simple, lead to non-trivial results including finite- size effects in a phase transition. The third approach that was proposed in this paper attempts at incorporat- ing the excluded volume effects in the framework of the Landau theory. We were not able to present an exact theory since it would require a detailed understanding of the end-to-end distribution of confined self-avoiding chains. To the best of our knowledge this problem is still not well explored. The simulations presented allowed us to evaluate the transition condition Eq. (21) which rep- resents the binodal line in the (H,L/N) plane. It is of interest to extend the simulations in order to locate the spinodal lines where one of the states looses stability, and to construct the full phase diagram. It is also possible to explore the properties of metastable states and their life- times controlled by the barrier heights. It is clear from the results on the distribution of the order parameter, Fig. 14, that the PERM algorithm experiences difficul- ties with sampling the configurations belonging to the escaped state, especially for long chains. The escaped branch of the distribution is cut-off quite sharply, which means that the important set of configurations charac- terized by larger stretching degree in the stem is vastly underrepresented. This is a generic problem that one en- counters when dealing with first-order transitions when the properties of the phases differ significantly. In our case, the PERM algorithm based on chain growth tech- nique is perfectly tuned to generate homogeneous config- urations of imprisoned chains but fails with strongly in- homogeneous escaped configurations. It is worth noting that a naive determination of the jumps in the average order parameter would have lead one to a wrong con- clusion that the jump disappears in the thermodynamic limit. Again, we expect this to be a generic problem when simulating weak first-order transitions. The most reliable analysis of the nature of the transition would require a detailed examination of the order parameter distribution. Acknowledgements We are grateful to the Deutsche Forschungsgemein- schaft (DFG) for financial support: H.-P.H. was sup- ported under grant NO SFB 625/A3, while L.I.K. and A.M.S. received partial support under grants NO 436 RUS 113/863/0 and RFBR 05-03-32003-a. H.-P.H. thanks P. Grassberger and W. Paul for very helpful dis- cussions. Appendix In this appendix, we discuss the finite-size behavior in the fluctuations in the number of imprisoned monomers Nimp in more detail. Following the technique of finite-size scaling analysis for first-order transitions as described in [27], we write down the probability distribution of the fraction of imprisoned monomers m = Nimp/N in the two-state model P (m) = δ(m−m1) e(t−ttr)a e(t−ttr)a + e−(t−ttr)a (m−m2) e−(t−ttr)a e(t−ttr)a + e−(t−ttr)a .(45) The first term accounts for the imprisoned state with m strictly equal to m1 = 1, while the second term describes the distribution ofm in the escaped state in the Gaussian approximation with the equilibrium average of m equal to m2 and dispersion σ0; t is the control parameter, ttr is its critical value at the transition point, and P (m) is normalized, P (m)dm = 1 (46) At the transition point t = ttr, P (m) = δ(m−m1) + (m−m2) which obeys the “equal-weight rule”, while for t 6= ttr the relative weight of the two states is exp [2(t− ttr)a]. The constant a−1 describes the range of t over which the transition is smeared out. For the Gaussian approx- imation to be meaningful the dispersion of m in the es- caped state, σ0, must be small compared to the difference ∆m = m1 − m2. Taking t = L/N and using Eqs. (30), (32), and (34) of the Landau theory, one expects the fol- lowing scaling: a−1 ∼ H2/3/L and σ20 ∼ H/L. Since the probability density is a sum of two contribu- tions, P (m) = P1(m) + P2(m), the k-th moment of m is defined by < mk >= mkP (m)dm =< mk >1 + < m k >2 (48) where < mk >1,2= mkP1,2(m)dm. Therefore, the first and second moment are given by < m >= m1p1 +m2p2 , (49) < m2 >= m21p1 + (m 2 + σ 0)p2 , (50) here p1 = e (t−ttr)a/2 cosh [(t− ttr)a] is the relative weight of the imprisoned state, and p2 = 1 − p1 is the relative weight of the escaped state. Instead of a δ-function singularity at t = ttr, the vari- ance of the fraction of imprisoned monomers in a finite system becomes < m2 > − < m >2= p1p2(∆m)2 + p2σ20 (51) which shows a smooth asymmetric peak close to t = ttr of approximate height ∆m2+σ20 . Here ∆m is the relative reduction in the number of imprisoned monomers at the transition point, for which the analytical Landau theory predicts a value of 0.055, see Eq. (37). The first term in Eq. (51) is symmetric with respect to the transition point since p1p2 = 1/(4 cosh 2 [(t− ttr)a]). The second term, however, is asymmetric, as it describes the intrinsic fluc- tuations in the escaped state. The resultant asymmetry is clearly seen in Fig. 17. We conclude that the full vari- ance of m is ill suited for a precise determination of the transition point. The situation is quite different if we analyze the vari- ances calculated with the partial probability densities P1(m) and P2(m) restricted to the imprisoned (escaped) configurations. In the simulations, the product Nσ21,2 was calculated. The variance due to the imprisoned con- figurations only (with m1 = 1) gives a perfectly symmet- ric curve as a function of the control parameter: Nσ21(m) = Nm 1p1p2 = 4 cosh2 [(t− ttr)a] with the peak value ofNtr/4 ≈ LH1/3/4. Numerical data presented in Fig. 18 supports this prediction with very high accuracy. The variance due to escaped configura- tions is somewhat modified by the intrinsic fluctuations 0.01 0.02 0.03 0.04 0.05 0.06 0.37 0.375 0.38 0.385 0.39 0.395 ∆Nimp/N 0.1 σ1(m) 0.1 σ2(m) 1-<m> FIG. 17: The square root of the variance σ1(m) for the im- prisoned states, σ2(m) for the escaped chains, 1− < m >, σ(m) of the chain either in an imprisoned state or in an es- caped state, and the difference ∆σ = σ1(m)− σ2(m) against L/N . 1000 2000 3000 0.37 0.375 0.38 0.385 0.39 L = 3200, H = 17 Eq. (53) A (L,H)1 (L,H)1 FIG. 18: Variance due to the imprisoned configuration multiplied by N , Nσ21(m), plotted against L/N for L = 3200 and H = 17. The solid curve is the best fit of Eq. (52), Nm21/4cosh 2 [(t− ttr)a], with the height of the peak A1(L,H) = N(m1) 2/4 ≈ 2099.74, the FWHM Γimp(L,H) ≈ 1.7627/a = 0.0035, and the position of the peak ttr,1 = (L/N)tr,1 = 0.3810. The FWHM are given by the distance between points on the curve shown at which the correspond- ing height reaches half height of the peaks (half maximum). in the escaped state Nσ22(m) = N(m 2p1p2 + p2σ ≈ Nm22p1p2 = N(1−∆m)2 4 cosh2 [(t− ttr)a] but in contrast to Eq. (51), the asymmetric term is al- ways negligible. Indeed, the coefficient with the sym- metric term, m22 = (1 − ∆m)2 is close to 1 while both quantities (∆m)2 and σ20 are quite small. In Fig. 17, 0.01 0.02 0.03 0.04 0 0.0003 0.0006 0.0009 0.0012 0.0015 0.0018 H = 129 H = 65 H = 33 H = 17 H = 9 FIG. 19: FWHM Γα(L,H) for the imprisoned state (α = 1) and for the escaped state (α = 2) against L−1. The dashed curves are a1,H(H/L) + b1,H(H/L) 2 and give the best fit of the data. Values of a1,H and b1,H are listed in Table IV. TABLE IV: Results of the coefficients a1,H , b1,H , a2,H , c2,H and d2,H for the curve fitting in Fig. 19 and 20. H a1,H b1,H a2,H c2,H d2,H 9 0.8221 0.7841 0.9940 0.2131 2.2802 17 0.6845 -2.0403 0.9845 0.0961 0.4540 33 0.5600 -2.3531 0.9725 0.0430 0.0570 65 0.4301 -1.3774 0.9545 0.0173 0.0095 129 0.3185 -0.7332 0.9220 0.0070 0.0017 we plot the full dispersion σ(m) (the square root of the full variance) that includes contributions from all config- urations, and partial dispersions σ1(m) and σ2(m) due to imprisoned and escaped configurations separately, the difference ∆σ = σ1(m) − σ2(m), as well as the aver- age fraction of escaped monomers, 1− < m >, as func- tions of L/N . It is clear that the full curve is strongly asymmetric in contrast to partial dispersion curves, in good agreement with the theoretical description above. On the other hand, the curve of ∆σ shows the same be- haviour as the curve of σ(m) near the transition point, and the heights of these two peaks correspond to the half size of the jump ∆m = m1 − m2 = ∆Nimp/N . By fitting the partial variances Nσ21(m) and Nσ as functions of L/N according to Eqs. (52) and (53), respectively, we obtain the full width at half-maximum (FWHM), Γ(L,H) = 2arccosh( 2)/a, the height of the peak Aα(L,H) = Nm α/4, and the transition point ttr,α = (L/N)tr,α for α = 1 (imprisoned configurations) and α = 2 (escaped configurations). One example of the curve fitting for L = 3200 and H = 17 is shown in Fig. 18. Note that the peak height and the transition point are related to the theoretical prediction A1(L,H) ≈ LH1/3/4 with very high accuracy. (a) (b) 0.0005 0.001 0.0015 0.002 0.0025 0 0.0003 0.0006 0.0009 0.0012 0.0015 H = 9 H = 17 H = 33 H = 65 H = 129 0.0005 0.001 0.0015 0.002 0.0025 0.003 0 0.0003 0.0006 0.0009 0.0012 0.0015 H = 9 H = 17 H = 33 H = 65 H = 129 FIG. 20: Inverse of the height of the peaks for the imprisoned state (a) A−11 (L,H), and for the escaped state (b) A 2 (L,H), plotted against L−1. The dashed curves are (a) a2,H(4H −1/3/L) and (b) c2,H(H/L) + d2,H(H/L) 2, and give the best fit of the data. Values of a2,H , c2,H , and d2,H are listed in Table IV. 0.05 0 0.03 0.06 0.09 0.12 0.15 H = 33 H = 65 H = 129 0.54H/L+0.058 Eq. (37) FIG. 21: The relative reduction in the number of imprisoned monomers ∆m, plotted against H/L. Results of Γα(L,H), Aα(L,H) for α = 1 and for α = 2, and ttr,α are shown in Fig. 19, 20 and 12. In Fig. 19, we see that the full widths Γα(L,H) for α = 1 and for α = 2 are overlapped with each other, and Γα → 0 as 1/L → 0 by fitting the data using a1,H(H/L) + b1,H(H/L) 2. In Fig. 20, the inverse of the height A−11 (L,H) → 0 as 1/L → 0 by fitting the data using a1,H(4H−1/3/L) and c1,H(H/L)+d1,H(H/L) 2. Since Γα → 0, and A−1α → 0 as 1/L → 0, i.e. a delta function, a sharp phase transition occurs in the thermodynamic limit. It is a strong indica- tion [27] that the transition is first-order like. 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An end-grafted flexible polymer chain in 3d space between two pistons undergoes an abrupt transition from a confined coil to a flower-like conformation when the number of monomers in the chain, N, reaches a critical value. In 2d geometry, excluded volume interactions between monomers of a chain confined inside a strip of finite length 2L transform the coil conformation into a linear string of blobs. However, the blob picture raises questions on the nature of this escape transition. To check the theoretical predictions based on the blob picture we study 2d single polymer chains with excluded volume interactions and with one end grafted in the middle of a strip of length 2L and width H by simulating self-avoiding walks on a square lattice with the pruned-enriched-Rosenbluth method (PERM). We estimate the free energy, the end-to-end distance, the number of imprisoned monomers, the order parameter, and its distribution. It is shown that in the thermodynamic limit of large N and L but finite L/N, there is a small but finite jump in several average characteristics, including the order parameter. We also present a theoretical description based on the Landau free energy approach, which is in good agreement with the simulation results. Both simulation results and the analytical theory indicate that the 2d escape transition is a weak first-order phase transition.

What is the order of 2D polymer escape transition? Hsiao-Ping Hsu and Kurt Binder Institut für Physik, Johannes Gutenberg-Universität Mainz D-55099 Mainz, Staudinger Weg 7, Germany Leonid I. Klushin American University of Beirut, Department of Physics, Beirut, Lebanon Alexander M. Skvortsov Chemical-Pharmaceutical Academy, Prof. Popova 14, 197022 St. Petersburg, Russia. (Dated: August 11, 2021) An end-grafted flexible polymer chain in 3d space between two pistons undergoes an abrupt transition from a confined coil to a flower-like conformation when the number of monomers in the chain, N , reaches a critical value. In 2d geometry, excluded volume interactions between monomers of a chain confined inside a strip of finite length 2L transform the coil conformation into a linear string of blobs. However, the blob picture raises questions on the nature of this escape transition. To check the theoretical predictions based on the blob picture we study 2d single polymer chains with excluded volume interactions and with one end grafted in the middle of a strip of length 2L and width H by simulating self-avoiding walks on a square lattice with the pruned-enriched-Rosenbluth method (PERM). We estimate the free energy, the end-to-end distance, the number of imprisoned monomers, the order parameter, and its distribution. It is shown that in the thermodynamic limit of large N and L but finite L/N , there is a small but finite jump in several average characteristics, including the order parameter. We also present a theoretical description based on the Landau free energy approach, which is in good agreement with the simulation results. Both simulation results and the analytical theory indicate that the 2d escape transition is a weak first-order phase transition. I. INTRODUCTION A phenomenon that was called escape transition oc- curs upon progressive squeezing an end-grafted polymer chain between two pistons and has attracted great inter- est [1-16]. At weak deformation the chain is compressed uniformly into a relatively thick pan-cake conformation. Beyond certain critical compression, the chain configu- ration changes abruptly. One part of the chain forms a stem stretching from the grafting point to the piston edge, while the rest of the segments form a coiled crown outside the piston, thus escaping from the region under- neath the piston. An abrupt change from one state to another implies a first order transition. Various aspects of this problem were investigated: The escape transition of compressed polymer mushrooms in 3d space was inves- tigated thoroughly by scaling theory [1], numerical cal- culations [2, 3, 4], and computer modeling under good solvent [5], and theta solvent [6] conditions. The escape transition of star polymers was discussed in [7], and the escape transition of di-block-copolymers was considered in [8]. The influence of the curvature of the pistons was investigated in [9, 10], and the effect of adsorption be- tween the polymer chain and the surface of the piston was considered in [11]. A comparison between Monte Carlo simulations and experimental results by atomic force-electrochemical microscopy was recently presented in [12]. A rigorous analytical theory for the equilibrium and kinetic aspects of the escape transition for a Gaus- sian chain was constructed in [13, 14]. Metastability ef- fects, negative compressibility and the nonequivalence of the escape transition in two conjugate ensembles were analyzed for the same model in [15, 16]. The reason for studying the escape transition is that it gives the possibility to understand the phenomenon of a very unconventional phase transition. The concept of a phase transition requires taking a thermodynamic limit. For standard low-molecular weight systems as well as for macromolecular systems in condensed bulk matter finite size effects are usually negligible. In contrast to that, phase transitions at the level of a single macromolecule e.g., the coil-globule transition [17], or polymer adsorp- tion at an interface [18, 19] - do not have any analogies in the physics of low molecular mass systems. A sin- gle macromolecule always consists of a finite number of monomers N : computer modeling rarely deals with N larger than 104 so that finite-size effects in the single- molecule phase transitions are the rule rather than the exception. The situation is much more complicated in the case of the escape transition. It was shown [1] that the escape transition point (critical compression) depends on the relation between the chain length Na (a is the dis- tance between neighboring monomers) and the piston ra- dius L. Therefore, to analyze the escape transition in the thermodynamic limit, it is necessary to take both Na → ∞ and L → ∞ but Na/L = const. The physics of phase transitions is generally known to be strongly affected by the spatial dimensionality. For phase transitions at the level of a single macromolecule the spatial dimensionality is important because excluded volume effects are especially large for polymeric chains in two dimensions [20]. An ideal chain without excluded http://arxiv.org/abs/0704.1857v2 volume interactions retains a Gaussian coil conformation with the lateral size ∼ N1/2 even when it is confined in a 2d strip. There is a very pronounced difference between this state and the partially escaped state with a strongly stretched stem of size L ∼ Na. On the contrary, a 2d polymer chain with excluded volume interactions con- fined in a strip is already strongly elongated with the size ∼ Na . It is not clear whether the difference between the confined and the escaped states is large enough to result in a phase transition. To analyze the excluded volume effects of the escape transition, we study a flexible polymer chain containing N links of length a (N monomers) grafted in the middle of a strip of length 2L and width H , under good solvent conditions. Schematic drawings of a polymer chain in an imprisoned state and in an escaped state are shown in Fig. 1. First, we summarize the results of the escape transition for a 2d Gaussian ideal chain in Sec. II, and then we give the theoretical predictions based on the blob picture in Sec. III. In an experimental setup, the escape transition is driven by changing the piston separation H , while in a blob picture, the escape transition is studied by changing the chain length N or the strip length L at fixed H which is also the size of a blob. Comparisons between the escape transition behavior of the Gaussian chain model and that of the blob model in the thermo- dynamic limit are given in Sec. IV. In Sec. V we present our results from Monte Carlo (MC) simulations with the pruned-enriched Rosenbluth method (PERM) [21] In Sec. VI we provide a theoretical description based on the Landau free energy approach which is compared with MC results. A summary and discussions are given in Sec. VII. Detailed analyses of the variances of the im- prisoned monomers are given in the Appendix. II. ESCAPE FOR A 2D GAUSSIAN CHAIN For the escape transition of a Gaussian chain, a closed- form of the exact partition function was obtained earlier in [14]. The asymptotic form of the free energy F has two branches: = Fimp imprisoned state = Fesc escaped state . (1) Here d is the dimensionality of space and a factor of kBT is absorbed in the free energy throughout the paper. It was shown by direct numerical comparison that the sim- ple asymptotic expressions provide a very accurate de- scription. The two branches meet at the transition point which is given by , with d = 2 . (2) The average lateral forces acting on the grafting point are obtained by ∂F (N,L,H) n "blobs" crown FIG. 1: Schematic drawings of a flexible polymer chain of length N grafted in the middle of the strip of length 2L and width H , in a blob picture: (a) As the chain is imprisoned inside the strip, it forms a sequence of nb blobs. (b) As the chain length is larger than the maximum chain length N∗ of a chain in an imprisoned state, the chain partially escapes from the strip and forms an escaped state. A escaped state consists of a “stem” containing N∗ monomers and a “crown” containing N −N∗ monomers. which is related to the work required to pull a chain end into confined space by a unit distance. The average com- pression force fH = − ∂F (N,L,H) is related to the work of compression. It was shown earlier [14] that the average fraction of impris- oned monomers is given by the derivative Nimp = ∂F (N,L,H)/∂u of the free energy with respect to the effective confining potential u = (πa/2H) The average end-to-end distance per monomer is: N−1/2 imprisoned state +N−1/2 (1−Nimp/N)1/2 escaped state The Landau order parameter s for the escape transition was introduced in [14] as a stretching degree of the chain rN/N for the imprisoned state and as the stretching de- gree of the stem L/nimp for the escaped state. Note that the order parameter is defined for an instantaneous configuration with a given end-to-end distance rN and the number of imprisoned monomers nimp which may be quite different from the equilibrium average values R = 〈rN 〉 and Nimp = 〈nimp〉. The Landau free energy is defined as a function of the order parameter. The Lan- dau free energy for the imprisoned state is just the free energy of a coil as a function of its free end position, which has a standard parabolic form: Φimp(s) = N . (6) At s = simpeq = 0, the system is in equilibrium and the corresponding equilibrium free energy, i.e., the depth of the minimum, is Φimp(0) = (N/2d)(πa/H) 2. For the escaped state, Φesc(s) = . (7) The position of the minimum is at s = sesceq = πa/dH, and the corresponding depth is Φesc(s eq ) = πL/H . The binodal is determined by the condition that the two min- ima are equally deep, i.e. Φimp(s eq ) = Φesc(s eq ), which leads to the transition point described by Eq. (2). At the transition point, the average order parameter S =< s > jumps from simpeq to s eq . The Landau function allows one to analyze metastable states and to define the two lines where either one of the metastable minima vanishes [16]. Theoretical predictions for the escape transition of a Gaussian chain are summarized in Table I and shown in Fig 2. Detailed discussion and comparison with the prediction from the blob picture for 2d polymer chains are given in Sec. IV. III. BLOB PICTURE OF A 2D ESCAPE A free chain in d = 2, has an average end-to-end dis- tance given by (a is the distance between neighboring monomers) [17-20] RF = aN 3/4 (8) here the prefactors of order unity are neglected through- out. Based on the blob picture we have a cigar of blobs (nb blobs in total) in the confined situation. Thus the average end-to-end distance is R = nb(2rb) = nbH (9) where rb is the blob radius. Within a blob, self-avoiding walks (SAW) statistics holds, so if g monomers belong to a blob H = ag3/4 = 2rb , g = (H/a) 4/3 . (10) Since every monomer of a chain in an imprisoned state must be in a blob we furthermore have N = gnb = nb(H/a) 4/3 , nb = N(H/a) −4/3 . (11) This yields, together with Eq. (9), the formula for the end-to-end distance R/a = N(H/a)−1/3 (12) If H is of the same order as RF, Eq. (8), one finds R/a = N3/4, giving a smooth crossover to mushroom behavior, as expected. The free energy excess of the chain in an imprisoned state (in units of kBT ), relative to an unconfined mush- room, is simply the number of blobs, nb , Fimp = nb = N(H/a) −4/3 (13) We now define N = N∗ as the maximum chain length of an imprisoned chain. Then for N > N∗ the chain con- sists of a “stem” containing N∗ imprisoned monomers and an escaped “crown” comprising the rest N − N∗ monomers (Fig. 1b). Thus, we find N∗ from the con- dition that R becomes equal to L at the transition point, using Eq. (12) N∗ = (L/a)(H/a)1/3 . (14) Since the free energy of an unconfined mushroom is taken as zero reference point, the “crown” does not contribute to the excess free energy of the escaped chain, which is hence due to the stem only: Fesc = N ∗(H/a)−4/3 = L/H . (15) The average end-to-end distance of the escaped chain (in axial direction parallel to the confining boundaries) hence becomes Resc = L+ a(N −N∗)3/4 . (16) Equations (13) and (15) show that the free energy as a function ofN for fixed H and L, consists of two branches, i.e., Fimp for the imprisoned state (N < N ∗) and Fesc for the escaped state (N > N∗), meeting at N = N∗. The lateral and the compression forces are obtained from the free energy by using Eqs. (3) and (4). We use the same definition of the order parameter s as that for the Gaussian chain, so the average order parameter S S =< s >= R/Na imprisoned state L/N∗a = (H/a)−1/3 escaped state From Eqs. (12), (14), and (17), we find that the order parameter does not show any discontinuity at the tran- sition, but simply stays constant, i.e. S = (H/a)−1/3. Results of the theoretical predictions are listed in Table I and also shown in Fig. 2. IV. COMPARISON OF THE GAUSSIAN AND BLOB PICTURES In the thermodynamic limit N → ∞, L → ∞, L/N remains as a nontrivial variable along with H . In Fig. 2, theoretical predictions of the Gaussian chain model and of the blob model are shown by dotted and solid lines, respectively. The strip width, H , is fixed, and the ratio L/N is varied. The chain is in an imprisoned state if L is larger than the corresponding critical value, the Gaussian chain model or for the blob model, and in the escaped state for for L ]. All curves show only the scaling behavior disregarding nu- merical coefficients of order one, and the bond length a is taken as a unit length. In both models, the free energy per monomer is given by piecewise linear functions of L/N . For the escaped ~LN( ) f /NH −1/2 N N /N FIG. 2: Theoretical predictions for various averaged chain characteristics plotted against L/N at constant strip width H : (a) the free energy per monomer F/N , (b) the lateral force fL, (c) the compression force per monomer fH/N , (d) the fraction of imprisoned monomers Nimp/N , (e) the order parameter S, (f) the end-to-end distance per monomer R/N , . Gaussian model results are shown by dotted lines, blob model results - by solid lines. The chain is in an imprisoned state for L ], and in an escaped state for L ] for the Gaussian chain model [blob picture]. TABLE I: Theoretical predictions for the average values of the free energy per monomer F/N , the lateral force fL, the compression force per monomer fH/N , the fraction of imprisoned monomers Nimp/N , the order parameter S, and the reduced end-to-end distance R/Na based on the Gaussian chain model and the blob picture. Characteristics Imprisoned Escaped of the chain state state Gaussian Blob Gaussian Blob )2 ( a )4/3 π fL 0 0 )7/3 π 1 1 2 )1/3 L )1/3 π N−1/2 N−1/2 N−1/4 state, the slope of F/N vs. L/N is the same up to numer- ical coefficients of order 1, and scales as H−1. This slope has the meaning of the lateral force acting on the grafting point. It is proportional to the inverse size of a blob which is defined purely by confinement width H irrespective of whether excluded volume interactions are present or not. The free energy per monomer in the imprisoned state is independent of L/N , and scales as the inverse number of monomers in one blob, g−1 = (a/H) , where ν is the Flory exponent. The transition points in the two models may be quite far apart since they scale differently with H . In the setup where L/N is fixed and the piston sepa- ration is decreasing, the blob model predicts the transi- tion to happen at lower compression, as compared to the Gaussian chain. In both models the lateral force jumps from H−1 to zero at the respective transition point. As for compression forces, they are strongly affected by ex- cluded volume interactions (Fig. 2c). The difference in the plateau values for the imprisoned state reflects the lower compressibility of the self-avoiding walk as com- pared to the Gaussian chain. This corresponds directly to the plateau values of the free energy in Fig. 2a. In the Gaussian chain model, the compression force jumps at the transition point by a factor of 2, while the jump in the blob model is by a factor of 4/3. Figures 2a - 2c suggest that the behavior of both models is fundamentally the same and characteristic of a first-order phase transition. However, the next 3 graphs demonstrate qualitatively different predictions of the Gaussian and the blob models. For the Gaussian chain model, a transition from the confined state to the escaped state is accompanied by a jump in the average number of imprisoned monomers. At the transition point, one half of the total number of monomers are ejected outside to form a crown, see Fig. 2d. In contrast to that, the blob model predicts a smooth change without a jump. This directly affects the behavior of the average order parameter: while in the Gaussian chain model there is a pronounced jump, the blob model suggests that the order parameter does not change at all. The large constant value of S in the blob model is due to the cigar-shape conformation of the chain in an imprisoned state which is identical to the shape of the stem in the escaped state. The behavior of the average end-to-end distance is also qualitatively different for the two models. The plateau value for the imprisoned Gaussian chain, R/N = N−1/2, is characteristic of the ideal coil, while for the elongated cigar, this ratio is N -independent. The linear part of the curves corresponding to the escaped state simply repre- sents the dominant contribution of the stem to the overall end-to-end distance, R ≈ L irrespective of the model. At very low values of L/N corrections due to the crown size become large, as shown in Fig 2f. The size of the Gaus- sian chain demonstrates a jump at the transition point consistent with a strong conformation change accompa- nying the first-order transition. In the blob model, the chain size does not have any jump. It is clear that the predictions of the blob model pre- sented in Fig. 2 contain some internal contradictions. On the one hand, the picture of the two branches of the free energy meeting at some angle suggests a first order tran- sition. The jumps in the lateral and compression forces are a simple consequence of that. On the other hand, nothing dramatic happens to the chain conformation in the blob picture: the change from a completely confined state to a state with a small escaped tail is continuous, as evidenced in Figs 2e and 2f. The presence of discontinu- ity in the slope of the fraction of imprisoned monomers suggests that the transition should be classified as second order. A crucial question to ask is whether one can identify two distinct separate states with a bimodal distribution of some appropriate order parameter. This we address by employing a Monte-Carlo simulation of 2d self-avoiding chains undergoing the escape transition. V. MONTE CARLO SIMULATION Single polymer chains grafted in the middle of a strip of length 2L and width H are described by SAWs of N steps FIG. 3: Schematic drawing of a polymer chain growing as a self-avoiding walk inside a finite strip and grafted at (x = 0, y = 0). Monomers are allowed to sit on the lattice sites except for the lattice sites representing the walls {−L ≤ x ≤ L , y = 0} and {−L ≤ x ≤ L , y = H}. The first monomer is attached with a bond to the grafting site marked by a cross. Lengths are measured in units of the lattice spacing. 1000 0 30 60 90 120 150 N3/4/H H = 17 H = 33 H = 65 H = 129 1.944 NH-4/3 FIG. 4: (a) Free energy relative to a free chain, F (N,L,H) = − ln[(Z(N,L,H)/Z0(N)], plotted against N 3/4H−1 for L = 6400 and H = 17, 33, 65, and 129. The dashed curve is Fimp(N,L,H) = 1.944NH −4/3 and gives the best fit of the data. on a square lattice between two hard walls with distance H as shown in Fig. 3. Monomers are supposed to sit on lattice sites but they are forbidden to sit on the two walls, i.e. {−L ≤ x ≤ L , y = 0} and {−L ≤ x ≤ L , y = H}. For our simulations we employ PERM and PERMwith k- step Markovian anticipation as described in Ref. [21, 22]. PERM is a biased chain growth algorithm with popula- tion control. Polymer chains are built like random walks by adding one monomer at each step. Thus it has the advantage of estimating the partition sum and counting the imprisoned monomers directly. We simulate 2d SAWs starting at the grafting point of the strip of length L = 800, L = 1600, 3200, and 6400. The width of the strip is varied from H = 5 to H = 129. Depending on the chosen sizes of L andH , the total chain length is varied from 2500 to 50000 in order to cover the transition region. 1000 10 100 1000 L = 800 L = 1600 L = 3200 L = 6400 2.03 L/H FIG. 5: The log-log plot of the excess free energy of the escaped chain, Fesc(N,L,H), plotted against L/H for vari- ous values of L and H . The dashed line is Fesc(N,L,H) = 2.03L/H and gives the best fit of the data. 0.02 0.04 0.06 0.08 0 0.2 0.4 0.6 0.8 1 H = 17 H = 28 H = 33 H = 65 L = 800 L = 1600 L = 3200 L = 6400 2.03 L/(NH) FIG. 6: Free energy relative to a free chain divided by N , F (N,L,H)/N , plotted against L/N for various values of L and H . The dashed line extrapolated to zero is 2.03L/(NH). A. Free energy Let us first discuss the scaling behavior of the free en- ergy. The partition sum of a free SAW in infinite volume for N → ∞ scales as Z0(N) = µ γ−1 (18) with µ∞ being the critical fugacity per monomer, and with γ = 43/32 being a universal exponent [18]. As chains are still confined in a strip of width H , one should expect the scaling laws of the excess free energy including the crossover from the region of wide strips, RF << H , (RF ∼ Nν is the Flory radius with ν = 3/4 in d = 2), to the region of narrow strips, RF >> H , where chains are stretched. Since the length of the strip is finite, here we are more interested in another expected crossover behavior of the excess free energy from an im- prisoned and stretched chain state to an escaped state. Therefore, we plot the excess free energy F (N,L,H) = − ln(Z(N,L,H)/Z0(N)) against N3/4H−1 with the pre- cise estimate of µ∞ = 0.37905228 [23] in Fig. 4. The partition sum Z(N,L,H) is the total number of possible configurations of SAW of N steps partially confined in a strip of length 2L and of width H , which is estimated directly in the simulation. In Fig. 4, the sharp crossover behavior from the imprisoned states to the escaped states is indeed seen as N increases for a fixed value of H . The excess free energy of the escaped chain is independent of N , as predicted in a blob picture by Eqs. (13) and (15). The best fit of the free energy for imprisoned state is given by Fimp(N,L,H) ∼ 1.944(2)NH−4/3 . (19) It is in perfect agreement with the previous estimation in Ref. [22], where the fugacity per monomer scales as µH − µ∞ ∼ 0.737H−4/3 and hence the free energy of the chains of size N in the imprisoned state scales as Fimp ∼ 0.737µ∞ NH −4/3 ≈ 1.944NH−4/3. Values of the excess free energy of the escaped chain, Fesc(N,L,H) are determined by the horizontal curves shown in Fig. 4. Results for L = 800, 1600, 3200 and 6400 are shown in Fig. 5, where we obtain Fesc(N,L,H) = 2.03(3)L/H . (20) As H becomes comparable to L, i.e. L/H ∼ O(10), the data points deviate slightly from the straight line, indicating that there are further finite size corrections for small L/H . In order to compare with the theoreti- cal prediction shown in Fig. 2a, we plot F (N,L,H)/N against L/N for various values of L and H in Fig. 6. In the escaped regime, these straight lines extrapolated to L/N = 0 are indeed described by Eq. (20) very well. With conventional Monte Carlo simulations it is difficult to estimate the partition sum precisely. With PERM we do have very precise estimates of Z(N,L,H), and there- fore we can obtain the lateral and the compression forces by differentiating the estimated free energy, Eqs. (19) and (20), with respect to L and H respectively. B. Average characteristics of polymer chains The most straightforward way to understand how the conformations of the polymer chains change as they un- dergo the escape transition is to estimate the end-to-end distance, < x >. In Fig. 7, we plot the end-to-end dis- tance per monomer< x > /N versus L/N . As long as the polymer chains are imprisoned the curves are horizontal, i.e. the degree of chain stretching is constant, and < x > increases linearly with N as shown in Ref. [22]. As the width of the strip, H , decreases, the chains are stretched more. As L/N decreases, we see that there is a jump 0.15 0.45 0 0.15 0.3 0.45 0.6 0.75 0.9 H = 5 H = 9 H = 17 H = 33 L = 800 L = 1600 L = 3200 L = 6400 FIG. 7: Average end-to-end distance divided by N , < x > /N , plotted against L/N for various values of L and H . The dashed line is < x > /N = L/N . 0 0.1 0.2 0.3 0.4 0.5 H = 9 H = 17 H = 33 H = 64 H = 128 FIG. 8: Average fraction of the imprisoned monomers, < Nimp > /N , plotted against L/N for L = 800 and various values of H . The jump becomes more prominent as H de- creases. The ends of the extrapolated straight dashed lines to L/N = 0 all approach zero. in each curve to another branch where < x >= L This means that the chain stretching is abruptly increased so that the chain reaches the edge of the strip, indicating that the transition to a partially escaped conformation is first-order like. This is in contrast to the smooth behav- ior predicted by the blob picture and shown in Fig. 2f. A jump-wise change in the chain stretching suggests a similar change in the average number of imprisoned monomers, < Nimp >. In Fig. 8, we plot the fraction of imprisoned monomers < Nimp > /N versus L/N for L = 800. As long as the chain is imprisoned, < Nimp > /N = 1. With the increase in the number of monomers, N , each curve indeed develops a jump and the jump becomes more pronounced for smaller H . For a fixed L and N → ∞, < Nimp > /N → 0 in accord with both theoretical 0 0.15 0.3 0.45 0.6 0.75 0.9 H = 5 H = 9 H = 17 H = 33 L = 800 L = 1600 L = 3200 L = 6400 FIG. 9: Average order parameter S plotted against L/N for various values of L and H . models. In Fig. 9, we plot the average of the order parameter S, versus L/N . We see clear jumps between the two states. For a given value of H , the order parameter behaves as a step function. The data seem to suggest that the mag- nitude of the jump decreases with the size of the system, L. A detailed analysis of the distributions of s will be presented in the next section. We will see that the ap- parent decrease in the jump is due to poor sampling of the escaped state. C. Transition points For our simulations the transition points can be deter- mined by analyzing three quantities: (1) free energy, (2) variance of the number of imprisoned monomers Nimp, and (3) variance of the end-to-end distance x. Since at the transition point the free energy of the imprisoned state is equal to the free energy of the escaped state, Eqs. (19) and (20) give the following relation between L, N and H at the transition point: = 1.04(2)H1/3 . (21) This shows that a polymer chain of sizeN can be confined in a strip by tuning the length L or the width H of the strip. The abrupt change scenarios of < x > /N and < Nimp > /N shown in Fig. 7 and 8 indicate a phase tran- sition but it is difficult to locate precisely the transition point. It is clear that all the chain configurations can be di- vided into two subsets: imprisoned and escaped. Far from the transition point, only one subset is important in defining the average characteristics, but in the vicinity of the transition point both subsets contribute, as shown 0.37 0.375 0.38 0.385 0.39 0.395 FIG. 10: Average fraction of the imprisoned monomers < Nimp > /N (solid line) and two partial contributions, < Nimp >1 /N due to confined configurations, and < Nimp >2 /N due to escaped configurations (dotted lines) as functions of L/N near the transition point (L/N)tr = 0.381 for L = 3200 and H = 17. in Fig. 10. The average < Nimp > can be rewritten as < Nimp > = NimpWN (C1) + NimpWN (C2) WN (C1) + WN (C2) = < Nimp >1 + < Nimp >2 (22) where C1 (C2) denotes the imprisoned (escaped) con- figurations, WN (C1) (WN (C2)) are the total weights of the chain for obtaining the configuration C1 (C2), and < . . . >α denotes the partial contributions due to the imprisoned configurations (α = 1) or the escaped config- uration (α = 2). Similarly, < N2imp >=< N imp >1 + < N imp >2 . (23) Finally, the variance of Nimp, σ 2(Nimp) =< N imp > − < Nimp > 2 can be expressed as follows, σ2(Nimp) = σ 1(Nimp)+σ 2(Nimp)−2 < Nimp >1< Nimp >2 where σ21,2(Nimp) =< N imp >1,2 − < Nimp >21,2. It is shown in the Appendix that the variances of partial con- tributions to the total number of imprisoned monomers, σ21(Nimp) and σ 2(Nimp) are much better suited for locat- ing the transition point than the full variance σ2(Nimp) because of the very asymmetric behavior of the latter. We present σ21(Nimp)/N and σ 2(Nimp)/N as functions of L/N in Fig. 11 for various values of H and L. It is clear that for a fixed width H , the peaks become sharper as the length L of the strip increases. The escape transition points are identified with the positions of the peaks of σ21(Nimp) and σ 2(Nimp) as determined by a curve fitting. Values of the transition points, (L/N)tr,1 and (L/N)tr,2 are the same to the third digit for fixed values of L and H , so the transition point is taken as an average of them, i.e. (L/N)tr = ((L/N)tr,1 + (L/N)tr,2)/2. Results of (L/N)tr obtained by this method are listed in Table II and presented in Fig. 12. A more detailed discussion are given in the Appendix. A similar method was also used for determining the transition point from the variance of the end-to-end distance x. Results are also listed in Ta- ble IIand presented in Fig. 12. All the values of (N/L)tr are plotted in Fig. 12 against H1/3. The best fit gives = 1.025(35)H1/3 (25) which is consistent with the estimation of Eq. (21) within the error bar. VI. LANDAU THEORY: DISTRIBUTION OF THE ORDER PARAMETER A. Analytical theory The approach based on the Landau free energy is per- fectly suited for analyzing the relevant states in the es- cape problem, including the metastable states. In the Landau theory, all the configurations are first subdivided into subsets associated with a given value of the order parameter, s , and summation is performed separately within each subset. The full partition function can be obtained then by integrating over the order parameter: Z = exp(−F ) = ds exp(−Φ(s)) (26) where Φ(s) is the Landau free energy function, i.e. the non-equilibrium free energy taken as a function of the order parameter. In the vicinity of the first order tran- sition point, the Landau free energy is expected to have two minima (one stable and the other metastable). Our analysis will be based on finding the metastable minima and the associated thermodynamic characteristics. The proper choice of the order parameter is not always ob- vious, nor are there any standard recipes for making it. One criterion is quite clear: the average value of the or- der parameter should allow one to distinguish between two phases. For a first-order transition, the average or- der parameter changes jump-wise. We require that the properly chosen order parameter changes continuously as the system evolves from a metastable state, through the transition state at the top of the barrier, and eventu- ally falls into the equilibrium minimum. We have shown earlier [13] that these criteria are satisfied if the order parameter is defined as the chain stretching in the con- fined coil state, s = r/(Na) where r is the instantaneous end-to-end distance of the chain of N monomers, or as the stretching of the stem only in the flower conforma- tion: s = L/(na), where n is the number of monomers in the stem. In analogy to the Gaussian case, the Landau function consists of two branches that have to be intro- duced separately. As the chain is in an imprisoned state, (a) (b) 1500 3000 4500 6000 7500 9000 0.1 0.2 0.3 0.4 0.5 H=129 L = 800 L = 1600 L = 3200 L = 6400 1500 3000 4500 6000 7500 9000 0.1 0.2 0.3 0.4 0.5 H=129 L = 800 L = 1600 L = 3200 L = 6400 FIG. 11: Variances of the number of imprisoned monomers divided by N , (a) σ21(Nimp)/N , for the imprisoned state, and (b) σ22(Nimp)/N , for the escaped state, plotted against L/N . The height of peaks increases with L for a fixed value of H . TABLE II: Values of the transition points, (L/N)tr, determined from the analysis of the variances σ 1(Nimp) and σ 1(x) for the imprisoned states, and from the variances σ22(Nimp) and σ 2(x) for the escaped states. (L/N)tr,Nimp (L/N)tr,x H L = 800 1600 3200 6400 L = 800 1600 3200 6400 9 0.4790 0.4766 0.4703 0.4649 0.4790 0.4766 0.4723 0.4674 17 0.3831 0.3846 0.3810 0.3754 0.3829 0.3836 0.3810 0.3754 33 0.3027 0.3060 0.3071 0.3036 0.3024 0.3078 0.3070 0.3036 65 0.2363 0.2412 0.2434 0.2455 0.2357 0.2409 0.2433 0.2455 129 0.1816 0.1880 0.1916 0.1932 0.1803 0.1877 0.1914 0.1930 the Landau free energy is directly expressed in terms of the distribution of the end-to-end distance. There exists no closed-form formula for such a distribution of confined chains with excluded volume interactions. However, the distribution of the gyration radius for 3d chains confined in a tube was studied analytically and numerically in [24]. It was proposed that the free energy of a confined chain with a given gyration radius rg can be presented as a sum of two terms: F (rg) = N Acα +B , (27) where c is the segment volume concentration expressed as a function of the gyration radius and the confinement geometry, α and δ are linked to the space dimension d and the Flory exponent ν by α = (νd− 1)−1 and δ = (1− ν)−1. The first term describes the concentration effects in the des Cloizeaux [25] form, the second term is the Pincus [26] scaling form of the stretching free energy, and A and B are model-dependent numerical coefficients of order unity. Instead of rg, we use the same ansatz, Eq. (27), to describe the end-to-end distance distribution by taking c = Na2/rH , α = 2, and δ = 4. The free energy of the chain in an imprisoned state as a function of s is hence given by Φimp(s) = N , s ≤ L . (28) Since we prefer to keep the basic scaling formula of the Landau free energy in order to provide predictions in a simple analytical form, here we are not going to consider the further logarithmic correction terms as shown in [24]. In the thermodynamic limit, the average value of the order parameter for the imprisoned state, Simp, is found by locating the minimum of Φimp(s), i.e. dΦimp(s)/ds = 0 at s = simpeq , and hence Simp = s eq = (A/2B) 1/6(a/H)1/3 . (29) The minimum of the Landau free energy gives the free energy for the imprisoned state at equilibrium Fimp = Φimp(Simp) = 3B N . (30) 1.5 3 4.5 6 7.5 L = 800, Nimp L = 1600, Nimp L = 3200, Nimp L = 6400, Nimp L = 800, x L = 1600, x L = 3200, x L = 6400, x FIG. 12: Transition points (N/L)tr versus H . The dashed line is (N/L)tr = 1.025(35)H 1/3 and gives the best fit of the data. Compared with Eq. (13), this is indeed the correct scal- ing of the free energy. The end-to-end distance at equi- librium is found as RN = NaSimp = (A/2B) 1/6(H/a)−1/3Na , (31) which is consistent with the result of the blob model, Eq. (12). As the chain is in an escaped state, the formula of the free energy function is identical to Eq. (28), but corrected for the fact that only the n monomers that are part of the stem contribute: Φesc(s) = n s−3 +Bs3 , s ≥ L .(32) The average value of the order parameter in the escaped state, Sesc, is found by locating the minimum of Φesc(s) and is given by Sesc = s eq = (A/B) 1/6(a/H)1/3 (33) Thus, the free energy of the escaped chain at equilibrium Fesc = Φesc(Sesc) = 2(AB) The transition point is found from the condition that the two minima of the Landau free energy function are of equal depth. Using Eqs. (30) and (34) we get It is interesting to calculate the size of jumps implied by the Landau theory in the order parameter, the im- prisoned monomers and the end-to-end distance at the N /Nimpr (3a/2 )(A/B) (H/a) L/N S/Sesceq 5/3 1/6 −1/3 (3a/2 )(A/B) (H/a) L/N 5/3 1/6 −1/3 FIG. 13: Based on the Landau theory, the theoretical predic- tions of the average values of (a) the fraction of imprisoned monomers Nimp/N , and (b) the order parameter S, are plot- ted against L/N . The chain is in an imprisoned state for L/N > (3a/25/3)(A/B)1/6(a/H)1/3, and in an escaped state for L/N < (3a/25/3)(A/B)1/6(a/H)1/3. transition. Using Eqs. (29) and (33) we immediately get the reduced jump of the order parameter Sesc − Simp = 1− 2−1/6 ≈ 0.1091 , (36) which is independent of H and the coefficients A and B. For an imprisoned state, < Nimp >= N by defini- tion, while for the coexisting escaped state with the same choice of H , L, and N we have only < Nimp >= L/Sesc monomers. From Eqs. (33) and (35), we obtain the rela- tive reduction in the number of imprisoned monomers ∆Nimp N − L/Sesc ≈ 0.055 . (37) This number has a simple meaning of the fraction of the chain escaping out of the confinement at the transition point. It is much smaller than 1/2 in the Gaussian chain model, but non-zero in contrast to the blob model. Fi- nally, the reduced jump of the end-to-end distance is ob- tained by combining Eqs. (31) and (35) ≈ 0.0572 . (38) Eqs. (36)-(38) show that the sizes of jumps in S, < Nimp > /N and RN/L are universal quantities. Results for the average order parameter S and the average frac- tion of imprisoned monomers < Nimp > /N predicted by the Landau theory are shown in Fig.13. Comparing with the numerical results shown in Fig. 8 and Fig.9, we see that the Landau theory a good qualitative agreement. The predicted free energy of the chain at equilibrium, Eqs. (30) and (34) follow the same scaling behavior as obtained by the MC simulations shown in Eqs. (19) and (20). This allows us to identify the numerical values of the constants A and B for our model: A ≈ 1.057 and B ≈ 0.975. B. Numerical comparisons Here we focus on the results of the Landau free energy of polymer chains partially confined in a strip of width H = 28 and of length L = 800, 1600, 3200, and 6400. Since PERM gives the possibility to estimate directly the partition sum and the properly normalized histograms, the Landau free energy as a function of s, Φ(N,L,H, s), is given by Φ(N,L,H, s) = − ln P (N,L,H, s) Z0(N) where P (N,L,H, s) = walks δs,s′ is the histogram of s, and the partition sum of the partially confined chains can be written as Z(N,L,H) = P (N,L,H, s) (40) in accordance with Eq. (26). In Fig. 14, we plot four sets of results of the Landau free energy per monomer Φ(N,L,H, s)/N versus the order parameter s for L = 800, 1600, 3200 and 6400. Since the transition point is near H1/3, the histograms are obtained for N/L = 3.05, 3.10 and 3.15 for each set. The predicted analytical re- sults of ΦP (s) = Φimp(s) for the imprisoned state and ΦP (s) = Φesc(s) for the escaped state, given by Eqs. (28) and (32) are also shown for comparison. On the left-hand side of the branch points, due to the finite-size effect, we see that the excess free energy for the imprisoned state (the minimum of the curve) at s = s eq,L converges to the predicted value (the minimum of the curve ΦP (s)) of polymer chains confined in an infinite strip at s = simpeq slowly as L increases but s eq,L is slightly larger than s as L → ∞. The difference between those curves corre- sponding to the different ratio N/L is almost invisible for a fixed value of L as predicted by Eq. (28). On the right-hand side of the branch points, we see that only those curves for L = 800 finally develop a parabola-like behavior with fluctuations and they are more concave than those curves predicted by Eq. (32). It shows that PERM has difficulties to sample configurations in the es- caped regime as L increases and gives an explanation why we should not trust the size of those jumps that appear in Fig. 9 too much. However, one can easily overlook the existence of two minima in such a delicate situation. With PERM, at least we are able to give evidence for this two minimum picture of the first-order like transi- tion. We also see that additional finite-size correction terms should be taken into account for the theoretical predictions in Eqs. (28) and (32). 0.023 0.024 0.025 0.026 0.027 0.25 0.3 0.35 0.4 0.45 N/L=3.05 N/L=3.10 N/L=3.15 L= 800 L=1600 L=3200 L=6400 ΦP(s)/N FIG. 14: The Landau free energy divided by N , Φ(N,L,H, s)/N , plotted against s for various values of L and H = 28. The predicted Landau free energy functions, ΦP (s) = Φimp, Eq. (28), in the imprisoned regime and ΦP (s) = Φesc, Eq. (32), in the escaped regime are also plotted (dashed lines). Taking the results for L = 6400 as a reference, we plot the same data but shift all other curves by some constants, c0,L = −0.00235, −0.00109, −0.00044 for L = 800, 1600, and 3200 to make the three branch points for N/L = 3.05, 3.10, and 3.15 coincide with each other in Fig. 15. According to the prediction by Eq. (28), we should expect that the four curves for different values of L overlap with each other in the imprisoned regime. In fact, it is not the case but the difference between these curves decreases as L increases, and finally they will converge to one curve as L becomes very large. In the escaped regime, surprisingly, we see that those curves corresponding to different L all overlap with each other for a fixed ratio of N/L as predicted by Eq. (32). Although the lack of data for larger L, precludes very strong conclusions, we may assume that these curves all show the same behavior as the curve for L = 800, and do further analysis. In order to determine the transition point and extract an accurate value for the jump in the order parameter from simulations, we use two parabolic functions gimp(s) and gesc(s) to fit the numerical data in the imprisoned and escaped regimes, respectively: gimp(s) = a1,L(s− simpeq,L) 2 + c1,L (41) gesc(s) = a2(s− sesceq,L)2 + c2 + b2 where a1,L, c1,L, a2, c2, b2, s eq,L, and s eq,L are deter- mined by curve fitting, and results are shown in Table III and Fig. 15. From the condition of equal depth of minima gimp(s = s eq,L) = gesc(s = s eq,L), (43) 0.023 0.024 0.025 0.25 0.3 0.35 0.4 0.45 N/L=3.05 N/L=3.10 N/L=3.15 L = 6400 L = 3200 L = 1600 L = 800 FIG. 15: The Landau free energy divided by N , Φ(N,L,H, s)/N , plotted against s for various values of L and H = 28. The two minima of Φ(N,L,H, s)/N are determined by fitting g(s) = gimp(s), Eq. (41), in the imprisoned regime, and g(s) = gesc(s), Eq. (42), in the escaped regime, going through those lower points around the two minima, respec- tively. 0.06 0.08 0.12 0.14 0.16 0.18 0 0.0003 0.0006 0.0009 0.0012 0.0015 curve fitting estimation Eq. (36) FIG. 16: The reduced jump of the order parameter ∆S/Sesc plotted against L−1. we obtain the transition points (N/L)tr = 3.13(2), 3.10(2), 3.09(1), and 3.08(1) for L = 800, 1600, 3200, and 6400, respectively, which are in perfect agreement with the results given by free energy, Eq. (21), and the results given by the variance of the end-to-end distance and the imprisoned monomers, Eq. (25). The values for the reduced jump of the order parame- sesceq,L − s sesceq,L obtained by the curve fitting are plotted in Fig 16 against L−1 together with the direct estimates in the simulations TABLE III: Results of the coefficients a1,L, c1,L, a2, c2, b2, eq,L, and s eq,L for the curve fitting in Fig. 15. L a1,L c1,L s eq,L a2 c2 b2 s 800 0.6281 0.02277 0.2945 0.6518 0.0497 -0.0086 0.3510 1600 0.6336 0.02303 0.2986 3200 0.6675 0.02313 0.3007 6400 0.6872 0.02320 0.3017 and the prediction by the analytical theory, Eq. (36). We see that ∆S/Sesc decreases as L increase. As L → ∞, it remains finite and the value is slightly larger than the predicted value by the analytical theory. However, in view of the numerical uncertainties of our curve fitting we consider that the predictions of the analytical theory and the results by the MC simulations agree with each other quite well. VII. SUMMARY AND DISCUSSION In this paper we attack the problem of the 2d-escape transition by combining several approaches. We first compare two simple pictures of the transition predicted for Gaussian chains and by a blob model. This compar- ison is useful from a general pedagogical point of view since the two models are in a sense complimentary: each captures some essential features of the phenomenon while failing in some other aspects. Both models are attrac- tive because of their clarity, and although mathemati- cally simple, lead to non-trivial results including finite- size effects in a phase transition. The third approach that was proposed in this paper attempts at incorporat- ing the excluded volume effects in the framework of the Landau theory. We were not able to present an exact theory since it would require a detailed understanding of the end-to-end distribution of confined self-avoiding chains. To the best of our knowledge this problem is still not well explored. The simulations presented allowed us to evaluate the transition condition Eq. (21) which rep- resents the binodal line in the (H,L/N) plane. It is of interest to extend the simulations in order to locate the spinodal lines where one of the states looses stability, and to construct the full phase diagram. It is also possible to explore the properties of metastable states and their life- times controlled by the barrier heights. It is clear from the results on the distribution of the order parameter, Fig. 14, that the PERM algorithm experiences difficul- ties with sampling the configurations belonging to the escaped state, especially for long chains. The escaped branch of the distribution is cut-off quite sharply, which means that the important set of configurations charac- terized by larger stretching degree in the stem is vastly underrepresented. This is a generic problem that one en- counters when dealing with first-order transitions when the properties of the phases differ significantly. In our case, the PERM algorithm based on chain growth tech- nique is perfectly tuned to generate homogeneous config- urations of imprisoned chains but fails with strongly in- homogeneous escaped configurations. It is worth noting that a naive determination of the jumps in the average order parameter would have lead one to a wrong con- clusion that the jump disappears in the thermodynamic limit. Again, we expect this to be a generic problem when simulating weak first-order transitions. The most reliable analysis of the nature of the transition would require a detailed examination of the order parameter distribution. Acknowledgements We are grateful to the Deutsche Forschungsgemein- schaft (DFG) for financial support: H.-P.H. was sup- ported under grant NO SFB 625/A3, while L.I.K. and A.M.S. received partial support under grants NO 436 RUS 113/863/0 and RFBR 05-03-32003-a. H.-P.H. thanks P. Grassberger and W. Paul for very helpful dis- cussions. Appendix In this appendix, we discuss the finite-size behavior in the fluctuations in the number of imprisoned monomers Nimp in more detail. Following the technique of finite-size scaling analysis for first-order transitions as described in [27], we write down the probability distribution of the fraction of imprisoned monomers m = Nimp/N in the two-state model P (m) = δ(m−m1) e(t−ttr)a e(t−ttr)a + e−(t−ttr)a (m−m2) e−(t−ttr)a e(t−ttr)a + e−(t−ttr)a .(45) The first term accounts for the imprisoned state with m strictly equal to m1 = 1, while the second term describes the distribution ofm in the escaped state in the Gaussian approximation with the equilibrium average of m equal to m2 and dispersion σ0; t is the control parameter, ttr is its critical value at the transition point, and P (m) is normalized, P (m)dm = 1 (46) At the transition point t = ttr, P (m) = δ(m−m1) + (m−m2) which obeys the “equal-weight rule”, while for t 6= ttr the relative weight of the two states is exp [2(t− ttr)a]. The constant a−1 describes the range of t over which the transition is smeared out. For the Gaussian approx- imation to be meaningful the dispersion of m in the es- caped state, σ0, must be small compared to the difference ∆m = m1 − m2. Taking t = L/N and using Eqs. (30), (32), and (34) of the Landau theory, one expects the fol- lowing scaling: a−1 ∼ H2/3/L and σ20 ∼ H/L. Since the probability density is a sum of two contribu- tions, P (m) = P1(m) + P2(m), the k-th moment of m is defined by < mk >= mkP (m)dm =< mk >1 + < m k >2 (48) where < mk >1,2= mkP1,2(m)dm. Therefore, the first and second moment are given by < m >= m1p1 +m2p2 , (49) < m2 >= m21p1 + (m 2 + σ 0)p2 , (50) here p1 = e (t−ttr)a/2 cosh [(t− ttr)a] is the relative weight of the imprisoned state, and p2 = 1 − p1 is the relative weight of the escaped state. Instead of a δ-function singularity at t = ttr, the vari- ance of the fraction of imprisoned monomers in a finite system becomes < m2 > − < m >2= p1p2(∆m)2 + p2σ20 (51) which shows a smooth asymmetric peak close to t = ttr of approximate height ∆m2+σ20 . Here ∆m is the relative reduction in the number of imprisoned monomers at the transition point, for which the analytical Landau theory predicts a value of 0.055, see Eq. (37). The first term in Eq. (51) is symmetric with respect to the transition point since p1p2 = 1/(4 cosh 2 [(t− ttr)a]). The second term, however, is asymmetric, as it describes the intrinsic fluc- tuations in the escaped state. The resultant asymmetry is clearly seen in Fig. 17. We conclude that the full vari- ance of m is ill suited for a precise determination of the transition point. The situation is quite different if we analyze the vari- ances calculated with the partial probability densities P1(m) and P2(m) restricted to the imprisoned (escaped) configurations. In the simulations, the product Nσ21,2 was calculated. The variance due to the imprisoned con- figurations only (with m1 = 1) gives a perfectly symmet- ric curve as a function of the control parameter: Nσ21(m) = Nm 1p1p2 = 4 cosh2 [(t− ttr)a] with the peak value ofNtr/4 ≈ LH1/3/4. Numerical data presented in Fig. 18 supports this prediction with very high accuracy. The variance due to escaped configura- tions is somewhat modified by the intrinsic fluctuations 0.01 0.02 0.03 0.04 0.05 0.06 0.37 0.375 0.38 0.385 0.39 0.395 ∆Nimp/N 0.1 σ1(m) 0.1 σ2(m) 1-<m> FIG. 17: The square root of the variance σ1(m) for the im- prisoned states, σ2(m) for the escaped chains, 1− < m >, σ(m) of the chain either in an imprisoned state or in an es- caped state, and the difference ∆σ = σ1(m)− σ2(m) against L/N . 1000 2000 3000 0.37 0.375 0.38 0.385 0.39 L = 3200, H = 17 Eq. (53) A (L,H)1 (L,H)1 FIG. 18: Variance due to the imprisoned configuration multiplied by N , Nσ21(m), plotted against L/N for L = 3200 and H = 17. The solid curve is the best fit of Eq. (52), Nm21/4cosh 2 [(t− ttr)a], with the height of the peak A1(L,H) = N(m1) 2/4 ≈ 2099.74, the FWHM Γimp(L,H) ≈ 1.7627/a = 0.0035, and the position of the peak ttr,1 = (L/N)tr,1 = 0.3810. The FWHM are given by the distance between points on the curve shown at which the correspond- ing height reaches half height of the peaks (half maximum). in the escaped state Nσ22(m) = N(m 2p1p2 + p2σ ≈ Nm22p1p2 = N(1−∆m)2 4 cosh2 [(t− ttr)a] but in contrast to Eq. (51), the asymmetric term is al- ways negligible. Indeed, the coefficient with the sym- metric term, m22 = (1 − ∆m)2 is close to 1 while both quantities (∆m)2 and σ20 are quite small. In Fig. 17, 0.01 0.02 0.03 0.04 0 0.0003 0.0006 0.0009 0.0012 0.0015 0.0018 H = 129 H = 65 H = 33 H = 17 H = 9 FIG. 19: FWHM Γα(L,H) for the imprisoned state (α = 1) and for the escaped state (α = 2) against L−1. The dashed curves are a1,H(H/L) + b1,H(H/L) 2 and give the best fit of the data. Values of a1,H and b1,H are listed in Table IV. TABLE IV: Results of the coefficients a1,H , b1,H , a2,H , c2,H and d2,H for the curve fitting in Fig. 19 and 20. H a1,H b1,H a2,H c2,H d2,H 9 0.8221 0.7841 0.9940 0.2131 2.2802 17 0.6845 -2.0403 0.9845 0.0961 0.4540 33 0.5600 -2.3531 0.9725 0.0430 0.0570 65 0.4301 -1.3774 0.9545 0.0173 0.0095 129 0.3185 -0.7332 0.9220 0.0070 0.0017 we plot the full dispersion σ(m) (the square root of the full variance) that includes contributions from all config- urations, and partial dispersions σ1(m) and σ2(m) due to imprisoned and escaped configurations separately, the difference ∆σ = σ1(m) − σ2(m), as well as the aver- age fraction of escaped monomers, 1− < m >, as func- tions of L/N . It is clear that the full curve is strongly asymmetric in contrast to partial dispersion curves, in good agreement with the theoretical description above. On the other hand, the curve of ∆σ shows the same be- haviour as the curve of σ(m) near the transition point, and the heights of these two peaks correspond to the half size of the jump ∆m = m1 − m2 = ∆Nimp/N . By fitting the partial variances Nσ21(m) and Nσ as functions of L/N according to Eqs. (52) and (53), respectively, we obtain the full width at half-maximum (FWHM), Γ(L,H) = 2arccosh( 2)/a, the height of the peak Aα(L,H) = Nm α/4, and the transition point ttr,α = (L/N)tr,α for α = 1 (imprisoned configurations) and α = 2 (escaped configurations). One example of the curve fitting for L = 3200 and H = 17 is shown in Fig. 18. Note that the peak height and the transition point are related to the theoretical prediction A1(L,H) ≈ LH1/3/4 with very high accuracy. (a) (b) 0.0005 0.001 0.0015 0.002 0.0025 0 0.0003 0.0006 0.0009 0.0012 0.0015 H = 9 H = 17 H = 33 H = 65 H = 129 0.0005 0.001 0.0015 0.002 0.0025 0.003 0 0.0003 0.0006 0.0009 0.0012 0.0015 H = 9 H = 17 H = 33 H = 65 H = 129 FIG. 20: Inverse of the height of the peaks for the imprisoned state (a) A−11 (L,H), and for the escaped state (b) A 2 (L,H), plotted against L−1. The dashed curves are (a) a2,H(4H −1/3/L) and (b) c2,H(H/L) + d2,H(H/L) 2, and give the best fit of the data. Values of a2,H , c2,H , and d2,H are listed in Table IV. 0.05 0 0.03 0.06 0.09 0.12 0.15 H = 33 H = 65 H = 129 0.54H/L+0.058 Eq. (37) FIG. 21: The relative reduction in the number of imprisoned monomers ∆m, plotted against H/L. Results of Γα(L,H), Aα(L,H) for α = 1 and for α = 2, and ttr,α are shown in Fig. 19, 20 and 12. In Fig. 19, we see that the full widths Γα(L,H) for α = 1 and for α = 2 are overlapped with each other, and Γα → 0 as 1/L → 0 by fitting the data using a1,H(H/L) + b1,H(H/L) 2. In Fig. 20, the inverse of the height A−11 (L,H) → 0 as 1/L → 0 by fitting the data using a1,H(4H−1/3/L) and c1,H(H/L)+d1,H(H/L) 2. Since Γα → 0, and A−1α → 0 as 1/L → 0, i.e. a delta function, a sharp phase transition occurs in the thermodynamic limit. It is a strong indica- tion [27] that the transition is first-order like. 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Noname manuscript No. (will be inserted by the editor) Detecting the Most Distant (z>7) Objects with ALMA Fabian Walter and Chris Carilli Received: date / Accepted: date Abstract Detecting and studying objects at the high- est redshifts, out to the end of Cosmic Reionization at z>7, is clearly a key science goal of ALMA. ALMA will in principle be able to detect objects in this red- shift range both from high-J (J>7) CO transitions and emission from ionized carbon, [CII], which is one of the main cooling lines of the ISM. ALMA will even be able to resolve this emission for individual targets, which will be one of the few ways to determine dynamical masses for systems in the Epoch of Reionization. We discuss some of the current problems regarding the detection and characterization of objects at high redshifts and how ALMA will eliminate most (but not all) of them. 1 Introduction: The highest redshift galaxies In recent years, deep narrow band surveys have revealed a major population of Lyman Alpha Emitters (LAE) out to very high redshifts (e.g. Hu et al. 2002, Kurk et al. 2004, Stern et al. 2005, Murayama et al. 2007). In particular, Taniguchi et al. (2005) report the detec- tion of 9 spectroscopically confirmed LAE at redshifts of z∼6.6 in the Subaru Deep Field (currently, the pub- lished LAE redshift record holder is at z=6.98, Iye et al. 2006). The mere presence of Lyman alpha emission in these sources provides strong evidence that they are undergoing bursts of star formation: the star formation It is our pleasure to thank our collaborators on this project: Frank Bertoldi, Dominik Riechers, Pierre Cox, Roberto Maiolino and Axel Weiß. F. Walter Max Planck Insitut für Astronomie Heidelberg, E-mail: wal- ter@mpia.de C. Carilli National Radio Astronomy Observatory, E-mail: ccar- illi@nrao.edu rates of individual objects are ∼10M⊙ yr −1 (based on their FUV luminosities) and their redshifts place them well within the end of cosmic reionization. They also appear to be very numerous: Tanaguchi et al. 2005 find ∼30 LAEs in only a quarter degree field (e.g., compared to ∼10 QSOs at z>6 which are distributed over a quar- ter of the sky! Fan et al. 2004). This implies that LAEs may play an important role in reionizing the universe at z>6 (for a review see Fan, Carilli & Keating, 2006). In- vestigating the physical properties of these sources are thus of great interest and ALMA will play a critical role in studying these objects, as discussed in the following. 2 Interstellar Medium: CO vs. [CII] emission 2.1 Carbon Monoxide (CO) Constraining the properties of the molecular gas in ob- jects at the end of cosmic reionization is clearly of key importance as such observations 1) will measure the available ’fuel’ for star formation, 2) will help to con- strain the dynamical mass of the system and will thus 3) allow to put these objects in an evolutionary con- text for early galaxy formation. Typically, at low and high redshifts, CO emission is used as a tracer for the molecular gas phase (e.g. review by Solomon & Vanden Bout 2004). It is important to keep in mind though, that, at the highest z, only the very high rotational lines of CO will be observable with ALMA. E.g. even in the lowest (currently funded) frequency band of ALMA (band 3, 84–119GHz), only CO transitions with J>7 (i.e., CO(7–6), CO(8–7), etc.) will be observable at z>7. This is graphically illustrated in Fig. 1 where we plot ALMA’s ‘CO discovery space’ (i.e., which line can be observed at which redshift using which ALMA band). The high–J transitions correspond to highly excited gas http://arxiv.org/abs/0704.1858v1 0 5 10 15 20 Redshift Band 4 (125-163 GHz) Band 3 ( 84-119 GHz) Band 4 (125-163 GHz) Band 6 (211-275 GHz) Band 7 (275-370 GHz) Band 8 (385-500 GHz) Band 9 (602-720 GHz) Band 10 (787-950 GHz) <− [CII] Fig. 1 ALMA CO ‘discocery space’: The horizontal lines indicate which CO transition (plotted on the y–axis) can be observed with which ALMA band as a function of redshift (plotted on the x– axis). For objects with z>7, only the higher–J CO transitions can be observed with ALMA. The [CII] ‘discovery space’ is also indicated. (either due to high kinetic temperatures, high densities, or both) which may not be excited in normal starform- ing environments. This is shown in Fig. 2 (taken from Weiss et al. 2005) where measured CO line strengths (as a function of J, this is sometimes referred to as CO line ladders/SEDs) are plotted for a number of key sources. What is immediately obvious from this plot is that most objects have sharply decreasing CO line strengths be- yond J> 8, in particular starforming systems such as NGC253, or the sub–millimeter galaxy plotted in this diagram (the quasars appear to be more excited, but their CO line SED still turns over at J∼7, for an excep- tional object see APM07279, Weiss et al. 2007). This comparison immediately implies that emission from the CO molecule will typically be very difficult to observe with ALMA at z>7 as the observable lines will simply not be excited. 2.2 Ionized Carbon ([CII]) to the Rescue! An alternative tracer of the interstellar medium is one of the main cooling line of the ISM, the 2P3/2 → 2P1/2 fine–structure line of C+ (or [CII]). In brief, the [CII] line is expected to be much stronger than any of the CO lines. Given its high frequency (157.74µm, corre- sponding to 1900.54GHz) [CII] studies in the local uni- verse are limited to airborne or satellite missions (e.g. Stacey et al. 1991, Malhotra et al. 1997, Madden et Fig. 2 Comparison of various normalized (by their CO(1–0) flux density) CO line SEDs at low and high redshift (figure taken from Weiss et al. 2005). The CO line SEDs decline rapidly beyond J=6–8. al. 1997). These studies have demonstrated that this single line can indeed carry a good fraction of the total infrared luminosity (LFIR) of an entire galaxy. In the lo- cal universe, the ratio LCII/LFIR has been found to be 2–5×10−4 in the case of ULIRGS (e.g. Gerin & Phillips 2000), but is more like 5–10×10−3 in more typical star- forming galaxies (for a discussion on possible reasons for the supressed ratio in ULIRGs see, e.g., Luhman et al. 1998). Notably, the ratio has been found to be 1% or even higher in low metallicity environments. E.g., in the low- metallicity galaxy IC10, LCII/LFIR reaches values as high as 4%, with an average value of 2% (Madden et al. 1997, see Israel et al. 1996 for a similar result for the LMC). This is the reason why it has long been ar- gued (e.g., Stark 1997) that observation of the [CII] line of prestine systems at the highest redshifts will likely be the key to study molecular gas in the earliest star- forming systems, in particular in the era of ALMA. The ALMA [CII] ‘discovery space’ is also indicated in Fig. 1. 3 Expected [CII] line strengths At the redshifts of the LAEs, the [CII] line is shifted to the 1mm band of ALMA (band 6, 211–275GHz). [CII] emission has recently been successfully detected using the IRAM 30m in the highest redshift quasar J1148+5251 at z=6.42 (Maiolino et al. 2005, see Fig. 3). The noteable difference between J1148+5251 and the z>6 LAEs is that the ratio LCII/LFIR has been found to be very low (∼5×10−4) in J1148+5251, i.e. in perfect Fig. 3 Top: First detection of [CII] at high redshift in the z=6.42 QSO J1148+5251 (Maiolino et al. 2005). Bottom: brightest CO transition (J=6) in the same source (Bertoldi et al. 2003, Walter et al. 2003). Note that the [CII] line is brighter by a factor of ∼5. agreement with studies of low redshift ULIRGs that show a central AGN. On the contrary, the LAE are presumably pure starbursts (no evidence for an AGN is found, Taniguchi et al. 2005) and they likely have lower metallicities compared to the highly overdense regions in which the luminous quasars are supposedly present. All these arguments point towards a LCII/LFIR ratio in LAE that is close to what is found for nearby normal galaxies, or perhaps even for the metal–poor dwarf galaxies (i.e., around 1% or even higher). In other words, the [CII] luminosity of the LAE may well be an order of magnitude stronger (for a given IR luminos- ity) than what has been found in the z=6.4 QSO. In the following we present a quick back–of–the envelope calculation based on the detected [CII] line strength in J 1148+5251 (∼10 mJy) which has a SFR of a few 1000 M⊙ yr −1. This SFR is more than two order of magni- tudes higher than the SFR found in a typical LAE, but as the LCII/LFIR may be higher by an order of magni- tude in the LAEs, the expected [CII] line strength of the LAE may be as high as 1mJy. Such a line should be easily detectable with ALMA at high significance in a few hours. 4 Resolving the ISM Detecting the [CII] (or CO) emission is critical to esti- mate the reservoir of the (molecular) gas in these early systems. A second step is then to spatially resolve the molecular gas distribution. In particular, given the typ- RIGHT ASCENSION (J2000) 11 48 16.70 16.68 16.66 16.64 16.62 16.60 52 51 50.8 Fig. 4 High–resolution CO image of the z=6.42 QSO J 1148+5251 obtained at the VLA (Walter et al. 2004). The res- olution achieved in these observations (0.15′′, corresponding to ∼1 kpc) will be routinely reached with ALMA. ical diameters of galaxies of many kpc, a linear resolu- tion of ∼ 1 kpc is needed to resolve the structure of the underlying galaxy. Such measurements are needed 1) to get an estimate for the size of the galaxy (and thus a better estimate for the dynamical mass), 2) to resolve potentially merging systems and 3) to better constrain the physical properties of the gas (e.g., by measuring the brightness temperature of the hosts). A linear res- olution of 1 kpc corresponds to a resolution of 0.15′′ at the redshifts under consideration (1′′ ∼ 5.8 kpc at z = 6). Such observations can then in turn be used to constrain the predictions by CDM simulations of early galaxy formation, and, if a large sample was available, put limits on the frequency of mergers at high redshift. In addition, such studies can be used to constrain the possible redshift-evolution of the MBH–σv relation in high–z quasars. Such observations will clearly be feasi- ble with ALMA in the extended arrays. High–resolution CO imaging is already possible with the current genera- tion of telescopes: we have used the VLA to resolve the molecular gas in the host galaxy of the z=6.42 QSO J1148+5251 (see Fig. 4, Walter et al. 2004). 5 The case for ALMA band 5 As a technical note: the ALMA redshift coverage for the [CII] line is not ideal as its frequency lies between the CO(17–16) and CO(16–15) transition (see Fig. 1). One concern is that the critical redshift range (8<z<10.5) is currently not fully covered: This frequency range corre- sponds to the ALMA band 5 which is only partly funded by the European Union as part of the Sixth Frame- work Programme (FP6) for up to 8 antennas. Clearly, it would be highly desireable to equip as many ALMA antennas with band 5 receivers as possible. 6 Concluding remarks ALMA observations of the [CII] line will play a fun- damental role in studying the youngest galaxies in the Epoch of Cosmic Reionization at z>7. Given the ex- pected line strengths it should be possible to resolve these galaxies in the [CII] line emission on kpc scales. Such measurements would not only constrain the sizes but would also help to derive the dynamical masses in these early starforming systems. Given the typical CO excitation in starforming galaxies (i.e. drop in excita- tion around the J∼6 transitions), ALMA will likely act as a [CII]– rather than a CO–machine for objects at these extreme redshifts. References 1. Bertoldi, F., Carilli, C. L., Cox, P., Fan, X., et al. 2003, A&A, 406, L55 2. Fan, X., Hennawi, J. F., Richards, G. T., et al. 2004, AJ, 128, 3. Fan, X., Carilli, C.L., Keating, B., 2006, ARA&A, 44, 415 4. Gerin, M., & Phillips, T. G. 2000, ApJ, 537, 644 5. Hu, E. M., Cowie, L. L., McMahon, R. G., et al. 2002, ApJ, 568, L75 6. Israel, F. P., Maloney, P. R., Geis, N., Herrmann, F., Madden, S. C., Poglitsch, A., Stacey, G.J., 1996, ApJ, 465, 738 7. Iye, M., Ota, K., Kashikawa, N., Furusawa, H., Hashimoto, T., Hattori, T., Matsuda, Y., Morokuma, T., Ouchi, M., Shi- masaku, K., 2006, Nature, 443, 186 8. Kurk, J. D., Cimatti, A., di Serego Alighieri, et al. 2004, A&A, 422, L13 9. Luhman, M. L., et al. 1998, ApJL, 504, L11 10. Madden, S. C., Poglitsch, A., Geis, N., Stacey, G. J., & Townes, C. H. 1997, ApJ, 483, 200 11. Maiolino, R., Cox, P., Caelli, P., et al. 2005, astro-ph/0508064 12. Malhotra, S., et al. 1997, ApJL, 491, L27 13. Murayama, T., Taniguchi, Y., Scoville, N.Z., et al. 2007, ApJS, astro-ph/0702458 14. Stacey, G. J., Geis, N., Genzel, R., Lugten, J. B., Poglitsch, A., Sternberg, A., & Townes, C. H. 1991, ApJ, 373, 423 15. Stark, A. A. 1997, ApJ, 481, 587 16. Stern, D., Yost, S. A., Eckart, M. E., Harrison, F. A., Helfand, D. J., Djorgovski, S. G., Malhotra, S., & Rhoads, J. E. 2005 , ApJ, 619, 12 17. Taniguchi, Y., Ajiki, M., Nagao, T. et al. 2005, PASJ, 57, 18. Walter, F., Bertoldi, F., Carilli, C. L., et al. 2003, Nature, 424, 406 19. Walter, F., Carilli, C., Bertoldi, F., et al. 2004, ApJ, 615, L17 20. Weiss, A., Downes, D., Neri, R., Walter, F., Henkel, C., Wilner, D.J., Wagg, J., Wiklind, T., 2007, A&A, in press (astro–ph/0702669) http://arxiv.org/abs/astro-ph/0508064 http://arxiv.org/abs/astro-ph/0702458 http://arxiv.org/abs/astro--ph/0702669 Introduction: The highest redshift galaxies Interstellar Medium: CO vs. [CII] emission Expected [CII] line strengths Resolving the ISM The case for ALMA band 5 Concluding remarks

Detecting and studying objects at the highest redshifts, out to the end of Cosmic Reionization at z>7, is clearly a key science goal of ALMA. ALMA will in principle be able to detect objects in this redshift range both from high-J (J>7) CO transitions and emission from ionized carbon, [CII], which is one of the main cooling lines of the ISM. ALMA will even be able to resolve this emission for individual targets, which will be one of the few ways to determine dynamical masses for systems in the Epoch of Reionization. We discuss some of the current problems regarding the detection and characterization of objects at high redshifts and how ALMA will eliminate most (but not all) of them.

Introduction: The highest redshift galaxies In recent years, deep narrow band surveys have revealed a major population of Lyman Alpha Emitters (LAE) out to very high redshifts (e.g. Hu et al. 2002, Kurk et al. 2004, Stern et al. 2005, Murayama et al. 2007). In particular, Taniguchi et al. (2005) report the detec- tion of 9 spectroscopically confirmed LAE at redshifts of z∼6.6 in the Subaru Deep Field (currently, the pub- lished LAE redshift record holder is at z=6.98, Iye et al. 2006). The mere presence of Lyman alpha emission in these sources provides strong evidence that they are undergoing bursts of star formation: the star formation It is our pleasure to thank our collaborators on this project: Frank Bertoldi, Dominik Riechers, Pierre Cox, Roberto Maiolino and Axel Weiß. F. Walter Max Planck Insitut für Astronomie Heidelberg, E-mail: wal- ter@mpia.de C. Carilli National Radio Astronomy Observatory, E-mail: ccar- illi@nrao.edu rates of individual objects are ∼10M⊙ yr −1 (based on their FUV luminosities) and their redshifts place them well within the end of cosmic reionization. They also appear to be very numerous: Tanaguchi et al. 2005 find ∼30 LAEs in only a quarter degree field (e.g., compared to ∼10 QSOs at z>6 which are distributed over a quar- ter of the sky! Fan et al. 2004). This implies that LAEs may play an important role in reionizing the universe at z>6 (for a review see Fan, Carilli & Keating, 2006). In- vestigating the physical properties of these sources are thus of great interest and ALMA will play a critical role in studying these objects, as discussed in the following. 2 Interstellar Medium: CO vs. [CII] emission 2.1 Carbon Monoxide (CO) Constraining the properties of the molecular gas in ob- jects at the end of cosmic reionization is clearly of key importance as such observations 1) will measure the available ’fuel’ for star formation, 2) will help to con- strain the dynamical mass of the system and will thus 3) allow to put these objects in an evolutionary con- text for early galaxy formation. Typically, at low and high redshifts, CO emission is used as a tracer for the molecular gas phase (e.g. review by Solomon & Vanden Bout 2004). It is important to keep in mind though, that, at the highest z, only the very high rotational lines of CO will be observable with ALMA. E.g. even in the lowest (currently funded) frequency band of ALMA (band 3, 84–119GHz), only CO transitions with J>7 (i.e., CO(7–6), CO(8–7), etc.) will be observable at z>7. This is graphically illustrated in Fig. 1 where we plot ALMA’s ‘CO discovery space’ (i.e., which line can be observed at which redshift using which ALMA band). The high–J transitions correspond to highly excited gas http://arxiv.org/abs/0704.1858v1 0 5 10 15 20 Redshift Band 4 (125-163 GHz) Band 3 ( 84-119 GHz) Band 4 (125-163 GHz) Band 6 (211-275 GHz) Band 7 (275-370 GHz) Band 8 (385-500 GHz) Band 9 (602-720 GHz) Band 10 (787-950 GHz) <− [CII] Fig. 1 ALMA CO ‘discocery space’: The horizontal lines indicate which CO transition (plotted on the y–axis) can be observed with which ALMA band as a function of redshift (plotted on the x– axis). For objects with z>7, only the higher–J CO transitions can be observed with ALMA. The [CII] ‘discovery space’ is also indicated. (either due to high kinetic temperatures, high densities, or both) which may not be excited in normal starform- ing environments. This is shown in Fig. 2 (taken from Weiss et al. 2005) where measured CO line strengths (as a function of J, this is sometimes referred to as CO line ladders/SEDs) are plotted for a number of key sources. What is immediately obvious from this plot is that most objects have sharply decreasing CO line strengths be- yond J> 8, in particular starforming systems such as NGC253, or the sub–millimeter galaxy plotted in this diagram (the quasars appear to be more excited, but their CO line SED still turns over at J∼7, for an excep- tional object see APM07279, Weiss et al. 2007). This comparison immediately implies that emission from the CO molecule will typically be very difficult to observe with ALMA at z>7 as the observable lines will simply not be excited. 2.2 Ionized Carbon ([CII]) to the Rescue! An alternative tracer of the interstellar medium is one of the main cooling line of the ISM, the 2P3/2 → 2P1/2 fine–structure line of C+ (or [CII]). In brief, the [CII] line is expected to be much stronger than any of the CO lines. Given its high frequency (157.74µm, corre- sponding to 1900.54GHz) [CII] studies in the local uni- verse are limited to airborne or satellite missions (e.g. Stacey et al. 1991, Malhotra et al. 1997, Madden et Fig. 2 Comparison of various normalized (by their CO(1–0) flux density) CO line SEDs at low and high redshift (figure taken from Weiss et al. 2005). The CO line SEDs decline rapidly beyond J=6–8. al. 1997). These studies have demonstrated that this single line can indeed carry a good fraction of the total infrared luminosity (LFIR) of an entire galaxy. In the lo- cal universe, the ratio LCII/LFIR has been found to be 2–5×10−4 in the case of ULIRGS (e.g. Gerin & Phillips 2000), but is more like 5–10×10−3 in more typical star- forming galaxies (for a discussion on possible reasons for the supressed ratio in ULIRGs see, e.g., Luhman et al. 1998). Notably, the ratio has been found to be 1% or even higher in low metallicity environments. E.g., in the low- metallicity galaxy IC10, LCII/LFIR reaches values as high as 4%, with an average value of 2% (Madden et al. 1997, see Israel et al. 1996 for a similar result for the LMC). This is the reason why it has long been ar- gued (e.g., Stark 1997) that observation of the [CII] line of prestine systems at the highest redshifts will likely be the key to study molecular gas in the earliest star- forming systems, in particular in the era of ALMA. The ALMA [CII] ‘discovery space’ is also indicated in Fig. 1. 3 Expected [CII] line strengths At the redshifts of the LAEs, the [CII] line is shifted to the 1mm band of ALMA (band 6, 211–275GHz). [CII] emission has recently been successfully detected using the IRAM 30m in the highest redshift quasar J1148+5251 at z=6.42 (Maiolino et al. 2005, see Fig. 3). The noteable difference between J1148+5251 and the z>6 LAEs is that the ratio LCII/LFIR has been found to be very low (∼5×10−4) in J1148+5251, i.e. in perfect Fig. 3 Top: First detection of [CII] at high redshift in the z=6.42 QSO J1148+5251 (Maiolino et al. 2005). Bottom: brightest CO transition (J=6) in the same source (Bertoldi et al. 2003, Walter et al. 2003). Note that the [CII] line is brighter by a factor of ∼5. agreement with studies of low redshift ULIRGs that show a central AGN. On the contrary, the LAE are presumably pure starbursts (no evidence for an AGN is found, Taniguchi et al. 2005) and they likely have lower metallicities compared to the highly overdense regions in which the luminous quasars are supposedly present. All these arguments point towards a LCII/LFIR ratio in LAE that is close to what is found for nearby normal galaxies, or perhaps even for the metal–poor dwarf galaxies (i.e., around 1% or even higher). In other words, the [CII] luminosity of the LAE may well be an order of magnitude stronger (for a given IR luminos- ity) than what has been found in the z=6.4 QSO. In the following we present a quick back–of–the envelope calculation based on the detected [CII] line strength in J 1148+5251 (∼10 mJy) which has a SFR of a few 1000 M⊙ yr −1. This SFR is more than two order of magni- tudes higher than the SFR found in a typical LAE, but as the LCII/LFIR may be higher by an order of magni- tude in the LAEs, the expected [CII] line strength of the LAE may be as high as 1mJy. Such a line should be easily detectable with ALMA at high significance in a few hours. 4 Resolving the ISM Detecting the [CII] (or CO) emission is critical to esti- mate the reservoir of the (molecular) gas in these early systems. A second step is then to spatially resolve the molecular gas distribution. In particular, given the typ- RIGHT ASCENSION (J2000) 11 48 16.70 16.68 16.66 16.64 16.62 16.60 52 51 50.8 Fig. 4 High–resolution CO image of the z=6.42 QSO J 1148+5251 obtained at the VLA (Walter et al. 2004). The res- olution achieved in these observations (0.15′′, corresponding to ∼1 kpc) will be routinely reached with ALMA. ical diameters of galaxies of many kpc, a linear resolu- tion of ∼ 1 kpc is needed to resolve the structure of the underlying galaxy. Such measurements are needed 1) to get an estimate for the size of the galaxy (and thus a better estimate for the dynamical mass), 2) to resolve potentially merging systems and 3) to better constrain the physical properties of the gas (e.g., by measuring the brightness temperature of the hosts). A linear res- olution of 1 kpc corresponds to a resolution of 0.15′′ at the redshifts under consideration (1′′ ∼ 5.8 kpc at z = 6). Such observations can then in turn be used to constrain the predictions by CDM simulations of early galaxy formation, and, if a large sample was available, put limits on the frequency of mergers at high redshift. In addition, such studies can be used to constrain the possible redshift-evolution of the MBH–σv relation in high–z quasars. Such observations will clearly be feasi- ble with ALMA in the extended arrays. High–resolution CO imaging is already possible with the current genera- tion of telescopes: we have used the VLA to resolve the molecular gas in the host galaxy of the z=6.42 QSO J1148+5251 (see Fig. 4, Walter et al. 2004). 5 The case for ALMA band 5 As a technical note: the ALMA redshift coverage for the [CII] line is not ideal as its frequency lies between the CO(17–16) and CO(16–15) transition (see Fig. 1). One concern is that the critical redshift range (8<z<10.5) is currently not fully covered: This frequency range corre- sponds to the ALMA band 5 which is only partly funded by the European Union as part of the Sixth Frame- work Programme (FP6) for up to 8 antennas. Clearly, it would be highly desireable to equip as many ALMA antennas with band 5 receivers as possible. 6 Concluding remarks ALMA observations of the [CII] line will play a fun- damental role in studying the youngest galaxies in the Epoch of Cosmic Reionization at z>7. Given the ex- pected line strengths it should be possible to resolve these galaxies in the [CII] line emission on kpc scales. Such measurements would not only constrain the sizes but would also help to derive the dynamical masses in these early starforming systems. Given the typical CO excitation in starforming galaxies (i.e. drop in excita- tion around the J∼6 transitions), ALMA will likely act as a [CII]– rather than a CO–machine for objects at these extreme redshifts. References 1. Bertoldi, F., Carilli, C. L., Cox, P., Fan, X., et al. 2003, A&A, 406, L55 2. Fan, X., Hennawi, J. F., Richards, G. T., et al. 2004, AJ, 128, 3. Fan, X., Carilli, C.L., Keating, B., 2006, ARA&A, 44, 415 4. Gerin, M., & Phillips, T. G. 2000, ApJ, 537, 644 5. Hu, E. M., Cowie, L. L., McMahon, R. G., et al. 2002, ApJ, 568, L75 6. Israel, F. P., Maloney, P. R., Geis, N., Herrmann, F., Madden, S. C., Poglitsch, A., Stacey, G.J., 1996, ApJ, 465, 738 7. Iye, M., Ota, K., Kashikawa, N., Furusawa, H., Hashimoto, T., Hattori, T., Matsuda, Y., Morokuma, T., Ouchi, M., Shi- masaku, K., 2006, Nature, 443, 186 8. Kurk, J. D., Cimatti, A., di Serego Alighieri, et al. 2004, A&A, 422, L13 9. Luhman, M. L., et al. 1998, ApJL, 504, L11 10. Madden, S. C., Poglitsch, A., Geis, N., Stacey, G. J., & Townes, C. H. 1997, ApJ, 483, 200 11. Maiolino, R., Cox, P., Caelli, P., et al. 2005, astro-ph/0508064 12. Malhotra, S., et al. 1997, ApJL, 491, L27 13. Murayama, T., Taniguchi, Y., Scoville, N.Z., et al. 2007, ApJS, astro-ph/0702458 14. Stacey, G. J., Geis, N., Genzel, R., Lugten, J. B., Poglitsch, A., Sternberg, A., & Townes, C. H. 1991, ApJ, 373, 423 15. Stark, A. A. 1997, ApJ, 481, 587 16. Stern, D., Yost, S. A., Eckart, M. E., Harrison, F. A., Helfand, D. J., Djorgovski, S. G., Malhotra, S., & Rhoads, J. E. 2005 , ApJ, 619, 12 17. Taniguchi, Y., Ajiki, M., Nagao, T. et al. 2005, PASJ, 57, 18. Walter, F., Bertoldi, F., Carilli, C. L., et al. 2003, Nature, 424, 406 19. Walter, F., Carilli, C., Bertoldi, F., et al. 2004, ApJ, 615, L17 20. Weiss, A., Downes, D., Neri, R., Walter, F., Henkel, C., Wilner, D.J., Wagg, J., Wiklind, T., 2007, A&A, in press (astro–ph/0702669) http://arxiv.org/abs/astro-ph/0508064 http://arxiv.org/abs/astro-ph/0702458 http://arxiv.org/abs/astro--ph/0702669 Introduction: The highest redshift galaxies Interstellar Medium: CO vs. [CII] emission Expected [CII] line strengths Resolving the ISM The case for ALMA band 5 Concluding remarks

WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP TADEUSZ PYTLIK AND RYSZARD SZWARC Abstract. Radial convolution operators on free groups with non- negative kernel of weak type (2, 2) and of restricted weak type (2, 2) are characterized. Estimates of weak type (p, p) are obtained as well for 1 < p < 2. 1. Introduction A discrete group G is called amenable if there exists a linear func- tional m on ℓ∞ (G) such that (1) inf f(x) ≤ m(f) ≤ sup f(x), (2) m(xf) = m(f), where xf(y) = f(x −1y). m is called a left invariant mean. Then the functionalM(f) = m(m(fx)) satisfies (1), (2) and is also right invariant, where fx(y) = f(yx). Let G be a discrete group. Consider a symmetric probability measure µ on G, i.e. µ(x)δx, µ(x) ≥ 0, µ(x) = 1, µ(x−1) = µ(x). The left convolution operator λ(µ) with µ is bounded on ℓ2(G) and ‖λ(µ)(f)‖2 = ‖µ ∗ f‖2 ≤ ‖f‖2, f ∈ ℓ 2(G). Indeed ‖µ ∗ f‖2 = µ(x)[δx ∗ f ] µ(x)‖δx ∗ f‖2 = ‖f‖2. 2000 Mathematics Subject Classification. Primary 43A15, Secondary 43A07, 43A15. Key words and phrases. free group, convolution operators, weak type, restricted weak type. The second author was supported by European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389 and by MNiSW Grant N201 054 32/4285. http://arxiv.org/abs/0704.1859v1 2 TADEUSZ PYTLIK AND RYSZARD SZWARC Thus ‖λ(µ)‖2→2 ≤ 1. Kesten [5] showed that a discrete group G is amenable iff for any symmetric probability measure µ on G we have ‖λ(µ)‖2→2 = 1. He showed that G is amenable if condition is satisfied for one measure µ such that supp µ generates G algebraically. In particular let G be gen- erated by g1, g2, . . . , gk and µ = (δgi + δg−1 ). Then G is amenable if and only if ‖λ(µ)‖2→2 = 1. In [4] Følner came up with another property equivalent to amenabil- ity. We say that a discrete group G satisfies the Følner condition if for any number ε > 0 and any finite set K ⊂ G there exists a finite set N ⊂ G such that (1) |xN △N | < ε|N |, x ∈ K. In other words N is almost K invariant. He showed that G is amenable if and only if the Følner condition holds. Assume that G is amenable. Let µ be a probability measure with finite support K. For ε = η2 > 0 choose N so as to satisfy (1). Then ‖µ ∗ χN − χN‖2 = µ(x)[χxN − χN ] µ(x)‖χxN − χN‖2 µ(x)‖χxN△N‖2 = µ(x)|xN △N |1/2 ≤ η|N |1/2 = η‖χN‖2. Therefore 〈µ ∗ χN , χN〉ℓ2(G) = 〈χN , χN〉ℓ2(G) + 〈µ ∗ χN − χN , χN〉ℓ2(G) ≥ (1− η)‖χN‖ which implies (2) sup N,M−finite 〈µ ∗ χN , χM〉 ‖χN‖2‖χM‖2 = 1 = ‖λ(µ)‖2→2. The same holds (with the same proof) for any 1 < p < +∞, i.e. (3) sup N,M−finite 〈µ ∗ χN , χM〉 ‖χN‖p‖χM‖p′ = 1 = ‖λ(µ)‖p→p, where p′ = p/(p− 1). We will use the notion of Lp,q spaces, which have been introduced by Lorentz (see [1]). Consider a general σ-finite measure space (Ω, ω) WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 3 and 1 < p < +∞. For f ∈ Lp(Ω, ω) and t > 0 we have tp ω{x : |f(x)| > t} ≤ |f(x)|pdω(x). Functions for which the left hand side is bounded form a linear space Lp,∞(Ω, ω) = f : sup tpω{x : |f(x)| > t} < +∞ called the weak Lp space. This space contains Lp(Ω, ω). For p′ = p/(p − 1) the predual of Lp ′,∞(Ω, ω) with respect to the standard inner product is denoted by Lp,1(Ω, ω). We have Lp,1(Ω, ω) ⊂ Lp(Ω, ω) ⊂ Lp,∞(Ω, ω). For p > 1 these spaces are normed. Any linear operator mapping Lp into itself is called of strong type (p, p). Linear operators T mapping Lp(Ω, ω) into Lp,∞(Ω, ω) are called of weak type (p.p), while those which map Lp,1(Ω, ω) into Lp,∞(Ω, ω) are called of restricted weak type (p, p). We will use the following facts. A linear operator T is bounded from Lp,1 into a Banach space X if and only if (4) ‖T‖L(p,1)→X = sup ‖TχE‖X ‖χE‖p < +∞. A linear operator T is bounded from Lp,1 into Lp,∞ if and only if (5) ‖T‖(p,1)→(p,∞) = sup E,F⊂Ω |〈TχE, χF 〉| ‖χE‖p‖χF‖p′ < +∞. Using this and duality between spaces L(p ′,1) and L(p,∞) we obtain (6) ‖T‖p→(p,∞) = ‖T ∗‖(p′,1)→p′ = sup ‖T ∗χE‖p′ ‖χE‖p′ The equalities (2) and (3) can be interpreted as follows. If the group G is discrete and amenable and µ is a symmetric probability measure on G, then ‖λ(µ)‖p→p = ‖λ(µ)‖(p′,1)→p′ = λ(µ)‖p→(p,∞) = ‖λ(µ)‖(p,1)→(p,∞) = 1. Hence for these groups convolution operators with nonnegative func- tions of strong type (p, p), of weak type (p, p) and of restricted weak type (p, p) coincide for any 1 < p < ∞. The situation is entirely different for nonamenable groups. Only special examples have been studied. It has been shown [9] that for p = 2 and G = Fk, the free group on k generators, k ≥ 2, there exist nonnegative functions f on G such that ‖λ(f)‖2→(2,∞) is finite 4 TADEUSZ PYTLIK AND RYSZARD SZWARC while ‖λ(f)‖2→2 is infinite, i.e. there exist convolution operators with nonnegative functions of weak type (2,2) which are not of strong type (2,2). The same has been shown for 1 < p < 2 [10]. These functions f can be chosen to be radial, i.e. constant on elements of the group G of the same length. It is an open problem if these results remain true for any discrete nonamenable group. In this work will focus on G = Fk. We are going to determine all non- negative radial functions f on G such that λ(f) is of weak type (2,2), as well those f for which λ(f) is of restricted weak type (2,2). In par- ticular we prove that these spaces are different. Next we will turn our attention to the case 1 < p < 2. By using interpolation machinery, du- ality and the results for p = 2 we will be able to determine nonnegative radial functions f for which λ(f) is of weak type (p, p). In this way we obtain a simpler proof of the upper estimate of ‖λ(f)‖p→(p,∞) obtained in [3]. Our method does not rely on deep theorems on representation theory 2. Radial convolution operators of weak type (2, 2) Let Fk = gp{g1, g2, . . . , gk} be a free group on k ≥ 2 generators. The group consists of reduced words in generators and their inverses. This representation is unique. The number of letters in reduced form defines length function on Fk. Let χn denote the indicator function of words of length n. There are 2k(2k − 1)n−1 such words. as we have 2k choices for the first letter and 2k − 1 choices for every consecutive one. Let q = 2k − 1. The next theorem generalizes the estimate for ‖λ(χn)‖2→(2,∞) given in [9]. Theorem 1. Let f = n=0 fnχn. The operator λ(f) is of weak type (2, 2) if A(f) := n,m=0 |fn||fm|q −(n+m)/2{1 + min(n,m)} < +∞. Moreover if fn ≥ 0 the condition is necessary and A(f) ≤ ‖λ(f)‖22→(2,∞) ≤ 4A(f). Proof. By (7), instead of estimating ‖λ(f)‖2→(2,∞) we may estimate ‖λ(f)‖(2,1)→2, which (see (4)) is equivalent to ‖f ∗ χE‖2 |E|1/2 WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 5 We have ‖f ∗ χE‖ 2 = 〈f ∗ f ∗ χE∗, χE〉 = n,m=0 fnfm〈χn ∗ χm ∗ χE , χE〉. Simple calculation gives that for n ≥ 1 we have χn ∗ χm = q n−1δmn χ0 + k=|n−m| k≡n+m mod 2 q(n+m−k)/2 χk. Clearly χ0 ∗ χ0 = χ0. Therefore χn ∗ χm ≤ 2 k=|n−m| k≡n+m mod 2 q(n+m−k)/2χk. Hence ‖f ∗ χE‖ 2 ≤ 2 n,m=0 fnfm q (n+m)/2 k=|n−m| k≡n+m mod 2 q−k/2 〈χk ∗ χE , χE〉. Lemma 1. 〈χk ∗ χE, χE〉 ≤ 2q [k/2]|E|. Proof. Define an operator Pk by the rule 〈Pkδx, δy〉 = 〈χk ∗ δx, δy〉 if |x| ≥ |y| 0 if |x| < |y|. 〈χk ∗ δx, δy〉 ≤ 〈Pkδx, δy〉+ 〈δx, Pkδy〉. This implies 〈χk ∗ χE , χE〉 ≤ 2〈PkχE , χE〉 ≤ 2‖PkχE‖1 ≤ 2|E| sup ‖Pkδx‖1 Pkδx = |w|=k |wx|≤|x| Let w = w1w2 where |w1| ≤ |w2| ≤ (k + 1)/2. The conditions |w| = k and |wx| ≤ |x| imply that w2 is determined by the first [(k + 1)/2] letters of x. Hence we have as many terms in the sum as choices for w1, i.e. at most q [k/2]. Thus ‖Pkδx‖1 ≤ q [k/2]. Therefore 〈χk ∗ χE, χE〉 ≤ 2q [k/2]|E|. 6 TADEUSZ PYTLIK AND RYSZARD SZWARC Lemma 1 implies that ‖f ∗ χE‖ n,m=0 |fn||fm|q (n+m)/2 k=|n−m| k≡n+m mod 2 n,m=0 |fn||fm|q (n+m)/2{1 + min(m,n)}. We obtain the upper estimate ‖λ(f)‖22→(2,∞) ≤ 4 n,m=0 |fn||fm|q (n+m)/2{1 + min(m,n)}. On the other hand if fn ≥ 0 we have ‖λ(f)‖22→(2,∞) ≥ q + 1 q−2k‖f ∗ χ2k‖ fn(χn ∗ χ2k) l=|n−2k| l≡n mod 2 q(n+2k−l)/2 χl q−l/2χl n=|2k−l| n≡l mod 2 n=|2k−l| n≡l mod 2 n=2k−l n≡l mod 2 n,m=0 fnfmq (n+m)/2 l=max(2k−n,2k−m) l≡n≡m mod 2 Considering even or odd values of m and n gives ‖λ(f)‖22→(2,∞) ≥ n,m=0 f2nf2mq n+m{1 + min(n,m)}, ‖λ(f)‖22→(2,∞) ≥ n,m=0 f2n+1f2m+1q n+m+1{1 + min(n,m)}. Since k is arbitrary ‖λ(f)‖22→(2,∞) ≥ n,m=0 n≡m mod 2 fnfmq (n+m)/2{1 + min(n,m)}. WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 7 This implies ‖λ(f)‖22→(2,∞) ≥ n,m=0 fnfmq (n+m)/2{1 + min(n,m)}, because the matrix a(n,m) = 1 + min(n,m) is positive definite. � Theorem 2. For n ≥ 0 there holds ‖λ(χn‖(2,1)→(2,∞) ≤ cq Proof. We have ‖λ(χn‖(2,1)→(2,∞) = sup E,F⊂Fr 〈χn ∗ χE, χF 〉 |E|1/2|F |1/2 The proof will be completed if we show (8) 〈χn ∗ χE , χF 〉 ≤ cq n/2|E|1/2|F |1/2. We will prove (8) by modification of the argument used in the proof of Lemma 1. Fix α ∈ R. Let Qαn denote the operator defined by the rule 〈Qαnδx, δy〉 = 〈χn ∗ δx, δy〉 if |x| ≥ q 0 if |x| < qα|y|. 〈χn ∗ δx, δy〉 ≤ 〈Q nδx, δy〉+ 〈δx, Q n δy〉. This implies 〈χn ∗ χE , χF 〉 ≤ ‖Q nχE‖1 + ‖Q n χF‖1 ≤ |E| sup ‖Qαnδx‖1 + |F | sup ‖Q−αn δx‖1(9) Qαnδx = |w|=n |wx|≤q−α|x| Let w = w2w1 where |w1| = [n/2] + [α] and |w2| = n− [n/2]− [α]. The conditions |w| = n and |wx| ≤ q−α|x| imply that w1 is determined by the first [n/2] + [α] letters of x. Hence we have as many terms in the sum as choices for w2, i.e. at most q n−[n/2]−[α]. Thus (10) ‖Qαnδx‖1 ≤ q 3/2q−αqn/2. Similarly ‖Q−αn δx‖1 ≤ q 3/2qαqn/2. Hence by (9) we get 〈χn ∗ χE , χF 〉 ≤ q 3/2qn/2{q−α |E|+ qα |F |}. 8 TADEUSZ PYTLIK AND RYSZARD SZWARC Choosing α = (log |E| − log |F |)/(2 log q) gives 〈χn ∗ χE , χF 〉 ≤ 2q 3/2qn/2|E|1/2|F |1/2. Theorem 3. Let f = n=0 fnχn and fn ≥ 0. The operator λ(f) is of restricted weak type (2, 2) if and only if f ∈ L(2,1). Proof. By Theorem 2 we have ‖λ(χn)‖(2,1)→(2,∞) ≤ Cq for some constant C > 0. Let f = n=0 fnχn. Then triangle inequality yields ‖f‖(2,1)→(2,∞) ≤ C By [8, Lemma 1] n/2 ≈ ‖f‖(2,1). On the other hand for fn ≥ 0 we have ‖f‖(2,1)→(2,∞) ≥ C sup q−(n+m)/2 〈f ∗ χn, χm〉 = C sup q−(n+m)/2 〈f, χm ∗ χn〉 ≥ C k=|n−m| k≡n+m mod 2 qk/2fk. Taking m = n or m = n+ 1 and letting n tend to infinity gives ‖f‖(2,1)→(2,∞) ≥ C q2k/2f2k, ‖f‖(2,1)→(2,∞) ≥ C q(2k+1)/2f2k+1. Therefore k=0 q k/2fk < +∞, i.e. f ∈ L (2,1) by (11). � 3. Weak type (p, p) for 1 < p < 2 Part of the next theorem, namely the first inequality is known from [3]. Actually it has been simply observed there that the inequality fol- lows by applying multilinear interpolation theorem to Pytlik’s estimate for ‖ fnλ(χn)‖p→p given in [8]. We will reprove the second inequal- ity by applying the same interpolation theorem to restricted weak type estimates given in the previous section. In this way we skip p → p WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 9 estimates whose proof as given in [8] is tricky, and later proof given in [3] makes use of advanced representation theory. Theorem 4. For 1 < p < 2 and for f = n=0 fnχn we have ‖λ(f)‖p→(p,∞) ≤ C‖f‖(p,p′). Moreover if f ≥ 0 then c‖f‖(p,p′) ≤ ‖λ(f)‖p→(p,∞). Proof. The subscript r will denote the subspace of radial functions, i.e. functions of the form n=0 fnχn, where fn are complex coefficients. By Theorem 3 we have L (2,1) r ∗ L (2,1) ⊂ L(2,∞). On the other hand L1r ∗ L 1 ⊂ L1. By multilinear interpolation theorem [1, 3.13.5, p. 76] we get L (p,s) r ∗ L (p,t) ⊂ L(p,u) where 1 ≤ p < 2 and 1 + 1/u = 1/s+ 1/t. Taking u = ∞, t = p and s = p′ gives L (p,p′) r ∗ L p ⊂ L(p,∞). This gives the first inequality. On the other hand for f = n=0 fnχn by (4) and by duality (6) we ‖λ(f)‖p→(p,∞) = ‖λ(f)‖(p′,1)→p′ ≥ c sup q−n/p ‖f ∗ χn‖p′. Similarly as in the proof of Theorem 4 we obtain f ∗ χn ≥ q(n−l)/2 m=|n−l| m≡l+n mod 2 qm/2fm Hence q−n‖f ∗ χn‖ ′(n−l)/2ql−n m=n−l m≡l+n mod 2 qm/2fm q(n−l)(p n−l = Taking supremum with respect to n and raising to the power 1/p′ give ‖λ(f)‖p→(p,∞) ≥ c np′/p )1/p′ Since the norm of f = n=0 fnχn in L (p,p′) r is equivalent to n=0 f np′/p )1/p′ the second inequality is proved. � 10 TADEUSZ PYTLIK AND RYSZARD SZWARC 4. Other estimates Theorem 5. For 1 ≤ s ≤ 2 ≤ t ≤ ∞ we have cn1−1/s+1/tqn/2 ≤ ‖λ(χn)‖(2,s)→(2,t) ≤ Cn 1−1/s+1/tqn/2. Proof. In order to get the second inequality we use only interpolation. First observe that the inequality is valid for s = 2, t = ∞ by Theorem 4 and for s = t = 2 by [2, 7]. Hence by complex interpolation of the Lorentz spaces it is valid for s = 2, t ≥ 2. Next it is valid for s = 1, t = ∞ by Theorem 3 and for s = t = 2. Hence by complex interpolation it is valid for 1 ≤ s ≤ 2, t = s′. Now we can use again complex interpolation to get the conclusion for 1 ≤ s ≤ 2 ≤ t ≤ ∞. The estimate from below can be obtained from ‖λ(χn)‖(2,s)→(2,t) ≥ ‖χn ∗ f‖(2,t) ‖f‖(2,s) where f = k=0 q −k/2χk. � Theorems 1, 2 and 5 suggest the following. Conjecture. Let f = n=0 fnχn ≥ 0. Then for 1 ≤ s ≤ 2 the operator λ(f) maps L(2,s) into L(2,∞) if and only if n,m=0 fnfmq −(n+m)/2{1 + min(n1/s , m1/s )} < +∞. References [1] J. Bergh, J. Löfström, Interpolation spaces. An introduction. Grundlehren der mathematischen Wissenschaften 223, Springer-Verlag, Berlin-Heidelberg-New York (1976). [2] J. M. Cohen, Operator norms on free groups, Boll. Unione Mat. Ital. VI. Ser. B 1(1982), 1055-1065. [3] M. Cowling, S. Meda, A. G. Setti, Alberto, Invariant operators on function spaces on homogeneous trees, Colloq. Math. 80 (1999), 53–61. [4] E. Følner, On groups with full Banach mean value, Math. Scand. 3 (1955), 243–254. [5] H. Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. [6] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354. [7] T. Pytlik, Radial functions on free groups and a decomposition of the regu- lar representation into irreducible components, J. Reine Angew. Math. 326 (1981),124-135. [8] T. Pytlik, Radial convolutors on free groups, Stud. Math. 78 (1984), 179–183. [9] R. Szwarc, Convolution operators of weak type (2,2) which are not of strong type (2,2), Proc. Am. Math. Soc. 87 (1983), 695–698. WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 11 [10] R. Szwarc, Convolution operators of weak type (p, p) which are not of strong type (p, p), Proc. Am. Math. Soc. 89 (1983), 184-185. Institute of Mathematics, University of Wroc law, pl. Grunwaldzki 2/4, 50-384 Wroc law, Poland Institute of Mathematics and Computer Science, University of Opole, ul. Oleska 48, 45-052 Opole, Poland E-mail address : szwarc2@gmail.com 1. Introduction 2. Radial convolution operators of weak type (2,2) 3. Weak type (p,p) for 1<p<2 4. Other estimates References

Radial convolution operators on free groups with nonnegative kernel of weak type $(2,2)$ and of restricted weak type $(2,2)$ are characterized. Estimates of weak type $(p,p)$ are obtained as well for $1<p<2.$

Introduction A discrete group G is called amenable if there exists a linear func- tional m on ℓ∞ (G) such that (1) inf f(x) ≤ m(f) ≤ sup f(x), (2) m(xf) = m(f), where xf(y) = f(x −1y). m is called a left invariant mean. Then the functionalM(f) = m(m(fx)) satisfies (1), (2) and is also right invariant, where fx(y) = f(yx). Let G be a discrete group. Consider a symmetric probability measure µ on G, i.e. µ(x)δx, µ(x) ≥ 0, µ(x) = 1, µ(x−1) = µ(x). The left convolution operator λ(µ) with µ is bounded on ℓ2(G) and ‖λ(µ)(f)‖2 = ‖µ ∗ f‖2 ≤ ‖f‖2, f ∈ ℓ 2(G). Indeed ‖µ ∗ f‖2 = µ(x)[δx ∗ f ] µ(x)‖δx ∗ f‖2 = ‖f‖2. 2000 Mathematics Subject Classification. Primary 43A15, Secondary 43A07, 43A15. Key words and phrases. free group, convolution operators, weak type, restricted weak type. The second author was supported by European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389 and by MNiSW Grant N201 054 32/4285. http://arxiv.org/abs/0704.1859v1 2 TADEUSZ PYTLIK AND RYSZARD SZWARC Thus ‖λ(µ)‖2→2 ≤ 1. Kesten [5] showed that a discrete group G is amenable iff for any symmetric probability measure µ on G we have ‖λ(µ)‖2→2 = 1. He showed that G is amenable if condition is satisfied for one measure µ such that supp µ generates G algebraically. In particular let G be gen- erated by g1, g2, . . . , gk and µ = (δgi + δg−1 ). Then G is amenable if and only if ‖λ(µ)‖2→2 = 1. In [4] Følner came up with another property equivalent to amenabil- ity. We say that a discrete group G satisfies the Følner condition if for any number ε > 0 and any finite set K ⊂ G there exists a finite set N ⊂ G such that (1) |xN △N | < ε|N |, x ∈ K. In other words N is almost K invariant. He showed that G is amenable if and only if the Følner condition holds. Assume that G is amenable. Let µ be a probability measure with finite support K. For ε = η2 > 0 choose N so as to satisfy (1). Then ‖µ ∗ χN − χN‖2 = µ(x)[χxN − χN ] µ(x)‖χxN − χN‖2 µ(x)‖χxN△N‖2 = µ(x)|xN △N |1/2 ≤ η|N |1/2 = η‖χN‖2. Therefore 〈µ ∗ χN , χN〉ℓ2(G) = 〈χN , χN〉ℓ2(G) + 〈µ ∗ χN − χN , χN〉ℓ2(G) ≥ (1− η)‖χN‖ which implies (2) sup N,M−finite 〈µ ∗ χN , χM〉 ‖χN‖2‖χM‖2 = 1 = ‖λ(µ)‖2→2. The same holds (with the same proof) for any 1 < p < +∞, i.e. (3) sup N,M−finite 〈µ ∗ χN , χM〉 ‖χN‖p‖χM‖p′ = 1 = ‖λ(µ)‖p→p, where p′ = p/(p− 1). We will use the notion of Lp,q spaces, which have been introduced by Lorentz (see [1]). Consider a general σ-finite measure space (Ω, ω) WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 3 and 1 < p < +∞. For f ∈ Lp(Ω, ω) and t > 0 we have tp ω{x : |f(x)| > t} ≤ |f(x)|pdω(x). Functions for which the left hand side is bounded form a linear space Lp,∞(Ω, ω) = f : sup tpω{x : |f(x)| > t} < +∞ called the weak Lp space. This space contains Lp(Ω, ω). For p′ = p/(p − 1) the predual of Lp ′,∞(Ω, ω) with respect to the standard inner product is denoted by Lp,1(Ω, ω). We have Lp,1(Ω, ω) ⊂ Lp(Ω, ω) ⊂ Lp,∞(Ω, ω). For p > 1 these spaces are normed. Any linear operator mapping Lp into itself is called of strong type (p, p). Linear operators T mapping Lp(Ω, ω) into Lp,∞(Ω, ω) are called of weak type (p.p), while those which map Lp,1(Ω, ω) into Lp,∞(Ω, ω) are called of restricted weak type (p, p). We will use the following facts. A linear operator T is bounded from Lp,1 into a Banach space X if and only if (4) ‖T‖L(p,1)→X = sup ‖TχE‖X ‖χE‖p < +∞. A linear operator T is bounded from Lp,1 into Lp,∞ if and only if (5) ‖T‖(p,1)→(p,∞) = sup E,F⊂Ω |〈TχE, χF 〉| ‖χE‖p‖χF‖p′ < +∞. Using this and duality between spaces L(p ′,1) and L(p,∞) we obtain (6) ‖T‖p→(p,∞) = ‖T ∗‖(p′,1)→p′ = sup ‖T ∗χE‖p′ ‖χE‖p′ The equalities (2) and (3) can be interpreted as follows. If the group G is discrete and amenable and µ is a symmetric probability measure on G, then ‖λ(µ)‖p→p = ‖λ(µ)‖(p′,1)→p′ = λ(µ)‖p→(p,∞) = ‖λ(µ)‖(p,1)→(p,∞) = 1. Hence for these groups convolution operators with nonnegative func- tions of strong type (p, p), of weak type (p, p) and of restricted weak type (p, p) coincide for any 1 < p < ∞. The situation is entirely different for nonamenable groups. Only special examples have been studied. It has been shown [9] that for p = 2 and G = Fk, the free group on k generators, k ≥ 2, there exist nonnegative functions f on G such that ‖λ(f)‖2→(2,∞) is finite 4 TADEUSZ PYTLIK AND RYSZARD SZWARC while ‖λ(f)‖2→2 is infinite, i.e. there exist convolution operators with nonnegative functions of weak type (2,2) which are not of strong type (2,2). The same has been shown for 1 < p < 2 [10]. These functions f can be chosen to be radial, i.e. constant on elements of the group G of the same length. It is an open problem if these results remain true for any discrete nonamenable group. In this work will focus on G = Fk. We are going to determine all non- negative radial functions f on G such that λ(f) is of weak type (2,2), as well those f for which λ(f) is of restricted weak type (2,2). In par- ticular we prove that these spaces are different. Next we will turn our attention to the case 1 < p < 2. By using interpolation machinery, du- ality and the results for p = 2 we will be able to determine nonnegative radial functions f for which λ(f) is of weak type (p, p). In this way we obtain a simpler proof of the upper estimate of ‖λ(f)‖p→(p,∞) obtained in [3]. Our method does not rely on deep theorems on representation theory 2. Radial convolution operators of weak type (2, 2) Let Fk = gp{g1, g2, . . . , gk} be a free group on k ≥ 2 generators. The group consists of reduced words in generators and their inverses. This representation is unique. The number of letters in reduced form defines length function on Fk. Let χn denote the indicator function of words of length n. There are 2k(2k − 1)n−1 such words. as we have 2k choices for the first letter and 2k − 1 choices for every consecutive one. Let q = 2k − 1. The next theorem generalizes the estimate for ‖λ(χn)‖2→(2,∞) given in [9]. Theorem 1. Let f = n=0 fnχn. The operator λ(f) is of weak type (2, 2) if A(f) := n,m=0 |fn||fm|q −(n+m)/2{1 + min(n,m)} < +∞. Moreover if fn ≥ 0 the condition is necessary and A(f) ≤ ‖λ(f)‖22→(2,∞) ≤ 4A(f). Proof. By (7), instead of estimating ‖λ(f)‖2→(2,∞) we may estimate ‖λ(f)‖(2,1)→2, which (see (4)) is equivalent to ‖f ∗ χE‖2 |E|1/2 WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 5 We have ‖f ∗ χE‖ 2 = 〈f ∗ f ∗ χE∗, χE〉 = n,m=0 fnfm〈χn ∗ χm ∗ χE , χE〉. Simple calculation gives that for n ≥ 1 we have χn ∗ χm = q n−1δmn χ0 + k=|n−m| k≡n+m mod 2 q(n+m−k)/2 χk. Clearly χ0 ∗ χ0 = χ0. Therefore χn ∗ χm ≤ 2 k=|n−m| k≡n+m mod 2 q(n+m−k)/2χk. Hence ‖f ∗ χE‖ 2 ≤ 2 n,m=0 fnfm q (n+m)/2 k=|n−m| k≡n+m mod 2 q−k/2 〈χk ∗ χE , χE〉. Lemma 1. 〈χk ∗ χE, χE〉 ≤ 2q [k/2]|E|. Proof. Define an operator Pk by the rule 〈Pkδx, δy〉 = 〈χk ∗ δx, δy〉 if |x| ≥ |y| 0 if |x| < |y|. 〈χk ∗ δx, δy〉 ≤ 〈Pkδx, δy〉+ 〈δx, Pkδy〉. This implies 〈χk ∗ χE , χE〉 ≤ 2〈PkχE , χE〉 ≤ 2‖PkχE‖1 ≤ 2|E| sup ‖Pkδx‖1 Pkδx = |w|=k |wx|≤|x| Let w = w1w2 where |w1| ≤ |w2| ≤ (k + 1)/2. The conditions |w| = k and |wx| ≤ |x| imply that w2 is determined by the first [(k + 1)/2] letters of x. Hence we have as many terms in the sum as choices for w1, i.e. at most q [k/2]. Thus ‖Pkδx‖1 ≤ q [k/2]. Therefore 〈χk ∗ χE, χE〉 ≤ 2q [k/2]|E|. 6 TADEUSZ PYTLIK AND RYSZARD SZWARC Lemma 1 implies that ‖f ∗ χE‖ n,m=0 |fn||fm|q (n+m)/2 k=|n−m| k≡n+m mod 2 n,m=0 |fn||fm|q (n+m)/2{1 + min(m,n)}. We obtain the upper estimate ‖λ(f)‖22→(2,∞) ≤ 4 n,m=0 |fn||fm|q (n+m)/2{1 + min(m,n)}. On the other hand if fn ≥ 0 we have ‖λ(f)‖22→(2,∞) ≥ q + 1 q−2k‖f ∗ χ2k‖ fn(χn ∗ χ2k) l=|n−2k| l≡n mod 2 q(n+2k−l)/2 χl q−l/2χl n=|2k−l| n≡l mod 2 n=|2k−l| n≡l mod 2 n=2k−l n≡l mod 2 n,m=0 fnfmq (n+m)/2 l=max(2k−n,2k−m) l≡n≡m mod 2 Considering even or odd values of m and n gives ‖λ(f)‖22→(2,∞) ≥ n,m=0 f2nf2mq n+m{1 + min(n,m)}, ‖λ(f)‖22→(2,∞) ≥ n,m=0 f2n+1f2m+1q n+m+1{1 + min(n,m)}. Since k is arbitrary ‖λ(f)‖22→(2,∞) ≥ n,m=0 n≡m mod 2 fnfmq (n+m)/2{1 + min(n,m)}. WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 7 This implies ‖λ(f)‖22→(2,∞) ≥ n,m=0 fnfmq (n+m)/2{1 + min(n,m)}, because the matrix a(n,m) = 1 + min(n,m) is positive definite. � Theorem 2. For n ≥ 0 there holds ‖λ(χn‖(2,1)→(2,∞) ≤ cq Proof. We have ‖λ(χn‖(2,1)→(2,∞) = sup E,F⊂Fr 〈χn ∗ χE, χF 〉 |E|1/2|F |1/2 The proof will be completed if we show (8) 〈χn ∗ χE , χF 〉 ≤ cq n/2|E|1/2|F |1/2. We will prove (8) by modification of the argument used in the proof of Lemma 1. Fix α ∈ R. Let Qαn denote the operator defined by the rule 〈Qαnδx, δy〉 = 〈χn ∗ δx, δy〉 if |x| ≥ q 0 if |x| < qα|y|. 〈χn ∗ δx, δy〉 ≤ 〈Q nδx, δy〉+ 〈δx, Q n δy〉. This implies 〈χn ∗ χE , χF 〉 ≤ ‖Q nχE‖1 + ‖Q n χF‖1 ≤ |E| sup ‖Qαnδx‖1 + |F | sup ‖Q−αn δx‖1(9) Qαnδx = |w|=n |wx|≤q−α|x| Let w = w2w1 where |w1| = [n/2] + [α] and |w2| = n− [n/2]− [α]. The conditions |w| = n and |wx| ≤ q−α|x| imply that w1 is determined by the first [n/2] + [α] letters of x. Hence we have as many terms in the sum as choices for w2, i.e. at most q n−[n/2]−[α]. Thus (10) ‖Qαnδx‖1 ≤ q 3/2q−αqn/2. Similarly ‖Q−αn δx‖1 ≤ q 3/2qαqn/2. Hence by (9) we get 〈χn ∗ χE , χF 〉 ≤ q 3/2qn/2{q−α |E|+ qα |F |}. 8 TADEUSZ PYTLIK AND RYSZARD SZWARC Choosing α = (log |E| − log |F |)/(2 log q) gives 〈χn ∗ χE , χF 〉 ≤ 2q 3/2qn/2|E|1/2|F |1/2. Theorem 3. Let f = n=0 fnχn and fn ≥ 0. The operator λ(f) is of restricted weak type (2, 2) if and only if f ∈ L(2,1). Proof. By Theorem 2 we have ‖λ(χn)‖(2,1)→(2,∞) ≤ Cq for some constant C > 0. Let f = n=0 fnχn. Then triangle inequality yields ‖f‖(2,1)→(2,∞) ≤ C By [8, Lemma 1] n/2 ≈ ‖f‖(2,1). On the other hand for fn ≥ 0 we have ‖f‖(2,1)→(2,∞) ≥ C sup q−(n+m)/2 〈f ∗ χn, χm〉 = C sup q−(n+m)/2 〈f, χm ∗ χn〉 ≥ C k=|n−m| k≡n+m mod 2 qk/2fk. Taking m = n or m = n+ 1 and letting n tend to infinity gives ‖f‖(2,1)→(2,∞) ≥ C q2k/2f2k, ‖f‖(2,1)→(2,∞) ≥ C q(2k+1)/2f2k+1. Therefore k=0 q k/2fk < +∞, i.e. f ∈ L (2,1) by (11). � 3. Weak type (p, p) for 1 < p < 2 Part of the next theorem, namely the first inequality is known from [3]. Actually it has been simply observed there that the inequality fol- lows by applying multilinear interpolation theorem to Pytlik’s estimate for ‖ fnλ(χn)‖p→p given in [8]. We will reprove the second inequal- ity by applying the same interpolation theorem to restricted weak type estimates given in the previous section. In this way we skip p → p WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 9 estimates whose proof as given in [8] is tricky, and later proof given in [3] makes use of advanced representation theory. Theorem 4. For 1 < p < 2 and for f = n=0 fnχn we have ‖λ(f)‖p→(p,∞) ≤ C‖f‖(p,p′). Moreover if f ≥ 0 then c‖f‖(p,p′) ≤ ‖λ(f)‖p→(p,∞). Proof. The subscript r will denote the subspace of radial functions, i.e. functions of the form n=0 fnχn, where fn are complex coefficients. By Theorem 3 we have L (2,1) r ∗ L (2,1) ⊂ L(2,∞). On the other hand L1r ∗ L 1 ⊂ L1. By multilinear interpolation theorem [1, 3.13.5, p. 76] we get L (p,s) r ∗ L (p,t) ⊂ L(p,u) where 1 ≤ p < 2 and 1 + 1/u = 1/s+ 1/t. Taking u = ∞, t = p and s = p′ gives L (p,p′) r ∗ L p ⊂ L(p,∞). This gives the first inequality. On the other hand for f = n=0 fnχn by (4) and by duality (6) we ‖λ(f)‖p→(p,∞) = ‖λ(f)‖(p′,1)→p′ ≥ c sup q−n/p ‖f ∗ χn‖p′. Similarly as in the proof of Theorem 4 we obtain f ∗ χn ≥ q(n−l)/2 m=|n−l| m≡l+n mod 2 qm/2fm Hence q−n‖f ∗ χn‖ ′(n−l)/2ql−n m=n−l m≡l+n mod 2 qm/2fm q(n−l)(p n−l = Taking supremum with respect to n and raising to the power 1/p′ give ‖λ(f)‖p→(p,∞) ≥ c np′/p )1/p′ Since the norm of f = n=0 fnχn in L (p,p′) r is equivalent to n=0 f np′/p )1/p′ the second inequality is proved. � 10 TADEUSZ PYTLIK AND RYSZARD SZWARC 4. Other estimates Theorem 5. For 1 ≤ s ≤ 2 ≤ t ≤ ∞ we have cn1−1/s+1/tqn/2 ≤ ‖λ(χn)‖(2,s)→(2,t) ≤ Cn 1−1/s+1/tqn/2. Proof. In order to get the second inequality we use only interpolation. First observe that the inequality is valid for s = 2, t = ∞ by Theorem 4 and for s = t = 2 by [2, 7]. Hence by complex interpolation of the Lorentz spaces it is valid for s = 2, t ≥ 2. Next it is valid for s = 1, t = ∞ by Theorem 3 and for s = t = 2. Hence by complex interpolation it is valid for 1 ≤ s ≤ 2, t = s′. Now we can use again complex interpolation to get the conclusion for 1 ≤ s ≤ 2 ≤ t ≤ ∞. The estimate from below can be obtained from ‖λ(χn)‖(2,s)→(2,t) ≥ ‖χn ∗ f‖(2,t) ‖f‖(2,s) where f = k=0 q −k/2χk. � Theorems 1, 2 and 5 suggest the following. Conjecture. Let f = n=0 fnχn ≥ 0. Then for 1 ≤ s ≤ 2 the operator λ(f) maps L(2,s) into L(2,∞) if and only if n,m=0 fnfmq −(n+m)/2{1 + min(n1/s , m1/s )} < +∞. References [1] J. Bergh, J. Löfström, Interpolation spaces. An introduction. Grundlehren der mathematischen Wissenschaften 223, Springer-Verlag, Berlin-Heidelberg-New York (1976). [2] J. M. Cohen, Operator norms on free groups, Boll. Unione Mat. Ital. VI. Ser. B 1(1982), 1055-1065. [3] M. Cowling, S. Meda, A. G. Setti, Alberto, Invariant operators on function spaces on homogeneous trees, Colloq. Math. 80 (1999), 53–61. [4] E. Følner, On groups with full Banach mean value, Math. Scand. 3 (1955), 243–254. [5] H. Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. [6] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354. [7] T. Pytlik, Radial functions on free groups and a decomposition of the regu- lar representation into irreducible components, J. Reine Angew. Math. 326 (1981),124-135. [8] T. Pytlik, Radial convolutors on free groups, Stud. Math. 78 (1984), 179–183. [9] R. Szwarc, Convolution operators of weak type (2,2) which are not of strong type (2,2), Proc. Am. Math. Soc. 87 (1983), 695–698. WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 11 [10] R. Szwarc, Convolution operators of weak type (p, p) which are not of strong type (p, p), Proc. Am. Math. Soc. 89 (1983), 184-185. Institute of Mathematics, University of Wroc law, pl. Grunwaldzki 2/4, 50-384 Wroc law, Poland Institute of Mathematics and Computer Science, University of Opole, ul. Oleska 48, 45-052 Opole, Poland E-mail address : szwarc2@gmail.com 1. Introduction 2. Radial convolution operators of weak type (2,2) 3. Weak type (p,p) for 1<p<2 4. Other estimates References

Draft version November 1, 2018 Preprint typeset using LATEX style emulateapj v. 7/8/03 HOW MERGERS MAY AFFECT THE MASS SCALING RELATION BETWEEN GRAVITATIONALLY BOUND SYSTEMS Chien Y. Peng1,2 (Received 2003 March 13) Draft version November 1, 2018 ABSTRACT Supermassive black hole (BH) masses (MBH) are strongly correlated with galaxy stellar bulge masses (Mbulge) and there are several ideas to explain the origin of this relationship. This study isolates the role of galaxy mergers from considerations of other detailed physics to more clearly show how a linear BH–galaxy mass relation (MBH-Mgal) can naturally emerge regardless of how primordial BHs were seeded inside galaxies, if the galaxy mass function declines with increasing mass. Under this circumstance, the MBH-Mgal relation is a passive attractor that eventually converges to a tight linear relation because of two basic statistical effects: a central limit-like tendency for galaxy mergers which is much stronger for major mergers than for minor mergers, and a convergence towards a linear relation that is due mainly to minor mergers. A curious consequence of this thought experiment is that, if galaxy bulges are formed by major mergers, then merging statistics naturally show that MBH would correlate more strongly with bulge dominated galaxies, because of stronger central-seeking tendencies, than with disk dominated galaxies. Even if some other physics is ultimately responsible for causing a linear MBH-Mbulge relationship, this thought experiment shows that, counter to intuition, random merging of galaxies tends to strengthen rather than weaken a pre-existing, linear, correlation. This idea may be generalized to other gravitationally bound systems (dark matter halo, compact nuclear objects) that retain their physical identities after experiencing mergers. Subject headings: galaxies: bulges — galaxies: formation — galaxies: evolution – galaxies: statistics 1. INTRODUCTION In recent years, there have been several surpris- ing discoveries of fundamental scaling relations be- tween supermassive black hole masses (MBH) and large scale galaxy bulge properties: stellar veloc- ity dispersion σ∗, bulge mass Mbulge, the pro- file slope of galaxies, and the inner core ra- dius (Gebhardt et al. 2000; Ferrarese & Merritt 2000; Kormendy & Richstone 1995; Magorrian et al. 1998; Ho 1999; Kormendy & Gebhardt 2001; Graham et al. 2001; Marconi & Hunt 2003; Häring & Rix 2004; Barth et al. 2005; Lauer et al. 2006; Greene & Ho 2006; Woo et al. 2006). At masses lower than Mbulge. 1010M⊙, the cen- tral object with which galaxies correlate may be either in- termediate mass BHs in dwarf galaxies (Filippenko & Ho 2003; Barth et al. 2004, 2005; Greene & Ho 2006) or central massive star clusters (Ferrarese et al. 2006; Graham & Driver 2007). The small amount of intrin- sic scatter between black holes and bulges is often inter- preted to suggest a causal connection between the two — that the growth of one might somehow regulate the other (e.g. Silk & Rees 1998; Di Matteo et al. 2005). When did the fundamental BH scaling relationships come about? At higher redshifts, observations are still in the early stages, complicated in part by the chal- lenges of measuring the BH mass and the bulge prop- erties in the same galaxy. However, recent evidence from the study of quasar host galaxies indicates that a fundamental MBH-Mbulge correlation might have been present as early as z ∼ 4 (Peng et al. 2006a,b). Fur- 1 Space Telescope Science Institute, 3700 San Martin Drive, Bal- timore, MD 21218; cyp@stsci.edu. 2 STScI (Giacconi) Fellow thermore, there also appears to be an evolution in the MBH-Mbulge ratio by a factor of ∼ 4 in the same studies, which points to the possibility that the BH masses may have matured more quickly than their surrounding stellar bulge mass in the past. In addition, other observations that use [Oiii] and CO emission line widths to infer the gravitational potential of quasar bulges (Shields et al. 2003, 2006; Salviander et al. 2006, 2007; Ho 2007) sug- gest similar trends3, and residual traces of evolution re- main detectable even below z = 1 (Treu et al. 2004, 2007; Woo et al. 2006). Despite the general agreements ob- servationally (but, see Li et al. 2006; Borys et al. 2005), there remain several weaknesses that still complicate the interpretation: a significant factor of 2 systematic un- certainty in the normalization of the BH mass calibra- tion, the possibility that the normalization of the virial BH mass estimate (Kaspi et al. 2000; Vestergaard 2002; Onken et al. 2004; Kaspi et al. 2005; Greene & Ho 2005; Peterson & Bentz 2006; Vestergaard & Peterson 2006) may evolve with time, because of the fact that the bulge masses are not directly measured, and the possibility that the host galaxy mass may be biased low in high redshift quasars (Lauer et al. 2007) because of the steep decline in the luminosity function of galaxies. With re- gard to the latter selection bias, it is worthwhile to note, however, that low redshift (z . 0.3) quasars host galax- ies (McLure & Dunlop 2001), which have similarly high MBH andMgal to the high redshift objects in Peng et al. (2006a,b), do not exhibit a mass dependent bias even when the scatter is large. Despite these uncertainties, the finding of an existing strong MBH-Mbulge correla- 3 A lack of evolution in the stellar velocity dispersion im- plies that bulge mass decreases with look-back time (see, e.g, Robertson et al. 2006b). http://arxiv.org/abs/0704.1860v5 2 PENG tion at z & 1 is likely to be more secure. Controversies about the evolution aside, it is not only puzzling that the MBH-bulge correlations should ex- ist, but that they would have a small intrinsic scatter of 0.3 dex in MBH and that the correlation with the bulge mass is practically linear (Marconi & Hunt 2003; Häring & Rix 2004; Lauer et al. 2006). These curious facts have received wide theoretical attention over the years and could be explained in a number of ways. For instance, both the MBH-Mbulge and MBH-σ∗ relation can be produced by gas accretion onto a nuclear disk (Burkert & Silk 2001; Cen 2007) or turbulent dissipa- tion of gas (Escala 2006) followed by star formation, by a combination of BH accretion, galaxy mergers, and star formation (Li et al. 2006; Kauffmann & Haehnelt 2000; Haehnelt & Kauffmann 2000), by gravitational collapse of inner parts of a galaxy that forms the bulge from a rotating isothermal sphere (Adams et al. 2001, 2003), by stellar capture (Miralda-Escudé & Kollmeier 2005) in the accretion disk. The effect of dissipation in galaxy merging on the BH and fundamental plane scaling rela- tions has also been visited: Ciotti & van Albada (2001) speculated, and Robertson et al. (2006a) show, that dis- sipational mergers can produce the MBH-σ∗ slope and maintain the fundamental plane of elliptical galaxies. On the other hand, whereas dissipationlessmergers of ellipti- cal galaxies also tend to preserve the observed fundamen- tal plane (Boylan-Kolchin et al. 2006; Robertson et al. 2006a), dissipationless mergers of disk galaxies tend to produce a relation that more closely parallels the virial plane. Despite the aforementioned models, the explanation that has spawned immense activity is the theory of feed- back from an active galactic nucleus (AGN). The AGN feedback idea rests observations that quasars typically radiate at a fixed fraction (10%−100%) of the Edding- ton luminosity. If such radiation is produced by a massive enough BH, this energy budget is in principle sufficient to quench star formation and terminate the BH growth itself (e.g. Silk & Rees 1998; Di Matteo et al. 2005; Cattaneo et al. 2005; Springel et al. 2005; Hopkins et al. 2007, 2006a, and references therein). This idea has been incorporated into cosmological merger simulations (e.g. Granato et al. 2004; Fontanot et al. 2006; Croton et al. 2006), and is seen to have profound possibilities for ex- plaining a wide array of other galaxy evolution puzzles, including the evolution in the galaxy and quasar luminos- ity functions, mass functions, star formation rates, the X-ray background, and the bimodality of galaxy colors (e.g. see Hopkins et al. 2006a,b, and references therein). Even though AGN feedback is promising for explaining many aspects of galaxy evolution, it is possible that some other mechanism (or mechanisms) may either have signif- icant influence on the scaling relation between MBH and Mbulge, or in fact be more fundamentally the cause. It is therefore worthwhile to look for such factors and to fully study their effects. Dating back years before AGN feed- back physics became a popular idea, one such fundamen- tal factor considered by many was galaxy merging (e.g. Kauffmann & Haehnelt 2000; Haehnelt & Kauffmann 2000; Ciotti & van Albada 2001; Nipoti et al. 2003; Islam et al. 2003, 2004; Volonteri et al. 2003). Even without feedback, those simulations already find it pos- sible to produce the BH scaling relations, albeit to dif- fering degrees of agreement with the observations. How- ever, the reason why a tight linear correlation should emerge is not immediately apparent even when consider- ing no other physics besides merging. In fact, as accorded correctly by intuition, a correlation indeed cannot arise spontaneously from chaotic combinations of galaxies and black hole masses in general. As will be shown below, what does make a tight correlation emerge is the fact that the galaxy mass function decreases with increasing mass. This circumstance brings about a number of inter- esting implications that will be explored in future stud- ies. In this study, the modest goal is to show that when the focus is switched from the BH-bulge relation, to un- derstanding the more general BH-galaxy relation, some insights may be gained into understanding the growth and evolution of the MBH-Mbulge relation itself. This study therefore revisits the issue of galaxy merg- ing from the standpoint of a thought experiment. This toy model identifies three basic reasons for why the MBH-Mbulge relation may appear the way it does, even if the BH masses and their host galaxy masses were completely uncorrelated initially, or if they started out with a steep powerlaw correlation. The over-arching premise is that the galaxy mass function has a Schechter (1976) powerlaw form, so that there is a decline in galaxy number density with increasing mass, especially above masses M∗. When this circumstance is met, it can be shown that a “linear attractor” and a central limit- like tendency can work efficiently to produce a linear MBH-Mgal relation, and to reduce the scatter over time. However, while it is tempting to generalize this result to the MBH-σ∗ relation, it is not as simple to do be- cause it is not clear how the stellar velocity dispersion scales during galaxy mergers, something that depends on physics not considered in this study (e.g. dissipational vs. dissipationless mergers, star formation, AGN feed- back – see Robertson et al. 2006b; Hopkins et al. 2007; Di Matteo et al. 2007; Sijacki et al. 2007). The main emphasis of the current study is thus only to present a pure statistical exercise, and as such will not invoke merger trees or external physics – be- sides which many and much more sophisticated mod- els have already been run (e.g. Volonteri et al. 2003; Islam et al. 2004; Granato et al. 2004; Fontanot et al. 2006; Robertson et al. 2006b; Ciotti et al. 2007). In a sense, this work examines a common thread shared by all previous cosmological simulations. While it is tempt- ing to invoke realism by introducing detailed physics from the get-go, e.g. star formation, BH accretion, and AGN feedback, isolating the effects of simple statistics enables a cleaner exposition of why the convergence be- havior should be expected. As such, the current study is not a critique of other, more physical, models which can also explain the same correlation, or to pass criti- cal judgment about which is more or less relevant. In- stead, the main message of this study is that, regard- less of what other physics may ultimately produce the MBH-Mbulge relation or weaken it, an existing corre- lation should strengthen if galaxies continue to merge thereafter, whether by major or by minor mergers. In other words, merging alone, in the absence of all other physics, is a sufficient condition to bring about a tight, linear, MBH-Mbulge relation over time, and is always “pulling behind the scenes” to bring about such a corre- MASS SCALING RELATIONS AND GALAXY MERGERS 3 NGAL M2 �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� Fig. 1.— (a) Two examples of initial correlations between MBH and Mgal. (b) An arbitrary number distribution of MBH at each mass slice M1 or M2, where µ is the mean value of the distribution. lation. The issue of interest to follow up is to what extent this, or other as yet identified mechanisms, may matter in the end, and to predict more realistic scatter in the MBH-Mgal relation (as opposed to only MBH-Mbulge) under this hypothesis using more realistic merger trees and physics. We will address this subject in a followup study. The following discussion will begin by presenting a heuristic view to explain why a linear MBH-Mgal re- lation is a natural consequence of random merging (Sec- tion 2), followed by Monte-Carlo simulations to illustrate the effect (Section 3), and lastly by a discussion and con- clusion. In much of the discussion below, the relationship under consideration is more generally the MBH-Mgal re- lation, whereas BHs are thought to correlate more tightly with the spheroid component of Mgal, that is Mbulge. It will be seen that the tighter correlation between MBH and Mbulge is a special case of the MBH-Mgal relation and can be understood in the same framework if this hy- pothesis represents the dominant mechanism by which BHs correlate with galaxy masses. 2. HEURISTIC PICTURES 2.1. The Central Limit Theorem of Galaxy Mergers As galaxies undergo merging, it can be shown that the scatter in the MBH-Mgal relation diminishes with increasing number of mergers as a consequence of the central limit theorem (see Appendix). To see this most easily, consider first major mergers, which by definition occur between galaxies of roughly equal mass, often de- fined to be within the range M1/M2 ≤ 4 (Figure 1a). Their similarity in mass also means that the number dis- tribution (i.e. mass function) ofMBH at a specific galaxy mass, M1, is similar to that at M2, i.e. they are drawn from parent distributions that have roughly the same shape (Figure 1b). However, the mean (µ) of the distri- butions might be offset by an amount that depends on the steepness of the MBH-Mgal correlation: the steeper the correlation (Figure 1a, upper ellipse), the larger is the offset. Furthermore, consider that in the limit of no initial MBH-Mgal correlation (Figure 1a, lower ellipse), µ is constant with galaxy mass. The average of any two BH masses randomly drawn from the parent distribu- tion (e.g. Fig. 1b) therefore has a tendency toward the mean value µ of the parent, by the central limit theorem. This tendency also means that the resulting BH mass distribution from mergers, obtained by sum- ming BHs drawn from the same distribution, will have a smaller log-normal scatter, σ(log(µ)) = σ(µ)/µ, than the original log-normal distribution, because the fractional mass error, σ 〈MBH,1 + MBH,2〉 / 〈MBH,1 + MBH,2〉, is smaller, which means the scatter in the MBH vs. Mgal relation, logarithmic on both axes, decreases. In the instance where the initial MBH mass distribu- tion (Figure 1b) is normal, with a scatter σBH,init, the resulting scatter of all galaxies that have undergone one full cycle of major mergers is σ (log µBH,merge) = σ (log µBH,init) / 2 (see Appendix). Therefore an en- semble of galaxies that has undergone Nmaj mergers should have a scatter in the MBH-Mgal relation that is reduced by ∼ 20.5Nmaj, compared with the initial rela- tion. However, a log-normal MBH distribution will have a different convergence rate. Clearly, this central limit theorem behavior applies to a finite parent distribution of any shape, but the size of the scatter and the rate at which the scatter decreases both depend on the shape of the distribution and the steepness of the initial MBH-Mgal correlation. In the situation where the correlation between MBH and Mgal is steep (e.g. Fig. 1a, upper ellipse) the effective cu- mulative mass function of BHs (Figure 1b) residing in galaxies with M2 ≤Mgal ≤ M1 has a wider σBH,init than if the MBH-Mgal relation is shallow (Fig. 1a, lower ellipse). For the same reason, minor mergers also have wider σBH,init, as the galaxy mass differences are larger, than do major mergers. Therefore, galaxies that have only undergone major mergers would produce an MBH- Mgal relation that is significantly tighter than for galax- ies that have only experienced minor mergers, for the same merger rate, and if the initial mass correlation is not flat. This effect is seen in the Monte-Carlo simula- tions below. 2.2. Galaxy Merging From a Schechter Mass Distribution If the mass density of galaxies follows a Schechter pow- erlaw form (Schechter 1976), Φ(M) = Φ0 , (1) 4 PENG ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������B log (M )gal ) a) ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� log (M )gal ) b) Fig. 2.— (a) No initial correlation in the MBH-Mgal relation. A galaxy at location 1 is more likely to merge with another galaxy at a much lower Mgal than with one that is comparable to itself (due to the bottom heavy mass function), but roughly of comparable MBH (due to non-correlation between MBH-Mgal). Thus the net evolutionary vector is steep. In contrast, a galaxy at location 2 is likely to merge with another that is comparable in both MBH and Mgal to itself, so the evolution vector is shallower. (b) An initial, strong powerlaw, correlation in the MBH-Mgal relation. A galaxy at location 1 is more likely to merge with another galaxy comparable in Mgal but at a much lower MBH than itself. Thus, the net evolutionary vector is shallow. In contrast, a galaxy at location 2 is likely to merge with another of comparable in MBH and Mgal to itself, so the evolution vector is steeper. then it is possible also to show that a linear MBH-Mgal relation naturally emerges over time, so that the relation, MBH(z) = Γ(z)Mgal(z)β, (2) eventually takes on β = 1. The value of Γ, which lo- cally is measured to be Γ(0) ∼ 1/800 for bulges (e.g. Häring & Rix 2004), is otherwise arbitrary in the discus- sion below. Γ is degenerate with respect to assumptions about the initial scatter of the MBH-Mgal relation, the initial slope β, and the initial normalization Γ(∞), for which there are currently insufficient observational con- straints; it will not be addressed further in this study. The other assumption used here is that the probability for two galaxies to merge comes from Monte-Carlo sam- pling of a Schechter mass function (Eq. 1). In actuality, galaxies do not merge randomly, especially at late times. However, complete randomness is only used to facilitate the discussion, and is not a pre-requisite, since the rea- soning depends only on the fact that minor mergers oc- cur more frequently than major mergers. This assump- tion does mean that, depending on the relative balance of major vs. minor mergers, the effects described here, namely convergence toward linearity versus central-limit behavior, may be more relevant at some epoch in time than at others The reason that a linear correlation emerges through galaxy mergers is illustrated in Figure 2. Figure 2a shows the situation in which the initial BH and galaxy mass distributions are completely uncorrelated, so that β = 0. The lower part of the diagram shows hypothetical mass functions with two different “faint end” slopes, A and B, which will be individually considered in the Monte-Carlo simulation below. If there is no correlation between the MBH and Mgal, then the ratio MBH/Mgal will be, on average, larger for low mass galaxies than for high mass galaxies as can be seen by comparing the MBH/Mgal ra- tio at any two locations, for example, those labeled “1” and “2.” Therefore, as galaxies merge, a massive object at the extreme end of the mass function, located at posi- tion 1, on average, is more likely to merge with another having a much larger MBH/Mgal, thereby evolving the merger product in a steep upward direction, as exagger- ated by the vertical arrow. In contrast, an object at position 2 is likely to merge with objects comparable in both MBH and Mgal to itself, so the net evolutionary vector has a shallower slope. Therefore, the cumulative effect of mergers along the mass spectrum is to steepen the massive end of the MBH-Mgal relation relative to the lower extreme, even as the lower end grows in MBH and Mgal on average. In the other extreme, Figure 2b, if the primordial rela- tion between the BH and galaxy masses is steep, corre- sponding to β ≥ 1, the opposite behavior occurs. Galax- ies at location 1 generally have a larger MBH/Mgal ra- tio than galaxies that have lower mass. Therefore, the MBH/Mgal ratio for massive galaxies would tend to de- crease through mergers. The net effect on massive galax- ies is to evolve the merger remnant more quickly to the right on average than a lower mass galaxy at location 2. Because of a mirror symmetry between Figures 2a and 2b, the natural equilibrium state of the MBH-Mgal re- lation is at β = 1, so that further merging of galaxies would evolve remnants along the linear relation, with a constant ratio Γ=MBH/Mgal. Also because of the con- MASS SCALING RELATIONS AND GALAXY MERGERS 5 vergence toward this “attractor” state the scatter in the relationship would necessarily decrease over time through galaxy merging. Lastly, it is worth mentioning that the presence of a break in the galaxy mass function at M∗ is a sufficient, but not necessary, condition for convergence toward lin- earity. A pure powerlaw with M∗= ∞ would produce a similar behavior, however, the convergence is slower for flatter slopes (α → −1). Furthermore, while the con- vergence behaviors just described is quite strong for a Schechter mass function with α = −1, the convergence would fail for a pure powerlaw with the same slope, be- cause of a lack of a break in the mass function. 3. MONTE CARLO SIMULATIONS To illustrate the idea discussed above, and to quantify how quickly a linear MBH-Mgal relation might emerge, it is useful to consider several numerical simulations for the situations shown in Figure 2. For each of the two scenarios, no-correlation (Fig. 2a) and steep correlation (Fig. 2b), it is also instructive to consider two differ- ent initial mass functions, A and B, shown in the lower half of Figure 2. The two powerlaw slopes explored be- low are α = −1.5 and α = −0.5, respectively. These choices are motivated by observations of the luminosity functions for high redshift galaxies (e.g. Gabasch et al. 2006; Giallongo et al. 2005) under the assumption that light traces mass. 3.1. Simulation Set-up and Definitions Definition of the number of major and minor mergers. — As implied in Section 2, how closely a galaxy lies to the linear part of the MBH-Mgal relation, and how tight the final scatter is for an ensemble of galaxies, will depend on the cumulative merger history. Therefore, the most useful way to understand the simulation results is to define the number of major mergers, Nmaj, as the cumulative sum of all such events over the entire tree for a given galaxy, not just in the most immediate, that is, the main, branch. For example, even if a galaxy has never experienced a single major merger on the main branch in its lifetime, it could still lie close to the final, linear part of the MBH-Mgal relation because the progenitors of the main branch, and their progenitors, and so forth, could have experienced a number of major mergers. Evolution of the mass function. — One issue to con- sider is how the galaxy mass function might evolve, and whether the path of evolution might affect the final con- vergence. The effect of galaxy masses growing with time is to both increase M∗ and steepen the “faint end slope” (α) of the Schechter function. As galaxy populations grow in mass and number density, the rate of change in M∗ and α would affect the rate of convergence to the final MBH-Mgal normalization, slope, and scatter. Pre- dicting the rate of change in the MBH-Mgal relation re- quires realistic merger trees and accounting for other de- tailed physics such as feedback, which will be addressed in a followup study. For the current purposes of show- ing that convergence toward a linear MBH-Mgal relation does naturally occur, it suffices to consider two scenarios: a replenishment scenario, in which the Schechter mass function is continuously, and randomly, replenished as galaxies merge, and a depletion scenario, in which no new galaxies are formed to take the place of those that have merged. Combined with considerations about the initial mass function slopes, the simulations will have covered the gamut of sensible possibilities and conditions that may be present at early and late cosmic epochs. Initial scatter of the MBH-Mgal relation and the initial mass function M∗. — In the simulation, the distribution of galaxies is first drawn randomly from the Schechter mass function, after which a BH mass is assigned, fol- lowing Equation 2, by drawing from a log-normal Gaus- sian distribution with a generous Gaussian dispersion of σBH = 2 dex, i.e. a scatter of a factor 100 in mass. The exact choice of the initial scatter is directly propor- tional to, but otherwise only partially determines, the final scatter in the MBH-Mgal relation. Other factors that determine the final scatter depend on how long the simulation runs, and on the initial value of M∗. Cur- rently, there are some observational constraint on the rate of mergers, which will be considered in a followup study. In this study, the results are merely normalized ar- bitrarily to match the final MBH-Mgal relation observed today. The simulation “clock.” — The progress of time is not well defined in Monte-Carlo simulations, so it is useful to define merging cycles for the sake of keeping track of the simulation progress. Each full cycle is defined as be- ing complete after the number of merger events equals the number of galaxies present at the beginning of that particular cycle. Galaxies that are produced or merged retain their states for the following cycle. Because some galaxies may merge multiple times by being drawn re- peatedly, not all galaxies will be involved in mergers af- ter each full cycle. The exact definition of the simulation clock is unimportant, as it is only the relative number of major vs. minor mergers on average that determines the degree of convergence toward a linear MBH-Mgal rela- tion, where a major merger is defined as having a mass ratio of at most 4:1. 3.2. Replenishment Scenarios The replenishment scenario is one of the two simple ways considered to emulate the progress of galaxy evo- lution. Here, by definition, the rate of galaxy mergers equals the rate of galaxy number production. The way a galaxy is newly produced is by being selected randomly from an initial mass function, parameterized by α and M∗, which does not evolve with time. In contrast, galax- ies “grow” only by merging with another member in the galaxy pool existing at the time, and hierarchical merg- ing is the only avenue for mass growth. Therefore, as galaxies merge, the cumulative mass function does un- dergo evolution. However, the total number of galax- ies remains constant because of the 1:1 ratio of merg- ing:replenishment. Figures 3 and 4 illustrate the results for initial con- ditions β = 0 (i.e. no MBH-Mgal correlation) and β = 2 (steep MBH-Mgal correlation), respectively, show- ing only a small subset of the data points. In each figure, two different initial mass functions, α = −0.5 (Figs. 3a, 4a) and α = −1.5 (Figs. 3b, 4b) are considered. In each of the Figures, the initial distribution of the MBH-Mgal relation (or lack thereof) is shown with crosses. The open colored data points show the MBH-Mgal development of galaxies that have undergone Nmaj ≥ 5 major merger episodes, after 10 (blue triangles), 100 (green squares), 6 PENG Fig. 3.— No initial correlation (β = 0) in the MBH-Mgal relation, in the replenishment scenario. The black crosses represent the initial distribution of points, and the solid line shows the local MBH-Mgal relationship from Häring & Rix (2004) – it is not a fit to the data points. The colored data points represent objects that have undergone at least five major merger episodes after 10 (blue triangles), 100 (green squares), and 1000 (red circles) complete “merging cycles.” The crosses are the primordial distribution, corresponding to the initial mass function. The shaded region illustrates the locus of all points after 1000 cycles; the density of points doubles with each contour level. The cumulative histograms after the corresponding merger sequences are shown below the data points. (a) An initial Schechter powerlaw slope of α = −0.5. (b) An initial Schechter powerlaw slope of α = −1.5. Fig. 4.— Similar to Figure 3, except for a steep initial correlation (β = 2) in the MBH-Mgal relation. See Figure 3 for details. MASS SCALING RELATIONS AND GALAXY MERGERS 7 Fig. 5.— Similar to Figure 3b, showing the effect of central-limit tendencies with increasing number of major mergers for galaxies after 1000 merger cycles. The colored data points illustrate objects that have undergone 1 ≤ Nmaj ≤ 4, (blue triangles), 5 ≤ Nmaj ≤ 14 (green squares), Nmaj > 14 (red circles) major mergers. The greyscale contours shows the locus of all the points. Fig. 6.— Similar to Figure 3, except the BH mass is drawn from a Schechter law of α = −1.5 instead of a Gaussian distribution. See Figure 3 for details. and 1000 (red circles) merger cycles have transpired. These data points effectively illustrate the progress of the MBH-Mgal evolution for objects that might be mor- phologically identified as early-type galaxies of each cy- cle. For clarity, the contour levels represent the locus of points after 1000 merger cycles, and the levels are spaced at multiples of 2 in density. The luminosity functions of the galaxy pool at the end of the merger cycles are shown in the lower half of each diagram in corresponding col- ors and locale in mass. Lastly, a linear reference line is overplotted in the Figures with normalization given by R0=800 (Häring & Rix 2004), and the simulations are scaled/shifted arbitrarily to match; it is not a fit to the data points. As shown in Figures 3 and 4, the convergence towards a tight linear relation is fairly quick. After five major merger episodes a linear relation starts to emerge regard- less of the initial conditions of the mass function or the form of the MBH-Mgal correlation. One reason for this quick convergence is the central-limit behavior of major mergers which is shown in Figure 5, in which the increas- ing number of mergers is represented by different symbols and shades. The one notable case where the convergence toward linearity is slower than the other scenarios is Fig- ure 4b, where the effect is only evident at 1010.5 M⊙ or greater, even as the scatter has decreased markedly. In general, if the MBH-Mgal correlation is steep initially, the tail at low mass remains steep after a large number of major mergers has occurred, even as the massive end converges toward linearity. Lastly, the qualitative convergence effects do not de- pend on the assumption about the distribution of BH mass at each galaxy mass. Figure 6 shows an example that is in direct analog to Figure 3b, except that the BH mass is instead drawn from a Schechter mass function with α = −1.5. 3.3. Depletion Scenarios The other extreme of the merger simulations is to con- sider what effect galaxy depletion from a finite reservoir has on the MBH-Mgal relation. Because the number density of galaxies builds up over time, the depletion scenario is expected to not be realistic. Nevertheless, it is useful for illustrating how the MBH-Mgal convergence is affected by a different evolution in the mass function as compared with the replenishment scenario. The depletion scenarios are constructed by creating a large sample of 5× 105 objects, initially having no corre- lation between BH and galaxy masses (Figure 7) or with a β = 2 correlation between the two (Figure 8). The BH masses are assigned to the galaxies with a log-normal distribution of dispersion σ = 2 centered around Equa- tion 2. In each scenario, galaxies are created to have initial mass functions of α = −0.5 (Figures 7a and 8a) or α = −1.5 (Figures 7b, 8b). Then, as galaxies merge, no new ones are created to replace them. As a consequence, the mass function evolves by growing in M∗, the num- ber density decreases, and a sharp truncation develops at low masses (see lower half of Figures 7 and 8). As the number of merging cycles increases, the scatter de- creases quickly and converges toward a linear relation, as illustrated by the solid line. Once again, as shown in Fig- ure 8 (especially 8b), the convergence is much slower for steep α and steep β compared with other scenarios. And while the convergence trends are noticeable, because of a dearth of minor galaxies with which to merge at late times (red circles), the slope is virtually “frozen in,” and the subsequent convergence is due mostly to the central- limit theorem. 4. DISCUSSION AND CONCLUSION This study has revisited the issue of how galaxy merg- ing may affect the MBH-Mgal scaling relation from the standpoint of basic mass addition and statistics, thereby clearly isolating the merger cause from other de- 8 PENG Fig. 7.— No initial correlation (β = 0) in the MBH-Mgal relation, depletion scenario. The contours represent the initial distribution of points, and a solid line shows the local MBH-Mgal relationship from Häring & Rix (2004) – it is not a fit to the data points. The colored data points represent objects that have undergone at least 1 major merger episodes after 1 complete merging cycles (blue triangles), 10 (green squares), and 14 (red circles). The cumulative histograms after the corresponding merger sequences are shown below the data points. a) An initial Schechter powerlaw slope of α = −0.5. b) An initial Schechter powerlaw slope of α = −1.5. Fig. 8.— Similar to Figure 7, except for a steep initial correlation (β = 2) in the MBH-Mgal relation. See Figure 7 for details. MASS SCALING RELATIONS AND GALAXY MERGERS 9 tailed physics that must otherwise affect galaxy evolu- tion. Through Monte-Carlo simulations, a tight, linear, MBH-Mgal correlation appears to emerge when galaxies have undergone five or more major mergers (along the entire tree, not just the main branch), and many minor ones, for practically all reasonable initial correlations be- tweenMBH andMgal, or a lack of one. The main reasons for these behaviors are seen to be the following: 1. The galaxy mass function decreases with increasing mass. 2. Major mergers have a strong central-limit ten- dency, so that regardless of the initial MBH-Mgal correlation, the scatter should decrease with an increasing number of events. While this ten- dency also acts on minor mergers, the drive toward smaller scatter is weaker because minor mergers oc- cur between galaxies that are vastly discrepant in both galaxy and BH mass as compared with major mergers, by definition. The corollary is that the steeper a correlation between MBH (y-axis) and Mgal, the stronger the central-limit tendency for major mergers compared with minor. However, major mergers alone are not enough to cause the MBH-Mgal relation to converge to linearity over time because the ratio MBH/Mgal is not changed much. 3. Minor mergers are primarily responsible for causing the MBH-Mgal relation to converge toward a linear — that is, MBH = ΓMgal— relation because the mass function of galaxies follows a Schechter pow- erlaw. Without minor mergers, the MBH-Mbulge relation can be “frozen” to a slope that is not necessarily linear. This linear attractor causes a convergence toward a tighter MBH-Mgal relation; however, it is less efficient at reducing the scatter compared with the central-limit seeking tendency of major mergers, as shown in Figure 5. It is curious that galaxy merging itself might produce a linear MBH-Mgal relation. However, a natural ques- tion that does arise is, “When is the merging statistics presented in this study relevant?” On the surface, it is easy to conclude that because the reasoning refers to a two component model it ought to apply to “dry” mergers, but perhaps not to a three component model involving stars, gas, and BH. Thus, the implication is also that it ought not apply to galaxies undergoing gas- rich mergers, that is, early cosmic history. However, it is not clear that such a skepticism is warranted. For example, in the entire discussion thus far, the abscissa, Mgal, might just as well refer to Mgal=Mstellar+Mgas, instead of just Mstellar. If BHs do not grow much by accretion and that the gas does not get removed from the definition of Mgal during mergers, then the MBH- Mgal correlation can emerge from statistical merging. The argument holds true even if Mgas transforms arbi- trarily into Mstellar, as long as the sum is conserved. In the limit where BHs do grow most of their mass dur- ing AGN accretion, as might be implied by Soltan (e.g. 1982); Yu & Tremaine (e.g. 2002), so that ∆MBH∝Mgas and ∆MBH≫MBH, then the correlation between MBH and Mgal comes out by construction rather than by sta- tistical merging. However, statistical reasoning would still be a “supporting actor” to reduce the scatter and to forcibly steer the MBH-Mgal relation in the preferred linear direction. Likewise, even if BH growth, or other physics (e.g. gravitational radiation, three body BH ejection – Merritt et al. 2004; Volonteri & Perna 2005; Ciotti et al. 2007, and references therein), were a “heat- ing” source, that is, one that randomizes a tight linear MBH-Mbulge relation, the linear and central limit attrac- tors would cause a re-convergence if galaxies continue to merge thereafter by both major and minor mergers. In summary, while it is entirely possible that the MBH- Mbulge relation has origins outside of basic statistics, galaxy merger statistics can still affect the final outcome of a MBH-Mgal correlation in both the scatter and the slope. In any event, statistical reasoning is a fundamen- tally robust explanation for why random galaxy merging does not corrupt a pre-existing MBH-Mbulge relation, which is important to bear in mind in the context of the MBH-Mbulge or MBH-Mgal relation in a hierarchically forming universe. While the MBH-Mbulge relation might have other origins, it is nonetheless interesting and revealing to follow through the consequences of statistical merg- ing. For instance, simple statistics naturally explains why black holes appear to correlate most strongly with galaxy bulges, rather than more generally with a galaxy as a whole, which might include a stellar disk (Kormendy & Gebhardt 2001): bulge masses, assum- ing they were assembled through major mergers, have a stronger central-seeking tendency than disk galaxies, whose growth history might involve more minor merger events. As such, the MBH-Mbulge relation is a special case of a more fundamental MBH-Mgal relation. Revers- ing the argument, the observational fact thatMBH corre- lates most strongly with bulge masses, coupled with the central-limit theorem reasoning, implies that the merger trees of elliptical galaxies were more dominated by ma- jor merger events than were disk galaxies. Conversely, the fact that the scatter in the MBH-Mgal relation is observed to be much larger for disk dominated galaxies implies, statistically, that their progenitors, and progen- itors thereof, have undergone more minor mergers. The possibility that a more fundamental correlation is between MBH and Mgal (rather than Mbulge) also has practical implications for what slope and scatter would be measured by observations. First, because the slope changes with mass even for objects that experienced the same number of major mergers (e.g. Figures 3 and 7), the deviation from linearity and the intrinsic scatter will depend exactly on how the data are cut. Simply defining a sample of objects based on a mass selection cut will bias one’s measurement of the slope and scatter. Fur- thermore, defining a sample based on morphology criteria may also implicitly preselect samples that have certain major vs. minor merger histories. Observationally, it is therefore crucial, when comparing intrinsic scatter and slope of the MBH-Mbulge relation to be specific about sample selection parameter space, morphology, bulge- to-disk ratios, or other criteria, lest the conclusions be caused by subtle but trivial selection biases. Another consequence of this thought experiment is that the ratio Γ=MBH/Mbulge approaches an asymp- 10 PENG totic value with time from having a smaller ratio in the past. On the surface, this appears contrary to the findings of Peng et al. (2006a,b); Woo et al. (2006); Shields et al. (2003, 2006) based on quasar host galaxy studies that the ratio Γ decreases over time. If the quasar host galaxy studies are correct and are not sig- nificantly affected by biases pointed out by Lauer et al. (2007), then some other physics not considered here is responsible for causing a decline in the normalization of Γ with time (e.g. see Croton 2006; Hopkins et al. 2007; Fontanot et al. 2006). For instance, the abscissa is am- biguous about what mass Mgal corresponds. If gas mass is a significant fraction of a galaxy’s mass, then form- ing stars out of the gas reservoir would decrease Γ over time, if the abscissa Mgal represents the galaxy’s stellar bulge mass. Secular growth of galaxy bulges by accret- ing stars in galaxy disks would also decrease Γ, at the expense of increasing the scatter. Major mergers of pure stellar bulges, however, would not cause Γ to decrease over time. In hindsight, the results of this study could have been anticipated from Islam et al. (2003, 2004); Ciotti et al. (2007), given that the initial conditions used in those studies are a special case of this one where the initial BH scatter σBH → 0 (Figure 3a or 3b) (M. Volonteri and L. Ciotti 2007, private communication). Just as rel- evant, Croton (2006); Ciotti et al. (2007) show that once the MBH-Mbulge relation is in place, it is fairly imper- vious to being randomized by galaxy merging. And be- cause Islam et al. (2003) also uses realistic cosmological merger trees, they confirm that the arguments presented here ought to remain relevant. However, the reasons be- hind the MBH-Mbulge convergence behavior are difficult to extract from previous studies because of the use of pri- ors, the use of identical BH seeds, the inclusion of other physics, and the focus on only the BH-bulge coevolution (i.e. major mergers). The latter, especially, is worth examining further, because the prior that one chooses about whether the BH correlates with just its bulge or with the entire galaxy can lead to differing interpreta- tions. In particular, one conclusion from Islam et al. (2003, 2004) is that the MBH-Mbulge relation converges to a slightly non-linear slope of β = 0.9; hence they reason that other physics, perhaps BH growth through accre- tion of gas, is required in order to increase the slope closer to linearity. The reasoning presented in the cur- rent study, however, would stipulate that linearity is an asymptotic outcome of mergers, but deviations from lin- earity come from the possibility that the low mass galax- ies have not yet achieved the asymptotic limit, because of a weaker convergence. At low masses, the slope de- viates from unity in either direction depending on the initial mass function of the galaxies (e.g. compare Fig- ure 3a with 3b), on the mass cut of the study, and on the relative incidence of minor versus major mergers. An interesting consequence to consider is how the MBH-Mgal relation might differ between high and low density galaxy clusters. However, one of the unrealis- tic side-effects of using Monte-Carlo simulations to de- termine merger rates is that the normalization of the MBH-Mbulge relation is the same in all density regimes. This is because the normalization factor, Γ, depends only on the ratio of major to minor merger events which, in the Monte-Carlo universe, is not affected by a simple rescaling of the mass function. However, in the real uni- verse, the relative rates of major and minor mergers can change with density and, as such, may result in different normalization and scatter in the MBH-Mgal relation. Lastly, because of the ambiguity in what Mgal corre- sponds, depending on whether it refers to the total stellar mass, gas mass, dark matter halo mass, or a combination thereof, the degree of scatter and linearity would clearly differ, as a result of different initial mass functions and merger histories. Because the scenarios considered above depend on a linear addition of masses, the arguments therefore may not apply to gas masses that are not grav- itationally bound to a galaxy. These and other issues will be addressed in a future study, which will incorporate the use of realistic merger trees. ACKNOWLEDGMENT Over the course of this study, I have greatly en- joyed lively discussions with many friends and colleagues, including Jenny Greene, Hans-Walter Rix, Luis Ho, Eric Bell, Michael Santos, Robert Kennicutt, Rachel Somerville, Aaron Barth, Christy Tremonti, Darren Cro- ton, Tommaso Treu, and David Koo. I also thank Jim Rose, Wayne Christiansen, Marta Volonteri, Luca Ciotti, and Alister Graham for comments and discussions on past and future studies. This study greatly benefited from discussions with Phil Hopkins on issues related to AGN feedback over the past several months. I also grate- fully acknowledge insightful comments and suggestions from the referee, and Science Editor Chung-Pei Ma, and the support of STScI through the Institute (Giacconi) Fellowship Program. APPENDIX This appendix shows that theMBH-Mbulge relation follows a central-limit-like behavior when galaxies undergo major mergers. Specifically, this means that if the initial parent distribution of progenitorMBH that undergoes merging is, for simplicity, normally distributed about a mean BH mass µ, thus having a logarithmic dispersion σ(log(µinit)), then a new distribution ofMBHs after merging will have a log-normal dispersion that scales as: σ (log (µmerge)) ∼ σ(log(µinit))/ First, the mean, µmerge, of the resulting BH distribution after two BHs merge from the initial parent distribution is: µmerge = 〈MBH,1 +MBH,2〉 , (1) where MBH,1 and MBH,2 are drawn from the same parent distribution for major galaxy mergers. Then, log (µmerge) = log 〈MBH,1 +MBH,2〉 , (2) MASS SCALING RELATIONS AND GALAXY MERGERS 11 From propagation of errors the log-normal error is: σ(log(x)) = σ(x)/x, then, σ (log (µmerge)) = σ 〈MBH,1 +MBH,2〉 〈MBH,1 +MBH,2〉 , (3) By definition, a distribution obtained by averaging the mass of merging BH pairs is a normal distribution with a mean of the initial parent distribution, µinit: µinit = 〈MBH,1 +MBH,2〉 µmerge . (4) Because M,1 and M,2 are drawn from a normalized distribution around a parent mean µ, the new distribution of σ(µmerge) ≡ σ 〈M,1 + M,2〉 is: σ (µmerge) ∼ σ(µinit)√ × 2. (5) Substituting Eqs. A4 and A5 into A3 yields: σ (log(µmerge)) ∼ σ(µinit) µinit . (6) Using the fact that, σ(log(µ)) = σ(µ)/µ, Equation A6 becomes: σ (log(µmerge)) ∼ σ (log(µinit))√ . (7) REFERENCES Adams, F. C., Graff, D. S., Mbonye, M., & Richstone, D. O. 2003, ApJ, 591, 125 Adams, F. C., Graff, D. S., & Richstone, D. O. 2001, ApJ, 551, Barth, A. J., Greene, J. E., & Ho, L. 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Supermassive black hole (BH) masses (MBH) are strongly correlated with galaxy stellar bulge masses (Mbulge) and there are several ideas to explain the origin of this relationship. This study isolates the role of galaxy mergers from considerations of other detailed physics to more clearly show how a linear BH-galaxy mass relation (MBH-Mgal) can naturally emerge regardless of how primordial BHs were seeded inside galaxies, if the galaxy mass function declines with increasing mass. Under this circumstance, the MBH-Mgal relation is a passive attractor that eventually converges to a tight linear relation because of two basic statistical effects: a central limit-like tendency for galaxy mergers which is much stronger for major mergers than minor mergers, and a convergence toward a linear relation that is due mainly to minor mergers. A curious consequence of this thought experiment is that, if galaxy bulges are formed by major mergers, then merging statistics naturally show that MBH would correlate more strongly with bulge dominated galaxies, because of stronger central-seeking tendencies, than with disk dominated galaxies. Even if some other physics is ultimately responsible for causing a linear MBH-Mbulge relationship, this thought experiment shows that, counter to intuition, random merging of galaxies that harbor random BH masses tends to strengthen rather than weaken a pre-existing, linear, correlation. This idea may be generalized to other gravitationally bound systems (dark matter halo, compact nuclear objects) that retain their physical identities after experiencing mergers.

Draft version November 1, 2018 Preprint typeset using LATEX style emulateapj v. 7/8/03 HOW MERGERS MAY AFFECT THE MASS SCALING RELATION BETWEEN GRAVITATIONALLY BOUND SYSTEMS Chien Y. Peng1,2 (Received 2003 March 13) Draft version November 1, 2018 ABSTRACT Supermassive black hole (BH) masses (MBH) are strongly correlated with galaxy stellar bulge masses (Mbulge) and there are several ideas to explain the origin of this relationship. This study isolates the role of galaxy mergers from considerations of other detailed physics to more clearly show how a linear BH–galaxy mass relation (MBH-Mgal) can naturally emerge regardless of how primordial BHs were seeded inside galaxies, if the galaxy mass function declines with increasing mass. Under this circumstance, the MBH-Mgal relation is a passive attractor that eventually converges to a tight linear relation because of two basic statistical effects: a central limit-like tendency for galaxy mergers which is much stronger for major mergers than for minor mergers, and a convergence towards a linear relation that is due mainly to minor mergers. A curious consequence of this thought experiment is that, if galaxy bulges are formed by major mergers, then merging statistics naturally show that MBH would correlate more strongly with bulge dominated galaxies, because of stronger central-seeking tendencies, than with disk dominated galaxies. Even if some other physics is ultimately responsible for causing a linear MBH-Mbulge relationship, this thought experiment shows that, counter to intuition, random merging of galaxies tends to strengthen rather than weaken a pre-existing, linear, correlation. This idea may be generalized to other gravitationally bound systems (dark matter halo, compact nuclear objects) that retain their physical identities after experiencing mergers. Subject headings: galaxies: bulges — galaxies: formation — galaxies: evolution – galaxies: statistics 1. INTRODUCTION In recent years, there have been several surpris- ing discoveries of fundamental scaling relations be- tween supermassive black hole masses (MBH) and large scale galaxy bulge properties: stellar veloc- ity dispersion σ∗, bulge mass Mbulge, the pro- file slope of galaxies, and the inner core ra- dius (Gebhardt et al. 2000; Ferrarese & Merritt 2000; Kormendy & Richstone 1995; Magorrian et al. 1998; Ho 1999; Kormendy & Gebhardt 2001; Graham et al. 2001; Marconi & Hunt 2003; Häring & Rix 2004; Barth et al. 2005; Lauer et al. 2006; Greene & Ho 2006; Woo et al. 2006). At masses lower than Mbulge. 1010M⊙, the cen- tral object with which galaxies correlate may be either in- termediate mass BHs in dwarf galaxies (Filippenko & Ho 2003; Barth et al. 2004, 2005; Greene & Ho 2006) or central massive star clusters (Ferrarese et al. 2006; Graham & Driver 2007). The small amount of intrin- sic scatter between black holes and bulges is often inter- preted to suggest a causal connection between the two — that the growth of one might somehow regulate the other (e.g. Silk & Rees 1998; Di Matteo et al. 2005). When did the fundamental BH scaling relationships come about? At higher redshifts, observations are still in the early stages, complicated in part by the chal- lenges of measuring the BH mass and the bulge prop- erties in the same galaxy. However, recent evidence from the study of quasar host galaxies indicates that a fundamental MBH-Mbulge correlation might have been present as early as z ∼ 4 (Peng et al. 2006a,b). Fur- 1 Space Telescope Science Institute, 3700 San Martin Drive, Bal- timore, MD 21218; cyp@stsci.edu. 2 STScI (Giacconi) Fellow thermore, there also appears to be an evolution in the MBH-Mbulge ratio by a factor of ∼ 4 in the same studies, which points to the possibility that the BH masses may have matured more quickly than their surrounding stellar bulge mass in the past. In addition, other observations that use [Oiii] and CO emission line widths to infer the gravitational potential of quasar bulges (Shields et al. 2003, 2006; Salviander et al. 2006, 2007; Ho 2007) sug- gest similar trends3, and residual traces of evolution re- main detectable even below z = 1 (Treu et al. 2004, 2007; Woo et al. 2006). Despite the general agreements ob- servationally (but, see Li et al. 2006; Borys et al. 2005), there remain several weaknesses that still complicate the interpretation: a significant factor of 2 systematic un- certainty in the normalization of the BH mass calibra- tion, the possibility that the normalization of the virial BH mass estimate (Kaspi et al. 2000; Vestergaard 2002; Onken et al. 2004; Kaspi et al. 2005; Greene & Ho 2005; Peterson & Bentz 2006; Vestergaard & Peterson 2006) may evolve with time, because of the fact that the bulge masses are not directly measured, and the possibility that the host galaxy mass may be biased low in high redshift quasars (Lauer et al. 2007) because of the steep decline in the luminosity function of galaxies. With re- gard to the latter selection bias, it is worthwhile to note, however, that low redshift (z . 0.3) quasars host galax- ies (McLure & Dunlop 2001), which have similarly high MBH andMgal to the high redshift objects in Peng et al. (2006a,b), do not exhibit a mass dependent bias even when the scatter is large. Despite these uncertainties, the finding of an existing strong MBH-Mbulge correla- 3 A lack of evolution in the stellar velocity dispersion im- plies that bulge mass decreases with look-back time (see, e.g, Robertson et al. 2006b). http://arxiv.org/abs/0704.1860v5 2 PENG tion at z & 1 is likely to be more secure. Controversies about the evolution aside, it is not only puzzling that the MBH-bulge correlations should ex- ist, but that they would have a small intrinsic scatter of 0.3 dex in MBH and that the correlation with the bulge mass is practically linear (Marconi & Hunt 2003; Häring & Rix 2004; Lauer et al. 2006). These curious facts have received wide theoretical attention over the years and could be explained in a number of ways. For instance, both the MBH-Mbulge and MBH-σ∗ relation can be produced by gas accretion onto a nuclear disk (Burkert & Silk 2001; Cen 2007) or turbulent dissipa- tion of gas (Escala 2006) followed by star formation, by a combination of BH accretion, galaxy mergers, and star formation (Li et al. 2006; Kauffmann & Haehnelt 2000; Haehnelt & Kauffmann 2000), by gravitational collapse of inner parts of a galaxy that forms the bulge from a rotating isothermal sphere (Adams et al. 2001, 2003), by stellar capture (Miralda-Escudé & Kollmeier 2005) in the accretion disk. The effect of dissipation in galaxy merging on the BH and fundamental plane scaling rela- tions has also been visited: Ciotti & van Albada (2001) speculated, and Robertson et al. (2006a) show, that dis- sipational mergers can produce the MBH-σ∗ slope and maintain the fundamental plane of elliptical galaxies. On the other hand, whereas dissipationlessmergers of ellipti- cal galaxies also tend to preserve the observed fundamen- tal plane (Boylan-Kolchin et al. 2006; Robertson et al. 2006a), dissipationless mergers of disk galaxies tend to produce a relation that more closely parallels the virial plane. Despite the aforementioned models, the explanation that has spawned immense activity is the theory of feed- back from an active galactic nucleus (AGN). The AGN feedback idea rests observations that quasars typically radiate at a fixed fraction (10%−100%) of the Edding- ton luminosity. If such radiation is produced by a massive enough BH, this energy budget is in principle sufficient to quench star formation and terminate the BH growth itself (e.g. Silk & Rees 1998; Di Matteo et al. 2005; Cattaneo et al. 2005; Springel et al. 2005; Hopkins et al. 2007, 2006a, and references therein). This idea has been incorporated into cosmological merger simulations (e.g. Granato et al. 2004; Fontanot et al. 2006; Croton et al. 2006), and is seen to have profound possibilities for ex- plaining a wide array of other galaxy evolution puzzles, including the evolution in the galaxy and quasar luminos- ity functions, mass functions, star formation rates, the X-ray background, and the bimodality of galaxy colors (e.g. see Hopkins et al. 2006a,b, and references therein). Even though AGN feedback is promising for explaining many aspects of galaxy evolution, it is possible that some other mechanism (or mechanisms) may either have signif- icant influence on the scaling relation between MBH and Mbulge, or in fact be more fundamentally the cause. It is therefore worthwhile to look for such factors and to fully study their effects. Dating back years before AGN feed- back physics became a popular idea, one such fundamen- tal factor considered by many was galaxy merging (e.g. Kauffmann & Haehnelt 2000; Haehnelt & Kauffmann 2000; Ciotti & van Albada 2001; Nipoti et al. 2003; Islam et al. 2003, 2004; Volonteri et al. 2003). Even without feedback, those simulations already find it pos- sible to produce the BH scaling relations, albeit to dif- fering degrees of agreement with the observations. How- ever, the reason why a tight linear correlation should emerge is not immediately apparent even when consider- ing no other physics besides merging. In fact, as accorded correctly by intuition, a correlation indeed cannot arise spontaneously from chaotic combinations of galaxies and black hole masses in general. As will be shown below, what does make a tight correlation emerge is the fact that the galaxy mass function decreases with increasing mass. This circumstance brings about a number of inter- esting implications that will be explored in future stud- ies. In this study, the modest goal is to show that when the focus is switched from the BH-bulge relation, to un- derstanding the more general BH-galaxy relation, some insights may be gained into understanding the growth and evolution of the MBH-Mbulge relation itself. This study therefore revisits the issue of galaxy merg- ing from the standpoint of a thought experiment. This toy model identifies three basic reasons for why the MBH-Mbulge relation may appear the way it does, even if the BH masses and their host galaxy masses were completely uncorrelated initially, or if they started out with a steep powerlaw correlation. The over-arching premise is that the galaxy mass function has a Schechter (1976) powerlaw form, so that there is a decline in galaxy number density with increasing mass, especially above masses M∗. When this circumstance is met, it can be shown that a “linear attractor” and a central limit- like tendency can work efficiently to produce a linear MBH-Mgal relation, and to reduce the scatter over time. However, while it is tempting to generalize this result to the MBH-σ∗ relation, it is not as simple to do be- cause it is not clear how the stellar velocity dispersion scales during galaxy mergers, something that depends on physics not considered in this study (e.g. dissipational vs. dissipationless mergers, star formation, AGN feed- back – see Robertson et al. 2006b; Hopkins et al. 2007; Di Matteo et al. 2007; Sijacki et al. 2007). The main emphasis of the current study is thus only to present a pure statistical exercise, and as such will not invoke merger trees or external physics – be- sides which many and much more sophisticated mod- els have already been run (e.g. Volonteri et al. 2003; Islam et al. 2004; Granato et al. 2004; Fontanot et al. 2006; Robertson et al. 2006b; Ciotti et al. 2007). In a sense, this work examines a common thread shared by all previous cosmological simulations. While it is tempt- ing to invoke realism by introducing detailed physics from the get-go, e.g. star formation, BH accretion, and AGN feedback, isolating the effects of simple statistics enables a cleaner exposition of why the convergence be- havior should be expected. As such, the current study is not a critique of other, more physical, models which can also explain the same correlation, or to pass criti- cal judgment about which is more or less relevant. In- stead, the main message of this study is that, regard- less of what other physics may ultimately produce the MBH-Mbulge relation or weaken it, an existing corre- lation should strengthen if galaxies continue to merge thereafter, whether by major or by minor mergers. In other words, merging alone, in the absence of all other physics, is a sufficient condition to bring about a tight, linear, MBH-Mbulge relation over time, and is always “pulling behind the scenes” to bring about such a corre- MASS SCALING RELATIONS AND GALAXY MERGERS 3 NGAL M2 �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� Fig. 1.— (a) Two examples of initial correlations between MBH and Mgal. (b) An arbitrary number distribution of MBH at each mass slice M1 or M2, where µ is the mean value of the distribution. lation. The issue of interest to follow up is to what extent this, or other as yet identified mechanisms, may matter in the end, and to predict more realistic scatter in the MBH-Mgal relation (as opposed to only MBH-Mbulge) under this hypothesis using more realistic merger trees and physics. We will address this subject in a followup study. The following discussion will begin by presenting a heuristic view to explain why a linear MBH-Mgal re- lation is a natural consequence of random merging (Sec- tion 2), followed by Monte-Carlo simulations to illustrate the effect (Section 3), and lastly by a discussion and con- clusion. In much of the discussion below, the relationship under consideration is more generally the MBH-Mgal re- lation, whereas BHs are thought to correlate more tightly with the spheroid component of Mgal, that is Mbulge. It will be seen that the tighter correlation between MBH and Mbulge is a special case of the MBH-Mgal relation and can be understood in the same framework if this hy- pothesis represents the dominant mechanism by which BHs correlate with galaxy masses. 2. HEURISTIC PICTURES 2.1. The Central Limit Theorem of Galaxy Mergers As galaxies undergo merging, it can be shown that the scatter in the MBH-Mgal relation diminishes with increasing number of mergers as a consequence of the central limit theorem (see Appendix). To see this most easily, consider first major mergers, which by definition occur between galaxies of roughly equal mass, often de- fined to be within the range M1/M2 ≤ 4 (Figure 1a). Their similarity in mass also means that the number dis- tribution (i.e. mass function) ofMBH at a specific galaxy mass, M1, is similar to that at M2, i.e. they are drawn from parent distributions that have roughly the same shape (Figure 1b). However, the mean (µ) of the distri- butions might be offset by an amount that depends on the steepness of the MBH-Mgal correlation: the steeper the correlation (Figure 1a, upper ellipse), the larger is the offset. Furthermore, consider that in the limit of no initial MBH-Mgal correlation (Figure 1a, lower ellipse), µ is constant with galaxy mass. The average of any two BH masses randomly drawn from the parent distribu- tion (e.g. Fig. 1b) therefore has a tendency toward the mean value µ of the parent, by the central limit theorem. This tendency also means that the resulting BH mass distribution from mergers, obtained by sum- ming BHs drawn from the same distribution, will have a smaller log-normal scatter, σ(log(µ)) = σ(µ)/µ, than the original log-normal distribution, because the fractional mass error, σ 〈MBH,1 + MBH,2〉 / 〈MBH,1 + MBH,2〉, is smaller, which means the scatter in the MBH vs. Mgal relation, logarithmic on both axes, decreases. In the instance where the initial MBH mass distribu- tion (Figure 1b) is normal, with a scatter σBH,init, the resulting scatter of all galaxies that have undergone one full cycle of major mergers is σ (log µBH,merge) = σ (log µBH,init) / 2 (see Appendix). Therefore an en- semble of galaxies that has undergone Nmaj mergers should have a scatter in the MBH-Mgal relation that is reduced by ∼ 20.5Nmaj, compared with the initial rela- tion. However, a log-normal MBH distribution will have a different convergence rate. Clearly, this central limit theorem behavior applies to a finite parent distribution of any shape, but the size of the scatter and the rate at which the scatter decreases both depend on the shape of the distribution and the steepness of the initial MBH-Mgal correlation. In the situation where the correlation between MBH and Mgal is steep (e.g. Fig. 1a, upper ellipse) the effective cu- mulative mass function of BHs (Figure 1b) residing in galaxies with M2 ≤Mgal ≤ M1 has a wider σBH,init than if the MBH-Mgal relation is shallow (Fig. 1a, lower ellipse). For the same reason, minor mergers also have wider σBH,init, as the galaxy mass differences are larger, than do major mergers. Therefore, galaxies that have only undergone major mergers would produce an MBH- Mgal relation that is significantly tighter than for galax- ies that have only experienced minor mergers, for the same merger rate, and if the initial mass correlation is not flat. This effect is seen in the Monte-Carlo simula- tions below. 2.2. Galaxy Merging From a Schechter Mass Distribution If the mass density of galaxies follows a Schechter pow- erlaw form (Schechter 1976), Φ(M) = Φ0 , (1) 4 PENG ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������B log (M )gal ) a) ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� log (M )gal ) b) Fig. 2.— (a) No initial correlation in the MBH-Mgal relation. A galaxy at location 1 is more likely to merge with another galaxy at a much lower Mgal than with one that is comparable to itself (due to the bottom heavy mass function), but roughly of comparable MBH (due to non-correlation between MBH-Mgal). Thus the net evolutionary vector is steep. In contrast, a galaxy at location 2 is likely to merge with another that is comparable in both MBH and Mgal to itself, so the evolution vector is shallower. (b) An initial, strong powerlaw, correlation in the MBH-Mgal relation. A galaxy at location 1 is more likely to merge with another galaxy comparable in Mgal but at a much lower MBH than itself. Thus, the net evolutionary vector is shallow. In contrast, a galaxy at location 2 is likely to merge with another of comparable in MBH and Mgal to itself, so the evolution vector is steeper. then it is possible also to show that a linear MBH-Mgal relation naturally emerges over time, so that the relation, MBH(z) = Γ(z)Mgal(z)β, (2) eventually takes on β = 1. The value of Γ, which lo- cally is measured to be Γ(0) ∼ 1/800 for bulges (e.g. Häring & Rix 2004), is otherwise arbitrary in the discus- sion below. Γ is degenerate with respect to assumptions about the initial scatter of the MBH-Mgal relation, the initial slope β, and the initial normalization Γ(∞), for which there are currently insufficient observational con- straints; it will not be addressed further in this study. The other assumption used here is that the probability for two galaxies to merge comes from Monte-Carlo sam- pling of a Schechter mass function (Eq. 1). In actuality, galaxies do not merge randomly, especially at late times. However, complete randomness is only used to facilitate the discussion, and is not a pre-requisite, since the rea- soning depends only on the fact that minor mergers oc- cur more frequently than major mergers. This assump- tion does mean that, depending on the relative balance of major vs. minor mergers, the effects described here, namely convergence toward linearity versus central-limit behavior, may be more relevant at some epoch in time than at others The reason that a linear correlation emerges through galaxy mergers is illustrated in Figure 2. Figure 2a shows the situation in which the initial BH and galaxy mass distributions are completely uncorrelated, so that β = 0. The lower part of the diagram shows hypothetical mass functions with two different “faint end” slopes, A and B, which will be individually considered in the Monte-Carlo simulation below. If there is no correlation between the MBH and Mgal, then the ratio MBH/Mgal will be, on average, larger for low mass galaxies than for high mass galaxies as can be seen by comparing the MBH/Mgal ra- tio at any two locations, for example, those labeled “1” and “2.” Therefore, as galaxies merge, a massive object at the extreme end of the mass function, located at posi- tion 1, on average, is more likely to merge with another having a much larger MBH/Mgal, thereby evolving the merger product in a steep upward direction, as exagger- ated by the vertical arrow. In contrast, an object at position 2 is likely to merge with objects comparable in both MBH and Mgal to itself, so the net evolutionary vector has a shallower slope. Therefore, the cumulative effect of mergers along the mass spectrum is to steepen the massive end of the MBH-Mgal relation relative to the lower extreme, even as the lower end grows in MBH and Mgal on average. In the other extreme, Figure 2b, if the primordial rela- tion between the BH and galaxy masses is steep, corre- sponding to β ≥ 1, the opposite behavior occurs. Galax- ies at location 1 generally have a larger MBH/Mgal ra- tio than galaxies that have lower mass. Therefore, the MBH/Mgal ratio for massive galaxies would tend to de- crease through mergers. The net effect on massive galax- ies is to evolve the merger remnant more quickly to the right on average than a lower mass galaxy at location 2. Because of a mirror symmetry between Figures 2a and 2b, the natural equilibrium state of the MBH-Mgal re- lation is at β = 1, so that further merging of galaxies would evolve remnants along the linear relation, with a constant ratio Γ=MBH/Mgal. Also because of the con- MASS SCALING RELATIONS AND GALAXY MERGERS 5 vergence toward this “attractor” state the scatter in the relationship would necessarily decrease over time through galaxy merging. Lastly, it is worth mentioning that the presence of a break in the galaxy mass function at M∗ is a sufficient, but not necessary, condition for convergence toward lin- earity. A pure powerlaw with M∗= ∞ would produce a similar behavior, however, the convergence is slower for flatter slopes (α → −1). Furthermore, while the con- vergence behaviors just described is quite strong for a Schechter mass function with α = −1, the convergence would fail for a pure powerlaw with the same slope, be- cause of a lack of a break in the mass function. 3. MONTE CARLO SIMULATIONS To illustrate the idea discussed above, and to quantify how quickly a linear MBH-Mgal relation might emerge, it is useful to consider several numerical simulations for the situations shown in Figure 2. For each of the two scenarios, no-correlation (Fig. 2a) and steep correlation (Fig. 2b), it is also instructive to consider two differ- ent initial mass functions, A and B, shown in the lower half of Figure 2. The two powerlaw slopes explored be- low are α = −1.5 and α = −0.5, respectively. These choices are motivated by observations of the luminosity functions for high redshift galaxies (e.g. Gabasch et al. 2006; Giallongo et al. 2005) under the assumption that light traces mass. 3.1. Simulation Set-up and Definitions Definition of the number of major and minor mergers. — As implied in Section 2, how closely a galaxy lies to the linear part of the MBH-Mgal relation, and how tight the final scatter is for an ensemble of galaxies, will depend on the cumulative merger history. Therefore, the most useful way to understand the simulation results is to define the number of major mergers, Nmaj, as the cumulative sum of all such events over the entire tree for a given galaxy, not just in the most immediate, that is, the main, branch. For example, even if a galaxy has never experienced a single major merger on the main branch in its lifetime, it could still lie close to the final, linear part of the MBH-Mgal relation because the progenitors of the main branch, and their progenitors, and so forth, could have experienced a number of major mergers. Evolution of the mass function. — One issue to con- sider is how the galaxy mass function might evolve, and whether the path of evolution might affect the final con- vergence. The effect of galaxy masses growing with time is to both increase M∗ and steepen the “faint end slope” (α) of the Schechter function. As galaxy populations grow in mass and number density, the rate of change in M∗ and α would affect the rate of convergence to the final MBH-Mgal normalization, slope, and scatter. Pre- dicting the rate of change in the MBH-Mgal relation re- quires realistic merger trees and accounting for other de- tailed physics such as feedback, which will be addressed in a followup study. For the current purposes of show- ing that convergence toward a linear MBH-Mgal relation does naturally occur, it suffices to consider two scenarios: a replenishment scenario, in which the Schechter mass function is continuously, and randomly, replenished as galaxies merge, and a depletion scenario, in which no new galaxies are formed to take the place of those that have merged. Combined with considerations about the initial mass function slopes, the simulations will have covered the gamut of sensible possibilities and conditions that may be present at early and late cosmic epochs. Initial scatter of the MBH-Mgal relation and the initial mass function M∗. — In the simulation, the distribution of galaxies is first drawn randomly from the Schechter mass function, after which a BH mass is assigned, fol- lowing Equation 2, by drawing from a log-normal Gaus- sian distribution with a generous Gaussian dispersion of σBH = 2 dex, i.e. a scatter of a factor 100 in mass. The exact choice of the initial scatter is directly propor- tional to, but otherwise only partially determines, the final scatter in the MBH-Mgal relation. Other factors that determine the final scatter depend on how long the simulation runs, and on the initial value of M∗. Cur- rently, there are some observational constraint on the rate of mergers, which will be considered in a followup study. In this study, the results are merely normalized ar- bitrarily to match the final MBH-Mgal relation observed today. The simulation “clock.” — The progress of time is not well defined in Monte-Carlo simulations, so it is useful to define merging cycles for the sake of keeping track of the simulation progress. Each full cycle is defined as be- ing complete after the number of merger events equals the number of galaxies present at the beginning of that particular cycle. Galaxies that are produced or merged retain their states for the following cycle. Because some galaxies may merge multiple times by being drawn re- peatedly, not all galaxies will be involved in mergers af- ter each full cycle. The exact definition of the simulation clock is unimportant, as it is only the relative number of major vs. minor mergers on average that determines the degree of convergence toward a linear MBH-Mgal rela- tion, where a major merger is defined as having a mass ratio of at most 4:1. 3.2. Replenishment Scenarios The replenishment scenario is one of the two simple ways considered to emulate the progress of galaxy evo- lution. Here, by definition, the rate of galaxy mergers equals the rate of galaxy number production. The way a galaxy is newly produced is by being selected randomly from an initial mass function, parameterized by α and M∗, which does not evolve with time. In contrast, galax- ies “grow” only by merging with another member in the galaxy pool existing at the time, and hierarchical merg- ing is the only avenue for mass growth. Therefore, as galaxies merge, the cumulative mass function does un- dergo evolution. However, the total number of galax- ies remains constant because of the 1:1 ratio of merg- ing:replenishment. Figures 3 and 4 illustrate the results for initial con- ditions β = 0 (i.e. no MBH-Mgal correlation) and β = 2 (steep MBH-Mgal correlation), respectively, show- ing only a small subset of the data points. In each figure, two different initial mass functions, α = −0.5 (Figs. 3a, 4a) and α = −1.5 (Figs. 3b, 4b) are considered. In each of the Figures, the initial distribution of the MBH-Mgal relation (or lack thereof) is shown with crosses. The open colored data points show the MBH-Mgal development of galaxies that have undergone Nmaj ≥ 5 major merger episodes, after 10 (blue triangles), 100 (green squares), 6 PENG Fig. 3.— No initial correlation (β = 0) in the MBH-Mgal relation, in the replenishment scenario. The black crosses represent the initial distribution of points, and the solid line shows the local MBH-Mgal relationship from Häring & Rix (2004) – it is not a fit to the data points. The colored data points represent objects that have undergone at least five major merger episodes after 10 (blue triangles), 100 (green squares), and 1000 (red circles) complete “merging cycles.” The crosses are the primordial distribution, corresponding to the initial mass function. The shaded region illustrates the locus of all points after 1000 cycles; the density of points doubles with each contour level. The cumulative histograms after the corresponding merger sequences are shown below the data points. (a) An initial Schechter powerlaw slope of α = −0.5. (b) An initial Schechter powerlaw slope of α = −1.5. Fig. 4.— Similar to Figure 3, except for a steep initial correlation (β = 2) in the MBH-Mgal relation. See Figure 3 for details. MASS SCALING RELATIONS AND GALAXY MERGERS 7 Fig. 5.— Similar to Figure 3b, showing the effect of central-limit tendencies with increasing number of major mergers for galaxies after 1000 merger cycles. The colored data points illustrate objects that have undergone 1 ≤ Nmaj ≤ 4, (blue triangles), 5 ≤ Nmaj ≤ 14 (green squares), Nmaj > 14 (red circles) major mergers. The greyscale contours shows the locus of all the points. Fig. 6.— Similar to Figure 3, except the BH mass is drawn from a Schechter law of α = −1.5 instead of a Gaussian distribution. See Figure 3 for details. and 1000 (red circles) merger cycles have transpired. These data points effectively illustrate the progress of the MBH-Mgal evolution for objects that might be mor- phologically identified as early-type galaxies of each cy- cle. For clarity, the contour levels represent the locus of points after 1000 merger cycles, and the levels are spaced at multiples of 2 in density. The luminosity functions of the galaxy pool at the end of the merger cycles are shown in the lower half of each diagram in corresponding col- ors and locale in mass. Lastly, a linear reference line is overplotted in the Figures with normalization given by R0=800 (Häring & Rix 2004), and the simulations are scaled/shifted arbitrarily to match; it is not a fit to the data points. As shown in Figures 3 and 4, the convergence towards a tight linear relation is fairly quick. After five major merger episodes a linear relation starts to emerge regard- less of the initial conditions of the mass function or the form of the MBH-Mgal correlation. One reason for this quick convergence is the central-limit behavior of major mergers which is shown in Figure 5, in which the increas- ing number of mergers is represented by different symbols and shades. The one notable case where the convergence toward linearity is slower than the other scenarios is Fig- ure 4b, where the effect is only evident at 1010.5 M⊙ or greater, even as the scatter has decreased markedly. In general, if the MBH-Mgal correlation is steep initially, the tail at low mass remains steep after a large number of major mergers has occurred, even as the massive end converges toward linearity. Lastly, the qualitative convergence effects do not de- pend on the assumption about the distribution of BH mass at each galaxy mass. Figure 6 shows an example that is in direct analog to Figure 3b, except that the BH mass is instead drawn from a Schechter mass function with α = −1.5. 3.3. Depletion Scenarios The other extreme of the merger simulations is to con- sider what effect galaxy depletion from a finite reservoir has on the MBH-Mgal relation. Because the number density of galaxies builds up over time, the depletion scenario is expected to not be realistic. Nevertheless, it is useful for illustrating how the MBH-Mgal convergence is affected by a different evolution in the mass function as compared with the replenishment scenario. The depletion scenarios are constructed by creating a large sample of 5× 105 objects, initially having no corre- lation between BH and galaxy masses (Figure 7) or with a β = 2 correlation between the two (Figure 8). The BH masses are assigned to the galaxies with a log-normal distribution of dispersion σ = 2 centered around Equa- tion 2. In each scenario, galaxies are created to have initial mass functions of α = −0.5 (Figures 7a and 8a) or α = −1.5 (Figures 7b, 8b). Then, as galaxies merge, no new ones are created to replace them. As a consequence, the mass function evolves by growing in M∗, the num- ber density decreases, and a sharp truncation develops at low masses (see lower half of Figures 7 and 8). As the number of merging cycles increases, the scatter de- creases quickly and converges toward a linear relation, as illustrated by the solid line. Once again, as shown in Fig- ure 8 (especially 8b), the convergence is much slower for steep α and steep β compared with other scenarios. And while the convergence trends are noticeable, because of a dearth of minor galaxies with which to merge at late times (red circles), the slope is virtually “frozen in,” and the subsequent convergence is due mostly to the central- limit theorem. 4. DISCUSSION AND CONCLUSION This study has revisited the issue of how galaxy merg- ing may affect the MBH-Mgal scaling relation from the standpoint of basic mass addition and statistics, thereby clearly isolating the merger cause from other de- 8 PENG Fig. 7.— No initial correlation (β = 0) in the MBH-Mgal relation, depletion scenario. The contours represent the initial distribution of points, and a solid line shows the local MBH-Mgal relationship from Häring & Rix (2004) – it is not a fit to the data points. The colored data points represent objects that have undergone at least 1 major merger episodes after 1 complete merging cycles (blue triangles), 10 (green squares), and 14 (red circles). The cumulative histograms after the corresponding merger sequences are shown below the data points. a) An initial Schechter powerlaw slope of α = −0.5. b) An initial Schechter powerlaw slope of α = −1.5. Fig. 8.— Similar to Figure 7, except for a steep initial correlation (β = 2) in the MBH-Mgal relation. See Figure 7 for details. MASS SCALING RELATIONS AND GALAXY MERGERS 9 tailed physics that must otherwise affect galaxy evolu- tion. Through Monte-Carlo simulations, a tight, linear, MBH-Mgal correlation appears to emerge when galaxies have undergone five or more major mergers (along the entire tree, not just the main branch), and many minor ones, for practically all reasonable initial correlations be- tweenMBH andMgal, or a lack of one. The main reasons for these behaviors are seen to be the following: 1. The galaxy mass function decreases with increasing mass. 2. Major mergers have a strong central-limit ten- dency, so that regardless of the initial MBH-Mgal correlation, the scatter should decrease with an increasing number of events. While this ten- dency also acts on minor mergers, the drive toward smaller scatter is weaker because minor mergers oc- cur between galaxies that are vastly discrepant in both galaxy and BH mass as compared with major mergers, by definition. The corollary is that the steeper a correlation between MBH (y-axis) and Mgal, the stronger the central-limit tendency for major mergers compared with minor. However, major mergers alone are not enough to cause the MBH-Mgal relation to converge to linearity over time because the ratio MBH/Mgal is not changed much. 3. Minor mergers are primarily responsible for causing the MBH-Mgal relation to converge toward a linear — that is, MBH = ΓMgal— relation because the mass function of galaxies follows a Schechter pow- erlaw. Without minor mergers, the MBH-Mbulge relation can be “frozen” to a slope that is not necessarily linear. This linear attractor causes a convergence toward a tighter MBH-Mgal relation; however, it is less efficient at reducing the scatter compared with the central-limit seeking tendency of major mergers, as shown in Figure 5. It is curious that galaxy merging itself might produce a linear MBH-Mgal relation. However, a natural ques- tion that does arise is, “When is the merging statistics presented in this study relevant?” On the surface, it is easy to conclude that because the reasoning refers to a two component model it ought to apply to “dry” mergers, but perhaps not to a three component model involving stars, gas, and BH. Thus, the implication is also that it ought not apply to galaxies undergoing gas- rich mergers, that is, early cosmic history. However, it is not clear that such a skepticism is warranted. For example, in the entire discussion thus far, the abscissa, Mgal, might just as well refer to Mgal=Mstellar+Mgas, instead of just Mstellar. If BHs do not grow much by accretion and that the gas does not get removed from the definition of Mgal during mergers, then the MBH- Mgal correlation can emerge from statistical merging. The argument holds true even if Mgas transforms arbi- trarily into Mstellar, as long as the sum is conserved. In the limit where BHs do grow most of their mass dur- ing AGN accretion, as might be implied by Soltan (e.g. 1982); Yu & Tremaine (e.g. 2002), so that ∆MBH∝Mgas and ∆MBH≫MBH, then the correlation between MBH and Mgal comes out by construction rather than by sta- tistical merging. However, statistical reasoning would still be a “supporting actor” to reduce the scatter and to forcibly steer the MBH-Mgal relation in the preferred linear direction. Likewise, even if BH growth, or other physics (e.g. gravitational radiation, three body BH ejection – Merritt et al. 2004; Volonteri & Perna 2005; Ciotti et al. 2007, and references therein), were a “heat- ing” source, that is, one that randomizes a tight linear MBH-Mbulge relation, the linear and central limit attrac- tors would cause a re-convergence if galaxies continue to merge thereafter by both major and minor mergers. In summary, while it is entirely possible that the MBH- Mbulge relation has origins outside of basic statistics, galaxy merger statistics can still affect the final outcome of a MBH-Mgal correlation in both the scatter and the slope. In any event, statistical reasoning is a fundamen- tally robust explanation for why random galaxy merging does not corrupt a pre-existing MBH-Mbulge relation, which is important to bear in mind in the context of the MBH-Mbulge or MBH-Mgal relation in a hierarchically forming universe. While the MBH-Mbulge relation might have other origins, it is nonetheless interesting and revealing to follow through the consequences of statistical merg- ing. For instance, simple statistics naturally explains why black holes appear to correlate most strongly with galaxy bulges, rather than more generally with a galaxy as a whole, which might include a stellar disk (Kormendy & Gebhardt 2001): bulge masses, assum- ing they were assembled through major mergers, have a stronger central-seeking tendency than disk galaxies, whose growth history might involve more minor merger events. As such, the MBH-Mbulge relation is a special case of a more fundamental MBH-Mgal relation. Revers- ing the argument, the observational fact thatMBH corre- lates most strongly with bulge masses, coupled with the central-limit theorem reasoning, implies that the merger trees of elliptical galaxies were more dominated by ma- jor merger events than were disk galaxies. Conversely, the fact that the scatter in the MBH-Mgal relation is observed to be much larger for disk dominated galaxies implies, statistically, that their progenitors, and progen- itors thereof, have undergone more minor mergers. The possibility that a more fundamental correlation is between MBH and Mgal (rather than Mbulge) also has practical implications for what slope and scatter would be measured by observations. First, because the slope changes with mass even for objects that experienced the same number of major mergers (e.g. Figures 3 and 7), the deviation from linearity and the intrinsic scatter will depend exactly on how the data are cut. Simply defining a sample of objects based on a mass selection cut will bias one’s measurement of the slope and scatter. Fur- thermore, defining a sample based on morphology criteria may also implicitly preselect samples that have certain major vs. minor merger histories. Observationally, it is therefore crucial, when comparing intrinsic scatter and slope of the MBH-Mbulge relation to be specific about sample selection parameter space, morphology, bulge- to-disk ratios, or other criteria, lest the conclusions be caused by subtle but trivial selection biases. Another consequence of this thought experiment is that the ratio Γ=MBH/Mbulge approaches an asymp- 10 PENG totic value with time from having a smaller ratio in the past. On the surface, this appears contrary to the findings of Peng et al. (2006a,b); Woo et al. (2006); Shields et al. (2003, 2006) based on quasar host galaxy studies that the ratio Γ decreases over time. If the quasar host galaxy studies are correct and are not sig- nificantly affected by biases pointed out by Lauer et al. (2007), then some other physics not considered here is responsible for causing a decline in the normalization of Γ with time (e.g. see Croton 2006; Hopkins et al. 2007; Fontanot et al. 2006). For instance, the abscissa is am- biguous about what mass Mgal corresponds. If gas mass is a significant fraction of a galaxy’s mass, then form- ing stars out of the gas reservoir would decrease Γ over time, if the abscissa Mgal represents the galaxy’s stellar bulge mass. Secular growth of galaxy bulges by accret- ing stars in galaxy disks would also decrease Γ, at the expense of increasing the scatter. Major mergers of pure stellar bulges, however, would not cause Γ to decrease over time. In hindsight, the results of this study could have been anticipated from Islam et al. (2003, 2004); Ciotti et al. (2007), given that the initial conditions used in those studies are a special case of this one where the initial BH scatter σBH → 0 (Figure 3a or 3b) (M. Volonteri and L. Ciotti 2007, private communication). Just as rel- evant, Croton (2006); Ciotti et al. (2007) show that once the MBH-Mbulge relation is in place, it is fairly imper- vious to being randomized by galaxy merging. And be- cause Islam et al. (2003) also uses realistic cosmological merger trees, they confirm that the arguments presented here ought to remain relevant. However, the reasons be- hind the MBH-Mbulge convergence behavior are difficult to extract from previous studies because of the use of pri- ors, the use of identical BH seeds, the inclusion of other physics, and the focus on only the BH-bulge coevolution (i.e. major mergers). The latter, especially, is worth examining further, because the prior that one chooses about whether the BH correlates with just its bulge or with the entire galaxy can lead to differing interpreta- tions. In particular, one conclusion from Islam et al. (2003, 2004) is that the MBH-Mbulge relation converges to a slightly non-linear slope of β = 0.9; hence they reason that other physics, perhaps BH growth through accre- tion of gas, is required in order to increase the slope closer to linearity. The reasoning presented in the cur- rent study, however, would stipulate that linearity is an asymptotic outcome of mergers, but deviations from lin- earity come from the possibility that the low mass galax- ies have not yet achieved the asymptotic limit, because of a weaker convergence. At low masses, the slope de- viates from unity in either direction depending on the initial mass function of the galaxies (e.g. compare Fig- ure 3a with 3b), on the mass cut of the study, and on the relative incidence of minor versus major mergers. An interesting consequence to consider is how the MBH-Mgal relation might differ between high and low density galaxy clusters. However, one of the unrealis- tic side-effects of using Monte-Carlo simulations to de- termine merger rates is that the normalization of the MBH-Mbulge relation is the same in all density regimes. This is because the normalization factor, Γ, depends only on the ratio of major to minor merger events which, in the Monte-Carlo universe, is not affected by a simple rescaling of the mass function. However, in the real uni- verse, the relative rates of major and minor mergers can change with density and, as such, may result in different normalization and scatter in the MBH-Mgal relation. Lastly, because of the ambiguity in what Mgal corre- sponds, depending on whether it refers to the total stellar mass, gas mass, dark matter halo mass, or a combination thereof, the degree of scatter and linearity would clearly differ, as a result of different initial mass functions and merger histories. Because the scenarios considered above depend on a linear addition of masses, the arguments therefore may not apply to gas masses that are not grav- itationally bound to a galaxy. These and other issues will be addressed in a future study, which will incorporate the use of realistic merger trees. ACKNOWLEDGMENT Over the course of this study, I have greatly en- joyed lively discussions with many friends and colleagues, including Jenny Greene, Hans-Walter Rix, Luis Ho, Eric Bell, Michael Santos, Robert Kennicutt, Rachel Somerville, Aaron Barth, Christy Tremonti, Darren Cro- ton, Tommaso Treu, and David Koo. I also thank Jim Rose, Wayne Christiansen, Marta Volonteri, Luca Ciotti, and Alister Graham for comments and discussions on past and future studies. This study greatly benefited from discussions with Phil Hopkins on issues related to AGN feedback over the past several months. I also grate- fully acknowledge insightful comments and suggestions from the referee, and Science Editor Chung-Pei Ma, and the support of STScI through the Institute (Giacconi) Fellowship Program. APPENDIX This appendix shows that theMBH-Mbulge relation follows a central-limit-like behavior when galaxies undergo major mergers. Specifically, this means that if the initial parent distribution of progenitorMBH that undergoes merging is, for simplicity, normally distributed about a mean BH mass µ, thus having a logarithmic dispersion σ(log(µinit)), then a new distribution ofMBHs after merging will have a log-normal dispersion that scales as: σ (log (µmerge)) ∼ σ(log(µinit))/ First, the mean, µmerge, of the resulting BH distribution after two BHs merge from the initial parent distribution is: µmerge = 〈MBH,1 +MBH,2〉 , (1) where MBH,1 and MBH,2 are drawn from the same parent distribution for major galaxy mergers. 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Journal of Nonlinear Mathematical Physics Volume *, Number * (20**), 1–?? Article Analycity and smoothing effect for the coupled system of equations of Korteweg - de Vries type with a single point singularity Mauricio Sepúlveda a and Octavio Vera Villagrán b a Departamento de Ingenieŕıa Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. E-mail: mauricio@ing-mat.udec.cl b Departamento de Matemática, Universidad del Bı́o-Bı́o, Collao 1202, Casilla 5-C, Concepción, Chile. E-mail: overa@ubiobio.cl Received Month *, 200*; Accepted in Revised Form Month *, 200* Abstract We study that a solution of the initial value problem associated for the coupled system of equations of Korteweg - de Vries type which appears as a model to describe the strong interaction of weakly nonlinear long waves, has analyticity in time and smooth- ing effect up to real analyticity if the initial data only has a single point singularity at x = 0. Keywords and phrases: Evolution equations, Gevrey class, Bourgain space, smoothing ef- fect. Mathematics Subject Classification: 35Q53 1 Introduction We consider the following coupled system of equations of Korteweg - de Vries type ũt + ũxxx + a3 ṽxxx + ũ ũx + a1 ṽ ṽx + a2 (ũ ṽ)x = 0, x, t ∈ R (1.1) b1 ṽt + ṽxxx + b2 a3 ũxxx + ṽ ṽx + b2 a2 ũ ũx + b2 a1 (ũ ṽ)x = 0, (1.2) ũ(x, 0) = ũ0(x), ṽ(x, 0) = ṽ0(x). (1.3) where ũ = ũ(x, t), ṽ = ṽ(x, t) are real-valued functions of the variables x and t and a1, a2, a3, b1, b2 are real constants with b1 > 0 and b2 > 0. The original coupled system is ũt + ũxxx + a3 ṽxxx + ũ p ũx + a1 ṽ p ṽx + a2 (ũ p ṽ)x = 0, x, t ∈ R (1.4) b1 ṽt + ṽxxx + b2 a3 ũxxx + ṽ p ṽx + b2 a2 ũ p ũx + b2 a1 (ũ ṽ p)x = 0 (1.5) ũ(x, 0) = ũ0(x), ṽ(x, 0) = ṽ0(x) (1.6) Copyright c© 200* by M Sepúlveda and O Vera http://arxiv.org/abs/0704.1861v1 2 M Sepúlveda and O Vera where ũ = ũ(x, t), ṽ = ṽ(x, t) are real-valued functions of the variables x and t and a1, a2, a3, b1, b2 are real constants with b1 > 0 and b2 > 0. The power p is an integer larger than or equal to one. The system (1.4)-(1.6) has the structure of a pair of Korteweg - de Vries equations coupled through both dispersive and nonlinear effects. In the case p = 1, the system (1.4)-(1.6) was derived by Gear and Grimshaw [9] as a model to describe the strong interaction of weakly nonlinear, long waves. Mathematical results on the system (1.4)-(1.6) were given by J. Bona et al. [5]. They proved that (1.4)-(1.6) is globally well posed in Hs(R)×Hs(R) for any s ≥ 1 provided |a3| < 1/ b2. The system (1.4)-(1.6) has been intensively studied by several authors (see [2, 3, 5, 7, 23] and the references therein). We have the following conservation laws E1(ũ) = ũ dx , E2(ṽ) = ṽ dx , E3(ũ, ṽ) = (b2ũ 2 + b1ṽ 2)dx (1.7) The time-invariance of the functionals E1 and E2 expresses the property that the mass of each mode separately is conserved during interaction, while that of E3 is an expression of the conservation of energy for the system of two models taken as a whole. The solutions of (1.4)-(1.6) satisfy an additional conservation law which is revealed by the time-invariance of the functional b2 ũ x + ṽ x + 2b2a3ũxṽx − b2 − b2a2ũ2ṽ − b2a2ũ2ṽ − b2a1ũṽ2 − The functional E4 is a Hamiltonian for the system (1.4)-(1.6) and if b2a 3 < 1, φ4 will be seen to provide an a priori estimate for the solutions (ũ, ṽ) of (1.4)-(1.6) in the space H1(R)×H1(R). Furthermore, the linearization of (1.1)-(1.3) about the rest state can be reduced to two, linear Korteweg - de Vries equations by a process of diagonalization. Using this remark and the smoothing properties (in both the temporal and spatial variables) for the linear Korteweg - de Vries derived by Kato [13, 15], Kenig, Ponce and Vega [18, 19] it will be shown that (1.4)-(1.6) is locally well-posed in Hs(R) × Hs(R) for any s ≥ 1 whenever b2a3 6= 1. This result was improved by J. M. Ash et al. [1] showing that the system (1.1)-(1.3) is globally well-posed in L2(R) × L2(R) provided that b2a3 6= 1. In 2004, F. Linares and M. Panthee [21] improve this result showing that the system (1.1)- (1.3) is locally well-posed in Hs(R) × Hs(R) for s > −3/4 and globally well-posed in Hs(R)×Hs(R) for s > −3/10 under some conditions on the coefficients, indeed for a3 = 0 and b1 = b2. Following the idea W. Craig et al. [6], it is shown in [23] that C ∞ solutions (ũ( · , t), ṽ( · , t)) to (1.1)-(1.3) are obtained for t > 0 if the initial data (ũ(x, 0), ṽ(x, 0)) belong to a suitable Sobolev space satisfying resonable conditions as |x| → ∞. Since (1.1)- (1.3) is a coupled system of Korteweg-de Vries equations, it is natural to ask whether it has a smoothing effect up to real analyticity if the initial data only has a single point singularity at x = 0 as the known results for the scalar case of a single Korteweg -de Vries equation. Using the scaling argument we can have an insight to this question. In this paper our purpose is to prove the analyticity in time of solutions to (1.1)-(1.3) without regularity assumption on the initial data improving those obtained in [23]. Our main tool is the generator of dilation P = 3 t ∂t + x ∂x. which almost commutes with the linear Korteweg-de Vries operator L = ∂t + ∂x. Indeed [L, P ] = 3L. A typical example of initial data satisfying the assumption of the above theorem is the Dirac delta measure, since (xk ∂x) kδ(x) = (−1)k k! δ(x). The other example of the data is p. v. 1 , where Analycity for the coupled system of KdV equations 3 p. v. denotes the Cauchy principal value. Linear combination of those distributions with analytic Hs data satisfying the assumption is also possible. In this sense, the Dirac delta measure adding the soliton initial data can be taken as an initial datum. Using the operator K = x · ▽ + 2 i t ∂t it was proved the Gevrey smoothing effect in space variable [8]. Indeed, it was shown that, if the initial data belongs to a Gevrey class of order 2, then solutions of some nonlinear Schrödinger equations become analytic in the space variable for t 6= 0. For the Korteweg-de Vries equations version of the generator of dilation is also useful to study the analyticity in time and the Gevrey effect in the space variables for solutions [8]. This paper is organized as follows: In section 2 we have the reduction of the problem and we outline briefly the notation, terminology to be used subsequently and results that will be used several times. In section 3 we prove a theorem of existence and well-posedness of the solutions. In section 4 we prove the following theorem: Theorem 1.1. Suppose that the initial data (ũ0, ṽ0) ∈ Hs(R) × Hs(R), s > −3/4 and A0, A1 > 0 such that ||(x ∂x)kũ0||Hs(R) < +∞ : ||(x ∂x)kũ0||Hs(R) < +∞. (1.8) Then for some b ∈ (1/2, 7/12), there exist T = T (||ũ0||Hs(R), ||ṽ0||Hs(R)) and a unique solution of (1.1)-(1.3) in a certain time (−T, T ) and the solution (ũ, ṽ) is time locally well-posed, i. e., the solution continuously depends on the initial data. Moreover, the solution (ũ, ṽ) is analytic at any point (x, t) ∈ R× {(−T, 0) ∪ (0, T )}. Corollary 1.1. Let s > −3/4, b ∈ (1/2, 7/12). Suppose that the initial data (ũ0, ṽ0) ∈ Hs(R)×Hs(R), and A0, A1 > 0 such that (k!)3 ||(x ∂x)kũ0||Hs(R) < +∞ : (k!)3 ||(x ∂x)kũ0||Hs(R) < +∞.(1.9) Then there exists a unique solution (ũ, ṽ) ∈ C((−T, T ), Hs(R))∩Xsb×C((−T, T ), Hs(R))∩ Xsb to the coupled system of Korteweg- de Vries equation (1.1)-(1.3) for a certain (−T, T ) and for any t ∈ (−T, 0)∪(0, T ), the pair (ũ, ṽ) are analytic functions in the space variable and for x ∈ R, ũ(x, · ) and ṽ(x, · ) are Gevrey 3 as function of the time variable. Remark 1.1. In Theorem 1.1 and Corollary 1.2, the assumption on the initial data implies analyticity and Gevrey 3 regularity except at the origin respectively. In this sense, those results state that the singularity at the origin immediately disappears after t > 0 or t < 0, up to analyticity. Remark 1.2. The crucial part for obtaining a full regularity is to gain the L2(R2) regular- ity of the solutions (uk, vk) from the negative order Sobolev space. This part is obtained in Proposition 4.1 in Section 4. We utilize a three steps recurrence argument for treating the nonlinearity appearing in the right hand side of t ∂3xuk = − Puk + x ∂xuk + tB k(u, u) + tB k(v, v) + tB k(u, v) (1.10) t ∂3xvk = − Pvk + x ∂xvk + t C k(u, u) + t C k(v, v) + t C k(u, v). (1.11) 4 M Sepúlveda and O Vera Then step by step, we obtain the pointwise analytic estimates t∈[t0−ǫ, t0+ǫ] ||∂mt ∂lxu||H1(x0−ǫ, x0+ǫ) ≤ cA 1 (m+ l)!, l, m = 0, 1, 2, . . . (1.12) t∈[t0−ǫ, t0+ǫ] ||∂mt ∂lxv||H1(x0−ǫ, x0+ǫ) ≤ cA 2 (m+ l)!, l, m = 0, 1, 2, . . . (1.13) Since initially we do not know whether the solution belong to even L2(R2) we should men- tion that the local well-posedness is essentially important for our argument and therefore it merely satisfies the coupled system equations in the sense of distribution. 2 Reduction of the Problem and Preliminary Results As mentioned in the introduction we consider the following coupled system of equations of Korteweg - de Vries type (1.1)-(1.3). If a3 = 0 there is no coupling in the dispersive terms. Let us assume that a3 6= 0. We are interested in decoupling the dispersive terms in the system (1.1)-(1.3). For this, let a23 b2 6= 1. We consider the associated linear system Wt +AWxxx = 0, W (x, 0) =W0(x) (2.1) where , A = a3 b2 The eigenvalues of A are given by 4 b2 a  (2.2) 4 b2 a  (2.3) which are distinct since b1 > 0, b2 > 0 and a3 6= 0. Our assumption a23 b2 6= 1 guarantees that α± 6= 0. Thus we can write the system (1.1)-(1.3) in a matrix form as in [21]. After we make the change of scale ũ(x, t) = u(α + x, t) and ṽ(x, t) = v(α − x, t). Then we obtain the system of equations ut + uxxx + a uux + b v vx + c (u v)x = 0, x, t ∈ R (2.4) vt + vxxx + ã u ux + b̃ v vx + c̃ (u v)x = 0, (2.5) u(x, 0) = u0(x), v(x, 0) = v0(x) (2.6) where a, b, c and ã, b̃, c̃ are constant. Remark 2.1. Notice that the nonlinear terms involving the functions u and v are not evaluated at the same point. Therefore those terms are not local anymore. Analycity for the coupled system of KdV equations 5 For s, b ∈ R define the spaces Xsb and Xsb−1 to be the completion of the Schwartz space S(R2) with respect to the norms ||u||Xs (1 + |τ − ξ3|)2b (1 + |ξ|)2s |û(ξ, τ)|2 dξ dτ ||u||Xs (1 + |τ − ξ3|)2(b−1) (1 + |ξ|)2s |û(ξ, τ)|2 dξ dτ where Xsb = {u ∈ S ′(R2) : ||u||Xsb <∞}. Let Fx and Fx, t be the Fourier transform in the x and (x, t) variables respectively. The Riesz operator Dx is defined by Dx = F−1ξ |ξ| Fx. The fractional derivative is defined by < Dx > s = F−1ξ < ξ > s Fx = F−1ξ (1 + |ξ| 2)s/2 Fx < Dx, t > s = F−1 < |ξ|+ |τ | >s Fx, t For < · >= (1 + | · |2)1/2, we have i) || · ||Hb(R:Hr(R)) = || < Dt >b< Dx >r · ||L2x, t(R2). ii) Hs(R) = {u ∈ S ′(R) : < Dx >s u ∈ L2(R)}. iii) || · ||Hs(R) = || < Dx >s · ||L2(R). Remark 2.2. With the above notation we obtain a) ||u||Hsx(R) = || < ξ > s û ||L2(R). b) ||u||L2t (R:Hrx(R)) = || < ξ > r û ||L2(R2). c) || < Dx >s u||L2(R) = ||u||Hs(R). d) || < Dt >b< Dx >r u||L2x, t(R2) = ||u||Hbt (R:Hrx(R)). e) || < Dx, t >s u||L2t (R:Hrx(R)) = || < ξ > r< |ξ|+ |τ | >s û(ξ, τ)||L2(R2). We consider the following operators: L = ∂t + ∂ x and J = x − 3 t ∂2x then [L, J ] ≡ LJ−J L = 0.We introduce the ”generator of dilation” P = 3 t ∂t+x ∂x for the linear part of the coupled system (2.4)-(2.6) and the ”localized dilation operator” P0 = 3 t0 ∂t+x0 ∂x. By employing a localization argument, we look at the operator P as a vector field P0 = 3 t0 ∂t+x0 ∂x near a fixed point (x0, t0) ∈ R×{(−T, 0)∪ (0, T )}. Since P0 is a directional derivative toward to (x0, t), we introduce another operator L30 = t0 ∂3x which plays the role of a non-tangential vector field to P0. Since P0 and L0 are linearly independent, the space and time derivative can be covered by those operator. The main reason why we choose L0 is because the corresponding variable coefficients operator L3 = t ∂3x can be treated via the equations (1.10)-(1.11) and a cut-off procedure enables us to handle the right hand side of those. Remark 2.3. For L and P we have the following properties: a) [L, P ] ≡ LP = (P + 3)L. b) LP k = (P + 3)kL. 6 M Sepúlveda and O Vera c) (P + 3)k∂x = ∂x(P + 2) d) (P + 3)k∂3x = ∂ e) P0 P = P P0 + 3P0 − 2x0 ∂x. Notation. The summation k=k1+k2+k3 0≤k1, k2, k3≤k is simply abbreviated by k=k1+k2+k3 Let P ku = uk, then ku) + ∂3x(P ku) = LP ku = (P + 3)kLu = (P + 3)k(∂tu+ ∂ = −(P + 3)k 2) + c ∂x(u v) (P + 3)k∂x(u (P + 3)k∂x(v 2)− c (P + 3)k∂x(u v) ∂x(P + 2) k(u2)− ∂x(P + 2) k(v2)− c ∂x(P + 2)k(u v). Noting that (P + 2)ku = 2k−jP ju. Hence B1k(u, u) = − ∂x(P + 2) k(u2) (P + 2)mu · P k−mu 2m−j P ju · P k−mu (m− j)! j! (k −m)! 2m−j P ju · P k−mu k=k1+k2+k3 k1! k2! k3! 2k1 ∂x (uk2 · uk3) . (2.7) In a similar way B2k(v, v) = − ∂x(P + 2) k(v2) = − b k′1! k · vk′ . (2.8) B3k(u, v) = c ∂x(P + 2) k(u v) = − c k=k′′ k′′1 ! k 2 ! k · vk′′ . (2.9) Analycity for the coupled system of KdV equations 7 Therefore ku) + ∂3x(P k=k1+k2+k3 k1! k2! k3! 2k1 ∂x (uk2 · uk3)− k′1! k · vk′ k=k′′ k′′1 ! k 2 ! k · vk′′ = B1k(u, u) +B k(v, v) +B k(u, v). (2.10) Performing similar calculations as above we obtain kv) + ∂3x(P k=k1+k2+k3 k1! k2! k3! 2k1 ∂x (uk2 · uk3)− k′1! k · vk′ k=k′′ k′′1 ! k 2 ! k · vk′′ = C1k(u, u) + C k(v, v) + C k(u, v). (2.11) The above nonlinear terms maintain the bilinear structure like that of the original coupled system of equations of KdV type, since Leibniz’s rule can be applied for operations of P. Now, each uk and vk satisfies the following system of equations ∂tuk + ∂ xuk = B k(u, u) +B k(v, v) +B k(u, v) ≡ Bk (2.12) ∂tvk + ∂ xvk = C k(u, u) + C k(v, v) + C k(u, v) ≡ Ck (2.13) uk(x, 0) = (x ∂x) ku0(x) ≡ uk0(x), vk(x, 0) = (x ∂x)kv0(x) ≡ vk0 (x). (2.14) In order to obtain a well-posedness result for the system (2.12)-(2.14) we use Duhamel’s principle and we study the following system of integral equations equivalent to the system (2.12)-(2.14) ψ(t)uk = ψ(t)V (t)u 0 − ψ(t) V (t− t′)ψT (t′)Bk(t′) dt′ (2.15) ψ(t) vk = ψ(t)V (t) v 0 − ψ(t) V (t− t′)ψT (t′)Ck(t′) dt′ (2.16) where V (t) = e−t ∂ x is the unitary group associated with the linear problem and ψ(t) ∈ C∞0 (R), 0 ≤ ψ ≤ 1 is a cut-off function such that ψ(t) = 1, if |t| < 1 0, if |t| > 2 and ψT (t) = ψ(t/T ) The following results are going to be used several times in the rest of this paper. 8 M Sepúlveda and O Vera Lemma 2.1 ([16]). . Let s ∈ R, a, a′ ∈ (0, 1/2), b ∈ (1/2, 1) and δ < 1. Then for any k = 0, 1, 2, . . . , we have ||ψδφk||Xs−a ≤ c δ (a−a′)/4(1−a′) ||φk||Xs , (2.17) ||ψδ V (t)φk||Xs ≤ c δ1/2−b||φk||Hs(R), (2.18)∣∣∣∣ ∣∣∣∣ψδ V (t− t′)Fk(t′) dt′ ≤ c δ1/2−b ||Fk||Xs . (2.19) Lemma 2.2 ([16]). . Let s > −3/4, b, b′ ∈ (1/2, 7/12) with b < b′. Then for any k, l = 0, 1, 2, . . . we have ||∂x(uk vl)||Xs ≤ c ||vk||Xs ||vl||Xs . (2.20) Lemma 2.3 ([12]). . Let s < 0, b ∈ (1/2, 7/12) and ψ = ψ(x, t) be a smooth cut- off function such that the support of ψ is in B2(0) and ψ = 1 on B1(0). We set ψǫ = ψ((x− x0)/ǫ, (t− t0)/ǫ). Then for f ∈ Xsb , we have ||ψǫ f ||Xs ≤ c ǫ−|s|−5|b|||ψǫ||X|s|+2 |b| ||f || s+2 |b| , (2.21) where the constant c is independent of ǫ and f. Lemma 2.4 ([12]). . Let P be the generator of the dilation and Dx, t be an operator defined by F−1 < |τ | + |ξ| > Fx, t. We fix an arbitrary point (x0, t0) ∈ R × {(−T, 0) ∪ (0, T )}. 1) Suppose that b ∈ (0, 1], r ∈ (−∞, 0] and g ∈ Xrb−1 with supp g ⊂ B2ǫ(x0, t0) and t∂3xg, P 3g ∈ Xrb−1. If ǫ > 0 is sufficiently small, then we have || < Dx, t >3b g||L2(R:Hr(R)) ≤ c ||g||Xr + ||t∂3xg||Xrb−1 + ||P 3g||Xr (2.22) where the constant c = c(x0, t0, ǫ). 2) If g ∈ Hµ−3(R2) with supp g ⊂ B2ǫ(x0, t0) and t∂3xg, P 3g ∈ Hµ−3(R2). Then for small ǫ, we have || < Dx, t >µ g||L2(R2) ≤ c ||g||Hµ−3(R2) + ||t∂3xg||Hµ−3(R2) + ||P 3g||Hµ−3(R2) (2.23) where the constant c = c(x0, t0, ǫ). Lemma 2.5 ([12]). . Let 0 ≤ s, r ≤ n/2 with n/2 ≤ s + r and suppose that f ∈ Hs(Rn) and g ∈ Hr(Rn). Then for any σ < s+ r − n/2, we have f g ∈ Hσ(Rn) and ||f g||Hσ(Rn) ≤ c(ǫ) ||f ||Hs(Rn)||g||Hr(Rn), (2.24) where ǫ = s+ r − n/2− σ. Corollary 2.1 ([12]). . For 1/2 < b < 1 and −3/4 < s < 0, we have ||ψ f || ≤ c ||f ||Xs (2.25) where ψ ∈ C∞0 (R2) and c is independent of f. Analycity for the coupled system of KdV equations 9 Lemma 2.6 ([12]). . Let ψ(x) be a smooth cut-off function in C∞0 ((−2, 2)) with ψ(x) = 1 on (−1, 1). We set ψǫ = ψ(x/ǫ) for 0 < ǫ < 1. Then for r ≤ 0, and f ∈ Hr, we have ||ψǫ f ||Hr(R) ≤ c ǫ−δ||f ||Hr(R) if − 1/2 ≤ r ≤ 0 c ǫ1/2+r||f ||Hr(R) if r < −1/2 where δ > 0 is an arbitrary small constant and c is independent of ǫ. Throughout this paper c is a generic constant, not necessarily the same at each occasion (it will change from line to line), which depends in an increasing way on the indicated quantities. 3 Existence and Well-Posedness We firstly solve the following (slightly general) system of equations ∂tuk + ∂ xuk = B k(u, u) +B k(v, v) +B k(u, v) ≡ Bk (3.1) ∂tvk + ∂ xvk = C k(u, u) + C k(v, v) + C k(u, v) ≡ Ck (3.2) uk(x, 0) = (x ∂x) ku0(x) ≡ uk0(x) , vk(x, 0) = (x ∂x)kv0(x) ≡ vk0 (x) (3.3) where Bk and Ck are as above. Definition 3.1. Let f = (f0, f1, . . . , fk) denotes the infinity series of distributions and define AA0(Xsb ) ≡ f = (f0, f1, . . . , fk), fi ∈ Xsb , (i = 0, 1, 2 . . .) such that ||f ||AA0 (Xsb ) < +∞ where ||f ||AA0 (Xsb ) ≡ ||fk||Xs Similarly, for u0 = {u00, u10, . . . , uk0, . . . } and v0 = {v00 , v10 , . . . , vk0 , . . . } we set ||u0||AA0 (Hs(R)) ≡ ||uk0 ||Hs(R) and ||v0||AA0 (Hs(R)) ≡ ||vk0 ||Hs(R) respectively. Remark 3.1. Each solution of the coupled system of Korteweg de Vries equations is accompanied by the following estimate ||P ku||Xs ≤ cAk0 k!, and ||P kv||Xsb ≤ cA 1 k!, k = 0, 1, 2, . . . Theorem 3.1. Let −3/4 < s, b ∈ (1/2, 7/12). Suppose that uk0 , vk0 ∈ Hs(R)(k = 0, 1, 2, . . .) and satisfies ||u0||AA0 (Xsb ) = ||uk0 ||Hs(R) < +∞ and ||v0||AA0 (Xsb ) = ||vk0 ||Hs(R) < +∞. 10 M Sepúlveda and O Vera Then there exist T = T (||uk0 ||Hs(R), ||vk0 ||Hs(R)) and a unique solution u = (u0, u1, . . .) and v = v(v0, v1, . . .) of the system (3.1)-(3.3) with uk, vk ∈ C((−T, T ) : Hs(R)) ∩Xsb and ||uk||Xs (R) < +∞, ||vk||Xs (R) < +∞. Moreover, the map (uk0 , v 0 ) → (u(t), v(t)) is Lipschitz continuous, i. e., ||u(t)− ũ(t)||AA0 (Xsb ) + ||u(t)− ũ(t)||C((−T, T ): Hs(R)) ≤ c(T ) ||u0 − ũ0||AA0 (Hs(R)) ||v(t) − ṽ(t)||AA0 (Xsb ) + ||v(t) − ṽ(t)||C((−T, T ): Hs(R)) ≤ c(T ) ||v0 − ṽ0||AA0 (Hs(R)). Proof. For given (u0, v0) ∈ AA0(Hs(R))×AA0(Hs(R)) and b > 1/2, let us define, HR1, R2 = (u, v) ∈ AA0(Xsb )×AA0(Xsb ) : ||u||AA0 (Xsb ) ≤ R1, ||v||AA0 (Xsb ) ≤ R2 where R1 = 2 c0 ||u0||AA0 (Hs(R)) and R2 = 2 c0 ||v0||AA0 (Hs(R)). Then HR1, R2 is a complete metric space with norm ||(u, v)||HR1, R2 = ||u||AA0 (Xsb ) + ||v||AA0 (Xsb ). Without loss of generality, we may assume that that R1 > 1 and R2 > 1. For (u, v) ∈ HR1, R2 , let us define the maps, Φku0(u, v) = ψ(t)V (t)u 0 − ψ(t) V (t− t′)ψT (t′)Bk(t′) dt′ (3.4) Ψkv0(u, v) = ψ(t)V (t) v 0 − ψ(t) V (t− t′)ψT (t′)Ck(t′) dt′. (3.5) We prove that Φ × Ψ maps HR1, R2 into HR1, R2 and it is a contraction. In fact, using lemma 2.1 and lemma 2.2 we have ||Φku0(u, v)||Xsb = ||ψ(t)V (t)u 0 ||Xsb + ∣∣∣∣ψ(t) V (t− t′)ψT (t′)Bk(t′) dt′ ≤ c0 ||uk0 ||Hs(R) + c1 T µ ||Bk||Xs ≤ c0 ||uk0 ||Hs(R) + c1 T µ k=k1+k2+k3 k1! k2! k3! 2k1 ||uk2 ||Xsb ||uk3 ||Xsb + c1 T k′1! k 1 ||vk′ ||vk′ + c1 T k=k′′ k′′1 ! k 2 ! k 1 ||uk′′ ||vk′′ Analycity for the coupled system of KdV equations 11 Applying a sum over k we have ||Φku0(u, v)||Xsb ||uk0 ||Hs(R) + c1 T µ k=k1+k2+k3 k1! k2! k3! 2k1 ||uk2 ||Xsb ||uk3 ||Xsb + c1 T k′1! k 1 ||vk′ ||vk′ + c1 T k=k′′ k′′1 ! k 2 ! k 1 ||uk′′ ||vk′′ ≤ c0 ||u0||AA0 (Hs(R)) + c1 T k=k1+k2+k3 ||uk2 ||Xsb ||uk3 ||Xsb + c1 T ||vk′ ||vk′ + c1 T k=k′′ k′′2 ! ||uk′′ k′′3 ! ||vk′′ ≤ c0 ||u0||AA0 (Hs(R)) + c1 T ||uk2 ||Xsb ||uk3 ||Xsb + c1 T ||vk′ ||vk′ + c1 T k′′1 ! k′′2 ! ||uk′′ k′′3 ! ||vk′′ = c0 ||u0||AA0 (Hs(R)) + c1 T e2A0 ||u||2AA0 (Xsb ) + c1 T e2A0 ||v||2AA0 (Xsb ) + c1 T µ c e2A0 ||u||AA0 (Xsb ) ||v||AA0 (Xsb ). Hence, choosing d = max{a/2, b/2, c} we have ||Φu0(u, v)||AA0 (Xsb ) ≤ c0 ||u0||AA0 (Hs(R)) + c1 T µ d e2A0 ||u||2AA0 (Xsb ) + ||v|| AA0 (X ) + ||u||AA0 (Xsb ) ||v||AA0 (Xsb ) ≤ c0 ||u0||AA0 (Hs(R)) + c1 d T µ e2A0 ||u||2AA0 (Xsb ) + ||v|| AA0 (X . (3.6) In a similar way, choosing d̃ = max{ã/2, b̃/2, c̃} we have ||Ψv0(u, v)||AA0 (Xsb ) ≤ c0 ||v0||AA0 (Hs(R)) + c2 d̃ T µ e2A0 ||u||2AA0 (Xsb ) + ||v|| AA0 (X .(3.7) 12 M Sepúlveda and O Vera If we choose T such that T µ ≤ 3 max{c1, c2} (R1 +R2)2 Then we obtain in (3.6) and (3.7) ||Φu0(u, v)||AA0 (Xsb ) ≤ R1 and ||Ψv0(u, v)||AA0 (Xsb ) ≤ R2. Therefore, (Φu0 , Ψv0) ∈ HR1, R2 .We show that Φu0×Ψv0 : (u, v) → (Φu0(u, v), Ψv0(u, v)) is a contraction. Let (u, v), (ũ, ṽ) ∈ HR1, R2 , then as above we get for d = max{a/2, b/2, c} ||Φu0(u, v)− Φu0(ũ, ṽ)||AA0 (Xsb ) c1 d T µ e2A0 (R1 +R2) ||u− ũ||AA0 (Xsb ) + ||v − ṽ||AA0 (Xsb ) . (3.8) In a similar way, choosing d̃ = max{ã/2, b̃/2, c̃} we have ||Ψv0(u, v)−Ψv0(ũ, ṽ)||AA0 (Xsb ) c2 d̃ T µ e2A0 (R1 +R2) ||u− ũ||AA0 (Xsb ) + ||v − ṽ||AA0 (Xsb ) . (3.9) Choosing T µ small enough, such that T µ ≤ 1 6 max{c1, c2} (R1 +R2)2 we obtain ||Φu0(u, v)− Φu0(ũ, ṽ)||AA0 (Xsb ) ≤ ||u− ũ||AA0 (Xsb ) + ||v − ṽ||AA0 (Xsb ) . (3.10) In a similar way ||Ψv0(u, v)−Ψv0(ũ, ṽ)||AA0 (Xsb ) ≤ ||u− ũ||AA0 (Xsb ) + ||v − ṽ||AA0 (Xsb ) . (3.11) Therefore the map Φu0 × Ψv0 is a contraction and we obtain a unique fixed point (u, v) which solves the initial value problem (3.1)-(3.3) for T < T µ. The rest of the proof follows a standard argument. Corollary 3.1. Let −3/4 < s, b ∈ (1/2, 7/12). Suppose that (x ∂x)ku0, (x ∂x)kv0 ∈ Hs(R)(k = 0, 1, 2, . . .) and that ||uk0 ||Hs(R) < +∞ and ||vk0 ||Hs(R) < +∞. Then there exist T = T (||uk0 ||Hs(R), ||vk0 ||Hs(R)) and a unique solution (u, v) of the coupled system equations KdV type (1.1)-(1.3) with u, v ∈ C((−T, T ) : Hs(R)) ∩Xsb and ||P ku||Xs (R) < +∞, ||P kv||Xs (R) < +∞. Analycity for the coupled system of KdV equations 13 Moreover, the map (u0, v0) → (u(t), v(t)) is Lipschitz continuous in the following sense: ||P ku(t)− P kũ(t)||Xs + ||P ku(t)− P kũ(t)||C((−T, T ): Hs(R)) ≤ c(T ) ||(x ∂x)k(u0 − ũ0)||Hs(R) ||v(t) − ṽ(t)||Xs + ||v(t)− ṽ(t)||C((−T, T ): Hs(R)) ≤ c(T ) ||(x ∂x)k(v0 − ṽ0)||Hs(R). 4 The main result In this section we prove the analyticity of the solution obtained in the previous section. We treat the solution uk ≡ P ku and vk ≡ P kv as if they satisfy the coupled system of equations (3.1)-(3.3) in the classical sense. This can be justified by a proper approximation procedure. The following results are going to be used in this section. Let (x0, t0) be arbitrarily taken in R×{(−T, 0)∪(0, T )}. By ψ(x, t) we denote a smooth cut-off function in C∞0 (B1(0)) and ψǫ = ψ((x − x0)/ǫ, (t− t0)/ǫ). Let ψ be a smooth cut-off function around the freezing point (x0, t0) with suppψ ⊂ C∞0 (Bǫ(x0, t0)). Proposition 4.1. For the cut-off function ψ defined above, there exists a positive constant c and A such that ||ψ P ku||L2x, t(R2) ≤ cA k (k!)2, k = 0, 1, 2, . . . (4.1) ||ψ P kv||L2x, t(R2) ≤ cA k (k!)2, k = 0, 1, 2, . . . (4.2) Proof. Using (2.22) with r = s− 1, we obtain || < Dx, t >3b ψP ku||L2t (R:Hs−1x (R)) ≤ c ||ψuk||Xs−1 + ||t ∂3x(ψuk)||Xs−1 + ||P 3(ψuk)||Xs−1 .(4.3) Each term in (4.3) is estimated separately. For the first term in the right hand side we use Lemma 2.3. Indeed, ||ψ uk||Xs−1 ≤ ||ψ uk||Xs ≤ c ||ψ|| |s|+2|b−1| |b−1| ||uk||Xs ≤ c(ψ)Ak1 k!. k = 0, 1, 2, . . .(4.4) The third term is estimated again using Corollary 2.6. ||P 3(ψ uk)||Xs−1 l (l − 3)! ||(P 3−lψ)P luk||Xsb−1 ≤ c(ψ) l (l − 3)! ||P luk||Xs l (l − 3)! ||P k+lu||Xs Ak+l1 (k + l)! ≤ cAk2 k!. k = 0, 1, 2, . . . (4.5) 14 M Sepúlveda and O Vera For the second term, we use (3.1) to reduce the third derivative in space to the dilation operator P. Since the generator of dilation is Puk = 3 t ∂tuk + x ∂xuk we obtain t ∂tuk = Puk − x ∂xuk. (4.6) Multiplying (3.1) by ψ t, we have ψ t ∂tuk + ψ t ∂ xuk = ψ tBk. (4.7) Replacing (4.6) in (4.7) we obtain ψ t ∂3xuk = − ψ Puk + ψ x∂xuk + ψ tBk. (4.8) hence ||ψ t ∂3xuk||Xs−1 ||ψ Puk||Xs−1 ||ψ x∂xuk||Xs−1 + ||ψ tBk||Xs−1 = F1 + F2 + F3. (4.9) Using the assumption in the Theorem, we have ||ψ Puk||Xs−1 ≤ c ||ψ|| ||P k+1u||Xs ≤ c ||P k+1u||Xs ≤ cAk+13 (k + 1)! ≤ cA 4 k!. (4.10) Similarly, we obtain ||ψ x∂xuk||Xs−1 ||∂x(ψ xuk)||Xs−1 ||∂x(ψ x)uk)||Xs−1 ||∂x(ψ x vk)||Xs + c ||∂x(ψ x)||X−s ||uk||Xs ||ψ x||Xs ||uk||Xsb + c ||∂x(ψ x)||X−s1−b ||uk||X ||ψ x||Xs + ||∂x(ψ x)||X−s Ak5k! ≤ cAk6k!. (4.11) Using Lemma 2.3 and 2.2, we have F3 = ||ψ tBk||Xs−1 ≤ c ||ψ|| ||B1k +B2k +B3k||Xsb−1 ||B1k ||Xsb−1 + ||B k ||Xsb−1 + ||B k||Xsb−1 Analycity for the coupled system of KdV equations 15 Then replacing B1k, B k and B k in (2.7), (2.8) and (2.9) we deduce F3 ≤ c k=k1+k2+k3 k1! k2! k3! 2k1 ||uk2 ||Xsb ||uk3 ||Xsb + c k′1! k 1 ||vk′ ||vk′ k=k′′ k′′1 ! k 2 ! k 1 ||uk′′ ||vk′′ k=k1+k2+k3 k1! k2! k3! 2k1 Ak27 · k2! A 7 · k3! + c k′1! k 8 · k 8 · k k=k′′ k′′1 ! k 2 ! k 9 · k 2 ! A 10 · k k=k1+k2+k3 2k1 A k2+k3 7 + c k=k′′ k′′1 ! 9 · A ≤ c k! Ak7 k−k1∑ 2k1 A− k17 + c k! A + c k! k=k′′ k′′1 ! ≤ c k! Ak7 k−k1∑ + c k! Ak8 + c k! k=k′′ k′′1 ! 9 · A ≤ c e2/A7 Ak7 · k! + c e2/A8 Ak8 · k! + c k! k=k′′ k′′1 ! e2/A7 + e2/A8 A11 · k! + c k! k=k′′ k′′1 ! 9 · A 10 . k = 0, 1, 2, . . . (4.12) Hence, from (4.10), (4.11) and (4.12) in (4.9) we obtain that there exists a positive constant c and A11 such that ||ψ t ∂3xuk||Xs−1 ≤ cA11 · k! + c k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . .(4.13) On the other hand, using ∂3x(ψ · f) = ψ · ∂3xf + 3 ∂2x(∂xψ · f)− 3 ∂x(∂2xψ · f) + ∂3xψ · f we have that ||t ∂3x(ψ · uk)||Xs−1 ≤ ||t ψ · ∂3xuk||Xs−1 + 3 ||∂2x(t ∂xψ · uk)||Xs−1 + 3 ||∂x(t ∂2xψ · uk)||Xs−1 + ||t ∂3xψ · uk||Xs−1 . (4.14) 16 M Sepúlveda and O Vera Using Lemma 2.2 and Lemma 2.3 we obtain ||∂2x(t ∂xψ · uk)||Xs−1 ≤ ||∂x(t ∂xψ · uk)||Xsb−1 ≤ c ||t ∂xψ||Xsb ||uk||Xsb ≤ cAk10 k! (4.15) ||∂x(t ∂2xψ · uk)||Xs−1 ≤ ||∂x(t ∂2xψ · uk)||Xsb−1 ≤ c ||t ∂ xψ||Xsb ||uk||Xsb ≤ cAk11 k! (4.16) ||t ∂3xψ · uk||Xs−1 ≤ c || < Dx, t >3/2 t ∂3xψ||X|s|+2|b−1| ||uk||Xsb−1 ≤ c ||uk||Xsb ≤ cAk12 k!. (4.17) Hence, replacing (4.13), (4.15),(4.16) and (4.17) in (4.14) we obtain that there exists a constant c and A14 such that ||t ∂3x(ψuk)||Xs−1 ≤ cAk14 · k! + c k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . .(4.18) Therefore, replacing (4.4), (4.5) and (4.18) in (4.3) we obtain that there exists a constant c and A15 such that || < Dx, t >3b ψ uk||L2t (R:Hs−1x (R)) ≤ cAk15 · k! + c k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.19) In a similar way, we obtain that there exists a constant c and A16 such that || < Dx, t >3b ψ vk||L2t (R:Hs−1x (R)) ≤ cAk16 · k! + c k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.20) Adding (4.19) and (4.20) we have || < Dx, t >3b ψ uk||L2t (R:Hs−1x (R)) + || < Dx, t > 3b ψ vk||L2t (R:Hs−1x (R)) ≤ cAk15 · k! + cAk16 · k! + c k! k=k′′ k′′1 ! 1 2 ·Ak ≤ c (Ak15 +Ak16) · k! + c k! k=k′′ k′′1 ! 1 2 ·Ak ≤ cAk17 · k! + c k! k=k′′ k′′1 ! 1 2 ·Ak 10 . (4.21) Analycity for the coupled system of KdV equations 17 We estimate the last term on the right hand side of (4.21) k=k′′ k′′1 ! 1 2 · Ak 9 · A (m− j)! 2(m−j) 2 ·Aj9 · A ≤ Ak10 (m− j)! ≤ Ak10 ≤ Ak10 +Ak10 ≤ Ak10 k! +Ak10 k! ≤ eA29/4 Ak10 k! + e4/A 10 Ak10 k! ≤ c Ak10 k!. (4.22) Replacing (4.22) in (4.21) we obtain || < Dx, t >3b ψ uk||L2t (R:Hs−1x (R)) + || < Dx, t > 3b ψ vk||L2t (R:Hs−1x (R)) ≤ cAk17 · k! + cAk19 · (k!)2 ≤ cAk17 · (k!)2 + cAk19 · (k!)2 ≤ cAk20 · (k!)2 (4.23) and the result follows. Remark 4.1. a) For simplicity, we only illustrate the conclusion for the case s ≥ −1/2−δ with b = 1/2 + δ/3 (for small δ > 0) and the case s = −3/4 + δ and b = 7/12 − δ/3. If s = −1/2− δ with b = 1/2+ δ/3, the initial data can involve Dirac’s delta measure δ0 and the latter is the critical case of the local well-posedness. b) The following inequality is simple to verify in both cases, ||ψ uk||L2x, t(R2) ≤ || < Dx > 3 b (ψ uk)||L2t (R: Hs−1x (R)) ≤ c || < Dx, t > 3 b (ψ uk)||L2t (R: Hs−1x (R)). Proposition 4.2. Under the same assumptions as in Proposition 4.1, there exist positive constants c and A such that ||ψ P ku||H7/2(R2) ≤ cA k (k!)2, k = 0, 1, 2, . . . (4.24) ||ψ P kv||H7/2(R2) ≤ cA k (k!)2, k = 0, 1, 2, . . . (4.25) Proof. We apply Lemma 2.4 to ψ uk ≡ ψ P ku with b = 1 and r = 0. || < Dx, t >3 ψ P ku||L2(R:L2x(R)) ||ψ uk||L2(R:L2x(R)) + ||t∂ x(ψ uk)||L2(R:L2x(R)) + ||P 3(ψ uk)||L2(R:L2x(R)) .(4.26) 18 M Sepúlveda and O Vera Therefore, if we wish to estimate the second term in the right hand side of (4.26) with the aid of the equation (2.12) ψ t ∂3xuk = − ψ Puk + ψ x∂xuk + t ψ Bk it is necessary to estimate ||ψ uk||L2t (R:H1x(R)) which is not yet obtained. Hence, we start from the lower regularity setting, i. e., applying (2.23) in Lemma 2.4 to ψ uk with µ = 1/2. Let ψ1 be a smaller size of smooth cut-off function with ψ1 ≤ ψ and ψ1 = 1 around (x0, t0). Applying (2.23) a ψ uk = ψ P ku with µ = 1/2 we have || < Dx, t >3 ψ1 P ku||H− 5/2(R2) ≤ c || < Dx, t > 3 ψ1 P ku||L2(R2) ||ψ1uk||H− 5/2(R2) + ||t∂ x(ψ1uk)||H− 5/2(R2) + ||P 3(ψ1uk)||H− 5/2(R2) . (4.27) The first term on the right hand side of (4.27) has already been estimated. For the third term we have ||P 3(ψ1uk)||H− 5/2(R2) ≤ ||P 3(ψ1uk)||L2x, t(R2) l!(3− l)! ||(P 3−lψ1)(P luk)||L2x, t(R2) l!(3− l)! ||P 3−lψ1||L∞x, t(R2)||P luk||L2x, t(R2) l!(3− l)! ||P k+lu||L2x, t(R2) Ak+l1 k! ≤ cA 2k! ≤ cAk2(k!)2. (4.28) For the second term on the right side hand we use the same idea of the remark above, using the dilation operator P. Indeed, ||t ∂3x(ψ1 uk)||H−5/2 ≤ ||ψ1 t ∂ xuk||H−5/2(R2) + 3 ||∂ x(t ∂xψ1 · uk)||H−5/2(R2) + 3 ||∂x(t ∂2xψ1 · uk)||H−5/2(R2) + ||t (∂ xψ1)uk||H−5/2(R2). (4.29) The last three term are bounded by the following: ||∂xψ1||L∞x, t(R2) + ||∂ xψ1||L∞x, t(R2) + ||∂ xψ1||L∞x, t(R2) ||ψ uk||L2x, t(R2) ≤ cAk3 k! ≤ cAk3 (k!)2. (4.30) On the other hand, using ||ψ1 t ∂3xuk||H−5/2(R2) ≤ ||ψ1 Puk||L2(R:L2x(R)) + ||xψ1 ∂xuk||H−5/2(R2) + ||t ψ1 Bk||H−5/2(R2) = F1 + F2 + F3. (4.31) F1 ≤ c ||ψ1||L∞x, t(R2) ||ψ P k+1u||L2x, t(R2) ≤ c ||ψ P k+1v||L2x, t(R2) ≤ cAk+14 (k + 1)! ≤ cA 5 k! ≤ cAk5 (k!)2, (4.32) Analycity for the coupled system of KdV equations 19 F2 ≤ ||xψ1 ∂xvk||L2(R:H−1x (R)) ≤ ||∂x(xψ1 vk)||L2(R:H−1x (R)) + ||∂x(xψ1)ψ vk||L2(R:H−1x (R)) ≤ ||xψ1 vk||L2x, t(R2) + ||∂x(xψ1)||L∞x, t(R2)||ψ vk||L2x, t(R2) ||xψ1||L∞x, t(R2) + ||∂x(xψ1))||L∞x, t(R2) ||ψ vk||L2x, t(R2) ≤ cAk6 k! ≤ cAk6 (k!)2. (4.33) Using Lemma 2.5(case σ = −5/2, s = 5, r = −5/2) F3 = ||t ψ1 Bk||H−5/2(R2) ≤ c1 ||ψ1||H5(R2)||ψ 2 Bx||H−5/2(R2) and replacing Bk by (2.10), we have F3 ≤ c1 k=k1+k2+k3 k1! k2! k3! 2k1 ||ψ uk2ψ uk3 ||H−3/2(R2) k′1! k 1 ||ψ vk′ ψ vk′ ||H−3/2(R2) + c1 |c| k=k′′ k′′1 ! k 2 ! k 1 ||ψ uk′′ ψ vk′′ ||H−3/2(R2) k=k1+k2+k3 k1! k2! k3! 2k1 ||ψ uk2 ||L2(R2)||ψ uk3 ||L2(R2) k′1! k 1 ||ψ vk′ ||L2(R2)||ψ vk′ ||L2(R2) + c1 |c| k=k′′ k′′1 ! k 2 ! k 1 ||ψ uk′′ ||L2(R2)||ψ vk′′ ||L2(R2) k=k1+k2+k3 k1! k2! k3! 2k1 Ak27 k2!A 7 k3! k′1! k + c1 |c| k=k′′ k′′1 ! k 2 ! k k=k1+k2+k3 Ak2+k37 + c1 + c1 |c| k! k=k′′ k′′1 ! 20 M Sepúlveda and O Vera and then F3 ≤ c1 k!Ak7 k−k1∑ A− k17 + c1 k!Ak8 + c1 |c| k! k=k′′ k′′1 ! e2/A7 Ak7 (k + 1)! + c1 e3/A8 Ak8 (k + 1)! + c1 |c| k! k=k′′ k′′1 ! 10 . (4.34) Replacing (4.30), (4.35) and (4.29) in (4.31) we obtain ||ψ1 t ∂3xuk||H−5/2(R2) ≤ c2Ak11 k! + c1 |c| k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.35) Replacing (4.30) and (4.35) in (4.29) ||t ∂3x(ψ1 uk)||H−5/2(R2) ≤ c3Ak12 k! + c1 |c| k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.36) Now replacing (4.28) and (4.36) in (4.27) we obtain || < Dx, t >3 ψ uk||H− 5/2(R2) ≤ c4Ak13 k! + c1 |c| k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.37) In particular ||ψ uk||H1/2(R2) ≤ c5Ak14 k! + c1 |c| k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.38) Using a similar argument as above for || < Dx, t >3 ψ P ku||H− 3/2(R2) with µ = 3/2 in (2.23) and replacing the support of the cut-off function ψǫ we obtain ||ψ uk||H3/2(R2) ≤ c5Ak14 k! + c1 |c| k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.39) Analycity for the coupled system of KdV equations 21 In a similar way we have ||ψ vk||H3/2(R2) ≤ c5Ak15 k! + c1 |c̃| k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.40) Adding (4.39) with (4.40) and performing straightforward calculations as (4.22) we obtain ||ψ uk||H3/2(R2) + ||ψ vk||H3/2(R2) ≤ C A k (k!)2, k = 0, 1, 2, . . . (4.41) To obtain the estimate for ||ψ P ku||H7/2(R2) and ||ψ P kv||H7/2(R2) we repeat the above method with µ = 7/2. Proposition 4.3. Suppose that ||ψ uk||H7/2(R2) ≤ cA 1 (k!) 2, k = 0, 1, 2, . . . (4.42) ||ψ vk||H7/2(R2) ≤ cA 2 (k!) 2, k = 0, 1, 2, . . . (4.43) then we have t∈[t0−ǫ, t0+ǫ] ||(t1/3∂x)P ku||H1(x0−ǫ, x0+ǫ) ≤ c1A 3 [ (k + l)!] 2, k, l = 0, 1, 2, . . .(4.44) t∈[t0−ǫ, t0+ǫ] ||(t1/3∂x)P kv||H1(x0−ǫ, x0+ǫ) ≤ c1A 4 [ (k + l)! ] 2, k, l = 0, 1, 2, . . .(4.45) where ǫ > 0 is so small that ψ ≡ 1 near I = (x0 − ǫ, x0 + ǫ)× (t0 − ǫ, t0 + ǫ). Proof. Let It0 = (t0 − ǫ, t0 + ǫ) and Ix0 = (x0 − ǫ, x0 + ǫ), then we have I = Ix0 × It0 . For any fixed t ∈ Ix0 , let L = t1/3∂x. We show that for some positive constants c and A0 the following inequality holds ||LlP ku||H1x(Ix0 ) ≤ cA 0 [ (k + l)! ] 2, ∀ k, ∀ l = 0, 1, 2, . . . (4.46) Now, let use induction over l. By the trace theorem, we have ||LlP ku||H1x(Ix0 ) ≤ ||t l/3 ∂lxP ku(t)||H1x(Ix0) ≤ (t0 + ǫ) l/3||∂lxP ku||H3/2(Ix0×It0) ≤ (t0 + ǫ)l/3||P ku||H7/2(Ix0×It0) ≤ (t0 + ǫ) l/3||ψ P ku||H7/2(R2) ≤ (t0 + ǫ)l/3 c1Ak1 k! ≤ (t0 + ǫ)l/3 c1Ak+l0 (k + l) ≤ (t0 + ǫ)l/3 c1Ak+l0 [ (k + l)! ] 2. (4.47) where we take c = (t0 + ǫ) l/3c1 and A0 = max{1, A1}. Hence, in the case l = 0, 1, 2, it is easy to show that (4.46) follows directly from the assumption. 22 M Sepúlveda and O Vera Now, we assume that (4.46) is true to l ≥ 2. Applying P k to the equation (2.4), we have ku) + ∂3x(P ku) = LP ku = (P + 3)kLu = (P + 3)k(∂tu+ ∂ = −(P + 3)k 2) + c ∂x(u v) (P + 3)k∂x(u (P + 3)k∂x(v 2)− c (P + 3)k∂x(u v) ∂x(P + 2) k(u2)− ∂x(P + 2) k(v2)− c ∂x(P + 2)k(u v) such that t ∂t(P ku) + t ∂3x(P ku) = − a t ∂x(P + 2) k(u2)− b t ∂x(P + 2) k(v2)− c t ∂x(P + 2)k(u v).(4.48) Moreover, P = 3 t ∂t + x ∂x. Then t ∂t(P ku) = P k+1u− x ∂x(P ku). (4.49) Replacing (4.49) in (4.48) we obtain L3P ku = t ∂3x(P ku) = − P k+1u+ x ∂x(P t ∂x(P + 2) k(u2)− t ∂x(P + 2) k(v2)− c t ∂x(P + 2)k(u v). (4.50) Hence, applying Ll−2 we have ||Ll+1P ku||H1x(Ix0 ) = ||L l−2L3P ku||H1x(Ix0) ||Ll−2 P k+1u||H1x(Ix0 ) + ||Ll−2 x ∂x(P ku)||H1x(Ix0 ) ||tLl−2 ∂x(P + 2)k(u2)||H1x(Ix0) + ||tLl−2 ∂x(P + 2)k(v2)||H1x(Ix0) + |c| ||tLl−2 ∂x(P + 2)k(u v)||H1x(Ix0 ) = F1 + F2 + F3 + F4 + F5. (4.51) Using the induction assumption, we obtain k+l+1 14 (k + l + 1)!. (4.52) We estimate the term Ll−2(x ∂x) for l ≥ 3. Let r = l − 2, then we estimate Lr(x ∂x) for r ≥ 1. ∂rx(x ∂x) = ∂r−kx (x ) · ∂kx( ∂x ). (4.53) ∂r−kx (x ) = 1 if k = r − 1 0 if k ≤ r − 2 Analycity for the coupled system of KdV equations 23 then in (4.53) we obtain ∂rx(x ∂x) = r ∂ x ( ∂x ) + x ∂ x( ∂x ) = r ∂ x + x ∂x( ∂ = (l − 2) ∂(l−2)x + x ∂x( ∂(l−2)x ), that is, Ll−2(x ∂x) = x ∂xLl−2 + (l − 2)Ll−2, for l ≥ 3. For F2 we have F2 ≤ ||x ∂xLl−2P ku||H1x(Ix0 ) + (l − 2) ||L l−2P ku||H1x(Ix0 ) ≤ ||x t−1/3Ll−1P ku||H1x(Ix0 ) + (l − 2) ||L l−2P ku||H1x(Ix0) ≤ c (t0 − ǫ) (|x0|+ ǫ+ 1) ||Ll−1P ku||H1x(Ix0 ) + (l − 2) ||L l−2P ku||H1x(Ix0 ) ≤ (t0 − ǫ)−1/3(|x0 + ǫ+ 1) c1 Ak+l−114 (k + l − 1)! + c1A k+l−1 14 (l − 2) (k + l − 1)! k+l+1 14 (k + l + 1)! (4.54) where we take A14 larger than (t0 − ǫ)−1/3(|x0|+ ǫ+ 1) and 3. Using that (L = t1/3 ∂3x) tLl−2∂x = t t(l−2)/3∂(l−2)x ∂x = t t−1/3 t(l−1)/3∂(l−1)x = t2/3 Ll−1, we have ||t2/3 Ll−1 (P + 2)k(u2)||H1x(Ix0 ) (t0 + ǫ) l−1=l1+l2 k=k1+k2+k3 (l − 1)! l1!l2! k1!k2!k3! × c2 ||Ll1P k1u||H1x(Ix0) ||L l2P k2u||H1x(Ix0). Using the induction assumption (t0 + ǫ) l−1=l1+l2 k=k1+k2+k3 1 k! (l − 1)! (l1 + k1)! l1! k1! (l2 + k2)! l2! k2! Ak+l−114 (t0 + ǫ) 2/3 c2 c 1 (l + k − 1)!Ak+l−114 l−1=l1+l2 k=k1+k2+k3 × (l1 + k1)! l1! k1! (l2 + k2)! l2! k2! k! (l − 1)! (l + k − 1)! Using that l−1=l1+l2 k=k1+k2+k3 (l1 + k1)! l1! k1! (l2 + k2)! l2! k2! k! (l − 1)! (l + k − 1)! ≤ e2(l + k)! we obtain F3 ≤ (t0 + ǫ)2/3 c2 c31 e2 (l + k)!Ak+l−114 ≤ k+l+1 14 (k + l + 1)! (4.55) where we take A14 larger than (t0 − ǫ)−1/3 c2 c21 e2, and 3. In a similar way k+l+1 15 (k + l + 1)! (4.56) 24 M Sepúlveda and O Vera where we take A15 larger than (t0 − ǫ)−1/3 c4 c23 e2, and 3. Finally, in a similar way k+l+1 16 (k + l + 1)! (4.57) where we take A16 larger than (t0 − ǫ)−1/3 c6 c25 e2, and 3. Therefore, from (4.52), (4.54), (4.55), (4.56) and (4.57) we obtain ||Ll+1P ku||H1x(Ix0) ≤ c7A k+l+1 17 (k + l + 1)!. (4.58) In a similar way, we obtain ||Ll+1P kv||H1x(Ix0) ≤ c7A k+l+1 17 (k + l + 1)!, (4.59) and the result follows. Proposition 4.4. Suppose that there exists a positive constants c1, c2 and A14, A15 such t∈[t0−ǫ, t0+ǫ] ||∂lxP ku||H1x(x0−ǫ, x0+ǫ) ≤ c1A 14 [ (k + l)! ] 2, k, l = 0, 1, 2, . . . (4.60) t∈[t0−ǫ, t0+ǫ] ||∂lxP kv||H1x(x0−ǫ, x0+ǫ) ≤ c2A 15 [ (k + l)! ] 2, k, l = 0, 1, 2, . . . (4.61) Then we have respectively t∈[t0−ǫ, t0+ǫ] ||∂mt ∂lxu||H1x(x0−ǫ, x0+ǫ) ≤ c3A 16 [ (m+ l)! ] 2, m, l = 0, 1, 2, . . . (4.62) t∈[t0−ǫ, t0+ǫ] ||∂mt ∂lxv||H1x(x0−ǫ, x0+ǫ) ≤ c4A 17 [ (m+ l)! ] 2, m, l = 0, 1, 2, . . . (4.63) where c3, c4 and A16, A17 only depend on c1, c2 and A14, A15, respectively and ǫ, (x0, t0). Proof. Using the idea of Proposition 4.3, we fix t ∈ Ix0 . First we show that for some positive constants c3, A16 and B16 ||(x ∂x)m ∂lxP kv||H1x(Ix0) ≤ c3A k+m+l 16(k +m+ l)!, k, m, l = 0, 1, 2, . . . (4.64) We use induction. Suppose that (4.64) is true for m. ||(x ∂x)m+1 ∂lxP kv||H1x(Ix0) = ||(x ∂x) (x ∂x)m ∂lxP kv||H1x(Ix0) ≤ (|x0|+ ǫ+ 1) ||(x ∂x + I)m ∂l+1x P kv||H1x(Ix0 ) ≤ c(|x0|, ǫ) ||(x ∂x)j ∂l+1x P kv||H1x(Ix0 ) k+l+j+1 16(k + l + j + 1)! ≤ c3Ak+l+m+116 B 16(k + l +m+ 1)! (A16 B16) −(m−j) (m− j)! (k + l + j + 1)! (k + l +m+ 1)! ≤ e−A16 B16c3Ak+l+m+116 B 16(k + l +m+ 1)! (4.65) Analycity for the coupled system of KdV equations 25 where we take B16 so large that B16 ≥ max{|x0|+ǫ+1, 1}.We show that for some positive constants c4, A17 we have ||(t ∂t)m ∂lxu||H1x(Ix0 ) ≤ c4A 17 (l +m)!, l, m = 0, 1, 2, . . . Using that t ∂t = (P − x ∂x), we obtain ||(t ∂t)m ∂lxu||H1x(Ix0 ) = 3 −m ||(P − x ∂x)m ∂lxu||H1x(Ix0 ) ≤ 3−m m=j1+j2 j1! j2! ||(x ∂x)j1 P j2∂lxu||H1x(Ix0 ) ≤ 3−m m=j1+j2 j1! j2! ||(x ∂x)j1 ∂lx(P − l)j2u||H1x(Ix0) ≤ 3−m m=j1+j2+j3 j1! j2! j3! lj3 ||(x ∂x)j1 ∂lxP j2u||H1x(Ix0). where we replace j2 into j2 + j3. Now, using the induction hypothesis we have (with B17 ≥ A16B16) ||(t ∂t)m ∂lxu||H1x(Ix0 ) ≤ 3−m m=j1+j2+j3 j1! j2! j3! lj3 c3B j1+j2+l 17 (j1 + j2 + l)! ≤ 3−m c3Bm+l17 (m+ l)! m=j1+j2+j3 j1! j2! j3! (j1 + j2 + l)! (m+ l)! , (4.66) Observing that lj3 (j1 + j2 + l)! (m+ l)! ≤ 1, we obtain in (4.66) ||(t ∂t)m ∂lxu||H1x(Ix0 ) ≤ 3 −m c3 (2 +B mBl+m17 (l +m)! ≤ c4Al+m17 (l +m)! where we take A17 = max{B17, 3−1B17 (2 +B−117 )}. We show that for some positive con- stants c4, A18 and B18 we have ||(t ∂t)j ∂mt ∂lxu||H1x(Ix0 ) ≤ c4A j+m+l 18 B18(j +m+ l)!, j, l, m = 0, 1, 2, . . . (4.67) Induction in m. ||(t ∂t)j ∂m+1t ∂lxu||H1x(Ix0 ) ≤ ||∂t(t ∂t − I) m ∂mt ∂ xu||H1x(Ix0) = t−1 ||t ∂t(t ∂t − I)j ∂mt ∂lxu||H1x(Ix0) ≤ (t0 − ǫ)−1 ||(t ∂t)j1+1 ∂mt ∂lxu||H1x(Ix0 ). 26 M Sepúlveda and O Vera Using the induction hypothesis ||(t ∂t)j ∂m+1t ∂lxu||H1x(Ix0 ) ≤ (t0 − ǫ)−1 j1+l+m+1 18 (j1 + l +m+ 1)! = c4 (t0 − ǫ)−1Aj1+l+m+118 B 18 (j1 + l +m+ 1)! −(j−j1 (j − j1)! (j1 +m+ l + 1)! (j − j1)! (j +m+ l + 1)! = c4 (t0 − ǫ)−1 e−A18 Aj+l+m+118 B 18 (j + l +m+ 1)! ≤ c4 Aj+l+m+118 B 18 (j + l +m+ 1)! where we take B18 larger than (t0− ǫ)−1 e−A18 . Finally, we choose j = 0 in (4.67) and take c2 = c4 and A15 = A18B18. The result of analyticity follows. Acknowledgments This work has been supported by Fondap in Applied Mathematics (Project # 15000001), CNPq/CONICYT Project, # 490987/2005-2 (Brazil) and # 2005-075 (Chile). 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Introduction Reduction of the Problem and Preliminary Results Existence and Well-Posedness The main result

We study that a solution of the initial value problem associated for the coupled system of equations of Korteweg - de Vries type which appears as a model to describe the strong interaction of weakly nonlinear long waves, has analyticity in time and smoothing effect up to real analyticity if the initial data only has a single point singularity at $x=0.$

Introduction We consider the following coupled system of equations of Korteweg - de Vries type ũt + ũxxx + a3 ṽxxx + ũ ũx + a1 ṽ ṽx + a2 (ũ ṽ)x = 0, x, t ∈ R (1.1) b1 ṽt + ṽxxx + b2 a3 ũxxx + ṽ ṽx + b2 a2 ũ ũx + b2 a1 (ũ ṽ)x = 0, (1.2) ũ(x, 0) = ũ0(x), ṽ(x, 0) = ṽ0(x). (1.3) where ũ = ũ(x, t), ṽ = ṽ(x, t) are real-valued functions of the variables x and t and a1, a2, a3, b1, b2 are real constants with b1 > 0 and b2 > 0. The original coupled system is ũt + ũxxx + a3 ṽxxx + ũ p ũx + a1 ṽ p ṽx + a2 (ũ p ṽ)x = 0, x, t ∈ R (1.4) b1 ṽt + ṽxxx + b2 a3 ũxxx + ṽ p ṽx + b2 a2 ũ p ũx + b2 a1 (ũ ṽ p)x = 0 (1.5) ũ(x, 0) = ũ0(x), ṽ(x, 0) = ṽ0(x) (1.6) Copyright c© 200* by M Sepúlveda and O Vera http://arxiv.org/abs/0704.1861v1 2 M Sepúlveda and O Vera where ũ = ũ(x, t), ṽ = ṽ(x, t) are real-valued functions of the variables x and t and a1, a2, a3, b1, b2 are real constants with b1 > 0 and b2 > 0. The power p is an integer larger than or equal to one. The system (1.4)-(1.6) has the structure of a pair of Korteweg - de Vries equations coupled through both dispersive and nonlinear effects. In the case p = 1, the system (1.4)-(1.6) was derived by Gear and Grimshaw [9] as a model to describe the strong interaction of weakly nonlinear, long waves. Mathematical results on the system (1.4)-(1.6) were given by J. Bona et al. [5]. They proved that (1.4)-(1.6) is globally well posed in Hs(R)×Hs(R) for any s ≥ 1 provided |a3| < 1/ b2. The system (1.4)-(1.6) has been intensively studied by several authors (see [2, 3, 5, 7, 23] and the references therein). We have the following conservation laws E1(ũ) = ũ dx , E2(ṽ) = ṽ dx , E3(ũ, ṽ) = (b2ũ 2 + b1ṽ 2)dx (1.7) The time-invariance of the functionals E1 and E2 expresses the property that the mass of each mode separately is conserved during interaction, while that of E3 is an expression of the conservation of energy for the system of two models taken as a whole. The solutions of (1.4)-(1.6) satisfy an additional conservation law which is revealed by the time-invariance of the functional b2 ũ x + ṽ x + 2b2a3ũxṽx − b2 − b2a2ũ2ṽ − b2a2ũ2ṽ − b2a1ũṽ2 − The functional E4 is a Hamiltonian for the system (1.4)-(1.6) and if b2a 3 < 1, φ4 will be seen to provide an a priori estimate for the solutions (ũ, ṽ) of (1.4)-(1.6) in the space H1(R)×H1(R). Furthermore, the linearization of (1.1)-(1.3) about the rest state can be reduced to two, linear Korteweg - de Vries equations by a process of diagonalization. Using this remark and the smoothing properties (in both the temporal and spatial variables) for the linear Korteweg - de Vries derived by Kato [13, 15], Kenig, Ponce and Vega [18, 19] it will be shown that (1.4)-(1.6) is locally well-posed in Hs(R) × Hs(R) for any s ≥ 1 whenever b2a3 6= 1. This result was improved by J. M. Ash et al. [1] showing that the system (1.1)-(1.3) is globally well-posed in L2(R) × L2(R) provided that b2a3 6= 1. In 2004, F. Linares and M. Panthee [21] improve this result showing that the system (1.1)- (1.3) is locally well-posed in Hs(R) × Hs(R) for s > −3/4 and globally well-posed in Hs(R)×Hs(R) for s > −3/10 under some conditions on the coefficients, indeed for a3 = 0 and b1 = b2. Following the idea W. Craig et al. [6], it is shown in [23] that C ∞ solutions (ũ( · , t), ṽ( · , t)) to (1.1)-(1.3) are obtained for t > 0 if the initial data (ũ(x, 0), ṽ(x, 0)) belong to a suitable Sobolev space satisfying resonable conditions as |x| → ∞. Since (1.1)- (1.3) is a coupled system of Korteweg-de Vries equations, it is natural to ask whether it has a smoothing effect up to real analyticity if the initial data only has a single point singularity at x = 0 as the known results for the scalar case of a single Korteweg -de Vries equation. Using the scaling argument we can have an insight to this question. In this paper our purpose is to prove the analyticity in time of solutions to (1.1)-(1.3) without regularity assumption on the initial data improving those obtained in [23]. Our main tool is the generator of dilation P = 3 t ∂t + x ∂x. which almost commutes with the linear Korteweg-de Vries operator L = ∂t + ∂x. Indeed [L, P ] = 3L. A typical example of initial data satisfying the assumption of the above theorem is the Dirac delta measure, since (xk ∂x) kδ(x) = (−1)k k! δ(x). The other example of the data is p. v. 1 , where Analycity for the coupled system of KdV equations 3 p. v. denotes the Cauchy principal value. Linear combination of those distributions with analytic Hs data satisfying the assumption is also possible. In this sense, the Dirac delta measure adding the soliton initial data can be taken as an initial datum. Using the operator K = x · ▽ + 2 i t ∂t it was proved the Gevrey smoothing effect in space variable [8]. Indeed, it was shown that, if the initial data belongs to a Gevrey class of order 2, then solutions of some nonlinear Schrödinger equations become analytic in the space variable for t 6= 0. For the Korteweg-de Vries equations version of the generator of dilation is also useful to study the analyticity in time and the Gevrey effect in the space variables for solutions [8]. This paper is organized as follows: In section 2 we have the reduction of the problem and we outline briefly the notation, terminology to be used subsequently and results that will be used several times. In section 3 we prove a theorem of existence and well-posedness of the solutions. In section 4 we prove the following theorem: Theorem 1.1. Suppose that the initial data (ũ0, ṽ0) ∈ Hs(R) × Hs(R), s > −3/4 and A0, A1 > 0 such that ||(x ∂x)kũ0||Hs(R) < +∞ : ||(x ∂x)kũ0||Hs(R) < +∞. (1.8) Then for some b ∈ (1/2, 7/12), there exist T = T (||ũ0||Hs(R), ||ṽ0||Hs(R)) and a unique solution of (1.1)-(1.3) in a certain time (−T, T ) and the solution (ũ, ṽ) is time locally well-posed, i. e., the solution continuously depends on the initial data. Moreover, the solution (ũ, ṽ) is analytic at any point (x, t) ∈ R× {(−T, 0) ∪ (0, T )}. Corollary 1.1. Let s > −3/4, b ∈ (1/2, 7/12). Suppose that the initial data (ũ0, ṽ0) ∈ Hs(R)×Hs(R), and A0, A1 > 0 such that (k!)3 ||(x ∂x)kũ0||Hs(R) < +∞ : (k!)3 ||(x ∂x)kũ0||Hs(R) < +∞.(1.9) Then there exists a unique solution (ũ, ṽ) ∈ C((−T, T ), Hs(R))∩Xsb×C((−T, T ), Hs(R))∩ Xsb to the coupled system of Korteweg- de Vries equation (1.1)-(1.3) for a certain (−T, T ) and for any t ∈ (−T, 0)∪(0, T ), the pair (ũ, ṽ) are analytic functions in the space variable and for x ∈ R, ũ(x, · ) and ṽ(x, · ) are Gevrey 3 as function of the time variable. Remark 1.1. In Theorem 1.1 and Corollary 1.2, the assumption on the initial data implies analyticity and Gevrey 3 regularity except at the origin respectively. In this sense, those results state that the singularity at the origin immediately disappears after t > 0 or t < 0, up to analyticity. Remark 1.2. The crucial part for obtaining a full regularity is to gain the L2(R2) regular- ity of the solutions (uk, vk) from the negative order Sobolev space. This part is obtained in Proposition 4.1 in Section 4. We utilize a three steps recurrence argument for treating the nonlinearity appearing in the right hand side of t ∂3xuk = − Puk + x ∂xuk + tB k(u, u) + tB k(v, v) + tB k(u, v) (1.10) t ∂3xvk = − Pvk + x ∂xvk + t C k(u, u) + t C k(v, v) + t C k(u, v). (1.11) 4 M Sepúlveda and O Vera Then step by step, we obtain the pointwise analytic estimates t∈[t0−ǫ, t0+ǫ] ||∂mt ∂lxu||H1(x0−ǫ, x0+ǫ) ≤ cA 1 (m+ l)!, l, m = 0, 1, 2, . . . (1.12) t∈[t0−ǫ, t0+ǫ] ||∂mt ∂lxv||H1(x0−ǫ, x0+ǫ) ≤ cA 2 (m+ l)!, l, m = 0, 1, 2, . . . (1.13) Since initially we do not know whether the solution belong to even L2(R2) we should men- tion that the local well-posedness is essentially important for our argument and therefore it merely satisfies the coupled system equations in the sense of distribution. 2 Reduction of the Problem and Preliminary Results As mentioned in the introduction we consider the following coupled system of equations of Korteweg - de Vries type (1.1)-(1.3). If a3 = 0 there is no coupling in the dispersive terms. Let us assume that a3 6= 0. We are interested in decoupling the dispersive terms in the system (1.1)-(1.3). For this, let a23 b2 6= 1. We consider the associated linear system Wt +AWxxx = 0, W (x, 0) =W0(x) (2.1) where , A = a3 b2 The eigenvalues of A are given by 4 b2 a  (2.2) 4 b2 a  (2.3) which are distinct since b1 > 0, b2 > 0 and a3 6= 0. Our assumption a23 b2 6= 1 guarantees that α± 6= 0. Thus we can write the system (1.1)-(1.3) in a matrix form as in [21]. After we make the change of scale ũ(x, t) = u(α + x, t) and ṽ(x, t) = v(α − x, t). Then we obtain the system of equations ut + uxxx + a uux + b v vx + c (u v)x = 0, x, t ∈ R (2.4) vt + vxxx + ã u ux + b̃ v vx + c̃ (u v)x = 0, (2.5) u(x, 0) = u0(x), v(x, 0) = v0(x) (2.6) where a, b, c and ã, b̃, c̃ are constant. Remark 2.1. Notice that the nonlinear terms involving the functions u and v are not evaluated at the same point. Therefore those terms are not local anymore. Analycity for the coupled system of KdV equations 5 For s, b ∈ R define the spaces Xsb and Xsb−1 to be the completion of the Schwartz space S(R2) with respect to the norms ||u||Xs (1 + |τ − ξ3|)2b (1 + |ξ|)2s |û(ξ, τ)|2 dξ dτ ||u||Xs (1 + |τ − ξ3|)2(b−1) (1 + |ξ|)2s |û(ξ, τ)|2 dξ dτ where Xsb = {u ∈ S ′(R2) : ||u||Xsb <∞}. Let Fx and Fx, t be the Fourier transform in the x and (x, t) variables respectively. The Riesz operator Dx is defined by Dx = F−1ξ |ξ| Fx. The fractional derivative is defined by < Dx > s = F−1ξ < ξ > s Fx = F−1ξ (1 + |ξ| 2)s/2 Fx < Dx, t > s = F−1 < |ξ|+ |τ | >s Fx, t For < · >= (1 + | · |2)1/2, we have i) || · ||Hb(R:Hr(R)) = || < Dt >b< Dx >r · ||L2x, t(R2). ii) Hs(R) = {u ∈ S ′(R) : < Dx >s u ∈ L2(R)}. iii) || · ||Hs(R) = || < Dx >s · ||L2(R). Remark 2.2. With the above notation we obtain a) ||u||Hsx(R) = || < ξ > s û ||L2(R). b) ||u||L2t (R:Hrx(R)) = || < ξ > r û ||L2(R2). c) || < Dx >s u||L2(R) = ||u||Hs(R). d) || < Dt >b< Dx >r u||L2x, t(R2) = ||u||Hbt (R:Hrx(R)). e) || < Dx, t >s u||L2t (R:Hrx(R)) = || < ξ > r< |ξ|+ |τ | >s û(ξ, τ)||L2(R2). We consider the following operators: L = ∂t + ∂ x and J = x − 3 t ∂2x then [L, J ] ≡ LJ−J L = 0.We introduce the ”generator of dilation” P = 3 t ∂t+x ∂x for the linear part of the coupled system (2.4)-(2.6) and the ”localized dilation operator” P0 = 3 t0 ∂t+x0 ∂x. By employing a localization argument, we look at the operator P as a vector field P0 = 3 t0 ∂t+x0 ∂x near a fixed point (x0, t0) ∈ R×{(−T, 0)∪ (0, T )}. Since P0 is a directional derivative toward to (x0, t), we introduce another operator L30 = t0 ∂3x which plays the role of a non-tangential vector field to P0. Since P0 and L0 are linearly independent, the space and time derivative can be covered by those operator. The main reason why we choose L0 is because the corresponding variable coefficients operator L3 = t ∂3x can be treated via the equations (1.10)-(1.11) and a cut-off procedure enables us to handle the right hand side of those. Remark 2.3. For L and P we have the following properties: a) [L, P ] ≡ LP = (P + 3)L. b) LP k = (P + 3)kL. 6 M Sepúlveda and O Vera c) (P + 3)k∂x = ∂x(P + 2) d) (P + 3)k∂3x = ∂ e) P0 P = P P0 + 3P0 − 2x0 ∂x. Notation. The summation k=k1+k2+k3 0≤k1, k2, k3≤k is simply abbreviated by k=k1+k2+k3 Let P ku = uk, then ku) + ∂3x(P ku) = LP ku = (P + 3)kLu = (P + 3)k(∂tu+ ∂ = −(P + 3)k 2) + c ∂x(u v) (P + 3)k∂x(u (P + 3)k∂x(v 2)− c (P + 3)k∂x(u v) ∂x(P + 2) k(u2)− ∂x(P + 2) k(v2)− c ∂x(P + 2)k(u v). Noting that (P + 2)ku = 2k−jP ju. Hence B1k(u, u) = − ∂x(P + 2) k(u2) (P + 2)mu · P k−mu 2m−j P ju · P k−mu (m− j)! j! (k −m)! 2m−j P ju · P k−mu k=k1+k2+k3 k1! k2! k3! 2k1 ∂x (uk2 · uk3) . (2.7) In a similar way B2k(v, v) = − ∂x(P + 2) k(v2) = − b k′1! k · vk′ . (2.8) B3k(u, v) = c ∂x(P + 2) k(u v) = − c k=k′′ k′′1 ! k 2 ! k · vk′′ . (2.9) Analycity for the coupled system of KdV equations 7 Therefore ku) + ∂3x(P k=k1+k2+k3 k1! k2! k3! 2k1 ∂x (uk2 · uk3)− k′1! k · vk′ k=k′′ k′′1 ! k 2 ! k · vk′′ = B1k(u, u) +B k(v, v) +B k(u, v). (2.10) Performing similar calculations as above we obtain kv) + ∂3x(P k=k1+k2+k3 k1! k2! k3! 2k1 ∂x (uk2 · uk3)− k′1! k · vk′ k=k′′ k′′1 ! k 2 ! k · vk′′ = C1k(u, u) + C k(v, v) + C k(u, v). (2.11) The above nonlinear terms maintain the bilinear structure like that of the original coupled system of equations of KdV type, since Leibniz’s rule can be applied for operations of P. Now, each uk and vk satisfies the following system of equations ∂tuk + ∂ xuk = B k(u, u) +B k(v, v) +B k(u, v) ≡ Bk (2.12) ∂tvk + ∂ xvk = C k(u, u) + C k(v, v) + C k(u, v) ≡ Ck (2.13) uk(x, 0) = (x ∂x) ku0(x) ≡ uk0(x), vk(x, 0) = (x ∂x)kv0(x) ≡ vk0 (x). (2.14) In order to obtain a well-posedness result for the system (2.12)-(2.14) we use Duhamel’s principle and we study the following system of integral equations equivalent to the system (2.12)-(2.14) ψ(t)uk = ψ(t)V (t)u 0 − ψ(t) V (t− t′)ψT (t′)Bk(t′) dt′ (2.15) ψ(t) vk = ψ(t)V (t) v 0 − ψ(t) V (t− t′)ψT (t′)Ck(t′) dt′ (2.16) where V (t) = e−t ∂ x is the unitary group associated with the linear problem and ψ(t) ∈ C∞0 (R), 0 ≤ ψ ≤ 1 is a cut-off function such that ψ(t) = 1, if |t| < 1 0, if |t| > 2 and ψT (t) = ψ(t/T ) The following results are going to be used several times in the rest of this paper. 8 M Sepúlveda and O Vera Lemma 2.1 ([16]). . Let s ∈ R, a, a′ ∈ (0, 1/2), b ∈ (1/2, 1) and δ < 1. Then for any k = 0, 1, 2, . . . , we have ||ψδφk||Xs−a ≤ c δ (a−a′)/4(1−a′) ||φk||Xs , (2.17) ||ψδ V (t)φk||Xs ≤ c δ1/2−b||φk||Hs(R), (2.18)∣∣∣∣ ∣∣∣∣ψδ V (t− t′)Fk(t′) dt′ ≤ c δ1/2−b ||Fk||Xs . (2.19) Lemma 2.2 ([16]). . Let s > −3/4, b, b′ ∈ (1/2, 7/12) with b < b′. Then for any k, l = 0, 1, 2, . . . we have ||∂x(uk vl)||Xs ≤ c ||vk||Xs ||vl||Xs . (2.20) Lemma 2.3 ([12]). . Let s < 0, b ∈ (1/2, 7/12) and ψ = ψ(x, t) be a smooth cut- off function such that the support of ψ is in B2(0) and ψ = 1 on B1(0). We set ψǫ = ψ((x− x0)/ǫ, (t− t0)/ǫ). Then for f ∈ Xsb , we have ||ψǫ f ||Xs ≤ c ǫ−|s|−5|b|||ψǫ||X|s|+2 |b| ||f || s+2 |b| , (2.21) where the constant c is independent of ǫ and f. Lemma 2.4 ([12]). . Let P be the generator of the dilation and Dx, t be an operator defined by F−1 < |τ | + |ξ| > Fx, t. We fix an arbitrary point (x0, t0) ∈ R × {(−T, 0) ∪ (0, T )}. 1) Suppose that b ∈ (0, 1], r ∈ (−∞, 0] and g ∈ Xrb−1 with supp g ⊂ B2ǫ(x0, t0) and t∂3xg, P 3g ∈ Xrb−1. If ǫ > 0 is sufficiently small, then we have || < Dx, t >3b g||L2(R:Hr(R)) ≤ c ||g||Xr + ||t∂3xg||Xrb−1 + ||P 3g||Xr (2.22) where the constant c = c(x0, t0, ǫ). 2) If g ∈ Hµ−3(R2) with supp g ⊂ B2ǫ(x0, t0) and t∂3xg, P 3g ∈ Hµ−3(R2). Then for small ǫ, we have || < Dx, t >µ g||L2(R2) ≤ c ||g||Hµ−3(R2) + ||t∂3xg||Hµ−3(R2) + ||P 3g||Hµ−3(R2) (2.23) where the constant c = c(x0, t0, ǫ). Lemma 2.5 ([12]). . Let 0 ≤ s, r ≤ n/2 with n/2 ≤ s + r and suppose that f ∈ Hs(Rn) and g ∈ Hr(Rn). Then for any σ < s+ r − n/2, we have f g ∈ Hσ(Rn) and ||f g||Hσ(Rn) ≤ c(ǫ) ||f ||Hs(Rn)||g||Hr(Rn), (2.24) where ǫ = s+ r − n/2− σ. Corollary 2.1 ([12]). . For 1/2 < b < 1 and −3/4 < s < 0, we have ||ψ f || ≤ c ||f ||Xs (2.25) where ψ ∈ C∞0 (R2) and c is independent of f. Analycity for the coupled system of KdV equations 9 Lemma 2.6 ([12]). . Let ψ(x) be a smooth cut-off function in C∞0 ((−2, 2)) with ψ(x) = 1 on (−1, 1). We set ψǫ = ψ(x/ǫ) for 0 < ǫ < 1. Then for r ≤ 0, and f ∈ Hr, we have ||ψǫ f ||Hr(R) ≤ c ǫ−δ||f ||Hr(R) if − 1/2 ≤ r ≤ 0 c ǫ1/2+r||f ||Hr(R) if r < −1/2 where δ > 0 is an arbitrary small constant and c is independent of ǫ. Throughout this paper c is a generic constant, not necessarily the same at each occasion (it will change from line to line), which depends in an increasing way on the indicated quantities. 3 Existence and Well-Posedness We firstly solve the following (slightly general) system of equations ∂tuk + ∂ xuk = B k(u, u) +B k(v, v) +B k(u, v) ≡ Bk (3.1) ∂tvk + ∂ xvk = C k(u, u) + C k(v, v) + C k(u, v) ≡ Ck (3.2) uk(x, 0) = (x ∂x) ku0(x) ≡ uk0(x) , vk(x, 0) = (x ∂x)kv0(x) ≡ vk0 (x) (3.3) where Bk and Ck are as above. Definition 3.1. Let f = (f0, f1, . . . , fk) denotes the infinity series of distributions and define AA0(Xsb ) ≡ f = (f0, f1, . . . , fk), fi ∈ Xsb , (i = 0, 1, 2 . . .) such that ||f ||AA0 (Xsb ) < +∞ where ||f ||AA0 (Xsb ) ≡ ||fk||Xs Similarly, for u0 = {u00, u10, . . . , uk0, . . . } and v0 = {v00 , v10 , . . . , vk0 , . . . } we set ||u0||AA0 (Hs(R)) ≡ ||uk0 ||Hs(R) and ||v0||AA0 (Hs(R)) ≡ ||vk0 ||Hs(R) respectively. Remark 3.1. Each solution of the coupled system of Korteweg de Vries equations is accompanied by the following estimate ||P ku||Xs ≤ cAk0 k!, and ||P kv||Xsb ≤ cA 1 k!, k = 0, 1, 2, . . . Theorem 3.1. Let −3/4 < s, b ∈ (1/2, 7/12). Suppose that uk0 , vk0 ∈ Hs(R)(k = 0, 1, 2, . . .) and satisfies ||u0||AA0 (Xsb ) = ||uk0 ||Hs(R) < +∞ and ||v0||AA0 (Xsb ) = ||vk0 ||Hs(R) < +∞. 10 M Sepúlveda and O Vera Then there exist T = T (||uk0 ||Hs(R), ||vk0 ||Hs(R)) and a unique solution u = (u0, u1, . . .) and v = v(v0, v1, . . .) of the system (3.1)-(3.3) with uk, vk ∈ C((−T, T ) : Hs(R)) ∩Xsb and ||uk||Xs (R) < +∞, ||vk||Xs (R) < +∞. Moreover, the map (uk0 , v 0 ) → (u(t), v(t)) is Lipschitz continuous, i. e., ||u(t)− ũ(t)||AA0 (Xsb ) + ||u(t)− ũ(t)||C((−T, T ): Hs(R)) ≤ c(T ) ||u0 − ũ0||AA0 (Hs(R)) ||v(t) − ṽ(t)||AA0 (Xsb ) + ||v(t) − ṽ(t)||C((−T, T ): Hs(R)) ≤ c(T ) ||v0 − ṽ0||AA0 (Hs(R)). Proof. For given (u0, v0) ∈ AA0(Hs(R))×AA0(Hs(R)) and b > 1/2, let us define, HR1, R2 = (u, v) ∈ AA0(Xsb )×AA0(Xsb ) : ||u||AA0 (Xsb ) ≤ R1, ||v||AA0 (Xsb ) ≤ R2 where R1 = 2 c0 ||u0||AA0 (Hs(R)) and R2 = 2 c0 ||v0||AA0 (Hs(R)). Then HR1, R2 is a complete metric space with norm ||(u, v)||HR1, R2 = ||u||AA0 (Xsb ) + ||v||AA0 (Xsb ). Without loss of generality, we may assume that that R1 > 1 and R2 > 1. For (u, v) ∈ HR1, R2 , let us define the maps, Φku0(u, v) = ψ(t)V (t)u 0 − ψ(t) V (t− t′)ψT (t′)Bk(t′) dt′ (3.4) Ψkv0(u, v) = ψ(t)V (t) v 0 − ψ(t) V (t− t′)ψT (t′)Ck(t′) dt′. (3.5) We prove that Φ × Ψ maps HR1, R2 into HR1, R2 and it is a contraction. In fact, using lemma 2.1 and lemma 2.2 we have ||Φku0(u, v)||Xsb = ||ψ(t)V (t)u 0 ||Xsb + ∣∣∣∣ψ(t) V (t− t′)ψT (t′)Bk(t′) dt′ ≤ c0 ||uk0 ||Hs(R) + c1 T µ ||Bk||Xs ≤ c0 ||uk0 ||Hs(R) + c1 T µ k=k1+k2+k3 k1! k2! k3! 2k1 ||uk2 ||Xsb ||uk3 ||Xsb + c1 T k′1! k 1 ||vk′ ||vk′ + c1 T k=k′′ k′′1 ! k 2 ! k 1 ||uk′′ ||vk′′ Analycity for the coupled system of KdV equations 11 Applying a sum over k we have ||Φku0(u, v)||Xsb ||uk0 ||Hs(R) + c1 T µ k=k1+k2+k3 k1! k2! k3! 2k1 ||uk2 ||Xsb ||uk3 ||Xsb + c1 T k′1! k 1 ||vk′ ||vk′ + c1 T k=k′′ k′′1 ! k 2 ! k 1 ||uk′′ ||vk′′ ≤ c0 ||u0||AA0 (Hs(R)) + c1 T k=k1+k2+k3 ||uk2 ||Xsb ||uk3 ||Xsb + c1 T ||vk′ ||vk′ + c1 T k=k′′ k′′2 ! ||uk′′ k′′3 ! ||vk′′ ≤ c0 ||u0||AA0 (Hs(R)) + c1 T ||uk2 ||Xsb ||uk3 ||Xsb + c1 T ||vk′ ||vk′ + c1 T k′′1 ! k′′2 ! ||uk′′ k′′3 ! ||vk′′ = c0 ||u0||AA0 (Hs(R)) + c1 T e2A0 ||u||2AA0 (Xsb ) + c1 T e2A0 ||v||2AA0 (Xsb ) + c1 T µ c e2A0 ||u||AA0 (Xsb ) ||v||AA0 (Xsb ). Hence, choosing d = max{a/2, b/2, c} we have ||Φu0(u, v)||AA0 (Xsb ) ≤ c0 ||u0||AA0 (Hs(R)) + c1 T µ d e2A0 ||u||2AA0 (Xsb ) + ||v|| AA0 (X ) + ||u||AA0 (Xsb ) ||v||AA0 (Xsb ) ≤ c0 ||u0||AA0 (Hs(R)) + c1 d T µ e2A0 ||u||2AA0 (Xsb ) + ||v|| AA0 (X . (3.6) In a similar way, choosing d̃ = max{ã/2, b̃/2, c̃} we have ||Ψv0(u, v)||AA0 (Xsb ) ≤ c0 ||v0||AA0 (Hs(R)) + c2 d̃ T µ e2A0 ||u||2AA0 (Xsb ) + ||v|| AA0 (X .(3.7) 12 M Sepúlveda and O Vera If we choose T such that T µ ≤ 3 max{c1, c2} (R1 +R2)2 Then we obtain in (3.6) and (3.7) ||Φu0(u, v)||AA0 (Xsb ) ≤ R1 and ||Ψv0(u, v)||AA0 (Xsb ) ≤ R2. Therefore, (Φu0 , Ψv0) ∈ HR1, R2 .We show that Φu0×Ψv0 : (u, v) → (Φu0(u, v), Ψv0(u, v)) is a contraction. Let (u, v), (ũ, ṽ) ∈ HR1, R2 , then as above we get for d = max{a/2, b/2, c} ||Φu0(u, v)− Φu0(ũ, ṽ)||AA0 (Xsb ) c1 d T µ e2A0 (R1 +R2) ||u− ũ||AA0 (Xsb ) + ||v − ṽ||AA0 (Xsb ) . (3.8) In a similar way, choosing d̃ = max{ã/2, b̃/2, c̃} we have ||Ψv0(u, v)−Ψv0(ũ, ṽ)||AA0 (Xsb ) c2 d̃ T µ e2A0 (R1 +R2) ||u− ũ||AA0 (Xsb ) + ||v − ṽ||AA0 (Xsb ) . (3.9) Choosing T µ small enough, such that T µ ≤ 1 6 max{c1, c2} (R1 +R2)2 we obtain ||Φu0(u, v)− Φu0(ũ, ṽ)||AA0 (Xsb ) ≤ ||u− ũ||AA0 (Xsb ) + ||v − ṽ||AA0 (Xsb ) . (3.10) In a similar way ||Ψv0(u, v)−Ψv0(ũ, ṽ)||AA0 (Xsb ) ≤ ||u− ũ||AA0 (Xsb ) + ||v − ṽ||AA0 (Xsb ) . (3.11) Therefore the map Φu0 × Ψv0 is a contraction and we obtain a unique fixed point (u, v) which solves the initial value problem (3.1)-(3.3) for T < T µ. The rest of the proof follows a standard argument. Corollary 3.1. Let −3/4 < s, b ∈ (1/2, 7/12). Suppose that (x ∂x)ku0, (x ∂x)kv0 ∈ Hs(R)(k = 0, 1, 2, . . .) and that ||uk0 ||Hs(R) < +∞ and ||vk0 ||Hs(R) < +∞. Then there exist T = T (||uk0 ||Hs(R), ||vk0 ||Hs(R)) and a unique solution (u, v) of the coupled system equations KdV type (1.1)-(1.3) with u, v ∈ C((−T, T ) : Hs(R)) ∩Xsb and ||P ku||Xs (R) < +∞, ||P kv||Xs (R) < +∞. Analycity for the coupled system of KdV equations 13 Moreover, the map (u0, v0) → (u(t), v(t)) is Lipschitz continuous in the following sense: ||P ku(t)− P kũ(t)||Xs + ||P ku(t)− P kũ(t)||C((−T, T ): Hs(R)) ≤ c(T ) ||(x ∂x)k(u0 − ũ0)||Hs(R) ||v(t) − ṽ(t)||Xs + ||v(t)− ṽ(t)||C((−T, T ): Hs(R)) ≤ c(T ) ||(x ∂x)k(v0 − ṽ0)||Hs(R). 4 The main result In this section we prove the analyticity of the solution obtained in the previous section. We treat the solution uk ≡ P ku and vk ≡ P kv as if they satisfy the coupled system of equations (3.1)-(3.3) in the classical sense. This can be justified by a proper approximation procedure. The following results are going to be used in this section. Let (x0, t0) be arbitrarily taken in R×{(−T, 0)∪(0, T )}. By ψ(x, t) we denote a smooth cut-off function in C∞0 (B1(0)) and ψǫ = ψ((x − x0)/ǫ, (t− t0)/ǫ). Let ψ be a smooth cut-off function around the freezing point (x0, t0) with suppψ ⊂ C∞0 (Bǫ(x0, t0)). Proposition 4.1. For the cut-off function ψ defined above, there exists a positive constant c and A such that ||ψ P ku||L2x, t(R2) ≤ cA k (k!)2, k = 0, 1, 2, . . . (4.1) ||ψ P kv||L2x, t(R2) ≤ cA k (k!)2, k = 0, 1, 2, . . . (4.2) Proof. Using (2.22) with r = s− 1, we obtain || < Dx, t >3b ψP ku||L2t (R:Hs−1x (R)) ≤ c ||ψuk||Xs−1 + ||t ∂3x(ψuk)||Xs−1 + ||P 3(ψuk)||Xs−1 .(4.3) Each term in (4.3) is estimated separately. For the first term in the right hand side we use Lemma 2.3. Indeed, ||ψ uk||Xs−1 ≤ ||ψ uk||Xs ≤ c ||ψ|| |s|+2|b−1| |b−1| ||uk||Xs ≤ c(ψ)Ak1 k!. k = 0, 1, 2, . . .(4.4) The third term is estimated again using Corollary 2.6. ||P 3(ψ uk)||Xs−1 l (l − 3)! ||(P 3−lψ)P luk||Xsb−1 ≤ c(ψ) l (l − 3)! ||P luk||Xs l (l − 3)! ||P k+lu||Xs Ak+l1 (k + l)! ≤ cAk2 k!. k = 0, 1, 2, . . . (4.5) 14 M Sepúlveda and O Vera For the second term, we use (3.1) to reduce the third derivative in space to the dilation operator P. Since the generator of dilation is Puk = 3 t ∂tuk + x ∂xuk we obtain t ∂tuk = Puk − x ∂xuk. (4.6) Multiplying (3.1) by ψ t, we have ψ t ∂tuk + ψ t ∂ xuk = ψ tBk. (4.7) Replacing (4.6) in (4.7) we obtain ψ t ∂3xuk = − ψ Puk + ψ x∂xuk + ψ tBk. (4.8) hence ||ψ t ∂3xuk||Xs−1 ||ψ Puk||Xs−1 ||ψ x∂xuk||Xs−1 + ||ψ tBk||Xs−1 = F1 + F2 + F3. (4.9) Using the assumption in the Theorem, we have ||ψ Puk||Xs−1 ≤ c ||ψ|| ||P k+1u||Xs ≤ c ||P k+1u||Xs ≤ cAk+13 (k + 1)! ≤ cA 4 k!. (4.10) Similarly, we obtain ||ψ x∂xuk||Xs−1 ||∂x(ψ xuk)||Xs−1 ||∂x(ψ x)uk)||Xs−1 ||∂x(ψ x vk)||Xs + c ||∂x(ψ x)||X−s ||uk||Xs ||ψ x||Xs ||uk||Xsb + c ||∂x(ψ x)||X−s1−b ||uk||X ||ψ x||Xs + ||∂x(ψ x)||X−s Ak5k! ≤ cAk6k!. (4.11) Using Lemma 2.3 and 2.2, we have F3 = ||ψ tBk||Xs−1 ≤ c ||ψ|| ||B1k +B2k +B3k||Xsb−1 ||B1k ||Xsb−1 + ||B k ||Xsb−1 + ||B k||Xsb−1 Analycity for the coupled system of KdV equations 15 Then replacing B1k, B k and B k in (2.7), (2.8) and (2.9) we deduce F3 ≤ c k=k1+k2+k3 k1! k2! k3! 2k1 ||uk2 ||Xsb ||uk3 ||Xsb + c k′1! k 1 ||vk′ ||vk′ k=k′′ k′′1 ! k 2 ! k 1 ||uk′′ ||vk′′ k=k1+k2+k3 k1! k2! k3! 2k1 Ak27 · k2! A 7 · k3! + c k′1! k 8 · k 8 · k k=k′′ k′′1 ! k 2 ! k 9 · k 2 ! A 10 · k k=k1+k2+k3 2k1 A k2+k3 7 + c k=k′′ k′′1 ! 9 · A ≤ c k! Ak7 k−k1∑ 2k1 A− k17 + c k! A + c k! k=k′′ k′′1 ! ≤ c k! Ak7 k−k1∑ + c k! Ak8 + c k! k=k′′ k′′1 ! 9 · A ≤ c e2/A7 Ak7 · k! + c e2/A8 Ak8 · k! + c k! k=k′′ k′′1 ! e2/A7 + e2/A8 A11 · k! + c k! k=k′′ k′′1 ! 9 · A 10 . k = 0, 1, 2, . . . (4.12) Hence, from (4.10), (4.11) and (4.12) in (4.9) we obtain that there exists a positive constant c and A11 such that ||ψ t ∂3xuk||Xs−1 ≤ cA11 · k! + c k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . .(4.13) On the other hand, using ∂3x(ψ · f) = ψ · ∂3xf + 3 ∂2x(∂xψ · f)− 3 ∂x(∂2xψ · f) + ∂3xψ · f we have that ||t ∂3x(ψ · uk)||Xs−1 ≤ ||t ψ · ∂3xuk||Xs−1 + 3 ||∂2x(t ∂xψ · uk)||Xs−1 + 3 ||∂x(t ∂2xψ · uk)||Xs−1 + ||t ∂3xψ · uk||Xs−1 . (4.14) 16 M Sepúlveda and O Vera Using Lemma 2.2 and Lemma 2.3 we obtain ||∂2x(t ∂xψ · uk)||Xs−1 ≤ ||∂x(t ∂xψ · uk)||Xsb−1 ≤ c ||t ∂xψ||Xsb ||uk||Xsb ≤ cAk10 k! (4.15) ||∂x(t ∂2xψ · uk)||Xs−1 ≤ ||∂x(t ∂2xψ · uk)||Xsb−1 ≤ c ||t ∂ xψ||Xsb ||uk||Xsb ≤ cAk11 k! (4.16) ||t ∂3xψ · uk||Xs−1 ≤ c || < Dx, t >3/2 t ∂3xψ||X|s|+2|b−1| ||uk||Xsb−1 ≤ c ||uk||Xsb ≤ cAk12 k!. (4.17) Hence, replacing (4.13), (4.15),(4.16) and (4.17) in (4.14) we obtain that there exists a constant c and A14 such that ||t ∂3x(ψuk)||Xs−1 ≤ cAk14 · k! + c k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . .(4.18) Therefore, replacing (4.4), (4.5) and (4.18) in (4.3) we obtain that there exists a constant c and A15 such that || < Dx, t >3b ψ uk||L2t (R:Hs−1x (R)) ≤ cAk15 · k! + c k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.19) In a similar way, we obtain that there exists a constant c and A16 such that || < Dx, t >3b ψ vk||L2t (R:Hs−1x (R)) ≤ cAk16 · k! + c k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.20) Adding (4.19) and (4.20) we have || < Dx, t >3b ψ uk||L2t (R:Hs−1x (R)) + || < Dx, t > 3b ψ vk||L2t (R:Hs−1x (R)) ≤ cAk15 · k! + cAk16 · k! + c k! k=k′′ k′′1 ! 1 2 ·Ak ≤ c (Ak15 +Ak16) · k! + c k! k=k′′ k′′1 ! 1 2 ·Ak ≤ cAk17 · k! + c k! k=k′′ k′′1 ! 1 2 ·Ak 10 . (4.21) Analycity for the coupled system of KdV equations 17 We estimate the last term on the right hand side of (4.21) k=k′′ k′′1 ! 1 2 · Ak 9 · A (m− j)! 2(m−j) 2 ·Aj9 · A ≤ Ak10 (m− j)! ≤ Ak10 ≤ Ak10 +Ak10 ≤ Ak10 k! +Ak10 k! ≤ eA29/4 Ak10 k! + e4/A 10 Ak10 k! ≤ c Ak10 k!. (4.22) Replacing (4.22) in (4.21) we obtain || < Dx, t >3b ψ uk||L2t (R:Hs−1x (R)) + || < Dx, t > 3b ψ vk||L2t (R:Hs−1x (R)) ≤ cAk17 · k! + cAk19 · (k!)2 ≤ cAk17 · (k!)2 + cAk19 · (k!)2 ≤ cAk20 · (k!)2 (4.23) and the result follows. Remark 4.1. a) For simplicity, we only illustrate the conclusion for the case s ≥ −1/2−δ with b = 1/2 + δ/3 (for small δ > 0) and the case s = −3/4 + δ and b = 7/12 − δ/3. If s = −1/2− δ with b = 1/2+ δ/3, the initial data can involve Dirac’s delta measure δ0 and the latter is the critical case of the local well-posedness. b) The following inequality is simple to verify in both cases, ||ψ uk||L2x, t(R2) ≤ || < Dx > 3 b (ψ uk)||L2t (R: Hs−1x (R)) ≤ c || < Dx, t > 3 b (ψ uk)||L2t (R: Hs−1x (R)). Proposition 4.2. Under the same assumptions as in Proposition 4.1, there exist positive constants c and A such that ||ψ P ku||H7/2(R2) ≤ cA k (k!)2, k = 0, 1, 2, . . . (4.24) ||ψ P kv||H7/2(R2) ≤ cA k (k!)2, k = 0, 1, 2, . . . (4.25) Proof. We apply Lemma 2.4 to ψ uk ≡ ψ P ku with b = 1 and r = 0. || < Dx, t >3 ψ P ku||L2(R:L2x(R)) ||ψ uk||L2(R:L2x(R)) + ||t∂ x(ψ uk)||L2(R:L2x(R)) + ||P 3(ψ uk)||L2(R:L2x(R)) .(4.26) 18 M Sepúlveda and O Vera Therefore, if we wish to estimate the second term in the right hand side of (4.26) with the aid of the equation (2.12) ψ t ∂3xuk = − ψ Puk + ψ x∂xuk + t ψ Bk it is necessary to estimate ||ψ uk||L2t (R:H1x(R)) which is not yet obtained. Hence, we start from the lower regularity setting, i. e., applying (2.23) in Lemma 2.4 to ψ uk with µ = 1/2. Let ψ1 be a smaller size of smooth cut-off function with ψ1 ≤ ψ and ψ1 = 1 around (x0, t0). Applying (2.23) a ψ uk = ψ P ku with µ = 1/2 we have || < Dx, t >3 ψ1 P ku||H− 5/2(R2) ≤ c || < Dx, t > 3 ψ1 P ku||L2(R2) ||ψ1uk||H− 5/2(R2) + ||t∂ x(ψ1uk)||H− 5/2(R2) + ||P 3(ψ1uk)||H− 5/2(R2) . (4.27) The first term on the right hand side of (4.27) has already been estimated. For the third term we have ||P 3(ψ1uk)||H− 5/2(R2) ≤ ||P 3(ψ1uk)||L2x, t(R2) l!(3− l)! ||(P 3−lψ1)(P luk)||L2x, t(R2) l!(3− l)! ||P 3−lψ1||L∞x, t(R2)||P luk||L2x, t(R2) l!(3− l)! ||P k+lu||L2x, t(R2) Ak+l1 k! ≤ cA 2k! ≤ cAk2(k!)2. (4.28) For the second term on the right side hand we use the same idea of the remark above, using the dilation operator P. Indeed, ||t ∂3x(ψ1 uk)||H−5/2 ≤ ||ψ1 t ∂ xuk||H−5/2(R2) + 3 ||∂ x(t ∂xψ1 · uk)||H−5/2(R2) + 3 ||∂x(t ∂2xψ1 · uk)||H−5/2(R2) + ||t (∂ xψ1)uk||H−5/2(R2). (4.29) The last three term are bounded by the following: ||∂xψ1||L∞x, t(R2) + ||∂ xψ1||L∞x, t(R2) + ||∂ xψ1||L∞x, t(R2) ||ψ uk||L2x, t(R2) ≤ cAk3 k! ≤ cAk3 (k!)2. (4.30) On the other hand, using ||ψ1 t ∂3xuk||H−5/2(R2) ≤ ||ψ1 Puk||L2(R:L2x(R)) + ||xψ1 ∂xuk||H−5/2(R2) + ||t ψ1 Bk||H−5/2(R2) = F1 + F2 + F3. (4.31) F1 ≤ c ||ψ1||L∞x, t(R2) ||ψ P k+1u||L2x, t(R2) ≤ c ||ψ P k+1v||L2x, t(R2) ≤ cAk+14 (k + 1)! ≤ cA 5 k! ≤ cAk5 (k!)2, (4.32) Analycity for the coupled system of KdV equations 19 F2 ≤ ||xψ1 ∂xvk||L2(R:H−1x (R)) ≤ ||∂x(xψ1 vk)||L2(R:H−1x (R)) + ||∂x(xψ1)ψ vk||L2(R:H−1x (R)) ≤ ||xψ1 vk||L2x, t(R2) + ||∂x(xψ1)||L∞x, t(R2)||ψ vk||L2x, t(R2) ||xψ1||L∞x, t(R2) + ||∂x(xψ1))||L∞x, t(R2) ||ψ vk||L2x, t(R2) ≤ cAk6 k! ≤ cAk6 (k!)2. (4.33) Using Lemma 2.5(case σ = −5/2, s = 5, r = −5/2) F3 = ||t ψ1 Bk||H−5/2(R2) ≤ c1 ||ψ1||H5(R2)||ψ 2 Bx||H−5/2(R2) and replacing Bk by (2.10), we have F3 ≤ c1 k=k1+k2+k3 k1! k2! k3! 2k1 ||ψ uk2ψ uk3 ||H−3/2(R2) k′1! k 1 ||ψ vk′ ψ vk′ ||H−3/2(R2) + c1 |c| k=k′′ k′′1 ! k 2 ! k 1 ||ψ uk′′ ψ vk′′ ||H−3/2(R2) k=k1+k2+k3 k1! k2! k3! 2k1 ||ψ uk2 ||L2(R2)||ψ uk3 ||L2(R2) k′1! k 1 ||ψ vk′ ||L2(R2)||ψ vk′ ||L2(R2) + c1 |c| k=k′′ k′′1 ! k 2 ! k 1 ||ψ uk′′ ||L2(R2)||ψ vk′′ ||L2(R2) k=k1+k2+k3 k1! k2! k3! 2k1 Ak27 k2!A 7 k3! k′1! k + c1 |c| k=k′′ k′′1 ! k 2 ! k k=k1+k2+k3 Ak2+k37 + c1 + c1 |c| k! k=k′′ k′′1 ! 20 M Sepúlveda and O Vera and then F3 ≤ c1 k!Ak7 k−k1∑ A− k17 + c1 k!Ak8 + c1 |c| k! k=k′′ k′′1 ! e2/A7 Ak7 (k + 1)! + c1 e3/A8 Ak8 (k + 1)! + c1 |c| k! k=k′′ k′′1 ! 10 . (4.34) Replacing (4.30), (4.35) and (4.29) in (4.31) we obtain ||ψ1 t ∂3xuk||H−5/2(R2) ≤ c2Ak11 k! + c1 |c| k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.35) Replacing (4.30) and (4.35) in (4.29) ||t ∂3x(ψ1 uk)||H−5/2(R2) ≤ c3Ak12 k! + c1 |c| k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.36) Now replacing (4.28) and (4.36) in (4.27) we obtain || < Dx, t >3 ψ uk||H− 5/2(R2) ≤ c4Ak13 k! + c1 |c| k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.37) In particular ||ψ uk||H1/2(R2) ≤ c5Ak14 k! + c1 |c| k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.38) Using a similar argument as above for || < Dx, t >3 ψ P ku||H− 3/2(R2) with µ = 3/2 in (2.23) and replacing the support of the cut-off function ψǫ we obtain ||ψ uk||H3/2(R2) ≤ c5Ak14 k! + c1 |c| k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.39) Analycity for the coupled system of KdV equations 21 In a similar way we have ||ψ vk||H3/2(R2) ≤ c5Ak15 k! + c1 |c̃| k! k=k′′ k′′1 ! 10 , k = 0, 1, 2, . . . (4.40) Adding (4.39) with (4.40) and performing straightforward calculations as (4.22) we obtain ||ψ uk||H3/2(R2) + ||ψ vk||H3/2(R2) ≤ C A k (k!)2, k = 0, 1, 2, . . . (4.41) To obtain the estimate for ||ψ P ku||H7/2(R2) and ||ψ P kv||H7/2(R2) we repeat the above method with µ = 7/2. Proposition 4.3. Suppose that ||ψ uk||H7/2(R2) ≤ cA 1 (k!) 2, k = 0, 1, 2, . . . (4.42) ||ψ vk||H7/2(R2) ≤ cA 2 (k!) 2, k = 0, 1, 2, . . . (4.43) then we have t∈[t0−ǫ, t0+ǫ] ||(t1/3∂x)P ku||H1(x0−ǫ, x0+ǫ) ≤ c1A 3 [ (k + l)!] 2, k, l = 0, 1, 2, . . .(4.44) t∈[t0−ǫ, t0+ǫ] ||(t1/3∂x)P kv||H1(x0−ǫ, x0+ǫ) ≤ c1A 4 [ (k + l)! ] 2, k, l = 0, 1, 2, . . .(4.45) where ǫ > 0 is so small that ψ ≡ 1 near I = (x0 − ǫ, x0 + ǫ)× (t0 − ǫ, t0 + ǫ). Proof. Let It0 = (t0 − ǫ, t0 + ǫ) and Ix0 = (x0 − ǫ, x0 + ǫ), then we have I = Ix0 × It0 . For any fixed t ∈ Ix0 , let L = t1/3∂x. We show that for some positive constants c and A0 the following inequality holds ||LlP ku||H1x(Ix0 ) ≤ cA 0 [ (k + l)! ] 2, ∀ k, ∀ l = 0, 1, 2, . . . (4.46) Now, let use induction over l. By the trace theorem, we have ||LlP ku||H1x(Ix0 ) ≤ ||t l/3 ∂lxP ku(t)||H1x(Ix0) ≤ (t0 + ǫ) l/3||∂lxP ku||H3/2(Ix0×It0) ≤ (t0 + ǫ)l/3||P ku||H7/2(Ix0×It0) ≤ (t0 + ǫ) l/3||ψ P ku||H7/2(R2) ≤ (t0 + ǫ)l/3 c1Ak1 k! ≤ (t0 + ǫ)l/3 c1Ak+l0 (k + l) ≤ (t0 + ǫ)l/3 c1Ak+l0 [ (k + l)! ] 2. (4.47) where we take c = (t0 + ǫ) l/3c1 and A0 = max{1, A1}. Hence, in the case l = 0, 1, 2, it is easy to show that (4.46) follows directly from the assumption. 22 M Sepúlveda and O Vera Now, we assume that (4.46) is true to l ≥ 2. Applying P k to the equation (2.4), we have ku) + ∂3x(P ku) = LP ku = (P + 3)kLu = (P + 3)k(∂tu+ ∂ = −(P + 3)k 2) + c ∂x(u v) (P + 3)k∂x(u (P + 3)k∂x(v 2)− c (P + 3)k∂x(u v) ∂x(P + 2) k(u2)− ∂x(P + 2) k(v2)− c ∂x(P + 2)k(u v) such that t ∂t(P ku) + t ∂3x(P ku) = − a t ∂x(P + 2) k(u2)− b t ∂x(P + 2) k(v2)− c t ∂x(P + 2)k(u v).(4.48) Moreover, P = 3 t ∂t + x ∂x. Then t ∂t(P ku) = P k+1u− x ∂x(P ku). (4.49) Replacing (4.49) in (4.48) we obtain L3P ku = t ∂3x(P ku) = − P k+1u+ x ∂x(P t ∂x(P + 2) k(u2)− t ∂x(P + 2) k(v2)− c t ∂x(P + 2)k(u v). (4.50) Hence, applying Ll−2 we have ||Ll+1P ku||H1x(Ix0 ) = ||L l−2L3P ku||H1x(Ix0) ||Ll−2 P k+1u||H1x(Ix0 ) + ||Ll−2 x ∂x(P ku)||H1x(Ix0 ) ||tLl−2 ∂x(P + 2)k(u2)||H1x(Ix0) + ||tLl−2 ∂x(P + 2)k(v2)||H1x(Ix0) + |c| ||tLl−2 ∂x(P + 2)k(u v)||H1x(Ix0 ) = F1 + F2 + F3 + F4 + F5. (4.51) Using the induction assumption, we obtain k+l+1 14 (k + l + 1)!. (4.52) We estimate the term Ll−2(x ∂x) for l ≥ 3. Let r = l − 2, then we estimate Lr(x ∂x) for r ≥ 1. ∂rx(x ∂x) = ∂r−kx (x ) · ∂kx( ∂x ). (4.53) ∂r−kx (x ) = 1 if k = r − 1 0 if k ≤ r − 2 Analycity for the coupled system of KdV equations 23 then in (4.53) we obtain ∂rx(x ∂x) = r ∂ x ( ∂x ) + x ∂ x( ∂x ) = r ∂ x + x ∂x( ∂ = (l − 2) ∂(l−2)x + x ∂x( ∂(l−2)x ), that is, Ll−2(x ∂x) = x ∂xLl−2 + (l − 2)Ll−2, for l ≥ 3. For F2 we have F2 ≤ ||x ∂xLl−2P ku||H1x(Ix0 ) + (l − 2) ||L l−2P ku||H1x(Ix0 ) ≤ ||x t−1/3Ll−1P ku||H1x(Ix0 ) + (l − 2) ||L l−2P ku||H1x(Ix0) ≤ c (t0 − ǫ) (|x0|+ ǫ+ 1) ||Ll−1P ku||H1x(Ix0 ) + (l − 2) ||L l−2P ku||H1x(Ix0 ) ≤ (t0 − ǫ)−1/3(|x0 + ǫ+ 1) c1 Ak+l−114 (k + l − 1)! + c1A k+l−1 14 (l − 2) (k + l − 1)! k+l+1 14 (k + l + 1)! (4.54) where we take A14 larger than (t0 − ǫ)−1/3(|x0|+ ǫ+ 1) and 3. Using that (L = t1/3 ∂3x) tLl−2∂x = t t(l−2)/3∂(l−2)x ∂x = t t−1/3 t(l−1)/3∂(l−1)x = t2/3 Ll−1, we have ||t2/3 Ll−1 (P + 2)k(u2)||H1x(Ix0 ) (t0 + ǫ) l−1=l1+l2 k=k1+k2+k3 (l − 1)! l1!l2! k1!k2!k3! × c2 ||Ll1P k1u||H1x(Ix0) ||L l2P k2u||H1x(Ix0). Using the induction assumption (t0 + ǫ) l−1=l1+l2 k=k1+k2+k3 1 k! (l − 1)! (l1 + k1)! l1! k1! (l2 + k2)! l2! k2! Ak+l−114 (t0 + ǫ) 2/3 c2 c 1 (l + k − 1)!Ak+l−114 l−1=l1+l2 k=k1+k2+k3 × (l1 + k1)! l1! k1! (l2 + k2)! l2! k2! k! (l − 1)! (l + k − 1)! Using that l−1=l1+l2 k=k1+k2+k3 (l1 + k1)! l1! k1! (l2 + k2)! l2! k2! k! (l − 1)! (l + k − 1)! ≤ e2(l + k)! we obtain F3 ≤ (t0 + ǫ)2/3 c2 c31 e2 (l + k)!Ak+l−114 ≤ k+l+1 14 (k + l + 1)! (4.55) where we take A14 larger than (t0 − ǫ)−1/3 c2 c21 e2, and 3. In a similar way k+l+1 15 (k + l + 1)! (4.56) 24 M Sepúlveda and O Vera where we take A15 larger than (t0 − ǫ)−1/3 c4 c23 e2, and 3. Finally, in a similar way k+l+1 16 (k + l + 1)! (4.57) where we take A16 larger than (t0 − ǫ)−1/3 c6 c25 e2, and 3. Therefore, from (4.52), (4.54), (4.55), (4.56) and (4.57) we obtain ||Ll+1P ku||H1x(Ix0) ≤ c7A k+l+1 17 (k + l + 1)!. (4.58) In a similar way, we obtain ||Ll+1P kv||H1x(Ix0) ≤ c7A k+l+1 17 (k + l + 1)!, (4.59) and the result follows. Proposition 4.4. Suppose that there exists a positive constants c1, c2 and A14, A15 such t∈[t0−ǫ, t0+ǫ] ||∂lxP ku||H1x(x0−ǫ, x0+ǫ) ≤ c1A 14 [ (k + l)! ] 2, k, l = 0, 1, 2, . . . (4.60) t∈[t0−ǫ, t0+ǫ] ||∂lxP kv||H1x(x0−ǫ, x0+ǫ) ≤ c2A 15 [ (k + l)! ] 2, k, l = 0, 1, 2, . . . (4.61) Then we have respectively t∈[t0−ǫ, t0+ǫ] ||∂mt ∂lxu||H1x(x0−ǫ, x0+ǫ) ≤ c3A 16 [ (m+ l)! ] 2, m, l = 0, 1, 2, . . . (4.62) t∈[t0−ǫ, t0+ǫ] ||∂mt ∂lxv||H1x(x0−ǫ, x0+ǫ) ≤ c4A 17 [ (m+ l)! ] 2, m, l = 0, 1, 2, . . . (4.63) where c3, c4 and A16, A17 only depend on c1, c2 and A14, A15, respectively and ǫ, (x0, t0). Proof. Using the idea of Proposition 4.3, we fix t ∈ Ix0 . First we show that for some positive constants c3, A16 and B16 ||(x ∂x)m ∂lxP kv||H1x(Ix0) ≤ c3A k+m+l 16(k +m+ l)!, k, m, l = 0, 1, 2, . . . (4.64) We use induction. Suppose that (4.64) is true for m. ||(x ∂x)m+1 ∂lxP kv||H1x(Ix0) = ||(x ∂x) (x ∂x)m ∂lxP kv||H1x(Ix0) ≤ (|x0|+ ǫ+ 1) ||(x ∂x + I)m ∂l+1x P kv||H1x(Ix0 ) ≤ c(|x0|, ǫ) ||(x ∂x)j ∂l+1x P kv||H1x(Ix0 ) k+l+j+1 16(k + l + j + 1)! ≤ c3Ak+l+m+116 B 16(k + l +m+ 1)! (A16 B16) −(m−j) (m− j)! (k + l + j + 1)! (k + l +m+ 1)! ≤ e−A16 B16c3Ak+l+m+116 B 16(k + l +m+ 1)! (4.65) Analycity for the coupled system of KdV equations 25 where we take B16 so large that B16 ≥ max{|x0|+ǫ+1, 1}.We show that for some positive constants c4, A17 we have ||(t ∂t)m ∂lxu||H1x(Ix0 ) ≤ c4A 17 (l +m)!, l, m = 0, 1, 2, . . . Using that t ∂t = (P − x ∂x), we obtain ||(t ∂t)m ∂lxu||H1x(Ix0 ) = 3 −m ||(P − x ∂x)m ∂lxu||H1x(Ix0 ) ≤ 3−m m=j1+j2 j1! j2! ||(x ∂x)j1 P j2∂lxu||H1x(Ix0 ) ≤ 3−m m=j1+j2 j1! j2! ||(x ∂x)j1 ∂lx(P − l)j2u||H1x(Ix0) ≤ 3−m m=j1+j2+j3 j1! j2! j3! lj3 ||(x ∂x)j1 ∂lxP j2u||H1x(Ix0). where we replace j2 into j2 + j3. Now, using the induction hypothesis we have (with B17 ≥ A16B16) ||(t ∂t)m ∂lxu||H1x(Ix0 ) ≤ 3−m m=j1+j2+j3 j1! j2! j3! lj3 c3B j1+j2+l 17 (j1 + j2 + l)! ≤ 3−m c3Bm+l17 (m+ l)! m=j1+j2+j3 j1! j2! j3! (j1 + j2 + l)! (m+ l)! , (4.66) Observing that lj3 (j1 + j2 + l)! (m+ l)! ≤ 1, we obtain in (4.66) ||(t ∂t)m ∂lxu||H1x(Ix0 ) ≤ 3 −m c3 (2 +B mBl+m17 (l +m)! ≤ c4Al+m17 (l +m)! where we take A17 = max{B17, 3−1B17 (2 +B−117 )}. We show that for some positive con- stants c4, A18 and B18 we have ||(t ∂t)j ∂mt ∂lxu||H1x(Ix0 ) ≤ c4A j+m+l 18 B18(j +m+ l)!, j, l, m = 0, 1, 2, . . . (4.67) Induction in m. ||(t ∂t)j ∂m+1t ∂lxu||H1x(Ix0 ) ≤ ||∂t(t ∂t − I) m ∂mt ∂ xu||H1x(Ix0) = t−1 ||t ∂t(t ∂t − I)j ∂mt ∂lxu||H1x(Ix0) ≤ (t0 − ǫ)−1 ||(t ∂t)j1+1 ∂mt ∂lxu||H1x(Ix0 ). 26 M Sepúlveda and O Vera Using the induction hypothesis ||(t ∂t)j ∂m+1t ∂lxu||H1x(Ix0 ) ≤ (t0 − ǫ)−1 j1+l+m+1 18 (j1 + l +m+ 1)! = c4 (t0 − ǫ)−1Aj1+l+m+118 B 18 (j1 + l +m+ 1)! −(j−j1 (j − j1)! (j1 +m+ l + 1)! (j − j1)! (j +m+ l + 1)! = c4 (t0 − ǫ)−1 e−A18 Aj+l+m+118 B 18 (j + l +m+ 1)! ≤ c4 Aj+l+m+118 B 18 (j + l +m+ 1)! where we take B18 larger than (t0− ǫ)−1 e−A18 . Finally, we choose j = 0 in (4.67) and take c2 = c4 and A15 = A18B18. The result of analyticity follows. Acknowledgments This work has been supported by Fondap in Applied Mathematics (Project # 15000001), CNPq/CONICYT Project, # 490987/2005-2 (Brazil) and # 2005-075 (Chile). 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Introduction Reduction of the Problem and Preliminary Results Existence and Well-Posedness The main result

Smoothing properties for the higher order nonlinear Schrödinger equation with constant coefficients Mauricio Sepúlveda ∗ Octavio Vera Villagrán.† Abstract We study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear Schrödinger equation with constant coefficients which appears as a model for propagation of pulse in optical fiber. Keywords and phrases: Evolution equations, weighted Sobolev space, gain in regularity. Mathematics Subject Classification: 35Q53, 47J35 1 Introduction We consider the initial value problem i ut + ω uxx + i β uxxx + |u| 2 u = 0 x, t ∈ R u(x, 0) = u0(x) where ω, β ∈ R, β 6= 0 and u = u(x, t) is a complex valued function. The above equation is a particular case of the equation i ut + ω uxx + i β uxxx + γ |u| 2 u+ i δ |u|2 ux + i ǫ u 2 ux = 0 x, t ∈ R u(x, 0) = u0(x) where ω, β, γ, δ are real numbers with β 6= 0. This equation was first proposed by A. Hasegawa and Y. Kodama [13] as a model for the propagation of a signal in an optic fiber (see also [20]). The equation (Q) can be reduced to other well known equations. For instance, setting ω = 1, β = δ = ǫ = 0 in (Q) we have the semilinear Schrödinger equation, i. e., i ut + uxx + γ |u| 2 u = 0. (Q1) If we let β = γ = 0 and ω = 1 in (Q), we obtain the derivative nonlinear Schrödinger equation i ut + uxx + i δ |u| 2 ux + i ǫ u 2 ux = 0. (Q2) Letting α = γ = ǫ = 0 in (Q), the equation that arises is the complex modified Korteweg-de Vries equation, i ut + i β uxxx + i δ |u| 2 ux = 0. (Q3) The initial value problem for the equations (Q1), (Q2) and (Q3) has been extensively studied in the last few years. See, for instance, [1, 2, 3, 5, 6, 8, 9, 17, 18, 26, 27] and references therein. In 1992, C. Laurey [22] considered the equation (Q) and proved local well-posedness of the initial value problem associated for data in Hs(R), s > 3/4, and global well-posedness in Hs(R), s ≥ 1. In 1997, G. Staffilani [28] for (Q) ∗Departamento de Ingenieŕıa Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. mauricio@ing- mat.udec.cl †Departamento de Matemática, Universidad del B́ıo-B́ıo, Collao 1202, Casilla 5-C, Concepción, Chile. overa@ubiobio.cl http://arxiv.org/abs/0704.1862v1 established local well-posedness for data in Hs(R), s ≥ 1/4 improving Laurey’s result. A similar result was given in [5, 6] with w(t), β(t) real functions. Our aim in this paper, is to study gain in regularity for the equation (P ). Specifically, we prove conditions on (P ) for which initial data u0 possessing sufficient decay at infinity and minimal amount of regularity will lead to a unique solution u(t) ∈ C∞(R) for 0 < t < T, where T is the existence time of the solution. We are not considering the equation (Q) because of the technique used here, we shall see that the last two terms in (Q) are not outstanding in the main inequality, indeed the two last terms are observed in the last two terms in the main inequality. In 1986, N. Hayashi et al. [13] showed that for the nonlinear Schrödinger equation (NLS): i ut + uxx = λ |u| p−1 u, (x, t) ∈ R × R with initial condition u(x, 0) = u0(x), x ∈ R and a certain assump- tion on λ and p, all solutions of finite energy are smooth for t 6= 0 provided the initial functions in H1(R)(or on L2(R)) decay sufficiently fast as |x| → ∞. The main tool is the operator J defined by Ju = ei x 2/4 t (2 i t) ∂x(e − i x2/4 t u) = (x + 2 i t ∂x)u which has the remarkable property that it commutes with the operator L defined by L = (i ∂t + ∂ x), namely LJ − JL = [L, J ] = 0. For the Korteweg-de Vries type equation (KdV), J. C. Saut and M. Temam [26] remarked that a so- lution u cannot gain or lose regularity. They showed that if u(x, 0) = u0(x) ∈ H s(R) for s ≥ 2, then u( · , t) ∈ Hs(R) for all t > 0. For the KdV equation on the line, Kato [17] motivated by work of Cohen [11] showed that if u(x, 0) = u0(x) ∈ L b ≡ H 2(R) ∩ L2(ebx dx)(b > 0) then the solution u(x, t) of the KdV equation becomes C∞ for all t > 0. A main ingredient in the proof was the fact that formally the semi-group S(t) = e−∂ x in L2b(R) is equivalent to Sb(t) = e − t (∂x−b) in L2(R) when t > 0. One would be inclined to believe that this was a special property of the KdV equation. However, his is not the case. The effect is due to the dispersive nature of the linear part of the equation. Kruzkov and Faminskii [21] proved that u(x, 0) = u0(x) ∈ L 2(R) such that xα u0(x) ∈ L 2((0, +∞)), the weak solution of the KdV equation, has l-continuous space derivatives for all t > 0 if l < 2α. The proof of this result is based on the asymptotic behavior of the Airy function and its derivatives, and on the smoothing effect of the KdV equation which was found in [17, 21]. While the proof of Kato appears to depend on special a priori estimates, some of this mystery has been solved by the result of local gain of finite regularity for various others linear and nonlinear dispersive equations due to Ginibre and Velo [12] and others. However, all of them require growth conditions on the nonlinear term. In 1992, W. Craig, T. Kappeler and W. Strauss [8, 9] proved for the fully nonlinear KdV equation ut+ f(uxxx, uxx, ux, u, x, t) = 0, x ∈ R, t > 0 and certain additional assumption over f that C solutions u(x, t) are obtained for all t > 0 if the initial data u0(x) decays faster than polynomially on R+ = {x ∈ R : x > 0} and has certain initial Sobolev regularity. Following this idea, H. Cai [4] stud- ied the nonlinear equation of KdV-type of the form ut + uxxx + a(x, , t) f(uxx, ux, u, x, t) = 0, where a(x, t) is positive and bounded, obtaining the same conclusion. Subsequent works were given by O. Vera [30, 31, 32, 33] for a nonlinear dispersive evolution equation, a KdV-Burgers type equation and for KdV- Kawahara type equation, respectively. In more than one spatial dimension, J. Levandosky [23], proved infinite gain in regularity results for nonlinear third-order equations. While [8] included local smoothing results for some mth-order dispersive equation in n spatial dimension, their results and the techniques are different from those presented by Levandosky. First, they consider equations with only a mild solution and Levandosky considers equations with very general nonlinearities including a fully nonlinear equation of the form ut + f(D 3u, D2u, Du, u, x, t) = 0, u(x, y, 0) = u0(x, y). Secondly, they indicate local gain in finite regularity and Levandosky proved complementary results showing the relationship between the decay at infinity of the initial data and the amount of gain in regularity. More specifically, it is proved a condition under which an equation of the form ut + a uxxx + b uxxy + c uxyy + d uyyy + f(D 2u, Du, u, x, t) = 0, u(x, y, 0) = u0(x, y), where a, b, c, d are assumed constant. Indeed, Levandosky proved sufficient conditions on this equation for which a solution u will experience an infinite gain in regularity. Specifically, prove conditions for which initial data u0(x, y) possessing sufficient decay at infinity and a minimal amount of regularity will lead to a unique solution u(t) ∈ C∞(R2) for T ∗ where T ∗ is the existence time of solutions. According to the characteristics of equations (P ) and considering the particular cases (Q1) and (Q2) we could hope that the (P ) equation have gain in regularity following the steps of N. Hayashi et al. [13] or W. Craig et al. [8]. In our problem, the initial idea is to apply the technique given by N. Hayashi et al. [13, 14] to obtain gain in regularity. Firstly, using straightforward calculus we can see that the equation (P ) has conservation of the energy, i. e., ||u||L2(R) = ||u0||L2(R). On the other hand, we look for estimates for ux that will help to obtain a priori estimates, basically to obtain estimates in L∞(R). Indeed, differentiating in the x-variable the equation (P ) we have i ux t + i β uxxxx + ω uxxx + (|u| 2)x u+ |u| 2 ux = 0, (1.1) and multiplying (1.1) by ux i ux ux t + i β ux uxxxx + ω ux uxxx + (|u| 2)x u ux + |u| 2 |ux| 2 = 0 − i ux ux t − i β ux uxxxx + ω ux uxxx + (|u| 2)x uux + |u| 2 |ux| 2 = 0. (applying conjugate) Subtracting and integrating over x ∈ R, we have 2dx+ i β ux uxxxxdx+ i β ux uxxxxdx + 2 i ω Im ux uxxxdx+ 2 i Im (|u|2)x u uxdx = 0. Performing integration by parts and straightforward calculations we obtain 2dx+ 2 Im (|u|2)x u uxdx = 0 (E1) where ||ux|| L2(R) + 2 Im u2 u2xdx = 0 (E2) or integrating by parts the second term in (E1) we obtain ||ux|| L2(R) − 2 Im |u|2 u uxxdx = 0. (E3) Thus it is not possible to estimate in H1(R), because it appears a second term with two derivatives. The reason of having an estimate in the derivative is related to Sobolev embedding. In one spatial dimension we have the embedding H1(R) →֒ L∞(R). It seems that the term i β uxxx is crucial. It makes the two ”top” terms look like KdV equation; that is, ut + uxxx + . . . . Of course, the solution is complex, so that the equation is like two coupled real KdV equations. This was our motivation to obtain gain in regularity using the idea of W. Craig et al. [8]. We prove conditions on (P ) for which initial data u0(x) possessing sufficient decay at infinity and a minimal amount of regularity will lead to a unique solution u(t) ∈ C∞(R) for t > 0. We use a technique of nonlinear multipliers, generalizing Kato’s original method, together with ideas of Craig and Goodman [7] All the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property. To mention only a few, the KdV, Benjamin-Ono, intermediate long wave, various Boussinesq, and Schrödinger equation are included. This paper is organized as follows: Section 2 outlines briefly the notation and terminology to be used subsequently. In section 3 we prove the main inequality. In section 4 we prove an important a priori estimate. In section 5 we prove a basic-local-in-time existence and uniqueness theorem. In section 6 we prove a basic global existence theorem. In section 7 we develop a series of estimates for solutions of equations (P ) in weighted Sobolev norms. These provide a starting point for the a priori gain of regularity. In section 8 we prove the following theorem: Theorem 1.1(Main Theorem). Let |ω| < 3 β, T > 0 and u(x, t) be a solution of (P ) in the region R× [0, T ] such that u ∈ L∞([0, T ] : H3(W0 L 0)) (1.2) for some L ≥ 2. Then u ∈ L∞([0, T ] : H3+l(Wσ, L−l, l)) ∩ L 2([0, T ] : H4+l(Wσ, L−l−1, l)) (1.3) for all 0 ≤ l ≤ L− 1 and all σ > 0. Remark. We consider the Gauge transformation u(x, t) = ei d2 x+i d3 t v (x− d1 t, t) ≡ e θ v (η, ξ) (1.4) where θ = i d2 x+ i d3 t, η = x− d1 t and ξ = t. Then ut = i d3 e θ v − d1 e θ vη + e θ vξ : ux = i d2 e θ v + eθ vη uxx = − d θ v + 2 i d2 e θ vη + e θ vη η : uxxx = − i d θ v − 3 d22 e θ vη + 3 i d2 e θ vηη + e θ vηηη. Replacing in (Q) we have − d3 e θ v − i d1 e θ vη + i e θ vξ − ω d θ v + 2 i ω d2 e θ vη + ω e θ vηη β d33 e θ v − 3 i β d22 e θ vη − 3 β d2 e θ vηη + i β e θ vηηη + γ |v| 2 eθ v − δ d2 |v| 2 eθ v + i δ |v|2 eθ vη + ǫ d2 e θ v2v + i ǫ eθ v2 vη = 0 where i vξ + (ω − 3 β d2) vηη + i β vηηη + (2 i ω d2 − 3 i β d 2 − i d1 + i δ |v| 2 + i ǫ v2) vη (β d32 − ω d 2 − d3 + γ |v| 2 − δ d2 |v| 2) v + ǫ d2 v 2v = 0 : d2 = : d3 = − 2ω3 27 β2 . (1.5) This way in (Q) we obtain i vξ + i β vηηη + i (δ |v| 2 + ǫ v2) vη + |v|2v + v2v = 0, but v2 v = v v v = |v|2v, then using the Gauge transformation we have the equivalent problem to (Q) i vξ + i β vηηη + i δ |v| 2 vη + i ǫ v 2 vη + γ + ǫ δ − ω δ |v|2v = 0 η, ξ ∈ R v(η, 0) = e− i ηu0(η). Here, rescaling the equation, we take β = 1. i vt + i vxxx + i δ |v| 2 vx + i ǫ v 2 vx + γ + ǫ δ − ω δ |v|2v = 0 x, t ∈ R v(x, 0) = e− i xu0(x). The above Gauge transformation is a bicontinuous map from Lp([0, T ] : Hs(Wσ i k)) to itself, as far as 0 < T < +∞ and p, s, σ, i, k used in this paper. With this, the assumption |ω| < 3 β imposed in Theorem 1.1 can be removed. 2 Preliminaries We consider the initial value problem i ut + ω uxx + i β uxxx + |u| 2 u = 0, x, t ∈ R u(x, 0) = u0(x) where ω, β ∈ R, β 6= 0 and u = u(x, t) is a complex valued function. Notation. We write ∂ = ∂/∂x, ∂t = ∂/∂t and we abbreviate uj = ∂ Definition 2.1. A function ξ = ξ(x, t) belongs to the weight class Wσ i k if it is a positive C ∞ func- tion on R× [0, T ], ∂ξ > 0 and there are constant cj , 0 ≤ j ≤ 5 such that 0 < c1 ≤ t − k e−σ x ξ(x, t) ≤ c2 ∀ x < −1, 0 < t < T. (2.1) 0 < c3 ≤ t − k x− i ξ(x, t) ≤ c4 ∀ x > 1, 0 < t < T. (2.2)( t | ∂tξ | + | ∂ /ξ ≤ c5 ∀ (x, t) ∈ R× [0, T ], ∀ j ∈ N. (2.3) Remark. We shall always take σ ≥ 0, i ≥ 1 and k ≥ 0. Example. Let ξ(x) = 1 + e−1/x for x > 0 1 for x ≤ 0 then ξ ∈ W0 i 0. Definition 2.2. Let N be a positive integer. By HN (Wσ i k) we denote the Sobolev space on R with a weight; that is, with the norm ||v||2HN (Wσ i k) = |∂jv(x)|2 ξ(x, t) dx < +∞ for any ξ ∈ Wσ i k and 0 < t < T., Even though the norm depends on ξ, all such choices leads to equivalent norms. Remark. Hs(Wσ i k) depends on t (because ξ = ξ(x, t)). Lemma 2.1. (See [4]) For ξ ∈ Wσ i 0 and σ ≥ 0, i ≥ 0, there exists a constant c > 0 such that, for u ∈ H1(Wσ i 0), ||ξ u2|| ≤ c |u|2 + |∂u|2 Lemma 2.2(The Gagliardo-Nirenberg inequality). Let q, r be any real numbers satisfying 1 ≤ q, r ≤ ∞ and let j and m be nonnegative integers such that j ≤ m. Then ||∂ju||Lp(R) ≤ c ||∂ mu||aLr(R) ||u|| Lq(R) where 1 = j + a (1−a) for all a in the interval j ≤ a ≤ 1, and M is a positive constant depending only on m, j, q, r and a. Definition 2.3. By L2([0, T ] : HN (Wσ i k)) we denote the space of functions v(x, t) with the norm (N integer positive) ||v||2L2([0, T ]:HN (Wσ i k)) = ||v(x, t)||2HN (Wσ i k)dt < +∞ Remark. The usual Sobolev space is HN (R) = HN (W0 0 0) without a weight. Remark. We shall derive the a priori estimates assuming that the solution is C∞, bounded as x → −∞, and rapidly decreasing as x → +∞, together with all of its derivatives. Considering the above notation, the higher order nonlinear Schrödinger equation can be written as i ut + i β u3 + ω u2 + |u| 2 u = 0, x, t ∈ R (2.4) where ω, β ∈ R, β 6= 0 and u = u(x, t) is a complex valued function. Throughout this paper c is a generic constant, not necessarily the same at each occasion(it will change from line to line), which depends in an increasing way on the indicated quantities. In this part, we only consider the case t > 0. The case t < 0 can be treated analogously. 3 Main Inequality Lemma 3.1. Let |ω| < 3 β. Let u be a solution of (2.4) with enough Sobolev regularity (for instance, u ∈ HN(R), N ≥ α+ 3), then ξ |uα| η |uα+1| θ |uα| Rαdx ≤ 0 (3.1) where η = (3 β − |ω|) ∂ξ for |ω| < 3 β θ = − [ ∂tξ + β ∂ 3ξ + |ω| ∂ξ + c0 ξ ] where c0 = ||u|| L∞(R) and Rα = Rα(|uα|, |uα−1|, . . .). Proof. Differentiating (2.4) α-times (for α ≥ 0) over x ∈ R leads to i uα t + i β uα+3 + ω uα+2 + (|u| 2)α u+ (|u|2)α−m um + |u| 2 uα = 0. (3.2) Let ξ = ξ(x, t), then multiplying (3.2) by ξ uα we have i ξ uα uα t + i β ξ uα uα+3 + ω ξ uα uα+2 + (|u| 2)α ξ u uα (|u|2)α−m ξ um uα + ξ |u| 2 |uα| 2 = 0 − i ξ uα uα t − i β ξ uα uα+3 + ω ξ uα uα+2 + (|u| 2)α ξ u uα (|u|2)α−m ξ um uα + ξ |u| 2 |uα| 2 = 0. (applying conjugate) Subtracting and integrating over x ∈ R we have ξ |uα| 2dx+ i β ξ uα uα+3dx+ i β ξ uα uα+3dx− i ξt |uα| ξ uα uα+2dx − ω ξ uα uα+2dx+ 2 i Im ξ (|u|2)α u uαdx + 2 i ξ (|u|2)α−m um uαdx = 0. (3.3) We estimate the second term integrating by parts ξ uα uα+3dx = ∂2ξ uα uα+1dx+ 2 ∂ξ |uα+1| ξ uα+2 uα+1dx. The other terms are calculated in a similar way. Hence, replacing in (3.3) and performing straightforward calculations we obtain ξ |uα| 2dx+ i β ∂2ξ uα uα+1dx+ 2 i β ∂ξ |uα+1| + i β ξ uα+2 uα+1dx+ i β ∂2ξ uα uα+1dx+ i β ∂ξ |uα+1| − i β ξ uα+1 uα+2dx− ω ∂ξ uα uα+1dx− ω ξ |uα+1| ∂ξ uα uα+1dx+ ω ξ |uα+1| 2dx− i ∂tξ |uα| + 2 i Im ξ (|u|2)α u uαdx+ 2 i ξ (|u|2)α−m um uαdx = 0 ξ |uα| 2dx− β ∂3ξ |uα| 2dx+ 3 β ∂ξ |uα+1| 2dx − 2ω Im ∂ξ uα uα+1dx ∂tξ |uα| 2dx+ 2 Im ξ (|u|2)α u uαdx+ 2 ξ (|u|2)α−m um uαdx = 0 hence ξ |uα| 2dx− β ∂3ξ |uα| 2dx+ 3 β ∂ξ |uα+1| 2dx+ 2 Im (|u|2)α ξ u uαdx ∂tξ |uα| 2dx+ 2 ξ (|u|2)α−m um uαdx = 2ω Im ∂ξ uα uα+1dx ≤ |ω| ∂ξ |uα| 2dx+ |ω| ∂ξ |uα+1| therefore ξ |uα| [ 3 β − |ω| ] ∂ξ |uα+1| [ ∂tξ + β ∂ 3ξ + |ω| ∂ξ ] |uα| + 2 Im (|u|2)α ξ u uαdx+ 2 ξ (|u|2)α−m um uαdx ≤ 0. (3.4) (|u|2)α = (u u )α = uα−k uk = uuα + uα−k uk + u uα (|u|2)α u uα = |u| 2|uα| uα−k uk u uα + u 2 u2α thus, ( |u|2)α ξ u uαdx = 2 ξ uα−k uk u uαdx+ 2 Im ξ u2 u2αdx ξ |uα−k| |uk| |u| |uα|dx+ 2 ξ |u|2 |uα| ξ |uα−k| |uk| |u| |uα|dx+ 2 ||u|| L∞(R) ξ |uα| ≤ 2 ||u||L∞(R) ξ |uα−k| |uk| |uα|dx+ 2 ||u|| L∞(R) ξ |uα| 2dx (3.5) hence, in (3.4) we have ξ |uα| [3 β − |ω| ] ∂ξ |uα+1| [∂tξ + β ∂ 3ξ + |ω| ∂ξ + c0 ξ ] |uα| − 2 c ξ |uα−k| |uk| |uα|dx− 2 ξ |(|u|2)α−m| |um| |uα|dx ≤ 0. Therefore, using straightforward calculations we obtain the main inequality ξ |uα| η |uα+1| θ |uα| Rαdx ≤ 0 where η = (3 β − |ω| ) ∂ξ for |ω| < 3 β θ = − [ ∂tξ + β ∂ 3ξ + |ω| ∂ξ + c0 ξ ] where c0 = ||u|| L∞(R) and Rα = Rα(|uα|, |uα−1|, . . .). Remark. In (3.4) using Young’s estimate and assuming that β > 0 we have 2ω Im uα uα+1 dx ≤ 2 dx+ 2 β |uα+1| 2 dx. Then, in (3.4) we obtain ξ |uα| 2dx− β ∂3ξ |uα| 2dx+ β ∂ξ |uα+1| 2dx + 2 Im (|u|2)α ξ u uαdx ∂tξ |uα| 2dx+ 2 ξ (|u|2)α−m um uαdx = 2ω Im ∂ξ uα uα+1dx and the assumption that |ω| < 3 β can be removed. Lemma 3.2. For η ∈ Wσ i k an arbitrary weight function and |ω| < 3 β, there exists ξ ∈ Wσ, i+1, k that satisfies η = (3 β − |ω|) ∂ξ for |ω| < 3 β. (3.6) Indeed, we have (3 β − |ω|) η(y, t) dy. (3.7) Lemma 3.3. The expression Rα in the inequality of Lemma 3.1 is a sum of terms of the form ξ uν1 uν2 uα (3.8) where 1 ≤ ν1 ≤ ν2 ≤ α and ν1 + ν2 = α (3.9) Proof. It follows from (3.5). 4 An a priori estimate We show now a fundamental a priori estimate used for a basic local-in-time existence theorem. We con- struct a mapping Z : L∞([0, T ] : Hs(R)) 7−→ L∞([0, T ] : Hs(R)) with the property: Given u(n) = Z(u(n−1)) and essupt∈[0, T ]||u (n−1)||s ≤ c0 then essupt∈[0, T ]||u (n)||s ≤ c0, where s and c0 > 0 are constants. This property tells us that Z : Bc0(0) 7−→ Bc0(0) where Bc0(0) = {v(x, t) : ||v( · , t)||s ≤ c0} is a ball in L ∞([0, T ] : Hs(R)). To guarantee this property, we will appeal to an a priori estimate which is the main object of this section. Differentiating (2.4) two times leads to i ∂tu2 + i β u5 + ω u4 + (|u| 2)2 u+ 2 (|u| 2)1 u1 + |u| 2 u2 = 0. (4.1) Let u = ∧v where ∧ = (I − ∂2)−1. Hence u = (I − ∂2)−1v then u− u2 = v where ∂tu2 = − vt + ut. Replacing in (4.1) we have − i vt + i β ∧ v5 + ω ∧ v4 + (| ∧ v| 2)2 ∧ v + 2 (| ∧ v| 2)1 ∧ v1 + | ∧ v|2 ∧ v2 − (i β ∧ v3 + ω ∧ v2 + | ∧ v| 2 ∧ v) = 0. (4.2) The (4.2) equation is linearized by substituting a new variable z in each coefficient: − i vt + i β ∧ v5 + ω ∧ v4 + (| ∧ z| 2)2 ∧ v + 2 (| ∧ z| 2)1 ∧ v1 + | ∧ z|2 ∧ v2 − (i β ∧ v3 + ω ∧ v2 + | ∧ z| 2 ∧ v) = 0. (4.3) The linear equation which is to be solved at each iteration is of the form i ∂tv = i β ∧ v 5 + ω ∧ v 4 − i β ∧ v 3 − ω ∧ v 2 + b (1) (4.4) where b(1) = (| ∧ z|2)2 ∧ v+2 (| ∧ z| 2)1 ∧ v1 + | ∧ z| 2 ∧ v2 − |∧ z| 2 ∧ v. Equation (4.4) is a linear equation at each iteration which can be solved in any interval of time in which the coefficient is defined. We consider the following lemma that will help us setting up the iteration scheme. Lemma 4.1. Let |ω| < 3 β. Given initial data u0(x) ∈ H ∞(R) = N≥0 H N (R) there exists a unique solution of (4.4) where b(1) is a smooth bounded coefficient with z ∈ H∞(R). The solution is defined in any time interval in which the coefficient is defined. Proof. Let T > 0 be arbitrary and M > 0 a constant. Let Γ = ξ ( i ∂t − i β ∧ ∂ 5 − ω ∧ ∂4 + i β ∧ ∂3 + ω ∧ ∂2 ) then in (4.4) we have Γu = ξ b(1). We consider the bilinear form B : D ×D 7−→ R, B(u, v) =< u, v >= Im e−Mt u v dx dt where D = {u ∈ C∞0 (R× [0, T ]) : u(x, 0) = 0 }. We have Γu · u = i ξ u ut − i β ξ u ∧ u5 − ω ξ u ∧ u4 + i β ξ u ∧ u3 + ω ξ u ∧ u2 Γu · u = − i ξ u ut + i β ξ u ∧ u5 − ω ξ u ∧ u4 − i β ξ u ∧ u3 + ω ξ u ∧ u2. (applying conjugate) Subtracting and integrating over x ∈ R we have 2 i Im Γu · udx = i ∂t ξ |u|2dx− i ∂tξ |u| 2dx− i β ξ u ∧ u5dx− i β ξ u ∧ u5dx ξ u ∧ u4dx+ ω ξ u ∧ u4dx+ i β ξ u ∧ u3dx+ i β ξ u ∧ u3dx ξ u ∧ u2dx− ω ξ u ∧ u2dx. Each term is treated separately, integrating by parts ξ u ∧ u5dx = ξ ∧ (I − ∂2)u ∧ u5dx = ξ ∧ u ∧ u5dx− ξ ∧ u2 ∧ u5dx ∂4ξ ∧ u ∧ u1dx+ ∂3ξ | ∧ u1| 2dx− 3 ∂2ξ ∧ u1 ∧ u2dx− 2 ∂ξ | ∧ u2| ξ ∧ u2 ∧ u3dx− ∂2ξ ∧ u2 ∧ u3dx− ∂ξ | ∧ u3| ξ ∧ u3 ∧ u4dx. The other terms are calculates in a similar way. Then 2 i Im Γu · udx = i ∂t ξ |u|2dx− i ∂tξ |u| 2dx− i β ∂4ξ ∧ u ∧ u1dx− i β ∂3ξ | ∧ u1| + 3 i β ∂2ξ ∧ u1 ∧ u2dx+ 2 i β ∂ξ | ∧ u2| 2dx− i β ξ ∧ u2 ∧ u3dx + i β ∂2ξ ∧ u2 ∧ u3dx+ i β ∂ξ | ∧ u3| 2dx− i β ξ ∧ u3 ∧ u4dx − i β ∂4ξ ∧ u ∧ u1dx − i β ∂3ξ | ∧ u1| 2dx+ 3 i β ∂2ξ ∧ u1 ∧ u2dx + 2 i β ∂ξ | ∧ u2| 2dx− i β ξ ∧ u2 ∧ u3dx+ i β ∂2ξ ∧ u2 ∧ u3dx + 2 i β ∂ξ | ∧ u3| 2dx+ i β ξ ∧ u3 ∧ u4dx+ ω ∂3ξ ∧ u ∧ u1dx ∂2ξ | ∧ u1| 2dx− 2ω ∂ξ ∧ u1 ∧ u2dx− ω ξ | ∧ u2| ∂ξ ∧ u2 ∧ u3dx− ω ξ | ∧ u3| 2dx − ω ∂3ξ ∧ u ∧ u1dx ∂2ξ | ∧ u1| 2dx+ 2ω ∂ξ ∧ u1 ∧ u2dx+ ω ξ | ∧ u2| ∂ξ ∧ u2 ∧ u3dx+ ω ξ | ∧ u3| 2dx + i β ∂2ξ ∧ u ∧ u1dx + i β ∂ξ | ∧ u1| 2dx− i β ξ ∧ u1 ∧ u2dx− i β ξ ∧ u2 ∧ u3dx + i β ∂2ξ ∧ u ∧ u1dx + i β ∂ξ | ∧ u1| 2dx− i β ξ ∧ u1 ∧ u2dx − i β ξ ∧ u2 ∧ u3dx− ω ∂ξ ∧ u ∧ u1dx− ω ξ | ∧ u1| 2dx− ω ξ | ∧ u2| ∂ξ ∧ u ∧ u1dx+ ω ξ | ∧ u1| 2dx+ ω ξ | ∧ u2| hence 2 i Im Γu · udx = i ∂t ξ |u|2dx− i ∂tξ |u| 2dx− i β ∂4ξ (| ∧ u|2)1dx − 2 i β ∂3ξ | ∧ u1| 2dx+ 3 i β ∂2ξ (| ∧ u1| 2)1dx + 4 i β ∂ξ | ∧ u2| − i β ξ (| ∧ u2| 2)1dx+ i β ∂2ξ (| ∧ u2| 2)1dx+ 3 i β ∂ξ | ∧ u3| +2 i ω Im ∂3ξ ∧ u ∧ u1dx − 4 i ω Im ∂ξ ∧ u1 ∧ u2dx − 2 i ω Im ∂ξ ∧ u2 ∧ u3dx+ i β ∂2ξ (| ∧ u|2)1dx+ 2 i β ∂ξ | ∧ u1| − i β ξ (| ∧ u1| 2)1dx− i β ξ (| ∧ u2| 2)1dx− 2ω Im ∂ξ ∧ u ∧ u1dx then, adding similar terms and cutting the letter i we obtain Γu · u dx = ∂t ξ |u|2dx− ∂tξ |u| 2dx + β ∂5ξ | ∧ u|2dx− 5 β ∂3ξ | ∧ u1| + 6 β ∂ξ | ∧ u2| 2dx− β ∂3ξ | ∧ u2| 2dx+ 3 β ∂ξ | ∧ u3| + 2ω Im ∂3ξ ∧ u ∧ u1dx− 4ω Im ∂ξ ∧ u1 ∧ u2dx− 2ω Im ∂ξ ∧ u2 ∧ u3dx ∂3ξ | ∧ u|2dx+ 3 β ∂ξ | ∧ u1| 2dx − 2ω Im ∂ξ ∧ u ∧ u1dx ∂ξ | ∧ u3| 2dx+ |ω| ∂ξ | ∧ u2| 2dx+ 2 |ω| ∂ξ | ∧ u1| 2dx+ 2 |ω| ∂ξ | ∧ u2| + |ω| ∂ξ | ∧ u|2dx+ |ω| ∂ξ | ∧ u1| 2dx+ |ω| |∂3ξ| | ∧ u|2dx + |ω| |∂3ξ| | ∧ u1| 2dx + ∂tξ |u| 2dx+ 2 Im Γu · udx ξ |u|2dx+ 3 β ∂ξ | ∧ u3| 2dx− β ∂3ξ | ∧ u2| 2dx+ 6 β ∂ξ | ∧ u2| − 5 β ∂3ξ | ∧ u1| 2dx+ 3 β ∂ξ | ∧ u1| 2dx+ β ∂5ξ | ∧ u|2dx − β ∂3ξ | ∧ u|2dx where 3 |ω| ∂ξ | ∧ u2| 2dx+ |ω| [|∂3ξ|+ 3 ∂ξ] | ∧ u1| + |ω| [|∂3ξ|+ ∂ξ + ∂tξ] | ∧ u| 2dx+ 2 Im Γu · udx ξ |u|2dx + [3 β − |ω|] ∂ξ | ∧ u3| 2dx− β ∂3ξ | ∧ u2| + 6 β ∂ξ | ∧ u2| 2dx− 5 β ∂3ξ | ∧ u1| 2dx+ 3 β ∂ξ | ∧ u1| ∂5ξ | ∧ u|2dx− β ∂3ξ | ∧ u|2dx ξ |u|2dx + β [−∂3ξ + 5∂ξ] | ∧ u2| [−5 ∂3ξ + 3∂ξ] | ∧ u1| 2dx+ β [∂3ξ − ∂3ξ] | ∧ u|2dx using (2.3), ∧un = (I − (I − ∂ 2))∧ un−2 = ∧un−2 − un−2 for n a positive integer and standard estimates we obtain Γu · u dx ≥ ∂t ξ |u|2 dx− c ξ |u|2 dx. Multiply this equation by e−Mt, and integrate with respect to t for t ∈ [0, T ] and u ∈ D e−Mt Γu · udx dt ≥ ξ |u|2dx dt− c ξ e−Mt |u|2dx dt = e−Mt ξ |u|2dx ξ e−Mt |u|2dx dt− c ξ e−Mt |u|2dx dt = e−Mt ξ(x, T ) |u(x, T )|2dx+M ξ e−Mt |u|2dx dt − c ξ e−Mt |u|2dx dt. < Γu, u >= Im e−Mt Γu · udx dt ≥ e−Mt ξ(x, T ) |u(x, T )|2dx+ (M − c) ξ e−Mt |u|2dx dt ξ e−Mt |u|2dx dt provided that M is chosen large enough. Then < Γu, u >≥< u, u >, for all u ∈ D. Let Γ∗ be the formal adjoint of Γ defined by Γ∗ = ξ(−i ∂t − i β ∧ ∂ 5 − ω ∧ ∂4 + i β ∧ ∂3 + ω ∧ ∂2). Let D∗ = {w ∈ C∞0 (R× [0, T ]) : w(x, T ) = 0 }. In a similar way we prove that < Γ∗w, w > ≥ < w, w >, ∀ w ∈ D∗. ¿From this equation, we have that Γ∗ is one-one. Therefore, < Γ∗w, Γ∗v > is an inner product on D∗. We denote by X the completion of D∗ with respect to this inner product. By Riesz’s Representation Theo- rem, there exists a unique solution V ∈ X, such that for any w ∈ D∗, < ξb(1), w >=< Γ∗V, Γ∗w > where we use that ξ b(1) ∈ X. Then if v = Γ∗V we have < v, Γ∗w >=< ξb(1), w > or < Γ∗w, v >=< w, ξb(1) > . Hence, v = Γ∗V is a weak solution of Γv = ξb(1) with v ∈ L2(R× [0, T ]) ≃ L2([0, T ] : L2(R)). Remark. To obtain higher regularity of the solution, we repeat the proof with higher derivatives. It is a standard approximation procedure to obtain a result for general initial data. The next step is to estimate the corresponding solutions v = v(x, t) of the equation (4.3) via the coeffi- cients of that equation. The following estimate is related to the existence of solutions theorem. Lemma 4.2. Let |ω| < 3 β and 0 < γ1 ≤ ξ ≤ γ2, with γ2, γ2 real constants. Let v, z ∈ C k([0, +∞) : HN(R)) for all k, N which satisfy (4.3). For each integer α there exist positive nondecreasing functions G and F such that for all t ≥ 0 ξ |vα| 2dx ≤ G(||z||λ) ||v|| α + F (||z||α) (4.5) where || · ||α is the norm in H α(R) and λ = max{1, α}. Proof. Differentiating α-times the equation (4.3), for some α ≥ 0 we have − i ∂tvα + i β ∧ vα+5 + ω ∧ vα+4 − i β ∧ vα+3 + h(j) ∧ vj + (|z| 2)α+2 ∧ v + p(∧zα+1, . . .) = 0 (4.6) where h(j) is a smooth function depending on | ∧ z|2, . . . with i = 2 + α − j. For α ≥ 2, p(∧zα+1, . . .) depends at most linearly on ∧zα+1, while for α = 2, p(∧zα+1, . . .) depends at most quadratically on ∧zα+1. We multiply equation (4.6) by ξ vα and integrate over x ∈ R ξ vα ∂tvαdx+ i β ξ vα ∧ vα+5dx+ ω ξ vα ∧ vα+4dx− i β ξ vα ∧ vα+3dx ξ vα ∧ vjdx+ ξ (|z|2)α+2vα ∧ vdx+ ξ vαp(∧zα+1, . . .)dx = 0 and applying conjugate ξ vα ∂tvαdx− i β ξ vα ∧ vα+5dx+ ω ξ vα ∧ vα+4dx+ i β ξ vα ∧ vα+3dx ξ vα ∧ vjdx+ ξ (|z|2)α+2vα ∧ vdx+ ξ vαp(∧zα+1, . . .)dx = 0. Subtracting, it follows that − i ∂t ξ |vα| 2dx+ i ∂tξ |vα| 2dx+ i β ξ vα ∧ vα+5dx+ i β ξ vα ∧ vα+5dx ξ vα ∧ vα+4dx− ω ξ vα ∧ vα+4dx− i β ξ vα ∧ vα+3dx− i β ξ vα ∧ vα+3dx ξ vα ∧ vjdx− ξ vα ∧ vjdx+ ξ (|z|2)α+2vα ∧ vdx (4.7) ξ (|z|2)α+2vα ∧ v dx+ ξ vαp(∧zα+1, . . .) dx− ξ vα p(∧zα+1, . . .)dx = 0. Each term is treated separately, integrating by parts ξ vα ∧ vα+5dx = ξ ∧ (I − ∂2)vα ∧ vα+5dx ξ ∧ vα ∧ vα+5dx− ξ ∧ vα+2 ∧ vα+5dx ∂4ξ ∧ vα ∧ vα+1dx+ ∂3ξ | ∧ vα+1| 2dx− 3 ∂2ξ ∧ vα+1 ∧ vα+2dx ∂ξ | ∧ vα+2| ξ ∧ vα+2 ∧ vα+3dx− ∂2ξ ∧ vα+2 ∧ vα+3dx ∂ξ | ∧ vα+3| ξ ∧ vα+4 ∧ vα+3dx. The other terms are calculated in a similar way. Hence in (4.7) we have performing straightforward calculations as above ξ |vα| ∂tξ |vα| 2dx − β ∂5ξ | ∧ vα| 2dx+ 2 β ∂3ξ | ∧ vα+1| ∂3ξ | ∧ vα+1| 2dx− 4 β ∂ξ | ∧ vα+2| 2dx− β ∂ξ | ∧ vα+2| ∂2ξ | ∧ vα+2| 2dx− 3 β ∂ξ | ∧ vα+3| 2 dx− 2ω Im ∂3ξ ∧ vα ∧ vα+1dx + 4ω Im ∂ξ ∧ vα+1 ∧ vα+2dx+ 2ω Im ∂ξ ∧ vα+2 ∧ vα+3dx + 2 β Im ∂ξ ∧ vα ∧ vα+2dx+ 2 β Im ξ ∧ vα+1 ∧ vα+2dx ∂ξ | ∧ vα+2| 2dx+ 2 h(j) Im ξ vα ∧ vjdx + 2 Im ξ (|z|2)α+2 vα ∧ vdx+ 2 Im ξ vαp(∧zα+1, . . .) dx = 0 ξ |vα| ∂tξ |vα| 2dx − 3 β ∂ξ | ∧ vα+3| 2dx+ β ∂2ξ | ∧ vα+2| − 6 β ∂ξ | ∧ vα+2| 2dx+ 5 β ∂3ξ | ∧ vα+1| 2dx− β ∂5ξ | ∧ vα| = − 2ω Im ∂ξ ∧ vα+2 ∧ vα+3dx− 4ω Im ∂ξ ∧ vα+1 ∧ vα+2dx − 2 β Im ξ ∧ vα+1 ∧ vα+2dx− 2 β Im ∂ξ ∧ vα ∧ vα+2dx +2ω Im ∂3ξ ∧ vα ∧ vα+1dx− 2 h(j) Im ξ vα ∧ vjdx − 2 Im ξ (|z|2)α+2vα ∧ vdx− 2 Im ξ vαp(∧zα+1, . . .) dx hence, ξ |vα| ∂tξ |vα| 2dx+ 3 β ∂ξ | ∧ vα+3| 2dx− β ∂2ξ | ∧ vα+2| ∂ξ | ∧ vα+2| 2dx− 5 β ∂3ξ | ∧ vα+1| 2dx+ β ∂5ξ | ∧ vα| = 2ω Im ∂ξ ∧ vα+2 ∧ vα+3dx+ 4ω Im ∂ξ ∧ vα+1 ∧ vα+2dx +2 β Im ξ ∧ vα+1 ∧ vα+2dx + 2 β Im ∂ξ ∧ vα ∧ vα+2dx − 2ω Im ∂3ξ ∧ vα ∧ vα+1dx+ 2 h(j) Im ξ vα ∧ vjdx + 2 Im ξ (|z|2)α+2vα ∧ vdx+ 2 Im ξ vαp(∧zα+1, . . .) dx ≤ |ω| ∂ξ | ∧ vα+2| 2 dx+ |ω| ∂ξ| ∧ vα+3| 2dx+ 2 |ω| ∂ξ | ∧ vα+1| + 2 |ω| ∂ξ| ∧ vα+2| 2dx+ |β| ξ | ∧ vα+1| 2dx+ |β| ξ | ∧ vα+2| + |β| ∂ξ| ∧ vα| 2dx+ |β| ∂ξ| ∧ vα+2| 2dx+ |ω| ∂3ξ| ∧ vα| + |ω| ∂3ξ| ∧ vα+1| 2dx + 2 ∣∣∣∣∣∣ ξ vα ∧ vjdx ∣∣∣∣∣∣ ξ (|z|2)α+2 vα ∧ vdx ξ vαp(∧zα+1, . . .) dx where ξ |vα| (3 β − |ω|)∂ξ | ∧ vα+3| 2dx + [β ∂2ξ − 6 β ∂ξ + 3 |ω| ∂ξ + |β| ∂ξ + |β| ξ] | ∧ vα+2| [5β∂3ξ + |ω|∂3ξ + 2 |ω|∂ξ + |β| ξ] | ∧ vα+1| [∂tξ + β ∂ 5ξ + |ω| ∂3ξ + |β| ∂ξ] | ∧ vα| ∣∣∣∣∣∣ ξ vα ∧ vjdx ∣∣∣∣∣∣ ξ (|z|2)α+2 vα ∧ vdx ∣∣∣∣+ 2 ξ vαp(∧zα+1, . . .) dx ∣∣∣∣ . using that |ω| < 3 β we have that the first term in the right hand side of the above expression is not positive. Hence, ξ |vα| [β ∂2ξ − 6 β ∂ξ + 3 |ω| ∂ξ + |β| ∂ξ + |β| ξ] | ∧ vα+2| [5 β ∂3ξ + |ω| ∂3ξ + 2 |ω| ∂ξ + |β| ξ] | ∧ vα+1| 2dx + [∂tξ + β ∂ 5ξ + |ω| ∂3ξ + |β| ∂ξ] | ∧ vα| ∣∣∣∣∣∣ ξ vα ∧ vjdx ∣∣∣∣∣∣ ξ (|z|2)α+2 vα ∧ v dx ∣∣∣∣+ 2 ξ vαp(∧zα+1, . . .) dx ∣∣∣∣ . Using that ∧vn = ∧vn−2 − vn−2 and a standard estimate, the lemma follows. 5 Uniqueness and Existence of a Local Solution In this section, we study the uniqueness and the existence of local strong solutions in the Sobolev space HN(R) for N ≥ 3 for the problem (2.4). To establish the existence of strong solutions for (2.4) we use the a priori estimate together with an approximation procedure. Theorem 5.1(Uniqueness). Let |ω| < 3 β, u0(x) ∈ H N (R) with N ≥ 3 and 0 < T < +∞. Then there is at most one strong solution u ∈ L∞([0, T ] : HN(R)) of (2.4) with initial data u(x, 0) = u0(x). Proof. Assume that u, v ∈ L∞([0, T ] : HN(R)) are two solutions of (2.4) with ut, vt ∈ L ∞([0, T ] : HN−3(R)), and with the same initial data. Then i (u− v)t + i β (u− v)3 + ω (u− v)2 + |u| 2 u− |v|2 v = 0 (5.1) with (u− v)(x, 0) = 0. By (5.1) i (u− v)t + i β (u − v)3 + ω (u − v)2 + |u| 2 (u− v) + (|u|2 − |v|2) v = 0 i (u− v)t + i β (u− v)3 + ω (u− v)2 + |u| 2 (u− v) + (|u| − |v|) (|u|+ |v|) v = 0. (5.2) Multiplying (5.2) by ξ(u − v) we have i ξ (u− v) (u − v)t + i β ξ (u − v) (u− v)3 + α ξ (u− v) (u − v)2 + |u|2 |u− v|2 + ξ (u− v) (|u| − |v|) (|u|+ |v|) v = 0. − i ξ (u− v) (u − v)t − i β ξ (u− v) (u − v)3 + α ξ (u − v) (u− v)2 + |u|2 |u− v|2 + ξ (u− v) (|u| − |v|) (|u|+ |v|) v = 0. (applying conjugate) Subtracting and integrating over x ∈ R we obtain ξ |u− v|2dx− i ∂tξ |u− v| 2dx+ i β ξ (u− v) (u− v)3dx + i β ξ (u − v) (u− v)3dx+ ω ξ (u − v) (u− v)2dx ξ (u− v) (u − v)2dx+ 2 i Im ξ (u − v) (|u| − |v|) (|u|+ |v|) v dx = 0 (5.3) Each term is treated separately, integrating by parts ξ (u− v) (u− v)3dx ∂2ξ (u− v) (u− v)1dx+ 2 ∂ξ |(u − v)1| ξ (u− v)1 (u− v)2dx. The other terms are calculated in a similar way. Hence in (5.3) we have ξ |u− v|2dx− i ∂tξ |u− v| 2dx+ i β ∂2ξ (u− v) (u − v)1dx +2 i β ∂ξ |(u − v)1| 2dx+ i β ξ (u − v)1 (u − v)2dx + i β ∂2ξ (u− v) (u − v)1dx + i β ∂ξ |(u− v)1| 2dx− i β ξ (u− v)1 (u− v)2dx− ω ∂ξ (u − v) (u− v)1dx ξ |(u− v)1| 2dx+ ω ∂ξ (u− v) (u − v)1dx+ ω ξ |(u − v)1| +2 i Im ξ (u− v) (|u| − |v|) (|u|+ |v|) v dx = 0 ξ |u− v|2dx− i ∂tξ |u− v| 2dx + i β ∂2ξ (|u − v|2)1dx+ 3 i β ∂ξ |(u − v)1| − 2 i ω Im ∂ξ (u− v) (u− v)1dx+ 2 i Im ξ (u− v) (|u| − |v|) (|u|+ |v|) v dx = 0 if and only if ξ |u− v|2dx− ∂tξ |u− v| 2 dx+ β ∂2ξ (|u− v|2)1dx + 3 β ∂ξ |(u− v)1| = 2ω Im ∂ξ (u− v) (u− v)1dx− 2 Im ξ (u− v) (|u| − |v|) (|u|+ |v|) v dx ≤ |ω| ∂ξ |u− v|2dx+ |ω| ∂ξ |(u− v)1| 2dx+ 2 ξ |u− v| | |u| − |v| | (|u|+ |v|) |v| dx. Using that | |u| − |v| | ≤ |u− v|, (2.3) and standard estimates, we have ξ |u− v|2dx+ [3 β − |ω| ] ∂ξ |(u− v)1| 2dx ≤ c ξ |u− v|2dx. Integrating in t ∈ [0, T ], using the fact that (u− v) vanishes at t = 0 and Gronwall’s inequality it follows that u = v. This proves the uniqueness of the solution. We construct the mapping Z : L∞([0, T ] : Hs(R)) 7−→ L∞([0, T ] : Hs(R)) where the initial condi- tion is given by u(n)(x, 0) = u0(x) and the first approximation is given by u(0) = u0(x) u(n) = Z(u(n−1)) n ≥ 1, where u(n−1) is in place of z in equation (4.3) and u(n) is in place of v which is the solution of equation (4.3). That is − i u t + i β ∧ u 5 + ω ∧ u 4 + (| ∧ u (n−1)|2)2 ∧ u (n) + 2 (| ∧ u(n−1)|2)1 ∧ u + | ∧ u(n−1)|2 ∧ u 2 − (i β ∧ u 3 + ω ∧ u 2 + | ∧ u (n−1)|2 ∧ u(n)) = 0. By Lemma 4.1, u(n) exists and is unique in C((0, +∞) : HN (R)). A choice of c0 and the use of the a priori estimate in Section 4 shows that Z : Bc0(0) 7−→ Bc0(0) where Bc0(0) is a bounded ball in L∞([0, T ] : Hs(R)). Theorem 5.2(Local solution). Let |ω| < 3 β and N an integer ≥ 3. If u0(x) ∈ H N (R), then there is T > 0 and u such that u is a strong solution of (2.4), u ∈ L∞([0, T ] : HN (R)) and u(x, 0) = u0(x). Proof. We prove that for u0(x) ∈ H ∞(R) = k≥0 H k(R) there exists a solution u ∈ L∞([0, T ] : HN(R)) with initial data u(x, 0) = u0(x) where the time of existence T > 0 only depends on the norm of u0(x). We define a sequence of approximations to equation (4.3) as t = i β ∧ v 5 + ω ∧ v 4 − i β ∧ v 3 − ω ∧ v 2 + | ∧ v (n−1)|2 ∧ v + O[ (| ∧ v(n−1)|2)2, (| ∧ v (n−1)|2)1, . . .) ] (5.4) where the initial condition is v(n)(x, 0) = u0(x)−∂ 2u0(x). The first approximation is given by v (0)(x, 0) = u0(x)− ∂ 2u0(x). Equation (5.4) is a linear equation at each iteration which can be solved in any interval of time in which the coefficients are defined. This is shown in Lemma 4.1. By Lemma 4.2, it follows that ξ |v(n)α | 2dx ≤ G(||v(n−1)||λ) ||v (n)||2α + F (||v (n−1)||α). (5.5) Choose α = 1 and let c ≥ ||u0−∂ 2u0||1 ≥ ||u0||3. For each iterate n, ||v (n)( · , t)|| is continuous in t ∈ [0, T ] and ||v(n)( · , 0)|| < c. Define c0 = c2+1. Let T 0 be the maximum time such that ||v (k)( · , t)||1 ≤ c3 for 0 ≤ t ≤ T 0 , 0 ≤ k ≤ n. Integrating (5.5) over [0, t] we have that for 0 ≤ t ≤ T 0 and j = 0, 1 ||v(n−1)||1 ||v(n)||2jds+ ||v(n−1)||j It follows that ξ(x, t)|v j (x, t)| 2dx ≤ ξ(x, 0)|v j (x, 0)| ||v(n−1)||1 ||v(n)||2jds ||v(n−1)||j hence j (x, t)| 2dx ≤ ξ(x, t)|v j (x, t)| ξ(x, 0)|v j (x, 0)| ||v(n−1)||1 ||v(n)||2jds ||v(n−1)||j 2dx ≤ j (x, 0)| G(c3) c23 t+ F (c3) and we obtain for j = 0, 1 that ||v(n)||1 ≤ G(c0) c20 t+ F (c0) Claim. T 0 does not approach to 0. On the contrary, assume that T 0 → 0. Since ||v (n)( · , t)|| is continuous for t ≥ 0, there exists τ ∈ [0, T ] such that ||v(k)( · , t)||1 = c0 for 0 ≤ τ ≤ T 0 , 0 ≤ k ≤ n. Then c20 ≤ G(c0) c20 T F (c0) as n → ∞, we have c2 + 1 c2 then 4 γ21 c4 + 1 ≤ 0 which is a contradiction. Consequently T 0 6→ 0. Choosing T = T (c) sufficiently small, and T not depending on n, one concludes that ||v(n)||1 ≤ C (5.6) for 0 ≤ t ≤ T. This shows that T 0 ≥ T. Hence, from (5.6) we imply that there exists a subsequence v(nj) ≡ v(n) such that ⇀ v weakly on L∞([0, T ] : H1(R)). (5.7) Claim. u = ∧v is a solution. In the linearized equation (5.4) we have 5 = ∧(I − (I − ∂ 3 = ∧v 3 − v 3 = ∂ 1︸ ︷︷ ︸ ∈L2(R) )− ∂2(v 1 )︸ ︷︷ ︸ ∈H−2(R) ∈ H−2(R). Since ∧ = (I − ∂2)−1 is bounded in H1(R), ∧v 5 belongs to H −2(R). v(n) is still bounded in L∞([0, T ] : H1(R)) →֒ L2([0, T ] : H1(R)) and since ∧ : L2(R) → H2(R) is a bounded operator, || ∧ v 1 ||H2(R) ≤ c ||v 1 ||L2(R) ≤ c ||v 1 ||H1(R). Consequently, ∧v 1 is bounded in L 2([0, T ] : H2(R)) →֒ L2([0, T ] : L2(R)). It follows that ∂2(∧v 1 ) is bounded in L2([0, T ] : H−2(R)), and 5 is bounded in L 2([0, T ] : H−2(R)). (5.8) Similarly, the other terms are bounded. By (5.4), v t is a sum of terms each of which is the product of a coefficient, uniformly bounded on n and a function in L2([0, T ] : H−2(R)) uniformly bounded on n such that v t is bounded in L 2([0, T ] : H−2(R)). On the other hand, H1loc(R) loc (R) →֒ H −4(R). By Lions-Aubin’s compactness Theorem [24] there is a subsequence v(nj) ≡ v(n) such that v(n) → v strongly on L2([0, T ] : H loc (R)). Hence, for a subsequence v (nj) ≡ v(n), we have v(n) → v a. e. in L2([0, T ] : loc (R)). Moreover, from (5.8), ∧v 5 ⇀ ∧v5 weakly in L 2([0, T ] : H−2(R)). Similarly, ∧v 2 ⇀ ∧v2 weakly in L2([0, T ] : H−2(R)). Since ||∧v(n)||H2(R) ≤ c ||v (n)||L2(R) ≤ c ||v (n)||H1(R) ≤ c ||v (n)||H1/2(R) and v(n) → v strongly on L2([0, T ] : H loc (R)) then ∧v (n) → ∧v strongly in L2([0, T ] : H2loc(R)). Thus, the fifth term on the right hand side of (5.4), | ∧ v(n−1)|2 ∧ v 2 ⇀ | ∧ v| 2 ∧ v2 weakly in L 2([0, T ] : L1loc(R)) as ∧v 2 ⇀ ∧v2 weakly in L 2([0, T ] : H−2(R)) and | ∧v(n−1)|2 → |∧v|2 strongly on L2([0, T ] : H2loc(R)). Similarly, the other terms in (5.4) converge to their limits, implying v t ⇀ vt weakly in L 2([0, T ] : L1loc(R)). Passing to the limit i vt = ∂ 2(i β ∧ v3 + ω ∧ v2 + | ∧ v| 2 ∧ v)− (i β ∧ v3 + ω ∧ v2 + | ∧ v| 2 ∧ v) = −(I − ∂2)(i β ∧ v3 + ω ∧ v2 + | ∧ v| 2 ∧ v). Thus i vt + (I − ∂ 2)(i β ∧ v3 + ω ∧ v2 + | ∧ v| 2 ∧ v) = 0. This way, we have (2.4) for u = ∧v. Now, we prove that there exists a solution of (2.4) with u ∈ L∞([0, T ] : HN (R)) and N ≥ 4, where T de- pends only on the norm of u0 in H 3(R). We already know that there is a solution u ∈ L∞([0, T ] : H3(R)). It is suffices to show that the approximating sequence v(n) is bounded in L∞([0, T ] : HN−2(R)). Taking α = N − 2 and considering (5.5) for α ≥ 2, we define cN−2 = ||u0(·)||N + 1. Let T N−3 be the largest time such that ||v(k)( · , t)||α ≤ cN−3 for 0 ≤ t ≤ T N−3, 0 ≤ k ≤ n. Integrating (5.5) over [0, t], for 0 ≤ t ≤ T N−3, we have ξ |v(n)α | ||v(n−1)||α ||v(n)||2αds+ ||v(n−1)||α It follows that ξ(x, t) |v(n)α | 2dx ≤ ξ(x, 0) |v(n)α (x, 0)| ||v(n−1)||α ||v(n)||2αds ||v(n−1)||α hence |v(n)α | 2dx ≤ ξ |v(n)α | 2dx ≤ ξ(x, 0) |v(n)α (x, 0)| ||v(n−1)||α ||v(n)||2αds ||v(n−1)||α |v(n)α | 2dx ≤ |v(n)α (x, 0)| 2dx + G(cN−3) c2N−3 t+ F (cN−3) ||v(n)α (x, 0)|| G(cN−3) c2N−3 t+ F (cN−3) ||u(x, 0)||2N + G(cN−3) c2N−3 t+ F (cN−3) and we obtain ||v(n)α ( · , t)|| αdx ≤ ||u(x, 0)||2N + G(cN−3) c2N−3 t+ F (cN−3) Claim. T N−3 does not approach to 0. On the contrary, assume that T N−3 → 0. Since ||v (n)( · , t)|| is continuous for t ≥ 0, there exists τ ∈ [0, TN−3] such that ||v (k)( · , τ)||α = cN−3 for 0 ≤ τ ≤ T (n), 0 ≤ k ≤ n. Then c2N−3 ≤ ||u(x, 0)||2N + G(cN−3) c2N−3 T N−3 + F (cN−3) as n → +∞, and we have ||u(x, 0)||2N + 1 ||u(x, 0)||2N then 4 γ21 ||u(x, 0)||4N + 1 ≤ 0 which is a contradiction. Then T N−3 6→ 0. By choosing TN−3 = TN−3(||u(x, 0)|| N ) sufficiently small, and TN−3 not depending on n, we conclude that ||v(n)( · , t)||2α ≤ c N−3 for all 0 ≤ t ≤ TN−3. (5.9) This shows that T N−3 ≥ TN−3. Thus, v ∈ L∞([0, TN−3] : H α(R)) ≡ L∞([0, TN−3] : H N−2(R)). Now, denote by 0 ≤ T ∗N−3 ≤ +∞ the maximal number such that for all 0 < t ≤ T N−3, u = ∧v ∈ L∞([0, t] : HN (R)). In particular, TN−3 ≤ T N−3 for all N ≥ 4. Thus, T can be chosen depending only on the norm of u0 in H 3(R). Approximating u0 by {u 0 } ∈ C 0 (R) such that ||u0 − u 0 ||HN (R) → 0 as j → +∞. Let uj be a solution of (2.4) with u(j)(x, 0) = u 0 . According to the above argument, there exists T which is independent on n but depending only on supj ||u 0 || such that u (j) there exists on [0, T ] and a subsequence u(j) −→ u in L∞([0, T ] : HN(R)). As a consequence of Theorem 5.1 and 5.2 and its proof, one obtains the following result. Corollary 5.3. Let |ω| < 3 β and let u0 ∈ H N (R) with N ≥ 3 such that u 0 → u0 in H N (R). Let u and u(j) be the corresponding unique solutions given by Theorems 5.1 and 5.2 in L∞([0, T ] : HN(R)) with T depending only on supj ||u 0 ||H3(R) such that ⇀ u weakly on L∞([0, T ] : HN(R)), u(j) → u strongly on L2([0, T ] : HN+1(R)). 6 Existence of Global Solutions Here, we will try to extend the local solution u ∈ L∞([0, T ] : HN (W0 i 0)) of (2.4) obtained in Theo- rem 5.2 to t ≥ 0. A standard way to obtain these extensions consists into deducing global estimations for the HN (W0 i 0)-norm of u in terms of the H N (W0 i 0)-norm of u(x, 0) = u0(x). These estimations are frequently based on conservation laws which contain the L2-norm of the solution and their spatial derivatives. It is not possible to do the same to give a solution of the problem of global existence because the difficulty here is that the weight depends on the x and t variables. To solve our problem we follow a different method using Leibniz’s rule like in the proof of Theorem 3.1 of Bona and Saut [3]. Theorem 6.1. For |ω| < 3 β there exists a global solution to (2.4) in the space Hs(R)∩HN (W0 i 0) with N integer ≥ 3 and s ≥ 2. Proof. The first part was proved in [3]. Differentiating (2.4) α-times (for α ≥ 0) over x ∈ R leads i uα t + i β uα+3 + ω uα+2 + (|u| 2)α u+ (|u|2)α−m um + |u| 2 uα = 0. (6.1) Let ξ = ξ(x, t), then multiplying (6.1) by ξ uα we have i ξ uα uα t + i β ξ uα uα+3 + ω ξ uα uα+2 + (|u| 2)α ξ u uα (|u|2)α−m ξ um uα + ξ |u| 2 |uα| 2 = 0 − i ξ uα uα t − i β ξ uα uα+3 + ω ξ uα uα+2 + (|u| 2)α ξ u uα (|u|2)α−m ξ um uα + ξ |u| 2 |uα| 2 = 0. (applying conjugate) Subtracting and integrating over x ∈ R we have ξ |uα| 2dx+ i β ξ uα uα+3dx+ i β ξ uα uα+3dx+ ω ξ uα uα+2dx (6.2) ξ uα uα+2dx+ 2 i Im ξ (|u|2)α u uαdx+ 2 i ξ (|u|2)α−m um uαdx = 0. Each term is calculated separately, integrating by parts in the second term we have ξ uα uα+3dx = ∂2ξ uα uα+1dx+ 2 ∂ξ |uα+1| ξ uα+2 uα+1dx. The other terms are calculated in a similar way. Hence in (6.2) ξ |uα| 2dx− β ∂3ξ |uα| 2dx+ 3 β ∂ξ |uα+1| 2dx − 2ω Im ∂ξ uα uα+1dx ∂tξ |uα| 2dx+ 2 Im ξ (|u|2)α u uαdx+ 2 ξ (|u|2)α−m um uαdx = 0 such that ξ |uα| 2dx− β ∂3ξ |uα| 2dx+ 3 β ∂ξ |uα+1| 2dx+ 2 Im (|u|2)α ξ u uαdx ∂tξ |uα| 2dx+ 2 ξ (|u|2)α−m um uαdx = 2α Im ∂ξ uα uα+1dx ≤ |ω| ∂ξ |uα| 2dx+ |ω| ∂ξ |uα+1| Hence ξ |uα| [ 3 β − |ω| ] ∂ξ |uα+1| [ ∂tξ + β ∂ 3ξ + |ω| ∂ξ ] |uα| + 2 Im (|u|2)α ξ u uαdx+ 2 ξ (|u|2)α−mum uαdx ≤ 0. (6.3) (|u|2)α = (u u )α = uα−k uk = uuα + uα−k uk + u uα (|u|2)α u uα = |u| 2 |uα| uα−k uk u uα + u 2 u2α hence ( |u|2 )α ξ u uαdx = 2 ξ uα−k uk u uαdx+ 2 Im ξ u2 u2αdx ξ |uα−k| |uk| |u| |uα|dx+ 2 ξ |u|2 |uα| ξ |uα−k| |uk| |u| |uα|dx+ 2 ||u|| L∞(R) ξ |uα| ≤ 2 ||u||L∞(R) ξ |uα−k| |uk| |uα|dx+ 2 ||u|| L∞(R) ξ |uα| 2dx (6.4) hence in (6.3) we have ξ |uα| [3 β − |ω| ] ∂ξ |uα+1| 2dx ≤ [∂tξ + β ∂ 3ξ + |ω| ∂ξ + c ξ ] |uα| ξ |uα−k| |uk| |u| |uα|dx− 2 ξ (|u|2)α−m um uαdx. Using (2.3), Gagliardo-Nirenberg’s inequality and standard estimates we get ξ |uα| 2dx+ [3 β − |ω| ] ∂ξ |uα+1| 2 dx ≤ c ξ |uα| 2dx. (6.5) Integrating (6.5) in t ∈ [0, Tmax = T ] we obtain ξ |uα| 2dx+ [3 β − |ω| ] ∂ξ |uα+1| 2dx ds ≤ ||u0(x)|| ξ |uα| where ξ |uα| 2dx ≤ ||u0(x)|| ξ |uα| Using Gronwall’s inequality ξ |uα| 2dx ≤ ||u0(x)|| c t ≤ ||u0(x)|| it follows that ξ |uα| 2dx ≤ c = c(T, ||u0(x)|| Then for any T = Tmax > 0 there exists c = c(T, ||u0(x)|| α) such that ||u||2α + [3 β − |ω| ] ∂ξ |uα+1| 2dx ds ≤ c. This concludes the proof. 7 Persistence Theorem As a starting point for the a priori gain of regularity results that will be discussed in the next section, we need to develop some estimates for solutions of the equation (2.4) in weighted Sobolev norms. The existence of these weighted estimates is often called the persistence of a property of the initial data u0. We show that if u0 ∈ H 3(R)∩HL(W0 i 0) for L ≥ 0, i ≥ 1, then the solution u( · , t) evolves in H L(W0 i 0) for t ∈ [0, T ]. The time interval of that persistence is at least as long as the interval guaranteed by the existence Theorem 5.2. Theorem 7.1 (Persistence). Let |ω| < 3 β and let i ≥ 1 and L ≥ 0 be non-negative integers, 0 < T < +∞. Assume that u is the solution to (2.4) in L∞([0, T ] : H3(R)) with initial data u0(x) = u(x, 0) ∈ H 3(R). If u0(x) ∈ H L(W0 i 0) then u ∈ L∞ [0, T ] : H3(R) ∩HL(W0 i 0) (7.1) |∂L+1u(x, t)|2 η dx dt < +∞ (7.2) where σ is arbitrary, η ∈ Wσ i 0 for i ≥ 1. Proof. We use induction on α. Let u ∈ L∞ [0, T ] : H3(R) ∩Hα(W0 i 0) for 0 ≤ α ≤ L. We derive formally some a priori estimate for the solution where the bound, involves only the norms of u in L∞([0, T ] : H3(R)) and the norms of u0 in H 3(W0 i 0). We do this by approximating u(x, t) through smooth solutions and the weight functions by smooth bounded functions. By Theorem 5.2, we have u(x, t) ∈ L∞([0, T ] : HN (R)) with N = max{L, 3}. In particular, uj(x, t) ∈ L ∞([0, T ]×R) for 0 ≤ j ≤ N − 1. To obtain (7.1) and (7.2) there are two ways of approximation. We approximate general solutions by smooth solutions, and we approximate general weight functions by bounded weight functions. The first of these procedure has already been discussed, so we shall concentrate on the second. Given a smooth weight function η(x) ∈ Wσ, i−1, 0 with σ > 0, we take a sequence η ν(x) of smooth bounded weight functions approximating η(x) from below, uniformly on any half line (−∞, c). Define the weight functions for the α-th induction step as (3 β − |ω|) ην(y, t) dy then the ξν are bounded weight functions which approximate a desired weight function ξ ∈ W0 i 0 from below, uniformly on a compact set. For α = 0, multiplying (2.4) by ξν u, we have i ξν uut + i β ξν uu3 + ω ξν u u2 + ξν |u| 4 = 0 − i ξν u ut − i β ξν u u3 + ω ξν u u2 + ξν |u| 4 = 0. (applying conjugate) Subtracting and integrating over x ∈ R we have ξν |u| 2dx − i ∂tξν |u| 2dx+ i β ξν uu3dx+ i β ξν u u3dx ξν uu2dx− ω ξν u u2dx = 0. (7.3) Each term is treated separately, integrating by parts in the third term we have ξν u u3dx = ∂2ξν uu1dx+ 2 ∂ξν |u1| ξν u2 u1dx. The other terms are calculated in a similar way. Hence in (7.3) we have ξν |u| ∂tξν |u| 2dx− β ∂3ξν |u| 2dx+ 3 β ∂ξν |u1| = 2ω Im ∂ξν u u1dx ≤ |ω| ∂ξν |u| 2dx+ |ω| ∂ξν |u1| Then, using (2.3) we obtain ξν |u| [3 β − |ω|] ∂ξν |u1| [∂tξν + β ∂ 3ξν + |ω| ∂ξν ] |u| 2dx ≤ c ξν |u| ξν |u| 2dx ≤ c ξν |u| We apply Gronwall’s Lemma to conclude that ξν |u| 2dx ≤ c(T, ||u0||). (7.4) for 0 ≤ t ≤ T, and c not depending on β > 0, the weighted estimate remains true for β → 0. Now, we assume that the result is true for (α − 1) and we prove that it is true for α. To prove this, we start from the main inequality (3.1) with ξ and η given by ξν and ην respectively. ξν |uα| ην |uα+1| θν |uα| Rαdx ≤ 0 where ην = (3β − |ω| ) ∂ξν for |ω| < 3 β θν = − [ ∂tξν + β ∂ 3ξν + |ω| ∂ξν + c0 ξν ] where c0 = ||u|| L∞(R) Rα = Rα(|uα|, |uα−1|, . . . ) ξν |uα| 2dx + ην |uα+1| 2dx ≤ − θν |uα| 2dx − ∣∣∣∣− θν |uα| ∣∣∣∣ ≤ |θν | |uα| |Rα|dx. Using (2.3) in the first part of the right hand side we obtain θν |uα| 2dx ≤ c ξν |uα| ξν |uα| ην |uα+1| 2dx ≤ c ξν |uα| |Rα|dx. (7.5) According to (3.8), Rα dx contains a term of the form ξν uν1 uν2 uαdx. (7.6) We estimate the term ξν uν1 uν2 uα dx for ν1 + ν2 = α. (7.7) Let ν2 ≤ α− 2. Integrating by parts one time in (7.7) we have ξν uν1 uν2 uα dx = − ∂ξν uν1 uν2 uα−1 dx− ξν uν1+1 uν2 uα−1 dx ξν uν1 uν2+1 uα−1 dx. We estimates the first term in the right hand side in (7.7). Using Holder’s inequality and standard estimates we obtain ξν |uν2+1| ξν |uν2 | )1/2](∫ ξν |uα−1| (7.8) where (7.8) is bounded by hypothesis. The other terms are estimates in a similar way. Now suppose that ν1 = ν2 = α− 1, then in (7.7) we have ξν uα−1 uα−1 uαdx, hence ξν |uα−1| 2 uαdx ∣∣∣∣ ≤ ||uα−1||L∞(R) ξν |uα−1| )1/2 (∫ ξν |uα| where ||uα−1||L∞(R) is bounded by hypothesis, and the estimate is complete. In a similar way we estimate all the other terms of Rα. Using these estimates in (7.5) and applying Gronwall’s argument, we obtain for 0 ≤ t ≤ T ξν |uα| ην |uα+1| 2dx ≤ c0 e ξν |∂ αu0(x)| 2dx+ 1 where c0 and c1 are independent of ν and such that letting the parameter ν → 0 the desired estimate (7.2) is obtained. 8 Main Theorem In this section we state and prove our main theorem, which states that if the initial data u(x, 0) decays faster than polynomially on R+ = {x ∈ R : x > 0} and possesses certain initial Sobolev regularity, then the solution u(x, t) ∈ C∞ for all t > 0. If η is an arbitrary weight function in Wσ i k, then by Lemma 3.2, there exists ξ ∈ Wσ, i+1, k which satisfies (3.1). For the main theorem, we take 4 ≤ α ≤ L+ 2. For α ≤ L+ 4, we take η ∈ Wσ, L−α−2, α−3 =⇒ ξ ∈ Wσ, L−α−3, α−3. (8.1) Lemma 8.1(Estimate of error terms). Let 4 ≤ α ≤ L+2 and the weight functions be chosen as in (8.1), ∣∣∣∣∣ (θ |uα| 2 +Rα)dx dt ∣∣∣∣∣ ≤ c, (8.2) where c depends only on the norms of u in L∞([0, T ] : Hβ(Wσ, L−β+3, β−3)) ∩ L 2([0, T ] : Hβ+1(Wσ, L−β+2, β−3)) for 3 ≤ β ≤ α− 1, and the norms of u in L∞([0, T ] : H3(W0 L 0)). Proof. We must estimate both Rα and θ. We begin with a term in Rα of the form ξ |uν1 | |uν2 | |uα| (8.3) assuming that ν1 ≤ α− 2. By the induction hypothesis, u is bounded in L∞([0, T ] : Hβ(Wσ, L−(β−3)+, (β−3)+)) for 0 ≤ β ≤ α − 1. By Lemma 2.1, ζ |uβ| 2 < +∞ (8.4) for 0 ≤ β ≤ α − 2 and ζ ∈ Wσ, L−(β−2)+, (β−2)+ . We estimate |uν1 | using (8.4). We estimate |uν2 | and |uα| using the weighted L 2 bounds ζ |uν2 | 2dx dt < +∞ for ζ ∈ Wσ, L−(ν2−3)+, (ν2−4)+ (8.5) and the same with ν2 replaced by α. It suffices to check the powers to t, the powers of x as x → +∞ and the exponential of x as x → −∞. For x > 1. In the (8.3) term, the factor ξ contributed according to (8.1) ξ(x, t) = tα−3 x(L−α+3) t−(α−3) x−(L−α+3)ξ(x, t) ≤ c2 t α−3 x(L−α+3) (using(2.3)) then ξ |uν1 | |uν2 | |uα| ≤ c2 t α−3 x(L−α+3)|uν1 | |uν2 | |uα|. Moreover |uν1 | |uν2 | |uα| = t (ν1−2) L−(ν1−2) −(ν1−2) (L−(ν1−2) 2 |uν1 | × (ν2−4) L−(ν2−3) −(ν2−4) (L−(ν2−3) 2 |uν2 | × (α−4)+ L−(α−3)+ −(α−4)+ (L−(α−3)+) 2 |uα|. tt follows that ξ |uν1 | |uν2 | |uα| ≤ c2 t M xT t (ν1−2) L−(ν1−2) 2 |uν1 | t (ν2−4) L−(ν2−3) 2 |uν2 | t (α−4)+ L−(α−3)+ 2 |uα| (8.6) where M = α− 3− (ν1 − 2) (ν2 − 4) (α− 4)+ T = (L− α+ 3)− (L − (α− 3)+)− (L− (ν2 − 3) (L − ν1 − 2) Claim. M ≥ 0 is large enough, that the extra power of t can be omitted 2M = 2α− 6− (ν1 − 2) + − (ν2 − 4) + − (α− 4)+ = α− 2− (ν1 − 2) + − (ν2 − 4) = α− 2− ν1 + 2− ν2 + 4 = α+ 4− (ν1 + ν2) = α+ 4− α = 4 ≥ 0. Claim. T ≤ 0 is such that the extra power of t can be omitted. 2T = 2L− 2α+ 6− L+ (α− 3)+ − L+ (ν2 − 3) + − L+ (ν1 − 2) = −L− α+ ν1 + ν2 − 2 = −L− α+ α− 2 = −(L+ 2) ≤ 0. Now, we study the behavior as x → −∞. Since each factor uνj (j = 1, 2) must grow slower that an exponential eσ ′ |x| and ξ decays as an exponential e−σ |x|, we simply need to choose the appropriate rela- tionship σ and σ′ at each induction step. The analysis will be completed with the case where ν1 ≥ α− 1. Then, in (3.9), if 2(α − 1) ≤ α, but α ≥ 3. So this possibility is impossible. For x < 1 the estimate is similar, except for an exponential weight. The analysis of all terms of Rα is estimated in a similar form. This completes the estimate of Rα. Now, we estimate the term θ |uα| 2 where θ is given in (3.1). We have that θ involves derivatives of u only up to order one, and hence, θ |uα| 2 is a sum of terms of the same type which we have already encountered in Rα. So, its integral can be bounded in the same type. Indeed, (3.1) shows that θ depends on ξt, ∂ and derivatives of lower order. By using (3.6) we have the claim. Theorem 8.2(Main Theorem). Let |ω| < 3 β, T > 0 and u(x, t) be a solution of (2.4) in the region R× [0, T ] such that u ∈ L∞([0, T ] : H3(W0 L 0)) (8.7) for some L ≥ 2. Then u ∈ L∞([0, T ] : H3+l(Wσ, L−l, l)) ∩ L 2([0, T ] : H4+l(Wσ, L−l−1, l)) (8.8) for all 0 ≤ l ≤ L− 1 and all σ > 0. Remark. If the assumption (8.7) holds for all L ≥ 2, the solution is infinitely differentiable in the x- variable. ¿From (2.4) we have that the solution is C∞ in both variables. We are also quantifying the gain of each derivative by the degree of vanishing of the initial data at infinity. Proof. We use induction on α. For α = 3, let u be a solution of (2.4) satisfying (8.7). Therefore, ut ∈ L ∞([0, T ] : L2(W0 L 0)) where u ∈ L ∞([0, T ] : H3(W0 L 0)) and ut ∈ L ∞([0, T ] : L2(W0 L 0)). Then u ∈ C([0, T ] : L2(W0 L 0)) ∩ Cw([0, T ] : H 3(W0 L 0)). Hence, u : [0, T ] 7−→ H 3(W0 L 0) is a weakly continuous function. In particular, u( · , t) ∈ H3(W0 L 0) for all t. Let t0 ∈ (0, T ) and u( · , t0) ∈ H 3(W0 L 0), then there are {u 0 } ⊆ C 0 (R) such that u 0 ( · ) → u( · , t0) in H 3(W0 L 0). Let u(n)(x, t) be a unique solution of (2.4) with u(n)(x, t0) = u 0 . Then by Theorem 5.1 and 5.2, there exists u in a time interval [t0, t0 + δ] where δ > 0 does not depend on n and u is a unique solution of (2.4), u(n) ∈ L∞([t0, t0 + δ] : H 3(W0 L 0)) with u (n)(x, t0) ≡ u 0 (x) → u(x, t0) ≡ u0(x) in H 3(W0 L 0). Now, by Theorem 7.1, we have u(n) ∈ L∞([t0, t0 + δ] : H 3(W0 L 0)) ∩ L 2([t0, t0 + δ] : H 4(Wσ, L−1, 0)) with a bound that depends only on the norm of u 0 in H 3(W0 L 0). Furthermore, Theorem 7.1 guarantees the non-uniform bounds [t0, t0+δ] (1 + |x+|) k | ∂αu(n)(x, t) | < +∞ for each n, k and α. The main inequality (3.1) and the estimate (8.2) are therefore valid for each u(n) in the interval [t0, t0 + δ]. η may be chosen arbitrarily in its weight class (8.1) and then ξ is defined by (3.7) and the constant c1, c2, c3, c4 are independent of n. From (3.1) and (8.1) we have [t0, t0+δ] ξ |u(n)α | ∫ t0+δ 2dx ≤ c (8.9) where by (8.2), c is independent of n. The estimate (8.9) is proved by induction for α = 3, 4, 5, . . . Thus u(n) is also bounded in L∞([t0, t0 + δ] : H α(Wσ, L−α+3, α−3)) ∩ L 2([t0, t0 + δ] : H α+1(Wσ, L−α+2, α−3)) (8.10) for α ≥ 3. Since u(n) → u in L∞([t0, t0 + δ] : H 3(W0 L 0)). By Corollary 5.3 it follows that u belongs to the space (8.10). Since δ is fixed, this result is valid over the whole interval [0, T ]. References [1] E. Bisognin, V. Visognin and O. Vera. Stabilization of solutions for the higher order nonlinear Schrödinger equation with localized damping. EJDE. Vol. 2007, 6(2007) 1-18. [2] V. Bisognin and O. Vera. 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Existence and uniqueness for a nonlinear dispersive equation, Divulgaciones Cient́ıficas, Vol 10, 2, 2002, 131-148. [32] O. Vera. Gain of regularity for a nonlinear dispersive equation, Series A: Mathematical Sciences, 9, 2003, 53-62. [33] O. Vera. Gain of regularity for a Korteweg-de Vries-Kawahara type equation, EJDE, Vol 2004, 71, 2004, 1-24. Introduction Preliminaries Main Inequality An a priori estimate Uniqueness and Existence of a Local Solution Existence of Global Solutions Persistence Theorem Main Theorem

We study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear Schr\"odinger equation with constant coefficients which appears as a model for propagation of pulse in optical fiber.

Introduction We consider the initial value problem i ut + ω uxx + i β uxxx + |u| 2 u = 0 x, t ∈ R u(x, 0) = u0(x) where ω, β ∈ R, β 6= 0 and u = u(x, t) is a complex valued function. The above equation is a particular case of the equation i ut + ω uxx + i β uxxx + γ |u| 2 u+ i δ |u|2 ux + i ǫ u 2 ux = 0 x, t ∈ R u(x, 0) = u0(x) where ω, β, γ, δ are real numbers with β 6= 0. This equation was first proposed by A. Hasegawa and Y. Kodama [13] as a model for the propagation of a signal in an optic fiber (see also [20]). The equation (Q) can be reduced to other well known equations. For instance, setting ω = 1, β = δ = ǫ = 0 in (Q) we have the semilinear Schrödinger equation, i. e., i ut + uxx + γ |u| 2 u = 0. (Q1) If we let β = γ = 0 and ω = 1 in (Q), we obtain the derivative nonlinear Schrödinger equation i ut + uxx + i δ |u| 2 ux + i ǫ u 2 ux = 0. (Q2) Letting α = γ = ǫ = 0 in (Q), the equation that arises is the complex modified Korteweg-de Vries equation, i ut + i β uxxx + i δ |u| 2 ux = 0. (Q3) The initial value problem for the equations (Q1), (Q2) and (Q3) has been extensively studied in the last few years. See, for instance, [1, 2, 3, 5, 6, 8, 9, 17, 18, 26, 27] and references therein. In 1992, C. Laurey [22] considered the equation (Q) and proved local well-posedness of the initial value problem associated for data in Hs(R), s > 3/4, and global well-posedness in Hs(R), s ≥ 1. In 1997, G. Staffilani [28] for (Q) ∗Departamento de Ingenieŕıa Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. mauricio@ing- mat.udec.cl †Departamento de Matemática, Universidad del B́ıo-B́ıo, Collao 1202, Casilla 5-C, Concepción, Chile. overa@ubiobio.cl http://arxiv.org/abs/0704.1862v1 established local well-posedness for data in Hs(R), s ≥ 1/4 improving Laurey’s result. A similar result was given in [5, 6] with w(t), β(t) real functions. Our aim in this paper, is to study gain in regularity for the equation (P ). Specifically, we prove conditions on (P ) for which initial data u0 possessing sufficient decay at infinity and minimal amount of regularity will lead to a unique solution u(t) ∈ C∞(R) for 0 < t < T, where T is the existence time of the solution. We are not considering the equation (Q) because of the technique used here, we shall see that the last two terms in (Q) are not outstanding in the main inequality, indeed the two last terms are observed in the last two terms in the main inequality. In 1986, N. Hayashi et al. [13] showed that for the nonlinear Schrödinger equation (NLS): i ut + uxx = λ |u| p−1 u, (x, t) ∈ R × R with initial condition u(x, 0) = u0(x), x ∈ R and a certain assump- tion on λ and p, all solutions of finite energy are smooth for t 6= 0 provided the initial functions in H1(R)(or on L2(R)) decay sufficiently fast as |x| → ∞. The main tool is the operator J defined by Ju = ei x 2/4 t (2 i t) ∂x(e − i x2/4 t u) = (x + 2 i t ∂x)u which has the remarkable property that it commutes with the operator L defined by L = (i ∂t + ∂ x), namely LJ − JL = [L, J ] = 0. For the Korteweg-de Vries type equation (KdV), J. C. Saut and M. Temam [26] remarked that a so- lution u cannot gain or lose regularity. They showed that if u(x, 0) = u0(x) ∈ H s(R) for s ≥ 2, then u( · , t) ∈ Hs(R) for all t > 0. For the KdV equation on the line, Kato [17] motivated by work of Cohen [11] showed that if u(x, 0) = u0(x) ∈ L b ≡ H 2(R) ∩ L2(ebx dx)(b > 0) then the solution u(x, t) of the KdV equation becomes C∞ for all t > 0. A main ingredient in the proof was the fact that formally the semi-group S(t) = e−∂ x in L2b(R) is equivalent to Sb(t) = e − t (∂x−b) in L2(R) when t > 0. One would be inclined to believe that this was a special property of the KdV equation. However, his is not the case. The effect is due to the dispersive nature of the linear part of the equation. Kruzkov and Faminskii [21] proved that u(x, 0) = u0(x) ∈ L 2(R) such that xα u0(x) ∈ L 2((0, +∞)), the weak solution of the KdV equation, has l-continuous space derivatives for all t > 0 if l < 2α. The proof of this result is based on the asymptotic behavior of the Airy function and its derivatives, and on the smoothing effect of the KdV equation which was found in [17, 21]. While the proof of Kato appears to depend on special a priori estimates, some of this mystery has been solved by the result of local gain of finite regularity for various others linear and nonlinear dispersive equations due to Ginibre and Velo [12] and others. However, all of them require growth conditions on the nonlinear term. In 1992, W. Craig, T. Kappeler and W. Strauss [8, 9] proved for the fully nonlinear KdV equation ut+ f(uxxx, uxx, ux, u, x, t) = 0, x ∈ R, t > 0 and certain additional assumption over f that C solutions u(x, t) are obtained for all t > 0 if the initial data u0(x) decays faster than polynomially on R+ = {x ∈ R : x > 0} and has certain initial Sobolev regularity. Following this idea, H. Cai [4] stud- ied the nonlinear equation of KdV-type of the form ut + uxxx + a(x, , t) f(uxx, ux, u, x, t) = 0, where a(x, t) is positive and bounded, obtaining the same conclusion. Subsequent works were given by O. Vera [30, 31, 32, 33] for a nonlinear dispersive evolution equation, a KdV-Burgers type equation and for KdV- Kawahara type equation, respectively. In more than one spatial dimension, J. Levandosky [23], proved infinite gain in regularity results for nonlinear third-order equations. While [8] included local smoothing results for some mth-order dispersive equation in n spatial dimension, their results and the techniques are different from those presented by Levandosky. First, they consider equations with only a mild solution and Levandosky considers equations with very general nonlinearities including a fully nonlinear equation of the form ut + f(D 3u, D2u, Du, u, x, t) = 0, u(x, y, 0) = u0(x, y). Secondly, they indicate local gain in finite regularity and Levandosky proved complementary results showing the relationship between the decay at infinity of the initial data and the amount of gain in regularity. More specifically, it is proved a condition under which an equation of the form ut + a uxxx + b uxxy + c uxyy + d uyyy + f(D 2u, Du, u, x, t) = 0, u(x, y, 0) = u0(x, y), where a, b, c, d are assumed constant. Indeed, Levandosky proved sufficient conditions on this equation for which a solution u will experience an infinite gain in regularity. Specifically, prove conditions for which initial data u0(x, y) possessing sufficient decay at infinity and a minimal amount of regularity will lead to a unique solution u(t) ∈ C∞(R2) for T ∗ where T ∗ is the existence time of solutions. According to the characteristics of equations (P ) and considering the particular cases (Q1) and (Q2) we could hope that the (P ) equation have gain in regularity following the steps of N. Hayashi et al. [13] or W. Craig et al. [8]. In our problem, the initial idea is to apply the technique given by N. Hayashi et al. [13, 14] to obtain gain in regularity. Firstly, using straightforward calculus we can see that the equation (P ) has conservation of the energy, i. e., ||u||L2(R) = ||u0||L2(R). On the other hand, we look for estimates for ux that will help to obtain a priori estimates, basically to obtain estimates in L∞(R). Indeed, differentiating in the x-variable the equation (P ) we have i ux t + i β uxxxx + ω uxxx + (|u| 2)x u+ |u| 2 ux = 0, (1.1) and multiplying (1.1) by ux i ux ux t + i β ux uxxxx + ω ux uxxx + (|u| 2)x u ux + |u| 2 |ux| 2 = 0 − i ux ux t − i β ux uxxxx + ω ux uxxx + (|u| 2)x uux + |u| 2 |ux| 2 = 0. (applying conjugate) Subtracting and integrating over x ∈ R, we have 2dx+ i β ux uxxxxdx+ i β ux uxxxxdx + 2 i ω Im ux uxxxdx+ 2 i Im (|u|2)x u uxdx = 0. Performing integration by parts and straightforward calculations we obtain 2dx+ 2 Im (|u|2)x u uxdx = 0 (E1) where ||ux|| L2(R) + 2 Im u2 u2xdx = 0 (E2) or integrating by parts the second term in (E1) we obtain ||ux|| L2(R) − 2 Im |u|2 u uxxdx = 0. (E3) Thus it is not possible to estimate in H1(R), because it appears a second term with two derivatives. The reason of having an estimate in the derivative is related to Sobolev embedding. In one spatial dimension we have the embedding H1(R) →֒ L∞(R). It seems that the term i β uxxx is crucial. It makes the two ”top” terms look like KdV equation; that is, ut + uxxx + . . . . Of course, the solution is complex, so that the equation is like two coupled real KdV equations. This was our motivation to obtain gain in regularity using the idea of W. Craig et al. [8]. We prove conditions on (P ) for which initial data u0(x) possessing sufficient decay at infinity and a minimal amount of regularity will lead to a unique solution u(t) ∈ C∞(R) for t > 0. We use a technique of nonlinear multipliers, generalizing Kato’s original method, together with ideas of Craig and Goodman [7] All the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property. To mention only a few, the KdV, Benjamin-Ono, intermediate long wave, various Boussinesq, and Schrödinger equation are included. This paper is organized as follows: Section 2 outlines briefly the notation and terminology to be used subsequently. In section 3 we prove the main inequality. In section 4 we prove an important a priori estimate. In section 5 we prove a basic-local-in-time existence and uniqueness theorem. In section 6 we prove a basic global existence theorem. In section 7 we develop a series of estimates for solutions of equations (P ) in weighted Sobolev norms. These provide a starting point for the a priori gain of regularity. In section 8 we prove the following theorem: Theorem 1.1(Main Theorem). Let |ω| < 3 β, T > 0 and u(x, t) be a solution of (P ) in the region R× [0, T ] such that u ∈ L∞([0, T ] : H3(W0 L 0)) (1.2) for some L ≥ 2. Then u ∈ L∞([0, T ] : H3+l(Wσ, L−l, l)) ∩ L 2([0, T ] : H4+l(Wσ, L−l−1, l)) (1.3) for all 0 ≤ l ≤ L− 1 and all σ > 0. Remark. We consider the Gauge transformation u(x, t) = ei d2 x+i d3 t v (x− d1 t, t) ≡ e θ v (η, ξ) (1.4) where θ = i d2 x+ i d3 t, η = x− d1 t and ξ = t. Then ut = i d3 e θ v − d1 e θ vη + e θ vξ : ux = i d2 e θ v + eθ vη uxx = − d θ v + 2 i d2 e θ vη + e θ vη η : uxxx = − i d θ v − 3 d22 e θ vη + 3 i d2 e θ vηη + e θ vηηη. Replacing in (Q) we have − d3 e θ v − i d1 e θ vη + i e θ vξ − ω d θ v + 2 i ω d2 e θ vη + ω e θ vηη β d33 e θ v − 3 i β d22 e θ vη − 3 β d2 e θ vηη + i β e θ vηηη + γ |v| 2 eθ v − δ d2 |v| 2 eθ v + i δ |v|2 eθ vη + ǫ d2 e θ v2v + i ǫ eθ v2 vη = 0 where i vξ + (ω − 3 β d2) vηη + i β vηηη + (2 i ω d2 − 3 i β d 2 − i d1 + i δ |v| 2 + i ǫ v2) vη (β d32 − ω d 2 − d3 + γ |v| 2 − δ d2 |v| 2) v + ǫ d2 v 2v = 0 : d2 = : d3 = − 2ω3 27 β2 . (1.5) This way in (Q) we obtain i vξ + i β vηηη + i (δ |v| 2 + ǫ v2) vη + |v|2v + v2v = 0, but v2 v = v v v = |v|2v, then using the Gauge transformation we have the equivalent problem to (Q) i vξ + i β vηηη + i δ |v| 2 vη + i ǫ v 2 vη + γ + ǫ δ − ω δ |v|2v = 0 η, ξ ∈ R v(η, 0) = e− i ηu0(η). Here, rescaling the equation, we take β = 1. i vt + i vxxx + i δ |v| 2 vx + i ǫ v 2 vx + γ + ǫ δ − ω δ |v|2v = 0 x, t ∈ R v(x, 0) = e− i xu0(x). The above Gauge transformation is a bicontinuous map from Lp([0, T ] : Hs(Wσ i k)) to itself, as far as 0 < T < +∞ and p, s, σ, i, k used in this paper. With this, the assumption |ω| < 3 β imposed in Theorem 1.1 can be removed. 2 Preliminaries We consider the initial value problem i ut + ω uxx + i β uxxx + |u| 2 u = 0, x, t ∈ R u(x, 0) = u0(x) where ω, β ∈ R, β 6= 0 and u = u(x, t) is a complex valued function. Notation. We write ∂ = ∂/∂x, ∂t = ∂/∂t and we abbreviate uj = ∂ Definition 2.1. A function ξ = ξ(x, t) belongs to the weight class Wσ i k if it is a positive C ∞ func- tion on R× [0, T ], ∂ξ > 0 and there are constant cj , 0 ≤ j ≤ 5 such that 0 < c1 ≤ t − k e−σ x ξ(x, t) ≤ c2 ∀ x < −1, 0 < t < T. (2.1) 0 < c3 ≤ t − k x− i ξ(x, t) ≤ c4 ∀ x > 1, 0 < t < T. (2.2)( t | ∂tξ | + | ∂ /ξ ≤ c5 ∀ (x, t) ∈ R× [0, T ], ∀ j ∈ N. (2.3) Remark. We shall always take σ ≥ 0, i ≥ 1 and k ≥ 0. Example. Let ξ(x) = 1 + e−1/x for x > 0 1 for x ≤ 0 then ξ ∈ W0 i 0. Definition 2.2. Let N be a positive integer. By HN (Wσ i k) we denote the Sobolev space on R with a weight; that is, with the norm ||v||2HN (Wσ i k) = |∂jv(x)|2 ξ(x, t) dx < +∞ for any ξ ∈ Wσ i k and 0 < t < T., Even though the norm depends on ξ, all such choices leads to equivalent norms. Remark. Hs(Wσ i k) depends on t (because ξ = ξ(x, t)). Lemma 2.1. (See [4]) For ξ ∈ Wσ i 0 and σ ≥ 0, i ≥ 0, there exists a constant c > 0 such that, for u ∈ H1(Wσ i 0), ||ξ u2|| ≤ c |u|2 + |∂u|2 Lemma 2.2(The Gagliardo-Nirenberg inequality). Let q, r be any real numbers satisfying 1 ≤ q, r ≤ ∞ and let j and m be nonnegative integers such that j ≤ m. Then ||∂ju||Lp(R) ≤ c ||∂ mu||aLr(R) ||u|| Lq(R) where 1 = j + a (1−a) for all a in the interval j ≤ a ≤ 1, and M is a positive constant depending only on m, j, q, r and a. Definition 2.3. By L2([0, T ] : HN (Wσ i k)) we denote the space of functions v(x, t) with the norm (N integer positive) ||v||2L2([0, T ]:HN (Wσ i k)) = ||v(x, t)||2HN (Wσ i k)dt < +∞ Remark. The usual Sobolev space is HN (R) = HN (W0 0 0) without a weight. Remark. We shall derive the a priori estimates assuming that the solution is C∞, bounded as x → −∞, and rapidly decreasing as x → +∞, together with all of its derivatives. Considering the above notation, the higher order nonlinear Schrödinger equation can be written as i ut + i β u3 + ω u2 + |u| 2 u = 0, x, t ∈ R (2.4) where ω, β ∈ R, β 6= 0 and u = u(x, t) is a complex valued function. Throughout this paper c is a generic constant, not necessarily the same at each occasion(it will change from line to line), which depends in an increasing way on the indicated quantities. In this part, we only consider the case t > 0. The case t < 0 can be treated analogously. 3 Main Inequality Lemma 3.1. Let |ω| < 3 β. Let u be a solution of (2.4) with enough Sobolev regularity (for instance, u ∈ HN(R), N ≥ α+ 3), then ξ |uα| η |uα+1| θ |uα| Rαdx ≤ 0 (3.1) where η = (3 β − |ω|) ∂ξ for |ω| < 3 β θ = − [ ∂tξ + β ∂ 3ξ + |ω| ∂ξ + c0 ξ ] where c0 = ||u|| L∞(R) and Rα = Rα(|uα|, |uα−1|, . . .). Proof. Differentiating (2.4) α-times (for α ≥ 0) over x ∈ R leads to i uα t + i β uα+3 + ω uα+2 + (|u| 2)α u+ (|u|2)α−m um + |u| 2 uα = 0. (3.2) Let ξ = ξ(x, t), then multiplying (3.2) by ξ uα we have i ξ uα uα t + i β ξ uα uα+3 + ω ξ uα uα+2 + (|u| 2)α ξ u uα (|u|2)α−m ξ um uα + ξ |u| 2 |uα| 2 = 0 − i ξ uα uα t − i β ξ uα uα+3 + ω ξ uα uα+2 + (|u| 2)α ξ u uα (|u|2)α−m ξ um uα + ξ |u| 2 |uα| 2 = 0. (applying conjugate) Subtracting and integrating over x ∈ R we have ξ |uα| 2dx+ i β ξ uα uα+3dx+ i β ξ uα uα+3dx− i ξt |uα| ξ uα uα+2dx − ω ξ uα uα+2dx+ 2 i Im ξ (|u|2)α u uαdx + 2 i ξ (|u|2)α−m um uαdx = 0. (3.3) We estimate the second term integrating by parts ξ uα uα+3dx = ∂2ξ uα uα+1dx+ 2 ∂ξ |uα+1| ξ uα+2 uα+1dx. The other terms are calculated in a similar way. Hence, replacing in (3.3) and performing straightforward calculations we obtain ξ |uα| 2dx+ i β ∂2ξ uα uα+1dx+ 2 i β ∂ξ |uα+1| + i β ξ uα+2 uα+1dx+ i β ∂2ξ uα uα+1dx+ i β ∂ξ |uα+1| − i β ξ uα+1 uα+2dx− ω ∂ξ uα uα+1dx− ω ξ |uα+1| ∂ξ uα uα+1dx+ ω ξ |uα+1| 2dx− i ∂tξ |uα| + 2 i Im ξ (|u|2)α u uαdx+ 2 i ξ (|u|2)α−m um uαdx = 0 ξ |uα| 2dx− β ∂3ξ |uα| 2dx+ 3 β ∂ξ |uα+1| 2dx − 2ω Im ∂ξ uα uα+1dx ∂tξ |uα| 2dx+ 2 Im ξ (|u|2)α u uαdx+ 2 ξ (|u|2)α−m um uαdx = 0 hence ξ |uα| 2dx− β ∂3ξ |uα| 2dx+ 3 β ∂ξ |uα+1| 2dx+ 2 Im (|u|2)α ξ u uαdx ∂tξ |uα| 2dx+ 2 ξ (|u|2)α−m um uαdx = 2ω Im ∂ξ uα uα+1dx ≤ |ω| ∂ξ |uα| 2dx+ |ω| ∂ξ |uα+1| therefore ξ |uα| [ 3 β − |ω| ] ∂ξ |uα+1| [ ∂tξ + β ∂ 3ξ + |ω| ∂ξ ] |uα| + 2 Im (|u|2)α ξ u uαdx+ 2 ξ (|u|2)α−m um uαdx ≤ 0. (3.4) (|u|2)α = (u u )α = uα−k uk = uuα + uα−k uk + u uα (|u|2)α u uα = |u| 2|uα| uα−k uk u uα + u 2 u2α thus, ( |u|2)α ξ u uαdx = 2 ξ uα−k uk u uαdx+ 2 Im ξ u2 u2αdx ξ |uα−k| |uk| |u| |uα|dx+ 2 ξ |u|2 |uα| ξ |uα−k| |uk| |u| |uα|dx+ 2 ||u|| L∞(R) ξ |uα| ≤ 2 ||u||L∞(R) ξ |uα−k| |uk| |uα|dx+ 2 ||u|| L∞(R) ξ |uα| 2dx (3.5) hence, in (3.4) we have ξ |uα| [3 β − |ω| ] ∂ξ |uα+1| [∂tξ + β ∂ 3ξ + |ω| ∂ξ + c0 ξ ] |uα| − 2 c ξ |uα−k| |uk| |uα|dx− 2 ξ |(|u|2)α−m| |um| |uα|dx ≤ 0. Therefore, using straightforward calculations we obtain the main inequality ξ |uα| η |uα+1| θ |uα| Rαdx ≤ 0 where η = (3 β − |ω| ) ∂ξ for |ω| < 3 β θ = − [ ∂tξ + β ∂ 3ξ + |ω| ∂ξ + c0 ξ ] where c0 = ||u|| L∞(R) and Rα = Rα(|uα|, |uα−1|, . . .). Remark. In (3.4) using Young’s estimate and assuming that β > 0 we have 2ω Im uα uα+1 dx ≤ 2 dx+ 2 β |uα+1| 2 dx. Then, in (3.4) we obtain ξ |uα| 2dx− β ∂3ξ |uα| 2dx+ β ∂ξ |uα+1| 2dx + 2 Im (|u|2)α ξ u uαdx ∂tξ |uα| 2dx+ 2 ξ (|u|2)α−m um uαdx = 2ω Im ∂ξ uα uα+1dx and the assumption that |ω| < 3 β can be removed. Lemma 3.2. For η ∈ Wσ i k an arbitrary weight function and |ω| < 3 β, there exists ξ ∈ Wσ, i+1, k that satisfies η = (3 β − |ω|) ∂ξ for |ω| < 3 β. (3.6) Indeed, we have (3 β − |ω|) η(y, t) dy. (3.7) Lemma 3.3. The expression Rα in the inequality of Lemma 3.1 is a sum of terms of the form ξ uν1 uν2 uα (3.8) where 1 ≤ ν1 ≤ ν2 ≤ α and ν1 + ν2 = α (3.9) Proof. It follows from (3.5). 4 An a priori estimate We show now a fundamental a priori estimate used for a basic local-in-time existence theorem. We con- struct a mapping Z : L∞([0, T ] : Hs(R)) 7−→ L∞([0, T ] : Hs(R)) with the property: Given u(n) = Z(u(n−1)) and essupt∈[0, T ]||u (n−1)||s ≤ c0 then essupt∈[0, T ]||u (n)||s ≤ c0, where s and c0 > 0 are constants. This property tells us that Z : Bc0(0) 7−→ Bc0(0) where Bc0(0) = {v(x, t) : ||v( · , t)||s ≤ c0} is a ball in L ∞([0, T ] : Hs(R)). To guarantee this property, we will appeal to an a priori estimate which is the main object of this section. Differentiating (2.4) two times leads to i ∂tu2 + i β u5 + ω u4 + (|u| 2)2 u+ 2 (|u| 2)1 u1 + |u| 2 u2 = 0. (4.1) Let u = ∧v where ∧ = (I − ∂2)−1. Hence u = (I − ∂2)−1v then u− u2 = v where ∂tu2 = − vt + ut. Replacing in (4.1) we have − i vt + i β ∧ v5 + ω ∧ v4 + (| ∧ v| 2)2 ∧ v + 2 (| ∧ v| 2)1 ∧ v1 + | ∧ v|2 ∧ v2 − (i β ∧ v3 + ω ∧ v2 + | ∧ v| 2 ∧ v) = 0. (4.2) The (4.2) equation is linearized by substituting a new variable z in each coefficient: − i vt + i β ∧ v5 + ω ∧ v4 + (| ∧ z| 2)2 ∧ v + 2 (| ∧ z| 2)1 ∧ v1 + | ∧ z|2 ∧ v2 − (i β ∧ v3 + ω ∧ v2 + | ∧ z| 2 ∧ v) = 0. (4.3) The linear equation which is to be solved at each iteration is of the form i ∂tv = i β ∧ v 5 + ω ∧ v 4 − i β ∧ v 3 − ω ∧ v 2 + b (1) (4.4) where b(1) = (| ∧ z|2)2 ∧ v+2 (| ∧ z| 2)1 ∧ v1 + | ∧ z| 2 ∧ v2 − |∧ z| 2 ∧ v. Equation (4.4) is a linear equation at each iteration which can be solved in any interval of time in which the coefficient is defined. We consider the following lemma that will help us setting up the iteration scheme. Lemma 4.1. Let |ω| < 3 β. Given initial data u0(x) ∈ H ∞(R) = N≥0 H N (R) there exists a unique solution of (4.4) where b(1) is a smooth bounded coefficient with z ∈ H∞(R). The solution is defined in any time interval in which the coefficient is defined. Proof. Let T > 0 be arbitrary and M > 0 a constant. Let Γ = ξ ( i ∂t − i β ∧ ∂ 5 − ω ∧ ∂4 + i β ∧ ∂3 + ω ∧ ∂2 ) then in (4.4) we have Γu = ξ b(1). We consider the bilinear form B : D ×D 7−→ R, B(u, v) =< u, v >= Im e−Mt u v dx dt where D = {u ∈ C∞0 (R× [0, T ]) : u(x, 0) = 0 }. We have Γu · u = i ξ u ut − i β ξ u ∧ u5 − ω ξ u ∧ u4 + i β ξ u ∧ u3 + ω ξ u ∧ u2 Γu · u = − i ξ u ut + i β ξ u ∧ u5 − ω ξ u ∧ u4 − i β ξ u ∧ u3 + ω ξ u ∧ u2. (applying conjugate) Subtracting and integrating over x ∈ R we have 2 i Im Γu · udx = i ∂t ξ |u|2dx− i ∂tξ |u| 2dx− i β ξ u ∧ u5dx− i β ξ u ∧ u5dx ξ u ∧ u4dx+ ω ξ u ∧ u4dx+ i β ξ u ∧ u3dx+ i β ξ u ∧ u3dx ξ u ∧ u2dx− ω ξ u ∧ u2dx. Each term is treated separately, integrating by parts ξ u ∧ u5dx = ξ ∧ (I − ∂2)u ∧ u5dx = ξ ∧ u ∧ u5dx− ξ ∧ u2 ∧ u5dx ∂4ξ ∧ u ∧ u1dx+ ∂3ξ | ∧ u1| 2dx− 3 ∂2ξ ∧ u1 ∧ u2dx− 2 ∂ξ | ∧ u2| ξ ∧ u2 ∧ u3dx− ∂2ξ ∧ u2 ∧ u3dx− ∂ξ | ∧ u3| ξ ∧ u3 ∧ u4dx. The other terms are calculates in a similar way. Then 2 i Im Γu · udx = i ∂t ξ |u|2dx− i ∂tξ |u| 2dx− i β ∂4ξ ∧ u ∧ u1dx− i β ∂3ξ | ∧ u1| + 3 i β ∂2ξ ∧ u1 ∧ u2dx+ 2 i β ∂ξ | ∧ u2| 2dx− i β ξ ∧ u2 ∧ u3dx + i β ∂2ξ ∧ u2 ∧ u3dx+ i β ∂ξ | ∧ u3| 2dx− i β ξ ∧ u3 ∧ u4dx − i β ∂4ξ ∧ u ∧ u1dx − i β ∂3ξ | ∧ u1| 2dx+ 3 i β ∂2ξ ∧ u1 ∧ u2dx + 2 i β ∂ξ | ∧ u2| 2dx− i β ξ ∧ u2 ∧ u3dx+ i β ∂2ξ ∧ u2 ∧ u3dx + 2 i β ∂ξ | ∧ u3| 2dx+ i β ξ ∧ u3 ∧ u4dx+ ω ∂3ξ ∧ u ∧ u1dx ∂2ξ | ∧ u1| 2dx− 2ω ∂ξ ∧ u1 ∧ u2dx− ω ξ | ∧ u2| ∂ξ ∧ u2 ∧ u3dx− ω ξ | ∧ u3| 2dx − ω ∂3ξ ∧ u ∧ u1dx ∂2ξ | ∧ u1| 2dx+ 2ω ∂ξ ∧ u1 ∧ u2dx+ ω ξ | ∧ u2| ∂ξ ∧ u2 ∧ u3dx+ ω ξ | ∧ u3| 2dx + i β ∂2ξ ∧ u ∧ u1dx + i β ∂ξ | ∧ u1| 2dx− i β ξ ∧ u1 ∧ u2dx− i β ξ ∧ u2 ∧ u3dx + i β ∂2ξ ∧ u ∧ u1dx + i β ∂ξ | ∧ u1| 2dx− i β ξ ∧ u1 ∧ u2dx − i β ξ ∧ u2 ∧ u3dx− ω ∂ξ ∧ u ∧ u1dx− ω ξ | ∧ u1| 2dx− ω ξ | ∧ u2| ∂ξ ∧ u ∧ u1dx+ ω ξ | ∧ u1| 2dx+ ω ξ | ∧ u2| hence 2 i Im Γu · udx = i ∂t ξ |u|2dx− i ∂tξ |u| 2dx− i β ∂4ξ (| ∧ u|2)1dx − 2 i β ∂3ξ | ∧ u1| 2dx+ 3 i β ∂2ξ (| ∧ u1| 2)1dx + 4 i β ∂ξ | ∧ u2| − i β ξ (| ∧ u2| 2)1dx+ i β ∂2ξ (| ∧ u2| 2)1dx+ 3 i β ∂ξ | ∧ u3| +2 i ω Im ∂3ξ ∧ u ∧ u1dx − 4 i ω Im ∂ξ ∧ u1 ∧ u2dx − 2 i ω Im ∂ξ ∧ u2 ∧ u3dx+ i β ∂2ξ (| ∧ u|2)1dx+ 2 i β ∂ξ | ∧ u1| − i β ξ (| ∧ u1| 2)1dx− i β ξ (| ∧ u2| 2)1dx− 2ω Im ∂ξ ∧ u ∧ u1dx then, adding similar terms and cutting the letter i we obtain Γu · u dx = ∂t ξ |u|2dx− ∂tξ |u| 2dx + β ∂5ξ | ∧ u|2dx− 5 β ∂3ξ | ∧ u1| + 6 β ∂ξ | ∧ u2| 2dx− β ∂3ξ | ∧ u2| 2dx+ 3 β ∂ξ | ∧ u3| + 2ω Im ∂3ξ ∧ u ∧ u1dx− 4ω Im ∂ξ ∧ u1 ∧ u2dx− 2ω Im ∂ξ ∧ u2 ∧ u3dx ∂3ξ | ∧ u|2dx+ 3 β ∂ξ | ∧ u1| 2dx − 2ω Im ∂ξ ∧ u ∧ u1dx ∂ξ | ∧ u3| 2dx+ |ω| ∂ξ | ∧ u2| 2dx+ 2 |ω| ∂ξ | ∧ u1| 2dx+ 2 |ω| ∂ξ | ∧ u2| + |ω| ∂ξ | ∧ u|2dx+ |ω| ∂ξ | ∧ u1| 2dx+ |ω| |∂3ξ| | ∧ u|2dx + |ω| |∂3ξ| | ∧ u1| 2dx + ∂tξ |u| 2dx+ 2 Im Γu · udx ξ |u|2dx+ 3 β ∂ξ | ∧ u3| 2dx− β ∂3ξ | ∧ u2| 2dx+ 6 β ∂ξ | ∧ u2| − 5 β ∂3ξ | ∧ u1| 2dx+ 3 β ∂ξ | ∧ u1| 2dx+ β ∂5ξ | ∧ u|2dx − β ∂3ξ | ∧ u|2dx where 3 |ω| ∂ξ | ∧ u2| 2dx+ |ω| [|∂3ξ|+ 3 ∂ξ] | ∧ u1| + |ω| [|∂3ξ|+ ∂ξ + ∂tξ] | ∧ u| 2dx+ 2 Im Γu · udx ξ |u|2dx + [3 β − |ω|] ∂ξ | ∧ u3| 2dx− β ∂3ξ | ∧ u2| + 6 β ∂ξ | ∧ u2| 2dx− 5 β ∂3ξ | ∧ u1| 2dx+ 3 β ∂ξ | ∧ u1| ∂5ξ | ∧ u|2dx− β ∂3ξ | ∧ u|2dx ξ |u|2dx + β [−∂3ξ + 5∂ξ] | ∧ u2| [−5 ∂3ξ + 3∂ξ] | ∧ u1| 2dx+ β [∂3ξ − ∂3ξ] | ∧ u|2dx using (2.3), ∧un = (I − (I − ∂ 2))∧ un−2 = ∧un−2 − un−2 for n a positive integer and standard estimates we obtain Γu · u dx ≥ ∂t ξ |u|2 dx− c ξ |u|2 dx. Multiply this equation by e−Mt, and integrate with respect to t for t ∈ [0, T ] and u ∈ D e−Mt Γu · udx dt ≥ ξ |u|2dx dt− c ξ e−Mt |u|2dx dt = e−Mt ξ |u|2dx ξ e−Mt |u|2dx dt− c ξ e−Mt |u|2dx dt = e−Mt ξ(x, T ) |u(x, T )|2dx+M ξ e−Mt |u|2dx dt − c ξ e−Mt |u|2dx dt. < Γu, u >= Im e−Mt Γu · udx dt ≥ e−Mt ξ(x, T ) |u(x, T )|2dx+ (M − c) ξ e−Mt |u|2dx dt ξ e−Mt |u|2dx dt provided that M is chosen large enough. Then < Γu, u >≥< u, u >, for all u ∈ D. Let Γ∗ be the formal adjoint of Γ defined by Γ∗ = ξ(−i ∂t − i β ∧ ∂ 5 − ω ∧ ∂4 + i β ∧ ∂3 + ω ∧ ∂2). Let D∗ = {w ∈ C∞0 (R× [0, T ]) : w(x, T ) = 0 }. In a similar way we prove that < Γ∗w, w > ≥ < w, w >, ∀ w ∈ D∗. ¿From this equation, we have that Γ∗ is one-one. Therefore, < Γ∗w, Γ∗v > is an inner product on D∗. We denote by X the completion of D∗ with respect to this inner product. By Riesz’s Representation Theo- rem, there exists a unique solution V ∈ X, such that for any w ∈ D∗, < ξb(1), w >=< Γ∗V, Γ∗w > where we use that ξ b(1) ∈ X. Then if v = Γ∗V we have < v, Γ∗w >=< ξb(1), w > or < Γ∗w, v >=< w, ξb(1) > . Hence, v = Γ∗V is a weak solution of Γv = ξb(1) with v ∈ L2(R× [0, T ]) ≃ L2([0, T ] : L2(R)). Remark. To obtain higher regularity of the solution, we repeat the proof with higher derivatives. It is a standard approximation procedure to obtain a result for general initial data. The next step is to estimate the corresponding solutions v = v(x, t) of the equation (4.3) via the coeffi- cients of that equation. The following estimate is related to the existence of solutions theorem. Lemma 4.2. Let |ω| < 3 β and 0 < γ1 ≤ ξ ≤ γ2, with γ2, γ2 real constants. Let v, z ∈ C k([0, +∞) : HN(R)) for all k, N which satisfy (4.3). For each integer α there exist positive nondecreasing functions G and F such that for all t ≥ 0 ξ |vα| 2dx ≤ G(||z||λ) ||v|| α + F (||z||α) (4.5) where || · ||α is the norm in H α(R) and λ = max{1, α}. Proof. Differentiating α-times the equation (4.3), for some α ≥ 0 we have − i ∂tvα + i β ∧ vα+5 + ω ∧ vα+4 − i β ∧ vα+3 + h(j) ∧ vj + (|z| 2)α+2 ∧ v + p(∧zα+1, . . .) = 0 (4.6) where h(j) is a smooth function depending on | ∧ z|2, . . . with i = 2 + α − j. For α ≥ 2, p(∧zα+1, . . .) depends at most linearly on ∧zα+1, while for α = 2, p(∧zα+1, . . .) depends at most quadratically on ∧zα+1. We multiply equation (4.6) by ξ vα and integrate over x ∈ R ξ vα ∂tvαdx+ i β ξ vα ∧ vα+5dx+ ω ξ vα ∧ vα+4dx− i β ξ vα ∧ vα+3dx ξ vα ∧ vjdx+ ξ (|z|2)α+2vα ∧ vdx+ ξ vαp(∧zα+1, . . .)dx = 0 and applying conjugate ξ vα ∂tvαdx− i β ξ vα ∧ vα+5dx+ ω ξ vα ∧ vα+4dx+ i β ξ vα ∧ vα+3dx ξ vα ∧ vjdx+ ξ (|z|2)α+2vα ∧ vdx+ ξ vαp(∧zα+1, . . .)dx = 0. Subtracting, it follows that − i ∂t ξ |vα| 2dx+ i ∂tξ |vα| 2dx+ i β ξ vα ∧ vα+5dx+ i β ξ vα ∧ vα+5dx ξ vα ∧ vα+4dx− ω ξ vα ∧ vα+4dx− i β ξ vα ∧ vα+3dx− i β ξ vα ∧ vα+3dx ξ vα ∧ vjdx− ξ vα ∧ vjdx+ ξ (|z|2)α+2vα ∧ vdx (4.7) ξ (|z|2)α+2vα ∧ v dx+ ξ vαp(∧zα+1, . . .) dx− ξ vα p(∧zα+1, . . .)dx = 0. Each term is treated separately, integrating by parts ξ vα ∧ vα+5dx = ξ ∧ (I − ∂2)vα ∧ vα+5dx ξ ∧ vα ∧ vα+5dx− ξ ∧ vα+2 ∧ vα+5dx ∂4ξ ∧ vα ∧ vα+1dx+ ∂3ξ | ∧ vα+1| 2dx− 3 ∂2ξ ∧ vα+1 ∧ vα+2dx ∂ξ | ∧ vα+2| ξ ∧ vα+2 ∧ vα+3dx− ∂2ξ ∧ vα+2 ∧ vα+3dx ∂ξ | ∧ vα+3| ξ ∧ vα+4 ∧ vα+3dx. The other terms are calculated in a similar way. Hence in (4.7) we have performing straightforward calculations as above ξ |vα| ∂tξ |vα| 2dx − β ∂5ξ | ∧ vα| 2dx+ 2 β ∂3ξ | ∧ vα+1| ∂3ξ | ∧ vα+1| 2dx− 4 β ∂ξ | ∧ vα+2| 2dx− β ∂ξ | ∧ vα+2| ∂2ξ | ∧ vα+2| 2dx− 3 β ∂ξ | ∧ vα+3| 2 dx− 2ω Im ∂3ξ ∧ vα ∧ vα+1dx + 4ω Im ∂ξ ∧ vα+1 ∧ vα+2dx+ 2ω Im ∂ξ ∧ vα+2 ∧ vα+3dx + 2 β Im ∂ξ ∧ vα ∧ vα+2dx+ 2 β Im ξ ∧ vα+1 ∧ vα+2dx ∂ξ | ∧ vα+2| 2dx+ 2 h(j) Im ξ vα ∧ vjdx + 2 Im ξ (|z|2)α+2 vα ∧ vdx+ 2 Im ξ vαp(∧zα+1, . . .) dx = 0 ξ |vα| ∂tξ |vα| 2dx − 3 β ∂ξ | ∧ vα+3| 2dx+ β ∂2ξ | ∧ vα+2| − 6 β ∂ξ | ∧ vα+2| 2dx+ 5 β ∂3ξ | ∧ vα+1| 2dx− β ∂5ξ | ∧ vα| = − 2ω Im ∂ξ ∧ vα+2 ∧ vα+3dx− 4ω Im ∂ξ ∧ vα+1 ∧ vα+2dx − 2 β Im ξ ∧ vα+1 ∧ vα+2dx− 2 β Im ∂ξ ∧ vα ∧ vα+2dx +2ω Im ∂3ξ ∧ vα ∧ vα+1dx− 2 h(j) Im ξ vα ∧ vjdx − 2 Im ξ (|z|2)α+2vα ∧ vdx− 2 Im ξ vαp(∧zα+1, . . .) dx hence, ξ |vα| ∂tξ |vα| 2dx+ 3 β ∂ξ | ∧ vα+3| 2dx− β ∂2ξ | ∧ vα+2| ∂ξ | ∧ vα+2| 2dx− 5 β ∂3ξ | ∧ vα+1| 2dx+ β ∂5ξ | ∧ vα| = 2ω Im ∂ξ ∧ vα+2 ∧ vα+3dx+ 4ω Im ∂ξ ∧ vα+1 ∧ vα+2dx +2 β Im ξ ∧ vα+1 ∧ vα+2dx + 2 β Im ∂ξ ∧ vα ∧ vα+2dx − 2ω Im ∂3ξ ∧ vα ∧ vα+1dx+ 2 h(j) Im ξ vα ∧ vjdx + 2 Im ξ (|z|2)α+2vα ∧ vdx+ 2 Im ξ vαp(∧zα+1, . . .) dx ≤ |ω| ∂ξ | ∧ vα+2| 2 dx+ |ω| ∂ξ| ∧ vα+3| 2dx+ 2 |ω| ∂ξ | ∧ vα+1| + 2 |ω| ∂ξ| ∧ vα+2| 2dx+ |β| ξ | ∧ vα+1| 2dx+ |β| ξ | ∧ vα+2| + |β| ∂ξ| ∧ vα| 2dx+ |β| ∂ξ| ∧ vα+2| 2dx+ |ω| ∂3ξ| ∧ vα| + |ω| ∂3ξ| ∧ vα+1| 2dx + 2 ∣∣∣∣∣∣ ξ vα ∧ vjdx ∣∣∣∣∣∣ ξ (|z|2)α+2 vα ∧ vdx ξ vαp(∧zα+1, . . .) dx where ξ |vα| (3 β − |ω|)∂ξ | ∧ vα+3| 2dx + [β ∂2ξ − 6 β ∂ξ + 3 |ω| ∂ξ + |β| ∂ξ + |β| ξ] | ∧ vα+2| [5β∂3ξ + |ω|∂3ξ + 2 |ω|∂ξ + |β| ξ] | ∧ vα+1| [∂tξ + β ∂ 5ξ + |ω| ∂3ξ + |β| ∂ξ] | ∧ vα| ∣∣∣∣∣∣ ξ vα ∧ vjdx ∣∣∣∣∣∣ ξ (|z|2)α+2 vα ∧ vdx ∣∣∣∣+ 2 ξ vαp(∧zα+1, . . .) dx ∣∣∣∣ . using that |ω| < 3 β we have that the first term in the right hand side of the above expression is not positive. Hence, ξ |vα| [β ∂2ξ − 6 β ∂ξ + 3 |ω| ∂ξ + |β| ∂ξ + |β| ξ] | ∧ vα+2| [5 β ∂3ξ + |ω| ∂3ξ + 2 |ω| ∂ξ + |β| ξ] | ∧ vα+1| 2dx + [∂tξ + β ∂ 5ξ + |ω| ∂3ξ + |β| ∂ξ] | ∧ vα| ∣∣∣∣∣∣ ξ vα ∧ vjdx ∣∣∣∣∣∣ ξ (|z|2)α+2 vα ∧ v dx ∣∣∣∣+ 2 ξ vαp(∧zα+1, . . .) dx ∣∣∣∣ . Using that ∧vn = ∧vn−2 − vn−2 and a standard estimate, the lemma follows. 5 Uniqueness and Existence of a Local Solution In this section, we study the uniqueness and the existence of local strong solutions in the Sobolev space HN(R) for N ≥ 3 for the problem (2.4). To establish the existence of strong solutions for (2.4) we use the a priori estimate together with an approximation procedure. Theorem 5.1(Uniqueness). Let |ω| < 3 β, u0(x) ∈ H N (R) with N ≥ 3 and 0 < T < +∞. Then there is at most one strong solution u ∈ L∞([0, T ] : HN(R)) of (2.4) with initial data u(x, 0) = u0(x). Proof. Assume that u, v ∈ L∞([0, T ] : HN(R)) are two solutions of (2.4) with ut, vt ∈ L ∞([0, T ] : HN−3(R)), and with the same initial data. Then i (u− v)t + i β (u− v)3 + ω (u− v)2 + |u| 2 u− |v|2 v = 0 (5.1) with (u− v)(x, 0) = 0. By (5.1) i (u− v)t + i β (u − v)3 + ω (u − v)2 + |u| 2 (u− v) + (|u|2 − |v|2) v = 0 i (u− v)t + i β (u− v)3 + ω (u− v)2 + |u| 2 (u− v) + (|u| − |v|) (|u|+ |v|) v = 0. (5.2) Multiplying (5.2) by ξ(u − v) we have i ξ (u− v) (u − v)t + i β ξ (u − v) (u− v)3 + α ξ (u− v) (u − v)2 + |u|2 |u− v|2 + ξ (u− v) (|u| − |v|) (|u|+ |v|) v = 0. − i ξ (u− v) (u − v)t − i β ξ (u− v) (u − v)3 + α ξ (u − v) (u− v)2 + |u|2 |u− v|2 + ξ (u− v) (|u| − |v|) (|u|+ |v|) v = 0. (applying conjugate) Subtracting and integrating over x ∈ R we obtain ξ |u− v|2dx− i ∂tξ |u− v| 2dx+ i β ξ (u− v) (u− v)3dx + i β ξ (u − v) (u− v)3dx+ ω ξ (u − v) (u− v)2dx ξ (u− v) (u − v)2dx+ 2 i Im ξ (u − v) (|u| − |v|) (|u|+ |v|) v dx = 0 (5.3) Each term is treated separately, integrating by parts ξ (u− v) (u− v)3dx ∂2ξ (u− v) (u− v)1dx+ 2 ∂ξ |(u − v)1| ξ (u− v)1 (u− v)2dx. The other terms are calculated in a similar way. Hence in (5.3) we have ξ |u− v|2dx− i ∂tξ |u− v| 2dx+ i β ∂2ξ (u− v) (u − v)1dx +2 i β ∂ξ |(u − v)1| 2dx+ i β ξ (u − v)1 (u − v)2dx + i β ∂2ξ (u− v) (u − v)1dx + i β ∂ξ |(u− v)1| 2dx− i β ξ (u− v)1 (u− v)2dx− ω ∂ξ (u − v) (u− v)1dx ξ |(u− v)1| 2dx+ ω ∂ξ (u− v) (u − v)1dx+ ω ξ |(u − v)1| +2 i Im ξ (u− v) (|u| − |v|) (|u|+ |v|) v dx = 0 ξ |u− v|2dx− i ∂tξ |u− v| 2dx + i β ∂2ξ (|u − v|2)1dx+ 3 i β ∂ξ |(u − v)1| − 2 i ω Im ∂ξ (u− v) (u− v)1dx+ 2 i Im ξ (u− v) (|u| − |v|) (|u|+ |v|) v dx = 0 if and only if ξ |u− v|2dx− ∂tξ |u− v| 2 dx+ β ∂2ξ (|u− v|2)1dx + 3 β ∂ξ |(u− v)1| = 2ω Im ∂ξ (u− v) (u− v)1dx− 2 Im ξ (u− v) (|u| − |v|) (|u|+ |v|) v dx ≤ |ω| ∂ξ |u− v|2dx+ |ω| ∂ξ |(u− v)1| 2dx+ 2 ξ |u− v| | |u| − |v| | (|u|+ |v|) |v| dx. Using that | |u| − |v| | ≤ |u− v|, (2.3) and standard estimates, we have ξ |u− v|2dx+ [3 β − |ω| ] ∂ξ |(u− v)1| 2dx ≤ c ξ |u− v|2dx. Integrating in t ∈ [0, T ], using the fact that (u− v) vanishes at t = 0 and Gronwall’s inequality it follows that u = v. This proves the uniqueness of the solution. We construct the mapping Z : L∞([0, T ] : Hs(R)) 7−→ L∞([0, T ] : Hs(R)) where the initial condi- tion is given by u(n)(x, 0) = u0(x) and the first approximation is given by u(0) = u0(x) u(n) = Z(u(n−1)) n ≥ 1, where u(n−1) is in place of z in equation (4.3) and u(n) is in place of v which is the solution of equation (4.3). That is − i u t + i β ∧ u 5 + ω ∧ u 4 + (| ∧ u (n−1)|2)2 ∧ u (n) + 2 (| ∧ u(n−1)|2)1 ∧ u + | ∧ u(n−1)|2 ∧ u 2 − (i β ∧ u 3 + ω ∧ u 2 + | ∧ u (n−1)|2 ∧ u(n)) = 0. By Lemma 4.1, u(n) exists and is unique in C((0, +∞) : HN (R)). A choice of c0 and the use of the a priori estimate in Section 4 shows that Z : Bc0(0) 7−→ Bc0(0) where Bc0(0) is a bounded ball in L∞([0, T ] : Hs(R)). Theorem 5.2(Local solution). Let |ω| < 3 β and N an integer ≥ 3. If u0(x) ∈ H N (R), then there is T > 0 and u such that u is a strong solution of (2.4), u ∈ L∞([0, T ] : HN (R)) and u(x, 0) = u0(x). Proof. We prove that for u0(x) ∈ H ∞(R) = k≥0 H k(R) there exists a solution u ∈ L∞([0, T ] : HN(R)) with initial data u(x, 0) = u0(x) where the time of existence T > 0 only depends on the norm of u0(x). We define a sequence of approximations to equation (4.3) as t = i β ∧ v 5 + ω ∧ v 4 − i β ∧ v 3 − ω ∧ v 2 + | ∧ v (n−1)|2 ∧ v + O[ (| ∧ v(n−1)|2)2, (| ∧ v (n−1)|2)1, . . .) ] (5.4) where the initial condition is v(n)(x, 0) = u0(x)−∂ 2u0(x). The first approximation is given by v (0)(x, 0) = u0(x)− ∂ 2u0(x). Equation (5.4) is a linear equation at each iteration which can be solved in any interval of time in which the coefficients are defined. This is shown in Lemma 4.1. By Lemma 4.2, it follows that ξ |v(n)α | 2dx ≤ G(||v(n−1)||λ) ||v (n)||2α + F (||v (n−1)||α). (5.5) Choose α = 1 and let c ≥ ||u0−∂ 2u0||1 ≥ ||u0||3. For each iterate n, ||v (n)( · , t)|| is continuous in t ∈ [0, T ] and ||v(n)( · , 0)|| < c. Define c0 = c2+1. Let T 0 be the maximum time such that ||v (k)( · , t)||1 ≤ c3 for 0 ≤ t ≤ T 0 , 0 ≤ k ≤ n. Integrating (5.5) over [0, t] we have that for 0 ≤ t ≤ T 0 and j = 0, 1 ||v(n−1)||1 ||v(n)||2jds+ ||v(n−1)||j It follows that ξ(x, t)|v j (x, t)| 2dx ≤ ξ(x, 0)|v j (x, 0)| ||v(n−1)||1 ||v(n)||2jds ||v(n−1)||j hence j (x, t)| 2dx ≤ ξ(x, t)|v j (x, t)| ξ(x, 0)|v j (x, 0)| ||v(n−1)||1 ||v(n)||2jds ||v(n−1)||j 2dx ≤ j (x, 0)| G(c3) c23 t+ F (c3) and we obtain for j = 0, 1 that ||v(n)||1 ≤ G(c0) c20 t+ F (c0) Claim. T 0 does not approach to 0. On the contrary, assume that T 0 → 0. Since ||v (n)( · , t)|| is continuous for t ≥ 0, there exists τ ∈ [0, T ] such that ||v(k)( · , t)||1 = c0 for 0 ≤ τ ≤ T 0 , 0 ≤ k ≤ n. Then c20 ≤ G(c0) c20 T F (c0) as n → ∞, we have c2 + 1 c2 then 4 γ21 c4 + 1 ≤ 0 which is a contradiction. Consequently T 0 6→ 0. Choosing T = T (c) sufficiently small, and T not depending on n, one concludes that ||v(n)||1 ≤ C (5.6) for 0 ≤ t ≤ T. This shows that T 0 ≥ T. Hence, from (5.6) we imply that there exists a subsequence v(nj) ≡ v(n) such that ⇀ v weakly on L∞([0, T ] : H1(R)). (5.7) Claim. u = ∧v is a solution. In the linearized equation (5.4) we have 5 = ∧(I − (I − ∂ 3 = ∧v 3 − v 3 = ∂ 1︸ ︷︷ ︸ ∈L2(R) )− ∂2(v 1 )︸ ︷︷ ︸ ∈H−2(R) ∈ H−2(R). Since ∧ = (I − ∂2)−1 is bounded in H1(R), ∧v 5 belongs to H −2(R). v(n) is still bounded in L∞([0, T ] : H1(R)) →֒ L2([0, T ] : H1(R)) and since ∧ : L2(R) → H2(R) is a bounded operator, || ∧ v 1 ||H2(R) ≤ c ||v 1 ||L2(R) ≤ c ||v 1 ||H1(R). Consequently, ∧v 1 is bounded in L 2([0, T ] : H2(R)) →֒ L2([0, T ] : L2(R)). It follows that ∂2(∧v 1 ) is bounded in L2([0, T ] : H−2(R)), and 5 is bounded in L 2([0, T ] : H−2(R)). (5.8) Similarly, the other terms are bounded. By (5.4), v t is a sum of terms each of which is the product of a coefficient, uniformly bounded on n and a function in L2([0, T ] : H−2(R)) uniformly bounded on n such that v t is bounded in L 2([0, T ] : H−2(R)). On the other hand, H1loc(R) loc (R) →֒ H −4(R). By Lions-Aubin’s compactness Theorem [24] there is a subsequence v(nj) ≡ v(n) such that v(n) → v strongly on L2([0, T ] : H loc (R)). Hence, for a subsequence v (nj) ≡ v(n), we have v(n) → v a. e. in L2([0, T ] : loc (R)). Moreover, from (5.8), ∧v 5 ⇀ ∧v5 weakly in L 2([0, T ] : H−2(R)). Similarly, ∧v 2 ⇀ ∧v2 weakly in L2([0, T ] : H−2(R)). Since ||∧v(n)||H2(R) ≤ c ||v (n)||L2(R) ≤ c ||v (n)||H1(R) ≤ c ||v (n)||H1/2(R) and v(n) → v strongly on L2([0, T ] : H loc (R)) then ∧v (n) → ∧v strongly in L2([0, T ] : H2loc(R)). Thus, the fifth term on the right hand side of (5.4), | ∧ v(n−1)|2 ∧ v 2 ⇀ | ∧ v| 2 ∧ v2 weakly in L 2([0, T ] : L1loc(R)) as ∧v 2 ⇀ ∧v2 weakly in L 2([0, T ] : H−2(R)) and | ∧v(n−1)|2 → |∧v|2 strongly on L2([0, T ] : H2loc(R)). Similarly, the other terms in (5.4) converge to their limits, implying v t ⇀ vt weakly in L 2([0, T ] : L1loc(R)). Passing to the limit i vt = ∂ 2(i β ∧ v3 + ω ∧ v2 + | ∧ v| 2 ∧ v)− (i β ∧ v3 + ω ∧ v2 + | ∧ v| 2 ∧ v) = −(I − ∂2)(i β ∧ v3 + ω ∧ v2 + | ∧ v| 2 ∧ v). Thus i vt + (I − ∂ 2)(i β ∧ v3 + ω ∧ v2 + | ∧ v| 2 ∧ v) = 0. This way, we have (2.4) for u = ∧v. Now, we prove that there exists a solution of (2.4) with u ∈ L∞([0, T ] : HN (R)) and N ≥ 4, where T de- pends only on the norm of u0 in H 3(R). We already know that there is a solution u ∈ L∞([0, T ] : H3(R)). It is suffices to show that the approximating sequence v(n) is bounded in L∞([0, T ] : HN−2(R)). Taking α = N − 2 and considering (5.5) for α ≥ 2, we define cN−2 = ||u0(·)||N + 1. Let T N−3 be the largest time such that ||v(k)( · , t)||α ≤ cN−3 for 0 ≤ t ≤ T N−3, 0 ≤ k ≤ n. Integrating (5.5) over [0, t], for 0 ≤ t ≤ T N−3, we have ξ |v(n)α | ||v(n−1)||α ||v(n)||2αds+ ||v(n−1)||α It follows that ξ(x, t) |v(n)α | 2dx ≤ ξ(x, 0) |v(n)α (x, 0)| ||v(n−1)||α ||v(n)||2αds ||v(n−1)||α hence |v(n)α | 2dx ≤ ξ |v(n)α | 2dx ≤ ξ(x, 0) |v(n)α (x, 0)| ||v(n−1)||α ||v(n)||2αds ||v(n−1)||α |v(n)α | 2dx ≤ |v(n)α (x, 0)| 2dx + G(cN−3) c2N−3 t+ F (cN−3) ||v(n)α (x, 0)|| G(cN−3) c2N−3 t+ F (cN−3) ||u(x, 0)||2N + G(cN−3) c2N−3 t+ F (cN−3) and we obtain ||v(n)α ( · , t)|| αdx ≤ ||u(x, 0)||2N + G(cN−3) c2N−3 t+ F (cN−3) Claim. T N−3 does not approach to 0. On the contrary, assume that T N−3 → 0. Since ||v (n)( · , t)|| is continuous for t ≥ 0, there exists τ ∈ [0, TN−3] such that ||v (k)( · , τ)||α = cN−3 for 0 ≤ τ ≤ T (n), 0 ≤ k ≤ n. Then c2N−3 ≤ ||u(x, 0)||2N + G(cN−3) c2N−3 T N−3 + F (cN−3) as n → +∞, and we have ||u(x, 0)||2N + 1 ||u(x, 0)||2N then 4 γ21 ||u(x, 0)||4N + 1 ≤ 0 which is a contradiction. Then T N−3 6→ 0. By choosing TN−3 = TN−3(||u(x, 0)|| N ) sufficiently small, and TN−3 not depending on n, we conclude that ||v(n)( · , t)||2α ≤ c N−3 for all 0 ≤ t ≤ TN−3. (5.9) This shows that T N−3 ≥ TN−3. Thus, v ∈ L∞([0, TN−3] : H α(R)) ≡ L∞([0, TN−3] : H N−2(R)). Now, denote by 0 ≤ T ∗N−3 ≤ +∞ the maximal number such that for all 0 < t ≤ T N−3, u = ∧v ∈ L∞([0, t] : HN (R)). In particular, TN−3 ≤ T N−3 for all N ≥ 4. Thus, T can be chosen depending only on the norm of u0 in H 3(R). Approximating u0 by {u 0 } ∈ C 0 (R) such that ||u0 − u 0 ||HN (R) → 0 as j → +∞. Let uj be a solution of (2.4) with u(j)(x, 0) = u 0 . According to the above argument, there exists T which is independent on n but depending only on supj ||u 0 || such that u (j) there exists on [0, T ] and a subsequence u(j) −→ u in L∞([0, T ] : HN(R)). As a consequence of Theorem 5.1 and 5.2 and its proof, one obtains the following result. Corollary 5.3. Let |ω| < 3 β and let u0 ∈ H N (R) with N ≥ 3 such that u 0 → u0 in H N (R). Let u and u(j) be the corresponding unique solutions given by Theorems 5.1 and 5.2 in L∞([0, T ] : HN(R)) with T depending only on supj ||u 0 ||H3(R) such that ⇀ u weakly on L∞([0, T ] : HN(R)), u(j) → u strongly on L2([0, T ] : HN+1(R)). 6 Existence of Global Solutions Here, we will try to extend the local solution u ∈ L∞([0, T ] : HN (W0 i 0)) of (2.4) obtained in Theo- rem 5.2 to t ≥ 0. A standard way to obtain these extensions consists into deducing global estimations for the HN (W0 i 0)-norm of u in terms of the H N (W0 i 0)-norm of u(x, 0) = u0(x). These estimations are frequently based on conservation laws which contain the L2-norm of the solution and their spatial derivatives. It is not possible to do the same to give a solution of the problem of global existence because the difficulty here is that the weight depends on the x and t variables. To solve our problem we follow a different method using Leibniz’s rule like in the proof of Theorem 3.1 of Bona and Saut [3]. Theorem 6.1. For |ω| < 3 β there exists a global solution to (2.4) in the space Hs(R)∩HN (W0 i 0) with N integer ≥ 3 and s ≥ 2. Proof. The first part was proved in [3]. Differentiating (2.4) α-times (for α ≥ 0) over x ∈ R leads i uα t + i β uα+3 + ω uα+2 + (|u| 2)α u+ (|u|2)α−m um + |u| 2 uα = 0. (6.1) Let ξ = ξ(x, t), then multiplying (6.1) by ξ uα we have i ξ uα uα t + i β ξ uα uα+3 + ω ξ uα uα+2 + (|u| 2)α ξ u uα (|u|2)α−m ξ um uα + ξ |u| 2 |uα| 2 = 0 − i ξ uα uα t − i β ξ uα uα+3 + ω ξ uα uα+2 + (|u| 2)α ξ u uα (|u|2)α−m ξ um uα + ξ |u| 2 |uα| 2 = 0. (applying conjugate) Subtracting and integrating over x ∈ R we have ξ |uα| 2dx+ i β ξ uα uα+3dx+ i β ξ uα uα+3dx+ ω ξ uα uα+2dx (6.2) ξ uα uα+2dx+ 2 i Im ξ (|u|2)α u uαdx+ 2 i ξ (|u|2)α−m um uαdx = 0. Each term is calculated separately, integrating by parts in the second term we have ξ uα uα+3dx = ∂2ξ uα uα+1dx+ 2 ∂ξ |uα+1| ξ uα+2 uα+1dx. The other terms are calculated in a similar way. Hence in (6.2) ξ |uα| 2dx− β ∂3ξ |uα| 2dx+ 3 β ∂ξ |uα+1| 2dx − 2ω Im ∂ξ uα uα+1dx ∂tξ |uα| 2dx+ 2 Im ξ (|u|2)α u uαdx+ 2 ξ (|u|2)α−m um uαdx = 0 such that ξ |uα| 2dx− β ∂3ξ |uα| 2dx+ 3 β ∂ξ |uα+1| 2dx+ 2 Im (|u|2)α ξ u uαdx ∂tξ |uα| 2dx+ 2 ξ (|u|2)α−m um uαdx = 2α Im ∂ξ uα uα+1dx ≤ |ω| ∂ξ |uα| 2dx+ |ω| ∂ξ |uα+1| Hence ξ |uα| [ 3 β − |ω| ] ∂ξ |uα+1| [ ∂tξ + β ∂ 3ξ + |ω| ∂ξ ] |uα| + 2 Im (|u|2)α ξ u uαdx+ 2 ξ (|u|2)α−mum uαdx ≤ 0. (6.3) (|u|2)α = (u u )α = uα−k uk = uuα + uα−k uk + u uα (|u|2)α u uα = |u| 2 |uα| uα−k uk u uα + u 2 u2α hence ( |u|2 )α ξ u uαdx = 2 ξ uα−k uk u uαdx+ 2 Im ξ u2 u2αdx ξ |uα−k| |uk| |u| |uα|dx+ 2 ξ |u|2 |uα| ξ |uα−k| |uk| |u| |uα|dx+ 2 ||u|| L∞(R) ξ |uα| ≤ 2 ||u||L∞(R) ξ |uα−k| |uk| |uα|dx+ 2 ||u|| L∞(R) ξ |uα| 2dx (6.4) hence in (6.3) we have ξ |uα| [3 β − |ω| ] ∂ξ |uα+1| 2dx ≤ [∂tξ + β ∂ 3ξ + |ω| ∂ξ + c ξ ] |uα| ξ |uα−k| |uk| |u| |uα|dx− 2 ξ (|u|2)α−m um uαdx. Using (2.3), Gagliardo-Nirenberg’s inequality and standard estimates we get ξ |uα| 2dx+ [3 β − |ω| ] ∂ξ |uα+1| 2 dx ≤ c ξ |uα| 2dx. (6.5) Integrating (6.5) in t ∈ [0, Tmax = T ] we obtain ξ |uα| 2dx+ [3 β − |ω| ] ∂ξ |uα+1| 2dx ds ≤ ||u0(x)|| ξ |uα| where ξ |uα| 2dx ≤ ||u0(x)|| ξ |uα| Using Gronwall’s inequality ξ |uα| 2dx ≤ ||u0(x)|| c t ≤ ||u0(x)|| it follows that ξ |uα| 2dx ≤ c = c(T, ||u0(x)|| Then for any T = Tmax > 0 there exists c = c(T, ||u0(x)|| α) such that ||u||2α + [3 β − |ω| ] ∂ξ |uα+1| 2dx ds ≤ c. This concludes the proof. 7 Persistence Theorem As a starting point for the a priori gain of regularity results that will be discussed in the next section, we need to develop some estimates for solutions of the equation (2.4) in weighted Sobolev norms. The existence of these weighted estimates is often called the persistence of a property of the initial data u0. We show that if u0 ∈ H 3(R)∩HL(W0 i 0) for L ≥ 0, i ≥ 1, then the solution u( · , t) evolves in H L(W0 i 0) for t ∈ [0, T ]. The time interval of that persistence is at least as long as the interval guaranteed by the existence Theorem 5.2. Theorem 7.1 (Persistence). Let |ω| < 3 β and let i ≥ 1 and L ≥ 0 be non-negative integers, 0 < T < +∞. Assume that u is the solution to (2.4) in L∞([0, T ] : H3(R)) with initial data u0(x) = u(x, 0) ∈ H 3(R). If u0(x) ∈ H L(W0 i 0) then u ∈ L∞ [0, T ] : H3(R) ∩HL(W0 i 0) (7.1) |∂L+1u(x, t)|2 η dx dt < +∞ (7.2) where σ is arbitrary, η ∈ Wσ i 0 for i ≥ 1. Proof. We use induction on α. Let u ∈ L∞ [0, T ] : H3(R) ∩Hα(W0 i 0) for 0 ≤ α ≤ L. We derive formally some a priori estimate for the solution where the bound, involves only the norms of u in L∞([0, T ] : H3(R)) and the norms of u0 in H 3(W0 i 0). We do this by approximating u(x, t) through smooth solutions and the weight functions by smooth bounded functions. By Theorem 5.2, we have u(x, t) ∈ L∞([0, T ] : HN (R)) with N = max{L, 3}. In particular, uj(x, t) ∈ L ∞([0, T ]×R) for 0 ≤ j ≤ N − 1. To obtain (7.1) and (7.2) there are two ways of approximation. We approximate general solutions by smooth solutions, and we approximate general weight functions by bounded weight functions. The first of these procedure has already been discussed, so we shall concentrate on the second. Given a smooth weight function η(x) ∈ Wσ, i−1, 0 with σ > 0, we take a sequence η ν(x) of smooth bounded weight functions approximating η(x) from below, uniformly on any half line (−∞, c). Define the weight functions for the α-th induction step as (3 β − |ω|) ην(y, t) dy then the ξν are bounded weight functions which approximate a desired weight function ξ ∈ W0 i 0 from below, uniformly on a compact set. For α = 0, multiplying (2.4) by ξν u, we have i ξν uut + i β ξν uu3 + ω ξν u u2 + ξν |u| 4 = 0 − i ξν u ut − i β ξν u u3 + ω ξν u u2 + ξν |u| 4 = 0. (applying conjugate) Subtracting and integrating over x ∈ R we have ξν |u| 2dx − i ∂tξν |u| 2dx+ i β ξν uu3dx+ i β ξν u u3dx ξν uu2dx− ω ξν u u2dx = 0. (7.3) Each term is treated separately, integrating by parts in the third term we have ξν u u3dx = ∂2ξν uu1dx+ 2 ∂ξν |u1| ξν u2 u1dx. The other terms are calculated in a similar way. Hence in (7.3) we have ξν |u| ∂tξν |u| 2dx− β ∂3ξν |u| 2dx+ 3 β ∂ξν |u1| = 2ω Im ∂ξν u u1dx ≤ |ω| ∂ξν |u| 2dx+ |ω| ∂ξν |u1| Then, using (2.3) we obtain ξν |u| [3 β − |ω|] ∂ξν |u1| [∂tξν + β ∂ 3ξν + |ω| ∂ξν ] |u| 2dx ≤ c ξν |u| ξν |u| 2dx ≤ c ξν |u| We apply Gronwall’s Lemma to conclude that ξν |u| 2dx ≤ c(T, ||u0||). (7.4) for 0 ≤ t ≤ T, and c not depending on β > 0, the weighted estimate remains true for β → 0. Now, we assume that the result is true for (α − 1) and we prove that it is true for α. To prove this, we start from the main inequality (3.1) with ξ and η given by ξν and ην respectively. ξν |uα| ην |uα+1| θν |uα| Rαdx ≤ 0 where ην = (3β − |ω| ) ∂ξν for |ω| < 3 β θν = − [ ∂tξν + β ∂ 3ξν + |ω| ∂ξν + c0 ξν ] where c0 = ||u|| L∞(R) Rα = Rα(|uα|, |uα−1|, . . . ) ξν |uα| 2dx + ην |uα+1| 2dx ≤ − θν |uα| 2dx − ∣∣∣∣− θν |uα| ∣∣∣∣ ≤ |θν | |uα| |Rα|dx. Using (2.3) in the first part of the right hand side we obtain θν |uα| 2dx ≤ c ξν |uα| ξν |uα| ην |uα+1| 2dx ≤ c ξν |uα| |Rα|dx. (7.5) According to (3.8), Rα dx contains a term of the form ξν uν1 uν2 uαdx. (7.6) We estimate the term ξν uν1 uν2 uα dx for ν1 + ν2 = α. (7.7) Let ν2 ≤ α− 2. Integrating by parts one time in (7.7) we have ξν uν1 uν2 uα dx = − ∂ξν uν1 uν2 uα−1 dx− ξν uν1+1 uν2 uα−1 dx ξν uν1 uν2+1 uα−1 dx. We estimates the first term in the right hand side in (7.7). Using Holder’s inequality and standard estimates we obtain ξν |uν2+1| ξν |uν2 | )1/2](∫ ξν |uα−1| (7.8) where (7.8) is bounded by hypothesis. The other terms are estimates in a similar way. Now suppose that ν1 = ν2 = α− 1, then in (7.7) we have ξν uα−1 uα−1 uαdx, hence ξν |uα−1| 2 uαdx ∣∣∣∣ ≤ ||uα−1||L∞(R) ξν |uα−1| )1/2 (∫ ξν |uα| where ||uα−1||L∞(R) is bounded by hypothesis, and the estimate is complete. In a similar way we estimate all the other terms of Rα. Using these estimates in (7.5) and applying Gronwall’s argument, we obtain for 0 ≤ t ≤ T ξν |uα| ην |uα+1| 2dx ≤ c0 e ξν |∂ αu0(x)| 2dx+ 1 where c0 and c1 are independent of ν and such that letting the parameter ν → 0 the desired estimate (7.2) is obtained. 8 Main Theorem In this section we state and prove our main theorem, which states that if the initial data u(x, 0) decays faster than polynomially on R+ = {x ∈ R : x > 0} and possesses certain initial Sobolev regularity, then the solution u(x, t) ∈ C∞ for all t > 0. If η is an arbitrary weight function in Wσ i k, then by Lemma 3.2, there exists ξ ∈ Wσ, i+1, k which satisfies (3.1). For the main theorem, we take 4 ≤ α ≤ L+ 2. For α ≤ L+ 4, we take η ∈ Wσ, L−α−2, α−3 =⇒ ξ ∈ Wσ, L−α−3, α−3. (8.1) Lemma 8.1(Estimate of error terms). Let 4 ≤ α ≤ L+2 and the weight functions be chosen as in (8.1), ∣∣∣∣∣ (θ |uα| 2 +Rα)dx dt ∣∣∣∣∣ ≤ c, (8.2) where c depends only on the norms of u in L∞([0, T ] : Hβ(Wσ, L−β+3, β−3)) ∩ L 2([0, T ] : Hβ+1(Wσ, L−β+2, β−3)) for 3 ≤ β ≤ α− 1, and the norms of u in L∞([0, T ] : H3(W0 L 0)). Proof. We must estimate both Rα and θ. We begin with a term in Rα of the form ξ |uν1 | |uν2 | |uα| (8.3) assuming that ν1 ≤ α− 2. By the induction hypothesis, u is bounded in L∞([0, T ] : Hβ(Wσ, L−(β−3)+, (β−3)+)) for 0 ≤ β ≤ α − 1. By Lemma 2.1, ζ |uβ| 2 < +∞ (8.4) for 0 ≤ β ≤ α − 2 and ζ ∈ Wσ, L−(β−2)+, (β−2)+ . We estimate |uν1 | using (8.4). We estimate |uν2 | and |uα| using the weighted L 2 bounds ζ |uν2 | 2dx dt < +∞ for ζ ∈ Wσ, L−(ν2−3)+, (ν2−4)+ (8.5) and the same with ν2 replaced by α. It suffices to check the powers to t, the powers of x as x → +∞ and the exponential of x as x → −∞. For x > 1. In the (8.3) term, the factor ξ contributed according to (8.1) ξ(x, t) = tα−3 x(L−α+3) t−(α−3) x−(L−α+3)ξ(x, t) ≤ c2 t α−3 x(L−α+3) (using(2.3)) then ξ |uν1 | |uν2 | |uα| ≤ c2 t α−3 x(L−α+3)|uν1 | |uν2 | |uα|. Moreover |uν1 | |uν2 | |uα| = t (ν1−2) L−(ν1−2) −(ν1−2) (L−(ν1−2) 2 |uν1 | × (ν2−4) L−(ν2−3) −(ν2−4) (L−(ν2−3) 2 |uν2 | × (α−4)+ L−(α−3)+ −(α−4)+ (L−(α−3)+) 2 |uα|. tt follows that ξ |uν1 | |uν2 | |uα| ≤ c2 t M xT t (ν1−2) L−(ν1−2) 2 |uν1 | t (ν2−4) L−(ν2−3) 2 |uν2 | t (α−4)+ L−(α−3)+ 2 |uα| (8.6) where M = α− 3− (ν1 − 2) (ν2 − 4) (α− 4)+ T = (L− α+ 3)− (L − (α− 3)+)− (L− (ν2 − 3) (L − ν1 − 2) Claim. M ≥ 0 is large enough, that the extra power of t can be omitted 2M = 2α− 6− (ν1 − 2) + − (ν2 − 4) + − (α− 4)+ = α− 2− (ν1 − 2) + − (ν2 − 4) = α− 2− ν1 + 2− ν2 + 4 = α+ 4− (ν1 + ν2) = α+ 4− α = 4 ≥ 0. Claim. T ≤ 0 is such that the extra power of t can be omitted. 2T = 2L− 2α+ 6− L+ (α− 3)+ − L+ (ν2 − 3) + − L+ (ν1 − 2) = −L− α+ ν1 + ν2 − 2 = −L− α+ α− 2 = −(L+ 2) ≤ 0. Now, we study the behavior as x → −∞. Since each factor uνj (j = 1, 2) must grow slower that an exponential eσ ′ |x| and ξ decays as an exponential e−σ |x|, we simply need to choose the appropriate rela- tionship σ and σ′ at each induction step. The analysis will be completed with the case where ν1 ≥ α− 1. Then, in (3.9), if 2(α − 1) ≤ α, but α ≥ 3. So this possibility is impossible. For x < 1 the estimate is similar, except for an exponential weight. The analysis of all terms of Rα is estimated in a similar form. This completes the estimate of Rα. Now, we estimate the term θ |uα| 2 where θ is given in (3.1). We have that θ involves derivatives of u only up to order one, and hence, θ |uα| 2 is a sum of terms of the same type which we have already encountered in Rα. So, its integral can be bounded in the same type. Indeed, (3.1) shows that θ depends on ξt, ∂ and derivatives of lower order. By using (3.6) we have the claim. Theorem 8.2(Main Theorem). Let |ω| < 3 β, T > 0 and u(x, t) be a solution of (2.4) in the region R× [0, T ] such that u ∈ L∞([0, T ] : H3(W0 L 0)) (8.7) for some L ≥ 2. Then u ∈ L∞([0, T ] : H3+l(Wσ, L−l, l)) ∩ L 2([0, T ] : H4+l(Wσ, L−l−1, l)) (8.8) for all 0 ≤ l ≤ L− 1 and all σ > 0. Remark. If the assumption (8.7) holds for all L ≥ 2, the solution is infinitely differentiable in the x- variable. ¿From (2.4) we have that the solution is C∞ in both variables. We are also quantifying the gain of each derivative by the degree of vanishing of the initial data at infinity. Proof. We use induction on α. For α = 3, let u be a solution of (2.4) satisfying (8.7). Therefore, ut ∈ L ∞([0, T ] : L2(W0 L 0)) where u ∈ L ∞([0, T ] : H3(W0 L 0)) and ut ∈ L ∞([0, T ] : L2(W0 L 0)). Then u ∈ C([0, T ] : L2(W0 L 0)) ∩ Cw([0, T ] : H 3(W0 L 0)). Hence, u : [0, T ] 7−→ H 3(W0 L 0) is a weakly continuous function. In particular, u( · , t) ∈ H3(W0 L 0) for all t. Let t0 ∈ (0, T ) and u( · , t0) ∈ H 3(W0 L 0), then there are {u 0 } ⊆ C 0 (R) such that u 0 ( · ) → u( · , t0) in H 3(W0 L 0). Let u(n)(x, t) be a unique solution of (2.4) with u(n)(x, t0) = u 0 . Then by Theorem 5.1 and 5.2, there exists u in a time interval [t0, t0 + δ] where δ > 0 does not depend on n and u is a unique solution of (2.4), u(n) ∈ L∞([t0, t0 + δ] : H 3(W0 L 0)) with u (n)(x, t0) ≡ u 0 (x) → u(x, t0) ≡ u0(x) in H 3(W0 L 0). Now, by Theorem 7.1, we have u(n) ∈ L∞([t0, t0 + δ] : H 3(W0 L 0)) ∩ L 2([t0, t0 + δ] : H 4(Wσ, L−1, 0)) with a bound that depends only on the norm of u 0 in H 3(W0 L 0). Furthermore, Theorem 7.1 guarantees the non-uniform bounds [t0, t0+δ] (1 + |x+|) k | ∂αu(n)(x, t) | < +∞ for each n, k and α. The main inequality (3.1) and the estimate (8.2) are therefore valid for each u(n) in the interval [t0, t0 + δ]. η may be chosen arbitrarily in its weight class (8.1) and then ξ is defined by (3.7) and the constant c1, c2, c3, c4 are independent of n. From (3.1) and (8.1) we have [t0, t0+δ] ξ |u(n)α | ∫ t0+δ 2dx ≤ c (8.9) where by (8.2), c is independent of n. The estimate (8.9) is proved by induction for α = 3, 4, 5, . . . Thus u(n) is also bounded in L∞([t0, t0 + δ] : H α(Wσ, L−α+3, α−3)) ∩ L 2([t0, t0 + δ] : H α+1(Wσ, L−α+2, α−3)) (8.10) for α ≥ 3. Since u(n) → u in L∞([t0, t0 + δ] : H 3(W0 L 0)). By Corollary 5.3 it follows that u belongs to the space (8.10). Since δ is fixed, this result is valid over the whole interval [0, T ]. References [1] E. Bisognin, V. Visognin and O. Vera. Stabilization of solutions for the higher order nonlinear Schrödinger equation with localized damping. EJDE. Vol. 2007, 6(2007) 1-18. [2] V. Bisognin and O. Vera. 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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN–PH–EP/2007–009 (revised author list) 2 April 2007 Double spin asymmetry in exclusive ρ0 muoproduction at COMPASS Abstract The longitudinal double spin asymmetry A 1 for exclusive leptoproduction of ρ mesons, µ + N → µ + N + ρ, is studied using the COMPASS 2002 and 2003 data. The measured reaction is incoherent exclusive ρ0 production on polarised deuterons. The Q2 and x dependence of A 1 is presented in a wide kinematical range 3 · 10−3 < Q2 < 7 (GeV/c)2 and 5 · 10−5 < x < 0.05. The presented results are the first measurements of A 1 at small Q 2 (Q2 < 0.1 (GeV/c)2) and small x (x < 3 · 10−3). The asymmetry is in general compatible with zero in the whole kinematical range. (to be submitted to Eur. Phys. J. C) http://arxiv.org/abs/0704.1863v2 The COMPASS Collaboration M. Alekseev29), V.Yu. Alexakhin8), Yu. Alexandrov18), G.D. Alexeev8), A. Amoroso29), A. Arbuzov8), B. Bade lek30), F. Balestra29), J. Ball25), G. Baum1), J. Barth4), Y. Bedfer25), C. Bernet25), R. Bertini29), M. Bettinelli19), R. Birsa28), J. Bisplinghoff3), P. Bordalo15,a), F. Bradamante28), A. Bravar16), A. Bressan28), G. Brona30), E. Burtin25), M.P. Bussa29), A. Chapiro27), M. Chiosso29), A. Cicuttin27), M. Colantoni29,b), S. Costa29), M.L. Crespo27), N. d’Hose25), S. Dalla Torre28), S. Das7), S.S. Dasgupta6), R. De Masi20), N. Dedek19), O.Yu. Denisov29,c), L. Dhara7), V. Diaz27), A.M. Dinkelbach20), S.V. Donskov24), V.A. Dorofeev24), N. Doshita21), V. Duic28), W. Dünnweber19), P.D. Eversheim3), W. Eyrich9), M. Fabro28), M. Faessler19), V. Falaleev11), A. Ferrero29), L. Ferrero29), M. Finger22), M. Finger jr.8), H. Fischer10), C. Franco15), J. Franz10), J.M. Friedrich20), V. Frolov29,c), R. Garfagnini29), F. Gautheron1), O.P. Gavrichtchouk8), R. Gazda30), S. Gerassimov18,20), R. Geyer19), M. Giorgi28), B. Gobbo28), S. Goertz2,4), A.M. Gorin24), S. Grabmüller20), O.A. Grajek30), A. Grasso29), B. Grube20), R. Gushterski8), A. Guskov8), F. Haas20), J. Hannappel4), D. von Harrach16), T. Hasegawa17), J. Heckmann2), S. Hedicke10), F.H. Heinsius10), R. Hermann16), C. Heß2), F. Hinterberger3), M. von Hodenberg10), N. Horikawa21,d), S. Horikawa21), C. Ilgner19), A.I. Ioukaev8), S. Ishimoto21), O. Ivanov8), Yu. Ivanshin8), T. Iwata21,32), R. Jahn3), A. Janata8), P. Jasinski16), R. Joosten3), N.I. Jouravlev8), E. Kabuß16), D. Kang10), B. Ketzer20), G.V. Khaustov24), Yu.A. Khokhlov24), Yu. Kisselev1,2), F. Klein4), K. Klimaszewski30), S. Koblitz16), J.H. Koivuniemi13), V.N. Kolosov24), E.V. Komissarov8), K. Kondo21), K. Königsmann10), I. Konorov18,20), V.F. Konstantinov24), A.S. Korentchenko8), A. Korzenev16,c), A.M. Kotzinian8,29), N.A. Koutchinski8), O. Kouznetsov8,25), N.P. Kravchuk8), A. Kral23), Z.V. Kroumchtein8), R. Kuhn20), F. Kunne25), K. Kurek30), M.E. Ladygin24), M. Lamanna11,28), J.M. Le Goff25), A.A. Lednev24), A. Lehmann9), J. Lichtenstadt26), T. Liska23), I. Ludwig10), A. Maggiora29), M. Maggiora29), A. Magnon25), G.K. Mallot11), A. Mann20), C. Marchand25), J. Marroncle25), A. Martin28), J. Marzec31), F. Massmann3), T. Matsuda17), A.N. Maximov8), W. Meyer2), A. Mielech28,30), Yu.V. Mikhailov24), M.A. Moinester26), A. Mutter10,16), O. Nähle3), A. Nagaytsev8), T. Nagel20), J. Nassalski30), S. Neliba23), F. Nerling10), S. Neubert20), D.P. Neyret25), V.I. Nikolaenko24), K. Nikolaev8), A.G. Olshevsky8), M. Ostrick4), A. Padee31), P. Pagano28), S. Panebianco25), R. Panknin4), D. Panzieri29,b), S. Paul20), B. Pawlukiewicz-Kaminska30) , D.V. Peshekhonov8), V.D. Peshekhonov8), G. Piragino29), S. Platchkov25), J. Pochodzalla16), J. Polak14), V.A. Polyakov24), J. Pretz4), S. Procureur25), C. Quintans15), J.-F. Rajotte19), V. Rapatsky8), S. Ramos15,a), G. Reicherz2), A. Richter9),F. Robinet25), E. Rocco28,29), E. Rondio30), A.M. Rozhdestvensky8) , D.I. Ryabchikov24), V.D. Samoylenko24), A. Sandacz30), H. Santos15), M.G. Sapozhnikov8), S. Sarkar7), I.A. Savin8), P. Schiavon28), C. Schill10), L. Schmitt20), P. Schönmeier9), W. Schröder9), O.Yu. Shevchenko8), H.-W. Siebert12,16), L. Silva15), L. Sinha7), A.N. Sissakian8), M. Slunecka8), G.I. Smirnov8), S. Sosio29), F. Sozzi28), V.P. Sugonyaev24), A. Srnka5), F. Stinzing9), M. Stolarski30,10), M. Sulc14), R. Sulej31), N. Takabayashi21), V.V. Tchalishev8), S. Tessaro28), F. Tessarotto28), A. Teufel9), L.G. Tkatchev8), G. Venugopal3), M. Virius23), N.V. Vlassov8), A. Vossen10), R. Webb9), E. Weise3), Q. Weitzel20), R. Windmolders4), S. Wirth9), W. Wíslicki30), K. Zaremba31), M. Zavertyaev18), E. Zemlyanichkina8), J. Zhao16), R. Ziegler3), and A. Zvyagin19) 1) Universität Bielefeld, Fakultät für Physik, 33501 Bielefeld, Germanye) 2) Universität Bochum, Institut für Experimentalphysik, 44780 Bochum, Germanye) 3) Universität Bonn, Helmholtz-Institut für Strahlen- und Kernphysik, 53115 Bonn, Germanye) 4) Universität Bonn, Physikalisches Institut, 53115 Bonn, Germanye) 5) Institute of Scientific Instruments, AS CR, 61264 Brno, Czech Republicf) 6) Burdwan University, Burdwan 713104, Indiah) 7) Matrivani Institute of Experimental Research & Education, Calcutta-700 030, Indiai) 8) Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia 9) Universität Erlangen–Nürnberg, Physikalisches Institut, 91054 Erlangen, Germanye) 10) Universität Freiburg, Physikalisches Institut, 79104 Freiburg, Germanye) 11) CERN, 1211 Geneva 23, Switzerland 12) Universität Heidelberg, Physikalisches Institut, 69120 Heidelberg, Germanye) 13) Helsinki University of Technology, Low Temperature Laboratory, 02015 HUT, Finland and University of Helsinki, Helsinki Institute of Physics, 00014 Helsinki, Finland 14) Technical University in Liberec, 46117 Liberec, Czech Republicf) 15) LIP, 1000-149 Lisbon, Portugalg) 16) Universität Mainz, Institut für Kernphysik, 55099 Mainz, Germanye) 17) University of Miyazaki, Miyazaki 889-2192, Japanj) 18) Lebedev Physical Institute, 119991 Moscow, Russia 19) Ludwig-Maximilians-Universität München, Department für Physik, 80799 Munich, Germanye) 20) Technische Universität München, Physik Department, 85748 Garching, Germanye) 21) Nagoya University, 464 Nagoya, Japanj) 22) Charles University, Faculty of Mathematics and Physics, 18000 Prague, Czech Republicf) 23) Czech Technical University in Prague, 16636 Prague, Czech Republicf) 24) State Research Center of the Russian Federation, Institute for High Energy Physics, 142281 Protvino, Russia 25) CEA DAPNIA/SPhN Saclay, 91191 Gif-sur-Yvette, France 26) Tel Aviv University, School of Physics and Astronomy, 69978 Tel Aviv, Israelk) 27) ICTP–INFN MLab Laboratory, 34014 Trieste, Italy 28) INFN Trieste and University of Trieste, Department of Physics, 34127 Trieste, Italy 29) INFN Turin and University of Turin, Physics Department, 10125 Turin, Italy 30) So ltan Institute for Nuclear Studies and Warsaw University, 00-681 Warsaw, Polandl) 31) Warsaw University of Technology, Institute of Radioelectronics, 00-665 Warsaw, Polandm) 32) Yamagata University, Yamagata, 992-8510 Japanj) a) Also at IST, Universidade Técnica de Lisboa, Lisbon, Portugal b) Also at University of East Piedmont, 15100 Alessandria, Italy c) On leave of absence from JINR Dubna d) Also at Chubu University, Kasugai, Aichi, 487-8501 Japan e) Supported by the German Bundesministerium für Bildung und Forschung f) Suppported by Czech Republic MEYS grants ME492 and LA242 g) Supported by the Portuguese FCT - Fundação para a Ciência e Tecnologia grants POCTI/FNU/49501/2002 and POCTI/FNU/50192/2003 h) Supported by DST-FIST II grants, Govt. of India i) Supported by the Shailabala Biswas Education Trust j) Supported by the Ministry of Education, Culture, Sports, Science and Technology, Japan; Daikou Foundation and Yamada Foundation k) Supported by the Israel Science Foundation, founded by the Israel Academy of Sciences and Human- ities l) Supported by KBN grant nr 621/E-78/SPUB-M/CERN/P-03/DZ 298 2000, nr 621/E- 78/SPB/CERN/P-03/DWM 576/2003-2006, and by MNII reasearch funds for 2005–2007 m) Supported by KBN grant nr 134/E-365/SPUB-M/CERN/P-03/DZ299/2000 1 Introduction In this paper we present results on the longitudinal double spin asymmetry A 1 for exclusive incoherent ρ0 production in the scattering of high energy muons on nucleons. The experiment was carried out at CERN by the COMPASS collaboration using the 160 GeV muon beam and the large 6LiD polarised target. The studied reaction is µ + N → µ′ + ρ0 + N ′, (1) where N is a quasi-free nucleon from the polarised deuterons. The reaction (1) can be described in terms of the virtual photoproduction process γ∗ + N → ρ0 + N ′. (2) The reaction (2) can be regarded as a fluctuation of the virtual photon into a quark- antiquark pair (in partonic language), or an off-shell vector meson (in Vector Meson Dominance model), which then scatters off the target nucleon resulting in the production of an on-shell vector meson. At high energies this is predominantly a diffractive process and plays an important role in the investigation of Pomeron exchange and its interpretation in terms of multiple gluon exchange. Most of the presently available information on the spin structure of reaction (2) stems from the ρ0 spin density matrix elements, which are obtained from the analysis of angular distributions of ρ0 production and decay [1]. Experimental results on ρ0 spin density matrix elements come from various experiments [2–6] including the preliminary results from COMPASS [7]. The emerging picture of the spin structure of the considered process is the following. At low photon virtuality Q2 the cross section by transverse virtual photons σT dominates, while the relative contribution of the cross section by longitudinal photons σL rapidly increases with Q2. At Q2 of about 2 (GeV/c)2 both components become comparable and at a larger Q2 the contribution of σL becomes dominant and continues to grow, although at lower rate than at low Q2. Approximately, the so called s-channel helicity conservation (SCHC) is valid, i.e. the helicity of the vector meson is the same as the helicity of the parent virtual photon. The data indicate that the process can be described approximately by the exchange in the t-channel of an object with natural parity P . Small deviations from SCHC are observed, also at the highest energies, whose origin is still to be understood. An interesting suggestion was made in Ref. [8] that at high energies the magnitudes of various helicity amplitudes for the reaction (2) may shed a light on the spin-orbital momentum structure of the vector meson. A complementary information can be obtained from measurements of the double spin cross section asymmetry, when the information on both the beam and target polari- sation is used. The asymmetry is defined as σ1/2 − σ3/2 σ1/2 + σ3/2 , (3) where σ1/2(3/2) stands for the cross sections of the reaction (2) and the subscripts denote the total virtual photon–nucleon angular momentum component along the virtual photon direction. In the following we will also use the asymmetry ALL which is defined for reaction (1) as the asymmetry of muon–nucleon cross sections for antiparallel and parallel beam and target longitudinal spin orientations. In the Regge approach [9] the longitudinal double spin asymmetry A 1 can arise due to the interference of amplitudes for exchange in the t-channel of Reggeons with natural parity (Pomeron, ρ, ω, f , A2 ) with amplitudes for Reggeons with unnatural parity (π,A1). No significant asymmetry is expected when only a non-perturbative Pomeron is exchanged because it has small spin-dependent couplings as found from hadron-nucleon data for cross sections and polarisations. Similarly, in the approach of Fraas [10], assuming approximate validity of SCHC, the spin asymmetry A 1 arises from the interference between parts of the helicity amplitudes for transverse photons corresponding to the natural and unnatural parity exchanges in the t channel. While a measurable asymmetry can arise even from a small contribution of the unnatural parity exchange, the latter may remain unmeasurable in the cross sections. A significant unnatural-parity contribution may indicate an exchange of certain Reggeons like π, A1 or in partonic terms an exchange of qq̄ pairs. In the same reference a theoretical prediction for A 1 was presented, which is based on the description of forward exclusive ρ0 leptoproduction and inclusive inelastic lepton- nucleon scattering by the off-diagonal Generalised Vector Meson Dominance (GVMD) model, applied to the case of polarised lepton–nucleon scattering. At the values of Bjorken variable x < 0.2, with additional assumptions [11], A 1 can be related to the A1 asymmetry for inclusive inelastic lepton scattering at the same x as 1 + (A1)2 . (4) This prediction is consistent with the HERMES results for both the proton and deuteron targets, although with rather large errors. In perturbative QCD, there exists a general proof of factorisation [12] for exclu- sive vector meson production by longitudinal photons. It allows a decomposition of the full amplitude for reaction (2) into three components: a hard scattering amplitude for the exchange of quarks or gluons, a distribution amplitude for the meson and the non- perturbative description of the target nucleon in terms of the generalised parton distri- butions (GPDs), which are related to the internal structure of the nucleon. No similar proof of factorisation exists for transverse virtual photons, and as a consequence the in- terpretation of A 1 in perturbative QCD is not possible at leading twist. However, a model including higher twist effects proposed by Martin et al. [13] describes the behaviour of both σL as well as of σT reasonably well. An extension of this model by Ryskin [14] for the spin dependent cross sections allows to relate A 1 to the spin dependent GPDs of gluons and quarks in the nucleon. The applicability of this model is limited to the range Q2 ≥ 4 (GeV/c)2. More recently another pQCD-inspired model involving GPDs has been proposed by Goloskokov and Kroll [15,16]. The non-leading twist asymmetry ALL results from the interference between the dominant GPD Hg and the helicity-dependent GPD H̃g. The asymmetry is estimated to be of the order k2T H̃g/(Q 2Hg), where kT is the transverse momentum of the quark and the antiquark. Up to now little experimental information has been available on the double spin asymmetries for exclusive leptoproduction of vector mesons. The first observation of a non- zero asymmetry A 1 in polarised electron–proton deep-inelastic scattering was reported by the HERMES experiment [11]. In the deep inelastic region (0.8 < Q2 < 3 (GeV/c)2) the measured asymmetry is equal to 0.23 ± 0.14 (stat) ± 0.02 (syst) [17], with little dependence on the kinematical variables. In contrast, for the ‘quasi-real photoproduction’ data, with 〈Q2〉 = 0.13 (GeV/c)2, the asymmetry for the proton target is consistent with zero. On the other hand the measured asymmetry A 1 for the polarised deuteron target and the asymmetry A 1 for exclusive production of φ meson either on polarised protons or deuterons are consistent with zero both in the deep inelastic and in the quasi-real photoproduction regions [17]. The HERMES result indicating a non-zero A 1 for the proton target differs from the unpublished result of similar measurements by the SMC experiment [18] at comparable values of Q2 but at about three times higher values of the photon-nucleon centre of mass energy W , i.e. at smaller x. The SMC measurements of ALL in several bins of Q 2 are consistent with zero for both proton and deuteron targets. 2 The experimental set-up The experiment [19] was performed with the high intensity positive muon beam from the CERN M2 beam line. The µ+ beam intensity is 2·108 per spill of 4.8 s with a cycle time of 16.8 s. The average beam energy is 160 GeV and the momentum spread is σp/p = 0.05. The momentum of each beam muon is measured upstream of the experimental area in a beam momentum station consisting of several planes of scintillator strips or scintillating fibres with a dipole magnet in between. The precision of the momentum determination is typically ∆p/p ≤ 0.003. The µ+ beam is naturally polarised by the weak decays of the parent hadrons. The polarisation of the muon varies with its energy and the average polarisation is −0.76. The beam traverses the two cells of the polarised target, each 60 cm long, 3 cm in diameter and separated by 10 cm, which are placed one after the other. The target cells are filled with 6LiD which is used as polarised deuteron target material and is longitudinally polarised by dynamic nuclear polarisation (DNP). The two cells are polarised in opposite directions so that data from both spin directions are recorded at the same time. The typical values of polarisation are about 0.50. A mixture of liquid 3He and 4He, used to refrigerate the target, and a small amount of heavier nuclei are also present in the target. The spin directions in the two target cells are reversed every 8 hours by rotating the direction of the magnetic field in the target. In this way fluxes and acceptances cancel in the calculation of spin asymmetries, provided that the ratio of acceptances of the two cells remains unchanged after the reversal. The COMPASS spectrometer is designed to reconstruct the scattered muons and the produced hadrons in wide momentum and angular ranges. It is divided in two stages with two dipole magnets, SM1 and SM2. The first magnet, SM1, accepts charged particles of momenta larger than 0.4 GeV/c, and the second one, SM2, those larger than 4 GeV/c. The angular acceptance of the spectrometer is limited by the aperture of the polarised target magnet. For the upstream end of the target it is ±70 mrad. To match the expected particle flux at various locations in the spectrometer, COM- PASS uses various tracking detectors. Small-angle tracking is provided by stations of scintillating fibres, silicon detectors, micromesh gaseous chambers and gas electron mul- tiplier chambers. Large-angle tracking devices are multiwire proportional chambers, drift chambers and straw detectors. Muons are identified in large-area mini drift tubes and drift tubes placed downstream of hadron absorbers. Hadrons are detected by two large iron-scintillator sampling calorimeters installed in front of the absorbers and shielded to avoid electromagnetic contamination. The identification of charged particles is possible with a RICH detector, although in this paper we have not utilised the information from the RICH. The data recording system is activated by various triggers indicating the presence of a scattered muon and/or an energy deposited by hadrons in the calorimeters. In addition to the inclusive trigger, in which the scattered muon is identified by coincidence signals in the trigger hodoscopes, several semi-inclusive triggers were used. They select events fulfilling the requirement to detect the scattered muon together with the energy deposited in the hadron calorimeters exceeding a given threshold. In 2003 the acceptance was further extended towards high Q2 values by the addition of a standalone calorimetric trigger in which no condition is set for the scattered muon. The COMPASS trigger system allows us to cover a wide range of Q2, from quasi-real photoproduction to deep inelastic interactions. A more detailed description of the COMPASS apparatus can be found in Ref. [19] 3 Event sample For the present analysis the whole data sample taken in 2002 and 2003 with the longitudinally polarised target is used. For an event to be accepted for further analysis it is required to originate in the target, have a reconstructed beam track, a scattered muon track, and only two additional tracks of oppositely charged hadrons associated to the primary vertex. The fluxes of beam muons passing through each target cell are equalised using appropriate cuts on the position and angle of the beam tracks. The charged pion mass hypothesis is assigned to each hadron track and the invariant mass of two pions, mππ, calculated. A cut on the invariant mass of two pions, 0.5 < mππ < 1 GeV/c2, is applied to select the ρ0. As slow recoil target particles are not detected, in order to select exclusive events we use the cut on the missing energy, −2.5 < Emiss < 2.5 GeV, and on the transverse momentum of ρ0 with respect to the direction of virtual photon, p2t < 0.5 (GeV/c) 2. Here Emiss = (M X−M2p )/2Mp, where MX is the missing mass of the unobserved recoiling system and Mp is the proton mass. Coherent interactions on the target nuclei are removed by a cut p2t > 0.15 (GeV/c) 2. To avoid large corrections for acceptance and misidentification of events, additional cuts ν > 30 GeV and Eµ′ > 20 GeV are applied. The distributions of mππ, Emiss and p t are shown in Fig. 1. Each plot is obtained applying all cuts except those corresponding to the displayed variable. On the left top panel of Fig. 1 a clear peak of the ρ0 resonance, centred at 770 MeV/c2, is visible on the top of the small contribution of background of the non-resonant π+π− pairs. Also the skewing of the resonance peak towards smaller values of mππ, due to an interference with the non-resonant background, is noticeable. A small bump below 0.4 GeV/c2 is due to assignment of the charged pion mass to the kaons from decays of φ mesons. The mass cuts eliminate the non-resonant background outside of the ρ0 peak, as well as the contribution of φ mesons. On the right top panel of the figure the peak at Emiss ≈ 0 is the signal of exclusive ρ0 production. The width of the peak, σ ≈ 1.1 GeV, is due to the spectrometer resolution. Non-exclusive events, where in addition to the recoil nucleon other undetected hadrons are produced, appear at Emiss > 0. Due to the finite resolution, however, they are not resolved from the exclusive peak. This background consists of two components: the double- diffractive events where additionally to ρ0 an excited nucleon state is produced in the nucleon vertex of reaction (2), and events with semi-inclusive ρ0 production, in which other hadrons are produced but escape detection. The p2t distribution shown on the bottom panel of the figure indicates a contribution from coherent production on target nuclei at small p2t values. A three-exponential fit to this distribution was performed, which indicates also a contribution of non-exclusive background increasing with p2t . Therefore to select the sample of exclusive incoherent ρ [GeV/cππm 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 [GeV]missE -4 -2 0 2 4 6 8 10 12 14 10000 [(GeV/c)2tp 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 1: Distributions of mππ (top left), Emiss (top right) and p t (bottom) for the exclu- sive sample. The arrows show cuts imposed on each variable to define the final sample. production, the aforementioned p2t cuts, indicated by arrows, were applied. After all selections the final sample consists of about 2.44 million events. The dis- tributions of Q2, x and W are shown in Fig. 2. The data cover a wide range in Q2 and x which extends towards the small values by almost two orders of magnitude compared to the similar studies reported in Ref. [17]. The sharp edge of the W distribution at the low W values is a consequence of the cut applied on ν. For this sample 〈W 〉 is equal to 10.2 GeV and 〈p2t 〉 = 0.27(GeV/c)2. 4 Extraction of asymmetry A The cross section asymmetry ALL = (σ↑↓ − σ↑↑)/(σ↑↓ + σ↑↑) for reaction (1) , for antiparallel (↑↓) and parallel (↑↑) spins of the incoming muon and the target nucleon, is related to the virtual-photon nucleon asymmetry A ALL = D (A 1 + ηA 2) , (5) where the factors D and η depend on the event kinematics and A 2 is related to the interference cross section for exclusive production by longitudinal and transverse virtual photons. As the presented results extend into the range of very small Q2, the exact formulae for the depolarisation factor D and kinematical factor η [20] are used without neglecting terms proportional to the lepton mass squared m2. The depolarisation factor is given by D(y,Q2) = y [(1 + γ2y/2)(2 − y) − 2y2m2/Q2] y2(1 − 2m2/Q2)(1 + γ2) + 2(1 + R)(1 − y − γ2y2/4) , (6) [(GeV/c)2Q -410 -310 -210 -110 1 10 210 [(GeV/c)2Q -410 -310 -210 -110 1 10 210 -610 -510 -410 -310 -210 -110 1 W [GeV] 0 5 10 15 20 25 Figure 2: Distributions of the kinematical variables for the final sample: Q2 with linear and logarithmic vertical axis scale (top left and right panels respectively), x (bottom left), and the energy W (bottom right). where R = σL/σT , σL(T ) is the cross section for reaction (2) initiated by longitudinally (transversely) polarised virtual photons, the fraction of the muon energy lost y = ν/Eµ and γ2 = Q2/ν2. The kinematical factor η(y,Q2) is the same as for the inclusive asym- metry. The asymmetry A 2 obeys the positivity limit A R, analogous to the one for the inclusive case. For Q2 ≤ 0.1 (GeV/c)2 the ratio R for the reaction (2) is small, cf. Fig. 3, and the positivity limit constrains A 2 to small values. Although for larger Q 2 the ratio R for the process (2) increases with Q2, because of small values of η the product R is small in the whole Q2 range of our sample. Therefore the second term in Eq. 5 can be neglected, so that ALL, (7) and the effect of this approximation is included in the systematic uncertainty of A The number of events Ni collected from a given target cell in a given time interval is related to the spin-independent cross section σ̄ for reaction (2) and to the asymmetry Ni = aiφiniσ̄(1 + PBPTfDA 1), (8) where PB and PT are the beam and target polarisations, φi is the incoming muon flux, ai the acceptance for the target cell, ni corresponding number of target nucleons, and f the target dilution factor. The asymmetry is extracted from the data sets taken before and after a reversal of the target spin directions. The four relations of Eq. 8, corresponding to the two cells (u and d) and the two spin orientations (1 and 2) lead to a second- [(GeV/c)2Q -210 -110 1 10 Figure 3: The ratio R = σL/σT as a function of Q 2 measured in the E665 experiment. The curve is a fit to the data described in the text. order equation in A 1 for the ratio (Nu,1Nd,2/Nd,1Nu,2). Here fluxes cancel out as well as acceptances, if the ratio of acceptances for the two cells is the same before and after the reversal [21]. In order to minimise the statistical error all quantities used in the asymmetry calculation are evaluated event by event with the weight factor w = PBfD. The polarisation of the beam muon, PB, is obtained from a simulation of the beam line and parameterised as a function of the beam momentum. The target polarisation is not included in the event weight w because it may vary in time and generate false asymmetries. An average PT is used for each target cell and each spin orientation. The ratio R, which enters the formula for D and strongly depends on Q2 for reaction (2), was calculated on an event-by-event basis using the parameterisation R(Q2) = a0(Q 2)a1 , (9) with a0 = 0.66 ± 0.05, and a1 = 0.61 ± 0.09. The parameterisation was obtained by the Fermilab E665 experiment from a fit to their R measurements for exclusive ρ0 muopro- duction on protons [3]. These are shown in Fig. 3 together with the fitted Q2-dependence. The preliminary COMPASS results on R for the incoherent exclusive ρ0 production on the nucleon [7], which cover a broader kinematic region in Q2 , agree reasonably well with this parameterisation. The uncertainty of a0 and a1 is included in the systematic error of The dilution factor f gives the fraction of events of reaction (2) originating from nucleons in polarised deuterons inside the target material. It is calculated event-by-event using the formula f = C1 · f0 = C1 · nD + ΣAnA(σ̃A/σ̃D) . (10) Here nD and nA denote numbers of nucleons in deuteron and nucleus of atomic mass A in the target, and σ̃D and σ̃A are the cross sections per nucleon for reaction (2) occurring on the deuteron and on the nucleus of atomic mass A, respectively. The sum runs over all nuclei present in the COMPASS target. The factor C1 takes into account that there are two polarised deuterons in the 6LiD molecule, as the 6Li nucleus is in a first approximation composed of a deuteron and an α particle. The measurements of the σ̃A/σ̃D for incoherent exclusive ρ 0 production come from the NMC [2], E665 [22] and early experiments on ρ0 photoproduction [23]. They were [(GeV/c)2Q 0 2 4 6 8 10 12 14 photoproduction [(GeV/c)2Q -310 -210 -110 1 10 Figure 4: (Left) Parameter α of Eq. 11 as a function of Q2 (from Ref. [24]). The exper- imental points and the fitted curve are shown. See text for details. (Right) The dilution factor f as a function of Q2. fitted in Ref. [24] with the formula: σ̃A = σp ·Aα(Q 2)−1, with α(Q2) − 1 = − exp{−Q2/Q20}, (11) where σp is the cross section for reaction (2) on the free proton. The value of the fitted parameter Q20 is equal to 9 ± 3 (GeV/c)2. The measured values of the parameter α and the fitted curve α(Q2) are shown on the left panel of Fig. 4 taken from Ref. [24]. On the right panel of the figure the average value of f is plotted for the various Q2 bins used in the present analysis. The values of f are equal to about 0.36 in most of the Q2 range, rising to about 0.38 at the highest Q2. The radiative corrections (RC) have been neglected in the present analysis, in par- ticular in the calculation of f , because they are expected to be small for reaction (1). They were evaluated [25] to be of the order of 6% for the NMC exclusive ρ0 production analysis. The small values of RC are mainly due to the requirement of event exclusivity via cuts on Emiss and p t , which largely suppress the dominant external photon radiation. The internal (infrared and virtual) RC were estimated in Ref. [25] to be of the order of 5 Systematic errors The main systematic uncertainty of A 1 comes from an estimate of possible false asymmetries. In order to improve the accuracy of this estimate, in addition to the standard sample of incoherent events, a second sample was selected by changing the p2t cuts to 0 < p2t < 0.5 (GeV/c) 2, (12) and keeping all the remaining selections and cuts the same as for the ‘incoherent sample’. In the following it will be referred to as the ‘extended p2t sample’. Such an extension of the p2t range allows one to obtain a sample which is about five times larger than the incoherent sample. However, in addition to incoherent events such a sample contains a large fraction of events originating from coherent ρ0 production. Therefore, for the estimate of the dilution factor f a different nuclear dependence of the exclusive cross section was used, applicable for the sum of coherent and incoherent cross sections [2]. The physics asymmetries A 1 for both samples are consistent within statistical errors. Possible, false experimental asymmetries were searched for by modifying the se- lection of data sets used for the asymmetry calculation. The grouping of the data into configurations with opposite target-polarisation was varied from large samples, covering at most two weeks of data taking, into about 100 small samples, taken in time intervals of the order of 16 hours. A statistical test was performed on the distributions of asymmetries obtained from these small samples. In each of the Q2 and x bins the dispersion of the values of A 1 around their mean agrees with the statistical error. Time-dependent effects which would lead to a broadening of these distributions were thus not observed. Allowing the dispersion of A 1 to vary within its two standard deviations we obtain for each bin an upper bound for the systematic error arising from time-dependent effects σfalseA,tdep < 0.56 σstat. (13) Here σstat is the statistical error on A 1 for the extended p t sample. The uncertainty on the estimates of possible false asymmetries due to the time-dependent effects is the dominant contribution to the total systematic error in most of the kinematical region. Asymmetries for configurations where spin effects cancel out were calculated to check the cancellation of effects due to fluxes and acceptances. They were found compatible with zero within the statistical errors. Asymmetries obtained with different settings of the microwave (MW) frequency, used for DNP, were compared in order to test possible effects related to the orientation of the target magnetic field. The results for the extended p2t sample tend to show that there is a small difference between asymmetries for the two MW configurations. However, because the numbers of events of the data samples taken with each MW setting are approximately balanced, the effect of this difference on A negligible for the total sample. The systematic error on A 1 also contains an overall scale uncertainty of 6.5% due to uncertainties on PB and PT . The uncertainty of the parameterisation of R(Q 2) affects the depolarisation factor D. The uncertainty of the dilution factor f is mostly due to uncer- tainty of the parameter α(Q2) which takes into account nuclear effects in the incoherent ρ0 production. The neglect of the A 2 term mainly affects the highest bins of Q 2 and x. Another source of systematic errors is due to the contribution of the non-exclusive background to our sample. This background originates from two sources. First one is due to the production of ρ0 accompanied by the dissociation of the target nucleon, the second one is the production of ρ0 in inclusive scattering. In order to evaluate the amount of background in the sample of exclusive events it is necessary to determine the Emiss dependence for the non-exclusive background in the region under the exclusive peak (cf. Fig. 1 ). For this purpose complete Monte Carlo simulations of the experiment were used, with events generated by either the PYTHIA 6.2 or LEPTO 6.5.1 generators. Events generated with LEPTO come only from deep inelastic scattering and cover the range of Q2 > 0.5 (GeV/c)2. Those generated with PYTHIA cover the whole kinematical range of the experiment and include exclusive production of vector mesons and processes with diffractive excitation of the target nucleon or the vector meson, in addition to inelastic production. The generated MC events were reconstructed and selected for the analysis using the same procedure as for the data. In each bin of Q2 the Emiss distribution for the MC was normalised to the corresponding one for the data in the range of large Emiss > 7.5 GeV. Then the normalised MC distribution was used to estimate the number of background events under the exclusive peak in the data. The fraction of background events in the sample of incoherent exclusive ρ0 production was estimated to be about 0.12±0.06 in most of the kinematical range, except in the largest Q2 region, where it is about 0.24±0.12. The large uncertainties of these fractions reflect the differences between estimates from LEPTO and PYTHIA in the region where they overlap. In the case of PYTHIA the uncertainties on the cross sections for diffractive photo- and electroproduction of vector mesons also contribute. For events generated with PYTHIA the Emiss distributions for various physics processes could be studied separately. It was found that events of ρ0 production with an excitation of the target nucleon into N∗ resonances of small mass, M < 2 GeV/c2, cannot be resolved from the exclusive peak and therefore were not included in the estimates of number of background events. An estimate of the asymmetry A 1 for the background was obtained using a non- exclusive sample, which was selected with the standard cuts used in this analysis, except the cut on Emiss which was modified to Emiss > 2.5 GeV. In different high-Emiss bins A for this sample was found compatible with zero. Because no indication of a non-zero A 1 for the background was found, and also due to a large uncertainty of the estimated amount of background in the exclusive sample, no background corrections were made. Instead, the effect of background was treated as a source of systematic error. Its contribution to the total systematic error was not significant in most of the kinematical range, except for the highest Q2 and x. The total systematic error on A 1 was obtained as a quadratic sum of the errors from all discussed sources. Its values for each Q2 and x bin are given in Tables 1 and 2. The total systematic error amounts to about 40% of the statistical error for most of the kinematical range. Both errors become comparable in the highest bin of Q2. 6 Results The COMPASS results on A 1 are shown as a function of Q 2 and x in Fig. 5 and listed in Tables 1 and 2. The statistical errors are represented by vertical bars and the total systematic errors by shaded bands. [(GeV/c)2Q -310 -210 -110 1 10 -410 -310 -210 -110 Figure 5: A 1 as a function of Q 2 (left) and x (right) from the present analysis. Error bars correspond to statistical errors, while bands at the bottom represent the systematical errors. The wide range in Q2 covers four orders of magnitude from 3 · 10−3 to 7 (GeV/c)2. The domain in x which is strongly correlated with Q2, varies from 5 · 10−5 to about 0.05 (see Tables for more details). For the whole kinematical range the A 1 asymmetry measured by COMPASS is consistent with zero. As discussed in the introduction, this indicates that the role of unnatural parity exchanges, like π- or A1-Reggeon exchange, is Table 1: Asymmetry A 1 as a function of Q 2. Both the statistical errors (first) and the total systematic errors (second) are listed. Q2 range 〈Q2〉 [(GeV/c)2] 〈x〉 〈ν〉 [GeV] Aρ1 0.0004 − 0.005 0.0031 4.0 · 10−5 42.8 −0.030 ± 0.045 ± 0.014 0.005 − 0.010 0.0074 8.4 · 10−5 49.9 0.048 ± 0.038 ± 0.013 0.010 − 0.025 0.017 1.8 · 10−4 55.6 0.063 ± 0.026 ± 0.014 0.025 − 0.050 0.036 3.7 · 10−4 59.9 −0.035 ± 0.027 ± 0.009 0.05 − 0.10 0.072 7.1 · 10−4 62.0 −0.010 ± 0.028 ± 0.008 0.10 − 0.25 0.16 0.0016 62.3 −0.019 ± 0.029 ± 0.009 0.25 − 0.50 0.35 0.0036 60.3 0.016 ± 0.045 ± 0.014 0.5 − 1 0.69 0.0074 58.6 0.141 ± 0.069 ± 0.030 1 − 4 1.7 0.018 59.7 0.000 ± 0.098 ± 0.035 4 − 50 6.8 0.075 55.9 −0.85 ± 0.50 ± 0.39 small in that kinematical domain, which is to be expected if diffraction is the dominant process for reaction (2). In Fig. 6 the COMPASS results are compared to the HERMES results on A 1 ob- tained on a deuteron target [17]. Note that the lowest Q2 and x HERMES points, re- ferred to as ‘quasi-photoproduction’, come from measurements where the kinematics of the small-angle scattered electron was not measured but estimated from a MC simulation. This is in contrast to COMPASS, where scattered muon kinematics is measured even at the smallest Q2. [(GeV/c)2Q -310 -210 -110 1 10 COMPASS HERMES quasi-photoprod. (d) HERMES electroprod. (d) -410 -310 -210 -110 COMPASS HERMES quasi-photoprod. (d) HERMES electroprod. (d) Figure 6: A 1 as a function of Q 2 (left) and x (right) from the present analysis (circles) compared to HERMES results on the deuteron target (triangles). For the COMPASS results inner bars represent statistical errors, while the outer bars correspond to the total error. For the HERMES results vertical bars represent the quadratic sum of statistical and systematic errors. The curve represents the prediction explained in the text. The results from both experiments are consistent within errors. The kinematical range covered by the present analysis extends further towards small values of x and Q2 by almost two orders of magnitude. In each of the two experiments A 1 is measured at different average W , which is equal to about 10 GeV for COMPASS and 5 GeV for Table 2: Asymmetry A 1 as a function of x. Both the statistical errors (first) and the total systematic errors (second) are listed. x range 〈x〉 〈Q2〉 [(GeV/c)2] 〈ν〉 [GeV] Aρ1 8 · 10−6 − 1 · 10−4 5.8 · 10−5 0.0058 51.7 0.035 ± 0.026 ± 0.011 1 · 10−4 − 2.5 · 10−4 1.7 · 10−4 0.019 59.7 0.036 ± 0.024 ± 0.010 2.5 · 10−4 − 5 · 10−4 3.6 · 10−4 0.041 61.3 −0.039 ± 0.027 ± 0.012 5 · 10−4 − 0.001 7.1 · 10−4 0.082 60.8 −0.010 ± 0.030 ± 0.010 0.001 − 0.002 0.0014 0.16 58.6 −0.005 ± 0.036 ± 0.013 0.002 − 0.004 0.0028 0.29 54.8 0.032 ± 0.050 ± 0.019 0.004 − 0.01 0.0062 0.59 50.7 0.019 ± 0.069 ± 0.026 0.01 − 0.025 0.015 1.3 47.5 −0.03 ± 0.14 ± 0.06 0.025 − 0.8 0.049 3.9 43.8 −0.27 ± 0.38 ± 0.19 HERMES. Thus, no significant W dependence is observed for A 1 on an isoscalar nucleon target. The x dependence of the measured A 1 is compared in Fig. 6 to the prediction given by Eq. 4, which relates A 1 to the asymmetry A1 for the inclusive inelastic lepton- nucleon scattering. To produce the curve the inclusive asymmetry A1 was parameterised as A1(x) = (x α − γα) · (1 − e−βx) , where α = 1.158 ± 0.024, β = 125.1 ± 115.7 and γ = 0.0180 ± 0.0038. The values of the parameters have been obtained from a fit of A1(x) to the world data from polarised deuteron targets [26–31] including COMPASS measurements at very low Q2 and x [32]. Within the present accuracy the results on A are consistent with this prediction. In the highest Q2 bin, 〈Q2〉 = 6.8 (GeV/c)2, in the kinematical domain of applica- bility of pQCD-inspired models which relate the asymmetry to the spin-dependent GPDs for gluons and quarks (cf. Introduction), one can observe a hint of a possible nonzero asymmetry, although with a large error. It should be noted that in Ref. [18] a nega- tive value of ALL different from zero by about 2 standard deviations was reported at 〈Q2〉 = 7.7 (GeV/c)2. At COMPASS, including the data taken with the longitudinally polarised deuteron target in 2004 and 2006 will result in an increase of statistics by a factor of about three compared to the present paper, and thus may help to clarify the issue. For the whole Q2 range future COMPASS data, to be taken with the polarised proton target, would be very valuable for checking if the role of the flavour-blind exchanges is indeed dominant, as expected for the Pomeron-mediated process. 7 Summary The longitudinal double spin asymmetry A 1 for the diffractive muoproduction of ρ meson, µ + N → µ + N + ρ, has been measured by scattering longitudinally polarised muons off longitudinally polarised deuterons from the 6LiD target and selecting incoherent exclusive ρ0 production. The presented results for the COMPASS 2002 and 2003 data cover a range of energy W from about 7 to 15 GeV. The Q2 and x dependence of A 1 is presented in a wide kinematical range 3 · 10−3 ≤ Q2 ≤ 7 (GeV/c)2 and 5 · 10−5 ≤ x ≤ 0.05. These results extend the range in Q2 and x by two orders of magnitude down with respect to the existing data from HERMES. The asymmetry A 1 is compatible with zero in the whole x and Q 2 range. This may indicate that the role of unnatural parity exchanges like π- or A1-Reggeon exchange is small in that kinematical domain. The x dependence of measured A 1 is consistent with the prediction of Ref. [11] which relates A 1 to the asymmetry A1 for the inclusive inelastic lepton–nucleon scattering. 8 Acknowledgements We gratefully acknowledge the support of the CERN management and staff and the skill and effort of the technicians of our collaborating institutes. Special thanks are due to V. Anosov and V. Pesaro for their support during the installation and the running of the experiment. This work was made possible by the financial support of our funding agencies. References [1] K. Schilling and G. Wolf, Nucl. Phys. B61 (1973) 381. [2] NMC Collab., M. Arneodo et al., Nucl. Phys. B429 (1994) 503. [3] E665 Collab., M.R. Adams et al., Z. Phys. C74 (1997) 237. [4] ZEUS Collab., J. Breitweg et al., Eur. Phys. J. C12 (2000) 393. [5] H1 Collab., C. Adloff et al., Eur. Phys. J. C13 (2000) 371; H1 Collab., C. Adloff et al., Phys. Lett. B 539 (2002) 25. [6] HERMES Collab., K. Ackerstaff et al., Eur. Phys. J. C18 (2000) 303. [7] A. Sandacz (on behalf of the COMPASS Collaboration), Nucl. Phys. B 146 (Proc. Suppl.) (2005) 581. [8] I.P. Ivanov, N.N. Nikolaev, JETP Lett. C29 (1999) 294; I.P. Ivanov, Diffractive production of vector mesons in Deep Inelastic Scattering within kt-factorization approach, hep-ph/0303053. [9] S.I. Manaenkov, Regge description of spin-spin asymmetry in photon diffractive dis- sociation, Preprint DESY 99-016 (see also hep-ph/9903405). [10] H. Fraas, Nucl. Phys. B113 (1976) 532. [11] HERMES Collab., A. Airapetian et al., Phys. Lett. B513 (2001) 301. [12] J.C. Collins, L. Frankfurt and M. Strikman, Phys. Rev. D56 (1997) 2982. [13] A.D. Martin, M.G. Ryskin and T. Teubner, Phys. Rev. D55 (1997) 4329. [14] M.G. Ryskin, Phys. Atom. Nucl. 62 (1999) 315; Yad. Fiz. 62 (1999) 350. [15] S.V. Goloskokov and P. Kroll, Eur. Phys. J. C42 (2005) 281. [16] S.V. Goloskokov and P. Kroll, hep-ph/0611290. [17] HERMES Collab., A. Airapetian et al., Eur. Phys. J. C29 (2003) 171. [18] A. Tripet, Nucl. Phys. B79 (Proc. Suppl.) (1999) 529. [19] COMPASS Collab., P. Abbon et al., Nucl. Instrum. Meth. A577 (2007) 455. [20] J. Kiryluk, Ph.D. thesis, Warsaw University, 2000. [21] SMC Collab., D. Adams et al., Phys. Rev. 56 (1997) 5330. [22] E665 Collab., M.R. Adams et al., Phys. Rev. Lett. 74 (1995) 1525. [23] T. Bauer et al., Rev. Mod. Phys. 50 (1978) 261, Erratum: ibid., 51 (1979) 407. [24] A. Tripet, Ph.D. thesis, Universität Bielefeld, 2002. [25] K. Kurek, QED radiative corrections in exclusive ρ0 leptoproduction, preprint DESY- 96-209, June 1996 (see also hep-ph/9606240). [26] SMC Collab., B. Adeva et al., Phys. Rev. D58 (1998) 112001. [27] E143 Collab., K. Abe et al., Phys. Rev. D58 (1998) 112003. http://arxiv.org/abs/hep-ph/0303053 http://arxiv.org/abs/hep-ph/9903405 http://arxiv.org/abs/hep-ph/0611290 http://arxiv.org/abs/hep-ph/9606240 [28] E155 Collab., P.L. Anthony et al., Phys. Lett. B463 (1999) 339. [29] SMC Collab., B. Adeva et al., Phys. Rev. D60 (1999) 072004; Erratum: ibid., D62 (2000) 079902. [30] HERMES Collab., A. Airapetian et al., Phys. Rev. D75 (2007) 012007. [31] COMPASS Collab., V.Yu. Alexakhin et al., Phys. Lett. B647 (2007) 8. [32] COMPASS Collab., V.Yu. Alexakhin et al., Phys. Lett. B647 (2007) 330. Introduction The experimental set-up Event sample Extraction of asymmetry A1 Systematic errors Results Summary Acknowledgements

The longitudinal double spin asymmetry A_1^rho for exclusive leptoproduction of rho^0 mesons, mu + N -> mu + N + rho, is studied using the COMPASS 2002 and 2003 data. The measured reaction is incoherent exclusive rho^0 production on polarised deuterons. The Q^2 and x dependence of A_1^rho is presented in a wide kinematical range: 3x10^-3 < Q^2 < 7 (GeV/c)^2 and 5x10^-5 < x < 0.05. The presented results are the first measurements of A_1^rho at small Q2 (Q2 < 0.1 (GeV/c)^2) and small x (x < 3x10^-3). The asymmetry is in general compatible with zero in the whole kinematical range.

Introduction In this paper we present results on the longitudinal double spin asymmetry A 1 for exclusive incoherent ρ0 production in the scattering of high energy muons on nucleons. The experiment was carried out at CERN by the COMPASS collaboration using the 160 GeV muon beam and the large 6LiD polarised target. The studied reaction is µ + N → µ′ + ρ0 + N ′, (1) where N is a quasi-free nucleon from the polarised deuterons. The reaction (1) can be described in terms of the virtual photoproduction process γ∗ + N → ρ0 + N ′. (2) The reaction (2) can be regarded as a fluctuation of the virtual photon into a quark- antiquark pair (in partonic language), or an off-shell vector meson (in Vector Meson Dominance model), which then scatters off the target nucleon resulting in the production of an on-shell vector meson. At high energies this is predominantly a diffractive process and plays an important role in the investigation of Pomeron exchange and its interpretation in terms of multiple gluon exchange. Most of the presently available information on the spin structure of reaction (2) stems from the ρ0 spin density matrix elements, which are obtained from the analysis of angular distributions of ρ0 production and decay [1]. Experimental results on ρ0 spin density matrix elements come from various experiments [2–6] including the preliminary results from COMPASS [7]. The emerging picture of the spin structure of the considered process is the following. At low photon virtuality Q2 the cross section by transverse virtual photons σT dominates, while the relative contribution of the cross section by longitudinal photons σL rapidly increases with Q2. At Q2 of about 2 (GeV/c)2 both components become comparable and at a larger Q2 the contribution of σL becomes dominant and continues to grow, although at lower rate than at low Q2. Approximately, the so called s-channel helicity conservation (SCHC) is valid, i.e. the helicity of the vector meson is the same as the helicity of the parent virtual photon. The data indicate that the process can be described approximately by the exchange in the t-channel of an object with natural parity P . Small deviations from SCHC are observed, also at the highest energies, whose origin is still to be understood. An interesting suggestion was made in Ref. [8] that at high energies the magnitudes of various helicity amplitudes for the reaction (2) may shed a light on the spin-orbital momentum structure of the vector meson. A complementary information can be obtained from measurements of the double spin cross section asymmetry, when the information on both the beam and target polari- sation is used. The asymmetry is defined as σ1/2 − σ3/2 σ1/2 + σ3/2 , (3) where σ1/2(3/2) stands for the cross sections of the reaction (2) and the subscripts denote the total virtual photon–nucleon angular momentum component along the virtual photon direction. In the following we will also use the asymmetry ALL which is defined for reaction (1) as the asymmetry of muon–nucleon cross sections for antiparallel and parallel beam and target longitudinal spin orientations. In the Regge approach [9] the longitudinal double spin asymmetry A 1 can arise due to the interference of amplitudes for exchange in the t-channel of Reggeons with natural parity (Pomeron, ρ, ω, f , A2 ) with amplitudes for Reggeons with unnatural parity (π,A1). No significant asymmetry is expected when only a non-perturbative Pomeron is exchanged because it has small spin-dependent couplings as found from hadron-nucleon data for cross sections and polarisations. Similarly, in the approach of Fraas [10], assuming approximate validity of SCHC, the spin asymmetry A 1 arises from the interference between parts of the helicity amplitudes for transverse photons corresponding to the natural and unnatural parity exchanges in the t channel. While a measurable asymmetry can arise even from a small contribution of the unnatural parity exchange, the latter may remain unmeasurable in the cross sections. A significant unnatural-parity contribution may indicate an exchange of certain Reggeons like π, A1 or in partonic terms an exchange of qq̄ pairs. In the same reference a theoretical prediction for A 1 was presented, which is based on the description of forward exclusive ρ0 leptoproduction and inclusive inelastic lepton- nucleon scattering by the off-diagonal Generalised Vector Meson Dominance (GVMD) model, applied to the case of polarised lepton–nucleon scattering. At the values of Bjorken variable x < 0.2, with additional assumptions [11], A 1 can be related to the A1 asymmetry for inclusive inelastic lepton scattering at the same x as 1 + (A1)2 . (4) This prediction is consistent with the HERMES results for both the proton and deuteron targets, although with rather large errors. In perturbative QCD, there exists a general proof of factorisation [12] for exclu- sive vector meson production by longitudinal photons. It allows a decomposition of the full amplitude for reaction (2) into three components: a hard scattering amplitude for the exchange of quarks or gluons, a distribution amplitude for the meson and the non- perturbative description of the target nucleon in terms of the generalised parton distri- butions (GPDs), which are related to the internal structure of the nucleon. No similar proof of factorisation exists for transverse virtual photons, and as a consequence the in- terpretation of A 1 in perturbative QCD is not possible at leading twist. However, a model including higher twist effects proposed by Martin et al. [13] describes the behaviour of both σL as well as of σT reasonably well. An extension of this model by Ryskin [14] for the spin dependent cross sections allows to relate A 1 to the spin dependent GPDs of gluons and quarks in the nucleon. The applicability of this model is limited to the range Q2 ≥ 4 (GeV/c)2. More recently another pQCD-inspired model involving GPDs has been proposed by Goloskokov and Kroll [15,16]. The non-leading twist asymmetry ALL results from the interference between the dominant GPD Hg and the helicity-dependent GPD H̃g. The asymmetry is estimated to be of the order k2T H̃g/(Q 2Hg), where kT is the transverse momentum of the quark and the antiquark. Up to now little experimental information has been available on the double spin asymmetries for exclusive leptoproduction of vector mesons. The first observation of a non- zero asymmetry A 1 in polarised electron–proton deep-inelastic scattering was reported by the HERMES experiment [11]. In the deep inelastic region (0.8 < Q2 < 3 (GeV/c)2) the measured asymmetry is equal to 0.23 ± 0.14 (stat) ± 0.02 (syst) [17], with little dependence on the kinematical variables. In contrast, for the ‘quasi-real photoproduction’ data, with 〈Q2〉 = 0.13 (GeV/c)2, the asymmetry for the proton target is consistent with zero. On the other hand the measured asymmetry A 1 for the polarised deuteron target and the asymmetry A 1 for exclusive production of φ meson either on polarised protons or deuterons are consistent with zero both in the deep inelastic and in the quasi-real photoproduction regions [17]. The HERMES result indicating a non-zero A 1 for the proton target differs from the unpublished result of similar measurements by the SMC experiment [18] at comparable values of Q2 but at about three times higher values of the photon-nucleon centre of mass energy W , i.e. at smaller x. The SMC measurements of ALL in several bins of Q 2 are consistent with zero for both proton and deuteron targets. 2 The experimental set-up The experiment [19] was performed with the high intensity positive muon beam from the CERN M2 beam line. The µ+ beam intensity is 2·108 per spill of 4.8 s with a cycle time of 16.8 s. The average beam energy is 160 GeV and the momentum spread is σp/p = 0.05. The momentum of each beam muon is measured upstream of the experimental area in a beam momentum station consisting of several planes of scintillator strips or scintillating fibres with a dipole magnet in between. The precision of the momentum determination is typically ∆p/p ≤ 0.003. The µ+ beam is naturally polarised by the weak decays of the parent hadrons. The polarisation of the muon varies with its energy and the average polarisation is −0.76. The beam traverses the two cells of the polarised target, each 60 cm long, 3 cm in diameter and separated by 10 cm, which are placed one after the other. The target cells are filled with 6LiD which is used as polarised deuteron target material and is longitudinally polarised by dynamic nuclear polarisation (DNP). The two cells are polarised in opposite directions so that data from both spin directions are recorded at the same time. The typical values of polarisation are about 0.50. A mixture of liquid 3He and 4He, used to refrigerate the target, and a small amount of heavier nuclei are also present in the target. The spin directions in the two target cells are reversed every 8 hours by rotating the direction of the magnetic field in the target. In this way fluxes and acceptances cancel in the calculation of spin asymmetries, provided that the ratio of acceptances of the two cells remains unchanged after the reversal. The COMPASS spectrometer is designed to reconstruct the scattered muons and the produced hadrons in wide momentum and angular ranges. It is divided in two stages with two dipole magnets, SM1 and SM2. The first magnet, SM1, accepts charged particles of momenta larger than 0.4 GeV/c, and the second one, SM2, those larger than 4 GeV/c. The angular acceptance of the spectrometer is limited by the aperture of the polarised target magnet. For the upstream end of the target it is ±70 mrad. To match the expected particle flux at various locations in the spectrometer, COM- PASS uses various tracking detectors. Small-angle tracking is provided by stations of scintillating fibres, silicon detectors, micromesh gaseous chambers and gas electron mul- tiplier chambers. Large-angle tracking devices are multiwire proportional chambers, drift chambers and straw detectors. Muons are identified in large-area mini drift tubes and drift tubes placed downstream of hadron absorbers. Hadrons are detected by two large iron-scintillator sampling calorimeters installed in front of the absorbers and shielded to avoid electromagnetic contamination. The identification of charged particles is possible with a RICH detector, although in this paper we have not utilised the information from the RICH. The data recording system is activated by various triggers indicating the presence of a scattered muon and/or an energy deposited by hadrons in the calorimeters. In addition to the inclusive trigger, in which the scattered muon is identified by coincidence signals in the trigger hodoscopes, several semi-inclusive triggers were used. They select events fulfilling the requirement to detect the scattered muon together with the energy deposited in the hadron calorimeters exceeding a given threshold. In 2003 the acceptance was further extended towards high Q2 values by the addition of a standalone calorimetric trigger in which no condition is set for the scattered muon. The COMPASS trigger system allows us to cover a wide range of Q2, from quasi-real photoproduction to deep inelastic interactions. A more detailed description of the COMPASS apparatus can be found in Ref. [19] 3 Event sample For the present analysis the whole data sample taken in 2002 and 2003 with the longitudinally polarised target is used. For an event to be accepted for further analysis it is required to originate in the target, have a reconstructed beam track, a scattered muon track, and only two additional tracks of oppositely charged hadrons associated to the primary vertex. The fluxes of beam muons passing through each target cell are equalised using appropriate cuts on the position and angle of the beam tracks. The charged pion mass hypothesis is assigned to each hadron track and the invariant mass of two pions, mππ, calculated. A cut on the invariant mass of two pions, 0.5 < mππ < 1 GeV/c2, is applied to select the ρ0. As slow recoil target particles are not detected, in order to select exclusive events we use the cut on the missing energy, −2.5 < Emiss < 2.5 GeV, and on the transverse momentum of ρ0 with respect to the direction of virtual photon, p2t < 0.5 (GeV/c) 2. Here Emiss = (M X−M2p )/2Mp, where MX is the missing mass of the unobserved recoiling system and Mp is the proton mass. Coherent interactions on the target nuclei are removed by a cut p2t > 0.15 (GeV/c) 2. To avoid large corrections for acceptance and misidentification of events, additional cuts ν > 30 GeV and Eµ′ > 20 GeV are applied. The distributions of mππ, Emiss and p t are shown in Fig. 1. Each plot is obtained applying all cuts except those corresponding to the displayed variable. On the left top panel of Fig. 1 a clear peak of the ρ0 resonance, centred at 770 MeV/c2, is visible on the top of the small contribution of background of the non-resonant π+π− pairs. Also the skewing of the resonance peak towards smaller values of mππ, due to an interference with the non-resonant background, is noticeable. A small bump below 0.4 GeV/c2 is due to assignment of the charged pion mass to the kaons from decays of φ mesons. The mass cuts eliminate the non-resonant background outside of the ρ0 peak, as well as the contribution of φ mesons. On the right top panel of the figure the peak at Emiss ≈ 0 is the signal of exclusive ρ0 production. The width of the peak, σ ≈ 1.1 GeV, is due to the spectrometer resolution. Non-exclusive events, where in addition to the recoil nucleon other undetected hadrons are produced, appear at Emiss > 0. Due to the finite resolution, however, they are not resolved from the exclusive peak. This background consists of two components: the double- diffractive events where additionally to ρ0 an excited nucleon state is produced in the nucleon vertex of reaction (2), and events with semi-inclusive ρ0 production, in which other hadrons are produced but escape detection. The p2t distribution shown on the bottom panel of the figure indicates a contribution from coherent production on target nuclei at small p2t values. A three-exponential fit to this distribution was performed, which indicates also a contribution of non-exclusive background increasing with p2t . Therefore to select the sample of exclusive incoherent ρ [GeV/cππm 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 [GeV]missE -4 -2 0 2 4 6 8 10 12 14 10000 [(GeV/c)2tp 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 1: Distributions of mππ (top left), Emiss (top right) and p t (bottom) for the exclu- sive sample. The arrows show cuts imposed on each variable to define the final sample. production, the aforementioned p2t cuts, indicated by arrows, were applied. After all selections the final sample consists of about 2.44 million events. The dis- tributions of Q2, x and W are shown in Fig. 2. The data cover a wide range in Q2 and x which extends towards the small values by almost two orders of magnitude compared to the similar studies reported in Ref. [17]. The sharp edge of the W distribution at the low W values is a consequence of the cut applied on ν. For this sample 〈W 〉 is equal to 10.2 GeV and 〈p2t 〉 = 0.27(GeV/c)2. 4 Extraction of asymmetry A The cross section asymmetry ALL = (σ↑↓ − σ↑↑)/(σ↑↓ + σ↑↑) for reaction (1) , for antiparallel (↑↓) and parallel (↑↑) spins of the incoming muon and the target nucleon, is related to the virtual-photon nucleon asymmetry A ALL = D (A 1 + ηA 2) , (5) where the factors D and η depend on the event kinematics and A 2 is related to the interference cross section for exclusive production by longitudinal and transverse virtual photons. As the presented results extend into the range of very small Q2, the exact formulae for the depolarisation factor D and kinematical factor η [20] are used without neglecting terms proportional to the lepton mass squared m2. The depolarisation factor is given by D(y,Q2) = y [(1 + γ2y/2)(2 − y) − 2y2m2/Q2] y2(1 − 2m2/Q2)(1 + γ2) + 2(1 + R)(1 − y − γ2y2/4) , (6) [(GeV/c)2Q -410 -310 -210 -110 1 10 210 [(GeV/c)2Q -410 -310 -210 -110 1 10 210 -610 -510 -410 -310 -210 -110 1 W [GeV] 0 5 10 15 20 25 Figure 2: Distributions of the kinematical variables for the final sample: Q2 with linear and logarithmic vertical axis scale (top left and right panels respectively), x (bottom left), and the energy W (bottom right). where R = σL/σT , σL(T ) is the cross section for reaction (2) initiated by longitudinally (transversely) polarised virtual photons, the fraction of the muon energy lost y = ν/Eµ and γ2 = Q2/ν2. The kinematical factor η(y,Q2) is the same as for the inclusive asym- metry. The asymmetry A 2 obeys the positivity limit A R, analogous to the one for the inclusive case. For Q2 ≤ 0.1 (GeV/c)2 the ratio R for the reaction (2) is small, cf. Fig. 3, and the positivity limit constrains A 2 to small values. Although for larger Q 2 the ratio R for the process (2) increases with Q2, because of small values of η the product R is small in the whole Q2 range of our sample. Therefore the second term in Eq. 5 can be neglected, so that ALL, (7) and the effect of this approximation is included in the systematic uncertainty of A The number of events Ni collected from a given target cell in a given time interval is related to the spin-independent cross section σ̄ for reaction (2) and to the asymmetry Ni = aiφiniσ̄(1 + PBPTfDA 1), (8) where PB and PT are the beam and target polarisations, φi is the incoming muon flux, ai the acceptance for the target cell, ni corresponding number of target nucleons, and f the target dilution factor. The asymmetry is extracted from the data sets taken before and after a reversal of the target spin directions. The four relations of Eq. 8, corresponding to the two cells (u and d) and the two spin orientations (1 and 2) lead to a second- [(GeV/c)2Q -210 -110 1 10 Figure 3: The ratio R = σL/σT as a function of Q 2 measured in the E665 experiment. The curve is a fit to the data described in the text. order equation in A 1 for the ratio (Nu,1Nd,2/Nd,1Nu,2). Here fluxes cancel out as well as acceptances, if the ratio of acceptances for the two cells is the same before and after the reversal [21]. In order to minimise the statistical error all quantities used in the asymmetry calculation are evaluated event by event with the weight factor w = PBfD. The polarisation of the beam muon, PB, is obtained from a simulation of the beam line and parameterised as a function of the beam momentum. The target polarisation is not included in the event weight w because it may vary in time and generate false asymmetries. An average PT is used for each target cell and each spin orientation. The ratio R, which enters the formula for D and strongly depends on Q2 for reaction (2), was calculated on an event-by-event basis using the parameterisation R(Q2) = a0(Q 2)a1 , (9) with a0 = 0.66 ± 0.05, and a1 = 0.61 ± 0.09. The parameterisation was obtained by the Fermilab E665 experiment from a fit to their R measurements for exclusive ρ0 muopro- duction on protons [3]. These are shown in Fig. 3 together with the fitted Q2-dependence. The preliminary COMPASS results on R for the incoherent exclusive ρ0 production on the nucleon [7], which cover a broader kinematic region in Q2 , agree reasonably well with this parameterisation. The uncertainty of a0 and a1 is included in the systematic error of The dilution factor f gives the fraction of events of reaction (2) originating from nucleons in polarised deuterons inside the target material. It is calculated event-by-event using the formula f = C1 · f0 = C1 · nD + ΣAnA(σ̃A/σ̃D) . (10) Here nD and nA denote numbers of nucleons in deuteron and nucleus of atomic mass A in the target, and σ̃D and σ̃A are the cross sections per nucleon for reaction (2) occurring on the deuteron and on the nucleus of atomic mass A, respectively. The sum runs over all nuclei present in the COMPASS target. The factor C1 takes into account that there are two polarised deuterons in the 6LiD molecule, as the 6Li nucleus is in a first approximation composed of a deuteron and an α particle. The measurements of the σ̃A/σ̃D for incoherent exclusive ρ 0 production come from the NMC [2], E665 [22] and early experiments on ρ0 photoproduction [23]. They were [(GeV/c)2Q 0 2 4 6 8 10 12 14 photoproduction [(GeV/c)2Q -310 -210 -110 1 10 Figure 4: (Left) Parameter α of Eq. 11 as a function of Q2 (from Ref. [24]). The exper- imental points and the fitted curve are shown. See text for details. (Right) The dilution factor f as a function of Q2. fitted in Ref. [24] with the formula: σ̃A = σp ·Aα(Q 2)−1, with α(Q2) − 1 = − exp{−Q2/Q20}, (11) where σp is the cross section for reaction (2) on the free proton. The value of the fitted parameter Q20 is equal to 9 ± 3 (GeV/c)2. The measured values of the parameter α and the fitted curve α(Q2) are shown on the left panel of Fig. 4 taken from Ref. [24]. On the right panel of the figure the average value of f is plotted for the various Q2 bins used in the present analysis. The values of f are equal to about 0.36 in most of the Q2 range, rising to about 0.38 at the highest Q2. The radiative corrections (RC) have been neglected in the present analysis, in par- ticular in the calculation of f , because they are expected to be small for reaction (1). They were evaluated [25] to be of the order of 6% for the NMC exclusive ρ0 production analysis. The small values of RC are mainly due to the requirement of event exclusivity via cuts on Emiss and p t , which largely suppress the dominant external photon radiation. The internal (infrared and virtual) RC were estimated in Ref. [25] to be of the order of 5 Systematic errors The main systematic uncertainty of A 1 comes from an estimate of possible false asymmetries. In order to improve the accuracy of this estimate, in addition to the standard sample of incoherent events, a second sample was selected by changing the p2t cuts to 0 < p2t < 0.5 (GeV/c) 2, (12) and keeping all the remaining selections and cuts the same as for the ‘incoherent sample’. In the following it will be referred to as the ‘extended p2t sample’. Such an extension of the p2t range allows one to obtain a sample which is about five times larger than the incoherent sample. However, in addition to incoherent events such a sample contains a large fraction of events originating from coherent ρ0 production. Therefore, for the estimate of the dilution factor f a different nuclear dependence of the exclusive cross section was used, applicable for the sum of coherent and incoherent cross sections [2]. The physics asymmetries A 1 for both samples are consistent within statistical errors. Possible, false experimental asymmetries were searched for by modifying the se- lection of data sets used for the asymmetry calculation. The grouping of the data into configurations with opposite target-polarisation was varied from large samples, covering at most two weeks of data taking, into about 100 small samples, taken in time intervals of the order of 16 hours. A statistical test was performed on the distributions of asymmetries obtained from these small samples. In each of the Q2 and x bins the dispersion of the values of A 1 around their mean agrees with the statistical error. Time-dependent effects which would lead to a broadening of these distributions were thus not observed. Allowing the dispersion of A 1 to vary within its two standard deviations we obtain for each bin an upper bound for the systematic error arising from time-dependent effects σfalseA,tdep < 0.56 σstat. (13) Here σstat is the statistical error on A 1 for the extended p t sample. The uncertainty on the estimates of possible false asymmetries due to the time-dependent effects is the dominant contribution to the total systematic error in most of the kinematical region. Asymmetries for configurations where spin effects cancel out were calculated to check the cancellation of effects due to fluxes and acceptances. They were found compatible with zero within the statistical errors. Asymmetries obtained with different settings of the microwave (MW) frequency, used for DNP, were compared in order to test possible effects related to the orientation of the target magnetic field. The results for the extended p2t sample tend to show that there is a small difference between asymmetries for the two MW configurations. However, because the numbers of events of the data samples taken with each MW setting are approximately balanced, the effect of this difference on A negligible for the total sample. The systematic error on A 1 also contains an overall scale uncertainty of 6.5% due to uncertainties on PB and PT . The uncertainty of the parameterisation of R(Q 2) affects the depolarisation factor D. The uncertainty of the dilution factor f is mostly due to uncer- tainty of the parameter α(Q2) which takes into account nuclear effects in the incoherent ρ0 production. The neglect of the A 2 term mainly affects the highest bins of Q 2 and x. Another source of systematic errors is due to the contribution of the non-exclusive background to our sample. This background originates from two sources. First one is due to the production of ρ0 accompanied by the dissociation of the target nucleon, the second one is the production of ρ0 in inclusive scattering. In order to evaluate the amount of background in the sample of exclusive events it is necessary to determine the Emiss dependence for the non-exclusive background in the region under the exclusive peak (cf. Fig. 1 ). For this purpose complete Monte Carlo simulations of the experiment were used, with events generated by either the PYTHIA 6.2 or LEPTO 6.5.1 generators. Events generated with LEPTO come only from deep inelastic scattering and cover the range of Q2 > 0.5 (GeV/c)2. Those generated with PYTHIA cover the whole kinematical range of the experiment and include exclusive production of vector mesons and processes with diffractive excitation of the target nucleon or the vector meson, in addition to inelastic production. The generated MC events were reconstructed and selected for the analysis using the same procedure as for the data. In each bin of Q2 the Emiss distribution for the MC was normalised to the corresponding one for the data in the range of large Emiss > 7.5 GeV. Then the normalised MC distribution was used to estimate the number of background events under the exclusive peak in the data. The fraction of background events in the sample of incoherent exclusive ρ0 production was estimated to be about 0.12±0.06 in most of the kinematical range, except in the largest Q2 region, where it is about 0.24±0.12. The large uncertainties of these fractions reflect the differences between estimates from LEPTO and PYTHIA in the region where they overlap. In the case of PYTHIA the uncertainties on the cross sections for diffractive photo- and electroproduction of vector mesons also contribute. For events generated with PYTHIA the Emiss distributions for various physics processes could be studied separately. It was found that events of ρ0 production with an excitation of the target nucleon into N∗ resonances of small mass, M < 2 GeV/c2, cannot be resolved from the exclusive peak and therefore were not included in the estimates of number of background events. An estimate of the asymmetry A 1 for the background was obtained using a non- exclusive sample, which was selected with the standard cuts used in this analysis, except the cut on Emiss which was modified to Emiss > 2.5 GeV. In different high-Emiss bins A for this sample was found compatible with zero. Because no indication of a non-zero A 1 for the background was found, and also due to a large uncertainty of the estimated amount of background in the exclusive sample, no background corrections were made. Instead, the effect of background was treated as a source of systematic error. Its contribution to the total systematic error was not significant in most of the kinematical range, except for the highest Q2 and x. The total systematic error on A 1 was obtained as a quadratic sum of the errors from all discussed sources. Its values for each Q2 and x bin are given in Tables 1 and 2. The total systematic error amounts to about 40% of the statistical error for most of the kinematical range. Both errors become comparable in the highest bin of Q2. 6 Results The COMPASS results on A 1 are shown as a function of Q 2 and x in Fig. 5 and listed in Tables 1 and 2. The statistical errors are represented by vertical bars and the total systematic errors by shaded bands. [(GeV/c)2Q -310 -210 -110 1 10 -410 -310 -210 -110 Figure 5: A 1 as a function of Q 2 (left) and x (right) from the present analysis. Error bars correspond to statistical errors, while bands at the bottom represent the systematical errors. The wide range in Q2 covers four orders of magnitude from 3 · 10−3 to 7 (GeV/c)2. The domain in x which is strongly correlated with Q2, varies from 5 · 10−5 to about 0.05 (see Tables for more details). For the whole kinematical range the A 1 asymmetry measured by COMPASS is consistent with zero. As discussed in the introduction, this indicates that the role of unnatural parity exchanges, like π- or A1-Reggeon exchange, is Table 1: Asymmetry A 1 as a function of Q 2. Both the statistical errors (first) and the total systematic errors (second) are listed. Q2 range 〈Q2〉 [(GeV/c)2] 〈x〉 〈ν〉 [GeV] Aρ1 0.0004 − 0.005 0.0031 4.0 · 10−5 42.8 −0.030 ± 0.045 ± 0.014 0.005 − 0.010 0.0074 8.4 · 10−5 49.9 0.048 ± 0.038 ± 0.013 0.010 − 0.025 0.017 1.8 · 10−4 55.6 0.063 ± 0.026 ± 0.014 0.025 − 0.050 0.036 3.7 · 10−4 59.9 −0.035 ± 0.027 ± 0.009 0.05 − 0.10 0.072 7.1 · 10−4 62.0 −0.010 ± 0.028 ± 0.008 0.10 − 0.25 0.16 0.0016 62.3 −0.019 ± 0.029 ± 0.009 0.25 − 0.50 0.35 0.0036 60.3 0.016 ± 0.045 ± 0.014 0.5 − 1 0.69 0.0074 58.6 0.141 ± 0.069 ± 0.030 1 − 4 1.7 0.018 59.7 0.000 ± 0.098 ± 0.035 4 − 50 6.8 0.075 55.9 −0.85 ± 0.50 ± 0.39 small in that kinematical domain, which is to be expected if diffraction is the dominant process for reaction (2). In Fig. 6 the COMPASS results are compared to the HERMES results on A 1 ob- tained on a deuteron target [17]. Note that the lowest Q2 and x HERMES points, re- ferred to as ‘quasi-photoproduction’, come from measurements where the kinematics of the small-angle scattered electron was not measured but estimated from a MC simulation. This is in contrast to COMPASS, where scattered muon kinematics is measured even at the smallest Q2. [(GeV/c)2Q -310 -210 -110 1 10 COMPASS HERMES quasi-photoprod. (d) HERMES electroprod. (d) -410 -310 -210 -110 COMPASS HERMES quasi-photoprod. (d) HERMES electroprod. (d) Figure 6: A 1 as a function of Q 2 (left) and x (right) from the present analysis (circles) compared to HERMES results on the deuteron target (triangles). For the COMPASS results inner bars represent statistical errors, while the outer bars correspond to the total error. For the HERMES results vertical bars represent the quadratic sum of statistical and systematic errors. The curve represents the prediction explained in the text. The results from both experiments are consistent within errors. The kinematical range covered by the present analysis extends further towards small values of x and Q2 by almost two orders of magnitude. In each of the two experiments A 1 is measured at different average W , which is equal to about 10 GeV for COMPASS and 5 GeV for Table 2: Asymmetry A 1 as a function of x. Both the statistical errors (first) and the total systematic errors (second) are listed. x range 〈x〉 〈Q2〉 [(GeV/c)2] 〈ν〉 [GeV] Aρ1 8 · 10−6 − 1 · 10−4 5.8 · 10−5 0.0058 51.7 0.035 ± 0.026 ± 0.011 1 · 10−4 − 2.5 · 10−4 1.7 · 10−4 0.019 59.7 0.036 ± 0.024 ± 0.010 2.5 · 10−4 − 5 · 10−4 3.6 · 10−4 0.041 61.3 −0.039 ± 0.027 ± 0.012 5 · 10−4 − 0.001 7.1 · 10−4 0.082 60.8 −0.010 ± 0.030 ± 0.010 0.001 − 0.002 0.0014 0.16 58.6 −0.005 ± 0.036 ± 0.013 0.002 − 0.004 0.0028 0.29 54.8 0.032 ± 0.050 ± 0.019 0.004 − 0.01 0.0062 0.59 50.7 0.019 ± 0.069 ± 0.026 0.01 − 0.025 0.015 1.3 47.5 −0.03 ± 0.14 ± 0.06 0.025 − 0.8 0.049 3.9 43.8 −0.27 ± 0.38 ± 0.19 HERMES. Thus, no significant W dependence is observed for A 1 on an isoscalar nucleon target. The x dependence of the measured A 1 is compared in Fig. 6 to the prediction given by Eq. 4, which relates A 1 to the asymmetry A1 for the inclusive inelastic lepton- nucleon scattering. To produce the curve the inclusive asymmetry A1 was parameterised as A1(x) = (x α − γα) · (1 − e−βx) , where α = 1.158 ± 0.024, β = 125.1 ± 115.7 and γ = 0.0180 ± 0.0038. The values of the parameters have been obtained from a fit of A1(x) to the world data from polarised deuteron targets [26–31] including COMPASS measurements at very low Q2 and x [32]. Within the present accuracy the results on A are consistent with this prediction. In the highest Q2 bin, 〈Q2〉 = 6.8 (GeV/c)2, in the kinematical domain of applica- bility of pQCD-inspired models which relate the asymmetry to the spin-dependent GPDs for gluons and quarks (cf. Introduction), one can observe a hint of a possible nonzero asymmetry, although with a large error. It should be noted that in Ref. [18] a nega- tive value of ALL different from zero by about 2 standard deviations was reported at 〈Q2〉 = 7.7 (GeV/c)2. At COMPASS, including the data taken with the longitudinally polarised deuteron target in 2004 and 2006 will result in an increase of statistics by a factor of about three compared to the present paper, and thus may help to clarify the issue. For the whole Q2 range future COMPASS data, to be taken with the polarised proton target, would be very valuable for checking if the role of the flavour-blind exchanges is indeed dominant, as expected for the Pomeron-mediated process. 7 Summary The longitudinal double spin asymmetry A 1 for the diffractive muoproduction of ρ meson, µ + N → µ + N + ρ, has been measured by scattering longitudinally polarised muons off longitudinally polarised deuterons from the 6LiD target and selecting incoherent exclusive ρ0 production. The presented results for the COMPASS 2002 and 2003 data cover a range of energy W from about 7 to 15 GeV. The Q2 and x dependence of A 1 is presented in a wide kinematical range 3 · 10−3 ≤ Q2 ≤ 7 (GeV/c)2 and 5 · 10−5 ≤ x ≤ 0.05. These results extend the range in Q2 and x by two orders of magnitude down with respect to the existing data from HERMES. The asymmetry A 1 is compatible with zero in the whole x and Q 2 range. 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High purity bright single photon source J. S. Neergaard-Nielsen, B. Melholt Nielsen, H. Takahashi⋆, A. I. Vistnes† and E. S. Polzik QUANTOP, Danish National Research Foundation Center for Quantum Optics, Niels Bohr Institute, University of Copenhagen, DK 2100, Denmark ⋆ Permanent address: Department of Applied Physics, The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan, and National Institute of Information and Communications Technology, 4-2-1 Nukui-Kita, Koganei, Tokyo 184-8795, Japan † Permanent address: Department of Physics, University of Oslo, Postboks 1048, Blindern, 0316 Oslo, Norway jneergrd@nbi.dk Abstract: Using cavity-enhanced non-degenerate parametric down- conversion, we have built a frequency tunable source of heralded single photons with a narrow bandwidth of 8 MHz, making it compatible with atomic quantum memories. The photon state is 70% pure single photon as characterized by a tomographic measurement and reconstruction of the quantum state, revealing a clearly negative Wigner function. Furthermore, it has a spectral brightness of ∼1,500 photons/s per MHz bandwidth, making it one of the brightest single photon sources available. We also investigate the correlation function of the down-converted fields using a combination of two very distinct detection methods; photon counting and homodyne measurement. © 2021 Optical Society of America OCIS codes: (120.2920) Homodyning; (230.6080) Sources; (270.5290) Photon statistics. References and links 1. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002). 2. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001). 3. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurášek, and E. S. 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Orozco, “Time-dependent electric field fluctuations at the subphoton level,” Phys. Rev. A 66, 033807 (2002). 1. Introduction Pure single photon states produced efficiently and at a high rate are highly desirable for prac- tical implementations of various quantum information processing protocols, in particular in quantum cryptography [1], quantum computing with linear optics [2], and for testing quantum memories [3, 4]. The latter applications require at the same time compatibility with some kind of a quantum memory. Different approaches towards generation of a single photon state have been implemented in a number of physical systems. It should be noted that in many instances a source is claimed to be ”a single photon source” based just on the property of antibunch- ing, i.e., on the low rate of two-photon contribution compared to a single photon part. Such http://arxiv.org/abs/quant-ph/0609033 property should be combined with the likewise low contribution of the vacuum state, in order to claim a high-purity truly single photon source which is the aim of the present work. Single emitters usually suffer from low purity due to small collection efficiency for light. For example, quantum dot based sources [5, 6], color centres in diamond [7, 8], single molecules [9], or a single atom [10] have the detection efficiency/purity at best at a few percent level. Placing sin- gle emitters inside high-Q cavities improves the purity dramatically. However, even complex state-of-the-art experiments still have limited overall collection efficiency and thus low purity. The best results with quantum dots [11] show 8% collection efficiency/purity. Besides, quan- tum dots usually emit light in a several GHz bandwidth. The best efforts with cavity-QED with atoms or ions yield 30-40% efficiency just outside the cavity and the overall efficiency/purity at 10-20% level [12, 13, 14]. Recently atomic ensembles have been used to produce non-classical light [15, 16, 17], however, the light collection efficiency does not exceed a few percent even when atoms are placed inside a cavity [16]. Parametric down-conversion in free space non- linear crystals or waveguides [18, 19, 20, 21, 22, 23] has been widely used for generation of heralded photon pulses. The major disadvantage of parametric down-conversion is the random arrival time of the photons – the source is not deterministic. However, this is compensated by many attractive properties like well-defined wavelength, high collection efficiency, and non- cryogenic experimental setups. The standard pulsed, single-pass down-conversion process suf- fers from a limited photon generation rate which must be kept low to avoid pulses containing two photons, which is detrimental for quantum information applications. The bandwidth of the down-conversion is typically several nanometers, which means that the spectral brightness (the number of photons per MHz per second) is below one. This poses a serious limitation to the feasibility of interaction with atomic systems, where linewidths are on the MHz scale. To over- come this problem, the nonlinear crystal can be placed inside an optical cavity which serves to enhance the down-conversion process and limit the bandwidth of the output to that of the cav- ity [24]. Furthermore, the spatial field mode is defined by the cavity as well, so no additional spatial filtering is needed. Various studies on this type of setup have been performed, and the results do indeed show a marked increase in the attained spectral brightness [24, 25, 26]. In this paper we present our scheme for generation of heralded single photons with a very high purity and spectral brightness, and we perform homodyne tomography on these photons, which gives a complete image of the state of the source. Tomographic measurements of single photons have previously been performed in the pulsed regime [20, 27, 28] but not for continu- ously pumped systems. In overview, we operate an optical parametric oscillator (OPO) with a pump level which is far below the oscillation threshold; in effect this is just cavity-enhancement of the spontaneous parametric down-conversion of the nonlinear crystal. The ordinary phase matching bandwidth of the down-conversion process is several nm, but the cavity effectively inhibits down-conversion into frequencies which do not fulfill the resonance condition. Thus, the output of the OPO consists of several narrow-band frequency modes separated by the free spectral range (FSR) of the cavity. By appropriately filtering the output, we can obtain with high efficiency a single photon in a specific one of those modes conditioned on the detection of a trigger photon. The bandwidth of this photon will then be the cavity bandwidth which is very narrow compared to the phase matching bandwidth. 2. Experiment The OPO, as well as the rest of the setup, is depicted in Fig. 1. It is a bow-tie type cavity with a length of 81 cm corresponding to a FSR of 370 MHz. Centered between two 5 cm curva- ture mirrors is a 10 mm long PPKTP crystal which is periodically poled for noncritical phase matching around 860 nm. The output coupler has a transmission of T = 12.5%, and the total internal losses are L = 0.4%, giving a cavity HWHM bandwidth of γ1/2 = 2π 4.0 MHz and an Fig. 1. (Color online) Setup diagram. The second harmonic generator (SHG) pumps the optical parametric oscillator (OPO). The filter cavities should allow only a single mode (at frequency ω−) to reach the single photon counting avalanche photo diode (APD). Two acousto optic modulators (AOM) shift the main frequency to ω− and ω+ - the latter is used for the local oscillator (LO) of the homodyne measurement, the former for an alignment beam, which is used to bring all cavities resonant with ω− but which is blocked during measurement. escape efficiency ηesc = T/(T +L) = 0.97. With an effective nonlinearity ENL ≈ 0.020 W−1, the threshold pump power for oscillation is around Pthr = (T +L) 2/4ENL = 210 mW. The blue pump (430 nm) is generated by frequency doubling the main Ti:Sapph laser in a second har- monic generator (SHG) of similar geometry as the OPO, but with a KNbO3 crystal as the nonlinear medium. For single photon generation the pump should be rather weak to inhibit the population of higher photon numbers. The pumping strength is quantized as the pump pa- rameter ε = Pb/Pthr, where Pb is the blue pump power. This pump parameter is most easily inferred by observing the parametric gain, G= 1/(1−ε)2 of a beam of half the pump frequency seeded into the OPO. The frequency spectrum of the OPO is illustrated in Fig. 2. With no seed beam, the output field in the degenerate frequency mode (half pump frequency) is quadrature-squeezed vacuum, whereas the non-degenerate modes taken individually are thermal states. They are, however, pairwise correlated symmetrically around the degenerate frequency. In the weak pump regime this means that for each down-converted photon in the ω− mode one FSR below the degenerate frequency, there is a twin photon in the ω+ mode one FSR above. In the time domain, the field operator correlations for the two modes are given by [29]: 〈â±(t)â∓(t ′)〉 = λ 2 − µ2 e−µ|t−t e−λ |t−t 〈↱(t)â±(t ′)〉 = λ 2 − µ2 e−µ|t−t −λ |t−t′| 〈â±(t)â±(t ′)〉 = 〈↱(t)â∓(t ′)〉= 0 , λ = γ1/2(1+ ε) , µ = γ1/2(1− ε) . Fig. 2. (Color online) Schematic illustration of the frequency mode spectrum of the OPO (blue). The pump at frequency 2ω0 induces down-conversion into these and several other neighbouring modes. The ω− and ω+ modes are correlated, and they are separated on the first filter cavity which is resonant on ω− and reflects ω+ (red). Subsequent filters, of which one is depicted (orange), serves to further suppress uncorrelated modes in the trigger arm. Thus, if we can spatially separate the two frequency modes and detect the ω− photon on a single photon detector we have heralded the existence of an ω+ photon within a temporal mode determined by these correlations. This separation is done using an empty cavity which works as a frequency filter; the FSR is four times that of the OPO, so with the cavity resonant on ω−, the ω+ mode will be almost completely reflected. With this scheme the non-degenerate OPO has previously been used to produce highly quadrature entangled EPR beams [30]. Because of the wide phase matching bandwidth, many other modes than ω− will slip through the first filter cavity. Hence we need two more filter cavities with different FSR and a 0.3 nm interference filter on the way towards the photon counting avalanche photo diode (APD). If any photons uncorrelated with the ω+ photons arrive at the APD, the ω+ state conditioned on these “false” detections will be the original thermal state instead of a single photon. The spectral arrangement of these filters is illustrated in Fig. 2. The lengths of the cavities are 210 mm, 3.7 mm, and 12 mm, and the FWHM bandwidths are roughly 48 MHz, 270 MHz, and 96 MHz, respectively. To keep all cavities (OPO + filters) on resonance with the ω− frequency, we monitor the total transmission (the APD click rate) and keep it on maximum using individual error signals from each cavity obtained by dithering them at different frequencies. We recently employed the same series of filter cavities and APD to herald the generation of a photon subtracted squeezed vacuum state (a ”Schrödinger kitten“) [31]. With the ω− and ω+ modes thus separated and with the APD click heralding an ω+ photon, the existence of this photon must be confirmed. Instead of just measuring the arrival of the photons on another APD, we do a homodyne measurement of the field by mixing it on a 50/50 beam splitter with a strong local oscillator (LO) and subsequently recording the difference of the photocurrents measured in the two arms. The LO has been shifted by 370 MHz to the center frequency of the ω+ mode by sending part of the main laser beam through an AOM (acousto optic modulator). The detector employs two Hamamatsu photo diodes (special production of the S5971 type) with a specified quantum efficiency of 98%. It has a bandwidth of more than 100 MHz, and with 1.5 mW light on each diode the shot noise is 10 dB above the electronic noise floor. The output of the detector goes to a fast digital oscilloscope which samples the signal at 500 MS/s for a period of 2 µs around each APD trigger event. By repeating the state generation and measurement several thousand times, statistics about the quadrature distribution of the output state is build up. We scan the phase of the local oscillator to observe all quadrature phases, but as expected the distribution is completely phase invariant. In the post-processing of the recorded noise, we have to extract the conditional quadrature information from the thermal state background. This is done by applying a temporal mode function filter, fs(t), to the noise traces and afterwards integrate the traces over time. This leaves us with a single mode quadrature value corresponding to the operator âs = ηsâ+(t ′)+ 1−ηsâ+,vac(t ′) dt ′ , (2) where ηs is the total generation and detection efficiency of the signal, and the vacuum mode is added to maintain the commutator relations. For the very low gain regime (ε ≪ 1), the optimal field mode function for high single photon fidelity is simply the double-sided exponential [24, fs,opt(t) = γ1/2e −γ1/2|t−tc| , (3) with tc the time of the trigger event. For high gains the problem of finding the optimal mode function becomes somewhat more involved – see [32] – but this is not a big concern at the low gains at which we operate. Due to the filtering of the trigger photon, the correlations between trigger and signal will be smeared out, so our optimal mode function should be a bit wider and rounded off. However, since the narrowest trigger filter cavity has a bandwidth 6 times wider than that of the OPO, the effect is not very significant, and using just the first approximation to the optimal mode function above, we obtain fidelities quite similar to those obtained using more precise mode functions. The procedure of post-processing the homodyne photo current with temporal filtering is equivalent to performing the homodyne measurement with a pulsed LO of the same shape as the mode function. However, the shaping of the LO will have to be initiated by the trigger photon detection, and until the shaped LO is ready, the signal photon must be delayed so that the two fields reach the beam splitter simultaneously. See [33] for a detailed account of the problem of temporal/spectral mode matching in continuous-wave homodyning. 3. Analysis For the data presented in this paper, we performed a total number of 180,000 genera- tions/measurements of the single photon state. The measured parametric gain was about G ≈ 1.2, corresponding to a pump parameter ε ≈ 0.09 ≪ 1 (the effective blue pump power is ≈ 1.7 mW). Applying the mode function (3) to the noise traces, we get the quadrature dis- tribution shown in Fig. 3(a,b). A simple fit to the single photon quadrature distribution admixed with vacuum, η |〈q|1〉|2+(1−η)|〈q|0〉|2, shows that our data is consistent with a single photon state which has been detected with an efficiency of η = 62%. Based on the measured quadrature values and corresponding phases, we have reconstructed the density matrix and Wigner func- tion of the generated state, using the maximum likelihood tomographic reconstruction method [34]. The results are presented in Fig. 3(c,d). We see that our state consists almost entirely of n = 0 and n = 1 number states, with an n = 1 population of 61%. There is, however, a tiny contribution of the n = 2 and even higher number states. This is unavoidable in down- conversion based single photon sources; there is a finite probability that neighbouring photon pairs are produced so close to each other that they overlap, thus giving a higher average pho- ton number than 1 within the mode function. For very low gain these higher photon number components become insignificant, but at the same time, of course, the generation rate goes towards zero. With the current gain we have achieved a good compromise between genera- tion rate and low 2-photon contribution to the state. The average trigger detection rate in this measurement series was 12,800s−1. Corrected for the trigger photon losses – the APD detec- tion efficiency of 44%, the total trigger beam path transmission of 14%, and the OPO escape efficiency of 97% – the estimated photon production rate was Robserved ≈ 215,000s−1. This figure is close to the theoretically expected production rate in each cavity mode, which, from Fig. 3. (Color online) a) Part of the recorded quadrature data set with corresponding phases. b) Histogram of the distribution of all 180,000 conditional quadrature points (blue) and 40,000 vacuum points (red). The superimposed curves are the theoretical vacuum state distribution, and the single photon distribution fitted to the data with the total efficiency η as the only parameter. The fitted value is η = 0.625±0.002. The dashed curve is the ideal (η = 1) single photon distribution. c) The density matrix of the state, reconstructed via a maximum likelihood method, and in d) the corresponding Wigner function. eq. (1), is Rtheory = 〈↱(t)â±(t)〉= γ1/2ε2/(1− ε2)≈ 200,000s−1. Both of these numbers are, however, too uncertain to be the basis for an estimate of the number of false clicks. The given photon production rate corresponds to a spectral brightness of 1500 photons/s per MHz within the 8 MHz FWHM bandwidth. The inferred total detection efficiency, η = 62%, does not fit too well with the calculated value based on independent estimates of the various loss/efficiency contributions, which was as follows. The already mentioned escape efficiency of the OPO was 97%, the transmission towards the homodyne detector was 92%, and the visibility with the LO was 97% leading to an overlap efficiency of (97%)2. On top of these purely optical loss contributions come a spec- ified diode quantum efficiency of 98% and a contribution from the electronic noise of 91% (in the frequency range concerned, the electronic noise level is 10.5 dB below vacuum noise). In total, the estimated efficiency of generation and measurement is η = 75% – but this number is far from what we observe. A likely explanation for part of this discrepancy might be an in- sufficient suppression of the uncorrelated frequency modes in the series of trigger filters. As already mentioned, this would lead to a statistical admixture of the thermal state rather than vacuum. There is already a small amount of thermal state admixed due to the dark counts of the APD (∼ 100s−1), but since the thermal state for the low gain is almost indistinguishable from the vacuum, the effect of the thermal state admixture is basically identical to losses (vacuum admixture). Hence, it is also difficult to assert whether the discrepancy between expected and observed efficiency is due to insufficient filtering or unknown sources of loss. Such an addi- tional source of loss could possibly be the effect of low-frequency classical laser noise which is not completely balanced out in the homodyne setup, since the state selection done by the mode function integration includes all frequencies within the OPO bandwidth. A beginning diode saturation due to too high intensity on the tiny diodes might be another cause. Finally, the temporal mode function, chosen as (3), is not ideally matched to the single photon field and hence some vacuum is admixed to the state on this account. Any fluctuations in the arrival time of the photons would have the same effect. The Wigner function in Fig. 3(d) clearly has the shape of the single photon Fock state, al- though mixed with some vacuum. The negative dip has a value of W (0,0) = −0.070 – a clear signature of a non-classical state measured with high efficiency. If we correct the state for the purely measurement related losses (detector quantum efficiency and noise), we get a 70% pure state with a Wigner function dip of W (0,0) =−0.12. This state is what we obtain after mixing on the beam splitter with the local oscillator, and as such is the state which would be relevant for the storage in an atomic memory, where the quantum state to be stored must be mixed with a strong interaction field [3]. 4. Correlation function measurement Now we demonstrate how the cross correlation function between the two modes of the down- converted field can be extracted from the recorded data. Usually correlation functions are meas- ured via coincidence clicks on photon counting detectors. In [35] each photon counter is re- placed with a homodyne detection setup and the g(2)(τ) correlation function is calculated from the continuous frequency sideband measurements of the field quadratures. The scheme pre- sented here, which in the essence is similar to the work by Foster et al. [36], is a combination of these two approaches, where one mode is detected by a photon counter and the other by a homodyne setup. We use exactly the same setup and the same data as for the single photon generation and measurement. Figure 4 shows the point-wise variance of the 180,000 2µs long quadrature noise traces, together with a similar variance for the vacuum state. Before taking the variance, the traces have been low-pass filtered by a Lorentz-shaped filter with a 30 MHz cut-off. The increased variance of the conditioned state around the trigger time is evident and consistent with the expected quadrature variance of a single photon state which – in the ideal case – is 3/2 in the normalization where the vacuum variance is 1/2. The reason for the lower peak value is partly the limited detection efficiency, but also the effect of including frequency components far outside the OPO bandwidth where there is nothing but vacuum (a frequency filter much narrower than the 30 MHz would decrease the contribution from this vacuum and hence increase the variance, but it would also widen the temporal shape). This signal mode variance conditioned on a trigger photon detection at time tc is 〈∆q̂s(τ)2〉 〈â†t (tc)(q̂s(tc + τ))2ât(tc)〉 〈â†t ât〉 〈â†t (tc)â†s (tc + τ)âs(tc + τ)ât(tc)〉 〈â†t ât〉 , (4) where we used that the input states are Gaussian with zero mean, and that many correla- tion terms of the non-degenerate OPO are vanishing, cf. (1). The quadrature operator q̂ is defined as q̂ ≡ âe−iθ + â†eiθ 2, where the phase is made implicit due to the phase- invariance of the state. In order to calculate explicitly the expected variance, the signal and trigger modes, âs and ât , must include the detection efficiencies and any transformations – optical or electronic – applied to them, as done by the mode function in (2). For example, the filtering of the trigger field by the filter cavities must be taken into account. For uncorre- lated modes (for instance far away from tc), the variance reduces to the thermal state variance 〈∆q̂s(τ)2〉 uncond = 〈∆q̂s(τ)2〉 thermal = 1/2+ 〈â†s âs〉. The g ts (τ) cross-correlation function is Fig. 4. (Color online) a) Variance of the recorded quadrature noise traces for the signal conditioned on a trigger event at t = 0 (red) and for vacuum (black). The traces have been low-pass filtered with a bandwidth of 30 MHz to suppress most of the detector output which lies outside of the field bandwidth. b) The cross correlation function, g ts (t − tc) (with the trigger time tc = −29ns), calculated from the variances in a). Far away from the trigger time the value is 1, but the large values around tc demonstrate a strong correlation between the trigger and signal fields. The black curves are the theoretically expected functions for single photon state contents (versus thermal state) of 1, 0.8, and 0.6 (the lowest). now easily seen to be a simple expression of the quadrature variances ts (τ) ≡ 〈â†t (tc)â†s (tc + τ)âs(tc + τ)ât(tc)〉 〈â†t ât〉〈â†s âs〉 〈∆q̂s(τ)2〉 − 1/2 〈∆q̂s(τ)2〉|uncond − 1/2 . (5) In Fig. 4(b), this g ts function has been calculated from the variances in Fig. 4(a), where the thermal state variances have been calculated as the mean values of the traces far away from the trigger time. The expression (5) does no longer depend on the signal efficiency, which means that high frequency vacuum contributions play no role in the shape and size of the correlation function. In the figure are also plotted three expected g ts functions, calculated from a pump parameter ε = 0.09 and a statistical single photon content of 1, 0.8, and 0.6, respectively, where the remaining parts are made up of the thermal state. The optical filtering of the trigger mode (24 MHz, according to the bandwidth of the narrowest filter cavity) and the digital 30 MHz signal mode filtering have been included in these plots. In principle, since the correlation functions are independent on the signal detection efficiency but depend on the amount of thermal state admixture, it should be possible to find this amount by fitting these theoretical curves to the measurements. However, the uncertainty in the value of the thermal state variance (which is very close to 1/2) turns into a huge uncertainty in the derived g ts function – the error bars are so large that they are not displayed in the figure – so that this estimation is meaningless. More precise values would have been attainable if we had made a large number of measurements of the unconditioned thermal state. 5. Conclusion In conclusion, we have presented a high-purity and high-spectral brightness source of her- alded single photons, whose narrow bandwidth and clean spatial mode make it very suitable for interactions with an atomic memory. The generated single photons were characterized by homodyne tomography, showing a clearly negative Wigner function. Finally, we demonstrated how the correlation function between two fields can be measured by a combination of instan- taneous photon counting and continuous homodyne measurement. The 70% purity of the state demonstrated here should not be the limit. With some effort a purity of 90% is within reach for the present setup. This research has been funded in part by EU grants QAP and COVAQIAL. We acknowledge enlightening discussions with Anne E. B. Nielsen and Klaus Mølmer, and the kind assistance of Akira Furusawa in supplying us with the PPKTP crystal. Introduction Experiment Analysis Correlation function measurement Conclusion

Using cavity-enhanced non-degenerate parametric downconversion, we have built a frequency tunable source of heralded single photons with a narrow bandwidth of 8 MHz, making it compatible with atomic quantum memories. The photon state is 70% pure single photon as characterized by a tomographic measurement and reconstruction of the quantum state, revealing a clearly negative Wigner function. Furthermore, it has a spectral brightness of ~1,500 photons/s per MHz bandwidth, making it one of the brightest single photon sources available. We also investigate the correlation function of the down-converted fields using a combination of two very distinct detection methods; photon counting and homodyne measurement.

Introduction Pure single photon states produced efficiently and at a high rate are highly desirable for prac- tical implementations of various quantum information processing protocols, in particular in quantum cryptography [1], quantum computing with linear optics [2], and for testing quantum memories [3, 4]. The latter applications require at the same time compatibility with some kind of a quantum memory. Different approaches towards generation of a single photon state have been implemented in a number of physical systems. It should be noted that in many instances a source is claimed to be ”a single photon source” based just on the property of antibunch- ing, i.e., on the low rate of two-photon contribution compared to a single photon part. Such http://arxiv.org/abs/quant-ph/0609033 property should be combined with the likewise low contribution of the vacuum state, in order to claim a high-purity truly single photon source which is the aim of the present work. Single emitters usually suffer from low purity due to small collection efficiency for light. For example, quantum dot based sources [5, 6], color centres in diamond [7, 8], single molecules [9], or a single atom [10] have the detection efficiency/purity at best at a few percent level. Placing sin- gle emitters inside high-Q cavities improves the purity dramatically. However, even complex state-of-the-art experiments still have limited overall collection efficiency and thus low purity. The best results with quantum dots [11] show 8% collection efficiency/purity. Besides, quan- tum dots usually emit light in a several GHz bandwidth. The best efforts with cavity-QED with atoms or ions yield 30-40% efficiency just outside the cavity and the overall efficiency/purity at 10-20% level [12, 13, 14]. Recently atomic ensembles have been used to produce non-classical light [15, 16, 17], however, the light collection efficiency does not exceed a few percent even when atoms are placed inside a cavity [16]. Parametric down-conversion in free space non- linear crystals or waveguides [18, 19, 20, 21, 22, 23] has been widely used for generation of heralded photon pulses. The major disadvantage of parametric down-conversion is the random arrival time of the photons – the source is not deterministic. However, this is compensated by many attractive properties like well-defined wavelength, high collection efficiency, and non- cryogenic experimental setups. The standard pulsed, single-pass down-conversion process suf- fers from a limited photon generation rate which must be kept low to avoid pulses containing two photons, which is detrimental for quantum information applications. The bandwidth of the down-conversion is typically several nanometers, which means that the spectral brightness (the number of photons per MHz per second) is below one. This poses a serious limitation to the feasibility of interaction with atomic systems, where linewidths are on the MHz scale. To over- come this problem, the nonlinear crystal can be placed inside an optical cavity which serves to enhance the down-conversion process and limit the bandwidth of the output to that of the cav- ity [24]. Furthermore, the spatial field mode is defined by the cavity as well, so no additional spatial filtering is needed. Various studies on this type of setup have been performed, and the results do indeed show a marked increase in the attained spectral brightness [24, 25, 26]. In this paper we present our scheme for generation of heralded single photons with a very high purity and spectral brightness, and we perform homodyne tomography on these photons, which gives a complete image of the state of the source. Tomographic measurements of single photons have previously been performed in the pulsed regime [20, 27, 28] but not for continu- ously pumped systems. In overview, we operate an optical parametric oscillator (OPO) with a pump level which is far below the oscillation threshold; in effect this is just cavity-enhancement of the spontaneous parametric down-conversion of the nonlinear crystal. The ordinary phase matching bandwidth of the down-conversion process is several nm, but the cavity effectively inhibits down-conversion into frequencies which do not fulfill the resonance condition. Thus, the output of the OPO consists of several narrow-band frequency modes separated by the free spectral range (FSR) of the cavity. By appropriately filtering the output, we can obtain with high efficiency a single photon in a specific one of those modes conditioned on the detection of a trigger photon. The bandwidth of this photon will then be the cavity bandwidth which is very narrow compared to the phase matching bandwidth. 2. Experiment The OPO, as well as the rest of the setup, is depicted in Fig. 1. It is a bow-tie type cavity with a length of 81 cm corresponding to a FSR of 370 MHz. Centered between two 5 cm curva- ture mirrors is a 10 mm long PPKTP crystal which is periodically poled for noncritical phase matching around 860 nm. The output coupler has a transmission of T = 12.5%, and the total internal losses are L = 0.4%, giving a cavity HWHM bandwidth of γ1/2 = 2π 4.0 MHz and an Fig. 1. (Color online) Setup diagram. The second harmonic generator (SHG) pumps the optical parametric oscillator (OPO). The filter cavities should allow only a single mode (at frequency ω−) to reach the single photon counting avalanche photo diode (APD). Two acousto optic modulators (AOM) shift the main frequency to ω− and ω+ - the latter is used for the local oscillator (LO) of the homodyne measurement, the former for an alignment beam, which is used to bring all cavities resonant with ω− but which is blocked during measurement. escape efficiency ηesc = T/(T +L) = 0.97. With an effective nonlinearity ENL ≈ 0.020 W−1, the threshold pump power for oscillation is around Pthr = (T +L) 2/4ENL = 210 mW. The blue pump (430 nm) is generated by frequency doubling the main Ti:Sapph laser in a second har- monic generator (SHG) of similar geometry as the OPO, but with a KNbO3 crystal as the nonlinear medium. For single photon generation the pump should be rather weak to inhibit the population of higher photon numbers. The pumping strength is quantized as the pump pa- rameter ε = Pb/Pthr, where Pb is the blue pump power. This pump parameter is most easily inferred by observing the parametric gain, G= 1/(1−ε)2 of a beam of half the pump frequency seeded into the OPO. The frequency spectrum of the OPO is illustrated in Fig. 2. With no seed beam, the output field in the degenerate frequency mode (half pump frequency) is quadrature-squeezed vacuum, whereas the non-degenerate modes taken individually are thermal states. They are, however, pairwise correlated symmetrically around the degenerate frequency. In the weak pump regime this means that for each down-converted photon in the ω− mode one FSR below the degenerate frequency, there is a twin photon in the ω+ mode one FSR above. In the time domain, the field operator correlations for the two modes are given by [29]: 〈â±(t)â∓(t ′)〉 = λ 2 − µ2 e−µ|t−t e−λ |t−t 〈↱(t)â±(t ′)〉 = λ 2 − µ2 e−µ|t−t −λ |t−t′| 〈â±(t)â±(t ′)〉 = 〈↱(t)â∓(t ′)〉= 0 , λ = γ1/2(1+ ε) , µ = γ1/2(1− ε) . Fig. 2. (Color online) Schematic illustration of the frequency mode spectrum of the OPO (blue). The pump at frequency 2ω0 induces down-conversion into these and several other neighbouring modes. The ω− and ω+ modes are correlated, and they are separated on the first filter cavity which is resonant on ω− and reflects ω+ (red). Subsequent filters, of which one is depicted (orange), serves to further suppress uncorrelated modes in the trigger arm. Thus, if we can spatially separate the two frequency modes and detect the ω− photon on a single photon detector we have heralded the existence of an ω+ photon within a temporal mode determined by these correlations. This separation is done using an empty cavity which works as a frequency filter; the FSR is four times that of the OPO, so with the cavity resonant on ω−, the ω+ mode will be almost completely reflected. With this scheme the non-degenerate OPO has previously been used to produce highly quadrature entangled EPR beams [30]. Because of the wide phase matching bandwidth, many other modes than ω− will slip through the first filter cavity. Hence we need two more filter cavities with different FSR and a 0.3 nm interference filter on the way towards the photon counting avalanche photo diode (APD). If any photons uncorrelated with the ω+ photons arrive at the APD, the ω+ state conditioned on these “false” detections will be the original thermal state instead of a single photon. The spectral arrangement of these filters is illustrated in Fig. 2. The lengths of the cavities are 210 mm, 3.7 mm, and 12 mm, and the FWHM bandwidths are roughly 48 MHz, 270 MHz, and 96 MHz, respectively. To keep all cavities (OPO + filters) on resonance with the ω− frequency, we monitor the total transmission (the APD click rate) and keep it on maximum using individual error signals from each cavity obtained by dithering them at different frequencies. We recently employed the same series of filter cavities and APD to herald the generation of a photon subtracted squeezed vacuum state (a ”Schrödinger kitten“) [31]. With the ω− and ω+ modes thus separated and with the APD click heralding an ω+ photon, the existence of this photon must be confirmed. Instead of just measuring the arrival of the photons on another APD, we do a homodyne measurement of the field by mixing it on a 50/50 beam splitter with a strong local oscillator (LO) and subsequently recording the difference of the photocurrents measured in the two arms. The LO has been shifted by 370 MHz to the center frequency of the ω+ mode by sending part of the main laser beam through an AOM (acousto optic modulator). The detector employs two Hamamatsu photo diodes (special production of the S5971 type) with a specified quantum efficiency of 98%. It has a bandwidth of more than 100 MHz, and with 1.5 mW light on each diode the shot noise is 10 dB above the electronic noise floor. The output of the detector goes to a fast digital oscilloscope which samples the signal at 500 MS/s for a period of 2 µs around each APD trigger event. By repeating the state generation and measurement several thousand times, statistics about the quadrature distribution of the output state is build up. We scan the phase of the local oscillator to observe all quadrature phases, but as expected the distribution is completely phase invariant. In the post-processing of the recorded noise, we have to extract the conditional quadrature information from the thermal state background. This is done by applying a temporal mode function filter, fs(t), to the noise traces and afterwards integrate the traces over time. This leaves us with a single mode quadrature value corresponding to the operator âs = ηsâ+(t ′)+ 1−ηsâ+,vac(t ′) dt ′ , (2) where ηs is the total generation and detection efficiency of the signal, and the vacuum mode is added to maintain the commutator relations. For the very low gain regime (ε ≪ 1), the optimal field mode function for high single photon fidelity is simply the double-sided exponential [24, fs,opt(t) = γ1/2e −γ1/2|t−tc| , (3) with tc the time of the trigger event. For high gains the problem of finding the optimal mode function becomes somewhat more involved – see [32] – but this is not a big concern at the low gains at which we operate. Due to the filtering of the trigger photon, the correlations between trigger and signal will be smeared out, so our optimal mode function should be a bit wider and rounded off. However, since the narrowest trigger filter cavity has a bandwidth 6 times wider than that of the OPO, the effect is not very significant, and using just the first approximation to the optimal mode function above, we obtain fidelities quite similar to those obtained using more precise mode functions. The procedure of post-processing the homodyne photo current with temporal filtering is equivalent to performing the homodyne measurement with a pulsed LO of the same shape as the mode function. However, the shaping of the LO will have to be initiated by the trigger photon detection, and until the shaped LO is ready, the signal photon must be delayed so that the two fields reach the beam splitter simultaneously. See [33] for a detailed account of the problem of temporal/spectral mode matching in continuous-wave homodyning. 3. Analysis For the data presented in this paper, we performed a total number of 180,000 genera- tions/measurements of the single photon state. The measured parametric gain was about G ≈ 1.2, corresponding to a pump parameter ε ≈ 0.09 ≪ 1 (the effective blue pump power is ≈ 1.7 mW). Applying the mode function (3) to the noise traces, we get the quadrature dis- tribution shown in Fig. 3(a,b). A simple fit to the single photon quadrature distribution admixed with vacuum, η |〈q|1〉|2+(1−η)|〈q|0〉|2, shows that our data is consistent with a single photon state which has been detected with an efficiency of η = 62%. Based on the measured quadrature values and corresponding phases, we have reconstructed the density matrix and Wigner func- tion of the generated state, using the maximum likelihood tomographic reconstruction method [34]. The results are presented in Fig. 3(c,d). We see that our state consists almost entirely of n = 0 and n = 1 number states, with an n = 1 population of 61%. There is, however, a tiny contribution of the n = 2 and even higher number states. This is unavoidable in down- conversion based single photon sources; there is a finite probability that neighbouring photon pairs are produced so close to each other that they overlap, thus giving a higher average pho- ton number than 1 within the mode function. For very low gain these higher photon number components become insignificant, but at the same time, of course, the generation rate goes towards zero. With the current gain we have achieved a good compromise between genera- tion rate and low 2-photon contribution to the state. The average trigger detection rate in this measurement series was 12,800s−1. Corrected for the trigger photon losses – the APD detec- tion efficiency of 44%, the total trigger beam path transmission of 14%, and the OPO escape efficiency of 97% – the estimated photon production rate was Robserved ≈ 215,000s−1. This figure is close to the theoretically expected production rate in each cavity mode, which, from Fig. 3. (Color online) a) Part of the recorded quadrature data set with corresponding phases. b) Histogram of the distribution of all 180,000 conditional quadrature points (blue) and 40,000 vacuum points (red). The superimposed curves are the theoretical vacuum state distribution, and the single photon distribution fitted to the data with the total efficiency η as the only parameter. The fitted value is η = 0.625±0.002. The dashed curve is the ideal (η = 1) single photon distribution. c) The density matrix of the state, reconstructed via a maximum likelihood method, and in d) the corresponding Wigner function. eq. (1), is Rtheory = 〈↱(t)â±(t)〉= γ1/2ε2/(1− ε2)≈ 200,000s−1. Both of these numbers are, however, too uncertain to be the basis for an estimate of the number of false clicks. The given photon production rate corresponds to a spectral brightness of 1500 photons/s per MHz within the 8 MHz FWHM bandwidth. The inferred total detection efficiency, η = 62%, does not fit too well with the calculated value based on independent estimates of the various loss/efficiency contributions, which was as follows. The already mentioned escape efficiency of the OPO was 97%, the transmission towards the homodyne detector was 92%, and the visibility with the LO was 97% leading to an overlap efficiency of (97%)2. On top of these purely optical loss contributions come a spec- ified diode quantum efficiency of 98% and a contribution from the electronic noise of 91% (in the frequency range concerned, the electronic noise level is 10.5 dB below vacuum noise). In total, the estimated efficiency of generation and measurement is η = 75% – but this number is far from what we observe. A likely explanation for part of this discrepancy might be an in- sufficient suppression of the uncorrelated frequency modes in the series of trigger filters. As already mentioned, this would lead to a statistical admixture of the thermal state rather than vacuum. There is already a small amount of thermal state admixed due to the dark counts of the APD (∼ 100s−1), but since the thermal state for the low gain is almost indistinguishable from the vacuum, the effect of the thermal state admixture is basically identical to losses (vacuum admixture). Hence, it is also difficult to assert whether the discrepancy between expected and observed efficiency is due to insufficient filtering or unknown sources of loss. Such an addi- tional source of loss could possibly be the effect of low-frequency classical laser noise which is not completely balanced out in the homodyne setup, since the state selection done by the mode function integration includes all frequencies within the OPO bandwidth. A beginning diode saturation due to too high intensity on the tiny diodes might be another cause. Finally, the temporal mode function, chosen as (3), is not ideally matched to the single photon field and hence some vacuum is admixed to the state on this account. Any fluctuations in the arrival time of the photons would have the same effect. The Wigner function in Fig. 3(d) clearly has the shape of the single photon Fock state, al- though mixed with some vacuum. The negative dip has a value of W (0,0) = −0.070 – a clear signature of a non-classical state measured with high efficiency. If we correct the state for the purely measurement related losses (detector quantum efficiency and noise), we get a 70% pure state with a Wigner function dip of W (0,0) =−0.12. This state is what we obtain after mixing on the beam splitter with the local oscillator, and as such is the state which would be relevant for the storage in an atomic memory, where the quantum state to be stored must be mixed with a strong interaction field [3]. 4. Correlation function measurement Now we demonstrate how the cross correlation function between the two modes of the down- converted field can be extracted from the recorded data. Usually correlation functions are meas- ured via coincidence clicks on photon counting detectors. In [35] each photon counter is re- placed with a homodyne detection setup and the g(2)(τ) correlation function is calculated from the continuous frequency sideband measurements of the field quadratures. The scheme pre- sented here, which in the essence is similar to the work by Foster et al. [36], is a combination of these two approaches, where one mode is detected by a photon counter and the other by a homodyne setup. We use exactly the same setup and the same data as for the single photon generation and measurement. Figure 4 shows the point-wise variance of the 180,000 2µs long quadrature noise traces, together with a similar variance for the vacuum state. Before taking the variance, the traces have been low-pass filtered by a Lorentz-shaped filter with a 30 MHz cut-off. The increased variance of the conditioned state around the trigger time is evident and consistent with the expected quadrature variance of a single photon state which – in the ideal case – is 3/2 in the normalization where the vacuum variance is 1/2. The reason for the lower peak value is partly the limited detection efficiency, but also the effect of including frequency components far outside the OPO bandwidth where there is nothing but vacuum (a frequency filter much narrower than the 30 MHz would decrease the contribution from this vacuum and hence increase the variance, but it would also widen the temporal shape). This signal mode variance conditioned on a trigger photon detection at time tc is 〈∆q̂s(τ)2〉 〈â†t (tc)(q̂s(tc + τ))2ât(tc)〉 〈â†t ât〉 〈â†t (tc)â†s (tc + τ)âs(tc + τ)ât(tc)〉 〈â†t ât〉 , (4) where we used that the input states are Gaussian with zero mean, and that many correla- tion terms of the non-degenerate OPO are vanishing, cf. (1). The quadrature operator q̂ is defined as q̂ ≡ âe−iθ + â†eiθ 2, where the phase is made implicit due to the phase- invariance of the state. In order to calculate explicitly the expected variance, the signal and trigger modes, âs and ât , must include the detection efficiencies and any transformations – optical or electronic – applied to them, as done by the mode function in (2). For example, the filtering of the trigger field by the filter cavities must be taken into account. For uncorre- lated modes (for instance far away from tc), the variance reduces to the thermal state variance 〈∆q̂s(τ)2〉 uncond = 〈∆q̂s(τ)2〉 thermal = 1/2+ 〈â†s âs〉. The g ts (τ) cross-correlation function is Fig. 4. (Color online) a) Variance of the recorded quadrature noise traces for the signal conditioned on a trigger event at t = 0 (red) and for vacuum (black). The traces have been low-pass filtered with a bandwidth of 30 MHz to suppress most of the detector output which lies outside of the field bandwidth. b) The cross correlation function, g ts (t − tc) (with the trigger time tc = −29ns), calculated from the variances in a). Far away from the trigger time the value is 1, but the large values around tc demonstrate a strong correlation between the trigger and signal fields. The black curves are the theoretically expected functions for single photon state contents (versus thermal state) of 1, 0.8, and 0.6 (the lowest). now easily seen to be a simple expression of the quadrature variances ts (τ) ≡ 〈â†t (tc)â†s (tc + τ)âs(tc + τ)ât(tc)〉 〈â†t ât〉〈â†s âs〉 〈∆q̂s(τ)2〉 − 1/2 〈∆q̂s(τ)2〉|uncond − 1/2 . (5) In Fig. 4(b), this g ts function has been calculated from the variances in Fig. 4(a), where the thermal state variances have been calculated as the mean values of the traces far away from the trigger time. The expression (5) does no longer depend on the signal efficiency, which means that high frequency vacuum contributions play no role in the shape and size of the correlation function. In the figure are also plotted three expected g ts functions, calculated from a pump parameter ε = 0.09 and a statistical single photon content of 1, 0.8, and 0.6, respectively, where the remaining parts are made up of the thermal state. The optical filtering of the trigger mode (24 MHz, according to the bandwidth of the narrowest filter cavity) and the digital 30 MHz signal mode filtering have been included in these plots. In principle, since the correlation functions are independent on the signal detection efficiency but depend on the amount of thermal state admixture, it should be possible to find this amount by fitting these theoretical curves to the measurements. However, the uncertainty in the value of the thermal state variance (which is very close to 1/2) turns into a huge uncertainty in the derived g ts function – the error bars are so large that they are not displayed in the figure – so that this estimation is meaningless. More precise values would have been attainable if we had made a large number of measurements of the unconditioned thermal state. 5. Conclusion In conclusion, we have presented a high-purity and high-spectral brightness source of her- alded single photons, whose narrow bandwidth and clean spatial mode make it very suitable for interactions with an atomic memory. The generated single photons were characterized by homodyne tomography, showing a clearly negative Wigner function. Finally, we demonstrated how the correlation function between two fields can be measured by a combination of instan- taneous photon counting and continuous homodyne measurement. The 70% purity of the state demonstrated here should not be the limit. With some effort a purity of 90% is within reach for the present setup. This research has been funded in part by EU grants QAP and COVAQIAL. We acknowledge enlightening discussions with Anne E. B. Nielsen and Klaus Mølmer, and the kind assistance of Akira Furusawa in supplying us with the PPKTP crystal. Introduction Experiment Analysis Correlation function measurement Conclusion

arXiv:0704.1865v1 [math.CA] 14 Apr 2007 On L1-Convergence of Fourier Series Under MVBV Condition Dansheng Yu,∗Ping Zhou†and Songping Zhou‡ Abstract Let f ∈ L2π be a real-valued even function with its Fourier series a02 + an cosnx, and let Sn (f, x) , n ≥ 1, be the n-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence {an} is decreasing and lim an = 0, then ‖f − Sn(f)‖L = 0 if and only if lim an logn = 0. We weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV ) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper which gives the L1-convergence of a function f ∈ L2π in complex space. We also give results on L1-approximation of a function f ∈ L2π under the MVBV condition. 2000 Mathematics Subject Classification: 42A25, 41A50. Keywords: complex trigonometric series, L1 convergence, monotonicity, mean value bounded variation. 1 Introduction Let L2π be the space of all complex-valued integrable functions f(x) of period 2π equipped with the norm ‖f‖L = |f(x)|dx. ∗Supported in part by NSERC RCD grant of St. Francis Xavier University and in part by AARMS of Canada. †Supported by NSERC of Canada. ‡The third author’s research is done as a W. F. James Chair Professor of St. Francis Xavier University. His research is also supported in part by NSF of China under grant number 10471130. http://arxiv.org/abs/0704.1865v1 Denote the Fourier series of f ∈ L2π by f̂(k)eikx, (1) and its partial sum Sn(f, x) by f̂(k)eikx. When f(x) ∈ L2π is a real valued even function, then the Fourier series of f has the ak cos kx, (2) correspondingly, its partial sum Sn(f, x) is ak cos kx. The following two classical convergence results can be found in many monographs (see [1] and [9], for example): Result One: If a nonnegative sequence {bn}∞n=1 is decreasing and limn→∞ bn = 0, then the series bn sinnx converges uniformly if and only if lim nbn = 0. Result Two: Let f ∈ L2π be an even function and (2) be its Fourier series. If the sequence {an}∞n=0 is nonnegative, decreasing, and limn→∞ an = 0, then ‖f − Sn(f)‖L = 0 if and only if lim an logn = 0. These results have been generalized by weakening the monotone conditions of the coefficient sequences. They have also been generalized to the complex valued function spaces. The most recent generalizations of Result One can be found in [8] where the monotonic condition is finally weakened to the MVBV condition (Mean Value Bounded Variation condition, see Corollary 2 in Section 2 for definition), and it is proved to be the weakest possible condition we can have to replace the monotone condition in Result One. The process of generalizing Result Two can be found in many papers, for example, see [2] - [7]. In this paper, we will weaken the monotone condition in Result Two (and all its later generalized conditions, see [8] for the relations between these conditions), to theMV BV condition in the complex valued function spaces (see Definition 1 in Section 2) in Theorem 1, and give the generalization in real valued function spaces as a special case of Theorem 1 in Corollary 2. Like the important role that theMVBV condition plays in generalizing Result One, although we are not able to prove it here, we propose that Theorem 1 in Section 2 is the ultimate generalization of Result Two, i.e. the MVBV condition is also the weakest possible condition we can have to replace the monotone condition in Result Two. We also discuss, under the MVBV condition, the L1-approximation rate of a function f ∈ L2π in the last section. Throughout this paper, we always use C(x) to indicate a positive constant de- pending upon x only, and use C to indicates an absolute positive constant. They may have different values in different occurrences. 2 L1 convergence In this section, we first give the definition ofMV BV condition, or the classMVBV S, and then prove our main result on L1-convergence of the Fourier series of a complex valued function f(x) ∈ L2π whose coefficients form a sequence in the class MVBV S. Definition 1. Let c := {cn}∞n=0 be a sequence of complex numbers satisfying cn ∈ K(θ1) := {z : | arg z| ≤ θ1} for some θ1 ∈ [0, π/2) and all n = 0, 1, 2, · · · . If there is a number λ ≥ 2 such that |∆ck| := |ck+1 − ck| ≤ C(c) k=[λ−1m] holds for all m = 1, 2, · · · , then we say that the sequence c is a Mean Value Bounded Variation Sequence, i.e., c ∈ MVBVS, in complex sense, or the sequence c satisfies the MVBV condition. Our main result of this paper is: Theorem 1. Let f(x) ∈ L2π be a complex-valued function. If the Fourier coef- ficients f̂(n) of f satisfy that {f̂(n)}+∞n=0 ∈ MVBV S and lim sup |∆f̂(k)−∆f̂(−k)| log k = 0, (3) where ∆f̂(k) = f̂(k + 1)− f̂(k), ∆f̂(−k) = f̂(−k − 1)− f̂(−k), k ≥ 0. ‖f − Sn(f)‖L = 0 if and only if f̂(n) log |n| = 0. In order to prove Theorem 1, we present the following four lemmas. Lemma 1. Let {cn} ∈MVBV S, then for any given 1 < µ < 2, we have |∆ck| log k = O [λ−1n]≤k≤[λn] |ck| log k , n→ ∞, where the implicit constant depends only on the sequence {cn} and λ. For sufficiently large n, the lemma can be derived directly from the conditions that 1 < µ < 2 and {cn} ∈ MVBV S. Lemma 2. Let {f̂(n)} ∈ K(θ0) for some θ0 ∈ [0, π/2), then ∣f̂(n+ k) ∣ = O (‖f − Sn(f)‖L) for all n = 1, 2, · · · , where the implicit constant depends only on θ0. Proof. Write φ±n(x) := ei(k∓n)x − e−i(k±n)x It follows from a well-known inequality (e.g. see Theorem 2.5 in [6]) sin kx |φ±n(x)| ≤ 6 Hence (f(x)− Sn(f, x))φ±n(x)dx ≤ ‖f − Sn(f)‖L, and therefore f̂(n + k) = O(‖f − Sn(f)‖L). Now as {f̂(n)} ∈ K(θ0) for some θ0 ∈ [0, π/2) and for all n ≥ 1, we have ∣f̂(n + k) ∣ ≤ C(θ0) Ref̂(n + k) ≤ C(θ0) f̂(n+ k) = O(‖f − Sn(f)‖L). Lemma 3. ([5]). Write Dk(x) : = sin((2k + 1)x/2) 2 sin(x/2) D∗k(x) : = cos(x/2)−cos((2k+1)x/2) 2 sin(x/2) |x| ≤ 1/n, − cos((2k+1)x/2) 2 sin(x/2) 1/n ≤ |x| ≤ π, Ek(x) : = Dk(x) + iD k(x). For k = n, n + 1, · · · , 2n, we have Ek(±x)−Ek−1(±x) = e±ikx, (4) Ek(x) + Ek(−x) = 2Dk(x), (5) ‖Ek‖L + ‖Dk‖L = O(log k). (6) Lemma 4. Let {f̂(n)} ∈MVBV S. If lim ‖f − Sn(f)‖L = 0, then f̂(n) log n = 0. Proof. By the definition of MV BV S, we derive that for k = n, n + 1, · · · , 2n, |f̂(2n)| ≤ |∆f̂(j)|+ |f̂(k)| |∆f̂(j)|+ |f̂(k)| j=[λ−1k] |f̂(j)| + |f̂(k)|. Therefore, it follows that from the fact that log n ≤ C (λ) [(λ+1)−2n] j=[λ]+1 we have |f̂(2n)| logn ≤ C(λ)|f̂(2n)| [(λ+1)−2n] j=[λ]+1 ≤ C(λ) [(λ+1)−2n] j=[λ]+1 [λ(n+j)] k=[λ−1(n+j)] |f̂(k)|+ |f̂(n+ j)| [(λ+1)−2n] j=[λ]+1 [λ(n+j)] k=[λ−1(n+j)] |f̂(k)| +C(λ) [(λ+1)−2n] |f̂(n+ j)| = : I1 + I2. (7) By applying Lemma 2, we see that I2 ≤ C(λ, θ0)‖f − Sn(f)‖L. (8) We calculate I1 as follows (note that we may add more repeated terms in the right hand side of every inequality below): [(λ+1)−2n] j=[λ]+1 [λn]+[λj]+1 k=[λ−1n]+[λ−1j] |f̂(k)| ≤ C(λ) [(λ+1)−2n] j=[λ]+1 [(λ+1)2] [λn]+1 k=[λ−1n] ∣f̂ (m[λ−1j] + k) ≤ C(λ) [(λ+1)2] [(λ+1)−2n] j=[λ]+1 [λn]−[λ−1n]+1 ∣f̂ ([λ−1n] +m[λ−1j] + k) ≤ C(λ) [(λ+1)2] [λn]−[λ−1n]+1 (λ(λ+1)2) ∣f̂ ([λ−1n] + k + j) [(λ+1)2] [λn]−[λ−1n]+1 ∥f − S[λ−1n]+k(f) (by Lemma 2) [λn]−[λ−1n]+1 ∥f − S[λ−1n]+k(f) . (9) Finally, by combining (7) - (9) and the condition ‖f − Sn(f)‖L = 0, we get f̂(2n) logn = 0. A similar argument yields that |f̂(2n + 1)| logn = 0. This proves Lemma 4. We now come to the proof of Theorem 1. Proof of Theorem 1. Sufficiency. Given ε > 0, by (3), there is a 1 < µ < 2 such that ∣∆f̂(k)−∆f̂(−k) ∣ log k ≤ ε (10) holds for sufficiently large n > 0. Let τµn,n(f, x) := [µn]− n [µn]−1 Sk(f, x) be the Vallée Poussin sum of order n of f . Then we have ‖f − τµn,n(f)‖L = 0. (11) By (4), (5), and applying Abel transformation, we get τµn,n(f, x)− Sn(f, x) [µn]− n k=n+1 ([µn]− k) f̂(k)eikx + f̂(−k)e−ikx [µn]− n ([µn]− k) 2∆f̂(k)Dk(x)− (∆f̂(k)−∆f̂(−k))Ek(−x) [µn]− n [µn]−1 f̂(k + 1)Ek(x)− f̂(−k − 1)Ek(−x) f̂(n)En(x) + f̂(−n)En(−x) . (12) Thus, by (6) and Lemma 1, we have ‖f − Sn(f)‖L ≤ ‖f − τµn,n(f)‖L + ‖τµn,n(f)− Sn(f)‖L = ‖f − τµn,n(f)‖L +O ∣∆f̂(k) ∣ log k ∣∆f̂(k)−∆f̂(−k) ∣ log k n≤|k|≤[µn] |f̂(k)| log |k| = ‖f − τµn,n(f)‖L +O [λ−1n]≤|k|≤[λn] |f̂(k)| log |k| ∣∆f̂(k)−∆f̂(−k) ∣ log k  , (13) lim sup ‖f − Sn(f)‖L ≤ ε follows from (10), (11) and the condition that f̂(n) log |n| = 0. This implies that ‖f − Sn(f)‖L = 0. Necessity. Since {f̂(n)} ∈MV BV S, by applying Lemma 4, we have f̂(n) log n = 0. (14) In order to prove lim f̂(n) log |n| = 0, by applying (12) and (6), we see that for any given µ, 1 < µ < 2, ‖f̂(−n)En(−x)‖L ≤ ‖τµn,n(f)− Sn(f)‖L [µn]− n [µn]−1 f̂(−k − 1)Ek(−x) |∆f̂(k)−∆f̂(−k)| log k + ∣∆f̂(k) ∣ log k n≤k≤[µn] |f̂(k)| log k . (15) It is not difficult to see that [µn]−1 f̂(−k − 1)Ek(−x) = I +O n max n<k≤[µn] |f̂(−k)| where n−1≤|x|≤π 2 sin(x/2) [µn]−1 f̂(−k − 1)e i(2k+1)x Since the trigonometric function system is orthonormal, we have n−1≤|x|≤π [µn]−1 f̂(−k − 1)e i(2k+1)x sin2(x/2) k=n+1 |f̂(−k)|2 n max n≤k≤[µn] |f̂(−k)| which yields that [µn]− n [µn]−1 f̂(−k − 1)Ek(−x) n<k≤[µn] |f̂(−k)| . (16) By combining (11), (14) - (16), with Lemma 1 and the condition ‖f − Sn(f)‖L = 0, and the fact (since f ∈ L2π) that f̂(−n) = 0, we have for n→ ∞, ‖f̂(−n)En(−x)‖L ≤ |∆f̂(k)−∆f̂(−k)| log k +‖τµn,n(f)− Sn(f)‖L [λ−1n]≤k≤[λn] |f̂(k)| log k n<k≤[µn] |f̂(−k)| |∆f̂(k)−∆f̂(−k)| log k + o(1). (17) On the other hand, we have ‖f̂(−n)En(−x)‖L ≥ |f̂(−n)|‖Dn(x)‖L ≥ |f̂(−n)| logn. (18) Hence, from (17), (18), and (10), we have that |f̂(−n)| logn ≤ |∆f̂(k)−∆f̂(−k)| log k ≤ ε holds for sufficiently large n, which, together with (14), completes the proof of neces- sity. In view of Lemma 1, we can see that the condition (3) in Theorem 1 can be replaced by the following condition lim sup |∆f̂(−k)| log k = 0, and the proof of the result is easier. Therefore we have a corollary to Theorem 1. Corollary 1. Let f(x) ∈ L2π be a complex valued function. If both {f̂(n)}+∞n=0 ∈ MVBV S and {f̂(−n)}+∞n=0 ∈MVBV S, then ‖f − Sn(f)‖L = 0 if and only if f̂(n) log |n| = 0. If f(x) is a real valued function, then its Fourier coefficients f̂(n) and f̂(−n) are a pair of conjugate complex numbers. Consequently, {f̂(n)}+∞n=0 ∈ MVBV S if and only if {f̂(−n)}+∞n=0 ∈ MVBV S. Thus, we have the following generalization of the classical result (cf. Result Two in the introduction): Corollary 2. Let f(x) ∈ L2π be a real valued even function and (2) be its Fourier series. If A = {an}+∞n=0 ∈MV BV S in real sense, i.e. {an} is a nonnegative sequence, and there is a number λ ≥ 2 such that |∆ak| ≤ C(A) k=[λ−1m] for all n = 1, 2, . . . , then ‖f − Sn(f)‖L = 0 if and only if an logn = 0. 3 L1 Approximation Let En(f)L be the best approximation of a complex valued function f ∈ L2π by trigonometric polynomials of degree n in L1 norm, that is, En(f)L := inf We establish the corresponding L1−approximation theorem in a similar way to Theorem 1: Theorem 2. Let f(x) ∈ L2π be a complex valued function, {ψn} a decreasing sequence tending to zero with ψn ∼ ψ2n, (19) i.e., there exist positive constants C1 and C2, such that C1ψn ≤ ψ2n ≤ C2ψn. If both {f̂(n)}+∞n=0 ∈MV BV S and {f̂(−n)}+∞n=0 ∈MVBV S, then ‖f − Sn(f)‖L = O(ψn) (20) if and only if En(f)L = O(ψn) and f̂(n) log |n| = O(ψ|n|). (21) Proof. Under the condition of Theorem 2, we see from (13) in the proof of Theorem 1 that ‖f − Sn(f)‖L ≤ ‖f − τµn,n(f)‖L +O [λ−1n]≤|k|≤[λn] |f̂(k)| log |k| ≤ C(µ)En(f) +O [λ−1n]≤|k|≤[λn] |f̂(k)| log |k| thus (20) holds if (19) and (21) hold. Now if (20) holds, then En(f)L = O(ψn) ‖f − τµn,n(f)‖L = O(ψn). From (7) - (9) in the proof of Lemma 4 and condition (19), we have |f̂(n)| logn ≤ C(λ) [λn]−[λ−1n]+1 ∥f − S[λ−1n]+j(f) +C(λ)‖f − Sn(f)‖L = O(ψn). Since {f̂(−n)}+∞n=0 ∈ MVBV S, by a similar argument to (22), we also have ∣f̂(−n) ∣ logn = O(ψn). This completes the proof of Theorem 2. In particular, if we take ψn := (n+ 1)r f (r), n + 1 where r is a positive integer, and ω(f, t)L is the modulus of continuity of f in L norm, i.e. ω(f, t)L := max 0≤h≤t ‖f(x+ h)− f(x)‖L. By Theorem 2 and the Jackson theorem (e.g. see [6] or [9]) in L1−space, we imme- diately have Corollary 3. Let f(x) ∈ L2π be a complex valued function. If both {f̂(n)}+∞n=0 ∈ MVBV S and {f̂(−n)}+∞n=0 ∈MVBV S hold, then ‖f − Sn(f)‖L = O (n+ 1)r f (r), n + 1 if and only if f̂(n) log |n| = O (n+ 1)r f (r), n + 1 This corollary generalizes the corresponding results in [5] and [2]. References [1] P. R. Boas Jr., Integrability theorems for trigonometric transforms, Springer, Ergebnisse 38, Berlin 1967. [2] R. J. Le and S. P. Zhou, On L1 convergence of Fourier series of complex valued functions, Studia Sci. Math. Hungar., to appear. [3] V. B. Stanojevic, L1−convergence of Fourier series with complex quasimonotone coefficients, Proc. Amer. Math. Soc., 86(1982), 241-247. [4] V. B. Stanojevic, L1−convergence of Fourier series with O−regularly varying quasimonotone coefficients, J. Approx. Theory, 60(1990), 168-173. [5] T. F. Xie and S. P. Zhou, L1−approximation of Fourier series of complex valued functions, Proc. Royal Soc. Edinburg, 126A(1996), 343-353. [6] T. F. Xie and S. P. Zhou, Approximation Theory of Real Functions, Hangzhou University Press, 1998. [7] D. S. Yu and S. P. Zhou, A generalization of monotonicity condition and applica- tions, Acta Math. Hungar., to appear. [8] S. P. Zhou, P. Zhou and D. S. Yu, Ultimate generalization to monotonicity for uni- form convergence of trigonometric series, arXiv:math.CA/0611805 v1, November 27, 2006, preprint. [9] A. Zygmund, Trigonometric Series, 2nd. Ed., Vol.I, Cambridge Univ. Press, Cam- bridge, 1959. Dan Sheng Yu and Song Ping Zhou: Institute of Mathematics Zhejiang Sci-Tech University Xiasha Economic Development Area Hangzhou, Zhejiang 310018 China Department of Mathematics, Statitics & Computer Science St. Francis Xavier University Antigonish, Nova Scotia, Canada B2G 2W5 Email: dsyu@zjip.com (D. S. Yu) szhou@zjip.com (S. P. Zhou) Ping Zhou: Department of Mathematics, Statitics & Computer Science St. Francis Xavier University Antigonish, Nova Scotia, Canada B2G 2W5 Email: pzhou@stfx.ca

Let $f\in L_{2\pi}$ be a real-valued even function with its Fourier series $ \frac{a_{0}}{2}+\sum_{n=1}^{\infty}a_{n}\cos nx,$ and let $S_{n}(f,x), n\geq 1,$ be the $n$-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence $\{a_{n}\}$ is decreasing and $\lim\limits_{n\to \infty}a_{n}=0$, then $$ \lim\limits_{n\to \infty}\Vert f-S_{n}(f)\Vert_{L}=0 {if and only if} \lim\limits_{n\to \infty}a_{n}\log n=0. $$ We weaken the monotone condition in this classical result to the so-called mean value bounded variation ($MVBV$) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper which gives the $L^{1}$% -convergence of a function $f\in L_{2\pi}$ in complex space. We also give results on $L^{1}$-approximation of a function $f\in L_{2\pi}$ under the $% MVBV$ condition.

Introduction Let L2π be the space of all complex-valued integrable functions f(x) of period 2π equipped with the norm ‖f‖L = |f(x)|dx. ∗Supported in part by NSERC RCD grant of St. Francis Xavier University and in part by AARMS of Canada. †Supported by NSERC of Canada. ‡The third author’s research is done as a W. F. James Chair Professor of St. Francis Xavier University. His research is also supported in part by NSF of China under grant number 10471130. http://arxiv.org/abs/0704.1865v1 Denote the Fourier series of f ∈ L2π by f̂(k)eikx, (1) and its partial sum Sn(f, x) by f̂(k)eikx. When f(x) ∈ L2π is a real valued even function, then the Fourier series of f has the ak cos kx, (2) correspondingly, its partial sum Sn(f, x) is ak cos kx. The following two classical convergence results can be found in many monographs (see [1] and [9], for example): Result One: If a nonnegative sequence {bn}∞n=1 is decreasing and limn→∞ bn = 0, then the series bn sinnx converges uniformly if and only if lim nbn = 0. Result Two: Let f ∈ L2π be an even function and (2) be its Fourier series. If the sequence {an}∞n=0 is nonnegative, decreasing, and limn→∞ an = 0, then ‖f − Sn(f)‖L = 0 if and only if lim an logn = 0. These results have been generalized by weakening the monotone conditions of the coefficient sequences. They have also been generalized to the complex valued function spaces. The most recent generalizations of Result One can be found in [8] where the monotonic condition is finally weakened to the MVBV condition (Mean Value Bounded Variation condition, see Corollary 2 in Section 2 for definition), and it is proved to be the weakest possible condition we can have to replace the monotone condition in Result One. The process of generalizing Result Two can be found in many papers, for example, see [2] - [7]. In this paper, we will weaken the monotone condition in Result Two (and all its later generalized conditions, see [8] for the relations between these conditions), to theMV BV condition in the complex valued function spaces (see Definition 1 in Section 2) in Theorem 1, and give the generalization in real valued function spaces as a special case of Theorem 1 in Corollary 2. Like the important role that theMVBV condition plays in generalizing Result One, although we are not able to prove it here, we propose that Theorem 1 in Section 2 is the ultimate generalization of Result Two, i.e. the MVBV condition is also the weakest possible condition we can have to replace the monotone condition in Result Two. We also discuss, under the MVBV condition, the L1-approximation rate of a function f ∈ L2π in the last section. Throughout this paper, we always use C(x) to indicate a positive constant de- pending upon x only, and use C to indicates an absolute positive constant. They may have different values in different occurrences. 2 L1 convergence In this section, we first give the definition ofMV BV condition, or the classMVBV S, and then prove our main result on L1-convergence of the Fourier series of a complex valued function f(x) ∈ L2π whose coefficients form a sequence in the class MVBV S. Definition 1. Let c := {cn}∞n=0 be a sequence of complex numbers satisfying cn ∈ K(θ1) := {z : | arg z| ≤ θ1} for some θ1 ∈ [0, π/2) and all n = 0, 1, 2, · · · . If there is a number λ ≥ 2 such that |∆ck| := |ck+1 − ck| ≤ C(c) k=[λ−1m] holds for all m = 1, 2, · · · , then we say that the sequence c is a Mean Value Bounded Variation Sequence, i.e., c ∈ MVBVS, in complex sense, or the sequence c satisfies the MVBV condition. Our main result of this paper is: Theorem 1. Let f(x) ∈ L2π be a complex-valued function. If the Fourier coef- ficients f̂(n) of f satisfy that {f̂(n)}+∞n=0 ∈ MVBV S and lim sup |∆f̂(k)−∆f̂(−k)| log k = 0, (3) where ∆f̂(k) = f̂(k + 1)− f̂(k), ∆f̂(−k) = f̂(−k − 1)− f̂(−k), k ≥ 0. ‖f − Sn(f)‖L = 0 if and only if f̂(n) log |n| = 0. In order to prove Theorem 1, we present the following four lemmas. Lemma 1. Let {cn} ∈MVBV S, then for any given 1 < µ < 2, we have |∆ck| log k = O [λ−1n]≤k≤[λn] |ck| log k , n→ ∞, where the implicit constant depends only on the sequence {cn} and λ. For sufficiently large n, the lemma can be derived directly from the conditions that 1 < µ < 2 and {cn} ∈ MVBV S. Lemma 2. Let {f̂(n)} ∈ K(θ0) for some θ0 ∈ [0, π/2), then ∣f̂(n+ k) ∣ = O (‖f − Sn(f)‖L) for all n = 1, 2, · · · , where the implicit constant depends only on θ0. Proof. Write φ±n(x) := ei(k∓n)x − e−i(k±n)x It follows from a well-known inequality (e.g. see Theorem 2.5 in [6]) sin kx |φ±n(x)| ≤ 6 Hence (f(x)− Sn(f, x))φ±n(x)dx ≤ ‖f − Sn(f)‖L, and therefore f̂(n + k) = O(‖f − Sn(f)‖L). Now as {f̂(n)} ∈ K(θ0) for some θ0 ∈ [0, π/2) and for all n ≥ 1, we have ∣f̂(n + k) ∣ ≤ C(θ0) Ref̂(n + k) ≤ C(θ0) f̂(n+ k) = O(‖f − Sn(f)‖L). Lemma 3. ([5]). Write Dk(x) : = sin((2k + 1)x/2) 2 sin(x/2) D∗k(x) : = cos(x/2)−cos((2k+1)x/2) 2 sin(x/2) |x| ≤ 1/n, − cos((2k+1)x/2) 2 sin(x/2) 1/n ≤ |x| ≤ π, Ek(x) : = Dk(x) + iD k(x). For k = n, n + 1, · · · , 2n, we have Ek(±x)−Ek−1(±x) = e±ikx, (4) Ek(x) + Ek(−x) = 2Dk(x), (5) ‖Ek‖L + ‖Dk‖L = O(log k). (6) Lemma 4. Let {f̂(n)} ∈MVBV S. If lim ‖f − Sn(f)‖L = 0, then f̂(n) log n = 0. Proof. By the definition of MV BV S, we derive that for k = n, n + 1, · · · , 2n, |f̂(2n)| ≤ |∆f̂(j)|+ |f̂(k)| |∆f̂(j)|+ |f̂(k)| j=[λ−1k] |f̂(j)| + |f̂(k)|. Therefore, it follows that from the fact that log n ≤ C (λ) [(λ+1)−2n] j=[λ]+1 we have |f̂(2n)| logn ≤ C(λ)|f̂(2n)| [(λ+1)−2n] j=[λ]+1 ≤ C(λ) [(λ+1)−2n] j=[λ]+1 [λ(n+j)] k=[λ−1(n+j)] |f̂(k)|+ |f̂(n+ j)| [(λ+1)−2n] j=[λ]+1 [λ(n+j)] k=[λ−1(n+j)] |f̂(k)| +C(λ) [(λ+1)−2n] |f̂(n+ j)| = : I1 + I2. (7) By applying Lemma 2, we see that I2 ≤ C(λ, θ0)‖f − Sn(f)‖L. (8) We calculate I1 as follows (note that we may add more repeated terms in the right hand side of every inequality below): [(λ+1)−2n] j=[λ]+1 [λn]+[λj]+1 k=[λ−1n]+[λ−1j] |f̂(k)| ≤ C(λ) [(λ+1)−2n] j=[λ]+1 [(λ+1)2] [λn]+1 k=[λ−1n] ∣f̂ (m[λ−1j] + k) ≤ C(λ) [(λ+1)2] [(λ+1)−2n] j=[λ]+1 [λn]−[λ−1n]+1 ∣f̂ ([λ−1n] +m[λ−1j] + k) ≤ C(λ) [(λ+1)2] [λn]−[λ−1n]+1 (λ(λ+1)2) ∣f̂ ([λ−1n] + k + j) [(λ+1)2] [λn]−[λ−1n]+1 ∥f − S[λ−1n]+k(f) (by Lemma 2) [λn]−[λ−1n]+1 ∥f − S[λ−1n]+k(f) . (9) Finally, by combining (7) - (9) and the condition ‖f − Sn(f)‖L = 0, we get f̂(2n) logn = 0. A similar argument yields that |f̂(2n + 1)| logn = 0. This proves Lemma 4. We now come to the proof of Theorem 1. Proof of Theorem 1. Sufficiency. Given ε > 0, by (3), there is a 1 < µ < 2 such that ∣∆f̂(k)−∆f̂(−k) ∣ log k ≤ ε (10) holds for sufficiently large n > 0. Let τµn,n(f, x) := [µn]− n [µn]−1 Sk(f, x) be the Vallée Poussin sum of order n of f . Then we have ‖f − τµn,n(f)‖L = 0. (11) By (4), (5), and applying Abel transformation, we get τµn,n(f, x)− Sn(f, x) [µn]− n k=n+1 ([µn]− k) f̂(k)eikx + f̂(−k)e−ikx [µn]− n ([µn]− k) 2∆f̂(k)Dk(x)− (∆f̂(k)−∆f̂(−k))Ek(−x) [µn]− n [µn]−1 f̂(k + 1)Ek(x)− f̂(−k − 1)Ek(−x) f̂(n)En(x) + f̂(−n)En(−x) . (12) Thus, by (6) and Lemma 1, we have ‖f − Sn(f)‖L ≤ ‖f − τµn,n(f)‖L + ‖τµn,n(f)− Sn(f)‖L = ‖f − τµn,n(f)‖L +O ∣∆f̂(k) ∣ log k ∣∆f̂(k)−∆f̂(−k) ∣ log k n≤|k|≤[µn] |f̂(k)| log |k| = ‖f − τµn,n(f)‖L +O [λ−1n]≤|k|≤[λn] |f̂(k)| log |k| ∣∆f̂(k)−∆f̂(−k) ∣ log k  , (13) lim sup ‖f − Sn(f)‖L ≤ ε follows from (10), (11) and the condition that f̂(n) log |n| = 0. This implies that ‖f − Sn(f)‖L = 0. Necessity. Since {f̂(n)} ∈MV BV S, by applying Lemma 4, we have f̂(n) log n = 0. (14) In order to prove lim f̂(n) log |n| = 0, by applying (12) and (6), we see that for any given µ, 1 < µ < 2, ‖f̂(−n)En(−x)‖L ≤ ‖τµn,n(f)− Sn(f)‖L [µn]− n [µn]−1 f̂(−k − 1)Ek(−x) |∆f̂(k)−∆f̂(−k)| log k + ∣∆f̂(k) ∣ log k n≤k≤[µn] |f̂(k)| log k . (15) It is not difficult to see that [µn]−1 f̂(−k − 1)Ek(−x) = I +O n max n<k≤[µn] |f̂(−k)| where n−1≤|x|≤π 2 sin(x/2) [µn]−1 f̂(−k − 1)e i(2k+1)x Since the trigonometric function system is orthonormal, we have n−1≤|x|≤π [µn]−1 f̂(−k − 1)e i(2k+1)x sin2(x/2) k=n+1 |f̂(−k)|2 n max n≤k≤[µn] |f̂(−k)| which yields that [µn]− n [µn]−1 f̂(−k − 1)Ek(−x) n<k≤[µn] |f̂(−k)| . (16) By combining (11), (14) - (16), with Lemma 1 and the condition ‖f − Sn(f)‖L = 0, and the fact (since f ∈ L2π) that f̂(−n) = 0, we have for n→ ∞, ‖f̂(−n)En(−x)‖L ≤ |∆f̂(k)−∆f̂(−k)| log k +‖τµn,n(f)− Sn(f)‖L [λ−1n]≤k≤[λn] |f̂(k)| log k n<k≤[µn] |f̂(−k)| |∆f̂(k)−∆f̂(−k)| log k + o(1). (17) On the other hand, we have ‖f̂(−n)En(−x)‖L ≥ |f̂(−n)|‖Dn(x)‖L ≥ |f̂(−n)| logn. (18) Hence, from (17), (18), and (10), we have that |f̂(−n)| logn ≤ |∆f̂(k)−∆f̂(−k)| log k ≤ ε holds for sufficiently large n, which, together with (14), completes the proof of neces- sity. In view of Lemma 1, we can see that the condition (3) in Theorem 1 can be replaced by the following condition lim sup |∆f̂(−k)| log k = 0, and the proof of the result is easier. Therefore we have a corollary to Theorem 1. Corollary 1. Let f(x) ∈ L2π be a complex valued function. If both {f̂(n)}+∞n=0 ∈ MVBV S and {f̂(−n)}+∞n=0 ∈MVBV S, then ‖f − Sn(f)‖L = 0 if and only if f̂(n) log |n| = 0. If f(x) is a real valued function, then its Fourier coefficients f̂(n) and f̂(−n) are a pair of conjugate complex numbers. Consequently, {f̂(n)}+∞n=0 ∈ MVBV S if and only if {f̂(−n)}+∞n=0 ∈ MVBV S. Thus, we have the following generalization of the classical result (cf. Result Two in the introduction): Corollary 2. Let f(x) ∈ L2π be a real valued even function and (2) be its Fourier series. If A = {an}+∞n=0 ∈MV BV S in real sense, i.e. {an} is a nonnegative sequence, and there is a number λ ≥ 2 such that |∆ak| ≤ C(A) k=[λ−1m] for all n = 1, 2, . . . , then ‖f − Sn(f)‖L = 0 if and only if an logn = 0. 3 L1 Approximation Let En(f)L be the best approximation of a complex valued function f ∈ L2π by trigonometric polynomials of degree n in L1 norm, that is, En(f)L := inf We establish the corresponding L1−approximation theorem in a similar way to Theorem 1: Theorem 2. Let f(x) ∈ L2π be a complex valued function, {ψn} a decreasing sequence tending to zero with ψn ∼ ψ2n, (19) i.e., there exist positive constants C1 and C2, such that C1ψn ≤ ψ2n ≤ C2ψn. If both {f̂(n)}+∞n=0 ∈MV BV S and {f̂(−n)}+∞n=0 ∈MVBV S, then ‖f − Sn(f)‖L = O(ψn) (20) if and only if En(f)L = O(ψn) and f̂(n) log |n| = O(ψ|n|). (21) Proof. Under the condition of Theorem 2, we see from (13) in the proof of Theorem 1 that ‖f − Sn(f)‖L ≤ ‖f − τµn,n(f)‖L +O [λ−1n]≤|k|≤[λn] |f̂(k)| log |k| ≤ C(µ)En(f) +O [λ−1n]≤|k|≤[λn] |f̂(k)| log |k| thus (20) holds if (19) and (21) hold. Now if (20) holds, then En(f)L = O(ψn) ‖f − τµn,n(f)‖L = O(ψn). From (7) - (9) in the proof of Lemma 4 and condition (19), we have |f̂(n)| logn ≤ C(λ) [λn]−[λ−1n]+1 ∥f − S[λ−1n]+j(f) +C(λ)‖f − Sn(f)‖L = O(ψn). Since {f̂(−n)}+∞n=0 ∈ MVBV S, by a similar argument to (22), we also have ∣f̂(−n) ∣ logn = O(ψn). This completes the proof of Theorem 2. In particular, if we take ψn := (n+ 1)r f (r), n + 1 where r is a positive integer, and ω(f, t)L is the modulus of continuity of f in L norm, i.e. ω(f, t)L := max 0≤h≤t ‖f(x+ h)− f(x)‖L. By Theorem 2 and the Jackson theorem (e.g. see [6] or [9]) in L1−space, we imme- diately have Corollary 3. Let f(x) ∈ L2π be a complex valued function. If both {f̂(n)}+∞n=0 ∈ MVBV S and {f̂(−n)}+∞n=0 ∈MVBV S hold, then ‖f − Sn(f)‖L = O (n+ 1)r f (r), n + 1 if and only if f̂(n) log |n| = O (n+ 1)r f (r), n + 1 This corollary generalizes the corresponding results in [5] and [2]. References [1] P. R. Boas Jr., Integrability theorems for trigonometric transforms, Springer, Ergebnisse 38, Berlin 1967. [2] R. J. Le and S. P. Zhou, On L1 convergence of Fourier series of complex valued functions, Studia Sci. Math. Hungar., to appear. [3] V. B. Stanojevic, L1−convergence of Fourier series with complex quasimonotone coefficients, Proc. Amer. Math. Soc., 86(1982), 241-247. [4] V. B. Stanojevic, L1−convergence of Fourier series with O−regularly varying quasimonotone coefficients, J. Approx. Theory, 60(1990), 168-173. [5] T. F. Xie and S. P. Zhou, L1−approximation of Fourier series of complex valued functions, Proc. Royal Soc. Edinburg, 126A(1996), 343-353. [6] T. F. Xie and S. P. Zhou, Approximation Theory of Real Functions, Hangzhou University Press, 1998. [7] D. S. Yu and S. P. Zhou, A generalization of monotonicity condition and applica- tions, Acta Math. Hungar., to appear. [8] S. P. Zhou, P. Zhou and D. S. Yu, Ultimate generalization to monotonicity for uni- form convergence of trigonometric series, arXiv:math.CA/0611805 v1, November 27, 2006, preprint. [9] A. Zygmund, Trigonometric Series, 2nd. Ed., Vol.I, Cambridge Univ. Press, Cam- bridge, 1959. Dan Sheng Yu and Song Ping Zhou: Institute of Mathematics Zhejiang Sci-Tech University Xiasha Economic Development Area Hangzhou, Zhejiang 310018 China Department of Mathematics, Statitics & Computer Science St. Francis Xavier University Antigonish, Nova Scotia, Canada B2G 2W5 Email: dsyu@zjip.com (D. S. Yu) szhou@zjip.com (S. P. Zhou) Ping Zhou: Department of Mathematics, Statitics & Computer Science St. Francis Xavier University Antigonish, Nova Scotia, Canada B2G 2W5 Email: pzhou@stfx.ca

On Global Solution to the Klein-Gordon-Hartree Equation below Energy Space Changxing Miao1 and Junyong Zhang2 1Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R. China. E-mail: miao changxing@iapcm.ac.cn 2The Graduate School of China Academy of Engineering Physics P. O. Box 2101, Beijing, China, 100088 E-mail: zhangjunyong111@sohu.com Abstract In this paper, we consider the Cauchy problem for Klein-Gordon equation with a cubic convolution nonlinearity in R3. By making use of Bourgain’s method in conjunction with a precise Strichartz estimate of S.Klainerman and D.Tataru, we establish the Hs(s < 1) global well-posedness of the Cauchy problem for the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving at the pre- viously discussed conclusion, we obtain global solution for this non-scaling equation with small initial data in Hs0×Hs0−1 where s0 = γ6 but not −1, for this equation that we consider is a subconformal equation in some sense. In doing so a number of nonlinear prior estimates are already established by using Bony’s decomposition, flexibility of Klein-Gordon admissible pairs which are slightly different from that of wave equation and a commutator estimate. We establish this commutator estimate by exploiting cancellation property and utilizing Coifman and Meyer multilinear multiplier theorem. As far as we know, it seems that this is the first result on low regularity for this Klein-Gordon-Hartree equation. Key Words: Klein-Gordon-Hartree equation, Low regularity, Precise Strichartz estimate, Bony’s para-product decomposition, Coifman and Meyer multilinear multiplier theorem. AMS Classification: 35Q40, 35Q55, 47J35. 1 Introduction We study the following Cauchy problem for the Klein-Gordon-Hartree equation: �φ+ φ+ (|x|−γ ∗ |φ|2)φ = 0 in R×R3 φ|t=0 = φ0, ∂tφ|t=0 = φ1. (1.1) Here φ(t, x) is a complex valued function defined in space time R1+3, and � = ∂tt −∆. http://arxiv.org/abs/0704.1866v4 Recently the Cauchy problem (1.1) has been extensively studied in the case with initial data (φ0, φ1) ∈ H1(Rn) × L2(Rn). The well-posedness and the asymptotic be- havior of solution to the Cauchy problem (1.1) have been studied by G.P. Menzala and W.Strauss [16, 17]. The scattering theory of solution to (1.1) has been established in [23]. On the other hand, the time-dependent Schrödinger equation with interaction term (|x|−γ ∗ |φ|2)φ has also been extensively studied. Ginibre and Velo [11] gave the scattering theory of Hartree equation for the energy subcritical case. For the energy critical case and mass critical, one can refer to [20, 21] with radial initial data. Many authors [4, 9, 12, 18, 30] have studied the local well-posedness (as well as global well-posedness) in fractional Sobolev spaces for the Cauchy problem of general semilinear wave or Schrödinger equations under minimal regularity assumptions on the initial data. For example, Tao [30] established the sharp local well-posedness of nonlinear wave equation. Kenig, Ponce, and Vega [12] had established the global well- posedness under the energy norm for the Cauchy problem of nonlinear wave equations with rough initial data (in particular, in Ḣs(R3), 3 < s < 1 for cubic wave equation). They used the Fourier truncation method discovered by Bourgain [4]. And also [18] extended Kenig-Ponce-Vega’s result to the dimension n > 4. Recently, I. Gallagher and F. Planchon [9] presented a different proof of the result in [12] for 3 < s < 1. H. Bahouri and Jean-Yves Chemin [2] proved global well posedness for s = 3 by using a nonlinear interpolation method and logarithmic estimates from S. Klainermann and D. Tataru[14]. We also find Roy [26] obtains the global well-posedness for rough initial data in Ḣs, 2 < s < 1 by following the I-method [5] and scaling transformation. However, if one similarly deals with Klein-Gordon equation by using I-method, he or she may meet a problem caused by the lack of the scaling property. More studies and discussions on the low regularity of nonlinear wave or dispersive Schrödinger equations could be found in [4, 31]. However, as far as we know, very few authors are engaged in studying the global well-posedness of the Cauchy problems (1.1) with less regular initial data. It is natural to ask whether a similar or better result holds for the problem (1.1). This paper endeavors to find a global well-posedness solution to the Cauchy problem (1.1) with initial data (φ0, φ1) ∈ Hs(R3) ×Hs−1(R3) for some s > γ4 with γ ∈ (2, 3). Now we should remark some differences between (1.1) and cubic wave equation. If one views (1.1) as a wave equation by dropping the massive term and then makes some scaling analysis, we will find this nonlocal nonlinear term shares the scaling property of the nonlinearity |u| 5−γ u. One can check that k := 4 +1 < 3 when 2 < γ < min{n, 4} with n = 3 and this result shows that the equation which we consider is in subconformal case. To obtain the global well-posedness theory, some previous literatures also show the subconformal equations are slightly different from the superconformal ones. For instance, Lindblad and Sogge [15] [27] have shown the global existence and scattering theory for small data in a less regularity space for the superconformal case, while not for the subconformal case. Inspired by [9], we also split the initial data into low frequency part data in H1 and high frequency part data in Hs0 with a suitable s0. Since the problem (1.1) is global well-posed for large data in H1 and small data in Hs0 , one may be tempted to follow a general principle of nonlinear interpolation and claim the problem (1.1) is global well-posed between them. Compared with the cubic wave equation, speaking of the Strichartz estimate, we believe that the global solution with high frequency data should exist in H −1. It is well known that the Strichartz estimate is associated with scaling transform and it is scaling invariant. Unfortunately, the equation that we consider is a subconformal one, and its concentration effects take over scaling. Since the Strichartz estimate is applied to our subconformal equation, hence this brings about some loss to get a better result. In order to get a better result, one should establish an estimate which is conformal invariant. Fortunately, we can take 0 6 θ 6 1 as a parameter for the flexible admissible pairs (see Definition 2.3)to make the Strichartz estimate of Klein-Gordon more flexible than wave equation. This helps us to get a global solution with the high frequency data, at the cost of 0 6 θ = 6 −2 6 1 which weakens the Strichartz estimate and causes 2 < γ < 3. One can refer the detail in Section 3. We point out that it is easy to have the result for < s < 1 by rough Hölder’s inequality. But how to get our low bound < s < 1? A good way to think about this is via precise Strichartz estimate to obtain index s as low as possible. The nonlinearity including a formal negative derivative brings us some difficulties caused by the fact that the negative derivative acts on the low frequency part. And this leads us to restricts rather than s > max{1 }. At the end of this section, we also give some intuitive analysis to show our result is reasonable. As a limited case, our result recovers the result of [9, 12] when γ tends to 3. During the process of proving our key estimate Lemma 5.1, the nonlocal nonlin- earity brings about some essential difficulties when we try to make use of the precise Strichartz estimate. Compared with the general semilinear nonlinearity, the convo- lution nonlinearity not only essentially represents a negative derivation in it but also has a difference construction of nonlinearity. These differences and difficulties prevent us from obtaining directly our expected result s > by restricting the range of the parameter r. To overcome these difficulties, we firstly construct a commutator and establish this commutator estimate by exploiting cancellation property and utilizing Coifman and Meyer multilinear multiplier theorem and then go on our process through using precise Strichartz estimate. Now we state our main result: Theorem 1.1 Let < s < 1 with 2 < γ < 3. If (φ0, φ1) ∈ Hs(R3) ×Hs−1(R3), then there exists a unique global solution φ of (1.1) such that φ ∈ C(R+;Hs(R3)). We conclude this section by giving a sketch of the proof of Theorem 1.1 and one shall read more detailed information in the rest of this paper. Without loss of generality, we only consider φ as a real function for simplicity from now on. Since the problem (1.1) is global well-posed for large data in H1 and small data in Hs0 with s0 = , one may be tempted to follow a general principle of nonlinear interpolation and believe the problem (1.1) to be global well-posed between them, as well as the cubic defocusing wave equation [9]. To make sense of this heuristic, we proceed it in the following steps. Step 1. The purpose of this step is to show the global well-posedness for the high frequency part. We split the initial data: φi = (I− SJ)φi + SJφi = vi + ui i = 0, 1 where I is identity operator and SJ is Littlewood-Paley operator, referring to Section 2. It is easy to see that ‖u0‖H1 . 2J(1−s)‖φ0‖Hs , ‖u1‖L2 . ‖φ1‖L2 ‖v0‖Hβ . 2J(β−s)‖φ0‖Hs , ‖v1‖Hβ−1 . 2J(β−s)‖φ1‖Hs−1 for all β 6 s. Thus it follows that Eh,σ . 2J(σ−s)Es, for σ 6 s (1.2) Eℓ,1 . 2J(1−s)Es, for s 6 1, (1.3) where == ‖φ0‖Hs + ‖φ1‖Hs−1 , (1.4) == ‖v0‖Hσ + ‖v1‖Hσ−1 , (1.5) == ‖u0‖Hσ + ‖u1‖Hσ−1 . (1.6) Choosing J large enough, one can achieve Eh,s0 small enough, in other words, initial data of the following problem �v + v + (|x|−γ ∗ v2)v = 0 in R× R3, v|t=0 = v0, ∂tv|t=0 = v1 (1.7) is small enough inHs0(R3)×Hs0−1(R3) where s0 < s. Due to some technique difficulties and this equation is subconformal one, we are restricted to choose s0 = while not − 1 proposed by scaling analysis or γ proposed by conformal analysis. We will get a global well-posed solution to the Cauchy problem (1.7), see Section 3 for details. Step 2. In order to recover a solution to our problem (1.1), we solve a perturbed equation with large initial data in H1 × L2, �u+ u+ I(u2)u+ 2I(uv)u+ I(v2)u+ I(u2)v + 2I(uv)v = 0, u|t=0 = u0 ∂tu|t=0 = u1, (1.8) where the operator I is the operator (−∆) 2 . We will prove there exists a unique local solution to (1.8) in C([0, T ];H1). Step 3. To complete the proof of Theorem 1.1, the key is how to extend the local solution to a global solution. We should establish a priori bound on the energy of the local solution u. In fact, the energy estimate yields ‖u(t)‖2H1 + ‖ut(t)‖ R3×R3 |x− y|−γu2(y, t)u2(x, t)dydx ‖u0‖2H1 + ‖u1‖ R3×R3 |x− y|−γu20(y)u20(x)dydx I(v2)(x, τ)u(x, τ)∂τu(x, τ)dxdτ I(uv)(x, τ)v(x, τ)∂τu(x, τ)dxdτ I(u2)(x, τ)v(x, τ)∂τu(x, τ)dxdτ I(uv)(x, τ)u(x, τ)∂τu(x, τ)dxdτ ∣∣∣∣ . Let HT (u) := sup H(u)(t) where H(u)(t) ‖u(t)‖2H1 + ‖ut(t)‖2L2 + R3×R3 |x− y|−γu2(y, t)u2(x, t)dydx and then by making use of Hölder’s inequality and Sobolev embedding, it follows that HT (u) .H(u)(0) +HT (u) ‖v(τ)‖2 dτ +H T (u) ‖v(τ)‖ .H(u)(0) +HT (u)T 6 ‖v‖2 T (u)T 3 ‖v‖Xα .22J(1−s) +HT (u)T 6 22J(β−s) +H T (u)T 3 2J(α−s) where α = , β = and the space Xα is defined in the coming section. What we want to do is to control HT (u) for arbitrarily large T . As long as s > (α+1)/2 = by choosing J large enough, bootstrap argument yields HT (u) . 2 2J(1−s). One can see that, if s > , the argument is trivial, since the above mentioned result can be deduced from some rough estimates such as the Hölder estimate. On the other hand, since the scaling suggests us that X −1 is the lowest regularity space which v could belong to, it is tempting and reasonable to believe that the best result obtained by this method is s > ( − 1+1)/2 = γ instead of α by − 1. To obtain this optimal result s > , we adopt some more sophisticated tools such as precise Strichartz estimate, Bony’s paraproduct estimates and twice Bony’s decomposition. This result is achieved under an assumption of a core estimate which will be shown through the precise Strichartz estimate and a commutator estimate. The paper is organized as follows: In the coming section, we recall some nota- tions and recollect some well known results on Besov spaces in conjunction with the Littlewood-Paley theory which will be used in the course of the proofs. Meanwhile, we also introduce the precise Strichartz estimate. Section 3 provides the global well- posedness of original equation evoking the high frequency part of initial data in Hs0 . In Section 4, we prove prove a local well-posedness of perturbed equation with the low frequency of the initial data in H1 by the standard fixed point theorem. In Section 5, we give a energy estimate for the low frequency part provided an assumption the key estimate in Lemma 5.1. We extend the local well-posedness of the perturbed equation to globally well posed by the bootstrap argument in Section 6. In the final section, we prove our essential and key lemma by the precise Strichartz estimate, commutator estimate and Coifman and Meyer multiplier theorem. 2 Preliminaries In this section, we shall present some well-known facts on the Littlewood-Paley theory and introduce some notations, definitions and estimates which are needed in this paper. Let S(R3) be the Schwarz class of rapidly decreasing functions. Given f ∈ S(R3), its Fourier transform Ff = f̂ is defined by f̂(ξ) = (2π)− e−ix·ξf(x)dx, F−1f = f̂(−ξ). Choose two nonnegative radial functions χ, ϕ ∈ S(R3) supported respectively in B = {ξ ∈ R3, |ξ| 6 4 } and C = {ξ ∈ R3, 3 6 |ξ| 6 8 } such that χ(ξ) + ϕ(2−jξ) = 1, ξ ∈ R3, ϕ(2−jξ) = 1, ξ ∈ R3\{0}, supp ϕ(2−j ·) ∩ supp ϕ(2−j′ ·) = ∅, |j − j′| > 2, supp χ(·) ∩ supp ϕ(2−j ·) = ∅, j > 1. Now we are in position to define the the Littlewood-Paley operators Sj, Ṡj, △j and △̇j which are used to define Besov space. 0, j 6 −2, χ(ξ)û(ξ) , j = −1, (F−1ϕ)(2jy)u(x− y)dy, j > 0, j′6j−1 △j′u = 2jn (F−1χ)(2jy)u(x− y)dy, == 2jn (F−1ϕ)(2jy)u(x− y)dy, j ∈ Z, j′6j−1 △̇j′u. One easily shows that △̇j = Ṡj+1 − Ṡj for j ∈ Z and △−1 = S0, △̇j = △j , j > 0. Now we give the Littlewood-Paley’s description of the Besov spaces. Definition 2.1 Let s ∈ R, 1 6 p, q 6 ∞. The homogenous Besov space Ḃsp,q is defined Ḃsp,q = {f ∈ Z ′(R3) : ‖f‖Ḃsp,q < ∞}, where Ḃsp,q 2jsq‖△̇jf‖qp , for q < ∞, 2js‖△̇jf‖p, for q = ∞, and Z ′(R3) can be identified by the quotient space S ′/P with the space P of polynomials. Definition 2.2 Let s ∈ R, 1 6 p, q 6 ∞. The inhomogeneous Besov space Bsp,q is defined by Bsp,q = {f ∈ S ′(R3) : ‖f‖Bsp,q < ∞}, where ‖f‖Bsp,q = 2jsq‖△jf‖qp + ‖S0(f)‖p, for q < ∞, 2js‖△jf‖p + ‖S0(f)‖p, for q = ∞. If s > 0, then Bsp,q = L p ∩ Ḃsp,q and ‖f‖Bsp,q ≈ ‖f‖p + ‖f‖Ḃsp,q . We refer the reader to [1, 6, 22, 32] for details. In order to investigate the low regularity solution of the Cauchy problem (1.1), we require the use of the smoothing effect described by the Strichartz estimates and precise Strichartz estimates. For the purpose of conveniently making use of the Strichartz estimate, we introduce the admissible definition and the resolution space. Definition 2.3 We shall say that a pair (q, r) is admissible, for 0 6 θ 6 1, if q, r > 2, (q, r, θ) 6= (2,∞, 0) and 1 2 + θ 2 + θ Remark 2.1 The above admissible pairs in Definition 2.3 is more flexible than wave admissible pairs, since θ can vary from 0 to 1. Obviously, an admissible pair in Defini- tion 2.3 will become a wave admissible pair when θ = 0. When we consider the global existence for the high frequency part, we shall use θ = 6 − 2 since the equation that we consider is a subconformal one. The resolution space is defined in the following way based on the admissible defini- tion. Xµ(I) := 06θ61 where (I) := u : u ∈ (C ∩ L∞)(I;Hµ) ∩ Lq(I;Bσr,2) (q,r) is admissible, = (3 + θ)( ) + σ − µ We go on this section by recalling the classical Strichartz estimate and the precise Strichartz estimate. This kind of estimate goes back to Strichartz [29], and has been proved in its generality by Ginibre and Velo [10], and Keel and Tao [13]. The Strichartz estimates for the Klein-Gordon equation by using the above flexible admissible pairs can be found in [19]. Proposition 2.1 Let u be a solution of �u+ u = f in R× R3 with u|t=0 = u0, ∂tu|t=0 = u1. Then, for any admissible pairs (q1, r1) and (q2, r2), we have that ‖△ju‖Lq1 (Lr1 ) + 2−j‖∂t△ju‖Lq1 (Lr1 ) j( 3+θ − 3+θ (‖△ju0‖L2 + 2−j‖△ju1‖L2) j[(3+θ)(1− 1 −1]‖△jf‖ . (2.1) We shall see that the classical Strichartz estimates are not enough to control some nonlinearities, and this leads us to resort to the following precise Strichartz estimates which were established by S.Klainerman and D.Tataru[14]. Proposition 2.2 Let u be a solution of �u+ u = 0 with u|t=0 = u0, ∂tu|t=0 = u1. Assume that the supports of the Fourier transform of u0 and u1 are included in a ball B(ξj, h2 j) with |ξj | ∈ [2j−2, 2j+2] and h < 18 . Then we have that, for any admissible couple (q, r), ‖u‖Lq(Lr) + 2−j‖∂tu‖Lq(Lr) 6 C2 r (‖u0‖L2 + 2−j‖u1‖L2). (2.2) Let us recall the Hardy-Littlewood-Sobolev inequality [22, 28] and a proposition of contraction which is generalization of Picard’s theorem [6]. We denote operator I by Iu def== (−∆) 2 u = |x|−γ ∗ u, ‖Iu‖Lq(R3) 6 Cp,q‖u‖Lp(R3) (2.3) 0 < γ < 3, 1 < p < q < ∞, and 1 − 3− γ Proposition 2.3 Let X be a Banach space and let B : X ×X × · · · ×X → X be a m-linear continuous operator (m > 2) satisfying ‖B(u1, u2, · · · , um)‖X 6 M‖u1‖X‖u2‖X · · · ‖um‖X , ∀u1, u2, · · · , um ∈ X for some constant M > 0. Let ε > 0 be such that m(2ε)m−1M < 1. Then for every y ∈ X with ‖y‖X 6 ε the equation u = y +B(u, u, · · · , u) (2.4) has a unique solution u ∈ X satisfying that ‖u‖X 6 2ε. Moreover, the solution u continuously depends on y in the sense that, if ‖y1‖X 6 ε and v = y1 +B(v, v, · · · , v), ‖v‖X 6 2ε then ‖u− v‖X 6 1−m(2ε)m−1M ‖y − y1‖X . (2.5) For the sake of convenience, we conclude this section by giving some notations. The solution φ to the Cauchy problem (1.1) is given by the following integral equation: φ(t, x) = K̇(t)φ0 +K(t)φ1 +B(φ, φ, φ) = T φ where K(t) := sin(t I −∆)√ B(u1, u2, u3) := − K(t− τ)(|x|−γ ∗ (u1u2))u3dτ. Throughout this article we shall denote by the letter C all universal constant and ε > 0 is a arbitrary small data. We shall sometimes replace an inequality of the type f 6 Cg by f . g. Also, we shall denote by (cj)j∈Z any sequence of norm less than 1 in ℓ 2(Z). 3 Global existence for the high frequency part Let us consider the Cauchy problem with the high frequency data, �v + v + (|x|−γ ∗ v2)v = 0, (t, x) ∈ R× R3 v|t=0 = v0, ∂tv|t=0 = v1, x ∈ R3. (3.1) and then its integral formation becomes v(t, x) =K̇(t)v0(x) +K(t)v1(x)− K(t− τ)(|x|−γ ∗ v2)vdτ ==K̇(t)v0(x) +K(t)v1(x) +B(v, v, v). (3.2) Our goal in this section is to prove the global well-posedness of (3.1) or (3.2). More precisely, we have the following proposition: Proposition 3.1 Let s0 = and suppose that (v0, v1) ∈ Hµ×Hµ−1 for any 0 6 µ 6 1. There exists a constant ε0 > 0 such that if ‖v0‖Hs0 + ‖v1‖Hs0−1 6 ε0, then there exists a unique global solution v to (3.1) or (3.2) in Xs0(R)∩Xµ(R). More- over, ‖v‖Xµ 6 Cµ (‖v0‖Hµ + ‖v1‖Hµ−1) . Remark 3.1 We focus on µ = and µ = in the coming section. It is well known that the global existence theory for small initial data is a straightfor- ward result of nonlinear estimate, thus how to obtain a suitable nonlinear estimate is essential. Before proving this proposition, we make some analysis on nonlinear estimate. As mentioned in the introduction, the nonlocal nonlinearity shares the scaling with a subconformal nonlinearity when γ < 3 and this may bring some troubles when we make a choice of a suitable resolution space Xs0 . Take 0 6 θ 6 1 as a parameter in the flexible admissible pairs (see Definition 2.3), and we make analysis on the relationship between θ and s0. The Strichartz estimate, Hölder inequality and Hardy-Littlewood-Sobolev inequality imply that, for σ 6 0, ‖B(v, v, v)‖Xs0 6 ‖(|x|−γ ∗ |v|2)v‖ 6 ‖v‖ Lq2 (B ‖v‖2Lq3 (Lr3 ), with satisfying = (3 + θ)( ) + σ + s0 − 1 = (3 + θ)( )− σ − s0 = (3 + θ)( )− s0 − 1 + γθ We find the fact index s0 is increasing when the parameter θ increases. It is tempting to choose θ = 0 to get the smallest s0 = − 1 proposed by scaling. However, in addition the admissible condition implies that 6 (2 + θ)( 6 (2 + θ)( 6 (2 + θ)( then a direction computation gives that ) 6 (2 + θ)(2− 1 which yields that 6 1 + If we choose θ = 0, then we are forced to γ > 3 which contradict with our requirement γ < 3. But if we choose θ = 6 − 2 and then s0 = γ6 and we are allowed by 2 6 γ 6 3. Proof of Proposition 3.1 Thanks to Strichartz estimate, we have ‖B(v, v, v)‖Xµ 6 ‖(|x|−γ ∗ |v|2)v‖ 6 ‖v‖ Lq2 (B ‖v‖2Lq3 (Lr3 ), where 3 + γ (1− µ− σ), 1 3 + γ (1− µ− σ) 3(µ+ σ) 3 + γ − γ(µ+ σ) 3 + γ 2(3 + γ) 9 + 3γ − γ2 6(3 + γ) When 0 6 µ 6 1 , we choose σ = 0; while 1 < µ 6 1, we choose σ = 1 Thus, ‖B(v, v, v)‖Xµ 6 ‖v‖Xµ‖v‖2Xs0 . (3.3) Combining this nonlinear estimate, the Proposition 3.1 follows from a standard con- traction argument and small initial data condition. 4 Local existence for the low frequency part In this part, we shall study the following perturbed problem in R× R3: �u+ u+ I(u2)u+ 2I(uv)u+ I(v2)u+ I(u2)v + 2I(uv)v = 0 u|t=0 = u0 ∂tu|t=0 = u1. (4.1) Proposition 4.1 Let α = , β = and assume that v be in Xα ∩ Xβ and (u0, u1) ∈ H1 ×L2, then there exists a positive time T such that a unique solution u to (4.1) satisfying u ∈ C([0, T ];H1). Proof of the Proposition 4.1 In practice, solving (4.1) on [0, T ] is equivalent to solving the following integral equation u =K̇(t)u0 +K(t)u1 K(t− τ) I(u2)u+ 2I(uv)u+ I(v2)u+ I(u2)v + 2I(uv)v ,T̃ u. Using the Strichartz estimate, we have K(t− τ)I(u2)udτ . ‖I(u2)u‖L1 (L2). On one hand, we make use of Hölder’s inequality and Hardy-Littlewood-Sobolev in- equality to deduce that ‖I(u2)u‖L1 (L2) 6 C‖I(u2)‖ 6 C‖u‖3 6 CT‖u‖3L∞ H1 . (4.2) For the rest of terms, arguing similarly as above, it can be obtained that ‖I(uv)u‖L1 (L2) 6 C‖u‖2L∞ L6‖v‖ 3 ‖u‖2L∞ H1‖v‖Xα , (4.3) ‖I(v2)u‖L1 (L2) 6 C‖u‖L∞T L6‖v‖ 3 ‖u‖L∞ H1‖v‖2Xβ , (4.4) ‖I(u2)v‖L1 (L2) 6 CT 3 ‖u‖2L∞ H1‖v‖Xα , (4.5) ‖I(uv)v‖L1 (L2) 6 CT 3 ‖u‖L∞ H1‖v‖2Xβ . (4.6) A combination of (4.2), (4.3)-(4.6) and the Strichartz estimate in Proposition 2.1 lead to the estimate ‖u‖L∞ (H1) . ‖u0‖H1 + ‖u1‖L2 + T‖u‖3L∞ 3 ‖u‖2L∞ H1‖v‖Xα + T 3 ‖u‖L∞ H1‖v‖2Xβ . As long as choosing T is small enough, T̃ is a contraction mapping in ball B(0, 2CEℓ,1). By means of Picard’s fixed point argument we have an unique solution u to (4.1) in L∞([0, T ];H1). Therefore, Proposition 4.1 is proved by the standard argument. 5 Energy estimate for the low frequency part In order to extend the local solution to a global solution, we shall prove a prior estimate for the Hamiltonian of u in this section. Let us recall the definition of Hamiltonian of u defined by H(u)(t) ‖u(t)‖2H1 + ‖ut(t)‖2L2 + R3×R3 |x− y|−γu2(y, t)u2(x, t)dydx Similarly we give another notation of the energy of u, which is denoted by E(u)(t) ‖u(t)‖2H1 + ‖ut(t)‖2L2 . HT (u) == sup H(u)(t), ET (u) == sup E(u)(t). To extend the local existence to global existence, we have to do a number of nonlinear a priori estimates provided that ET (u) 6 2CH(u)(0), see Proposition 5.1 and Lemma 5.1. As a direct consequence of the above assumption, we get an important relationship between E(u) and Es defined in the introduction ET (u) . 2 2J(1−s)(E2s + E4s ) . 22J(1−s). (5.1) In fact, it follows from Hardy-Littlewood Sobolev inequality and the definition of u0 ‖(|x|−γ ∗ u20)u20‖L1 . ‖u0‖412 6 ‖S0φ0‖412 06j6J ‖△jφ0‖412 And then the right hand of the above inequality can controlled that as soon as 1 > s > by utilizing Bernstein inequality ‖S0φ0‖4L2 + 06j6J −s)2j4s‖△jφ0‖4L2 . 2 2J(1−s)E4s . From now on, we assume (5.1) to in our subsequence proof. Proposition 5.1 Assume that (u0, u1) ∈ H1×L2, then the following estimate holds for s0, α, β defined in Proposition 3.1 and Proposition 4.1, HT (u) .H(u)(0) + T 3 2−2J(s−β)ET (u) + T 3 2−J(4s−α−2s0−1)ET (u) −2J [s−( −2J [s−( + T2−2J(s− ET (u) for max{2, 1 } < r1 < 23−γ and 6 r2 < ∞. Proof. Multiplying (4.1) by ∂tu and integrating over x and t, we have ‖u(t)‖2H1 + ‖ut(t)‖ R3×R3 |x− y|−γu2(y, t)u2(x, t)dydx ‖u0‖2H1 + ‖u1‖ R3×R3 |x− y|−γu20(y)u20(x)dydx I(v2)(x, τ)u(x, τ)∂τu(x, τ)dxdτ I(uv)(x, τ)v(x, τ)∂τu(x, τ)dxdτ I(u2)(x, τ)v(x, τ)∂τu(x, τ)dxdτ I(uv)(x, τ)u(x, τ)∂τu(x, τ)dxdτ ∣∣∣∣ . By taking the supermum over t 6 T , we have HT (u) .H(u)(0) + ∥∥I(v2)u∂tu + ‖I(uv)v∂tu‖L1 I(u2)v∂tudxdt I(uv)u∂tudxdt =H(u)(0) + I + II + III + IV. (5.2) The proof is broken down into the following several steps. (i) Firstly, we estimate I and II. Making a similarly argument as (4.4) in the proof of Proposition 4.1, it can be obtained that I 6 ‖I(v2)u‖L1 L2‖ut‖L∞T L2 6 T 3 ET (u)‖v‖2Xβ , and then keeping in mind v has been estimated in Proposition 3.1, this together with (1.2) yields that I 6 T 3 ET (u)E2h,β 6 T 3 ET (u)2 −2J(s−β)E2s . (5.3) Arguing similarly, we easily get II 6 T 3 ET (u)2 −2J(s−β)E2s . (5.4) (ii) Secondly, we estimate the terms III and IV . As mentioned in the introduction, one can get the same type of estimate as above for the terms I and II, but that will lead to s > α , which is worse than the exponent given in the Theorem 1.1. To improve the lower bound on s, we have to utilize more precise estimate on III and IV . We first split III and IV into two different pieces, respectively. One can write v = vF +B(v, v, v), where vF is its free part and the other one comes from nonlinear term. For the nonlinear part, it follows from (3.3) that ‖B(v, v, v)‖Xα 6 ‖v‖Xα‖v‖2Xs0 . This along with (4.5), one can see that ‖I(u2)B(v, v, v)ut‖L1 L1 6 ‖I(u2)B(v, v, v)‖L1 L2‖ut‖L∞T L2 3 ‖u‖2L∞ H1‖B(v, v, v)‖Xα‖ut‖L∞T L2 3 ET (u) 2 ‖v‖Xα‖v‖2Xs0 Moreover, we get by (1.2), ‖I(u2)B(v, v, v)ut‖L1 L1 6 T T (u)2 −J(3s−α−2s0)E3s . (5.5) By the same way as leading to (5.5), we easily infer that ‖I(uB(v, v, v))uut‖L1 L1 6 T T (u)2 −J(3s−α−2s0)E3s . (5.6) Thus, it is sufficient to estimate these terms including free part vF since (5.5) and (5.6). The following lemma gives estimates for the nonlinearity including free part vF . Lemma 5.1 Let vF be a solution of the free Klein-Gordon equation, and u be such that ET (u) . 2 2J(1−s). Then, for max{2, 1 } < r1 < 23−γ and 6 r2 < ∞ I(u2)vFutdxdt ∣∣∣ . −2J [s−( −2J [s−( + T2−2J(s− ET (u), (5.7) I(uvF )uutdxdt ∣∣∣ . T −2J [s−( ET (u). (5.8) Hence these together with (5.5)-(5.6) yield that III + IV . T 3 ET (u)2 −J [4s−α−2s0−1]E4s + −2J [s−( −2J [s−( + T2−2J(s− ET (u). (5.9) Therefore, we complete the proof of Proposition 5.1 provided that we had proved Lemma 5.1, whose proof is postponed in the last section. 6 Proof of Theorem 1.1 Since the Cauchy problem (1.1) is split into equation (3.1) which is globally well-posed by choosing J enough to make Eh,s0 < ε0 and equation (4.1) which is locally well-posed (see Proposition 3.1 and Proposition 4.1), we have to show that the local solution to equation (4.1) can be extended globally. Let us denote T ∗J the maximum time of existence in Proposition 4.1. Theorem 1.1 will be proved if T ∗J = +∞. Let us consider TJ the supremum of the T < T J such that ET (u) 6 2CH(u)(0). Thus, for any T < TJ , Proposition 5.1 gives us that ET (u) 6 H(u)(0) C + C1T 3 2−2J(s−β)E2s + C2T 3 2−J(4s−α−2s0−1)E4s + C3T −2J [s−( )]E2s + C4T2 −2J(s− 1 )E2s + C5T −2J [s−( )]E2s By the assumption of Theorem 1.1 s > , one easily verifies that s > max if choosing r1 sufficiently close to and r2 large enough. We infer that TJ > T̃J if we choose T̃J such that = min 22J(s−β) 5C1E2s 24J(s− 5C2E4s 2J [s−( 5C3E2s ) 2r1 22J(s− 5C4E2s 2J [s−( 5C5E2s ) 2r2 By the definition of TJ , we get T J > T̃J . Obviously, T̃J tends to infinity when J tend to infinity. This completes the proof of Theorem 1.1. 7 Proof of Lemma 5.1 In order to make conveniently use of the precise Strichartz estimate on which mostly the following proof relies, we begin this section by introducing a family of balls of center ν )ν∈Λj,k of radius 2 k and a function χ ∈ C∞c (B(0, 1)) such that for j > 0 ∀ξ ∈ 2jC, ν∈Λj,k χ(2−k(ξ − ξj,kν )) = 1 and C−10 6 ν∈Λj,k χ2(2−k(ξ − ξj,kν )) 6 C0. Let us define that, for some constant c △νj,ka == F−1 ϕ(2−jξ)χ(2−k(ξ − ξj,kν )) â(ξ) △̃νj,ka == F−1 ϕ̃(2−jξ)χ(c2−k(ξ + ξj,kν )) â(ξ) As the support of the Fourier transform of a product belongs to the sum of the support of each Fourier transform, we have △ja = ν∈Λj,k △νj,ka, △jb = ν′∈Λj,k △ν′j,kb. In view of this fact that if k 6 j − 2 ν,ν′∈Λj,k △νj,ka△ν is vanish when ξ ν is close to ξ , without loss of generality, we can write △k(△ja△jb) ≈ △k ν∈Λj,k △νj,ka△̃νj,kb. (7.1) For the sake of convenience, we also fix the notation in this section that, for 0 6= f(t, x) ∈ L2TL2 ck = 2 ‖△kv0‖L2 + 2−k‖△kv1‖L2 E−1h,σ, c̃k = ‖△kf‖L2 ‖f‖L2 with σ = 1/2 + 1/r for 2 6 r < ∞. Proof of Lemma 5.1. We first prove (5.7). In view of the fact that v̂F only has high frequencies, Bony’s decomposition implies that there exists constant N0 such that I(u2)vFut = j>J−N0 Sj+2vF△jI(u2)ut + j>J−N0 Sj−1I(u2)△jvFut. (7.2) Since the negative derivative I acts on the high frequency for the former term while on the low frequency for the latter one, the first term is much better than the second one. We shall estimate the first term by using merely the Hölder inequality, Bernstein inequality and classical Strichartz estimates. Firstly, we see that, for 2 6 r < ∞ j>J−N0 ‖Sj+2vF△jI(u2)‖L2x . j>J−N0 j′6j+1 ‖△j′vF ‖L∞x ‖△jI(u 2)‖L2x j>J−N0 j′6j+1 r ‖△j′vF ‖Lr2j(γ− )‖u‖2L∞ Bernstein inequality and (2.1) in Proposition 2.1 with 1 for all 2 6 r < ∞ imply that j>J−N0 ‖Sj+2vF△jI(u2)‖L2x p ‖u‖2L∞ j>J−N0 j′6j+1 r ‖△j′vF‖Lp j(γ− 7 p ‖u‖2L∞ j>J−N0 2j(γ− j′6j+1 j′( 3 ‖△j′v0‖L2 + 2−j ′‖△j′v1‖L2 The right hand of the above inequality can be controlled by p ‖u‖2L∞ j>J−N0 2j(γ− j′6j+1 2 cj′Eh,σ and moreover it follows from (1.2), the definition of Eh,σ and Sobolev embedding that j>J−N0 ‖Sj+2vF△jI(u2)ut‖L1x pEh,σ‖u‖2L∞ j>J−N0 2j(γ−3)‖ut‖L∞ r 2−2J [s−( )]E2s ‖u‖2L∞ (H1), (7.3) for 4 6 r < ∞. Let us estimate the second term in (7.2) by the precise Strichartz estimates. Since this term contains that the negative derivative acts on the low frequency part Sj−1(u it leads to our new parameter r < 2 by some technique difficulties. Noting that Fourier-Plancherel formula and Hölder’s inequality, we can see that j>J−N0 Sj−1I(u2)△jvFutdxdt . j>J−N0 −16k6j−2 △kI(u2)△jvF△jutdxdt △kI(u2)△k k6j−2,J−N06j △jvF△jut . ‖u2‖ 2k(γ− k6j−2,J−N06j △jvF△jut (7.4) On one hand, we have J−N06j △jvF△jut ‖L2dt . j>J−N0 ‖△jvF△jut‖L1 j>J−N0 ‖△jvF ‖L∞ L2‖△jut‖L2 j>J−N0 2−sjcj c̃jEh,s‖ut‖L2t,x . If (7.4) is controlled by the term at k = −1, we can see that j>J−N0 ‖Sj−1I(u2)△jvF ‖L2 L2 . T 2 2−2J(s− )E2s ‖u‖L∞T (H1). (7.5) On the other hand, one denotes gk := △k k6j−2 △jvF△jut to estimate 2k(γ− )‖gk‖ Let us write that k6j−2 ν∈Λj,k △νj,kvF△jut. As the support of the Fourier transform of a product is included in the sum of the support of each Fourier transform, we obtain k6j−2 ν∈Λj,k △νj,kvF △̃νj,kut, as well as in (7.1). Using Hölder inequality, we get k6j−2 ν∈Λj,k ‖△νj,kvF ‖Lr‖△̃νj,kut‖L2 and the Cauchy-Schwarz inequality and the L2 quasi-orthogonality properties yield that k6j−2 ν∈Λj,k ‖△νj,kvF‖2Lr ν∈Λj,k ut‖2L2 k6j−2 ν∈Λj,k ‖△νj,kvF‖2Lr 2 ‖△jut‖L2 . (7.6) Precise Strichartz estimate implies that, for 1 with 2 6 r < ∞, r+2 ) 06k6j−2 2(k−j)( ν∈Λj,k ‖△νj,kv0‖2L2 2 + 2−j ν∈Λj,k ‖△νj,kv1‖2L2 ‖△jut‖L2t,x and observe the quasi-orthogonality properties again, this can be dominated by 06k6j−2 2(k−j)( ‖△jv0‖L2 + 2−j‖△jv1‖L2 ‖△jut‖L2t,x . Keeping the definitions of Eh,σ and cj in mind, one can see that r+2 ) p 2k( k6j−2 r cj c̃jEh,σE T (u) . T p 2k( )Eh,σE T (u). Therefore, we get that 2k(γ− )‖gk‖ r+2 ) 2k(γ−3)Eh,σE T (u) which implies nothing but j>J−N0 Sj−1I(u2)△jvFutdxdt ∣∣∣ . T 1− p Eh,σE T (u). (7.7) Finally, we get that, for 4 6 r < ∞ j>J−N0 Sj−1I(u2)△jvFutdxdt ∣∣∣ . T r 2−2J [s−( )]E2sET (u). However, although the r ranges 4 6 r < ∞, the above estimate still needs s > 3 to continue our proof. If we only consider the high frequency k > J , the (7.7) can be modified by j>J−N0 Sj−1I(u2)△jvFutdxdt ∣∣∣ . T 1− p 2J(γ−3)Eh,σE T (u) (7.8) and then we can obtain a better result j>J−N0 Sj−1I(u2)△jvFutdxdt ∣∣∣ . T r 2−2J [s−( )]E2sET (u), which implies the bad influence comes from the low frequency part and this is consist of the effect of negative derivative acts on the low frequency. But if we choose σ̃ = γ− 5 instead of σ, we can improve (7.8), at cost of restricting r such that max{2, 1 } < r < while not 2 6 r < ∞. Now we turn to details. It follows from similar argument r+2 ) p 2k( k6j−2 2−j(γ−3+ )cj c̃jEh,σ̃E T (u) where σ̃ = γ − 5 with 1 < 3− γ < 2 . We get 2k(γ− )‖gk‖ r+2 ) k6j−2 2(k−j)(γ−3+ )cj c̃jEh,σ̃E T (u) which implies nothing but j>J−N0 Sj−1I(u2)△jvFutdxdt ∣∣∣ . T 1− pEh,σ̃E T (u) by Young’s inequality. Note that σ̃ 6 γ < s when r sufficiently closes to 2 , therefore j>J−N0 Sj−1I(u2)△jvFutdxdt ∣∣∣ . T r 2−2J(s− )E2sET (u). Combining this with (7.3) and (7.5), we complete the proof of (5.7) by obtaining j>J−N0 Sj−1I(u2)△jvFutdxdt −2J [s−( −2J [s−( + T2−2J(s− E2sET (u) with max{2, 1 } < r1 < 23−γ and 6 r2 < ∞. We secondly prove (5.8) which is different from (5.7). To this end, we need to make Bony’s decomposition more than once and establish a commutator estimate, which helps us to complete our proof. In view of the fact that v̂F only has high frequencies again, it follows from Bony’s decomposition that there exists N0 such that I(uvF )uut = j>J−N0 Sj+2vF△ju uut + j>J−N0 △jvFSj−1u == I + II. (7.9) In order to estimate the term I, we split it into two pieces with N1 ≫ N0 > 0 j>J−N0 uut△kI Sj+2vF△ju j>J−N0 k6J−N1 uut△kI Sj+2vF△ju j>J−N0 k>J−N1 uut△kI Sj+2vF△ju == I1 + I2. The estimate of I1 is broken down into the following two cases. Case 1. 2 < γ 6 5 In this case, to our purpose, we obtain the following coarse estimate by Hölder’s inequality ‖I1‖L1x . j>J−N0 k6J−N1 Sj+2vF△ju ‖L3‖u‖L∞ L6‖ut‖L∞ j>J−N0 k6J−N1 2k(γ−2)‖△k Sj+2vF△ju ET (u) j>J−N0 k6J−N1 2k(γ−2)‖Sj+2vF‖L6‖△ju‖L2ET (u) k6J−N1 2k(γ−2) j>J−N0 2−j2j‖△ju‖L2 ‖△j′vF ‖L6ET (u). Choosing (p, r) such that 1 with 2 6 r 6 6, the Strichartz estimate yeilds ‖I1‖L1 k6J−N1 2k(γ−2) j>J−N0 j′( 3 ‖△j′v0‖L2 + 2−j ′‖△j′v1‖L2 T (u). Arguing similarly as before it yields that ‖I1‖L1 k6J−N1 2k(γ−2) j>J−N0 r cj′Eh,1/2E T (u) r 2−2J [s−( )]E2sET (u) with 2 6 r 6 6. If choose r = 6, one can easily check that γ 2 < γ 6 5 . Although this result is enough for us to prove the main theorem, we want to improve the result for this term by loosen the upper bound of r from 6 to ∞ through the precise Strichartz estimate. Arguing similarly as before, we have ‖I1‖L1x . j>J−N0 k6J−N1 Sj+2vF△ju ‖L3‖u‖L∞ L6‖ut‖L∞ j>J−N0 k6J−N1 2k(γ−3)2k 2r ‖△k Sj+2vF△ju ET (u) Since the Fourier transform of Sj−1vF△ju was supported in 2jC and k ≪ j, △k(Sj−1vF△ju) vanishes which implies △k Sj+2vF△ju △̃jvF△ju . As the support of the Fourier transform of a product is included in the sum of the support of each Fourier transform, we also have △̃jvF△ju ν,ν′∈Λj,k △νj,kvF△ν ν∈Λj,k △νj,kvF △̃νj,ku Choosing (p, r) such that 1 for 2 6 r < ∞, it follows from the Hölder inequality and L2 quasi-orthogonality properties that Sj+2vF△ju r+2 ) ν∈Λj,k ‖△νj,kvF‖Lr‖△̃νj,ku‖L2 ν∈Λj,k ‖△νj,kvF ‖2LpLr ν∈Λj,k ‖△νj,ku‖2L2 ν∈Λj,k ‖△νj,kvF ‖2LpLr ∥∥△ju Then the precise Strichartz estimate yields that ‖I1‖L1 k6J−N1 2k(γ−3)2k j>J−N0 2−j2j ∥∥△ju 2(k−j)( ν∈Λj,k ‖△νj,kv0‖2L2 2 + 2−j ν∈Λj,k ‖△νj,kv1‖2L2 ET (u). By the L2quasi-orthogonality properties, it gives that ‖I1‖L1 k6J−N1 2k(γ−3)2k j>J−N0 2−j2j ∥∥△ju ‖△jv0‖L2 + 2−j‖△jv1‖L2 ET (u). Utilizing the technique as before yields that ‖I1‖L1 k6J−N1 2k(γ−2+ j>J−N0 2−j(1+ )cjEh,σE T (u) r 2J(γ−3)Eh,σE T (u) . T r 2−2J [s−( )]E2sET (u), with 4 6 r < ∞. Case 2. 5 < γ < 3 In the this case, the fact γ− 5 > 0 helps us to obtain the the desirable result easily. Arguing similarly as before, we have ‖I1‖L1x . j>J−N0 k6J−N1 Sj+2vF△ju ‖L3‖u‖L∞ L6‖ut‖L∞ j>J−N0 k6J−N1 2k(γ−3)23k( Sj+2vF△ju ‖L2ET (u) j>J−N0 k6J−N1 2k(γ− )‖Sj+2vF‖L∞‖△ju‖L2ET (u). Choosing (p, r) such that 1 with 2 6 r < ∞, the Strichartz estimate yields ‖I1‖L1 k6J−N1 2k(γ− j>J−N0 2 cj′Eh,σE T (u) p 2J(γ− j>J−N0 2 Eh,σE T (u) r 2−2J [s−( )]E2sE T (u). Combining these two cases, we have shown that ‖I1‖L1 r 2−2J [s−( )]E2sET (u) (7.10) with 4 6 r < ∞. To control ‖I‖L1 , it remains to estimate ‖I2‖L1 . Compared with ‖I1‖L1 , since the negative derivative acts on the high frequency, the upper bound of ‖I2‖L1 is much easier to get. Here is the details: ‖I2‖L1x . j>J−N0 k>J−N1 Sj+2vF△ju ‖L3‖u‖L∞ L6‖ut‖L∞ j>J−N0 k>J−N1 2k(γ−3)‖Sj+2vF ‖L∞‖△ju‖L3ET (u). Choosing (p, r) such that 1 with 2 6 r < ∞ again, the Strichartz estimate yields ‖I2‖L1 k>J−N1 2k(γ−3) j>J−N0 2 cj′Eh,σE T (u) r 2−2J [s−( )]E2sET (u). Combining this with (7.10), we obtain that ‖I‖L1 r 2−2J [s−( )]E2sET (u) (7.11) for 4 6 r < ∞. To complete the proof the Lemma 5.1, it remains to estimate II. One can proceed this as above by Hölder’s inequality to estimate j>J−N0 2j(γ−3)‖△jvFSj−1u‖L3 ET (u). (7.12) Resorting to the Hölder inequality and the classical Strichartz estimate, one can obtain ‖II‖L1 L1 . T r 2−2J [s−( )]E2sET (u). with 2 6 r 6 6. One also can try to improve the result by using the precise Strichartz estimate as before, but it fails and merely obtain that ‖II‖L1 L1 . T r 2−2J [s−( )]E2sET (u). with 2 6 r 6 4. One can easily check that the result is worse than the desirable result because of the restriction of r. Compared with the second term in (7.2), the negative derivative acts on the high frequency part so that it is tempting to obtain a better result than that of (7.2). But △jvF is bound with Sj−1u by the operator I, and this structure prevents us from using efficiently the precise Strichartz estimate. If one first resort to the Hölder inequality, as shown in (7.12), he or she merely obtains a loss result because of the range restriction of r. To go around this difficulty, we first establish a commutator estimate through exploiting cancellation property. Now we turn to details. Our task is to estimate j>J−N0 I(△jvFSj−1u)uutdxdt In order to drag the Sj−1u out of the operator I, we construct uI(△jvF )Sj−1u and the triangle inequality yields that j>J−N0 I(△jvFSj−1u)uutdxdt ∣∣∣ 6 j>J−N0 ∥∥(I(△jvFSj−1u)− I(△jvF )Sj−1u j>J−N0 I(△jvF )Sj−1uuutdxdt We benefit from the cancellation when we deal with the first term. Since both the Fourier transformation of I(△jvFSj−1u) and I(△jvF )Sj−1u are supported in a ring sized 2j , the Hölder inequality and the Bernstein inequality lead to that ∥∥(I(△jvFSj−1u)− I(△jvF )Sj−1u 2 ‖I(△jvFSj−1u)− I(△jvF )Sj−1u‖L2x‖u‖L6 . Before estimating its right hand, we recall the Coifman and Meyer multiplier theorem. Consider an infinitely differentiable symbol m : Rnk 7→ C so that for all α ∈ Nnk and all ξ = (ξ1, ξ2, · · · , ξk) ∈ Rnk, there is a constant c(α) such that |∂αξ m(ξ)| 6 c(α)(1 + |ξ|)−|α|. (7.13) Define the multilinear operator T by [T (f1, · · · , fk)](x) = eix·(ξ1+···+ξk)m(ξ1, · · · , ξk)f̂1(ξ1), · · · , f̂k(ξk)dξ1 · · · dξk, (7.14) F [T (f1, · · · , fk)](ξ) = ξ=ξ1+···+ξk m(ξ1, · · · , ξk)f̂1(ξ1), · · · , f̂k(ξk)dξ1 · · · dξk−1. (7.15) Proposition 7.1 ([8],Page 179.) Suppose pj ∈ (1,∞), j = 1, · · · k, are such that 1p = + · · ·+ 1 6 1. Assume m(ξ1, · · · , ξk) a smooth symbol as in (7.13). Then there is a constant C = C(pi, n, k, c(α)) so that for all Schwarz class functions f1, · · · , fk, ‖[T (f1, · · · , fk)](x)‖Lp(Rn) 6 C‖f1‖Lp1 (Rn) · · · ‖fk‖Lpk (Rn). (7.16) Since the operator I is a convolution operator with kernel |x|−γ in R3, we can write F [I(△jvFSj−1u)−I(△jvF )Sj−1u](ξ) = ξ=ξ1+ξ2 |ξ1+ξ2|γ−3−|ξ1|γ−3 △̂jvF (ξ1)Ŝj−1u(ξ2)dξ2. By the mean value theorem, the right hand of the above formula becomes that ξ=ξ1+ξ2 |ξ1 + λξ2|γ−4 (ξ1 + λξ2) · ξ2 |ξ1 + λξ2| △̂jvF (ξ1)Ŝj−1u(ξ2)dξ2, for a certain λ ∈ [0, 1]. Moreover, we rewrite it as follow: ξ=ξ1+ξ2 m(ξ1, ξ2)f̂1(ξ1)f̂2(ξ2)dξ2, m(ξ1, ξ2) = (ξ1 + λξ2)|ξ1 + λξ2|γ−5|ξ1|4−γ , f1 = |∇|γ−4△jvF , f2 = ∇Sj−1u. Observe that |ξ1| > 2j−1 and 2j−2 > |ξ2|, we have that |ξ1+λξ2| ∼ |ξ1| > 2J−N0 . Hence, we can check that the symbol m(ξ1, ξ2) satisfies the estimate (7.13). Finally, it follows from Proposition 7.1 that ‖I(△jvFSj−1u)− I(△jvF )Sj−1u‖L2x . ‖f1‖Lrx‖f2‖ with 2 < r < ∞. After making use of the Bernstein inequality, the right hand can be controlled by 2j(γ−4+ )‖△jvF ‖Lrx‖∇u‖L2x . Keeping in mind j > J−N0 and recalling the definition of Eh,σ, the Strichartz estimate and a direct calculation of summing in j show that j>J−N0 2 2j(γ−4+ )‖△jvF ‖Lp j>J−N0 2j(γ−3+ −s)Eh,s. with 1 and 2 < r < ∞. Choosing r such that max{2, 1 } 6 r < ∞, we have j>J−N0 ∥∥(I(△jvFSj−1u)− I(△jvF )Sj−1u r 2−2J [s−( )]E2sET (u). (7.17) Now the rest of the paper devotes to estimate this term j>J−N0 I(△jvF )Sj−1uuutdxdt In order to use precise Strichartz estimate, we need to decompose this term by Bony’s para-product decomposition again, I(△jvF )Sj−1uuut = Sk−1(uSj−1u)△kI(△jvF )ut +△k(uSj−1u)Sk+2I(△jvF ) = II1 + II2. After decomposing this, the term II1 is similar to the second term in the (7.2) and the negative derivative acts on the high frequency △jvF leading to a better result than the second term in the (7.2). Thanks to Fourier-Plancherel formula and Hölder inequality, we obtain j>J−N0 II1dxdt ≈ j>J−N0 Sk−1(uSj−1u)△kI(△jvF )△kutdxdt j>J−N0 k′6k−2 △k′(uSj−1u)△kI(△jvF )△kutdxdt j>J−N0 △k′(uSj−1u)△k′ k′6k−2 (△kI(△jvF )△kut)dxdt j>J−N0 ‖uSj−1u‖ 2 ‖△k′ k′6k−2 (△kI(△jvF )△kut)‖L2 j>J−N0 ‖u‖2L∞H1 2 ‖△k′ k′6k−2 (△kI(△jvF )△kut)‖L2 On the other hand, one denotes gk′,j = △k′ k′6k−2 △kI(△jvF )△kut to estimate ∑ ′(− 1 )‖gk′,j‖ Let us write that gk′,j = k′6k−2 ν∈Λk,k′ △νk,k′I(△jvF )△kut As the support of the Fourier transform of a product is included in the sum of the support of each Fourier transform, we obtain gk′,j = k′6k−2 ν∈Λk,k′ △νk,k′I(△jvF )△̃νk,k′ut Using Hölder inequality, we get ‖gk′,j‖ k′6k−2 ν∈Λk,k′ ‖△νk,k′I(△jvF )‖Lr‖△̃νk,k′ut‖L2 6 2j(γ−3) k′6k−2 ν∈Λk,k′ ‖△νk,k′vF ‖2Lr ν∈Λk,k′ ‖△νk,k′ut‖2L2 6 2j(γ−3) k′6k−2 ν∈Λk,k′ ‖△νk,k′vF ‖2Lr 2‖△kut‖L2 the use of quasi-orthogonality properties is made in the last inequality. Precise Strichartz estimate and the quasi-orthogonality properties imply that ‖gk′,j‖ r+2 ) p 2j(γ−3) k′6k−2 ′−k)( 1 ν∈Λk,k′ ‖△νk,k′v0‖2L2 2 + 2−k ν∈Λk,k′ ‖△νk,k′v1‖2L2 ‖△kut‖L2 p 2j(γ−3) k′6k−2 ′−k)( 1 ‖△kv0‖L2 + 2−k‖△kv1‖L2 ‖△kut‖L2 with 1 for 2 6 r < ∞. Therefore ′(− 1 )‖gk′,j‖ r+2 ) p 2j(γ−3) k′6k−2 ′−k) 2 r ck c̃kEh,σE T (u). A direct computation shows that 2 ‖gk′,j‖L1 ′(− 1 )‖gk′,j‖ r+2 ) p 2j(γ−3)Eh,σE T (u). Hence, we have that j>J−N0 II1dxdt ∣∣∣ . 2−2J [s−( r E2sET (u) (7.18) with 4 6 r < ∞. Finally, we conclude this section by giving the estimate of II2. j>J−N0 II2dxdt ∣∣∣ . T j>J−N0 ‖△k(uSj−1u)Sk+1I(△jvF )‖L2 L2‖ut‖L∞T L2 j>J−N0 2j(γ−3) ‖△k(uSj−1u)‖L∞ r ‖△k′△jvF‖Lp T (u) j>J−N0 2j(γ−3) ‖△k(uSj−1u)‖L∞ 2 ck′Eh,σE T (u) j>J−N0 2j(γ−3) 2 ‖△k(uSj−1u)‖L∞ (k′−k) 1 2Eh,σE T (u) j>J−N0 2j(γ−3)‖ck′‖ℓ2(Z)‖2 2 ‖△k(uSj−1u)‖L∞ L2‖ℓ2(Z)‖2− 2 ‖ℓ2(N)Eh,σE T (u) p 2J(γ−3)E T (u)Eh,σ . 2 −2J [s−( r E2sET (u). (7.19) Collecting (7.18) and (7.19), we have been proved that j>J−N0 I(△jvF )Sj−1uuutdxdt ∣∣∣ . T r 2−2J [s−( )]E2sET (u), (7.20) with 4 6 r < ∞. Finally, we complete the proof of (5.8) by (7.11) and (7.20), hence it ends the proof of Lemma 5.1. Acknowledgements: The authors are grateful to Prof. J.Chemin for sending his lecture to us. The authors were partly supported by the NSF of China, No.10725102. References [1] Bergh J. and Löfstrom J., Interpolation spaces, An Introduction, New York: Springer-Verlag, 1976. [2] H.Bahouri and Jean-Yves Chemin On global well-posedness for defocusing cubic wave equation , Internat. Math. Res. Notices Volume 2006, Article ID 54873, Pages 1-12. [3] Jean Bourgain. Refinements of Strichartz’s inequality and applications to 2D-NLS with critical nonlinearity. Internat. Math. Res. Notices, 5:253-283, 1998. [4] J. Bourgain. The Global Solution of Nonlinear Schrödinger Equations, American Mathematical Society, Providence, 1999. 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[31] T.Tao, Nonlinear Dispersive Equations, local and global analysis, CBMS Regional Series in Mathematics, no.106, Amer. Math. Soc. Providence, RI,2006. [32] Triebel, H., Interpolation theory, function spaces, differential operators. North- Holland, Amsterdam, 1978. http://arxiv.org/abs/math/0612028 Introduction Preliminaries Global existence for the high frequency part Local existence for the low frequency part Energy estimate for the low frequency part Proof of Theorem ?? Proof of Lemma ?? References

In this paper, we consider the Cauchy problem for Klein-Gordon equation with a cubic convolution nonlinearity in $\R^3$. By making use of Bourgain's method in conjunction with a precise Strichartz estimate of S.Klainerman and D.Tataru, we establish the $H^s (s<1)$ global well-posedness of the Cauchy problem for the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving at the previously discussed conclusion, we obtain global solution for this non-scaling equation with small initial data in $H^{s_0}\times H^{s_0-1}$ where $s_0=\frac\gamma 6$ but not $\frac\gamma2-1$, for this equation that we consider is a subconformal equation in some sense. In doing so a number of nonlinear prior estimates are already established by using Bony's decomposition, flexibility of Klein-Gordon admissible pairs which are slightly different from that of wave equation and a commutator estimate. We establish this commutator estimate by exploiting cancellation property and utilizing Coifman and Meyer multilinear multiplier theorem. As far as we know, it seems that this is the first result on low regularity for this Klein-Gordon-Hartree equation.

Introduction We study the following Cauchy problem for the Klein-Gordon-Hartree equation: �φ+ φ+ (|x|−γ ∗ |φ|2)φ = 0 in R×R3 φ|t=0 = φ0, ∂tφ|t=0 = φ1. (1.1) Here φ(t, x) is a complex valued function defined in space time R1+3, and � = ∂tt −∆. http://arxiv.org/abs/0704.1866v4 Recently the Cauchy problem (1.1) has been extensively studied in the case with initial data (φ0, φ1) ∈ H1(Rn) × L2(Rn). The well-posedness and the asymptotic be- havior of solution to the Cauchy problem (1.1) have been studied by G.P. Menzala and W.Strauss [16, 17]. The scattering theory of solution to (1.1) has been established in [23]. On the other hand, the time-dependent Schrödinger equation with interaction term (|x|−γ ∗ |φ|2)φ has also been extensively studied. Ginibre and Velo [11] gave the scattering theory of Hartree equation for the energy subcritical case. For the energy critical case and mass critical, one can refer to [20, 21] with radial initial data. Many authors [4, 9, 12, 18, 30] have studied the local well-posedness (as well as global well-posedness) in fractional Sobolev spaces for the Cauchy problem of general semilinear wave or Schrödinger equations under minimal regularity assumptions on the initial data. For example, Tao [30] established the sharp local well-posedness of nonlinear wave equation. Kenig, Ponce, and Vega [12] had established the global well- posedness under the energy norm for the Cauchy problem of nonlinear wave equations with rough initial data (in particular, in Ḣs(R3), 3 < s < 1 for cubic wave equation). They used the Fourier truncation method discovered by Bourgain [4]. And also [18] extended Kenig-Ponce-Vega’s result to the dimension n > 4. Recently, I. Gallagher and F. Planchon [9] presented a different proof of the result in [12] for 3 < s < 1. H. Bahouri and Jean-Yves Chemin [2] proved global well posedness for s = 3 by using a nonlinear interpolation method and logarithmic estimates from S. Klainermann and D. Tataru[14]. We also find Roy [26] obtains the global well-posedness for rough initial data in Ḣs, 2 < s < 1 by following the I-method [5] and scaling transformation. However, if one similarly deals with Klein-Gordon equation by using I-method, he or she may meet a problem caused by the lack of the scaling property. More studies and discussions on the low regularity of nonlinear wave or dispersive Schrödinger equations could be found in [4, 31]. However, as far as we know, very few authors are engaged in studying the global well-posedness of the Cauchy problems (1.1) with less regular initial data. It is natural to ask whether a similar or better result holds for the problem (1.1). This paper endeavors to find a global well-posedness solution to the Cauchy problem (1.1) with initial data (φ0, φ1) ∈ Hs(R3) ×Hs−1(R3) for some s > γ4 with γ ∈ (2, 3). Now we should remark some differences between (1.1) and cubic wave equation. If one views (1.1) as a wave equation by dropping the massive term and then makes some scaling analysis, we will find this nonlocal nonlinear term shares the scaling property of the nonlinearity |u| 5−γ u. One can check that k := 4 +1 < 3 when 2 < γ < min{n, 4} with n = 3 and this result shows that the equation which we consider is in subconformal case. To obtain the global well-posedness theory, some previous literatures also show the subconformal equations are slightly different from the superconformal ones. For instance, Lindblad and Sogge [15] [27] have shown the global existence and scattering theory for small data in a less regularity space for the superconformal case, while not for the subconformal case. Inspired by [9], we also split the initial data into low frequency part data in H1 and high frequency part data in Hs0 with a suitable s0. Since the problem (1.1) is global well-posed for large data in H1 and small data in Hs0 , one may be tempted to follow a general principle of nonlinear interpolation and claim the problem (1.1) is global well-posed between them. Compared with the cubic wave equation, speaking of the Strichartz estimate, we believe that the global solution with high frequency data should exist in H −1. It is well known that the Strichartz estimate is associated with scaling transform and it is scaling invariant. Unfortunately, the equation that we consider is a subconformal one, and its concentration effects take over scaling. Since the Strichartz estimate is applied to our subconformal equation, hence this brings about some loss to get a better result. In order to get a better result, one should establish an estimate which is conformal invariant. Fortunately, we can take 0 6 θ 6 1 as a parameter for the flexible admissible pairs (see Definition 2.3)to make the Strichartz estimate of Klein-Gordon more flexible than wave equation. This helps us to get a global solution with the high frequency data, at the cost of 0 6 θ = 6 −2 6 1 which weakens the Strichartz estimate and causes 2 < γ < 3. One can refer the detail in Section 3. We point out that it is easy to have the result for < s < 1 by rough Hölder’s inequality. But how to get our low bound < s < 1? A good way to think about this is via precise Strichartz estimate to obtain index s as low as possible. The nonlinearity including a formal negative derivative brings us some difficulties caused by the fact that the negative derivative acts on the low frequency part. And this leads us to restricts rather than s > max{1 }. At the end of this section, we also give some intuitive analysis to show our result is reasonable. As a limited case, our result recovers the result of [9, 12] when γ tends to 3. During the process of proving our key estimate Lemma 5.1, the nonlocal nonlin- earity brings about some essential difficulties when we try to make use of the precise Strichartz estimate. Compared with the general semilinear nonlinearity, the convo- lution nonlinearity not only essentially represents a negative derivation in it but also has a difference construction of nonlinearity. These differences and difficulties prevent us from obtaining directly our expected result s > by restricting the range of the parameter r. To overcome these difficulties, we firstly construct a commutator and establish this commutator estimate by exploiting cancellation property and utilizing Coifman and Meyer multilinear multiplier theorem and then go on our process through using precise Strichartz estimate. Now we state our main result: Theorem 1.1 Let < s < 1 with 2 < γ < 3. If (φ0, φ1) ∈ Hs(R3) ×Hs−1(R3), then there exists a unique global solution φ of (1.1) such that φ ∈ C(R+;Hs(R3)). We conclude this section by giving a sketch of the proof of Theorem 1.1 and one shall read more detailed information in the rest of this paper. Without loss of generality, we only consider φ as a real function for simplicity from now on. Since the problem (1.1) is global well-posed for large data in H1 and small data in Hs0 with s0 = , one may be tempted to follow a general principle of nonlinear interpolation and believe the problem (1.1) to be global well-posed between them, as well as the cubic defocusing wave equation [9]. To make sense of this heuristic, we proceed it in the following steps. Step 1. The purpose of this step is to show the global well-posedness for the high frequency part. We split the initial data: φi = (I− SJ)φi + SJφi = vi + ui i = 0, 1 where I is identity operator and SJ is Littlewood-Paley operator, referring to Section 2. It is easy to see that ‖u0‖H1 . 2J(1−s)‖φ0‖Hs , ‖u1‖L2 . ‖φ1‖L2 ‖v0‖Hβ . 2J(β−s)‖φ0‖Hs , ‖v1‖Hβ−1 . 2J(β−s)‖φ1‖Hs−1 for all β 6 s. Thus it follows that Eh,σ . 2J(σ−s)Es, for σ 6 s (1.2) Eℓ,1 . 2J(1−s)Es, for s 6 1, (1.3) where == ‖φ0‖Hs + ‖φ1‖Hs−1 , (1.4) == ‖v0‖Hσ + ‖v1‖Hσ−1 , (1.5) == ‖u0‖Hσ + ‖u1‖Hσ−1 . (1.6) Choosing J large enough, one can achieve Eh,s0 small enough, in other words, initial data of the following problem �v + v + (|x|−γ ∗ v2)v = 0 in R× R3, v|t=0 = v0, ∂tv|t=0 = v1 (1.7) is small enough inHs0(R3)×Hs0−1(R3) where s0 < s. Due to some technique difficulties and this equation is subconformal one, we are restricted to choose s0 = while not − 1 proposed by scaling analysis or γ proposed by conformal analysis. We will get a global well-posed solution to the Cauchy problem (1.7), see Section 3 for details. Step 2. In order to recover a solution to our problem (1.1), we solve a perturbed equation with large initial data in H1 × L2, �u+ u+ I(u2)u+ 2I(uv)u+ I(v2)u+ I(u2)v + 2I(uv)v = 0, u|t=0 = u0 ∂tu|t=0 = u1, (1.8) where the operator I is the operator (−∆) 2 . We will prove there exists a unique local solution to (1.8) in C([0, T ];H1). Step 3. To complete the proof of Theorem 1.1, the key is how to extend the local solution to a global solution. We should establish a priori bound on the energy of the local solution u. In fact, the energy estimate yields ‖u(t)‖2H1 + ‖ut(t)‖ R3×R3 |x− y|−γu2(y, t)u2(x, t)dydx ‖u0‖2H1 + ‖u1‖ R3×R3 |x− y|−γu20(y)u20(x)dydx I(v2)(x, τ)u(x, τ)∂τu(x, τ)dxdτ I(uv)(x, τ)v(x, τ)∂τu(x, τ)dxdτ I(u2)(x, τ)v(x, τ)∂τu(x, τ)dxdτ I(uv)(x, τ)u(x, τ)∂τu(x, τ)dxdτ ∣∣∣∣ . Let HT (u) := sup H(u)(t) where H(u)(t) ‖u(t)‖2H1 + ‖ut(t)‖2L2 + R3×R3 |x− y|−γu2(y, t)u2(x, t)dydx and then by making use of Hölder’s inequality and Sobolev embedding, it follows that HT (u) .H(u)(0) +HT (u) ‖v(τ)‖2 dτ +H T (u) ‖v(τ)‖ .H(u)(0) +HT (u)T 6 ‖v‖2 T (u)T 3 ‖v‖Xα .22J(1−s) +HT (u)T 6 22J(β−s) +H T (u)T 3 2J(α−s) where α = , β = and the space Xα is defined in the coming section. What we want to do is to control HT (u) for arbitrarily large T . As long as s > (α+1)/2 = by choosing J large enough, bootstrap argument yields HT (u) . 2 2J(1−s). One can see that, if s > , the argument is trivial, since the above mentioned result can be deduced from some rough estimates such as the Hölder estimate. On the other hand, since the scaling suggests us that X −1 is the lowest regularity space which v could belong to, it is tempting and reasonable to believe that the best result obtained by this method is s > ( − 1+1)/2 = γ instead of α by − 1. To obtain this optimal result s > , we adopt some more sophisticated tools such as precise Strichartz estimate, Bony’s paraproduct estimates and twice Bony’s decomposition. This result is achieved under an assumption of a core estimate which will be shown through the precise Strichartz estimate and a commutator estimate. The paper is organized as follows: In the coming section, we recall some nota- tions and recollect some well known results on Besov spaces in conjunction with the Littlewood-Paley theory which will be used in the course of the proofs. Meanwhile, we also introduce the precise Strichartz estimate. Section 3 provides the global well- posedness of original equation evoking the high frequency part of initial data in Hs0 . In Section 4, we prove prove a local well-posedness of perturbed equation with the low frequency of the initial data in H1 by the standard fixed point theorem. In Section 5, we give a energy estimate for the low frequency part provided an assumption the key estimate in Lemma 5.1. We extend the local well-posedness of the perturbed equation to globally well posed by the bootstrap argument in Section 6. In the final section, we prove our essential and key lemma by the precise Strichartz estimate, commutator estimate and Coifman and Meyer multiplier theorem. 2 Preliminaries In this section, we shall present some well-known facts on the Littlewood-Paley theory and introduce some notations, definitions and estimates which are needed in this paper. Let S(R3) be the Schwarz class of rapidly decreasing functions. Given f ∈ S(R3), its Fourier transform Ff = f̂ is defined by f̂(ξ) = (2π)− e−ix·ξf(x)dx, F−1f = f̂(−ξ). Choose two nonnegative radial functions χ, ϕ ∈ S(R3) supported respectively in B = {ξ ∈ R3, |ξ| 6 4 } and C = {ξ ∈ R3, 3 6 |ξ| 6 8 } such that χ(ξ) + ϕ(2−jξ) = 1, ξ ∈ R3, ϕ(2−jξ) = 1, ξ ∈ R3\{0}, supp ϕ(2−j ·) ∩ supp ϕ(2−j′ ·) = ∅, |j − j′| > 2, supp χ(·) ∩ supp ϕ(2−j ·) = ∅, j > 1. Now we are in position to define the the Littlewood-Paley operators Sj, Ṡj, △j and △̇j which are used to define Besov space. 0, j 6 −2, χ(ξ)û(ξ) , j = −1, (F−1ϕ)(2jy)u(x− y)dy, j > 0, j′6j−1 △j′u = 2jn (F−1χ)(2jy)u(x− y)dy, == 2jn (F−1ϕ)(2jy)u(x− y)dy, j ∈ Z, j′6j−1 △̇j′u. One easily shows that △̇j = Ṡj+1 − Ṡj for j ∈ Z and △−1 = S0, △̇j = △j , j > 0. Now we give the Littlewood-Paley’s description of the Besov spaces. Definition 2.1 Let s ∈ R, 1 6 p, q 6 ∞. The homogenous Besov space Ḃsp,q is defined Ḃsp,q = {f ∈ Z ′(R3) : ‖f‖Ḃsp,q < ∞}, where Ḃsp,q 2jsq‖△̇jf‖qp , for q < ∞, 2js‖△̇jf‖p, for q = ∞, and Z ′(R3) can be identified by the quotient space S ′/P with the space P of polynomials. Definition 2.2 Let s ∈ R, 1 6 p, q 6 ∞. The inhomogeneous Besov space Bsp,q is defined by Bsp,q = {f ∈ S ′(R3) : ‖f‖Bsp,q < ∞}, where ‖f‖Bsp,q = 2jsq‖△jf‖qp + ‖S0(f)‖p, for q < ∞, 2js‖△jf‖p + ‖S0(f)‖p, for q = ∞. If s > 0, then Bsp,q = L p ∩ Ḃsp,q and ‖f‖Bsp,q ≈ ‖f‖p + ‖f‖Ḃsp,q . We refer the reader to [1, 6, 22, 32] for details. In order to investigate the low regularity solution of the Cauchy problem (1.1), we require the use of the smoothing effect described by the Strichartz estimates and precise Strichartz estimates. For the purpose of conveniently making use of the Strichartz estimate, we introduce the admissible definition and the resolution space. Definition 2.3 We shall say that a pair (q, r) is admissible, for 0 6 θ 6 1, if q, r > 2, (q, r, θ) 6= (2,∞, 0) and 1 2 + θ 2 + θ Remark 2.1 The above admissible pairs in Definition 2.3 is more flexible than wave admissible pairs, since θ can vary from 0 to 1. Obviously, an admissible pair in Defini- tion 2.3 will become a wave admissible pair when θ = 0. When we consider the global existence for the high frequency part, we shall use θ = 6 − 2 since the equation that we consider is a subconformal one. The resolution space is defined in the following way based on the admissible defini- tion. Xµ(I) := 06θ61 where (I) := u : u ∈ (C ∩ L∞)(I;Hµ) ∩ Lq(I;Bσr,2) (q,r) is admissible, = (3 + θ)( ) + σ − µ We go on this section by recalling the classical Strichartz estimate and the precise Strichartz estimate. This kind of estimate goes back to Strichartz [29], and has been proved in its generality by Ginibre and Velo [10], and Keel and Tao [13]. The Strichartz estimates for the Klein-Gordon equation by using the above flexible admissible pairs can be found in [19]. Proposition 2.1 Let u be a solution of �u+ u = f in R× R3 with u|t=0 = u0, ∂tu|t=0 = u1. Then, for any admissible pairs (q1, r1) and (q2, r2), we have that ‖△ju‖Lq1 (Lr1 ) + 2−j‖∂t△ju‖Lq1 (Lr1 ) j( 3+θ − 3+θ (‖△ju0‖L2 + 2−j‖△ju1‖L2) j[(3+θ)(1− 1 −1]‖△jf‖ . (2.1) We shall see that the classical Strichartz estimates are not enough to control some nonlinearities, and this leads us to resort to the following precise Strichartz estimates which were established by S.Klainerman and D.Tataru[14]. Proposition 2.2 Let u be a solution of �u+ u = 0 with u|t=0 = u0, ∂tu|t=0 = u1. Assume that the supports of the Fourier transform of u0 and u1 are included in a ball B(ξj, h2 j) with |ξj | ∈ [2j−2, 2j+2] and h < 18 . Then we have that, for any admissible couple (q, r), ‖u‖Lq(Lr) + 2−j‖∂tu‖Lq(Lr) 6 C2 r (‖u0‖L2 + 2−j‖u1‖L2). (2.2) Let us recall the Hardy-Littlewood-Sobolev inequality [22, 28] and a proposition of contraction which is generalization of Picard’s theorem [6]. We denote operator I by Iu def== (−∆) 2 u = |x|−γ ∗ u, ‖Iu‖Lq(R3) 6 Cp,q‖u‖Lp(R3) (2.3) 0 < γ < 3, 1 < p < q < ∞, and 1 − 3− γ Proposition 2.3 Let X be a Banach space and let B : X ×X × · · · ×X → X be a m-linear continuous operator (m > 2) satisfying ‖B(u1, u2, · · · , um)‖X 6 M‖u1‖X‖u2‖X · · · ‖um‖X , ∀u1, u2, · · · , um ∈ X for some constant M > 0. Let ε > 0 be such that m(2ε)m−1M < 1. Then for every y ∈ X with ‖y‖X 6 ε the equation u = y +B(u, u, · · · , u) (2.4) has a unique solution u ∈ X satisfying that ‖u‖X 6 2ε. Moreover, the solution u continuously depends on y in the sense that, if ‖y1‖X 6 ε and v = y1 +B(v, v, · · · , v), ‖v‖X 6 2ε then ‖u− v‖X 6 1−m(2ε)m−1M ‖y − y1‖X . (2.5) For the sake of convenience, we conclude this section by giving some notations. The solution φ to the Cauchy problem (1.1) is given by the following integral equation: φ(t, x) = K̇(t)φ0 +K(t)φ1 +B(φ, φ, φ) = T φ where K(t) := sin(t I −∆)√ B(u1, u2, u3) := − K(t− τ)(|x|−γ ∗ (u1u2))u3dτ. Throughout this article we shall denote by the letter C all universal constant and ε > 0 is a arbitrary small data. We shall sometimes replace an inequality of the type f 6 Cg by f . g. Also, we shall denote by (cj)j∈Z any sequence of norm less than 1 in ℓ 2(Z). 3 Global existence for the high frequency part Let us consider the Cauchy problem with the high frequency data, �v + v + (|x|−γ ∗ v2)v = 0, (t, x) ∈ R× R3 v|t=0 = v0, ∂tv|t=0 = v1, x ∈ R3. (3.1) and then its integral formation becomes v(t, x) =K̇(t)v0(x) +K(t)v1(x)− K(t− τ)(|x|−γ ∗ v2)vdτ ==K̇(t)v0(x) +K(t)v1(x) +B(v, v, v). (3.2) Our goal in this section is to prove the global well-posedness of (3.1) or (3.2). More precisely, we have the following proposition: Proposition 3.1 Let s0 = and suppose that (v0, v1) ∈ Hµ×Hµ−1 for any 0 6 µ 6 1. There exists a constant ε0 > 0 such that if ‖v0‖Hs0 + ‖v1‖Hs0−1 6 ε0, then there exists a unique global solution v to (3.1) or (3.2) in Xs0(R)∩Xµ(R). More- over, ‖v‖Xµ 6 Cµ (‖v0‖Hµ + ‖v1‖Hµ−1) . Remark 3.1 We focus on µ = and µ = in the coming section. It is well known that the global existence theory for small initial data is a straightfor- ward result of nonlinear estimate, thus how to obtain a suitable nonlinear estimate is essential. Before proving this proposition, we make some analysis on nonlinear estimate. As mentioned in the introduction, the nonlocal nonlinearity shares the scaling with a subconformal nonlinearity when γ < 3 and this may bring some troubles when we make a choice of a suitable resolution space Xs0 . Take 0 6 θ 6 1 as a parameter in the flexible admissible pairs (see Definition 2.3), and we make analysis on the relationship between θ and s0. The Strichartz estimate, Hölder inequality and Hardy-Littlewood-Sobolev inequality imply that, for σ 6 0, ‖B(v, v, v)‖Xs0 6 ‖(|x|−γ ∗ |v|2)v‖ 6 ‖v‖ Lq2 (B ‖v‖2Lq3 (Lr3 ), with satisfying = (3 + θ)( ) + σ + s0 − 1 = (3 + θ)( )− σ − s0 = (3 + θ)( )− s0 − 1 + γθ We find the fact index s0 is increasing when the parameter θ increases. It is tempting to choose θ = 0 to get the smallest s0 = − 1 proposed by scaling. However, in addition the admissible condition implies that 6 (2 + θ)( 6 (2 + θ)( 6 (2 + θ)( then a direction computation gives that ) 6 (2 + θ)(2− 1 which yields that 6 1 + If we choose θ = 0, then we are forced to γ > 3 which contradict with our requirement γ < 3. But if we choose θ = 6 − 2 and then s0 = γ6 and we are allowed by 2 6 γ 6 3. Proof of Proposition 3.1 Thanks to Strichartz estimate, we have ‖B(v, v, v)‖Xµ 6 ‖(|x|−γ ∗ |v|2)v‖ 6 ‖v‖ Lq2 (B ‖v‖2Lq3 (Lr3 ), where 3 + γ (1− µ− σ), 1 3 + γ (1− µ− σ) 3(µ+ σ) 3 + γ − γ(µ+ σ) 3 + γ 2(3 + γ) 9 + 3γ − γ2 6(3 + γ) When 0 6 µ 6 1 , we choose σ = 0; while 1 < µ 6 1, we choose σ = 1 Thus, ‖B(v, v, v)‖Xµ 6 ‖v‖Xµ‖v‖2Xs0 . (3.3) Combining this nonlinear estimate, the Proposition 3.1 follows from a standard con- traction argument and small initial data condition. 4 Local existence for the low frequency part In this part, we shall study the following perturbed problem in R× R3: �u+ u+ I(u2)u+ 2I(uv)u+ I(v2)u+ I(u2)v + 2I(uv)v = 0 u|t=0 = u0 ∂tu|t=0 = u1. (4.1) Proposition 4.1 Let α = , β = and assume that v be in Xα ∩ Xβ and (u0, u1) ∈ H1 ×L2, then there exists a positive time T such that a unique solution u to (4.1) satisfying u ∈ C([0, T ];H1). Proof of the Proposition 4.1 In practice, solving (4.1) on [0, T ] is equivalent to solving the following integral equation u =K̇(t)u0 +K(t)u1 K(t− τ) I(u2)u+ 2I(uv)u+ I(v2)u+ I(u2)v + 2I(uv)v ,T̃ u. Using the Strichartz estimate, we have K(t− τ)I(u2)udτ . ‖I(u2)u‖L1 (L2). On one hand, we make use of Hölder’s inequality and Hardy-Littlewood-Sobolev in- equality to deduce that ‖I(u2)u‖L1 (L2) 6 C‖I(u2)‖ 6 C‖u‖3 6 CT‖u‖3L∞ H1 . (4.2) For the rest of terms, arguing similarly as above, it can be obtained that ‖I(uv)u‖L1 (L2) 6 C‖u‖2L∞ L6‖v‖ 3 ‖u‖2L∞ H1‖v‖Xα , (4.3) ‖I(v2)u‖L1 (L2) 6 C‖u‖L∞T L6‖v‖ 3 ‖u‖L∞ H1‖v‖2Xβ , (4.4) ‖I(u2)v‖L1 (L2) 6 CT 3 ‖u‖2L∞ H1‖v‖Xα , (4.5) ‖I(uv)v‖L1 (L2) 6 CT 3 ‖u‖L∞ H1‖v‖2Xβ . (4.6) A combination of (4.2), (4.3)-(4.6) and the Strichartz estimate in Proposition 2.1 lead to the estimate ‖u‖L∞ (H1) . ‖u0‖H1 + ‖u1‖L2 + T‖u‖3L∞ 3 ‖u‖2L∞ H1‖v‖Xα + T 3 ‖u‖L∞ H1‖v‖2Xβ . As long as choosing T is small enough, T̃ is a contraction mapping in ball B(0, 2CEℓ,1). By means of Picard’s fixed point argument we have an unique solution u to (4.1) in L∞([0, T ];H1). Therefore, Proposition 4.1 is proved by the standard argument. 5 Energy estimate for the low frequency part In order to extend the local solution to a global solution, we shall prove a prior estimate for the Hamiltonian of u in this section. Let us recall the definition of Hamiltonian of u defined by H(u)(t) ‖u(t)‖2H1 + ‖ut(t)‖2L2 + R3×R3 |x− y|−γu2(y, t)u2(x, t)dydx Similarly we give another notation of the energy of u, which is denoted by E(u)(t) ‖u(t)‖2H1 + ‖ut(t)‖2L2 . HT (u) == sup H(u)(t), ET (u) == sup E(u)(t). To extend the local existence to global existence, we have to do a number of nonlinear a priori estimates provided that ET (u) 6 2CH(u)(0), see Proposition 5.1 and Lemma 5.1. As a direct consequence of the above assumption, we get an important relationship between E(u) and Es defined in the introduction ET (u) . 2 2J(1−s)(E2s + E4s ) . 22J(1−s). (5.1) In fact, it follows from Hardy-Littlewood Sobolev inequality and the definition of u0 ‖(|x|−γ ∗ u20)u20‖L1 . ‖u0‖412 6 ‖S0φ0‖412 06j6J ‖△jφ0‖412 And then the right hand of the above inequality can controlled that as soon as 1 > s > by utilizing Bernstein inequality ‖S0φ0‖4L2 + 06j6J −s)2j4s‖△jφ0‖4L2 . 2 2J(1−s)E4s . From now on, we assume (5.1) to in our subsequence proof. Proposition 5.1 Assume that (u0, u1) ∈ H1×L2, then the following estimate holds for s0, α, β defined in Proposition 3.1 and Proposition 4.1, HT (u) .H(u)(0) + T 3 2−2J(s−β)ET (u) + T 3 2−J(4s−α−2s0−1)ET (u) −2J [s−( −2J [s−( + T2−2J(s− ET (u) for max{2, 1 } < r1 < 23−γ and 6 r2 < ∞. Proof. Multiplying (4.1) by ∂tu and integrating over x and t, we have ‖u(t)‖2H1 + ‖ut(t)‖ R3×R3 |x− y|−γu2(y, t)u2(x, t)dydx ‖u0‖2H1 + ‖u1‖ R3×R3 |x− y|−γu20(y)u20(x)dydx I(v2)(x, τ)u(x, τ)∂τu(x, τ)dxdτ I(uv)(x, τ)v(x, τ)∂τu(x, τ)dxdτ I(u2)(x, τ)v(x, τ)∂τu(x, τ)dxdτ I(uv)(x, τ)u(x, τ)∂τu(x, τ)dxdτ ∣∣∣∣ . By taking the supermum over t 6 T , we have HT (u) .H(u)(0) + ∥∥I(v2)u∂tu + ‖I(uv)v∂tu‖L1 I(u2)v∂tudxdt I(uv)u∂tudxdt =H(u)(0) + I + II + III + IV. (5.2) The proof is broken down into the following several steps. (i) Firstly, we estimate I and II. Making a similarly argument as (4.4) in the proof of Proposition 4.1, it can be obtained that I 6 ‖I(v2)u‖L1 L2‖ut‖L∞T L2 6 T 3 ET (u)‖v‖2Xβ , and then keeping in mind v has been estimated in Proposition 3.1, this together with (1.2) yields that I 6 T 3 ET (u)E2h,β 6 T 3 ET (u)2 −2J(s−β)E2s . (5.3) Arguing similarly, we easily get II 6 T 3 ET (u)2 −2J(s−β)E2s . (5.4) (ii) Secondly, we estimate the terms III and IV . As mentioned in the introduction, one can get the same type of estimate as above for the terms I and II, but that will lead to s > α , which is worse than the exponent given in the Theorem 1.1. To improve the lower bound on s, we have to utilize more precise estimate on III and IV . We first split III and IV into two different pieces, respectively. One can write v = vF +B(v, v, v), where vF is its free part and the other one comes from nonlinear term. For the nonlinear part, it follows from (3.3) that ‖B(v, v, v)‖Xα 6 ‖v‖Xα‖v‖2Xs0 . This along with (4.5), one can see that ‖I(u2)B(v, v, v)ut‖L1 L1 6 ‖I(u2)B(v, v, v)‖L1 L2‖ut‖L∞T L2 3 ‖u‖2L∞ H1‖B(v, v, v)‖Xα‖ut‖L∞T L2 3 ET (u) 2 ‖v‖Xα‖v‖2Xs0 Moreover, we get by (1.2), ‖I(u2)B(v, v, v)ut‖L1 L1 6 T T (u)2 −J(3s−α−2s0)E3s . (5.5) By the same way as leading to (5.5), we easily infer that ‖I(uB(v, v, v))uut‖L1 L1 6 T T (u)2 −J(3s−α−2s0)E3s . (5.6) Thus, it is sufficient to estimate these terms including free part vF since (5.5) and (5.6). The following lemma gives estimates for the nonlinearity including free part vF . Lemma 5.1 Let vF be a solution of the free Klein-Gordon equation, and u be such that ET (u) . 2 2J(1−s). Then, for max{2, 1 } < r1 < 23−γ and 6 r2 < ∞ I(u2)vFutdxdt ∣∣∣ . −2J [s−( −2J [s−( + T2−2J(s− ET (u), (5.7) I(uvF )uutdxdt ∣∣∣ . T −2J [s−( ET (u). (5.8) Hence these together with (5.5)-(5.6) yield that III + IV . T 3 ET (u)2 −J [4s−α−2s0−1]E4s + −2J [s−( −2J [s−( + T2−2J(s− ET (u). (5.9) Therefore, we complete the proof of Proposition 5.1 provided that we had proved Lemma 5.1, whose proof is postponed in the last section. 6 Proof of Theorem 1.1 Since the Cauchy problem (1.1) is split into equation (3.1) which is globally well-posed by choosing J enough to make Eh,s0 < ε0 and equation (4.1) which is locally well-posed (see Proposition 3.1 and Proposition 4.1), we have to show that the local solution to equation (4.1) can be extended globally. Let us denote T ∗J the maximum time of existence in Proposition 4.1. Theorem 1.1 will be proved if T ∗J = +∞. Let us consider TJ the supremum of the T < T J such that ET (u) 6 2CH(u)(0). Thus, for any T < TJ , Proposition 5.1 gives us that ET (u) 6 H(u)(0) C + C1T 3 2−2J(s−β)E2s + C2T 3 2−J(4s−α−2s0−1)E4s + C3T −2J [s−( )]E2s + C4T2 −2J(s− 1 )E2s + C5T −2J [s−( )]E2s By the assumption of Theorem 1.1 s > , one easily verifies that s > max if choosing r1 sufficiently close to and r2 large enough. We infer that TJ > T̃J if we choose T̃J such that = min 22J(s−β) 5C1E2s 24J(s− 5C2E4s 2J [s−( 5C3E2s ) 2r1 22J(s− 5C4E2s 2J [s−( 5C5E2s ) 2r2 By the definition of TJ , we get T J > T̃J . Obviously, T̃J tends to infinity when J tend to infinity. This completes the proof of Theorem 1.1. 7 Proof of Lemma 5.1 In order to make conveniently use of the precise Strichartz estimate on which mostly the following proof relies, we begin this section by introducing a family of balls of center ν )ν∈Λj,k of radius 2 k and a function χ ∈ C∞c (B(0, 1)) such that for j > 0 ∀ξ ∈ 2jC, ν∈Λj,k χ(2−k(ξ − ξj,kν )) = 1 and C−10 6 ν∈Λj,k χ2(2−k(ξ − ξj,kν )) 6 C0. Let us define that, for some constant c △νj,ka == F−1 ϕ(2−jξ)χ(2−k(ξ − ξj,kν )) â(ξ) △̃νj,ka == F−1 ϕ̃(2−jξ)χ(c2−k(ξ + ξj,kν )) â(ξ) As the support of the Fourier transform of a product belongs to the sum of the support of each Fourier transform, we have △ja = ν∈Λj,k △νj,ka, △jb = ν′∈Λj,k △ν′j,kb. In view of this fact that if k 6 j − 2 ν,ν′∈Λj,k △νj,ka△ν is vanish when ξ ν is close to ξ , without loss of generality, we can write △k(△ja△jb) ≈ △k ν∈Λj,k △νj,ka△̃νj,kb. (7.1) For the sake of convenience, we also fix the notation in this section that, for 0 6= f(t, x) ∈ L2TL2 ck = 2 ‖△kv0‖L2 + 2−k‖△kv1‖L2 E−1h,σ, c̃k = ‖△kf‖L2 ‖f‖L2 with σ = 1/2 + 1/r for 2 6 r < ∞. Proof of Lemma 5.1. We first prove (5.7). In view of the fact that v̂F only has high frequencies, Bony’s decomposition implies that there exists constant N0 such that I(u2)vFut = j>J−N0 Sj+2vF△jI(u2)ut + j>J−N0 Sj−1I(u2)△jvFut. (7.2) Since the negative derivative I acts on the high frequency for the former term while on the low frequency for the latter one, the first term is much better than the second one. We shall estimate the first term by using merely the Hölder inequality, Bernstein inequality and classical Strichartz estimates. Firstly, we see that, for 2 6 r < ∞ j>J−N0 ‖Sj+2vF△jI(u2)‖L2x . j>J−N0 j′6j+1 ‖△j′vF ‖L∞x ‖△jI(u 2)‖L2x j>J−N0 j′6j+1 r ‖△j′vF ‖Lr2j(γ− )‖u‖2L∞ Bernstein inequality and (2.1) in Proposition 2.1 with 1 for all 2 6 r < ∞ imply that j>J−N0 ‖Sj+2vF△jI(u2)‖L2x p ‖u‖2L∞ j>J−N0 j′6j+1 r ‖△j′vF‖Lp j(γ− 7 p ‖u‖2L∞ j>J−N0 2j(γ− j′6j+1 j′( 3 ‖△j′v0‖L2 + 2−j ′‖△j′v1‖L2 The right hand of the above inequality can be controlled by p ‖u‖2L∞ j>J−N0 2j(γ− j′6j+1 2 cj′Eh,σ and moreover it follows from (1.2), the definition of Eh,σ and Sobolev embedding that j>J−N0 ‖Sj+2vF△jI(u2)ut‖L1x pEh,σ‖u‖2L∞ j>J−N0 2j(γ−3)‖ut‖L∞ r 2−2J [s−( )]E2s ‖u‖2L∞ (H1), (7.3) for 4 6 r < ∞. Let us estimate the second term in (7.2) by the precise Strichartz estimates. Since this term contains that the negative derivative acts on the low frequency part Sj−1(u it leads to our new parameter r < 2 by some technique difficulties. Noting that Fourier-Plancherel formula and Hölder’s inequality, we can see that j>J−N0 Sj−1I(u2)△jvFutdxdt . j>J−N0 −16k6j−2 △kI(u2)△jvF△jutdxdt △kI(u2)△k k6j−2,J−N06j △jvF△jut . ‖u2‖ 2k(γ− k6j−2,J−N06j △jvF△jut (7.4) On one hand, we have J−N06j △jvF△jut ‖L2dt . j>J−N0 ‖△jvF△jut‖L1 j>J−N0 ‖△jvF ‖L∞ L2‖△jut‖L2 j>J−N0 2−sjcj c̃jEh,s‖ut‖L2t,x . If (7.4) is controlled by the term at k = −1, we can see that j>J−N0 ‖Sj−1I(u2)△jvF ‖L2 L2 . T 2 2−2J(s− )E2s ‖u‖L∞T (H1). (7.5) On the other hand, one denotes gk := △k k6j−2 △jvF△jut to estimate 2k(γ− )‖gk‖ Let us write that k6j−2 ν∈Λj,k △νj,kvF△jut. As the support of the Fourier transform of a product is included in the sum of the support of each Fourier transform, we obtain k6j−2 ν∈Λj,k △νj,kvF △̃νj,kut, as well as in (7.1). Using Hölder inequality, we get k6j−2 ν∈Λj,k ‖△νj,kvF ‖Lr‖△̃νj,kut‖L2 and the Cauchy-Schwarz inequality and the L2 quasi-orthogonality properties yield that k6j−2 ν∈Λj,k ‖△νj,kvF‖2Lr ν∈Λj,k ut‖2L2 k6j−2 ν∈Λj,k ‖△νj,kvF‖2Lr 2 ‖△jut‖L2 . (7.6) Precise Strichartz estimate implies that, for 1 with 2 6 r < ∞, r+2 ) 06k6j−2 2(k−j)( ν∈Λj,k ‖△νj,kv0‖2L2 2 + 2−j ν∈Λj,k ‖△νj,kv1‖2L2 ‖△jut‖L2t,x and observe the quasi-orthogonality properties again, this can be dominated by 06k6j−2 2(k−j)( ‖△jv0‖L2 + 2−j‖△jv1‖L2 ‖△jut‖L2t,x . Keeping the definitions of Eh,σ and cj in mind, one can see that r+2 ) p 2k( k6j−2 r cj c̃jEh,σE T (u) . T p 2k( )Eh,σE T (u). Therefore, we get that 2k(γ− )‖gk‖ r+2 ) 2k(γ−3)Eh,σE T (u) which implies nothing but j>J−N0 Sj−1I(u2)△jvFutdxdt ∣∣∣ . T 1− p Eh,σE T (u). (7.7) Finally, we get that, for 4 6 r < ∞ j>J−N0 Sj−1I(u2)△jvFutdxdt ∣∣∣ . T r 2−2J [s−( )]E2sET (u). However, although the r ranges 4 6 r < ∞, the above estimate still needs s > 3 to continue our proof. If we only consider the high frequency k > J , the (7.7) can be modified by j>J−N0 Sj−1I(u2)△jvFutdxdt ∣∣∣ . T 1− p 2J(γ−3)Eh,σE T (u) (7.8) and then we can obtain a better result j>J−N0 Sj−1I(u2)△jvFutdxdt ∣∣∣ . T r 2−2J [s−( )]E2sET (u), which implies the bad influence comes from the low frequency part and this is consist of the effect of negative derivative acts on the low frequency. But if we choose σ̃ = γ− 5 instead of σ, we can improve (7.8), at cost of restricting r such that max{2, 1 } < r < while not 2 6 r < ∞. Now we turn to details. It follows from similar argument r+2 ) p 2k( k6j−2 2−j(γ−3+ )cj c̃jEh,σ̃E T (u) where σ̃ = γ − 5 with 1 < 3− γ < 2 . We get 2k(γ− )‖gk‖ r+2 ) k6j−2 2(k−j)(γ−3+ )cj c̃jEh,σ̃E T (u) which implies nothing but j>J−N0 Sj−1I(u2)△jvFutdxdt ∣∣∣ . T 1− pEh,σ̃E T (u) by Young’s inequality. Note that σ̃ 6 γ < s when r sufficiently closes to 2 , therefore j>J−N0 Sj−1I(u2)△jvFutdxdt ∣∣∣ . T r 2−2J(s− )E2sET (u). Combining this with (7.3) and (7.5), we complete the proof of (5.7) by obtaining j>J−N0 Sj−1I(u2)△jvFutdxdt −2J [s−( −2J [s−( + T2−2J(s− E2sET (u) with max{2, 1 } < r1 < 23−γ and 6 r2 < ∞. We secondly prove (5.8) which is different from (5.7). To this end, we need to make Bony’s decomposition more than once and establish a commutator estimate, which helps us to complete our proof. In view of the fact that v̂F only has high frequencies again, it follows from Bony’s decomposition that there exists N0 such that I(uvF )uut = j>J−N0 Sj+2vF△ju uut + j>J−N0 △jvFSj−1u == I + II. (7.9) In order to estimate the term I, we split it into two pieces with N1 ≫ N0 > 0 j>J−N0 uut△kI Sj+2vF△ju j>J−N0 k6J−N1 uut△kI Sj+2vF△ju j>J−N0 k>J−N1 uut△kI Sj+2vF△ju == I1 + I2. The estimate of I1 is broken down into the following two cases. Case 1. 2 < γ 6 5 In this case, to our purpose, we obtain the following coarse estimate by Hölder’s inequality ‖I1‖L1x . j>J−N0 k6J−N1 Sj+2vF△ju ‖L3‖u‖L∞ L6‖ut‖L∞ j>J−N0 k6J−N1 2k(γ−2)‖△k Sj+2vF△ju ET (u) j>J−N0 k6J−N1 2k(γ−2)‖Sj+2vF‖L6‖△ju‖L2ET (u) k6J−N1 2k(γ−2) j>J−N0 2−j2j‖△ju‖L2 ‖△j′vF ‖L6ET (u). Choosing (p, r) such that 1 with 2 6 r 6 6, the Strichartz estimate yeilds ‖I1‖L1 k6J−N1 2k(γ−2) j>J−N0 j′( 3 ‖△j′v0‖L2 + 2−j ′‖△j′v1‖L2 T (u). Arguing similarly as before it yields that ‖I1‖L1 k6J−N1 2k(γ−2) j>J−N0 r cj′Eh,1/2E T (u) r 2−2J [s−( )]E2sET (u) with 2 6 r 6 6. If choose r = 6, one can easily check that γ 2 < γ 6 5 . Although this result is enough for us to prove the main theorem, we want to improve the result for this term by loosen the upper bound of r from 6 to ∞ through the precise Strichartz estimate. Arguing similarly as before, we have ‖I1‖L1x . j>J−N0 k6J−N1 Sj+2vF△ju ‖L3‖u‖L∞ L6‖ut‖L∞ j>J−N0 k6J−N1 2k(γ−3)2k 2r ‖△k Sj+2vF△ju ET (u) Since the Fourier transform of Sj−1vF△ju was supported in 2jC and k ≪ j, △k(Sj−1vF△ju) vanishes which implies △k Sj+2vF△ju △̃jvF△ju . As the support of the Fourier transform of a product is included in the sum of the support of each Fourier transform, we also have △̃jvF△ju ν,ν′∈Λj,k △νj,kvF△ν ν∈Λj,k △νj,kvF △̃νj,ku Choosing (p, r) such that 1 for 2 6 r < ∞, it follows from the Hölder inequality and L2 quasi-orthogonality properties that Sj+2vF△ju r+2 ) ν∈Λj,k ‖△νj,kvF‖Lr‖△̃νj,ku‖L2 ν∈Λj,k ‖△νj,kvF ‖2LpLr ν∈Λj,k ‖△νj,ku‖2L2 ν∈Λj,k ‖△νj,kvF ‖2LpLr ∥∥△ju Then the precise Strichartz estimate yields that ‖I1‖L1 k6J−N1 2k(γ−3)2k j>J−N0 2−j2j ∥∥△ju 2(k−j)( ν∈Λj,k ‖△νj,kv0‖2L2 2 + 2−j ν∈Λj,k ‖△νj,kv1‖2L2 ET (u). By the L2quasi-orthogonality properties, it gives that ‖I1‖L1 k6J−N1 2k(γ−3)2k j>J−N0 2−j2j ∥∥△ju ‖△jv0‖L2 + 2−j‖△jv1‖L2 ET (u). Utilizing the technique as before yields that ‖I1‖L1 k6J−N1 2k(γ−2+ j>J−N0 2−j(1+ )cjEh,σE T (u) r 2J(γ−3)Eh,σE T (u) . T r 2−2J [s−( )]E2sET (u), with 4 6 r < ∞. Case 2. 5 < γ < 3 In the this case, the fact γ− 5 > 0 helps us to obtain the the desirable result easily. Arguing similarly as before, we have ‖I1‖L1x . j>J−N0 k6J−N1 Sj+2vF△ju ‖L3‖u‖L∞ L6‖ut‖L∞ j>J−N0 k6J−N1 2k(γ−3)23k( Sj+2vF△ju ‖L2ET (u) j>J−N0 k6J−N1 2k(γ− )‖Sj+2vF‖L∞‖△ju‖L2ET (u). Choosing (p, r) such that 1 with 2 6 r < ∞, the Strichartz estimate yields ‖I1‖L1 k6J−N1 2k(γ− j>J−N0 2 cj′Eh,σE T (u) p 2J(γ− j>J−N0 2 Eh,σE T (u) r 2−2J [s−( )]E2sE T (u). Combining these two cases, we have shown that ‖I1‖L1 r 2−2J [s−( )]E2sET (u) (7.10) with 4 6 r < ∞. To control ‖I‖L1 , it remains to estimate ‖I2‖L1 . Compared with ‖I1‖L1 , since the negative derivative acts on the high frequency, the upper bound of ‖I2‖L1 is much easier to get. Here is the details: ‖I2‖L1x . j>J−N0 k>J−N1 Sj+2vF△ju ‖L3‖u‖L∞ L6‖ut‖L∞ j>J−N0 k>J−N1 2k(γ−3)‖Sj+2vF ‖L∞‖△ju‖L3ET (u). Choosing (p, r) such that 1 with 2 6 r < ∞ again, the Strichartz estimate yields ‖I2‖L1 k>J−N1 2k(γ−3) j>J−N0 2 cj′Eh,σE T (u) r 2−2J [s−( )]E2sET (u). Combining this with (7.10), we obtain that ‖I‖L1 r 2−2J [s−( )]E2sET (u) (7.11) for 4 6 r < ∞. To complete the proof the Lemma 5.1, it remains to estimate II. One can proceed this as above by Hölder’s inequality to estimate j>J−N0 2j(γ−3)‖△jvFSj−1u‖L3 ET (u). (7.12) Resorting to the Hölder inequality and the classical Strichartz estimate, one can obtain ‖II‖L1 L1 . T r 2−2J [s−( )]E2sET (u). with 2 6 r 6 6. One also can try to improve the result by using the precise Strichartz estimate as before, but it fails and merely obtain that ‖II‖L1 L1 . T r 2−2J [s−( )]E2sET (u). with 2 6 r 6 4. One can easily check that the result is worse than the desirable result because of the restriction of r. Compared with the second term in (7.2), the negative derivative acts on the high frequency part so that it is tempting to obtain a better result than that of (7.2). But △jvF is bound with Sj−1u by the operator I, and this structure prevents us from using efficiently the precise Strichartz estimate. If one first resort to the Hölder inequality, as shown in (7.12), he or she merely obtains a loss result because of the range restriction of r. To go around this difficulty, we first establish a commutator estimate through exploiting cancellation property. Now we turn to details. Our task is to estimate j>J−N0 I(△jvFSj−1u)uutdxdt In order to drag the Sj−1u out of the operator I, we construct uI(△jvF )Sj−1u and the triangle inequality yields that j>J−N0 I(△jvFSj−1u)uutdxdt ∣∣∣ 6 j>J−N0 ∥∥(I(△jvFSj−1u)− I(△jvF )Sj−1u j>J−N0 I(△jvF )Sj−1uuutdxdt We benefit from the cancellation when we deal with the first term. Since both the Fourier transformation of I(△jvFSj−1u) and I(△jvF )Sj−1u are supported in a ring sized 2j , the Hölder inequality and the Bernstein inequality lead to that ∥∥(I(△jvFSj−1u)− I(△jvF )Sj−1u 2 ‖I(△jvFSj−1u)− I(△jvF )Sj−1u‖L2x‖u‖L6 . Before estimating its right hand, we recall the Coifman and Meyer multiplier theorem. Consider an infinitely differentiable symbol m : Rnk 7→ C so that for all α ∈ Nnk and all ξ = (ξ1, ξ2, · · · , ξk) ∈ Rnk, there is a constant c(α) such that |∂αξ m(ξ)| 6 c(α)(1 + |ξ|)−|α|. (7.13) Define the multilinear operator T by [T (f1, · · · , fk)](x) = eix·(ξ1+···+ξk)m(ξ1, · · · , ξk)f̂1(ξ1), · · · , f̂k(ξk)dξ1 · · · dξk, (7.14) F [T (f1, · · · , fk)](ξ) = ξ=ξ1+···+ξk m(ξ1, · · · , ξk)f̂1(ξ1), · · · , f̂k(ξk)dξ1 · · · dξk−1. (7.15) Proposition 7.1 ([8],Page 179.) Suppose pj ∈ (1,∞), j = 1, · · · k, are such that 1p = + · · ·+ 1 6 1. Assume m(ξ1, · · · , ξk) a smooth symbol as in (7.13). Then there is a constant C = C(pi, n, k, c(α)) so that for all Schwarz class functions f1, · · · , fk, ‖[T (f1, · · · , fk)](x)‖Lp(Rn) 6 C‖f1‖Lp1 (Rn) · · · ‖fk‖Lpk (Rn). (7.16) Since the operator I is a convolution operator with kernel |x|−γ in R3, we can write F [I(△jvFSj−1u)−I(△jvF )Sj−1u](ξ) = ξ=ξ1+ξ2 |ξ1+ξ2|γ−3−|ξ1|γ−3 △̂jvF (ξ1)Ŝj−1u(ξ2)dξ2. By the mean value theorem, the right hand of the above formula becomes that ξ=ξ1+ξ2 |ξ1 + λξ2|γ−4 (ξ1 + λξ2) · ξ2 |ξ1 + λξ2| △̂jvF (ξ1)Ŝj−1u(ξ2)dξ2, for a certain λ ∈ [0, 1]. Moreover, we rewrite it as follow: ξ=ξ1+ξ2 m(ξ1, ξ2)f̂1(ξ1)f̂2(ξ2)dξ2, m(ξ1, ξ2) = (ξ1 + λξ2)|ξ1 + λξ2|γ−5|ξ1|4−γ , f1 = |∇|γ−4△jvF , f2 = ∇Sj−1u. Observe that |ξ1| > 2j−1 and 2j−2 > |ξ2|, we have that |ξ1+λξ2| ∼ |ξ1| > 2J−N0 . Hence, we can check that the symbol m(ξ1, ξ2) satisfies the estimate (7.13). Finally, it follows from Proposition 7.1 that ‖I(△jvFSj−1u)− I(△jvF )Sj−1u‖L2x . ‖f1‖Lrx‖f2‖ with 2 < r < ∞. After making use of the Bernstein inequality, the right hand can be controlled by 2j(γ−4+ )‖△jvF ‖Lrx‖∇u‖L2x . Keeping in mind j > J−N0 and recalling the definition of Eh,σ, the Strichartz estimate and a direct calculation of summing in j show that j>J−N0 2 2j(γ−4+ )‖△jvF ‖Lp j>J−N0 2j(γ−3+ −s)Eh,s. with 1 and 2 < r < ∞. Choosing r such that max{2, 1 } 6 r < ∞, we have j>J−N0 ∥∥(I(△jvFSj−1u)− I(△jvF )Sj−1u r 2−2J [s−( )]E2sET (u). (7.17) Now the rest of the paper devotes to estimate this term j>J−N0 I(△jvF )Sj−1uuutdxdt In order to use precise Strichartz estimate, we need to decompose this term by Bony’s para-product decomposition again, I(△jvF )Sj−1uuut = Sk−1(uSj−1u)△kI(△jvF )ut +△k(uSj−1u)Sk+2I(△jvF ) = II1 + II2. After decomposing this, the term II1 is similar to the second term in the (7.2) and the negative derivative acts on the high frequency △jvF leading to a better result than the second term in the (7.2). Thanks to Fourier-Plancherel formula and Hölder inequality, we obtain j>J−N0 II1dxdt ≈ j>J−N0 Sk−1(uSj−1u)△kI(△jvF )△kutdxdt j>J−N0 k′6k−2 △k′(uSj−1u)△kI(△jvF )△kutdxdt j>J−N0 △k′(uSj−1u)△k′ k′6k−2 (△kI(△jvF )△kut)dxdt j>J−N0 ‖uSj−1u‖ 2 ‖△k′ k′6k−2 (△kI(△jvF )△kut)‖L2 j>J−N0 ‖u‖2L∞H1 2 ‖△k′ k′6k−2 (△kI(△jvF )△kut)‖L2 On the other hand, one denotes gk′,j = △k′ k′6k−2 △kI(△jvF )△kut to estimate ∑ ′(− 1 )‖gk′,j‖ Let us write that gk′,j = k′6k−2 ν∈Λk,k′ △νk,k′I(△jvF )△kut As the support of the Fourier transform of a product is included in the sum of the support of each Fourier transform, we obtain gk′,j = k′6k−2 ν∈Λk,k′ △νk,k′I(△jvF )△̃νk,k′ut Using Hölder inequality, we get ‖gk′,j‖ k′6k−2 ν∈Λk,k′ ‖△νk,k′I(△jvF )‖Lr‖△̃νk,k′ut‖L2 6 2j(γ−3) k′6k−2 ν∈Λk,k′ ‖△νk,k′vF ‖2Lr ν∈Λk,k′ ‖△νk,k′ut‖2L2 6 2j(γ−3) k′6k−2 ν∈Λk,k′ ‖△νk,k′vF ‖2Lr 2‖△kut‖L2 the use of quasi-orthogonality properties is made in the last inequality. Precise Strichartz estimate and the quasi-orthogonality properties imply that ‖gk′,j‖ r+2 ) p 2j(γ−3) k′6k−2 ′−k)( 1 ν∈Λk,k′ ‖△νk,k′v0‖2L2 2 + 2−k ν∈Λk,k′ ‖△νk,k′v1‖2L2 ‖△kut‖L2 p 2j(γ−3) k′6k−2 ′−k)( 1 ‖△kv0‖L2 + 2−k‖△kv1‖L2 ‖△kut‖L2 with 1 for 2 6 r < ∞. Therefore ′(− 1 )‖gk′,j‖ r+2 ) p 2j(γ−3) k′6k−2 ′−k) 2 r ck c̃kEh,σE T (u). A direct computation shows that 2 ‖gk′,j‖L1 ′(− 1 )‖gk′,j‖ r+2 ) p 2j(γ−3)Eh,σE T (u). Hence, we have that j>J−N0 II1dxdt ∣∣∣ . 2−2J [s−( r E2sET (u) (7.18) with 4 6 r < ∞. Finally, we conclude this section by giving the estimate of II2. j>J−N0 II2dxdt ∣∣∣ . T j>J−N0 ‖△k(uSj−1u)Sk+1I(△jvF )‖L2 L2‖ut‖L∞T L2 j>J−N0 2j(γ−3) ‖△k(uSj−1u)‖L∞ r ‖△k′△jvF‖Lp T (u) j>J−N0 2j(γ−3) ‖△k(uSj−1u)‖L∞ 2 ck′Eh,σE T (u) j>J−N0 2j(γ−3) 2 ‖△k(uSj−1u)‖L∞ (k′−k) 1 2Eh,σE T (u) j>J−N0 2j(γ−3)‖ck′‖ℓ2(Z)‖2 2 ‖△k(uSj−1u)‖L∞ L2‖ℓ2(Z)‖2− 2 ‖ℓ2(N)Eh,σE T (u) p 2J(γ−3)E T (u)Eh,σ . 2 −2J [s−( r E2sET (u). (7.19) Collecting (7.18) and (7.19), we have been proved that j>J−N0 I(△jvF )Sj−1uuutdxdt ∣∣∣ . T r 2−2J [s−( )]E2sET (u), (7.20) with 4 6 r < ∞. Finally, we complete the proof of (5.8) by (7.11) and (7.20), hence it ends the proof of Lemma 5.1. Acknowledgements: The authors are grateful to Prof. J.Chemin for sending his lecture to us. The authors were partly supported by the NSF of China, No.10725102. References [1] Bergh J. and Löfstrom J., Interpolation spaces, An Introduction, New York: Springer-Verlag, 1976. [2] H.Bahouri and Jean-Yves Chemin On global well-posedness for defocusing cubic wave equation , Internat. Math. Res. Notices Volume 2006, Article ID 54873, Pages 1-12. [3] Jean Bourgain. Refinements of Strichartz’s inequality and applications to 2D-NLS with critical nonlinearity. Internat. Math. Res. Notices, 5:253-283, 1998. [4] J. Bourgain. The Global Solution of Nonlinear Schrödinger Equations, American Mathematical Society, Providence, 1999. 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Parameter estimation for power-law distributions by maximum likelihood methods Theoretical Physics Preprint Parameter estimation for power-law distributions by maximum likelihood methods Heiko Bauke Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, United Kingdom∗ Distributions following a power-law are an ubiquitous phenomenon. Methods for determining the exponent of a power-law tail by graphical means are often used in practice but are intrinsically unreliable. Maximum likelihood estimators for the exponent are a mathematically sound alternative to graphical methods. http://arXiv.org/abs/0704.1867v2 Parameter estimation for power-law distributions by maximum likelihood methods Heiko Bauke Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, United Kingdom∗ (Dated: August 11, 2007) Distributions following a power-law are an ubiquitous phenomenon. Methods for determining the exponent of a power-law tail by graphical means are often used in practice but are intrinsically unreliable. Maximum likelihood estimators for the exponent are a mathematically sound alternative to graphical methods. PACS numbers: 02.50.Tt, 89.75.-k 1. Introduction The distribution of a discrete random variable is referred to as a distribution with a power-law tail if it falls as p(k) ∼ k−γ (1) for k ∈ N and k ≥ kmin. Power-laws are ubiquitous distribu- tions that can be found in many systems from different disci- plines, see [1, 2] and references therein for some examples. Experimental data of quantities that follow a power-law are usually very noisy; and therefore obtaining reliable estimates for the exponent γ is notoriously difficult. Estimates that are based on graphical methods are certainly used most often in practice. But simple graphical methods are intrinsically unre- liable and not able to establish a reliable estimate of the expo- nent γ . For that reason, the authors of [3] introduced an alternative approach based on a maximum likelihood estimator for the exponent γ . Unfortunately the authors concentrate on a rather idealized type of power-law distributions, namely p1(k;γ) = ζ (γ,1) with k ∈ N, where the normalization constant ζ (γ,1) is given by the Hurwitz-ζ -function which is defined for γ > 1 and a > ζ (γ,a) = (i+ a)γ . (3) The distribution (2) is characterized by one parameter only, and therefore all properties of this distribution (e. g. its mean) are determined solely by the exponent γ . In many applications the power-law (2) is too restrictive. If one states that a quantity follows a power-law, then this means usually that the tail (k ≥ kmin) of the distribution p(k) falls proportionally to k−γ . Probabilities p(k) for k < kmin may differ from the power-law and admit the possibility to tune the mean or other characteristics independently of γ . In some ∗E-mail:heiko.bauke@physics.ox.ac.uk situations probabilities p(k) may differ from a power-law for k ≥ kmax as well, e. g. the distribution may have an exponential cut-off. Therefore, I will generalize the maximum likelihood ap- proach introduced in [3] to distributions that follow a power- law within a certain range kmin ≤ k < kmax but differ from a power-law outside this range in an arbitrary way. Furthermore, I will give some statements about the large sample properties of the estimate of the power-law exponent and present a nu- merical procedure to identify the power-law regime of the dis- tribution p(k). But first, let us see what is wrong with popular graphical methods. 2. Trouble with graphical methods All graphical methods for estimating power-law exponents are based on a linear least squares fit of some empirical data points (x1,y1), (x2,y2),. . . , (xM,yM) to the function y(x) = a0 + a1x . (4) The linear least squares fit minimizes the residual (yi −a0 −a1xi)2 . (5) Estimates â0 and â1 of the parameters a0 and a1 are given by â0 = ∑Mi=1 yi ∑Mi=1 x ∑Mi=1 xi ∑Mi=1 yixi ∑Mi=1 x ∑Mi=1 xi â1 = ∑Mi=1 yixi ∑Mi=1 xi ∑Mi=1 yi ∑Mi=1 x ∑Mi=1 xi . (7) The ansatz for the residual (5) and derivation of (6) and (7) are based on several assumptions regarding the data points (xi,yi). It is assumed that there are no statistical uncertain- ties in xi, but yi may contain some statistical error. The errors in different yi are independent identically distributed random mailto:heiko.bauke@physics.ox.ac.uk variables with mean zero. In particular the standard devia- tion of the error is independent of xi. For various graphical methods for the estimation of the exponent of a power-law distribution these conditions are not met, leading to the poor performance of these methods. To illustrate the failure of graphical methods by a computer experiment N = 10000 random numbers mi had been drawn from distribution (2) with γ = 2.5 and an estimate γ̂ for the ex- ponent γ was determined by various graphical methods. The estimator is a random variable and its distribution depends on the method that has been used to obtain the estimate. Impor- tant measures of the quality of an estimator are its mean and its standard deviation. If the mean of the estimator equals the true exponent γ then the estimator is unbiased and estimators with a distribution that is concentrated around γ are desirable. For each graphical method a histogram of the distribution of the estimator was calculated to rate the quality of the estimator by repeating the numerical experiment 500 times. The most straight forward (and most unreliable) graphical approach is based on a plot of the empirical probability dis- tribution p̂(k) on a double-logarithmic scale. Introducing the indicator function I [·], which is one if the statement in the brackets is true and else zero, the empirical probability distri- bution is given by p̂(k) = I [mi = k] . (8) An estimate γ̂ for the power-law exponent γ is established by a least squares fit to (xi,yi) = (lnk, ln p̂(k)) for all k ∈ N with p̂(k) > 0, (9) γ̂ equals the estimate (7) for the slope, see Figure 1 a. Because the lack of data points in the tail of the empirical distribution this procedure underestimates systematically the exponent γ , see Table I. There are two ways to deal with the sparseness in the tail of the empirical distribution, logarithmic binning and consider- ing the empirical cumulative distribution P̂(k) instead of p̂(k). The cumulative probability distribution of (2) is defined by P(k) = ζ (γ,1) . (10) If p(k) has a power-law tail with exponent γ then P(k) follows approximately a power-law with exponent γ − 1 because for k ≫ 1 the distribution P(k) can be approximated by P(k) ≈ ζ (γ,1) (γ −1)ζ (γ,1) . (11) The empirical cumulative probability distribution is given by P̂(k) = I [mi ≥ k] . (12) It is less sensitive to the noise in the tail of the distribution and therefore a fit of (xi,yi) = (lnk, ln P̂(k)) for all k ∈ N with P̂(k) > 0 (13) Table I: Mean and standard deviation of the distribution of the esti- mate for the power-law exponent γ for various methods. All methods have been applied to the same data sets of random numbers from distribution (2) with γ = 2.5. See text for details. mean standard deviation method estimate of estimate fit on empirical distribution 1.597 0.167 fit on cumulative empirical distri- bution 2.395 0.304 fit on empirical distribution with logarithmic binninga 2.397 0.080 fit on cumulative empirical distri- bution with logarithmic binning 2.544 0.127 maximum likelihood 2.500 0.016 aIn [3] a similar experiment is reported. For a fit of the logarithmically binned empirical probability distribution the authors find a systematical bias of 29 %. I cannot reconstruct such a strong bias, instead I get a bias of 5 % only. Probably the quality of this method depends on the details of the binning procedure. to a straight line gives much better estimates for the exponent, see Figure 1 b. But there is still a small bias to too small values and the distribution of this estimate is rather broad, see Table I. Logarithmic binning reduces the noise in the tail of the em- pirical distributions p̂(k) and P̂(k) by merging data points into groups. By introducing the logarithmically scaled boundaries bi = roundc i with some c > 1 (14) (The function roundx rounds x to the nearest integer.) a linear least squares fit is performed to (xi,yi) = bi + bi+1 −1 bi+1−1 p̂(k) bi+1 −bi (xi,yi) = bi + bi+1 −1 bi+1−1 P̂(k) bi+1 −bi , (16) respectively. As a consequence of the binning the width of the distribution of the estimate γ̂ of the power-law exponent γ is reduced, see Figure 1 c, 1 d and Table I. According to the numerical experiments a fit of the logarithmically binned cumulative distribution gives the best results among graphical methods. It shows the smallest systematic bias. All the methods that have been considered so far have a common weakness. In the deviation of (6) and (7) it was as- sumed that the standard deviation of the distribution of the error in yi is the same for all data points (xi,yi). But this is obviously not the case. For fixed k the empirical distribution p̂(k) is a random variable with mean p1(k;γ) and standard deviation p1(k;γ)(1− p1(k;γ))/N. For the corresponding data on a logarithmic scale the standard deviation is approxi- mately given by the quotient p1(k;γ)(1− p1(k;γ))/N p1(k;γ) 1− p1(k;γ) N p1(k;γ) . (17) fit on the distribution p̂(k) fit on the cumulative distribution P̂(k) = ∑i≥k p̂(i) a) b) c) d) Figure 1: Comparison of various methods for estimating the exponent of a power-law. Each figure shows data for a single data set of N = 10000 samples drawn from distribution (2) with γ = 2.5. Insets present histograms of estimates for γ for 500 different data sets. A power-law distribution p1(k;γ) is a monotonically decreas- ing function and therefore (17) is an increasing function of k. Because the variation of the statistical error is not taken into account, the distribution of the estimate γ̂ is very broad. Methods that deal with the cumulative distribution have an additional weakness. Cumulation has the side-effect that the statistical errors in yi are not independent any more, which violates another assumption of the deviation of (6) and (7). To sum up, estimates of exponents of power-law distribu- tions based on a linear least squares fit are intrinsically inac- curate and lack a sound mathematical justification. 3. Maximum likelihood estimators Maximum likelihood estimators offer a solid alternative to graphical methods. Let p(k;θ ) denote a single parameter prob- ability distribution. The maximum likelihood estimator θ̂N for the unknown parameter based on a sample m1,m2, . . . ,mN of size N is given by θ̂N = argmax [L(θ )] = argmax [lnL(θ )] , (18) where L(θ ) = p(mi;θ ) (19) denotes the likelihood function. In the limit of asymptotically large samples and under some regularity conditions maximum likelihood estimators share some desirable features [5, 6]. • The estimator θ̂N exists and is unique. • The estimator θ̂N is consistent, that means for every ε > 0 |θ̂N −θ |< ε = 1 , (20) where P |θ̂N −θ |< ε denotes the probability that the dif- ference |θ̂N −θ | is less than ε . • The estimator θ̂N is asymptotically normal with mean θ and variance (∆θ̂N)2 ln p(k;θ ) , (21) where E [·] indicates the expectation value of the quantity in the brackets. • Maximum likelihood estimators have asymptotically mini- mal variance among all asymptotically unbiased estimators. One says, they are asymptotically efficient. 4. Maximum likelihood estimators for genuine power-laws The most general discrete genuine power-law distribution has a lower as well as an upper bound and is given by pkmin,kmax(k;γ) = ζ (γ,kmin,kmax) for k ∈ N with kmin ≤ k < kmax. Where the non-standard nota- ζ (γ,kmin,kmax) := ζ (γ,kmin)− ζ (γ,kmax) (23) has been introduced. If the upper bound is missing the distri- bution pkmin(k;γ) = ζ (γ,kmin) has to be considered for k ∈N with k ≥ kmin. The distributions (22) and (24) are generalizations of (2) and will be useful for the analysis of more general distributions that show a power- law behavior only in a certain range but have an arbitrary pro- file outside the power-law regime. This kind of distributions will be considered in section 5. The maximum likelihood estimator γ̂N for the parameter γ of the distribution (22) follows from (18) and is given by γ̂N = argmax −N lnζ (γ,kmin,kmax) or equivalently by the implicit equation ζ ′(γ̂N ,kmin,kmax) ζ (γ̂N ,kmin,kmax) lnmi = 0 , (26) which has to be solved numerically. The prime denotes the derivative with respect to γ . The asymptotic variance of this estimator γ̂N follows from (21) and equals (∆γ̂N)2 ζ (γ,kmin,kmax)2 ζ ′′(γ,kmin,kmax)ζ (γ,kmin,kmax)− ζ ′(γ,kmin,kmax)2 1 2 3 4 5 6 7 8 Figure 2: Asymptotic standard deviation for maximum likelihood esti- mators for the exponent of a power-law distribution (24). Figure 3: Empirical distribution of the maximum likelihood estima- tor (histogram) versus its theoretical asymptotic distribution, which is given by a normal distribution with mean γ = 2.5 and variance (27). The histogram has been obtained from the same data as in Figure 1. In the limit kmax → ∞ equations (25), (26), and (27) give the maximum likelihood estimator and the asymptotic variance of this estimator for power-law distributions lacking an upper cut-off (24). A graphical representation of the standard devia- tion (27) in the limit kmax → ∞ is given in Figure 2. For each fixed kmin the quantity ∆γ̂N N grows faster than linear with γ . Therefore the larger the exponent γ the larger the sample size that is necessary to get an estimate within a given error bound. If the maximum likelihood method is applied (assuming a distribution (24)) to the same data as in section 2, numeri- cal experiments show that the estimates for the exponent are much more precise. The estimate has no identifiable system- atic bias, the standard deviation of the distribution of the esti- mate is smaller by an order of magnitude compared to graphi- cal methods, see Table I and Figure 3. 5. Maximum likelihood method for general power-law distributions The maximum likelihood procedure outlined in section 4 can be generalized further to distributions p(k) that are no pure power-laws (22) or (24) but follow a power-law within a cer- tain finite range or follow a power-law in the whole tail of the distribution and have an arbitrary profile outside the power- law regime. The popurse of this section is to establish meth- ods for identifying the power-law regime and for estimating the exponent of the power-law regime without making special assumptions about the profile of the probability distribution beyond the power-law regime. The main problem for a generalization of the maximum likelihood approach is that there might be no good hypothe- sis for the profile of the probability distribution beyond the power-law regime. To overcome this difficulty the empirical data set is restricted to a window kcmin ≤ mi < kcmax. (The following discussion covers the case of a power-law tail distri- butions as well by setting kcmax = ∞.) Assuming that p(k) has a power-law profile for kcmin ≤ k < kcmax then the probabil- ity distribution of the restricted data set is given by (22) with kmin = kcmin and kmax = kcmax (or by (24) with kmin = kcmin) and some unknown exponent γ . This allows to estimate the power-law exponent by the application of the maximum like- lihood method on the restricted data set of size N′ as presented in section 4 without making a hypothesis about the profile of the probability distribution beyond the power-law regime. In order to apply the maximum likelihood method one has to determine the cut-off points kcmin and kcmax first. Here it has to be taken into account that if the window kcmin ≤ mi < kcmax is chosen too large the estimate γ̂ is systematically bi- ased, but on the other hand if it is too small the statistical error is larger than necessary. In some cases one can make conser- vative estimates for kcmin and kcmax by plotting the empirical probability distribution (8) on a double-logarithmic scale. An appropriate window can also be found by determining esti- mates γ̂N′(kcmin) as a function of the window and a χ2-test. Assuming the empirical data is drawn from a distribution with a power-law tail (no upper cut-off) the lower cut-off point kcmin can be determined in the following systematic way. By varying the parameter kcmin the maximum likelihood approach gives a sequence of estimates γ̂N′(kcmin). If kcmin is very large the estimate will be quite inaccurate because only a tiny frac- tion of the experimental data is taken into account; but the smaller the cut-off kcmin the more accurate the estimate of the exponent. If kcmin approaches the point from above (but is still above) where the probability distribution starts do differ from a power-law γ̂N′(kcmin) will give a very precise estimate for the exponent γ . On the other hand, if kcmin is too small the hypothesis that the (restricted) empirical data is drawn from a power-law distribution is violated which causes a significant change of the estimate of the power-law exponent. If the empirical data is drawn from a distribution having both a lower crossover point as well as an upper crossover point a sequence of estimates γ̂N′(kcmin) is determined by re- stricting the data to a sliding window kcmin ≤ mi < wkcmin = kcmax with w > 1. As long as the window lies completely within the power-law regime the maximum likelihood esti- mate obtained from the restricted data set will give a reliable estimate of the power-law exponent. If the window lies at least partly outside the power-law regime the estimate is systemati- cally biased. To illustrate the procedures outlined above I generated two data sets from two distributions having a power-law regime. The first data set of N = 10000 samples was drawn from a distribution with a power-law tail which is given by p(k) ∼ 5−2.5 for 1 ≤ k ≤ 5 k−2.5 for k > 5 . Plotting the sequence of estimates γ̂N′(kcmin) against the pa- rameter kcmin reveals the exponent γ = 2.5 as wells as the crossover point k = 5 very clearly, see Figure 4. The second data set of N = 100000 samples was drawn from a distribution with two crossover points, viz. p(k) ∼ 5−2.25 for 1 ≤ k ≤ 5 k−2.25 for 5 ≤ k ≤ 100 100−2.25e−0.05(k−100) for k > 100 . Figure 5 shows the sequence of estimates γ̂N′(kcmin) that had been determined from restricted data sets of samples within the sliding window kcmin ≤ mi < 5kcmin. This sequence ex- hibits a broad plateau that corresponds to the power-law expo- nent γ = 2.25. If the window does not lie completely inside c min Figure 4: Sequence of estimates γ̂N ′(kcmin) as a function of the cut- off kcmin for a data set of N = 10000 samples from distribution (28). Filled symbols mark where the χ2-test has rejected the hypothesis that the restricted data follows a power-law (24) with exponent γ = γ̂N ′(kcmin). An error probability of α = 0.001 was chosen. c min 2kc max Figure 5: Sequence of estimates γ̂N ′(kcmin) as a function of the lower cut-off kcmin for a data set of N = 100000 samples from distribution (29). Filled symbols mark where the χ2-test has rejected the hypoth- esis that the restricted data follows a power-law (22) with exponent γ = γ̂N ′(kcmin). An error probability of α = 0.001 and the window width w = 5 had been chosen. the power-law regime the estimate γ̂N′(kcmin) deviates system- atically from the known exponent. Apart from a visual inspection of the γ̂N′(kcmin) plot the crossover point(s) to the power-law regime can be determined by means of a χ2-test. To apply a χ2-test the data set has to be divided into some bins and the following binning turned out to be appropriate: The data is partitioned into a small number b, say b = 6, of bins. In the case of a distribution with a power- law tail this means each bin j collects n j items such that n1 = N ′ p̂(kcmin) q1 = pkcmin(kcmin; γ̂N′(kcmin)) (30) n2 = N ′ p̂(kcmin + 1) q2 = pkcmin(kcmin + 1; γ̂N′(kcmin)) and finally nb = N k=kcmin+b−1 p̂(k) qb = k=kcmin+b−1 pkcmin(k; γ̂N′ (kcmin)) , where q j denotes the probability that a data point falls into bin j under the assumption that the (restricted) data follows the power-law (24) with kmin = kcmin and the exponent γ = γ̂N′(kcmin). For distributions with a finite power-law regime the binning procedure can be carried out in a similar way. In this case the summation index in (32) is bounded by kcmin + b−1≤ k < kcmax and the probability pkcmin,kcmax(k; γ̂N′(kcmin)) has to be considered instead of pkcmin(k; γ̂N′(kcmin)). The test statistic of the χ2-test is given by (n j −N′q j)2 N′q j . (33) If the to kcmin ≤ mi < kcmax restricted data is given by the power-law (22) or (24) with kmin = kcmin, kmax = kcmax, and the exponent γ = γ̂N′(kcmin) then the statistic c2 follows asymptotically a χ2-distribution with ν = (b− 1) degrees of freedom, which is given by pχ2(x,ν) = xν/2−1e−x/2 2−ν/2 Γ(ν/2) . (34) Let χ2α be the (1−α)-quantile of the distribution (34). The hy- pothesis that the restricted data is given by the power-law (22) or (24), respectively, with kmin = kcmin, kmax = kcmax, and the exponent γ = γ̂N′(kcmin) is accepted with the error probability α if c2 ≤ χ2α . If the window kcmin ≤ mi < kcmax lies not com- pletely within the power-law regime this hypothesis will be rejected by the χ2-test and one can detect the upper crossover point as well as the lower crossover point (where the power- law loses its validity) in a reliable way, see Figure 4 and Fig- ure 5. 6. Computational remarks The normalizing factors of the probability distributions (22) and (24) are given by the Hurwitz-ζ -function. This func- tion is less common than other special functions and may not be available in the reader’s favorite statistical software pack- age but the GNU Scientific Library [7] offers an open source implementation of this function. A direct calculation of the Hurwitz-ζ -function by truncating the sum (3) gives unsatis- factory results. The maximum likelihood estimator of the exponent can be computed numerically either by solving (25) or (26). Equation (25) has the advantage that it can be solved without calculating derivatives of the Hurwitz-ζ -function [8], whereas the solu- tion of (26) involves its first derivative (e. g. bisection method) or even higher derivatives (e. g. Newton-Raphson method). An explicit implementation of these derivatives is often not available but may be calculated numerically. 7. Conclusion Methods based on a least squares fit are not suited to establish estimates for power-law distribution exponents because least squares fits rely on assumptions about the data set that are not fulfilled by empirical data from power-law distributions. In this paper maximum likelihood estimators have been intro- duced as a reliable alternative to graphical methods. These es- timators are asymptotically efficient and can be applied to data from a wide class of distributions having a power-law regime. The crossover points that separate the power-law regime from the rest of the distribution can be determined by a procedure based on a χ2-test. Finally I would like to mention that the idea to plot a se- quence of estimates γ̂N′(kcmin) as shown in Figure 4 is re- lated to so-called Hill plots [9, 10]. The Hill estimator is a maximum likelihood estimator for the inverse of the ex- ponent of the continuous Pareto distribution p(k) = (γ − 1)(k/kmin) −γ/kmin, see [10] for a detailed discussion. Acknowledgments Work sponsored by the European Community’s FP6 Infor- mation Society Technologies programme under contract IST- 001935, EVERGROW. [1] M.E.J. Newman, Contemporary Physics 46(5), 323 (2005) [2] E.F. Keller, BioEssays 27(10), 1060 (2005) [3] M.L. Goldstein, S.A. Morris, G.G. Yen, The European Physical Journal B 41(2), 255 (2004) [4] F. Ramsey, D. Schafer, The Statistical Sleuth: A Course in Meth- ods of Data Analysis, 2nd edn. (Duxbury Press, Pacific Grove, CA, 2002) [5] L.J. Bain, M. Engelhardt, Introduction to Probability and Math- ematical Statistics, Duxbury Classic Series, 2nd edn. (Duxbury Press, 2000) [6] Y. Pawitan, In all likelihood: statistical modelling and inference using likelihood, Oxford science publications (Oxford Univer- sity Press, 2001) [7] GNU Scientific Library, http://www.gnu.org/software/gsl/ [8] R.P. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, New Jersey, 1973) [9] B.M. Hill, Annals of Statistics 3(5), 1163 (1975) [10] H. Drees, L. de Haan, S. Resnick, Annals of Statistics 28(25), 254 (2000) http://www.gnu.org/software/gsl/

Distributions following a power-law are an ubiquitous phenomenon. Methods for determining the exponent of a power-law tail by graphical means are often used in practice but are intrinsically unreliable. Maximum likelihood estimators for the exponent are a mathematically sound alternative to graphical methods.

Introduction The distribution of a discrete random variable is referred to as a distribution with a power-law tail if it falls as p(k) ∼ k−γ (1) for k ∈ N and k ≥ kmin. Power-laws are ubiquitous distribu- tions that can be found in many systems from different disci- plines, see [1, 2] and references therein for some examples. Experimental data of quantities that follow a power-law are usually very noisy; and therefore obtaining reliable estimates for the exponent γ is notoriously difficult. Estimates that are based on graphical methods are certainly used most often in practice. But simple graphical methods are intrinsically unre- liable and not able to establish a reliable estimate of the expo- nent γ . For that reason, the authors of [3] introduced an alternative approach based on a maximum likelihood estimator for the exponent γ . Unfortunately the authors concentrate on a rather idealized type of power-law distributions, namely p1(k;γ) = ζ (γ,1) with k ∈ N, where the normalization constant ζ (γ,1) is given by the Hurwitz-ζ -function which is defined for γ > 1 and a > ζ (γ,a) = (i+ a)γ . (3) The distribution (2) is characterized by one parameter only, and therefore all properties of this distribution (e. g. its mean) are determined solely by the exponent γ . In many applications the power-law (2) is too restrictive. If one states that a quantity follows a power-law, then this means usually that the tail (k ≥ kmin) of the distribution p(k) falls proportionally to k−γ . Probabilities p(k) for k < kmin may differ from the power-law and admit the possibility to tune the mean or other characteristics independently of γ . In some ∗E-mail:heiko.bauke@physics.ox.ac.uk situations probabilities p(k) may differ from a power-law for k ≥ kmax as well, e. g. the distribution may have an exponential cut-off. Therefore, I will generalize the maximum likelihood ap- proach introduced in [3] to distributions that follow a power- law within a certain range kmin ≤ k < kmax but differ from a power-law outside this range in an arbitrary way. Furthermore, I will give some statements about the large sample properties of the estimate of the power-law exponent and present a nu- merical procedure to identify the power-law regime of the dis- tribution p(k). But first, let us see what is wrong with popular graphical methods. 2. Trouble with graphical methods All graphical methods for estimating power-law exponents are based on a linear least squares fit of some empirical data points (x1,y1), (x2,y2),. . . , (xM,yM) to the function y(x) = a0 + a1x . (4) The linear least squares fit minimizes the residual (yi −a0 −a1xi)2 . (5) Estimates â0 and â1 of the parameters a0 and a1 are given by â0 = ∑Mi=1 yi ∑Mi=1 x ∑Mi=1 xi ∑Mi=1 yixi ∑Mi=1 x ∑Mi=1 xi â1 = ∑Mi=1 yixi ∑Mi=1 xi ∑Mi=1 yi ∑Mi=1 x ∑Mi=1 xi . (7) The ansatz for the residual (5) and derivation of (6) and (7) are based on several assumptions regarding the data points (xi,yi). It is assumed that there are no statistical uncertain- ties in xi, but yi may contain some statistical error. The errors in different yi are independent identically distributed random mailto:heiko.bauke@physics.ox.ac.uk variables with mean zero. In particular the standard devia- tion of the error is independent of xi. For various graphical methods for the estimation of the exponent of a power-law distribution these conditions are not met, leading to the poor performance of these methods. To illustrate the failure of graphical methods by a computer experiment N = 10000 random numbers mi had been drawn from distribution (2) with γ = 2.5 and an estimate γ̂ for the ex- ponent γ was determined by various graphical methods. The estimator is a random variable and its distribution depends on the method that has been used to obtain the estimate. Impor- tant measures of the quality of an estimator are its mean and its standard deviation. If the mean of the estimator equals the true exponent γ then the estimator is unbiased and estimators with a distribution that is concentrated around γ are desirable. For each graphical method a histogram of the distribution of the estimator was calculated to rate the quality of the estimator by repeating the numerical experiment 500 times. The most straight forward (and most unreliable) graphical approach is based on a plot of the empirical probability dis- tribution p̂(k) on a double-logarithmic scale. Introducing the indicator function I [·], which is one if the statement in the brackets is true and else zero, the empirical probability distri- bution is given by p̂(k) = I [mi = k] . (8) An estimate γ̂ for the power-law exponent γ is established by a least squares fit to (xi,yi) = (lnk, ln p̂(k)) for all k ∈ N with p̂(k) > 0, (9) γ̂ equals the estimate (7) for the slope, see Figure 1 a. Because the lack of data points in the tail of the empirical distribution this procedure underestimates systematically the exponent γ , see Table I. There are two ways to deal with the sparseness in the tail of the empirical distribution, logarithmic binning and consider- ing the empirical cumulative distribution P̂(k) instead of p̂(k). The cumulative probability distribution of (2) is defined by P(k) = ζ (γ,1) . (10) If p(k) has a power-law tail with exponent γ then P(k) follows approximately a power-law with exponent γ − 1 because for k ≫ 1 the distribution P(k) can be approximated by P(k) ≈ ζ (γ,1) (γ −1)ζ (γ,1) . (11) The empirical cumulative probability distribution is given by P̂(k) = I [mi ≥ k] . (12) It is less sensitive to the noise in the tail of the distribution and therefore a fit of (xi,yi) = (lnk, ln P̂(k)) for all k ∈ N with P̂(k) > 0 (13) Table I: Mean and standard deviation of the distribution of the esti- mate for the power-law exponent γ for various methods. All methods have been applied to the same data sets of random numbers from distribution (2) with γ = 2.5. See text for details. mean standard deviation method estimate of estimate fit on empirical distribution 1.597 0.167 fit on cumulative empirical distri- bution 2.395 0.304 fit on empirical distribution with logarithmic binninga 2.397 0.080 fit on cumulative empirical distri- bution with logarithmic binning 2.544 0.127 maximum likelihood 2.500 0.016 aIn [3] a similar experiment is reported. For a fit of the logarithmically binned empirical probability distribution the authors find a systematical bias of 29 %. I cannot reconstruct such a strong bias, instead I get a bias of 5 % only. Probably the quality of this method depends on the details of the binning procedure. to a straight line gives much better estimates for the exponent, see Figure 1 b. But there is still a small bias to too small values and the distribution of this estimate is rather broad, see Table I. Logarithmic binning reduces the noise in the tail of the em- pirical distributions p̂(k) and P̂(k) by merging data points into groups. By introducing the logarithmically scaled boundaries bi = roundc i with some c > 1 (14) (The function roundx rounds x to the nearest integer.) a linear least squares fit is performed to (xi,yi) = bi + bi+1 −1 bi+1−1 p̂(k) bi+1 −bi (xi,yi) = bi + bi+1 −1 bi+1−1 P̂(k) bi+1 −bi , (16) respectively. As a consequence of the binning the width of the distribution of the estimate γ̂ of the power-law exponent γ is reduced, see Figure 1 c, 1 d and Table I. According to the numerical experiments a fit of the logarithmically binned cumulative distribution gives the best results among graphical methods. It shows the smallest systematic bias. All the methods that have been considered so far have a common weakness. In the deviation of (6) and (7) it was as- sumed that the standard deviation of the distribution of the error in yi is the same for all data points (xi,yi). But this is obviously not the case. For fixed k the empirical distribution p̂(k) is a random variable with mean p1(k;γ) and standard deviation p1(k;γ)(1− p1(k;γ))/N. For the corresponding data on a logarithmic scale the standard deviation is approxi- mately given by the quotient p1(k;γ)(1− p1(k;γ))/N p1(k;γ) 1− p1(k;γ) N p1(k;γ) . (17) fit on the distribution p̂(k) fit on the cumulative distribution P̂(k) = ∑i≥k p̂(i) a) b) c) d) Figure 1: Comparison of various methods for estimating the exponent of a power-law. Each figure shows data for a single data set of N = 10000 samples drawn from distribution (2) with γ = 2.5. Insets present histograms of estimates for γ for 500 different data sets. A power-law distribution p1(k;γ) is a monotonically decreas- ing function and therefore (17) is an increasing function of k. Because the variation of the statistical error is not taken into account, the distribution of the estimate γ̂ is very broad. Methods that deal with the cumulative distribution have an additional weakness. Cumulation has the side-effect that the statistical errors in yi are not independent any more, which violates another assumption of the deviation of (6) and (7). To sum up, estimates of exponents of power-law distribu- tions based on a linear least squares fit are intrinsically inac- curate and lack a sound mathematical justification. 3. Maximum likelihood estimators Maximum likelihood estimators offer a solid alternative to graphical methods. Let p(k;θ ) denote a single parameter prob- ability distribution. The maximum likelihood estimator θ̂N for the unknown parameter based on a sample m1,m2, . . . ,mN of size N is given by θ̂N = argmax [L(θ )] = argmax [lnL(θ )] , (18) where L(θ ) = p(mi;θ ) (19) denotes the likelihood function. In the limit of asymptotically large samples and under some regularity conditions maximum likelihood estimators share some desirable features [5, 6]. • The estimator θ̂N exists and is unique. • The estimator θ̂N is consistent, that means for every ε > 0 |θ̂N −θ |< ε = 1 , (20) where P |θ̂N −θ |< ε denotes the probability that the dif- ference |θ̂N −θ | is less than ε . • The estimator θ̂N is asymptotically normal with mean θ and variance (∆θ̂N)2 ln p(k;θ ) , (21) where E [·] indicates the expectation value of the quantity in the brackets. • Maximum likelihood estimators have asymptotically mini- mal variance among all asymptotically unbiased estimators. One says, they are asymptotically efficient. 4. Maximum likelihood estimators for genuine power-laws The most general discrete genuine power-law distribution has a lower as well as an upper bound and is given by pkmin,kmax(k;γ) = ζ (γ,kmin,kmax) for k ∈ N with kmin ≤ k < kmax. Where the non-standard nota- ζ (γ,kmin,kmax) := ζ (γ,kmin)− ζ (γ,kmax) (23) has been introduced. If the upper bound is missing the distri- bution pkmin(k;γ) = ζ (γ,kmin) has to be considered for k ∈N with k ≥ kmin. The distributions (22) and (24) are generalizations of (2) and will be useful for the analysis of more general distributions that show a power- law behavior only in a certain range but have an arbitrary pro- file outside the power-law regime. This kind of distributions will be considered in section 5. The maximum likelihood estimator γ̂N for the parameter γ of the distribution (22) follows from (18) and is given by γ̂N = argmax −N lnζ (γ,kmin,kmax) or equivalently by the implicit equation ζ ′(γ̂N ,kmin,kmax) ζ (γ̂N ,kmin,kmax) lnmi = 0 , (26) which has to be solved numerically. The prime denotes the derivative with respect to γ . The asymptotic variance of this estimator γ̂N follows from (21) and equals (∆γ̂N)2 ζ (γ,kmin,kmax)2 ζ ′′(γ,kmin,kmax)ζ (γ,kmin,kmax)− ζ ′(γ,kmin,kmax)2 1 2 3 4 5 6 7 8 Figure 2: Asymptotic standard deviation for maximum likelihood esti- mators for the exponent of a power-law distribution (24). Figure 3: Empirical distribution of the maximum likelihood estima- tor (histogram) versus its theoretical asymptotic distribution, which is given by a normal distribution with mean γ = 2.5 and variance (27). The histogram has been obtained from the same data as in Figure 1. In the limit kmax → ∞ equations (25), (26), and (27) give the maximum likelihood estimator and the asymptotic variance of this estimator for power-law distributions lacking an upper cut-off (24). A graphical representation of the standard devia- tion (27) in the limit kmax → ∞ is given in Figure 2. For each fixed kmin the quantity ∆γ̂N N grows faster than linear with γ . Therefore the larger the exponent γ the larger the sample size that is necessary to get an estimate within a given error bound. If the maximum likelihood method is applied (assuming a distribution (24)) to the same data as in section 2, numeri- cal experiments show that the estimates for the exponent are much more precise. The estimate has no identifiable system- atic bias, the standard deviation of the distribution of the esti- mate is smaller by an order of magnitude compared to graphi- cal methods, see Table I and Figure 3. 5. Maximum likelihood method for general power-law distributions The maximum likelihood procedure outlined in section 4 can be generalized further to distributions p(k) that are no pure power-laws (22) or (24) but follow a power-law within a cer- tain finite range or follow a power-law in the whole tail of the distribution and have an arbitrary profile outside the power- law regime. The popurse of this section is to establish meth- ods for identifying the power-law regime and for estimating the exponent of the power-law regime without making special assumptions about the profile of the probability distribution beyond the power-law regime. The main problem for a generalization of the maximum likelihood approach is that there might be no good hypothe- sis for the profile of the probability distribution beyond the power-law regime. To overcome this difficulty the empirical data set is restricted to a window kcmin ≤ mi < kcmax. (The following discussion covers the case of a power-law tail distri- butions as well by setting kcmax = ∞.) Assuming that p(k) has a power-law profile for kcmin ≤ k < kcmax then the probabil- ity distribution of the restricted data set is given by (22) with kmin = kcmin and kmax = kcmax (or by (24) with kmin = kcmin) and some unknown exponent γ . This allows to estimate the power-law exponent by the application of the maximum like- lihood method on the restricted data set of size N′ as presented in section 4 without making a hypothesis about the profile of the probability distribution beyond the power-law regime. In order to apply the maximum likelihood method one has to determine the cut-off points kcmin and kcmax first. Here it has to be taken into account that if the window kcmin ≤ mi < kcmax is chosen too large the estimate γ̂ is systematically bi- ased, but on the other hand if it is too small the statistical error is larger than necessary. In some cases one can make conser- vative estimates for kcmin and kcmax by plotting the empirical probability distribution (8) on a double-logarithmic scale. An appropriate window can also be found by determining esti- mates γ̂N′(kcmin) as a function of the window and a χ2-test. Assuming the empirical data is drawn from a distribution with a power-law tail (no upper cut-off) the lower cut-off point kcmin can be determined in the following systematic way. By varying the parameter kcmin the maximum likelihood approach gives a sequence of estimates γ̂N′(kcmin). If kcmin is very large the estimate will be quite inaccurate because only a tiny frac- tion of the experimental data is taken into account; but the smaller the cut-off kcmin the more accurate the estimate of the exponent. If kcmin approaches the point from above (but is still above) where the probability distribution starts do differ from a power-law γ̂N′(kcmin) will give a very precise estimate for the exponent γ . On the other hand, if kcmin is too small the hypothesis that the (restricted) empirical data is drawn from a power-law distribution is violated which causes a significant change of the estimate of the power-law exponent. If the empirical data is drawn from a distribution having both a lower crossover point as well as an upper crossover point a sequence of estimates γ̂N′(kcmin) is determined by re- stricting the data to a sliding window kcmin ≤ mi < wkcmin = kcmax with w > 1. As long as the window lies completely within the power-law regime the maximum likelihood esti- mate obtained from the restricted data set will give a reliable estimate of the power-law exponent. If the window lies at least partly outside the power-law regime the estimate is systemati- cally biased. To illustrate the procedures outlined above I generated two data sets from two distributions having a power-law regime. The first data set of N = 10000 samples was drawn from a distribution with a power-law tail which is given by p(k) ∼ 5−2.5 for 1 ≤ k ≤ 5 k−2.5 for k > 5 . Plotting the sequence of estimates γ̂N′(kcmin) against the pa- rameter kcmin reveals the exponent γ = 2.5 as wells as the crossover point k = 5 very clearly, see Figure 4. The second data set of N = 100000 samples was drawn from a distribution with two crossover points, viz. p(k) ∼ 5−2.25 for 1 ≤ k ≤ 5 k−2.25 for 5 ≤ k ≤ 100 100−2.25e−0.05(k−100) for k > 100 . Figure 5 shows the sequence of estimates γ̂N′(kcmin) that had been determined from restricted data sets of samples within the sliding window kcmin ≤ mi < 5kcmin. This sequence ex- hibits a broad plateau that corresponds to the power-law expo- nent γ = 2.25. If the window does not lie completely inside c min Figure 4: Sequence of estimates γ̂N ′(kcmin) as a function of the cut- off kcmin for a data set of N = 10000 samples from distribution (28). Filled symbols mark where the χ2-test has rejected the hypothesis that the restricted data follows a power-law (24) with exponent γ = γ̂N ′(kcmin). An error probability of α = 0.001 was chosen. c min 2kc max Figure 5: Sequence of estimates γ̂N ′(kcmin) as a function of the lower cut-off kcmin for a data set of N = 100000 samples from distribution (29). Filled symbols mark where the χ2-test has rejected the hypoth- esis that the restricted data follows a power-law (22) with exponent γ = γ̂N ′(kcmin). An error probability of α = 0.001 and the window width w = 5 had been chosen. the power-law regime the estimate γ̂N′(kcmin) deviates system- atically from the known exponent. Apart from a visual inspection of the γ̂N′(kcmin) plot the crossover point(s) to the power-law regime can be determined by means of a χ2-test. To apply a χ2-test the data set has to be divided into some bins and the following binning turned out to be appropriate: The data is partitioned into a small number b, say b = 6, of bins. In the case of a distribution with a power- law tail this means each bin j collects n j items such that n1 = N ′ p̂(kcmin) q1 = pkcmin(kcmin; γ̂N′(kcmin)) (30) n2 = N ′ p̂(kcmin + 1) q2 = pkcmin(kcmin + 1; γ̂N′(kcmin)) and finally nb = N k=kcmin+b−1 p̂(k) qb = k=kcmin+b−1 pkcmin(k; γ̂N′ (kcmin)) , where q j denotes the probability that a data point falls into bin j under the assumption that the (restricted) data follows the power-law (24) with kmin = kcmin and the exponent γ = γ̂N′(kcmin). For distributions with a finite power-law regime the binning procedure can be carried out in a similar way. In this case the summation index in (32) is bounded by kcmin + b−1≤ k < kcmax and the probability pkcmin,kcmax(k; γ̂N′(kcmin)) has to be considered instead of pkcmin(k; γ̂N′(kcmin)). The test statistic of the χ2-test is given by (n j −N′q j)2 N′q j . (33) If the to kcmin ≤ mi < kcmax restricted data is given by the power-law (22) or (24) with kmin = kcmin, kmax = kcmax, and the exponent γ = γ̂N′(kcmin) then the statistic c2 follows asymptotically a χ2-distribution with ν = (b− 1) degrees of freedom, which is given by pχ2(x,ν) = xν/2−1e−x/2 2−ν/2 Γ(ν/2) . (34) Let χ2α be the (1−α)-quantile of the distribution (34). The hy- pothesis that the restricted data is given by the power-law (22) or (24), respectively, with kmin = kcmin, kmax = kcmax, and the exponent γ = γ̂N′(kcmin) is accepted with the error probability α if c2 ≤ χ2α . If the window kcmin ≤ mi < kcmax lies not com- pletely within the power-law regime this hypothesis will be rejected by the χ2-test and one can detect the upper crossover point as well as the lower crossover point (where the power- law loses its validity) in a reliable way, see Figure 4 and Fig- ure 5. 6. Computational remarks The normalizing factors of the probability distributions (22) and (24) are given by the Hurwitz-ζ -function. This func- tion is less common than other special functions and may not be available in the reader’s favorite statistical software pack- age but the GNU Scientific Library [7] offers an open source implementation of this function. A direct calculation of the Hurwitz-ζ -function by truncating the sum (3) gives unsatis- factory results. The maximum likelihood estimator of the exponent can be computed numerically either by solving (25) or (26). Equation (25) has the advantage that it can be solved without calculating derivatives of the Hurwitz-ζ -function [8], whereas the solu- tion of (26) involves its first derivative (e. g. bisection method) or even higher derivatives (e. g. Newton-Raphson method). An explicit implementation of these derivatives is often not available but may be calculated numerically. 7. Conclusion Methods based on a least squares fit are not suited to establish estimates for power-law distribution exponents because least squares fits rely on assumptions about the data set that are not fulfilled by empirical data from power-law distributions. In this paper maximum likelihood estimators have been intro- duced as a reliable alternative to graphical methods. These es- timators are asymptotically efficient and can be applied to data from a wide class of distributions having a power-law regime. The crossover points that separate the power-law regime from the rest of the distribution can be determined by a procedure based on a χ2-test. Finally I would like to mention that the idea to plot a se- quence of estimates γ̂N′(kcmin) as shown in Figure 4 is re- lated to so-called Hill plots [9, 10]. The Hill estimator is a maximum likelihood estimator for the inverse of the ex- ponent of the continuous Pareto distribution p(k) = (γ − 1)(k/kmin) −γ/kmin, see [10] for a detailed discussion. Acknowledgments Work sponsored by the European Community’s FP6 Infor- mation Society Technologies programme under contract IST- 001935, EVERGROW. [1] M.E.J. Newman, Contemporary Physics 46(5), 323 (2005) [2] E.F. Keller, BioEssays 27(10), 1060 (2005) [3] M.L. Goldstein, S.A. Morris, G.G. Yen, The European Physical Journal B 41(2), 255 (2004) [4] F. Ramsey, D. Schafer, The Statistical Sleuth: A Course in Meth- ods of Data Analysis, 2nd edn. (Duxbury Press, Pacific Grove, CA, 2002) [5] L.J. Bain, M. Engelhardt, Introduction to Probability and Math- ematical Statistics, Duxbury Classic Series, 2nd edn. (Duxbury Press, 2000) [6] Y. Pawitan, In all likelihood: statistical modelling and inference using likelihood, Oxford science publications (Oxford Univer- sity Press, 2001) [7] GNU Scientific Library, http://www.gnu.org/software/gsl/ [8] R.P. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, New Jersey, 1973) [9] B.M. Hill, Annals of Statistics 3(5), 1163 (1975) [10] H. Drees, L. de Haan, S. Resnick, Annals of Statistics 28(25), 254 (2000) http://www.gnu.org/software/gsl/

THE WEIL REPRESENTATION AND HECKE OPERATORS FOR VECTOR VALUED MODULAR FORMS JAN HENDRIK BRUINIER AND OLIVER STEIN Abstract. We define Hecke operators on vector valued modular forms transforming with the Weil representation associated to a discriminant form. We describe the properties of the corresponding algebra of Hecke operators and study the action on modular forms. 1. Introduction Hecke operators are a fundamental tool in the study of modular forms. They can be used to obtain information on the arithmetic nature of the Fourier coefficients. They are vital for the definition of L-functions associated to modular forms and for understanding their properties. The theory of Hecke operators is well developed for scalar valued modular forms [Sh1]. In many recent works, vector valued modular forms associated to the Weil representation play an important role, see e.g. [Bo1], [Bo2], [Br], [McG], [Sch]. For instance, Borcherds uses them to provide a elegant description of the Fourier expansion of various theta liftings. The purpose of the present paper is to develop the foundations of a Hecke theory for such vector valued modular forms. The results can be used to associate an L-function to a vector valued modular form, essentially the standard L-function. We now describe the content of this paper in more detail. Let L be a non-degenerate even lattice of type (b+, b−) and level N . The modulo 1 reduction of the quadratic form on the dual lattice L′ defines a Q/Z-valued quadratic form on the discriminant group A = L′/L. To simplify the exposition, we assume throughout the introduction that the signature sig(L) = b+ − b− of L is even. In the body of the paper, both, odd and even signature is treated. The Weil representation associated to the discriminant form A is a unitary representation of Γ = SL2(Z) on the group ring C[A], ρA : Γ −→ U(C([A])), defined by (2.3), (2.4). It factors through the finite quotient S(N) := SL2(Z/NZ) ∼= Γ/Γ(N). Date: October 27, 2018. 2000 Mathematics Subject Classification. 11F27, 11F25. http://arxiv.org/abs/0704.1868v1 2 JAN H. BRUINIER AND OLIVER STEIN Let k ∈ Z. A holomorphic function f : H → C[A] is called a modular form of weight k and type ρA for the group Γ, if f(Mτ) = (cτ + d)kρA(M)f(τ) for all M = ( a bc d ) ∈ Γ, and f is holomorphic at the cusp ∞. We denote the vector space of such holomorphic modular forms by Mk,A, and write Sk,A for the subspace of cusp forms. In order to define Hecke operators on vector valued modular forms of type ρA, we need to extend the representation ρA to a sufficiently large subgroup of GL 2 (Q). A natural starting point is to try to extend ρA, viewed as a representation of S(N), to a representation of G(N) := GL2(Z/NZ). However, it was observed by E. Freitag that such an extension does not exist in general, see Example 3.1. Here we consider the subgroup {M ∈ G(N); det(M) ≡ � (mod N)} of matrices whose determinant is a square modulo N . It has the extension Q(N) = {(M, r) ∈ G(N)× U(N); det(M) ≡ r2 (mod N)}, where U(N) denotes the unit group of Z/NZ. The group Q(N) is isomorphic to S(N) × U(N). Consequently, we may extend the Weil representation to Q(N) by taking the tensor product of ρA on S(N) and a suitable character on U(N), see Proposition 3.3. If M is an element of G(N) whose determinant is a square modulo N , and r, r′ ∈ U(N) with det(M) ≡ r2 ≡ r′2 (mod N), then (M, r) and (M, r′) both belong to Q(N). We prove that the action of ρA(M, r) and ρA(M, r ′) on C[A] differ only by the action of an element of the orthogonal group O(A), see Proposition 3.5. This extension of the Weil representation can be used to define a Hecke operator T (M, r) on Mk,A for every pair (M, r), where M ∈ M2(Z) and r ∈ U(N) with det(M) ≡ r2 (mod N). We compute the action of these operators on the Fourier expansion of a modular form (see Section 4.1). They generalize the classical Hecke operators on scalar valued modular forms and Jacobi forms (see e.g. Remark 4.4 and Remark 4.11). In particular, for every positive integer m coprime to N we obtain a Hecke operator T (m2)∗ := T on Mk,A. These operators generate a commutative subalgebra of End(Mk,A), which is actu- ally already generated by the T (p2)∗ for p prime and coprime to N . The operators T (m2)∗ take cusp forms to cusp forms and are self-adjoint with respect to the Petersson scalar product (see Theorem 4.12). In particular, Sk,A has a basis of simultaneous eigenforms of all T (m2)∗ with (m,N) = 1. In Section 5 we extend the definition of the Hecke operators T (m2)∗ to all positive integers m, not necessarily coprime to N . This is done by defining the right-action on C[A] of a matrix α = by the same formula as in the case where m is coprime to N . Notice, that the corresponding linear map C[A] −→ C[A], eλ 7→ eλ |A α = emλ THE WEIL REPRESENTATION AND HECKE OPERATORS 3 is neither surjective nor injective in general. However, it still can be used to obtain an “action” of the double coset ΓαΓ, see Proposition 5.1 and Lemma 5.2. This suffices to define a corresponding Hecke operator T (m2)∗ on Mk,A, which is consistent with our earlier definition when m is coprime to N . For any positive integer m, the Hecke operator T (m2)∗ is self adjoint with respect to the Petersson scalar product. Moreover, if m and n are coprime, then T (m2)∗T (n2)∗ = T (m2n2)∗, see Theorem 5.6. Observe that for a prime p dividing N the local Hecke algebra, that is, the subalgebra of End(Mk,A) generated by the T (p 2ν)∗, is considerably more complicated than in the case where p is coprime to N . For instance, it is commutative if p is coprime to N , but in general non-commutative if p divides N . Let S be a finite set of primes and let NS be the product of the primes in S. Let f ∈ Sk,A be a common eigenform of all T (m2)∗ with (m,NS) = 1, so f |k,A T (m2)∗ = λm(f)f. We can use the above results to define an L-function associated to f by putting LS(s, f) = (m,NS)=1 λm(f)m It is easily seen that LS(s, f) converges for ℜ(s) sufficiently large. By Theorem 5.6, this L-function has an Euler product expansion. According to [Bö], it should be viewed as the standard L-function of f . It would be interesting to study the analytic properties of LS(s, f) in more detail. This could possibly be done by using a variant of the doubling method (see [Bö], [Ga], [PSR]) involving a Siegel Eisenstein series of genus 2 associated to the Weil representation of Sp(2,Z) on C[A2]. Moreover, it would be very interesting to develop a theory of new forms for the space Mk,A. One could try to associate an irreducible automorphic representation to a vector valued new form and study the properties of the resulting map. If the signature of L is odd, one can carry over the above results. However, one has to work with the metaplectic cover of Γ and with similar {±1}-extensions of S(N), G(N), and Q(N). In this case Mk,A vanishes unless k is half-integral. Following the argument of Shimura [Sh2], we show that the Hecke operator T (M, r) vanishes identically unless det(M) is the square of a rational number, see Proposition 4.9. The computation of the action of the Hecke operators on modular forms is more involved than in the even signature (i.e. integral weight) case, see Theorem 4.10. We thank E. Freitag for many valuable discussions on this paper. Moreover, we thank J. Funke for several useful comments. 2. Discriminant forms and the Weil representation Here we briefly summarize some facts on lattices, discriminant forms, and the Weil representation. See also [Bo1], [Bo2], [Br]. 4 JAN H. BRUINIER AND OLIVER STEIN Let L be a non-degenerate even lattice of type (b+, b−). We denote the bilinear form on L by (·, ·) and the associated quadratic form by x 7→ 1 x2 = 1 (x, x). We let sig(L) = b+−b− be the signature of L. We write L′ for the dual lattice of L, and denote by N the level of L, that is, the smallest positive integer such that N x2 ∈ Z for all x ∈ L′. The finite abelian group L′/L is called the discriminant group of L. Its order is equal to the absolute value of the Gram determinant of L. Recall that a discriminant form is a finite abelian group A together with a Q/Z-valued non-degenerate quadratic form x 7→ 1 x2, for x ∈ A (see [Ni]). If L is a non-degenerate even lattice then L′/L is a discriminant form where the quadratic form is given by the mod 1 reduction of the quadratic form on L′. Conversely, every discriminant form can be obtained in this way. The quadratic form on L′/L determines the signature of L modulo 8 by Milgram’s formula (see [MH] Appendix 4): (2.1) λ∈L′/L e(λ2/2) = |L′/L|e(sig(L)/8). Here and throughout we abbreviate e(z) = e2πiz for z ∈ C. We define the signature sig(A) ∈ Z/8Z of a discriminant form A to be the signature of any even lattice with that discriminant form. If A is a discriminant form, then we write An for the subgroup of elements that are n-th powers of elements of A. Moreover, we write An for the subgroup of elements of A whose order divides n. We have an exact sequence 0 −→ An −→ A −→ An −→ 0,(2.2) and An is the orthogonal complement of An. Let H = {τ ∈ C; ℑ(τ) > 0} by the complex upper half plane. We write G̃L 2 (R) for the metaplectic two-fold cover of GL+2 (R). The elements of this group are pairs (M,φ(τ)) where M = ( a bc d ) ∈ GL 2 (R) and φ : H → C is a holomorphic function with φ(τ)2 = cτ +d. The multiplication is defined by (M,φ(τ))(M ′, φ′(τ)) = (MM ′, φ(M ′τ)φ′(τ)). For g = (M,φ) ∈ G̃L 2 (R), we put det(g) = det(M). Moreover, if G is a subset of GL+2 (R), we write G̃ for its inverse image under the covering map. Throughout we write Γ = SL2(Z) for the full modular group. It is well known that the integral metaplectic group Γ̃ is generated by T = (( 1 10 1 ) , 1), and S = (( 1 0 ) , τ ). One has the relations S2 = (ST )3 = Z, where Z = ((−1 0 is the standard generator of the center of Γ̃. We now recall the Weil representation associated with a discriminant form A (see also [Bo1], [Bo2]). It is a representation of Γ̃ on the group algebra C[A]. We denote the standard basis elements of C[A] by eλ, λ ∈ A, and write 〈·, ·〉 for the standard scalar product (antilinear in the second entry) such that 〈eλ, eµ〉 = δλ,µ. The Weil representation ρA associated with the discriminant form A is the unitary representation of Γ̃ on the group THE WEIL REPRESENTATION AND HECKE OPERATORS 5 algebra C[A] defined by ρA(T )(eλ) = e(λ 2/2)eλ,(2.3) ρA(S)(eλ) = e(− sig(A)/8)√ e(−(λ, µ))eµ.(2.4) Note that ρA(Z)(eλ) = e(− sig(A)/4)e−λ.(2.5) The orthogonal group O(A) also acts on C[A] by ρA(h)(eλ) = eh−1λ(2.6) for h ∈ O(A), and the actions of Γ̃ and O(A) commute. If the signature of A is even, then (2.5) implies that Z2 acts trivially. Hence, the Weil representation factors through Γ. Moreover, it is trivial on the principal congruence subgroup Γ(N), where N is the level of A, i.e., the level of any even lattice L with L′/L = A (see e.g. [Eb], Chapter 3, Theorem 3.2). Therefore, ρA factors through the finite group S(N) := SL2(Z/NZ) ∼= Γ/Γ(N).(2.7) If the signature of A is odd, we notice that the level of A must be divisible by 4. This follows from the oddity formula ([CS] p. 383 (30)) which implies that A contains odd 2-adic Jordan components. On Γ(4) the metaplectic cover has the section s : Γ(4) −→ Γ̃(4), cτ + d given by the theta multiplier system. Here · denotes the principal branch of the holomor- phic square root. The same argument as at the end of the proof of Theorem 5.4 in [Bo2] implies that ρA is trivial on s(Γ(N)) and factors through the central extension of S(N) by {±1} given by S1(N) := Γ̃/s(Γ(N)).(2.8) We will also need the action of ρA on diagonal matrices in S(N). Following [McG], for integers a, d coprime to N such that ad ≡ 1 (mod N), we put Rd := ST dS−1T aST d.(2.9) It is easily checked that Rd = (M,φ) where M ≡ ( a 00 d ) (mod N). Lemma 2.1. (See [McG] Lemma 4.6.) For a, d as above we have ρA(Rd)eλ = gd(A) edλ.(2.10) Here gd(A) denotes the Gauss sum gd(A) = e(dλ2/2)(2.11) 6 JAN H. BRUINIER AND OLIVER STEIN and g(A) = g1(A). � Notice that by Milgram’s formula we have g(A) = |A|e(sig(A)/8). Moreover, one easily checks that |gd(A)| = |A|. If r ∈ Z is coprime to N , then we have gdr2(A) = gd(A). In particular, gd(A) = ga(A). Finally, Lemma 2.1 and the fact that ρA is a representation imply the relation gdr(A)g(A) gd(A)gr(A) 1, if sig(A) is even, ±1, if sig(A) is odd. (2.12) The following more general formula was given by Borcherds. Proposition 2.2. (See [Bo2] Theorem 5.4.) Let g = (( a bc d ) , cτ + d) ∈ Γ̃, and suppose that b and c are divisible by N . Then ρA(g)eλ = χA(g)edλ.(2.13) Here χA denotes the character of Γ̃0(N) defined in [Bo2] Theorem 5.4. � Lemma 2.3. Let U = ( 1 01 1 ) , τ + 1 ∈ Γ̃. The action of Um is given by m)eλ = µ,ν∈A −mµ2/2 + (µ, λ− ν) Proof. Since Um = ST−mS−1, this follows immediately from (2.3) and (2.4). � In many recent works vector valued modular forms associated to the Weil representation are considered (see e.g. [Bo1], [Bo2], [Br], [McG], [Sch]). Let k ∈ 1 Z, and let Γ′ ⊂ Γ̃ be a subgroup of finite index. A holomorphic function f : H → C[A] is called a modular form of weight k and type ρA for the group Γ ′, if f(Mτ) = φ(τ)2kρA(M,φ)f(τ) for all (M,φ) ∈ Γ′, and f is holomorphic at the cusps of Γ′. We denote the C-vector space of such holomorphic modular forms by Mk,A(Γ ′). Moreover, for the full modular group we put Mk,A = Mk,A(Γ̃). Formula (2.5) implies that Mk,A = {0} unless 2k ≡ sig(A) mod 2.(2.14) Recall that for f, g ∈ Mk,A(Γ′) the Petersson scalar product is defined by (f, g) = [Γ̃ : Γ′] 〈f(τ), g(τ)〉 yk dx dy .(2.15) Here x denotes the real part and y the imaginary part of τ ∈ H. The Petersson scalar product converges when f ⊗ g is a cusp form. THE WEIL REPRESENTATION AND HECKE OPERATORS 7 3. Extending the Weil representation In the classical theory of scalar valued modular forms Hecke operators play an impor- tant role (see e.g. [Sh1]). It is natural to try to define Hecke operators on vector valued modular forms of type ρA as well. This requires the extension of the representation ρA to a representation (of a sufficiently large subgroup) of G̃L 2 (Q). However, it is not obvious how this can be done. A natural starting point is to try to extend ρA, viewed as a representation of S(N) (respectively S1(N)), to a representation of (a double cover of) GL2(Z/NZ). However, it was observed by E. Freitag that such an extension does not exist in general. This follows from the following example. Example 3.1. Let d ≡ 1 (mod 4) be an integer such that p := |d| is a prime. We consider the ring of integers O in the quadratic field Q( d) of discriminant d. Together with the norm form, it is an even lattice of type (1, 1) if d > 0, and of type (2, 0) if d < 0. The dual lattice is 1√ O, the inverse of the different, and the corresponding discriminant form A can be identified with the finite field Fp together with the quadratic form x 7→ −1dx 2. The associated Weil representation ρA is a p-dimensional representation of S(p) = SL2(Fp) on C[A]. The action of the orthogonal group O(A) = {±1} splits C[A] into two S(p)-invariant subspaces C[A]+ = span{eλ + e−λ; λ ∈ A}, C[A]− = span{eλ − e−λ; λ ∈ A}. They have dimension p+1 , and p−1 , respectively. It follows from [NW], Theorem 4, that the corresponding representations of S(p) are irreducible. On the other hand, the character table of GL2(Fp) is well known, see e.g. [FH] §5.2. It has p− 1 one-dimensional representations, p− 1 irreducible p-dimensional representations, (p− 1)(p− 2)/2 irreducible (p+1)-dimensional representations, and (p2 − p)/2 irreducible (p− 1)-dimsional representations. Now assume that p ≥ 5 and that ρA has an extension ρ̃A to a representation of GL2(Fp). Because of the irreducibility of C[A]±, such an extension would have to be a p-dimensional irreducible representation of GL2(Fp). But these representations remain irreducible under restriction to SL2(Fp), see [FH], p.72 (2). We obtain a contradiction. Remark 3.2. In [McG], McGraw continues ρA to an action of GL2(Z/NZ). However, this action is not C-linear, causing serious difficulties when one tries to define Hecke operators. Here we consider a different group extension Q(N) of S(N) and show that ρA can be continued to a representation of Q(N). Together with the considerations of Section 5 this will suffice for many applications of Hecke operators; for instance, to define the standard L-function of a modular form of type ρA. Let A be a discriminant form as in the previous section, and let N be the level of A. We denote by U(N) the unit group of Z/NZ. We briefly write G(N) = GL2(Z/NZ) for the general linear group modulo N . The determinant homomorphism G(N) → U(N) gives 8 JAN H. BRUINIER AND OLIVER STEIN rise to the exact sequence 1 −→ S(N) −→ G(N) −→ U(N) −→ 1.(3.1) This sequence splits and G(N) can be viewed as a semidirect product of S(N) and U(N). 3.1. The case of even signature. Throughout this subsection we assume that sig(A) is even. Let Q(N) be the group Q(N) = {(M, r) ∈ G(N)× U(N); det(M) ≡ r2 (mod N)}(3.2) with the product defined component-wise. We have an exact sequence 1 −→ S(N) −→ Q(N) −→ U(N) −→ 1,(3.3) where S(N) → Q(N) is given by M 7→ (M, 1), and Q(N) → U(N) is given by (M, r) 7→ r. The latter homomorphism has the section U(N) −→ Q(N), r 7→ .(3.4) For (M, r) ∈ Q(N) the assignment (M, r) 7→ (M ( r 00 r ) , r) defines an isomorphism Q(N) ∼= S(N)× U(N). We consider the action of S(N) on C[A] by the Weil representation ρA. In view of (2.12), the assignment r 7→ g(A) gr(A) defines a character of U(N). We define a unitary representation of U(N) on C[A] by putting ρA(r)eλ = ρA gr(A) eλ(3.5) for r ∈ U(N). Proposition 3.3. The Weil representation ρA of the group S(N) extends to a unitary representation of Q(N) by (3.5). Proof. Since Q(N) = S(N) × U(N) we only have to check that the actions of S(N) and U(N) commute. This is obvious. � Remark 3.4. Clearly we could take any character of U(N) to define the action of the Weil representation on U(N). The above choice is compatible with the definition of the Weil representation on double cosets in Section 5. Moreover, it is compatible with the usual Hecke operators on scalar valued modular forms, see Remark 4.4 and Remark 4.11. The following proposition shows that the first entry of an element (M, r) ∈ Q(N) gives the “essential contribution” to the Weil representation. If we fix M then for the different choices of r ∈ U(N) the Weil representation ρA(M, r) differs by the action of an element of the orthogonal group O(A). Proposition 3.5. Let (M, r1), (M, r2) ∈ Q(N). Then h : A → A, λ 7→ r1r−12 λ is an orthogonal transformation in O(A), and ρA(M, r1)eλ = ρA(M, r2)ρA(h)eλ = ρA(M, r2)eh−1λ. THE WEIL REPRESENTATION AND HECKE OPERATORS 9 Proof. We have (M, r2) −1(M, r1) = (1, r1r 2 ), and t = r1r 2 ∈ U(N) has the property that t2 ≡ 1 (mod N). According to (2.10) and (3.5), the action of (1, t) is given by ρA(1, t)eλ = et−1λ. Since (tλ, tµ) = (t2λ, µ) = (λ, µ) for all λ, µ ∈ A, multiplication by t−1 is an orthogonal transformation. � Lemma 3.6. We have eλ = er−1λ,(3.6) eλ = erλ.(3.7) Proof. In Q(N) we have = (Rr−1 , 1) Therefore the fist formula follows from Lemma 2.1 and (3.5). The second formula follows similarly. � 3.2. The case of odd signature. Throughout this subsection we assume that sig(A) is odd. In this case the argument of the previous section only shows that the Weil repre- sentation extends to a projective representation of Q(N). More precisely, it extends to a group homomorphism Q(N) −→ U(C[A])/{±1}. This projective representation gives rise to a 2-cocycle of Q(N) with values in {±1}. The cocycle defines a central extension Q1(N) of Q(N) by {±1}. The group S1(N), see (2.8), can be identified with a subgroup of Q1(N) (see Section 4.2 for more details). Proposition 3.7. The assignment (3.5) defines an extension of the Weil representation to a projective representation of Q(N). It lifts to a unitary representation of Q1(N). 4. Hecke operators on vector valued modular forms We now use the results of Section 3 to define Hecke operators on vector valued modular forms of type ρA. We consider the groups G(N) = {M ∈ GL+2 (Q); ∃n ∈ Z with (n,N) = 1 such that nM ∈ M2(Z)(4.1) and (det(nM), N) = 1}, Q(N) = {(M, r) ∈ G(N)× U(N); det(M) ≡ r2 (mod N)}.(4.2) We view Γ as a subgroup of Q(N) by the embedding M 7→ (M, 1). We are interested in the action of the Hecke algebra of the pair (Q(N),Γ) on modular forms of type ρA and weight k. We have to distinguish the cases whether sig(A) is even or odd. 10 JAN H. BRUINIER AND OLIVER STEIN 4.1. The case of even signature. The composition of the reduction map Q(N) → Q(N) with the Weil representation ρA : Q(N) → U(C[A]) induces a unitary representation of Q(N) on C[A], which we will denote by ρA as well. This left action induces a corresponding right action by a |A (M, r) = ρA(M, r)−1a.(4.3) In view of (2.14) we only have to consider modular forms of integral weight k ∈ Z. Let f be a complex valued function on H. For M ∈ GL+2 (R) the Petersson slash operator is defined by (f |k M)(τ) = det(M)k/2j(M, τ)−kf(Mτ).(4.4) This defines a right action of G(N) on functions H → C. The center acts by multiplication with ±1. If f : H → C[A] is a function we write f = λ∈A fλ ⊗ eλ for its decomposition in components with respect to the standard basis of C[A]. The tensor product of the above two actions yields a right action of Q(N) on such functions, denoted by f |k,A (M, r) = fλ |k M eλ |A (M, r) .(4.5) Notice that a holomorphic function f : H → C[A] belongs to Mk,A, if and only if f |k,A M = f for all M ∈ Γ, and f is holomorphic at the cusp ∞. We consider the Hecke algebra of the pair (Q(N),Γ) in the sense of Shimura [Sh1]. If (M, r) ∈ Q(N), the corresponding double coset decomposes into a finite union of left cosets Γ · (M, r) · Γ = γ∈Γ\ΓMΓ Γ · (γ, r). Definition 4.1. For (M, r) ∈ Q(N) we define the corresponding Hecke operator T (M, r) on Mk,A by f |k,A T (M, r) = det(M)k/2−1 γ∈Γ\ΓMΓ f |k,A (γ, r), f ∈ Mk,A.(4.6) The usual argument now shows that f |k,A T (M, r) ∈ Mk,A. Hence T (M, r) defines an endomorphism of Mk,A. A modular form f ∈ Mk,A has a Fourier expansion of the form f(τ) = n∈Z+λ2/2 c(λ, n)e(nτ)⊗ eλ.(4.7) Theorem 4.2. Let p be a prime which is a square modulo N , and assume that r ∈ U(N) with p ≡ r2 (mod N). Let f ∈ Mk,A and denote the Fourier expansion as in (4.7). Then f |k,A T n∈Z+λ2/2 b(λ, n)e(nτ)⊗ eλ, THE WEIL REPRESENTATION AND HECKE OPERATORS 11 where b(λ, n) = c(rλ, pn) + pk−1c(λ/r, n/p). Here we understand that c(λ/r, n/p) = 0 if p ∤ n, i.e., if ordp(n) = 0. Proof. Using Lemma 3.6, the formula follows in the same way as in the scalar valued case. � Proposition 4.3. Let p be a prime coprime to N . Let f ∈ Mk,A and denote the Fourier expansion as in (4.7). Then f |k,A T n∈Z+λ2/2 b(λ, n)e(nτ)⊗ eλ, where b(λ, n) = c(pλ, p2n) + gp(A) pk−2(δp(n)− 1)c(λ, n) + p2k−2c(λ/p, n/p2), δp(n) = p, if p | n, 0, if p ∤ n. Moreover, we understand that c(λ/p, n/p2) = 0 if p2 ∤ n. Proof. We omit the proof, which is similar (but easier) to the proof of Theorem 4.10. � Remark 4.4. If |A| = ℓ is a prime, then Mk,A can be identified with the plus or minus subspace of the space Mk(ℓ, χℓ) of scalar valued modular forms for Γ0(ℓ) with nebentypus, see [BB]. Under this identification, the above Hecke operators correspond to the usual Hecke operators onMk(ℓ, χℓ). This follows by comparing the actions on Fourier expansions. For Hecke operators on Mk(ℓ, χℓ) see e.g. [Mi], Lemma 4.5.14. 4.2. The case of odd signature. We now assume that sig(A) is odd. In view of (2.14) we only have to consider modular forms of half integral weight k ∈ Z + 1/2. In this case, the problem arises that both, (4.3) and (4.4), only define projective actions of Q(N). To obtain honest actions, one has to consider appropriate central extensions. We begin by considering the action on C[A]. The composition of the natural reduction Q(N) → Q(N) with the projective Weil representation ρA : Q(N) → U(C[A])/{±1} induces a projective representation ρA : Q(N) −→ U(C[A])/{±1}, g 7→ ρA(g). If we choose for every g ∈ Q(N) a ρ̃A(g) ∈ U(C[A]) such that ρ̃A(g) 7→ ρA(g) under the projection to U(C[A])/{±1}, we obtain a 2-cocycle c with values in {±1} defined by ρ̃A(g1g2) = c(g1, g2)ρ̃A(g1)ρ̃A(g2) for g1, g2 ∈ Q(N). This cocycle gives rise to a central extension Q1(N) = Q(N)× {±1},(4.8) 12 JAN H. BRUINIER AND OLIVER STEIN where the multiplication is defined by (g1, t1)(g2, t2) = (g1g2, t1t2c(g1, g2) Here the cocycle condition c(g1g2, g3)c(g1, g2) = c(g1, g2g3)c(g2, g3) is equivalent to the associativity law for the above multiplication. We obtain a unitary representation Q1(N) −→ U(C[A])(4.9) by putting ρA(g, t) = tρ̃A(g). This left action induces a corresponding right action a |A (g, t) = ρA(g, t)−1a,(4.10) for (g, t) ∈ Q1(N) and a ∈ C[A]. Without loss of generality, for (M, 1) ∈ Γ× {1} ⊂ Q(N) we choose ρ̃A(M, 1) = ρA(M, j(M, τ)).(4.11) Then we have an injective homomorphism Γ̃ −→ Q1(N), (M,± j(M, τ)) 7→ (M, 1,±1).(4.12) Moreover, for a positive integer m coprime to N , we put eλ = em−1λ.(4.13) To define an action on functions we consider the metaplectic group G̃L 2 (R). It acts on functions f : H → C by (f |k (M,φ))(τ) = det(M)k/2φ(τ)−2kf(Mτ)(4.14) for (M,φ) ∈ G̃L 2 (R). In particular, we have an action of H̃(N) ⊂ G̃L 2 (R), where H(N) = {M ∈ G(N); det(M) is a square mod N} ⊂ GL+2 (R)(4.15) is the image of the projection of Q(N) to the first component. Notice that the cocycle of Q(N) given by the choice of ± j(M, τ) for (M, r) ∈ Q(N) and the cocycle c are not isomorphic (this follows for instance from Lemma 4.8 below). However, their restrictions to Γ̃ are isomorphic. To define an action on C[A]-valued functions in weight k we have to consider a combi- nation of the above extensions. We let Q2(N) be the group of tuples (M,φ, r, t), where g = (M,φ) ∈ H̃(N), and r ∈ U(N) with det(M) ≡ r2 (mod N), and t ∈ {±1}. The composition law is defined by (g1, r1, t1)(g2, r2, t2) = (g1g2, r1r2, t1t2c((M1, r1), (M2, r2)) −1)(4.16) for (gi, ri, ti) ∈ Q2(N) and gi = (Mi, φi). We denote by P : Q2 → H̃(N) the natural projection. It has the kernel {(1, 1, r, t) ∈ Q2; r2 ≡ 1 (N)}. THE WEIL REPRESENTATION AND HECKE OPERATORS 13 Over Γ̃ the projection P has the section L : Γ̃ −→ Q2, (M,± j(M, τ)) 7→ (M,± j(M, τ), 1,±1).(4.17) We write ∆ = L(Γ̃).(4.18) We define the Weil representation ρA on Q2(N) by composing the natural map to Q1(N) with the Weil representation on that group. For γ ∈ Γ̃ we have ρA(γ) = ρA(L(γ)).(4.19) The tensor product of the above two actions yields a right action of Q2(N) on functions f : H → C[A]. If we write f = λ∈A fλ ⊗ eλ for the decomposition in components with respect to the standard basis of C[A], the action is given by f |k,A (M,φ, r, t) = fλ |k (M,φ) eλ |A (M, r, t) (4.20) for (M,φ, r, t) ∈ Q2(N). A holomorphic function f : H → C[A] belongs to Mk,A, if and only if f |k,A (M,φ) = f for all (M,φ) ∈ Γ̃, and f is holomorphic at the cusp ∞. We may now define Hecke operators on modular forms of type ρA following Shimura [Sh2]. For α = (M,φ) ∈ H̃(N) and ξ = (α, r, t) ∈ Q2(N) we consider the double coset ∆ξ∆. If ∆ξ∆ = is a left coset decomposition, we define the Hecke operator T (ξ) by f |k,A T (ξ) = det(α)k/2−1 f |k,A ξi(4.21) for f ∈ Mk,A. It is easily seen that T (ξ) is independent of the choice of the coset represen- tatives and defines an endomorphism of Mk,A. We recall the following standard lemma. Lemma 4.5. Let the notation be as above. Then (∆ ∩ ξ−1∆ξ) · γi (γi ∈ ∆) is a disjoint left coset decomposition if and only if ∆ξ∆ = ∆ · ξγi is a disjoint left coset decomposition. � To compute the action of Hecke operators we have to compare the groups ∆ ∩ ξ−1∆ξ and L(Γ̃ ∩ α−1Γ̃α). Let α ∈ H̃(N) and ξ ∈ Q2(N) with P (ξ) = α. For γ ∈ Γ̃ ∩ α−1Γ̃α we L(αγα−1) = ξL(γ)ξ−1 · (1, 1, t(γ))(4.22) 14 JAN H. BRUINIER AND OLIVER STEIN with t(γ) ∈ {±1}. Here t(γ) is independent of the choice of ξ and is determined by the condition ρA(αγα −1) = t(γ)ρA(ξ)ρA(γ)ρA(ξ) −1.(4.23) Hence t defines a group homomorphism t : Γ̃ ∩ α−1Γ̃α −→ {±1}.(4.24) Lemma 4.6. Let the notation be as above. We have L(ker(t)) = ∆∩ ξ−1∆ξ. Moreover, if t is non-trivial, then f |k,A T (ξ) = 0 for all f ∈ Mk,A. Proof. The proof is analogous to Proposition 1.0 in [Sh2] � Lemma 4.7. Let the notation be as above. The homomorphism t is trivial if and only if P gives a bijective map of ∆ξ∆ onto Γ̃αΓ̃. Moreover, when this is the case then ∆ξ∆ = i ∆ξi (where ξi ∈ ∆ξ∆) is a disjoint union if and only if Γ̃αΓ̃ = i Γ̃P (ξi) is a disjoint union. Proof. The proof is analogous to Proposition 1.1 in [Sh2] � Lemma 4.8. Let m,n be positive integers coprime to N , α = ((m 00 n ) , n) ∈ H̃(N), and ξ = (α, r, t) ∈ Q2(N). Define t : Γ̃ ∩ α−1Γ̃α −→ {±1} as in (4.24). Then cτ + d Here the quadratic residue symbol is defined as in [Sh2]. Proof. Define Γ′ = Γ0(m) ∩ Γ0(n) ⊂ Γ. We first notice that Γ̃′ = Γ̃ ∩ α−1Γ̃α. Since t is a homomorphism it suffices to prove the assertion for a set of generators of Γ̃′. It is easily verified that Γ̃′ is generated by T n, Um, and Γ̃′ ∩ Γ̃00(N). For γ = T n ∈ Γ̃′ we have αγα−1 = Tm. We compute t(γ) using (4.23). There is a constant Cξ of modulus 1 such that ρA(ξ)eλ = Cξenr−1λ. Hence ρA(ξ)ρA(T n)ρA(ξ) eλ = C ξ ρA(ξ)ρA(T n)en−1rλ = C−1ξ e(n −1r2λ2/2)ρA(ξ)en−1rλ = e(n−1r2λ2/2)eλ = e(mλ2/2)eλ. Here n−1 in the exponentials means the inverse of n in U(N). On the other hand, we have m)eλ = e(mλ 2/2)eλ. Hence t(T n) = 1 in accordance with the formula. THE WEIL REPRESENTATION AND HECKE OPERATORS 15 For γ = Um ∈ Γ̃′ we have αγα−1 = Un. We find ρA(ξ)ρA(U m)ρA(ξ) eλ = C ξ ρA(ξ)ρA(U m)en−1rλ ρA(ξ) µ,ν∈A −mµ2/2 + (µ, n−1rλ− ν) µ,ν∈A −mµ2/2 + (n−1rµ, λ− ν) µ,ν∈A −nµ2/2 + (µ, λ− ν) This is equal to ρA(U n). Hence t(Um) = 1 in accordance with the formula. To compute t(γ) for γ ∈ Γ̃′ ∩ Γ̃00(N), we use the formula for the Weil representation of Proposition 2.2. Using the definition one easily checks that if γ ∈ Γ̃′ ∩ Γ̃00(N), we have χA(γ) = χA(αγα Therefore ρA(ξ)ρA(γ)ρA(ξ) eλ = C ξ ρA(ξ)ρA(γ)en−1rλ = χA(γ)edλ ρA(αγα −1)eλ. So t(γ) = as claimed. � Proposition 4.9. Let α = (M,φ) ∈ H̃(N) and ξ = (α, r, t) ∈ Q2(N). Then the Hecke operator T (ξ) on Mk,A vanishes identically unless det(M) is a square in Q. Proof. By multiplying with a positive integer we may assume without loss of generality that M has entries in Z. According to the elementary divisor theorem for Γ we may further assume that M = (m 00 n ) with positive integers m,n. So we may assume that Hence the assertion follows from Lemma 4.6 and Lemma 4.8. � We now study the relation between Hecke operators and Fourier coefficients. The fol- lowing theorem is the analogue of Proposition 4.3 in the odd signature case. It can be viewed as a generalization of Theorem 1.7 in [Sh2]. Theorem 4.10. Let p be a prime coprime to N , and put ∈ H̃(N), ξ = (α, p, 1) ∈ Q2(N). 16 JAN H. BRUINIER AND OLIVER STEIN Let f ∈ Mk,A and write f(τ) = n∈Z+λ2/2 c(λ, n)e(nτ)⊗ eλ, f |k,AT (ξ) = n∈Z+λ2/2 b(λ, n)e(nτ)⊗ eλ. b(λ, n) = c(pλ, p2n) + ǫ sig(A)+(−1|A|) |A|2sig(A) pk−3/2 c(λ, n) + p2k−2c(λ/p, n/p2). Here, for an odd integer d we put 1, if d ≡ 1 (mod 4), i, if d ≡ −1 (mod 4). Moreover, we understand that c(λ/p, n/p2) = 0 if p2 ∤ n. Proof. To compute f |k,AT (ξ), we need a set of representatives for ∆\∆ξ∆. In view of Lemma 4.7 and Lemma 4.8, the map Lξ : Γ̃αΓ̃ −→ ∆ξ∆, δ = γαγ′ 7→ Lξ(δ) := L(γ)ξL(γ′) is a bijection (where γ, γ′ ∈ Γ̃). Here we have Lξ(δ) = (δ, p, t), and t = t(δ) is uniquely determined by the condition ρA(δ, p, t) = ρA(γ)ρA(ξ)ρA(γ ′).(4.25) We have the disjoint left coset decomposition Γ̃αΓ̃ = Γ̃α ∪ h (p)∗ Γ̃βh ∪ b (p2) Γ̃γb, where Nsτ + p −Nps 1 −Npsτ + 1 −N p2 −Nτ + p2 1 −Nt N p2d Nτ + p2d and r, s ∈ Z are chosen such that pr − N2hs = 1, and d, t ∈ Z are chosen such that p2d+N2t = 1. Consequently, we obtain the disjoint left coset decomposition ∆ξ∆ = ∆ξ ∪ h (p)∗ ∆Lξ(βh) ∪ b (p2) ∆Lξ(γb).(4.26) THE WEIL REPRESENTATION AND HECKE OPERATORS 17 The action of Lξ(βh) and Lξ(γb) in the Weil representation can be computed by means of (4.25) and the above decompositions. Using Proposition 2.2 and the fact that ρA(ξ)eλ = ep−1λ by (4.13), we find that ρA(Lξ(γb))eλ = χA −N p2 −Nτ + p2 1 −Nt N p2d Nτ + p2d e(bλ2/2)epλ. Since d is a square modulo N2, it is a square modulo the square-free part of |A| and modulo 8. Therefore, a quick calculation shows that the character values are 1. Consequently, ρA(Lξ(γb))eλ = e(bλ 2/2)epλ.(4.27) In the same way we obtain ρA(Lξ(βh))eλ = ρA Nsτ + p ρA(ξ)eλ Nsτ + p ep−1λ Nsτ + p 1−(−1|A|)−sig(A) |A|2sig(A) Here in the last line we have used the explicit formula for χA. Since −N2hs ≡ 1 (mod p), we find ρA(Lξ(βh))eλ = ǫ 1−(−1|A|)−sig(A) |A|2sig(A) eλ.(4.28) Now we can compute the Fourier expansion of f |k,AT (ξ). We have f |k,AT (ξ) = pk−2f |k,Aξ + pk−2 h (p)∗ f |k,ALξ(βh) + pk−2 b (p2) f |k,ALξ(γb).(4.29) For the first summand we find pk−2f |k,Aξ = pk−2 fλ |k α eλ |A ξ = p2k−2 2τ)⊗ epλ. 18 JAN H. BRUINIER AND OLIVER STEIN For the second summand in (4.29) we get h (p)∗ f |k,ALξ(βh) = pk−2 h (p)∗ fλ |k βh eλ |A Lξ(βh) sig(A)+(−1|A|)−1 |A|2sig(A) h (p)∗ fλ(τ +Nh/p)⊗ eλ. By means of the formula for the Gauss sum h(p)∗ e(kh/p) = p we obtain h (p)∗ fλ(τ +Nh/p) = n∈Z+λ2/2 c(λ, n)e(nτ), and therefore h (p)∗ f |k,ALξ(βh) sig(A)+(−1|A|) |A|2sig(A) pk−3/2 n∈Z+λ2/2 c(λ, n)e(nτ)⊗ eλ. Finally, for the third summand in (4.29) we get b (p2) f |k,ALξ(γb) = pk−2 b (p2) fλ |k γb eλ |A Lξ(γb) = p−2 b (p2) e(−b(p−1λ)2/2)fλ(τ/p2 + b/p2)⊗ ep−1λ n∈Z+λ2/2 c(pλ, p2n)e(nτ)⊗ eλ. This concludes the proof of the theorem. � Remark 4.11. Let m be a positive integer. Let L be the lattice Z with the quadratic form x 7→ −mx2. Then L′ = 1 Z, andMk,L′/L is isomorphic to the space Jk+1/2,m of Jacobi forms of weight k+1/2 and index m (cf. [EZ], Theorem 5.1). Under this isomorphism the Hecke operator T (ξ) of Theorem 4.10 corresponds to the Hecke operator Tp on Jk+1/2,m defined in [EZ] §4 (3). This follows from Theorem 4.10 and [EZ], Theorem 4.5, by comparing the actions on Fourier expansions. Notice that we have in this particular case sig(A)+(−1|A|) |A|2sig(A) For the lattice L = Z with the quadratic form x 7→ mx2 the space Mk,L′/L is isomorphic to the space Jskewk+1/2,m of skew holomorphic Jacobi forms of weight k + 1/2 and index m defined in [Sk]. Again, the Hecke operators of Theorem 4.10 correspond to the usual Hecke operators on skew-holomorphic Jacobi forms. THE WEIL REPRESENTATION AND HECKE OPERATORS 19 4.3. A Hecke algebra on vector valued modular forms. Using the double coset actions of the previous section, we may define for every positive integer m coprime to N a Hecke operator T (m2)∗ : Mk,A → Mk,A, by f 7→ f |k,A T (m2)∗ = f |k,A T , if sig(A) is even, f |k,A T , 1, m, 1 , if sig(A) is odd. (4.30) In the case of even signature (that is integral weight), the operator T (m2)∗ differs from the usual Hecke operator T (m2) which is given by the sum of double cosets consisting of all integral matrices of determinant m2. This is the reason for our notation. In the case of odd signature (that is half-integral weight), the operator T (m2)∗ is analogous to the Hecke operator in [Sh2] on scalar valued modular forms. Theorem 4.12. The Hecke operators T (m2)∗ (for m coprime to N) generate a commu- tative subalgebra of End(Mk,A), which is actually already generated by the T (p 2)∗ for p prime and coprime to N . The operators T (m2)∗ take cusp forms to cusp forms and are self-adjoint with respect to the Petersson scalar product. Proof. Using the actions (4.5) and (4.20), this follows in the usual way from the properties of the abstract Hecke algebra of the pair (Q(N),Γ), respectively (Q2(N),∆). � 5. The Weil representation on double cosets We now want to define Hecke operators T (m2)∗ as in Section 4.3 for all positive integers m, not necessarily coprime to N . If m and N are not coprime, then the reduction of does not belong to GL2(Z/NZ). So we cannot use the results of the previous sections. However, it is still possible to extend the Weil representation ρA to the corresponding double coset in a compatible way, as we will see. Let m be a positive integer and α = ∈ G̃L 2 (R). We define a right action on C[A] by eλ |A α = emλ.(5.1) To lighten the notation, we will frequenty drop the subscript from the slash operator. Comparing with (3.6) and (4.13), we see that (5.1) is compatible with our earlier definition in the case that (m,N) = 1. Moreover, if δ = γαγ′ ∈ Γ̃αΓ̃, we put eλ | δ = eλ | γ | α | γ′.(5.2) We now show that this right action is well defined, that is, independent of the decomposition of δ. Proposition 5.1. Let δ = γαγ′ = γ1αγ 1 ∈ Γ̃αΓ̃ (where γ, γ′, γ1, γ′1 ∈ Γ̃). Then eλ | γ | α | γ′ = eλ | γ1 | α | γ′1. Proof. First, one easily shows that it suffices to prove the proposition in the case that γ′ = γ1 = 1. So we have δ = γα = αγ 1 and need to show that eλ | γ | α = eλ | α | γ′1.(5.3) 20 JAN H. BRUINIER AND OLIVER STEIN If we write γ = ( a bc d ) ,± cτ + d , then γ′1 = a b/m2 m2c d m2cτ + d . In particular, γ ∈ Γ̃0(m2) and γ′1 ∈ Γ̃0(m2). It suffices to prove (5.3) for γ in a set of generators of Γ̃0(m2). It is easily seen that Γ̃0(m2) is generated by Γ̃0(m2) ∩ Γ̃00(N), Tm , and U . For γ ∈ Γ̃0(m2) ∩ Γ̃00(N) the identity (5.3) immediately follows from Proposition 2.2. For γ = Tm it is easily verified as well. We now consider (5.3) for γ = U . Using Lemma 2.3, we see that the left hand side of (5.3) is equal to eλ | U | α = µ,ν∈A µ2/2 + (µ, λ− ν) emν . Using (2.2), in the sum over ν we write ν = ν ′/m + ν ′′ where ν ′ ∈ Am and ν ′′ ∈ Am. We obtain eλ | U | α = ν′∈Am ν′′∈Am µ2/2 + (µ, λ− ν ′/m− ν ′′) The sum over ν ′′ is equal to |Am| if µ ∈ Am and 0 otherwise. Hence eλ | U | α = µ2/2 + (µ, λ− ν/m) (mµ)2/2 + (mµ, λ− ν/m) On the other hand, the right hand side of (5.3) is equal to eλ | α | Um µ,ν∈A m2µ2/2 + (µ,mλ− ν) |A||Am| µ,ν∈A µ′∈Am m2µ2/2 + (µ+ µ′, mλ− ν) The sum over µ′ is equal to |Am| if ν ∈ Am and 0 otherwise. Hence eλ | α | Um m2µ2/2 + (µ,mλ− ν) (mµ)2/2 + (mµ, λ− ν/m) This concludes the proof of the proposition. � Lemma 5.2. Let δ ∈ Γ̃αΓ̃, and let γ, γ′ ∈ Γ̃. Then eλ | (γδγ′) = eλ | γ | δ | γ′. Proof. This follows immediately from the definition and Proposition 5.1. � THE WEIL REPRESENTATION AND HECKE OPERATORS 21 The element β = ∈ G̃L 2 (R) belongs to the double coset Γ̃αΓ̃. The following Proposition gives its action on C[A]. Proposition 5.3. We have eλ | β = eµ.(5.4) Moreover, for the standard scalar product on C[A] we have 〈a | α, b〉 = 〈a, b | β〉, a, b ∈ C[A].(5.5) Proof. The first assertion follows from the fact that β = SαS−1 and Lemma 5.2. The second assertion is verified by a straightforward computation. � Proposition 5.4. Let m,n be coprime positive integers, and put α = . Then for g ∈ Γ̃αΓ̃ and h ∈ Γ̃βΓ̃ we have eλ | g | h = eλ | (gh).(5.6) Proof. We write g = γαγ′ and h = δβδ′ with γ, γ′, δ, δ′ ∈ Γ̃. Since (m,n) = 1, a simple argument using the elementary divisor theorem shows that gh = ǫαβǫ′ for suitable ǫ, ǫ′ ∈ Γ̃. In view of Lemma 5.2 it suffices to prove the assertion in the case that γ = δ′ = 1. As a second reduction step, we now show that we may in addition assume that ǫ′ = 1. In fact, the identity gh = αγ′δβ = ǫαβǫ′ implies that δβǫ′−1 = γ′−1α−1ǫαβ. The matrix component of the left hand side has integral entries, hence the same is true for the right had side. Using the coprimality of m and n we may infer that δ̃ := γ′−1α−1ǫα belongs to Γ̃. We obtain δβ = δ̃βǫ′, (αγ′)(δ̃β) = ǫαβ. From the assertion in the ǫ′ = 1 case we get eλ | (αγ′) | (δ̃β) = eλ | (ǫαβ), which implies the assertion for arbitrary ǫ′. Finally we need to prove the claim in the case that γ = δ′ = ǫ′ = 1. So g = αγ′, h = δβ gh = αγ′δβ = ǫαβ. Since αγ′δ = ǫα, Lemma 5.2 implies that eλ | (αγ′δ) = eλ | (ǫα), eλ | (αγ′δ) | β = eλ | (ǫα) | β. Now the claim follows from Lemma 5.2 and the fact that eλ | α | β = eλ | (αβ). � 22 JAN H. BRUINIER AND OLIVER STEIN Definition 5.5. Let m be a positive integer and α = ∈ G̃L 2 (R). Let Γ̃ · α · Γ̃ = Γ̃ · δi be a disjoint left coset decomposition. We define the Hecke operator T (m2)∗ on modular forms f ∈ Mk,A by f 7→ f |k,A T (m2)∗ = mk−2 f |k,A δi = mk−2 fλ |k δi eλ |A δi Lemma 5.2 implies that the definition does not depend on the choice of the coset repre- sentatives. Notice that for m coprime to N , this definition agrees with the earlier definition in Section 4.3. Theorem 5.6. For any positive integer m, the Hecke operator T (m2)∗ is a linear operator on Mk,A taking cusp forms to cusp forms. It is self adjoint with respect to the Petersson scalar product. Moreover, if m,n are coprime, then T (m2)∗T (n2)∗ = T (m2n2)∗. Proof. 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Mathematisches Institut, Universität zu Köln, Weyertal 86–90, D-50931 Köln, Germany E-mail address : bruinier@math.uni-koeln.de E-mail address : ostein@math.uni-koeln.de 1. Introduction 2. Discriminant forms and the Weil representation 3. Extending the Weil representation 3.1. The case of even signature 3.2. The case of odd signature 4. Hecke operators on vector valued modular forms 4.1. The case of even signature 4.2. The case of odd signature 4.3. A Hecke algebra on vector valued modular forms 5. The Weil representation on double cosets References

We define Hecke operators on vector valued modular forms transforming with the Weil representation associated to a discriminant form. We describe the properties of the corresponding algebra of Hecke operators and study the action on modular forms.

Introduction Hecke operators are a fundamental tool in the study of modular forms. They can be used to obtain information on the arithmetic nature of the Fourier coefficients. They are vital for the definition of L-functions associated to modular forms and for understanding their properties. The theory of Hecke operators is well developed for scalar valued modular forms [Sh1]. In many recent works, vector valued modular forms associated to the Weil representation play an important role, see e.g. [Bo1], [Bo2], [Br], [McG], [Sch]. For instance, Borcherds uses them to provide a elegant description of the Fourier expansion of various theta liftings. The purpose of the present paper is to develop the foundations of a Hecke theory for such vector valued modular forms. The results can be used to associate an L-function to a vector valued modular form, essentially the standard L-function. We now describe the content of this paper in more detail. Let L be a non-degenerate even lattice of type (b+, b−) and level N . The modulo 1 reduction of the quadratic form on the dual lattice L′ defines a Q/Z-valued quadratic form on the discriminant group A = L′/L. To simplify the exposition, we assume throughout the introduction that the signature sig(L) = b+ − b− of L is even. In the body of the paper, both, odd and even signature is treated. The Weil representation associated to the discriminant form A is a unitary representation of Γ = SL2(Z) on the group ring C[A], ρA : Γ −→ U(C([A])), defined by (2.3), (2.4). It factors through the finite quotient S(N) := SL2(Z/NZ) ∼= Γ/Γ(N). Date: October 27, 2018. 2000 Mathematics Subject Classification. 11F27, 11F25. http://arxiv.org/abs/0704.1868v1 2 JAN H. BRUINIER AND OLIVER STEIN Let k ∈ Z. A holomorphic function f : H → C[A] is called a modular form of weight k and type ρA for the group Γ, if f(Mτ) = (cτ + d)kρA(M)f(τ) for all M = ( a bc d ) ∈ Γ, and f is holomorphic at the cusp ∞. We denote the vector space of such holomorphic modular forms by Mk,A, and write Sk,A for the subspace of cusp forms. In order to define Hecke operators on vector valued modular forms of type ρA, we need to extend the representation ρA to a sufficiently large subgroup of GL 2 (Q). A natural starting point is to try to extend ρA, viewed as a representation of S(N), to a representation of G(N) := GL2(Z/NZ). However, it was observed by E. Freitag that such an extension does not exist in general, see Example 3.1. Here we consider the subgroup {M ∈ G(N); det(M) ≡ � (mod N)} of matrices whose determinant is a square modulo N . It has the extension Q(N) = {(M, r) ∈ G(N)× U(N); det(M) ≡ r2 (mod N)}, where U(N) denotes the unit group of Z/NZ. The group Q(N) is isomorphic to S(N) × U(N). Consequently, we may extend the Weil representation to Q(N) by taking the tensor product of ρA on S(N) and a suitable character on U(N), see Proposition 3.3. If M is an element of G(N) whose determinant is a square modulo N , and r, r′ ∈ U(N) with det(M) ≡ r2 ≡ r′2 (mod N), then (M, r) and (M, r′) both belong to Q(N). We prove that the action of ρA(M, r) and ρA(M, r ′) on C[A] differ only by the action of an element of the orthogonal group O(A), see Proposition 3.5. This extension of the Weil representation can be used to define a Hecke operator T (M, r) on Mk,A for every pair (M, r), where M ∈ M2(Z) and r ∈ U(N) with det(M) ≡ r2 (mod N). We compute the action of these operators on the Fourier expansion of a modular form (see Section 4.1). They generalize the classical Hecke operators on scalar valued modular forms and Jacobi forms (see e.g. Remark 4.4 and Remark 4.11). In particular, for every positive integer m coprime to N we obtain a Hecke operator T (m2)∗ := T on Mk,A. These operators generate a commutative subalgebra of End(Mk,A), which is actu- ally already generated by the T (p2)∗ for p prime and coprime to N . The operators T (m2)∗ take cusp forms to cusp forms and are self-adjoint with respect to the Petersson scalar product (see Theorem 4.12). In particular, Sk,A has a basis of simultaneous eigenforms of all T (m2)∗ with (m,N) = 1. In Section 5 we extend the definition of the Hecke operators T (m2)∗ to all positive integers m, not necessarily coprime to N . This is done by defining the right-action on C[A] of a matrix α = by the same formula as in the case where m is coprime to N . Notice, that the corresponding linear map C[A] −→ C[A], eλ 7→ eλ |A α = emλ THE WEIL REPRESENTATION AND HECKE OPERATORS 3 is neither surjective nor injective in general. However, it still can be used to obtain an “action” of the double coset ΓαΓ, see Proposition 5.1 and Lemma 5.2. This suffices to define a corresponding Hecke operator T (m2)∗ on Mk,A, which is consistent with our earlier definition when m is coprime to N . For any positive integer m, the Hecke operator T (m2)∗ is self adjoint with respect to the Petersson scalar product. Moreover, if m and n are coprime, then T (m2)∗T (n2)∗ = T (m2n2)∗, see Theorem 5.6. Observe that for a prime p dividing N the local Hecke algebra, that is, the subalgebra of End(Mk,A) generated by the T (p 2ν)∗, is considerably more complicated than in the case where p is coprime to N . For instance, it is commutative if p is coprime to N , but in general non-commutative if p divides N . Let S be a finite set of primes and let NS be the product of the primes in S. Let f ∈ Sk,A be a common eigenform of all T (m2)∗ with (m,NS) = 1, so f |k,A T (m2)∗ = λm(f)f. We can use the above results to define an L-function associated to f by putting LS(s, f) = (m,NS)=1 λm(f)m It is easily seen that LS(s, f) converges for ℜ(s) sufficiently large. By Theorem 5.6, this L-function has an Euler product expansion. According to [Bö], it should be viewed as the standard L-function of f . It would be interesting to study the analytic properties of LS(s, f) in more detail. This could possibly be done by using a variant of the doubling method (see [Bö], [Ga], [PSR]) involving a Siegel Eisenstein series of genus 2 associated to the Weil representation of Sp(2,Z) on C[A2]. Moreover, it would be very interesting to develop a theory of new forms for the space Mk,A. One could try to associate an irreducible automorphic representation to a vector valued new form and study the properties of the resulting map. If the signature of L is odd, one can carry over the above results. However, one has to work with the metaplectic cover of Γ and with similar {±1}-extensions of S(N), G(N), and Q(N). In this case Mk,A vanishes unless k is half-integral. Following the argument of Shimura [Sh2], we show that the Hecke operator T (M, r) vanishes identically unless det(M) is the square of a rational number, see Proposition 4.9. The computation of the action of the Hecke operators on modular forms is more involved than in the even signature (i.e. integral weight) case, see Theorem 4.10. We thank E. Freitag for many valuable discussions on this paper. Moreover, we thank J. Funke for several useful comments. 2. Discriminant forms and the Weil representation Here we briefly summarize some facts on lattices, discriminant forms, and the Weil representation. See also [Bo1], [Bo2], [Br]. 4 JAN H. BRUINIER AND OLIVER STEIN Let L be a non-degenerate even lattice of type (b+, b−). We denote the bilinear form on L by (·, ·) and the associated quadratic form by x 7→ 1 x2 = 1 (x, x). We let sig(L) = b+−b− be the signature of L. We write L′ for the dual lattice of L, and denote by N the level of L, that is, the smallest positive integer such that N x2 ∈ Z for all x ∈ L′. The finite abelian group L′/L is called the discriminant group of L. Its order is equal to the absolute value of the Gram determinant of L. Recall that a discriminant form is a finite abelian group A together with a Q/Z-valued non-degenerate quadratic form x 7→ 1 x2, for x ∈ A (see [Ni]). If L is a non-degenerate even lattice then L′/L is a discriminant form where the quadratic form is given by the mod 1 reduction of the quadratic form on L′. Conversely, every discriminant form can be obtained in this way. The quadratic form on L′/L determines the signature of L modulo 8 by Milgram’s formula (see [MH] Appendix 4): (2.1) λ∈L′/L e(λ2/2) = |L′/L|e(sig(L)/8). Here and throughout we abbreviate e(z) = e2πiz for z ∈ C. We define the signature sig(A) ∈ Z/8Z of a discriminant form A to be the signature of any even lattice with that discriminant form. If A is a discriminant form, then we write An for the subgroup of elements that are n-th powers of elements of A. Moreover, we write An for the subgroup of elements of A whose order divides n. We have an exact sequence 0 −→ An −→ A −→ An −→ 0,(2.2) and An is the orthogonal complement of An. Let H = {τ ∈ C; ℑ(τ) > 0} by the complex upper half plane. We write G̃L 2 (R) for the metaplectic two-fold cover of GL+2 (R). The elements of this group are pairs (M,φ(τ)) where M = ( a bc d ) ∈ GL 2 (R) and φ : H → C is a holomorphic function with φ(τ)2 = cτ +d. The multiplication is defined by (M,φ(τ))(M ′, φ′(τ)) = (MM ′, φ(M ′τ)φ′(τ)). For g = (M,φ) ∈ G̃L 2 (R), we put det(g) = det(M). Moreover, if G is a subset of GL+2 (R), we write G̃ for its inverse image under the covering map. Throughout we write Γ = SL2(Z) for the full modular group. It is well known that the integral metaplectic group Γ̃ is generated by T = (( 1 10 1 ) , 1), and S = (( 1 0 ) , τ ). One has the relations S2 = (ST )3 = Z, where Z = ((−1 0 is the standard generator of the center of Γ̃. We now recall the Weil representation associated with a discriminant form A (see also [Bo1], [Bo2]). It is a representation of Γ̃ on the group algebra C[A]. We denote the standard basis elements of C[A] by eλ, λ ∈ A, and write 〈·, ·〉 for the standard scalar product (antilinear in the second entry) such that 〈eλ, eµ〉 = δλ,µ. The Weil representation ρA associated with the discriminant form A is the unitary representation of Γ̃ on the group THE WEIL REPRESENTATION AND HECKE OPERATORS 5 algebra C[A] defined by ρA(T )(eλ) = e(λ 2/2)eλ,(2.3) ρA(S)(eλ) = e(− sig(A)/8)√ e(−(λ, µ))eµ.(2.4) Note that ρA(Z)(eλ) = e(− sig(A)/4)e−λ.(2.5) The orthogonal group O(A) also acts on C[A] by ρA(h)(eλ) = eh−1λ(2.6) for h ∈ O(A), and the actions of Γ̃ and O(A) commute. If the signature of A is even, then (2.5) implies that Z2 acts trivially. Hence, the Weil representation factors through Γ. Moreover, it is trivial on the principal congruence subgroup Γ(N), where N is the level of A, i.e., the level of any even lattice L with L′/L = A (see e.g. [Eb], Chapter 3, Theorem 3.2). Therefore, ρA factors through the finite group S(N) := SL2(Z/NZ) ∼= Γ/Γ(N).(2.7) If the signature of A is odd, we notice that the level of A must be divisible by 4. This follows from the oddity formula ([CS] p. 383 (30)) which implies that A contains odd 2-adic Jordan components. On Γ(4) the metaplectic cover has the section s : Γ(4) −→ Γ̃(4), cτ + d given by the theta multiplier system. Here · denotes the principal branch of the holomor- phic square root. The same argument as at the end of the proof of Theorem 5.4 in [Bo2] implies that ρA is trivial on s(Γ(N)) and factors through the central extension of S(N) by {±1} given by S1(N) := Γ̃/s(Γ(N)).(2.8) We will also need the action of ρA on diagonal matrices in S(N). Following [McG], for integers a, d coprime to N such that ad ≡ 1 (mod N), we put Rd := ST dS−1T aST d.(2.9) It is easily checked that Rd = (M,φ) where M ≡ ( a 00 d ) (mod N). Lemma 2.1. (See [McG] Lemma 4.6.) For a, d as above we have ρA(Rd)eλ = gd(A) edλ.(2.10) Here gd(A) denotes the Gauss sum gd(A) = e(dλ2/2)(2.11) 6 JAN H. BRUINIER AND OLIVER STEIN and g(A) = g1(A). � Notice that by Milgram’s formula we have g(A) = |A|e(sig(A)/8). Moreover, one easily checks that |gd(A)| = |A|. If r ∈ Z is coprime to N , then we have gdr2(A) = gd(A). In particular, gd(A) = ga(A). Finally, Lemma 2.1 and the fact that ρA is a representation imply the relation gdr(A)g(A) gd(A)gr(A) 1, if sig(A) is even, ±1, if sig(A) is odd. (2.12) The following more general formula was given by Borcherds. Proposition 2.2. (See [Bo2] Theorem 5.4.) Let g = (( a bc d ) , cτ + d) ∈ Γ̃, and suppose that b and c are divisible by N . Then ρA(g)eλ = χA(g)edλ.(2.13) Here χA denotes the character of Γ̃0(N) defined in [Bo2] Theorem 5.4. � Lemma 2.3. Let U = ( 1 01 1 ) , τ + 1 ∈ Γ̃. The action of Um is given by m)eλ = µ,ν∈A −mµ2/2 + (µ, λ− ν) Proof. Since Um = ST−mS−1, this follows immediately from (2.3) and (2.4). � In many recent works vector valued modular forms associated to the Weil representation are considered (see e.g. [Bo1], [Bo2], [Br], [McG], [Sch]). Let k ∈ 1 Z, and let Γ′ ⊂ Γ̃ be a subgroup of finite index. A holomorphic function f : H → C[A] is called a modular form of weight k and type ρA for the group Γ ′, if f(Mτ) = φ(τ)2kρA(M,φ)f(τ) for all (M,φ) ∈ Γ′, and f is holomorphic at the cusps of Γ′. We denote the C-vector space of such holomorphic modular forms by Mk,A(Γ ′). Moreover, for the full modular group we put Mk,A = Mk,A(Γ̃). Formula (2.5) implies that Mk,A = {0} unless 2k ≡ sig(A) mod 2.(2.14) Recall that for f, g ∈ Mk,A(Γ′) the Petersson scalar product is defined by (f, g) = [Γ̃ : Γ′] 〈f(τ), g(τ)〉 yk dx dy .(2.15) Here x denotes the real part and y the imaginary part of τ ∈ H. The Petersson scalar product converges when f ⊗ g is a cusp form. THE WEIL REPRESENTATION AND HECKE OPERATORS 7 3. Extending the Weil representation In the classical theory of scalar valued modular forms Hecke operators play an impor- tant role (see e.g. [Sh1]). It is natural to try to define Hecke operators on vector valued modular forms of type ρA as well. This requires the extension of the representation ρA to a representation (of a sufficiently large subgroup) of G̃L 2 (Q). However, it is not obvious how this can be done. A natural starting point is to try to extend ρA, viewed as a representation of S(N) (respectively S1(N)), to a representation of (a double cover of) GL2(Z/NZ). However, it was observed by E. Freitag that such an extension does not exist in general. This follows from the following example. Example 3.1. Let d ≡ 1 (mod 4) be an integer such that p := |d| is a prime. We consider the ring of integers O in the quadratic field Q( d) of discriminant d. Together with the norm form, it is an even lattice of type (1, 1) if d > 0, and of type (2, 0) if d < 0. The dual lattice is 1√ O, the inverse of the different, and the corresponding discriminant form A can be identified with the finite field Fp together with the quadratic form x 7→ −1dx 2. The associated Weil representation ρA is a p-dimensional representation of S(p) = SL2(Fp) on C[A]. The action of the orthogonal group O(A) = {±1} splits C[A] into two S(p)-invariant subspaces C[A]+ = span{eλ + e−λ; λ ∈ A}, C[A]− = span{eλ − e−λ; λ ∈ A}. They have dimension p+1 , and p−1 , respectively. It follows from [NW], Theorem 4, that the corresponding representations of S(p) are irreducible. On the other hand, the character table of GL2(Fp) is well known, see e.g. [FH] §5.2. It has p− 1 one-dimensional representations, p− 1 irreducible p-dimensional representations, (p− 1)(p− 2)/2 irreducible (p+1)-dimensional representations, and (p2 − p)/2 irreducible (p− 1)-dimsional representations. Now assume that p ≥ 5 and that ρA has an extension ρ̃A to a representation of GL2(Fp). Because of the irreducibility of C[A]±, such an extension would have to be a p-dimensional irreducible representation of GL2(Fp). But these representations remain irreducible under restriction to SL2(Fp), see [FH], p.72 (2). We obtain a contradiction. Remark 3.2. In [McG], McGraw continues ρA to an action of GL2(Z/NZ). However, this action is not C-linear, causing serious difficulties when one tries to define Hecke operators. Here we consider a different group extension Q(N) of S(N) and show that ρA can be continued to a representation of Q(N). Together with the considerations of Section 5 this will suffice for many applications of Hecke operators; for instance, to define the standard L-function of a modular form of type ρA. Let A be a discriminant form as in the previous section, and let N be the level of A. We denote by U(N) the unit group of Z/NZ. We briefly write G(N) = GL2(Z/NZ) for the general linear group modulo N . The determinant homomorphism G(N) → U(N) gives 8 JAN H. BRUINIER AND OLIVER STEIN rise to the exact sequence 1 −→ S(N) −→ G(N) −→ U(N) −→ 1.(3.1) This sequence splits and G(N) can be viewed as a semidirect product of S(N) and U(N). 3.1. The case of even signature. Throughout this subsection we assume that sig(A) is even. Let Q(N) be the group Q(N) = {(M, r) ∈ G(N)× U(N); det(M) ≡ r2 (mod N)}(3.2) with the product defined component-wise. We have an exact sequence 1 −→ S(N) −→ Q(N) −→ U(N) −→ 1,(3.3) where S(N) → Q(N) is given by M 7→ (M, 1), and Q(N) → U(N) is given by (M, r) 7→ r. The latter homomorphism has the section U(N) −→ Q(N), r 7→ .(3.4) For (M, r) ∈ Q(N) the assignment (M, r) 7→ (M ( r 00 r ) , r) defines an isomorphism Q(N) ∼= S(N)× U(N). We consider the action of S(N) on C[A] by the Weil representation ρA. In view of (2.12), the assignment r 7→ g(A) gr(A) defines a character of U(N). We define a unitary representation of U(N) on C[A] by putting ρA(r)eλ = ρA gr(A) eλ(3.5) for r ∈ U(N). Proposition 3.3. The Weil representation ρA of the group S(N) extends to a unitary representation of Q(N) by (3.5). Proof. Since Q(N) = S(N) × U(N) we only have to check that the actions of S(N) and U(N) commute. This is obvious. � Remark 3.4. Clearly we could take any character of U(N) to define the action of the Weil representation on U(N). The above choice is compatible with the definition of the Weil representation on double cosets in Section 5. Moreover, it is compatible with the usual Hecke operators on scalar valued modular forms, see Remark 4.4 and Remark 4.11. The following proposition shows that the first entry of an element (M, r) ∈ Q(N) gives the “essential contribution” to the Weil representation. If we fix M then for the different choices of r ∈ U(N) the Weil representation ρA(M, r) differs by the action of an element of the orthogonal group O(A). Proposition 3.5. Let (M, r1), (M, r2) ∈ Q(N). Then h : A → A, λ 7→ r1r−12 λ is an orthogonal transformation in O(A), and ρA(M, r1)eλ = ρA(M, r2)ρA(h)eλ = ρA(M, r2)eh−1λ. THE WEIL REPRESENTATION AND HECKE OPERATORS 9 Proof. We have (M, r2) −1(M, r1) = (1, r1r 2 ), and t = r1r 2 ∈ U(N) has the property that t2 ≡ 1 (mod N). According to (2.10) and (3.5), the action of (1, t) is given by ρA(1, t)eλ = et−1λ. Since (tλ, tµ) = (t2λ, µ) = (λ, µ) for all λ, µ ∈ A, multiplication by t−1 is an orthogonal transformation. � Lemma 3.6. We have eλ = er−1λ,(3.6) eλ = erλ.(3.7) Proof. In Q(N) we have = (Rr−1 , 1) Therefore the fist formula follows from Lemma 2.1 and (3.5). The second formula follows similarly. � 3.2. The case of odd signature. Throughout this subsection we assume that sig(A) is odd. In this case the argument of the previous section only shows that the Weil repre- sentation extends to a projective representation of Q(N). More precisely, it extends to a group homomorphism Q(N) −→ U(C[A])/{±1}. This projective representation gives rise to a 2-cocycle of Q(N) with values in {±1}. The cocycle defines a central extension Q1(N) of Q(N) by {±1}. The group S1(N), see (2.8), can be identified with a subgroup of Q1(N) (see Section 4.2 for more details). Proposition 3.7. The assignment (3.5) defines an extension of the Weil representation to a projective representation of Q(N). It lifts to a unitary representation of Q1(N). 4. Hecke operators on vector valued modular forms We now use the results of Section 3 to define Hecke operators on vector valued modular forms of type ρA. We consider the groups G(N) = {M ∈ GL+2 (Q); ∃n ∈ Z with (n,N) = 1 such that nM ∈ M2(Z)(4.1) and (det(nM), N) = 1}, Q(N) = {(M, r) ∈ G(N)× U(N); det(M) ≡ r2 (mod N)}.(4.2) We view Γ as a subgroup of Q(N) by the embedding M 7→ (M, 1). We are interested in the action of the Hecke algebra of the pair (Q(N),Γ) on modular forms of type ρA and weight k. We have to distinguish the cases whether sig(A) is even or odd. 10 JAN H. BRUINIER AND OLIVER STEIN 4.1. The case of even signature. The composition of the reduction map Q(N) → Q(N) with the Weil representation ρA : Q(N) → U(C[A]) induces a unitary representation of Q(N) on C[A], which we will denote by ρA as well. This left action induces a corresponding right action by a |A (M, r) = ρA(M, r)−1a.(4.3) In view of (2.14) we only have to consider modular forms of integral weight k ∈ Z. Let f be a complex valued function on H. For M ∈ GL+2 (R) the Petersson slash operator is defined by (f |k M)(τ) = det(M)k/2j(M, τ)−kf(Mτ).(4.4) This defines a right action of G(N) on functions H → C. The center acts by multiplication with ±1. If f : H → C[A] is a function we write f = λ∈A fλ ⊗ eλ for its decomposition in components with respect to the standard basis of C[A]. The tensor product of the above two actions yields a right action of Q(N) on such functions, denoted by f |k,A (M, r) = fλ |k M eλ |A (M, r) .(4.5) Notice that a holomorphic function f : H → C[A] belongs to Mk,A, if and only if f |k,A M = f for all M ∈ Γ, and f is holomorphic at the cusp ∞. We consider the Hecke algebra of the pair (Q(N),Γ) in the sense of Shimura [Sh1]. If (M, r) ∈ Q(N), the corresponding double coset decomposes into a finite union of left cosets Γ · (M, r) · Γ = γ∈Γ\ΓMΓ Γ · (γ, r). Definition 4.1. For (M, r) ∈ Q(N) we define the corresponding Hecke operator T (M, r) on Mk,A by f |k,A T (M, r) = det(M)k/2−1 γ∈Γ\ΓMΓ f |k,A (γ, r), f ∈ Mk,A.(4.6) The usual argument now shows that f |k,A T (M, r) ∈ Mk,A. Hence T (M, r) defines an endomorphism of Mk,A. A modular form f ∈ Mk,A has a Fourier expansion of the form f(τ) = n∈Z+λ2/2 c(λ, n)e(nτ)⊗ eλ.(4.7) Theorem 4.2. Let p be a prime which is a square modulo N , and assume that r ∈ U(N) with p ≡ r2 (mod N). Let f ∈ Mk,A and denote the Fourier expansion as in (4.7). Then f |k,A T n∈Z+λ2/2 b(λ, n)e(nτ)⊗ eλ, THE WEIL REPRESENTATION AND HECKE OPERATORS 11 where b(λ, n) = c(rλ, pn) + pk−1c(λ/r, n/p). Here we understand that c(λ/r, n/p) = 0 if p ∤ n, i.e., if ordp(n) = 0. Proof. Using Lemma 3.6, the formula follows in the same way as in the scalar valued case. � Proposition 4.3. Let p be a prime coprime to N . Let f ∈ Mk,A and denote the Fourier expansion as in (4.7). Then f |k,A T n∈Z+λ2/2 b(λ, n)e(nτ)⊗ eλ, where b(λ, n) = c(pλ, p2n) + gp(A) pk−2(δp(n)− 1)c(λ, n) + p2k−2c(λ/p, n/p2), δp(n) = p, if p | n, 0, if p ∤ n. Moreover, we understand that c(λ/p, n/p2) = 0 if p2 ∤ n. Proof. We omit the proof, which is similar (but easier) to the proof of Theorem 4.10. � Remark 4.4. If |A| = ℓ is a prime, then Mk,A can be identified with the plus or minus subspace of the space Mk(ℓ, χℓ) of scalar valued modular forms for Γ0(ℓ) with nebentypus, see [BB]. Under this identification, the above Hecke operators correspond to the usual Hecke operators onMk(ℓ, χℓ). This follows by comparing the actions on Fourier expansions. For Hecke operators on Mk(ℓ, χℓ) see e.g. [Mi], Lemma 4.5.14. 4.2. The case of odd signature. We now assume that sig(A) is odd. In view of (2.14) we only have to consider modular forms of half integral weight k ∈ Z + 1/2. In this case, the problem arises that both, (4.3) and (4.4), only define projective actions of Q(N). To obtain honest actions, one has to consider appropriate central extensions. We begin by considering the action on C[A]. The composition of the natural reduction Q(N) → Q(N) with the projective Weil representation ρA : Q(N) → U(C[A])/{±1} induces a projective representation ρA : Q(N) −→ U(C[A])/{±1}, g 7→ ρA(g). If we choose for every g ∈ Q(N) a ρ̃A(g) ∈ U(C[A]) such that ρ̃A(g) 7→ ρA(g) under the projection to U(C[A])/{±1}, we obtain a 2-cocycle c with values in {±1} defined by ρ̃A(g1g2) = c(g1, g2)ρ̃A(g1)ρ̃A(g2) for g1, g2 ∈ Q(N). This cocycle gives rise to a central extension Q1(N) = Q(N)× {±1},(4.8) 12 JAN H. BRUINIER AND OLIVER STEIN where the multiplication is defined by (g1, t1)(g2, t2) = (g1g2, t1t2c(g1, g2) Here the cocycle condition c(g1g2, g3)c(g1, g2) = c(g1, g2g3)c(g2, g3) is equivalent to the associativity law for the above multiplication. We obtain a unitary representation Q1(N) −→ U(C[A])(4.9) by putting ρA(g, t) = tρ̃A(g). This left action induces a corresponding right action a |A (g, t) = ρA(g, t)−1a,(4.10) for (g, t) ∈ Q1(N) and a ∈ C[A]. Without loss of generality, for (M, 1) ∈ Γ× {1} ⊂ Q(N) we choose ρ̃A(M, 1) = ρA(M, j(M, τ)).(4.11) Then we have an injective homomorphism Γ̃ −→ Q1(N), (M,± j(M, τ)) 7→ (M, 1,±1).(4.12) Moreover, for a positive integer m coprime to N , we put eλ = em−1λ.(4.13) To define an action on functions we consider the metaplectic group G̃L 2 (R). It acts on functions f : H → C by (f |k (M,φ))(τ) = det(M)k/2φ(τ)−2kf(Mτ)(4.14) for (M,φ) ∈ G̃L 2 (R). In particular, we have an action of H̃(N) ⊂ G̃L 2 (R), where H(N) = {M ∈ G(N); det(M) is a square mod N} ⊂ GL+2 (R)(4.15) is the image of the projection of Q(N) to the first component. Notice that the cocycle of Q(N) given by the choice of ± j(M, τ) for (M, r) ∈ Q(N) and the cocycle c are not isomorphic (this follows for instance from Lemma 4.8 below). However, their restrictions to Γ̃ are isomorphic. To define an action on C[A]-valued functions in weight k we have to consider a combi- nation of the above extensions. We let Q2(N) be the group of tuples (M,φ, r, t), where g = (M,φ) ∈ H̃(N), and r ∈ U(N) with det(M) ≡ r2 (mod N), and t ∈ {±1}. The composition law is defined by (g1, r1, t1)(g2, r2, t2) = (g1g2, r1r2, t1t2c((M1, r1), (M2, r2)) −1)(4.16) for (gi, ri, ti) ∈ Q2(N) and gi = (Mi, φi). We denote by P : Q2 → H̃(N) the natural projection. It has the kernel {(1, 1, r, t) ∈ Q2; r2 ≡ 1 (N)}. THE WEIL REPRESENTATION AND HECKE OPERATORS 13 Over Γ̃ the projection P has the section L : Γ̃ −→ Q2, (M,± j(M, τ)) 7→ (M,± j(M, τ), 1,±1).(4.17) We write ∆ = L(Γ̃).(4.18) We define the Weil representation ρA on Q2(N) by composing the natural map to Q1(N) with the Weil representation on that group. For γ ∈ Γ̃ we have ρA(γ) = ρA(L(γ)).(4.19) The tensor product of the above two actions yields a right action of Q2(N) on functions f : H → C[A]. If we write f = λ∈A fλ ⊗ eλ for the decomposition in components with respect to the standard basis of C[A], the action is given by f |k,A (M,φ, r, t) = fλ |k (M,φ) eλ |A (M, r, t) (4.20) for (M,φ, r, t) ∈ Q2(N). A holomorphic function f : H → C[A] belongs to Mk,A, if and only if f |k,A (M,φ) = f for all (M,φ) ∈ Γ̃, and f is holomorphic at the cusp ∞. We may now define Hecke operators on modular forms of type ρA following Shimura [Sh2]. For α = (M,φ) ∈ H̃(N) and ξ = (α, r, t) ∈ Q2(N) we consider the double coset ∆ξ∆. If ∆ξ∆ = is a left coset decomposition, we define the Hecke operator T (ξ) by f |k,A T (ξ) = det(α)k/2−1 f |k,A ξi(4.21) for f ∈ Mk,A. It is easily seen that T (ξ) is independent of the choice of the coset represen- tatives and defines an endomorphism of Mk,A. We recall the following standard lemma. Lemma 4.5. Let the notation be as above. Then (∆ ∩ ξ−1∆ξ) · γi (γi ∈ ∆) is a disjoint left coset decomposition if and only if ∆ξ∆ = ∆ · ξγi is a disjoint left coset decomposition. � To compute the action of Hecke operators we have to compare the groups ∆ ∩ ξ−1∆ξ and L(Γ̃ ∩ α−1Γ̃α). Let α ∈ H̃(N) and ξ ∈ Q2(N) with P (ξ) = α. For γ ∈ Γ̃ ∩ α−1Γ̃α we L(αγα−1) = ξL(γ)ξ−1 · (1, 1, t(γ))(4.22) 14 JAN H. BRUINIER AND OLIVER STEIN with t(γ) ∈ {±1}. Here t(γ) is independent of the choice of ξ and is determined by the condition ρA(αγα −1) = t(γ)ρA(ξ)ρA(γ)ρA(ξ) −1.(4.23) Hence t defines a group homomorphism t : Γ̃ ∩ α−1Γ̃α −→ {±1}.(4.24) Lemma 4.6. Let the notation be as above. We have L(ker(t)) = ∆∩ ξ−1∆ξ. Moreover, if t is non-trivial, then f |k,A T (ξ) = 0 for all f ∈ Mk,A. Proof. The proof is analogous to Proposition 1.0 in [Sh2] � Lemma 4.7. Let the notation be as above. The homomorphism t is trivial if and only if P gives a bijective map of ∆ξ∆ onto Γ̃αΓ̃. Moreover, when this is the case then ∆ξ∆ = i ∆ξi (where ξi ∈ ∆ξ∆) is a disjoint union if and only if Γ̃αΓ̃ = i Γ̃P (ξi) is a disjoint union. Proof. The proof is analogous to Proposition 1.1 in [Sh2] � Lemma 4.8. Let m,n be positive integers coprime to N , α = ((m 00 n ) , n) ∈ H̃(N), and ξ = (α, r, t) ∈ Q2(N). Define t : Γ̃ ∩ α−1Γ̃α −→ {±1} as in (4.24). Then cτ + d Here the quadratic residue symbol is defined as in [Sh2]. Proof. Define Γ′ = Γ0(m) ∩ Γ0(n) ⊂ Γ. We first notice that Γ̃′ = Γ̃ ∩ α−1Γ̃α. Since t is a homomorphism it suffices to prove the assertion for a set of generators of Γ̃′. It is easily verified that Γ̃′ is generated by T n, Um, and Γ̃′ ∩ Γ̃00(N). For γ = T n ∈ Γ̃′ we have αγα−1 = Tm. We compute t(γ) using (4.23). There is a constant Cξ of modulus 1 such that ρA(ξ)eλ = Cξenr−1λ. Hence ρA(ξ)ρA(T n)ρA(ξ) eλ = C ξ ρA(ξ)ρA(T n)en−1rλ = C−1ξ e(n −1r2λ2/2)ρA(ξ)en−1rλ = e(n−1r2λ2/2)eλ = e(mλ2/2)eλ. Here n−1 in the exponentials means the inverse of n in U(N). On the other hand, we have m)eλ = e(mλ 2/2)eλ. Hence t(T n) = 1 in accordance with the formula. THE WEIL REPRESENTATION AND HECKE OPERATORS 15 For γ = Um ∈ Γ̃′ we have αγα−1 = Un. We find ρA(ξ)ρA(U m)ρA(ξ) eλ = C ξ ρA(ξ)ρA(U m)en−1rλ ρA(ξ) µ,ν∈A −mµ2/2 + (µ, n−1rλ− ν) µ,ν∈A −mµ2/2 + (n−1rµ, λ− ν) µ,ν∈A −nµ2/2 + (µ, λ− ν) This is equal to ρA(U n). Hence t(Um) = 1 in accordance with the formula. To compute t(γ) for γ ∈ Γ̃′ ∩ Γ̃00(N), we use the formula for the Weil representation of Proposition 2.2. Using the definition one easily checks that if γ ∈ Γ̃′ ∩ Γ̃00(N), we have χA(γ) = χA(αγα Therefore ρA(ξ)ρA(γ)ρA(ξ) eλ = C ξ ρA(ξ)ρA(γ)en−1rλ = χA(γ)edλ ρA(αγα −1)eλ. So t(γ) = as claimed. � Proposition 4.9. Let α = (M,φ) ∈ H̃(N) and ξ = (α, r, t) ∈ Q2(N). Then the Hecke operator T (ξ) on Mk,A vanishes identically unless det(M) is a square in Q. Proof. By multiplying with a positive integer we may assume without loss of generality that M has entries in Z. According to the elementary divisor theorem for Γ we may further assume that M = (m 00 n ) with positive integers m,n. So we may assume that Hence the assertion follows from Lemma 4.6 and Lemma 4.8. � We now study the relation between Hecke operators and Fourier coefficients. The fol- lowing theorem is the analogue of Proposition 4.3 in the odd signature case. It can be viewed as a generalization of Theorem 1.7 in [Sh2]. Theorem 4.10. Let p be a prime coprime to N , and put ∈ H̃(N), ξ = (α, p, 1) ∈ Q2(N). 16 JAN H. BRUINIER AND OLIVER STEIN Let f ∈ Mk,A and write f(τ) = n∈Z+λ2/2 c(λ, n)e(nτ)⊗ eλ, f |k,AT (ξ) = n∈Z+λ2/2 b(λ, n)e(nτ)⊗ eλ. b(λ, n) = c(pλ, p2n) + ǫ sig(A)+(−1|A|) |A|2sig(A) pk−3/2 c(λ, n) + p2k−2c(λ/p, n/p2). Here, for an odd integer d we put 1, if d ≡ 1 (mod 4), i, if d ≡ −1 (mod 4). Moreover, we understand that c(λ/p, n/p2) = 0 if p2 ∤ n. Proof. To compute f |k,AT (ξ), we need a set of representatives for ∆\∆ξ∆. In view of Lemma 4.7 and Lemma 4.8, the map Lξ : Γ̃αΓ̃ −→ ∆ξ∆, δ = γαγ′ 7→ Lξ(δ) := L(γ)ξL(γ′) is a bijection (where γ, γ′ ∈ Γ̃). Here we have Lξ(δ) = (δ, p, t), and t = t(δ) is uniquely determined by the condition ρA(δ, p, t) = ρA(γ)ρA(ξ)ρA(γ ′).(4.25) We have the disjoint left coset decomposition Γ̃αΓ̃ = Γ̃α ∪ h (p)∗ Γ̃βh ∪ b (p2) Γ̃γb, where Nsτ + p −Nps 1 −Npsτ + 1 −N p2 −Nτ + p2 1 −Nt N p2d Nτ + p2d and r, s ∈ Z are chosen such that pr − N2hs = 1, and d, t ∈ Z are chosen such that p2d+N2t = 1. Consequently, we obtain the disjoint left coset decomposition ∆ξ∆ = ∆ξ ∪ h (p)∗ ∆Lξ(βh) ∪ b (p2) ∆Lξ(γb).(4.26) THE WEIL REPRESENTATION AND HECKE OPERATORS 17 The action of Lξ(βh) and Lξ(γb) in the Weil representation can be computed by means of (4.25) and the above decompositions. Using Proposition 2.2 and the fact that ρA(ξ)eλ = ep−1λ by (4.13), we find that ρA(Lξ(γb))eλ = χA −N p2 −Nτ + p2 1 −Nt N p2d Nτ + p2d e(bλ2/2)epλ. Since d is a square modulo N2, it is a square modulo the square-free part of |A| and modulo 8. Therefore, a quick calculation shows that the character values are 1. Consequently, ρA(Lξ(γb))eλ = e(bλ 2/2)epλ.(4.27) In the same way we obtain ρA(Lξ(βh))eλ = ρA Nsτ + p ρA(ξ)eλ Nsτ + p ep−1λ Nsτ + p 1−(−1|A|)−sig(A) |A|2sig(A) Here in the last line we have used the explicit formula for χA. Since −N2hs ≡ 1 (mod p), we find ρA(Lξ(βh))eλ = ǫ 1−(−1|A|)−sig(A) |A|2sig(A) eλ.(4.28) Now we can compute the Fourier expansion of f |k,AT (ξ). We have f |k,AT (ξ) = pk−2f |k,Aξ + pk−2 h (p)∗ f |k,ALξ(βh) + pk−2 b (p2) f |k,ALξ(γb).(4.29) For the first summand we find pk−2f |k,Aξ = pk−2 fλ |k α eλ |A ξ = p2k−2 2τ)⊗ epλ. 18 JAN H. BRUINIER AND OLIVER STEIN For the second summand in (4.29) we get h (p)∗ f |k,ALξ(βh) = pk−2 h (p)∗ fλ |k βh eλ |A Lξ(βh) sig(A)+(−1|A|)−1 |A|2sig(A) h (p)∗ fλ(τ +Nh/p)⊗ eλ. By means of the formula for the Gauss sum h(p)∗ e(kh/p) = p we obtain h (p)∗ fλ(τ +Nh/p) = n∈Z+λ2/2 c(λ, n)e(nτ), and therefore h (p)∗ f |k,ALξ(βh) sig(A)+(−1|A|) |A|2sig(A) pk−3/2 n∈Z+λ2/2 c(λ, n)e(nτ)⊗ eλ. Finally, for the third summand in (4.29) we get b (p2) f |k,ALξ(γb) = pk−2 b (p2) fλ |k γb eλ |A Lξ(γb) = p−2 b (p2) e(−b(p−1λ)2/2)fλ(τ/p2 + b/p2)⊗ ep−1λ n∈Z+λ2/2 c(pλ, p2n)e(nτ)⊗ eλ. This concludes the proof of the theorem. � Remark 4.11. Let m be a positive integer. Let L be the lattice Z with the quadratic form x 7→ −mx2. Then L′ = 1 Z, andMk,L′/L is isomorphic to the space Jk+1/2,m of Jacobi forms of weight k+1/2 and index m (cf. [EZ], Theorem 5.1). Under this isomorphism the Hecke operator T (ξ) of Theorem 4.10 corresponds to the Hecke operator Tp on Jk+1/2,m defined in [EZ] §4 (3). This follows from Theorem 4.10 and [EZ], Theorem 4.5, by comparing the actions on Fourier expansions. Notice that we have in this particular case sig(A)+(−1|A|) |A|2sig(A) For the lattice L = Z with the quadratic form x 7→ mx2 the space Mk,L′/L is isomorphic to the space Jskewk+1/2,m of skew holomorphic Jacobi forms of weight k + 1/2 and index m defined in [Sk]. Again, the Hecke operators of Theorem 4.10 correspond to the usual Hecke operators on skew-holomorphic Jacobi forms. THE WEIL REPRESENTATION AND HECKE OPERATORS 19 4.3. A Hecke algebra on vector valued modular forms. Using the double coset actions of the previous section, we may define for every positive integer m coprime to N a Hecke operator T (m2)∗ : Mk,A → Mk,A, by f 7→ f |k,A T (m2)∗ = f |k,A T , if sig(A) is even, f |k,A T , 1, m, 1 , if sig(A) is odd. (4.30) In the case of even signature (that is integral weight), the operator T (m2)∗ differs from the usual Hecke operator T (m2) which is given by the sum of double cosets consisting of all integral matrices of determinant m2. This is the reason for our notation. In the case of odd signature (that is half-integral weight), the operator T (m2)∗ is analogous to the Hecke operator in [Sh2] on scalar valued modular forms. Theorem 4.12. The Hecke operators T (m2)∗ (for m coprime to N) generate a commu- tative subalgebra of End(Mk,A), which is actually already generated by the T (p 2)∗ for p prime and coprime to N . The operators T (m2)∗ take cusp forms to cusp forms and are self-adjoint with respect to the Petersson scalar product. Proof. Using the actions (4.5) and (4.20), this follows in the usual way from the properties of the abstract Hecke algebra of the pair (Q(N),Γ), respectively (Q2(N),∆). � 5. The Weil representation on double cosets We now want to define Hecke operators T (m2)∗ as in Section 4.3 for all positive integers m, not necessarily coprime to N . If m and N are not coprime, then the reduction of does not belong to GL2(Z/NZ). So we cannot use the results of the previous sections. However, it is still possible to extend the Weil representation ρA to the corresponding double coset in a compatible way, as we will see. Let m be a positive integer and α = ∈ G̃L 2 (R). We define a right action on C[A] by eλ |A α = emλ.(5.1) To lighten the notation, we will frequenty drop the subscript from the slash operator. Comparing with (3.6) and (4.13), we see that (5.1) is compatible with our earlier definition in the case that (m,N) = 1. Moreover, if δ = γαγ′ ∈ Γ̃αΓ̃, we put eλ | δ = eλ | γ | α | γ′.(5.2) We now show that this right action is well defined, that is, independent of the decomposition of δ. Proposition 5.1. Let δ = γαγ′ = γ1αγ 1 ∈ Γ̃αΓ̃ (where γ, γ′, γ1, γ′1 ∈ Γ̃). Then eλ | γ | α | γ′ = eλ | γ1 | α | γ′1. Proof. First, one easily shows that it suffices to prove the proposition in the case that γ′ = γ1 = 1. So we have δ = γα = αγ 1 and need to show that eλ | γ | α = eλ | α | γ′1.(5.3) 20 JAN H. BRUINIER AND OLIVER STEIN If we write γ = ( a bc d ) ,± cτ + d , then γ′1 = a b/m2 m2c d m2cτ + d . In particular, γ ∈ Γ̃0(m2) and γ′1 ∈ Γ̃0(m2). It suffices to prove (5.3) for γ in a set of generators of Γ̃0(m2). It is easily seen that Γ̃0(m2) is generated by Γ̃0(m2) ∩ Γ̃00(N), Tm , and U . For γ ∈ Γ̃0(m2) ∩ Γ̃00(N) the identity (5.3) immediately follows from Proposition 2.2. For γ = Tm it is easily verified as well. We now consider (5.3) for γ = U . Using Lemma 2.3, we see that the left hand side of (5.3) is equal to eλ | U | α = µ,ν∈A µ2/2 + (µ, λ− ν) emν . Using (2.2), in the sum over ν we write ν = ν ′/m + ν ′′ where ν ′ ∈ Am and ν ′′ ∈ Am. We obtain eλ | U | α = ν′∈Am ν′′∈Am µ2/2 + (µ, λ− ν ′/m− ν ′′) The sum over ν ′′ is equal to |Am| if µ ∈ Am and 0 otherwise. Hence eλ | U | α = µ2/2 + (µ, λ− ν/m) (mµ)2/2 + (mµ, λ− ν/m) On the other hand, the right hand side of (5.3) is equal to eλ | α | Um µ,ν∈A m2µ2/2 + (µ,mλ− ν) |A||Am| µ,ν∈A µ′∈Am m2µ2/2 + (µ+ µ′, mλ− ν) The sum over µ′ is equal to |Am| if ν ∈ Am and 0 otherwise. Hence eλ | α | Um m2µ2/2 + (µ,mλ− ν) (mµ)2/2 + (mµ, λ− ν/m) This concludes the proof of the proposition. � Lemma 5.2. Let δ ∈ Γ̃αΓ̃, and let γ, γ′ ∈ Γ̃. Then eλ | (γδγ′) = eλ | γ | δ | γ′. Proof. This follows immediately from the definition and Proposition 5.1. � THE WEIL REPRESENTATION AND HECKE OPERATORS 21 The element β = ∈ G̃L 2 (R) belongs to the double coset Γ̃αΓ̃. The following Proposition gives its action on C[A]. Proposition 5.3. We have eλ | β = eµ.(5.4) Moreover, for the standard scalar product on C[A] we have 〈a | α, b〉 = 〈a, b | β〉, a, b ∈ C[A].(5.5) Proof. The first assertion follows from the fact that β = SαS−1 and Lemma 5.2. The second assertion is verified by a straightforward computation. � Proposition 5.4. Let m,n be coprime positive integers, and put α = . Then for g ∈ Γ̃αΓ̃ and h ∈ Γ̃βΓ̃ we have eλ | g | h = eλ | (gh).(5.6) Proof. We write g = γαγ′ and h = δβδ′ with γ, γ′, δ, δ′ ∈ Γ̃. Since (m,n) = 1, a simple argument using the elementary divisor theorem shows that gh = ǫαβǫ′ for suitable ǫ, ǫ′ ∈ Γ̃. In view of Lemma 5.2 it suffices to prove the assertion in the case that γ = δ′ = 1. As a second reduction step, we now show that we may in addition assume that ǫ′ = 1. In fact, the identity gh = αγ′δβ = ǫαβǫ′ implies that δβǫ′−1 = γ′−1α−1ǫαβ. The matrix component of the left hand side has integral entries, hence the same is true for the right had side. Using the coprimality of m and n we may infer that δ̃ := γ′−1α−1ǫα belongs to Γ̃. We obtain δβ = δ̃βǫ′, (αγ′)(δ̃β) = ǫαβ. From the assertion in the ǫ′ = 1 case we get eλ | (αγ′) | (δ̃β) = eλ | (ǫαβ), which implies the assertion for arbitrary ǫ′. Finally we need to prove the claim in the case that γ = δ′ = ǫ′ = 1. So g = αγ′, h = δβ gh = αγ′δβ = ǫαβ. Since αγ′δ = ǫα, Lemma 5.2 implies that eλ | (αγ′δ) = eλ | (ǫα), eλ | (αγ′δ) | β = eλ | (ǫα) | β. Now the claim follows from Lemma 5.2 and the fact that eλ | α | β = eλ | (αβ). � 22 JAN H. BRUINIER AND OLIVER STEIN Definition 5.5. Let m be a positive integer and α = ∈ G̃L 2 (R). Let Γ̃ · α · Γ̃ = Γ̃ · δi be a disjoint left coset decomposition. We define the Hecke operator T (m2)∗ on modular forms f ∈ Mk,A by f 7→ f |k,A T (m2)∗ = mk−2 f |k,A δi = mk−2 fλ |k δi eλ |A δi Lemma 5.2 implies that the definition does not depend on the choice of the coset repre- sentatives. Notice that for m coprime to N , this definition agrees with the earlier definition in Section 4.3. Theorem 5.6. For any positive integer m, the Hecke operator T (m2)∗ is a linear operator on Mk,A taking cusp forms to cusp forms. It is self adjoint with respect to the Petersson scalar product. Moreover, if m,n are coprime, then T (m2)∗T (n2)∗ = T (m2n2)∗. Proof. The first assertion is a consequence of Lemma 5.2. The self adjointness follows from Proposition 5.3 along the standard argument for scalar valued modular forms (cf. [Bu], Theorem 1.4.3). The last assertion is a consequence of Proposition 5.4 and the correspond- ing property of the abstract Hecke algebra. � Remark 5.7. For a prime p dividing N the local Hecke algebra, that is, the subalgebra of End(Mk,A) generated by the T (p 2ν)∗, is considerably more complicated than in the case where p is coprime to N . For instance, it is commutative if p is coprime to N , but in general non-commutative if p divides N . References [Bö] S. Böcherer, Über die Funktionalgleichung automorpher L-Funktionen zur Siegelschen Modul- gruppe, J. Reine Angew. Math. 362 (1985), 146–168. [Bo1] R. Borcherds, Automorphic forms with singularities on Grassmannians, Inv. Math. 132 (1998), 491-562. [Bo2] R. Borcherds, Reflection groups of Lorentian lattices, Duke Math. J. 104 (2000), 319-366. [Br] J. H. Bruinier, Borcherds products on O(2, l) and Chern classes of Heegner divisors, Springer Lecture Notes in Mathematics 1780, Springer-Verlag (2002). [BB] J. H. Bruinier and M. Bundschuh, On Borcherds products associated with lattices of prime dis- criminant, Ramanujan J. 7 (2003), 49–61. [Bu] D. Bump, Automorphic Forms and Representations, Cambridge University Press (1998). [CS] J. H. Conway and H. J. Sloane, Sphere packings, lattices and groups. Third edition. Grundlehren der Mathematischen Wissenschaften, 290, Springer-Verlag, New York (1999). [Eb] W. Ebeling, Lattices and codes. A course partially based on lectures by F. Hirzebruch. Second revised edition. Advanced Lectures in Mathematics, Vieweg, Braunschweig (2002). [EZ] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progress in Math. 55, Birkhäuser (1985). [FH] W. Fulton and J. Harris, Representation Theory, Springer GTM 129, Springer Verlag (1991). THE WEIL REPRESENTATION AND HECKE OPERATORS 23 [Ga] P. Garrett, Pullbacks of Eisenstein series; Applications. In: Automorphic forms of several variables, Taniguchi Symposium, Katata, 1983, Birhäuser (1984). [McG] W. J. McGraw, The rationality of vector valued modular forms associated with the Weil represen- tation, Math. Ann. 326 (2003), 105–122. [MH] J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer-Verlag, New York-Heidelberg (1973). [Mi] T. Miyake, Modular forms, Springer, 1989. [Ni] V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177, 238. English translation in Mathematics of the U.S.S.R., Izvestia 14 (1980), 103–167. [NW] A. Nobs and J. Wolfart, Die irreduziblen Darstellungen der Gruppen SL2(Zp), insbesondere SL2(Z2). I. Teil, Comment Math. Helvetici 51 (1976), 491–526. [PSR] I. Piatetski-Shapiro, S. Rallis, L-functions for classical groups, Lecture Notes in Mathematics 1254, Springer-Verlag, Berlin (1987). [Sch] N. R. Scheithauer, On the classification of automorphic products and generalized Kac-Moody alge- bras, Invent. Math. 164 (2006), 641–678. [Sh1] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univer- sity Press, Princeton (1971). [Sh2] G. Shimura, On modular forms of half integral weight, Annals of Math. 97 (1973), 440-481. [Shin] T. Shintani, On the construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83-126. [Sk] N.-P. Skoruppa, Developments in the theory of Jacobi forms. In: Proceedings of the conference on automorphic funtions and their applications, Chabarovsk (eds.: N. Kuznetsov and V. Bykovsky), The USSR Academy of Science (1990), 167–185. (see also MPI-preprint 89-40, Bonn (1989).) Mathematisches Institut, Universität zu Köln, Weyertal 86–90, D-50931 Köln, Germany E-mail address : bruinier@math.uni-koeln.de E-mail address : ostein@math.uni-koeln.de 1. Introduction 2. Discriminant forms and the Weil representation 3. Extending the Weil representation 3.1. The case of even signature 3.2. The case of odd signature 4. Hecke operators on vector valued modular forms 4.1. The case of even signature 4.2. The case of odd signature 4.3. A Hecke algebra on vector valued modular forms 5. The Weil representation on double cosets References

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In the present article, we describe a method of introducing the harmonic potential into the Klein-Gordon equation, leading to genuine bound states. The eigenfunctions and eigenenergies are worked out explicitly.

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Microsoft Word - statia.doc Band in ARPES caused by photodissociation of Landau-Pekar polarons A.E.Myasnikova, E.N. Myasnikov South Federal University, 344090 Rostov-on-Don, Russia We consider decay of phonon condensate into phonons at photodissociation of the strong coupling large polaron (SCLP), when the charge carrier becomes free. Expression to calculate the band in ARPES caused by photodissociation of SCLP is obtained. The band in ARPES caused by photodissociation of strong-coupling large-radius polarons is a broad band with the shape determined by Poisson distribution. It can be structured or unstructured depending on the phonon dispersion since a distance between neighbouring lines comprising the band is the phonon energy. Half-width of the band is in the interval 1.3 - 1.7Ep, depending on the phonon energy. The band maximum is situated approximately at the electron energy Ephot-W-3.2Ep (where Ephot is the photon energy, W is work function), and its position does not depend on the electron wave vector direction. PACS numbers 71.38.Fp, 74.25.Gz, 79.60.Bm 1. Introduction We have shown earlier [1] that the strong-coupling large radius polaron (i.e. polarons that occur at strong electron-phonon interaction in a medium where the conduction band is not narrow) contains phonon condensate. (In other words, the polarization field in such polarons is in quantum-coherent state.) If the charge carrier is removed from the SCLP on its photodissociation the phonon condensate decays into phonons. As any Bose-condensate, the phonon condensate is a superposition of states with different number of quanta where summands are phased up to small quantum fluctuations. Therefore photodissociation of such polarons is accompanied by radiation of different number of phonons in each act. Decay of the phonon condensate at the SCLP photodissociation results in a wide band in the optical conductivity spectrum. The calculated band in the optical conductivity spectrum [2] has the maximum at about 4.2Ep and half-width about 2.5 Ep, where Ep is the polaron binding energy. The calculated band is in good conformity with mid-IR band in the optical conductivity spectra of complex oxides. The predicted ratio (≈4) of the maximum position of the band caused by the SCLP photoionization to the frequency of maximum of the band caused by phototransitions into polaron excited states calculated by other group [3] is also in good conformity with experiments. Obviously, decay of phonon condensate at the SCLP photodissociation should display itself as sufficiently wide band in photoemission too. The aim of the present study is to obtain an expression to calculate the band in ARPES spectrum caused by photodissociation of SCLP. 2. Photodissociation of the strong-coupling large polaron Following Emin [] we calculate the band in the optical conductivity spectrum caused by photodissociation of SCLP using the simplest treatment of the photoelectric effect. Therefore our result can be easily reconstructed in order to describe the band in ARPES caused by the SCLP photodissociation. So, the initial state of the system is a charge carrier in autolocalized state (coupled in the polarization potential well that represents a polarization field in coherent state. The polaron photodissociation occurs as a result of interaction of an electromagnetic wave of frequency Ω with the charge carrier in the polaron (the longitudinal field of the polarization in the polaron obviously does not interact with the transverse electromagnetic wave). The operator of the interaction has the form ( ) rQkA ie H *intˆ = , (1) where kh is the electron momentum, A is the amplitude of the vector potential of the electromagnetic field, related with its intensity I as it follows: cI hπ2/2AΩ= ; Q is the wave vector of the electromagnetic wave. According to Fermi golden rule a probability of transition of the system from the state i into the state f per unit time under the influence of the operator intĤ has the form ( )fiif EEiHfW −= δ , (2) where iE and fE are the energies of the initial and final states of the whole system. If the initial state is the ground state [1,2] of the polaron in an electromagnetic field of a frequency Ω then ( ) ( )∏−+= qdrri ββπβ exp17/ 3 (3) and iE = Ω+−− hWE p where Ep is the polaron binding energy, W is work function, and Pekar wave function for the carrier state in the polaron is used. To describe the final state we following Emin [3] use the simplest treatment of the photoelectric effect where the final state of the charge carrier is approximated as a free-carrier state. The vectors of possible final states of the phonon field are the eigen vectors { } ∏= qq νν of the non-shifted Hamiltonian ∑ += qqq bbH ph ωhˆ describing the states with the certain number of quanta qν in each harmonics. Thus, after the photodissociation the state (3) transforms into a state ( ) { }∏−= qkr νiLf exp2 , (4) provided the sum of qν (taking values 0 or 1) from the set { }qν yields a certain number ν . Hence, the energy of the final state is ωνh , if we neglect the dependence of ω on q. Thus, ( ) ⎟⎟ −−−Ω+−=− ωνδδ h WEEE pfi . (5) As the operator intĤ acts only on the electron variables, the matrix element of the transition has the form ( ) ( )∫ ∏− qqkrr derHiLdiHf r νβπβ β17/ˆexpˆ 3int2 . (6) Naturally, it contains the scalar product of the vector of a coherent state of the phonon field by a vector of its state with the certain number of phonons. After carrying out the integration in (18) the probability of the electron transition into a state with the wave vector with modulus k and direction in a spatial angle ϕθθ ddsin has the form ( ) ( ) ( )∏⋅ qqk kkQ ν ddLcm 3*, 17 where ( ) ( ) Ω== d 2 hh π ρ k (8) is the spectral density of the final carrier states with the wave vector directed in the body angle dΩ [3]. According to (5) the electron momentum kh and energy ε in the final state are related as follows: ( ) ( )ωνεε hhh −−Ω== pEmmk ** 22 . (9) According to the energy and the momentum conservation laws (Exp.(5) and 0qkQ += , where q0 is the wave vector of the phonon field after the polaron photoionization, qqq ν0 ) an experiment can measure only the probability (7) summarized over all possible sets { }qν having the same values of ∑= qνν and q0: ( ) ( ) ( )kkQkA qqk ρνββ ν ddLcm dW ∑∏⋅ 3*, 17 ,(10) Here symbol ν over Σ denotes that the sum is carried out over the sets{ }qν satisfying the condition νν =∑ q . Besides, there is not a set with 0=ν among the sets { }qν . Indeed, in such a case 00 =q , hence, Qk = , and 0== QAkA , i.e. the probability of a transition with appearance of such a set { }qν is zero. Summarized probability (10) contains as a multiplier the sum dP . (11) The sum (11) calculated in [2] with taking into account the fact that dk<<1 has the form: ==∑∏ edP )!1( qq (12) where ν is the average number of phonons radiated at the SCLP photodissociation [2]: ( ) 12 −= ων hpE . (13) To write the scalar product in (10) let us use ordinary for ARPES experiment geometry [4] where the wave vector Q of the incident photon lies in XZ plane of the coordinate system and makes angle ψ with z axes. The wave vector k of the electron is considered to have a component k lying in the XY plane of the coordinate system (and coinciding with the sample surface) and ⊥k in the perpendicular direction. Then ψεψϕ sin/2coscos)( 22* AkmAk −+= hkA (14) where the relation 22* /2 kmk −=⊥ hε is taken into account. Finally, the wave vector Q of the photon ordinarily used in ARPES experiments can be approximated as zero []. Thus, expression (10) takes the form ( ) ( ) kAkmk εψεψϕ βπ )!1()1( sin/2coscos 256 1 (15) Exp.(15) represents the probability of the polaron photodisociation with appearance of ν phonons and electron with the energy ε and wave vector k in the body angle dΩ around certain direction. The direction is determined by k projection on X-, Y- and Z-axes k cosφ, k sinφ, and 22* /2 kmk −=⊥ hε , respectively. In Exp.(15) k(ε) is determined by Exp.(9). Figs.1,2 demonstrate calculated by Exp.15 band in ARPES caused by photodissociation of SCLP with the binding energy 0.14 eV and 0.17 eV, respectively. The upper curves on both figures correspond to kx=1, ky=0, the lower curves correspond to kx=0, ky=0. For both figures the electron-phonon interaction constant α=6 (it determines the phonon energy for given Ep), W−Ωh =20eV and phonon dispersion is neglected. If do not neglect the phonon dispersion the points on Figs.1,2 will transform into “partial” bands, and the resulting summarized band can be structured or unstructured depending on the phonon dispersion since a distance between neighbouring lines comprising the band is the phonon energy. Half-width of the band is in the interval 1.3 - 1.7Ep, depending on the phonon energy. The band maximum is situated approximately at the electron energy Ephot-W-3.2Ep (where Ephot is the photon energy, W is work function), and its position does not depend on the electron wave vector direction. 0.5 1 1.5 2 2.5 19.3 19.4 19.5 19.6 19.7 19.8 g, eV Fig.1. Band in ARPES caused by photoionization of SCLP with Ep=0.14 eV in neglect of phonon dispersion, at α=6 and hΩ-W=20 eV. The upper and lower “curves” correspond to kx=1 and kx=0, respectively, ky=0. 2 4 6 8 19.2 19.3 19.4 19.5 19.6 19.7 Fig.2. Band in ARPES caused by photoionization of SCLP with Ep=0.17 eV in neglect of phonon dispersion, at α=6 and hΩ-W=20 eV. The upper and lower “curves” correspond to kx=1 and kx=0, respectively, ky=0. It is worth noting that intensity of the so-called “coherent” zero-phonon line (band) is equal to zero as it can be seen from Figs1,2. The reason of this is explained below Exp.(10). For example, we can calculate the polaron binding energies in YBa2Cu3O6+y, Nd2CuO4-y, La2CuO4+y and La2-xSrxCuO4+y from the position of maximum of the mid-infrared band in their optical conductivity spectra [5]. Then the maximum of the bands in ARPES of these substances caused by photodissociation of the polarons will be approximately at the electron energy (with respect to the energy Ephot-W) -0.48eV, -0.52eV, -0.44eV and -0.4eV, respectively. Table 1 condoptΩh ,eV Ep , eV ARPES maxΩh , eV, with respect to Ephot-W Yba2Cu3O6+y 0.62±0.05 ≈0.155 ≈0.48 Nd2CuO4-y 0.76±0.01 ≈0.17 ≈0.52 La2CuO4+y 0.6±0.02 ≈0.14 ≈0.44 La2-xSrxCuO4+y 0.53±0.05 ≈0.13 ≈0.4 To make comparison of the theoretical results with experiments Figs.3,4 demonstrate ARPES spectra of underdoped La2-xSrxCuO4 [6] and Nd2-xCexCuO4 [7], respectively. Fig.3. ARPES spectra of underdoped Fig.4. ARPES spectra of underdoped La2-xSrxCuO4 [6] Nd2-xCexCuO4 [7] 3. Temperature behavior of the band caused by SCLP photoionization Velocity of SCLP is limited by the minimum phase (or maximum group) velocity u of phonons participating in SCLP formation. (It is shown for phonon dispersion of the form 2 ku+= ωω in [8]). As a result temperature of SCLP thermal destruction is much lower than their binding energy Ep [9]. The temperature corresponding to double lowering of the polaron concentration due to their thermal destruction can be approximated as [10] 176.0 278.0 ⎟⎟ ET pc where m* is the bare carrier effective mass, u is the maximum group (and minimum phase) velocity of phonons interacting with the charge carrier, ∞−= εε /1/1 0c , 0ε is a static dielectric constant, ∞ε is a dielectric permittivity at high frequencies, p0 - «maximum» momentum of the carrier in the polaron [9]. An example of temperature dependence of SCLP concentration is demonstrated by Fig.5 [10]. Fig.5. Temperature dependence of polaron concentration (curve 1), concentration of free carriers with the momentums p<p0 (curve 2) and concentration of free carriers with the momentums p>p0 (curve 3) in a system with total carrier concentration n=1.3n0, polaron binding energy Ep=0.11eV, u=5*104cm/s. n0 is the maximum polaron concentration, 00 )2/()3/4(*2 hππpn = . Accordingly, integral intensity of the band caused by SCLP photodissociation will decrease as it occurs, e.g., in optical conductrivity spectra of 5233.0 OVNa−β shown by Fig.6 [11]. Fig.6. Optical conductivity spectra of 5233.0 OVNa−β [11]. References 1. E. N. Myasnikov, A.E. Myasnikova, and F. V. Kusmartsev, Phys. Rev. B 72, 224303 (2005). 2. E. N. Myasnikov, A.E. Myasnikova, Z.P. Mastropas, cond-mat/0703693. 3. D. Emin, Phys. Rev. B 48, 1369 (1993). 4. A. Damascelli, Z. Hussain and Z.X.Shen, Rev. Mod. Phys. 75, 473 (2003). 5. G. A. Thomas et al., Phys. Rev. B 45, 2474 (1992). 6. A. Ino et al., Phys. Rev. B 62, 4137 (2000). 7. N.P. Armitage et al., Phys. Rev. Lett. 88, 257001 (2002). 8. A. E. Myasnikova. Phys. Rev. B 52, 10457 (1995). 9. E. N. Myasnikov and A. E. Myasnikova, JETP 89, 746 (1999). 10. A. E. Myasnikova. Phys. Let. A 291, 439 (2001). 11. C. Presura et al., Phys. Rev. Lett. 90, 026402, (2003).

We consider decay of phonon condensate into phonons at photodissociation of the strong coupling large polaron (SCLP), when the charge carrier becomes free. Expression to calculate the band in ARPES caused by photodissociation of SCLP is obtained. The band in ARPES caused by photodissociation of strong-coupling large-radius polarons is a broad band with the shape determined by Poisson distribution. It can be structured or unstructured depending on the phonon dispersion since a distance between neighbouring lines comprising the band is the phonon energy. Half-width of the band is in the interval 1.3 - 1.7Ep, depending on the phonon energy. The band maximum is situated approximately at the electron energy Ephot-W-3.2Ep (where Ephot is the photon energy, W is work function), and its position does not depend on the electron wave vector direction.

Introduction We have shown earlier [1] that the strong-coupling large radius polaron (i.e. polarons that occur at strong electron-phonon interaction in a medium where the conduction band is not narrow) contains phonon condensate. (In other words, the polarization field in such polarons is in quantum-coherent state.) If the charge carrier is removed from the SCLP on its photodissociation the phonon condensate decays into phonons. As any Bose-condensate, the phonon condensate is a superposition of states with different number of quanta where summands are phased up to small quantum fluctuations. Therefore photodissociation of such polarons is accompanied by radiation of different number of phonons in each act. Decay of the phonon condensate at the SCLP photodissociation results in a wide band in the optical conductivity spectrum. The calculated band in the optical conductivity spectrum [2] has the maximum at about 4.2Ep and half-width about 2.5 Ep, where Ep is the polaron binding energy. The calculated band is in good conformity with mid-IR band in the optical conductivity spectra of complex oxides. The predicted ratio (≈4) of the maximum position of the band caused by the SCLP photoionization to the frequency of maximum of the band caused by phototransitions into polaron excited states calculated by other group [3] is also in good conformity with experiments. Obviously, decay of phonon condensate at the SCLP photodissociation should display itself as sufficiently wide band in photoemission too. The aim of the present study is to obtain an expression to calculate the band in ARPES spectrum caused by photodissociation of SCLP. 2. Photodissociation of the strong-coupling large polaron Following Emin [] we calculate the band in the optical conductivity spectrum caused by photodissociation of SCLP using the simplest treatment of the photoelectric effect. Therefore our result can be easily reconstructed in order to describe the band in ARPES caused by the SCLP photodissociation. So, the initial state of the system is a charge carrier in autolocalized state (coupled in the polarization potential well that represents a polarization field in coherent state. The polaron photodissociation occurs as a result of interaction of an electromagnetic wave of frequency Ω with the charge carrier in the polaron (the longitudinal field of the polarization in the polaron obviously does not interact with the transverse electromagnetic wave). The operator of the interaction has the form ( ) rQkA ie H *intˆ = , (1) where kh is the electron momentum, A is the amplitude of the vector potential of the electromagnetic field, related with its intensity I as it follows: cI hπ2/2AΩ= ; Q is the wave vector of the electromagnetic wave. According to Fermi golden rule a probability of transition of the system from the state i into the state f per unit time under the influence of the operator intĤ has the form ( )fiif EEiHfW −= δ , (2) where iE and fE are the energies of the initial and final states of the whole system. If the initial state is the ground state [1,2] of the polaron in an electromagnetic field of a frequency Ω then ( ) ( )∏−+= qdrri ββπβ exp17/ 3 (3) and iE = Ω+−− hWE p where Ep is the polaron binding energy, W is work function, and Pekar wave function for the carrier state in the polaron is used. To describe the final state we following Emin [3] use the simplest treatment of the photoelectric effect where the final state of the charge carrier is approximated as a free-carrier state. The vectors of possible final states of the phonon field are the eigen vectors { } ∏= qq νν of the non-shifted Hamiltonian ∑ += qqq bbH ph ωhˆ describing the states with the certain number of quanta qν in each harmonics. Thus, after the photodissociation the state (3) transforms into a state ( ) { }∏−= qkr νiLf exp2 , (4) provided the sum of qν (taking values 0 or 1) from the set { }qν yields a certain number ν . Hence, the energy of the final state is ωνh , if we neglect the dependence of ω on q. Thus, ( ) ⎟⎟ −−−Ω+−=− ωνδδ h WEEE pfi . (5) As the operator intĤ acts only on the electron variables, the matrix element of the transition has the form ( ) ( )∫ ∏− qqkrr derHiLdiHf r νβπβ β17/ˆexpˆ 3int2 . (6) Naturally, it contains the scalar product of the vector of a coherent state of the phonon field by a vector of its state with the certain number of phonons. After carrying out the integration in (18) the probability of the electron transition into a state with the wave vector with modulus k and direction in a spatial angle ϕθθ ddsin has the form ( ) ( ) ( )∏⋅ qqk kkQ ν ddLcm 3*, 17 where ( ) ( ) Ω== d 2 hh π ρ k (8) is the spectral density of the final carrier states with the wave vector directed in the body angle dΩ [3]. According to (5) the electron momentum kh and energy ε in the final state are related as follows: ( ) ( )ωνεε hhh −−Ω== pEmmk ** 22 . (9) According to the energy and the momentum conservation laws (Exp.(5) and 0qkQ += , where q0 is the wave vector of the phonon field after the polaron photoionization, qqq ν0 ) an experiment can measure only the probability (7) summarized over all possible sets { }qν having the same values of ∑= qνν and q0: ( ) ( ) ( )kkQkA qqk ρνββ ν ddLcm dW ∑∏⋅ 3*, 17 ,(10) Here symbol ν over Σ denotes that the sum is carried out over the sets{ }qν satisfying the condition νν =∑ q . Besides, there is not a set with 0=ν among the sets { }qν . Indeed, in such a case 00 =q , hence, Qk = , and 0== QAkA , i.e. the probability of a transition with appearance of such a set { }qν is zero. Summarized probability (10) contains as a multiplier the sum dP . (11) The sum (11) calculated in [2] with taking into account the fact that dk<<1 has the form: ==∑∏ edP )!1( qq (12) where ν is the average number of phonons radiated at the SCLP photodissociation [2]: ( ) 12 −= ων hpE . (13) To write the scalar product in (10) let us use ordinary for ARPES experiment geometry [4] where the wave vector Q of the incident photon lies in XZ plane of the coordinate system and makes angle ψ with z axes. The wave vector k of the electron is considered to have a component k lying in the XY plane of the coordinate system (and coinciding with the sample surface) and ⊥k in the perpendicular direction. Then ψεψϕ sin/2coscos)( 22* AkmAk −+= hkA (14) where the relation 22* /2 kmk −=⊥ hε is taken into account. Finally, the wave vector Q of the photon ordinarily used in ARPES experiments can be approximated as zero []. Thus, expression (10) takes the form ( ) ( ) kAkmk εψεψϕ βπ )!1()1( sin/2coscos 256 1 (15) Exp.(15) represents the probability of the polaron photodisociation with appearance of ν phonons and electron with the energy ε and wave vector k in the body angle dΩ around certain direction. The direction is determined by k projection on X-, Y- and Z-axes k cosφ, k sinφ, and 22* /2 kmk −=⊥ hε , respectively. In Exp.(15) k(ε) is determined by Exp.(9). Figs.1,2 demonstrate calculated by Exp.15 band in ARPES caused by photodissociation of SCLP with the binding energy 0.14 eV and 0.17 eV, respectively. The upper curves on both figures correspond to kx=1, ky=0, the lower curves correspond to kx=0, ky=0. For both figures the electron-phonon interaction constant α=6 (it determines the phonon energy for given Ep), W−Ωh =20eV and phonon dispersion is neglected. If do not neglect the phonon dispersion the points on Figs.1,2 will transform into “partial” bands, and the resulting summarized band can be structured or unstructured depending on the phonon dispersion since a distance between neighbouring lines comprising the band is the phonon energy. Half-width of the band is in the interval 1.3 - 1.7Ep, depending on the phonon energy. The band maximum is situated approximately at the electron energy Ephot-W-3.2Ep (where Ephot is the photon energy, W is work function), and its position does not depend on the electron wave vector direction. 0.5 1 1.5 2 2.5 19.3 19.4 19.5 19.6 19.7 19.8 g, eV Fig.1. Band in ARPES caused by photoionization of SCLP with Ep=0.14 eV in neglect of phonon dispersion, at α=6 and hΩ-W=20 eV. The upper and lower “curves” correspond to kx=1 and kx=0, respectively, ky=0. 2 4 6 8 19.2 19.3 19.4 19.5 19.6 19.7 Fig.2. Band in ARPES caused by photoionization of SCLP with Ep=0.17 eV in neglect of phonon dispersion, at α=6 and hΩ-W=20 eV. The upper and lower “curves” correspond to kx=1 and kx=0, respectively, ky=0. It is worth noting that intensity of the so-called “coherent” zero-phonon line (band) is equal to zero as it can be seen from Figs1,2. The reason of this is explained below Exp.(10). For example, we can calculate the polaron binding energies in YBa2Cu3O6+y, Nd2CuO4-y, La2CuO4+y and La2-xSrxCuO4+y from the position of maximum of the mid-infrared band in their optical conductivity spectra [5]. Then the maximum of the bands in ARPES of these substances caused by photodissociation of the polarons will be approximately at the electron energy (with respect to the energy Ephot-W) -0.48eV, -0.52eV, -0.44eV and -0.4eV, respectively. Table 1 condoptΩh ,eV Ep , eV ARPES maxΩh , eV, with respect to Ephot-W Yba2Cu3O6+y 0.62±0.05 ≈0.155 ≈0.48 Nd2CuO4-y 0.76±0.01 ≈0.17 ≈0.52 La2CuO4+y 0.6±0.02 ≈0.14 ≈0.44 La2-xSrxCuO4+y 0.53±0.05 ≈0.13 ≈0.4 To make comparison of the theoretical results with experiments Figs.3,4 demonstrate ARPES spectra of underdoped La2-xSrxCuO4 [6] and Nd2-xCexCuO4 [7], respectively. Fig.3. ARPES spectra of underdoped Fig.4. ARPES spectra of underdoped La2-xSrxCuO4 [6] Nd2-xCexCuO4 [7] 3. Temperature behavior of the band caused by SCLP photoionization Velocity of SCLP is limited by the minimum phase (or maximum group) velocity u of phonons participating in SCLP formation. (It is shown for phonon dispersion of the form 2 ku+= ωω in [8]). As a result temperature of SCLP thermal destruction is much lower than their binding energy Ep [9]. The temperature corresponding to double lowering of the polaron concentration due to their thermal destruction can be approximated as [10] 176.0 278.0 ⎟⎟ ET pc where m* is the bare carrier effective mass, u is the maximum group (and minimum phase) velocity of phonons interacting with the charge carrier, ∞−= εε /1/1 0c , 0ε is a static dielectric constant, ∞ε is a dielectric permittivity at high frequencies, p0 - «maximum» momentum of the carrier in the polaron [9]. An example of temperature dependence of SCLP concentration is demonstrated by Fig.5 [10]. Fig.5. Temperature dependence of polaron concentration (curve 1), concentration of free carriers with the momentums p<p0 (curve 2) and concentration of free carriers with the momentums p>p0 (curve 3) in a system with total carrier concentration n=1.3n0, polaron binding energy Ep=0.11eV, u=5*104cm/s. n0 is the maximum polaron concentration, 00 )2/()3/4(*2 hππpn = . Accordingly, integral intensity of the band caused by SCLP photodissociation will decrease as it occurs, e.g., in optical conductrivity spectra of 5233.0 OVNa−β shown by Fig.6 [11]. Fig.6. Optical conductivity spectra of 5233.0 OVNa−β [11]. References 1. E. N. Myasnikov, A.E. Myasnikova, and F. V. Kusmartsev, Phys. Rev. B 72, 224303 (2005). 2. E. N. Myasnikov, A.E. Myasnikova, Z.P. Mastropas, cond-mat/0703693. 3. D. Emin, Phys. Rev. B 48, 1369 (1993). 4. A. Damascelli, Z. Hussain and Z.X.Shen, Rev. Mod. Phys. 75, 473 (2003). 5. G. A. Thomas et al., Phys. Rev. B 45, 2474 (1992). 6. A. Ino et al., Phys. Rev. B 62, 4137 (2000). 7. N.P. Armitage et al., Phys. Rev. Lett. 88, 257001 (2002). 8. A. E. Myasnikova. Phys. Rev. B 52, 10457 (1995). 9. E. N. Myasnikov and A. E. Myasnikova, JETP 89, 746 (1999). 10. A. E. Myasnikova. Phys. Let. A 291, 439 (2001). 11. C. Presura et al., Phys. Rev. Lett. 90, 026402, (2003).

From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures A. Balaguera-Antoĺınez Max Planck Institut für Extraterrestrische Physik, Garching, Gliessenbachstrasse 1 D-85748, Garching, Germany E-mail: a-balagu@uniandes.edu.co M. Nowakowski Departamento de F́ısica, Universidad de los Andes,A.A. 4976, Bogotá, D.C., Colombia. E-mail: mnowakos@uniandes.edu.co Abstract. We explore the effects of background cosmology on large scale structures with non-spherical symmetry by using the concept of quasi-equilibrium which allows certain internal properties (e.g. angular velocity) of the bodies to change with time. In accordance with the discovery of the accelerated phase of the universe we model the cosmological background by two representative models: the ΛCDM Model and the Chaplygin Gas Model. We compare the effects of the two models on various properties of large astrophysical objects. Different equations of state are also invoked in the investigation. PACS numbers: 95.30.Sf, 98.62.Dm, 98.80.Jk, 98.52.Eh, 98.56.Ew http://arxiv.org/abs/0704.1871v2 From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 2 1. Introduction The interest in the impact of the cosmological expansion on bound astrophysical system can be traced back to the paper by Einstein and Straus [1]. Over the subsequent years different approaches have been used to gain some insight into the interplay between cosmology and astrophysical structures [2, 3, 4, 5, 6]. With the advent of the discovery of the accelerated Universe [7], it seems timely to pick up the topic once again and contrast the effects of different models which can explain the current accelerated phase. Indeed, in [8] such an examination was performed for phantom [9] and quintessence cosmologies [10]. In the present article we will use the same approach as [2] and [8] and apply it to models with a positive cosmological constant and Chaplygin gas models. These two models will serve us as representatives for cosmological models which can explain the current stage of acceleration. In addition we will also invoke a general model with variable equation of state of the form ωx = px/ρx (DECDM). The approach mentioned above consists in generalizing the Newtonian limit by allowing terms which are specific to the background cosmology. This is not limited to the inclusion of the cosmological constant. Notice that the effects of the background cosmology on bound systems are now in general time dependent. This is, however not in contradiction with the fact that we rightly consider most of the astrophysical structures to be in gravitational equilibrium. The time dependent background cosmology will not tear the system apart, unless its density is diluted, but change only, over cosmological time, the internal properties of the system (like inner velocities or angular velocity etc). As a mathematical tool we will make use of the virial theorem ([11] for different methods) by allowing certain internal properties to be time dependent. The virial theorem can be easily derived even in the presence of the cosmological background as the latter enters only the Newtonian limit (i.e. the Poisson equation for the gravitational potential which includes now a time dependent part). As shown in [12] Dark Energy can affect certain astrophysical static properties which have to do with either the virialization of the system [13] or the motion of test bodies [2]. For instance, the cosmological constant sets itself scales of distance (rΛ = Λ −1/2) ‡, time and mass, which are of the same order of magnitude as the radius of the observed universe, the age of the universe and the total mass of the universe (the so called coincidence problem). At the first glance it looks hopeless to expect any effects from Λ at astrophysical scales. This is so indeed as long as no other scale enters the theory. If the latter appears in the theory a combination of the large cosmological and the small internal scale can yield values of astrophysical relevance (e.g. (rsrΛ) 1/3 where rs is the Schwarzschild radius) [14]. Furthermore, general conditions for detecting the effects of variations of ’constants’ represented by scalar fields (just like the cosmological constant and quintessence models) at scales ranging from terrestial scales have been well established in [15, 16, 17]. Another enhancement mechanism of background cosmology emerges when we are dealing with non-spherically symmetric objects. In such case ‡ We work in the so-called geometrized units, i.e. GN = c = 1 From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 3 the ratio of two length scales to some positive power (bigger than one) goes often hand in hand with Λ. For this reason, in examining time dependent effects, we will mostly employ ellipsoidal configurations. For general consideration of equilibrium in the spherical case see [18] and [19], the quasi spherical case has been discussed in [20]. In this paper, we will entirely concentrate on effects of Dark Energy followed over cosmological times, past and future. We do not expect these effects to be large, however, we think that it is of some importance to probe into these matter, especially in view of the accelerated universe. In [21] we restricted ourselves to effects over small time scales. In this sense these two articles complement each other. 2. Local dynamics with background cosmology In standard cosmology, the universe is described as an ideal fluid which implies the properties of homogeneity and isotropy. Such properties are mathematically represented in Friedman-Robertson-Walker line element, which is written for a flat universe as ds2 = −dt2 + R2(t) (dr2 + r2dΩ2). Einstein field equations and the FRW line element yields the acceleration equation and the well known Friedman equation, written respectively Ṙ(t) = H(t)2 = πρb(t), R̈(t) π [ρb(t) + 3pb(t)] , (1) In conventional cosmology, the total energy density ρb is a contribution of a radiation, cold dark matter and dark energy. By neglecting interaction among these components, each one evolves through the conservation equation ρ̇ = −3H(ρ+ p), so that radiation scales as ρrad = ρrad(R = 1)R −4, cold dark matter scales as ρcdm = ρcdm(R = 1)R while the Dark Energy component associated with the cosmological constant is ρvac = constant. The major contribution in connection to the cosmological constant is approximately 70% of total energy density [7]. 2.1. Dark energy As pointed before, almost the 70% of the content of energy density in the unverse is ruled by a dark energy component, and astronomical observations imply that at the present time this dark energy component might be represented by the cosmological constant. That is, at the present time, the equation of state that displays the best fit with astronomical observation is simply pvac = −ρvac, i.e, ω = −1. Nevertheless, several models has been proposed in order to reproduce the present value of dark energy but maintaining a time dependent energy density. One generalizes the equation of state for the dark energy by px(R) = ωx(R)ρx(R). Dark energy then scales as ρx(R) = ρx(R = 1)a −f(R), where the function f(R) is defined as f(R) ≡ 3 ′) + 1 dR′, (2) From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 4 The model is constrained by requiring that at the present time the dark energy density reproduces the vacuum energy density associated with the cosmological constant, that is, ρx(R = 1) = ρvac. As pointed before, the case ωx = −1 corresponds to the cosmological constant ρx = ρvac = Λ/8π, while for 0 > ωx > −1 one stands in the Dark Energy realms. Models with ωx < −1 lead to future singularities in the so called Phantom regime [9]. The description of a dynamical dark energy is compatible with the description of inflationary models. In the DECDM model, the Friedmann equation and the acceleration equation are written as Ṙ(t) = H20Ωvach1(R) R̈(t) = −H20Ωvach2(R) = − πρvach2(R).(3) The functions h1,2(R) are given as h1(R) ≡ R−3 +R−f(R), (4) h2(R) ≡ R−3 +R−f(z) (1 + 3ωx(R)) 2.2. Chaplygin Gas The main stages the universe has passed through in the standard cosmology scenario can be reproduced by the introduction of the equation of state of an exotic ideal relativistic gas written as pch = κ1ρch − , (5) The relevance of this equation of state lies in three different facts: a) it violates the strong energy condition, which is necessary to obtain an accelerated phase at the present time b) it generates a well definite speed of sound, which is relevant for the process of structure formation and c) it unifies the early radiation or dark-matter behavior with the late dark energy dominance. The so-called pure Chaplygin gas is obtained with κ1 = 0 and γ = 1. The time evolution of the Chaplygin Gas-energy density is given from the integration of mass-energy conservation equation as ρch(R) = A+BR−nβ , (6) with n = 3(1 + κ1), β = γ + 1 and where A and B are integration constants given as B = ρch(R = 1) β − A, A = 0 , (7) with ρch(R = 1) = ρcrit as the Chaplygin Gas energy density at the present time (only for a flat universe). In order to write the integration constants as we have done, we have re-parametrized the Chaplygin Gas equation of state with κ2 = αnH 0 such that a finite age and a present accelerated phase for the Chaplygin gas universe is reached for . (8) From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 5 This bound can be derived after integration of the Friedmann equation H(R)2 = A+BR−nβ . (9) in order to determine the age of the universe. The age of the universe is given in terms of hypergeometric function as × 2F1 2nβ + n+ 4 with q ≡ 3ω + 7. For the pure Chaplygin gas one then obtains a finite age and the observed acceleration at the present time only if the parameter α is such that α3 < 10 which in turn implies a bound on the Chaplygin parameter κ2 < ρ crit. This warranties that the equation of state is ωch(today) > −1. The solution given in Eq.(6) allows us to describe the behavior of Chaplygin gas at different ages. For early times the κ1-term dominates and Chaplygin Gas behaves as ρch = B 1/βR−n. This scales as radiation for n = 4 (κ1 = 1/3). Hence we could in principle relate the integration constant B with cosmological parameters of the concordance model as B = ρ rad(R = 1). For κ1 = 0, the model displays a matter dominated epoch at early times with ρch ≈ B1/βR−3. In this case one can write the integration constant B = ρ mat(R = 1) and also the constant A as A = ρ crit(1 − Ω mat). Finally for large times, Chaplygin gas acquires a vacuum-like behavior with ρch(t → ∞) = A1/β = H20 . (11) This would lead to an effective vacuum energy density ρeffvac which can be associated to the current vacuum energy density ρvac via ρeffvac = ρvac With ρeffvac > ρvac, the effects of this effective vacuum energy density may be relevant in the Newtonian limit when we explore the equilibrium conditions of large scale structures. Let us concentrate on the case κ1 = 0, which does not reproduce an early radiation dominated era but still unifies an early dark matter dominance and a vacuum dominated era at large scale factors. The acceleration equation for this model reads as R̈(t) = − 4 πρch(R)ηch(R) (13) = − 4 A +BR(t)−3β 1− 3κ2 A +BR(t)−3β where ηch ≡ 1 + 3ωch and ωch = ωch(R) ≡ = −κ2ρch(R)−β (14) From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 6 1 1.5 2 2.5 3 3.5 4 Scale Factor β = 2 , α = 0.001 2 , 0.0006 2 , 0.0001 1 1.5 2 2.5 3 3.5 4 4.5 5 Scale factor −1e−14 −5e−15 5e−15 1e−14 1.5e−14 2e−14 Figure 1. Left: Equation of state for Chaplygin gas ωch = pch/ρch for different Chaplygin parameters. Right: Acceleration equation as a function of the (future) scale factor for different Chaplygin parameters. is the time dependent equation of state. In Fig. 1 we show the behavior of the equation of state ωch and the acceleration equation for different values of Chaplygin parameters. 2.3. Newtonian description The effects of the background can be explored through the Newtonian limit of field equations [22], from which one can derive a modified Poisson’s equation [2] ∇2Φ = 4πδρ− 3 , (15) where δρ is the overdensity that gives rise to the “gravitational contribution” of the potential. The total energy density within the clustered configuration ρ is a contribution of the background ρb and the collapsed fraction δρ. The background energy density depends on the model we are interested in: ρb = ρ(R)− δρ = ρcdm(R) DECDM ρch(R) Chaplygin Note that the last expression implies that in the DECDM model one has assumed that only the cold dark matter component is what collapses and forms vitalized structures, while in Chaplygin gas the perturbation is done on the total energy density. The solution From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 7 for the potential can be simplified as Φ(r, R) = Φgrav(r)− r2, Φgrav(r) = − δρ(r′) |r− r′| d3r′, (17) where one has neglected the terms associated to boundary conditions (the potential cannot be zero at infinity) [22]. The last term of Eq.(15) reduces to ρvac for ωx = −1 and negligible contribution from the cold dark matter component, that is, for low red-shifts. This limit represents the Newton-Hooke space time, which aside from the gravitational interaction has an external force of the Hooke form ∼ Λr2. Gravitational equilibrium is represented through Euler equation ρu̇ = −∇p − ρ∇Φ, where ρ is given by Eq.(16). The next generation of equilibrium equations comes from taking moments on Euler equation and derive the second order virial equation [12, 22, 23] d2Iik = 2Kik +Wgravik + 3Πik + Iik, (18) where Wgravik is the gravitational potential energy tensor, whose trace corresponds to the gravitational potential energy, Kik is the kinetic energy tensor, Iik is the moment of inertia tensor and Πik is the dispersion tensor. The trace of these quantities lead to the well known forms for gravitational potential energy: Wgrav = − ρri∂iΦgravd ρΦgravd 3r, (19) together with I ≡ 2d3r and Π ≡ 3r. As usual, the set of equilibrium equations is closed with the equation for mass conservation, energy conservation and an equation of state p = p(δρ, s). The tensor virial equation can be thought as a differential equation for the moment of inertia, which in turns is converted to a differential equation for the parameters determining the geometrical properties of the configuration. In order to explore the contribution from the background, let us write the virial equation (18) as d2Iik R2H2(R) 2Tik +Wgravik + Iik − . (20) 3. Dynamical equilibrium If we assume equilibrium via Ï ≈ 0, we obtain the virial theorem |Wgrav| = 2K + I, K = T + 3 Π. (21) Strictly speaking, a formal equilibrium configuration is never reached since the energy- like terms in (18) are time dependent. Nevertheless, we can use equation (21) by assuming that as long as R evolves in time, the configuration evolves through successive states of equilibrium. The expressions derived in the last section, specially (15) and (21) can be used for testing dark energy models on configurations in equilibrium. In this section, we will From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 8 concentrate on possible effects of the two cosmological models described above on cosmological structures with non-spherical symmetry, such as low density galaxies and galactic clusters. For ellipsoidal homogeneous configurations, the gravitational potential tensor and the moment of inertia tensor are written as Wgravik = − π2ρ(ρ− ρb)a1a2a3a2iAiδik = −2πδρAiIik, Iik = πρa1a2a3a i δik, (22) where the quantities Ai are functions of the eccentricities of each case: for an oblate configuration we have a1 = a2 > a3, e oblate = 1 − q23, while for prolate a1 = a2 < a3, e2prolate = 1− q−23 , with qi ≡ ai/a1. The functions Ai are given as [24]: A1 = A2 = 1− e2 arcsin e 1− e2 Oblate 1− e2 1− e2 1 + e Prolate 1− e2 1− e2 arcsin e Oblate 1− e2 1 + e Prolate In the limit e → 0 one has Ai → 2/3 for the prolate and oblate configurations. Using Eqs.(22), the potential energies are then written for the models of interest as W ik = Wgravik + Iik (25) π2ρ2a1a2a3a iAiδik − πηxρxIik + 2πρcdm π2ρ2a1a2a3a iAiδik + 2πρch for DECDM and Chaplygin Gas respectuvely, with ηx = 1+3ωx and ρx = ρvac(1+z) f(z). Note that the last term on the DECDM model vanishes in two different situations: the first is in the case of spherical symmetry, and when we neglect the contribution from the cold dark matter component with respect to the proper density of the system and the dark energy contribution. This approximation is only valid for very low red- shifts. At high red shifts we approach a matter dominated universe and hence the term proportional to ρcdm is relevant. 3.1. Oblate systems When considering oblate configurations (a1 = a2 > a3), we assume that the kinetic energy is due to constant rotation along the minor axis. We then may write the kinetic From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 9 energy tensor as Tik = Ω2rotIik − ΩrotiIkjΩrotj , (26) In order to determine the angular velocity, one may consider a non-zero isotropic dispersion tensor Πik = δikΠ. We then have Ω2rot = |Wgravxx | − |Wgravzz | 1− Izz , (27) which follows from eliminating the trace of the dispersion tensor from the tensor virial equations. Using Eq.(25), we can explicitly write for the angular velocity Ω2rot A1 −A3q23 × (28) 1− q23 2πδρ(A1 −A3q23) H20Ωvach2(R) DECDM πρch(R)ηch(R) Chaplygin In the Newton-Hooke space time with ωx = −1 (h2 → −1) this reduces to Ω2rot ≈ A1 − A3q23 − (1− q23). (29) which in turn reproduces the Maclaurin formula for ρvac = 0. 3.2. Prolate systems For prolate configurations (a1 = a2 < a3), we solve for the velocity dispersion of main components rather than for an angular velocity (rotation of prolate configurations, although rare, has been observed and some properties of this situation has been explored in [21]). Hence we use the virial theorem (21) with a the kinetic energy given by K = 1 ρ〈v2〉 d3r = 2 πρa21a3〈v2〉. (30) In analogy with Eq.(28), the virial theorem then implies for the velocity dispersion 2A1 + q 2 + q23 2πδρ(2A1 + q H20Ωvach2(R) πρch(R)ηch(R) for the DECDM and Chaplygin Gas model respectively. As in (28), one obtains a scale factor (or red-shift) dependence of the velocity dispersion through the contribution of the cosmological background. In the Newton-Hooke space-time one reduces to a21(2A1 + q 2 + q23 2A1 + q , (32) which in turn reduces for spherical symmetry to the known expression [12, 25] . (33) From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures10 0.6 1.6 2.6 Scale factor 0.985 0.995 0.6 1.6 2.6 Scale factor 0.9985 0.999 0.9995 e = 0.5 δρ = 100 ρvac δρ = 1000 ρvac Figure 2. Equation (34) for ∆v, for different densities and eccentricities in a ΛCDM model. The structure of the resulting expressions for the angular velocity and the velocity dispersion can be written in a similar way. Let ∆ measures the ratio of each velocities with respect to the velocities when there is not background contribution. We then may write ∆Ω,v ≡ Ω2rot Ω2rot(ρb = 0) 〈v2〉(ρb = 0) = 1− GΩ,v , (34) where the geometrical factor G is written for each case as 1− q23 A1 − q23A3 Oblate, rotational velocity 2 + q23 2A1 + q Prolate, velocity dispersion Returning to the oblate case, in a generalized Newton-Hooke space-time (i.e, with −1 < ω < 0 ), one observes that the Dark Energy component decreases the angular velocity (or the velocity dispersion in the prolate case) in order for the system to maintain equilibrium (since 1 + 3ωx < 0). At the present time (R = 1, z = 0) we get ∆Ω ∼ 1− 0.8g(e)(ρvac/δρ), where g(e) ≡ 4 (1− e2)1/2(3− 2e2) arcsin e− 3e(1− e2) . (36) An extreme situation is reached in the limit ∆Ω → 0, which implies Ω2 → 0. For e ∼ 0.8 this would require ζ ≡ 2ρvac/δρ ∼ 0.5, which is a very diluted configuration. For higher From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures11 0.6 1.6 2.6 3.6 Scale factor e = 0.5 0.6 1.6 2.6 3.6 Scale factor 0.995 0.997 0.999 1.001 1.003 1.005 1.007 1.009 δρ = 100ρcrit β = 2 α = 0.005 δρ = 1000ρcrit β = 2 α = 0.005 Figure 3. Equation (34) for ∆v, for different densities and eccentricities in a Chaplygin Gas model for typical values of Chaplygin parameters. eccentricities (say e ∼ 0.97) one would need a system with ζ ∼ 0.2 which is still a rather low value. In the ΛCDM model Ω = 0 is reached at a red-shift given by [26] zc = [(2Ωvac(ζg(e)− 2))/(ζg(e)Ωcdm)]1/3−1. For realistic examples as δρ ∼ 200ρcdm the value zc could be reached for e very close to 1. That is, the value zc requires extreme flat objects. In figures 2 and 3 we show the behavior of the ratio ∆v for different values of eccentricities and proper densities in the DECDM case with ωx = −1 and in the Chaplygin Gas model, respectively. The larger effects appears for low densities and large eccentricities, as expected. However, the fractional change is very low. For δρ = 102ρvac, the larger effect occurs for scale factors ≥ 2.6 with a change of 1%. In Fig. 4 we show the behavior of ∆Ω − 1 as a function of the red-shift for different densities and e = 0.9, 0.95 for three different values of the equation of state for Dark Energy. It is clear that the effects associated with a cosmological constant are stronger than the ones associated with other Dark Energy models, but in general those effects are small for realistic values of ζ as used in the figure. We would have to measure the angular velocity at different red-shifts very exactly to see an effect over a range of z. However, the difference between the models is more significant. From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures12 0 0.1 0.2 0.3 0.4 redshift z −0.002 −0.001 0.001 0.002 ω x = −1/3 ω x = −2/3 ω x = −1 0 0.2 0.4 redshift z −0.25 −0.15 −0.05 e = 0.9 δρ = 200 ρvac e = 0.95 δρ = 20 ρvac Figure 4. Equation (34) for ∆Ω − 1 in terms of the red-shift (up to a ∼ 0.7), for different densities and eccentricities in a DECDM model for typical values of equation of state. [26] 4. Dynamical evolution In this section we want to explore the virial equation with the cosmological background without insisting on the equilibrium condition (dynamical equilibrium). Even though we will not make extensive use of these new equations, we want to demonstrate that the equilibrium condition is not a necessary ingredient while studying the effects of different background cosmologies. The results are differential equation for the geometrical parameters of a given configuration. Our starting point is equation (20). In the first approach, we will concentrate on pressurless systems with homogeneous density. The problem is the reduced to determine a1 = a1(R) and a3 = a3(R). As pointed out before, Eq.(20) represents a set of two non-linear coupled second order equations for i = x, z d2Iii R2H2(R) 2Tii +Wgravii + Iii − . (37) where we have used Eq.(25). The derivatives of the moment of inertia tensor are given as follows πρa21a3ai = 2a−1i Iii, (38) d2Iii πρa21a3ai  = 2a−1i  Iii, From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures13 These expressions enclose our approximation: the volume of the system is constant even if the semi-axes may change, so that one can write the moment of inertia tensor as Iik = Ma2i δik, (39) where the mass M is M = Mb + δM = πa21a3ρb(R) + δM, δM = δρ d3r. (40) In order to deal with the kinetic energy tensor, we follow [27] and transform from proper coordinates to fixed co-moving coordinates via ri = Aiαxα, where Aiα is such that Aiα = aiδiα and hence the ellipsoid is characterized by the constraint δαβxαx β = 1. We have Tik = xαxβAd 3x, (41) where A = det(Aαβ) = a1a2a3. Using the definition of the moment of inertia tensor one Iik = ρAiαAkβA xαxβd 3x , (42) and hence Tik = A−1iα A Iik. (43) We then may write Tii = Iii = a−2i R2 Iii × H20Ωvach1(R) πρch(R) for the DECDM and Chaplygin Gas respectively. Equation (37) together with (38) are the differential equation to be solved. The functions Ai depend on the eccentricity of the system and hence they are also function of the semiaxis Ai(z) = Ai(a1(R), a3(R)). The differential equation can then be cast into the following form R2H2(R) 2Tii +Wii . (45) 5. Conclusion In this article we have examined the effects of the expansion of the Universe on certain quasi-static properties of large astrophysical bodies. We have done this by invoking a dynamical equilibrium i.e. the time dependent response of the astrophysical objects to the time dependent cosmological background is taken into account by allowing some internal properties to become epoch dependent. In spite of the fact that the expansion of the Universe is accelerated the effects are rather small, at the most few per cents over a large cosmological time stretch in the case of the Chaplygin gas model. However, a qualitative investigation, done in the present paper, seems mandatory to establish the From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures14 size of the effects. Different equations of state can be clearly distinguished theoretically, however, in practice this would require a good knowledge of the angular velocities at different redshifts. In spite of the smallness of the effects, the results clearly demonstrate that background cosmology affects, in principle, local properties of astrophysical bodies and furthermore, the details of these effects are model-dependent. In this paper, we focused on the the dynamical equilibrium. One can enlarge the concept of the influence of background cosmology by dropping the equilibrium condition and go over to a fully dynamical scenario. A first step in this direction was done on section 4. Supplementing these equation with Euler equation for the angular velocity, one has a full set of differential equations determining the time evolution of the parameters during (and after) virialization. We will come back to this point in future publications. References [1] Einstein, A. Straus E. G., Rev. Mod. Phys. 17, 2 and 3, 1945 [2] Noerdlinger P., Petrosian V., Astrophys. J, 168, 1 (1971) [3] Bona C., Stela J., Phys. Rev. D36, 2915 (1987) [4] Baker G., A., arXive:astro-ph/0112320 [5] MacVittie G., C., Mon.Not.Roy.Astron.Soc. 93, 325 (1933) [6] Nunes, N. J. and Mota, D. F., Mon. Not. Roy. Astron. Soc. 368:2 751 (2006), astro-ph/0409481 [7] Perlmutter S. et al., Nature 391, 51, (1998); Perlmutter S. et al., ApJ, 517, 565, (1999) [8] Nesseris S., Perivolaropoulos L., Phys. Rev. D70, 123129 (2004) [9] Nojiri S., Odinstov S. D., arXiv: hep-th/0505215; Godlowski W., Szydlowski M., arXiv: astro-ph/0507322. Polarski D., Ranquet A., arXiv: astro-ph/0507299. Hu W., Phys. Rev. D71, 047301 (2005). arXiv: astro-ph/0410680. [10] Barreiro, T., Copeland, E., Nunes, N. J. Phys. Rev. D 61, 127301 (2000); Steinhardt, P. J., Wang, L., Zlatev, I., Phys. Rev. D59, 123504 (1999); Albrecht, L.R., Skordis, K., Phys. Rev. Lett. 84 2076 (2000); L. Wang, R. R. Caldwell, J. P. Ostriker, Paul J. Steinhardt. ApJ. 530, 17-35 (2000), arXiv: astro-ph/9901388. [11] Cardoso, V. and Gaultieri, L., Class. Quantum Grav. 23, 7198 (2006) [12] Balaguera-Antoĺınez, A., Nowakowski, M., Astron. Astrophys 441, 23 (2005) [13] Mota, D. F. and van de Bruck, C., Astron. Astrophys. 421, 71 (2004) [14] Balaguera-Antoĺınez, A., Böhmer, C., Nowakowski, M, Class. Quantum Grav. 23, 485-496 (2006) [15] Douglas J. Shaw, John D. Barrow Phys.Lett. B639 596-599 (2006) [16] Douglas J. Shaw, John D. Barrow Phys.Lett. B73 123506 (2006) [17] Douglas J. Shaw, John D. Barrow Phys.Lett. B73 123505 (2006) [18] Boehmer, C. G., Gen. Rel. Grav 36, 1039 (2004) [19] Boehmer, C. G. and Harko, T., Phys. Rev. D71, 084026 (2005) [20] Debnath, U., Nath, S. and Chakraborty, S., Mon. Not. Astron. Soc. 369, 1961 (2006) [21] Balaguera-Antoĺınez, A., Mota, D., F., Nowakowski, M., Class. Quantum Grav. 23, 4497-4510 (2006) [22] Nowakowski, M., Int. J. Mod. Phys., D10, 649 (2001) [23] Nowakowski, M., Sanabria, J.-C., and Garcia, A., Phys. Rev. D66, 023003 (2002) [24] Binney, J & Tremaine, S., Galactic Dynamics, Princeton University Press, 1987 [25] Wang, L., and Steinhardt, P. J., ApJ 508, 483 (1998) [26] Balaguera-Antoĺınez, A., Nowakowski, M., arXive: astro-ph/0603624 [27] Peebles P.J.E., The Large-Scale structure of the Universe, Princeton University Press, 1980. http://arxiv.org/abs/astro-ph/0112320 http://arxiv.org/abs/astro-ph/0409481 http://arxiv.org/abs/hep-th/0505215 http://arxiv.org/abs/astro-ph/0507322 http://arxiv.org/abs/astro-ph/0507299 http://arxiv.org/abs/astro-ph/0410680 http://arxiv.org/abs/astro-ph/9901388 http://arxiv.org/abs/astro-ph/0603624 Introduction Local dynamics with background cosmology Dark energy Chaplygin Gas Newtonian description Dynamical equilibrium Oblate systems Prolate systems Dynamical evolution Conclusion

We explore the effects of background cosmology on large scale structures with non-spherical symmetry by using the concept of quasi-equilibrium which allows certain internal properties (e.g. angular velocity) of the bodies to change with time. In accordance with the discovery of the accelerated phase of the universe we model the cosmological background by two representative models: the $\Lambda$CDM Model and the Chaplygin Gas Model. We compare the effects of the two models on various properties of large astrophysical objects. Different equations of state are also invoked in the investigation.

Introduction The interest in the impact of the cosmological expansion on bound astrophysical system can be traced back to the paper by Einstein and Straus [1]. Over the subsequent years different approaches have been used to gain some insight into the interplay between cosmology and astrophysical structures [2, 3, 4, 5, 6]. With the advent of the discovery of the accelerated Universe [7], it seems timely to pick up the topic once again and contrast the effects of different models which can explain the current accelerated phase. Indeed, in [8] such an examination was performed for phantom [9] and quintessence cosmologies [10]. In the present article we will use the same approach as [2] and [8] and apply it to models with a positive cosmological constant and Chaplygin gas models. These two models will serve us as representatives for cosmological models which can explain the current stage of acceleration. In addition we will also invoke a general model with variable equation of state of the form ωx = px/ρx (DECDM). The approach mentioned above consists in generalizing the Newtonian limit by allowing terms which are specific to the background cosmology. This is not limited to the inclusion of the cosmological constant. Notice that the effects of the background cosmology on bound systems are now in general time dependent. This is, however not in contradiction with the fact that we rightly consider most of the astrophysical structures to be in gravitational equilibrium. The time dependent background cosmology will not tear the system apart, unless its density is diluted, but change only, over cosmological time, the internal properties of the system (like inner velocities or angular velocity etc). As a mathematical tool we will make use of the virial theorem ([11] for different methods) by allowing certain internal properties to be time dependent. The virial theorem can be easily derived even in the presence of the cosmological background as the latter enters only the Newtonian limit (i.e. the Poisson equation for the gravitational potential which includes now a time dependent part). As shown in [12] Dark Energy can affect certain astrophysical static properties which have to do with either the virialization of the system [13] or the motion of test bodies [2]. For instance, the cosmological constant sets itself scales of distance (rΛ = Λ −1/2) ‡, time and mass, which are of the same order of magnitude as the radius of the observed universe, the age of the universe and the total mass of the universe (the so called coincidence problem). At the first glance it looks hopeless to expect any effects from Λ at astrophysical scales. This is so indeed as long as no other scale enters the theory. If the latter appears in the theory a combination of the large cosmological and the small internal scale can yield values of astrophysical relevance (e.g. (rsrΛ) 1/3 where rs is the Schwarzschild radius) [14]. Furthermore, general conditions for detecting the effects of variations of ’constants’ represented by scalar fields (just like the cosmological constant and quintessence models) at scales ranging from terrestial scales have been well established in [15, 16, 17]. Another enhancement mechanism of background cosmology emerges when we are dealing with non-spherically symmetric objects. In such case ‡ We work in the so-called geometrized units, i.e. GN = c = 1 From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 3 the ratio of two length scales to some positive power (bigger than one) goes often hand in hand with Λ. For this reason, in examining time dependent effects, we will mostly employ ellipsoidal configurations. For general consideration of equilibrium in the spherical case see [18] and [19], the quasi spherical case has been discussed in [20]. In this paper, we will entirely concentrate on effects of Dark Energy followed over cosmological times, past and future. We do not expect these effects to be large, however, we think that it is of some importance to probe into these matter, especially in view of the accelerated universe. In [21] we restricted ourselves to effects over small time scales. In this sense these two articles complement each other. 2. Local dynamics with background cosmology In standard cosmology, the universe is described as an ideal fluid which implies the properties of homogeneity and isotropy. Such properties are mathematically represented in Friedman-Robertson-Walker line element, which is written for a flat universe as ds2 = −dt2 + R2(t) (dr2 + r2dΩ2). Einstein field equations and the FRW line element yields the acceleration equation and the well known Friedman equation, written respectively Ṙ(t) = H(t)2 = πρb(t), R̈(t) π [ρb(t) + 3pb(t)] , (1) In conventional cosmology, the total energy density ρb is a contribution of a radiation, cold dark matter and dark energy. By neglecting interaction among these components, each one evolves through the conservation equation ρ̇ = −3H(ρ+ p), so that radiation scales as ρrad = ρrad(R = 1)R −4, cold dark matter scales as ρcdm = ρcdm(R = 1)R while the Dark Energy component associated with the cosmological constant is ρvac = constant. The major contribution in connection to the cosmological constant is approximately 70% of total energy density [7]. 2.1. Dark energy As pointed before, almost the 70% of the content of energy density in the unverse is ruled by a dark energy component, and astronomical observations imply that at the present time this dark energy component might be represented by the cosmological constant. That is, at the present time, the equation of state that displays the best fit with astronomical observation is simply pvac = −ρvac, i.e, ω = −1. Nevertheless, several models has been proposed in order to reproduce the present value of dark energy but maintaining a time dependent energy density. One generalizes the equation of state for the dark energy by px(R) = ωx(R)ρx(R). Dark energy then scales as ρx(R) = ρx(R = 1)a −f(R), where the function f(R) is defined as f(R) ≡ 3 ′) + 1 dR′, (2) From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 4 The model is constrained by requiring that at the present time the dark energy density reproduces the vacuum energy density associated with the cosmological constant, that is, ρx(R = 1) = ρvac. As pointed before, the case ωx = −1 corresponds to the cosmological constant ρx = ρvac = Λ/8π, while for 0 > ωx > −1 one stands in the Dark Energy realms. Models with ωx < −1 lead to future singularities in the so called Phantom regime [9]. The description of a dynamical dark energy is compatible with the description of inflationary models. In the DECDM model, the Friedmann equation and the acceleration equation are written as Ṙ(t) = H20Ωvach1(R) R̈(t) = −H20Ωvach2(R) = − πρvach2(R).(3) The functions h1,2(R) are given as h1(R) ≡ R−3 +R−f(R), (4) h2(R) ≡ R−3 +R−f(z) (1 + 3ωx(R)) 2.2. Chaplygin Gas The main stages the universe has passed through in the standard cosmology scenario can be reproduced by the introduction of the equation of state of an exotic ideal relativistic gas written as pch = κ1ρch − , (5) The relevance of this equation of state lies in three different facts: a) it violates the strong energy condition, which is necessary to obtain an accelerated phase at the present time b) it generates a well definite speed of sound, which is relevant for the process of structure formation and c) it unifies the early radiation or dark-matter behavior with the late dark energy dominance. The so-called pure Chaplygin gas is obtained with κ1 = 0 and γ = 1. The time evolution of the Chaplygin Gas-energy density is given from the integration of mass-energy conservation equation as ρch(R) = A+BR−nβ , (6) with n = 3(1 + κ1), β = γ + 1 and where A and B are integration constants given as B = ρch(R = 1) β − A, A = 0 , (7) with ρch(R = 1) = ρcrit as the Chaplygin Gas energy density at the present time (only for a flat universe). In order to write the integration constants as we have done, we have re-parametrized the Chaplygin Gas equation of state with κ2 = αnH 0 such that a finite age and a present accelerated phase for the Chaplygin gas universe is reached for . (8) From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 5 This bound can be derived after integration of the Friedmann equation H(R)2 = A+BR−nβ . (9) in order to determine the age of the universe. The age of the universe is given in terms of hypergeometric function as × 2F1 2nβ + n+ 4 with q ≡ 3ω + 7. For the pure Chaplygin gas one then obtains a finite age and the observed acceleration at the present time only if the parameter α is such that α3 < 10 which in turn implies a bound on the Chaplygin parameter κ2 < ρ crit. This warranties that the equation of state is ωch(today) > −1. The solution given in Eq.(6) allows us to describe the behavior of Chaplygin gas at different ages. For early times the κ1-term dominates and Chaplygin Gas behaves as ρch = B 1/βR−n. This scales as radiation for n = 4 (κ1 = 1/3). Hence we could in principle relate the integration constant B with cosmological parameters of the concordance model as B = ρ rad(R = 1). For κ1 = 0, the model displays a matter dominated epoch at early times with ρch ≈ B1/βR−3. In this case one can write the integration constant B = ρ mat(R = 1) and also the constant A as A = ρ crit(1 − Ω mat). Finally for large times, Chaplygin gas acquires a vacuum-like behavior with ρch(t → ∞) = A1/β = H20 . (11) This would lead to an effective vacuum energy density ρeffvac which can be associated to the current vacuum energy density ρvac via ρeffvac = ρvac With ρeffvac > ρvac, the effects of this effective vacuum energy density may be relevant in the Newtonian limit when we explore the equilibrium conditions of large scale structures. Let us concentrate on the case κ1 = 0, which does not reproduce an early radiation dominated era but still unifies an early dark matter dominance and a vacuum dominated era at large scale factors. The acceleration equation for this model reads as R̈(t) = − 4 πρch(R)ηch(R) (13) = − 4 A +BR(t)−3β 1− 3κ2 A +BR(t)−3β where ηch ≡ 1 + 3ωch and ωch = ωch(R) ≡ = −κ2ρch(R)−β (14) From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 6 1 1.5 2 2.5 3 3.5 4 Scale Factor β = 2 , α = 0.001 2 , 0.0006 2 , 0.0001 1 1.5 2 2.5 3 3.5 4 4.5 5 Scale factor −1e−14 −5e−15 5e−15 1e−14 1.5e−14 2e−14 Figure 1. Left: Equation of state for Chaplygin gas ωch = pch/ρch for different Chaplygin parameters. Right: Acceleration equation as a function of the (future) scale factor for different Chaplygin parameters. is the time dependent equation of state. In Fig. 1 we show the behavior of the equation of state ωch and the acceleration equation for different values of Chaplygin parameters. 2.3. Newtonian description The effects of the background can be explored through the Newtonian limit of field equations [22], from which one can derive a modified Poisson’s equation [2] ∇2Φ = 4πδρ− 3 , (15) where δρ is the overdensity that gives rise to the “gravitational contribution” of the potential. The total energy density within the clustered configuration ρ is a contribution of the background ρb and the collapsed fraction δρ. The background energy density depends on the model we are interested in: ρb = ρ(R)− δρ = ρcdm(R) DECDM ρch(R) Chaplygin Note that the last expression implies that in the DECDM model one has assumed that only the cold dark matter component is what collapses and forms vitalized structures, while in Chaplygin gas the perturbation is done on the total energy density. The solution From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 7 for the potential can be simplified as Φ(r, R) = Φgrav(r)− r2, Φgrav(r) = − δρ(r′) |r− r′| d3r′, (17) where one has neglected the terms associated to boundary conditions (the potential cannot be zero at infinity) [22]. The last term of Eq.(15) reduces to ρvac for ωx = −1 and negligible contribution from the cold dark matter component, that is, for low red-shifts. This limit represents the Newton-Hooke space time, which aside from the gravitational interaction has an external force of the Hooke form ∼ Λr2. Gravitational equilibrium is represented through Euler equation ρu̇ = −∇p − ρ∇Φ, where ρ is given by Eq.(16). The next generation of equilibrium equations comes from taking moments on Euler equation and derive the second order virial equation [12, 22, 23] d2Iik = 2Kik +Wgravik + 3Πik + Iik, (18) where Wgravik is the gravitational potential energy tensor, whose trace corresponds to the gravitational potential energy, Kik is the kinetic energy tensor, Iik is the moment of inertia tensor and Πik is the dispersion tensor. The trace of these quantities lead to the well known forms for gravitational potential energy: Wgrav = − ρri∂iΦgravd ρΦgravd 3r, (19) together with I ≡ 2d3r and Π ≡ 3r. As usual, the set of equilibrium equations is closed with the equation for mass conservation, energy conservation and an equation of state p = p(δρ, s). The tensor virial equation can be thought as a differential equation for the moment of inertia, which in turns is converted to a differential equation for the parameters determining the geometrical properties of the configuration. In order to explore the contribution from the background, let us write the virial equation (18) as d2Iik R2H2(R) 2Tik +Wgravik + Iik − . (20) 3. Dynamical equilibrium If we assume equilibrium via Ï ≈ 0, we obtain the virial theorem |Wgrav| = 2K + I, K = T + 3 Π. (21) Strictly speaking, a formal equilibrium configuration is never reached since the energy- like terms in (18) are time dependent. Nevertheless, we can use equation (21) by assuming that as long as R evolves in time, the configuration evolves through successive states of equilibrium. The expressions derived in the last section, specially (15) and (21) can be used for testing dark energy models on configurations in equilibrium. In this section, we will From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 8 concentrate on possible effects of the two cosmological models described above on cosmological structures with non-spherical symmetry, such as low density galaxies and galactic clusters. For ellipsoidal homogeneous configurations, the gravitational potential tensor and the moment of inertia tensor are written as Wgravik = − π2ρ(ρ− ρb)a1a2a3a2iAiδik = −2πδρAiIik, Iik = πρa1a2a3a i δik, (22) where the quantities Ai are functions of the eccentricities of each case: for an oblate configuration we have a1 = a2 > a3, e oblate = 1 − q23, while for prolate a1 = a2 < a3, e2prolate = 1− q−23 , with qi ≡ ai/a1. The functions Ai are given as [24]: A1 = A2 = 1− e2 arcsin e 1− e2 Oblate 1− e2 1− e2 1 + e Prolate 1− e2 1− e2 arcsin e Oblate 1− e2 1 + e Prolate In the limit e → 0 one has Ai → 2/3 for the prolate and oblate configurations. Using Eqs.(22), the potential energies are then written for the models of interest as W ik = Wgravik + Iik (25) π2ρ2a1a2a3a iAiδik − πηxρxIik + 2πρcdm π2ρ2a1a2a3a iAiδik + 2πρch for DECDM and Chaplygin Gas respectuvely, with ηx = 1+3ωx and ρx = ρvac(1+z) f(z). Note that the last term on the DECDM model vanishes in two different situations: the first is in the case of spherical symmetry, and when we neglect the contribution from the cold dark matter component with respect to the proper density of the system and the dark energy contribution. This approximation is only valid for very low red- shifts. At high red shifts we approach a matter dominated universe and hence the term proportional to ρcdm is relevant. 3.1. Oblate systems When considering oblate configurations (a1 = a2 > a3), we assume that the kinetic energy is due to constant rotation along the minor axis. We then may write the kinetic From Global to Local Dynamics: Effects of the Expansion on Astrophysical Structures 9 energy tensor as Tik = Ω2rotIik − ΩrotiIkjΩrotj , (26) In order to determine the angular velocity, one may consider a non-zero isotropic dispersion tensor Πik = δikΠ. We then have Ω2rot = |Wgravxx | − |Wgravzz | 1− Izz , (27) which follows from eliminating the trace of the dispersion tensor from the tensor virial equations. Using Eq.(25), we can explicitly write for the angular velocity Ω2rot A1 −A3q23 × (28) 1− q23 2πδρ(A1 −A3q23) H20Ωvach2(R) DECDM πρch(R)ηch(R) Chaplygin In the Newton-Hooke space time with ωx = −1 (h2 → −1) this reduces to Ω2rot ≈ A1 − A3q23 − (1− q23). (29) which in turn reproduces the Maclaurin formula for ρvac = 0. 3.2. Prolate systems For prolate configurations (a1 = a2 < a3), we solve for the velocity dispersion of main components rather than for an angular velocity (rotation of prolate configurations, although rare, has been observed and some properties of this situation has been explored in [21]). Hence we use the virial theorem (21) with a the kinetic energy given by K = 1 ρ〈v2〉 d3r = 2 πρa21a3〈v2〉. (30) In analogy with Eq.(28), the virial theorem then implies for the velocity dispersion 2A1 + q 2 + q23 2πδρ(2A1 + q H20Ωvach2(R) πρch(R)ηch(R) for the DECDM and Chaplygin Gas model respectively. As in (28), one obtains a scale factor (or red-shift) dependence of the velocity dispersion through the contribution of the cosmological background. In the Newton-Hooke space-time one reduces to a21(2A1 + q 2 + q23 2A1 + q , (32) which in turn reduces for spherical symmetry to the known expression [12, 25] . (33) From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures10 0.6 1.6 2.6 Scale factor 0.985 0.995 0.6 1.6 2.6 Scale factor 0.9985 0.999 0.9995 e = 0.5 δρ = 100 ρvac δρ = 1000 ρvac Figure 2. Equation (34) for ∆v, for different densities and eccentricities in a ΛCDM model. The structure of the resulting expressions for the angular velocity and the velocity dispersion can be written in a similar way. Let ∆ measures the ratio of each velocities with respect to the velocities when there is not background contribution. We then may write ∆Ω,v ≡ Ω2rot Ω2rot(ρb = 0) 〈v2〉(ρb = 0) = 1− GΩ,v , (34) where the geometrical factor G is written for each case as 1− q23 A1 − q23A3 Oblate, rotational velocity 2 + q23 2A1 + q Prolate, velocity dispersion Returning to the oblate case, in a generalized Newton-Hooke space-time (i.e, with −1 < ω < 0 ), one observes that the Dark Energy component decreases the angular velocity (or the velocity dispersion in the prolate case) in order for the system to maintain equilibrium (since 1 + 3ωx < 0). At the present time (R = 1, z = 0) we get ∆Ω ∼ 1− 0.8g(e)(ρvac/δρ), where g(e) ≡ 4 (1− e2)1/2(3− 2e2) arcsin e− 3e(1− e2) . (36) An extreme situation is reached in the limit ∆Ω → 0, which implies Ω2 → 0. For e ∼ 0.8 this would require ζ ≡ 2ρvac/δρ ∼ 0.5, which is a very diluted configuration. For higher From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures11 0.6 1.6 2.6 3.6 Scale factor e = 0.5 0.6 1.6 2.6 3.6 Scale factor 0.995 0.997 0.999 1.001 1.003 1.005 1.007 1.009 δρ = 100ρcrit β = 2 α = 0.005 δρ = 1000ρcrit β = 2 α = 0.005 Figure 3. Equation (34) for ∆v, for different densities and eccentricities in a Chaplygin Gas model for typical values of Chaplygin parameters. eccentricities (say e ∼ 0.97) one would need a system with ζ ∼ 0.2 which is still a rather low value. In the ΛCDM model Ω = 0 is reached at a red-shift given by [26] zc = [(2Ωvac(ζg(e)− 2))/(ζg(e)Ωcdm)]1/3−1. For realistic examples as δρ ∼ 200ρcdm the value zc could be reached for e very close to 1. That is, the value zc requires extreme flat objects. In figures 2 and 3 we show the behavior of the ratio ∆v for different values of eccentricities and proper densities in the DECDM case with ωx = −1 and in the Chaplygin Gas model, respectively. The larger effects appears for low densities and large eccentricities, as expected. However, the fractional change is very low. For δρ = 102ρvac, the larger effect occurs for scale factors ≥ 2.6 with a change of 1%. In Fig. 4 we show the behavior of ∆Ω − 1 as a function of the red-shift for different densities and e = 0.9, 0.95 for three different values of the equation of state for Dark Energy. It is clear that the effects associated with a cosmological constant are stronger than the ones associated with other Dark Energy models, but in general those effects are small for realistic values of ζ as used in the figure. We would have to measure the angular velocity at different red-shifts very exactly to see an effect over a range of z. However, the difference between the models is more significant. From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures12 0 0.1 0.2 0.3 0.4 redshift z −0.002 −0.001 0.001 0.002 ω x = −1/3 ω x = −2/3 ω x = −1 0 0.2 0.4 redshift z −0.25 −0.15 −0.05 e = 0.9 δρ = 200 ρvac e = 0.95 δρ = 20 ρvac Figure 4. Equation (34) for ∆Ω − 1 in terms of the red-shift (up to a ∼ 0.7), for different densities and eccentricities in a DECDM model for typical values of equation of state. [26] 4. Dynamical evolution In this section we want to explore the virial equation with the cosmological background without insisting on the equilibrium condition (dynamical equilibrium). Even though we will not make extensive use of these new equations, we want to demonstrate that the equilibrium condition is not a necessary ingredient while studying the effects of different background cosmologies. The results are differential equation for the geometrical parameters of a given configuration. Our starting point is equation (20). In the first approach, we will concentrate on pressurless systems with homogeneous density. The problem is the reduced to determine a1 = a1(R) and a3 = a3(R). As pointed out before, Eq.(20) represents a set of two non-linear coupled second order equations for i = x, z d2Iii R2H2(R) 2Tii +Wgravii + Iii − . (37) where we have used Eq.(25). The derivatives of the moment of inertia tensor are given as follows πρa21a3ai = 2a−1i Iii, (38) d2Iii πρa21a3ai  = 2a−1i  Iii, From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures13 These expressions enclose our approximation: the volume of the system is constant even if the semi-axes may change, so that one can write the moment of inertia tensor as Iik = Ma2i δik, (39) where the mass M is M = Mb + δM = πa21a3ρb(R) + δM, δM = δρ d3r. (40) In order to deal with the kinetic energy tensor, we follow [27] and transform from proper coordinates to fixed co-moving coordinates via ri = Aiαxα, where Aiα is such that Aiα = aiδiα and hence the ellipsoid is characterized by the constraint δαβxαx β = 1. We have Tik = xαxβAd 3x, (41) where A = det(Aαβ) = a1a2a3. Using the definition of the moment of inertia tensor one Iik = ρAiαAkβA xαxβd 3x , (42) and hence Tik = A−1iα A Iik. (43) We then may write Tii = Iii = a−2i R2 Iii × H20Ωvach1(R) πρch(R) for the DECDM and Chaplygin Gas respectively. Equation (37) together with (38) are the differential equation to be solved. The functions Ai depend on the eccentricity of the system and hence they are also function of the semiaxis Ai(z) = Ai(a1(R), a3(R)). The differential equation can then be cast into the following form R2H2(R) 2Tii +Wii . (45) 5. Conclusion In this article we have examined the effects of the expansion of the Universe on certain quasi-static properties of large astrophysical bodies. We have done this by invoking a dynamical equilibrium i.e. the time dependent response of the astrophysical objects to the time dependent cosmological background is taken into account by allowing some internal properties to become epoch dependent. In spite of the fact that the expansion of the Universe is accelerated the effects are rather small, at the most few per cents over a large cosmological time stretch in the case of the Chaplygin gas model. However, a qualitative investigation, done in the present paper, seems mandatory to establish the From Global to Local Dynamics:Effects of the Expansion on Astrophysical Structures14 size of the effects. Different equations of state can be clearly distinguished theoretically, however, in practice this would require a good knowledge of the angular velocities at different redshifts. In spite of the smallness of the effects, the results clearly demonstrate that background cosmology affects, in principle, local properties of astrophysical bodies and furthermore, the details of these effects are model-dependent. In this paper, we focused on the the dynamical equilibrium. One can enlarge the concept of the influence of background cosmology by dropping the equilibrium condition and go over to a fully dynamical scenario. A first step in this direction was done on section 4. Supplementing these equation with Euler equation for the angular velocity, one has a full set of differential equations determining the time evolution of the parameters during (and after) virialization. We will come back to this point in future publications. References [1] Einstein, A. Straus E. G., Rev. Mod. Phys. 17, 2 and 3, 1945 [2] Noerdlinger P., Petrosian V., Astrophys. J, 168, 1 (1971) [3] Bona C., Stela J., Phys. Rev. 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Quantum Grav. 23, 7198 (2006) [12] Balaguera-Antoĺınez, A., Nowakowski, M., Astron. Astrophys 441, 23 (2005) [13] Mota, D. F. and van de Bruck, C., Astron. Astrophys. 421, 71 (2004) [14] Balaguera-Antoĺınez, A., Böhmer, C., Nowakowski, M, Class. Quantum Grav. 23, 485-496 (2006) [15] Douglas J. Shaw, John D. Barrow Phys.Lett. B639 596-599 (2006) [16] Douglas J. Shaw, John D. Barrow Phys.Lett. B73 123506 (2006) [17] Douglas J. Shaw, John D. Barrow Phys.Lett. B73 123505 (2006) [18] Boehmer, C. G., Gen. Rel. Grav 36, 1039 (2004) [19] Boehmer, C. G. and Harko, T., Phys. Rev. D71, 084026 (2005) [20] Debnath, U., Nath, S. and Chakraborty, S., Mon. Not. Astron. Soc. 369, 1961 (2006) [21] Balaguera-Antoĺınez, A., Mota, D., F., Nowakowski, M., Class. Quantum Grav. 23, 4497-4510 (2006) [22] Nowakowski, M., Int. J. Mod. Phys., D10, 649 (2001) [23] Nowakowski, M., Sanabria, J.-C., and Garcia, A., Phys. Rev. D66, 023003 (2002) [24] Binney, J & Tremaine, S., Galactic Dynamics, Princeton University Press, 1987 [25] Wang, L., and Steinhardt, P. J., ApJ 508, 483 (1998) [26] Balaguera-Antoĺınez, A., Nowakowski, M., arXive: astro-ph/0603624 [27] Peebles P.J.E., The Large-Scale structure of the Universe, Princeton University Press, 1980. http://arxiv.org/abs/astro-ph/0112320 http://arxiv.org/abs/astro-ph/0409481 http://arxiv.org/abs/hep-th/0505215 http://arxiv.org/abs/astro-ph/0507322 http://arxiv.org/abs/astro-ph/0507299 http://arxiv.org/abs/astro-ph/0410680 http://arxiv.org/abs/astro-ph/9901388 http://arxiv.org/abs/astro-ph/0603624 Introduction Local dynamics with background cosmology Dark energy Chaplygin Gas Newtonian description Dynamical equilibrium Oblate systems Prolate systems Dynamical evolution Conclusion

Evidence for nonmonotonic magnetic field penetration in a type-I superconductor V.F. Kozhevnikov1∗, C.V. Giuraniuc1, M.J. Van Bael1, K. Temst2, C. Van Haesendonck1, T.M. Mishonov3, T. Charlton4, R.M. Dalgliesh4, Yu.N. Khaidukov5, Yu.V. Nikitenko5, V.L. Aksenov5, V.N. Gladilin1,6, V.M. Fomin6, J.T. Devreese6, and J.O. Indekeu1 Laboratorium voor Vaste-Stoffysica en Magnetisme Katholieke Universiteit Leuven, 3001 Leuven, Belgium Instituut voor Kern- en Stralingsfysica, Katholieke Universiteit Leuven, 3001 Leuven, Belgium Department of Theoretical Physics, St Clement of Ohrid University of Sofia, 1164 Sofia, Bulgaria ISIS Science Division, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, United Kingdom Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia Theoretische Fysica van de Vaste Stoffen, Universiteit Antwerpen, 2020 Antwerpen, Belgium (Dated: October 24, 2021) Polarized neutron reflectometry (PNR) provides evidence that nonlocal electrodynamics governs the magnetic field penetration in an extreme low-κ superconductor. The sample is an indium film with a large elastic mean free path (11 µm) deposited on a silicon oxide wafer. It is shown that PNR can resolve the difference between the reflected neutron spin asymmetries predicted by the local and nonlocal theories of superconductivity. The experimental data support the nonlocal theory, which predicts a nonmonotonic decay of the magnetic field. PACS numbers: 74.20.-z, 74.25.Ha, 78.70.Nx In this paper we pose and answer experimentally the following fundamental questions. Are nonlocal electrody- namics effects measurable in superconductors? Can the nonmonotonic decay of magnetic field penetration pre- dicted by the nonlocal theory be observed? To what ex- tent can Polarized Neutron Reflectometry (PNR) resolve the difference between local and nonlocal diamagnetic re- sponses expected for strongly type-I superconductors? Nonlocality is a key concept of superconductivity the- ory, but its experimental verification is still not estab- lished. In the Meissner state, a magnetic field ap- plied parallel to the surface located at z = 0 causes the magnetic induction B(z) to penetrate over a depth λ ≡ B(0)−1 B(z)dz. In the London (local) limit, appropriate to most type-II superconductors, B(z) ∝ exp(−z/λL), where λL is the London penetration depth. In 1953, to explain the variation of λ in type-I supercon- ductor Sn caused by adding In, Pippard proposed that the current density is related to the average of the vector potential over a region of size ξ0 (the Pippard coherence length) [1]. The smaller the Ginzburg-Landau parameter κ, the more important this nonlocal effect. Nonlocal theory predicts that B(z) deviates from a simple exponential decay. B(z) is nonmonotonic and, moreover, changes sign at a specific depth [1]. In the pure limit (ξ0 ≪ ℓ, where ℓ is the elastic mean free path) B(z) is determined by the intrinsic parameters λL(T = 0) and ξ0, and by the temperature T . The devia- tion is most significant in “extreme” type-I superconduc- tors (κ ≪ 1/ 2), such as Al (κ ≈ 0.01) and In (0.06). For these the results of the Pippard theory are identi- cal to those of the Bardeen-Cooper-Schrieffer theory [2]. On the other hand, the local approximation is safe for κ > 1.5 [3], i.e., marginal nonlocal effects are predicted for some type-II superconductors such as Nb (κ ≈ 1). � �� ��� ��� ��� ��� ��� ��� ��� ������ ���������� � ���� FIG. 1: Magnetic-induction profiles B(z) in a semi-infinite In sample. The dashed (solid) line corresponds to the local (non- local) relation between current density and vector potential. B(z) in In, calculated in local and nonlocal approaches, in the pure limit with ξ0 = 0.38 µm and λL(0) = 0.025 µm [4], for T = 1.8 K, is shown in Fig. 1. Details about the formalism can be found in Ref. 3. In the nonlocal approach B(2λL)/B(0) ≈ 1/e. The sign reversal is ex- pected at z ≈ 5.5λL, and the amplitude of the reversed field is at most about 3% of the field at the surface. An observation of sign reversal was reported in Ref. 5. An external AC magnetic field H with amplitude up to 30 Oe was applied parallel to a hollow cylindrical Sn film about 2 µm thick, and a strongly attenuated (108 times) signal with reversed phase was detected inside the cylinder at T = 2.88 K and H ≈ 25 Oe. This phase difference was interpreted as a change of sign in the pen- etrating field. However, this is questionable because the phase difference drops back to zero at a larger (30 Oe) field, whereas the critical field Hc at 2.9 K is 115 Oe [6]. http://arxiv.org/abs/0704.1872v2 Nowadays B(z) can be measured directly using po- larized neutron reflectometry (PNR) [7] and low-energy muon spin rotation (LE-µSR) [8] techniques. We com- ment briefly on the latter before focusing on the former. In the LE-µSR technique positive muons polarized per- pendicularly to the applied field are implanted in a sam- ple over a distance determined by the muon energy. B(z) is obtained from measuring the Larmor precession fre- quency of the muon spins at stopping distance. In other words, the implanted muons serve as tiny sensors of the magnetic field inside the sample. However, in practice the muon precession is progressively damped with depth due to a broad distribution of stopping distances [9]. From our prospectus, this is the main difficulty in applying the LE-µSR technique to fields with a sharp profile. Recently, the LE-µSR technique was used to measure B(z) in Pb, Nb, and Ta [9]. Most interesting is the obser- vation of a non-exponential shape of B(z) for all studied metals. The nonlinearity of the semi-log plots for B(z) is marginal, which is exactly what should be expected theoretically in view of the fairly high κ of the studied samples. For example, κ of pure Nb (residual resistivity ratioRRR = 1600) is 1.3 at 3 K and 1.0 at 7 K [10]. How- ever, in Ref. 9 κ of less pure Nb (RRR=133) is reported to be 0.7 at 2.96 K and 7.6 K. This and some other in- consistencies with well established literature data suggest that the muon probing results may contain some hidden uncertainties. Therefore, additional experiments would be worthwhile, in particular on low-κ superconductors. The PNR technique is based on the change of the neu- tron index of refraction in a magnetized medium. When a collimated neutron beam polarized along the magnetic field is incident on a flat, laterally uniform sample under a grazing angle, its specular reflectivity R is determined by the profile of the neutron scattering potential below the surface. R is measured versus momentum transfer Q = 4π sin θ/λn, where θ is the angle of incidence and λn the neutron wavelength. The scattering potential con- sists of a nuclear and a magnetic part, which results in different reflectivities R+ and R− for neutrons with spins polarized parallel (up) and anti-parallel (down) to the applied field, respectively. Direct information about the sample magnetization is obtained by combining R+ and R−; the combination s = (R+ − R−)/(R+ + R−) is the spin asymmetry. B(z) can be found by fitting s(Q) data with s(Q) calculations based on theoretical models for B(z). PNR has been applied for measuring the penetra- tion depth and for detecting surface superconductivity in Nb [11, 12], high-Tc cuprates [13, 14, 15] and Pb [16, 17]. The nonlocal effect in B(z) measured with PNR was discussed in Refs. 11, 12, 16 and 17. Although some devi- ation from exponential decay was noticed in Refs. 12 and 16, no solid confirmation of the nonlocal theory was ob- tained. The authors of Ref. 17 correctly pointed out that experiments with low-κ type-I superconductors are de- sirable to verify nonlocality, but their overall conclusion was that PNR is incapable of detecting nonlocality in any superconductor. In this light it is very interesting to reassess the problem of nonlocality with state-of-the-art PNR applied to a low-κ material such as In, since during the last decade many neutron source facilities have sig- nificantly progressed in neutron flux, neutron optics and detector technology. The design of the sample for the PNR study is based on the following requirements. The irradiated surface must be flat and possess minimal possible roughness. The sample must be thick enough to have the same electro- magnetic properties as the bulk material. Degradation of the surface quality with increasing thickness limits the film thickness. Neutrons reflected back from the sub- strate should have a negligible effect on the reflectivity in a region close to the critical edge of total reflection, Qc, where the reflectivity is most sensitive to the magnetic properties. Two approaches can meet these requirements. One is to deposit a thick film on a flat substrate that reflects least. This can be achieved if the neutron refraction index of the substrate is larger than that of the sample. This approach was taken in the experiments on Nb [11, 12] and Pb [16, 17]. In fact, this was the only option, in view of the negligibly small absorption of neutrons in Nb and Pb. However, In is a strong absorber, which enables one to rely on substrates with a refractive index smaller than that of In, provided the thickness of the indium film is properly optimized. In this approach a second plateau or “hill”, associated with total reflection from the sample- substrate interface, is expected in the reflectivity curve R(Q). This should yield additional information about the sample structure. Modelling shows that an indium thickness of 2.5 µm is appropriate. Such a sample was fabricated in the present work. High purity indium (99.9999%) was deposited by ther- mal evaporation on the polished side of a silicon oxide wafer at room temperature. The substrate size was 2×2 cm2×1 mm. The base pressure and the evaporation rate were 4×10−8 mbar and 60-70 Å/s, respectively. The nominal film thickness, as recorded by a quartz monitor, was 2.5 µm. Several smaller area samples were simulta- neously fabricated for the film characterization. The root-mean-square (rms) surface roughness σ probed with an atomic force microscope (AFM) yielded 2.0, 6.7 and 8.0 nm at the scale of 1, 5 and 10 µm, re- spectively. A scan range up to 10 µm was not sufficient to reach saturation of the roughness. Consequently, 8.0 nm is a lower bound on the roughness at the scale of the neutron coherence length (≈ 100 µm [18]). In our simulations, effects due to surface roughness are mod- elled using Névot-Croce factors [7], where the roughness is characterized by σ; it was allowed to vary to fit the experimental data. Another parameter associated with the sample surface is the thickness of the indium oxide film. When exposed to air, In, like its neighbors in the Periodic Table, Al and Ga, instantly forms a protective oxide layer. A surface of indium in air remains lustrous for years. This suggests that the oxide layer is very thin, perhaps of the order of a few monolayers, and should not affect the neutron reflectivity. This is consistent with the negative result of Rutherford backscattering measurements performed on our sample: no oxide film has been detected. The electromagnetic properties of the sample were characterized by measurements of the DC magnetiza- tion M and the electrical resistivity. The shape of the M(H)-curves is typical for type-I superconductors. The obtained phase diagram Hc(T ) agrees well with the lit- erature data [6]. Tc of our sample (3.415 K) matches the tabulated value of 3.4145 K [19], and RRR = 540. Corre- spondingly, ℓ ≈ 11µm is much larger than ξ0. Therefore, our sample is a type-I superconductor in the pure limit. PNR experiments were performed both on the RE- MUR reflectometer [20] at the Joint Institute for Nuclear Research (Dubna) and on the CRISP instrument [21] at ISIS (Oxford). Both sets of measurements confirm that splitting of the R+(Q) and R−(Q) curves is achievable for our sample. The ISIS data, which are the most detailed, allow a quantitative analysis to which we now turn. CRISP operates with a spin-polarized polychromatic pulsed neutron beam. The angle of incidence and the instrumental resolution ∆Q/Q were set to 0.24 degrees and 3%, respectively. The reflectivity in the Meissner state was measured at T = 1.8 K and at magnetic fields of 77, 140, 166, and 194 Oe (Hc(1.8 K) = 205 Oe). The obtained data sets are shown in Fig. 2. The R(Q)-dependencies exhibit a hill caused by the total reflection from the substrate. The splitting between R+ and R− is clearly visible near Qc; different magnitudes of the error bars are due to different times of exposure. The data obtained at 77 and 166 Oe have the smallest statistical error and will be used for further discussion. The data for the reflectivity in the normal state are shown in Fig. 3. Solid curves are simulations, in which the sample was represented by a pure In film on a SiO2 substrate. In the simulations the angular beam resolution was allowed to vary due to the unknown uncertainties of the instrumental resolution and of the geometrical factor (as only part of the beam covers the sample). The simulation curve near Qc is mostly controlled by the resolution (see also [16]). The next segment, down to the foothill, is determined by the roughness of the sample surface. The location of the ascending part (0.011 < Q(Å−1) < 0.014) is governed by the film thick- ness. The curve segment following the hill is determined by the substrate scattering properties. No attempts were made to achieve a better fit for that segment, because there the spin asymmetry is indistinguishable from zero. The best fit (Fig. 3) was obtained for the model sample with σ = 14 nm and ∆Q/Q = 2.5 %. Fitting the ascend- ing part enables one to determine the film thickness in situ. The statistical error of the reflectivity data in this region being ± 5%, the thickness was found to be 2400 ± 30 nm, in agreement with the nominal thickness of 2.5 µm. These parameters were further used for simulating the spin asymmetry. Attempts to introduce an indium 0.008 0.010 0.012 0.014 spin up spin down 77 Oe 140 Oe 166 Oe 194 Oe momentum transfer (A-1) FIG. 2: Reflectivity of polarized neutrons measured in the Meissner state. Qc is the momentum transfer for total neu- tron reflection from the outer surface. The scale is shown for the data at H = 194 Oe; the other data have been shifted for clarity. 0.01 0.02 0.03 0.04 0.010 0.012 momentum transfer (Å-1) FIG. 3: Neutron reflectivity at T = 4.6 K. Curves 1 and 2 are simulations for a film thickness of 2.40 and 2.50 µm, respectively. The inset shows the data for the full range of Q-values. oxide layer on top of the sample yielded no reasonable fit for any appreciable thickness (> 1nm) of the oxide layer. This is consistent with our expectation that the indium oxide layer does not affect the neutron reflectivity. For our simulations of the reflectivity in the Meissner state, the magnetic field profiles shown in Fig. 1 were as- sumed on both sides of the sample. The spin asymmetry data for fields 77 Oe and 166 Oe, along with simulations 77 Oe 0.008 0.009 0.010 0.011 0.012 b 166 Oe momentum transfer (Å-1) FIG. 4: Spin asymmetry at T = 1.8 K and H = 77 Oe (a) and 166 Oe (b). The curves are simulations performed within the local (dashed line) and nonlocal (solid line) approaches. for the local and nonlocal field distributions, are shown in Fig. 4. For field 77 Oe (Fig. 4a), the results of the “nonlocal” simulation fit the experimental data somewhat better, but no clear discrimination between the local and nonlo- cal approaches is possible due to insufficient accuracy of the data at this field. A significantly clearer distinction is apparent for field 166 Oe due to the larger amplitude of s(Q). As can be seen from Fig. 4b, the quality of the fits, with B(z) calculated in the local and nonlocal ap- proaches, is different. The nonlocal simulation fits the experimental data definitely better. It is worth stress- ing that no adjustable parameters have been used for the simulations of spin asymmetry. In conclusion, nonlocal electrodynamics effects are measurable in extreme type-I superconductors. State-of- the-art PNR measurements performed on low-κ super- conductor In, combined with simulation, unambiguously support the nonlocal theory and at the same time demon- strate consistency with the literature data for λL(0) and ξ0. Consequently, evidence has been gathered for the nonmonotonic decay and sign reversal of the penetrating magnetic field predicted by the nonlocal electrodynamics approach. We thank A. Volodin for AFM, S. Vandezande for electrical conductivity, and A.P. Kobzev for Ruther- ford backscattering measurements. 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Polarized neutron reflectometry (PNR) provides evidence that nonlocal electrodynamics governs the magnetic field penetration in an extreme low-k superconductor. The sample is an indium film with a large elastic mean free path (11 mkm) deposited on a silicon oxide wafer. It is shown that PNR can resolve the difference between the reflected neutron spin asymmetries predicted by the local and nonlocal theories of superconductivity. The experimental data support the nonlocal theory, which predicts a nonmonotonic decay of the magnetic field.

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An Achievable Rate Region for Interference Channels with Conferencing Yi Cao and Biao Chen Department of EECS, Syracuse University, Syracuse, NY 13244 Email: ycao01@syr.edu, bichen@ecs.syr.edu Abstract— In this paper, we propose an achievable rate region for discrete memoryless interference channels with conferencing at the transmitter side. We employ superposition block Markov encoding, combined with simultaneous superposition coding, dirty paper coding, and random binning to obtain the achievable rate region. We show that, under respective conditions, the proposed achievable region reduces to Han and Kobayashi’s achievable region for interference channels, the capacity region for degraded relay channels, and the capacity region for the Gaussian vector broadcast channel. Numerical examples for the Gaussian case are given. Index terms — interference channels, dirty paper coding, superposition block Markov encoding, random binning. I. INTRODUCTION The capacity region of an interference channel (IC), where the information sources at the two transmitters are statistically independent, has been a long standing problem. Carleial was the first to use the superposition code idea [1] to obtain an inner bound for IC. This inner bound was later improved by Han and Kobayashi [2] who gave an achievable rate region that is the largest reported to this date. Recently, a simplified description of the Han-Kobayashi (HK) rate region for the general IC is derived by Chong-Motani-Garg in [3]. A related and less well investigated problem is when the information sources at the two transmitters are correlated, i.e., interference channel with common information (ICCI) [4], [5]. In [4], an achievable rate region, an outer bound, and a limiting expression for the capacity region were obtained. Later, the capacity region of this channel under strong interference was found in [5]. Recently, improved achievable regions for general ICCI [6], [7] and three new outer bounds for the capacity region of Gaussian ICCI [8] were proposed. However, all those results are based on the assumption that the common message is available noncausally. In this work, we investigate the problem of user cooperation in interference channels for the causal case. Here, each user not only transmits his own message to the intended receiver, but also serves as a relay to help transmit part of the other user’s message. We apply the superposition block Markov encoding, which was used previously for the relay channel [9] and for user cooperation in multiple access channels [10]. Our proposed achievable rate region is a generalized form of the HK region for IC [2], the capacity region of degraded relay channels [9], and the capacity region of the Gaussian vector broadcast channel (GVBC) [11]. This paper is organized as follows. In section II, we present the channel model and review some existing results. In section III, we propose an achievable region for general IC with transmitter conferencing. In section IV, numerical examples are used to compare the proposed region with the HK region and the capacity region of GVBC. We conclude in section V. II. PRELIMINARIES AND EXISTING RESULTS A. Definitions A memoryless discrete IC with conferencing (ICC) is de- noted by (X1,X2, p,Y1,Y2, Ỹ1, Ỹ2), where X1,X2 are two finite alphabet sets for the channel input, Y1,Y2 are two finite alphabet sets for the channel output, Ỹ1, Ỹ2 are two finite Fig. 1. Interference channel with conferencing at the transmitter side. alphabet sets for the received signals at the transmitters (which also serve as relays), and p is the channel transition prob- ability p(y1, y2, ỹ1, ỹ2|x1, x2). Here we assume the channel is memoryless and encoders 1 and 2 are allowed to depend only on their own messages and the past values of ỹ2 and ỹ1. Let M1 = {1, 2, · · ·,M1} and M2 = {1, 2, · · ·,M2} be the message sets of sender 1 and sender 2, respectively. Thus, for w1 ∈ M1 and w2 ∈ M2, the joint probability mass function of M1 ×M2 ×Xn1 ×Xn2 ×Yn1 ×Yn2 × Ỹn1 × Ỹn2 is given by p(w1, w2,x1,x2,y1,y2, ỹ1, ỹ2) = p(w1)p(w2) i=1 p(x1i|w1, ỹ21, · · ·, ỹ2i−1) ×p(x2i|w2, ỹ11, · · ·, ỹ1i−1)p(y1i, y2i, ỹ1i, ỹ2i|x1i, x2i) Suppose w1 ∈ M1 and w2 ∈ M2 are sent by transmitters 1 and 2 respectively, g1 and g2 are the decoding functions at receivers 1 and 2; the average probabilities of decoding error of this channel are defined as e,1 ≡ w1,w2 Pr(g1(Y1) 6= w1|w1, w2 sent) (2) e,2 ≡ w1,w2 Pr(g2(Y2) 6= w2|w1, w2 sent) (3) http://arxiv.org/abs/0704.1873v1 The capacity region of ICC is the closure of all rate pairs (R1, R2) such that P e,1 → 0, P e,2 → 0 as codeword length n→ ∞, where R1 = 1n logM1 and R2 = logM2. B. Existing Results 1) Chong-Motani-Garg recently derived a simplified de- scription of the HK region for IC [3], as summarized below. Proposition 1: Let P∗1 be the set of probability distributions P ∗1 (·) that factor as P ∗1 (q, u1, u2, x1, x2) = p(q)p(x1u1|q)p(x2u2|q). (4) For a fixed P ∗1 ∈ P∗1 , let RcHK(P ∗1 ) be the set of (R1, R2) sat- isfying (9)-(15) in Theorem 2 of [3]. Then P∗1 ∈P RcHK (P ∗1 ) is equivalent to the HK region. 2) The capacity of the degraded relay channel is given in proposition 2 [9]. Proposition 2: A relay channel consists of an input x1, a relay output y1, a channel output y, and a relay sender x2 (whose transmission is allowed to depend on the past symbols of y1). If y is a degraded form of y1 [9], then C = max p(x1,x2) min{I(X1, X2;Y ), I(X1;Y1|X2)}. (5) 3) The capacity region of GVBC is computed using a covariance matrix constraint on the inputs X = (X1, X2) of the form E[XXT ] ≤ S. In order to mimic the individual power constraints P1 and P2 on the two users for the vector case, the input covariance matrix S is of the form S = , for some − P1P2 ≤ c ≤ P1P2. Then, the capacity region of GVBC is given below [11]. Proposition 3: For each such S and all positive semi-definite matrices B and D, where B +D ≤ S, both rate pairs R1 ≤ 12 log |H1BH 1 +Q1| , R2 ≤ 12 log |H2(B+D)H 2 +Q2| |H2BH R1 ≤ 12 log |H1(B+D)H 1 +Q1| |H1DH 1 +Q1| , R2 ≤ 12 log |H2DH 2 +Q2| are achievable, where H1 = (1, a21) and H2 = (a12, 1). The convex hull of the union of these pairs over all possible S,B and D matrices is the capacity region of GVBC. III. MAIN RESULTS We first give a brief outline of our encoding-decoding strategy. We split each user’s message into two parts: M and W , where M is to be sent directly to the intended receiver, and W is the cooperative message to be sent to the receiver via the cooperation of the other user (relay). Our cooperation strategy is based on superposition block Markov encoding with the assumption that W can be perfectly decoded by the relay. The purpose of introducing M is to achieve a reasonable rate region (no less than IC without conferencing) even when the conferencing channel is poor. For the message M , we apply simultaneous superposition coding [2] and further split it into two parts: private message V and common message U . For the cooperation in transmitting W , we jointly consider B blocks, each of n symbols. Each user transmits a sequence of B − 1 messages w1, · · ·, wB−1 in B blocks, with no new message in the last block. Note that as B → ∞, (B − 1)/B is arbitrarily close to 1, hence the penalty on rate is negligible. Now we take user 1 as an example to show the whole process. Suppose there are 2nR13 codewords of W1 for user 1 to transmit. We establish a random partition by randomly throwing them into 2nR10 cells. This partition is made known to both transmitters and receivers. Suppose user 1 sends message w1,b−1 at block b−1. At the end of block b−1, following our assumption, user 2 can perfectly decode w1,b−1 and calculate the cell index s1b to which w1,b−1 belongs. At block b, both user 1 and 2 spend some power transmitting s1b. This provides the basis for cooperatively resolving the remaining Y1 uncertainty about w1,b−1. After decoding s1b at the end of block b, receiver Y1 intersects its ambiguity set D(y1(b− 1)) (i.e., the set of all codewords w that are jointly typical with y1,b−1 [9]) with cell s1b and gets the unique correct codeword w1,b−1 with a probability close to 1. Since both users 1 and 2 can perfectly decode each other’s messages W1, W2 and then calculate the corresponding cell indices S1, S2, we can employ dirty paper coding (DPC) to transmit V1, U1 and S1 treating S2 as a known interference at transmitter 1. Similarly at transmitter 2, we can transmit V2, U2 and S2 treating S1 as a known interference. Thus, introducing auxiliary random variables M1, N1, G1, H1 and M2, N2, G2, H2 for DPC, we summarize the achievable region for ICC in the theorem below. Thoerem 1: Let Z1 = (Y1, Y2, Ỹ1, Ỹ2, X1, X2,M1, N1, G1, H1, V2, U2,W2, S2, Q) and let P∗1 be the set of distribution on Z1 that can be factored into the form p(q)p(u2|q)p(w2|q)p(s2|q)p(v2|u2s2q) ×p(n1|s2q)p(g1|s2q)p(h1|s2q)p(m1|n1h1s2q) ×p(x1|m1g1s2q)p(x2|v2w2h1q)p(y1y2ỹ1ỹ2|x1x2) Let S(Z1) be the set of (R1, R2) such that R1 = R11+R12+ R13 and R2 = R22 +R21 +R23 satisfying: R11 ≤ L11 − I(M1;S2|N1H1Q) (9) R12 ≤ L12 − I(N1;S2|Q) (10) R13 ≤ L13 − I(G1;S2|Q) (11) R10 ≤ L10 − I(H1;S2|Q) (12) L11 ≤ I(Y1N1H1U2;M1|Q) (13) L11 + L12 ≤ I(Y1H1U2;M1N1|Q) (14) L11 + L10 ≤ I(Y1N1U2;M1H1|Q) (15) L11 +R21 ≤ I(Y1N1H1;M1U2|Q) (16) L11 + L12 + L10 ≤ I(Y1U2;M1N1H1|Q)) (17) L11 + L12 +R21 ≤ I(Y1H1;M1N1U2|Q) (18) L11 + L10 +R21 ≤ I(Y1N1;M1H1U2|Q) (19) L11 + L12 + L10 +R21 ≤ I(Y1;M1N1H1U2|Q) (20) L13 ≤ R10 + I(Y1M1N1H1U2;G1|Q) (21) L13 ≤ I(Ỹ1H1S2;G1|Q) (22) R22 ≤ I(Y2U2S2N1;V2|Q) (23) R22 +R21 ≤ I(Y2S2N1;V2U2|Q) (24) R22 +R20 ≤ I(Y2U2N1;V2S2|Q) (25) R22 + L12 ≤ I(Y2U2S2;V2N1|Q) (26) R22 +R21 +R20 ≤ I(Y2N1;V2U2S2|Q) (27) R22 +R21 + L12 ≤ I(Y2S2;V2U2N1|Q) (28) R22 +R20 + L12 ≤ I(Y2U2;V2S2N1|Q) (29) R22 +R21 +R20 + L12 ≤ I(Y2;V2U2S2N1|Q) (30) R23 ≤ R20 + I(Y2V2U2S2N1;W2|Q) (31) R23 ≤ I(Ỹ2H1S2;W2|Q) (32) Let R∗1 = S(Z1). Swap index 1 and 2 in all of the above statements and inequalities and we get R∗2 = S(Z2). Then the achievable region R∗ = convhull(R∗1 ∪R∗2) and the cardinality ||Q|| ≤ 33. Proof : We only need to prove the achievability of R∗1. Codebook Generation: Let q = (q(1), · · ·, q(n)) be a random sequence of Qn distributed according to t=1 p(q (t)). Generate 2nR21 i.i.d (independent and identically dis- tributed) codewords u2(j2) for common messages, 2 i.i.d codewords s2(l2) for cell indices, and 2 nR23 i.i.d codewords w2(k2) for cooperative messages according to t=1 p(u 2 |q(t)), t=1 p(s 2 |q(t)) and t=1 p(w 2 |q(t)), respectively. For each pair of (u2(j2), s2(l2)), generate 2nR22 i.i.d codewords v2(i2, j2, l2) for private messages ac- cording to t=1 p(v (t))1. Generate 2nL12 i.i.d codewords n1(η1) for common messages according to t=1 p(n 1 |q(t)) and randomly place them into 2nR12 bins2; generate 2nL10 i.i.d codewords h1(ω1) for cell indices accord- ing to t=1 p(h 1 |q(t)) and randomly place them into 2nR10 bins; generate 2nL13 i.i.d codewords g1(ψ1) for cooperative messages according to t=1 p(g 1 |q(t)) and randomly place them into 2nR13 bins. For each pair of (n1(η1),h1(ω1)), generate 2nL11 i.i.d codewords m1(ξ1, η1, ω1) for private mes- sages according to t=1 p(m (t)) and randomly place them into 2nR11 bins. To apply superposition block Markov encoding, we also need two random partitions. Randomly place the above gen- erated 2nR23 codewords w2(k2) into 2 nR20 cells, and those 2nL13 codewords g1(ψ1) into 2 nR10 cells. Encoding: In block b, user 2 wants to send new indices i2b, j2b and k2b. For cooperatively resolving the remaining Y2 uncertainty about w2(k2,b−1) in the previous block b − 1, it also sends the cell index of w2(k2,b−1), denoted by l2b. At the same time, user 1 wants to send new indices i1b, j1b, k1b and the cell index of g1(ψ1,b−1), denoted by l1b. Since user 1 can also perfectly calculate l2b at the end of block b − 1, it looks into bins j1b, k1b and l1b for codewords n1(η1b) and g1(ψ1b) and h1(ω1b) that are jointly typical with s2(l2b), 1Note that the codebook generation for the direct transmission part follows that of [3] instead of [2]. 2Random binning is used both for the superposition block Markov encoding (relay) part and for DPC. To distinguish, we use ”cell” when referring to superposition block Markov encoding and ”bin” when referring to DPC. respectively. For the previously found (n1(η1b),h1(ω1b)), encoder 1 looks into bin i1b for codeword m1(ξ1b, η1b, ω1b) such that (q, s2(l2b),n1(η1b),h1(ω1b),m1(ξ1b, η1b, ω1b)) are jointly typical. For the above bin searching, if there is more than one such codeword, pick the one with the smallest index; if there is no such codeword, declare an error. Then, user 1 sends x1 generated according to t=1 p(x 1 |m1(ξ1b, η1b, ω1b)(t)g1(ψ1b)(t)s2(l2b)(t)q(t)) and user 2 sends x2 generated according to t=1 p(x 2 |v2(i2b, j2b, l2b)(t)w2(k2b)(t)h1(ω1b)(t)q(t)). Decoding: User 2, as a relay to user 1, wants to correctly recover the new index k1b sent in block b. Since it already knows h1(ω1b) and s2(l2b) during encoding, it looks for all the sequences g1(ψ1), such that {q, s2(l2b),h1(ω1b),g1(ψ1), ỹ1b} ∈ A(n)ǫ (QS2H1G1Ỹ1) If those g1(ψ1) have the same bin index k1b, we declare k1b = k1b. Otherwise, we declare an error. On the other hand, user 1 determines the unique w2(k2b), such that {q, s2(l2b),h1(ω1b),w2(k2b), ỹ2b} ∈ A(n)ǫ (QS2H1W2Ỹ2). At the receiver side, we assume Y1 knows i1,b−1, j1,b−1, l1,b−1 and j2,b−1, and it can construct m1(ξ1,b−1, η1,b−1, ω1,b−1), n1(η1,b−1), h1(ω1,b−1) and u2(j2,b−1), which are jointly typical with y1,b−1. Now it wants to first decode bin indices i1b, j1b, l1b and the common message index j2b. It looks for m1(ξ1, η1, ω1),n1(η1),h1(ω1),u2(j2), such that {q,m1(ξ1, η1, ω1),n1(η1),h1(ω1),u2(j2),y1b} ∈ A(n)ǫ (QM1N1H1U2Y1). (35) If those sequences satisfying (35) have the same bin indices and message index respectively, we declare î1b = i1b, ĵ1b = j1b, l̂1b = l1b and ĵ2b = j2b. Otherwise, declare an error. Assuming cell index l1b is successfully decoded at Y1, then we declare k̂1,b−1 = k1,b−1 if those sequences g1(ψ1) ∈ C(l1b) ∩ D(y1(b − 1)) have the same bin index k1,b−1. Here C(l1b) denotes the set of g1(ψ1) in cell l1b, and D(y1(b− 1)) is the ambiguity set, i.e., sequences of g1(ψ1) such that {q,m1(ξ1,b−1, η1,b−1, ω1,b−1),n1(η1,b−1),h1(ω1,b−1), u2(j2,b−1),g1(ψ1),y1,b−1} ∈ A(n)ǫ (QM1N1H1U2G1Y1). (36) For Y2, the decoding process is the same and we skip the details. Analysis of error probability: We first consider P and we still use the story in block b. Let P0 denote the probability that there is no m in bin i1b, such that (q, s2(l2b),n1(η1b),h1(ω1b),m1(ξ1, η1b, ω1b)) are jointly typical. Then, P0 ≤ (1− 2−n(I(M1;S2|N1H1Q)+3ǫ))2 n(L11−R11) ≤ e−2 −n(I(M1;S2|N1H1Q)+3ǫ−L11+R11+1/n) So, (9) guarantees P0 → 0 as n→ ∞. Similarly, bounds (10)- (12) guarantee that encoder 1 can find codewords n1(η1b), g1(ψ1b) and h1(ω1b), which are jointly typical with s2(l2b), respectively. Now we calculate the error probability for user 2 (as a relay) to decode k1b. Denote the sent codeword g1(ψ1b) as g1(k1b, k ∗) since it is picked from bin k1b. k ∗ denotes the index of g1(ψ1b) in bin k1b. Let E1(k1, k) denote the event (33) and let P1 denote the probability for user 2 to make a decoding error. Then P1 ≡ Pr{Ec1(k1b, k∗) or k1 6=k1b E1(k1, k)} (39) ≤ Pr{Ec1(k1b, k∗)}+ k1 6=k1b,k Pr{E1(k1, k)}(40) k1 6=k1b,k Pr{E1(k1, k)} (41) For k1 6= k1b, we know Pr{E1(k1, k)} (qs2h1g1ỹ1b)∈A p(q)p(g1|q)p(s2h1ỹ1b|q) ≤ |A(n)ǫ |2−n(H(Q)−ǫ)2−n(H(G1|Q)−ǫ)2−n(H(S2H1Ỹ1|Q)−ǫ) ≤ 2−n(H(Q)+H(G1|Q)+H(S2H1Ỹ1|Q)−H(QG1S2H1Ỹ1)−4ǫ) ≤ 2−n(I(S2H1Ỹ1;G1|Q)−4ǫ) Therefore, P1 ≤ ǫ + 2−n(I(S2H1Ỹ1;G1|Q)−L13−4ǫ). ǫ can be arbitrarily small by letting n→ ∞. Thus, bound (22) assures P1 → 0 as n→ ∞. For the decoding of i1b, j1b, l1b and j2b by Y1, it is a direct application of the simultaneous superposition coding [2]. However, regarding our codebook generation scheme, particularly the construction of m1, we will get a somewhat simpler description, similar to that of Chong-Motani-Garg [3]. This leads to the bounds (13)-(20) and we skip the details here. Let E2(k1, k) denote the event that g1(k1, k) ∈ C(l1b) ∩ D(y1(b − 1)) and E21(k1, k) denote the event (36). We also define an indicator function I(k1, k). If g1(k1, k) satisfies (36), I(k1, k) = 1; otherwise, I(k1, k) = 0. The number of sequences in D(y1(b− 1)) with bin index k1 6= k1,b−1 is ||D(y1(b− 1)|| (42) k1 6=k1,b−1,k E(I(k1, k)) (43) k1 6=k1,b−1,k Pr{E21(k1, k)} (44) k1 6=k1,b−1,k 2−n(I(Y1M1N1H1U2;G1|Q)−4ǫ) (45) ≤ 2−n(I(Y1M1N1H1U2;G1|Q)−L13−4ǫ) (46) Now, let P2 denote the probability of error for Y1 to decode k1,b−1. Denote the actually sent codeword g1(ψ1,b−1) in block b− 1 as g1(k1,b−1, k∗). Then, P2 ≡ Pr{Ec2(k1,b−1, k∗) or k1 6=k1,b−1 E2(k1, k)}(47) k1 6=k1,b−1,k Pr{E2(k1, k)} (48) ≤ ǫ+ ||D(y1(b − 1)|| · 2−nR10 (49) ≤ ǫ+ 2−n(I(Y1M1N1H1U2;G1|Q)+R10−L13−4ǫ) (50) Bound (21) guarantees P2 → 0 as n → ∞. Thus, with bounds (9)-(22), it is guaranteed that Y1 will correctly decode i1b, j1b, l1b, j2b and k1,b−1 at the end of block b with a probability arbitrarily close to 1. Then, the information state of receiver Y1 propagates forward, yielding the total decoding error probability P e,1 → 0 as n → ∞. The analysis of P is similar to P e,1 , which leads to bounds (23)-(32). Thus, R∗1 is achievable. Proof of R∗2 is identical, hence R∗ is achievable for ICC via time sharing. The cardinality bound on Q, i.e., |Q| ≤ 33 is obtained by applying the Caratheodory Theorem to a set of inequalities that bound the rate pair (R1, R2), obtained via Fourier- Motzkin elimination to Eqs. (9)-(32). Q.E.D. Remarks: From the encoding-decoding strategy of the above theorem, the achievable region R∗ is actually a generalization of the HK region for IC, the capacity region of degraded relay channels, and the capacity region of GVBC. It reduces to those extreme cases under the conditions elaborated below. 1)When the conferencing channel between the two users is very poor, the bound in (22) and (32) can be very small. In this case, allocating power for cooperation will actually reduce the rates otherwise achievable via direct transmission. As a result, the encoders will not allocate any power to transmit W1 and W2, so W1 = W2 = S1 = S2 = 0 and G1 = G2 = H1 = H2 = 0. Then, both R∗1 and R∗2 reduce to the region in Proposition 1, which equals to the HK region. 2)When the conferencing channel between the two users is good enough, it is not necessary to transmit messages directly to the receiver, because cooperative transmission with the other user will always yield a better rate. In this case, the encoders will let V1 = V2 = U1 = U2 = 0 and M1 = M2 = N1 = N2 = 0. Now, if user 2 refrains from transmitting its own message and only serves as a relay to user 1 (i.e., W2 = S2 = G2 = H2 = 0), both R∗1 and R∗2 reduce to the capacity region of the degraded relay channel in Proposition 2. Similarly, if user 1 serves only as a relay to user 2, it also reduces to the capacity region of the degraded relay channel. 3)When the conferencing channel between the two users is ideal (i.e., the conferencing channel capacity is infinite), the bounds in (22) and (32) are no longer needed. So, W1 = W2 = G1 = G2 ≈ 0. Combining the result in case 2 that V1 = V2 = U1 = U2 = 0 and M1 =M2 = N1 = N2 = 0, we can easily check that R∗1 reduces to R1 ≤ I(Y1;H1|Q)− I(H1;S2|Q) (51) R2 ≤ I(Y2;S2|Q) (52) and R∗2 reduces to R1 ≤ I(Y1;S1|Q) (53) R2 ≤ I(Y2;H2|Q)− I(H2;S1|Q) (54) For the Gaussian case, the rate region (51)-(52) becomes (6) and (53)-(54) becomes (7). So, in this case, R∗ reduces to the capacity region of GVBC. 4)During the review process of this paper, we became aware of [12], which essentially tackles the same problem using a different approach. In [12], user cooperation results in a common information (in the sense of [4]) at the encoders and the decoder uses backward decoding (similar to that of [10]) instead of the random partitioning (i.e., binning) we use in our approach. Except for some extreme cases, it appears no subset relation can be established. The obtained achievable region in [12] is simpler as it does not involve a large number of auxiliary variables; however, the scheme in [12] is strictly suboptimal for certain extreme cases (e.g., degraded relay channels, IC with degraded message sets with weak interference, and MIMO BC) whereas our achievable region can be easily shown to be optimal in each of these cases. IV. NUMERICAL EXAMPLES The standard form of a Gaussian interference channel is: Y1 = X1 + a21X2 + Z1 (55) Y2 = a12X1 +X2 + Z2 (56) where Z1 and Z2 are arbitrarily correlated zero mean, unit variance Gaussian random variables. Suppose the power con- straints of X1 and X2 are P1 and P2, respectively. For the conferencing channel with perfect echo cancellation, we have Ỹ1 = K1X1 + Z̃1, Ỹ2 = K2X2 + Z̃2 (57) where Z̃1 and Z̃2 are both zero mean, unit variance Gaussian variables. By reciprocity, we assume the channel coefficient K1 = K2. Since the computation of R∗ is formidable, here we constrain all the inputs to be Gaussian distributed and set Q = φ in order to compare our region with G′ in (5.9) of [2] and the capacity region of GVBC. We denote this modified region as R. Consider, that for certain αt, βt, γt, θt, µt ∈ [0, 1], with αt+βt+γt+ θt+µt = 1, where t = 1, 2, the following hold: Ut ∼ N(0, βtPt),Wt ∼ N(0, γtPt), St ∼ N(0, 1) (58) Vt = V t + Ut + θtPtSt, where V t ∼ N(0, αtPt) (59) X1 = V1 +W1 + µ1P1S2, X2 = V2 +W2 + µ2P2S1 (60) After applying Fourier-Motzkin Elimination on those bounds (9)-(32), we find that for each set of (α1, β1, γ1, θ1, µ1) and (α2, β2, γ2, θ2, µ2), both S(Z1) and S(Z2) are delimited by straight lines of slope 0,− 1 ,−1,−2,∞ as in the original HK region. Exhausting all the parameters between [0, 1], and taking the convex hull of all those S(Z1) and S(Z2), we get the achievable region R for ICC in Fig.2. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 HK region K1=K2=1 K1=K2=4 Gaussian vector BC Fig. 2. Comparison of R with HK region and Gaussian vector broadcast channel capacity. P1 = 6, P2 = 1.5, a12 = a21 = 0.74 Remarks: 1) When there is no conferencing between the two users, the achievable region reduces to the HK region. When the quality of the conferencing channel improves, it increases our achievable region for ICC within the limit of the capacity of GVBC. 2) When the channel coefficient is K1 = K2 = 4, the region R is already very close to the upper bound; when K1 = K2 = 1, which is equal to the channel coefficient of the transmitter to the receiver, cooperation achieves a slightly better rate region than independent transmission. 3) For the channel coefficient K1 = K2 = 4, the corre- sponding relay channels (i.e., one of the users only serves as a relay) are degraded, thus the intercepts of the bound at both axes are the capacities of respective relay channels. REFERENCES [1] A.B. Carleial, “Interference Channels,” IEEE Trans. Inform. Theory, vol. 24, Jan. 1978. [2] T.Han and K.Kobayashi, “A new achievable rate region for the interference channel,” IEEE Trans. on Information Theory, vol. IT-27, pp. 49–60, Jan. 1981. [3] H. F. Chong, M. Motani, H. K. Garg, and H. E. Gamal, “On the Han- Kobayashi Region for the Interference Channel,” submitted to IEEE Trans. Inform. Theory, Aug. 2006. [4] H. H. Tan, “Two-User Interference Channels with Correlated Informa- tion Sources,” Information and Control, vol. 44, pp. 77–104, 1980. [5] I. Maric, R. D. Yates, and G. Kramer, “The Capacity Region of the Strong Interference Channel With Common Information,” in Asilomar Conference On Signals, Systems and Computers, Pacific Grove, CA, Nov. 2005. [6] Y. Cao, B. Chen, and J. Zhang, “A New Achievable Rate Region for Interference Channels with Common Information,” in proc. IEEE WCNC’07, Hong Kong, China, March 2007. [7] J. Jiang, Y. Xin, and H. Garg, “Interference Channels with Common Information,” submitted to IEEE Trans. Inform. Theory, Oct. 2006. [8] Y. Cao and B. Chen, “Outer Bounds On the Capacity Region of Gaussian Interference Channels with Common Information,” in submitted to IEEE Globecom’07, Washington DC, Nov. 2007. [9] T.M. Cover and A. El Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Inform. Theory, vol. 25, pp. 572–584, 1979. [10] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity: Part I System description,” IEEE Trans. Comm., vol. 51, pp. 1927–1938, Nov. 2003. [11] H. Weingarten, Y. Steinberg, and S. Shamai, “The Capacity Region of the Gaussian Multiple Input Multiple Output Broadcast Channel,” IEEE Trans. Inform. Theory, vol. 52, Sep. 2006. [12] D. Tuninetti, “On Interference Channels with Generalized Feedback (IFC-GF),” in ITA’07 Workshop, UCSD, Feb. 2007. Introduction Preliminaries and Existing Results Definitions Existing Results Main Results Numerical Examples References

In this paper, we propose an achievable rate region for discrete memoryless interference channels with conferencing at the transmitter side. We employ superposition block Markov encoding, combined with simultaneous superposition coding, dirty paper coding, and random binning to obtain the achievable rate region. We show that, under respective conditions, the proposed achievable region reduces to Han and Kobayashi achievable region for interference channels, the capacity region for degraded relay channels, and the capacity region for the Gaussian vector broadcast channel. Numerical examples for the Gaussian case are given.

Introduction Preliminaries and Existing Results Definitions Existing Results Main Results Numerical Examples References

K расчету распространения электромагнитных импульсов Electromagnetic Pulse Propagation over Nonuniform Earth Surface: Numerical Simulation Alexei V. Popov and Vladimir V. Kopeikin Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation IZMIRAN, Troitsk, Moscow region, 142190 Russia popov@izmiran.ru, kopeikin@izmiran.ru 1. Introduction Parabolic equation method proposed by Leontowich and Fock [1,2] is an efficient simulation approach to VHF propagation over the earth surface. Deep physical analysis and advanced mathematical methods [3,4] turned Leontovich’s PE into a universal tool of diffraction theory. Its applications go far beyond the initial problem circle – e. g. [5-8]. The key role in this development played the decisive turn to straightforward numerical techniques pioneered by Malyuzhinets and Tappert [9,10]. In radio wave propagation, PE was used first to derive explicit analytical formulae for the EM field strength in model environments. A simplification has been reached by introducing the impedance boundary condition (BC) [11]. Taking into account tropospheric refraction ducts required the use of sophisticated asymptotic methods [12]. Further development (almost exclusively towards numerical implementation) was aimed at refined PE modifications [13-15], account for irregular terrain [16], introducing artificial transparent boundaries [17,18] and nonlocal BC to describe rough interfaces [19]. A non-stationary PE counterpart and a finite-difference (FD) scheme for its solution have been proposed by Claerbout and applied to seismic problems [13]. Afterwards, this “time-domain parabolic equation” (TDPE) was used to calculate acoustic propagation in ocean [20]. At the same time, little attempts of using TDPE to simulate EM pulse propagation in realistic environments are known. In this paper, we consider computational aspects of EM pulse propagation along the nonuniform earth surface. For ultrawide-band pulses without carrier, TDPE results directly from the exact wave equation written in a narrow vicinity of the wave front. To solve it by finite differences, we introduce a time-domain analog of the impedance BC and a nonlocal BC of transparency reducing the open computational domain to a strip of finite width. Numerical examples demonstrate the influence of soil conductivity on the received pulse waveform which can be used in remote sensing. mailto:popov@izmiran.ru mailto:kopeikin@izmiran.ru For a high-frequency modulated EM pulse, TDPE arises as a convolution of PE solutions with the pulse envelope spectrum. In order to overcome computational difficulties, we develop an asymptotic approach based on the ray structure of the monochromatic wave field calculated at the carrier frequency. To accommodate complex-valued asymptotic solutions to the real initial condition we use the “analytic signal” approach introduced by Vainstein, Heyman and Felsen [21, 22]. An explicit solution of the time-domain transport equation reduces the computational procedure to numerical integration of standard PE at the carrier frequency and evaluation of a given 1D function in time domain. This diminishes computational expenses by 2-3 orders of magnitude and allows for pulsed wave field calculation in vast domains measured by tens of thousands wavelengths. As an example, we consider a problem of target altitude determination from overland radar data [23]. This work has been done in collaboration with the Institute for high-frequency technique (IHF), Stuttgart University. Preliminary results appeared as short papers [24,25], a Russian version has been published in [26]. We dedicate this publication to the memory of Leopold Benno Felsen. 2. Monochromatic wave propagation Omitting technical details and method refinements – see [12, 19], recall PE based scheme of monochromatic wave propagation over a smoothly nonuniform earth surface - Fig.1. )(xhz = 0 z 0 1 00 , = = σ ε x ε, σ )(xhz = Fig. 1. Elevated source illuminating smoothly rolling terrain (sketch). Horizontal magnetic component ),( zxHH y = satisfies Helmholtz equation 0~22 ε (1) with complex permittivity ωσπεε i4~ += , where σ is soil conductivity in the Gaussian units set. In the upper medium 0,1 == σε and the contact conditions at are )(xhz = , . (2) where xzn ∂∂+∂∂=∂∂ αα sincos , )('arctan xh=α . At large distances from the source the wave field is sought as a plane wave with slowly varying complex amplitude: [ )(exp),,(),,( tkxikzxutzxH ]ω−≈ (3) Here, λπω 2≡= ck is the wave number, and the complex “attenuation function” satisfies the Leontovich PE ),,( kzxu )(,02 2 ik >= (4) In 3D, divergence factor x1 must be added in (3). In this paper we use Gaussian initial condition ( ) ( −−−⎟⎟ exp),,0( zzikzz kzu β ) (5) corresponding to an exact solution of PE (4) = xzz kzxui 22 exp),,( (6) - skewly propagating Gaussian beam with initial width and wave front radius 0w 0ρ , determined by complex parameter ( ) 12000 21 += wkix ρ ; β being a small elevation angle. Impedance approximation is based on wave beam contraction when entering a denser dielectric medium. Standard Leontovich BC [11] 0,~ =−=∂ (7) results from the contact conditions (2) under the assumption of almost vertical propagation in the lower medium: ( )ε~exp),( ikzTzxH −≈− . For grazing angles (Fig. 2) this assumption breaks and a plane incident wave ( )[ ]ββ coscosexp),( zxikzxH −=+ with small 1<<β Fig. 2. Derivation of impedance BC for grazing angles. enters the half-space close to the total internal reflection angle 0<z ( )210 ~arccos εγ = : 0 sin~2 γε γγ +≈ . Hence ensue ( )[ ] ( )[ ] ( ( )[ ] ( )000 exp)( expexp)( sincos ~exp)(),( tgzxVtgzxikT ctgzxk tgzxikT zxikTtxH (8) In virtue of the superposition principle, Eq. (8) holds for an arbitrary paraxial wave packet with the corresponding slowly varying function . Eliminating the latter by differentiation and making use of (2), we obtain 0,0~~ 1~ ==+ ε (9) This modified impedance BC provides a more accurate approximation of the reflection coefficient, especially, in a vicinity of the Brewster angle ( ) 210 1~arcsin += εβ . Fig. 3 allows one to compare the exact Fresnel reflection coefficient cos~sin~ cos~sin~ =FR (10а) with the Leontovich approximation 1sin~ 1sin~ βLR (10b) and that resulting from the modified impedance BC (9) Fig.3. Comparison between exact Fresnel reflection coefficient and impedance approximations. ( ) ( ) ( ) 1~cos~sin~ 1~cos~sin~ εβεβε εβεβε βMR (10c) Taking into account the boundary tilts )(xh′ and using "parabolic" approximation (3), we derive a modified BC for the attenuation function : ),( zxu )(,0)('~ xhzuxhik (11) Contrary to the standard Leontovich BC (9), here it is not necessary to assume 1~ >>ε - Eq. (11) breaks only for 11~ <<−ε when nonlocal effects of wave interaction in both media are to be taken into account [19]. The impedance BC grants uniqueness of the solution of PE (4). In fact, calculating the energy flow through a vertical cross section one obtains ),()( dzzxuxI , 0),(~ −= hxu (12) which proves stability and uniqueness of the boundary value problem solution. Finite-difference methods of PE solution have been studied in early works by Malyuzhinets and coauthors [9, 27]. Further method development is described in monographs [5, 19]. We employ a six-point implicit FD scheme supplemented with the impedance BC (11) at and a discrete approximation of the nonlocal transparency BC [17,28] imposed at the artificial computational boundary )(zhz = maxzz = : ( ) ( ) maxmax , , (13) An example of simulated VHF propagation over irregular terrain is illustrated by Fig. 4. Fig. 4. VHF attenuation function over irregular earth surface. 3. Radio pulse propagation: Fourier synthesis A straightforward way to describe EM transients is to convolve monochromatic wave fields with the signal spectrum. In a 1D case, the propagating pulse is a superposition of plane waves [ ]∫ −= dkekFtxH tkkxi )()(~ ),( ω (14) In free space, kc=ω , and formula (14) yields a dispersion-less traveling wave −=−= dkekFsFxctFtxH iks)(~ )(),(),( (15) In a 2D environment, a natural generalization of the 1D solution (14) is a paraxial wave packet dkekzxukFtzxH ctxik )(),,()(~ ),,( − (16) where is a solution of the PE (4) at a fixed frequency ),,( kzxu kc=ω . The superposition (16) will approximate an exact solution of the wave equation if the spectrum )(~ xF is confined near a certain positive satisfying the PE applicability conditions: LDk ππ 220 >>>> where and are lateral and longitudinal characteristic scales of the problem. Consider a quasi-monochromatic pulse tctfctF 0cos)()( ω= with duration 02 ωπ>>T . Its Fourier transform ( ) ([ 00 ~~2 )(~ kkfkkfkF ++−= )], (17) where is the envelope spectrum, contains negative frequencies not described by PE (4). Introducing complex signal = dtectfckf tiω)()( )(ctf )exp()()( 0tictfctFc ω−= eliminates the second term in (17). Still, the remaining "positive" component ( )0 kkf − centered at may spread onto negative semi-axis. In order to avoid nonphysical effects of negative frequencies propagation, the "analytic signal" [21] can be used, defined as a one-side inverse Fourier transform of the truncated spectrum >++−= kkkfkkfkF kF (18) Thus, by definition, the analytic signal is a Cauchy type integral FdkekFsF , (19) regular in the lower half-plane 0Im <s . For real s, its real part coincides with whereas the imaginary part is given by Hilbert transform ( )∫ FsF V.P. )(Im (20) Introduction of the analytic signal violates the causality principle: the real signal is zero before the moment of switching on the transmitter while for . However, for a high-frequency radio pulse this discrepancy is small. So the analytic signal envelope defined as )(ctF 0)( ≠+ ctF 0<t )(sF + is close to but, contrary to the "naïve" complex signal , admits analytic continuation into complex domain, compatible with asymptotic propagation laws [22]. As an example, consider a modulated high-frequency pulse with )(sf )(sFc sksfsF 0cos)()( = the envelope Fig. 5. Modulated radio pulse waveform (21). Fig. 6. Analytic signal spectrum (18). 0),exp(sin sbsas sf , ibkkba 222 −−+ = (21) see Fig. 5. For its length is ab ≈ acT π~=Λ . The envelope spectrum )(kf has a peak at with 0=k Λ≈Δ π2k and tends to zero for ∞→k as ( )2kaO . Spectra )(~ kF and )(~ kF + are shown in Fig. 6; the analytic signal envelope )(~ sF + is plotted in Fig. 7a,b for real and complex arguments. (a) (b) Fig. 7. Analytic signal envelope of real (a) and complex (b) arguments. For a wave packet −++ = )(),,()(~ ),,( dkekzxukFtzxH ctxik , (22) )(0 0)(),,0( zikezAkzu Φ= the “initial” condition [ ] [ )()()(~ ),,0( 00 )(0 0 zctFzAekF tzH ctzik Φ−== + ] (23) describes an analytic signal with amplitude and initial delay )(ctF + )(0 zA czzt )()( 00 Φ= . Fig. 8. Example of a Fourier-synthesized propagating EM pulse. An example of modulated pulse propagation over smoothly rolling interface is depicted in Fig. 8. A sequence of snap-shots traces the evolution of the initial pulse envelope defined by (5), (21) due to the incident Gaussian beam divergence and reflection from the curved interface ),()( 00 kzuctf )(xhz = . It should be noted that Fourier synthesis is computationally efficient only for rather narrow-band pulses. In fact, for a good approximation of the convolution integral (22) one has to solve PE (4) for a set of wave numbers covering the spectral band kkkkk Δ+<<Δ− 00 , Λ≈Δ π2k with a small frequency step ( kk Δ<<δ ) and, to avoid phantom solutions in the given range xΔ , even more restrictive condition must be posed: xk Δ<< πδ 2 . Adequate simulation methods for wide-band EM pulse propagation are discussed in the following sections. 4. Time-domain PE and boundary conditions Straightforward derivation shows that if ( )kzxu ,, is a solution of PE (4), the transient wave packet (16) , as a function of variables ),,(),,( szxtzx Π≡Η xctszx −=,, , satisfies the Claerbout equation zsx ∂ (24) Equation (24), usually called "time-domain parabolic equation" (TDPE), has been obtained in [13] by formal substitution sik ∂∂= as well as by the reduction of the time-dependent wave equation (25) in a narrow vicinity of the paraxial wave front ctx = . Introduction of scaled variables Λ−=== )(,, xctDzLx ηςξ , where L, D are computational domain length and width, is spatial pulse length, yields Λ ςηξ ∂ , 1<<Λ== DLDν (26) Neglecting the small term results in TDPE (24).This derivation clarifies the nature of the “time-domain parabolic equation”: )( 2νO 1) It is a hyperbolic equation written in a traveling coordinate frame ; ),,( szx 2) TDPE does not describe the backward moving waves; 3) TDPE is a paraxial (narrow-angle) approximation valid in a narrow strip 1)( <<= νOLD ; 4) TDPE describes short pulses 1)( <<=Λ νOD whose length Λ is comparable with the wave front deviation from the plane ctx = (Fig. 9); 5) TDPE solutions are not necessary modulated high-frequency signals – they can represent short ultrawide-band pulses without carrier, e. g. a damped sinusoid (21). )(ctf Fig. 9. Derivation of TDPE (24). Here, a seeming contradiction may arise, as the spectral maximum of can lie in the vicinity of zero frequency, not described by PE (4). As a matter of fact, at small distances from the wave front )(Λ= Os the main part of the pulse energy is determined by the high- frequency edge of its spectrum Dak π2~ >> satisfying PE applicability conditions. To solve TDPE (24), an FD scheme of the 2nd order of accuracy has been proposed in [13]: ( ) ( )lllllllll rrrrrrrr ,,11,1,1 2,,11,1, )(4 nnnnznnnn zsx Π+Π+Π+Π∇ =Π+Π−Π−Π ΔΔ ++++++++ (27) Here, , { }mnn ll ,, Π=Π ,xnxn Δ= zmzm Δ= , ss Δ= ll ; ( ) 1,,1,,2 2 −+ Π+Π−Π=Π∇ mnmnmn nz llll . This equation is solved by zigzag marching in plane between boundary values ),( sx ( ) [ ] 0)0,,(,)()(,,0 00 =ΠΦ−=Π zxzctfzAsz (28) (given source and causality condition). At each marching step , a three-diagonal linear equation set arises for the unknown vector ),( mn 1,1 ++Π l n . In order to complete the boundary value problem, we have to add a correct BC taking into account soil properties and to find a way of the domain truncation without creating spurious reflections. Both problems are resolved by applying Fourier transform to the frequency-domain BC (11), (13). Consider a paraxial wave packet dkekzxukfszx iks− ∫=Π ),,()( , (29) satisfying the causality condition 0),,( =Π szx for 0<s . We rewrite the impedance BC (11) emphasizing the dependence of complex permittivity kciσπεε 4~ += on the wave number ck ω= : 0)(' , (30) where )1(2,4 −== επσεπσ cqcr . Multiplying Eq. (30) by and applying Fourier transform (29), we get dkkekzxukf xhshx iks),,()( ),,()('),,( . (31) Substituting here the inverse Fourier transform (32) ∫ ),,(),,()( ηη ηdezxkzxukf ik we obtain, by standard calculations, the following expression for the RHS of (31): dsNhxshx )(),,(),,( ; ⎥ −+= ∫ tqrrs qr qtIeqesN )( )()( (33) Thus, we have derived a nonlocal 2D boundary condition dsNhxshx xhshx )(),,(),,( ),,()('),,( ηηη (34) being an exact time-domain counterpart of the impedance BC (11). Its nonlocality is a consequence of interaction between two waves propagating along the earth surface with different phase velocities. The integral term kernel )( η−sN can be easily calculated for different ε and σ . For 15.3>ε , function monotonously tends to zero with increasing s – see Fig. 10. Fig. 10. Kernel of nonlocal impedance BC (34). It is interesting to note that in both limiting cases: perfectly conducting boundary )( ∞=σ and zero soil conductivity )0( =σ Eq. (34) reduces to a local BC. In the former case tends to a delta function, for . The integral in (34) limits to =∞→−= 1)(,)0( dssNqrN ),,( shx , the RHS vanishes, and we get a Neumann BC. In the opposite case )0( →σ the spatial scale of (length of the pulse “dispersion tail”) is growing but its absolute value is tending to zero, so only a local term remains. In a similar way, the time-domain generalization of the transparency BC (13) is derived which grants the absence of spurious reflections a from the artificial computational boundary . Applying to Eq. (13) the Fourier transform (29), (32) and denoting maxzz = ipk = , we get ( ) ( ) ( ) ∫∫∫ maxmax (35) Finally, evaluating the inner integral we obtain an elegant 2D boundary condition ( ) ( ) ( )( )ηξ ηξπ −−∂∂ ∫ ∫ sx ,, max max (36) symmetric with respect to the variables x, z , which could be expected from the symmetry of the TDPE (24). A simulated example of ultrawide-band EM pulse propagation over a nonuniform earth surface with soil parameters 10=ε , )01.0(109 17 мSs −⋅=σ is depicted in Fig. 11а. Evolution of the spatial amplitude distribution for a pulsed signal generated by a Gaussian source: ( )[ ]20200 exp)( wzzzA −−= with a skew curved wave front: )(2)()( 00 00 zzzzz −+−=Φ βρ is shown in a grey color scale. The initial pulse waveform is a damped sinusoid (21) with )(ctf ba = and spatial length ma 30==Λ π . (a) (b) Fig. 11. Propagation of ultrawide-band pulse (21) over nonuniform earth surface. Initial Gaussian beam parameters: ,80,3000 mmz == 0w 1.0,3000 −== βρ m . Soil conductivity: mS01.0=σ (a), 0=σ (b). Fig. 12. Received pulse waveform depending on soil conductivity: mS01.0=σ (solid line), mS001.0=σ (dashed line); initial pulse (dots). The snapshots clearly show the reflected pulse generation at the earth surface. The transparency BC (36) imposed at the height mz 500max = assures unimpeded radiation exit from the computational domain. Finite soil conductivity causes signal dispersion appearing in a certain delay of the reflected pulse. It is obvious from the comparison with a similar plot calculated for a model non-conducting soil 0=σ - Fig. 11b. A quantitative estimation of the effect can be made by means of Fig. 12 revealing a considerable dependence of the pulse waveform on the soil conductivity. This effect caused by the pulse penetration into the ground can be used for ecological monitoring (water pollution, earthquake precursors, etc.). 5. Hybrid TDPE and short high-frequency pulse propagation An important practical issue is overland propagation of short EM pulses with high- frequency carrier. Basically, having absolute stability, TDPE (24) is capable to describe wide- band radio pulse propagation. However, the computational expense is drastically growing with increasing carrier frequency. If 0ω considerably exceeds the spectral band of the signal it is useful to factor out the carrier: [ ])(exp),,(),,( 00 txkiszxUszx ω−=Π (37) and to consider the transient signal envelope satisfying a hybrid equation ),,( szxU ik , (38) combining the features of standard Leontovich PE (4) with Claerbout TDPE (24). Unfortunately, despite a relatively slow variations of in space-time, straightforward ),,( szxU numerical solution of this hybrid TDPE (HPE) entails considerable difficulties, as the large coefficient by 0k xU ∂∂ demands a dense computational grid. On the other hand, the presence of a big parameter allows us to construct an asymptotic solution of HPE (38) radically reducing the computational burden. In order to find a proper asymptotic Ansatz, consider monochromatic PE (4) at the carrier frequency ck00 =ω with the initial condition . We admit complex values of the eikonal to describe relatively narrow wave packets, like a Gaussian beam (6). For , we obtain an asymptotic solution 00)(),0( zikezAzu Φ= )(0 zΦ (39) ),(0 0),(),,( zxikezxAkzxu Φ= where satisfies a “parabolic” eikonal equation ),( zxΦ 1 2 =Φ+Φ zx (40) while slowly varying amplitude is governed by the paraxial transport equation ),( zxA =Φ+Φ+ AAA zzzzx (41) Eqs. (40)-(41), being an approximate form of the well-known laws of geometric optics (GO) [4], can be easily solved by the method of characteristics. Consider a particular solution of the eikonal equation (40) corresponding to a bundle of rays spreading from a central point : 0,0 zzx == zx −−+≈ =Φ 20 ),( (42) At a characteristic line xzz γ+= 0 we have xxx ),( γγ =Φ . The envelope of the family (42) solves the boundary value problem with an arbitrary initial condition . Define )(),0( 0 zz Φ=Φ [ ] xzzxzzx )( )()(, 0 0000 γγ +Φ=+Φ (43) By differentiating Eq. (43) with respect to x and we obtain 0z xx zzx ')'1(,2 2 γγγγγ +Φ′=Φ+=Φ+Φ (44) Function (43) will satisfy the eikonal equation (40) if the ray direction is matched with the local wave front tilt: )()( 000 zz Φ′=γ . Having constructed the eikonal ),( zxΦ we reduce the transport equation (41) to an ODE ( ) ( ) 0, 0 =+′+ ++ xzxA γ (45) with an evident integral xzzxA =+ (46) In a similar way, an asymptotic solution of the modified Claerbout equation (38) can be found. Substituting the Ansatz [ ]),(exp),,(),,( 0 zxikszxIszxU Φ= into Eq. (38) we obtain 0 =−+⎟ ⎛ Φ+Φ+Φ−+⎟ ⎛ +Φ+Φ− xszzzzzzsxxx IIIIIIikikk (47) The leading term ( )20kO disappears in virtue of the eikonal equation (40). Thus, to the accuracy ( )10−kO , a space-time transport equation arises for the slowly varying amplitude : ),,( szxI =Φ+Φ+Φ− IIII zzzzsxx (48) As the coefficients of Eq. (48) do not depend on s, it has a solution of the following form (49) [ ),(),(),,( zxsgzxAszxI Ψ−= ] Here, is a solution of the stationary transport equation (41) while is an arbitrary function of ),( zxA )(sg xcts −= , and satisfies a linear PDE ),( zxΨ xzzx Φ−=ΨΦ+Ψ (50) Solving Eq. (50) by characteristics one easily gets [ ] [ ] )()(,)(, 00000 zxzzxxzzx θγγ ++Φ=+Ψ (51) where )( 0zθ is an arbitrary function. So, the solution of the HPE (38) has asymptotic representation [ ] [ ] [ ] ),,()(),(),(exp),(),(~),,( 000 kzxuzzxsgzxikzxsgzxAszxU θ−Φ−=ΦΨ− (52) Here, is a solution of the standard Leontovich PE (4), and are its amplitude and eikonal, respectively; and ),,( 0kzxu ),( zxA ),( zxΦ )(sg )( 0zθ are arbitrary functions, and is to be found from the transcendental equation ),(0 zxz zxzz =Φ′+ )( 000 . Asymptotic solution (52) is a paraxial version of the space-time GO [29], the rays and wave fronts being defined numerically via parabolic equation. Functions and ),( zxA ),( zxΦ are generally complex- valued, so distinction between wave amplitude and complex “phase” is made solely on the basis of their different dependence on frequency. In particular, complex eikonal , defined as ),( zxΦ ),,(log),( 0 kkukki izx (53) is calculated from PE numerical solutions at two close frequencies ck 2,12,1 =ω . Physically, complex eikonal in Eq. (52) appears due to diffraction effects described by PE (4). An important consequence is the absence of singularities in the constructed asymptotic solution, as the "parabolic" rays do not produce caustics in the real space. Another effect caused by diffraction – pulse envelope distortion also is taken into account via complex values of the signal delay czx ),(ψ . Physical meaning of complex xcts −= is provided by the theory of analytic signal [22]. Arbitrary functions in Eq. (52) are determined by the initial conditions. In the simplest case the constructed transient (37) has the form [ ]),(),(),,(),,( zxxctFzxAxctzxtzxH Φ−−≈−Π≡ + (54) where and are complex amplitude and eikonal evolved from the initial , given by Eq. (23) and is the analytic signal (19) corresponding to the real signal . Physical meaning has the real part of the complex solution (54) or, from the practical point of view, its normalized envelope ),( zxA ),( zxΦ )(0 zA )(0 zΦ )(exp)()( 0 tictftcF ω−≈ )(ctf 2),,( szxH . In virtue of the superposition principle, a more general asymptotic solution can be constructed as a number of terms (54). That is a direct analogy with ordinary GO where the incident and reflected waves correspond to different ray families. An important practical example is radar pulse propagation over the earth surface when the direct and reflected from the ground pulses can be distinguished and used for target location [23]. Consider first a model example: a short pulse with carrier frequency and damped sinusoidal envelope (21), propagating over a slowly rolling boundary . Initial pulse parameters are: МHz2000 =f )(xhz = maba 9, ≈=Λ= π ; ,150 m=w ,01.0−=β m2000 =ρ . Stationary field distribution calculated by numerical integration of PE (4) at cfk 00 2π= produces a regular interference pattern – Fig. 13. It can be represented as a superposition of Fig. 13. Monochromatic attenuation function. Fig. 14. Reflected wave phase distribution. the incident Gaussian beam (6) with the reflected wave ),,( 0kzxui ir uukzxu −≡),,( 0 determined by the terrain and the impedance BC (11). Functions are given by the asymptotic solution (39)-(41), and eikonal )(xhz = ),(),,( zxzxA ii Φ ),( zxrΦ is reconstructed from the spatial phase distribution of the reflected wave – see Fig. 14. In accordance with such a monochromatic framework, a two-term asymptotic formula arises for the pulsed transient: [ ] [ ] [ ] [ ),(),(),(),( ),(),(),(),(~),,( ),(),( zxxctFzxeuzxxctFezxu zxxctFzxAzxxctFzxAtzxH ri Φ−−+Φ−−= Φ−−+Φ−− +Φ−+Φ− (55) Note that to find the amplitude and complex delay of the incident and reflected signals we need just to solve the standard PE in frequency domain at two close frequencies 02,1 ωω ≈ - see Eq. (53). In time domain, calculation reduces to the evaluation of an analytic function for the given argument values of interest. That radically diminishes the required computational resources compared with direct numerical integration of the TDPE (24). )(sF + Fig. 15. Received pulse envelope as function of receiver altitude. (a) (b) Fig. 16. Comparison between numerical solution of TDPE (24) (a) and asymptotic solution (55) (b). The envelope of the received analytic signal (55), as a function of xcts −= and the receiver height z, for is shown in Fig. 15. One can see profound interference minima near the earth surface and a good separation of the direct and reflected pulse to heights above 400 m. Figures 16a,b compare the asymptotic solution with direct numerical integration of the Claerbout TDPE (24). Qualitatively, they are almost identical. Some hardly seen discrepancy is due to a limited accuracy of the asymptotic solution and FD scheme (27). This comparison demonstrates the efficiency of the developed approach. Substantial acceleration of the numerical procedure (by around 200 times in this example) makes a good reason to use it in realistic conditions. kmx 7= As an example, we simulate an experimental situation [23]: radar pulse propagation between two aircrafts flying by parallel routs over an irregular terrain. The experiment [23] was aimed at simultaneous determination of the target range and altitude from the measured return times of the direct radar pulse and the echo signal from the earth surface. Our goal is to develop an efficient method of EM field calculations under conditions of multipath and signal distortion. At such large ranges )100( kmX = the Earth sphericity must be taken into account. For this purpose, a parabolic hump *2)( earthRxXx − has been added to the real terrain profile plotted in .9 of [23]. Atmospheric refraction has been considered by using the equivalent Earth radius earthearth RR 34 * = [12]. Global field strength distribution produced by the incident carrier wave at is depicted in Fig. 17a. Despite evident multipath character of the reflected wave (Fig. 17b) its eikonal МHz1410 =f ),( zxrΦ has a rather regular structure. Therefore, our PE based version of complex GO can be applied to simulate the averaged (a) (b) Fig. 17. Simulation of experiment [23]: global field strength (attenuation function) at , reflected wave (b). )(1410 aMHzf = parameters of the received radar pulse (the actually observed signal is a stochastic quantity with normal distribution [23]). Its envelope, as a function of the distance from the paraxial wave front xcts −= and the receiver height z, is depicted in Fig. 18. Fig. 18. Gaussian pulse envelope as function Fig. 19. Target height from reflected of relative delay and receiver altitude. pulse delay [23]. A Gaussian pulse waveform is chosen with the parameters corresponding to the experimental situation [23]: ,141,3.5 00 МHz== fkmz m75≈Λ . The direct and reflected pulses are distinctly separated for which allows one to reliably solve the triangle for target altitude determination - see Fig. 19 borrowed from [23]. Variability and statistics of kmz 5.4> 21,, RRR the simulated reflected pulse resemble the experimental plots presented in [23], and the calculated received pulsed signal envelope for a fixed receiver height kmz 3.5= (Fig. 20) Fig. 20. Received pulse envelope: asymptotic HPE solution (solid line), experiment [23] (dash), stochastic model calculation [23] (dots). agrees well both with the experimental data and the results of thorough statistical modeling [23]. The quantitative discrepancy in the reflected pulse amplitude does not exceed the inherent uncertainty due to the errors in terrain description. Acknowledgements This work was supported in part by a joint RFBR-DFG grant No 01-02-04003. The authors are grateful to Friedrich Landstorfer and Ningyan Zhu who initiated this research. References 1. M.A. Leontovich, A new method to solve problems of EM wave propagation over the earth surface (in Russian). USSR Academy of Sciences Trans., Physics Series, vol. 8, No. 1, pp. 16-22, 1944. 2. M.A. Leontovich, V.A. Fock, Solution of the problem of electromagnetic wave propagation along the Earth’s surface by the method of parabolic equation, J. Phus. USSR, vol. 10, pp. 13-23, 1946. 3. G.D. Malyuzhinets, Progress in understanding diffraction phenomena, Soviet. Phys. Uspekhi., vol. 69, pp.321-334, 1959. 4. V.M. Babič, V.S Buldyrev. Short-Wavelength Diffraction Theory: Asymptotic Methods. Springer-Verlag, Berlin, 1990 5. D. Lee, A.D. Pierce, E.-C. Shang, Parabolic equation development in the twentieth century, J. Comput. Acoustics, vol. 8, No. 4, 527-637, 2000. 6. L. A. Vainstein, Open resonators and open waveguides (in Russian). Soviet Radio, Moscow, 1966. 7. M.D. Feit, J.A. Fleck, Jr. Light propagation in graded-index fibers. Appl. Optics, vol. 17, pp. 3990-3998, 1978. 8. Yu.V. Kopylov, A.V. Popov, A.V. Vinogradov. Application of the parabolic wave equation to X-ray diffraction optics, Optics Communications, v. 118, pp. 619-636, 1995. 9. G.D. Malyuzhinets, A.V. Popov, Yu.N. Cherkashin. On the development of a computational method in diffraction theory. 3rd All-Union Symposium on Diffraction of Waves, Academy of Sciences, Moscow, 1964. 10. F.D. Tappert. The parabolic approximation method. Lecture Notes in Physics, v.70: Wave propagation and Underwater Acoustics, pp.224-287, Springer-Verlag, New York, 1977. 11. M.A. Leontovich. On approximate boundary conditions on the surface of well- conducting bodies (in Russian). Investigations on Radiowave Propagation, part 2, pp. 5-12. Academy of Sciences, Moscow, 1948 12. V.A. Fock, Electromagnetic Diffraction and Propagation Problems, Pergamon Press, 1965. 13. J. F. Claerbout. Fundamentals of Geophysical Data Processing with Applications to Petroleum Prospecting, McGraw-Hill, New York, 1976. 14. A.V.Popov, S.A. Hosiosky. On a generalized parabolic equation of diffraction theory (in Russian). J. Comp. Math. and Math. Phys., vol.17, No. 2, pp. 527-533, 1977. 15. E.A. Polyansky, On the relation between solutions of Helmholtz and SchrÖdinger type equations (in Russian), J. Comp. Math. Math. Phys., vol.2, No.1, pp. 241-249, 1972. 16. M.F. Levy. Parabolic equation modelling of propagation over irregular terrain. Electronics Letters, vol. 26, pp. 1153-1155, 1990. 17. V.A. Baskakov, A.V. Popov. Implementation of transparent boundaries for numerical solution of the Schroedinger equation, Wave Motion, vol.14, No.1, pp.123-128, 1991. 18. S.W. Marcus. A generalized impedance method for application of the parabolic approximation to underwater acoustics, J. Acoust. Soc. Am., vol. 89, pp.391-398, 1991. 19. M.F. Levy. Parabolic Equation Method for Electromagnetic Wave Propagation. IEE Electromagnetic Wave Series vol. 45, 2000. 20. M.D. Collins. The time-domain solution of the wide-angle parabolic equation including the effect of sediment dispersion. J. Acoust. Soc. Am., vol. 84, No. 6, pp. 2114-2125, 1988. 21. L.A. Vainstein, D.E. Vakman. Frequency Discrimination in Oscillation and Wave Theory (in Russian). Nauka, Moscow, 1983. 22. E. Heyman, L.B. Felsen. Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics. J. Opt. Soc. Am. A, vol. 18, No. 7, pp. 1588-1611, 2001. 23. L.M. Zurk. Experimental observation and statistics of multipath from terrain with application to overland height finding. IEEE Trans. Antennas Propag., vol. 47, No. 1, pp. 121-131, 1999. 24. A.V. Popov, V.V. Kopeikin, Ning Yan Zhu, F.M. Landstorfer. Modelling EM transient propagation over irregular dispersive boundary. Electronics Letters, vol. 38, No. 14, pp. 691-692, 2002. 25. A.V. Popov, V.V. Kopeikin, F.M. Landstorfer. Full-wave simulation of overland radar pulse propagation. Electronics Letters, vol. 39, No. 6, pp. 550-552, 2003. 26. A.V. Popov, V.V. Kopeikin. Prediction of electromagnetic pulse propagation along the earth surface (in Russian). Progress of Modern Radioelectronics, No. 1’2005, pp. 20-35. Radiotechnika, Moscow, 2005. 27. A.V. Popov. Solution of parabolic equation of diffraction theory by finite difference method. J. Comp. Math. and Math. Phys., vol.8, No. 5, pp. 1140-1144, 1968. 28. A.V. Popov. Accurate modeling of transparent boundaries in quasi-optics. Radio Science, vol. 31, No. 6, pp. 1781-1790, 1996. 29. V.M. Babič, V.S. Buldyrev, I.A. Molotkov. Space-Time Ray Method. Linear and Nonlinear Waves (in Russian). SPB University Press, St. Petersburg, 1995.

We simulate EM pulse propagation along the nonuniform earth surface using so called time-domain parabolic equation. To solve it by finite differences, we introduce a time-domain analog of the impedance boundary condition and a nonlocal BC of transparency reducing open computational domain to a strip of finite width. Numerical examples demonstrate influence of soil conductivity on the wide-band pulse waveform. For a high-frequency modulated EM pulse, we develop an asymptotic approach based on the ray structure of the monochromatic wave field at carrier frequency. This radically diminishes the computation costs and allows for pulsed wave field calculation in vast domains measured by tens of thousands wavelengths.

Introduction Parabolic equation method proposed by Leontowich and Fock [1,2] is an efficient simulation approach to VHF propagation over the earth surface. Deep physical analysis and advanced mathematical methods [3,4] turned Leontovich’s PE into a universal tool of diffraction theory. Its applications go far beyond the initial problem circle – e. g. [5-8]. The key role in this development played the decisive turn to straightforward numerical techniques pioneered by Malyuzhinets and Tappert [9,10]. In radio wave propagation, PE was used first to derive explicit analytical formulae for the EM field strength in model environments. A simplification has been reached by introducing the impedance boundary condition (BC) [11]. Taking into account tropospheric refraction ducts required the use of sophisticated asymptotic methods [12]. Further development (almost exclusively towards numerical implementation) was aimed at refined PE modifications [13-15], account for irregular terrain [16], introducing artificial transparent boundaries [17,18] and nonlocal BC to describe rough interfaces [19]. A non-stationary PE counterpart and a finite-difference (FD) scheme for its solution have been proposed by Claerbout and applied to seismic problems [13]. Afterwards, this “time-domain parabolic equation” (TDPE) was used to calculate acoustic propagation in ocean [20]. At the same time, little attempts of using TDPE to simulate EM pulse propagation in realistic environments are known. In this paper, we consider computational aspects of EM pulse propagation along the nonuniform earth surface. For ultrawide-band pulses without carrier, TDPE results directly from the exact wave equation written in a narrow vicinity of the wave front. To solve it by finite differences, we introduce a time-domain analog of the impedance BC and a nonlocal BC of transparency reducing the open computational domain to a strip of finite width. Numerical examples demonstrate the influence of soil conductivity on the received pulse waveform which can be used in remote sensing. mailto:popov@izmiran.ru mailto:kopeikin@izmiran.ru For a high-frequency modulated EM pulse, TDPE arises as a convolution of PE solutions with the pulse envelope spectrum. In order to overcome computational difficulties, we develop an asymptotic approach based on the ray structure of the monochromatic wave field calculated at the carrier frequency. To accommodate complex-valued asymptotic solutions to the real initial condition we use the “analytic signal” approach introduced by Vainstein, Heyman and Felsen [21, 22]. An explicit solution of the time-domain transport equation reduces the computational procedure to numerical integration of standard PE at the carrier frequency and evaluation of a given 1D function in time domain. This diminishes computational expenses by 2-3 orders of magnitude and allows for pulsed wave field calculation in vast domains measured by tens of thousands wavelengths. As an example, we consider a problem of target altitude determination from overland radar data [23]. This work has been done in collaboration with the Institute for high-frequency technique (IHF), Stuttgart University. Preliminary results appeared as short papers [24,25], a Russian version has been published in [26]. We dedicate this publication to the memory of Leopold Benno Felsen. 2. Monochromatic wave propagation Omitting technical details and method refinements – see [12, 19], recall PE based scheme of monochromatic wave propagation over a smoothly nonuniform earth surface - Fig.1. )(xhz = 0 z 0 1 00 , = = σ ε x ε, σ )(xhz = Fig. 1. Elevated source illuminating smoothly rolling terrain (sketch). Horizontal magnetic component ),( zxHH y = satisfies Helmholtz equation 0~22 ε (1) with complex permittivity ωσπεε i4~ += , where σ is soil conductivity in the Gaussian units set. In the upper medium 0,1 == σε and the contact conditions at are )(xhz = , . (2) where xzn ∂∂+∂∂=∂∂ αα sincos , )('arctan xh=α . At large distances from the source the wave field is sought as a plane wave with slowly varying complex amplitude: [ )(exp),,(),,( tkxikzxutzxH ]ω−≈ (3) Here, λπω 2≡= ck is the wave number, and the complex “attenuation function” satisfies the Leontovich PE ),,( kzxu )(,02 2 ik >= (4) In 3D, divergence factor x1 must be added in (3). In this paper we use Gaussian initial condition ( ) ( −−−⎟⎟ exp),,0( zzikzz kzu β ) (5) corresponding to an exact solution of PE (4) = xzz kzxui 22 exp),,( (6) - skewly propagating Gaussian beam with initial width and wave front radius 0w 0ρ , determined by complex parameter ( ) 12000 21 += wkix ρ ; β being a small elevation angle. Impedance approximation is based on wave beam contraction when entering a denser dielectric medium. Standard Leontovich BC [11] 0,~ =−=∂ (7) results from the contact conditions (2) under the assumption of almost vertical propagation in the lower medium: ( )ε~exp),( ikzTzxH −≈− . For grazing angles (Fig. 2) this assumption breaks and a plane incident wave ( )[ ]ββ coscosexp),( zxikzxH −=+ with small 1<<β Fig. 2. Derivation of impedance BC for grazing angles. enters the half-space close to the total internal reflection angle 0<z ( )210 ~arccos εγ = : 0 sin~2 γε γγ +≈ . Hence ensue ( )[ ] ( )[ ] ( ( )[ ] ( )000 exp)( expexp)( sincos ~exp)(),( tgzxVtgzxikT ctgzxk tgzxikT zxikTtxH (8) In virtue of the superposition principle, Eq. (8) holds for an arbitrary paraxial wave packet with the corresponding slowly varying function . Eliminating the latter by differentiation and making use of (2), we obtain 0,0~~ 1~ ==+ ε (9) This modified impedance BC provides a more accurate approximation of the reflection coefficient, especially, in a vicinity of the Brewster angle ( ) 210 1~arcsin += εβ . Fig. 3 allows one to compare the exact Fresnel reflection coefficient cos~sin~ cos~sin~ =FR (10а) with the Leontovich approximation 1sin~ 1sin~ βLR (10b) and that resulting from the modified impedance BC (9) Fig.3. Comparison between exact Fresnel reflection coefficient and impedance approximations. ( ) ( ) ( ) 1~cos~sin~ 1~cos~sin~ εβεβε εβεβε βMR (10c) Taking into account the boundary tilts )(xh′ and using "parabolic" approximation (3), we derive a modified BC for the attenuation function : ),( zxu )(,0)('~ xhzuxhik (11) Contrary to the standard Leontovich BC (9), here it is not necessary to assume 1~ >>ε - Eq. (11) breaks only for 11~ <<−ε when nonlocal effects of wave interaction in both media are to be taken into account [19]. The impedance BC grants uniqueness of the solution of PE (4). In fact, calculating the energy flow through a vertical cross section one obtains ),()( dzzxuxI , 0),(~ −= hxu (12) which proves stability and uniqueness of the boundary value problem solution. Finite-difference methods of PE solution have been studied in early works by Malyuzhinets and coauthors [9, 27]. Further method development is described in monographs [5, 19]. We employ a six-point implicit FD scheme supplemented with the impedance BC (11) at and a discrete approximation of the nonlocal transparency BC [17,28] imposed at the artificial computational boundary )(zhz = maxzz = : ( ) ( ) maxmax , , (13) An example of simulated VHF propagation over irregular terrain is illustrated by Fig. 4. Fig. 4. VHF attenuation function over irregular earth surface. 3. Radio pulse propagation: Fourier synthesis A straightforward way to describe EM transients is to convolve monochromatic wave fields with the signal spectrum. In a 1D case, the propagating pulse is a superposition of plane waves [ ]∫ −= dkekFtxH tkkxi )()(~ ),( ω (14) In free space, kc=ω , and formula (14) yields a dispersion-less traveling wave −=−= dkekFsFxctFtxH iks)(~ )(),(),( (15) In a 2D environment, a natural generalization of the 1D solution (14) is a paraxial wave packet dkekzxukFtzxH ctxik )(),,()(~ ),,( − (16) where is a solution of the PE (4) at a fixed frequency ),,( kzxu kc=ω . The superposition (16) will approximate an exact solution of the wave equation if the spectrum )(~ xF is confined near a certain positive satisfying the PE applicability conditions: LDk ππ 220 >>>> where and are lateral and longitudinal characteristic scales of the problem. Consider a quasi-monochromatic pulse tctfctF 0cos)()( ω= with duration 02 ωπ>>T . Its Fourier transform ( ) ([ 00 ~~2 )(~ kkfkkfkF ++−= )], (17) where is the envelope spectrum, contains negative frequencies not described by PE (4). Introducing complex signal = dtectfckf tiω)()( )(ctf )exp()()( 0tictfctFc ω−= eliminates the second term in (17). Still, the remaining "positive" component ( )0 kkf − centered at may spread onto negative semi-axis. In order to avoid nonphysical effects of negative frequencies propagation, the "analytic signal" [21] can be used, defined as a one-side inverse Fourier transform of the truncated spectrum >++−= kkkfkkfkF kF (18) Thus, by definition, the analytic signal is a Cauchy type integral FdkekFsF , (19) regular in the lower half-plane 0Im <s . For real s, its real part coincides with whereas the imaginary part is given by Hilbert transform ( )∫ FsF V.P. )(Im (20) Introduction of the analytic signal violates the causality principle: the real signal is zero before the moment of switching on the transmitter while for . However, for a high-frequency radio pulse this discrepancy is small. So the analytic signal envelope defined as )(ctF 0)( ≠+ ctF 0<t )(sF + is close to but, contrary to the "naïve" complex signal , admits analytic continuation into complex domain, compatible with asymptotic propagation laws [22]. As an example, consider a modulated high-frequency pulse with )(sf )(sFc sksfsF 0cos)()( = the envelope Fig. 5. Modulated radio pulse waveform (21). Fig. 6. Analytic signal spectrum (18). 0),exp(sin sbsas sf , ibkkba 222 −−+ = (21) see Fig. 5. For its length is ab ≈ acT π~=Λ . The envelope spectrum )(kf has a peak at with 0=k Λ≈Δ π2k and tends to zero for ∞→k as ( )2kaO . Spectra )(~ kF and )(~ kF + are shown in Fig. 6; the analytic signal envelope )(~ sF + is plotted in Fig. 7a,b for real and complex arguments. (a) (b) Fig. 7. Analytic signal envelope of real (a) and complex (b) arguments. For a wave packet −++ = )(),,()(~ ),,( dkekzxukFtzxH ctxik , (22) )(0 0)(),,0( zikezAkzu Φ= the “initial” condition [ ] [ )()()(~ ),,0( 00 )(0 0 zctFzAekF tzH ctzik Φ−== + ] (23) describes an analytic signal with amplitude and initial delay )(ctF + )(0 zA czzt )()( 00 Φ= . Fig. 8. Example of a Fourier-synthesized propagating EM pulse. An example of modulated pulse propagation over smoothly rolling interface is depicted in Fig. 8. A sequence of snap-shots traces the evolution of the initial pulse envelope defined by (5), (21) due to the incident Gaussian beam divergence and reflection from the curved interface ),()( 00 kzuctf )(xhz = . It should be noted that Fourier synthesis is computationally efficient only for rather narrow-band pulses. In fact, for a good approximation of the convolution integral (22) one has to solve PE (4) for a set of wave numbers covering the spectral band kkkkk Δ+<<Δ− 00 , Λ≈Δ π2k with a small frequency step ( kk Δ<<δ ) and, to avoid phantom solutions in the given range xΔ , even more restrictive condition must be posed: xk Δ<< πδ 2 . Adequate simulation methods for wide-band EM pulse propagation are discussed in the following sections. 4. Time-domain PE and boundary conditions Straightforward derivation shows that if ( )kzxu ,, is a solution of PE (4), the transient wave packet (16) , as a function of variables ),,(),,( szxtzx Π≡Η xctszx −=,, , satisfies the Claerbout equation zsx ∂ (24) Equation (24), usually called "time-domain parabolic equation" (TDPE), has been obtained in [13] by formal substitution sik ∂∂= as well as by the reduction of the time-dependent wave equation (25) in a narrow vicinity of the paraxial wave front ctx = . Introduction of scaled variables Λ−=== )(,, xctDzLx ηςξ , where L, D are computational domain length and width, is spatial pulse length, yields Λ ςηξ ∂ , 1<<Λ== DLDν (26) Neglecting the small term results in TDPE (24).This derivation clarifies the nature of the “time-domain parabolic equation”: )( 2νO 1) It is a hyperbolic equation written in a traveling coordinate frame ; ),,( szx 2) TDPE does not describe the backward moving waves; 3) TDPE is a paraxial (narrow-angle) approximation valid in a narrow strip 1)( <<= νOLD ; 4) TDPE describes short pulses 1)( <<=Λ νOD whose length Λ is comparable with the wave front deviation from the plane ctx = (Fig. 9); 5) TDPE solutions are not necessary modulated high-frequency signals – they can represent short ultrawide-band pulses without carrier, e. g. a damped sinusoid (21). )(ctf Fig. 9. Derivation of TDPE (24). Here, a seeming contradiction may arise, as the spectral maximum of can lie in the vicinity of zero frequency, not described by PE (4). As a matter of fact, at small distances from the wave front )(Λ= Os the main part of the pulse energy is determined by the high- frequency edge of its spectrum Dak π2~ >> satisfying PE applicability conditions. To solve TDPE (24), an FD scheme of the 2nd order of accuracy has been proposed in [13]: ( ) ( )lllllllll rrrrrrrr ,,11,1,1 2,,11,1, )(4 nnnnznnnn zsx Π+Π+Π+Π∇ =Π+Π−Π−Π ΔΔ ++++++++ (27) Here, , { }mnn ll ,, Π=Π ,xnxn Δ= zmzm Δ= , ss Δ= ll ; ( ) 1,,1,,2 2 −+ Π+Π−Π=Π∇ mnmnmn nz llll . This equation is solved by zigzag marching in plane between boundary values ),( sx ( ) [ ] 0)0,,(,)()(,,0 00 =ΠΦ−=Π zxzctfzAsz (28) (given source and causality condition). At each marching step , a three-diagonal linear equation set arises for the unknown vector ),( mn 1,1 ++Π l n . In order to complete the boundary value problem, we have to add a correct BC taking into account soil properties and to find a way of the domain truncation without creating spurious reflections. Both problems are resolved by applying Fourier transform to the frequency-domain BC (11), (13). Consider a paraxial wave packet dkekzxukfszx iks− ∫=Π ),,()( , (29) satisfying the causality condition 0),,( =Π szx for 0<s . We rewrite the impedance BC (11) emphasizing the dependence of complex permittivity kciσπεε 4~ += on the wave number ck ω= : 0)(' , (30) where )1(2,4 −== επσεπσ cqcr . Multiplying Eq. (30) by and applying Fourier transform (29), we get dkkekzxukf xhshx iks),,()( ),,()('),,( . (31) Substituting here the inverse Fourier transform (32) ∫ ),,(),,()( ηη ηdezxkzxukf ik we obtain, by standard calculations, the following expression for the RHS of (31): dsNhxshx )(),,(),,( ; ⎥ −+= ∫ tqrrs qr qtIeqesN )( )()( (33) Thus, we have derived a nonlocal 2D boundary condition dsNhxshx xhshx )(),,(),,( ),,()('),,( ηηη (34) being an exact time-domain counterpart of the impedance BC (11). Its nonlocality is a consequence of interaction between two waves propagating along the earth surface with different phase velocities. The integral term kernel )( η−sN can be easily calculated for different ε and σ . For 15.3>ε , function monotonously tends to zero with increasing s – see Fig. 10. Fig. 10. Kernel of nonlocal impedance BC (34). It is interesting to note that in both limiting cases: perfectly conducting boundary )( ∞=σ and zero soil conductivity )0( =σ Eq. (34) reduces to a local BC. In the former case tends to a delta function, for . The integral in (34) limits to =∞→−= 1)(,)0( dssNqrN ),,( shx , the RHS vanishes, and we get a Neumann BC. In the opposite case )0( →σ the spatial scale of (length of the pulse “dispersion tail”) is growing but its absolute value is tending to zero, so only a local term remains. In a similar way, the time-domain generalization of the transparency BC (13) is derived which grants the absence of spurious reflections a from the artificial computational boundary . Applying to Eq. (13) the Fourier transform (29), (32) and denoting maxzz = ipk = , we get ( ) ( ) ( ) ∫∫∫ maxmax (35) Finally, evaluating the inner integral we obtain an elegant 2D boundary condition ( ) ( ) ( )( )ηξ ηξπ −−∂∂ ∫ ∫ sx ,, max max (36) symmetric with respect to the variables x, z , which could be expected from the symmetry of the TDPE (24). A simulated example of ultrawide-band EM pulse propagation over a nonuniform earth surface with soil parameters 10=ε , )01.0(109 17 мSs −⋅=σ is depicted in Fig. 11а. Evolution of the spatial amplitude distribution for a pulsed signal generated by a Gaussian source: ( )[ ]20200 exp)( wzzzA −−= with a skew curved wave front: )(2)()( 00 00 zzzzz −+−=Φ βρ is shown in a grey color scale. The initial pulse waveform is a damped sinusoid (21) with )(ctf ba = and spatial length ma 30==Λ π . (a) (b) Fig. 11. Propagation of ultrawide-band pulse (21) over nonuniform earth surface. Initial Gaussian beam parameters: ,80,3000 mmz == 0w 1.0,3000 −== βρ m . Soil conductivity: mS01.0=σ (a), 0=σ (b). Fig. 12. Received pulse waveform depending on soil conductivity: mS01.0=σ (solid line), mS001.0=σ (dashed line); initial pulse (dots). The snapshots clearly show the reflected pulse generation at the earth surface. The transparency BC (36) imposed at the height mz 500max = assures unimpeded radiation exit from the computational domain. Finite soil conductivity causes signal dispersion appearing in a certain delay of the reflected pulse. It is obvious from the comparison with a similar plot calculated for a model non-conducting soil 0=σ - Fig. 11b. A quantitative estimation of the effect can be made by means of Fig. 12 revealing a considerable dependence of the pulse waveform on the soil conductivity. This effect caused by the pulse penetration into the ground can be used for ecological monitoring (water pollution, earthquake precursors, etc.). 5. Hybrid TDPE and short high-frequency pulse propagation An important practical issue is overland propagation of short EM pulses with high- frequency carrier. Basically, having absolute stability, TDPE (24) is capable to describe wide- band radio pulse propagation. However, the computational expense is drastically growing with increasing carrier frequency. If 0ω considerably exceeds the spectral band of the signal it is useful to factor out the carrier: [ ])(exp),,(),,( 00 txkiszxUszx ω−=Π (37) and to consider the transient signal envelope satisfying a hybrid equation ),,( szxU ik , (38) combining the features of standard Leontovich PE (4) with Claerbout TDPE (24). Unfortunately, despite a relatively slow variations of in space-time, straightforward ),,( szxU numerical solution of this hybrid TDPE (HPE) entails considerable difficulties, as the large coefficient by 0k xU ∂∂ demands a dense computational grid. On the other hand, the presence of a big parameter allows us to construct an asymptotic solution of HPE (38) radically reducing the computational burden. In order to find a proper asymptotic Ansatz, consider monochromatic PE (4) at the carrier frequency ck00 =ω with the initial condition . We admit complex values of the eikonal to describe relatively narrow wave packets, like a Gaussian beam (6). For , we obtain an asymptotic solution 00)(),0( zikezAzu Φ= )(0 zΦ (39) ),(0 0),(),,( zxikezxAkzxu Φ= where satisfies a “parabolic” eikonal equation ),( zxΦ 1 2 =Φ+Φ zx (40) while slowly varying amplitude is governed by the paraxial transport equation ),( zxA =Φ+Φ+ AAA zzzzx (41) Eqs. (40)-(41), being an approximate form of the well-known laws of geometric optics (GO) [4], can be easily solved by the method of characteristics. Consider a particular solution of the eikonal equation (40) corresponding to a bundle of rays spreading from a central point : 0,0 zzx == zx −−+≈ =Φ 20 ),( (42) At a characteristic line xzz γ+= 0 we have xxx ),( γγ =Φ . The envelope of the family (42) solves the boundary value problem with an arbitrary initial condition . Define )(),0( 0 zz Φ=Φ [ ] xzzxzzx )( )()(, 0 0000 γγ +Φ=+Φ (43) By differentiating Eq. (43) with respect to x and we obtain 0z xx zzx ')'1(,2 2 γγγγγ +Φ′=Φ+=Φ+Φ (44) Function (43) will satisfy the eikonal equation (40) if the ray direction is matched with the local wave front tilt: )()( 000 zz Φ′=γ . Having constructed the eikonal ),( zxΦ we reduce the transport equation (41) to an ODE ( ) ( ) 0, 0 =+′+ ++ xzxA γ (45) with an evident integral xzzxA =+ (46) In a similar way, an asymptotic solution of the modified Claerbout equation (38) can be found. Substituting the Ansatz [ ]),(exp),,(),,( 0 zxikszxIszxU Φ= into Eq. (38) we obtain 0 =−+⎟ ⎛ Φ+Φ+Φ−+⎟ ⎛ +Φ+Φ− xszzzzzzsxxx IIIIIIikikk (47) The leading term ( )20kO disappears in virtue of the eikonal equation (40). Thus, to the accuracy ( )10−kO , a space-time transport equation arises for the slowly varying amplitude : ),,( szxI =Φ+Φ+Φ− IIII zzzzsxx (48) As the coefficients of Eq. (48) do not depend on s, it has a solution of the following form (49) [ ),(),(),,( zxsgzxAszxI Ψ−= ] Here, is a solution of the stationary transport equation (41) while is an arbitrary function of ),( zxA )(sg xcts −= , and satisfies a linear PDE ),( zxΨ xzzx Φ−=ΨΦ+Ψ (50) Solving Eq. (50) by characteristics one easily gets [ ] [ ] )()(,)(, 00000 zxzzxxzzx θγγ ++Φ=+Ψ (51) where )( 0zθ is an arbitrary function. So, the solution of the HPE (38) has asymptotic representation [ ] [ ] [ ] ),,()(),(),(exp),(),(~),,( 000 kzxuzzxsgzxikzxsgzxAszxU θ−Φ−=ΦΨ− (52) Here, is a solution of the standard Leontovich PE (4), and are its amplitude and eikonal, respectively; and ),,( 0kzxu ),( zxA ),( zxΦ )(sg )( 0zθ are arbitrary functions, and is to be found from the transcendental equation ),(0 zxz zxzz =Φ′+ )( 000 . Asymptotic solution (52) is a paraxial version of the space-time GO [29], the rays and wave fronts being defined numerically via parabolic equation. Functions and ),( zxA ),( zxΦ are generally complex- valued, so distinction between wave amplitude and complex “phase” is made solely on the basis of their different dependence on frequency. In particular, complex eikonal , defined as ),( zxΦ ),,(log),( 0 kkukki izx (53) is calculated from PE numerical solutions at two close frequencies ck 2,12,1 =ω . Physically, complex eikonal in Eq. (52) appears due to diffraction effects described by PE (4). An important consequence is the absence of singularities in the constructed asymptotic solution, as the "parabolic" rays do not produce caustics in the real space. Another effect caused by diffraction – pulse envelope distortion also is taken into account via complex values of the signal delay czx ),(ψ . Physical meaning of complex xcts −= is provided by the theory of analytic signal [22]. Arbitrary functions in Eq. (52) are determined by the initial conditions. In the simplest case the constructed transient (37) has the form [ ]),(),(),,(),,( zxxctFzxAxctzxtzxH Φ−−≈−Π≡ + (54) where and are complex amplitude and eikonal evolved from the initial , given by Eq. (23) and is the analytic signal (19) corresponding to the real signal . Physical meaning has the real part of the complex solution (54) or, from the practical point of view, its normalized envelope ),( zxA ),( zxΦ )(0 zA )(0 zΦ )(exp)()( 0 tictftcF ω−≈ )(ctf 2),,( szxH . In virtue of the superposition principle, a more general asymptotic solution can be constructed as a number of terms (54). That is a direct analogy with ordinary GO where the incident and reflected waves correspond to different ray families. An important practical example is radar pulse propagation over the earth surface when the direct and reflected from the ground pulses can be distinguished and used for target location [23]. Consider first a model example: a short pulse with carrier frequency and damped sinusoidal envelope (21), propagating over a slowly rolling boundary . Initial pulse parameters are: МHz2000 =f )(xhz = maba 9, ≈=Λ= π ; ,150 m=w ,01.0−=β m2000 =ρ . Stationary field distribution calculated by numerical integration of PE (4) at cfk 00 2π= produces a regular interference pattern – Fig. 13. It can be represented as a superposition of Fig. 13. Monochromatic attenuation function. Fig. 14. Reflected wave phase distribution. the incident Gaussian beam (6) with the reflected wave ),,( 0kzxui ir uukzxu −≡),,( 0 determined by the terrain and the impedance BC (11). Functions are given by the asymptotic solution (39)-(41), and eikonal )(xhz = ),(),,( zxzxA ii Φ ),( zxrΦ is reconstructed from the spatial phase distribution of the reflected wave – see Fig. 14. In accordance with such a monochromatic framework, a two-term asymptotic formula arises for the pulsed transient: [ ] [ ] [ ] [ ),(),(),(),( ),(),(),(),(~),,( ),(),( zxxctFzxeuzxxctFezxu zxxctFzxAzxxctFzxAtzxH ri Φ−−+Φ−−= Φ−−+Φ−− +Φ−+Φ− (55) Note that to find the amplitude and complex delay of the incident and reflected signals we need just to solve the standard PE in frequency domain at two close frequencies 02,1 ωω ≈ - see Eq. (53). In time domain, calculation reduces to the evaluation of an analytic function for the given argument values of interest. That radically diminishes the required computational resources compared with direct numerical integration of the TDPE (24). )(sF + Fig. 15. Received pulse envelope as function of receiver altitude. (a) (b) Fig. 16. Comparison between numerical solution of TDPE (24) (a) and asymptotic solution (55) (b). The envelope of the received analytic signal (55), as a function of xcts −= and the receiver height z, for is shown in Fig. 15. One can see profound interference minima near the earth surface and a good separation of the direct and reflected pulse to heights above 400 m. Figures 16a,b compare the asymptotic solution with direct numerical integration of the Claerbout TDPE (24). Qualitatively, they are almost identical. Some hardly seen discrepancy is due to a limited accuracy of the asymptotic solution and FD scheme (27). This comparison demonstrates the efficiency of the developed approach. Substantial acceleration of the numerical procedure (by around 200 times in this example) makes a good reason to use it in realistic conditions. kmx 7= As an example, we simulate an experimental situation [23]: radar pulse propagation between two aircrafts flying by parallel routs over an irregular terrain. The experiment [23] was aimed at simultaneous determination of the target range and altitude from the measured return times of the direct radar pulse and the echo signal from the earth surface. Our goal is to develop an efficient method of EM field calculations under conditions of multipath and signal distortion. At such large ranges )100( kmX = the Earth sphericity must be taken into account. For this purpose, a parabolic hump *2)( earthRxXx − has been added to the real terrain profile plotted in .9 of [23]. Atmospheric refraction has been considered by using the equivalent Earth radius earthearth RR 34 * = [12]. Global field strength distribution produced by the incident carrier wave at is depicted in Fig. 17a. Despite evident multipath character of the reflected wave (Fig. 17b) its eikonal МHz1410 =f ),( zxrΦ has a rather regular structure. Therefore, our PE based version of complex GO can be applied to simulate the averaged (a) (b) Fig. 17. Simulation of experiment [23]: global field strength (attenuation function) at , reflected wave (b). )(1410 aMHzf = parameters of the received radar pulse (the actually observed signal is a stochastic quantity with normal distribution [23]). Its envelope, as a function of the distance from the paraxial wave front xcts −= and the receiver height z, is depicted in Fig. 18. Fig. 18. Gaussian pulse envelope as function Fig. 19. Target height from reflected of relative delay and receiver altitude. pulse delay [23]. A Gaussian pulse waveform is chosen with the parameters corresponding to the experimental situation [23]: ,141,3.5 00 МHz== fkmz m75≈Λ . The direct and reflected pulses are distinctly separated for which allows one to reliably solve the triangle for target altitude determination - see Fig. 19 borrowed from [23]. Variability and statistics of kmz 5.4> 21,, RRR the simulated reflected pulse resemble the experimental plots presented in [23], and the calculated received pulsed signal envelope for a fixed receiver height kmz 3.5= (Fig. 20) Fig. 20. Received pulse envelope: asymptotic HPE solution (solid line), experiment [23] (dash), stochastic model calculation [23] (dots). agrees well both with the experimental data and the results of thorough statistical modeling [23]. The quantitative discrepancy in the reflected pulse amplitude does not exceed the inherent uncertainty due to the errors in terrain description. Acknowledgements This work was supported in part by a joint RFBR-DFG grant No 01-02-04003. The authors are grateful to Friedrich Landstorfer and Ningyan Zhu who initiated this research. References 1. M.A. Leontovich, A new method to solve problems of EM wave propagation over the earth surface (in Russian). 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The time-domain solution of the wide-angle parabolic equation including the effect of sediment dispersion. J. Acoust. Soc. Am., vol. 84, No. 6, pp. 2114-2125, 1988. 21. L.A. Vainstein, D.E. Vakman. Frequency Discrimination in Oscillation and Wave Theory (in Russian). Nauka, Moscow, 1983. 22. E. Heyman, L.B. Felsen. Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics. J. Opt. Soc. Am. A, vol. 18, No. 7, pp. 1588-1611, 2001. 23. L.M. Zurk. Experimental observation and statistics of multipath from terrain with application to overland height finding. IEEE Trans. Antennas Propag., vol. 47, No. 1, pp. 121-131, 1999. 24. A.V. Popov, V.V. Kopeikin, Ning Yan Zhu, F.M. Landstorfer. Modelling EM transient propagation over irregular dispersive boundary. Electronics Letters, vol. 38, No. 14, pp. 691-692, 2002. 25. A.V. Popov, V.V. Kopeikin, F.M. Landstorfer. Full-wave simulation of overland radar pulse propagation. 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Simulation of Graphene Nanoribbon Field Effect Transistors Gianluca Fiori, Giuseppe Iannaccone Dipartimento di Ingegneria dell’Informazione : Elettronica, Informatica, Telecomunicazioni, Università di Pisa, Via Caruso 16, 56126 Pisa, Italy. email : g.fiori@iet.unipi.it; Tel. +39 050 2217638; Fax : + 39 050 2217522 May 28, 2018 DRAFT http://arxiv.org/abs/0704.1875v2 Abstract We present an atomistic three-dimensional simulation of graphene nanoribbon field effect transistors (GNR-FETs), based on the self-consistent solution of the 3D Poisson and Schrödinger equation with open boundary conditions within the non-equilibrium Green’s Function formalism and a tight-binding Hamiltonian. With respect to carbon nanotube FETs, GNR-FETs exhibit comparable performance, reduced sensitivity on the variability of channel chirality, and similar leakage problems due to band-to-band tunneling. Acceptable transistor performance requires prohibitive effective nanoribbon width of 1-2 nm and atomistic precision, that could in principle be obtained with periodic etch patterns or stress patterns. Keyworks: graphene, nanoribbon, NEGF, three-dimensional Poisson, Atomistic tight-binding Hamiltonian. May 28, 2018 DRAFT I. INTRODUCTION In the last decade, Carbon NanoTubes (CNT) have attracted extraordinary interest for their extremely interesting physical and electrical properties [1], and their potential as an alternative to silicon as channel material for transistors beyond CMOS technology [2]. Recent experiments by Novoselov et al. [3] demonstrated the possibility of fabricating stable single atomic layer graphene sheets, with remarkable electrical properties, that have brought new excitation to the field of carbon electronics. Two-dimensional graphene is a zero gap material, which makes it not suitable for transistor applications. Energy gap can however be induced by means of lateral confinement [4], realized for example by lithography definition of narrow graphene stripes, the so-called graphene nanoribbons. Experiments on graphene-based devices [5] and Graphene NanoRibbon FETs [6] (GNR-FETs) have appeared only very recently, and demonstrate limited capability to modulate the conductance of a graphene channel at room temperature. The main problem is the need to fabricate extremely narrow nanowires (of the order of 1 nm) with atomic precision to obtain an energy gap adequate for room temperature operation. Since at the moment the fabrication technology is at its very first steps, computer simulations can be very useful to provide physical insights of GNR-FETs and to estimate the attainable performance. Recent theoretical works have shown that graphene nanoribbons have an energy gap which has an oscillating behavior as a function of width, with average roughly proportional to the inverse width, and that edge states play a very important role in inhibiting the existence of fully metallic nanoribbons [7]. Such behavior cannot be reproduced if one does not consider edge effects [8]. Also from the simulation point of view, research on GNR-FETs is at an embryonic stage: the only works available in the literature [9], [10] are based on a semiclassical analytical top-of-the-barrier model. For short-channel transistors only a three-dimensional simulation is suitable for an accurate evaluation of the electrostatics and of intraband and interband tunneling. To this purpose, we have developed a code for the simulation of GNR-FETs, based on the Non-Equilibrium Green’s Function formalism (NEGF), with a tight-binding Hamiltonian built from pz orbital basis set in the real space, which has been included in our in-house three-dimensional device simulator NANOTCAD ViDES [11]. We will show that GNR-FETs have performance comparable with CNT-FETs, and that can be greatly affected by the channel width and edge roughness. II. PHYSICAL MODEL AND RESULTS Our approach is based on the self-consistent solution of the three-dimensional Poisson and Schrödinger equations with open boundary conditions [12], which is able to take into account fully ballistic transport, in order to outline the higher limits of device performance, as well as elastic scattering due to line edge roughness. The Hamiltonian is taken from [7], in which edge states at the nanoribbon lateral ends have been considered. In this work we refer to (N ,0) armchair graphene nanoribbons, which consist of an unrolled (n,0) zig-zag nanotube with N = 2n. May 28, 2018 DRAFT The considered double-gate GNR-FETs have the structure depicted in the inset of Fig. 1. The gates are metallic, the oxide thickness tox is equal to 1 nm, the channel is 15 nm long, and W is the channel width. The source and drain extensions are 10 nm long, and are doped with a molar fraction of fully ionized donors f = 5 × 10−3. The spacing between parallel GNRs is 4 nm. In Fig. 1 the transfer characteristics of a (12,0) GNR-FET (W=1.37 nm) for drain-to-source voltage VDS of 0.1 and 0.5 V are shown, and compared to those of a (16,0) CNT-FET with the same geometry (same tox, L and device spacing), whose energy gap (Egap) is close to that of the GNR-FET and equal to 0.6 eV. Good control of the channel by the gate potential is shown at VDS= 0.1 V, since the sub-threshold swing (S) for the GNR-FET and the CNT-FET are 64 and 68 mV/dec, respectively. For VDS= 0.5 V we observe a pronounced degradation of S, with S=191 mV/dec for the GNR-FET and almost 250 mV/dec for the CNT-FET. This has to be imputed to Hole-Induced Barrier Lowering (HIBL) [12] : in the sub-threshold regime, when sufficiently high VDS is applied, confined states in the valence band of the channel align with the occupied states in the drain, leading to band-to-band injection of holes in the channel. If only elastic band-to-band tunneling can occur (as assumed in our simulation) the excess of holes in the channel lowers the channel potential, increasing the off current and S, as shown in Fig. 1: the lower the energy gap and the higher the VDS , the higher the HIBL effect. HIBL is more pronounced in CNT-FETs than in GNR-FETs, because the conduction band of CNTs is double degenerate and therefore CNTs have twice the density of states of GNRs with the same gap. If, on the other hand, inelastic band-to-band tunneling or Schockley-Read-Hall mechanisms are relevant, holes can recombine with electrons at the source and, instead of HIBL, we observe a leakage current from source to drain due to gate-induced drain leakage (GIDL) [13]. In strong inversion, the transconductance gm at VDS=0.1 V is 3600 and 6100 µS/µm for the GNR-FET and the CNT-FET, respectively, whereas at VDS=0.5 V, we obtain gm=4800 µS/µm for the GNR and a gm=8760 µS/µm for the CNT. The advantage of CNT-FETs is due to the double degeneracy of the conduction band in carbon nanotubes. It is known that a variability of the chirality of fabricated CNTs yields metallic nanotubes useless for transistor applications. For GNRs this problem is mitigated, since all GNRs are semiconducting. In order to investigate quantitatively the effect of a finite fabrication tolerance on the width of GNRs, we have computed the transfer characteristics of GNR-FETs with different chiralities : (12,0), (14,0) and (16,0). As can be seen in Fig. 2a, the three devices behave as transistors, but show very different behavior, even if they differ by only one carbon atom along the channel width. The problem is that the gap is still largely dependent on the chirality : the (16,0) GNR (W=1.87 nm) has the larger gap (Egap = 0.71 eV), while the (14,0) (W=1.62 nm) has the smallest gap (Egap = 0.13 eV). As a consequence, the (16,0) device show the best gate control over the channel potential, while the (14,0) the worst : the energy gap is so small that elastic band-to-band tunneling occurs at the source and current is dominated by GIDL. Such problem is reduced if rough edges are considered. We have considered the impact of line edge roughness in a (16,0) GNR-FET device, by randomly decoupling carbon atoms on the lateral boundaries of the GNR. The transfer May 28, 2018 DRAFT characteristic for one example is shown in Fig. 2 (dashed line). Since the channel consists of several hundreds of rings, the rough GNR behaves as a GNR with an intermediate effective gap. More statistical simulations would be needed to assess the dispersion of the electrical characteristics, but the typical GNRs is probably long enough to provide sufficient averaging to suppress inter device dispersion. Rough edge scattering strongly affects the on- current and the transconductance suppressing it by about 30% with respect to fully ballistic transistors. Additional suppression in realistic GNR-FETs can be due to defects, ionized impurities and phonon scattering. From the above simulations, it is clear that lateral confinement way beyond state-of-the-art etching capabilities would be needed to obtain adequate Egap. We also found that electrostatic periodic potential modulation with a peak-to-peak value of few Volts is not sufficient to induce the required gap of few hundreds mV. In order to evaluate whether a periodic strain pattern can allow to engineer the GNR gap, we have computed the energy gap in a (24,0) GNR (W=5.86 nm), multiplying the overlap integral of the element of the Hamiltonian in correspondence of the couple of atoms in the middle of the GNR by a ”strain factor” σ : σ is larger than one for compressive strain, and smaller than one for tensile strain. As can be seen in Fig. 3, compressive strain seems to be able to increase the energy gap of the nanowire by a significant amount. Of course, we can only suggest to experimentalists to evaluate the option. III. CONCLUSIONS In this work, a simulation study of GNR-FETs has been performed by means of the self-consistent solution of the 3D Poisson and Schrödinger equation with open boundary conditions, within the NEGF formalism. Edge states have been considered at the lateral ends of the nanoribbon using the model proposed by [7]. GNR-FETs exhibit performance similar to CNT-FETs, also showing significant band-to-band tunneling when small gap devices are considered and large VDS is applied. GNR-FETs are more robust than CNT-FETs with respect to variability of the channel chirality, and edge roughness seems to play an useful averaging effect. Finally we suggest that periodic strain could in principle represent an alternative to etching for inducing an energy gap in graphene. ACKNOWLEDGMENT Authors thank Prof. Massimo Macucci for suggestions and fruitful discussions. Support from the European Science Foundation EUROCORES Programme Fundamentals of Nanoelectronics, through funding from the CNR (awarded to IEEIIT-PISA) and the EC Sixth Framework Programme, under project Dewint (Contract N. ERAS-CT- 2003-980409) is gratefully acknowledged. May 28, 2018 DRAFT REFERENCES [1] A. Javey, J. Guo, Q. Wang, M. Lundstrom and H. Dai, “Ballistic carbon nanotube field-effect transistors”, Nature, Vol. 424, No. 9593, pp.654-657, 2003. [2] International Technology Roadmap for Semiconductor 2005, Available : http://public.itrs.net. [3] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films”, Science, Vol. 306, No. 5696, pp. 666-669, 2004. [4] M. Y. Han, B. Özyilmaz, Y. Zhang, and P. Kim, “Energy Band Gap Engineering of Graphene Nanoribbons”, available at : http://arxiv.org/abs/cond-mat/0702511. [5] M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, “A Graphene Field-Effect Device”, Electr. Dev. Lett., Vol. 28, No. 4, pp. 282-284, 2007. [6] Z.Chen, Y-M. Lin, M. J. Rooks and P. Avouris, “Graphene Nano-Ribbon Electronics”, available at : http://arxiv.org/abs/cond-mat/0701599. [7] Y.W. Son, M.L. Cohen, and S.G. Louie, “Energy gaps in graphene nanoribbons”, Phys. Rev. Lett., Vol. 97, pp.216803, 2006. [8] S. Reich, J. Maultzsch, C. Thomsen, P. Ordejon, “Tight-binding description of graphene”, Phys. Rev. B, Vol. 66, p. 035412, 2002. [9] Y. Ouyang, Y. Yoon, J.K. Fodor, and J. Guo, “Comparison of performance limits for carbon nanoribbon and carbon nanotube transistors”, Appl. Phys. Lett., Vol. 89, pp.203107-203109, 2006. [10] G.Liang, N. Neophytou, D.E. Nikonov, M.S. Lundstrom, “Performance projections for ballistic graphene nanoribbon field-effect transistors”, IEEE Trans. Electr. Dev., Vol. 54, pp.677-682, 2007. [11] G. Fiori, G. Iannaccone ”Code for the 3D Simulation of Nanoscale Semiconductor Devices, Including Drift-Diffusion and Ballistic Transport in 1D and 2D Subbands, and 3D Tunneling”, Journal of Computational Electronics, Vol. 4, pp. 63-66, 2005. [12] G. Fiori, G. Iannaccone and G. Klimeck, “A Three-Dimensional Simulation Study of the Performance of Carbon Nanotube Field-Effect Transistors With Doped Reservoirs and Realistic Geometry” IEEE Trans. Electr. Dev., Vol. 53, pp.1782-1788, 2006. [13] J. Chen, T. Y. Chan, I. C. Chen, P. K. Ko and C. Hu, “Subbreakdown drain leakage current in MOSFET”, IEEE El. Dev. Lett., Vol. 8 pp.515-517, 1987. May 28, 2018 DRAFT http://public.itrs.net http://arxiv.org/abs/cond-mat/0702511 http://arxiv.org/abs/cond-mat/0701599 Fig. 1. Transfer characteristics of double-gate CNT and GNR-FETs, with doped source and drain reservoirs, with channel length equal to 15 nm, oxide thickness tox equal to 1 nm and channel width W=1.37 nm. The lateral space is equal to 2 nm. In the inset, a sketch of the GNR-FET is shown. Fig. 2. Transfer characteristics in the logarithmic a) and linear b) scale of GNR-FETs with different chiralities (12,0), (14,0) and (16,0) (channel width W equal to 1.37 nm, 1.62 nm and 1.87 nm, respectively), for VDS=0.1 V. The transfer characteristic for the (16,0) GNR-FET, when roughness at the lateral edge of the GNR is considered is also shown (dashed line). In the inset, a sketch of the graphene nanoribbon is shown, where randomly decoupled atoms have been highlighted (thick lines). Fig. 3. Energy gap of a (24,0) GNR, when tensile and compressive strain is considered in correspondence of the middle of the nanoribbon, as a function of the strain factor by which the Hamiltonians elements of the strained carbon atoms are multiplied. May 28, 2018 DRAFT 0 0.2 0.4 0.6 0.8 gate voltage (V) = 0.1 V = 0.5 V ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� 10 nm tox = 1 nm L = 15 nm double gate FIG. 1 Gianluca Fiori, Giuseppe Iannaccone Electron Device Letters May 28, 2018 DRAFT 0 0.2 0.4 0.6 0.8 gate voltage (V) GNR (12,0) GNR (14,0) GNR (16,0) GNR (16,0) with line edge roughness 0 0.2 0.4 0.6 0.8 gate voltage (V) = 0.1 V a) b) FIG. 2 Gianluca Fiori, Giuseppe Iannaccone Electron Device Letters May 28, 2018 DRAFT 0.5 1.0 1.5 2.0 strain factor (24,0) GNR tensile strain compressive strain FIG. 3 Gianluca Fiori, Giuseppe Iannaccone Electron Device Letters May 28, 2018 DRAFT Introduction Physical model and Results Conclusions References

We present an atomistic three-dimensional simulation of graphene nanoribbon field effect transistors (GNR-FETs), based on the self-consistent solution of the 3D Poisson and Schroedinger equation with open boundary conditions within the non-equilibrium Green's Function formalism and a tight-binding hamiltonian. With respect to carbon nanotube FETs, GNR-FETs exhibit comparable performance, reduced sensitivity on the variability of channel chirality, and similar leakage problems due to band-to-band tunneling. Acceptable transistor performance requires effective nanoribbon width of 1-2 nm, that could be obtained with periodic etching patterns or stress patterns.

Introduction Physical model and Results Conclusions References

Stars and the holographic upper bound on gravitational action Scott Funkhouser National Oceanic and Atmospheric Administration, 2234 South Hobson Ave., Charleston, SC, 29405-2413 ABSTRACT The holographic upper bound on entropy is applied to the gravitational action associated with the non-relativistic contraction of a nebula. A critical radius is identified, as a function of the initial radius and mass, for which the number of bits associated with the action would equal the maximum number of bits allowed to the body. The gravitational action of a typical star approximately saturates the holographic bound, perhaps suggesting a physical link between holographic principles and astrophysical processes. Consider an isolated, cold, spherical nebula of gas whose initial radius is r0 and whose mass is M. The nebula may have some non-vanishing, uniform rotational angular momentum whose magnitude is J. Let the body be subject only to internal forces, and let the mutual gravitational attraction among the constituent particles be, initially, the only significant force. (At some point the contraction may end when another force or effective force becomes significant with respect to the gravitational binding force.) At some initial time t=0 let the body begin to contract gravitationally, and let the rotational angular momentum and mass of the body remain constant. There is associated with the contraction a characteristic quantity of action. In this present work the characteristic action of a contracting nebula is examined in scenarios where relativistic effects are not important. (Numerical or geometrical coefficients of order near 100 are also unimportant here.) Consider times t in which the characteristic radius r of the nebula is much smaller than r0. The magnitude U of the gravitational potential energy of the contracting body will have changed by an amount ΔU ~ GM , (1) where G is the Newtonian gravitational coupling. The elapsed time Δt=t between the beginning of the contraction and any time t when r << r0 is approximately , (2) where g ~ GM /r0 2 is the initial gravitational acceleration. The action A associated with the contraction is well represented by ΔUΔt, and it follows from (1) and (2) that A is given by GM 3r0 , (3) r << r0 . Whereas principles of quantum mechanics establish a minimum possible scale of action, the thermodynamics of black holes and holographic considerations lead to a maximum. The holographic upper bound on the number N(R) of degrees of freedom that the contents of a sphere of radius R could occupy is given by N(R) ~ 2 , (4) where lP is the Planck length. The maximum entropy of the system and the maximum number of bits of information that could be registered by the system are both proportional to N(R). Since the minimum possible quantum of action is the Planck quantum h, there must be associated with any quantity B of action at least B /h registered bits. The maximum action a(R) allowed to the contents of a sphere of radius R is therefore given a(R) ~ hN(R) ~ R2c 3 . (5) The holographic limit on action leads to a critical condition for the contracting nebula. As the body contracts and r decreases, the action (3) increases while the upper bound (5) decreases. The gravitational action of the body would be equal to the maximum action allowed to the body if the radius r were to become as small as the critical radius rc that satisfies the condition GM 3r0 , (6) and thus . (7) The critical action Ac of the body when r=rc is given by Ac ~ Mr0c . (8) It is instructive to note that the right side of (8) is the holographic upper bound on the rotational angular momentum of a mass M and radius r0 [2]. If rotational angular momentum is conserved during contraction then the Mr0c represents the maximum possible rotational angular momentum of the body in its final state as well. As a nebula of mass M and initial radius r0 contracts there must be some mechanism that prevents the body from contracting to radii smaller than r=rc. Note that there is no known mechanism by which holographic principles alone may generate some reaction by which a violation of the holographic bound is prevented. The final, characteristic radius of a contracting body such as the nebula in the present scenario is determined, in general, by one of three primary processes: (a) The body becomes virialized, whereby further contraction is prevented by centrifugal tendencies. (b) The density of the body becomes of the order the characteristic atomic density, whereby contraction is halted by inter-atomic repulsion. (c) The thermal radiation pressure within the body becomes as great as the gravitational pressure. If violation of the holographic upper bound is to be prevented by such mechanisms then there must be some implicit physical connections among astrophysical processes, the physical parameters of astronomical structures and holographic principles. Consider a star whose mass is of the order 1030kg that forms from the gravitational contraction of a nebula whose initial characteristic radius is of the order 1013m, or hundreds of astronomical units. The critical radius at which the action associated with the gravitational contraction will saturate the holographic upper bound is of the order 108m, which is of the order the characteristic radius of a solar-mass star. Stars may therefore constitute a class of bodies for which the holographic upper bound is nearly saturated by the gravitational action. Typical stars are also remarkable since they represent the unique situation in which all three of the processes (a) – (c), listed above, are comparably significant in preventing further gravitational contraction. References: [1] Lloyd, S., Nature, 406, 1047-1054 (2000) [2] Hod, S., Phys. Rev. D 61, 024018 (2000)

The holographic upper bound on entropy is applied to the gravitational action associated with the non-relativistic contraction of a nebula. A critical radius is identified, as a function of the initial radius and mass, for which the number of bits associated with the action would equal the maximum number of bits allowed to the body. The gravitational action of a typical star approximately saturates the holographic bound, perhaps suggesting a physical link between holographic principles and astrophysical processes.

Stars and the holographic upper bound on gravitational action Scott Funkhouser National Oceanic and Atmospheric Administration, 2234 South Hobson Ave., Charleston, SC, 29405-2413 ABSTRACT The holographic upper bound on entropy is applied to the gravitational action associated with the non-relativistic contraction of a nebula. A critical radius is identified, as a function of the initial radius and mass, for which the number of bits associated with the action would equal the maximum number of bits allowed to the body. The gravitational action of a typical star approximately saturates the holographic bound, perhaps suggesting a physical link between holographic principles and astrophysical processes. Consider an isolated, cold, spherical nebula of gas whose initial radius is r0 and whose mass is M. The nebula may have some non-vanishing, uniform rotational angular momentum whose magnitude is J. Let the body be subject only to internal forces, and let the mutual gravitational attraction among the constituent particles be, initially, the only significant force. (At some point the contraction may end when another force or effective force becomes significant with respect to the gravitational binding force.) At some initial time t=0 let the body begin to contract gravitationally, and let the rotational angular momentum and mass of the body remain constant. There is associated with the contraction a characteristic quantity of action. In this present work the characteristic action of a contracting nebula is examined in scenarios where relativistic effects are not important. (Numerical or geometrical coefficients of order near 100 are also unimportant here.) Consider times t in which the characteristic radius r of the nebula is much smaller than r0. The magnitude U of the gravitational potential energy of the contracting body will have changed by an amount ΔU ~ GM , (1) where G is the Newtonian gravitational coupling. The elapsed time Δt=t between the beginning of the contraction and any time t when r << r0 is approximately , (2) where g ~ GM /r0 2 is the initial gravitational acceleration. The action A associated with the contraction is well represented by ΔUΔt, and it follows from (1) and (2) that A is given by GM 3r0 , (3) r << r0 . Whereas principles of quantum mechanics establish a minimum possible scale of action, the thermodynamics of black holes and holographic considerations lead to a maximum. The holographic upper bound on the number N(R) of degrees of freedom that the contents of a sphere of radius R could occupy is given by N(R) ~ 2 , (4) where lP is the Planck length. The maximum entropy of the system and the maximum number of bits of information that could be registered by the system are both proportional to N(R). Since the minimum possible quantum of action is the Planck quantum h, there must be associated with any quantity B of action at least B /h registered bits. The maximum action a(R) allowed to the contents of a sphere of radius R is therefore given a(R) ~ hN(R) ~ R2c 3 . (5) The holographic limit on action leads to a critical condition for the contracting nebula. As the body contracts and r decreases, the action (3) increases while the upper bound (5) decreases. The gravitational action of the body would be equal to the maximum action allowed to the body if the radius r were to become as small as the critical radius rc that satisfies the condition GM 3r0 , (6) and thus . (7) The critical action Ac of the body when r=rc is given by Ac ~ Mr0c . (8) It is instructive to note that the right side of (8) is the holographic upper bound on the rotational angular momentum of a mass M and radius r0 [2]. If rotational angular momentum is conserved during contraction then the Mr0c represents the maximum possible rotational angular momentum of the body in its final state as well. As a nebula of mass M and initial radius r0 contracts there must be some mechanism that prevents the body from contracting to radii smaller than r=rc. Note that there is no known mechanism by which holographic principles alone may generate some reaction by which a violation of the holographic bound is prevented. The final, characteristic radius of a contracting body such as the nebula in the present scenario is determined, in general, by one of three primary processes: (a) The body becomes virialized, whereby further contraction is prevented by centrifugal tendencies. (b) The density of the body becomes of the order the characteristic atomic density, whereby contraction is halted by inter-atomic repulsion. (c) The thermal radiation pressure within the body becomes as great as the gravitational pressure. If violation of the holographic upper bound is to be prevented by such mechanisms then there must be some implicit physical connections among astrophysical processes, the physical parameters of astronomical structures and holographic principles. Consider a star whose mass is of the order 1030kg that forms from the gravitational contraction of a nebula whose initial characteristic radius is of the order 1013m, or hundreds of astronomical units. The critical radius at which the action associated with the gravitational contraction will saturate the holographic upper bound is of the order 108m, which is of the order the characteristic radius of a solar-mass star. Stars may therefore constitute a class of bodies for which the holographic upper bound is nearly saturated by the gravitational action. Typical stars are also remarkable since they represent the unique situation in which all three of the processes (a) – (c), listed above, are comparably significant in preventing further gravitational contraction. References: [1] Lloyd, S., Nature, 406, 1047-1054 (2000) [2] Hod, S., Phys. Rev. D 61, 024018 (2000)

NEW VERSIONS OF SCHUR-WEYL DUALITY STEPHEN DOTY Abstract. After reviewing classical Schur-Weyl duality, we present some other contexts which enjoy similar features, relating to Brauer algebras and classical groups. 1. Classical Schur-Weyl duality 1.1. Schur’s double-centralizer result. Consider the vector space V = Cn. The symmetric group Sr acts naturally on its r-fold tensor power V ⊗r, by permuting the tensor positions. This action obviously commutes with the natural action of GLn = GLn(C), acting by matrix multiplication in each tensor position. So we have a CGLn-CSn bimod- ule structure on V ⊗r. (Here CG denotes the group algebra of a group G.) In 1927, Schur [Sc] proved that the image of each group algebra under its representation equals the full centralizer algebra for the other action. More precisely, if we name the representations as follows (1) CGLn −−−→ End(V ⊗r) ←−−− CSr then we have equalities ρ(CGLn) = EndSr(V ⊗r)(2) σ(CSr) = EndGLn(V ⊗r).(3) (Here, for a given set S operating on a vector space T through linear endomorphisms, EndS(T ) denotes the set of linear endomrphisms of T commuting with each endomorphism coming from S.) Results of Carter-Lusztig [CL] and J.A. Green [G] (and others) show that all the above statements remain true if one replaces C by an arbitrary infinite field K. These notes are based on a lecture, various versions of which I have given in the past year, in a number of locations, including Stuttgart, Birmingham, Queen Mary (London), Lancaster, Manchester, Oxford, and Cambridge. I’m grateful to the organizers of those events for the opportunity to present these ideas. http://arxiv.org/abs/0704.1877v1 2 STEPHEN DOTY 1.2. Schur algebras. The finite-dimensional algebra in (2) above, for any K, is known as the Schur algebra, and often denoted by SK(n, r) or simply S(n, r). The Schur algebra “sees” the part of the ratio- nal representation theory of the algebraic group GLn(K) occurring (in some appropriate sense) in V ⊗r. More precisely, there is an equiva- lence between r-homogeneous polynomial representations of GLn(K) and SK(n, r)-modules. In characteristic 0, those representations (as r varies) determine all finite-dimensional rational representations, while in positive characteristic they still provide a tremendous amount of information. The representation σ in (1) is faithful if n > r, so σ induces an isomorphism (4) KSr ≃ EndGLn(V ⊗r) = EndSK(n,r)(V ⊗r) (n > r). This leads to intimate connections between polynomial representations of GLn(K) and representations of KSr, a theme that has been ex- ploited by many authors in recent years. Perhaps the most dramatic example of this is the result of Erdmann [E] (building on previous work of Donkin [Do3] and Ringel [R]) which shows that knowing decompo- sition numbers for all symmetric groups in positive characteristic will determine the decomposition numbers for general linear groups in the same characteristic. Conversely, James [Ja] had already shown that the decomposition matrix for a symmetric group is a submatrix of the decomposition matrix for an appropriate Schur algebra. Thus the (still open) general problem of determining the modular characters of sym- metric groups is equivalent to the similar problem for general linear groups (over infinite fields). 1.3. The enveloping algebra approach. Return to the basic setup, over C. One may differentiate the action of the Lie group GLn(C) to obtain an action of its Lie algebra gln. Replacing the representation ρ in (1) by its derivative representation dρ : U(gln)→ End(V ⊗r) leads to the following alternative statement of Schur’s result: dρ(U(gln)) = EndSr(V ⊗r)(5) σ(CSr) = Endgln(V ⊗r).(6) In particular, the Schur algebra (over C) is a homomorphic image of U(gln). All of this works over an arbitrary integral domain K if we replace U(gln) by its “hyperalgebra” UK := K⊗ZUZ obtained by change of ring from a suitable Z-form of U(gln); see [Do1]. (One can adapt the Kostant Z-form, originally defined for the enveloping algebra of a semisimple Lie algebra, to the reductive gln.) NEW VERSIONS OF SCHUR-WEYL DUALITY 3 1.4. The quantum case. Jimbo [Ji] extended the results of 1.3 to the quantum case (where the quantum parameter is not a root of unity). One needs to replace Sr by the Iwahori-Hecke algebra H(Sr) and re- place U(gln) by the quantized enveloping algebra U(gln). The analogue of the Schur algebra in this context is known as the q-Schur algebra, of- ten denoted by S(n, r) or Sq(n, r). Dipper and James [DJ] have shown that q-Schur algebras are fundamental for the modular representation theory of finite general linear groups. As many authors have observed, the picture in 1.1 can also be quan- tized. For that one needs a suitable quantization of the coordinate algebra of the algebraic group GLn. There is a completely different (geometric) construction of q-Schur algebras given in [BLM]. 1.5. Integral forms. The Schur algebras SC(n, r) admit an integral form SZ(n, r) such that SK(n, r) ≃ K ⊗Z SZ(n, r) for any field K. In fact SZ(n, r) is simply the image of UZ (see 1.3) under the surjective homomorphism U(gln) → SC(n, r). Similarly, the quantum Schur al- gebra SQ(v)(n, r) admits an integral form defining all specializations via base change. One needs to replace Z by A = Z[v, v−1]; then the integral form SA(n, r) is the image of the Lusztig A-form UA under the surjection U(gln) → SQ(v)(n, r). (To match this up with various specializations in the literature, one often has to take q = v2.) 1.6. Generators and relations. Recently, in joint work with Gi- aquinto (see [DG]), a very simple set of elements generating the kernel of the surjection U(gln) → SC(n, r) was found. A very similar set of elements generates the kernel of the surjection U(gln) → SQ(v)(n, r). These elements are expressible entirely in terms of the Chevalley gener- ators for the zero part of U(gln) or U(gln). Thus we obtain a presenta- tion of SC(n, r) and SQ(v)(n, r) by generators and relations, compatible with the usual Serre (Drinfeld-Jimbo) presentation of U(gln) (resp., U(gln)). As a result, we find a certain subset of the integral PBW- basis for U(gln) or U(gln) the image of which gives an integral basis for SZ(n, r) or SA(n, r). This basis yields a similar basis in any spe- cialization. Moreover, a subset of it provides a new integral basis of H(Sn). 2. The Brauer algebra From now on I will assume, unless stated otherwise, that the under- lying field is C (it could just as well be any field of characteristic zero). 4 STEPHEN DOTY One expects that many statements will be valid over an arbitrary infi- nite field, via some appropriate integral form, similar to what happens in type A. 2.1. The algebra B r . Let R be a commutative ring, and consider the free R[x]-module B r with basis consisting of all r-diagrams. An r-diagram is an (undirected) graph on 2r vertices and r edges such that each vertex is incident to precisely one edge. One usually thinks of the vertices as arranged in two rows of r each, the top and bottom rows. (See Figure 1.) Edges connecting two vertices in the same row (different rows) are called horizontal (resp., vertical). We can compose two such diagrams D1, D2 by identifying the bottom row of vertices in the first diagram with the top row of vertices in the second diagram. The result is a graph with a certain number, δ(D1, D2), of interior loops. After removing the interior loops and the identified vertices, retaining the edges and remaining vertices, we obtain a new r-diagram D1 ◦D2, the composite diagram. Multiplication of r-diagrams is defined by the rule D1 ·D2 = x δ(D1,D2)(D1 ◦D2). One can check that this multiplication makes B r into an associative algebra; this is the Brauer algebra. (See Figures 1–3 for an illustration of the multiplication in the Brauer algebra.) Note that if we take x = 1 then the set of r-diagrams is a monoid under diagram composition, and B r is simply the semigroup algebra of that monoid. Figure 1. Two Brauer diagrams D1, D2 for r = 5. For any x the group algebra R[x]Sr may be identified with the sub- algebra of B r spanned by the diagrams containing only vertical edges. Such Brauer diagrams provide a graphical depiction of permutations. The group algebra R[x]Sr of Sr also appears as a quotient of B r , the quotient by the two-sided ideal spanned by all diagrams containing at least one horizontal edge. NEW VERSIONS OF SCHUR-WEYL DUALITY 5 Figure 2. Computing the composite of D1 and D2. Figure 3. The composite diagram D1 ◦D2. Label the vertices in each row of an r-diagram by the indices 1, . . . , r. For any 1 6 i 6= j 6 r let ci,j be the r-diagram with horizontal edges connecting vertices i, j on the top and bottom rows. All other vertices in the diagram ci,j are vertical, connecting vertex k on the top and bottom rows, for all k 6= i, j. Brauer observed that B r is generated by the permutation diagrams together with just one of the ci,j. 2.2. Schur-Weyl duality. Brauer [Br] introduced the algebra B 1936 to describe the invariants of symplectic and orthogonal groups acting on V ⊗r. (Brauer’s conventions were slightly different; we are here following the approach of Hanlon and Wales [HW], who pointed out that B r is isomorphic with the algebra defined by Brauer to deal with the symplectic case.) Let G be Spn or On, where n is even in the first instance. By restricting the action ρ considered in 1.1 we have an action of G on V ⊗r. One can extend the action of Sr to an action of r (over C) on V ⊗r, where ǫ = −1 if G = Spn and ǫ = 1 if G = On. To do this, it is enough to specify the action of the diagram ci,j. This acts on V ⊗r as one of Weyl’s contraction maps contracting in tensor positions i and j. So we have (commuting) representations (7) CG −−−→ End(V ⊗r) ←−−− B 6 STEPHEN DOTY which satisfy Schur-Weyl duality; i.e., the image of each representation equals the full centralizer algebra of the other action: ρ(CG) = End (V ⊗r)(8) σ(B(ǫn)r ) = EndG(V ⊗r).(9) The algebras in equality (8) are the symplectic and orthogonal Schur algebras (see [Do2], [D1], [D2]). If n > r − 1 the representation σ in (7) is faithful [Bro]; thus it induces an isomorphism B r ≃ EndG(V 2.3. Schur-Weyl duality in type D. In type Dn/2 (n even) the orthogonal group On is not connected, and contains the connected semisimple group SOn (special orthogonal group) as subgroup of in- dex 2. In order to handle this situation, Brauer (see also [Gr]) defined a larger algebra D r , spanned by the usual r-diagrams previously de- fined, together with certain partial r-diagrams on 2r vertices and r−n edges, in which n vertices in each of the top and bottom rows are not incident to any edge, and showed that the action of B r can be ex- tended to an action of this larger algebra D r on V ⊗r. Thus we have representations (10) CSOn −−−→ End(V ⊗r) ←−−− D Brauer showed that the actions of SOn and D r on V ⊗r satisfy Schur- Weyl duality: ρ(CSOn) = EndD(ǫn)r (V ⊗r)(11) σ(D(ǫn)r ) = EndSOn(V ⊗r).(12) The algebra in (11) is a second Schur algebra in type D, a proper subalgebra of the algebra End (V ⊗r) appearing in (8) above. 2.4. Generators and relations. One can formulate the above state- ments of Schur-Weyl duality using enveloping algebras, analogous to 1.3. This leads to a presentation (see [DGS]) of the symplectic and orthogonal Schur algebras which is compatible with (a slight modifi- cation of) the usual Serre presentation of the enveloping algebra U(g), where g = spn (n even) or son. 2.5. The quantum case. There is a q-version of the Schur-Weyl du- ality considered in this section, although not as developed as in type A. One needs to replace the Brauer algebra by its q-analogue, the Birman- Murakami-Wenzl (BMW) algebra (see [BW], [Mu]), and replace the NEW VERSIONS OF SCHUR-WEYL DUALITY 7 enveloping algebra by a suitable quantized enveloping algebra. One can think of the BMW algebra in terms of Kauffman’s tangle monoid; see [Ka], [HR], [MW]. (Roughly speaking, tangles are replacements for Brauer diagrams, in which one keeps track of under and over cross- ings, subject to certain natural relations.) There are applications of the BMW algebra to knot theory, as one might imagine. This leads to a q-analogue of the symplectic Schur algebras, in par- ticular, which have been studied by Oehms [Oe]. To the best of my knowledge, a q-analogue of the larger algebra D (n even) considered in 2.3 remains to be formulated. 3. The walled Brauer algebra 3.1. The algebra B r,s . This algebra was defined in 1994 in [BCHLLS]. It is the subalgebra of B r+s spanned by the set of (r, s)-diagrams. By definition, an (r, s)-diagram is an (r+ s)-diagram in which we imagine a wall separating the first r from the last s columns of vertices, such that: (a) all horizontal edges cross the wall; (b) no vertical edges cross the wall. An edge crosses the wall if its two vertices lie on opposite sides of the wall. The multiplication in B r,s is that of B Label the vertices on the top and bottom rows of an (r, s)-diagram by the numbers 1, . . . , r to the left of the wall and −1, . . . ,−s to the right of the wall. Let ci,−j (1 6 i 6 r; 1 6 j 6 s) be the diagram with a horizontal edge connecting vertices i and −j on the top row and the same on the bottom row, and with all other edges connecting vertex k (k 6= i,−j) in the top and bottom rows. It is easy to see that the walled Brauer algebra is generated by the permutations it contains along with just one of the ci,−j. (Note that ci,−j is the (r+ s)-diagram denoted by ci,r+j in 2.1.) 3.2. Dimension. What is the dimension of B r,s ? One way to answer that question is to consider the map, flip, from (r + s)-diagrams to (r+s)-diagrams, defined by interchanging the top and bottom vertices to the right of the imaginary wall. For example, Figure 4 shows a (4, 2)-diagram (to the left) and its corresponding 6-diagram, obtained from the left diagram by applying flip. Note that flip is involutary: applying it twice gives the original diagram back again. 8 STEPHEN DOTY Figure 4. A (4, 2)-diagram and its corresponding per- mutation, after applying flip. One easily checks that the map flip carries (r, s)-diagrams bijectively onto the set of (r+ s)-diagrams with all edges vertical. Such diagrams correspond with permutations of r+s objects, so the dimension of B is (r + s)!. 3.3. Another view of B r,s . The above correspondence between (r, s)- diagrams and permutations gives another way to think of the multipli- cation in B r,s . Given two (r, s)-diagrams D1, D2 let D 2 be their corresponding permutations obtained by applying flip. Define a new (rather bizarre) composition on permutations as follows. Given any two permutation diagrams D′1, D 2 (with r+ s columns of vertices) identify the first r vertices of the bottom row of D′1 with the first r vertices of the top row of D′2, and identify the last s vertices of the top row of D′1 with the last s vertices of the bottom row of D 2. After removing loops and identified vertices this gives a new permutation diagram D′3 in which the vertices in the top (resp., bottom) row are the remaining top (bottom) row vertices from the original diagrams. Let δ(D′1, D 2) be the number of loops removed in computing the com- posite permutation diagram D′3. Define multiplication of permutation diagrams by the rule D′1 ·D 2 = x δ(D′1,D 2)D′3 In other words, we are multiplying permutations by composing maps “on the right” on one side of the wall, and “on the left” on the other side (roughly speaking). For example, Figure 5 below shows the com- putation of the composite diagram in the walled Brauer algebra (left column) and the computation in terms of the corresponding permuta- tions (right column). Figure 6 shows the resulting diagrams after the single loop and identified vertices have been removed. One can check that D′3 coincides with (D1 ◦ D2) ′ and δ(D1, D2) = δ(D′1, D 2). In other words, flip defines an algebra isomorphism between NEW VERSIONS OF SCHUR-WEYL DUALITY 9 Figure 5. Composition of diagrams and permutations. The diagrams on the left correspond under flip with the permutations on the right. Figure 6. The corresponding diagrams resulting from Figure 5. The two diagrams correspond under flip. the algebra B r,s and the algebra B̃ r,s spanned by permutation diagrams with the multiplication defined above. Note that in particular B̃ r,0 ≃ R[x]Sr and B̃ 0,s ≃ (R[x]Ss) 3.4. Schur-Weyl duality. Consider mixed tensor space V r,s := V ⊗r⊗ V ∗⊗s, where V ∗ is the usual linear dual space of V . Mixed tensor space is naturally a module for GLn, and one obtains an action of B r,s on V r,s simply by restricting the action of B r+s, which acts the same on V r,s as it does on V ⊗(r+s), since on restriction to On we have V ≃ V Thus we have the following commutative diagram −−−→ End(V r,s) ←−−− B yι′ −−−→ End(V ⊗(r+s)) ←−−− B 10 STEPHEN DOTY in which the vertical maps ι, ι′ are inclusion. By [BCHLLS], the actions of GLn and B r,s on V r,s in the first row of the diagram satisfy Schur- Weyl duality: ρ(CGLn) = EndB(n)r,s (V r,s)(14) σ(B(n)r,s ) = EndGLn(V r,s).(15) The algebra in (14) is another Schur algebra S(n; r, s) in type A, stud- ied in [DD]. These Schur algebras provide us with a new family of quasihereditary algebras, generalizing the classical Schur algebras, since S(n; r, 0) ≃ S(n, r). In fact, the S(n; r, s) provide a new class of gen- eralized Schur algebras in the sense of Donkin [Do1]. For fixed n, the family of S(n; r, s)-modules as r, s vary constitutes the family of all rational representations of GLn. Whence the name rational Schur alge- bras for the S(n; r, s). When n > r + s, the representation σ in the top row of (13) above is faithful, so induces an isomorphism B r,s ≃ EndGLn(V r,s). 3.5. The quantum case. Quantizations of the walled Brauer algebra have been defined and studied in work of Halverson [Ha], Leduc [Le], Kosuda-Murakami [KM], and Kosuda [K]. 4. The deranged algebra 4.1. The problem. One might wonder if there are versions of Schur- Weyl duality in which the natural module V is replaced by some other representation. Perhaps the first choice would be to replace V with the adjoint module, i.e., an algebraic group acting on its Lie algebra via the adjoint representation. The simplest instance of this would be type A, where we consider the module sl⊗rn as an SLn-module, and ask for its centralizer algebra EndSLn(sl n ) = EndGLn(sl n ). (It makes no difference whether we regard sln as module for SLn or for GLn. We look at sl⊗rn rather than gl n since gln is not simple as a GLn-module or SLn-module.) 4.2. Relation with Brauer algebras. Even though this is a ques- tion about type A, its solution is intimately connected with the walled Brauer algebra. Here is a brief outline of the solution to this problem, recently obtained in [BD]. The main idea is to utilize the decomposition gln = sln ⊕ C (as SLn or GLn module) to write (16) gl⊗rn = sl 06t<r sl NEW VERSIONS OF SCHUR-WEYL DUALITY 11 where the left-hand-side identifies with V r,r = V ⊗r ⊗ V ∗⊗r via the natural isomorphism V ⊗ V ∗ ≃ End(V ) ≃ gln. Thus it follows that our desired tensor space sl⊗rn is isomorphic with a direct summand (as a GLn or SLn-module) of the mixed tensor space V r,r, and for n > 2r the centralizing algebra C = EndGLn(sl n ) is obtainable as a certain subalgebra eEndGLn(V r,r)e where e is the idempotent corresponding to the projection onto the summand sl⊗rn . Thus (17) C ≃ eB(n)r,r e (n > 2r) where e = i(1 − n −1ci,−i) (notation of 3.1). The algebra in (17) has a basis consisting of all elements of the form eDe as D ranges over the set of (r, r)-diagrams with no horizontal edges connecting i to −i. Such diagrams correspond under flip, after inverting the signs labeling the bottom row, with derangements of 2r objects. (A derangement is a permutation having no fixed points.) Let N(k) = the number of derangements of k objects, then we have by the Inclusion-Exclusion Principle (see [St, 2.2.1 Example]) (18) N(k) = (−1)k−j which is the nearest integer to k!/e (e = 2.7182818 . . . ). Thus the dimension of the centralizer algebra eB r,r e is given by N(2r). Because of this connection with derangements, the algebra eB r,r e is known as the deranged algebra. In [BD] explicit formulas are obtained for the number of times a given simple module L appears as a summand of sl⊗rn . In particular, it is shown that N(r) = the multiplicity of the trivial module in sl⊗rn N(r − 1) = the multiplicity of sln in sl demonstrating that the combinatorics of derangement numbers is in- herent in this theory. 4.3. Schur-Weyl duality. At least when n > 2r, the actions of GLn and C on sl⊗rn satisfy Schur-Weyl duality: ρ(CGLn) = EndeB(n)r,r e (sl⊗rn )(19) σ(eB(n)r,r e) = EndGLn(sl n ).(20) This will almost certainly hold for all n, r. 12 STEPHEN DOTY References [BLM] Bĕılinson, A.A.; Lusztig, G.; MacPherson, R. A geometric setting for the quantum deformation of GLn, Duke Math. J. 61 (1990), 655–677. [BCHLLS] Benkart, G.; Chakrabarti, M.; Halverson, T.; Leduc, R.; Lee, C.; Stroomer, J., Tensor product representations of general linear groups and their connections with Brauer algebras, J. Algebra 166 (1994), 529–567. [BD] Benkart, G.; Doty, S., Derangements and tensor powers of adjoint modules for sln, J. Algebraic Combin. 16 (2002), 31–42. [BW] Birman, J.S.; Wenzl, H., Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), 249–273. [Br] Brauer, R., On algebras which are connected with the semisimple continuous groups, Annals of Math. 38 (1937), 857–872. [Bro] Brown, William P., An algebra related to the orthogonal group, Michigan Math. J. 3 (1955), 1–22; The semisimplicity of ωnf , Ann. of Math. 63 (1956), 324–335. [CL] Carter, R.W.; Lusztig, G., On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242. [Do1] Donkin, S., On Schur algebras and related algebras I, J. Algebra 104 (1986), 310–328. [Do2] Donkin, S., Good filtrations of rational modules for reductive groups, in: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), pp. 69–80, Proc. Sympos. Pure Math., 47, Part 1, Amer. Math. Soc., Providence, RI, 1987. [Do3] Donkin, S., On tilting modules for algebraic groups, Math. Z. 212 (1993), 39–60. [DD] Dipper, R.; Doty, S., Rational Schur algebras, in preparation. [D1] Doty, S., Polynomial representations, algebraic monoids, and Schur algebras of classical type, J. Pure Appl. Algebra 123 (1998), 165–199. [D2] Doty, S., Representation theory of reductive normal algebraic monoids, Trans. Amer. Math. Soc. 351 (1999), 2539–2551. [DG] Doty, S.; Giaquinto, A., Presenting Schur algebras, Internat. Math. Research Notices 2002, 1907–1944. [DGS] Doty, S.; Giaquinto, A.; Sullivan, J., Presenting Schur algebras in types B, C, D, in preparation. [DJ] Dipper, R.; James, G., The q-Schur algebra, Proc. London Math. Soc. (3) 59 (1989), 23–50; q-tensor space and q-Weyl modules, Trans. Amer. Math. Soc. 327 (1991), 251–282. [E] Erdmann, K., Decomposition numbers for symmetric groups and composition factors of Weyl modules, J. Algebra 180 (1996), 316–320. [G] Green, J. A., Polynomial representations of GLn, Lecture Notes in Math. 830, Springer-Verlag, Berlin-New York, 1980. NEW VERSIONS OF SCHUR-WEYL DUALITY 13 [Gr] Grood, C., Brauer algebras and centralizer algebras for SO(2n,C), J. Algebra 222 (1999), 678–707. [Ha] Halverson, T., Characters of the centralizer algebras of mixed tensor representations of GL(r,C) and the quantum group Uq(gl(r,C)), Pacific J. Math. 174 (1996), 359–410. [HR] Halverson, T.; Ram, A., Characters of algebras containing a Jones basic construction: the Temperley-Lieb, Okasa, Brauer, and Birman-Wenzl algebras, Adv. Math. 116 (1995), 263–321. [HW] Hanlon, P.; Wales, D., On the decomposition of Brauer’s centralizer algebras, J. Algebra 121 (1989), 409–445; Eigenvalues connected with Brauer’s centralizer algebras, J. Algebra 121 (1989), 446–476; Computing the discriminants of Brauer’s centralizer algebras, Math. Comp. 54 (1990), 771–796. [Oe] Oehms, S., Symplektische q-Schur-Algebren, Dissertation, Universität Stuttgart, Stuttgart, 1997. Berichte aus der Mathematik, Verlag Shaker, Aachen, 1997. [Ja] James, G.D., The decomposition of tensors over fields of prime characteristic, Math. Z. 172 (1980), 161–178. [Ji] Jimbo, M., A q-analogue of U(gl(N + 1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247–252. [Ka] Kauffman, L.H., An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), 417–471. [K] Kosuda, M., Representation of q-analogue of rational Brauer algebras, Tsukuba J. Math. 21 (1997), 707–728. [KM] Kosuda, M.; Murakami, J., The centralizer algebras of mixed tensor representations of Uq(gln) and the HOMFLY polynomial of links, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), 148–151; Centralizer algebras of the mixed tensor representations of quantum group Uq(gl(n,C)), Osaka J. Math. 30 (1993), 475–507. [Le] Leduc, R., A two-parameter version of the centralizer algebra of the mixed tensor representations of the general linear group and quantum general linear group, Ph.D. Dissertation, University of Wisconsin-Madison 1994. [MW] Morton, H.R.; Wassermann, A.J., A basis for the Birman-Wenzl algebra, Preprint 1989, with foreword and references updated Jan. 2000. [http://www.liv.ac.uk/∼su14/knotprints.html] [Mu] Murakami, J., The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), 745–758; The representations of the q-analogue of Brauer’s centralizer algebras and the Kauffman polynomial of links, Publ. Res. Inst. Math. Sci. 26 (1990), 935–945. [R] Ringel, C., The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), 209–223. http://www.liv.ac.uk/~su14/knotprints.html 14 STEPHEN DOTY [Sc] Schur, I., Über die rationalen Darstellungen der allgemeinen linearen Gruppe, (1927). Reprinted in Schur, I., Gesammelte Abhandlungen, Vol. III, pp. 68–85, Springer-Verlag, Berlin, 1973. [St] Stanley, R.P., Enumerative combinatorics Vol. 1, Cambridge Studies in Advanced Math., 49, Cambridge Univ. Press, Cambridge, 1997. Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60626 U.S.A. E-mail address : doty@math.luc.edu 1. Classical Schur-Weyl duality 1.1. Schur's double-centralizer result 1.2. Schur algebras 1.3. The enveloping algebra approach 1.4. The quantum case 1.5. Integral forms 1.6. Generators and relations 2. The Brauer algebra 2.1. The algebra Br(x) 2.2. Schur-Weyl duality 2.3. Schur-Weyl duality in type D 2.4. Generators and relations 2.5. The quantum case 3. The walled Brauer algebra 3.1. The algebra Br,s(x) 3.2. Dimension 3.3. Another view of Br,s(x) 3.4. Schur-Weyl duality 3.5. The quantum case 4. The deranged algebra 4.1. The problem 4.2. Relation with Brauer algebras 4.3. Schur-Weyl duality References

After reviewing classical Schur-Weyl duality, we present some other contexts which enjoy similar features, relating to Brauer algebras and classical groups.

NEW VERSIONS OF SCHUR-WEYL DUALITY STEPHEN DOTY Abstract. After reviewing classical Schur-Weyl duality, we present some other contexts which enjoy similar features, relating to Brauer algebras and classical groups. 1. Classical Schur-Weyl duality 1.1. Schur’s double-centralizer result. Consider the vector space V = Cn. The symmetric group Sr acts naturally on its r-fold tensor power V ⊗r, by permuting the tensor positions. This action obviously commutes with the natural action of GLn = GLn(C), acting by matrix multiplication in each tensor position. So we have a CGLn-CSn bimod- ule structure on V ⊗r. (Here CG denotes the group algebra of a group G.) In 1927, Schur [Sc] proved that the image of each group algebra under its representation equals the full centralizer algebra for the other action. More precisely, if we name the representations as follows (1) CGLn −−−→ End(V ⊗r) ←−−− CSr then we have equalities ρ(CGLn) = EndSr(V ⊗r)(2) σ(CSr) = EndGLn(V ⊗r).(3) (Here, for a given set S operating on a vector space T through linear endomorphisms, EndS(T ) denotes the set of linear endomrphisms of T commuting with each endomorphism coming from S.) Results of Carter-Lusztig [CL] and J.A. Green [G] (and others) show that all the above statements remain true if one replaces C by an arbitrary infinite field K. These notes are based on a lecture, various versions of which I have given in the past year, in a number of locations, including Stuttgart, Birmingham, Queen Mary (London), Lancaster, Manchester, Oxford, and Cambridge. I’m grateful to the organizers of those events for the opportunity to present these ideas. http://arxiv.org/abs/0704.1877v1 2 STEPHEN DOTY 1.2. Schur algebras. The finite-dimensional algebra in (2) above, for any K, is known as the Schur algebra, and often denoted by SK(n, r) or simply S(n, r). The Schur algebra “sees” the part of the ratio- nal representation theory of the algebraic group GLn(K) occurring (in some appropriate sense) in V ⊗r. More precisely, there is an equiva- lence between r-homogeneous polynomial representations of GLn(K) and SK(n, r)-modules. In characteristic 0, those representations (as r varies) determine all finite-dimensional rational representations, while in positive characteristic they still provide a tremendous amount of information. The representation σ in (1) is faithful if n > r, so σ induces an isomorphism (4) KSr ≃ EndGLn(V ⊗r) = EndSK(n,r)(V ⊗r) (n > r). This leads to intimate connections between polynomial representations of GLn(K) and representations of KSr, a theme that has been ex- ploited by many authors in recent years. Perhaps the most dramatic example of this is the result of Erdmann [E] (building on previous work of Donkin [Do3] and Ringel [R]) which shows that knowing decompo- sition numbers for all symmetric groups in positive characteristic will determine the decomposition numbers for general linear groups in the same characteristic. Conversely, James [Ja] had already shown that the decomposition matrix for a symmetric group is a submatrix of the decomposition matrix for an appropriate Schur algebra. Thus the (still open) general problem of determining the modular characters of sym- metric groups is equivalent to the similar problem for general linear groups (over infinite fields). 1.3. The enveloping algebra approach. Return to the basic setup, over C. One may differentiate the action of the Lie group GLn(C) to obtain an action of its Lie algebra gln. Replacing the representation ρ in (1) by its derivative representation dρ : U(gln)→ End(V ⊗r) leads to the following alternative statement of Schur’s result: dρ(U(gln)) = EndSr(V ⊗r)(5) σ(CSr) = Endgln(V ⊗r).(6) In particular, the Schur algebra (over C) is a homomorphic image of U(gln). All of this works over an arbitrary integral domain K if we replace U(gln) by its “hyperalgebra” UK := K⊗ZUZ obtained by change of ring from a suitable Z-form of U(gln); see [Do1]. (One can adapt the Kostant Z-form, originally defined for the enveloping algebra of a semisimple Lie algebra, to the reductive gln.) NEW VERSIONS OF SCHUR-WEYL DUALITY 3 1.4. The quantum case. Jimbo [Ji] extended the results of 1.3 to the quantum case (where the quantum parameter is not a root of unity). One needs to replace Sr by the Iwahori-Hecke algebra H(Sr) and re- place U(gln) by the quantized enveloping algebra U(gln). The analogue of the Schur algebra in this context is known as the q-Schur algebra, of- ten denoted by S(n, r) or Sq(n, r). Dipper and James [DJ] have shown that q-Schur algebras are fundamental for the modular representation theory of finite general linear groups. As many authors have observed, the picture in 1.1 can also be quan- tized. For that one needs a suitable quantization of the coordinate algebra of the algebraic group GLn. There is a completely different (geometric) construction of q-Schur algebras given in [BLM]. 1.5. Integral forms. The Schur algebras SC(n, r) admit an integral form SZ(n, r) such that SK(n, r) ≃ K ⊗Z SZ(n, r) for any field K. In fact SZ(n, r) is simply the image of UZ (see 1.3) under the surjective homomorphism U(gln) → SC(n, r). Similarly, the quantum Schur al- gebra SQ(v)(n, r) admits an integral form defining all specializations via base change. One needs to replace Z by A = Z[v, v−1]; then the integral form SA(n, r) is the image of the Lusztig A-form UA under the surjection U(gln) → SQ(v)(n, r). (To match this up with various specializations in the literature, one often has to take q = v2.) 1.6. Generators and relations. Recently, in joint work with Gi- aquinto (see [DG]), a very simple set of elements generating the kernel of the surjection U(gln) → SC(n, r) was found. A very similar set of elements generates the kernel of the surjection U(gln) → SQ(v)(n, r). These elements are expressible entirely in terms of the Chevalley gener- ators for the zero part of U(gln) or U(gln). Thus we obtain a presenta- tion of SC(n, r) and SQ(v)(n, r) by generators and relations, compatible with the usual Serre (Drinfeld-Jimbo) presentation of U(gln) (resp., U(gln)). As a result, we find a certain subset of the integral PBW- basis for U(gln) or U(gln) the image of which gives an integral basis for SZ(n, r) or SA(n, r). This basis yields a similar basis in any spe- cialization. Moreover, a subset of it provides a new integral basis of H(Sn). 2. The Brauer algebra From now on I will assume, unless stated otherwise, that the under- lying field is C (it could just as well be any field of characteristic zero). 4 STEPHEN DOTY One expects that many statements will be valid over an arbitrary infi- nite field, via some appropriate integral form, similar to what happens in type A. 2.1. The algebra B r . Let R be a commutative ring, and consider the free R[x]-module B r with basis consisting of all r-diagrams. An r-diagram is an (undirected) graph on 2r vertices and r edges such that each vertex is incident to precisely one edge. One usually thinks of the vertices as arranged in two rows of r each, the top and bottom rows. (See Figure 1.) Edges connecting two vertices in the same row (different rows) are called horizontal (resp., vertical). We can compose two such diagrams D1, D2 by identifying the bottom row of vertices in the first diagram with the top row of vertices in the second diagram. The result is a graph with a certain number, δ(D1, D2), of interior loops. After removing the interior loops and the identified vertices, retaining the edges and remaining vertices, we obtain a new r-diagram D1 ◦D2, the composite diagram. Multiplication of r-diagrams is defined by the rule D1 ·D2 = x δ(D1,D2)(D1 ◦D2). One can check that this multiplication makes B r into an associative algebra; this is the Brauer algebra. (See Figures 1–3 for an illustration of the multiplication in the Brauer algebra.) Note that if we take x = 1 then the set of r-diagrams is a monoid under diagram composition, and B r is simply the semigroup algebra of that monoid. Figure 1. Two Brauer diagrams D1, D2 for r = 5. For any x the group algebra R[x]Sr may be identified with the sub- algebra of B r spanned by the diagrams containing only vertical edges. Such Brauer diagrams provide a graphical depiction of permutations. The group algebra R[x]Sr of Sr also appears as a quotient of B r , the quotient by the two-sided ideal spanned by all diagrams containing at least one horizontal edge. NEW VERSIONS OF SCHUR-WEYL DUALITY 5 Figure 2. Computing the composite of D1 and D2. Figure 3. The composite diagram D1 ◦D2. Label the vertices in each row of an r-diagram by the indices 1, . . . , r. For any 1 6 i 6= j 6 r let ci,j be the r-diagram with horizontal edges connecting vertices i, j on the top and bottom rows. All other vertices in the diagram ci,j are vertical, connecting vertex k on the top and bottom rows, for all k 6= i, j. Brauer observed that B r is generated by the permutation diagrams together with just one of the ci,j. 2.2. Schur-Weyl duality. Brauer [Br] introduced the algebra B 1936 to describe the invariants of symplectic and orthogonal groups acting on V ⊗r. (Brauer’s conventions were slightly different; we are here following the approach of Hanlon and Wales [HW], who pointed out that B r is isomorphic with the algebra defined by Brauer to deal with the symplectic case.) Let G be Spn or On, where n is even in the first instance. By restricting the action ρ considered in 1.1 we have an action of G on V ⊗r. One can extend the action of Sr to an action of r (over C) on V ⊗r, where ǫ = −1 if G = Spn and ǫ = 1 if G = On. To do this, it is enough to specify the action of the diagram ci,j. This acts on V ⊗r as one of Weyl’s contraction maps contracting in tensor positions i and j. So we have (commuting) representations (7) CG −−−→ End(V ⊗r) ←−−− B 6 STEPHEN DOTY which satisfy Schur-Weyl duality; i.e., the image of each representation equals the full centralizer algebra of the other action: ρ(CG) = End (V ⊗r)(8) σ(B(ǫn)r ) = EndG(V ⊗r).(9) The algebras in equality (8) are the symplectic and orthogonal Schur algebras (see [Do2], [D1], [D2]). If n > r − 1 the representation σ in (7) is faithful [Bro]; thus it induces an isomorphism B r ≃ EndG(V 2.3. Schur-Weyl duality in type D. In type Dn/2 (n even) the orthogonal group On is not connected, and contains the connected semisimple group SOn (special orthogonal group) as subgroup of in- dex 2. In order to handle this situation, Brauer (see also [Gr]) defined a larger algebra D r , spanned by the usual r-diagrams previously de- fined, together with certain partial r-diagrams on 2r vertices and r−n edges, in which n vertices in each of the top and bottom rows are not incident to any edge, and showed that the action of B r can be ex- tended to an action of this larger algebra D r on V ⊗r. Thus we have representations (10) CSOn −−−→ End(V ⊗r) ←−−− D Brauer showed that the actions of SOn and D r on V ⊗r satisfy Schur- Weyl duality: ρ(CSOn) = EndD(ǫn)r (V ⊗r)(11) σ(D(ǫn)r ) = EndSOn(V ⊗r).(12) The algebra in (11) is a second Schur algebra in type D, a proper subalgebra of the algebra End (V ⊗r) appearing in (8) above. 2.4. Generators and relations. One can formulate the above state- ments of Schur-Weyl duality using enveloping algebras, analogous to 1.3. This leads to a presentation (see [DGS]) of the symplectic and orthogonal Schur algebras which is compatible with (a slight modifi- cation of) the usual Serre presentation of the enveloping algebra U(g), where g = spn (n even) or son. 2.5. The quantum case. There is a q-version of the Schur-Weyl du- ality considered in this section, although not as developed as in type A. One needs to replace the Brauer algebra by its q-analogue, the Birman- Murakami-Wenzl (BMW) algebra (see [BW], [Mu]), and replace the NEW VERSIONS OF SCHUR-WEYL DUALITY 7 enveloping algebra by a suitable quantized enveloping algebra. One can think of the BMW algebra in terms of Kauffman’s tangle monoid; see [Ka], [HR], [MW]. (Roughly speaking, tangles are replacements for Brauer diagrams, in which one keeps track of under and over cross- ings, subject to certain natural relations.) There are applications of the BMW algebra to knot theory, as one might imagine. This leads to a q-analogue of the symplectic Schur algebras, in par- ticular, which have been studied by Oehms [Oe]. To the best of my knowledge, a q-analogue of the larger algebra D (n even) considered in 2.3 remains to be formulated. 3. The walled Brauer algebra 3.1. The algebra B r,s . This algebra was defined in 1994 in [BCHLLS]. It is the subalgebra of B r+s spanned by the set of (r, s)-diagrams. By definition, an (r, s)-diagram is an (r+ s)-diagram in which we imagine a wall separating the first r from the last s columns of vertices, such that: (a) all horizontal edges cross the wall; (b) no vertical edges cross the wall. An edge crosses the wall if its two vertices lie on opposite sides of the wall. The multiplication in B r,s is that of B Label the vertices on the top and bottom rows of an (r, s)-diagram by the numbers 1, . . . , r to the left of the wall and −1, . . . ,−s to the right of the wall. Let ci,−j (1 6 i 6 r; 1 6 j 6 s) be the diagram with a horizontal edge connecting vertices i and −j on the top row and the same on the bottom row, and with all other edges connecting vertex k (k 6= i,−j) in the top and bottom rows. It is easy to see that the walled Brauer algebra is generated by the permutations it contains along with just one of the ci,−j. (Note that ci,−j is the (r+ s)-diagram denoted by ci,r+j in 2.1.) 3.2. Dimension. What is the dimension of B r,s ? One way to answer that question is to consider the map, flip, from (r + s)-diagrams to (r+s)-diagrams, defined by interchanging the top and bottom vertices to the right of the imaginary wall. For example, Figure 4 shows a (4, 2)-diagram (to the left) and its corresponding 6-diagram, obtained from the left diagram by applying flip. Note that flip is involutary: applying it twice gives the original diagram back again. 8 STEPHEN DOTY Figure 4. A (4, 2)-diagram and its corresponding per- mutation, after applying flip. One easily checks that the map flip carries (r, s)-diagrams bijectively onto the set of (r+ s)-diagrams with all edges vertical. Such diagrams correspond with permutations of r+s objects, so the dimension of B is (r + s)!. 3.3. Another view of B r,s . The above correspondence between (r, s)- diagrams and permutations gives another way to think of the multipli- cation in B r,s . Given two (r, s)-diagrams D1, D2 let D 2 be their corresponding permutations obtained by applying flip. Define a new (rather bizarre) composition on permutations as follows. Given any two permutation diagrams D′1, D 2 (with r+ s columns of vertices) identify the first r vertices of the bottom row of D′1 with the first r vertices of the top row of D′2, and identify the last s vertices of the top row of D′1 with the last s vertices of the bottom row of D 2. After removing loops and identified vertices this gives a new permutation diagram D′3 in which the vertices in the top (resp., bottom) row are the remaining top (bottom) row vertices from the original diagrams. Let δ(D′1, D 2) be the number of loops removed in computing the com- posite permutation diagram D′3. Define multiplication of permutation diagrams by the rule D′1 ·D 2 = x δ(D′1,D 2)D′3 In other words, we are multiplying permutations by composing maps “on the right” on one side of the wall, and “on the left” on the other side (roughly speaking). For example, Figure 5 below shows the com- putation of the composite diagram in the walled Brauer algebra (left column) and the computation in terms of the corresponding permuta- tions (right column). Figure 6 shows the resulting diagrams after the single loop and identified vertices have been removed. One can check that D′3 coincides with (D1 ◦ D2) ′ and δ(D1, D2) = δ(D′1, D 2). In other words, flip defines an algebra isomorphism between NEW VERSIONS OF SCHUR-WEYL DUALITY 9 Figure 5. Composition of diagrams and permutations. The diagrams on the left correspond under flip with the permutations on the right. Figure 6. The corresponding diagrams resulting from Figure 5. The two diagrams correspond under flip. the algebra B r,s and the algebra B̃ r,s spanned by permutation diagrams with the multiplication defined above. Note that in particular B̃ r,0 ≃ R[x]Sr and B̃ 0,s ≃ (R[x]Ss) 3.4. Schur-Weyl duality. Consider mixed tensor space V r,s := V ⊗r⊗ V ∗⊗s, where V ∗ is the usual linear dual space of V . Mixed tensor space is naturally a module for GLn, and one obtains an action of B r,s on V r,s simply by restricting the action of B r+s, which acts the same on V r,s as it does on V ⊗(r+s), since on restriction to On we have V ≃ V Thus we have the following commutative diagram −−−→ End(V r,s) ←−−− B yι′ −−−→ End(V ⊗(r+s)) ←−−− B 10 STEPHEN DOTY in which the vertical maps ι, ι′ are inclusion. By [BCHLLS], the actions of GLn and B r,s on V r,s in the first row of the diagram satisfy Schur- Weyl duality: ρ(CGLn) = EndB(n)r,s (V r,s)(14) σ(B(n)r,s ) = EndGLn(V r,s).(15) The algebra in (14) is another Schur algebra S(n; r, s) in type A, stud- ied in [DD]. These Schur algebras provide us with a new family of quasihereditary algebras, generalizing the classical Schur algebras, since S(n; r, 0) ≃ S(n, r). In fact, the S(n; r, s) provide a new class of gen- eralized Schur algebras in the sense of Donkin [Do1]. For fixed n, the family of S(n; r, s)-modules as r, s vary constitutes the family of all rational representations of GLn. Whence the name rational Schur alge- bras for the S(n; r, s). When n > r + s, the representation σ in the top row of (13) above is faithful, so induces an isomorphism B r,s ≃ EndGLn(V r,s). 3.5. The quantum case. Quantizations of the walled Brauer algebra have been defined and studied in work of Halverson [Ha], Leduc [Le], Kosuda-Murakami [KM], and Kosuda [K]. 4. The deranged algebra 4.1. The problem. One might wonder if there are versions of Schur- Weyl duality in which the natural module V is replaced by some other representation. Perhaps the first choice would be to replace V with the adjoint module, i.e., an algebraic group acting on its Lie algebra via the adjoint representation. The simplest instance of this would be type A, where we consider the module sl⊗rn as an SLn-module, and ask for its centralizer algebra EndSLn(sl n ) = EndGLn(sl n ). (It makes no difference whether we regard sln as module for SLn or for GLn. We look at sl⊗rn rather than gl n since gln is not simple as a GLn-module or SLn-module.) 4.2. Relation with Brauer algebras. Even though this is a ques- tion about type A, its solution is intimately connected with the walled Brauer algebra. Here is a brief outline of the solution to this problem, recently obtained in [BD]. The main idea is to utilize the decomposition gln = sln ⊕ C (as SLn or GLn module) to write (16) gl⊗rn = sl 06t<r sl NEW VERSIONS OF SCHUR-WEYL DUALITY 11 where the left-hand-side identifies with V r,r = V ⊗r ⊗ V ∗⊗r via the natural isomorphism V ⊗ V ∗ ≃ End(V ) ≃ gln. Thus it follows that our desired tensor space sl⊗rn is isomorphic with a direct summand (as a GLn or SLn-module) of the mixed tensor space V r,r, and for n > 2r the centralizing algebra C = EndGLn(sl n ) is obtainable as a certain subalgebra eEndGLn(V r,r)e where e is the idempotent corresponding to the projection onto the summand sl⊗rn . Thus (17) C ≃ eB(n)r,r e (n > 2r) where e = i(1 − n −1ci,−i) (notation of 3.1). The algebra in (17) has a basis consisting of all elements of the form eDe as D ranges over the set of (r, r)-diagrams with no horizontal edges connecting i to −i. Such diagrams correspond under flip, after inverting the signs labeling the bottom row, with derangements of 2r objects. (A derangement is a permutation having no fixed points.) Let N(k) = the number of derangements of k objects, then we have by the Inclusion-Exclusion Principle (see [St, 2.2.1 Example]) (18) N(k) = (−1)k−j which is the nearest integer to k!/e (e = 2.7182818 . . . ). Thus the dimension of the centralizer algebra eB r,r e is given by N(2r). Because of this connection with derangements, the algebra eB r,r e is known as the deranged algebra. In [BD] explicit formulas are obtained for the number of times a given simple module L appears as a summand of sl⊗rn . In particular, it is shown that N(r) = the multiplicity of the trivial module in sl⊗rn N(r − 1) = the multiplicity of sln in sl demonstrating that the combinatorics of derangement numbers is in- herent in this theory. 4.3. Schur-Weyl duality. At least when n > 2r, the actions of GLn and C on sl⊗rn satisfy Schur-Weyl duality: ρ(CGLn) = EndeB(n)r,r e (sl⊗rn )(19) σ(eB(n)r,r e) = EndGLn(sl n ).(20) This will almost certainly hold for all n, r. 12 STEPHEN DOTY References [BLM] Bĕılinson, A.A.; Lusztig, G.; MacPherson, R. A geometric setting for the quantum deformation of GLn, Duke Math. J. 61 (1990), 655–677. 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Z. 208 (1991), 209–223. http://www.liv.ac.uk/~su14/knotprints.html 14 STEPHEN DOTY [Sc] Schur, I., Über die rationalen Darstellungen der allgemeinen linearen Gruppe, (1927). Reprinted in Schur, I., Gesammelte Abhandlungen, Vol. III, pp. 68–85, Springer-Verlag, Berlin, 1973. [St] Stanley, R.P., Enumerative combinatorics Vol. 1, Cambridge Studies in Advanced Math., 49, Cambridge Univ. Press, Cambridge, 1997. Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60626 U.S.A. E-mail address : doty@math.luc.edu 1. Classical Schur-Weyl duality 1.1. Schur's double-centralizer result 1.2. Schur algebras 1.3. The enveloping algebra approach 1.4. The quantum case 1.5. Integral forms 1.6. Generators and relations 2. The Brauer algebra 2.1. The algebra Br(x) 2.2. Schur-Weyl duality 2.3. Schur-Weyl duality in type D 2.4. Generators and relations 2.5. The quantum case 3. The walled Brauer algebra 3.1. The algebra Br,s(x) 3.2. Dimension 3.3. Another view of Br,s(x) 3.4. Schur-Weyl duality 3.5. The quantum case 4. The deranged algebra 4.1. The problem 4.2. Relation with Brauer algebras 4.3. Schur-Weyl duality References

Orbital-Free Density Functional Theory: Kinetic Potentials and Ab Initio Local Pseudopotentials Jeng-Da Chai∗1 and John D. Weeks1, 2 Institute for Physical Science and Technology, and Department of Chemistry and Biochemistry, University of Maryland, College Park, MD 20742 (Dated: November 1, 2018) In the density functional (DF) theory of Kohn and Sham, the kinetic energy of the ground state of a system of noninteracting electrons in a general external field is calculated using a set of orbitals. Orbital free methods attempt to calculate this directly from the electron density by approximating the universal but unknown kinetic energy density functional. However simple local approximations are inaccurate and it has proved very difficult to devise generally accurate nonlocal approximations. We focus instead on the kinetic potential, the functional derivative of the kinetic energy DF, which appears in the Euler equation for the electron density. We argue that the kinetic potential is more local and more amenable to simple physically motivated approximations in many relevant cases, and describe two pathways by which the value of the kinetic energy can be efficiently calculated. We propose two nonlocal orbital free kinetic potentials that reduce to known exact forms for both slowly varying and rapidly varying perturbations and also reproduce exact results for the linear response of the density of the homogeneous system to small perturbations. A simple and systematic approach for generating accurate and weak ab initio local pseudopotentials which produce a smooth slowly varying valence component of the electron density is proposed for use in orbital free DF calculations of molecules and solids. The use of these local pseudopotentials further minimizes the possible errors from the kinetic potentials. Our theory yields results for the total energies and ionization energies of atoms, and for the shell structure in the atomic radial density profiles that are in very good agreement with calculations using the full Kohn-Sham theory. I. INTRODUCTION Density-functional theory (DFT) has become one of the most powerful tools for investigating the electronic structure of large complex systems. In principle, as shown by Hohenberg and Kohn [1], the exact ground state energy of a system of N electrons can be formally written as a functional E[ρ] of only the electron density ρ(r), a function of three variables, and the external field Vext(r). Determining the energy and other ground state properties from such an approach could dramatically reduce the computational cost for large systems when compared with traditional quantum chemistry methods, which deal with wavefunctions involving coordinates of all N electrons [2, 3]. Kohn and Sham (KS) [4, 5] showed that E[ρ] can be usefully partitioned into the following set of terms: E[ρ] = Ts[ρ] + EH [ρ] + Exc[ρ] + ρ(r)Vext(r)dr. (1) Here Ts[ρ] is the noninteracting kinetic energy density functional (KEDF), which gives the kinetic energy of a model system of N noninteracting electrons in a self- consistent field chosen so that the ground state density ∗E-mail: jdchai@berkeley.edu. Present address: Molecular Foundry, Materials Sciences Division, Lawrence Berkeley National Laboratory, and Department of Chemistry, University of Califor- nia, Berkeley, California 94720 equals ρ(r), EH [ρ] ≡ ρ(r)ρ(r′) |r− r′| drdr′ (2) is the classical electron-electron potential energy (Hartree energy) and Exc[ρ] is the exchange-correlation energy (in- cluding the difference between the interacting and non- interacting kinetic energy and the difference between the quantum and classical electron-electron potential en- ergy). The last term on the right of Eq. (1) is the only term that depends explicitly on the external potential Vext(r). Atomic units are used throughout the paper. If all these functionals were known, then the density ρ(r) could be obtained from the variational principle (Eu- ler equation) associated with minimizing Eq. (1): µ = VTs(r; [ρ]) + Veff (r; [ρ]), (3) and the total energy of the inhomogeneous system could then be determined from the energy functional E[ρ]. All other physical quantities related to the ground-state den- sity could also be computed. Here µ is the chemical po- tential (the Lagrange multiplier associated with the nor- malization condition ρ(r)dr = N), and Veff (r; [ρ]) is an effective one-body potential defined by Veff (r; [ρ]) ≡ δρ(r) EH [ρ] + Exc[ρ] + ρ(r)Vext(r)dr = VH(r; [ρ]) + Vxc(r; [ρ]) + Vext(r), (4) where VH(r; [ρ]) ≡ δEH [ρ]/δρ(r) = ρ(r′) |r− r′| dr′ (5) http://arxiv.org/abs/0704.1878v1 is the Hartree potential, and Vxc(r; [ρ]) ≡ δExc[ρ]/δρ(r) is the exchange-correlation potential. Similarly we interpret VTs(r; [ρ]) ≡ δTs[ρ]/δρ(r) (6) as the kinetic potential (KP) arising from the KEDF [6]. Further progress requires an accurate determination of the noninteracting kinetic energy, whose magnitude is much larger than the exchange-correlation energy. The initial development of DFT as a practical computational method was made possible by KS’s realization that the numerical value of the noninteracting kinetic energy can be exactly calculated, not directly from the density itself using Ts[ρ], but by introducing a set of N one-electron wave functions (orbitals) satisfying the N coupled KS equations that describe the model system [4, 5]. Research could then focus on determining the remain- ing small term Exc[ρ]. Here even local density approx- imations have often proved useful. Through the efforts of many workers we now have generally accurate expres- sions for Exc[ρ]. Using these along with the KS orbitals to calculate the kinetic energy, one can accurately cal- culate both the total energy E[ρ] and the ground-state density ρ(r) for a wide variety of systems. However, the use of the KS orbitals usually generates a relatively expensive O(N3) scaling of computational cost with the number of electrons. While this scaling is much better than that of most standard methods that include correlation energy, calculations for large systems remain problematic. This remaining bottleneck could be removed if there were an accurate treatment of the kinetic energy in terms of the electron density only [2, 3, 7]. To that end there has been considerable effort in- vested in developing “orbital-free”density functional the- ory (OF-DFT) by making direct approximations for Ts[ρ] [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. While earlier simple local density approximations for Ts[ρ] like those used in the Thomas-Fermi (TF) model [21] are very inaccurate, there have been two main advances in recent work that offer prospects for significant improvements. The first is the introduction of nonlocal KEDFs that reproduce known exact results for very slowly varying or very rapidly varying fields and give the exact linear re- sponse (LR) of the density of the uniform model system to small perturbations. Similar ideas have been success- fully applied to classical nonuniform fluids [22]. The sec- ond advance is to focus not on the total density but on the smaller and more slowly varying valence electron den- sity as described by a weak pseudopotential acting only on the valence electrons. While conventional pseudopo- tential methods use orbitals, recently developed ab ini- tio local pseudopotential (AILPS) methods determine the unique local one-body potential producing a given target valence density by solving the KS equations inversely, us- ing the one-to-one mapping between density and poten- tials in DFT [23]. For OF-DFT with LR-based KEDF’s, the use of pseudopotentials not only can reduce the com- putational cost, but also can improve its accuracy, since the system will be closer to the LR regime where Ts[ρ] is designed to be accurate. [24, 25]. Indeed, very promising results using such OF-DFT methods have been obtained for a variety of nearly free-electron-like metals. However, existing KEDF’s have not yet achieved chem- ical accuracy for systems with localized and more rapidly varying electron densities like molecules or for covalent or ionic solids. The main problem is that the exact Ts[ρ] is highly nonlocal, and we have little idea of the func- tional form of the nonlocality for densities far from the LR regime. It has proved very difficult to understand what errors an approximate nonlocal Ts[ρ] will produce in the density as determined by the Euler equation with a general Vext(r). We explore here a different way to attack this basic problem. The exact Ts[ρ] can be formally obtained from VTs(r; [ρ]) by functional integration over density changes in all regions of space [6, 18]. Because of this integration Ts[ρ] is a more nonlocal functional of the density than is VTs(r; [ρ]). More detailed arguments arriving at this same conclusion have been recently presented [26]. Since most problems in devising accurate approximations for Ts[ρ] have arisen from the nonlocality, this suggests it could be worthwhile to try to develop approximations for the KP VTs(r; [ρ]) itself. To illustrate this point, Chai and Weeks [7] added a simple gradient correction to the original local TF KP for atoms [21], with a coefficient chosen to reproduce the exact boundary condition of exponential decay of the electron density far from the nucleus. Though quanti- tative results were not obtained, the resulting modified Thomas-Fermi (MTF) model gave energies for atoms and for closed-shell diatomic molecules that showed notable improvements when compared to the original TF and re- lated gradient corrected KEDF models. However, the local gradient correction used in the MTF KP cannot re- produce the oscillatory atomic shell structure and it does not satisfy the exact LR behavior in the homogeneous limit. It is clear that nonlocality even in the KP must be taken into account to achieve more accurate results. We propose here new nonlocal approximations for the KP using ideas similar to those employed for the nonlocal KEDFs. These new KPs satisfy the exact LR condition in the uniform limit, and reproduce known exact limiting forms of VTs(r; [ρ]) both for very slowly varying and very rapidly varying perturbations. As will become clear, the nonlocality in our KP is determined by the requirement that LR is exactly satisfied, and it is much easier to en- sure that LR holds for the KP than it is for analogous KEDF models. We believe this level of nonlocality in the KP may suffice in many cases when used in conjunction with AILPS methods to describe slowly varying valence density components closer to the LR regime. The remainder of this paper is organized as follow. Section II will discuss some general pathways connect- ing Ts[ρ] and VTs(r; [ρ]). Section III will describe limit- ing forms of the KEDF and KP for slowly varying and rapidly varying perturbations, and discuss LR theory, an exact theory for the response of density of the uniform electron gas to small perturbations. Section IV will de- velop two nonlocal KPs incorporating both the correct limiting forms of the exact KP and the exact LR of the free-electron gas. Section V will compare the numerical results of the present method for atoms with the KS-DFT and other KEDFs, both for all-electron calculations and for valence electrons using the AILPS. We find that the use of AILPS indeed reduces errors arising from nonlocal- ity in these approximate KPs or KEDFs, which give very accurate results for the relatively slowly varying valence densities. Our conclusions are given in Section VI. II. PATHWAYS FROM VTs(r; [ρ]) TO Ts[ρ] If Ts[ρ] is known, VTs(r; [ρ]) can be simply computed by functional differentiation. However, there is no unique way of determining Ts[ρ] from a given VTs(r; [ρ]). Many possible pathways can be used to construct Ts[ρ] by func- tional integration of VTs(r; [ρ]) [27, 28]. If the exact VTs(r; [ρ]) is used and the integration is carried out ex- actly, then all pathways would give the same exact result for Ts[ρ]. However, when an approximate VTs(r; [ρ]) is used, different pathways will give different results for the kinetic energy. But this “thermodynamic inconsistency” is small if reasonably good approximations are used, since the integration tends to smooth out local errors that may exist in the density [7, 28]. More problematic is the fact that most pathways re- quire additional results for partially coupled systems as the external field or density perturbation is gradually turned on, which adds to the computational burden. In particular, most earlier work has used a “potential en- ergy pathway”, where the external potential is scaled by a coupling parameter [7, 27]. The kinetic energy can then be found by subtracting the potential energy (calculated from the potential energy density functionals) from the total energy. However, this pathway is expensive, since one has to solve the Euler equation (3) for each partially coupled V λext(r) (with the same µ), to determine the cor- responding ρλ. A. Herring’s Pathway However, Herring showed there is a particular path- way arising from exact scaling relations between the non- interacting kinetic energy Ts[ρ] with respect to the coor- dinate r in ρ(r) where very simple results involving only the final density can sometimes be found [6, 18]. If the coordinate r is scaled to αr, the normalized scaled den- sity is ρα(r) = α 3ρ(αr). It is easy to show that the exact Ts[ρ] then obeys α2Ts[ρ] = Ts[ρα]. (7) For isolated systems, such as atoms and molecules, the density and its derivatives to all order vanish far from the nuclei. For such systems, when Eq. (7) is differentiated with respect to α, and the partial derivative is evaluated at α = 1, we find the formally exact result Ts[ρ] = VTs(r; [ρ])∇ · (rρ(r))dr. (8) Therefore, once the kinetic potential VTs(r; [ρ]) is known for some given ρ(r), the numerical value of Ts[ρ] can then be immediately determined from Eq. (8). Since there is no need to perform a coupling parameter integra- tion over the change of density or potential, this scheme is not only fast, but also numerically reliable. The final form of Eq. (8) is essentially the viral theorem, and is directly related to the force on molecules [29, 30, 31, 32]. Note that this simple and exact pathway holds only for the noninteracting Ts[ρ] [31], which again shows the virtues of the KS partitioning of the total energy. We will use Eq. (8) as the basic pathway to determine the numer- ical value of Ts[ρ] from a given approximate VTs(r; [ρ]) for most calculations in this paper. However, Eq. (8) does not hold for extended solid-state systems because of nonvanishing boundary terms, and thus far we have not found an exact and simple way of including them. Fortunately, there is another class of computationally efficient“density pathways”that can be used for extended systems, as we now show. Density pathways can also be used for atomic and molecular systems to check the accuracy of the VTs used, since results using the exact VTs would be independent of path [27, 28]. B. Density Pathways The change in the kinetic energy can be formally re- lated to a coupling parameter integration, where the den- sity changes from some known value at λ = 0 to the final density at λ = 1: Ts[ρ] = Tλ=0 + drVTs(r; [ρλ]) ∂ρλ(r) In most cases a simple linear density pathway will suffice. Here the density ρ(r) is linearly scaled by a coupling pa- rameter λ from some uniform reference density ρ0 natu- rally chosen to be the uniform electron density N/V in extended systems: ρλ(r) = ρ0 + λ[ρ(r) − ρ0]. (10) Then Eq. (9) becomes Ts[ρ] = Tλ=0 + drVTs(r; [ρλ])[ρ(r) − ρ0] (11) Here Tλ=0 is the kinetic energy of the uniform system, i.e., the Thomas-Fermi kinetic energy TTF [ρ0]. For ex- tended systems, where the Herring’s pathway cannot be used, this density pathway appears to be a good way to compute T . Other density pathways, like the square-root pathway introduced by Chen and Weeks [28] to describe nonuniform hard sphere fluids, can be defined, and have proved useful in certain applications, but we do not con- sider them here. Note from Eq. (10) that ρλ(r) depends only on the fi- nal density, so evaluation of Eq. (11) is straightforward and this pathway is computationally efficient. Unlike the potential energy pathway where the external potential is scaled, there is no need to solve the Euler equation (3) for its corresponding external potential V λext(r) at each λ. However, for isolated systems, where ρ0 = 0, this path- way is likely to be less accurate than Herring’s pathway, since it does not automatically satisfy the virial theorem. III. EXACT LIMITS AND LINEAR RESPONSE THEORY Although the exact Ts[ρ] is still unknown, several limit- ing forms have been discovered for particular density dis- tributions. These provide important cornerstones that can be used to construct accurate KEDFs and KPs in many cases, as will be seen below. In particular, the Thomas-Fermi (TF) KEDF [21] is known to be exact for a uniform system: TTF [ρ] = CF ρ5/3(r)dr, (12) where CF = (3π2)2/3. The TF KEDF TTF [ρ] is de- rived by local use of uniform free-electron gas model, and is exact for a system with an infinite number of electrons. The corresponding expression for the TF KP is VTF (r; [ρ]) ≡ δTTF [ρ]/δρ(r) = 2/3(r). (13) This depends only on the local value of ρ2/3(r) and thus formally is more local than the TF KEDF, whose func- tional dependence on ρ involves the density at all r. Of course in this simple case the functional integration of Eq. (13) can be carried out exactly to yield Eq. (12), but this cannot be done in general and the nonlocality of Ts[ρ] has proved problematic. Results for nonuniform systems are best described in Fourier space. For a very slowly varying perturbation of the density, the second-order gradient expansion is exact [33]. It is easy to see that results correct to second order at small wavevectors are given by TTF 1 W [ρ] ≡ TTF [ρ] + TW [ρ], (14) where TW [ρ] ≡ |▽ρ(r)|2 dr (15) is the von Weizsäcker (W) KEDF [34]. TW [ρ] is exact for a system with one or two electrons, or where the density can be accurately described by a sin- gle orbital. Moreover it has been argued [8, 9, 10, 12] that TW [ρ] gives the correct leading order term for a rapidly varying perturbation with only high wavevector compo- nents and that the next order correction is reproduced TW− 3 TF [ρ] ≡ TW [ρ]− TTF [ρ]. (16) The W KP is VW (r; [ρ]) ≡ δTW [ρ]/δρ(r) = |▽ρ(r)|2 ρ2(r) ▽2ρ(r) If we represent the full density by an effective single or- bital function ψ(r), ρ(r) = |ψ(r)|2 (18) then the W KP can be written in a compact form that will later prove useful: VW (r; [ρ]) = − ▽2ψ(r) 2ψ(r) . (19) Finally, the linear response of the density of a uniform non-interacting electron gas with density ρ0 to a small perturbation δV (k) = ǫke ik·r is exactly known [35], δρ(k) = χL(q)δV (k). (20) q ≡ k/2kF (21) is a dimensionless wavevector, where kF ≡ (3π 1/3 (22) is the Fermi wavevector (FWV) and k ≡ |k|. The LR function χL(q) has the form χL(q) = − F−1L (q) 1− q2 1 + q , (23) where FL(q) ≡ 1− q2 1 + q has been called the Lindhard function [10]. It is known that the weak logarithmic singularity at q = 1 in F−1L (q) is responsible for Friedel oscillations, and may also be important for the appearance of atomic shell structure. This singularity further divides the Lindhard function into two branches in Fourier space: the low- momentum (q < 1 ) or the low-q (LQ) branch, and the high-momentum (q > 1) or the high-q (HQ) branch [10]. Figure 1: Linear response functions of a uniform system of noninteracting Fermions as given by the TF, W, and MTF (see Ref. [7]) models. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 exact The dimensionless response function arising from the TF KEDF is FTF (q) = 1, and that from the W KEDF is FW (q) = 3q 2 [36]. Clearly, no linear combination of the TF and the W KEDFs can reproduce the exact Lindhard function in Eq. (24). This has the following two limits [10], FL(q) = 1 + q2/3 +O(q4) q ≪ 1 3q2 − 3/5 +O(q−2) q ≫ 1 It should be noted that the expansions for both the low-q and high-q limits are correct to all orders in per- turbation theory, but valid only in the appropriate limits in Fourier space. On the other hand, the LR theory is valid for all wavevectors, but is only accurate for small perturbations. Therefore, the regime where the response functions of the two limiting KEDFs deviate from the exact LR function gives an indication of the range of wavevectors where the two limiting forms are inaccurate. As shown in Fig. 1, the response function F−1TF (q) has no momentum dependence and is only exact at q = 0. The response function F−1W (q) is exact asymptotically at high q, and remains fairly accurate for q & 2, but is di- vergent in the low-q branch, and fails completely for the nearly uniform electron gas. In contrast, the MTF model [7] gives a reasonably accurate average description of the exact response function, especially in the important re- gion near the singularity at q = 1. IV. CONSTRUCTION OF NONLOCAL KINETIC POTENTIALS A. Kinetic Energy Density Functionals TTFλW [ρ] and TWλTF [ρ] Simple linear combinations of the two limiting KEDF’s in Eqs. (12) and (15), such as the TFλW KEDF [33, 37, 38, 39] TTFλW [ρ] ≡ TTF [ρ] + λTw[ρ] (26) and the WλTF KEDF [40, 41, 42, 43] TWλTF [ρ] ≡ TW [ρ] + λTTF [ρ] (27) have been widely studied for several decades. The value of the parameter λ was either determined empirically for getting good atomic energy or obtained by some semi- classical arguments. The advantage of these approaches is the ability to gen- erate a family of simple KEDF’s easily. It has been shown empirically that the TF1/5W model can give good val- ues for atomic energies, but the predicted density profiles are generally not very accurate, both near and far away from the nucleus. The TTFλW [ρ] and TWλTF [ρ] function- als give the correct leading term in the density response to a slowly-varying perturbation and a rapidly-varying perturbation respectively, and with particular choices of λ as in Eqs. (14) and (16) they can reproduce the next order term. Unfortunately, they then will have an in- correct leading term in the opposite limit, unless λ = 1. However it has been shown that TTFW [ρ] with λ = 1 always overestimates the exact Ts[ρ] for various systems [10]. Finally, none of these functionals can reproduce the exact response function FL(q) in the homogeneous limit. Since these models fail to satisfy all the known limits, and nonlocality in Ts[ρ] is not correctly described, it is also not surprising that atomic shell structure is missing in these approaches. B. Combining TF and W Kinetic Potentials We argue that it may be more profitable to take advan- tage of known limiting forms of the KP, rather than the KEDF, and develop approximations for the more local VTs(r; [ρ]) directly. Again we can rely on known results in the linear response regime when the density variations are not too large. From Eqs. (14) and (16), the following linear combina- tions of the TF KP and the W KP in Eqs. (13) and (17) can reproduce exact results to second order for very small and very large wavevector perturbations respectively: VTs(r; [ρ]) ≈ VTF (r; [ρ]) + VW (r; [ρ]) q ≪ 1 VW (r; [ρ])− VTF (r; [ρ]) q ≫ 1 Since VTF (r; [ρ]) and VW (r; [ρ]) are the only components up to second order of the two exact limiting forms of the KP, we can combine them in analogy to the TFλW and WλTF models and arrive at generalized KPs. However, instead of combining them using a fixed pa- rameter λ, it seems natural to represent them in Fourier space and allow a wavevector dependence in λ = λ(q) to connect the limiting forms. The λ(q) can then be chosen in a very simple way so that the exact LR function is Figure 2: Weight function f̂(q) for the HQ and LQ KPs. 0 0.5 1 1.5 2 2.5 3 reproduced for a uniform system with density ρ0. In this way the LR function bridges the exact limits at large and small wavevectors, and if the theory is applied to weak perturbations in the linear response regime for interme- diate wavevectors we can expect very accurate results. Here, we derive such generalized KPs based on the KP for the WλTF model. C. HQ Kinetic Potential In analogy to the WλTF model in Eq. (27) we look for a kinetic potential of the form V 0HQ(k) = VW (k) + λHQ(q) VTF (k) = VTF (k) + VW (k) + f̂(q) VTF (k), (29) where q = k/2kF (30) is a dimensionless wavevector normalized by the FWV kF in Eq. (22) of a uniform reference system with density ρ0 and f̂(q) = λHQ(q) − 1. The superscript 0 in V indicates use of a uniform reference system. For a small perturbation, we can linearize the V 0HQ(k) in Eq. (29). Requiring that it satisfy LR exactly then determines the weight function f̂(q) as f̂(q) = FL(q)− 3q 2 − 1. (31) See Fig. 2. We refer to Eq. (29) with Eq. (31) as the HQ KPmodel. It reproduces the correct high-q limit in Eq. (28) up to the second order. However, unlike Eq. (16), it also satisfies the correct low-q limit to leading order and gives exact results for all q in the linear response regime. Inverse Fourier transform of Eq. (29) then gives V 0HQ(r; [ρ], kF ) = VTF (r; [ρ]) + VW (r; [ρ]) f(|r− r′|; kF ) VTF (r ′; [ρ])dr′. This expression is directly useful for extended systems where a reasonable ρ0 can be defined. For isolated sys- tems such as atoms and molecules where the density van- ishes far from the nuclei, it seems natural to replace kF in Eq. (32) by the local Fermi wavevector (LFWV) kF (r) ≡ (3π 2ρ(r))1/3, (33) though errors may be introduced for rapidly varying den- sity distributions. Using Eq. (13), this yields the general form of our proposed HQ kinetic potential: VHQ(r; [ρ], kF (r)) = VTF (r; [ρ]) + VW (r; [ρ]) f(|r− r′|; kF (r)) ρ 2/3(r′)dr′. Note that the last term in Eq. (34) is most easily com- puted in Fourier space as 3(2π)3 f̂(k/2kF (r))ρ 2/3(k)e−ik·rdk, (35) As can been seen in Eq. (31), the weight function f̂(k/2kF (r)) is determined analytically. Unlike the LR- based KEDF approaches, no first-order differential equa- tion is needed to solve for the weight function in Fourier space. For atomic systems where the LFWV is used, the convolution in Eq. (35) must be carried out numerically, which will lead to a quadratic scaling of the HQ model (and related LQ model described below) in the numberN of electrons. For extended systems where one can expand about the local density ρ0, one can use fast Fourier trans- forms (FFT’s) for a much more efficient computation of this integral [44]. D. LQ Kinetic Potential In analogy to the TFλW model in Eq. (26), we could similarly generate a KP that is accurate to second order at low q while still reproducing the leading term at high q. However this is numerically less useful because the analogue of Eq. (35) involves the Fourier transform of VW (r ′; [ρ]), which cannot be simply expressed in terms of the density. Instead, by empirically taking a properly chosen component of the density outside the integral we find that Eq. (34) can be modified to produce a new LQ KP that is accurate to second order at low q and first order at high q [45]: VLQ(r; [ρ], kF (r)) = VTF (r; [ρ]) + VW (r; [ρ]) 1/6(r) f(|r− r′|; kF (r)) ρ 1/2(r′)dr′ Extending these ideas we have constructed a modified KP that satisfies LR everywhere and is accurate to sec- ond order at both low and high q [45]. However the functional form is much more complicated, and little ad- ditional accuracy is gained from the improved behavior at very small or very large wavevectors, since all forms use LR to interpolate for intermediate wavevectors, and this is where most errors arise in practice. Thus we will report results here only for the HQ and LQ models. V. RESULTS FOR ATOMS USING THE HQ AND LQ MODELS For completeness and to compare to earlier work, we first briefly discuss all-electron calculations using the pro- posed HQ and LQ model KPs and the full atomic poten- tials. We then describe results using AILPS methods. These are compared with the KS-DFT, the TFλW mod- els, and the CAT model introduced by Chacón, Alvarel- los, and Tarazona [12, 13, 14]. The CAT model is a LR-based KEDF method, which gives some indication of shell structure. We employed the latest version, which uses a nonlocal two-body Fermi wave vector with a pre- scribed functional form depending on an empirical pa- rameter β = 1/2 (defined in Eq. (3) of Ref. [14]). This caused the numerical calculations [46] to be consider- ably more costly than those of the LQ or HQ models, which used the local Fermi wave vector as in Eq. (33). All calculations are spin-restricted and use the local den- sity approximation (LDA) [47, 48, 49] for the exchange- correlation functional. A. All-electron calculations All-electron calculations consider the density response to the large and rapidly varying nuclear potential. Since the system is far from the linear response regime, quanti- tative results from the HQ and LQ models (or from LR- based KEDF methods) cannot be expected. However, by incorporating exact results for very large and very small wavevectors, these models do correct major deficiencies of the purely local TF model (which, e.g., predicts an infinite density at the nucleus!) and even give some qual- itative indications of atomic shell structure. The numerical method use the Pauli kinetic potential V P (r; [ρ]) [50, 51], defined as V P (r; [ρ]) ≡ VTs(r; [ρ]) − VW (r; [ρ]). (37) Since VW (r; [ρ]) is the exact KP for a system where the density can be accurately described by a single orbital, if V P (r; [ρ]) is omitted, one would essentially obtain the ground state density of the corresponding Boson system, where all the electrons are in the same orbital. If we represent the full density by a single orbital function ψ(r), so that ρ(r) and VW (r; [ρ]) can be written in the forms of Eq. (18) and Eq. (19) respectively, we can then combine V P (r; [ρ]) with the one-body potential Veff (r; [ρ]) in Eq. (3), and derive a Schrödinger-like equation for the Bose orbital ψ(r), ▽2 + Veff (r; [ρ]) + V P (r; [ρ]) ψ(r) = µψ(r). (38) Table I: Atomic energy E using the KS, LQ, HQ, CAT and the TFλW models in all-electron calculations. MAPE, the mean absolute percentage error (relative to the KS method) of various OF models are given at the bottom of their respective columns. KS LQ HQ CAT TF1/5W TFW He −2.834 −2.565 −2.437 −2.675 −2.911 −1.559 Ne −128.2 −134.3 −126.6 −126.2 −129.5 −86.40 Ar −525.9 −545.9 −512.2 −515.1 −526.2 −375.5 Kr −2750 −2805 −2621 −2712 −2748 −2099 Xe −7229 −7306 −6844 −7141 −7214 −5701 Be −14.45 −14.39 −13.64 −14.11 −14.71 −8.699 Mg −199.1 −207.9 −195.7 −195.2 −200.0 −136.4 C −37.42 −38.97 −36.85 −37.25 −38.41 −24.01 N −54.02 −56.71 −53.59 −53.84 −55.39 −35.33 O −74.47 −78.39 −74.02 −74.08 −76.11 −49.39 Si −288.2 −300.4 −282.5 −282.2 −288.9 −200.5 P −339.9 −354.1 −332.7 −332.8 −340.6 −238.3 S −396.7 −412.8 −387.7 −388.5 −397.3 −279.9 MAPE 4.06% 3.42% 1.82% 1.10% 32.0% Table II: Electron density at the nucleus ρ(0), using the KS, LQ, HQ, CAT and the TFλW models in all-electron calcula- tions. MAPE, the mean absolute percentage error (relative to the KS method) of various OF models are given at the bottom of their respective columns. KS LQ HQ CAT TF1/5W TFW He 3.525 3.088 2.742 3.600 18.23 0.9515 Ne 614.5 576.6 517.6 613.2 2596 169.6 Ar 3819 3642 3282 3812 1.548 × 104 1093 Be 34.86 30.49 27.17 33.75 158.2 8.952 Mg 1086 1024 920.9 1083 4519 303.0 C 126.0 113.3 101.2 122.8 547.9 33.07 N 203.9 185.6 166.1 200.0 876.7 54.24 O 308.6 284.1 254.6 304.6 1317 83.19 Si 1754 1662 1495 1749 7218 493.9 P 2173 2062 1857 2167 8901 614.5 S 2654 2523 2272 2647 1.083 × 104 753.5 MAPE 7.61% 17.2% 1.14% 331% 72.6% In other words, Veff (r; [ρ]) + V P (r; [ρ]) is now the one- body effective potential for the corresponding Boson sys- tem with the same electron density. This reduction of an N -fermion problem to a Boson form is widely imple- mented in OF-DFT due to its numerical stability and its easy implementation using existing KS-DFT codes [10]. The associated Pauli potentials for the HQ and LQ models are immediately obtained by subtraction of the W KP from Eq. (34) and Eq. (36) respectively. The stan- dard finite difference method for solving Euler equations for the TFW models [52] are implemented for the LQ and HQ models, and the nonlocal terms are evaluated by Fourier transforms. The choices of radial grids for both of the real and Fourier space and other detailed numerical methods are given in Ref. [7]. The kinetic energy for HQ and LQ models is computed using the Herring pathway in Eq. (8). As shown in Table I, the atomic energy calculated by Figure 3: (Color online) Radial density r2ρ(r) of the Kr atom using the KS method, the LQ and HQ models, the CAT model, and the TFλW models (see the inset) with the full nuclear potential. 0 0.5 1 1.5 2 0 0.5 1 1.5 2 TF1/5W the energy-optimized TF1/5W model is very close to the KS-DFT, and outperforms all the LR-based models, and other TFλW models. In Table II, we compare the elec- tron density at the nucleus ρ(0) for various models. The TF1/5Wmodel overestimates ρ(0) by about a factor of 4, while the TFW model underestimates it by about 30%. The predicted values of ρ(0) for all the LR-based mod- els are very close to the KS results, and are much better than the TFλW models. In Fig. (3), we compare the radial density distribu- tion r2ρ(r) of the LQ and HQ models to that predicted by other theories for the Kr atom. Both the TF1/5W and TFW models predict smooth and structureless ra- dial density profiles. Using the full Coulombic potential, all the LR-based models can predict an incipient shell structure for heavy atoms (Z & 30), and these results are typical. Since the potential is certainly far beyond the LR regime, these qualitative results with some sug- gestion of shell structure are about as good as could be hoped for. The surprisingly good total energies given in Table I for the TF1/5W model and the LR-based models shows that averaged thermodynamic properties are less sensitive to errors in the KP than is the density profile. The difference in the results for the LQ, HQ and CAT models indicates that the LR-based OF theory is being used outside its range of validity. As shown below, we gain a significant improvement by using the AILPS to deal with these difficulties. B. Ab initio local pseudopotential calculations As discussed earlier, the use of pseudopotentials in non- local LR-based OF-DFT can improve the accuracy of the theory because the weaker pseudopotential is more nearly in the LR regime, where the theory is designed to be ac- Figure 4: The smooth target density ρ̃v(r) from Eq. (40), with parameters given in Table IV for the Si pseudoatom used in the inverse-KS process, and the valence density ρv(r) pre- dicted by the LQ and HQ models using the Vps(r) (see Fig. 5) corresponding to ρ̃v(r). The arrow indicates the location of rc. 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.5 1 1.5 2 2.5 3 smooth Figure 5: The AILPS Vps(r) for Si generated by the target density ρ̃v(r) in Fig. 4. 0 0.5 1 1.5 2 2.5 3 curate. Our proposed HQ and LQ models can be used with any existing AILPS. However, since we want to as- sess the performance of these models for a wide class of atomic systems, we describe here a new method for de- termining reasonable AILPS for general atomic systems. These pseudopotentials will be used in all our calculations and can be transfered to other molecular and solid state environments, but we expect (and find in cases where comparison can be made) little change if other reason- able AILPS are used. Because of the one-to-one mapping between the effec- tive one-body potential acting on a system of N electrons and the electron density in the ground-state configura- tion, it is possible to obtain a unique local one-body po- tential that generates a given target density ρ̃(r) by using Figure 6: (Color online) Radial valence density r2ρv(r) of the Si atom using the KS method and various models using AILPS. Parameters used for constructing this reference sys- tem are shown in Table IV. The arrow indicates the location of rc. Inset: The corresponding radial total density r 2ρ(r), which is dominated by the core component for r < rc. 0.05 0.15 0.25 0.5 1 1.5 2 2.5 3 TF1/5W 0 0.5 1 1.5 2 a KS orbital-based method in an inverse way [23]. To con- struct an AILPS for a given atom we separate the total electron density ρ(r) into a “core density” ρ̃c(r), which is supposed not to vary significantly in other molecular or solid state environments, and the target“valence density” ρ̃v(r) where ρ(r) = ρ̃v(r) + ρ̃c(r). (39) Because DFT requires only the electron density, we can take a more general view of what is meant by the core and valence components than is used in most orbital-based methods. Here, we directly construct a smooth target valence density for the Nv = N − Nc valence electrons, with Nc chosen to be the number of electrons in the noble gas configuration. Our proposed target valence density ρ̃v(r) for atoms equals the full KS density ρKS(r) outside a core of radius rc, and is designed to be small and slowly varying inside rc. The functional form we take is ρ̃v(r) = tρKS(rc) + a0r q exp[−rp(a r2)] r ≤ rc ρKS(r) r > rc Figure 4 gives an example of ρ̃v(r) for Si that will be discussed in more detail below. We find most results are insensitive to the details of our fitting procedure. Param- eter values for a variety of atomic systems are given in the Appendix in Table IV along with the physical and technical considerations that guided our choice of this particular form for ρ̃v(r). The Appendix also discusses some of the general issues that arise in using these atomic AILPS in other environments. The local pseudopotential is directly related to the effective one body potential that reproduces ρ̃v(r) ex- actly when using the full KS theory. Following previous work [24, 25], for a given ρ̃v(r), the inverse-KS equations are solved to get the effective one-body screened poten- tial Vscr(r). The desired ab initio local pseudopotential Vps(r) is then obtained by subtracting the Hartree po- tential and the exchange-correlation potential: Vps(r) = Vscr(r)− VH(r; [ρ̃v])− Vxc(r; [ρ̃v]). (41) This relatively expensive procedure to determine Vps(r) requires the use of orbitals. However it needs to be done only once for each atom, and the resulting Vps(r) can then be used in a variety of other environments if the atomic core densities remain essentially constant. Once suitable Vps(r) have been determined (by this or other means) they can be incorporated in Vext(r) in dif- ferent ways, depending on the particular system of inter- est. OF-DFT theory can then be used to determine the valence density ρv(r) in direct analogy to the all-electron calculations for the full atomic potential in Eq. (38): ▽2 + VH(r; [ρv]) + Vxc(r; [ρv]) +Vext(r) + VP (r; [ρv])}ψv(r) = µψv(r) (42) where ρv(r) = |ψv(r)| 2. (43) A simple and direct test of OF-DFT is to use Eqs. (42) and (43) for the same atomic system for which Vps(r) was constructed. Thus we take Vext(r) = Vps(r) for a given atom as input data in Eq. (42). The valence density ρv(r) predicted by the OF KPs is determined from Eqs. (42) and (43), and can be directly compared to the exact target density ρ̃v(r) for this atomic system given by the full KS theory. This is illustrated in Fig. 4, which shows the input target valence density ρ̃v(r) for Si used in the inverse- KS process. The total KS density ρKS(r) equals ρ̃v(r) for r ≥ rc, indicated by the arrow in Fig. 4, and then increases rapidly for r < rc, reaching a very large value at the nucleus, ρKS(0) = 1754. In contrast, the proposed target valence density ρ̃v(r) remains small and relatively slowly varying inside the core, with ρ̃v(0) = 0.005621. Also shown in Fig. 4 are the predicted valence densities ρv(r) for Si given by the LQ and HQ models. Because of the relatively weak Vps(r) and slowly varying valence den- sity, both the LQ and HQ KP models predict results very close to those given by the exact KS treatment of the ki- netic energy, and perform markedly better than they did for the all-electron calculations using the full Coulomb potential. Fig. 5 shows the corresponding AILPS gener- ated by the inverse KS procedure. It is much smaller in the core region than the full atomic potential and more likely to be accurately treated by LR-based methods. The radial valence density r2ρv(r) of the Si and Ar atoms predicted by the various methods are shown in Figs. 6 and 7. The consistency of our OF theory when pseudopotentials are used is illustrated by the similar- ity of the density predicted by the HQ and LQ models. Figure 7: (Color online) Same as in Fig. 6 but for the Ar atom. 0.5 1 1.5 2 2.5 3 TF1/5W 0 0.5 1 1.5 2 The slight deviations from the KS-DFT results for Ar in the valence region in Fig. 7 are among the largest we en- countered for all atoms tested, and could be due to the relatively large number of valence electrons (8) compared to core electrons (10). Further errors may arise from the LFWV approximation in Eq. (33). Once ρv(r) has been determined using OF-DFT, it can be added to the known input core density ρ̃c(r) to obtain the predicted total density, since the basic assumption of our AILPS is that the core density remains unchanged in different chemical environments. The core density is defined in Eqs. (39) and (40). The inset in Fig. 6 shows that the HQ and LQ treatment of the valence density for the Si atom does not produce noticeable errors in the total density, as expected, and the shell structure remains in excellent agreement with the full KS-DFT calculations. As a further test of OF-DFT we can compare the en- ergy for the valence density given by the various methods to the exact valence energy for the target valence density determined by the inverse KS method. Table III gives the valence energy values for the KS-DFT, and the LQ and the HQ models. As can been seen, both the LQ and HQ models give very good agreement with KS-DFT and perform significantly better than the other models. These results show that for this class of relatively weak pseudopotentials the OF treatment of the KP is quite satisfactory. To test of the transferability of the present AILPS, we also performed calculations for positive ions. Fig. 8 shows that the ionization energies of various atoms calculated using KS-DFT and the full atomic densities and those from the KS-DFT using the valence densities with AILPS are very similar. Therefore, the present AILPS are quite transferable to these positive ions. The ionization energy for models using the AILPS is obtained by subtracting the valence energies for systems with Nv and Nv − 1 electrons. Since the core electrons are assumed to be unaltered in different chemical envi- Table III: The total valence energy Ev[ρv] using the KS method, the LQ and HQ models, the CAT model, and the TFλW models. MAE, the mean absolute error (relative to the KS method) of various OF models are given at the bot- tom of their respective columns. Parameters used in Eq. (40) for such systems are given in Table IV. KS LQ HQ CAT TF1/5W TFW Be −0.9914 −0.8955 −0.8950 −0.9583 −1.214 −0.7786 C −6.134 −6.080 −6.100 −6.345 −7.761 −5.266 N −11.04 −11.06 −11.09 −11.32 −13.93 −9.462 O −18.01 −18.09 −18.09 −18.19 −22.44 −15.30 Si −3.771 −3.738 −3.750 −3.869 −4.467 −3.350 P −6.474 −6.432 −6.455 −6.582 −7.385 −5.756 S −10.20 −10.10 −10.14 −10.24 −11.27 −9.023 Ar −21.37 −20.84 −20.91 −20.85 −22.40 −18.56 MAE 0.119 0.103 0.184 1.610 1.312 ronments, the atomic core energy is a constant that can- cels here or in other similar applications to molecules and solids. Limitations of the LQ and HQ models are more evident here, but they do capture the overall periodicity of the ionization energies well, and perform significantly better than the other models. VI. CONCLUSION In summary, we propose two nonlocal OF KPs that satisfy exact limits for small and large wavevector per- turbations and reproduce the exact LR function in the homogeneous limit. These are the same limits that sev- eral current KEDFs are designed to satisfy. However, because of the more local nature of the KP, it is much easier to satisfy these conditions for the KP than for the KEDF, and there may be other physical and technical advantages arising from use the more local KP. In general, there is no reason to believe that any LR- based OF-DFT should work well for arbitrary systems where the model potentials are far beyond the LR regime. However, most chemical processes involve changes of va- lence electron densities, which can often be described by a weak AILPS. Thus the use of a LR-based OF-KP to- gether with AILPS for atomic systems at least seems well justified. The small and relatively slowly varying ρv(r) also provides some justification for our use of the local FWV kF (r) in Eq. (33). When the AILPS is used, the valence densities given by the LQ and HQ KPs are close to those given by the KS method. Thus the particular integration pathway used to get the total energy value becomes unimportant. The simple pathway in Eq. (8) is especially useful, since no coupling parameter integration is needed. The proposed models are not only conceptually simple, but also exact for a model system with a weak poten- tial and a slowly-varying density. The appearance of the atomic shell structure was found to be very sensitive to the accuracy of the proposed KPs. The LR-based LQ and Figure 8: (Color online) Ionization energies (shown in Hartree) of the first and the second row atoms using the full KS method, and various models using AILPS. The mean abso- lute errors (relative to the full KS method) of various models using AILPS are: KS (0.1 eV), LQ (1.8 eV), HQ (2.1 eV), CAT (3.9 eV), TFW (3.1 eV), and TF1/5W (4.3 eV). Ioniza- tion energies using the TF1/5W model are not shown in the figure due to its relatively poor performance. 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 KSfull HQ KPs give at best only qualitative indications of shell structure for full atomic systems, though total energies are surprisingly good. Still better results for atoms and ions can be found by focusing on the valence density as determined by a relatively weak AILPS. While these re- sults seem promising, improved KPs are needed and fur- ther investigation is required to see if these ideas can be usefully applied to other relevant systems like molecules and solids. Some initial results along these lines will be reported elsewhere. Acknowledgments This work has been supported by the NSF Grant CHE01-11104, and by the NSF-MRSEC at the University of Maryland under Grant No. DMR 00-80008. J.D.C. acknowledges the support from the UMCP Graduate School Fellowship, the IPST Alexander Fellowship, and the CHPH Bolck Grant Supplemental Fellowship. We are grateful to Prof. Emily Carter and members of her group for many helpful discussions and comments on an earlier version of this paper. Table IV: Parameters used in Eq. (A.1) for the target valence density of various atoms. Here, p = q = 6, and t = 0.1 are used for all systems. The Vps(r) generated from these parameterized ρ̃v(r) can then be used in OF-DFT. Nv a0 a1 a2 rc Li 1 2.983 × 10−4 0.05260 −6.560 × 10−3 2.135 Be 2 0.02655 0.7078 −0.2130 1.370 B 3 0.8032 5.855 −3.531 0.9714 C 4 11.53 30.17 −31.20 0.7429 N 5 98.80 112.8 −180.2 0.5978 O 6 592.7 338.9 −779.8 0.4981 F 7 2750 871.4 −2744 0.4256 Ne 8 1.052 × 104 1993 −8267 0.3707 Na 1 5.234 × 10−5 9.840 × 10−3 −6.610 × 10−4 2.904 Mg 2 9.805 × 10−4 0.04445 −5.002 × 10−3 2.233 Al 3 6.164 × 10−3 0.1273 −0.02056 1.861 Si 4 0.02942 0.3119 −0.06861 1.593 P 5 0.1149 0.6864 −0.1979 1.390 S 6 0.3841 1.390 −0.5100 1.231 Cl 7 1.133 2.632 −1.201 1.103 Ar 8 3.018 4.714 −2.624 0.9985 Appendix: AB INITIO LOCAL PSEUDOPOTENTIALS As discussed above, our proposed target valence den- sity for atoms has the following form: ρ̃v(r) = tρKS(rc) + a0r q exp[−rp(a r2)] r ≤ rc ρKS(r) r > rc (A.1) Here p and q are taken as even integers. The larger they are, the smaller and more slowly varying is the va- lence density near r = 0 but the sharper is the peak near the core radius rc. As a compromise, we take here p = q = 6, which generates relatively slowly varying local pseudopotentials Vps(r). For applications in different en- vironments, such as molecules or crystals, the core size rc has to be small to maintain transferability of the atomic core density. For this reason, we force ρ̃v(r = 0) to be small by taking a small t. If t = 0, the strict vanish- ing of the valence density near the nucleus would require a very repulsive Vps, which is certainly undesirable for the LR-based OF-DFT. However, if t is too large, there will exist a long oscillatory tail outside the core in the corresponding Vps(r). This is an undesirable feature for transferability to other environments, as will be discussed below. These two points constrain the value of t and we use here t = 0.1 for all the atomic systems considered. The four parameters a0, a1, a2, and rc are determined by requiring continuity of the function ρ̃v(r) and its first two derivatives at r = rc, and by satisfying the normal- ization condition: Nv = 4π ρ̃v(r)r 2dr. (A.2) Here we used the standard noble gas cores to determine Figure 9: The inverse KS procedure generates very small oscillations in the tail of rVps(r) for Si (shown here) and other atoms. The two points where rVps(r) = −4 for Si are r1 = 2.336 and r2 = 4.576 or r = 2.139. The arrow indi- cates the location of r2. To achieve good transferability (see text), this rVps(r) is modified by setting rVps(r) = −4 when r ≥ r2 = 4.576. See Fig. 5 for a large scale view. -4.03 -4.02 -4.01 -3.99 -3.98 -3.97 -3.96 -3.95 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 Nv = N −Nc, though other choices could in principle be made. To construct our local pseudopotential Vps(r) for atoms, we first solve the KS equations for an atom with the full Coulomb potential. With the KS density ρKS(r) and Nc determined, this construction ensures that as r → rc, the associated core density smoothly approaches zero as O(| r − rc | After generating the parameterized target valence den- sity ρ̃v(r), the set of inverse-KS equations are solved to obtain the corresponding one-body screened poten- tial Vscr(r). In principle, the AILPS Vps(r) is then given by Eq. (41). Using this, we find with an acceptable Nc that essentially the same density profiles outside rc are predicted by the LQ and HQ models for a wide range of choices of p, q, and t. This would be expected if most fea- tures of the resulting set of model potentials are within the different regimes accurately described by the OF KP. These results clearly show that OF-DFT can give accu- rate results for this class of physically relevant and rela- tively weak and slowly varying model potentials. However, we found that the rVps(r) constructed in this way for group III to group VIII elements deviates from −Zv outside the core by a small and long range oscilla- tion. For the 2s22px atoms, the maximum amplitude of the oscillation is about 0.18. For the 3s23px atoms, it ranges from 0.003 for Al to 0.037 for Ar. Similar oscil- latory tails were observed by Wang and Stott [25], and are thought to arise from the inability of a local pseu- dopotential to represent both s and p orbitals of the cor- responding nonlocal pseudopotentials. These oscillations are so small in magnitude that they have almost no effect on the density profile or energy of atoms, but they can cause transferability problems when used for solids [53]. Following Wang and Stott’s approach, we replace Vps(r) by −Zv/r at the largest point where rVps(r) = −Zv (see Fig. (9)). The new Vps(r) can then be used for both atomic and solid state calculations. 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In the density functional (DF) theory of Kohn and Sham, the kinetic energy of the ground state of a system of noninteracting electrons in a general external field is calculated using a set of orbitals. Orbital free methods attempt to calculate this directly from the electron density by approximating the universal but unknown kinetic energy density functional. However simple local approximations are inaccurate and it has proved very difficult to devise generally accurate nonlocal approximations. We focus instead on the kinetic potential, the functional derivative of the kinetic energy DF, which appears in the Euler equation for the electron density. We argue that the kinetic potential is more local and more amenable to simple physically motivated approximations in many relevant cases, and describe two pathways by which the value of the kinetic energy can be efficiently calculated. We propose two nonlocal orbital free kinetic potentials that reduce to known exact forms for both slowly varying and rapidly varying perturbations and also reproduce exact results for the linear response of the density of the homogeneous system to small perturbations. A simple and systematic approach for generating accurate and weak ab-initio local pseudopotentials which produce a smooth slowly varying valence component of the electron density is proposed for use in orbital free DF calculations of molecules and solids. The use of these local pseudopotentials further minimizes the possible errors from the kinetic potentials. Our theory yields results for the total energies and ionization energies of atoms, and for the shell structure in the atomic radial density profiles that are in very good agreement with calculations using the full Kohn-Sham theory.

Orbital-Free Density Functional Theory: Kinetic Potentials and Ab Initio Local Pseudopotentials Jeng-Da Chai∗1 and John D. Weeks1, 2 Institute for Physical Science and Technology, and Department of Chemistry and Biochemistry, University of Maryland, College Park, MD 20742 (Dated: November 1, 2018) In the density functional (DF) theory of Kohn and Sham, the kinetic energy of the ground state of a system of noninteracting electrons in a general external field is calculated using a set of orbitals. Orbital free methods attempt to calculate this directly from the electron density by approximating the universal but unknown kinetic energy density functional. However simple local approximations are inaccurate and it has proved very difficult to devise generally accurate nonlocal approximations. We focus instead on the kinetic potential, the functional derivative of the kinetic energy DF, which appears in the Euler equation for the electron density. We argue that the kinetic potential is more local and more amenable to simple physically motivated approximations in many relevant cases, and describe two pathways by which the value of the kinetic energy can be efficiently calculated. We propose two nonlocal orbital free kinetic potentials that reduce to known exact forms for both slowly varying and rapidly varying perturbations and also reproduce exact results for the linear response of the density of the homogeneous system to small perturbations. A simple and systematic approach for generating accurate and weak ab initio local pseudopotentials which produce a smooth slowly varying valence component of the electron density is proposed for use in orbital free DF calculations of molecules and solids. The use of these local pseudopotentials further minimizes the possible errors from the kinetic potentials. Our theory yields results for the total energies and ionization energies of atoms, and for the shell structure in the atomic radial density profiles that are in very good agreement with calculations using the full Kohn-Sham theory. I. INTRODUCTION Density-functional theory (DFT) has become one of the most powerful tools for investigating the electronic structure of large complex systems. In principle, as shown by Hohenberg and Kohn [1], the exact ground state energy of a system of N electrons can be formally written as a functional E[ρ] of only the electron density ρ(r), a function of three variables, and the external field Vext(r). Determining the energy and other ground state properties from such an approach could dramatically reduce the computational cost for large systems when compared with traditional quantum chemistry methods, which deal with wavefunctions involving coordinates of all N electrons [2, 3]. Kohn and Sham (KS) [4, 5] showed that E[ρ] can be usefully partitioned into the following set of terms: E[ρ] = Ts[ρ] + EH [ρ] + Exc[ρ] + ρ(r)Vext(r)dr. (1) Here Ts[ρ] is the noninteracting kinetic energy density functional (KEDF), which gives the kinetic energy of a model system of N noninteracting electrons in a self- consistent field chosen so that the ground state density ∗E-mail: jdchai@berkeley.edu. Present address: Molecular Foundry, Materials Sciences Division, Lawrence Berkeley National Laboratory, and Department of Chemistry, University of Califor- nia, Berkeley, California 94720 equals ρ(r), EH [ρ] ≡ ρ(r)ρ(r′) |r− r′| drdr′ (2) is the classical electron-electron potential energy (Hartree energy) and Exc[ρ] is the exchange-correlation energy (in- cluding the difference between the interacting and non- interacting kinetic energy and the difference between the quantum and classical electron-electron potential en- ergy). The last term on the right of Eq. (1) is the only term that depends explicitly on the external potential Vext(r). Atomic units are used throughout the paper. If all these functionals were known, then the density ρ(r) could be obtained from the variational principle (Eu- ler equation) associated with minimizing Eq. (1): µ = VTs(r; [ρ]) + Veff (r; [ρ]), (3) and the total energy of the inhomogeneous system could then be determined from the energy functional E[ρ]. All other physical quantities related to the ground-state den- sity could also be computed. Here µ is the chemical po- tential (the Lagrange multiplier associated with the nor- malization condition ρ(r)dr = N), and Veff (r; [ρ]) is an effective one-body potential defined by Veff (r; [ρ]) ≡ δρ(r) EH [ρ] + Exc[ρ] + ρ(r)Vext(r)dr = VH(r; [ρ]) + Vxc(r; [ρ]) + Vext(r), (4) where VH(r; [ρ]) ≡ δEH [ρ]/δρ(r) = ρ(r′) |r− r′| dr′ (5) http://arxiv.org/abs/0704.1878v1 is the Hartree potential, and Vxc(r; [ρ]) ≡ δExc[ρ]/δρ(r) is the exchange-correlation potential. Similarly we interpret VTs(r; [ρ]) ≡ δTs[ρ]/δρ(r) (6) as the kinetic potential (KP) arising from the KEDF [6]. Further progress requires an accurate determination of the noninteracting kinetic energy, whose magnitude is much larger than the exchange-correlation energy. The initial development of DFT as a practical computational method was made possible by KS’s realization that the numerical value of the noninteracting kinetic energy can be exactly calculated, not directly from the density itself using Ts[ρ], but by introducing a set of N one-electron wave functions (orbitals) satisfying the N coupled KS equations that describe the model system [4, 5]. Research could then focus on determining the remain- ing small term Exc[ρ]. Here even local density approx- imations have often proved useful. Through the efforts of many workers we now have generally accurate expres- sions for Exc[ρ]. Using these along with the KS orbitals to calculate the kinetic energy, one can accurately cal- culate both the total energy E[ρ] and the ground-state density ρ(r) for a wide variety of systems. However, the use of the KS orbitals usually generates a relatively expensive O(N3) scaling of computational cost with the number of electrons. While this scaling is much better than that of most standard methods that include correlation energy, calculations for large systems remain problematic. This remaining bottleneck could be removed if there were an accurate treatment of the kinetic energy in terms of the electron density only [2, 3, 7]. To that end there has been considerable effort in- vested in developing “orbital-free”density functional the- ory (OF-DFT) by making direct approximations for Ts[ρ] [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. While earlier simple local density approximations for Ts[ρ] like those used in the Thomas-Fermi (TF) model [21] are very inaccurate, there have been two main advances in recent work that offer prospects for significant improvements. The first is the introduction of nonlocal KEDFs that reproduce known exact results for very slowly varying or very rapidly varying fields and give the exact linear re- sponse (LR) of the density of the uniform model system to small perturbations. Similar ideas have been success- fully applied to classical nonuniform fluids [22]. The sec- ond advance is to focus not on the total density but on the smaller and more slowly varying valence electron den- sity as described by a weak pseudopotential acting only on the valence electrons. While conventional pseudopo- tential methods use orbitals, recently developed ab ini- tio local pseudopotential (AILPS) methods determine the unique local one-body potential producing a given target valence density by solving the KS equations inversely, us- ing the one-to-one mapping between density and poten- tials in DFT [23]. For OF-DFT with LR-based KEDF’s, the use of pseudopotentials not only can reduce the com- putational cost, but also can improve its accuracy, since the system will be closer to the LR regime where Ts[ρ] is designed to be accurate. [24, 25]. Indeed, very promising results using such OF-DFT methods have been obtained for a variety of nearly free-electron-like metals. However, existing KEDF’s have not yet achieved chem- ical accuracy for systems with localized and more rapidly varying electron densities like molecules or for covalent or ionic solids. The main problem is that the exact Ts[ρ] is highly nonlocal, and we have little idea of the func- tional form of the nonlocality for densities far from the LR regime. It has proved very difficult to understand what errors an approximate nonlocal Ts[ρ] will produce in the density as determined by the Euler equation with a general Vext(r). We explore here a different way to attack this basic problem. The exact Ts[ρ] can be formally obtained from VTs(r; [ρ]) by functional integration over density changes in all regions of space [6, 18]. Because of this integration Ts[ρ] is a more nonlocal functional of the density than is VTs(r; [ρ]). More detailed arguments arriving at this same conclusion have been recently presented [26]. Since most problems in devising accurate approximations for Ts[ρ] have arisen from the nonlocality, this suggests it could be worthwhile to try to develop approximations for the KP VTs(r; [ρ]) itself. To illustrate this point, Chai and Weeks [7] added a simple gradient correction to the original local TF KP for atoms [21], with a coefficient chosen to reproduce the exact boundary condition of exponential decay of the electron density far from the nucleus. Though quanti- tative results were not obtained, the resulting modified Thomas-Fermi (MTF) model gave energies for atoms and for closed-shell diatomic molecules that showed notable improvements when compared to the original TF and re- lated gradient corrected KEDF models. However, the local gradient correction used in the MTF KP cannot re- produce the oscillatory atomic shell structure and it does not satisfy the exact LR behavior in the homogeneous limit. It is clear that nonlocality even in the KP must be taken into account to achieve more accurate results. We propose here new nonlocal approximations for the KP using ideas similar to those employed for the nonlocal KEDFs. These new KPs satisfy the exact LR condition in the uniform limit, and reproduce known exact limiting forms of VTs(r; [ρ]) both for very slowly varying and very rapidly varying perturbations. As will become clear, the nonlocality in our KP is determined by the requirement that LR is exactly satisfied, and it is much easier to en- sure that LR holds for the KP than it is for analogous KEDF models. We believe this level of nonlocality in the KP may suffice in many cases when used in conjunction with AILPS methods to describe slowly varying valence density components closer to the LR regime. The remainder of this paper is organized as follow. Section II will discuss some general pathways connect- ing Ts[ρ] and VTs(r; [ρ]). Section III will describe limit- ing forms of the KEDF and KP for slowly varying and rapidly varying perturbations, and discuss LR theory, an exact theory for the response of density of the uniform electron gas to small perturbations. Section IV will de- velop two nonlocal KPs incorporating both the correct limiting forms of the exact KP and the exact LR of the free-electron gas. Section V will compare the numerical results of the present method for atoms with the KS-DFT and other KEDFs, both for all-electron calculations and for valence electrons using the AILPS. We find that the use of AILPS indeed reduces errors arising from nonlocal- ity in these approximate KPs or KEDFs, which give very accurate results for the relatively slowly varying valence densities. Our conclusions are given in Section VI. II. PATHWAYS FROM VTs(r; [ρ]) TO Ts[ρ] If Ts[ρ] is known, VTs(r; [ρ]) can be simply computed by functional differentiation. However, there is no unique way of determining Ts[ρ] from a given VTs(r; [ρ]). Many possible pathways can be used to construct Ts[ρ] by func- tional integration of VTs(r; [ρ]) [27, 28]. If the exact VTs(r; [ρ]) is used and the integration is carried out ex- actly, then all pathways would give the same exact result for Ts[ρ]. However, when an approximate VTs(r; [ρ]) is used, different pathways will give different results for the kinetic energy. But this “thermodynamic inconsistency” is small if reasonably good approximations are used, since the integration tends to smooth out local errors that may exist in the density [7, 28]. More problematic is the fact that most pathways re- quire additional results for partially coupled systems as the external field or density perturbation is gradually turned on, which adds to the computational burden. In particular, most earlier work has used a “potential en- ergy pathway”, where the external potential is scaled by a coupling parameter [7, 27]. The kinetic energy can then be found by subtracting the potential energy (calculated from the potential energy density functionals) from the total energy. However, this pathway is expensive, since one has to solve the Euler equation (3) for each partially coupled V λext(r) (with the same µ), to determine the cor- responding ρλ. A. Herring’s Pathway However, Herring showed there is a particular path- way arising from exact scaling relations between the non- interacting kinetic energy Ts[ρ] with respect to the coor- dinate r in ρ(r) where very simple results involving only the final density can sometimes be found [6, 18]. If the coordinate r is scaled to αr, the normalized scaled den- sity is ρα(r) = α 3ρ(αr). It is easy to show that the exact Ts[ρ] then obeys α2Ts[ρ] = Ts[ρα]. (7) For isolated systems, such as atoms and molecules, the density and its derivatives to all order vanish far from the nuclei. For such systems, when Eq. (7) is differentiated with respect to α, and the partial derivative is evaluated at α = 1, we find the formally exact result Ts[ρ] = VTs(r; [ρ])∇ · (rρ(r))dr. (8) Therefore, once the kinetic potential VTs(r; [ρ]) is known for some given ρ(r), the numerical value of Ts[ρ] can then be immediately determined from Eq. (8). Since there is no need to perform a coupling parameter integra- tion over the change of density or potential, this scheme is not only fast, but also numerically reliable. The final form of Eq. (8) is essentially the viral theorem, and is directly related to the force on molecules [29, 30, 31, 32]. Note that this simple and exact pathway holds only for the noninteracting Ts[ρ] [31], which again shows the virtues of the KS partitioning of the total energy. We will use Eq. (8) as the basic pathway to determine the numer- ical value of Ts[ρ] from a given approximate VTs(r; [ρ]) for most calculations in this paper. However, Eq. (8) does not hold for extended solid-state systems because of nonvanishing boundary terms, and thus far we have not found an exact and simple way of including them. Fortunately, there is another class of computationally efficient“density pathways”that can be used for extended systems, as we now show. Density pathways can also be used for atomic and molecular systems to check the accuracy of the VTs used, since results using the exact VTs would be independent of path [27, 28]. B. Density Pathways The change in the kinetic energy can be formally re- lated to a coupling parameter integration, where the den- sity changes from some known value at λ = 0 to the final density at λ = 1: Ts[ρ] = Tλ=0 + drVTs(r; [ρλ]) ∂ρλ(r) In most cases a simple linear density pathway will suffice. Here the density ρ(r) is linearly scaled by a coupling pa- rameter λ from some uniform reference density ρ0 natu- rally chosen to be the uniform electron density N/V in extended systems: ρλ(r) = ρ0 + λ[ρ(r) − ρ0]. (10) Then Eq. (9) becomes Ts[ρ] = Tλ=0 + drVTs(r; [ρλ])[ρ(r) − ρ0] (11) Here Tλ=0 is the kinetic energy of the uniform system, i.e., the Thomas-Fermi kinetic energy TTF [ρ0]. For ex- tended systems, where the Herring’s pathway cannot be used, this density pathway appears to be a good way to compute T . Other density pathways, like the square-root pathway introduced by Chen and Weeks [28] to describe nonuniform hard sphere fluids, can be defined, and have proved useful in certain applications, but we do not con- sider them here. Note from Eq. (10) that ρλ(r) depends only on the fi- nal density, so evaluation of Eq. (11) is straightforward and this pathway is computationally efficient. Unlike the potential energy pathway where the external potential is scaled, there is no need to solve the Euler equation (3) for its corresponding external potential V λext(r) at each λ. However, for isolated systems, where ρ0 = 0, this path- way is likely to be less accurate than Herring’s pathway, since it does not automatically satisfy the virial theorem. III. EXACT LIMITS AND LINEAR RESPONSE THEORY Although the exact Ts[ρ] is still unknown, several limit- ing forms have been discovered for particular density dis- tributions. These provide important cornerstones that can be used to construct accurate KEDFs and KPs in many cases, as will be seen below. In particular, the Thomas-Fermi (TF) KEDF [21] is known to be exact for a uniform system: TTF [ρ] = CF ρ5/3(r)dr, (12) where CF = (3π2)2/3. The TF KEDF TTF [ρ] is de- rived by local use of uniform free-electron gas model, and is exact for a system with an infinite number of electrons. The corresponding expression for the TF KP is VTF (r; [ρ]) ≡ δTTF [ρ]/δρ(r) = 2/3(r). (13) This depends only on the local value of ρ2/3(r) and thus formally is more local than the TF KEDF, whose func- tional dependence on ρ involves the density at all r. Of course in this simple case the functional integration of Eq. (13) can be carried out exactly to yield Eq. (12), but this cannot be done in general and the nonlocality of Ts[ρ] has proved problematic. Results for nonuniform systems are best described in Fourier space. For a very slowly varying perturbation of the density, the second-order gradient expansion is exact [33]. It is easy to see that results correct to second order at small wavevectors are given by TTF 1 W [ρ] ≡ TTF [ρ] + TW [ρ], (14) where TW [ρ] ≡ |▽ρ(r)|2 dr (15) is the von Weizsäcker (W) KEDF [34]. TW [ρ] is exact for a system with one or two electrons, or where the density can be accurately described by a sin- gle orbital. Moreover it has been argued [8, 9, 10, 12] that TW [ρ] gives the correct leading order term for a rapidly varying perturbation with only high wavevector compo- nents and that the next order correction is reproduced TW− 3 TF [ρ] ≡ TW [ρ]− TTF [ρ]. (16) The W KP is VW (r; [ρ]) ≡ δTW [ρ]/δρ(r) = |▽ρ(r)|2 ρ2(r) ▽2ρ(r) If we represent the full density by an effective single or- bital function ψ(r), ρ(r) = |ψ(r)|2 (18) then the W KP can be written in a compact form that will later prove useful: VW (r; [ρ]) = − ▽2ψ(r) 2ψ(r) . (19) Finally, the linear response of the density of a uniform non-interacting electron gas with density ρ0 to a small perturbation δV (k) = ǫke ik·r is exactly known [35], δρ(k) = χL(q)δV (k). (20) q ≡ k/2kF (21) is a dimensionless wavevector, where kF ≡ (3π 1/3 (22) is the Fermi wavevector (FWV) and k ≡ |k|. The LR function χL(q) has the form χL(q) = − F−1L (q) 1− q2 1 + q , (23) where FL(q) ≡ 1− q2 1 + q has been called the Lindhard function [10]. It is known that the weak logarithmic singularity at q = 1 in F−1L (q) is responsible for Friedel oscillations, and may also be important for the appearance of atomic shell structure. This singularity further divides the Lindhard function into two branches in Fourier space: the low- momentum (q < 1 ) or the low-q (LQ) branch, and the high-momentum (q > 1) or the high-q (HQ) branch [10]. Figure 1: Linear response functions of a uniform system of noninteracting Fermions as given by the TF, W, and MTF (see Ref. [7]) models. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 exact The dimensionless response function arising from the TF KEDF is FTF (q) = 1, and that from the W KEDF is FW (q) = 3q 2 [36]. Clearly, no linear combination of the TF and the W KEDFs can reproduce the exact Lindhard function in Eq. (24). This has the following two limits [10], FL(q) = 1 + q2/3 +O(q4) q ≪ 1 3q2 − 3/5 +O(q−2) q ≫ 1 It should be noted that the expansions for both the low-q and high-q limits are correct to all orders in per- turbation theory, but valid only in the appropriate limits in Fourier space. On the other hand, the LR theory is valid for all wavevectors, but is only accurate for small perturbations. Therefore, the regime where the response functions of the two limiting KEDFs deviate from the exact LR function gives an indication of the range of wavevectors where the two limiting forms are inaccurate. As shown in Fig. 1, the response function F−1TF (q) has no momentum dependence and is only exact at q = 0. The response function F−1W (q) is exact asymptotically at high q, and remains fairly accurate for q & 2, but is di- vergent in the low-q branch, and fails completely for the nearly uniform electron gas. In contrast, the MTF model [7] gives a reasonably accurate average description of the exact response function, especially in the important re- gion near the singularity at q = 1. IV. CONSTRUCTION OF NONLOCAL KINETIC POTENTIALS A. Kinetic Energy Density Functionals TTFλW [ρ] and TWλTF [ρ] Simple linear combinations of the two limiting KEDF’s in Eqs. (12) and (15), such as the TFλW KEDF [33, 37, 38, 39] TTFλW [ρ] ≡ TTF [ρ] + λTw[ρ] (26) and the WλTF KEDF [40, 41, 42, 43] TWλTF [ρ] ≡ TW [ρ] + λTTF [ρ] (27) have been widely studied for several decades. The value of the parameter λ was either determined empirically for getting good atomic energy or obtained by some semi- classical arguments. The advantage of these approaches is the ability to gen- erate a family of simple KEDF’s easily. It has been shown empirically that the TF1/5W model can give good val- ues for atomic energies, but the predicted density profiles are generally not very accurate, both near and far away from the nucleus. The TTFλW [ρ] and TWλTF [ρ] function- als give the correct leading term in the density response to a slowly-varying perturbation and a rapidly-varying perturbation respectively, and with particular choices of λ as in Eqs. (14) and (16) they can reproduce the next order term. Unfortunately, they then will have an in- correct leading term in the opposite limit, unless λ = 1. However it has been shown that TTFW [ρ] with λ = 1 always overestimates the exact Ts[ρ] for various systems [10]. Finally, none of these functionals can reproduce the exact response function FL(q) in the homogeneous limit. Since these models fail to satisfy all the known limits, and nonlocality in Ts[ρ] is not correctly described, it is also not surprising that atomic shell structure is missing in these approaches. B. Combining TF and W Kinetic Potentials We argue that it may be more profitable to take advan- tage of known limiting forms of the KP, rather than the KEDF, and develop approximations for the more local VTs(r; [ρ]) directly. Again we can rely on known results in the linear response regime when the density variations are not too large. From Eqs. (14) and (16), the following linear combina- tions of the TF KP and the W KP in Eqs. (13) and (17) can reproduce exact results to second order for very small and very large wavevector perturbations respectively: VTs(r; [ρ]) ≈ VTF (r; [ρ]) + VW (r; [ρ]) q ≪ 1 VW (r; [ρ])− VTF (r; [ρ]) q ≫ 1 Since VTF (r; [ρ]) and VW (r; [ρ]) are the only components up to second order of the two exact limiting forms of the KP, we can combine them in analogy to the TFλW and WλTF models and arrive at generalized KPs. However, instead of combining them using a fixed pa- rameter λ, it seems natural to represent them in Fourier space and allow a wavevector dependence in λ = λ(q) to connect the limiting forms. The λ(q) can then be chosen in a very simple way so that the exact LR function is Figure 2: Weight function f̂(q) for the HQ and LQ KPs. 0 0.5 1 1.5 2 2.5 3 reproduced for a uniform system with density ρ0. In this way the LR function bridges the exact limits at large and small wavevectors, and if the theory is applied to weak perturbations in the linear response regime for interme- diate wavevectors we can expect very accurate results. Here, we derive such generalized KPs based on the KP for the WλTF model. C. HQ Kinetic Potential In analogy to the WλTF model in Eq. (27) we look for a kinetic potential of the form V 0HQ(k) = VW (k) + λHQ(q) VTF (k) = VTF (k) + VW (k) + f̂(q) VTF (k), (29) where q = k/2kF (30) is a dimensionless wavevector normalized by the FWV kF in Eq. (22) of a uniform reference system with density ρ0 and f̂(q) = λHQ(q) − 1. The superscript 0 in V indicates use of a uniform reference system. For a small perturbation, we can linearize the V 0HQ(k) in Eq. (29). Requiring that it satisfy LR exactly then determines the weight function f̂(q) as f̂(q) = FL(q)− 3q 2 − 1. (31) See Fig. 2. We refer to Eq. (29) with Eq. (31) as the HQ KPmodel. It reproduces the correct high-q limit in Eq. (28) up to the second order. However, unlike Eq. (16), it also satisfies the correct low-q limit to leading order and gives exact results for all q in the linear response regime. Inverse Fourier transform of Eq. (29) then gives V 0HQ(r; [ρ], kF ) = VTF (r; [ρ]) + VW (r; [ρ]) f(|r− r′|; kF ) VTF (r ′; [ρ])dr′. This expression is directly useful for extended systems where a reasonable ρ0 can be defined. For isolated sys- tems such as atoms and molecules where the density van- ishes far from the nuclei, it seems natural to replace kF in Eq. (32) by the local Fermi wavevector (LFWV) kF (r) ≡ (3π 2ρ(r))1/3, (33) though errors may be introduced for rapidly varying den- sity distributions. Using Eq. (13), this yields the general form of our proposed HQ kinetic potential: VHQ(r; [ρ], kF (r)) = VTF (r; [ρ]) + VW (r; [ρ]) f(|r− r′|; kF (r)) ρ 2/3(r′)dr′. Note that the last term in Eq. (34) is most easily com- puted in Fourier space as 3(2π)3 f̂(k/2kF (r))ρ 2/3(k)e−ik·rdk, (35) As can been seen in Eq. (31), the weight function f̂(k/2kF (r)) is determined analytically. Unlike the LR- based KEDF approaches, no first-order differential equa- tion is needed to solve for the weight function in Fourier space. For atomic systems where the LFWV is used, the convolution in Eq. (35) must be carried out numerically, which will lead to a quadratic scaling of the HQ model (and related LQ model described below) in the numberN of electrons. For extended systems where one can expand about the local density ρ0, one can use fast Fourier trans- forms (FFT’s) for a much more efficient computation of this integral [44]. D. LQ Kinetic Potential In analogy to the TFλW model in Eq. (26), we could similarly generate a KP that is accurate to second order at low q while still reproducing the leading term at high q. However this is numerically less useful because the analogue of Eq. (35) involves the Fourier transform of VW (r ′; [ρ]), which cannot be simply expressed in terms of the density. Instead, by empirically taking a properly chosen component of the density outside the integral we find that Eq. (34) can be modified to produce a new LQ KP that is accurate to second order at low q and first order at high q [45]: VLQ(r; [ρ], kF (r)) = VTF (r; [ρ]) + VW (r; [ρ]) 1/6(r) f(|r− r′|; kF (r)) ρ 1/2(r′)dr′ Extending these ideas we have constructed a modified KP that satisfies LR everywhere and is accurate to sec- ond order at both low and high q [45]. However the functional form is much more complicated, and little ad- ditional accuracy is gained from the improved behavior at very small or very large wavevectors, since all forms use LR to interpolate for intermediate wavevectors, and this is where most errors arise in practice. Thus we will report results here only for the HQ and LQ models. V. RESULTS FOR ATOMS USING THE HQ AND LQ MODELS For completeness and to compare to earlier work, we first briefly discuss all-electron calculations using the pro- posed HQ and LQ model KPs and the full atomic poten- tials. We then describe results using AILPS methods. These are compared with the KS-DFT, the TFλW mod- els, and the CAT model introduced by Chacón, Alvarel- los, and Tarazona [12, 13, 14]. The CAT model is a LR-based KEDF method, which gives some indication of shell structure. We employed the latest version, which uses a nonlocal two-body Fermi wave vector with a pre- scribed functional form depending on an empirical pa- rameter β = 1/2 (defined in Eq. (3) of Ref. [14]). This caused the numerical calculations [46] to be consider- ably more costly than those of the LQ or HQ models, which used the local Fermi wave vector as in Eq. (33). All calculations are spin-restricted and use the local den- sity approximation (LDA) [47, 48, 49] for the exchange- correlation functional. A. All-electron calculations All-electron calculations consider the density response to the large and rapidly varying nuclear potential. Since the system is far from the linear response regime, quanti- tative results from the HQ and LQ models (or from LR- based KEDF methods) cannot be expected. However, by incorporating exact results for very large and very small wavevectors, these models do correct major deficiencies of the purely local TF model (which, e.g., predicts an infinite density at the nucleus!) and even give some qual- itative indications of atomic shell structure. The numerical method use the Pauli kinetic potential V P (r; [ρ]) [50, 51], defined as V P (r; [ρ]) ≡ VTs(r; [ρ]) − VW (r; [ρ]). (37) Since VW (r; [ρ]) is the exact KP for a system where the density can be accurately described by a single orbital, if V P (r; [ρ]) is omitted, one would essentially obtain the ground state density of the corresponding Boson system, where all the electrons are in the same orbital. If we represent the full density by a single orbital function ψ(r), so that ρ(r) and VW (r; [ρ]) can be written in the forms of Eq. (18) and Eq. (19) respectively, we can then combine V P (r; [ρ]) with the one-body potential Veff (r; [ρ]) in Eq. (3), and derive a Schrödinger-like equation for the Bose orbital ψ(r), ▽2 + Veff (r; [ρ]) + V P (r; [ρ]) ψ(r) = µψ(r). (38) Table I: Atomic energy E using the KS, LQ, HQ, CAT and the TFλW models in all-electron calculations. MAPE, the mean absolute percentage error (relative to the KS method) of various OF models are given at the bottom of their respective columns. KS LQ HQ CAT TF1/5W TFW He −2.834 −2.565 −2.437 −2.675 −2.911 −1.559 Ne −128.2 −134.3 −126.6 −126.2 −129.5 −86.40 Ar −525.9 −545.9 −512.2 −515.1 −526.2 −375.5 Kr −2750 −2805 −2621 −2712 −2748 −2099 Xe −7229 −7306 −6844 −7141 −7214 −5701 Be −14.45 −14.39 −13.64 −14.11 −14.71 −8.699 Mg −199.1 −207.9 −195.7 −195.2 −200.0 −136.4 C −37.42 −38.97 −36.85 −37.25 −38.41 −24.01 N −54.02 −56.71 −53.59 −53.84 −55.39 −35.33 O −74.47 −78.39 −74.02 −74.08 −76.11 −49.39 Si −288.2 −300.4 −282.5 −282.2 −288.9 −200.5 P −339.9 −354.1 −332.7 −332.8 −340.6 −238.3 S −396.7 −412.8 −387.7 −388.5 −397.3 −279.9 MAPE 4.06% 3.42% 1.82% 1.10% 32.0% Table II: Electron density at the nucleus ρ(0), using the KS, LQ, HQ, CAT and the TFλW models in all-electron calcula- tions. MAPE, the mean absolute percentage error (relative to the KS method) of various OF models are given at the bottom of their respective columns. KS LQ HQ CAT TF1/5W TFW He 3.525 3.088 2.742 3.600 18.23 0.9515 Ne 614.5 576.6 517.6 613.2 2596 169.6 Ar 3819 3642 3282 3812 1.548 × 104 1093 Be 34.86 30.49 27.17 33.75 158.2 8.952 Mg 1086 1024 920.9 1083 4519 303.0 C 126.0 113.3 101.2 122.8 547.9 33.07 N 203.9 185.6 166.1 200.0 876.7 54.24 O 308.6 284.1 254.6 304.6 1317 83.19 Si 1754 1662 1495 1749 7218 493.9 P 2173 2062 1857 2167 8901 614.5 S 2654 2523 2272 2647 1.083 × 104 753.5 MAPE 7.61% 17.2% 1.14% 331% 72.6% In other words, Veff (r; [ρ]) + V P (r; [ρ]) is now the one- body effective potential for the corresponding Boson sys- tem with the same electron density. This reduction of an N -fermion problem to a Boson form is widely imple- mented in OF-DFT due to its numerical stability and its easy implementation using existing KS-DFT codes [10]. The associated Pauli potentials for the HQ and LQ models are immediately obtained by subtraction of the W KP from Eq. (34) and Eq. (36) respectively. The stan- dard finite difference method for solving Euler equations for the TFW models [52] are implemented for the LQ and HQ models, and the nonlocal terms are evaluated by Fourier transforms. The choices of radial grids for both of the real and Fourier space and other detailed numerical methods are given in Ref. [7]. The kinetic energy for HQ and LQ models is computed using the Herring pathway in Eq. (8). As shown in Table I, the atomic energy calculated by Figure 3: (Color online) Radial density r2ρ(r) of the Kr atom using the KS method, the LQ and HQ models, the CAT model, and the TFλW models (see the inset) with the full nuclear potential. 0 0.5 1 1.5 2 0 0.5 1 1.5 2 TF1/5W the energy-optimized TF1/5W model is very close to the KS-DFT, and outperforms all the LR-based models, and other TFλW models. In Table II, we compare the elec- tron density at the nucleus ρ(0) for various models. The TF1/5Wmodel overestimates ρ(0) by about a factor of 4, while the TFW model underestimates it by about 30%. The predicted values of ρ(0) for all the LR-based mod- els are very close to the KS results, and are much better than the TFλW models. In Fig. (3), we compare the radial density distribu- tion r2ρ(r) of the LQ and HQ models to that predicted by other theories for the Kr atom. Both the TF1/5W and TFW models predict smooth and structureless ra- dial density profiles. Using the full Coulombic potential, all the LR-based models can predict an incipient shell structure for heavy atoms (Z & 30), and these results are typical. Since the potential is certainly far beyond the LR regime, these qualitative results with some sug- gestion of shell structure are about as good as could be hoped for. The surprisingly good total energies given in Table I for the TF1/5W model and the LR-based models shows that averaged thermodynamic properties are less sensitive to errors in the KP than is the density profile. The difference in the results for the LQ, HQ and CAT models indicates that the LR-based OF theory is being used outside its range of validity. As shown below, we gain a significant improvement by using the AILPS to deal with these difficulties. B. Ab initio local pseudopotential calculations As discussed earlier, the use of pseudopotentials in non- local LR-based OF-DFT can improve the accuracy of the theory because the weaker pseudopotential is more nearly in the LR regime, where the theory is designed to be ac- Figure 4: The smooth target density ρ̃v(r) from Eq. (40), with parameters given in Table IV for the Si pseudoatom used in the inverse-KS process, and the valence density ρv(r) pre- dicted by the LQ and HQ models using the Vps(r) (see Fig. 5) corresponding to ρ̃v(r). The arrow indicates the location of rc. 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.5 1 1.5 2 2.5 3 smooth Figure 5: The AILPS Vps(r) for Si generated by the target density ρ̃v(r) in Fig. 4. 0 0.5 1 1.5 2 2.5 3 curate. Our proposed HQ and LQ models can be used with any existing AILPS. However, since we want to as- sess the performance of these models for a wide class of atomic systems, we describe here a new method for de- termining reasonable AILPS for general atomic systems. These pseudopotentials will be used in all our calculations and can be transfered to other molecular and solid state environments, but we expect (and find in cases where comparison can be made) little change if other reason- able AILPS are used. Because of the one-to-one mapping between the effec- tive one-body potential acting on a system of N electrons and the electron density in the ground-state configura- tion, it is possible to obtain a unique local one-body po- tential that generates a given target density ρ̃(r) by using Figure 6: (Color online) Radial valence density r2ρv(r) of the Si atom using the KS method and various models using AILPS. Parameters used for constructing this reference sys- tem are shown in Table IV. The arrow indicates the location of rc. Inset: The corresponding radial total density r 2ρ(r), which is dominated by the core component for r < rc. 0.05 0.15 0.25 0.5 1 1.5 2 2.5 3 TF1/5W 0 0.5 1 1.5 2 a KS orbital-based method in an inverse way [23]. To con- struct an AILPS for a given atom we separate the total electron density ρ(r) into a “core density” ρ̃c(r), which is supposed not to vary significantly in other molecular or solid state environments, and the target“valence density” ρ̃v(r) where ρ(r) = ρ̃v(r) + ρ̃c(r). (39) Because DFT requires only the electron density, we can take a more general view of what is meant by the core and valence components than is used in most orbital-based methods. Here, we directly construct a smooth target valence density for the Nv = N − Nc valence electrons, with Nc chosen to be the number of electrons in the noble gas configuration. Our proposed target valence density ρ̃v(r) for atoms equals the full KS density ρKS(r) outside a core of radius rc, and is designed to be small and slowly varying inside rc. The functional form we take is ρ̃v(r) = tρKS(rc) + a0r q exp[−rp(a r2)] r ≤ rc ρKS(r) r > rc Figure 4 gives an example of ρ̃v(r) for Si that will be discussed in more detail below. We find most results are insensitive to the details of our fitting procedure. Param- eter values for a variety of atomic systems are given in the Appendix in Table IV along with the physical and technical considerations that guided our choice of this particular form for ρ̃v(r). The Appendix also discusses some of the general issues that arise in using these atomic AILPS in other environments. The local pseudopotential is directly related to the effective one body potential that reproduces ρ̃v(r) ex- actly when using the full KS theory. Following previous work [24, 25], for a given ρ̃v(r), the inverse-KS equations are solved to get the effective one-body screened poten- tial Vscr(r). The desired ab initio local pseudopotential Vps(r) is then obtained by subtracting the Hartree po- tential and the exchange-correlation potential: Vps(r) = Vscr(r)− VH(r; [ρ̃v])− Vxc(r; [ρ̃v]). (41) This relatively expensive procedure to determine Vps(r) requires the use of orbitals. However it needs to be done only once for each atom, and the resulting Vps(r) can then be used in a variety of other environments if the atomic core densities remain essentially constant. Once suitable Vps(r) have been determined (by this or other means) they can be incorporated in Vext(r) in dif- ferent ways, depending on the particular system of inter- est. OF-DFT theory can then be used to determine the valence density ρv(r) in direct analogy to the all-electron calculations for the full atomic potential in Eq. (38): ▽2 + VH(r; [ρv]) + Vxc(r; [ρv]) +Vext(r) + VP (r; [ρv])}ψv(r) = µψv(r) (42) where ρv(r) = |ψv(r)| 2. (43) A simple and direct test of OF-DFT is to use Eqs. (42) and (43) for the same atomic system for which Vps(r) was constructed. Thus we take Vext(r) = Vps(r) for a given atom as input data in Eq. (42). The valence density ρv(r) predicted by the OF KPs is determined from Eqs. (42) and (43), and can be directly compared to the exact target density ρ̃v(r) for this atomic system given by the full KS theory. This is illustrated in Fig. 4, which shows the input target valence density ρ̃v(r) for Si used in the inverse- KS process. The total KS density ρKS(r) equals ρ̃v(r) for r ≥ rc, indicated by the arrow in Fig. 4, and then increases rapidly for r < rc, reaching a very large value at the nucleus, ρKS(0) = 1754. In contrast, the proposed target valence density ρ̃v(r) remains small and relatively slowly varying inside the core, with ρ̃v(0) = 0.005621. Also shown in Fig. 4 are the predicted valence densities ρv(r) for Si given by the LQ and HQ models. Because of the relatively weak Vps(r) and slowly varying valence den- sity, both the LQ and HQ KP models predict results very close to those given by the exact KS treatment of the ki- netic energy, and perform markedly better than they did for the all-electron calculations using the full Coulomb potential. Fig. 5 shows the corresponding AILPS gener- ated by the inverse KS procedure. It is much smaller in the core region than the full atomic potential and more likely to be accurately treated by LR-based methods. The radial valence density r2ρv(r) of the Si and Ar atoms predicted by the various methods are shown in Figs. 6 and 7. The consistency of our OF theory when pseudopotentials are used is illustrated by the similar- ity of the density predicted by the HQ and LQ models. Figure 7: (Color online) Same as in Fig. 6 but for the Ar atom. 0.5 1 1.5 2 2.5 3 TF1/5W 0 0.5 1 1.5 2 The slight deviations from the KS-DFT results for Ar in the valence region in Fig. 7 are among the largest we en- countered for all atoms tested, and could be due to the relatively large number of valence electrons (8) compared to core electrons (10). Further errors may arise from the LFWV approximation in Eq. (33). Once ρv(r) has been determined using OF-DFT, it can be added to the known input core density ρ̃c(r) to obtain the predicted total density, since the basic assumption of our AILPS is that the core density remains unchanged in different chemical environments. The core density is defined in Eqs. (39) and (40). The inset in Fig. 6 shows that the HQ and LQ treatment of the valence density for the Si atom does not produce noticeable errors in the total density, as expected, and the shell structure remains in excellent agreement with the full KS-DFT calculations. As a further test of OF-DFT we can compare the en- ergy for the valence density given by the various methods to the exact valence energy for the target valence density determined by the inverse KS method. Table III gives the valence energy values for the KS-DFT, and the LQ and the HQ models. As can been seen, both the LQ and HQ models give very good agreement with KS-DFT and perform significantly better than the other models. These results show that for this class of relatively weak pseudopotentials the OF treatment of the KP is quite satisfactory. To test of the transferability of the present AILPS, we also performed calculations for positive ions. Fig. 8 shows that the ionization energies of various atoms calculated using KS-DFT and the full atomic densities and those from the KS-DFT using the valence densities with AILPS are very similar. Therefore, the present AILPS are quite transferable to these positive ions. The ionization energy for models using the AILPS is obtained by subtracting the valence energies for systems with Nv and Nv − 1 electrons. Since the core electrons are assumed to be unaltered in different chemical envi- Table III: The total valence energy Ev[ρv] using the KS method, the LQ and HQ models, the CAT model, and the TFλW models. MAE, the mean absolute error (relative to the KS method) of various OF models are given at the bot- tom of their respective columns. Parameters used in Eq. (40) for such systems are given in Table IV. KS LQ HQ CAT TF1/5W TFW Be −0.9914 −0.8955 −0.8950 −0.9583 −1.214 −0.7786 C −6.134 −6.080 −6.100 −6.345 −7.761 −5.266 N −11.04 −11.06 −11.09 −11.32 −13.93 −9.462 O −18.01 −18.09 −18.09 −18.19 −22.44 −15.30 Si −3.771 −3.738 −3.750 −3.869 −4.467 −3.350 P −6.474 −6.432 −6.455 −6.582 −7.385 −5.756 S −10.20 −10.10 −10.14 −10.24 −11.27 −9.023 Ar −21.37 −20.84 −20.91 −20.85 −22.40 −18.56 MAE 0.119 0.103 0.184 1.610 1.312 ronments, the atomic core energy is a constant that can- cels here or in other similar applications to molecules and solids. Limitations of the LQ and HQ models are more evident here, but they do capture the overall periodicity of the ionization energies well, and perform significantly better than the other models. VI. CONCLUSION In summary, we propose two nonlocal OF KPs that satisfy exact limits for small and large wavevector per- turbations and reproduce the exact LR function in the homogeneous limit. These are the same limits that sev- eral current KEDFs are designed to satisfy. However, because of the more local nature of the KP, it is much easier to satisfy these conditions for the KP than for the KEDF, and there may be other physical and technical advantages arising from use the more local KP. In general, there is no reason to believe that any LR- based OF-DFT should work well for arbitrary systems where the model potentials are far beyond the LR regime. However, most chemical processes involve changes of va- lence electron densities, which can often be described by a weak AILPS. Thus the use of a LR-based OF-KP to- gether with AILPS for atomic systems at least seems well justified. The small and relatively slowly varying ρv(r) also provides some justification for our use of the local FWV kF (r) in Eq. (33). When the AILPS is used, the valence densities given by the LQ and HQ KPs are close to those given by the KS method. Thus the particular integration pathway used to get the total energy value becomes unimportant. The simple pathway in Eq. (8) is especially useful, since no coupling parameter integration is needed. The proposed models are not only conceptually simple, but also exact for a model system with a weak poten- tial and a slowly-varying density. The appearance of the atomic shell structure was found to be very sensitive to the accuracy of the proposed KPs. The LR-based LQ and Figure 8: (Color online) Ionization energies (shown in Hartree) of the first and the second row atoms using the full KS method, and various models using AILPS. The mean abso- lute errors (relative to the full KS method) of various models using AILPS are: KS (0.1 eV), LQ (1.8 eV), HQ (2.1 eV), CAT (3.9 eV), TFW (3.1 eV), and TF1/5W (4.3 eV). Ioniza- tion energies using the TF1/5W model are not shown in the figure due to its relatively poor performance. 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 KSfull HQ KPs give at best only qualitative indications of shell structure for full atomic systems, though total energies are surprisingly good. Still better results for atoms and ions can be found by focusing on the valence density as determined by a relatively weak AILPS. While these re- sults seem promising, improved KPs are needed and fur- ther investigation is required to see if these ideas can be usefully applied to other relevant systems like molecules and solids. Some initial results along these lines will be reported elsewhere. Acknowledgments This work has been supported by the NSF Grant CHE01-11104, and by the NSF-MRSEC at the University of Maryland under Grant No. DMR 00-80008. J.D.C. acknowledges the support from the UMCP Graduate School Fellowship, the IPST Alexander Fellowship, and the CHPH Bolck Grant Supplemental Fellowship. We are grateful to Prof. Emily Carter and members of her group for many helpful discussions and comments on an earlier version of this paper. Table IV: Parameters used in Eq. (A.1) for the target valence density of various atoms. Here, p = q = 6, and t = 0.1 are used for all systems. The Vps(r) generated from these parameterized ρ̃v(r) can then be used in OF-DFT. Nv a0 a1 a2 rc Li 1 2.983 × 10−4 0.05260 −6.560 × 10−3 2.135 Be 2 0.02655 0.7078 −0.2130 1.370 B 3 0.8032 5.855 −3.531 0.9714 C 4 11.53 30.17 −31.20 0.7429 N 5 98.80 112.8 −180.2 0.5978 O 6 592.7 338.9 −779.8 0.4981 F 7 2750 871.4 −2744 0.4256 Ne 8 1.052 × 104 1993 −8267 0.3707 Na 1 5.234 × 10−5 9.840 × 10−3 −6.610 × 10−4 2.904 Mg 2 9.805 × 10−4 0.04445 −5.002 × 10−3 2.233 Al 3 6.164 × 10−3 0.1273 −0.02056 1.861 Si 4 0.02942 0.3119 −0.06861 1.593 P 5 0.1149 0.6864 −0.1979 1.390 S 6 0.3841 1.390 −0.5100 1.231 Cl 7 1.133 2.632 −1.201 1.103 Ar 8 3.018 4.714 −2.624 0.9985 Appendix: AB INITIO LOCAL PSEUDOPOTENTIALS As discussed above, our proposed target valence den- sity for atoms has the following form: ρ̃v(r) = tρKS(rc) + a0r q exp[−rp(a r2)] r ≤ rc ρKS(r) r > rc (A.1) Here p and q are taken as even integers. The larger they are, the smaller and more slowly varying is the va- lence density near r = 0 but the sharper is the peak near the core radius rc. As a compromise, we take here p = q = 6, which generates relatively slowly varying local pseudopotentials Vps(r). For applications in different en- vironments, such as molecules or crystals, the core size rc has to be small to maintain transferability of the atomic core density. For this reason, we force ρ̃v(r = 0) to be small by taking a small t. If t = 0, the strict vanish- ing of the valence density near the nucleus would require a very repulsive Vps, which is certainly undesirable for the LR-based OF-DFT. However, if t is too large, there will exist a long oscillatory tail outside the core in the corresponding Vps(r). This is an undesirable feature for transferability to other environments, as will be discussed below. These two points constrain the value of t and we use here t = 0.1 for all the atomic systems considered. The four parameters a0, a1, a2, and rc are determined by requiring continuity of the function ρ̃v(r) and its first two derivatives at r = rc, and by satisfying the normal- ization condition: Nv = 4π ρ̃v(r)r 2dr. (A.2) Here we used the standard noble gas cores to determine Figure 9: The inverse KS procedure generates very small oscillations in the tail of rVps(r) for Si (shown here) and other atoms. The two points where rVps(r) = −4 for Si are r1 = 2.336 and r2 = 4.576 or r = 2.139. The arrow indi- cates the location of r2. To achieve good transferability (see text), this rVps(r) is modified by setting rVps(r) = −4 when r ≥ r2 = 4.576. See Fig. 5 for a large scale view. -4.03 -4.02 -4.01 -3.99 -3.98 -3.97 -3.96 -3.95 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 Nv = N −Nc, though other choices could in principle be made. To construct our local pseudopotential Vps(r) for atoms, we first solve the KS equations for an atom with the full Coulomb potential. With the KS density ρKS(r) and Nc determined, this construction ensures that as r → rc, the associated core density smoothly approaches zero as O(| r − rc | After generating the parameterized target valence den- sity ρ̃v(r), the set of inverse-KS equations are solved to obtain the corresponding one-body screened poten- tial Vscr(r). In principle, the AILPS Vps(r) is then given by Eq. (41). Using this, we find with an acceptable Nc that essentially the same density profiles outside rc are predicted by the LQ and HQ models for a wide range of choices of p, q, and t. This would be expected if most fea- tures of the resulting set of model potentials are within the different regimes accurately described by the OF KP. These results clearly show that OF-DFT can give accu- rate results for this class of physically relevant and rela- tively weak and slowly varying model potentials. However, we found that the rVps(r) constructed in this way for group III to group VIII elements deviates from −Zv outside the core by a small and long range oscilla- tion. For the 2s22px atoms, the maximum amplitude of the oscillation is about 0.18. For the 3s23px atoms, it ranges from 0.003 for Al to 0.037 for Ar. Similar oscil- latory tails were observed by Wang and Stott [25], and are thought to arise from the inability of a local pseu- dopotential to represent both s and p orbitals of the cor- responding nonlocal pseudopotentials. These oscillations are so small in magnitude that they have almost no effect on the density profile or energy of atoms, but they can cause transferability problems when used for solids [53]. Following Wang and Stott’s approach, we replace Vps(r) by −Zv/r at the largest point where rVps(r) = −Zv (see Fig. (9)). The new Vps(r) can then be used for both atomic and solid state calculations. 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Lower bounds in some power sum problems Johan Andersson∗ October 29, 2018 Abstract We prove that for j ≥ 0 one has that |zk|=1 ν=1,...,n2+j (1 + j)(n − 1) 2(j + n2) . (*) This improves upon former estimates. Our proof will use Fejér kernels. We also prove corresponding results for non pure power sums, and the pure power sum estimate − o(1) n ≤ inf |zk|=1 ν=1,...,⌊αn2⌋ for constants α > 1. This improves further on (*) when j ≥ 2n2. 1 Introduction The power sum method of Turán (see Turán [Tur84] or Montgomery [Mon94] Chapter 5) allows us to obtain lower bounds for power sums ν=N(n),...,M(n) |g(ν)|, (1) where g(ν) = k (2) ∗Department of Mathematics, Stockholm University, SE-10691, Sweden. johana@math.su.se http://arxiv.org/abs/0704.1879v1 for zk and bk complex numbers, and M(n) −N(n) ≥ n a function of n. We will henceforth assume that the bk ≥ 0 are positive. In particular we are interested in the case of pure power sums (bk = 1) S(ν) = zνk , (3) and the minimum norm mink |zk| = 1. We will also assume that N(n) = 1. In this case a number of results have been proved. ν=1,...,n |S(ν)| ≥ 1, (Turán [Tur69]) ν=1,...,2nm−m(m+1)+1 |S(ν)| ≥ m. (1 ≤ m ≤ n) (Andersson [And96]) Under the min norm it seems reasonable that the minimal systems (z1, . . . , zn) which minimize the expressions actually lies on, or very close to the unit circle. This has been difficult to prove and in fact in that case, when the zk are uni- modular, Newman, Cassels and Szalay have independently proved the stronger result 1≤ν≤cn cn− n+ 1 . ([Tur84], Theorem 7.3) 2 One sided bounds We will denote g(ν) = bke(θkν), (4) where θk are real numbers and bk > 0. We will let A = g(0) = bk, and B = In this section we will furthermore assume that g is real valued. In particular this implies that g(ν) = g(−ν). (5) We will also let g+(ν) = g(ν), g(ν) > 0, 0, otherwise, and g−(ν) = g(ν), g(ν) < 0, 0, otherwise. It is clear that g(ν) = g+(ν) + g−(ν), |g(ν)| = g+(ν)− g−(ν). (6) Our method of proof will use the Fejér kernel Fm+1(x) = 1− |ν| e(νx). (7) The Fejér kernel can be written as Fm+1(x) = sin π(m+ 1)x sin πx , (8) and is thus non negative. We will let g+(ν)>0 1≤ν≤m , and β = g−(ν)<0 1≤ν≤m . (9) From the representation (8) it follows that Fm+1(0) = m+1. From this it is clear , (10) and thus also α + β ≤ 1. (11) 2.1 The first method Lemma 1. One has that |g(ν)|2 ≥ (m+ 1)B − A Proof. We have that 1− |ν| |g(ν)|2 = k,l=1 bkblFm+1(θk − θl), which by the contribution of the diagonal k = l, and the non negativity of the Fejér kernel, eq. (8) implies that |g(ν)|2 ≥ b2kFm+1(0). The result follows by subtracting the term ν = 0 and using equation (5). Lemma 2. Suppose that |g(v)| ≤ M for ν = 1, . . . , m. Then g+(ν) ≥ B(m+ 1)− AM − A2 Proof. Since g+(ν) = (|g(ν)| + g(ν))/2 and obviously |g(ν)| ≥ |g(ν)|2/M when ν 6= 0 we have that g+(ν) ≥ |g(ν)|2 + g(ν) The first term can be estimated by Lemma 1 and gives the contribution B(m+ 1)− A2 The second term can be investigated by use of the Fejér kernel. By the non negativity of the Fejér kernel, equation (8) we have that 1− |ν| g(ν) ≥ 0 By the fact that g(0) = A and using equation (5) we find that g(ν) ≥ −A , (12) which gives the remaining contribution to our Lemma. We now prove the following Theorem. Theorem 1. Suppose that |g(ν)| ≤ M for ν = 1, . . . , m. Then one has that ν=1,...,m g+(ν) ≥ B(m+ 1)−AM − A2 Proof. This follows from Lemma 2 and equation (10). 2.2 An improvement for large values of m As m tends to infinity Theorem 1 will give us ν=1,...,m g+(ν) ≥ B − o(1), We will prove a stronger results which allows us to obtain ν=1,...,m g+(ν) ≥ M + 2− − o(1). This will give sharper results when M ≍ B and for large m. Theorem 2. Suppose that |g(ν)| ≤ M for ν = 1, . . . , m. If B(m + 1) − A2 − mM2 ≥ 0 then one has that ν=1,...,m g+(ν) ≥ B(m+ 1)− A In case B(m+ 1)−A2 −mM2 ≤ 0 one has that ν=1,...,m g+(ν) ≥ M + 2× B(m+ 1)− A2 −mM2 B(m+ 1)− A2 − AM under the assumption that the denominator in the last fraction is positive. Proof. By equations (9) and (11) it is clear that g−(ν) ≥ −M(1 − α). (13) By equation (6) and the fact that |g(ν)| ≤ M we get the inequality g+(ν) ≥ |g(ν)| + g−(ν). By combining this with equation (13) we see that g+(ν) ≥ 2 |g(ν)|2 −M(1 − α). which by Lemma 1 can be estimated by B(m+ 1)−A2 −M(1 − α). This together with the definition of α, equation (9) implies that g+(ν) ≥ B(m+ 1)−A2 −M(1− α) B(m+ 1)−A2 −mM2 for some ν = 1, . . . , m. We see that if B(m+1)−A2−mM2 ≥ 0 then the function is decreasing in α and the minimum over 0 < α ≤ 1 is attained for α = 1. This gives us case 1. In the case when B(m + 1) − A2 − mM2 ≤ 0 the function is increasing in α and we use the following estimate B(m+ 1)− AM −A2 which follows from Lemma 2 to obtain a lower bound. By putting this value in the right hand side of (14) we obtain a lower bound and we obtain the second part of our theorem. We remark that we also need that B(m+ 1)−AM −A2 is positive since otherwise we get an α < 0. 3 A lower bound for power sums We will now use our one sided theorems to obtain improved lower bounds for the absolute values of power sums. Theorem 3. Let bνk, A = B 1 − B2, and B = B22 − B4. Then one has that ν=1,...,m |g(ν)|2 ≥ B2 + B(1 + 1/m) AB2 + A . (15) One also has that ν=1,...,m |g(ν)|2 ≥ 2B2 − 2× A2 − B +mB4 B(m+ 1)− AB2 − A2 when m ≥ (B −A2)/B4 and both the numerator and the denominator in the last fraction is positive (this is true for m sufficiently large). Proof. Let g(ν) be defined by equation (4). Then |g(ν)|2 = b2k + cke(λkν), = B2 + h(ν) where ck = bibj and λk = θi − θj for i 6= j. It is clear that h(ν) is real valued and hence we can use the methods of section 2. Let us now assume that |h(ν)| ≤ B2 for ν = 1, . . . , m. By using Theorem 1 with M = B2 we have that there exist a ν with ν = 1, . . . , m such that g+(ν) ≥ B(m+ 1)− AB2 −A This implies (15) in case |h(ν)| ≤ B2. We have by the definition of B that B(m+ 1)− A2 −mB22 = B − A2 −mB4 which is negative if m ≥ (B − A2)/B4, and it follows from Theorem 2 with M = B2 that h(ν) ≥ B2 − 2A2 − 2B + 2mB4 B(m+ 1)−AB2 − A2 for some ν = 1, . . . , m. This implies (16) in case |h(ν)| ≤ B2. Let us now assume that |h(ν)| > B2 for some ν = 1, . . . , m. Since |g(ν)|2 = B2 + h(ν) ≥ 0 this means that h(ν) > B2 and |g(ν)|2 ≥ 2B2. We see that this implies (15) since 2B2 ≥ B2+(1+1/m)B/(2B2) and the third term on the right hand side in (15) is negative. Likewise it implies (16) since the last term on the right hand side in (16) is negative. 4 The pure power sum case In the pure power sum case we have that Bk = n, A = B = n 2 − n in Theorem 3 and it follows that Corollary 1. One has that (i) max ν=1,...,m |S(ν)|2 ≥ n+ (−1 + n)(1 +m− n2) (ii) max ν=1,...,m |S(ν)|2 ≥ 2n− 2(1 +m− 2n 2 + n3) (−1 + n)(1 +m− n2) . (m > n2) Corollary 1 (i) improves upon known results for m bigger than n2. In fact it is convenient to write m = n2 + j and we obtain Theorem 4. One has for j ≥ 0 that ν=1,...,n2+j |S(ν)| ≥ (1 + j)(n− 1) 2(j + n2) For j = 0 and in the case of unimodular numbers zk it improves slightly on the general lower bound ν=1,...,n2 |S(ν)| ≥ in Turán’s problem 10 from Andersson [And96]. We obtain Corollary 2. One has that |zk|=1 ν=1,...,n2 |S(ν)| ≥ Another result where the lower bound follows from Theorem 4 and the upper bound follows from Montgomery’s construction (see Montgomery [Mon94] page 101. Example 6.) is the following Corollary 3. Suppose that n + 1 is a prime number. One then has that 2n− 1 2(n2 + n− 1) ≤ inf |zk|=1 ν=1,...,n2+n−1 |S(ν)| ≤ n+ 1. We remark that the lower bound holds in general. We see that this approxi- mately halves the previous gap between the upper and lower bound. For further discussions of explicit constructions that yields similar upper bounds in power sum problems, see our paper Andersson [Anda]. In our paper Andersson [Andb] page 17, we considered functions Λ that fulfills n(Λ(α)− o(1)) ≤ inf |zk|=1 ν=1,...,⌊αn2⌋ |S(ν)|. We proved that we can choose Λ(α) = 1 for α > 0 and furthermore that for 0 < α ≤ 1 we proved that it is the best possible. We asked whether the function must be identically 1 or must be bounded. While we can not answer if there exist such an unbounded function it follows from Corollary 1 that we can choose a Λ such that limα→∞ Λ(α) = 2. More specifically we obtain the following Theorem. Theorem 5. Let α ≥ 1 be a constant. One then has that Φ(α)− o(1) n ≤ inf |zk|=1 ν=1,...,⌊αn2⌋ |S(ν)| ≤ ⌈α⌉+ o(1) where Φ(α) = , 1 ≤ α ≤ 3, , α ≥ 3. Proof. The lower bound for 1 ≤ α ≤ 3 follows from Corollary 1 (i) with m = ⌊αn2⌋. The lower bound for 3 ≤ α follows from Corollary 1 (ii) with m = ⌊αn2⌋. The upper bound follows from Theorem 6 in Andersson [Andb]. In particular this will give us − o(1) n ≤ inf |zk|=1 ν=1,...,2n2 |S(ν)| ≤ 2 + o(1) We see that the lower and upper bounds are not the same and we do not yet have the true asymptotic. This contrasts to the case when we take the maximum over the interval ν = 1, . . . , n2 where we proved (see Andersson [Andb]) |zk|=1 ν=1,...,n2 |S(ν)| ∼ References [Anda] Johan Andersson. Explicit solutions to certain inf max problems from Turán power sum theory, arXiv:math/0607238v2 [math.NT]. To appear in Indagationes Mathematicae. [Andb] Johan Andersson. Turán’s problem 10 revisited, arXiv:math/0609271v1 [math.NT]. [And96] Johan Andersson. On some power sum problems of Turán and Erdős. Acta Math. Hungar., 70(4):305–316, 1996. [Mon94] Hugh L. Montgomery. Ten lectures on the interface between analytic number theory and harmonic analysis, volume 84 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1994. [Tur69] Paul Turán. On a certain limitation of eigenvalues of matrices. Aequa- tiones Math., 2:184–189, 1969. [Tur84] Paul Turán. On a new method of analysis and its applications. Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1984. With the assistance of G. Halász and J. Pintz, With a foreword by Vera T. Sós, A Wiley-Interscience Publication. Introduction One sided bounds The first method An improvement for large values of m A lower bound for power sums The pure power sum case

We study the power sum problem max_{v=1,...,m} | sum_{k=1}^n z_k^v | and by using features of Fejer kernels we give new lower bounds in the case of unimodular complex numbers z_k and m cn^2 for constants c>1.

Introduction The power sum method of Turán (see Turán [Tur84] or Montgomery [Mon94] Chapter 5) allows us to obtain lower bounds for power sums ν=N(n),...,M(n) |g(ν)|, (1) where g(ν) = k (2) ∗Department of Mathematics, Stockholm University, SE-10691, Sweden. johana@math.su.se http://arxiv.org/abs/0704.1879v1 for zk and bk complex numbers, and M(n) −N(n) ≥ n a function of n. We will henceforth assume that the bk ≥ 0 are positive. In particular we are interested in the case of pure power sums (bk = 1) S(ν) = zνk , (3) and the minimum norm mink |zk| = 1. We will also assume that N(n) = 1. In this case a number of results have been proved. ν=1,...,n |S(ν)| ≥ 1, (Turán [Tur69]) ν=1,...,2nm−m(m+1)+1 |S(ν)| ≥ m. (1 ≤ m ≤ n) (Andersson [And96]) Under the min norm it seems reasonable that the minimal systems (z1, . . . , zn) which minimize the expressions actually lies on, or very close to the unit circle. This has been difficult to prove and in fact in that case, when the zk are uni- modular, Newman, Cassels and Szalay have independently proved the stronger result 1≤ν≤cn cn− n+ 1 . ([Tur84], Theorem 7.3) 2 One sided bounds We will denote g(ν) = bke(θkν), (4) where θk are real numbers and bk > 0. We will let A = g(0) = bk, and B = In this section we will furthermore assume that g is real valued. In particular this implies that g(ν) = g(−ν). (5) We will also let g+(ν) = g(ν), g(ν) > 0, 0, otherwise, and g−(ν) = g(ν), g(ν) < 0, 0, otherwise. It is clear that g(ν) = g+(ν) + g−(ν), |g(ν)| = g+(ν)− g−(ν). (6) Our method of proof will use the Fejér kernel Fm+1(x) = 1− |ν| e(νx). (7) The Fejér kernel can be written as Fm+1(x) = sin π(m+ 1)x sin πx , (8) and is thus non negative. We will let g+(ν)>0 1≤ν≤m , and β = g−(ν)<0 1≤ν≤m . (9) From the representation (8) it follows that Fm+1(0) = m+1. From this it is clear , (10) and thus also α + β ≤ 1. (11) 2.1 The first method Lemma 1. One has that |g(ν)|2 ≥ (m+ 1)B − A Proof. We have that 1− |ν| |g(ν)|2 = k,l=1 bkblFm+1(θk − θl), which by the contribution of the diagonal k = l, and the non negativity of the Fejér kernel, eq. (8) implies that |g(ν)|2 ≥ b2kFm+1(0). The result follows by subtracting the term ν = 0 and using equation (5). Lemma 2. Suppose that |g(v)| ≤ M for ν = 1, . . . , m. Then g+(ν) ≥ B(m+ 1)− AM − A2 Proof. Since g+(ν) = (|g(ν)| + g(ν))/2 and obviously |g(ν)| ≥ |g(ν)|2/M when ν 6= 0 we have that g+(ν) ≥ |g(ν)|2 + g(ν) The first term can be estimated by Lemma 1 and gives the contribution B(m+ 1)− A2 The second term can be investigated by use of the Fejér kernel. By the non negativity of the Fejér kernel, equation (8) we have that 1− |ν| g(ν) ≥ 0 By the fact that g(0) = A and using equation (5) we find that g(ν) ≥ −A , (12) which gives the remaining contribution to our Lemma. We now prove the following Theorem. Theorem 1. Suppose that |g(ν)| ≤ M for ν = 1, . . . , m. Then one has that ν=1,...,m g+(ν) ≥ B(m+ 1)−AM − A2 Proof. This follows from Lemma 2 and equation (10). 2.2 An improvement for large values of m As m tends to infinity Theorem 1 will give us ν=1,...,m g+(ν) ≥ B − o(1), We will prove a stronger results which allows us to obtain ν=1,...,m g+(ν) ≥ M + 2− − o(1). This will give sharper results when M ≍ B and for large m. Theorem 2. Suppose that |g(ν)| ≤ M for ν = 1, . . . , m. If B(m + 1) − A2 − mM2 ≥ 0 then one has that ν=1,...,m g+(ν) ≥ B(m+ 1)− A In case B(m+ 1)−A2 −mM2 ≤ 0 one has that ν=1,...,m g+(ν) ≥ M + 2× B(m+ 1)− A2 −mM2 B(m+ 1)− A2 − AM under the assumption that the denominator in the last fraction is positive. Proof. By equations (9) and (11) it is clear that g−(ν) ≥ −M(1 − α). (13) By equation (6) and the fact that |g(ν)| ≤ M we get the inequality g+(ν) ≥ |g(ν)| + g−(ν). By combining this with equation (13) we see that g+(ν) ≥ 2 |g(ν)|2 −M(1 − α). which by Lemma 1 can be estimated by B(m+ 1)−A2 −M(1 − α). This together with the definition of α, equation (9) implies that g+(ν) ≥ B(m+ 1)−A2 −M(1− α) B(m+ 1)−A2 −mM2 for some ν = 1, . . . , m. We see that if B(m+1)−A2−mM2 ≥ 0 then the function is decreasing in α and the minimum over 0 < α ≤ 1 is attained for α = 1. This gives us case 1. In the case when B(m + 1) − A2 − mM2 ≤ 0 the function is increasing in α and we use the following estimate B(m+ 1)− AM −A2 which follows from Lemma 2 to obtain a lower bound. By putting this value in the right hand side of (14) we obtain a lower bound and we obtain the second part of our theorem. We remark that we also need that B(m+ 1)−AM −A2 is positive since otherwise we get an α < 0. 3 A lower bound for power sums We will now use our one sided theorems to obtain improved lower bounds for the absolute values of power sums. Theorem 3. Let bνk, A = B 1 − B2, and B = B22 − B4. Then one has that ν=1,...,m |g(ν)|2 ≥ B2 + B(1 + 1/m) AB2 + A . (15) One also has that ν=1,...,m |g(ν)|2 ≥ 2B2 − 2× A2 − B +mB4 B(m+ 1)− AB2 − A2 when m ≥ (B −A2)/B4 and both the numerator and the denominator in the last fraction is positive (this is true for m sufficiently large). Proof. Let g(ν) be defined by equation (4). Then |g(ν)|2 = b2k + cke(λkν), = B2 + h(ν) where ck = bibj and λk = θi − θj for i 6= j. It is clear that h(ν) is real valued and hence we can use the methods of section 2. Let us now assume that |h(ν)| ≤ B2 for ν = 1, . . . , m. By using Theorem 1 with M = B2 we have that there exist a ν with ν = 1, . . . , m such that g+(ν) ≥ B(m+ 1)− AB2 −A This implies (15) in case |h(ν)| ≤ B2. We have by the definition of B that B(m+ 1)− A2 −mB22 = B − A2 −mB4 which is negative if m ≥ (B − A2)/B4, and it follows from Theorem 2 with M = B2 that h(ν) ≥ B2 − 2A2 − 2B + 2mB4 B(m+ 1)−AB2 − A2 for some ν = 1, . . . , m. This implies (16) in case |h(ν)| ≤ B2. Let us now assume that |h(ν)| > B2 for some ν = 1, . . . , m. Since |g(ν)|2 = B2 + h(ν) ≥ 0 this means that h(ν) > B2 and |g(ν)|2 ≥ 2B2. We see that this implies (15) since 2B2 ≥ B2+(1+1/m)B/(2B2) and the third term on the right hand side in (15) is negative. Likewise it implies (16) since the last term on the right hand side in (16) is negative. 4 The pure power sum case In the pure power sum case we have that Bk = n, A = B = n 2 − n in Theorem 3 and it follows that Corollary 1. One has that (i) max ν=1,...,m |S(ν)|2 ≥ n+ (−1 + n)(1 +m− n2) (ii) max ν=1,...,m |S(ν)|2 ≥ 2n− 2(1 +m− 2n 2 + n3) (−1 + n)(1 +m− n2) . (m > n2) Corollary 1 (i) improves upon known results for m bigger than n2. In fact it is convenient to write m = n2 + j and we obtain Theorem 4. One has for j ≥ 0 that ν=1,...,n2+j |S(ν)| ≥ (1 + j)(n− 1) 2(j + n2) For j = 0 and in the case of unimodular numbers zk it improves slightly on the general lower bound ν=1,...,n2 |S(ν)| ≥ in Turán’s problem 10 from Andersson [And96]. We obtain Corollary 2. One has that |zk|=1 ν=1,...,n2 |S(ν)| ≥ Another result where the lower bound follows from Theorem 4 and the upper bound follows from Montgomery’s construction (see Montgomery [Mon94] page 101. Example 6.) is the following Corollary 3. Suppose that n + 1 is a prime number. One then has that 2n− 1 2(n2 + n− 1) ≤ inf |zk|=1 ν=1,...,n2+n−1 |S(ν)| ≤ n+ 1. We remark that the lower bound holds in general. We see that this approxi- mately halves the previous gap between the upper and lower bound. For further discussions of explicit constructions that yields similar upper bounds in power sum problems, see our paper Andersson [Anda]. In our paper Andersson [Andb] page 17, we considered functions Λ that fulfills n(Λ(α)− o(1)) ≤ inf |zk|=1 ν=1,...,⌊αn2⌋ |S(ν)|. We proved that we can choose Λ(α) = 1 for α > 0 and furthermore that for 0 < α ≤ 1 we proved that it is the best possible. We asked whether the function must be identically 1 or must be bounded. While we can not answer if there exist such an unbounded function it follows from Corollary 1 that we can choose a Λ such that limα→∞ Λ(α) = 2. More specifically we obtain the following Theorem. Theorem 5. Let α ≥ 1 be a constant. One then has that Φ(α)− o(1) n ≤ inf |zk|=1 ν=1,...,⌊αn2⌋ |S(ν)| ≤ ⌈α⌉+ o(1) where Φ(α) = , 1 ≤ α ≤ 3, , α ≥ 3. Proof. The lower bound for 1 ≤ α ≤ 3 follows from Corollary 1 (i) with m = ⌊αn2⌋. The lower bound for 3 ≤ α follows from Corollary 1 (ii) with m = ⌊αn2⌋. The upper bound follows from Theorem 6 in Andersson [Andb]. In particular this will give us − o(1) n ≤ inf |zk|=1 ν=1,...,2n2 |S(ν)| ≤ 2 + o(1) We see that the lower and upper bounds are not the same and we do not yet have the true asymptotic. This contrasts to the case when we take the maximum over the interval ν = 1, . . . , n2 where we proved (see Andersson [Andb]) |zk|=1 ν=1,...,n2 |S(ν)| ∼ References [Anda] Johan Andersson. Explicit solutions to certain inf max problems from Turán power sum theory, arXiv:math/0607238v2 [math.NT]. To appear in Indagationes Mathematicae. [Andb] Johan Andersson. Turán’s problem 10 revisited, arXiv:math/0609271v1 [math.NT]. [And96] Johan Andersson. On some power sum problems of Turán and Erdős. Acta Math. Hungar., 70(4):305–316, 1996. [Mon94] Hugh L. Montgomery. Ten lectures on the interface between analytic number theory and harmonic analysis, volume 84 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1994. [Tur69] Paul Turán. On a certain limitation of eigenvalues of matrices. Aequa- tiones Math., 2:184–189, 1969. [Tur84] Paul Turán. On a new method of analysis and its applications. Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1984. With the assistance of G. Halász and J. Pintz, With a foreword by Vera T. Sós, A Wiley-Interscience Publication. Introduction One sided bounds The first method An improvement for large values of m A lower bound for power sums The pure power sum case

arXiv:0704.1880v2 [cond-mat.stat-mech] 19 Apr 2007 Quantum Quenches in Extended Systems Pasquale Calabrese1 and John Cardy2 Dipartimento di Fisica dell’Università di Pisa and INFN, Pisa, Italy Institute for Theoretical Physics, University of Amsterdam, 1018 XE Amsterdam, The Netherlands. and Oxford University, Rudolf Peierls Centre for Theoretical Physics, Oxford, United Kingdom All Souls College, Oxford, United Kingdom (Dated: October 23, 2018) We study in general the time-evolution of correlation functions in a extended quantum system after the quench of a parameter in the hamiltonian. We show that correlation functions in d dimensions can be extracted using methods of boundary critical phenomena in d + 1 dimensions. For d = 1 this allows to use the powerful tools of conformal field theory in the case of critical evolution. Several results are obtained in generic dimension in the gaussian (mean-field) approximation. These predictions are checked against the real-time evolution of some solvable models that allows also to understand which features are valid beyond the critical evolution. All our findings may be explained in terms of a picture generally valid, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate with a finite speed through the system. Furthermore we show that the long-time results can be interpreted in terms of a generalized Gibbs ensemble. We discuss some open questions and possible future developments. I. INTRODUCTION Suppose that an extended quantum system in d dimensions (for example a quantum spin system), is prepared at time t = 0 in a pure state |ψ0〉 which is the ground state of some hamiltonian H0 (or, more generally, in a thermal state at a temperature less than the mass gap to the first excited state). For times t > 0 the system evolves unitarily according to the dynamics given by a different hamiltonian H , which may be related to H0 by varying a parameter such as an external field. This variation, or quench, is supposed to be carried out over a time scale much less than the inverse mass gap. How does the state |ψ(t)〉 = e−iHt|ψ0〉 evolve? For a finite number of degrees of freedom, the system generically shows a periodic (or quasiperiodic) behavior with a period that typically increases when the number of degrees of freedom grows. This is the well-known phenomenon of quantum recurrence. However, in the thermodynamic limit this is no longer necessarily the case, and the natural question arises as to whether the system (or, rather, a macroscopically large subsystem) reaches a stationary state for very large times. To attack this question we consider the simpler question of how the correlation functions, expectation values of products of local observables, evolve and whether they reach constant values for large times. This problem has its own theoretical interest, being a first step towards the understanding of equilibration in quantum systems. On the same fundamental level we can also ask whether the approach can provide a new tool for the characterization of the collective excitations in strongly correlated systems. However, until recently it has been considered a largely academic question, because the time scales over which most condensed matter systems can evolve coherently without coupling to the local environment are far too short, and the effects of dissipation and noise are inescapable. Recent developments of experimental tools for studying the behavior of ultra-cold atoms have revised completely this negative attitude. In fact, thanks to the phenomenon of Feshbach resonance it is possible to tune the coupling of an interacting system to essentially any value and in extremely short times. In addition, the coupling to dissipative degrees of freedom can be much weaker than in ordinary condensed matter systems. This has allowed the observation, among other things, of the collapse and revival of a Bose-Einstein condensate [1] and of the quenching of a spinor condensate [2]. Furthermore, using highly anisotropic optical lattices, it has been possible to build and study essentially one-dimensional systems [3], and, of direct interest for this paper, the coherent non-equilibrium dynamics of these integrable models has been measured [4]. (Other interesting experiments are considered in Refs. [5].) These striking experimental results have largely motivated the development of new numerical methods to study non-equilibrium dynamics, the most successful one being the time-dependent density matrix renormalization group (DMRG) [6]. On the purely theoretical side these kind of questions were first considered (as far as we are aware) in the seventies in the context of the quantum Ising-XY model in Refs. [7, 8] (see also [9–14] for recent developments). However, in the very last few years, after the previously mentioned experimental progress, the study of quantum quenches has been pursued in a systematic manner and a large number of results is nowadays available. To quote only a few examples, the models considered include several realizations of one-dimensional (1D) Bose gases [15–25], Luttinger liquids [26, 27], coupled 1D condensates [28], strongly correlated 1D fermions [29], and mean field fermion condensates [30, 31]. A http://arxiv.org/abs/0704.1880v2 closely related topic, that will not be considered here, concerns the formation of defects when crossing a critical point with changing the external parameter at a fixed rate, i.e. the so-called quantum Kibble-Zurek mechanism [32]. Most of these papers concern the exact or approximate solutions of very specific models. These are often strongly relevant, since in many cases they can be directly compared with existing experimental results or give very accurate predictions for future investigations. However, from the study of specific models it is difficult to draw general con- clusions on the physics of quantum quenches. There have been at least two notable exceptions. In Ref. [15] it was conjectured that the asymptotic state at very large times can be described by a generalized Gibbs ensemble (for more details we refer the reader to section VI where this conjecture will be explicitly discussed in relation to some of our findings). In Ref. [33] we studied the time evolution after a quench in general, exploiting the path integral approach and the well-known mapping of the quantum problem to a classical one in d + 1 dimensions. The translational in- variance in the (imaginary) time direction is explicitly broken and thus the initial state plays the role of a boundary condition. This limits the range of applicability of field-theoretical methods, but when the hamiltonian H is at or close to a quantum critical point we could use the renormalization group (RG) theory of boundary critical behavior (see, e.g., [34]). From this point of view, particularly powerful analytic results are available for d = 1 because then the 1 + 1-dimensional problem is described asymptotically by a boundary conformal field theory (BCFT)[35, 36]. The aim of this paper is twofold: on the one hand we give a detailed description of the methods and the results reported briefly in Ref. [33], and on the other we generalize these results and give some new physical insights. Several results, in fact, appear here for the first time. The paper is organized as follows. In Sec. II we introduce the path- integral formalism that is applied in the following section III to one-dimensional critical systems by means of CFT. In Sec. IV we consider two simple 1D models whose non-equilibrium dynamics is exactly solvable to check the correctness of the previous results and to extend our findings to gapped and lattice systems. In Sec. V we generalize the method to higher dimensions. In the last section VI we analyze all our results and explain most of the general findings in quantum quenches by means of simple physical arguments. We broadly discuss some open questions and possible future developments. II. PATH INTEGRAL FORMULATION AND SURFACE CRITICALITY Let us consider a lattice quantum theory in d space dimensions. The lattice spacing is a, and the lattice variables are labelled by a discrete vector variable r. Time is considered to be continuous. The dynamics of the theory is described by the hamiltonian H . Suppose we prepare this system in a state |ψ0〉 that is not an eigenstate of H and unitarily evolve it according to H . The expectation value of a local operator O({ri}) at time t is 〈O(t, {ri})〉 = 〈ψ0|eiHtO({ri})e−iHt|ψ0〉 . (1) We modify this time-dependent expectation value as 〈O(t, {ri})〉 = Z−1〈ψ0|eiHt−ǫHO({ri})e−iHt−ǫH |ψ0〉 , (2) where we have included damping factors e−ǫH in such a way as to make the path integral representation of the expectation value absolutely convergent. The normalization factor Z = 〈ψ0|e−2ǫH |ψ0〉 ensures that the expectation value of the identity is one. At the end of the calculation we shall set ǫ to zero. Eq. (2) may be represented by an analytically continued path integral in imaginary time over the field variables φ(τ, r), with initial and final values weighted by the matrix elements with |ψ0〉: [dφ(τ, r)]〈ψ0|φ(τ2, r)〉〈φ(τ1 , r)|ψ0〉 e− L[φ]dτ where L[φ]dτ is the (euclidean) action. The operator O is inserted at τ = 0, and the width of the slab is 2ǫ. τ1 and τ2 should be considered as real numbers during the calculation, and only at the end should they be continued to their effective values ±ǫ− it. In this way we have reduced the real-time non-equilibrium evolution of a d dimensional systems to the thermodynamics of a d + 1 field theory in a slab geometry with the initial state |ψ0〉 playing the role of boundary condition at both the borders of the slab. The validity of our results relies on the technical assumption that the leading asymptotic behavior given by field theory, which applies to the Euclidean region (large imaginary times), may simply be analytically continued to find the behavior at large real time. Eq. (2) can be in principle used to describe the time evolution in any theory and for all possible initial conditions. Unfortunately, in confined geometries only a few field theories with very specific boundary conditions can be solved analytically in such a way to have results that can be continued from real to complex values. Some examples of these will be given in Sec. V. (An interesting case concerns integrable massive boundary field theories as for example the recent application to the dynamics of coupled 1D condensates in Ref. [28].) However, the treatment greatly simplifies if a system is at or close to a (quantum) phase transition. In fact, in this case, we can use the powerful tools of Renormalization Group (RG) theory of boundary critical phenomena (see, e.g. [34]). For example, in the case of a scalar order parameter (corresponding to the Ising universality class), the slab geometry of above is usually described by the action [34] S(φ) = (∂φ)2 + ddr c[φ(τ = τ1, r) + φ(τ = τ2, r)] , (4) where m2 measures the distance from criticality, g > 0 ensures the stability in the broken-symmetry phase, and c denotes the surface enhancement, i.e. the difference of the interactions on the surface with respect to the bulk. The value of c depends on the boundary conditions. For example, the choice c = +∞ forces the field to vanish on the boundary, thus corresponding to Dirichlet boundary conditions. On the other hand c = −∞ forces the field to diverge at the boundary. Finite values of c correspond to intermediate boundary conditions. According to the RG theory, the different boundary universality classes are characterized by the fixed point of c. A complete RG analysis (see, e.g.,[34]) shows that the possible fixed points of c are c∗ = ∞, 0,−∞, with c∗ = ±∞ stable and c∗ = 0 unstable. Any other value of c flows under RG transformations to ±∞. These universality classes are called ordinary (c∗ = ∞), special (c∗ = 0), and extraordinary (c∗ = −∞) [34]. Thus, for the purpose of extracting the asymptotic behavior, as long as |ψ0〉 is translationally invariant, we may replace it by the appropriate RG-invariant boundary state |ψ∗0〉 to which it flows. The difference may be taken into account, to leading order, by assuming that the RG-invariant boundary conditions are not imposed at τ = τ1 and τ2 but at τ = τ1 − τ0 and τ = τ2 + τ0. In the language of boundary critical behavior, τ0 is called the extrapolation length [34]. It characterizes the RG distance of the actual boundary state from the RG-invariant one. It is always necessary because scale-invariant boundary states are not in fact normalizable[36]. It is expected to be of the order of the typical time-scale of the dynamics near the ground state of H0, that is the inverse gap m 0 . (This can be checked explicitly for a free field theory, see later). The effect of introducing τ0 is simply to replace ǫ by ǫ + τ0. The limit ǫ→ 0+ can now safely be taken, so the width of the slab is then taken to be 2τ0. For simplicity in the calculations, in the following we will consider the equivalent slab geometry between τ = 0 and τ = 2τ0 with the operator O inserted at τ = τ0 + it. This is illustrated in the left part of Fig. 1. In fact, in this geometry we can consider products of operators at different times tj , by analytically continuing their labels to τ0 + itj . Let us discuss to what initial conditions the fixed points correspond. In the case of free boundary conditions, when the order parameter is unconstrained, the continuum limit forces it to vanish there, so it corresponds to the ordinary transition (c = ∞). For a lattice model, this corresponds to a completely disordered initial state (e.g. for the Ising chain in a transverse field corresponds to infinite field). In contrast for non-vanishing fixed boundary conditions (e.g. + or − for Ising-like systems), the continuum limit makes the order parameter to diverge at the boundary, thus corresponding to the extraordinary transition (c = −∞). III. ONE SPACE DIMENSION AND CONFORMAL FIELD THEORY In this section we specialize the methods just introduced to the case when H is at a quantum critical point whose long-distance behavior is given by a 1+1-dimensional CFT, with dispersion relation Ωk = v|k|. We set v = 1 in the following. RG-invariant boundary conditions then correspond to conformally invariant boundary states. In this case the correlation functions are accessible through the powerful tools of boundary CFT. �������������������������� �������������������������� �������������������������� �������������������������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� w w z FIG. 1: Left: Space-imaginary time region in (2). Imwi = τ , that will be analytically continued to τ → τ0 + it. Right: Conformal mapping of the left geometry to the upper half-plane (c.f. Eq. (5)). Note that argzi = θ = πτ/2τ0. The main property we will repeatedly use in the following is the relation of correlation functions of (primary) operators among two geometries connected by a conformal transformation. For example, the slab geometry of above is just a two-dimensional strip whose points are labelled by a complex number w = r + iτ with 0 < Imw < 2τ0. The strip can be obtained from the upper half-plane (UHP) Imz > 0 by the conformal mapping w(z) = log z , (5) with the images of points at the same imaginary time on the strip lying along argzi = θ = πτ/2τ0. In the case where O is a product of local primary scalar operators Φi(wi), the expectation value in the strip is related to the one in the UHP by the standard transformation Φi(wi)〉strip = |w′(zi)|−xi〈 Φi(zi)〉UHP , (6) where xi is the bulk scaling dimension of Φi. Note that the (eventual) expectation values of the Φi in the ground state of H are supposed to have been subtracted off. The asymptotic real time dependence is obtained via the analytic continuation τ → τ0 + it, and taking the limit t, rij ≫ τ0. In the following subsections we apply these methods to some specific cases. A. The one-point function In the UHP, the one-point function of a scalar primary field with bulk scaling dimension x is 〈Φ(z)〉UHP = AΦb [2Im(z)] −x, as a simple consequence of scaling invariance. The normalization factor AΦb is a non-universal ampli- tude. In CFT the normalizations are chosen in such a way that the bulk two-point functions have unit amplitude (i.e. 〈Φ(z1)Φ(z2)〉bulk = |z2 − z1|−2x). This choice fixes unambiguously the amplitude AΦb that turns out to depend both on the considered field Φ and on the boundary condition on the real axis b. It vanishes if Φ corresponds to an operator whose expectation value in |ψ0〉 vanishes, and thus 〈Φ(t)〉 = 0, for all times. When the primary field is not vanishing on the boundary, performing the conformal mapping (5) we obtain 〈Φ(w)〉strip = |w′(z)|−x〈Φ(z(w))〉UHP = AΦb sin(πτ/(2τ0)) that continued to real time τ = τ0 + it gives 〈Φ(t)〉 = AΦb cosh(πt/(2τ0)) ≃ AΦb e−xπt/2τ0 . (8) Thus the order parameter (and any other observables described by a primary field) decays exponentially in time to the ground-state value, with a non-universal relaxation time tOrel = 2τ0/xOπ. The ratio of the relaxation times of two different observables equals the inverse of the ratio of their scaling dimensions and it is then universal. The normalization factor AΦb is known for the simplest boundary universality classes [37]. In the case of Φ being the order parameter and the boundary condition is fixed (ψ0(x) = ∞) AΦb is 1 for the free boson and 21/4 for the Ising model. An important exception to this law is the local energy density (or any piece thereof). This corresponds to the tt component of the energy-momentum tensor Tµν . In CFT this is not a primary operator. Indeed, if it is normalized so that 〈Tµν〉UHP = 0, in the strip [38] 〈Ttt(r, τ)〉 = πc/24(2τ0)2 (where c is the central charge of the CFT) so that it does not decay in time. Of course this is to be expected since the dynamics conserves energy. A similar feature is expected to hold for other local densities corresponding to globally conserved quantities which commute with H , for example the total spin in isotropic models. B. The two-point function In the case of the one-point function, scaling invariance was enough to fix the functional dependence on the position in the UHP. However, the form of the two-point function depends explicitly on the boundary universality class and on the operator considered. In the following subsections we will consider the equal-time correlation function for the order parameter in the gaussian and in the Ising universality classes that are easily treated in full generality. At the quantum level they describe (among the other things) a chain of harmonic oscillators (explicitly considered in Sec. IVA) and the Ising model in a transverse field (whose real time evolution has been considered in Refs. [7–9, 11] and is briefly reviewed in Sec. IVB). Finally we will discuss the general form of the two-point function for asymptotically large time and distance, that can be obtained from general CFT arguments. 1. The gaussian model The content of this subsection has been already reported in Ref. [39], during the study of the time evolution of the entanglement entropy, that transforms like the two-point function of a primary field in a boundary gaussian theory [40]. We report it here for sake of completeness. For a free boson the two-point function in the UHP is [35] 〈Φ(z1)Φ(z2)〉UHP = z12̄z21̄ z12z1̄2̄z11̄z22̄ , (9) with zij = |zi − zj | and zk̄ = zk. Note that Φ is not the gaussian field θ(z), but its exponential Φ(z) = eiθ(z) (see Sec. IVA). Under the conformal mapping (5) we obtain the two-point function on the strip at imaginary time τ , at distance r apart 〈Φ(r, τ)Φ(0, τ)〉strip = |w′(z1)|−x|w′(z2)|−x〈Φ(z1(w))〈Φ(z2(w))〉UHP = cosh(πr/2τ0)− cos(πτ/τ0) 8 sinh2(πr/4τ0) sin 2(πτ/2τ0) , (10) that continued to real time τ = τ0 + it gives 〈Φ(r, t)Φ(0, t)〉 = cosh(πr/2τ0) + cosh(πt/τ0) 8 sinh2(πr/4τ0) cosh 2(πt/2τ0) . (11) In the case where r/τ0 and t/τ0 are large this simplifies to 〈Φ(r, t)Φ(0, t)〉 = (π/2τ0)2x eπr/2τ0 + eπt/τ0 eπr/2τ0 · eπt/τ0 e−xπt/τ0 for t < r/2 e−xπr/2τ0 for t > r/2 . (12) i.e. the two point function at fixed r decays exponentially in time up to t∗ = r/2 and then saturates to a value that depends exponentially on the separation. In the case of fixed initial condition, with one-point function given by Eq. (8), the connected correlation function is 〈Φ(r, t)Φ(0, t)〉conn = 〈Φ(r, t)Φ(0, t)〉 − 〈Φ(0, t)〉2 ∝ 0 for t < r/2 , e−xπr/2τ0 − e−xπt/τ0 for t > r/2 , i.e. correlations start developing at t∗ = r/2 and, being t ≫ τ0, at t∗ the connected two-point function almost immediately jumps to its asymptotic value. In the case of disordered initial conditions (ψ0(r) = 0), connected and full correlation functions are equal. 2. The Ising universality class For the Ising model the two-point function in the UHP is [35] 〈Φ(z1)Φ(z2)〉UHP = z12̄z21̄ z12z1̄2̄z11̄z22̄ F (η) , (14) where F (η) is given by F (η) = 1 + η1/2 ± 1− η1/2√ , (15) and η is the four-point ratio z11̄z22̄ z12̄z21̄ . (16) The sign ± depends on the boundary conditions. + corresponds to fixed boundary conditions and and − to disordered ones. The only difference with respect to the gaussian case is that we have also to map F (η) according to the conformal transformation (5). After simple algebra we have 2 sin2(πτ/2τ0) cosh(πr/2τ0)− cos(πτ/τ0) , (17) and so 〈Φ(r, τ)Φ(0, τ)〉strip = cosh(πr/2τ0)− cos(πτ/τ0) 8 sinh2(πr/4τ0) sin 2(πτ/2τ0) 2 sin(πτ/2τ0) cosh(πr/2τ0)− cos(πτ/τ0) 2 sin(πτ/2τ0) cosh(πr/2τ0)− cos(πτ/τ0) . (18) Analytically continuing to real time τ = τ0 + it we obtain 〈Φ(r, t)Φ(0, t)〉 = cosh(πr/2τ0) + cosh(πt/τ0) 8 sinh2(πr/4τ0) cosh 2(πt/2τ0) 2 cosh(πt/2τ0) cosh(πr/2τ0) + cosh(πt/τ0) 2 cosh(πt/2τ0) cosh(πr/2τ0) + cosh(πt/τ0) , (19) that for r/τ0 and t/τ0 much larger than 1 simplifies to (π/2τ0) 1/4 1√ eπr/2τ0 + eπt/τ0 eπr/2τ0 · eπt/τ0 eπt/2τ0√ eπr/2τ0 + eπt/τ0 πt/2τ0 eπr/2τ0 + eπt/τ0 . (20) Note that the exponential terms in the square root are always ≪ 1. Thus for fixed boundary condition we get the free boson result Eq. (12) with x = 1/8. For the connected part we need to subtract 〈Φ(0, t)〉2 given by Eq. (8) with AΦ+ = 2 1/4. We finally obtain 〈Φ(r, t)Φ(0, t)〉conn = 〈Φ(r, t)Φ(0, t)〉 − 〈Φ(0, t)〉2 ∝ 0 for t < r/2 , e−πr/16τ0 − e−πt/8τ0 for t > r/2 . Thus, also for the Ising model with fixed boundary conditions, connected correlations start developing at t = t∗ = r/2. In the case of disordered initial condition, we have 〈Φ(r, t)Φ(0, t)〉 ∝ eπr/2τ0 + eπt/τ0 eπr/2τ0 · eπt/τ0 eπt/2τ0√ eπr/2τ0 + eπt/τ0 e−π(r−3/2t)/4τ0 for t < r/2 , e−πr/16τ0 for t > r/2 , resulting in an exponential space dependence even for t < r/2 (clearly in this case the connected correlation function equals the full one). 3. The general two point-function From the results reported for the gaussian and Ising models, it is now relatively simple to understand the general properties of the time dependence of the two-point function in the very general case. The two-point function in the half-plane has the general form [35] 〈Φ(z1)Φ(z2)〉UHP = z12̄z21̄ z12z1̄2̄z11̄z22̄ F (η) , (23) ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ FIG. 2: Left: Space-time region for the correlation functions at different times. Right: Conformal mapping to the upper half-plane. where the function F (η) depends explicitly on the considered model. Under the conformal map to the strip we know that the first part of Eq. (23) transforms according to Eq. (10). Thus we need only to map F (η) that, in the general case, is an unknown function. However, during the study of the Ising model we showed that the analytical continuation of η for t, r ≫ τ0 is η ∼ e πt/τ0 eπr/2τ0 + eπt/τ0 . (24) Thus for t < r/2 we have η ∼ eπ(t−r/2)/τ0 ≪ 1 and in the opposite case t > r/2 we have η ∼ 1. As a consequence to have the asymptotic behavior of the two-point function we only need to know the behavior close to η ∼ 0 (i.e. the behavior close to the surface) and for η ∼ 1 (i.e. deep in the bulk). Fortunately they are both exactly known. Indeed when η ∼ 1 the two points are deep in the bulk, meaning F (1) = 1. Instead for η ≪ 1, from the short-distance expansion, we have F (η) ≃ (AΦb )2ηxb , (25) where xb is the boundary scaling dimension of the leading boundary operator to which Φ couples and A b is the bulk-boundary operator product expansion coefficient that equals the one introduced in Eq. (8) [see e.g. Ref. [37]]. All the previous observations and the explicit calculations of the previous sections lead for t > r/2 to 〈Φ(r, t)Φ(0, t)〉 ∝ e−xπr/2τ0 , (26) while for t < r/2 we get 〈Φ(r, t)Φ(0, t)〉 ∝ (AΦb )2e−xπt/τ0 × eπxb(t−r/2)/τ0 . (27) Note that if 〈Φ〉 6= 0, xb = 0 and the last factor is absent. The leading term is then just 〈Φ〉2. Thus the leading term in the connected two-point function vanishes for t < r/2, and its first non-vanishing contributions is given by subleading terms either in F or in the bulk-boundary short-distance expansion. It is very interesting that we only have to know the behavior as η → 0 and 1 to get the results we need for large r and t. However, we stress that only a complete calculation (as those performed in the preceding sections) gives the full analytic structure of the CFT result needed to justify the analytical continuation from imaginary to real time. Moreover, the behavior within a distance O(τ0) of the horizon r = 2t depends on the detailed form of F . C. Correlations functions at different times Let us consider the case of the two-point function calculated at different real times 〈Φ(r, t)Φ(0, s)〉. This is again obtained by mapping the imaginary time strip to the UHP, but in this case the two points are w1 = r + iτ1 and w2 = 0+ iτ2, that, at the end of the calculation, must be analytically continued to τ1 = τ0 + it and τ2 = τ0 + is. See Fig. 2 for a pictorial representation of the space-time domain in the strip and the resulting mapping to the UHP. Let us start the discussion with the free boson. The distances appearing in Eq. (9) are z212 = 1 + e πr/τ0 − 2eπr/2τ0 cos(θ1 − θ2) , z11̄ = 2 sin θ1 , z212̄ = 1 + e πr/τ0 − 2eπr/2τ0 cos(θ1 + θ2) , z22̄ = 2eπr/2τ0 sin θ2 , (28) where θi = πτi/2τ0. Thus the correlation function on the strip is 〈Φ(r, τ1)Φ(0, τ2)〉strip = |w′(z1)|−x|w′(z2)|−x〈Φ(z1(w))〈Φ(z2(w))〉UHP = (29) cosh(πr/2τ0)− cos(π(τ1 + τ2)/2τ0) 4 sin(πτ1/2τ0) sin(πτ2/2τ0)(cosh(πr/2τ0)− cos(π(τ1 − τ2)/2τ0)) that for τ1 = τ2 reduces to Eq. (10) as it should. Continuing to real times and considering r, t, s, |t − s| ≫ τ0 we obtain τ0 + e (t+s) (t+s) τ0 + e |t−s| e−xπ(t+s)/4τ0 for r > t+ s , e−xπr/4τ0 for t− s < r < t+ s , e−xπ|t−s|/4τ0 for r < |t− s| . Following the line sketched in the previous subsection, it is easy to generalize this result to the most general CFT. In the case of a theory with fixed initial conditions (i.e., 〈Φ〉 6= 0) the asymptotic result is the same as before, with only the crossover points being affected by the precise expression for F (η). Instead, in the case where 〈Φ〉 = 0, the first case gains an additional factor e−πxb(t+s−r)/4τ0 . Note that the autocorrelation function (i.e. r = 0) has only an exponential dependence on the time separation t− s and does not exhibit aging in this regime. D. Evolution with boundaries We now consider the case of time evolution of a half-chain with some boundary condition at r = 0. For simplicity we assume that the (conformal) boundary condition is of the same kind of the initial boundary condition (for example we fix all the spins at t = 0 and at the boundary r = 0 to point in the same direction). The space-time region we have to consider is depicted in Fig. 3. If different initial and boundary were considered, one needs to insert boundary conditions changing operators at the corners of the figure. The w plane is mapped into the UHP by z(w) = sin , (31) with the corners at ±τ0 mapped to ±1. The mapping of w1 is z1 ≡ z(w1) = z(−τ0 + τ1 + ir) = − cos(πτ1/2τ0) cosh(πr/2τ0) + i sin(πτ1/2τ0) sinh(πr/2τ0) . (32) In the z plane the 1-point function is 〈Φ(z1)〉UHP ∝ |Imz1|−x → [sin(πτ1/2τ0) sinh(πr/2τ0)]−x , (33) |w′(z1)|2 = |1− z2| cosh(πr/τ0)− cos(πτ1/τ0) . (34) Thus on the strip we have 〈Φ(w1)〉strip = |w′(z1)|−x〈Φ(w(z1))〉UHP ∝ sin2(πτ1/2τ0) sinh 2(πr/2τ0) cosh(πr/τ0)− cos(πτ1/τ0) ]−x/2 , (35) that continued to real time τ1 = it is 〈Φ(t, r)〉 ∝ cosh(πt/τ0) + cosh(πr/τ0) cosh(πt/2τ0)2 sinh 2(πr/2τ0) , (36) and for t, r ≫ τ0 simplifies to 〈Φ(t, r) ∝ eπr/τ0 + eπt/τ0 eπr/τ0 · eπt/τ0 e−πxt/2τ0 for t < r , e−πxr/2τ0 for t > r . Note that in this case the characteristic time is t∗ = r and not r/2. This explains also why the entanglement entropy of a semi-infinite chain with free boundary condition at x = 0, has characteristic time t∗ = r as firstly noted in Ref. [41]. �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� ���������������� ���������������� ���������������� ���������������� ���������������� ���������������� 0 1−1 FIG. 3: Left: Space-time region for the one-point function in a boundary (at r = 0) geometry. Note w = τ + ir. Right: Conformal mapping to the upper-half plane, c.f. Eq. (31). E. Discussion and interpretation of the CFT results All the correlation functions calculated so far display two very general features: first there is a sharp horizon (or light-cone) effect at t = t∗ = r/2 (or r) resulting in a behavior before and after t∗ completely different; second the asymptotic long-time correlation functions are the same as those at finite temperature βeff = 4τ0. The light-cone effect is a very general phenomenon and and will be discussed in section VI. In the following sections we will point out that also the “effective temperature” is a general phenomenon, but it has some specific CFT features that are worthy of comment. In fact, it is very easy to understand the technical reason why we find an effective temperature despite the fact that we are studying a pure state at T = 0. The finite temperature correlations can be calculated by studying the field theory on a cylinder of circumference β = 1/T . In CFT a cylinder can be obtained by mapping the complex plane with the logarithmic transformation β/(2π) log z. The form for the two-point function in the slab depends in general on the function F (η) –cf. Eq. (15)– but when we analytically continue and take the limit of large real time, we find that effectively the points are far from the boundary, i.e. at η = 1. Thus we get the same result as we would get if we conformally transformed from the full plane to a cylinder, and from Eq. (5) the effective temperature is βeff = 4τ0. A similar argument can be worked out for the multi-point functions as well. An addtional comment concerns what we expect for correlation functions of general operators, not only primary. We have clearly seen that the local energy density does not relax, of course, and this is consistent with its not being primary. But, we also know that at finite temperature 〈Ttt〉β = πc/6β2, that is perfectly compatible with the previous result with βeff = 4τ0. Furthermore, for large real times, the two-point function of Ttt − 〈T 〉tt does behave as though it was at finite temperature. This means, in particular, that the energy fluctuations in a large but finite volume (the specific heat) are the same as those at finite temperature. Thus one is tempted to extend this finite temperature interpretation to non-primary operators, and indeed it is the case. In fact, in the argument of above for the equivalence of the long-time correlations and finite T , there are essentially three steps. First, we need to write down the form of the correlation function in the half-plane, but this only depends on special conformal transformation and so is valid for any quasi-primary operator like Tµν . Second we have to transform from the slab to the half-plane: for non-primary operators this can have some anomalous term. Finally we need to compare the large limit of this to what one would get transforming directly from the cylinder to the full plane. However, the two conformal transformations we are comparing are both logarithmic (in one case 2τ0/π log z, in the other case β/2π log z), so the anomalous terms should be the same. Thus the two correlations functions are the same. This argument works for all quasi-primary operators. Then, since we can get all the non-quasiprimaries by considering successive operator product expansions with the stress tensor, it also works for all operators. IV. EXACT REAL-TIME DYNAMICS IN SIMPLE INTEGRABLE CHAINS The results of the previous section rely on the technical assumption that the leading asymptotic behavior given by CFT, which applies to the euclidean region (large imaginary times), may simply be analytically continued to find the behavior at large real time. While such procedures have been shown to give the correct behavior for the time-dependent correlations in equilibrium, it is important to check them in specific solvable cases for non-equilibrium evolution. Thus, as a complement to the CFT calculations, in this section we consider the real-time evolution of two simple analytically tractable models. We solve the dynamics of a chain of coupled harmonic oscillators and we review and re-analyze some known results for the Ising-XY chain in a transverse magnetic field. Beyond providing examples of the CFT results (with central charge c = 1 and 1/2 respectively) in the critical case, these models allow us to take into account the effects of a finite mass-gap and of the lattice. A. The chain of harmonic oscillators The simplest model with an exactly solvable non-equilibrium dynamics is surely a chain of coupled harmonic oscillators with hamiltonian π2r +m 2φ2r + ω2j (φr+j − φr)2 . (38) We introduce a coupling more general than simple nearest-neighbor hopping so as to allow for a general dispersion relation below. For simplicity we also assume periodic boundary conditions. ϕn and πn are the position and the momentum operators of the n-th oscillator, with equal time commuting relations [ϕm, πn] = iδnm [ϕn, ϕm] = [πn, πm] = 0 . (39) The hamiltonian can be written in diagonal form H(m) = k ΩkA kAk with modes (Ωkϕk + iπk) , (40) (Ωkϕ−k − iπ−k) , (41) Ω2k = m 2 + 2 ωj (1− cos(2πkj/N)) . (42) Note that we use the same symbols for the operators and their Fourier transforms (ϕk = 1/ n=0 e 2πikn/Nϕn and analogously for πk). We consider the scenario in which the system is prepared in a state |ψ0〉, that is ground-state of H(m0), and at the time t = 0 the mass is quenched to a different value m 6= m0. We use the notation Ω0k for the dispersion relation for t < 0 and the Ωk for the one for t > 0. Since 〈ψ0|ϕn|ψ0〉 = 0, the expectation value of the field ϕn vanishes at any time. This example in fact corresponds to the quench from the disordered phase in the language of the previous section. Thus we concentrate our attention on the two-point function 〈ψ(t)|ϕnϕ0|ψ(t)〉 = 〈ψ0|ϕHn (t)ϕH0 (t)|ψ0〉 , (43) where we introduced the operator in the Heisenberg picture ϕHn (t), whose time evolution is given by ϕHn (t) = ei(pkn−Ωkt)Ak + e −i(pkn−Ωkt)A , (44) where pk = 2πk/N . Accordingly, the product of the two fields is ϕHn (t)ϕ 0 (t) = ΩkΩk′ ei(pkn−Ωkt)Ak + e −i(pkn−Ωkt)A e−iΩk′ tAk′ + e . (45) In order to have the time dependent two-point function we need the expectation values of the bilinear combinations of A’s on the initial state, that is annihilated by the A0k. Thus it is enough to write A’s as functions of A0’s, i.e. ≡ ckA0k + dkA†0−k , (46) +A0−k ≡ ckA†0k + dkA0−k , (47) leading to (we understand 〈·〉 = 〈ψ0| · |ψ0〉) 〈AkAk′ 〉 = 〈(ckA0k + dkA†0−k)(ck′A0k′ + dk′A 0−k′ )〉 = ckdk′〈A0kA 0−k′〉 = ckdkδk,−k′ , (48) 〈AkA†k′ 〉 = 〈(ckA0k + dkA 0−k)(ck′A 0k′ + dk′A0−k′ )〉 = ckck′〈A0kA 0k′ 〉 = c kδk,k′ , (49) 〈A†kAk′ 〉 = 〈(ckA 0k + dkA0−k)(ck′A0k′ + dk′A 0−k′ )〉 = dkdk′〈A0−kA 0−k′〉 = d kδk,k′ , (50) 〈A†kA k′ 〉 = 〈(ckA 0k + dkA0−k)(ck′A 0k′ + dk′A0−k′ )〉 = dkck′〈A0−kA 0k′ 〉 = ckdkδk,−k′ . (51) Finally we arrive at 〈ϕHn (t)ϕH0 (t)〉 = ckdke i(pkn−2Ωkt) + c2ke ipkn + d2ke −ipkn + ckdke −i(pkn−2Ωkt) , (52) which, in the thermodynamic limit (N → ∞), may be written as 〈ϕHr (t)ϕH0 (t)〉 − 〈ϕHr (0)ϕH0 (0)〉 = (Ω20k − Ω2k)(1 − cos(2Ωkt)) Ω2kΩ0k , (53) where the integral is on th first Brillouin zone |k| < π/a. Note that for t = 0 and for m = m0 this two-point function reduces to the static one, as it should. This result can also be found by integrating the Heisenberg equations of motion for each mode. For future reference it is also useful to write down explicitly the Fourier transform known as momentum distribution function ρ(k) = (Ω20k +Ω k)− (Ω20k − Ω2k) cos(2Ωkt) Ω2kΩ0k . (54) Note that when considering correlation functions in momentum space, a long-time limit does not exist and we need to take the time average, in contrast to what happens in real space. 1. The continuum limit In Eq. (52) everything is completely general and applies to any chain with finite lattice spacing. Let us know discuss the continuum limit that is achieved by sending N → ∞ in such a way that (1/N) dp/(2π), pn → p, and Ω2k → Ω2p = m2 + p2. In this limit the correlation function becomes (∆m2 = m20 −m2) G(r, t) ≡ 〈ϕHr (t)ϕH0 (t)〉 = −∆m2 cos(2 p2 +m2t) +m2 +m20 + 2p (m2 + p2) m20 + p . (55) Since a closed form for such integral is quite difficult to write down in the most general case, we will only consider some particular cases. Let us first consider the conformal evolution (m = 0) from an initial state with a very large mass m0 → ∞. This should reproduce the CFT result previous section for all the time t, since the correlations of the initial state shrink to a point. We have G(r, t) = m0 1− cos(2pt) 0 for t < r/2 , (2t− r) for t > r/2 . To compare such result with the conformal result given by Eq. (12), we have to keep in mind that the primary field is not the gaussian one ϕH(r, t), but its imaginary exponential. Thus we need 〈eiqϕH (r,t)e−iqϕH (0,t)〉, with an arbitrary q. Despite of the apparent complexity of such correlator, it is very simple to obtain it using the standard property of gaussian integrals 〈eiqϕ H (r,t)e−iqϕ H(0,t)〉 = e−q 2〈(ϕH(r,t)−ϕH(0,t))2〉/2 = eq 2(G(r,t)−G(0,t)) , (57) leading to 〈eiqϕ H (r,t)e−iqϕ H (0,t)〉 = 2m0t for t < r/2 , 2m0r/2 for t > r/2 , 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 0 1 2 FIG. 4: Left: G(r, t)/m0 given by Eq. (60) as function of t, at fixed r = 1. Three different values of m0 = 10, 3, 1 (from the bottom to the top) are shown. Inset: Lattice effects showing the cos 4t oscillations on top of the continuum result. Right: G(r, t)/m0 given by the numerical integral of Eq. (62) as function of t, at fixed m, r = 1. It is compared with the asymptotic behavior for 0 < 2t− r ≪ m−1 and for t ≫ r. that is exactly the same of Eq. (12) with xΦ ∝ q2 and τ0 ∝ m−10 , confirming that τ0 is just proportional to the correlation in the initial state, as its interpretation in terms of the extrapolation length suggests. The case m = 0 and m0 finite corresponds to a conformal evolution from a generic state with correlations propor- tional to m−10 . Thus we expect the CFT result Eq. (12) to be true for asymptotic large times and separations. From Eq. (55), the two-point function of the gaussian field is G(r, t) = −m20 cos(2pt) +m20 + 2p2 m20 + p , (59) that can be written as 2πG(r, t) = 2K0(m0r) + f(m0r) − f(m0(r − 2t)) + f(m0(r + 2t)) , (60) where K0(y) is the modified Bessel function and f(y) = 1 + G2113 0 1 1/2 , (61) with G2113 the Meijer G-function (see e.g. [42]). Note that f(x) is characterized by f ′′(x) = −K0(x) and f(0) = f ′(0) = 0. In the limit m0 → ∞, it is easy to show that G(r, t) reduces to the previous result. In the left panel of Fig. 4 , we report G(r, t) as function of t at fixed r = 1 for m0 = 10, 3, 1. It is evident that a finite m0 results in smoothing the curve close to t = r/2 and giving an offset in zero. Both the effects are more pronounced as m0 decreases. For large t, independently on m0 6= 0, we have G(r, t) = t + O(t0), confirming that the CFT result is correct for asymptotic large times. The case with arbitrary m and m0 is quite cumbersome to be worked out analytically and not really illuminating. For this reason we concentrate here on the massive evolution from a state with m0 → ∞. In this case the correlation function reads G(r, t) = m0 1− cos(2 p2 +m2t) (m2 + p2) . (62) In the limit t→ ∞ the cosine term averages to zero, giving G(r, t = ∞) = m0 (m2 + p2) . (63) To understand the time dependence, let us first note that, despite the presence of a square-root, the integrand of Eq. (62) is analytic in p, since the square-root is the argument of the even cosine function. Thus we can make the integral in the complex plane and use the Cauchy theorem. As long as r > 2t, the behavior for p→ i∞ is dominated by eipr and we can safely close the contour path in the upper half-plane where the residue at p = im is zero. As a consequence G(r, t < r/2) = 0. For t > r/2, the integral is more difficult, since we can not close the contour in the upper half-plane, because the cosine is “larger” than eipr for p→ i∞. However, the approach to the asymptotic value is easily worked out. In fact, for t≫ r, the term eipr in the integral (62) can be approximated with 1, since it is slowly oscillating. Thus we have G(r, t ≫ r)−G(r, t = ∞) ≃ −m0 cos(2 p2 +m2t) (m2 + p2) = −m0fm(t) , (64) where fm(t) = 1/(2m) − t1F2(1/2; 1, 3/2;−m2t2), that satisfies f ′m(t) = −J0(2mt) and fm(0) = 1/(2m). Note that fm(t) is just the result for r = 0. In the complementary region 0 < 2t−r ≪ m−1, the integral in Eq. (62) is dominated by the modes with p≫ m, and so it can be described by the conformal result. In the right panel of Fig. 4, we plot the time-dependence of G(r, t) at fixed r as obtained by numerically integrating Eq. (62) for m = r = 1. The plot shows that the conformal result t − r/2 describes the behavior close to t = r/2, while for larger times the asymptotic expression gives an excellent approximation. We finally note that G(r, t) at fixed t displays spatial oscillations (for r < 2t, else it vanishes), as it can be simply realized by a stationary phase argument. 2. Lattice effects Another advantage of this simple model is that the effects of the lattice can be easily understood. In fact, by a stationary phase argument, the dominant contribution to (53) in the limits of large t and r comes from where, the group velocity vk ≡ Ω′k = r/2t, independently of the explicit form of Ωk. Consequently, the two-point function of a gapless model with dispersion relation Ωk = 2 sink/2 differs from the continuum limit previously derived for t > r/2vm, where vm is the maximum group velocity. In particular G(r, t) receives a contribution from the slowest mode (k = π with vπ = 0) whose effect is to add to Eq. (56) fast oscillations going as cos(2Ωπt) = cos 4t. The resulting G(r, t) is plotted in the inset of Fig. 4 in the gapless case. Thus, lattice effects play a more important role in the cases where the asymptotic result vanishes. A typical example is the energy density that in this model is proportional to (ϕr+1 − ϕr)2. The continuum calculation would just give zero (the mean-energy is conserved), but the approach to this value is governed by lattice effects and is dominated by the smallest group velocity that comes from the zone boundary at |k| = π. In fact, just with a trivial calculation we 〈(ϕr+1 − ϕr)2〉 = 2(〈ϕ20〉 − 〈ϕ0ϕ1〉) = (1− cos k) (1− cos 2Ωkt) 2 sin2 k/2 (1− cos 2Ωkt) ∝ J0(4t) ∼ t−1/2 cos 4t , (65) that displays the typical cos 4t from the zone boundary at |k| = π. The lattice dispersion relations are sensitive to the microscopical details of the model and consequently quantities like the energy-density show a dependence on this. For example, a lattice massless fermion has dispersion relation Ωk = 2 sink/2 and energy density given by sin 2 k. In this case the time evolution of the energy density is sin2 k (1− cos 2Ωkt) 2 sin2 k/2 , (66) that for large times goes like t−3/2 cos 4t, resulting in a different power law compared to before. If the system is quenched to a gapped H with Ω2k = m 2 + 2(1 − cos k) the maximum group velocity corresponds to a non-zero wave number. This gives rise to spatial oscillations in the correlation function (this is true also in the continuum limit as already discussed in the previous subsection). B. The Ising-XY chain in a transverse magnetic field The most studied one dimensional quantum spin model is the so-called XY chain in a transverse field, defined by the hamiltonian (1 + γ) σxj σ j+1 + (1− γ) j+1 − hσz , (67) where σ x,y,z i are the Pauli matrices, γ is called anisotropy parameter, and h is the applied external transverse field. It is well known that for any γ 6= 0 the model undergoes a phase transition at h = 1, that is in the universality class of the Ising model (defined by γ = 1). For simplicity we will just consider the Ising case, other values of γ 6= 0 being equivalent. We consider the non-equilibrium unitary dynamics that follows from a quench of the magnetic field at t = 0 from h0 to h1 6= h0. Earlier works on this subject by Barouch et al. date back to seventies [7, 8]. Exploiting the mapping of this model onto a free fermion, the time evolution of the transverse magnetization (that is not the order parameter) was obtained exactly [7]: mz(t) = (h0 − h1) sin2 k cos(2ǫ1t)− (cos k − h1)[(cos k − h0)(cos k − h1) + sin2 k] , (68) where ǫi = ǫ(hi) with ǫ(h) = sin2 k + (h− cos k)2. In Ref. [7] it was shown that (for non-exceptional parameters) the approach to the asymptotic value for t→ ∞ is of the form t−3/2 cos 4t. This is simply shown in the case of h0 = ∞ and h1 = 1, when the integral simplifies and we obtain (see also [9]) mz(t) = sin2 k cos(2ǫ1t) 4 sin2 k/2 J1(4t) , (69) where J1 is the Bessel function of the first kind whose asymptotic expansion for large argument is J1(x) ∼ 2/πx cos(x+ π/4). To understand this result we should keep in mind that the transverse magnetization is not the order parameter: it corresponds to the product of two “disorder parameters” at neighbor sites on the dual lattice. Thus it must have the symmetry of an energy operator, for which CFT just predicts a constant asymptotic result. The power law term t−3/2 cos(4t) is just the lattice correction to the asymptotic. This correction clearly shows the fermionic nature of the model on the basis of what discussed in the previous section. The asymptotic result of the two-point function for t → ∞ has been studied in Ref. [11]. Calling Gn = 〈Φ(n,∞)Φ(0,∞)〉, in the case h0 = ∞ it has been found 2nhn1 , forh1 ≥ 1 , cos[n arccos(h1)] , forh1 ≤ 1 , instead for h0 = 0 , forh1 ≥ 1 , (n+ 1) ln , forh1 ≤ 1 , that for large n decay exponentially with n. It has been shown that Gn is decaying exponentially with n for general h0 and h1 [11], although closed forms are not available. The exponential decay is the prediction of CFT that (maybe surprisingly) applies even far from the critical point h1 = 1. The time dependence of two-point function has been studied in Ref. [9] by means of exact diagonalization of the model with open boundary condition at the two ends r = 0, L. The results of interest for this paper are • The connected two-point function of σz in the thermodynamic limit and at the critical point is 〈σzr (t)σz0(t)〉c = J2r(4t) 2 − 1 J2r+1(4t)J2r−1(4t) , (72) which is valid both for h0 = 0,∞. Neglecting fast oscillations, 〈σzr (t)σz0(t)〉c increases as r2 for r < 2t and then it drops almost immediately to 0. • The autocorrelation function (not connected) of σz at different times is 〈σz0(t1)σz0(t2)〉 = J20 (2t2 − 2t1)− [f(t2 + t1)± g(t2 − t1)] , (73) where f(x) = J2(2x) + J0(2x), g(x) = J2(2x)− J0(2x) and the sign + (−) refers to h0 = 0 (h0 = ∞). Let us comment on these results in view of the general understanding we found so far. For t < t∗ all the connected correlation functions are zero, in agreement with CFT. All the oscillation terms of the asymptotic form cos(4t) are, as we discussed in the previous section, a lattice effect. Concerning the σz correlator, the r 2 dependence for t > r/2 is a consequence of the fact that σz is not primary. The same is true for the two-time correlations function of σz that also decays as a power law of t− s for large times, instead of the exponential prediction by CFT for primary field. V. HIGHER DIMENSIONS Until now we just considered one-dimensional systems. Despite the fact that in low dimensions the effect of fluctu- ations is more pronounced making the physics highly non-trivial it is desirable to have results in higher dimensions as well. The method presented in Sec. II to obtain the non-equilibrium dynamics of a quantum model close to a critical point from the critical behavior of a system confined in a slab geometry applies to generic dimension d through the study of the hamiltonian (4). Its analysis proceeds via field-theoretical RG that may provide the all scaling quantities of the model in an expansion close to the upper critical dimensions (u.c.d.), that is D = d+ 1 = 4. Above the u.c.d. mean-field (or gaussian) results are exact, with logarithmic correction at the u.c.d.. For dimensions lower than the u.c.d., the scaling quantities are obtained as series in ǫ = 4 − (d + 1). Thus for the time-evolution problem the simple mean-field solution represents an exact scaling result (a part log corrections) for the physically relevant three-dimensional case. An alternative method to attack analytically the hamiltonian (4) is to consider an N component field φ and taking the limit N → ∞, but this will not be employed here. A. Dirichlet boundary conditions: the two-point function The D-dimensional slab geometry with Dirichlet boundary conditions has been the subject of several investigations. The two-point function has been calculated at the first order in ǫ expansion in Ref. [43]. The gaussian two-point function, with partial Fourier transform in the parallel directions reads [43] G(p, z1, z2) = e−b|z1−z2| − e−b(z1+z2) + e −b(z1−z2) + e−b(z2−z1) − e−b(z1+z2) − eb(z1+z2) e2bL − 1 , (74) with b = p2 +m2. We are interested in the case where L = 2τ0, z1 = z2 = τ that we will analytically continue to τ → τ0 + it, and for computational simplicity we will restrict to the massless case m = 0. Thus in real space and imaginary time, we have (p = |p|) G(r, τ) = (2π)3 e−ip·r 1− e−2pτ + 2(1− cosh(2pτ)) e4pτ0 − 1 (2π)2 1− e−2pτ + 2(1− cosh(2pτ)) e4pτ0 − 1 d(cos θ)eipr cos θ = (2π)2r dp sin pr 1− e−2pτ + 2(1− cosh(2pτ)) e4pτ0 − 1 . (75) This integral can be performed by making the sum over all the residues coming from the denominator e4pτ0 − 1. The calculation is rather involved, but the final result is very simple: G(r, τ) = coth(πr/4τ0) sin 2(πτ/2τ0) cosh(πr/2τ0)− cos(πτ/τ0) . (76) Continuing to real time τ = τ0 + it we obtain G(r, t) = coth(πr/4τ0) cosh 2(πt/2τ0) cosh(πr/2τ0) + cosh(πt/τ0) , (77) that for t, r ≫ τ0 simplifies to G(r, t) ≃ 1 eπt/τ0 eπt/τ0 + eπr/2τ0 eπ(t−r/2)/2τ0/r for t < r/2 , 1/r for t > r/2 . We recall that this result in d = 3 is exact a part log corrections. Thus the basic structure of the two-point function in 3D is the same as in 1D, with a characteristic time t∗ = r/2. Using the result in Ref. [43] it is in principle possible to calculate the correlation functions for d < 3 in the ǫ expansion framework. However, this requires the analytical continuation of complicated functions, resulting in a quite cumbersome algebra, as we shall see in the following for a simpler observable. Eq. (74) can be used in principle to determine the gaussian behavior in any d < 3. Unfortunately the integral one gets is not analytically tractable for d 6= 1, 3. A possible strategy would be to perform the analytic continuation before of the integral and then evaluate it through a saddle-point approximation. It is straightforward to show that the result obtained in this way is equivalent to what we discuss in section VD where we remand for the analysis of 1 < d < 3. B. Dirichlet boundary conditions: a non-trivial one-point function In the case of Dirichlet boundary conditions, the order parameter profile in the slab geometry is trivially vanishing. However not all the one-point expectation values are zero. Let us consider as a typical example the operator O = φ2 that has been calculated at the first order in ǫ expansion in Ref. [44]. It has the scaling form 〈O(z, L)〉 ≃ L−d+1/νH(z/L) , (79) where ν is the correlation length exponent. In D = 4, the function H(x) is [44] H(x) = sin2(πx) , (80) that, using L = 2τ0 and continuing to z = τ = τ0 + it, leads to the real time evolution 〈O(t)〉 − 〈O(t = ∞)〉 ≃ τ−20 (πt/2τ0) ∼ τ−20 e−πt/2τ0 , (81) i.e. 〈O(t)〉 approaches its asymptotic value exponentially, as it as been found for all primary operators in d = 1. The one-loop result for the O(N) model in D = 4− ǫ dimensions is [44] H(x) = sin2(πx) 1 + ǫ N + 2 N + 8 sinπx − ǫ[ζ′(2, x) + ζ′(2, 1− x)] + const , (82) where const stands for terms do not depend on x and ζ′(α, x) = ∂αζ(α, x) and ζ is the generalized Riemann function. This example clearly shows that within the ǫ expansion, the scaling functions contain logarithmic contributions that originate from the expansion of power law as e.g. sinǫ πx = 1 + ǫ ln sinπx. To have a function with good analytical structure to perform the real time continuation, it is desirable to “exponentiate” such logarithms. In Ref. [44] the exponentiation procedure leaded to H(x) = sin(πx) )2−1/ν [ζ(D − 2, x) + ζ(D − 2, 1− x)− 2ζ(D − 2)] (2 − 1/ν) sin(πx) )D−2−1/ν , (83) with ν = 1/2 + (N + 2)/(N + 8) ǫ/4. Performing the analytical continuation we have the sum of two exponentials, and in any dimension d < 3 the second one has a largest “relaxation time” that hence is dominating for large t. Considering only the second term we have 〈O(t ≫ τ0)〉 − 〈O(t = ∞)〉 ∝ e−(d−1−1/ν)πt/2τ0 . (84) Note that the subleading term e−(2−1/ν)πt/2τ0 is multiplied by a log t term arising from the ζ function. This example put forward the idea that (at least for Dirichlet boundary conditions) the exponential relaxation of the one-point functions is not only a property of one-dimensional systems, but holds in any dimension (with eventually log corrections) with relaxation times related to the scaling dimensions of the operator. C. Fixed boundary conditions: the one-point function The case of the extraordinary transition, that corresponds to fixed boundary conditions, has been considered in Ref. [45]. The magnetization profile can be written as [45] φ(z) = dn(2K4z/L) sn(2K4z/L) , (85) where K4 = K(k4), where K(k) is the elliptic integral, k4 the elliptic modulus that in terms of the parameter of the model is φ2L2 = (2K4) 2(2k24 − 1), and sn(x) and dn(x) and the Jacobi functions. Continuing to real time, and using the properties of the Jacobi functions we obtain φ(t) = 1− k4cn(K4t/τ0, 1− k4) , (86) that, contrarily to all the other cases we have considered, is oscillating and not exponentially decreasing. To our knowledge there are no result in the ǫ-expansion for the magnetization profile. However, using “local- functional methods” [46] it has been obtained an approximate profile in D = 3 that involves, as in mean-field theory, Jacobi elliptic functions. It can be easily continued to real time via z → τ0 + it and again one finds an oscillating behavior with time. The method exploited in Ref. [46] can be used in any dimensions, obtaining an always an oscillating behavior with a period that diverges as D approaches 2, recovering Eq. (7). All these calculations are essentially mean field and we do not know how the inclusion of fluctuations changes them. It can possible that for the extraordinary transition (i.e. fixed initial conditions), the exponential decay founds at D = 2 is more an exception rather than a rule, because of the simple analytic structure of the trigonometric functions in the complex plane. Another possibility is that fluctuations destroy these oscillations. Only a complete analytical calculation (e.g. in large N) can help in understanding this point. D. A real-time solvable model As for the one-dimensional case, it is worth to check the results coming from the analytical continuation of large imaginary-time with exactly solvable models. The simplest (and probably one of the few) model solvable in generic dimension is the generalization of the hamiltonian (38) to a d-dimensional hypercubic lattice. The solution of such model proceeds via Fourier transform as in one dimension. The final result is simply give by Eq. (53) with the replacement dk → ddk, i.e. 〈ϕr(t)ϕ0(t)〉 − 〈ϕr(0)ϕ0(0)〉 = (2π)d eik·r (Ω20k − Ω2k)(1 − cos(2Ωkt)) Ω2kΩ0k . (87) First of all let us note that this expression in virtually identical to Eq. (75) if we take τ0 ∝ m−10 → 0. In fact, taking τ0 → 0 in Eq. (75), only the third term matters, since it is O(τ−10 ) relative to the first two. Taking z = r + it and b = Ωp we get G(p) ∝ τ−10 Ω−2p (1− cos 2Ωpt) that is Eq. (87) for m0 → ∞. To understand the general features let us consider in details the conformal evolution (Ωk = v|k|) from a disordered state (Ω0k = m0). The derivative of the two-point function is (k = |k|) ∂t〈ϕr(t)ϕ0(t)〉 = 2m0Im (2π)d ei(k·r−2kvt) . (88) Except for d = 1, this can be done analytically only in d = 3, where we can write it as dθ sin θeik(r cos θ−2vt) ∼ sin(kr)/r e2ikvt ∼ (1/r)δ(vt− r/2) . (89) Integrating with respect to t we get zero for t < r/2v and 1/r for t > r/2v. For general d we have ∂t〈ϕr(t)ϕ0(t)〉 ∝ 2m0Im kd−2dk dθ(sin θ)d−2eik(r cos θ−2vt) . (90) By a saddle point argument, we can assume that the dominant behavior comes from θ close to zero, so the θ integral gives dθθd−2e−ikrθ 2/2 ∼ (kr)−(d−1)/2 , (91) leading to (kr)(d−1)/2 ei(k(r−2vt)) ∼ r−(d−1)/2(2vt− r)−(d−1)/2Θ(vt− r/2) . (92) Integrating with respect to t, we get zero for t < r/2v, as expected by causality, and r−(d−1)/2(2vt− r)(3−d)/2 , (93) for t > r/2v. It is interesting that this gaussian correlation function blows up at large t only for d < 3, when we expect the fluctuations to become important. It would be interesting to study this in the φ4 theory for large N , by replacing φ4 by 3〈φ2〉φ2 in the usual way, where now 〈φ2〉 depends on t and is calculated self-consistently. VI. PHYSICAL INTERPRETATION AND DISCUSSION In this paper we studied in general the non-equilibrium unitary dynamics that follows a sudden quantum quench. We showed that if the hamiltonian H governing the time evolution is at a critical point, while H0 (i.e. the one for t < 0) is not, the expectation value of a class of operators (primary ones in CFT) relaxes to the ground-state value exponentially in time with universal ratio of decaying constants. We also found that connected two-point functions of operators at distance r are vanishing for t < r/2v, while for t > r/2v reach exponentially fast a value that depends exponentially on the separation, in contrast with the power laws typical of equilibrium configuration. We also considered the real-time dynamics of simple exactly solvable models and we found that several of the typical characteristics of the critical points still hold. Roughly speaking, critical points are not special as far as quenching dynamics is concerned. In fact, also for gapped systems, connected correlation functions vanish (or are strongly suppressed) for t < r/2v and for asymptotic large times resemble those at finite temperature despite the fact that the whole system is in a pure state. Several other examples in the recent literature (see e.g. [9, 15, 24, 26, 39, 41]) gives further evidence that these two effects are actually true in general, at least in the realm of exactly solvable models considered so far. In the following we give a simple interpretation of these two features separately trying to understanding their physical origin. A. The horizon effect The qualitative, and many of the quantitative, features found for the time evolution of correlation functions may be understood physically on the basis of a picture we first introduced in Ref. [39] to describe the time evolution of the entanglement entropy. Later we generalized it to correlation functions in Ref. [33] and it has been largely adopted thereafter [2, 24, 26, 41]. We emphasize that such scheme is not an ab initio calculation but rather a simplified picture which allows us to explain physically our findings. The initial state |ψ0〉 has an (extensively) high energy relative to the ground state of the hamiltonian H which governs the subsequent time evolution, and therefore acts as a source of quasiparticle excitations. Those quasi-particles originating from closely separated points (roughly within the correlation length ξ0 = m 0 of the ground state of H0) are quantum entangled and particles emitted from far different points are incoherent. If the quasiparticle dispersion relation is E = Ωk, the classical velocity is vk = ∇kΩk. We assume that there is a maximum allowed speed vm = maxk |vk|. A quasiparticle of momentum k produced at r is therefore at r+vkt at time t, ignoring scattering effects. This is the only physical assumption of the argument. Scattering effects are not present in the theories considered so far, but as evident from the argument outlined below they can play a role for only t > r/2vm (allowing, perhaps, for a renormalization of vm by the interactions). These free quasi-particles have two distinct effects. Firstly, incoherent quasi-particles arriving a given point r from well-separated sources cause relaxation of (most) local observables at r towards their ground state expectation values. (An exception is the local energy density which of course is conserved.) Secondly, entangled quasi-particles arriving at the same time t at points with separation |r| ≫ ξ0 induce quantum correlations between local observables. In the case where they travel at a unique speed v (as in CFT), therefore, there is a sharp “horizon” or light-cone effect: the connected correlations do not change from their initial values until time t ∼ |r|/2v. In the CFT case this horizon effect is rounded off in a (calculable) manner over the region t−|r|/2v ∼ τ0, since quasi-particles remain entangled over this distance scale. After this they rapidly saturate to time-independent values. For large separations (but still ≪ 2vt), these decay exponentially ∼ exp(−πx|r|/2τ0). Thus, while the generic one-point functions relax to their ground-state values (we recall in CFT this relaxation is exponential ∼ exp(−πxvt/τ0)), the correlation functions do not, because, at quantum criticality, these would have a power law dependence. Of course, this is to be expected since the mean energy is much higher than that of the ground state, and it does not relax. This simple argument also explains why for the case of a semi-infinite chain the relevant time scale is r/v rather than r/2v, since one of the two particles arriving in r has been reflected from the end of the chain. This has been also stressed in Ref. [41], in the study of the time dependence of entanglement entropy of finite chains with open boundary conditions. All our results are consistent with this picture as long as the quasi-particles are assumed to all propagate at the same speed, resulting from a “conformal” dispersion relation Ωk = v|k|. However, it is very simple to generalize this picture to different dispersion relations, taking into account that each particle propagates at group velocity vk ≡ Ω′k appropriate to the wave number k. In this case the horizon effect first occurs at time t ∼ |r|/2vm, where vm is the maximum group velocity. If vm occurs at a non-zero wave number, it gives rise to spatial oscillations in the correlation function. Thus again connected correlation function are expected to be strongly suppressed for t < t∗ and start developing only after t∗. In the case of a general dispersion relation we do not have a proof, beyond the stationary phase approximation, that the connected correlation functions remain constant up to this time, but one would expect it on the grounds of causality. (The proof in the case of a relativistic dispersion relation uses Lorentz invariance in an essential way.) However, because there are also quasi-particles moving at speeds less than vm, the approach to the asymptotic behavior at late times is less abrupt. In fact, for a lattice dispersion relation where Ω′k vanishes at the zone boundary, the approach to the limit is slow, as an inverse power of t. A similar result applies to the 1-point functions. This is consistent with the exact results obtained here and elsewhere. B. The large time limit and the generalized Gibbs ensemble The existence and the understanding of the asymptotic state resulting from the evolution from an arbitrary state is one of the most-interesting problem in statistical mechanics. A robust theory able to predict this state ab-initio still does not exist. A currently popular idea is that for late times the system (or rather macroscopically large subsystems) ‘look like’ they are in a thermal state, despite the fact that the actual state of the whole system is pure. A common belief is that a region of dimension r can be thermalized by the infinitely large rest of the system which acts as a bath (see e.g. [24, 39, 47]). But this intriguing idea is not sufficient to give the value of the resulting effective temperature. A major step toward the clarification of the properties of the asymptotic state has been made by Rigol et al. [15]. In fact, it was conjectured that if the asymptotic stationary state exists, it is given by a generalized Gibbs ensemble obtained by maximizing the entropy S = −Tr ρ log ρ, subject to all the constraints imposed by the dynamics [15]. Consequently, denoting with Im a maximal set of commuting and linearly independent integrals of motion, the density matrix is ρ = Z−1e− λmIm , Z = Tr e− λmIm . (94) We note that such a density matrix describes a pure state only if the model under consideration is integrable, i.e. if the number of integral of motions equals the number degrees of freedom. If there are not enough integrals of motion, ρ corresponds to a mixed state and it is not clear to us to which extent it can describe the pure state resulting from the time-evolution. The values of the Lagrange multipliers λm are fixed by the initial conditions: Tr Imρ = 〈Im〉t=0 . (95) In the following we will take this generalized Gibbs ensemble as a postulate and we will show how it nicely and naturally explains the “effective temperature” effect observed for large times. However we stress that there is still no proof for this assumption that, to our opinion, cannot be considered on the same fundamental level as the thermal Gibbs ensemble. Let us consider the chain of harmonic oscillators of the previous section as a typical example. We will soon see that most of the features are quite general. In this case the natural choice for an infinite set of integral of motion is the number of particles with momentum k, i.e. nk = A kAk. Most other observables can be written in terms of these, i.e. k Ωknk. Consequently the expectation value is given by 〈O〉t=∞ = TrOρ = TrOZ−1e− λknk , (96) that can be seen as a thermal density matrix with a k dependent effective temperature given by βeff(k)Ωk = λk . (97) Thus an effective temperature already appeared. Note that this state can still be pure because such a temperature is k dependent. To fix λk we need 〈nk〉t=0. From Eq. (50) we get 〈nk〉t=0 = 〈A†kAk〉t=0 = d . (98) The calculation then proceeds as for a thermal distribution 〈nk〉ρ = Trnkρ = − lnZ , with Z = Tr e− λknk = e−λknk = 1− e−λk , (99) so that 〈nk〉ρ = ln(1− e−λk) = 1 eλk − 1 . (100) From this eλk = 1 + d−2k and βeff(k) = ln(1 + d−2k ) . (101) At finite temperature the correlation function in momentum space is 〈ϕkϕ−k〉β = 〈 (Ak +A k)(A−k +A −k)〉β = 〈A†kA−k〉β + 〈AkA 1 + e−βΩk 1− e−βΩk , (102) that substituting the previous result for βeff gives ρ(k)t=∞ = 〈ϕkϕ−k〉t=∞ = (1 + 2d2k) = Ω2k +Ω Ω2kΩ0k , (103) that is exactly the time-average of Eq. (54) reproducing, after Fourier transforming, the well defined correlation in real space for t→ ∞. So this generalized Gibbs ensemble correctly reproduces the exact diagonalization of the model and gives also few insights more. In fact, in the limit m0 → ∞ the effective temperature is independent from k and m obtaining βeff = 4/m0, explaining a posteriori the simplicity of the results in this case. Note instead that for arbitrary m0 and m = 0, i.e. conformal evolution, βeff(k) is a function of k. In this case, the large distance properties of correlation functions are described by the mode with k = 0, for which, independently of m0 we get βeff(k = 0,m = 0) = 4/m0, consistently with the previous findings and general expectations. Furthermore we can conclude that the large r asymptotic behavior is always governed by the effective temperature βeff(k = 0) = 2(log(|m0 − m|/(m0 + m))/m. Another interesting feature is that βeff(k = 0) gives the asymptotic behavior of the correlation functions of all those observables that are effectively coupled with the zero-mode. In the opposite case the relevant temperature is the largest βeff(k) with the k mode coupled with the observable. Another interesting feature is that on a very rough basis one can be tempted to assume that that βeff(k) is directly related to the excess of energy of the mode k generated by the quench, but this is not the case. In fact 〈H〉t=0 = k Ωkd k that is different from β eff , being the same only in the limit d−2k → 0, i.e. m0 → ∞. Clearly the same reasoning of before applies every time we consider a model that can be diagonalized in momentum space with a proper choice of the quasi-particles, i.e. every time that H = k ΩkA kAk for some Ak. This means that the excitations are non-interacting. As far as we are aware all the applications of the generalized Gibbs ensemble to the date only concern this kind of models [15, 26], but there are few numerical hints suggesting that can be true more generically [29]. Now we outline how we imagine the generalized Gibbs ensemble given by Eq. (94) could be used to justify the effective temperature scenario for large time for any integrable system, i.e. with a complete set of integrals of motion. The hamiltonian can be written as H = m amIm, with some am eventually zero. Thus one can think to an m- dependent effective temperature βeff(m) = λm/am, but this temperature does not give directly the behavior of the correlation function for large distance because in general the integral of motions are not diagonal in k-space. Thus we can only conjecture that the correlation functions of a given operator O(r) are governed by the largest βeff(m) with m among the integral of motions to which O(r) effectively couples. We stress that this is only a crude argument and we are still not able to put it on a firmer basis. C. Discussion and open questions We presented a quite complete picture on the time evolution of a quantum system after a sudden quench of one hamiltonian parameter. Despite the fact that a lot of work has been done, still more is left for future investigation. The first problem that must be addressed is the real-time dynamics of effectively interacting systems. In this direction Bethe Ansatz solvable models like the Lieb-Liniger gas and Heisenberg spin chains are among the best candidates for an analytical approach. This would clarify how a non-trivial two-body scattering matrix can modify the time-evolution (only inside the light-cone) of the correlation functions and whether the crude argument we outlined for an “effective temperature” for the long-time state is valid. To clarify this point also the study of (boundary) integrable massive field theories can be of some help. Another approach, currently under investigation[48], is the direct analysis of the perturbative expansion for the correlation functions in a λφ4 field theory. This has some simplifying features in the large m0 limit which may allow it to be resummed to all orders. It is very important to get non-trivial results for such a non-integrable interacting theory. Numerical computations for non-integrable systems should also be performed to understand if and eventually to which extent the generalized Gibbs ensemble picture is valid beyond integrability. Some numerics concerning this point are already available [22, 23, 29] but still a clear scenario is not emerged. Among non-integrable models special care must be given to disordered systems because of the non existence of a speed of sound as a consequence of Anderson localization [49]. Some insights can be obtained from those models whose equilibrium behavior can be analytically obtained by means of the strong-disorder renormalization-group [50]. Even in this case time-dependent DMRG can help a general understanding. The time-evolution of the entanglement entropy [41] already revealed that in these systems there is no sign of the light-cone (as easily predictable because of the absence of speed of sound) and the effective motion of “quasi-particles” is more diffusive rather than ballistic. Furthermore the results at the largest available times does not look thermal at all, but this can be also due to the time window accessible with numerics. Another very interesting question is what happens at finite temperature and under what conditions a system can equilibrate. It should be relatively simple to generalize the results for all the “quasi-free” models already considered at T = 0. Work in this direction is in progress, but it will be almost impossible to consider the same problem for more complicated models. Finally it is also important to understand the role played by the initial state. We assumed always a translational invariant one with short range correlations. Thus one natural modification consists in taking a state that is only locally different from the actual ground-state. This problem is known in the literature as a “local quench” and it has already been considered to some extent (see e.g. [51–53]), but still a general picture as the one outlined here for global quenches does not exist (this is also interesting for eventual connections with quantum impurity problems, see [54] as a review). Another natural modification of the initial state is one with long-range correlations, such as a critical one and let it evolve with another critical hamiltonian. This has been done for the Luttinger liquid [26] and the results show the typical functional dependence of a light-cone scenario (i.e. everything depends only on x± 2vt) but the long-time correlations decays as power-laws, with exponents that are different from equilibrium ones and can be predicted by the generalized Gibbs ensemble. 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We study in general the time-evolution of correlation functions in a extended quantum system after the quench of a parameter in the hamiltonian. We show that correlation functions in d dimensions can be extracted using methods of boundary critical phenomena in d+1 dimensions. For d=1 this allows to use the powerful tools of conformal field theory in the case of critical evolution. Several results are obtained in generic dimension in the gaussian (mean-field) approximation. These predictions are checked against the real-time evolution of some solvable models that allows also to understand which features are valid beyond the critical evolution. All our findings may be explained in terms of a picture generally valid, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate with a finite speed through the system. Furthermore we show that the long-time results can be interpreted in terms of a generalized Gibbs ensemble. We discuss some open questions and possible future developments.

arXiv:0704.1880v2 [cond-mat.stat-mech] 19 Apr 2007 Quantum Quenches in Extended Systems Pasquale Calabrese1 and John Cardy2 Dipartimento di Fisica dell’Università di Pisa and INFN, Pisa, Italy Institute for Theoretical Physics, University of Amsterdam, 1018 XE Amsterdam, The Netherlands. and Oxford University, Rudolf Peierls Centre for Theoretical Physics, Oxford, United Kingdom All Souls College, Oxford, United Kingdom (Dated: October 23, 2018) We study in general the time-evolution of correlation functions in a extended quantum system after the quench of a parameter in the hamiltonian. We show that correlation functions in d dimensions can be extracted using methods of boundary critical phenomena in d + 1 dimensions. For d = 1 this allows to use the powerful tools of conformal field theory in the case of critical evolution. Several results are obtained in generic dimension in the gaussian (mean-field) approximation. These predictions are checked against the real-time evolution of some solvable models that allows also to understand which features are valid beyond the critical evolution. All our findings may be explained in terms of a picture generally valid, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate with a finite speed through the system. Furthermore we show that the long-time results can be interpreted in terms of a generalized Gibbs ensemble. We discuss some open questions and possible future developments. I. INTRODUCTION Suppose that an extended quantum system in d dimensions (for example a quantum spin system), is prepared at time t = 0 in a pure state |ψ0〉 which is the ground state of some hamiltonian H0 (or, more generally, in a thermal state at a temperature less than the mass gap to the first excited state). For times t > 0 the system evolves unitarily according to the dynamics given by a different hamiltonian H , which may be related to H0 by varying a parameter such as an external field. This variation, or quench, is supposed to be carried out over a time scale much less than the inverse mass gap. How does the state |ψ(t)〉 = e−iHt|ψ0〉 evolve? For a finite number of degrees of freedom, the system generically shows a periodic (or quasiperiodic) behavior with a period that typically increases when the number of degrees of freedom grows. This is the well-known phenomenon of quantum recurrence. However, in the thermodynamic limit this is no longer necessarily the case, and the natural question arises as to whether the system (or, rather, a macroscopically large subsystem) reaches a stationary state for very large times. To attack this question we consider the simpler question of how the correlation functions, expectation values of products of local observables, evolve and whether they reach constant values for large times. This problem has its own theoretical interest, being a first step towards the understanding of equilibration in quantum systems. On the same fundamental level we can also ask whether the approach can provide a new tool for the characterization of the collective excitations in strongly correlated systems. However, until recently it has been considered a largely academic question, because the time scales over which most condensed matter systems can evolve coherently without coupling to the local environment are far too short, and the effects of dissipation and noise are inescapable. Recent developments of experimental tools for studying the behavior of ultra-cold atoms have revised completely this negative attitude. In fact, thanks to the phenomenon of Feshbach resonance it is possible to tune the coupling of an interacting system to essentially any value and in extremely short times. In addition, the coupling to dissipative degrees of freedom can be much weaker than in ordinary condensed matter systems. This has allowed the observation, among other things, of the collapse and revival of a Bose-Einstein condensate [1] and of the quenching of a spinor condensate [2]. Furthermore, using highly anisotropic optical lattices, it has been possible to build and study essentially one-dimensional systems [3], and, of direct interest for this paper, the coherent non-equilibrium dynamics of these integrable models has been measured [4]. (Other interesting experiments are considered in Refs. [5].) These striking experimental results have largely motivated the development of new numerical methods to study non-equilibrium dynamics, the most successful one being the time-dependent density matrix renormalization group (DMRG) [6]. On the purely theoretical side these kind of questions were first considered (as far as we are aware) in the seventies in the context of the quantum Ising-XY model in Refs. [7, 8] (see also [9–14] for recent developments). However, in the very last few years, after the previously mentioned experimental progress, the study of quantum quenches has been pursued in a systematic manner and a large number of results is nowadays available. To quote only a few examples, the models considered include several realizations of one-dimensional (1D) Bose gases [15–25], Luttinger liquids [26, 27], coupled 1D condensates [28], strongly correlated 1D fermions [29], and mean field fermion condensates [30, 31]. A http://arxiv.org/abs/0704.1880v2 closely related topic, that will not be considered here, concerns the formation of defects when crossing a critical point with changing the external parameter at a fixed rate, i.e. the so-called quantum Kibble-Zurek mechanism [32]. Most of these papers concern the exact or approximate solutions of very specific models. These are often strongly relevant, since in many cases they can be directly compared with existing experimental results or give very accurate predictions for future investigations. However, from the study of specific models it is difficult to draw general con- clusions on the physics of quantum quenches. There have been at least two notable exceptions. In Ref. [15] it was conjectured that the asymptotic state at very large times can be described by a generalized Gibbs ensemble (for more details we refer the reader to section VI where this conjecture will be explicitly discussed in relation to some of our findings). In Ref. [33] we studied the time evolution after a quench in general, exploiting the path integral approach and the well-known mapping of the quantum problem to a classical one in d + 1 dimensions. The translational in- variance in the (imaginary) time direction is explicitly broken and thus the initial state plays the role of a boundary condition. This limits the range of applicability of field-theoretical methods, but when the hamiltonian H is at or close to a quantum critical point we could use the renormalization group (RG) theory of boundary critical behavior (see, e.g., [34]). From this point of view, particularly powerful analytic results are available for d = 1 because then the 1 + 1-dimensional problem is described asymptotically by a boundary conformal field theory (BCFT)[35, 36]. The aim of this paper is twofold: on the one hand we give a detailed description of the methods and the results reported briefly in Ref. [33], and on the other we generalize these results and give some new physical insights. Several results, in fact, appear here for the first time. The paper is organized as follows. In Sec. II we introduce the path- integral formalism that is applied in the following section III to one-dimensional critical systems by means of CFT. In Sec. IV we consider two simple 1D models whose non-equilibrium dynamics is exactly solvable to check the correctness of the previous results and to extend our findings to gapped and lattice systems. In Sec. V we generalize the method to higher dimensions. In the last section VI we analyze all our results and explain most of the general findings in quantum quenches by means of simple physical arguments. We broadly discuss some open questions and possible future developments. II. PATH INTEGRAL FORMULATION AND SURFACE CRITICALITY Let us consider a lattice quantum theory in d space dimensions. The lattice spacing is a, and the lattice variables are labelled by a discrete vector variable r. Time is considered to be continuous. The dynamics of the theory is described by the hamiltonian H . Suppose we prepare this system in a state |ψ0〉 that is not an eigenstate of H and unitarily evolve it according to H . The expectation value of a local operator O({ri}) at time t is 〈O(t, {ri})〉 = 〈ψ0|eiHtO({ri})e−iHt|ψ0〉 . (1) We modify this time-dependent expectation value as 〈O(t, {ri})〉 = Z−1〈ψ0|eiHt−ǫHO({ri})e−iHt−ǫH |ψ0〉 , (2) where we have included damping factors e−ǫH in such a way as to make the path integral representation of the expectation value absolutely convergent. The normalization factor Z = 〈ψ0|e−2ǫH |ψ0〉 ensures that the expectation value of the identity is one. At the end of the calculation we shall set ǫ to zero. Eq. (2) may be represented by an analytically continued path integral in imaginary time over the field variables φ(τ, r), with initial and final values weighted by the matrix elements with |ψ0〉: [dφ(τ, r)]〈ψ0|φ(τ2, r)〉〈φ(τ1 , r)|ψ0〉 e− L[φ]dτ where L[φ]dτ is the (euclidean) action. The operator O is inserted at τ = 0, and the width of the slab is 2ǫ. τ1 and τ2 should be considered as real numbers during the calculation, and only at the end should they be continued to their effective values ±ǫ− it. In this way we have reduced the real-time non-equilibrium evolution of a d dimensional systems to the thermodynamics of a d + 1 field theory in a slab geometry with the initial state |ψ0〉 playing the role of boundary condition at both the borders of the slab. The validity of our results relies on the technical assumption that the leading asymptotic behavior given by field theory, which applies to the Euclidean region (large imaginary times), may simply be analytically continued to find the behavior at large real time. Eq. (2) can be in principle used to describe the time evolution in any theory and for all possible initial conditions. Unfortunately, in confined geometries only a few field theories with very specific boundary conditions can be solved analytically in such a way to have results that can be continued from real to complex values. Some examples of these will be given in Sec. V. (An interesting case concerns integrable massive boundary field theories as for example the recent application to the dynamics of coupled 1D condensates in Ref. [28].) However, the treatment greatly simplifies if a system is at or close to a (quantum) phase transition. In fact, in this case, we can use the powerful tools of Renormalization Group (RG) theory of boundary critical phenomena (see, e.g. [34]). For example, in the case of a scalar order parameter (corresponding to the Ising universality class), the slab geometry of above is usually described by the action [34] S(φ) = (∂φ)2 + ddr c[φ(τ = τ1, r) + φ(τ = τ2, r)] , (4) where m2 measures the distance from criticality, g > 0 ensures the stability in the broken-symmetry phase, and c denotes the surface enhancement, i.e. the difference of the interactions on the surface with respect to the bulk. The value of c depends on the boundary conditions. For example, the choice c = +∞ forces the field to vanish on the boundary, thus corresponding to Dirichlet boundary conditions. On the other hand c = −∞ forces the field to diverge at the boundary. Finite values of c correspond to intermediate boundary conditions. According to the RG theory, the different boundary universality classes are characterized by the fixed point of c. A complete RG analysis (see, e.g.,[34]) shows that the possible fixed points of c are c∗ = ∞, 0,−∞, with c∗ = ±∞ stable and c∗ = 0 unstable. Any other value of c flows under RG transformations to ±∞. These universality classes are called ordinary (c∗ = ∞), special (c∗ = 0), and extraordinary (c∗ = −∞) [34]. Thus, for the purpose of extracting the asymptotic behavior, as long as |ψ0〉 is translationally invariant, we may replace it by the appropriate RG-invariant boundary state |ψ∗0〉 to which it flows. The difference may be taken into account, to leading order, by assuming that the RG-invariant boundary conditions are not imposed at τ = τ1 and τ2 but at τ = τ1 − τ0 and τ = τ2 + τ0. In the language of boundary critical behavior, τ0 is called the extrapolation length [34]. It characterizes the RG distance of the actual boundary state from the RG-invariant one. It is always necessary because scale-invariant boundary states are not in fact normalizable[36]. It is expected to be of the order of the typical time-scale of the dynamics near the ground state of H0, that is the inverse gap m 0 . (This can be checked explicitly for a free field theory, see later). The effect of introducing τ0 is simply to replace ǫ by ǫ + τ0. The limit ǫ→ 0+ can now safely be taken, so the width of the slab is then taken to be 2τ0. For simplicity in the calculations, in the following we will consider the equivalent slab geometry between τ = 0 and τ = 2τ0 with the operator O inserted at τ = τ0 + it. This is illustrated in the left part of Fig. 1. In fact, in this geometry we can consider products of operators at different times tj , by analytically continuing their labels to τ0 + itj . Let us discuss to what initial conditions the fixed points correspond. In the case of free boundary conditions, when the order parameter is unconstrained, the continuum limit forces it to vanish there, so it corresponds to the ordinary transition (c = ∞). For a lattice model, this corresponds to a completely disordered initial state (e.g. for the Ising chain in a transverse field corresponds to infinite field). In contrast for non-vanishing fixed boundary conditions (e.g. + or − for Ising-like systems), the continuum limit makes the order parameter to diverge at the boundary, thus corresponding to the extraordinary transition (c = −∞). III. ONE SPACE DIMENSION AND CONFORMAL FIELD THEORY In this section we specialize the methods just introduced to the case when H is at a quantum critical point whose long-distance behavior is given by a 1+1-dimensional CFT, with dispersion relation Ωk = v|k|. We set v = 1 in the following. RG-invariant boundary conditions then correspond to conformally invariant boundary states. In this case the correlation functions are accessible through the powerful tools of boundary CFT. �������������������������� �������������������������� �������������������������� �������������������������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� w w z FIG. 1: Left: Space-imaginary time region in (2). Imwi = τ , that will be analytically continued to τ → τ0 + it. Right: Conformal mapping of the left geometry to the upper half-plane (c.f. Eq. (5)). Note that argzi = θ = πτ/2τ0. The main property we will repeatedly use in the following is the relation of correlation functions of (primary) operators among two geometries connected by a conformal transformation. For example, the slab geometry of above is just a two-dimensional strip whose points are labelled by a complex number w = r + iτ with 0 < Imw < 2τ0. The strip can be obtained from the upper half-plane (UHP) Imz > 0 by the conformal mapping w(z) = log z , (5) with the images of points at the same imaginary time on the strip lying along argzi = θ = πτ/2τ0. In the case where O is a product of local primary scalar operators Φi(wi), the expectation value in the strip is related to the one in the UHP by the standard transformation Φi(wi)〉strip = |w′(zi)|−xi〈 Φi(zi)〉UHP , (6) where xi is the bulk scaling dimension of Φi. Note that the (eventual) expectation values of the Φi in the ground state of H are supposed to have been subtracted off. The asymptotic real time dependence is obtained via the analytic continuation τ → τ0 + it, and taking the limit t, rij ≫ τ0. In the following subsections we apply these methods to some specific cases. A. The one-point function In the UHP, the one-point function of a scalar primary field with bulk scaling dimension x is 〈Φ(z)〉UHP = AΦb [2Im(z)] −x, as a simple consequence of scaling invariance. The normalization factor AΦb is a non-universal ampli- tude. In CFT the normalizations are chosen in such a way that the bulk two-point functions have unit amplitude (i.e. 〈Φ(z1)Φ(z2)〉bulk = |z2 − z1|−2x). This choice fixes unambiguously the amplitude AΦb that turns out to depend both on the considered field Φ and on the boundary condition on the real axis b. It vanishes if Φ corresponds to an operator whose expectation value in |ψ0〉 vanishes, and thus 〈Φ(t)〉 = 0, for all times. When the primary field is not vanishing on the boundary, performing the conformal mapping (5) we obtain 〈Φ(w)〉strip = |w′(z)|−x〈Φ(z(w))〉UHP = AΦb sin(πτ/(2τ0)) that continued to real time τ = τ0 + it gives 〈Φ(t)〉 = AΦb cosh(πt/(2τ0)) ≃ AΦb e−xπt/2τ0 . (8) Thus the order parameter (and any other observables described by a primary field) decays exponentially in time to the ground-state value, with a non-universal relaxation time tOrel = 2τ0/xOπ. The ratio of the relaxation times of two different observables equals the inverse of the ratio of their scaling dimensions and it is then universal. The normalization factor AΦb is known for the simplest boundary universality classes [37]. In the case of Φ being the order parameter and the boundary condition is fixed (ψ0(x) = ∞) AΦb is 1 for the free boson and 21/4 for the Ising model. An important exception to this law is the local energy density (or any piece thereof). This corresponds to the tt component of the energy-momentum tensor Tµν . In CFT this is not a primary operator. Indeed, if it is normalized so that 〈Tµν〉UHP = 0, in the strip [38] 〈Ttt(r, τ)〉 = πc/24(2τ0)2 (where c is the central charge of the CFT) so that it does not decay in time. Of course this is to be expected since the dynamics conserves energy. A similar feature is expected to hold for other local densities corresponding to globally conserved quantities which commute with H , for example the total spin in isotropic models. B. The two-point function In the case of the one-point function, scaling invariance was enough to fix the functional dependence on the position in the UHP. However, the form of the two-point function depends explicitly on the boundary universality class and on the operator considered. In the following subsections we will consider the equal-time correlation function for the order parameter in the gaussian and in the Ising universality classes that are easily treated in full generality. At the quantum level they describe (among the other things) a chain of harmonic oscillators (explicitly considered in Sec. IVA) and the Ising model in a transverse field (whose real time evolution has been considered in Refs. [7–9, 11] and is briefly reviewed in Sec. IVB). Finally we will discuss the general form of the two-point function for asymptotically large time and distance, that can be obtained from general CFT arguments. 1. The gaussian model The content of this subsection has been already reported in Ref. [39], during the study of the time evolution of the entanglement entropy, that transforms like the two-point function of a primary field in a boundary gaussian theory [40]. We report it here for sake of completeness. For a free boson the two-point function in the UHP is [35] 〈Φ(z1)Φ(z2)〉UHP = z12̄z21̄ z12z1̄2̄z11̄z22̄ , (9) with zij = |zi − zj | and zk̄ = zk. Note that Φ is not the gaussian field θ(z), but its exponential Φ(z) = eiθ(z) (see Sec. IVA). Under the conformal mapping (5) we obtain the two-point function on the strip at imaginary time τ , at distance r apart 〈Φ(r, τ)Φ(0, τ)〉strip = |w′(z1)|−x|w′(z2)|−x〈Φ(z1(w))〈Φ(z2(w))〉UHP = cosh(πr/2τ0)− cos(πτ/τ0) 8 sinh2(πr/4τ0) sin 2(πτ/2τ0) , (10) that continued to real time τ = τ0 + it gives 〈Φ(r, t)Φ(0, t)〉 = cosh(πr/2τ0) + cosh(πt/τ0) 8 sinh2(πr/4τ0) cosh 2(πt/2τ0) . (11) In the case where r/τ0 and t/τ0 are large this simplifies to 〈Φ(r, t)Φ(0, t)〉 = (π/2τ0)2x eπr/2τ0 + eπt/τ0 eπr/2τ0 · eπt/τ0 e−xπt/τ0 for t < r/2 e−xπr/2τ0 for t > r/2 . (12) i.e. the two point function at fixed r decays exponentially in time up to t∗ = r/2 and then saturates to a value that depends exponentially on the separation. In the case of fixed initial condition, with one-point function given by Eq. (8), the connected correlation function is 〈Φ(r, t)Φ(0, t)〉conn = 〈Φ(r, t)Φ(0, t)〉 − 〈Φ(0, t)〉2 ∝ 0 for t < r/2 , e−xπr/2τ0 − e−xπt/τ0 for t > r/2 , i.e. correlations start developing at t∗ = r/2 and, being t ≫ τ0, at t∗ the connected two-point function almost immediately jumps to its asymptotic value. In the case of disordered initial conditions (ψ0(r) = 0), connected and full correlation functions are equal. 2. The Ising universality class For the Ising model the two-point function in the UHP is [35] 〈Φ(z1)Φ(z2)〉UHP = z12̄z21̄ z12z1̄2̄z11̄z22̄ F (η) , (14) where F (η) is given by F (η) = 1 + η1/2 ± 1− η1/2√ , (15) and η is the four-point ratio z11̄z22̄ z12̄z21̄ . (16) The sign ± depends on the boundary conditions. + corresponds to fixed boundary conditions and and − to disordered ones. The only difference with respect to the gaussian case is that we have also to map F (η) according to the conformal transformation (5). After simple algebra we have 2 sin2(πτ/2τ0) cosh(πr/2τ0)− cos(πτ/τ0) , (17) and so 〈Φ(r, τ)Φ(0, τ)〉strip = cosh(πr/2τ0)− cos(πτ/τ0) 8 sinh2(πr/4τ0) sin 2(πτ/2τ0) 2 sin(πτ/2τ0) cosh(πr/2τ0)− cos(πτ/τ0) 2 sin(πτ/2τ0) cosh(πr/2τ0)− cos(πτ/τ0) . (18) Analytically continuing to real time τ = τ0 + it we obtain 〈Φ(r, t)Φ(0, t)〉 = cosh(πr/2τ0) + cosh(πt/τ0) 8 sinh2(πr/4τ0) cosh 2(πt/2τ0) 2 cosh(πt/2τ0) cosh(πr/2τ0) + cosh(πt/τ0) 2 cosh(πt/2τ0) cosh(πr/2τ0) + cosh(πt/τ0) , (19) that for r/τ0 and t/τ0 much larger than 1 simplifies to (π/2τ0) 1/4 1√ eπr/2τ0 + eπt/τ0 eπr/2τ0 · eπt/τ0 eπt/2τ0√ eπr/2τ0 + eπt/τ0 πt/2τ0 eπr/2τ0 + eπt/τ0 . (20) Note that the exponential terms in the square root are always ≪ 1. Thus for fixed boundary condition we get the free boson result Eq. (12) with x = 1/8. For the connected part we need to subtract 〈Φ(0, t)〉2 given by Eq. (8) with AΦ+ = 2 1/4. We finally obtain 〈Φ(r, t)Φ(0, t)〉conn = 〈Φ(r, t)Φ(0, t)〉 − 〈Φ(0, t)〉2 ∝ 0 for t < r/2 , e−πr/16τ0 − e−πt/8τ0 for t > r/2 . Thus, also for the Ising model with fixed boundary conditions, connected correlations start developing at t = t∗ = r/2. In the case of disordered initial condition, we have 〈Φ(r, t)Φ(0, t)〉 ∝ eπr/2τ0 + eπt/τ0 eπr/2τ0 · eπt/τ0 eπt/2τ0√ eπr/2τ0 + eπt/τ0 e−π(r−3/2t)/4τ0 for t < r/2 , e−πr/16τ0 for t > r/2 , resulting in an exponential space dependence even for t < r/2 (clearly in this case the connected correlation function equals the full one). 3. The general two point-function From the results reported for the gaussian and Ising models, it is now relatively simple to understand the general properties of the time dependence of the two-point function in the very general case. The two-point function in the half-plane has the general form [35] 〈Φ(z1)Φ(z2)〉UHP = z12̄z21̄ z12z1̄2̄z11̄z22̄ F (η) , (23) ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ FIG. 2: Left: Space-time region for the correlation functions at different times. Right: Conformal mapping to the upper half-plane. where the function F (η) depends explicitly on the considered model. Under the conformal map to the strip we know that the first part of Eq. (23) transforms according to Eq. (10). Thus we need only to map F (η) that, in the general case, is an unknown function. However, during the study of the Ising model we showed that the analytical continuation of η for t, r ≫ τ0 is η ∼ e πt/τ0 eπr/2τ0 + eπt/τ0 . (24) Thus for t < r/2 we have η ∼ eπ(t−r/2)/τ0 ≪ 1 and in the opposite case t > r/2 we have η ∼ 1. As a consequence to have the asymptotic behavior of the two-point function we only need to know the behavior close to η ∼ 0 (i.e. the behavior close to the surface) and for η ∼ 1 (i.e. deep in the bulk). Fortunately they are both exactly known. Indeed when η ∼ 1 the two points are deep in the bulk, meaning F (1) = 1. Instead for η ≪ 1, from the short-distance expansion, we have F (η) ≃ (AΦb )2ηxb , (25) where xb is the boundary scaling dimension of the leading boundary operator to which Φ couples and A b is the bulk-boundary operator product expansion coefficient that equals the one introduced in Eq. (8) [see e.g. Ref. [37]]. All the previous observations and the explicit calculations of the previous sections lead for t > r/2 to 〈Φ(r, t)Φ(0, t)〉 ∝ e−xπr/2τ0 , (26) while for t < r/2 we get 〈Φ(r, t)Φ(0, t)〉 ∝ (AΦb )2e−xπt/τ0 × eπxb(t−r/2)/τ0 . (27) Note that if 〈Φ〉 6= 0, xb = 0 and the last factor is absent. The leading term is then just 〈Φ〉2. Thus the leading term in the connected two-point function vanishes for t < r/2, and its first non-vanishing contributions is given by subleading terms either in F or in the bulk-boundary short-distance expansion. It is very interesting that we only have to know the behavior as η → 0 and 1 to get the results we need for large r and t. However, we stress that only a complete calculation (as those performed in the preceding sections) gives the full analytic structure of the CFT result needed to justify the analytical continuation from imaginary to real time. Moreover, the behavior within a distance O(τ0) of the horizon r = 2t depends on the detailed form of F . C. Correlations functions at different times Let us consider the case of the two-point function calculated at different real times 〈Φ(r, t)Φ(0, s)〉. This is again obtained by mapping the imaginary time strip to the UHP, but in this case the two points are w1 = r + iτ1 and w2 = 0+ iτ2, that, at the end of the calculation, must be analytically continued to τ1 = τ0 + it and τ2 = τ0 + is. See Fig. 2 for a pictorial representation of the space-time domain in the strip and the resulting mapping to the UHP. Let us start the discussion with the free boson. The distances appearing in Eq. (9) are z212 = 1 + e πr/τ0 − 2eπr/2τ0 cos(θ1 − θ2) , z11̄ = 2 sin θ1 , z212̄ = 1 + e πr/τ0 − 2eπr/2τ0 cos(θ1 + θ2) , z22̄ = 2eπr/2τ0 sin θ2 , (28) where θi = πτi/2τ0. Thus the correlation function on the strip is 〈Φ(r, τ1)Φ(0, τ2)〉strip = |w′(z1)|−x|w′(z2)|−x〈Φ(z1(w))〈Φ(z2(w))〉UHP = (29) cosh(πr/2τ0)− cos(π(τ1 + τ2)/2τ0) 4 sin(πτ1/2τ0) sin(πτ2/2τ0)(cosh(πr/2τ0)− cos(π(τ1 − τ2)/2τ0)) that for τ1 = τ2 reduces to Eq. (10) as it should. Continuing to real times and considering r, t, s, |t − s| ≫ τ0 we obtain τ0 + e (t+s) (t+s) τ0 + e |t−s| e−xπ(t+s)/4τ0 for r > t+ s , e−xπr/4τ0 for t− s < r < t+ s , e−xπ|t−s|/4τ0 for r < |t− s| . Following the line sketched in the previous subsection, it is easy to generalize this result to the most general CFT. In the case of a theory with fixed initial conditions (i.e., 〈Φ〉 6= 0) the asymptotic result is the same as before, with only the crossover points being affected by the precise expression for F (η). Instead, in the case where 〈Φ〉 = 0, the first case gains an additional factor e−πxb(t+s−r)/4τ0 . Note that the autocorrelation function (i.e. r = 0) has only an exponential dependence on the time separation t− s and does not exhibit aging in this regime. D. Evolution with boundaries We now consider the case of time evolution of a half-chain with some boundary condition at r = 0. For simplicity we assume that the (conformal) boundary condition is of the same kind of the initial boundary condition (for example we fix all the spins at t = 0 and at the boundary r = 0 to point in the same direction). The space-time region we have to consider is depicted in Fig. 3. If different initial and boundary were considered, one needs to insert boundary conditions changing operators at the corners of the figure. The w plane is mapped into the UHP by z(w) = sin , (31) with the corners at ±τ0 mapped to ±1. The mapping of w1 is z1 ≡ z(w1) = z(−τ0 + τ1 + ir) = − cos(πτ1/2τ0) cosh(πr/2τ0) + i sin(πτ1/2τ0) sinh(πr/2τ0) . (32) In the z plane the 1-point function is 〈Φ(z1)〉UHP ∝ |Imz1|−x → [sin(πτ1/2τ0) sinh(πr/2τ0)]−x , (33) |w′(z1)|2 = |1− z2| cosh(πr/τ0)− cos(πτ1/τ0) . (34) Thus on the strip we have 〈Φ(w1)〉strip = |w′(z1)|−x〈Φ(w(z1))〉UHP ∝ sin2(πτ1/2τ0) sinh 2(πr/2τ0) cosh(πr/τ0)− cos(πτ1/τ0) ]−x/2 , (35) that continued to real time τ1 = it is 〈Φ(t, r)〉 ∝ cosh(πt/τ0) + cosh(πr/τ0) cosh(πt/2τ0)2 sinh 2(πr/2τ0) , (36) and for t, r ≫ τ0 simplifies to 〈Φ(t, r) ∝ eπr/τ0 + eπt/τ0 eπr/τ0 · eπt/τ0 e−πxt/2τ0 for t < r , e−πxr/2τ0 for t > r . Note that in this case the characteristic time is t∗ = r and not r/2. This explains also why the entanglement entropy of a semi-infinite chain with free boundary condition at x = 0, has characteristic time t∗ = r as firstly noted in Ref. [41]. �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� ���������������� ���������������� ���������������� ���������������� ���������������� ���������������� 0 1−1 FIG. 3: Left: Space-time region for the one-point function in a boundary (at r = 0) geometry. Note w = τ + ir. Right: Conformal mapping to the upper-half plane, c.f. Eq. (31). E. Discussion and interpretation of the CFT results All the correlation functions calculated so far display two very general features: first there is a sharp horizon (or light-cone) effect at t = t∗ = r/2 (or r) resulting in a behavior before and after t∗ completely different; second the asymptotic long-time correlation functions are the same as those at finite temperature βeff = 4τ0. The light-cone effect is a very general phenomenon and and will be discussed in section VI. In the following sections we will point out that also the “effective temperature” is a general phenomenon, but it has some specific CFT features that are worthy of comment. In fact, it is very easy to understand the technical reason why we find an effective temperature despite the fact that we are studying a pure state at T = 0. The finite temperature correlations can be calculated by studying the field theory on a cylinder of circumference β = 1/T . In CFT a cylinder can be obtained by mapping the complex plane with the logarithmic transformation β/(2π) log z. The form for the two-point function in the slab depends in general on the function F (η) –cf. Eq. (15)– but when we analytically continue and take the limit of large real time, we find that effectively the points are far from the boundary, i.e. at η = 1. Thus we get the same result as we would get if we conformally transformed from the full plane to a cylinder, and from Eq. (5) the effective temperature is βeff = 4τ0. A similar argument can be worked out for the multi-point functions as well. An addtional comment concerns what we expect for correlation functions of general operators, not only primary. We have clearly seen that the local energy density does not relax, of course, and this is consistent with its not being primary. But, we also know that at finite temperature 〈Ttt〉β = πc/6β2, that is perfectly compatible with the previous result with βeff = 4τ0. Furthermore, for large real times, the two-point function of Ttt − 〈T 〉tt does behave as though it was at finite temperature. This means, in particular, that the energy fluctuations in a large but finite volume (the specific heat) are the same as those at finite temperature. Thus one is tempted to extend this finite temperature interpretation to non-primary operators, and indeed it is the case. In fact, in the argument of above for the equivalence of the long-time correlations and finite T , there are essentially three steps. First, we need to write down the form of the correlation function in the half-plane, but this only depends on special conformal transformation and so is valid for any quasi-primary operator like Tµν . Second we have to transform from the slab to the half-plane: for non-primary operators this can have some anomalous term. Finally we need to compare the large limit of this to what one would get transforming directly from the cylinder to the full plane. However, the two conformal transformations we are comparing are both logarithmic (in one case 2τ0/π log z, in the other case β/2π log z), so the anomalous terms should be the same. Thus the two correlations functions are the same. This argument works for all quasi-primary operators. Then, since we can get all the non-quasiprimaries by considering successive operator product expansions with the stress tensor, it also works for all operators. IV. EXACT REAL-TIME DYNAMICS IN SIMPLE INTEGRABLE CHAINS The results of the previous section rely on the technical assumption that the leading asymptotic behavior given by CFT, which applies to the euclidean region (large imaginary times), may simply be analytically continued to find the behavior at large real time. While such procedures have been shown to give the correct behavior for the time-dependent correlations in equilibrium, it is important to check them in specific solvable cases for non-equilibrium evolution. Thus, as a complement to the CFT calculations, in this section we consider the real-time evolution of two simple analytically tractable models. We solve the dynamics of a chain of coupled harmonic oscillators and we review and re-analyze some known results for the Ising-XY chain in a transverse magnetic field. Beyond providing examples of the CFT results (with central charge c = 1 and 1/2 respectively) in the critical case, these models allow us to take into account the effects of a finite mass-gap and of the lattice. A. The chain of harmonic oscillators The simplest model with an exactly solvable non-equilibrium dynamics is surely a chain of coupled harmonic oscillators with hamiltonian π2r +m 2φ2r + ω2j (φr+j − φr)2 . (38) We introduce a coupling more general than simple nearest-neighbor hopping so as to allow for a general dispersion relation below. For simplicity we also assume periodic boundary conditions. ϕn and πn are the position and the momentum operators of the n-th oscillator, with equal time commuting relations [ϕm, πn] = iδnm [ϕn, ϕm] = [πn, πm] = 0 . (39) The hamiltonian can be written in diagonal form H(m) = k ΩkA kAk with modes (Ωkϕk + iπk) , (40) (Ωkϕ−k − iπ−k) , (41) Ω2k = m 2 + 2 ωj (1− cos(2πkj/N)) . (42) Note that we use the same symbols for the operators and their Fourier transforms (ϕk = 1/ n=0 e 2πikn/Nϕn and analogously for πk). We consider the scenario in which the system is prepared in a state |ψ0〉, that is ground-state of H(m0), and at the time t = 0 the mass is quenched to a different value m 6= m0. We use the notation Ω0k for the dispersion relation for t < 0 and the Ωk for the one for t > 0. Since 〈ψ0|ϕn|ψ0〉 = 0, the expectation value of the field ϕn vanishes at any time. This example in fact corresponds to the quench from the disordered phase in the language of the previous section. Thus we concentrate our attention on the two-point function 〈ψ(t)|ϕnϕ0|ψ(t)〉 = 〈ψ0|ϕHn (t)ϕH0 (t)|ψ0〉 , (43) where we introduced the operator in the Heisenberg picture ϕHn (t), whose time evolution is given by ϕHn (t) = ei(pkn−Ωkt)Ak + e −i(pkn−Ωkt)A , (44) where pk = 2πk/N . Accordingly, the product of the two fields is ϕHn (t)ϕ 0 (t) = ΩkΩk′ ei(pkn−Ωkt)Ak + e −i(pkn−Ωkt)A e−iΩk′ tAk′ + e . (45) In order to have the time dependent two-point function we need the expectation values of the bilinear combinations of A’s on the initial state, that is annihilated by the A0k. Thus it is enough to write A’s as functions of A0’s, i.e. ≡ ckA0k + dkA†0−k , (46) +A0−k ≡ ckA†0k + dkA0−k , (47) leading to (we understand 〈·〉 = 〈ψ0| · |ψ0〉) 〈AkAk′ 〉 = 〈(ckA0k + dkA†0−k)(ck′A0k′ + dk′A 0−k′ )〉 = ckdk′〈A0kA 0−k′〉 = ckdkδk,−k′ , (48) 〈AkA†k′ 〉 = 〈(ckA0k + dkA 0−k)(ck′A 0k′ + dk′A0−k′ )〉 = ckck′〈A0kA 0k′ 〉 = c kδk,k′ , (49) 〈A†kAk′ 〉 = 〈(ckA 0k + dkA0−k)(ck′A0k′ + dk′A 0−k′ )〉 = dkdk′〈A0−kA 0−k′〉 = d kδk,k′ , (50) 〈A†kA k′ 〉 = 〈(ckA 0k + dkA0−k)(ck′A 0k′ + dk′A0−k′ )〉 = dkck′〈A0−kA 0k′ 〉 = ckdkδk,−k′ . (51) Finally we arrive at 〈ϕHn (t)ϕH0 (t)〉 = ckdke i(pkn−2Ωkt) + c2ke ipkn + d2ke −ipkn + ckdke −i(pkn−2Ωkt) , (52) which, in the thermodynamic limit (N → ∞), may be written as 〈ϕHr (t)ϕH0 (t)〉 − 〈ϕHr (0)ϕH0 (0)〉 = (Ω20k − Ω2k)(1 − cos(2Ωkt)) Ω2kΩ0k , (53) where the integral is on th first Brillouin zone |k| < π/a. Note that for t = 0 and for m = m0 this two-point function reduces to the static one, as it should. This result can also be found by integrating the Heisenberg equations of motion for each mode. For future reference it is also useful to write down explicitly the Fourier transform known as momentum distribution function ρ(k) = (Ω20k +Ω k)− (Ω20k − Ω2k) cos(2Ωkt) Ω2kΩ0k . (54) Note that when considering correlation functions in momentum space, a long-time limit does not exist and we need to take the time average, in contrast to what happens in real space. 1. The continuum limit In Eq. (52) everything is completely general and applies to any chain with finite lattice spacing. Let us know discuss the continuum limit that is achieved by sending N → ∞ in such a way that (1/N) dp/(2π), pn → p, and Ω2k → Ω2p = m2 + p2. In this limit the correlation function becomes (∆m2 = m20 −m2) G(r, t) ≡ 〈ϕHr (t)ϕH0 (t)〉 = −∆m2 cos(2 p2 +m2t) +m2 +m20 + 2p (m2 + p2) m20 + p . (55) Since a closed form for such integral is quite difficult to write down in the most general case, we will only consider some particular cases. Let us first consider the conformal evolution (m = 0) from an initial state with a very large mass m0 → ∞. This should reproduce the CFT result previous section for all the time t, since the correlations of the initial state shrink to a point. We have G(r, t) = m0 1− cos(2pt) 0 for t < r/2 , (2t− r) for t > r/2 . To compare such result with the conformal result given by Eq. (12), we have to keep in mind that the primary field is not the gaussian one ϕH(r, t), but its imaginary exponential. Thus we need 〈eiqϕH (r,t)e−iqϕH (0,t)〉, with an arbitrary q. Despite of the apparent complexity of such correlator, it is very simple to obtain it using the standard property of gaussian integrals 〈eiqϕ H (r,t)e−iqϕ H(0,t)〉 = e−q 2〈(ϕH(r,t)−ϕH(0,t))2〉/2 = eq 2(G(r,t)−G(0,t)) , (57) leading to 〈eiqϕ H (r,t)e−iqϕ H (0,t)〉 = 2m0t for t < r/2 , 2m0r/2 for t > r/2 , 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 0 1 2 FIG. 4: Left: G(r, t)/m0 given by Eq. (60) as function of t, at fixed r = 1. Three different values of m0 = 10, 3, 1 (from the bottom to the top) are shown. Inset: Lattice effects showing the cos 4t oscillations on top of the continuum result. Right: G(r, t)/m0 given by the numerical integral of Eq. (62) as function of t, at fixed m, r = 1. It is compared with the asymptotic behavior for 0 < 2t− r ≪ m−1 and for t ≫ r. that is exactly the same of Eq. (12) with xΦ ∝ q2 and τ0 ∝ m−10 , confirming that τ0 is just proportional to the correlation in the initial state, as its interpretation in terms of the extrapolation length suggests. The case m = 0 and m0 finite corresponds to a conformal evolution from a generic state with correlations propor- tional to m−10 . Thus we expect the CFT result Eq. (12) to be true for asymptotic large times and separations. From Eq. (55), the two-point function of the gaussian field is G(r, t) = −m20 cos(2pt) +m20 + 2p2 m20 + p , (59) that can be written as 2πG(r, t) = 2K0(m0r) + f(m0r) − f(m0(r − 2t)) + f(m0(r + 2t)) , (60) where K0(y) is the modified Bessel function and f(y) = 1 + G2113 0 1 1/2 , (61) with G2113 the Meijer G-function (see e.g. [42]). Note that f(x) is characterized by f ′′(x) = −K0(x) and f(0) = f ′(0) = 0. In the limit m0 → ∞, it is easy to show that G(r, t) reduces to the previous result. In the left panel of Fig. 4 , we report G(r, t) as function of t at fixed r = 1 for m0 = 10, 3, 1. It is evident that a finite m0 results in smoothing the curve close to t = r/2 and giving an offset in zero. Both the effects are more pronounced as m0 decreases. For large t, independently on m0 6= 0, we have G(r, t) = t + O(t0), confirming that the CFT result is correct for asymptotic large times. The case with arbitrary m and m0 is quite cumbersome to be worked out analytically and not really illuminating. For this reason we concentrate here on the massive evolution from a state with m0 → ∞. In this case the correlation function reads G(r, t) = m0 1− cos(2 p2 +m2t) (m2 + p2) . (62) In the limit t→ ∞ the cosine term averages to zero, giving G(r, t = ∞) = m0 (m2 + p2) . (63) To understand the time dependence, let us first note that, despite the presence of a square-root, the integrand of Eq. (62) is analytic in p, since the square-root is the argument of the even cosine function. Thus we can make the integral in the complex plane and use the Cauchy theorem. As long as r > 2t, the behavior for p→ i∞ is dominated by eipr and we can safely close the contour path in the upper half-plane where the residue at p = im is zero. As a consequence G(r, t < r/2) = 0. For t > r/2, the integral is more difficult, since we can not close the contour in the upper half-plane, because the cosine is “larger” than eipr for p→ i∞. However, the approach to the asymptotic value is easily worked out. In fact, for t≫ r, the term eipr in the integral (62) can be approximated with 1, since it is slowly oscillating. Thus we have G(r, t ≫ r)−G(r, t = ∞) ≃ −m0 cos(2 p2 +m2t) (m2 + p2) = −m0fm(t) , (64) where fm(t) = 1/(2m) − t1F2(1/2; 1, 3/2;−m2t2), that satisfies f ′m(t) = −J0(2mt) and fm(0) = 1/(2m). Note that fm(t) is just the result for r = 0. In the complementary region 0 < 2t−r ≪ m−1, the integral in Eq. (62) is dominated by the modes with p≫ m, and so it can be described by the conformal result. In the right panel of Fig. 4, we plot the time-dependence of G(r, t) at fixed r as obtained by numerically integrating Eq. (62) for m = r = 1. The plot shows that the conformal result t − r/2 describes the behavior close to t = r/2, while for larger times the asymptotic expression gives an excellent approximation. We finally note that G(r, t) at fixed t displays spatial oscillations (for r < 2t, else it vanishes), as it can be simply realized by a stationary phase argument. 2. Lattice effects Another advantage of this simple model is that the effects of the lattice can be easily understood. In fact, by a stationary phase argument, the dominant contribution to (53) in the limits of large t and r comes from where, the group velocity vk ≡ Ω′k = r/2t, independently of the explicit form of Ωk. Consequently, the two-point function of a gapless model with dispersion relation Ωk = 2 sink/2 differs from the continuum limit previously derived for t > r/2vm, where vm is the maximum group velocity. In particular G(r, t) receives a contribution from the slowest mode (k = π with vπ = 0) whose effect is to add to Eq. (56) fast oscillations going as cos(2Ωπt) = cos 4t. The resulting G(r, t) is plotted in the inset of Fig. 4 in the gapless case. Thus, lattice effects play a more important role in the cases where the asymptotic result vanishes. A typical example is the energy density that in this model is proportional to (ϕr+1 − ϕr)2. The continuum calculation would just give zero (the mean-energy is conserved), but the approach to this value is governed by lattice effects and is dominated by the smallest group velocity that comes from the zone boundary at |k| = π. In fact, just with a trivial calculation we 〈(ϕr+1 − ϕr)2〉 = 2(〈ϕ20〉 − 〈ϕ0ϕ1〉) = (1− cos k) (1− cos 2Ωkt) 2 sin2 k/2 (1− cos 2Ωkt) ∝ J0(4t) ∼ t−1/2 cos 4t , (65) that displays the typical cos 4t from the zone boundary at |k| = π. The lattice dispersion relations are sensitive to the microscopical details of the model and consequently quantities like the energy-density show a dependence on this. For example, a lattice massless fermion has dispersion relation Ωk = 2 sink/2 and energy density given by sin 2 k. In this case the time evolution of the energy density is sin2 k (1− cos 2Ωkt) 2 sin2 k/2 , (66) that for large times goes like t−3/2 cos 4t, resulting in a different power law compared to before. If the system is quenched to a gapped H with Ω2k = m 2 + 2(1 − cos k) the maximum group velocity corresponds to a non-zero wave number. This gives rise to spatial oscillations in the correlation function (this is true also in the continuum limit as already discussed in the previous subsection). B. The Ising-XY chain in a transverse magnetic field The most studied one dimensional quantum spin model is the so-called XY chain in a transverse field, defined by the hamiltonian (1 + γ) σxj σ j+1 + (1− γ) j+1 − hσz , (67) where σ x,y,z i are the Pauli matrices, γ is called anisotropy parameter, and h is the applied external transverse field. It is well known that for any γ 6= 0 the model undergoes a phase transition at h = 1, that is in the universality class of the Ising model (defined by γ = 1). For simplicity we will just consider the Ising case, other values of γ 6= 0 being equivalent. We consider the non-equilibrium unitary dynamics that follows from a quench of the magnetic field at t = 0 from h0 to h1 6= h0. Earlier works on this subject by Barouch et al. date back to seventies [7, 8]. Exploiting the mapping of this model onto a free fermion, the time evolution of the transverse magnetization (that is not the order parameter) was obtained exactly [7]: mz(t) = (h0 − h1) sin2 k cos(2ǫ1t)− (cos k − h1)[(cos k − h0)(cos k − h1) + sin2 k] , (68) where ǫi = ǫ(hi) with ǫ(h) = sin2 k + (h− cos k)2. In Ref. [7] it was shown that (for non-exceptional parameters) the approach to the asymptotic value for t→ ∞ is of the form t−3/2 cos 4t. This is simply shown in the case of h0 = ∞ and h1 = 1, when the integral simplifies and we obtain (see also [9]) mz(t) = sin2 k cos(2ǫ1t) 4 sin2 k/2 J1(4t) , (69) where J1 is the Bessel function of the first kind whose asymptotic expansion for large argument is J1(x) ∼ 2/πx cos(x+ π/4). To understand this result we should keep in mind that the transverse magnetization is not the order parameter: it corresponds to the product of two “disorder parameters” at neighbor sites on the dual lattice. Thus it must have the symmetry of an energy operator, for which CFT just predicts a constant asymptotic result. The power law term t−3/2 cos(4t) is just the lattice correction to the asymptotic. This correction clearly shows the fermionic nature of the model on the basis of what discussed in the previous section. The asymptotic result of the two-point function for t → ∞ has been studied in Ref. [11]. Calling Gn = 〈Φ(n,∞)Φ(0,∞)〉, in the case h0 = ∞ it has been found 2nhn1 , forh1 ≥ 1 , cos[n arccos(h1)] , forh1 ≤ 1 , instead for h0 = 0 , forh1 ≥ 1 , (n+ 1) ln , forh1 ≤ 1 , that for large n decay exponentially with n. It has been shown that Gn is decaying exponentially with n for general h0 and h1 [11], although closed forms are not available. The exponential decay is the prediction of CFT that (maybe surprisingly) applies even far from the critical point h1 = 1. The time dependence of two-point function has been studied in Ref. [9] by means of exact diagonalization of the model with open boundary condition at the two ends r = 0, L. The results of interest for this paper are • The connected two-point function of σz in the thermodynamic limit and at the critical point is 〈σzr (t)σz0(t)〉c = J2r(4t) 2 − 1 J2r+1(4t)J2r−1(4t) , (72) which is valid both for h0 = 0,∞. Neglecting fast oscillations, 〈σzr (t)σz0(t)〉c increases as r2 for r < 2t and then it drops almost immediately to 0. • The autocorrelation function (not connected) of σz at different times is 〈σz0(t1)σz0(t2)〉 = J20 (2t2 − 2t1)− [f(t2 + t1)± g(t2 − t1)] , (73) where f(x) = J2(2x) + J0(2x), g(x) = J2(2x)− J0(2x) and the sign + (−) refers to h0 = 0 (h0 = ∞). Let us comment on these results in view of the general understanding we found so far. For t < t∗ all the connected correlation functions are zero, in agreement with CFT. All the oscillation terms of the asymptotic form cos(4t) are, as we discussed in the previous section, a lattice effect. Concerning the σz correlator, the r 2 dependence for t > r/2 is a consequence of the fact that σz is not primary. The same is true for the two-time correlations function of σz that also decays as a power law of t− s for large times, instead of the exponential prediction by CFT for primary field. V. HIGHER DIMENSIONS Until now we just considered one-dimensional systems. Despite the fact that in low dimensions the effect of fluctu- ations is more pronounced making the physics highly non-trivial it is desirable to have results in higher dimensions as well. The method presented in Sec. II to obtain the non-equilibrium dynamics of a quantum model close to a critical point from the critical behavior of a system confined in a slab geometry applies to generic dimension d through the study of the hamiltonian (4). Its analysis proceeds via field-theoretical RG that may provide the all scaling quantities of the model in an expansion close to the upper critical dimensions (u.c.d.), that is D = d+ 1 = 4. Above the u.c.d. mean-field (or gaussian) results are exact, with logarithmic correction at the u.c.d.. For dimensions lower than the u.c.d., the scaling quantities are obtained as series in ǫ = 4 − (d + 1). Thus for the time-evolution problem the simple mean-field solution represents an exact scaling result (a part log corrections) for the physically relevant three-dimensional case. An alternative method to attack analytically the hamiltonian (4) is to consider an N component field φ and taking the limit N → ∞, but this will not be employed here. A. Dirichlet boundary conditions: the two-point function The D-dimensional slab geometry with Dirichlet boundary conditions has been the subject of several investigations. The two-point function has been calculated at the first order in ǫ expansion in Ref. [43]. The gaussian two-point function, with partial Fourier transform in the parallel directions reads [43] G(p, z1, z2) = e−b|z1−z2| − e−b(z1+z2) + e −b(z1−z2) + e−b(z2−z1) − e−b(z1+z2) − eb(z1+z2) e2bL − 1 , (74) with b = p2 +m2. We are interested in the case where L = 2τ0, z1 = z2 = τ that we will analytically continue to τ → τ0 + it, and for computational simplicity we will restrict to the massless case m = 0. Thus in real space and imaginary time, we have (p = |p|) G(r, τ) = (2π)3 e−ip·r 1− e−2pτ + 2(1− cosh(2pτ)) e4pτ0 − 1 (2π)2 1− e−2pτ + 2(1− cosh(2pτ)) e4pτ0 − 1 d(cos θ)eipr cos θ = (2π)2r dp sin pr 1− e−2pτ + 2(1− cosh(2pτ)) e4pτ0 − 1 . (75) This integral can be performed by making the sum over all the residues coming from the denominator e4pτ0 − 1. The calculation is rather involved, but the final result is very simple: G(r, τ) = coth(πr/4τ0) sin 2(πτ/2τ0) cosh(πr/2τ0)− cos(πτ/τ0) . (76) Continuing to real time τ = τ0 + it we obtain G(r, t) = coth(πr/4τ0) cosh 2(πt/2τ0) cosh(πr/2τ0) + cosh(πt/τ0) , (77) that for t, r ≫ τ0 simplifies to G(r, t) ≃ 1 eπt/τ0 eπt/τ0 + eπr/2τ0 eπ(t−r/2)/2τ0/r for t < r/2 , 1/r for t > r/2 . We recall that this result in d = 3 is exact a part log corrections. Thus the basic structure of the two-point function in 3D is the same as in 1D, with a characteristic time t∗ = r/2. Using the result in Ref. [43] it is in principle possible to calculate the correlation functions for d < 3 in the ǫ expansion framework. However, this requires the analytical continuation of complicated functions, resulting in a quite cumbersome algebra, as we shall see in the following for a simpler observable. Eq. (74) can be used in principle to determine the gaussian behavior in any d < 3. Unfortunately the integral one gets is not analytically tractable for d 6= 1, 3. A possible strategy would be to perform the analytic continuation before of the integral and then evaluate it through a saddle-point approximation. It is straightforward to show that the result obtained in this way is equivalent to what we discuss in section VD where we remand for the analysis of 1 < d < 3. B. Dirichlet boundary conditions: a non-trivial one-point function In the case of Dirichlet boundary conditions, the order parameter profile in the slab geometry is trivially vanishing. However not all the one-point expectation values are zero. Let us consider as a typical example the operator O = φ2 that has been calculated at the first order in ǫ expansion in Ref. [44]. It has the scaling form 〈O(z, L)〉 ≃ L−d+1/νH(z/L) , (79) where ν is the correlation length exponent. In D = 4, the function H(x) is [44] H(x) = sin2(πx) , (80) that, using L = 2τ0 and continuing to z = τ = τ0 + it, leads to the real time evolution 〈O(t)〉 − 〈O(t = ∞)〉 ≃ τ−20 (πt/2τ0) ∼ τ−20 e−πt/2τ0 , (81) i.e. 〈O(t)〉 approaches its asymptotic value exponentially, as it as been found for all primary operators in d = 1. The one-loop result for the O(N) model in D = 4− ǫ dimensions is [44] H(x) = sin2(πx) 1 + ǫ N + 2 N + 8 sinπx − ǫ[ζ′(2, x) + ζ′(2, 1− x)] + const , (82) where const stands for terms do not depend on x and ζ′(α, x) = ∂αζ(α, x) and ζ is the generalized Riemann function. This example clearly shows that within the ǫ expansion, the scaling functions contain logarithmic contributions that originate from the expansion of power law as e.g. sinǫ πx = 1 + ǫ ln sinπx. To have a function with good analytical structure to perform the real time continuation, it is desirable to “exponentiate” such logarithms. In Ref. [44] the exponentiation procedure leaded to H(x) = sin(πx) )2−1/ν [ζ(D − 2, x) + ζ(D − 2, 1− x)− 2ζ(D − 2)] (2 − 1/ν) sin(πx) )D−2−1/ν , (83) with ν = 1/2 + (N + 2)/(N + 8) ǫ/4. Performing the analytical continuation we have the sum of two exponentials, and in any dimension d < 3 the second one has a largest “relaxation time” that hence is dominating for large t. Considering only the second term we have 〈O(t ≫ τ0)〉 − 〈O(t = ∞)〉 ∝ e−(d−1−1/ν)πt/2τ0 . (84) Note that the subleading term e−(2−1/ν)πt/2τ0 is multiplied by a log t term arising from the ζ function. This example put forward the idea that (at least for Dirichlet boundary conditions) the exponential relaxation of the one-point functions is not only a property of one-dimensional systems, but holds in any dimension (with eventually log corrections) with relaxation times related to the scaling dimensions of the operator. C. Fixed boundary conditions: the one-point function The case of the extraordinary transition, that corresponds to fixed boundary conditions, has been considered in Ref. [45]. The magnetization profile can be written as [45] φ(z) = dn(2K4z/L) sn(2K4z/L) , (85) where K4 = K(k4), where K(k) is the elliptic integral, k4 the elliptic modulus that in terms of the parameter of the model is φ2L2 = (2K4) 2(2k24 − 1), and sn(x) and dn(x) and the Jacobi functions. Continuing to real time, and using the properties of the Jacobi functions we obtain φ(t) = 1− k4cn(K4t/τ0, 1− k4) , (86) that, contrarily to all the other cases we have considered, is oscillating and not exponentially decreasing. To our knowledge there are no result in the ǫ-expansion for the magnetization profile. However, using “local- functional methods” [46] it has been obtained an approximate profile in D = 3 that involves, as in mean-field theory, Jacobi elliptic functions. It can be easily continued to real time via z → τ0 + it and again one finds an oscillating behavior with time. The method exploited in Ref. [46] can be used in any dimensions, obtaining an always an oscillating behavior with a period that diverges as D approaches 2, recovering Eq. (7). All these calculations are essentially mean field and we do not know how the inclusion of fluctuations changes them. It can possible that for the extraordinary transition (i.e. fixed initial conditions), the exponential decay founds at D = 2 is more an exception rather than a rule, because of the simple analytic structure of the trigonometric functions in the complex plane. Another possibility is that fluctuations destroy these oscillations. Only a complete analytical calculation (e.g. in large N) can help in understanding this point. D. A real-time solvable model As for the one-dimensional case, it is worth to check the results coming from the analytical continuation of large imaginary-time with exactly solvable models. The simplest (and probably one of the few) model solvable in generic dimension is the generalization of the hamiltonian (38) to a d-dimensional hypercubic lattice. The solution of such model proceeds via Fourier transform as in one dimension. The final result is simply give by Eq. (53) with the replacement dk → ddk, i.e. 〈ϕr(t)ϕ0(t)〉 − 〈ϕr(0)ϕ0(0)〉 = (2π)d eik·r (Ω20k − Ω2k)(1 − cos(2Ωkt)) Ω2kΩ0k . (87) First of all let us note that this expression in virtually identical to Eq. (75) if we take τ0 ∝ m−10 → 0. In fact, taking τ0 → 0 in Eq. (75), only the third term matters, since it is O(τ−10 ) relative to the first two. Taking z = r + it and b = Ωp we get G(p) ∝ τ−10 Ω−2p (1− cos 2Ωpt) that is Eq. (87) for m0 → ∞. To understand the general features let us consider in details the conformal evolution (Ωk = v|k|) from a disordered state (Ω0k = m0). The derivative of the two-point function is (k = |k|) ∂t〈ϕr(t)ϕ0(t)〉 = 2m0Im (2π)d ei(k·r−2kvt) . (88) Except for d = 1, this can be done analytically only in d = 3, where we can write it as dθ sin θeik(r cos θ−2vt) ∼ sin(kr)/r e2ikvt ∼ (1/r)δ(vt− r/2) . (89) Integrating with respect to t we get zero for t < r/2v and 1/r for t > r/2v. For general d we have ∂t〈ϕr(t)ϕ0(t)〉 ∝ 2m0Im kd−2dk dθ(sin θ)d−2eik(r cos θ−2vt) . (90) By a saddle point argument, we can assume that the dominant behavior comes from θ close to zero, so the θ integral gives dθθd−2e−ikrθ 2/2 ∼ (kr)−(d−1)/2 , (91) leading to (kr)(d−1)/2 ei(k(r−2vt)) ∼ r−(d−1)/2(2vt− r)−(d−1)/2Θ(vt− r/2) . (92) Integrating with respect to t, we get zero for t < r/2v, as expected by causality, and r−(d−1)/2(2vt− r)(3−d)/2 , (93) for t > r/2v. It is interesting that this gaussian correlation function blows up at large t only for d < 3, when we expect the fluctuations to become important. It would be interesting to study this in the φ4 theory for large N , by replacing φ4 by 3〈φ2〉φ2 in the usual way, where now 〈φ2〉 depends on t and is calculated self-consistently. VI. PHYSICAL INTERPRETATION AND DISCUSSION In this paper we studied in general the non-equilibrium unitary dynamics that follows a sudden quantum quench. We showed that if the hamiltonian H governing the time evolution is at a critical point, while H0 (i.e. the one for t < 0) is not, the expectation value of a class of operators (primary ones in CFT) relaxes to the ground-state value exponentially in time with universal ratio of decaying constants. We also found that connected two-point functions of operators at distance r are vanishing for t < r/2v, while for t > r/2v reach exponentially fast a value that depends exponentially on the separation, in contrast with the power laws typical of equilibrium configuration. We also considered the real-time dynamics of simple exactly solvable models and we found that several of the typical characteristics of the critical points still hold. Roughly speaking, critical points are not special as far as quenching dynamics is concerned. In fact, also for gapped systems, connected correlation functions vanish (or are strongly suppressed) for t < r/2v and for asymptotic large times resemble those at finite temperature despite the fact that the whole system is in a pure state. Several other examples in the recent literature (see e.g. [9, 15, 24, 26, 39, 41]) gives further evidence that these two effects are actually true in general, at least in the realm of exactly solvable models considered so far. In the following we give a simple interpretation of these two features separately trying to understanding their physical origin. A. The horizon effect The qualitative, and many of the quantitative, features found for the time evolution of correlation functions may be understood physically on the basis of a picture we first introduced in Ref. [39] to describe the time evolution of the entanglement entropy. Later we generalized it to correlation functions in Ref. [33] and it has been largely adopted thereafter [2, 24, 26, 41]. We emphasize that such scheme is not an ab initio calculation but rather a simplified picture which allows us to explain physically our findings. The initial state |ψ0〉 has an (extensively) high energy relative to the ground state of the hamiltonian H which governs the subsequent time evolution, and therefore acts as a source of quasiparticle excitations. Those quasi-particles originating from closely separated points (roughly within the correlation length ξ0 = m 0 of the ground state of H0) are quantum entangled and particles emitted from far different points are incoherent. If the quasiparticle dispersion relation is E = Ωk, the classical velocity is vk = ∇kΩk. We assume that there is a maximum allowed speed vm = maxk |vk|. A quasiparticle of momentum k produced at r is therefore at r+vkt at time t, ignoring scattering effects. This is the only physical assumption of the argument. Scattering effects are not present in the theories considered so far, but as evident from the argument outlined below they can play a role for only t > r/2vm (allowing, perhaps, for a renormalization of vm by the interactions). These free quasi-particles have two distinct effects. Firstly, incoherent quasi-particles arriving a given point r from well-separated sources cause relaxation of (most) local observables at r towards their ground state expectation values. (An exception is the local energy density which of course is conserved.) Secondly, entangled quasi-particles arriving at the same time t at points with separation |r| ≫ ξ0 induce quantum correlations between local observables. In the case where they travel at a unique speed v (as in CFT), therefore, there is a sharp “horizon” or light-cone effect: the connected correlations do not change from their initial values until time t ∼ |r|/2v. In the CFT case this horizon effect is rounded off in a (calculable) manner over the region t−|r|/2v ∼ τ0, since quasi-particles remain entangled over this distance scale. After this they rapidly saturate to time-independent values. For large separations (but still ≪ 2vt), these decay exponentially ∼ exp(−πx|r|/2τ0). Thus, while the generic one-point functions relax to their ground-state values (we recall in CFT this relaxation is exponential ∼ exp(−πxvt/τ0)), the correlation functions do not, because, at quantum criticality, these would have a power law dependence. Of course, this is to be expected since the mean energy is much higher than that of the ground state, and it does not relax. This simple argument also explains why for the case of a semi-infinite chain the relevant time scale is r/v rather than r/2v, since one of the two particles arriving in r has been reflected from the end of the chain. This has been also stressed in Ref. [41], in the study of the time dependence of entanglement entropy of finite chains with open boundary conditions. All our results are consistent with this picture as long as the quasi-particles are assumed to all propagate at the same speed, resulting from a “conformal” dispersion relation Ωk = v|k|. However, it is very simple to generalize this picture to different dispersion relations, taking into account that each particle propagates at group velocity vk ≡ Ω′k appropriate to the wave number k. In this case the horizon effect first occurs at time t ∼ |r|/2vm, where vm is the maximum group velocity. If vm occurs at a non-zero wave number, it gives rise to spatial oscillations in the correlation function. Thus again connected correlation function are expected to be strongly suppressed for t < t∗ and start developing only after t∗. In the case of a general dispersion relation we do not have a proof, beyond the stationary phase approximation, that the connected correlation functions remain constant up to this time, but one would expect it on the grounds of causality. (The proof in the case of a relativistic dispersion relation uses Lorentz invariance in an essential way.) However, because there are also quasi-particles moving at speeds less than vm, the approach to the asymptotic behavior at late times is less abrupt. In fact, for a lattice dispersion relation where Ω′k vanishes at the zone boundary, the approach to the limit is slow, as an inverse power of t. A similar result applies to the 1-point functions. This is consistent with the exact results obtained here and elsewhere. B. The large time limit and the generalized Gibbs ensemble The existence and the understanding of the asymptotic state resulting from the evolution from an arbitrary state is one of the most-interesting problem in statistical mechanics. A robust theory able to predict this state ab-initio still does not exist. A currently popular idea is that for late times the system (or rather macroscopically large subsystems) ‘look like’ they are in a thermal state, despite the fact that the actual state of the whole system is pure. A common belief is that a region of dimension r can be thermalized by the infinitely large rest of the system which acts as a bath (see e.g. [24, 39, 47]). But this intriguing idea is not sufficient to give the value of the resulting effective temperature. A major step toward the clarification of the properties of the asymptotic state has been made by Rigol et al. [15]. In fact, it was conjectured that if the asymptotic stationary state exists, it is given by a generalized Gibbs ensemble obtained by maximizing the entropy S = −Tr ρ log ρ, subject to all the constraints imposed by the dynamics [15]. Consequently, denoting with Im a maximal set of commuting and linearly independent integrals of motion, the density matrix is ρ = Z−1e− λmIm , Z = Tr e− λmIm . (94) We note that such a density matrix describes a pure state only if the model under consideration is integrable, i.e. if the number of integral of motions equals the number degrees of freedom. If there are not enough integrals of motion, ρ corresponds to a mixed state and it is not clear to us to which extent it can describe the pure state resulting from the time-evolution. The values of the Lagrange multipliers λm are fixed by the initial conditions: Tr Imρ = 〈Im〉t=0 . (95) In the following we will take this generalized Gibbs ensemble as a postulate and we will show how it nicely and naturally explains the “effective temperature” effect observed for large times. However we stress that there is still no proof for this assumption that, to our opinion, cannot be considered on the same fundamental level as the thermal Gibbs ensemble. Let us consider the chain of harmonic oscillators of the previous section as a typical example. We will soon see that most of the features are quite general. In this case the natural choice for an infinite set of integral of motion is the number of particles with momentum k, i.e. nk = A kAk. Most other observables can be written in terms of these, i.e. k Ωknk. Consequently the expectation value is given by 〈O〉t=∞ = TrOρ = TrOZ−1e− λknk , (96) that can be seen as a thermal density matrix with a k dependent effective temperature given by βeff(k)Ωk = λk . (97) Thus an effective temperature already appeared. Note that this state can still be pure because such a temperature is k dependent. To fix λk we need 〈nk〉t=0. From Eq. (50) we get 〈nk〉t=0 = 〈A†kAk〉t=0 = d . (98) The calculation then proceeds as for a thermal distribution 〈nk〉ρ = Trnkρ = − lnZ , with Z = Tr e− λknk = e−λknk = 1− e−λk , (99) so that 〈nk〉ρ = ln(1− e−λk) = 1 eλk − 1 . (100) From this eλk = 1 + d−2k and βeff(k) = ln(1 + d−2k ) . (101) At finite temperature the correlation function in momentum space is 〈ϕkϕ−k〉β = 〈 (Ak +A k)(A−k +A −k)〉β = 〈A†kA−k〉β + 〈AkA 1 + e−βΩk 1− e−βΩk , (102) that substituting the previous result for βeff gives ρ(k)t=∞ = 〈ϕkϕ−k〉t=∞ = (1 + 2d2k) = Ω2k +Ω Ω2kΩ0k , (103) that is exactly the time-average of Eq. (54) reproducing, after Fourier transforming, the well defined correlation in real space for t→ ∞. So this generalized Gibbs ensemble correctly reproduces the exact diagonalization of the model and gives also few insights more. In fact, in the limit m0 → ∞ the effective temperature is independent from k and m obtaining βeff = 4/m0, explaining a posteriori the simplicity of the results in this case. Note instead that for arbitrary m0 and m = 0, i.e. conformal evolution, βeff(k) is a function of k. In this case, the large distance properties of correlation functions are described by the mode with k = 0, for which, independently of m0 we get βeff(k = 0,m = 0) = 4/m0, consistently with the previous findings and general expectations. Furthermore we can conclude that the large r asymptotic behavior is always governed by the effective temperature βeff(k = 0) = 2(log(|m0 − m|/(m0 + m))/m. Another interesting feature is that βeff(k = 0) gives the asymptotic behavior of the correlation functions of all those observables that are effectively coupled with the zero-mode. In the opposite case the relevant temperature is the largest βeff(k) with the k mode coupled with the observable. Another interesting feature is that on a very rough basis one can be tempted to assume that that βeff(k) is directly related to the excess of energy of the mode k generated by the quench, but this is not the case. In fact 〈H〉t=0 = k Ωkd k that is different from β eff , being the same only in the limit d−2k → 0, i.e. m0 → ∞. Clearly the same reasoning of before applies every time we consider a model that can be diagonalized in momentum space with a proper choice of the quasi-particles, i.e. every time that H = k ΩkA kAk for some Ak. This means that the excitations are non-interacting. As far as we are aware all the applications of the generalized Gibbs ensemble to the date only concern this kind of models [15, 26], but there are few numerical hints suggesting that can be true more generically [29]. Now we outline how we imagine the generalized Gibbs ensemble given by Eq. (94) could be used to justify the effective temperature scenario for large time for any integrable system, i.e. with a complete set of integrals of motion. The hamiltonian can be written as H = m amIm, with some am eventually zero. Thus one can think to an m- dependent effective temperature βeff(m) = λm/am, but this temperature does not give directly the behavior of the correlation function for large distance because in general the integral of motions are not diagonal in k-space. Thus we can only conjecture that the correlation functions of a given operator O(r) are governed by the largest βeff(m) with m among the integral of motions to which O(r) effectively couples. We stress that this is only a crude argument and we are still not able to put it on a firmer basis. C. Discussion and open questions We presented a quite complete picture on the time evolution of a quantum system after a sudden quench of one hamiltonian parameter. Despite the fact that a lot of work has been done, still more is left for future investigation. The first problem that must be addressed is the real-time dynamics of effectively interacting systems. In this direction Bethe Ansatz solvable models like the Lieb-Liniger gas and Heisenberg spin chains are among the best candidates for an analytical approach. This would clarify how a non-trivial two-body scattering matrix can modify the time-evolution (only inside the light-cone) of the correlation functions and whether the crude argument we outlined for an “effective temperature” for the long-time state is valid. To clarify this point also the study of (boundary) integrable massive field theories can be of some help. Another approach, currently under investigation[48], is the direct analysis of the perturbative expansion for the correlation functions in a λφ4 field theory. This has some simplifying features in the large m0 limit which may allow it to be resummed to all orders. It is very important to get non-trivial results for such a non-integrable interacting theory. Numerical computations for non-integrable systems should also be performed to understand if and eventually to which extent the generalized Gibbs ensemble picture is valid beyond integrability. Some numerics concerning this point are already available [22, 23, 29] but still a clear scenario is not emerged. Among non-integrable models special care must be given to disordered systems because of the non existence of a speed of sound as a consequence of Anderson localization [49]. Some insights can be obtained from those models whose equilibrium behavior can be analytically obtained by means of the strong-disorder renormalization-group [50]. Even in this case time-dependent DMRG can help a general understanding. The time-evolution of the entanglement entropy [41] already revealed that in these systems there is no sign of the light-cone (as easily predictable because of the absence of speed of sound) and the effective motion of “quasi-particles” is more diffusive rather than ballistic. Furthermore the results at the largest available times does not look thermal at all, but this can be also due to the time window accessible with numerics. Another very interesting question is what happens at finite temperature and under what conditions a system can equilibrate. It should be relatively simple to generalize the results for all the “quasi-free” models already considered at T = 0. Work in this direction is in progress, but it will be almost impossible to consider the same problem for more complicated models. Finally it is also important to understand the role played by the initial state. We assumed always a translational invariant one with short range correlations. Thus one natural modification consists in taking a state that is only locally different from the actual ground-state. This problem is known in the literature as a “local quench” and it has already been considered to some extent (see e.g. [51–53]), but still a general picture as the one outlined here for global quenches does not exist (this is also interesting for eventual connections with quantum impurity problems, see [54] as a review). Another natural modification of the initial state is one with long-range correlations, such as a critical one and let it evolve with another critical hamiltonian. This has been done for the Luttinger liquid [26] and the results show the typical functional dependence of a light-cone scenario (i.e. everything depends only on x± 2vt) but the long-time correlations decays as power-laws, with exponents that are different from equilibrium ones and can be predicted by the generalized Gibbs ensemble. 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Statistical Properties of Many Particle Eigenfunctions Eric J. Heller1, 2, ∗ and Brian R. Landry2 Department of Physics, Harvard University, Cambridge, MA 02138 Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 (Dated: November 12, 2018) Wavefunction correlations and density matrices for few or many particles are derived from the properties of semiclassical energy Green functions. Universal features of fixed energy (microcanoni- cal) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as N → ∞. This arises through a little known asymptotic limit of Bessel functions. Constraints due to symmetries, boundaries, and collisions between particles can be included. I. INTRODUCTION The standard tools of quantum chaos investigations include random matrix theory and periodic orbit theory (Gutzwiller trace formula), the Van Vleck-Morette-Gutzwiller propagator, and many techniques and phenomena derived from these approaches. Standing somewhat to the side as an inspired insight is Berry’s conjecture, which loosely stated is the idea that as ~ → 0 eigenstates will be indistinguishable from superpositions of infinitely many (local) plane waves with random amplitude, direction, and phase, but with fixed wavelength appropriate to the local kinetic energy. In two dimensions, these assumptions result in strictly Gaussian statistics of the eigenfunctions and the autocorrelation function 〈ψ∗(~x)ψ(~x+ ~R)〉 = J0(ka) where k is the local wavenumber and |~R| = a. The Berry random plane wave (RPW)[1] hypothesis is free of any specific dynamical information, except fixed total energy, which defines the “ensemble” (i.e. microcanonical). The perspective developed here suggests that by extending the RPW hypothesis we can conveniently accommodate many other constraints, incorporating information about real systems. In fact this program has already begun, with Berry’s inclusion of the presence of nearby hard walls[2], and Bies and Heller’s soft boundary results[3], and multiple hard walls[4]. Related work by Urbina and Richter[5] and one of us [6] may also be viewed in this light. The idea of random waves subject to constraints is not confined to one particle in two dimensions. Indeed Berry gave the N - dimensional formula for free particles in his 1977 paper[7]. Since the underlying idea in the RPW hypothesis is uniform randomness within a quantum context, i.e. the underpinning of quantum statistical mechanics, we must encounter some familiar territory as the RPW hypothesis is extended to the large N limit. In 1994, Srednicki had suggested that the Berry random wave hypothesis was indeed a foundation for quantum statistical mechanics[8], and showed that the appropriate canonical ensemble was reached for large N , depending on particle statistics. The present paper shows more specifically what happens as the number of particles increases, through a nonstandard and apparently unpublished asymptotic form for Bessel functions (we have not been able to find it in the literature, although it “ought” to be there), which encodes the equivalence of the canonical and microcanonical ensembles of statistical mechanics. In making the connections to quantum statistical mechanics one also needs procedures for incorporating constraints, which are an essential aspect of the theory. Thus our procedures for generalizing the RPW to include constraints, mentioned above, is an essential new feature, since the constrained eigenstates are no longer random in Berry’s (and Srednicki’s) original sense. Given a continuum at energy E, such as in an enclosure with walls very far away, we can perform the average over all random waves as a trace, i.e. 〈ψ∗(~x)ψ(~x′)〉 = Tr [δ(E −H)|~x〉〈~x′| ] , (1) which immediately yields Berry’s result, apart from normalization which we choose differently here. However a trace over a basis is independent of any unitary transformation on that basis, so it does not matter whether we use a trace over a complete set of random waves or simple local plane waves; both give J0(ka) for the case of one free particle in two dimensions. In this way the imaginary part of the retarded Green’s function − 1 Im [G+(E)] = δ(E−H) becomes central, formally convenient, and equivalent to Berry’s RPW hypothesis. ∗Electronic address: heller@physics.harvard.edu http://arxiv.org/abs/0704.1881v1 mailto:heller@physics.harvard.edu II. PRELIMINARIES We begin by reviewing well known formalism to establish context and notation. The Green function completely characterizes a quantum system, whether it is interacting or not, or has few or many degrees of freedom. The retarded Green function G+, i.e. G+ = P − iπδ(E −H), (2) where P stands for the principal value of the integral, is the basis for wavefunction statistics and density matrix information, through the follow relations, with a convenient choice of normalization: < ψ(x)ψ∗(x′) > = − 1 Im〈x|G+|x′〉/ρ(E) (3) = 〈x|δ(E −H)|x′〉/ρ(E) (4) where ρ(E) = Tr[δ(E −H)] (5) and where < · · · > stands for the average over the degeneracies. We take these degeneracies to be of dimension up to ND − 1, where N is the number of particles and D the spatial dimension each particle lives in. (We use boldface notation, e.g. x for the N ∗ D degrees of freedom.) If true degeneracies do not exist in a particular system, we can artificially open the system up to a continuum. For example, a two dimensional closed billiard does not have a degeneracy, but it acquires one if we open a hole in it and let it communicate with the outside unbounded 2D space. Of course this changes the billiard properties, and the size of the hole might be problematic, but in fact we shall never really have to open a system up in this way. The quantity δ(E−H) then implies the average over all scattering wavefunctions at fixed energy E. There are other interpretations which can be put on the average correlation < ψ(x)ψ∗(x′) >; for example we can imagine a large number of potentals which differ in some far away place, and in a way so as to all have an eigenvalue at a particular energy. Then, the average has the interpretation of the average over this “disorder” ensemble. A slightly different procedure is advocated by Richter et. al., wherein an energy average is taken[5]. Another interpretation can be applied to individual eigenstates in a closed system, assuming they are at least locally uniform in their properties, by taking the average over different points of origin x. This is particularly appropriate when the analogous classical system is chaotic, as mentioned above [1]. We will be evaluating the Green functions semiclassically in what follows, restricting the time over which the contributing trajectories propagate. The wavefunction correlation is equal to the coordinate space matrix element of the constant energy density matrix: < ψ(x)ψ∗(x′) >= 〈x|δ(E −H)|x′〉/ρ(E) = ρ(x,x′, E) (6) Reduced density matrices can also be derived from wavefunction correlations ; e.g. ρ̃(~x1, ~x 1, E) = d~x2d~x3 · · · d~xN ρ(~x1, ~x2, · · · ; ~x′1, ~x2, · · · ;E), (7) the one particle reduced density matrix. We can approach the correlations via Fourier transform from the time domain, since δ(E −H) = 1 eiEt/~e−iHt/~ dt. (8) Thus the statistics, density matrices and correlations are derivable without further averaging by knowing the time propagator. In the following, we define the Green function propagator G(x,x′, t) and the retarded Green function propagator G+(x,x′, t) as G(x,x′, t) = 〈x|e−iHt/~|x′〉 G+(x,x′, t) = Θ(t)〈x|e−iHt/~|x′〉 (9) where Θ(t) is the Heavyside step function Θ(t) = 0, t < 0, Θ(t) = 1, t > 0. It is very rewarding to expand the propagator in semiclassical terms, involving short time (zero length) and longer trajectories. We take Gdirect(x,x + r, t) = 〈x| exp[−iHt/~]|x+r〉, the very short time semiclassical propagator, which for N particles each in D dimensions reads Gdirect(x,x + r, t) ≈ 2πi~t )ND/2 2/2~t−iV (x+ r )t/~ (10) where r2 = |r|2. It is not difficult to cast the Fourier transform of this short time version to fit the definition of a Hankel function, G+cl(x,x + r, E) = 2πi~t )ND/2 2/2~t−iV (x+ r )t/~eiEt/~ dt = − im d (kr) (11) where d = ND/2−1, k = k(x+r/2, E) andH(1)d (kr) = Jd(kr)+iNd(kr) is the Hankel function of order d, and Jd is the regular Bessel function of order d. The wavevector k varies with the local potential, i.e. ~2k(x, E)2/2m = E − V (x). Here, using only the extreme short time version of the propagator, we must suppose r is not large compared to significant changes in the potential, but this restriction can be removed by using the full semiclassical propagator rather than the short time version. For the case of one particle in two dimensions, d = 0, and we recover Berry’s original result for one particle in 2D, 〈ψ∗(~x)ψ(~x + ~r)〉 ∝ J0(kr). According to the short time approximation, for any N , < ψ(x)ψ∗(x+ r) >≈ − G+cl(x,x + r, E) Jd(kr) (12) where k = k(x, E). This result includes interparticle correlations through the potential V (x) and the spatial depen- dence of k = k(x, E); the diagonal r = 0 limit (following section) is equivalent to classical statistical mechanics. The implications of this for the nondiagonal short time Green’s function are intriguing. The way r is defined, it does not matter whether one particle is off diagonal (xi 6= xi′ ) or several or all of them. For given r, the Green’s function will be the same, apart from changes in the potential V (x + r/2). It is interesting that although the short time Green function is manifestly semiclassical, the energy form, e.g. Eq. 12 is obtained by exact Fourier transform of the semiclassical propagator, rather than by stationary phase. III. DIAGONAL LIMIT The diagonal (r → 0) N body Green function is obtained using the asymptotic form Jd(kr) = Γ(d+ 1) we obtain G+cl(x,x, E) Γ(d+ 1) where the second form uses Stirling’s approximation, n! ∼ nne−n 2πn, and is appropriate below when we consider large N . We note that this behaves as k2d ∼ (E−V (~x))d. This factor is familiar from the computation of the classical density of states. Tracing over all ~x results in Γ(d+ 1) δ(E −Hcl(p,x)) = ρcl(E) (15) i.e. the classical density of states. The association of the short time propagator with the classical Hamiltonian and classical density of states is well known. The Berry RPW hypothesis, the short time propagator, and the classical or Weyl (sometimes called Thomas-Fermi) term in the quantum density of states are all closely related. The quantum spacial integral is over all coordinates, so how does the classical partition function emerge if the classical integral is only over classically allowed coordinates? For forbidden positions, k is imaginary and can be written as say iκ. An identity for Hankel functions can then be used (in+1H n (ix) = Kn(x)) to show that the green function is real so that the imaginary part is zero, explaining why the integral is only over classically allowed positions. As long as r = 0 (i.e. diagonal Green’s function) the results obtained within the short time propagator approximation for any quantity in the presence of a potential (including interparticle potentials such as atom-atom interactions) will be purely classical. Since we will be discussing the equivalence of the results from the different ensembles for r 6= 0, it is useful to recall how the classical coordinate space densities in the different ensembles can be shown to coincide since this corresponds to the r = 0 case. The normalized phase space density in the microcanonical ensemble and the phase space density in the canonical ensemble are given by ρcl(p,x, E) = ρcl(E) δ(E −Hcl(p,x)) (16) ρcl(p,x, β) = Qcl(β) e−βHcl(p,x) (17) respectively. The density of states and partition function are of course the normalization factors so that ρcl(E) = dxdp δ(E −Hcl(p,x)) (18) Qcl(β) = dxdp e−βHcl(p,x) (19) Integrating each phase space density over momentum space allows us to compare the coordinate space densities: ρcl(x, E) = dx p2d ρcl(x, β) = e−βV (x) dx e−βV (x) with p = 2m(E − V (x)). Using the relationship between E and β, E − 〈V 〉 = ND , where 〈V 〉 is the ensemble average of the potential in one of the statistical ensembles, the coordinate space density becomes p2d = (2m(d+ 1)/β)d (〈V 〉 − V (x)) β In the limit N → ∞ ( d→ ∞) this is p2d = (2m(d+ 1)/β)de(〈V 〉−V (x))β (23) dx p2d e−V (x)β dx e−V (x)β This is one of the standard ways of establishing a connection between the ensembles[9]. Since the diagonal Green’s function gives classical results we can use it to study classical properties. For example, we can inquire about the average two particle spacing distribution ρE(r12) or the probability density for a single particle PE(~x1) starting with the short time semiclassical Green’s function and the results will coincide with classical microcanonical statistical mechanics. This statement holds for all N . Similarly, in the large N limit the canonical ensemble results for these quantities must emerge. This point becomes more interesting for the non-diagonal case, considered next. IV. LINK TO THE CANONICAL ENSEMBLE A. Bessel functions become Gaussians As yet we have found nothing too surprising or useful beyond standard classical statistical mechanics. This changes when we consider the large N limit for the non-diagonal Green’s function, r 6= 0. Taking the large N limit of Eq. 12, 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 FIG. 1: As N increases, the combination 1 Jd(x), where d = ND/2 − 1, approaches a Gaussian. This is the key link between the quantum microcanonical and canonical ensembles. we are confronted with a new question about Bessel functions. The large d limit of Jd(x) is indeed well known, but this is not yet sufficient for our purposes. It reads Jd(kr) (kr)d 2d Γ(d+ 1) This is the standard formula given in the usual references. Eq. 25 should be the first term in a power seres for Jd(kr) in kr. Another standard result is the power series expansion, valid for all d and kr: Jd(kr) = (−1)m m!Γ(m+ d+ 1) )2m+d We actually require a different asymptotic result. What make our demands unusual is that, assuming we want the energy to increase in proportion to the number of particles (appropriate to many applications of the large N limit), then k ∼ d; this means that for fixed r the combination (kr) is increasing as d as d → ∞. If the argument of the Bessel function increases without bound along with it’s order, some new considerations come into play. We find the desired form using Eq. 26, after summing a series recognized as that of a Gaussian Taylor expansion, (kr)d Jd(kr) = 2d d! −k2r2 4(d+ 1) 2d d! 2r2/(4(d+1)), (27) where again ~2k2/2m = E − V (x). Note that as d → ∞, the argument of the Gaussian holds fixed because of the factor of d + 1 in the denominator of that argument. Figure 1 illustrates the convergence to the Gaussian as N increases. The asymptotic limit in Equation 27 is not in the usual references, although related results have been given for N-bead polymer random chain end-to-end distributions[10]. The connection between the path integral for the propagator and polymer chains is well known[11]. It is interesting that a Gaussian emerges from Bessel functions in the large N limit. We can put Eq. 27 together with Eq. 12 and Eq. 4, and express the result, as N → ∞, < ψ(x)ψ∗(x+ r) > = ρ(x,x′, E) → 1 2π~2d! 2r2/4(d+1). (28) For noninteracting particles moving in zero potential but confined to volume V the short time approximation becomes exact and k is constant. For this system the wavefunction correlation becomes < ψ(x)ψ∗(x + r) > = ρ(x,x′, E) → 1 2r2/4(d+1). (29) Something familiar is emerging, here derived in the unfamiliar context of fixed energy (microcanonical ensemble). For comparison we recall the standard result for the ideal gas at temperature T [12]: 〈x|e−βH |x+ r〉 Tr[e−βH ] = ρ(x,x′, β) = 2/λ2 (30) where λ = h/ 2πmκT is the thermal wavelength. Indeed for the free particle case, k is fixed by E and 〈K〉 = D/2NκT = ~2k2/2m, where K is the kinetic energy and κ is Boltzmann’s constant, 2r2/4(d+1) = e−πr 2/λ2 . (31) The canonical ensemble result for the propagator has “dropped out” of the asymptotic large N limit of a micro- canonical Green function, at least for noninteracting particles, and an unusual asymptotic form for the Bessel function has emerged as the link. With some caveats, the statement δ(E −H) ∼ e−βH (32) has meaning in the large N limit, where it is understood E grows as N , and a temperature extracted. At a qualitative level, Eq. 32 merely expresses the known equivalence of the ensembles. In the case of an interaction potential, the relation between E and temperature is of course problematical. B. Interacting Particles - Short Time Limit We can say more about interacting particles using only the short time propagator introduced above. Longer time events will be discussed in Sec. VI. The short-time approximation to the correlation function for large N , which is equal to the matrix elements of the density operator in coordinate space using our normalization, (Eq. 28) is given by ρcl(x,x ′, E) = 2π~2d! 2r2/4(d+1) (33) with ~k = 2m(E − V (x+x′ )) and r = |x − x′|. Again, the Gaussian form of this expression arises from the asymptotic limit of the Bessel function. In the interacting case this can again be brought into the same form as the equivalent expression at constant temperature: ρcl(x,x ′, β) = 2πβ~2 +V ( x+x In order to make the connection we must identify the energy with a certain temperature. This relationship between E and β is E − 〈V 〉 = ND where 〈V 〉 is the ensemble average of the potential in one of the statistical ensembles. Using this relationship in Eq. 33 gives ρcl(x,x ′, E) = 2π~2d! 2~2β e m(〈V 〉−V )r2 2~2(d+1) (36) In order for Eq. 36 to be equivalent to Eq. 34 the term with 〈V 〉−V must be negligible. This is true for configurations of particles which possess the typical (and vastly most probable) sum total kinetic energy for all the particles. Since the typical total kinetic energy is by far the most probable, nearly all points in configuration space lead to small values of 〈V 〉 − V , and that term is negligible almost always. The remaining terms in Eq. 36 and Eq. 34 are shown to be the same by the equivalence of the classical ensembles as shown in Sec. III. It is also telling to trace over the coordinates of all but one of the interacting particles, given by a coordinate ~y. We thus seek the reduced density matrix, diagonal or off diagonal in ~y. The trace will over many coordinates be overwhelmingly dominated (in the large N limit) by the most probable total kinetic energy for all the particles. Then we find G(~y, ~y′, β) ∼ λ−3N−2e−πr 2/λ2 (37) where r2 = |~y − ~y′|2 and λ = h/ 2πmκT . Thus the quantum mechanical single particle Green function and density matrix make sense as their imaginary time counterparts in the N → ∞ limit, in accordance with well known results for the canonical ensemble. C. Large N limit and Boltzmann averaged Green functions Even though it is a necessary consequence of the equivalence of the ensembles, it is interesting to establish the generality of the Boltzmann average over the energy of a noninteracting subsystem in the following way. Suppose N −M particles are no longer interacting with the remaining M particles, but their states are correlated by having been in contact in the past with the total energy fixed at E. In the time domain and in an obvious notation we have G+N (y, z;y ′, z′, t) = i~ G+N−M (y,y ′, t)G+M (z, z ′, t) (38) Then the Fourier convolution theorem can be applied to the Fourier transform into the energy domain, i.e. G+N (y, z;y ′, z′, E) = G+N−M (y,y ′, E − E′)G+M (z, z ′, E′) dE′ (39) which incidentally leads to some rather unlikely looking identities for Bessel functions; the reader may easily generate them. Our purpose is served if, focussing on the subsystem of M particles, we trace over the N −M y coordinates. This gives Try[G N−M (E − E ′)] ∼ lim Γ(dN−M + 1) )dN−M Γ(dN−M ) πdN−M+1|y′ − y|2dN−M times a volume factor, in the case of an ideal gas. The second term is not a function of E′. Therefore the integral of it times GM (z, z ′, E) is proportional to δ(z′ − z). So long as z 6= z′ that term is zero. Neglecting all unimportant (for this argument) factors this leaves Try[G N−M (E − E ′)] ∝ (E − E′)dN−M = EdN−M )dN−M ∼ EdN−M e−βE with of course β = 1/κT . In arriving at Eq. 41 we used E = D NκT for the case of particles embedded in D dimensions. Finally we arrive at Try[G N (E)] ∝ G+M (z, z ′, E′) dE′ = G+M (z, z ′, β) (42) in the large N limit. This establishes the generality of the Boltzmann average over the subsystem energy for large N . This discussion establishes again the connection between the canonical and microcanonical ensembles, however in a way not involving the Bessel functions and their asymptotic form, so it is less general than other results in this paper valid for any N . D. Stationary phase canonical limit It is also possible to recover the Gaussian form in Eq. 28 by carrying out the integral in Eq. 11 by stationary phase, provided the real factor involving t in the denominator is taken into the exponent, as −ND/2 log t i.e. G+cl(x,x+ r, E) = )ND/2 2/2~t−iV (x+ r )t/~+iEt/~−ND/2 log t dt. (43) The complex stationary phase point t∗ in the large N limit becomes t∗ = −iND~/(2(E−V )) , yielding the same result as in Eq. 28, with ~2k(x, E)2/2m = E − V (x), and making this another route between the quantum microcanonical and canonical ensembles. Since the positions are arbitrary we cannot however identify the average kinetic energy with E − V , and thus without further averaging we cannot associate t∗ with any inverse temperature. It is interesting nonetheless that there is a complex time t∗ appropriate to every position x, even if that time is not related to the temperature. For an ideal gas the stationary phase time is t∗ = −i~/κT = −iβ~, after making the identification E = ND/2kT . A discussion about traces over most of the coordinates and the recovery of the usual temperature through 〈K〉 = D/2NkT proceeds as in Sec. IVB. V. CONSTRAINTS In the large N limit the ergodic hypothesis is strongly motivated, but statistical mechanics does not pre-suppose that ergodicity is unchecked; rather constraints are always present, such as walls and boundaries which control volume. Ergodicity is then defined with respect to these constraints. The guiding idea in this paper, i.e. the extended Berry RPW hypothesis, is that eigenstates of the full system are “as random as possible, subject to prior constraints”. In this way thermodynamic constraints arise naturally. The real time, real energy (microcanonical ) semiclassical Green function approach not only automatically generates the averages required to get appropriate wavefunction statistics, it also provides a natural way to include many constraints such as walls, symmetries, and even the existence of collisions between particles by going beyond the short time limit term to include returning (not necessarily periodic) trajectories. The semiclassical Ansatz for these extended problems in the presence of constraints is G(x,x′, t) ≈ Gdirect(x,x′, t) + Gj(x,x ′, t) (44) where Gj(x,x + r, t) is a semiclassical (Van Vleck-Morette-Gutzwiller) Green function, Gj(x,x ′; t) = )ND/2 ∣ ∂2Sj(x,x ′; t) ∂x∂x′ iSj(x,x ′; t)/~− corresponding to the jth trajectory contributing to the path from x to x+r, and Gdirect(x,x+r, t) is given by Eq. 10. The short time term Gdirect(x,x+ r, t), is singled out as the shortest contributing trajectory: supposing r to be small compared to distances to walls etc., we still have a short time, ballistic trajectory as quite distinct from trajectories which have traveled some distance away and come back. There are cases where this separation is not clean; for such cases we can adjust notation accordingly. Note that since a trace over all position is not being taken, there is no appearance semiclassically of periodic orbits as the only surviving contributors. “Closed” orbits however can play a large role semiclassically, a fact recognized long ago by Delos[13]. A. N particles and a wall A very useful example is provided by a plane Dirichlet wall felt by all the particles (e.g. ψ(~x1, ~x2, · · ·~xN ) = 0 for yi = 0, i = 1, · · ·N), as in a gas confined by a rigid container. The Green function and eigenfunctions must vanish if one or more particles approaches this wall. We can use the method of images, generalized to N particles, if the particles are noninteracting. (The interacting case can in principle be handled by semiclassical trajectory techniques which we bring up in the next section.) The Green function Gwall(x,x ′) will consist of the shortest distance contribution for which all particles take a direct path from x to x′, plus paths where one particle has bounced off the wall, paths where two particles have, etc. These histories are included automatically if we apply the symmetrization operator which imposes the image reflections. This operator can be written (1 −Ri) = 1− RiRj − · · · (46) where Ri is the operator for reflection about the y = 0 axis for the i th particle. Applied to the Green function G(x,x + r, t), considered as a function of the coordinates in x in the absence of the wall, R yields the series Gwall(x,x ′, t) = Gdirect(x,x ′, t)− Gi(x,x ′, t) + Gij(x,x ′, t)− · · · (47) where Gi(x,x ′, t) corresponds to the ith particle getting from ~xi to ~x i by bouncing off the wall while the others take direct paths, etc. The Fourier transform gives an analogous equation for Gwall(x,x ′, E). The effect of the symmetrization is to create Green function sources reflected across the wall and given proper sign, in the manner familiar from the method of images. The short time path is shown by the direct path solid line in Fig 2, corresponding to the term Gst(x,x ′, t). The bounce path is equivalent to a source reflected across the wall with an opposite sign, i.e. the method of images. Define G+st(x,x+ r, E) Jd(kr) (kr)d ≡ a(k)Fd(kr) (48) FIG. 2: A short and a bouncing path for a particle propagating near a wall. The bounce contribution, if viewed by the image method, is equivalent to a contribution of opposite sign coming from the reflected point ~xR with the wall removed. G+wall(x,x ′, E) = a(k) Fd(kr)− Fd(kri) + Fd(krij)− · · ·  . (49) This is the general result for any N . It would appear to be difficult to take it further, since all the distances, e.g. rij = m 6=i,j |~xm − ~x′m|2 + |~xRi − ~x′i|2 + |~xRj − ~x′j |2, (50) where ~xRj is the reflected j th particle coordinates, involve square roots. However if we use the large N asymptotic form, we find, using Fd(kr) → exp[−k2r2/4(d+ 1)]/2dd!, Im [Gwall(x,x ′, E)] = i − e−γ(r 1− e−γ∆ where γ = k2/4(d + 1) = π/λ2 and ∆2i = (r 2 − r2i . Since ri is the “direct” distance from ~xi to ~x′i, (see Fig 2), ∆2i records the distance change upon reflection of the i th particle. We note that ∆2i (and thus the Green function) vanishes as any particle approaches a wall in either x or x′. It is also simple to see that the single particle density ρ(~x) in this noninteracting case becomes, for large N , ρ(~x) = ρ0(1− e−4γx ) (52) where x is the distance to the wall and ρ0 is the density far from the wall. The formulas Eq. 49 and Eq. 51 generalize Berry’s result[14] for the wavefunction squared of one particle in two dimensions near a wall, namely 〈|ψ(~x)|2〉 = 1− J0(k|~xR − ~x|) d~x (1− J0(k|~xR − ~x|)) . (53) The Gaussian we get for large N has a very simple interpretation. First we note that for noninteracting systems in the canonical ensemble we can write the total density matrix as a product of one particle density matrices. This is essentially the form of Eq. 51, since we can write each one particle density matrix as ρ(~x, ~x′, β) = e−γ|~x−~x ′|2/N 1− e−γ(|~xR−~x′|2−|~x−~x′|2) 1− e−γ|~xR−~x|2 1− e−γ|~xR−~x|2 1− e−γ|~xR−~x|2 ) (54) FIG. 3: The particle symmetry or antisymmetry condition is equivalent to requiring mirror symmetry or antisymmetry across the ~xi = ~xj (hyper)plane. This corresponds to having additional contributions from the images of the particles reflected over the symmetry planes. where the second form is the diagonal element. However Eq. 54 also arises as the density matrix obtained from the Boltzmann average of Berry’s result; i.e. averaging the fixed energy results over a canonical distribution of energies, as can be seen from the integral 1− J0(k|~xR − ~x|) 2k2/2mdk k e−β~ 2k2/2mdk 1− e−m|~x R−~x|2/2β~2 For D = 2 and N = 1 a Boltzmann average yields the Gaussian. Indeed this necessarily holds in any number of dimensions; i.e. the appropriate Boltzmann average of Jd(kr)/(kr) d must yield a Gaussian for any d. In the thermodynamic N → ∞ limit for noninteracting particles, each particle separately is Boltzmann distributed over energy, so the result must be the same as a Boltzmann average of the one particle results for any dimension D and for any constraints. B. Symmetries - Fermions and Bosons Particle symmetry is an essential part of the many body problem. It’s effect, like other symmetries, is to generate permutations where the distances have changed due to particle exchange. Figure 3 shows this effect graphically. It is gratifying to see directly that permutations which induce large new distances (coming from remote pairs of particles, where “remote” is a relative term depending on the temperature) make little contribution. Consider N noninteracting Fermions or Bosons; we wish to compute the reduced density matrix for two Fermions or Bosons. This is a well known result for N → ∞[12]. The symmetric or antisymmetric Green function is GS/A(x,x+ r, E) = Hd(krn) (kr)d where rn = | ~x1 − ~xp1 ′|2 + · · ·+ | ~xN − ~xpN ′|2, {p1, · · · , pN} is the nth permutation of {1, · · · , N}, and ǫn = 1 if the parity of the permutation is even and ǫn = ±1 if the parity of the permutation is odd (with the upper sign for bosons and the lower sign for fermions). 〈ψ∗( ~x1 · · · ~xN )ψ( ~x1 · · · ~xN )〉 = − GS/A(x,x+ r, E) ρ(E)N ! Jd(krn) (kr)d In the limit that N is large, this becomes 〈ψ∗( ~x1 · · · ~xN )ψ( ~x1 · · · ~xN )〉 = ρ(E)N ! 2π~2d! 2r2n/4(d+1) (58) The diagonal component of this with the rn’s written out explicitly is 〈ψ∗( ~x1 · · · ~xN )ψ( ~x1 · · · ~xN )〉 = 2ρ(E)N !π~2d! )d N ! −k2(~x1−~xp1) 2/4(d+1) · · · e−k 2(~xN−~xpN) 2/4(d+1) (59) Up to the normalization constant this is the constant temperature density matrix for N noninteracting fermions or bosons: 〈ψ∗( ~x1 · · · ~xN )ψ( ~x1 · · · ~xN )〉 = 2ρ(E)N !π~2d! )d N ! −m(~x1−~xp1) 2/2β~2 · · · e−m(~xN−~xpN ) 2/2β~2 (60) Again the identification E = D NκT was used. This can be rewritten as an integral over wavevectors: 〈|ψ(x)|2〉 = A d~k1 · · · d~kN e−β~ 2/2m+i~k1·(~x1−~xp1) · · · e−β~ 2/2m+i~kN ·(~xN−~xpN ) (61) where A = m 2ρ(E)N !π~2d! is the normalization constant. Rearranging gives 〈|ψ(x)|2〉 = A d~k1 · · · d~kN e−β~ 2+···+kN 2)/mei( ~k1−~kp1)·~x1 · · · ei(~kN−~kpN )·~xN (62) If the volume that the particles are confined to is large but finite, 〈|ψ(x)|2〉d~x3...d~xN = AV N−2 d~k e−β~ 2/2mei( ~k1−~kp1)·~x1ei( ~k2−~kp2)·~x2δ~k3,~kp3 · · · δ~kN ,~kpN (63) For fermions if the wavevector of any two particles are the same the term is killed by the term with the wavevectors reversed in accordance with the Pauli principle. This leaves only two terms 〈|ψ(x)|2〉d~x3 · · · d~xN = AV N−2 dk e−β~ 2/2mei( ~k1−~kp1)·~x1ei( ~k2−~kp2)·~x2 (64) For bosons there are also only two types of terms, but each is multiplied by the same factor since like terms are added together. Either way, carrying out the integral over k, 〈|ψ(x)|2〉d~x3 · · · d~xN = 1± e−m( ~x1− ~x2)2/β~2 d~x1d~x2 1± e−m( ~x1− ~x2)2/β~2 ) (65) This is the well known result for the density of two noninteracting fermions or bosons. VI. SCATTERING A hard wall is a potential energy feature which induces a boundary condition, requiring the wavefunction or Green function to vanish as the wall is approached. Softer potentials do not induce fixed boundary conditions and require a different treatment. A potential may still however be thought of as a constraint: we consider waves as random as possible subject to the existence of a potential, be it fixed or interparticle. In practice this means we return to the Green function formulation used throughout. Consider a soft repulsive or attractive potential somewhere in a noninteracting gas. Assuming no boundaries, mutually noninteracting particles can interact with the potential 0 or 1 times. (We assume for simplicity that the potential is short ranged. Because of the ergodicity assumption inherent to the random wave hypothesis, the presence of remote walls would actually make no difference.) This circumstance develops along lines very similar to the wall, except that we cannot use the method of images. It illustrates the use of the full semiclassical propagator within this formalism. Eq. 46 and Eq. 47 both hold, with the effect of Ri changed to mean “the i th particle takes the path from initial to final coordinates in which it deflects from the potential, if such a path exists classically”. For N particles, there is a “direct” term in Eq. 47 where no particle interacts with the potential, N terms where one of them does, etc. We have, in the simple case shown in Fig. 4, and in analogy with Eq. 47, G(x,x′, t) = Gdirect(x,x ′, t) + Gbounce,i(x,x ′, t) + Gbounce,i,j(x,x ′, t) + · · · (66) with Gdirect(x,x ′, t) given by Eq. 10, and e.g. Gbounce,i(z,yi, z+ r,y i, t) ≈ (N−1)D ∂2Si(yi,y i; t) ∂yi∂y 2/2~t−iV (z+ r )t/~+iSi(yi,y ;t)/~− 2 (67) Considering this term where only the ith particle with coordinate yi interacts with the potential, we have N − 1 “spectator” z particles, and the propagator becomes a product of the noninteracting Green function for N − 1 particles and a more complicated Van Vleck semiclassical term for the colliding particle. The noninteracting part contributes a term (N − 1)D/2 log t in the exponent along with the one particle classical action of the ith particle. For sufficiently large N , and tracing over the z particles, this factor leads again to the usual time condition t∗ = −iβ~ and a thermal average of the one particle energy Green function under the Fourier transform from time to energy, as in Equation 42: G(y,y′, E) ≈ G(y,y′, β) = Gdirect(y,y′, β) + Gbounce,i(y,y ′, β) + Gbounce,i,j(y,y ′, β) + · · · (68) t∗ = −iβ~ becomes the imaginary time over which the action for the y coordinates are evaluated. VII. CONCLUSION Starting with Berry’s random plane wave conjecture for chaotic Hamiltonian systems, we have followed it’s impli- cations for moderate and large numbers of particles N . In the large N limit we have necessarily arrived at some familiar territory in statistical mechanics. We have adopted a Green function, semiclassical perspective, arriving at a Gaussian-Bessel function asymptotic result for energy Green functions, providing an analytic connection between the quantum microcanonical and canonical ensembles. We have extended the incorporation of constraints into the random wave hypothesis, considering several types of constraints, including walls and interparticle collisions. Indeed the guid- ing perspective has been to make quantum waves “as random as possible subject to known prior constraints”. This must ultimately be equivalent to the ergodic hypothesis of quantum statistical mechanics. The nonstandard methods and perspective used here may possibly lead to new avenues of inquiry, and it is our hope that the semiclassical approach might permit new ways of treating strongly interacting systems. The next stage in the development of this approach is to consider short ranged potentials between particles, i.e. interparticle collisions. The first corrections to the free particle limit involve binary collisions, which can be computed semiclassically or using a delta potential appropriate to s-wave scatterers. Again the effect of the other particles will be to provide a thermal reservoir which essentially averages the Green function over a thermal distribution of energies (if N is sufficiently large). We save this for a future paper, where we hope to examine specific potentials and derive two particle radial distribution functions. FIG. 4: A short ballistic and a colliding path both lead to the same final point for a particle propagating near a localized repulsive potential. The colliding path cannot be treated by the short time approximation; rather, a Van Vleck Green function is required. In this term, all but the ith particle remain in place. Acknowledgments We thank Adam Wasserman for helpful discussions, and the National Science Foundation under grant NSF-CHE- 0073544. [1] Berry M V 1983 in Chaotic Behaviour of Deterministic Systems ed G Iooss G, R Helleman and R Stora (New York: North-Holland) p 171 [2] Berry M V 2002 J. Phys. A: Math. Gen. 35 3025 [3] Bies W E and Heller E J 2002 J. Phys. A: Math. Gen. 35 5673 [4] Bies W E, Lepore N, and Heller E J 2003 J. Phys. A: Math. Gen. 36 1605 [5] Urbina J D and Richter K 2003 J. Phys. A: Math. Gen. 36 L495 [6] Heller E J 2006 Mol. Phys. 104 1207 [7] Berry M V 1977 J. Phys. A: Math Gen. 10 2083 [8] Srednicki M 1994 Phys. Rev. E 50 888 [9] Jancel R 1969 Foundations of Classical and Quantum Statistical Mechanics (Oxford: Pergamon) [10] Kleinert H M 1995 Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics 2nd edition (Singapore: World Scientific) [11] Chandler D and Wolynes P G 1981J. Chem. Phys. 74 4078 [12] Pathria R K 1996 Statistical Mechanics 2nd edition (Oxford: Butterworth-Heineman) [13] Wang D M and Delos J B 2001 Phys. Rev. A 63 043409 [14] Berry M V 2002 J. Phys. A 35 3025 Introduction Preliminaries Diagonal limit Link to the canonical ensemble Bessel functions become Gaussians Interacting Particles - Short Time Limit Large N limit and Boltzmann averaged Green functions Stationary phase canonical limit Constraints N particles and a wall Symmetries - Fermions and Bosons Scattering Conclusion References

Wavefunction correlations and density matrices for few or many particles are derived from the properties of semiclassical energy Green functions. Universal features of fixed energy (microcanonical) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as the number of particles approaches infinity. This arises through a little known asymptotic limit of Bessel functions. Constraints due to symmetries, boundaries, and collisions between particles can be included.

Introduction Preliminaries Diagonal limit Link to the canonical ensemble Bessel functions become Gaussians Interacting Particles - Short Time Limit Large N limit and Boltzmann averaged Green functions Stationary phase canonical limit Constraints N particles and a wall Symmetries - Fermions and Bosons Scattering Conclusion References

The power of choice in network growth Raissa M. D’Souza,1, 2 Paul L. Krapivsky,3 and Cristopher Moore4, 2 1Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616 2and the Santa Fe Institute, Santa Fe, NM 87501 3Department of Physics, Boston University, Boston, MA 02215 4Computer Science Department, University of New Mexico, Albuquerque, NM 87131 The “power of choice” has been shown to radically alter the behavior of a number of randomized algorithms. Here we explore the effects of choice on models of tree and network growth. In our models each new node has k randomly chosen contacts, where k > 1 is a constant. It then attaches to whichever one of these contacts is most desirable in some sense, such as its distance from the root or its degree. Even when the new node has just two choices, i.e., when k = 2, the resulting network can be very different from a random graph or tree. For instance, if the new node attaches to the contact which is closest to the root of the tree, the distribution of depths changes from Poisson to a traveling wave solution. If the new node attaches to the contact with the smallest degree, the degree distribution is closer to uniform than in a random graph, so that with high probability there are no nodes in the network with degree greater than O(log log N). Finally, if the new node attaches to the contact with the largest degree, we find that the degree distribution is a power law with exponent −1 up to degrees roughly equal to k, with an exponential cutoff beyond that; thus, in this case, we need k � 1 to see a power law over a wide range of degrees. PACS numbers: 89.75.Hc,02.50.Ey,05.40.-a I. FORMULATION OF THE MODEL Over the past decade, the “power of choice” has emerged as a theme in research on optimization and ran- domized algorithms [1, 2, 3, 4]. Consider a random de- cision process. Typically at each step of the process a decision is reached by choosing one outcome at random and accepting this choice. Now, rather then one random alternative being presented at each decision point, let a small set of randomly generated alternatives be pre- sented, and let the best one be selected. It has been shown that with as few as two alternatives at each deci- sion point, the resulting properties of the process can be radically altered. This was first explored in the context of load-balancing the allocation of jobs arriving at ran- dom times to a batch of processors. With as few as two choices, the maximum load on any one processor drops dramatically from O(logN) to O(log logN). Increasing the number of choices beyond two only improves this by a constant factor, illustrating the “power of two choices.” Here we explore the effect of choice on random network growth. Perhaps the simplest way to build a growing ran- dom network is to attach each new node to an existing node which is chosen uniformly at random. This pro- cess generates random recursive trees which have been studied in great detail (see e.g. [5, 6, 7, 8] and references therein). Here we discuss a simple generalization: for each new node we choose k > 1 existing ‘contact’ nodes uniformly at random, select the ‘best’ one according to some definition, and connect the new node to it. This creates a random tree [9] whose statistics may be very different from those of a random recursive tree. We have to define, of course, the ‘quality’ of the node so that we can choose the best one. One natural defi- nition of quality in a tree is distance to the root — the closer to the root, the better, so that the new node at- taches to whichever one of its contacts is closest to the root (and, if more than one contact has this smallest dis- tance, we choose one of them randomly). This could cor- respond, for instance, to someone joining a hierarchical organization, and choosing to become a daughter node of whichever one of their k contacts is highest up in the hierarchy. Another natural definition is to measure quality by degree of the contact node: for instance, to attach the new node to the contact node with highest degree, again breaking ties randomly. Note that this is very different from the preferential attachment process [10], where the contact is selected from the entire graph with probabil- ity proportional to its degree. This latter process requires complete knowledge of the degree of all existing nodes. In contrast, our model assumes that the new node pos- sesses only a small amount of local information, namely, the degrees of a small number of potential contacts. This brings us to another motivation for this work: the desire to understand the effects of limited, local information on network growth. For the smallest-depth model, we find a marked differ- ence in behavior for k ≥ 2 versus k = 1. The measure of interest in this case is the depth distribution (the frac- tion of nodes at each depth j). For k = 1, i.e., a random recursive tree, this distribution is Poisson. For k ≥ 2, the same Poisson distribution is observed for distances close to the root, however for larger distances the depth distribution obeys a traveling wave solution. We also con- sider using maximal depth, rather than minimal depth, as the contact node selection criterion and find a similar traveling-wave solution. For the highest-degree model, we find that the degree distribution decays exponentially for degree i > k. For i < k the degree distribution exhibits power-law like be- havior, thus in order to observe a power law for any sub- stantial regime requires k � 1. In other words, a large amount of (overhead/state/knowledge of the system) is required to achieve a power law distribution. Finally, in analogy to the above-referenced works on load balancing, the lowest-degree model achieves a de- gree distribution which is very close to uniform, in which the maximum degree in the entire graph is O(log logN) as opposed to the maximum degree in a Poisson distri- bution, which is roughly O(logN). II. SMALLEST DEPTH Let N be the total number of nodes and Dj(N) be the number of nodes at distance j from the root. By defini- tion, D0(N) ≡ 1, since the root is distance 0 from itself. Thus D0(N) is a deterministic quantity, while Dj(N) with 1 ≤ j < N are random variables. We shall focus on their averages Qj(N) ≡ 〈Dj(N)〉. An average value provides a good description of a random variable when it is large and hence fluctuations are relatively small; we will see that this is indeed correct for D1(N). To set the stage we begin in Sect. II A with the sim- pler case of random recursive trees, for which everything is already known (see e.g. [11]). We then consider the influence of 2 or more choices in Sect. II B. A. Random recursive trees and depth The quantity Dj grows each time a node at distance j − 1 is selected as the contact node. The average depth distribution thus satisfies the master equation [12] Qj(N + 1) = Qj(N) + Qj−1(N) . (1) This equation is exact and it applies even for j=0 if we set Q−1(N) ≡ 0. Using the recursive nature of (1), we first solve for Q1(N), then Q2(N), etc. This gives Qj(N + 1) = 1≤m1<···<mj≤N m1 × · · · ×mj . (2) Equivalently, we can recast the j-fold sums into simple sums, although the results look less neat. For example, Q1(N) = HN−1 (3a) Q2(N) = (HN−1) 2 −H(2)N−1 where H(p)N = 1≤n≤N n −p are harmonic numbers. The asymptotic behaviors of HN ≡ H N , H N , and other harmonic numbers are well-known [13], and the resulting asymptotics of the depth distribution are Q1(N + 1) = lnN + γ + + · · · Q2(N + 1) = (lnN)2 + γ lnN + + · · · , where γ ≈ 0.577 is the Euler-Mascheroni constant. Anal- ogous results hold for Qj(N) for larger j. If we merely want to establish the leading asymptotic behavior, we can replace the summation in (2) by inte- gration. This leads to the simple result Qj(N)→ (lnN)j showing that in the limit N → ∞, the depth distribu- tion is Poisson with mean lnN . Alternatively, we can derive (4) within a continuum approach by replacing fi- nite differences by derivatives in the N →∞ limit of (1). This procedure recasts discrete master equations into dif- ferential equations Qj−1 (5) Solving (5) one recovers (4). The normalization requirement j≥0Dj(N) = N im- plies the sum rule for the averages∑ Qj(N) = N (6) The continuum approximation (4) agrees with the sum rule (2) implying that it well approximates the depth distribution in the entire range. We therefore use it to find the depth of the recursive random tree. The depth is defined as the maximal jmax. The criterion Qjmax = 1 leads to an estimate [11] jmax = e lnN (7) It is possible to derive this result within the exact (dis- crete) approach and to determine the fluctuations of jmax. However, for our purposes (7) is sufficient. B. The model with k = 2 choices Now suppose the new node has k = 2 choices. In this case, we have Dj(N + 1) = Dj(N) + 1 if the two contact nodes have minimum depth j−1, or equivalently, if both of them have depth at least j−1, but if they do not both have depth greater than j − 1. The probability of this is i≥j−1 = N−2 D2j−1 + 2Dj−1∑  . (8) This leads to the exact recurrence Qj(N + 1) = Qj(N) +N D2j−1 + 2Dj−1 Unfortunately, this is not very helpful since the average of the product of random quantities differs from the product of their averages, viz. 〈DiDj〉 6= 〈Di〉〈Dj〉. One can, of course, write down an exact recurrence for 〈DiDj〉, but this involves third order moments 〈DiDjDk〉, and so on. Thus the hierarchical nature of the governing equations does not allow us to obtain complete and rigorous results as is possible for the case k = 1. The cases of j = 1, 2 are exceptional and one can de- termine Q1 and Q2 analytically. For j = 1 the analysis is especially simple since D0 = 1, i≥1Di = N−1, and the growth rate (8) simplifies to [1+2(N−1)]/N2. Therefore the average number of the neighbors of the root grows ac- cording to an exact and closed recurrence Q1(N + 1) = Q1(N) + 2N − 1 Solving (9) subject to Q1(1) = 0 yields Q1(N) = 2n− 1 = 2HN−1 −H N−1 (10) Similarly for j = 2 we use relation i≥2Di = N−1−D1 and obtain Q2(N + 1) = Q2(N) + 2 N − 1 Q1(N)− 〈D21(N)〉 To obtain a closed recurrence for Q2 we need to deter- mine 〈D21(N)〉, the average of the square of the number of neighbors of the root. Then (8) leads to D1(N + 1) = D1(N) + 1 prob N−2(2N − 1) D1(N) prob 1−N−2(2N − 1) Squaring this equation and averaging we obtain 〈D21(N + 1)〉 = 2N − 1 〈D21(N)〉 2N − 1 [〈D21(N)〉+ 2Q1(N) + 1] = 〈D21(N)〉+ 2 2N − 1 Q1(N) + 2N − 1 Rather than directly solving this recurrence, we can use it together with (9) to establish a simpler recurrence for the variance V1(N) = 〈D21(N)〉 − 〈D1(N)〉2. We find V1(N + 1) = V1(N) + 2N − 1 2N − 1 which is readily solved to give V1(N + 1) = 2HN − 5H N + 4H Thus 〈D21〉 6= 〈D1〉2, yet the variance is asymptotically 2 lnN and therefore fluctuations of the random variable D1 are indeed small compared to its average which grows as 2 lnN , see (10). We determined 〈D21(N)〉 = V1(N) +Q21(N) and there- fore Q2 satisfies a closed solvable recurrence (11). The solution reads Q2(N) = [Q1(N)] 2n− 1 V1(n) + [Q1(n)]2 +Q1(n) For j ≥ 3, the problem becomes genuinely hierarchi- cal and intractable. If we are seeking only the leading behavior, however, we can proceed. When N � 1 and j is sufficiently small, namely such that i≤j Qi � N , we can replace the sum i≥j Di by N and the growth rate (8) by 2Dj−1/N . Thus we arrive at a set of differ- ential equations . (13) Solving these equations we obtain Qj(N) = (2 lnN)j . (14) We check the validity of this approximation by substitut- ing it back into our assumption i≤j Qi � N which we used in the derivation of (13). This suggests that (14) holds when j < v lnN (i.e., for small distances from the root) where v is the smallest positive root of = 1 . (15) We can write v in terms of Lambert’s function W (x), defined as the root of WeW = x: v = −1/W−1(−1/2e) (16) where W−1 denotes the −1st branch of the Lambert func- tion. Numerically, v = 0.373365... When j ≥ v lnN we cannot use (14). However, as long as Qj is much larger than 1, let us assume that the fluctuations in Dj are small. In that case we can replace averages 〈DjDk〉 by QjQk, and in this regime we obtain = N−2 Q2j−1 + 2Qj−1∑  . (17) It is convenient to introduce the cumulative variable Qi (18) that is, the average fraction of nodes whose depth is at least j. Summing (17) over all i ≥ j we arrive at a neat recurrence Nqj = q j−1 . (19) The form of this equation suggests the introduction of a new ‘time’ variable t = lnN . (20) This transformation recasts (19) into = −qj + q2j−1 (21) which should be solved subject to the step function initial condition: qj(0) = 1 for j ≤ 0 and qj(0) = 0 for j > 0. Equation (21) has appeared in various contexts (see e.g. [14]) and while it is unsolvable, an asymptotic be- havior of its solution is understood. In the long time limit, the solution approaches a ‘traveling wave’ form, qj(t)→ q(j − vt) . (22) Plugging (22) into (21) one finds that q(x) satisfies = q(x)− q(x− 1)2 . (23) The boundary conditions are q(−∞) = 1, q(+∞) = 0 . (24) The boundary-value problem (23)–(24) is still intractable analytically. However, the velocity v can be determined even without a complete solution for q(x). The method relies on the analysis of the tail region x → −∞. One notices that (23) admits an exponential solution in this region, 1− q(x) ∝ eλx as x→ −∞ . (25) Plugging this into (23) shows that the velocity v is related to λ via the dispersion relation [14] 1− 2e−λ The maximum of v = v(λ) is given by (15) and it oc- curs at the largest positive root λ of the transcendental equation 2(1 + λ) = eλ. This is λ = −1−W−1(−1/2e) (27) or numerically, λ = 1.67835... Comparing with (16), we see that λ and v are related as follows, λ = −1 + 1/v . (28) Strictly speaking, one can only assert that velocity does not exceed the maximum of (26). However, the so-called selection principle tells us that this extremal value is realized for any initial conditions which vanish sufficiently rapidly at infinity. The selection principle has been rigorously proven for a few nonlinear parabolic par- tial differential equations. Yet heuristic arguments and numerical evidence indicate that the its range of appli- cability is much broader. This is reviewed in [15] in the context of partial differential equations and in [16] in the context of difference equations. Thus there is a sharp front at depth jfront ≈ vt = v lnN to leading order, where the depth of most nodes in the tree is concentrated. Furthermore, the width of this front remains finite even in the limitN →∞. It is also possible to compute the sub-leading correction to the position of the front [14], giving an improved estimate of its location: jfront ≈ v lnN + ln lnN . (29) To estimate the maximum depth jmax, it is necessary to bound the tail of q(x) in the positive direction x→ +∞. To do this, note that by definition q(x) is monotonically decreasing, and by (23) this implies that q(x) ≤ q(x− 1)2 and therefore that this tail is doubly exponential, q(x) ∝ e−A·2 for some constant A > 0. Setting q(x) = 1/N then gives the estimate jmax ≈ jfront + ln lnN minus a constant C = lnA/ ln 2. As shown in Fig. 1, (29) and (31) are indeed excellent estimates of the average and maximum depth respectively. FIG. 1: The average depth (circles) and maximum depth (crosses) of a tree with k = 2, averaged over 103 indepdent trials for each value of N , and (dashed) the expressions (29) and (31) for jfront and jmax respectively. C. The effect of choice At first sight, it seems that having two choices instead of one does not qualitatively affect the outcome, since the depth distributions (4) and (14) both seem Poissonian, and both have typical depth O(log n). This is, however, an illusion. First of all, the distribution (4) for random recursive trees is indeed Poissonian while (14) is valid only for j < v lnN . Secondly, while both types of trees have depth O(log n), choice causes the depth to be much more concentrated. This is easiest to see if we consider the cumulative depth distribution (18). For random re- cursive trees, qj(t) is asymptotically qj(t) = j − t where erfc(z) is the error function erfc(z) = dη e−η qj(t) = 1 j − t� − 0 j − t� + The boundary layer where q changes from one to zero is not a true front as its width grows with ‘time’ as lnN . On the other hand, for the model with choice the cu- mulative depth distribution has a traveling wave shape with a front of constant width. Thus qj(t) = 1 j − jfront � −1 0 j − jfront � +1 D. Multiple choices What if the new node has more than two choices? The cases with k ≥ 3 (with k constant) are morally similar to the k = 2 case: the cumulative depth distribution obeys the differential equation = −qj + qkj−1 . (34) Transforming this to qj(t) = q(j−vt) as before, we obtain = q(x)− q(x− 1)k . (35) The solution is again a traveling wave, whose velocity v depends on k. Assuming the selection principle, v is the smallest positive root of = 1 (36) which can be written in terms of Lambert’s function as v = −1/W−1(−1/ke) . (37) Asymptotically, as k grows we have ln ke+ ln ln ke ln ln k . (38) A more precise estimate for jfront is again given by (29), with λ given by (28). For j � jfront, (4) and (14) gener- alize to Qj(N) = (k lnN)j . (39) Finally, the tail of q(x) is doubly exponential, q(x) ≈ e−Ak and the maximum depth is given by jmax ≈ jfront + ln lnN . (41) III. LARGEST DEPTH We pause here to consider a model in which we reverse our definition of the ‘better’ node, and attach each new node to the contact node which is furthest from the root. If k = 2, then we have Dj(N + 1) = Dj(N) + 1 whenever the maximum depth of the two nodes is j − 1, and this occurs with probability (j−1∑ = N−2 D2j−1 + 2Dj−1 . (42) For instance, the average number of the neighbors of the root grows according to Q1(N + 1) = Q1(N) + and therefore Q1(N) = H N−1 . (44) Thus the average number of neighbors of the root does not diverge as in the smallest depth model, but instead approaches the constant ζ(2) = π2/6. Generally, the behavior of Qj(N) for small j is very different from (14), viz. for j = O(1) the average number of nodes of depth j remains finite in the N →∞ limit. Therefore in contrast with the smallest depth model, the quantities Dj(∞) are not self-averaging when j = O(1) and their averages do not characterize them. Yet, the probability distribution P (s) = Prob[D1(∞) = s] (45) can be determined. For instance, D1(2) = 1 and the probability that the root still has one neighbor when the network size reaches N is Prob[D1(N) = 1] = and therefore P (1) = . (46) Proceeding with this line of reasoning one obtains P (s+ 1) = 2≤n1<···<ns (n21 − 1)× · · · × (n2s − 1) which can be expressed as a sum involving the zeta func- tion at positive integers. However, even though the Dj are not self-averaging, there are many similarities between the smallest depth model and this one. In particular, the cumulative depth distribution has a traveling wave shape (22). Indeed, af- ter several mappings [16] the model becomes identical to one which has appeared in studies of collision processes in gases [17], fragmentation processes [18], and other prob- lems [14]. If we define the cumulative variable as Qi , (47) then writing qj(t) = q(j − vt) gives (19), (21) and (23) again, but now with the boundary conditions q(−∞) = 0, q(+∞) = 1 . (48) With these boundary conditions, (23) admits a solution whose tail in the positive direction is exponential, 1− q(x) ∝ e−µx as x→ +∞ (49) and the dispersion relation is now 2eµ − 1 . (50) The selection principle now suggests that v is the mini- mum of (50). This is the larger of the two real roots of the transcendental equation (15), which is v = 4.31107... A more precise estimate of jfront is jfront = v lnN − ln lnN (51) where µ = 0.768039... is the larger root of 2(1−µ) = e−µ. More generally, for k > 2 the velocity v is the larger real root of (36), or v = −1/W1(−1/ke) (52) which, as k grows, approaches v ≈ ke− 1 . (53) The position of the front is given by (51) with µ = 1− 1/v . (54) Finally, since the tail of q(x) in the positive direction is given by (49), setting qj = 1 − 1/N gives the following estimate of the maximum depth, jmax = jfront + lnN . (55) Note that, unlike the minimum depth model, jmax−jfront is O(logN) instead of O(log logN), since the tail (49) is exponential rather than doubly exponential. IV. HIGHEST DEGREE We now consider a model in which quality is measured not by depth, but by the degree of the contact node — the higher the degree, the better. As we will show below, in this case the degree distribution exhibits a power law up to degree j ∼ k, beyond which it decays exponen- tially. Therefore, in this model we need a large number of choices, k � 1, in order to observe a power law over a wide range of degrees. A. Recurrence for the degree distribution We start by writing a master equation for the degree distribution of the network. We add one node at each step, so at time t there are t nodes in the network. Let Ni(t) be the number of nodes which have degree i at time t, and let Ci(t) = j=1Nj(t) be the corre- sponding total number of nodes of degree i or less at time t. Normalizing these numbers, let ai(t) = Ni(t)/t be the fraction of nodes which have degree i, and let ci(t) = j=1 aj(t) = Ci(t)/t be the corresponding cu- mulative distribution. At each iteration, we choose k contact nodes at random from the t existing nodes, and connect the new node to the contact node of highest degree, with ties broken ran- domly. The evolution of the expected cumulative degree distribution can can be written as, for all i ≥ 1, Ci(t+ 1) = Ci(t) + 1− ci(t) k − ci−1(t)k , (56) since Ci increases by 1 for each new node added, and decreases precisely when the new node connects to a node of degree i. This latter event occurs when all k nodes have degree i or less, but not all have degree i−1 or less. Writing ci(t) = Ci(t)/t and making the assumption that a steady-state limit exists, we obtain the recurrence ci = 1− (cki − c i−1) . (57) We note that in the case k = 1, where there is no choice, the solution to (57) is simply ci = 1− 2−i and ai = 2−i (58) which is the degree distribution of a random recursive tree. B. The model with k ≥ 2 choices We are particularly interested in the behavior for small k. Recall that the “power of choice” comes from situa- tions where results vary dramatically if k = 2 rather than k = 1. For k ≥ 2 we can solve (57) analytically only in the regime i � 1 as discussed in detail below. Yet, for k = 2, equation (57) is very easy to solve numerically as it reduces to the quadratic equation: c2i + ci − (1 + c i−1) = 0. (59) Figure 2 is a plot of the degree distribution, ai, for k = 1 and k = 2. Recall ai = ci − ci−1. The data points are from a numerical simulation with k = 2, grown to size 1×106 nodes. Note the excellent agreement. Though the distribution for k = 2 decays less slowly than k = 1 both exhibit exponential decay, thus the nature of the solution is not altered with such minor amounts of choice. From numerical simulation with k ≥ 2 we find different behaviors for i > k than for i < k (see Fig. 3). For degree i > k we observe ai ∼ exp(−i/k). For i < k we observe what appears to be a power law in that regime, ai ∼ k−γ , with γ ≈ 1.5. The largest k we simulated was k = 32, hence the “power law” regime is quite small. Rather than computer simulation, we can look at the asymptotic limits of (57) and arrive at these similar results in the limit i � 1 and k � 1. Note, the asymptotic limit will give γ = 1, and we can attribute the difference with numerical results to finite size effects in simulation. C. Asymptotic limits In the asymptotic regime i � 1 we write ci = 1 − �i and assume that �i � 1. To first order, cki = 1 − k�i. Simplifying (57), we find (k+ 1)�i = k�i−1 and therefore 1− ci = Ak k + 1 when i� 1, (60) where Ak is a constant depending on k. We argue below Ak ∼ k−1 as k →∞ (61) In the rest of this section we always assume that k � 1. Let us start with nodes of degree one (which are often called ‘leaves’). In this case we have c1 = a1 and equa- tion (57) reduces to a1 = 1− ak1 . (62) 5 10 15 20 deg, i FIG. 2: The degree distribution, ai, for the highest degree model, for both k = 1 and k = 2. The points at data from numerical simulation of the model with k = 2. ● ● ●●●● ●●●●●●●●● ●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●● 1 2 5 10 20 50 100 deg, i o k=16 a(i) ~ i^(−1.5) a(i) ~ exp(−i/16) FIG. 3: Numerical simulation results for k = 16. Note that for i < k we observe ai ∼ i−1.5, while for i > k we observe i ∼ e−i/k. Writing a1 = 1− and assuming that W � k yields ak1 = e−W . This allows us to recast (62) into WeW = k (64) so W is Lambert’s function W (k). For large k, we have W (k) ≈ ln k, justifying our assumption that W � k. Thus almost all nodes are leaves: the fraction of nodes whose degree exceeds one is 1−a1 = W (k)/k ≈ (ln k)/k. Analyzing (57) for i = 2, 3, . . . one finds that the fol- lowing ansatz is useful: ci = 1− W − wi Plugging (65) into (57) we obtain 1 + ewi−1 − ewi = W−1wi (66) Since W →∞ as k →∞, Eq. (66) simplifies to 1 + ewi−1 − ewi = 0 (67) whose solution (satisfying w1 = 0) is wi = ln i. Plugging this to (65) we find that ai = ci − ci−1 is given by ai = k −1 ln when 2 ≤ i� k (68) The upper bound i� k is necessary since we can use (67) instead of (66) only when wi �W which is equivalent to ln i� ln k. Note that we can further simplify (68) when i� 1, viz. when 1� i� k (69) Thus up to a crossover at i = k, the degree distribution exhibits an algebraic behavior ai ∼ i−1 with unusually small exponent. The derivation of (60) actually holds when i� k. Us- ing (60) we compute ai = ci − ci−1 to give ai = k k + 1 when i� k (70) The regions of the validity of (69) and (70) do not for- mally overlap. It is reasonable to assume, however, that they remain qualitatively correct. Then from Eq. (69) we obtain ak ∼ k−2 while Eq. (70) leads to ak ∼ k−1Ak. Matching this values we confirm the announced asymp- totic of the amplitude, Eq. (61). Furthermore, we find k + 1 ≈ e−i/k when 1� k � i. (71) V. LOWEST DEGREE There are situations where one wants to ensure that all nodes have low degree, for instance consider the case of load-balancing discussed in Sec. I. Thus the final variant we consider is when an incoming node connects to the target node of lowest degree. A. Recurrence for the degree distribution As in Sec. IV, we begin by writing the master equa- tion for the degree distribution of the network. Again let Ni(t) be the number of nodes which have degree i at time t, and now let Ci(t) = j≥iNj(t) be the cor- responding total number of nodes of degree i or greater at time t. Normalizing, let ai(t) = Ni/t and let ci(t) =∑ j≥i aj(t) = Ci(t)/t be the complementary cumulative distribution. At each iteration, we choose k contact nodes at ran- dom from the t existing nodes, and connect the new node to the contact node of lowest degree, with ties broken randomly. The evolution of the expected complementary cumulative degree distribution can can be written, for all i > 1, as Ci(t+ 1) = Ci(t) + ci−1(t) k − ci(t)k , (72) since Ci increases precisely when the new node connects to a node of degree i − 1. This event occurs when all k nodes have degree i − 1 or greater, but not all have degree i or greater. Writing ci(t) = Ci(t)/t and making the assumption that a steady-state limit exists, we obtain the recurrence ci = c i−1 − c i . (73) We note that in the case k = 1, where there is no choice, the solution to (73) is simply ci = 2 −(i−1) and ai = 2 −i (74) which, as (58), is the degree distribution of a random recursive tree. B. The model with k ≥ 2 choices For k = 2, (73) is very easy to solve numerically as it reduces to the quadratic equation: c2i + ci − c i−1 = 0 . (75) Figure 4 is a plot of the degree distribution, ai, for k = 1 and k = 2. Recall here, ai = ci − ci+1. The data points are from a numerical simulation with k = 2, grown to size 1 × 106 nodes. Note the excellent agreement. With minor choice, the degree distribution is radically altered. For all k ≥ 2 we can show the upper bound on the maximum degree is O(log logN) using a method similar to that in [19]. From (73), for i ≥ 3 we obtain the upper bound, ci ≤ cki−1, and by recursion: ci ≤ cki−1 ≤ c 2 , (76) where K = k(i−2). Since c2 < 1, ci decreases doubly- exponentially. To find imax, the typical largest degree present after addition of N nodes, we set ci = 1/N . Solv- ing this relation we find: imax ≤ logk log1/c2 N = O(log logN). (77) 5 10 15 20 deg, i FIG. 4: The degree distribution, ai, for the lowest degree model, for both k = 1 and k = 2. The points at data from numerical simulation of the model with k = 2. VI. DISCUSSION We explore the “power of choice” in network growth by introducing a minimalist generalization of random re- cursive trees. At each decision point k > 1 choices are presented and the most desirable one selected. If the cri- teria is to minimize or maximize network depth, a small amount of choice has a dramatic effect. For k = 1 the depth distribution decays with a Poisson behavior. For k ≥ 2 this Poisson decay is seen for distances close to the root, but for further distances, the depth distribution obeys a traveling wave behavior. If the criteria instead involves node degree, we must distinguish the maximum degree model from the minimum degree one. For mini- mum degree, choice has a dramatic effect. Going from k = 1 to k = 2 the degree distribution changes from geo- metric decay to double-exponential decay (and hence the maximum degree observed in the network changes from O(logN) to O(log logN)). In contrast, for maximum de- gree, a large number of choices, k � 1, must be allowed before a change from the k = 1 behavior is observed. The degree distribution decays exponentially for all small val- ues of k. Once k � 1 a power law distribution results for nodes of degree i < k, while for nodes of degree i > k the distribution decays exponentially. We established many results about the depth distri- bution. Some of them are exact, others (namely the assumption that the maximum allowed value of veloc- ity is realized, employed at the end of Sec. II B) utilize a selection principle which is not rigorously established for (21). There is no doubt of the validity of this principle in a broad range of contexts, and there is firm numerical support of all analytical results derived herein. Acknowledgments. P.L.K. is thankful to CNLS (Los Alamos National Laboratory) for hospitality during the initial stage of this research, and to Renaud Lambiotte for interesting correspondence. C.M. is supported by NSF grant CCF-0524613 and ARO contract W911NF-04-R- [1] Y. Azar, A. Z. Broder, A. R. Karlin, and E. Upfal. In Proc. 26th ACM Symp. Theory of Computing, 593–602 (1994). [2] Y. Azar, A. Z. Broder, A. R. Karlin, and E. Upfal, SIAM J. Comp. 29, 180–200 (1999). [3] M. Adler, S. Chakarabarti, M. Mitzenmacher, and L. Rasmussen, Rand. Struct. Alg. 13, 159–188 (1998). [4] M. Mitzenmacher and E. Upfal, Probability and Comput- ing : Randomized Algorithms and Probabilistic Analysis (Cambridge University Press, New York, 2005). [5] R. T. Smythe and H. Mahmoud, Theory Probab. Math. Statist. 51, 1–27 (1995). [6] M. Drmota and B. Gittenberger, Random Struct. Alg. 10, 421–451 (1997); M. Drmota and H.-K. Hwang, Adv. Appl. Probab. 37, 321–341 (2005). [7] P. L. Krapivsky and S. Redner, Phys. Rev. Lett. 89, 258703 (2002); (cond-mat/0207370). [8] S. Janson, Alea Lat. Am. J. Probab. Math. Stat. 1 347– 366 (2006); (math/0509471). [9] We always assume that we start with a tree; we further assume (if not stated otherwise) that the initial network is a single node which is the root. [10] A. L. Barabási and R. Albert, Science 286, 509–512 (1999). [11] P. L. Krapivsky and S. Redner, Phys. Rev. E 63, 066123 (2001); (cond-mat/0011094). [12] Equation (1) appears in many contexts, e.g. for networks with copying: P. L. Krapivsky and S. Redner, Phys. Rev. E 71, 036118 (2005); (cond-mat/0410379). [13] R. L. Graham, D. E. Knuth, and O. Patashnik, Con- crete Mathematics : A Foundation for Computer Science (Reading, Mass.: Addison-Wesley, 1989). [14] E. Ben-Naim, P. L. Krapivsky, and S. N. Majumdar, Phys. Rev. E 64, 035101 (2001); (cond-mat/0105309). [15] W. van Saarloos, Phys. Rep. 386, 29 (2003). [16] S. N. Majumdar and P. L. Krapivsky, Physica A 318, 161–170 (2003); (cond-mat/0205581). [17] R. van Zon, H. van Beijeren, and Ch. Dellago, Phys. Rev. Lett. 80, 2035–2038 (1998). [18] P. L. Krapivsky and S. N. Majumdar, Phys. Rev. Lett. 85, 5492–5495 (2000). [19] M. Mitzenmacher, IEEE Trans. Paral. Dist. Sys. bf 12, 1094–1104 (2001). http://arxiv.org/abs/cond-mat/0207370 http://arxiv.org/abs/math/0509471 http://arxiv.org/abs/cond-mat/0011094 http://arxiv.org/abs/cond-mat/0410379 http://arxiv.org/abs/cond-mat/0105309 http://arxiv.org/abs/cond-mat/0205581 Formulation of the model Smallest Depth Random recursive trees and depth The model with k=2 choices The effect of choice Multiple choices Largest Depth Highest Degree Recurrence for the degree distribution The model with k 2 choices Asymptotic limits Lowest Degree Recurrence for the degree distribution The model with k2 choices Discussion References

The "power of choice" has been shown to radically alter the behavior of a number of randomized algorithms. Here we explore the effects of choice on models of tree and network growth. In our models each new node has k randomly chosen contacts, where k > 1 is a constant. It then attaches to whichever one of these contacts is most desirable in some sense, such as its distance from the root or its degree. Even when the new node has just two choices, i.e., when k=2, the resulting network can be very different from a random graph or tree. For instance, if the new node attaches to the contact which is closest to the root of the tree, the distribution of depths changes from Poisson to a traveling wave solution. If the new node attaches to the contact with the smallest degree, the degree distribution is closer to uniform than in a random graph, so that with high probability there are no nodes in the network with degree greater than O(log log N). Finally, if the new node attaches to the contact with the largest degree, we find that the degree distribution is a power law with exponent -1 up to degrees roughly equal to k, with an exponential cutoff beyond that; thus, in this case, we need k >> 1 to see a power law over a wide range of degrees.

The power of choice in network growth Raissa M. D’Souza,1, 2 Paul L. Krapivsky,3 and Cristopher Moore4, 2 1Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616 2and the Santa Fe Institute, Santa Fe, NM 87501 3Department of Physics, Boston University, Boston, MA 02215 4Computer Science Department, University of New Mexico, Albuquerque, NM 87131 The “power of choice” has been shown to radically alter the behavior of a number of randomized algorithms. Here we explore the effects of choice on models of tree and network growth. In our models each new node has k randomly chosen contacts, where k > 1 is a constant. It then attaches to whichever one of these contacts is most desirable in some sense, such as its distance from the root or its degree. Even when the new node has just two choices, i.e., when k = 2, the resulting network can be very different from a random graph or tree. For instance, if the new node attaches to the contact which is closest to the root of the tree, the distribution of depths changes from Poisson to a traveling wave solution. If the new node attaches to the contact with the smallest degree, the degree distribution is closer to uniform than in a random graph, so that with high probability there are no nodes in the network with degree greater than O(log log N). Finally, if the new node attaches to the contact with the largest degree, we find that the degree distribution is a power law with exponent −1 up to degrees roughly equal to k, with an exponential cutoff beyond that; thus, in this case, we need k � 1 to see a power law over a wide range of degrees. PACS numbers: 89.75.Hc,02.50.Ey,05.40.-a I. FORMULATION OF THE MODEL Over the past decade, the “power of choice” has emerged as a theme in research on optimization and ran- domized algorithms [1, 2, 3, 4]. Consider a random de- cision process. Typically at each step of the process a decision is reached by choosing one outcome at random and accepting this choice. Now, rather then one random alternative being presented at each decision point, let a small set of randomly generated alternatives be pre- sented, and let the best one be selected. It has been shown that with as few as two alternatives at each deci- sion point, the resulting properties of the process can be radically altered. This was first explored in the context of load-balancing the allocation of jobs arriving at ran- dom times to a batch of processors. With as few as two choices, the maximum load on any one processor drops dramatically from O(logN) to O(log logN). Increasing the number of choices beyond two only improves this by a constant factor, illustrating the “power of two choices.” Here we explore the effect of choice on random network growth. Perhaps the simplest way to build a growing ran- dom network is to attach each new node to an existing node which is chosen uniformly at random. This pro- cess generates random recursive trees which have been studied in great detail (see e.g. [5, 6, 7, 8] and references therein). Here we discuss a simple generalization: for each new node we choose k > 1 existing ‘contact’ nodes uniformly at random, select the ‘best’ one according to some definition, and connect the new node to it. This creates a random tree [9] whose statistics may be very different from those of a random recursive tree. We have to define, of course, the ‘quality’ of the node so that we can choose the best one. One natural defi- nition of quality in a tree is distance to the root — the closer to the root, the better, so that the new node at- taches to whichever one of its contacts is closest to the root (and, if more than one contact has this smallest dis- tance, we choose one of them randomly). This could cor- respond, for instance, to someone joining a hierarchical organization, and choosing to become a daughter node of whichever one of their k contacts is highest up in the hierarchy. Another natural definition is to measure quality by degree of the contact node: for instance, to attach the new node to the contact node with highest degree, again breaking ties randomly. Note that this is very different from the preferential attachment process [10], where the contact is selected from the entire graph with probabil- ity proportional to its degree. This latter process requires complete knowledge of the degree of all existing nodes. In contrast, our model assumes that the new node pos- sesses only a small amount of local information, namely, the degrees of a small number of potential contacts. This brings us to another motivation for this work: the desire to understand the effects of limited, local information on network growth. For the smallest-depth model, we find a marked differ- ence in behavior for k ≥ 2 versus k = 1. The measure of interest in this case is the depth distribution (the frac- tion of nodes at each depth j). For k = 1, i.e., a random recursive tree, this distribution is Poisson. For k ≥ 2, the same Poisson distribution is observed for distances close to the root, however for larger distances the depth distribution obeys a traveling wave solution. We also con- sider using maximal depth, rather than minimal depth, as the contact node selection criterion and find a similar traveling-wave solution. For the highest-degree model, we find that the degree distribution decays exponentially for degree i > k. For i < k the degree distribution exhibits power-law like be- havior, thus in order to observe a power law for any sub- stantial regime requires k � 1. In other words, a large amount of (overhead/state/knowledge of the system) is required to achieve a power law distribution. Finally, in analogy to the above-referenced works on load balancing, the lowest-degree model achieves a de- gree distribution which is very close to uniform, in which the maximum degree in the entire graph is O(log logN) as opposed to the maximum degree in a Poisson distri- bution, which is roughly O(logN). II. SMALLEST DEPTH Let N be the total number of nodes and Dj(N) be the number of nodes at distance j from the root. By defini- tion, D0(N) ≡ 1, since the root is distance 0 from itself. Thus D0(N) is a deterministic quantity, while Dj(N) with 1 ≤ j < N are random variables. We shall focus on their averages Qj(N) ≡ 〈Dj(N)〉. An average value provides a good description of a random variable when it is large and hence fluctuations are relatively small; we will see that this is indeed correct for D1(N). To set the stage we begin in Sect. II A with the sim- pler case of random recursive trees, for which everything is already known (see e.g. [11]). We then consider the influence of 2 or more choices in Sect. II B. A. Random recursive trees and depth The quantity Dj grows each time a node at distance j − 1 is selected as the contact node. The average depth distribution thus satisfies the master equation [12] Qj(N + 1) = Qj(N) + Qj−1(N) . (1) This equation is exact and it applies even for j=0 if we set Q−1(N) ≡ 0. Using the recursive nature of (1), we first solve for Q1(N), then Q2(N), etc. This gives Qj(N + 1) = 1≤m1<···<mj≤N m1 × · · · ×mj . (2) Equivalently, we can recast the j-fold sums into simple sums, although the results look less neat. For example, Q1(N) = HN−1 (3a) Q2(N) = (HN−1) 2 −H(2)N−1 where H(p)N = 1≤n≤N n −p are harmonic numbers. The asymptotic behaviors of HN ≡ H N , H N , and other harmonic numbers are well-known [13], and the resulting asymptotics of the depth distribution are Q1(N + 1) = lnN + γ + + · · · Q2(N + 1) = (lnN)2 + γ lnN + + · · · , where γ ≈ 0.577 is the Euler-Mascheroni constant. Anal- ogous results hold for Qj(N) for larger j. If we merely want to establish the leading asymptotic behavior, we can replace the summation in (2) by inte- gration. This leads to the simple result Qj(N)→ (lnN)j showing that in the limit N → ∞, the depth distribu- tion is Poisson with mean lnN . Alternatively, we can derive (4) within a continuum approach by replacing fi- nite differences by derivatives in the N →∞ limit of (1). This procedure recasts discrete master equations into dif- ferential equations Qj−1 (5) Solving (5) one recovers (4). The normalization requirement j≥0Dj(N) = N im- plies the sum rule for the averages∑ Qj(N) = N (6) The continuum approximation (4) agrees with the sum rule (2) implying that it well approximates the depth distribution in the entire range. We therefore use it to find the depth of the recursive random tree. The depth is defined as the maximal jmax. The criterion Qjmax = 1 leads to an estimate [11] jmax = e lnN (7) It is possible to derive this result within the exact (dis- crete) approach and to determine the fluctuations of jmax. However, for our purposes (7) is sufficient. B. The model with k = 2 choices Now suppose the new node has k = 2 choices. In this case, we have Dj(N + 1) = Dj(N) + 1 if the two contact nodes have minimum depth j−1, or equivalently, if both of them have depth at least j−1, but if they do not both have depth greater than j − 1. The probability of this is i≥j−1 = N−2 D2j−1 + 2Dj−1∑  . (8) This leads to the exact recurrence Qj(N + 1) = Qj(N) +N D2j−1 + 2Dj−1 Unfortunately, this is not very helpful since the average of the product of random quantities differs from the product of their averages, viz. 〈DiDj〉 6= 〈Di〉〈Dj〉. One can, of course, write down an exact recurrence for 〈DiDj〉, but this involves third order moments 〈DiDjDk〉, and so on. Thus the hierarchical nature of the governing equations does not allow us to obtain complete and rigorous results as is possible for the case k = 1. The cases of j = 1, 2 are exceptional and one can de- termine Q1 and Q2 analytically. For j = 1 the analysis is especially simple since D0 = 1, i≥1Di = N−1, and the growth rate (8) simplifies to [1+2(N−1)]/N2. Therefore the average number of the neighbors of the root grows ac- cording to an exact and closed recurrence Q1(N + 1) = Q1(N) + 2N − 1 Solving (9) subject to Q1(1) = 0 yields Q1(N) = 2n− 1 = 2HN−1 −H N−1 (10) Similarly for j = 2 we use relation i≥2Di = N−1−D1 and obtain Q2(N + 1) = Q2(N) + 2 N − 1 Q1(N)− 〈D21(N)〉 To obtain a closed recurrence for Q2 we need to deter- mine 〈D21(N)〉, the average of the square of the number of neighbors of the root. Then (8) leads to D1(N + 1) = D1(N) + 1 prob N−2(2N − 1) D1(N) prob 1−N−2(2N − 1) Squaring this equation and averaging we obtain 〈D21(N + 1)〉 = 2N − 1 〈D21(N)〉 2N − 1 [〈D21(N)〉+ 2Q1(N) + 1] = 〈D21(N)〉+ 2 2N − 1 Q1(N) + 2N − 1 Rather than directly solving this recurrence, we can use it together with (9) to establish a simpler recurrence for the variance V1(N) = 〈D21(N)〉 − 〈D1(N)〉2. We find V1(N + 1) = V1(N) + 2N − 1 2N − 1 which is readily solved to give V1(N + 1) = 2HN − 5H N + 4H Thus 〈D21〉 6= 〈D1〉2, yet the variance is asymptotically 2 lnN and therefore fluctuations of the random variable D1 are indeed small compared to its average which grows as 2 lnN , see (10). We determined 〈D21(N)〉 = V1(N) +Q21(N) and there- fore Q2 satisfies a closed solvable recurrence (11). The solution reads Q2(N) = [Q1(N)] 2n− 1 V1(n) + [Q1(n)]2 +Q1(n) For j ≥ 3, the problem becomes genuinely hierarchi- cal and intractable. If we are seeking only the leading behavior, however, we can proceed. When N � 1 and j is sufficiently small, namely such that i≤j Qi � N , we can replace the sum i≥j Di by N and the growth rate (8) by 2Dj−1/N . Thus we arrive at a set of differ- ential equations . (13) Solving these equations we obtain Qj(N) = (2 lnN)j . (14) We check the validity of this approximation by substitut- ing it back into our assumption i≤j Qi � N which we used in the derivation of (13). This suggests that (14) holds when j < v lnN (i.e., for small distances from the root) where v is the smallest positive root of = 1 . (15) We can write v in terms of Lambert’s function W (x), defined as the root of WeW = x: v = −1/W−1(−1/2e) (16) where W−1 denotes the −1st branch of the Lambert func- tion. Numerically, v = 0.373365... When j ≥ v lnN we cannot use (14). However, as long as Qj is much larger than 1, let us assume that the fluctuations in Dj are small. In that case we can replace averages 〈DjDk〉 by QjQk, and in this regime we obtain = N−2 Q2j−1 + 2Qj−1∑  . (17) It is convenient to introduce the cumulative variable Qi (18) that is, the average fraction of nodes whose depth is at least j. Summing (17) over all i ≥ j we arrive at a neat recurrence Nqj = q j−1 . (19) The form of this equation suggests the introduction of a new ‘time’ variable t = lnN . (20) This transformation recasts (19) into = −qj + q2j−1 (21) which should be solved subject to the step function initial condition: qj(0) = 1 for j ≤ 0 and qj(0) = 0 for j > 0. Equation (21) has appeared in various contexts (see e.g. [14]) and while it is unsolvable, an asymptotic be- havior of its solution is understood. In the long time limit, the solution approaches a ‘traveling wave’ form, qj(t)→ q(j − vt) . (22) Plugging (22) into (21) one finds that q(x) satisfies = q(x)− q(x− 1)2 . (23) The boundary conditions are q(−∞) = 1, q(+∞) = 0 . (24) The boundary-value problem (23)–(24) is still intractable analytically. However, the velocity v can be determined even without a complete solution for q(x). The method relies on the analysis of the tail region x → −∞. One notices that (23) admits an exponential solution in this region, 1− q(x) ∝ eλx as x→ −∞ . (25) Plugging this into (23) shows that the velocity v is related to λ via the dispersion relation [14] 1− 2e−λ The maximum of v = v(λ) is given by (15) and it oc- curs at the largest positive root λ of the transcendental equation 2(1 + λ) = eλ. This is λ = −1−W−1(−1/2e) (27) or numerically, λ = 1.67835... Comparing with (16), we see that λ and v are related as follows, λ = −1 + 1/v . (28) Strictly speaking, one can only assert that velocity does not exceed the maximum of (26). However, the so-called selection principle tells us that this extremal value is realized for any initial conditions which vanish sufficiently rapidly at infinity. The selection principle has been rigorously proven for a few nonlinear parabolic par- tial differential equations. Yet heuristic arguments and numerical evidence indicate that the its range of appli- cability is much broader. This is reviewed in [15] in the context of partial differential equations and in [16] in the context of difference equations. Thus there is a sharp front at depth jfront ≈ vt = v lnN to leading order, where the depth of most nodes in the tree is concentrated. Furthermore, the width of this front remains finite even in the limitN →∞. It is also possible to compute the sub-leading correction to the position of the front [14], giving an improved estimate of its location: jfront ≈ v lnN + ln lnN . (29) To estimate the maximum depth jmax, it is necessary to bound the tail of q(x) in the positive direction x→ +∞. To do this, note that by definition q(x) is monotonically decreasing, and by (23) this implies that q(x) ≤ q(x− 1)2 and therefore that this tail is doubly exponential, q(x) ∝ e−A·2 for some constant A > 0. Setting q(x) = 1/N then gives the estimate jmax ≈ jfront + ln lnN minus a constant C = lnA/ ln 2. As shown in Fig. 1, (29) and (31) are indeed excellent estimates of the average and maximum depth respectively. FIG. 1: The average depth (circles) and maximum depth (crosses) of a tree with k = 2, averaged over 103 indepdent trials for each value of N , and (dashed) the expressions (29) and (31) for jfront and jmax respectively. C. The effect of choice At first sight, it seems that having two choices instead of one does not qualitatively affect the outcome, since the depth distributions (4) and (14) both seem Poissonian, and both have typical depth O(log n). This is, however, an illusion. First of all, the distribution (4) for random recursive trees is indeed Poissonian while (14) is valid only for j < v lnN . Secondly, while both types of trees have depth O(log n), choice causes the depth to be much more concentrated. This is easiest to see if we consider the cumulative depth distribution (18). For random re- cursive trees, qj(t) is asymptotically qj(t) = j − t where erfc(z) is the error function erfc(z) = dη e−η qj(t) = 1 j − t� − 0 j − t� + The boundary layer where q changes from one to zero is not a true front as its width grows with ‘time’ as lnN . On the other hand, for the model with choice the cu- mulative depth distribution has a traveling wave shape with a front of constant width. Thus qj(t) = 1 j − jfront � −1 0 j − jfront � +1 D. Multiple choices What if the new node has more than two choices? The cases with k ≥ 3 (with k constant) are morally similar to the k = 2 case: the cumulative depth distribution obeys the differential equation = −qj + qkj−1 . (34) Transforming this to qj(t) = q(j−vt) as before, we obtain = q(x)− q(x− 1)k . (35) The solution is again a traveling wave, whose velocity v depends on k. Assuming the selection principle, v is the smallest positive root of = 1 (36) which can be written in terms of Lambert’s function as v = −1/W−1(−1/ke) . (37) Asymptotically, as k grows we have ln ke+ ln ln ke ln ln k . (38) A more precise estimate for jfront is again given by (29), with λ given by (28). For j � jfront, (4) and (14) gener- alize to Qj(N) = (k lnN)j . (39) Finally, the tail of q(x) is doubly exponential, q(x) ≈ e−Ak and the maximum depth is given by jmax ≈ jfront + ln lnN . (41) III. LARGEST DEPTH We pause here to consider a model in which we reverse our definition of the ‘better’ node, and attach each new node to the contact node which is furthest from the root. If k = 2, then we have Dj(N + 1) = Dj(N) + 1 whenever the maximum depth of the two nodes is j − 1, and this occurs with probability (j−1∑ = N−2 D2j−1 + 2Dj−1 . (42) For instance, the average number of the neighbors of the root grows according to Q1(N + 1) = Q1(N) + and therefore Q1(N) = H N−1 . (44) Thus the average number of neighbors of the root does not diverge as in the smallest depth model, but instead approaches the constant ζ(2) = π2/6. Generally, the behavior of Qj(N) for small j is very different from (14), viz. for j = O(1) the average number of nodes of depth j remains finite in the N →∞ limit. Therefore in contrast with the smallest depth model, the quantities Dj(∞) are not self-averaging when j = O(1) and their averages do not characterize them. Yet, the probability distribution P (s) = Prob[D1(∞) = s] (45) can be determined. For instance, D1(2) = 1 and the probability that the root still has one neighbor when the network size reaches N is Prob[D1(N) = 1] = and therefore P (1) = . (46) Proceeding with this line of reasoning one obtains P (s+ 1) = 2≤n1<···<ns (n21 − 1)× · · · × (n2s − 1) which can be expressed as a sum involving the zeta func- tion at positive integers. However, even though the Dj are not self-averaging, there are many similarities between the smallest depth model and this one. In particular, the cumulative depth distribution has a traveling wave shape (22). Indeed, af- ter several mappings [16] the model becomes identical to one which has appeared in studies of collision processes in gases [17], fragmentation processes [18], and other prob- lems [14]. If we define the cumulative variable as Qi , (47) then writing qj(t) = q(j − vt) gives (19), (21) and (23) again, but now with the boundary conditions q(−∞) = 0, q(+∞) = 1 . (48) With these boundary conditions, (23) admits a solution whose tail in the positive direction is exponential, 1− q(x) ∝ e−µx as x→ +∞ (49) and the dispersion relation is now 2eµ − 1 . (50) The selection principle now suggests that v is the mini- mum of (50). This is the larger of the two real roots of the transcendental equation (15), which is v = 4.31107... A more precise estimate of jfront is jfront = v lnN − ln lnN (51) where µ = 0.768039... is the larger root of 2(1−µ) = e−µ. More generally, for k > 2 the velocity v is the larger real root of (36), or v = −1/W1(−1/ke) (52) which, as k grows, approaches v ≈ ke− 1 . (53) The position of the front is given by (51) with µ = 1− 1/v . (54) Finally, since the tail of q(x) in the positive direction is given by (49), setting qj = 1 − 1/N gives the following estimate of the maximum depth, jmax = jfront + lnN . (55) Note that, unlike the minimum depth model, jmax−jfront is O(logN) instead of O(log logN), since the tail (49) is exponential rather than doubly exponential. IV. HIGHEST DEGREE We now consider a model in which quality is measured not by depth, but by the degree of the contact node — the higher the degree, the better. As we will show below, in this case the degree distribution exhibits a power law up to degree j ∼ k, beyond which it decays exponen- tially. Therefore, in this model we need a large number of choices, k � 1, in order to observe a power law over a wide range of degrees. A. Recurrence for the degree distribution We start by writing a master equation for the degree distribution of the network. We add one node at each step, so at time t there are t nodes in the network. Let Ni(t) be the number of nodes which have degree i at time t, and let Ci(t) = j=1Nj(t) be the corre- sponding total number of nodes of degree i or less at time t. Normalizing these numbers, let ai(t) = Ni(t)/t be the fraction of nodes which have degree i, and let ci(t) = j=1 aj(t) = Ci(t)/t be the corresponding cu- mulative distribution. At each iteration, we choose k contact nodes at random from the t existing nodes, and connect the new node to the contact node of highest degree, with ties broken ran- domly. The evolution of the expected cumulative degree distribution can can be written as, for all i ≥ 1, Ci(t+ 1) = Ci(t) + 1− ci(t) k − ci−1(t)k , (56) since Ci increases by 1 for each new node added, and decreases precisely when the new node connects to a node of degree i. This latter event occurs when all k nodes have degree i or less, but not all have degree i−1 or less. Writing ci(t) = Ci(t)/t and making the assumption that a steady-state limit exists, we obtain the recurrence ci = 1− (cki − c i−1) . (57) We note that in the case k = 1, where there is no choice, the solution to (57) is simply ci = 1− 2−i and ai = 2−i (58) which is the degree distribution of a random recursive tree. B. The model with k ≥ 2 choices We are particularly interested in the behavior for small k. Recall that the “power of choice” comes from situa- tions where results vary dramatically if k = 2 rather than k = 1. For k ≥ 2 we can solve (57) analytically only in the regime i � 1 as discussed in detail below. Yet, for k = 2, equation (57) is very easy to solve numerically as it reduces to the quadratic equation: c2i + ci − (1 + c i−1) = 0. (59) Figure 2 is a plot of the degree distribution, ai, for k = 1 and k = 2. Recall ai = ci − ci−1. The data points are from a numerical simulation with k = 2, grown to size 1×106 nodes. Note the excellent agreement. Though the distribution for k = 2 decays less slowly than k = 1 both exhibit exponential decay, thus the nature of the solution is not altered with such minor amounts of choice. From numerical simulation with k ≥ 2 we find different behaviors for i > k than for i < k (see Fig. 3). For degree i > k we observe ai ∼ exp(−i/k). For i < k we observe what appears to be a power law in that regime, ai ∼ k−γ , with γ ≈ 1.5. The largest k we simulated was k = 32, hence the “power law” regime is quite small. Rather than computer simulation, we can look at the asymptotic limits of (57) and arrive at these similar results in the limit i � 1 and k � 1. Note, the asymptotic limit will give γ = 1, and we can attribute the difference with numerical results to finite size effects in simulation. C. Asymptotic limits In the asymptotic regime i � 1 we write ci = 1 − �i and assume that �i � 1. To first order, cki = 1 − k�i. Simplifying (57), we find (k+ 1)�i = k�i−1 and therefore 1− ci = Ak k + 1 when i� 1, (60) where Ak is a constant depending on k. We argue below Ak ∼ k−1 as k →∞ (61) In the rest of this section we always assume that k � 1. Let us start with nodes of degree one (which are often called ‘leaves’). In this case we have c1 = a1 and equa- tion (57) reduces to a1 = 1− ak1 . (62) 5 10 15 20 deg, i FIG. 2: The degree distribution, ai, for the highest degree model, for both k = 1 and k = 2. The points at data from numerical simulation of the model with k = 2. ● ● ●●●● ●●●●●●●●● ●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●● 1 2 5 10 20 50 100 deg, i o k=16 a(i) ~ i^(−1.5) a(i) ~ exp(−i/16) FIG. 3: Numerical simulation results for k = 16. Note that for i < k we observe ai ∼ i−1.5, while for i > k we observe i ∼ e−i/k. Writing a1 = 1− and assuming that W � k yields ak1 = e−W . This allows us to recast (62) into WeW = k (64) so W is Lambert’s function W (k). For large k, we have W (k) ≈ ln k, justifying our assumption that W � k. Thus almost all nodes are leaves: the fraction of nodes whose degree exceeds one is 1−a1 = W (k)/k ≈ (ln k)/k. Analyzing (57) for i = 2, 3, . . . one finds that the fol- lowing ansatz is useful: ci = 1− W − wi Plugging (65) into (57) we obtain 1 + ewi−1 − ewi = W−1wi (66) Since W →∞ as k →∞, Eq. (66) simplifies to 1 + ewi−1 − ewi = 0 (67) whose solution (satisfying w1 = 0) is wi = ln i. Plugging this to (65) we find that ai = ci − ci−1 is given by ai = k −1 ln when 2 ≤ i� k (68) The upper bound i� k is necessary since we can use (67) instead of (66) only when wi �W which is equivalent to ln i� ln k. Note that we can further simplify (68) when i� 1, viz. when 1� i� k (69) Thus up to a crossover at i = k, the degree distribution exhibits an algebraic behavior ai ∼ i−1 with unusually small exponent. The derivation of (60) actually holds when i� k. Us- ing (60) we compute ai = ci − ci−1 to give ai = k k + 1 when i� k (70) The regions of the validity of (69) and (70) do not for- mally overlap. It is reasonable to assume, however, that they remain qualitatively correct. Then from Eq. (69) we obtain ak ∼ k−2 while Eq. (70) leads to ak ∼ k−1Ak. Matching this values we confirm the announced asymp- totic of the amplitude, Eq. (61). Furthermore, we find k + 1 ≈ e−i/k when 1� k � i. (71) V. LOWEST DEGREE There are situations where one wants to ensure that all nodes have low degree, for instance consider the case of load-balancing discussed in Sec. I. Thus the final variant we consider is when an incoming node connects to the target node of lowest degree. A. Recurrence for the degree distribution As in Sec. IV, we begin by writing the master equa- tion for the degree distribution of the network. Again let Ni(t) be the number of nodes which have degree i at time t, and now let Ci(t) = j≥iNj(t) be the cor- responding total number of nodes of degree i or greater at time t. Normalizing, let ai(t) = Ni/t and let ci(t) =∑ j≥i aj(t) = Ci(t)/t be the complementary cumulative distribution. At each iteration, we choose k contact nodes at ran- dom from the t existing nodes, and connect the new node to the contact node of lowest degree, with ties broken randomly. The evolution of the expected complementary cumulative degree distribution can can be written, for all i > 1, as Ci(t+ 1) = Ci(t) + ci−1(t) k − ci(t)k , (72) since Ci increases precisely when the new node connects to a node of degree i − 1. This event occurs when all k nodes have degree i − 1 or greater, but not all have degree i or greater. Writing ci(t) = Ci(t)/t and making the assumption that a steady-state limit exists, we obtain the recurrence ci = c i−1 − c i . (73) We note that in the case k = 1, where there is no choice, the solution to (73) is simply ci = 2 −(i−1) and ai = 2 −i (74) which, as (58), is the degree distribution of a random recursive tree. B. The model with k ≥ 2 choices For k = 2, (73) is very easy to solve numerically as it reduces to the quadratic equation: c2i + ci − c i−1 = 0 . (75) Figure 4 is a plot of the degree distribution, ai, for k = 1 and k = 2. Recall here, ai = ci − ci+1. The data points are from a numerical simulation with k = 2, grown to size 1 × 106 nodes. Note the excellent agreement. With minor choice, the degree distribution is radically altered. For all k ≥ 2 we can show the upper bound on the maximum degree is O(log logN) using a method similar to that in [19]. From (73), for i ≥ 3 we obtain the upper bound, ci ≤ cki−1, and by recursion: ci ≤ cki−1 ≤ c 2 , (76) where K = k(i−2). Since c2 < 1, ci decreases doubly- exponentially. To find imax, the typical largest degree present after addition of N nodes, we set ci = 1/N . Solv- ing this relation we find: imax ≤ logk log1/c2 N = O(log logN). (77) 5 10 15 20 deg, i FIG. 4: The degree distribution, ai, for the lowest degree model, for both k = 1 and k = 2. The points at data from numerical simulation of the model with k = 2. VI. DISCUSSION We explore the “power of choice” in network growth by introducing a minimalist generalization of random re- cursive trees. At each decision point k > 1 choices are presented and the most desirable one selected. If the cri- teria is to minimize or maximize network depth, a small amount of choice has a dramatic effect. For k = 1 the depth distribution decays with a Poisson behavior. For k ≥ 2 this Poisson decay is seen for distances close to the root, but for further distances, the depth distribution obeys a traveling wave behavior. If the criteria instead involves node degree, we must distinguish the maximum degree model from the minimum degree one. For mini- mum degree, choice has a dramatic effect. Going from k = 1 to k = 2 the degree distribution changes from geo- metric decay to double-exponential decay (and hence the maximum degree observed in the network changes from O(logN) to O(log logN)). In contrast, for maximum de- gree, a large number of choices, k � 1, must be allowed before a change from the k = 1 behavior is observed. The degree distribution decays exponentially for all small val- ues of k. Once k � 1 a power law distribution results for nodes of degree i < k, while for nodes of degree i > k the distribution decays exponentially. We established many results about the depth distri- bution. Some of them are exact, others (namely the assumption that the maximum allowed value of veloc- ity is realized, employed at the end of Sec. II B) utilize a selection principle which is not rigorously established for (21). There is no doubt of the validity of this principle in a broad range of contexts, and there is firm numerical support of all analytical results derived herein. Acknowledgments. P.L.K. is thankful to CNLS (Los Alamos National Laboratory) for hospitality during the initial stage of this research, and to Renaud Lambiotte for interesting correspondence. C.M. is supported by NSF grant CCF-0524613 and ARO contract W911NF-04-R- [1] Y. Azar, A. Z. Broder, A. R. Karlin, and E. Upfal. In Proc. 26th ACM Symp. Theory of Computing, 593–602 (1994). [2] Y. Azar, A. Z. Broder, A. R. Karlin, and E. Upfal, SIAM J. Comp. 29, 180–200 (1999). [3] M. Adler, S. Chakarabarti, M. Mitzenmacher, and L. Rasmussen, Rand. Struct. Alg. 13, 159–188 (1998). [4] M. Mitzenmacher and E. Upfal, Probability and Comput- ing : Randomized Algorithms and Probabilistic Analysis (Cambridge University Press, New York, 2005). [5] R. T. Smythe and H. 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Sys. bf 12, 1094–1104 (2001). http://arxiv.org/abs/cond-mat/0207370 http://arxiv.org/abs/math/0509471 http://arxiv.org/abs/cond-mat/0011094 http://arxiv.org/abs/cond-mat/0410379 http://arxiv.org/abs/cond-mat/0105309 http://arxiv.org/abs/cond-mat/0205581 Formulation of the model Smallest Depth Random recursive trees and depth The model with k=2 choices The effect of choice Multiple choices Largest Depth Highest Degree Recurrence for the degree distribution The model with k 2 choices Asymptotic limits Lowest Degree Recurrence for the degree distribution The model with k2 choices Discussion References

Terahertz Time-Domain Magnetospectroscopy of a High-Mobility Two-Dimensional Electron Gas Xiangfeng Wang, David J. Hilton, Lei Ren, Daniel M. Mittleman, and Junichiro Kono∗ Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA John L. Reno Sandia National Laboratories, P. O. Box 5800, Albuquerque, New Mexico 87185, USA (Dated: October 27, 2018) We have observed cyclotron resonance in a high-mobility GaAs/AlGaAs two-dimensional electron gas by using the techniques of terahertz time-domain spectroscopy combined with magnetic fields. From this, we calculate the real and imaginary parts of the diagonal elements of the magnetocon- ductivity tensor, which in turn allows us to extract the concentration, effective mass, and scattering time of the electrons in the sample. We demonstrate the utility of ultrafast terahertz spectroscopy, which can recover the true linewidth of cyclotron resonance in a high-mobility (> 106 cm2 V−1 s−1) sample without being affected by the saturation effect. PACS numbers: 300.6500 Spectroscopy, time-resolved, 320.7130 Ultrafast processes in condensed matter, including semiconductors Quantum coherence is an important ingredient in mod- ern condensed matter physics as well as in emerging tech- nologies. The creation and manipulation of a coherent superposition of two or multiple quantum states is the subject of many current studies. An ultrahigh-mobility two-dimensional electron gas (2DEG) offers an ideal sys- tem for studying novel quantum coherent phenomena in a clean, solid-state environment. In particular, when one applies a magnetic field perpendicular to the 2DEG, the density of states splits into Landau levels, making a fully- tunable, atomic-like system. In addition, a variety of phe- nomena that occur in the 2DEG arising from carrier in- teractions, confinement, and disorder can make quantum coherent effects even more exotic than in atomic or molec- ular systems. However, there has been little success in performing coherent spectroscopy of Landau-quantized 2DEGs, although there is a long history of cyclotron res- onance (CR) studies of 2DEGs using Fourier-transform infrared (FTIR) spectroscopy [1, 2, 3, 4, 5, 6]. Terahertz (THz) time-domain magnetospec- troscopy [7], which combines conventional THz time-domain spectroscopy (THz-TDS) with a high magnetic field, has a number of inherent advantages compared to traditional FTIR techniques. THz-TDS directly measures both the amplitude and phase of the electric field E and allows for the simultaneous determination of the real and imaginary parts of the conductivity without using Kramers-Kronig techniques. Additionally, use of a temporally-gated detection scheme, common to THz-TDS techniques, significantly suppresses background thermal noise and results in an enhanced signal-to-noise ratio [8, 9]. THz-TDS was used earlier [10] to observe CR in rela- tively low-mobility (µe = 2.7 × 10 5 cm2V−1s−1) 2DEG ∗Electronic address: kono@rice.edu samples. In addition, THz-TDS has been successfully employed to study quantum coherent phenomena in a wide range of systems, including the rotational transi- tions of N2O molecules [11], intersubband transitions in semiconductor quantum wells [12], and surface plasmons propagating on metal-film hole arrays [13]. Here, we report the observation of long-lived, magnetic-field-dependent coherent oscillations in a high- mobility GaAs/AlGaAs 2DEG in a perpendicular mag- netic field. We explain our observations in terms of a co- herent superposition created by the incident THz pulse between the lowest unfilled Landau level and the highest filled Landau level. In addition, we determine elements of the complex magnetoconductivity tensor σ̃ as a func- tion of both frequency ν and magnetic field B, which in turn allows us to determine the cyclotron frequency νc, effective mass m ∗, and cyclotron resonance linewidth ∆νc (or the scattering time τ = 1/∆νc) as a function of B. Finally, we show that THz-TDS can overcome the “saturation effect” [14, 15] that often prevents FTIR- based techniques from determining the true linewithds of CR in high-mobility (µe > 10 6 cm2V−1s−1) 2DEGs. We successfully measured the linewidth as a function of temperature and magnetic field, which will be reported in detail elsewhere. Broadband, ultrashort THz pulses were generated and detected using a standard photoconductive antenna- receiver setup [8, 9]. Time delay was provided by an oscillating retro-reflector operating at 3 Hz; each data set was the averaged result of ∼ 800 scans. We used an Oxford superconducting magnet (SM-4000-10T) to pro- duce fields ranging from 0 to 1.4 T and temperatures from 1.5 K to 300 K. The sample studied in this experiment was a modulation-doped GaAs/AlGaAs single quantum well with an electron concentration of ne = 2.0 × 1011 cm−2 and mobility of µe = 3.7 × 10 6 cm2 V−1 s−1 at 4.2 K, determined through Shubnikov-de Haas and DC conductivity measurements. http://arxiv.org/abs/0704.1883v1 mailto:kono@rice.edu 20100-10-20 Time (ps) 1.28T FIG. 1: (color online) THz waveforms transmitted through a high-mobility 2DEG at 0 T (a) and at 1.28 T (b) at 2 K. The cyclotron oscillations induced by the magnetic field (c) are isolated by subtracting (a) from (b). In this experiment, the THz waveform is measured af- ter transmission through the sample in a B field from 0 T to 1.4 T. Figure 1 plots these waveforms at 0 T [trace (a)] and at a finite B (1.28 T) [trace (b)]. Trace (c) is the difference between the transmitted THz electric field at 1.28 T and 0 T highlighting the change to the THz trans- mission due to the B field (data enlarged 10 times), which shows the B-induced oscillations of the electric field of the THz pulse. We verify that the observed oscillations originate from the 2DEG and not a B-dependence of any of the optics in the experiment by first measuring the B- dependent THz transmission in the absence of the 2DEG in an otherwise identical configuration. Figure 2(a) shows similar oscillations induced in the transmitted THz wave- form from 0.7 T to 1.4 T whose frequency and decay time vary with B. Figure 3(a) shows the amplitude of the Fourier- transformed electric fields at 0 T and 1.28 T. A B-field- induced absorption, or a dip, is clearly seen in the 1.28 T spectrum. Figure 3(b) shows the magnitude of the com- plex transmission coefficient (T = ∣eiφ) at 1.28 T, while Fig. 3(c) shows the phase, φ. In our experiment, the THz pulse is linearly polarized (x̂), and we detect only one polarization component (x̂) after transmission through the sample. As a result, our measurement is dependent on the corresponding diagonal element of the magnetoconductivity tensor, σxx [1]. In order to extract the conductivity from the complex transmission coeffi- cient, we model this sample as a thin conducting sheet on a thick substrate with an index of refraction, n. In this approximation, the ratio of the Fourier transform of the waveforms at a finite B field, E , to the zero-field transmitted spectrum, E , is given by: 2Y + σxx ) , (1) 20100-10 Time (ps) 0.8 T 0.9 T 1.0 T 1.1 T 1.2 T 1.3 T 1.4 T 0.7 T (a) E |N +1> FIG. 2: (a) Time-domain cyclotron oscillations in a high- mobility 2DEG from 0.7 T to 1.4 T at 2 K. Traces are verti- cally offset for clarity. (b) A Landau-quantized 2DEG. DOS: density of states. 0.80.70.60.50.40.3 Frequency (THz) Experiment Theory 1.28 T σ''xx 0.80.70.60.50.40.3 Frequency (THz) 1.28 T 1.20.80.40.0 Frequency (THz) 1.28 T (a) 1.2 0.80.70.60.50.40.3 Frequency (THz) 1.28 T FIG. 3: (color online) (a) Amplitude of the transformed elec- tric fields at 0 T and 1.28 T. (b) Magnitude of the complex transmission coefficient at 1.28 T. (c) Phase of the transmis- sion coefficient. (d) Real (σ′ ) and imaginary (σ′′ ) parts of the magnetoconductivity tensor element σxx at 1.28 T. The trace is vertically offset. where Y = n/Z0 is the admittance of the GaAs substrate and Z0 = 377 Ω is the impedance of free space [8]. Fig- ure 3(d) highlights real (σ′xx) and imaginary (σ xx) parts of the extracted conductivity tensor element at 1.28 T. Due to the rotational symmetry of the system perpendic- ular to the plane of the 2DEG, we would expect the same result in the case of input polarization and detection both along ŷ, i.e., σxx = σyy [1]. We determine the cyclotron frequency νc (s −1), the cyclotron resonance linewidth ∆νc = 1/τ (s −1), and the magnitude of the conductivity σ0 (Ω −1) by fitting the results shown in Figs. 3(c) and 3(d). Both the real (σ′xx) and imaginary (σ′′xx) parts of the magnetoconductivity tensor element are fit by σxx = σ xx + iσ 1 + 2πi ν − νc . (2) A representative fit at 1.28 T is shown in Fig. 3(d), using σ0 = 0.0126 Ω −1, νc = 0.529 THz, and τ = 15.6 ps. An applied B field perpendicular to the 2DEG results in the formation of discrete Landau levels [see Fig. 2(b)] with an energy separation, ∆E, between the ∣N + 1〉 levels given by ∆E = ~ = hνc (3) where e is the electron charge and h is Planck’s constant. An incident THz wave with a photon energy equal to this separation coherently creates a superposition state between the highest filled Landau level, , and the low- est unfilled Landau level, ∣N +1 , as shown in Fig. 2(b). This results in an atomic-like two-level system; all other Landau levels are either completely filled or completely empty (as long as the THz field is sufficiently weak as in our experiment) and do not affect the transmission of the THz pulse. The observed damped oscillations in our ex- perimental data can thus be viewed as the free induction decay [16] of such coherently coupled Landau levels. Using the extracted value of νc and Eq. (3), we obtain a value of the effective mass of m∗ = 0.0676m0, where m0 = 9.11 × 10 −31 kg is the free electron mass. Also, using the extracted values of σ0 and τ , we can determine the value of the electron concentration of ne = 1.95 × 1011 cm−2, which is consistent with the concentration ob- tained from transport measurements (2.0 × 1011 cm−2). Finally, the extracted linewidth is a measure of the scat- tering mechanisms present in the sample at this temper- ature and B field. We have systematically studied the temperature and B dependence of ∆νc, which would al- low us to elucidate a detailed theoretical understanding of the physical origins of this linewidth and will be re- ported in detail eslewhere. In high-mobility samples, the apparent linewidth de- termined by FTIR measurements is much larger than the true linewidth, a phenomenon commonly referred to as the “saturation effect” [14, 15]. This results from the decrease in detectable transmission of the THz radiation over a broad spectral range; as the conductivity increases with either increasing carrier concentration or mobility, a spectral region exists with a finite width where, effec- tively, no transmission is permitted. The lack of a phase sensitive detection scheme in traditional FTIR techniques makes the direct determination of the complex conduc- tivity in this situation difficult; the broadened linewidth in this case could result either from the increase in mo- bility or concentration. Because of this saturation effect, almost no systematic linewidth studies exist for high- mobility 2DEG samples (> 106 cm2 V−1 s−1). In order to overcome the lack of phase sensitive detec- tion in FTIR measurements, different methods have been proposed. For example, measurement of the transmission coefficient of a 2DEG over a broad spectral range will per- mit the use of Kramers-Kronig techniques to calculate the phase at THz frequencies and determine the complex conductivity [17]. A second alternate method for deter- mining the complex conductivity assumes a Drude form for σ̃ and fits this to the measured intensity transmission coefficient; lack of a direct phase measurement makes this an ambiguous determination of ne and τ [14]. THz-TDS allows for the direct determination of the full complex conductivity of the sample without resort to Kramers-Kronig techniques and without an a priori as- sumption of the lineshape function. The increased signal- to-noise ratio inherent to the gated detection scheme allows for the determination of the lower transmitted THz signals that result from high-mobility and high- concentration 2DEGs. Second, the additional spectro- scopic information determined from the phase sensitive measurement removes the ambiguity between ne and τ . As a result, no assumption of lineshape is necessary in or- der to calculate the full complex conductivity. Employing this technique allows for the simultaneous determination of both ne and τ from the measurement of the transmit- ted THz waveform electric field. In summary, we have observed time-domain cyclotron resonance oscillations in a GaAs/AlGaAs 2DEG, which can be modeled as the decay of the coherent superpo- sition of two coupled Landau levels induced by the in- cident THz pulse. The real and imaginary parts of the conductivity are determined simultaneously at different magnetic fields without using Kramers-Kronig analysis. We show that our THz technique has many advantages for doing cyclotron resonance measurements, especially for high-mobility samples. This work was supported by the National Science Foundation (through Grant Nos. DMR-0134058 and DMR-0325474). Sandia is a multiprogram laboratory op- erated by Sandia Corporation, a Lockheed Martin Com- pany, for the National Nuclear Security Administration under Contract DE-AC04-94AL85000. [1] E. D. Palik and J. K. Furdyna, Rep. Prog. Phys. 33, 1193 (1970). [2] J. G. Mavroides, in Optical Properties of Solids, edited by F. Abeles (North-Holland, Amsterdam, 1972), pp. 351– [3] B. D. McCombe and R. J. Wagner, in Advances in Elec- tronics and Electron Physics, edited by L. Marton (Aca- demic Press, New York, 1975), vol. 37, pp. 1–78. [4] A. Petrou and B. D. McCombe, in Landau Level Spec- troscopy, edited by G. Landwehr and E. I. Rashba (Else- vier Science, Amsterdam, 1991), pp. 679–775. [5] R. J. Nicholas, in Handbook on Semiconductors, Vol. 2 “Optical Properties”, edited by M. Balkanski (Elsevier, Amsterdam, 1994), pp. 385–461. [6] J. Kono, in Methods in Materials Research, edited by E. N. Kaufmann et al. (John Wiley & Sons, New York, 2001), Unit 9b.2. [7] S. A. Crooker, Rev. Sci. Instrum. 73, 3258 (2002). [8] M. C. Nuss and J. Orenstein, in Millimeter and Submil- limeter Wave Spectroscopy of Solids, edited by G. Grüner (Springer-Verlag, Berlin, 1998), pp. 7–50. [9] D. M. Mittleman, ed., Sensing with Terahertz Radiation, (Springer, Berlin, 2003). [10] D. Some and A. V. Nurmikko, Appl. Phys. Lett. 65, 3377 (1994). [11] H. Harde, S. Keiding, and D. Grischkowsky, Phys. Rev. Lett. 66, 1834 (1991). [12] J. N. Heyman, R. Kersting, and K. Unterrainer, Appl. Phys. Lett. 72, 644 (1998). [13] D. Du and D. Grischkowsky, Phys. Rev. Lett. 93, 196804 (2004). [14] M. J. Chou and D. C. Tsui, Phys. Rev. B 37, 848 (1988). [15] S. A. Studenikin, M. Potemski, A. Sachrajda, M. Hillke, P. Pfeiffer, and K. W. West, Phys. Rev. B 71, 245313 (2005). [16] L. Allen and J. H. Eberly, Optical Resonance and Two- Level Atoms (Dover, New York, 1987). [17] H. Kuzmany, Solid-State Spectroscopy (Springer, Berlin, 1998).

We have observed cyclotron resonance in a high-mobility GaAs/AlGaAs two-dimensional electron gas by using the techniques of terahertz time-domain spectroscopy combined with magnetic fields. From this, we calculate the real and imaginary parts of the diagonal elements of the magnetoconductivity tensor, which in turn allows us to extract the concentration, effective mass, and scattering time of the electrons in the sample. We demonstrate the utility of ultrafast terahertz spectroscopy, which can recover the true linewidth of cyclotron resonance in a high-mobility ($>{10}^{6} \mathrm{cm^{2} V^{-1} s^{-1}}$) sample without being affected by the saturation effect.

Terahertz Time-Domain Magnetospectroscopy of a High-Mobility Two-Dimensional Electron Gas Xiangfeng Wang, David J. Hilton, Lei Ren, Daniel M. Mittleman, and Junichiro Kono∗ Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA John L. Reno Sandia National Laboratories, P. O. Box 5800, Albuquerque, New Mexico 87185, USA (Dated: October 27, 2018) We have observed cyclotron resonance in a high-mobility GaAs/AlGaAs two-dimensional electron gas by using the techniques of terahertz time-domain spectroscopy combined with magnetic fields. From this, we calculate the real and imaginary parts of the diagonal elements of the magnetocon- ductivity tensor, which in turn allows us to extract the concentration, effective mass, and scattering time of the electrons in the sample. We demonstrate the utility of ultrafast terahertz spectroscopy, which can recover the true linewidth of cyclotron resonance in a high-mobility (> 106 cm2 V−1 s−1) sample without being affected by the saturation effect. PACS numbers: 300.6500 Spectroscopy, time-resolved, 320.7130 Ultrafast processes in condensed matter, including semiconductors Quantum coherence is an important ingredient in mod- ern condensed matter physics as well as in emerging tech- nologies. The creation and manipulation of a coherent superposition of two or multiple quantum states is the subject of many current studies. An ultrahigh-mobility two-dimensional electron gas (2DEG) offers an ideal sys- tem for studying novel quantum coherent phenomena in a clean, solid-state environment. In particular, when one applies a magnetic field perpendicular to the 2DEG, the density of states splits into Landau levels, making a fully- tunable, atomic-like system. In addition, a variety of phe- nomena that occur in the 2DEG arising from carrier in- teractions, confinement, and disorder can make quantum coherent effects even more exotic than in atomic or molec- ular systems. However, there has been little success in performing coherent spectroscopy of Landau-quantized 2DEGs, although there is a long history of cyclotron res- onance (CR) studies of 2DEGs using Fourier-transform infrared (FTIR) spectroscopy [1, 2, 3, 4, 5, 6]. Terahertz (THz) time-domain magnetospec- troscopy [7], which combines conventional THz time-domain spectroscopy (THz-TDS) with a high magnetic field, has a number of inherent advantages compared to traditional FTIR techniques. THz-TDS directly measures both the amplitude and phase of the electric field E and allows for the simultaneous determination of the real and imaginary parts of the conductivity without using Kramers-Kronig techniques. Additionally, use of a temporally-gated detection scheme, common to THz-TDS techniques, significantly suppresses background thermal noise and results in an enhanced signal-to-noise ratio [8, 9]. THz-TDS was used earlier [10] to observe CR in rela- tively low-mobility (µe = 2.7 × 10 5 cm2V−1s−1) 2DEG ∗Electronic address: kono@rice.edu samples. In addition, THz-TDS has been successfully employed to study quantum coherent phenomena in a wide range of systems, including the rotational transi- tions of N2O molecules [11], intersubband transitions in semiconductor quantum wells [12], and surface plasmons propagating on metal-film hole arrays [13]. Here, we report the observation of long-lived, magnetic-field-dependent coherent oscillations in a high- mobility GaAs/AlGaAs 2DEG in a perpendicular mag- netic field. We explain our observations in terms of a co- herent superposition created by the incident THz pulse between the lowest unfilled Landau level and the highest filled Landau level. In addition, we determine elements of the complex magnetoconductivity tensor σ̃ as a func- tion of both frequency ν and magnetic field B, which in turn allows us to determine the cyclotron frequency νc, effective mass m ∗, and cyclotron resonance linewidth ∆νc (or the scattering time τ = 1/∆νc) as a function of B. Finally, we show that THz-TDS can overcome the “saturation effect” [14, 15] that often prevents FTIR- based techniques from determining the true linewithds of CR in high-mobility (µe > 10 6 cm2V−1s−1) 2DEGs. We successfully measured the linewidth as a function of temperature and magnetic field, which will be reported in detail elsewhere. Broadband, ultrashort THz pulses were generated and detected using a standard photoconductive antenna- receiver setup [8, 9]. Time delay was provided by an oscillating retro-reflector operating at 3 Hz; each data set was the averaged result of ∼ 800 scans. We used an Oxford superconducting magnet (SM-4000-10T) to pro- duce fields ranging from 0 to 1.4 T and temperatures from 1.5 K to 300 K. The sample studied in this experiment was a modulation-doped GaAs/AlGaAs single quantum well with an electron concentration of ne = 2.0 × 1011 cm−2 and mobility of µe = 3.7 × 10 6 cm2 V−1 s−1 at 4.2 K, determined through Shubnikov-de Haas and DC conductivity measurements. http://arxiv.org/abs/0704.1883v1 mailto:kono@rice.edu 20100-10-20 Time (ps) 1.28T FIG. 1: (color online) THz waveforms transmitted through a high-mobility 2DEG at 0 T (a) and at 1.28 T (b) at 2 K. The cyclotron oscillations induced by the magnetic field (c) are isolated by subtracting (a) from (b). In this experiment, the THz waveform is measured af- ter transmission through the sample in a B field from 0 T to 1.4 T. Figure 1 plots these waveforms at 0 T [trace (a)] and at a finite B (1.28 T) [trace (b)]. Trace (c) is the difference between the transmitted THz electric field at 1.28 T and 0 T highlighting the change to the THz trans- mission due to the B field (data enlarged 10 times), which shows the B-induced oscillations of the electric field of the THz pulse. We verify that the observed oscillations originate from the 2DEG and not a B-dependence of any of the optics in the experiment by first measuring the B- dependent THz transmission in the absence of the 2DEG in an otherwise identical configuration. Figure 2(a) shows similar oscillations induced in the transmitted THz wave- form from 0.7 T to 1.4 T whose frequency and decay time vary with B. Figure 3(a) shows the amplitude of the Fourier- transformed electric fields at 0 T and 1.28 T. A B-field- induced absorption, or a dip, is clearly seen in the 1.28 T spectrum. Figure 3(b) shows the magnitude of the com- plex transmission coefficient (T = ∣eiφ) at 1.28 T, while Fig. 3(c) shows the phase, φ. In our experiment, the THz pulse is linearly polarized (x̂), and we detect only one polarization component (x̂) after transmission through the sample. As a result, our measurement is dependent on the corresponding diagonal element of the magnetoconductivity tensor, σxx [1]. In order to extract the conductivity from the complex transmission coeffi- cient, we model this sample as a thin conducting sheet on a thick substrate with an index of refraction, n. In this approximation, the ratio of the Fourier transform of the waveforms at a finite B field, E , to the zero-field transmitted spectrum, E , is given by: 2Y + σxx ) , (1) 20100-10 Time (ps) 0.8 T 0.9 T 1.0 T 1.1 T 1.2 T 1.3 T 1.4 T 0.7 T (a) E |N +1> FIG. 2: (a) Time-domain cyclotron oscillations in a high- mobility 2DEG from 0.7 T to 1.4 T at 2 K. Traces are verti- cally offset for clarity. (b) A Landau-quantized 2DEG. DOS: density of states. 0.80.70.60.50.40.3 Frequency (THz) Experiment Theory 1.28 T σ''xx 0.80.70.60.50.40.3 Frequency (THz) 1.28 T 1.20.80.40.0 Frequency (THz) 1.28 T (a) 1.2 0.80.70.60.50.40.3 Frequency (THz) 1.28 T FIG. 3: (color online) (a) Amplitude of the transformed elec- tric fields at 0 T and 1.28 T. (b) Magnitude of the complex transmission coefficient at 1.28 T. (c) Phase of the transmis- sion coefficient. (d) Real (σ′ ) and imaginary (σ′′ ) parts of the magnetoconductivity tensor element σxx at 1.28 T. The trace is vertically offset. where Y = n/Z0 is the admittance of the GaAs substrate and Z0 = 377 Ω is the impedance of free space [8]. Fig- ure 3(d) highlights real (σ′xx) and imaginary (σ xx) parts of the extracted conductivity tensor element at 1.28 T. Due to the rotational symmetry of the system perpendic- ular to the plane of the 2DEG, we would expect the same result in the case of input polarization and detection both along ŷ, i.e., σxx = σyy [1]. We determine the cyclotron frequency νc (s −1), the cyclotron resonance linewidth ∆νc = 1/τ (s −1), and the magnitude of the conductivity σ0 (Ω −1) by fitting the results shown in Figs. 3(c) and 3(d). Both the real (σ′xx) and imaginary (σ′′xx) parts of the magnetoconductivity tensor element are fit by σxx = σ xx + iσ 1 + 2πi ν − νc . (2) A representative fit at 1.28 T is shown in Fig. 3(d), using σ0 = 0.0126 Ω −1, νc = 0.529 THz, and τ = 15.6 ps. An applied B field perpendicular to the 2DEG results in the formation of discrete Landau levels [see Fig. 2(b)] with an energy separation, ∆E, between the ∣N + 1〉 levels given by ∆E = ~ = hνc (3) where e is the electron charge and h is Planck’s constant. An incident THz wave with a photon energy equal to this separation coherently creates a superposition state between the highest filled Landau level, , and the low- est unfilled Landau level, ∣N +1 , as shown in Fig. 2(b). This results in an atomic-like two-level system; all other Landau levels are either completely filled or completely empty (as long as the THz field is sufficiently weak as in our experiment) and do not affect the transmission of the THz pulse. The observed damped oscillations in our ex- perimental data can thus be viewed as the free induction decay [16] of such coherently coupled Landau levels. Using the extracted value of νc and Eq. (3), we obtain a value of the effective mass of m∗ = 0.0676m0, where m0 = 9.11 × 10 −31 kg is the free electron mass. Also, using the extracted values of σ0 and τ , we can determine the value of the electron concentration of ne = 1.95 × 1011 cm−2, which is consistent with the concentration ob- tained from transport measurements (2.0 × 1011 cm−2). Finally, the extracted linewidth is a measure of the scat- tering mechanisms present in the sample at this temper- ature and B field. We have systematically studied the temperature and B dependence of ∆νc, which would al- low us to elucidate a detailed theoretical understanding of the physical origins of this linewidth and will be re- ported in detail eslewhere. In high-mobility samples, the apparent linewidth de- termined by FTIR measurements is much larger than the true linewidth, a phenomenon commonly referred to as the “saturation effect” [14, 15]. This results from the decrease in detectable transmission of the THz radiation over a broad spectral range; as the conductivity increases with either increasing carrier concentration or mobility, a spectral region exists with a finite width where, effec- tively, no transmission is permitted. The lack of a phase sensitive detection scheme in traditional FTIR techniques makes the direct determination of the complex conduc- tivity in this situation difficult; the broadened linewidth in this case could result either from the increase in mo- bility or concentration. Because of this saturation effect, almost no systematic linewidth studies exist for high- mobility 2DEG samples (> 106 cm2 V−1 s−1). In order to overcome the lack of phase sensitive detec- tion in FTIR measurements, different methods have been proposed. For example, measurement of the transmission coefficient of a 2DEG over a broad spectral range will per- mit the use of Kramers-Kronig techniques to calculate the phase at THz frequencies and determine the complex conductivity [17]. A second alternate method for deter- mining the complex conductivity assumes a Drude form for σ̃ and fits this to the measured intensity transmission coefficient; lack of a direct phase measurement makes this an ambiguous determination of ne and τ [14]. THz-TDS allows for the direct determination of the full complex conductivity of the sample without resort to Kramers-Kronig techniques and without an a priori as- sumption of the lineshape function. The increased signal- to-noise ratio inherent to the gated detection scheme allows for the determination of the lower transmitted THz signals that result from high-mobility and high- concentration 2DEGs. Second, the additional spectro- scopic information determined from the phase sensitive measurement removes the ambiguity between ne and τ . As a result, no assumption of lineshape is necessary in or- der to calculate the full complex conductivity. Employing this technique allows for the simultaneous determination of both ne and τ from the measurement of the transmit- ted THz waveform electric field. In summary, we have observed time-domain cyclotron resonance oscillations in a GaAs/AlGaAs 2DEG, which can be modeled as the decay of the coherent superpo- sition of two coupled Landau levels induced by the in- cident THz pulse. The real and imaginary parts of the conductivity are determined simultaneously at different magnetic fields without using Kramers-Kronig analysis. We show that our THz technique has many advantages for doing cyclotron resonance measurements, especially for high-mobility samples. This work was supported by the National Science Foundation (through Grant Nos. DMR-0134058 and DMR-0325474). Sandia is a multiprogram laboratory op- erated by Sandia Corporation, a Lockheed Martin Com- pany, for the National Nuclear Security Administration under Contract DE-AC04-94AL85000. [1] E. D. Palik and J. K. Furdyna, Rep. Prog. Phys. 33, 1193 (1970). [2] J. G. Mavroides, in Optical Properties of Solids, edited by F. Abeles (North-Holland, Amsterdam, 1972), pp. 351– [3] B. D. McCombe and R. J. Wagner, in Advances in Elec- tronics and Electron Physics, edited by L. Marton (Aca- demic Press, New York, 1975), vol. 37, pp. 1–78. [4] A. Petrou and B. D. McCombe, in Landau Level Spec- troscopy, edited by G. Landwehr and E. I. Rashba (Else- vier Science, Amsterdam, 1991), pp. 679–775. [5] R. J. Nicholas, in Handbook on Semiconductors, Vol. 2 “Optical Properties”, edited by M. Balkanski (Elsevier, Amsterdam, 1994), pp. 385–461. [6] J. Kono, in Methods in Materials Research, edited by E. N. Kaufmann et al. (John Wiley & Sons, New York, 2001), Unit 9b.2. [7] S. A. Crooker, Rev. Sci. Instrum. 73, 3258 (2002). [8] M. C. Nuss and J. Orenstein, in Millimeter and Submil- limeter Wave Spectroscopy of Solids, edited by G. Grüner (Springer-Verlag, Berlin, 1998), pp. 7–50. [9] D. M. Mittleman, ed., Sensing with Terahertz Radiation, (Springer, Berlin, 2003). [10] D. Some and A. V. Nurmikko, Appl. Phys. Lett. 65, 3377 (1994). [11] H. Harde, S. Keiding, and D. Grischkowsky, Phys. Rev. Lett. 66, 1834 (1991). [12] J. N. Heyman, R. Kersting, and K. Unterrainer, Appl. Phys. Lett. 72, 644 (1998). [13] D. Du and D. Grischkowsky, Phys. Rev. Lett. 93, 196804 (2004). [14] M. J. Chou and D. C. Tsui, Phys. Rev. B 37, 848 (1988). [15] S. A. Studenikin, M. Potemski, A. Sachrajda, M. Hillke, P. Pfeiffer, and K. W. West, Phys. Rev. B 71, 245313 (2005). [16] L. Allen and J. H. Eberly, Optical Resonance and Two- Level Atoms (Dover, New York, 1987). [17] H. Kuzmany, Solid-State Spectroscopy (Springer, Berlin, 1998).

arXiv:0704.1884v1 [math.LO] 14 Apr 2007 Coloring ordinals by reals Jörg Brendle and Sakaé Fuchino March 24, 2007 Abstract We study combinatorial principles we call Homogeneity Principle HP(κ) and Injectivity Principle IP(κ, λ) for regular κ > ℵ1 and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of Cs(κ) and Fs(κ) of I. Juhász, L. Soukup and Z. Szentmiklóssy [13]. Generalizing, their results, we show e.g. that IP(ℵ2,ℵ1) (hence also IP(ℵ2,ℵ2) as well as HP(ℵ2)) holds in a generic extension of a model of CH by Cohen forcing and IP(ℵ2,ℵ2) (hence also HP(ℵ2)) holds in a generic extension by countable support side-by-side prod- uct of Sacks or Prikry-Silver forcing (Corollary 4.8). We also show that the latter result is optimal (Theorem 5.2). Relations between these principles and their influence on the values of the variations b↑, bh, b∗, do of the bounding number b are studied. One of the consequences of HP(κ) besides Cs(κ) is that there is no pro- jective well-ordering of length κ on any subset of ωω. We construct a model in which there is no projective well-ordering of length ω2 on any subset of ωω (do = ℵ1 in our terminology) while b ∗ = ℵ2 (Theorem 6.4). 2000 Mathematical Subject Classification: 03E05 03E17 03E35 03E65 Keywords: Homogeneity Principle, Injectivity Principle, bounding number, projective well-ordering, Cohen forcing, Brendle-LaBerge forcing, Prikry-Silver forcing 0)The first author is partially supported by Grant-in-Aid for Scientific Research (C) 17540116, Japan Society for the Promotion of Science. The second author is partially supported by Chubu University grant 16IS55A. http://arxiv.org/abs/0704.1884v1 1 Introduction The Cohen model which is obtained by adding at least ℵ2 Cohen reals over a model of GCH was the first and simplest model for the negation of CH, and it is still one of the most important. A plethora of statements have been shown to be consistent with ZFC by adjoining Cohen reals, and it is therefore natural to look for axioms which hold in the Cohen model and from which many such statements can be decided, that is, axioms which capture as much as possible of the combinatorial structure of the Cohen extension. Something similar has been done for the iterated Sacks model by Ciesielski and Pawlikowski who devised the Covering Property Axiom CPA [2]. For Cohen models, several such axioms have been proposed in the past. Some of them are homogeneity type statements, that is, they assert that given at least ω2 many reals, many of them “look similar”. Examples are the combinatorial principles Cs(κ), Ĉs(κ), and Fs(κ) introduced by I. Juhász, L. Soukup and Z. Szentmiklóssy [13] who showed they hold in Cohen models (see Section 2 below for definitions). On the other hand, rather different-looking statements have been also investi- gated in connection with Cohen models, for example, the axiom WFN asserting that 〈P(ω),⊆〉 has the weak Freese-Nation Property (see [8], [10] and [5]). Here a partial ordering 〈P,≤〉 has the weak Freese-Nation Property if there is a mapping f : P → [P ]ℵ0 such that for all p, q ∈ P , p ≤P q holds if and only if there is an r ∈ f(p) ∩ f(q) such that p ≤P r ≤P q. In [8], it is shown that WFN holds in a Cohen model for adding ℵn Cohen reals for any n < ω. If we start e.g. from V = L then WFN holds even after adding any number of Cohen reals ([10]). In [5], it was shown that WFN implies many of the known combinatorial properties of Cohen models and so it may be seen as an axiomatization of the combinatorial structure of the Cohen extension. Since WFN can be reformulated in terms of elementary submodels, WFN as well as some closely related statements have come to be known as elementary submodel type axioms (see [12] for this). At first glance it seemed that there would be no connection between these two types of axioms except that they both hold in a Cohen model. Surprisingly enough though S. Shelah [17] showed that Cs(ℵ2) follows from the combinatorial principle he called Princ, which is a consequence of WFN. The proof can be easily recast to show that WFN implies Cs(κ) for all regular κ > ℵ1 (see [6] for more details). In this paper, we introduce some new principles of the homogeneity type, namely, the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ, λ) which are formulated in terms of homogeneity of colorings of ordinals below the cardinal κ by reals. We establish that these axioms hold in Cohen models and address the question in which other models these axioms hold as well. It turns out that, in fact, these principles seem to capture a good deal of the combinatorial features of models of set theory obtained by forcing by the side-by-side (finite or countable support) product of copies of a fixed relatively small partial ordering (see Theorem 4.3 and Corollary 4.8). Though the relation of these principles to WFN is not yet completely clear, our principles imply the principles of I. Juhász, L. Soukup and Z. Szentmiklóssy (Theorem 2.7) and thus can be seen as natural strengthenings of these principles. Our paper is organized as follows. In Section 2, we review the principles Cs(κ), Ĉs(κ) and Fs(κ) of I. Juhász, L. Soukup and Z. Szentmiklóssy, and introduce our principles HP(κ) and IP(κ, λ). Some basic facts in ZFC concerning these principles are also proved. In particular, we show that Cs(κ) and Ĉs(κ) follow from HP(κ) (Theorem 2.7), Fs(κ) follows from IP(κ,ℵ1) (Theorem 2.8) and HP(κ) follows from IP(κ, κ) (Theorem 2.9). After reviewing some cardinal invariants introduced in [6] which are variants of the bounding number b and the shrinking number b∗ in [3], we study in Section 3 the effect of the combinatorial principles Cs(κ), Ĉs(κ) and HP(κ) on the values of these cardinal invariants. In Section 4 we give a forcing construction of models of IP(κ, λ) (Theorem 4.3) and its applications (Corollary 4.8). The results in this section improve consistency results in [13]. As a further application of Theorem 4.3 we show in Section 5 the consistency of ¬IP(ℵ2,ℵ1) and IP(ℵ2,ℵ2). One of the consequences of HP(ℵ2) discussed in Section 3 is that there is no definable well-ordering of length ω2 on any subset of ωω (or do = ℵ1 in our notation). Refining a forcing extension of Brendle and LaBerge [1], we prove in Section 6 the consistency of do = ℵ1 with b ∗ = ℵ2 (Theorem 6.4). We also show that the model of do = ℵ1 and b ∗ = ℵ2 we construct in this section satisfies a strong negation of Cs(ℵ2). Section 7 is devoted to the consistency proof of the combinatorial principle used in the proof of Theorem 6.4. In Section 8, we summarize the consistency results obtained in this paper to- gether with other consistency results established by some previously known con- structions. We discuss also some open problems at the end of the section. 2 Combinatorial principles formulated in terms of coloring of ordinals by reals For any set X , let (2.1) ((X)) = {~x ∈ Xn : ~x is injective} and (2.2) ((X)) ((X)) Likewise, for any sets X0,...,Xn−1, let (2.3) ((X0, ...,Xn−1)) = {~x ∈ X0 × · · · ×Xn−1 : ~x is injective}. For a cardinal κ, the following principle Cs(κ) was introduced by I. Juhász, L. Soukup and Z. Szentmiklóssy in [13]. Cs(κ): For any matrix 〈aα,n : α ∈ κ, n ∈ ω〉 of subsets of ω and T ⊆ ω>ω, at least one of the following holds: (c0) there is a stationary S ⊆ κ such that n<|t| aαn,t(n) 6= ∅ for all t ∈ T and 〈α0, ...,α|t|−1〉 ∈ ((S)) (c1) there exist t ∈ T and stationary S0,...,S|t|−1 ⊆ κ such that n<|t| aαn,t(n) = ∅ for all 〈α0, ...,α|t|−1〉 ∈ ((S0, ...,S|t|−1)). For any cardinal κ it is easy to see that Cs(κ) holds if and only if Cs(cf κ) holds. Thus it is enough to consider Cs(κ) for regular κ. The corresponding assertion is also true for other combinatorial principles we are going to introduce in this section. Hence, in the rest of this section, we shall assume that κ is a regular cardinal unless mentioned otherwise. The combinatorial principle Ĉs(κ), a sort of dual of the principle Cs(κ), is also considered in [13]: Ĉs(κ): For any T ⊆ ω<ω and any matrix 〈aα,n : α < κ, n ∈ ω〉 of subsets of ω, at least one of the following holds: (ĉ0) there is a stationary S ⊆ κ such that | n<|t| aαn,t(n) | < ℵ0 for every t ∈ T and 〈α0, ...,α| t |−1〉 ∈ ((S)) (ĉ1) there exist t ∈ T and stationary S0,...,S|t|−1 ⊆ κ such that | n<|t| aαn,t(n) | = ℵ0 for every 〈α0, ...,α| t |−1〉 ∈ ((S0, ...,S|t|−1)). The following is easily seen: Lemma 2.1. (I. Juhász, L. Soukup and Z. Szentmiklóssy [13]) (a) Neither of Cs(ℵ1) and Ĉ s(ℵ1) holds. (b) Cs(κ) and Ĉs(κ) hold for any regular κ > 2ℵ0. Let us call a subset A of H(ℵ1) definable if there are a formula ϕ and a ∈ H(ℵ1) such that A = {x ∈ H(ℵ1) : 〈H(ℵ1),∈〉 |= ϕ(x, a)}. Note that for any n ∈ ω, A ⊆ Rn is projective if and only if it is definable in our sense. Note also since all elements of H(ℵ1) can be coded by elements of ωω we may assume that a as above is an element of ωω. In Theorem 2.7, we show that the following Homogeneity Principle HP(κ) im- plies both of Cs(κ) and Ĉs(κ). HP(κ): For any f : κ→ P(ω) and any definable A ⊆ ((P(ω))) , at least one of the following holds: (h0) there is a stationary S ⊆ κ such that ((f ′′S)) \ {∅} ⊆ A; (h1) there are k ∈ ω \ 1 and stationary S0,...,Sk−1 ⊆ κ such that ((f ′′S0, ..., f ′′Sk−1)) ∩ A = ∅. Note that P(ω) in the definition of HP(κ) above can be replaced by R, ωω, (P(ω))n or (ωω)n etc. since these spaces can be coded as definable subsets of P(ω) and vice versa. As for Cs(κ) (and Ĉs(κ)), it is enough to consider HP(κ) for regular κ. Lemma 2.1 is also true for HP(κ): Lemma 2.2. (a) HP(ℵ1) does not hold. (b) HP(κ) holds for any regular κ > 2ℵ0. Proof. (a): This follows from Lemma 2.1 and Theorem 2.7. (b): Let κ > 2ℵ0 be a regular cardinal. Suppose that f : κ → P(ω) and A are as in the definition of HP(κ). Then there is a stationary S ⊆ κ such that f ↾ S is constant. If (h0) in the definition of HP(κ) does not hold then we must have ((f ′′S)) ∩A = ∅ since ((f ′′S)) = ∅ for all n > 1. Hence (h1) holds with n = 1 and S0 = S. (Lemma 2.2) The following combinatorial principle Fs(κ) is also introduced in [13]: Fs(κ): For any T ⊆ ω<ω and any matrix 〈aα,n : α < κ, n ∈ ω〉 of subsets of ω, at least one of the following holds: (f 0) there is a stationary S ⊆ κ such that n<|t|aαn,t(n) : t ∈ T and 〈α0, ...,α| t |−1〉 ∈ ((S)) } | ≤ ℵ0 ; (f 1) there are t ∈ T and stationary S0,...,S|t|−1 ⊆ κ such that for every 〈α0, ...,α| t |−1〉, 〈β0, ...,β| t |−1〉 ∈ ((S0, ...,S|t|−1)), if αn 6= βn for all n < |t|, then n<|t| aαn,t(n) 6= n<|t| aβn,t(n). Lemma 2.3. (I. Juhász, L. Soukup and Z. Szentmiklóssy [13]) (a) Fs(ℵ1) does not hold. (b) Fs(κ) holds for every regular κ > 2ℵ0. (c) Fs(κ) implies Ĉs(κ). A combinatorial principle in terms of coloring of ordinals by reals corresponding naturally to Fs(κ) might be the following Injectivity Principle IP(κ, λ) for cardinals κ and λ with λ ≤ κ: IP(κ, λ): For any f : κ→ P(ω) and definable g : ((P(ω))) → P(ω), at least one of the following holds: (i0) there is a stationary S ⊆ κ such that | g ′′((f ′′S)) | < λ for every n ∈ ω; (i1) there are k ∈ ω\1 and stationary S0,...,Sk−1 ⊆ κ such that for any 〈x0, ...,xk−1〉, 〈y0, ..., yk−1〉 ∈ ((f ′′S0, ..., f ′′Sk−1)), if xn 6= yn for all n < k, then we have g(x0, ..., xk−1) 6= g(y0, ..., yk−1). Again here, we may replace P(ω) in the definition of IP(κ, κ) above by R, ωω, (P(ω))n or (ωω)n etc. Lemma 2.4. (a) For λ ≤ λ′ ≤ κ, IP(κ, λ) implies IP(κ, λ′). (b) IP(ℵ1,ℵ1) does not hold. Proof. (a): Immediate from the definition. (b): By Lemma 2.2, (a) and Theorem 2.9. (Lemma 2.4) IP(κ,ℵ0) for a regular cardinal κ is equivalent to the cardinal inequality 2 ℵ0 < κ. Proposition 2.5. For a regular cardinal κ the following are equivalent: (a) IP(κ,ℵ0) holds; (b) 2 ℵ0 < κ; (c) IP(κ, 2) holds. Proof. (a) ⇒ (b): Suppose that 2ℵ0 ≥ κ. We show that IP(κ,ℵ0) does not hold. Let f : κ → P(ω) be any injective mapping and g : ((P(ω))) → P(ω) be defined by g(∅) = ∅, g(〈x〉) = ∅ for all x ∈ P(ω) and g(〈x0, ..., xn−1〉) = min{m ∈ ω : m ∈ x0 6↔ m ∈ x1} for 〈x0, ..., xn−1〉 ∈ ((P(ω))) with n ≥ 2. Let S be any stationary subset of κ. Then | g ′′((f ′′S)) | ≥ ℵ0: Suppose not and let k ∈ ω be such that g ′′((f ′′S)) Since P(k + 1) is finite, there are α, β ∈ S, α 6= β such that f(α) ∩ (k + 1) = f(β) ∩ (k + 1). But then, by definition of g, it follows that g(〈f(α), f(β)〉) > k. This is a contradiction. Thus (i0) does not hold for these f and g. On the other hand, for arbitrary stationary subsets S0,...,Sn−1 of κ, as there are only countably many values of g, we can find 〈x0, ..., xn−1〉, 〈y0, ..., yn−1〉 ∈ ((f ′′S0, ..., f ′′Sn−1)) such that xi 6= yi for all i < n and g(〈x0, ..., xn−1〉) = g(〈y0, ..., yn−1〉). Thus (i1) neither holds. (b) ⇒ (c): Suppose 2ℵ0 < κ. For f : κ → P(ω) and g : ((P(ω))) → P(ω) as in the definition of IP(κ, 2), there is a stationary S ⊆ κ such that f is constant on S. This S witnesses that (i0) holds. (c) ⇒ (a): This follows from Lemma 2.4, (a). (Proposition 2.5) Corollary 2.6. IP(ℵ2,ℵ0) is equivalent to CH. IP(ℵ2,ℵ1) and IP(ℵ2,ℵ2) are thus the first two non-trivial instances of IP(κ, λ). For κ ≥ ℵ2, the principles introduced in this section and some other principles discussed in [6] can be put together in the following diagram: IP(κ,ℵ1) Fs(κ) HP(κ) Ĉs(κ) Cs(κ) Princ(κ, κ) SEP(κ, κ) Theorem 2.8 Theorem 2.9 Theorem 2.7 [13] [17] (see also [6]) IP(κ, κ) Theorem 2.7 fig. 1 In the rest of the section, we shall prove the implications indicated by the thick arrows in fig.1. Theorem 2.7. For a regular cardinal κ, HP(κ) implies both Cs(κ) and Ĉs(κ). Proof. We prove that HP(κ) implies Cs(κ). The other implication can be proved similarly. By Lemma 2.1, (b), we may assume that κ ≤ 2ℵ0. Let 〈ti : i ∈ ω〉 be an enumeration of ω>ω such that (2.4) | ti | ≤ i for all i < ω and let ι : P(ω) → P(ω)ω be a definable bijection. For each x ∈ P(ω) and i < ω, let (x)i denote the i’th component of ι(x). Suppose that T ⊆ ω>ω and A = 〈aα,n : α < κ, n ∈ ω〉 is a matrix of subsets of ω. We show that either (c0) or (c1) holds for these A and T . Let g : κ → P(ω) be a fixed injective mapping which exists by κ ≤ 2ℵ0 . Let f : κ→ P(ω) be defined by (2.5) f(α) = ι−1(〈a′α,n : n ∈ ω〉) where (2.6) a′α,n = g(α), if n = 0, aα,n−1, otherwise. Note that f is injective by “if n = 0” clause of (2.6). For i < ω, let (2.7) A∗i = {〈x0, ..., xi−1〉 ∈ ((P(ω))) n<| ti | (xn)ti(n)+1 6= ∅}, if ti ∈ T, ((P(ω))) , otherwise (2.8) A = A∗i . It is easy to see that A is definable noting that T ∈ H(ℵ1) and hence T can be used as a parameter in the definition of A. By HP(κ), we have either (h0) or (h1) for these A and f . Assume first that (h0) holds. Then there is a stationary S ⊆ κ such that ((f ′′S)) \ {∅} ⊆ A. We show that this S witnesses (c0) for T and A: for t ∈ T , let i ∈ ω be such that t = ti. By (2.4), we have | t | ≤ i. For s ∈ ((S)) | t | , let s′ ∈ ((S)) be an end-extension of s. Then 〈f(s′(0)), ..., f(s′(i−1))〉 ∈ ((f ′′S)) since f is injective. Hence 〈f(s′(0)), ..., f(s′(i − 1))〉 ∈ A∗i by the assumption on S. By (2.7), we have n<| ti | (f(s′(n)))ti(n)+1 = n<| ti | s′(n),ti(n)+1 n<| t | as(n),t(n). Thus T and A satisfy (c0). Assume now that (h1) holds. In this case, there are i ∈ ω and stationary S0,...,Si−1 ⊆ κ such that (2.9) ((f ′′S0, ..., f ′′Si−1)) ⊆ ((P(ω))) \ A∗i . Let t = ti. Then t ∈ T by (2.9) and “otherwise” clause of (2.7). For s ∈ ((S0, ...,S|t|−1)), let s ′ ∈ ((S0, ...,Si−1)) be an end extension of s. Then we have 〈f(s′(0)), ..., f(s′(i− 1))〉 ∈ ((f ′′S0, ..., f ′′Si−1)). It follows that 〈f(s′(0)), ..., f(s′(i− 1))〉 ∈ ((P(ω))) \ A∗i by (2.9). Hence, by (2.7), we have n<| t |(f(s ′(n)))t(n)+1 = n<| ti | s′(n),t(n)+1 n<| t | as(n),t(n). Thus, T and A satisfy (c1) in this case. The proof of Ĉs(κ) from HP(κ) is exactly like the proof above with (2.7) replaced (2.7) A∗i = {〈x0, ..., xi−1〉 ∈ ((P(ω))) n<| ti | (xn)ti(n)+1 | < ℵ0}, if ti ∈ T, ((P(ω))) , otherwise. (Theorem 2.7) HP(κ) also imply other variants of Cs(κ). For example, let ∗Cs(κ): For any matrix 〈aα,n : α ∈ κ, n ∈ ω〉 of subsets of ω and T ⊆ at least one of the following holds: (∗c0) there is a stationary S ⊆ κ such that n<|t| aαn,t(n) is infinite for all t ∈ T and 〈α0, ...,α|t|−1〉 ∈ ((S)) (∗c1) there exist t ∈ T and stationary S0,...,S|t|−1 ⊆ κ such that n<|t| aαn,t(n) is finite for all 〈α0, ...,α|t|−1〉 ∈ ((S0, ...,S|t|−1)). It is easy to see by a proof similar to that of Theorem 2.7 that HP(κ) implies ∗Cs(κ) as well. The following can also be proved similarly to Theorem 2.7. Theorem 2.8. IP(κ,ℵ1) implies F s(κ). Theorem 2.9. IP(κ, κ) implies HP(κ). Proof. Suppose that A ⊆ ((P(ω))) is definable and f : κ → P(ω). If f−1[{x}] is stationary for some x ∈ P(ω), then, either (h0) holds for S = f−1[{x}] or (h1) holds for n = 1 and S0 = f −1[{x}] depending on whether x ∈ A or not. Otherwise let g : ((P(ω))) → P(ω) be defined by (2.10) g(∅) = ∅; (2.11) g(〈x〉) = ∅ for all x ∈ P(ω) and (2.12) g(〈x0, ..., xn−1, x〉) = ∅, if 〈x0, ..., xn−1〉 ∈ A, x otherwise for all 〈x0, ..., xn−1, x〉 ∈ ((P(ω))) . If (i0) holds for this g with S as in (i0), then, by (2.12), we must have g ′′((f ′′S)) = {∅}. Hence ((f ′′S)) \ {∅} ⊆ A. On the other hand, if (i1) holds for some n < ω and S0 ,...,Sn−1, then we should have n ≥ 2 by (2.11) and g(〈x0 , ..., xn−2, xn−1〉) = xn−1 for all xi ∈ f ′′Si, i < n by (2.12). It follows that ((f ′′S0, ..., f ′′Sn−2)) ⊆ ((P(ω))) \ A by (2.12). (Theorem 2.9) 3 The bounding number and its variations In this section, we show that the combinatorial principles introduced in the last section make some of the cardinal invariants from [6] small. Adopting the notation of [6], we consider the following spectra of cardinal num- bers in connection with a partial ordering 〈P,≤〉; unbounded spectrum, hereditary unbounded spectrum and the spectrum of length of P : S(P ) = {|X | : X ⊆ P,X is unbounded in P, ∀B ∈ [X ]<|X |(B is bounded in P )} , h(P ) = {|X | : X ⊆ P, ∀B ⊆ X (B is bounded in P ↔ |B | < |X |)} , ↑(P ) = {cf(C) : C ⊆ P, C is an unbounded chain} . Clearly, we have (3.1) S↑(P ) ⊆ Sh(P ) ⊆ S(P ). For P = 〈ωω,≤∗〉, we shall simply write S↑, Sh and S in place of S↑(〈ωω,≤∗〉), h(〈ωω,≤∗〉) and S(〈ωω,≤∗〉), respectively. Recall that the bounding number b is defined by b = min{|X | : X ⊆ ωω is unbounded with respect to ≤∗}. The variant b∗ of b was introduced and studied in [3] and [14] where ∗ = min{κ : ∀X ⊆ ωω X is unbounded → ∃X ′ ∈ [X ]≤κ(X ′ is unbounded) b and b∗ can be characterized in terms of S↑, Sh and S as follows: Lemma 3.1. (a) b = minS↑ = minSh = minS. (b) b∗ = supS. In analogy to Lemma 3.1, (b), let (3.2) b↑ = supS↑, bh = supSh. Recall also that the dominating number d is defined as: d = min{|X | : X ⊆ ωω, X dominates ωω}. By (3.1) and Lemma 3.1, we have Lemma 3.2. b ≤ b↑ ≤ bh ≤ b∗ ≤ d. fig. 2 DO = {cf(otp(〈X,R ↾ X〉)) : X ⊆ ωω, R is a definable binary relation and R ∩X2 well orders X} do = supDO. By definition, S↑ ⊆ DO. Hence Lemma 3.3. b↑ ≤ do. Lemma 3.2 and Lemma 3.3 may be put together into the following diagram: b ≤ b↑ ≤ bh ≤ b∗ ≤ d fig. 3 If S↑ has a maximal element then we have b↑ = maxS↑. In such case we shall say that b↑ is attained. Also we shall say that b∗, bh or do is attained if the corresponding set has a maximal element. In the following, Reg denotes the class of regular cardinals. The following lemma can be proved similarly to Lemma 3.7, (c). Lemma 3.4. ([6]) Sh ∩ Reg ⊆ DO. Corollary 3.5. If Sh ∩ Reg is cofinal in Sh then bh ≤ do. Note that the condition “Sh ∩ Reg is cofinal in Sh” holds if 2ℵ0 < ℵω or if b is regular and attained. Under this condition, we can thus improve the diagram in fig.3 to the following: b ≤ b↑ ≤ bh ≤ b∗ ≤ d fig. 4 For an ideal I over a set X , non(I) and cov(I) denote, as usual, the uniformity and the covering number of I, respectively. More exactly non(I) = min{|A | : A ∈ P(X) \ I} and cov(I) = min{|A | : A ⊆ I, A = X}. meager and null denote the ideal of meager sets and the ideal of null sets (over R) respectively. Lemma 3.6. Suppose that I is an ideal over R with Borel basis. Then we have min{non(I), cov(I)} ≤ do. In particular, we have min{non(meager), cov(meager)} ≤ do and min{non(null), cov(null)} ≤ do. Proof. Suppose that I ⊆ P(R) is an ideal with a Borel basis and κ = min{non(I), cov(I)}. We can construct inductively a sequence 〈〈fα, gα〉 : α < κ〉 such that (3.3) fα, gα ∈ ωω for all α < κ; (3.4) gα codes a Borel set Xα ⊆ ωω such that Xα ∈ I and {fβ : β < α} ⊆ Xα ; (3.5) fα 6∈ Xβ for all α < κ. Note that (3.4) is possible by κ ≤ non(I) and (3.5) by κ ≤ cov(I). The sequence 〈〈fα, gα〉 : α < κ〉 is well ordered in order type κ by the definable ordering: 〈f ′, g′〉 ≤ 〈f, g〉 ⇔ f ′ is an element of the Borel set coded by g. It follows that κ ≤ do. (Lemma 3.6) The following lemma shows the relations of cardinal numbers b, b↑, bh, do to the combinatorial principles introduced in Section 2. Lemma 3.7. (a) (I. Juhász, L. Soukup and Z. Szentmiklóssy [13]) If there is a ≤∗-chain of length κ then ¬Cs(κ) and ¬Ĉs(κ). In particular, κ ∈ S↑ implies ¬Cs(κ) and ¬Ĉs(κ). (b) Cs(κ) (or Ĉs(κ)) implies b↑ ≤ κ. If b↑ is attained then Cs(κ) (or Ĉs(κ)) implies b↑ < κ. (c) If κ ≤ λ for some λ ∈ Sh with cf λ ≥ κ then ¬Cs(κ) and ¬Ĉs(κ). (d) If Sh ∩ Reg is cofinal in Sh then Cs(κ) (or Ĉs(κ)) implies bh ≤ κ. If bh is regular and attained then Cs(κ) (or Ĉs(κ)) implies bh < κ. (e) κ ∈ DO implies ¬HP(κ). (f) HP(κ) implies do ≤ κ. If do is attained then HP(κ) implies do < κ. Proof. (a): See [13]. (b): This follows from (a). (c): Suppose that κ ≤ λ ∈ Sh and κ ≤ cf λ. We show ¬Cs(κ). ¬Ĉs(κ) can be proved similarly from these assumptions. Let X ⊆ ωω with |X | = λ be as in the definition of Sh. Then we can find fα ∈ X and gα ∈ ωω for α < κ such that (3.6) fα ≤ ∗ gβ for all α < β < κ; (3.7) fβ 6≤ ∗ g+α for all α ≤ β < κ where g α is defined by g α (k) = gα(k) + 1 for all k ∈ ω. Note that (3.7) is possible since cf(|X |) ≥ κ. For α < κ, let gα,n ∈ ωω, n ∈ ω be such that (3.8) {gα,n : n ∈ ω} = {g ∈ ωω : g =∗ gα}. (3.9) aα,0 = {〈k, ℓ〉 ∈ ω 2 : ℓ ≤ fα(k)} and (3.10) aα,n+1 = {〈k, ℓ〉 ∈ ω 2 : ℓ > gα,n(k)} for all n ∈ ω. We show that A = 〈aα,n : α ∈ κ, n ∈ ω〉 with T = {〈0, n〉 : n ∈ ω \ 1} is a counter-example to Cs(κ). Suppose first that S ⊆ κ is stationary. For any α ∈ S, let β ∈ S be such that α < β. Then we have fα ≤ ∗ gβ by (3.6). Hence there is n ∈ ω such that fα ≤ gβ,n. By (3.9) and (3.10), it follows that aα,0 ∩ aβ,n+1 = ∅. This shows that 〈A, T 〉 6|= (c0). Suppose now that S0, S1 ⊆ κ are stationary and 〈0, n〉 ∈ T . By the definition of T , it follows that n ∈ ω\1. Let α ∈ S0 and β ∈ S1 be such that β < α. Then, by (3.7), we have fα 6≤ . Thus, by (3.9) and (3.10), it follows that aα,0 ∩ aβ,n 6= ∅. This shows that 〈A, T 〉 6|= (c1). (d): This follows easily from (c). (e): Suppose that κ ∈ DO and let 〈X,R〉 be such that X ⊆ P(ω), R is a projective binary relation and otp(〈X,R ∩ X2〉) = κ. Let f : κ → P(ω) be the mapping sending α < κ to the α’th element of X with respect to R. Let A = R ∪ k∈ω\{2}((P(ω))) Then it is easily seen that 〈f, A〉 6|= (h0) and 〈f, A〉 6|= (h1). (f): This follows from (d) since DO is downward closed. (Lemma 3.7) Corollary 3.8. (a) HP(κ) implies min{non(I), cov(I)} ≤ κ for any ideal I over R with Borel basis. In particular, it implies min{non(meager), cov(meager)} ≤ κ and min{non(null), cov(null)} ≤ κ. (b) If do is attained then HP(κ) implies min{non(I), cov(I)} < κ for all any I over R with Borel basis. In particular, it implies min{non(meager), cov(meager)} < κ and min{non(null), cov(null)} < κ. Proof. By Lemma 3.6 and Lemma 3.7, (f). (Corollary 3.8) Corollary 3.9. (a) Cs(ℵ2) (or Ĉ s(ℵ2)) implies b h = ℵ1. (b) HP(ℵ2) implies do = min{non(meager), cov(meager)} = min{non(null), cov(null)} = ℵ1. Proof. (a): By Lemma 3.7, (d). (b): By Lemma 3.7, (f) and Corollary 3.8. (Corollary 3.9) do = ℵ1 HP(ℵ2) IP(ℵ2,ℵ1) h = ℵ1b ↑ = ℵ1b = ℵ1 b ∗ = ℵ1 Cs(ℵ2) Ĉs(ℵ2) Fs(ℵ2) Corollary 3.9,(b) Corollary 3.9,(a) fig. 5 4 A forcing construction of models of IP(κ, λ) In this section, we shall prove that IP(κ, λ) holds in a generic extension by a homogeneous product of copies of a relatively small partial ordering (Theorem 4.3). Let us begin with definition of some notions needed for precise formulation of the theorem. For cardinals κ and µ, κ is said to be µ-inaccessible if κ is regular and λµ < κ holds for all λ < κ. Similarly, we say that κ is <µ-inaccessible if κ is regular and λ<µ < κ holds for all λ < κ. Thus, if µ is a successor cardinal, say µ = µ+0 , then κ is <µ-inaccessible if and only if κ is µ0-inaccessible. In our context, <µ-inaccessibility is relevant because of the following variant of the ∆-System Lemma of Erdős and Rado. For cardinals µ < κ, let Eκ≥µ = {α < κ : cf(α) ≥ µ} and let Eκµ , E ≤µ etc. be defined analogously. Theorem 4.1. (P. Erdős and R. Rado, see [13]) Suppose that κ is <µ-inaccessible and S ⊆ Eκ≥µ is stationary in κ. For any sequence 〈xα : α ∈ S〉 of sets of cardinality < µ there is a stationary S∗ ⊆ S such that 〈xα : α ∈ S ∗〉 form a ∆-system. For a sequence Pα, α < δ of posets and an ideal I ⊆ P(δ), we consider the I-support product Pα of Pα, α < δ here as being defined as (4.1) Pα = {f : f : D → Pα for some D ∈ I and f(α) ∈ Pα \ {1lPα} for all α ∈ D } with the ordering (4.2) f ≤QI g ⇔ dom(f) ⊇ dom(g) and f(α) ≤Pα g(α) for all α ∈ dom(g) for all f , g ∈ Pα. In particular, 1lQI = ∅ is the largest element of with respect to ≤QI α<δ Pα Though this definition of product of posets is different from the standard one, it gives a poset forcing equivalent to the product given by the standard definition. The present definition is chosen here for the sake of smoother treatment of p ↾ X , P ↾ X , G ↾ X etc. (see (4.5), (4.7) etc.) As usual, the ideal [δ]<ℵ0 is denoted by fin and Pα is called the finite support product of Pα, α < δ. I. Juhász and K. Kunen [12] proved the following theorem for µ = ℵ1 and I = [δ]<ℵ0 . Their proof also applies to the following slight generalization. Theorem 4.2. (I. Juhász and K. Kunen [12]) Suppose that P = Pα for some ideal I ⊆ P(δ), P satisfies the µ-c.c. and |Pα | ≤ 2 <µ for all α < δ. Then, for all <µ-inaccessible κ we have ‖– P “C s(κ) ”. Suppose that I ⊆ P(δ) is an ideal and P = Pα is an I-support product of posets Pα, α < δ. For p ∈ P, the support supp(p) of p is defined by (4.3) supp(p) = dom(p). We assume in the following that P-names are constructed just as in [15]. For a P-name ȧ, the support supp(ȧ) is defined by (4.4) supp(ȧ) = {supp(p) : 〈ḃ, p〉 ∈ tcl(ȧ) for some P-name ḃ}. For X ∈ P(δ) (not necessarily in I), let (4.5) P ↾ X = {p ↾ X : p ∈ P}. By (4.1) and since I is an ideal, we have (4.6) P ↾ X = {p ∈ P : supp(p) ⊆ X}. In particular, (4.7) P ↾ X ⊆ P. Furthermore, it is easy to see that P ↾ X ≤◦ P. Thus, if G is a (V,P)-generic filter then G ∩ (P ↾ X) is a (V,P ↾ X)-generic filter. We shall denote the generic filter G ∩ (P ↾ X) by GX . Note that a P-name ȧ is a P ↾ X-name if and only if supp(ȧ) ⊆ X . We shall call an I-support product P = Pα homogeneous if Pα ∼= Pβ for all α, β < δ and I is translation invariant, that is, I = {j ′′x : x ∈ I} for all bijections j : δ → δ. Note that if I is translation invariant then I = [δ] < λ for some λ. For a homogeneous P = Pα, we shall always assume that a commutative system iα,β : Pα → Pβ, α, β < δ of isomorphisms is fixed. With such a fixed system of isomorphisms, every bijection j : δ → δ induces an isomorphism j̃ : P defined by (4.8) dom(j̃(p)) = j ′′ dom(p) ; for α ∈ dom(j̃(p)), j̃(p)(α) = ij−1(α),α ◦ p ◦ j −1(α) for all p ∈ P. For notational simplicity we shall denote the isomorphism on P-names induced from j̃ also by j̃. Note that for P and j as above, p ∈ P, P-names ȧ0,..., ȧn−1 and a formula ϕ in the language of set theory LZF, we have (4.9) p ‖–P “ϕ(ȧ0, ..., ȧn−1) ” if and only if j̃(p) ‖–P “ϕ(j̃(ȧ0), ..., j̃(ȧn−1)) ”. We are now ready to formulate the main result of the present section: Theorem 4.3. Suppose that (4.10) λ is a regular uncountable cardinal with 2<λ = λ, µ ∈ {λ, λ+} and κ is a <λ-inaccessible cardinal. Let P = Pα be a homogeneous I-support product such that (4.11) I ⊆ [δ]<λ; (4.12) |Pα | ≤ λ for all α < δ and P satisfies the µ-c.c.; (4.13) P is proper. Then ‖– P “ IP(κ, µ) ” holds. The proof of Theorem 4.3 will be given after the following Lemmas 4.4 to 4.7. As in [15], a P-name ẋ of a subset of ω for a poset P is called a nice P-name if there are antichains Aẋ,n, n ∈ ω in P such that ẋ = {〈ň, p〉 : p ∈ Aẋ,n}. Note that, for such a name ẋ, we have supp(ẋ) = n∈ω Aẋ,n. It is easy to see that, for all P-names ẋ of subsets of ω, there is a nice P-name ẋ′ such that ‖–P “ ẋ = ẋ We say that a nice P-name of a subset of ω with Aẋ,n, n ∈ ω as above is slim if Aẋ,n is countable for all n < ω. The following lemmas are well-known: Lemma 4.4. Suppose that P is a proper poset and p ∈ P. For any P-name ẋ of a subset of ω, there are q ≤P p and a slim P-name ẋ ′ such that q ‖–P “ ẋ = ẋ Proof. By the remark above, we may assume without loss of generality that ẋ is a nice P-name. Let Aẋ,n, n ∈ ω be as above and ẏ be a P-name such that ‖–P “ ẏ = {s ∈ P : s ∈ ( n∈ω Aẋ,n) ∩ Ġ} ”. Then we have ‖– P “ ẏ is a countable subset of P ”. As P is proper there exist q ≤P p and countable y ⊆ P such that q ‖– P “ ẏ ⊆ y ”. Let ẋ ′ = {〈ň, s〉 : n ∈ ω, s ∈ Aẋ,n ∩ y}. These q and ẋ ′ are as desired. (Lemma 4.4) Lemma 4.5. Suppose that P = Pα is a κ-c.c. I-support product for an ideal I ⊆ [δ]<λ and κ is <λ-inaccessible. If S ⊆ Eκ≥λ is stationary and pα ∈ P for α ∈ S are such that supp(pα), α ∈ S form a ∆-system with the root R and there is p∗ ∈ P ↾ R such that pα ↾ R = p ∗ for all α ∈ S, then p∗ ‖– P “ {α ∈ S : pα ∈ Ġ} is stationary ” where Ġ denotes the standard P-name of a (V,P)-generic filter. Proof. By κ-c.c. of P, κ remains a regular cardinal in P-generic extensions. Let Ṡ be a P-name of {α ∈ S : pα ∈ Ġ}. Suppose that Ċ is a P-name of a club subset of κ and p ≤P p ∗. It is enough to show that there is a q ≤P p such that q ‖– P “ Ċ ∩ Ṡ 6= ∅ ”. Let θ be sufficiently large and let M ≺ H(θ) be such that (4.14) I, P, κ, Ċ, 〈pα : α ∈ S〉, p ∈M ; (4.15) |M | < κ ∩M < κ; (4.16) [M ]<λ ⊆M and (4.17) α∗ ∈ S where α∗ = κ ∩M . (4.16) is possible since κ is <λ-inaccessible. (4.17) is possible since S ⊆ Eκ≥λ and S is stationary in κ. Claim 4.5.1. ‖–P “α ∗ ∈ Ċ ”. ⊢ Since P satisfies the κ-c.c., we have H(θ) |= ∀α < κ ∃β ∈ κ \ α ( ‖–P “β ∈ Ċ ”). By (4.14), and elementarity of M it follows that M |= ∀α < κ ∃β ∈ κ \ α ( ‖–P “β ∈ Ċ ”). Thus ‖–P “ Ċ ∩ α ∗ is unbounded in α∗ ”. Since ‖–P “ Ċ is a club in κ ”, it follows that ‖–P “α ∗ ∈ Ċ ”. ⊣ (Claim 4.5.1) Claim 4.5.2. supp(pα∗) ∩M = R and supp(p) ∩ supp(pα∗) = R . ⊢ Suppose u = (supp(pα∗)∩M)\R 6= ∅. By (4.16), u ∈M . Hence by elementarity M |= ∃α < κ (u ⊆ supp(pα)). Let α ∈ κ ∩M be such that u ⊆ supp(pα). Then α < α∗ and R ∪ u ⊆ pα ∩ pα∗ . This is a contradiction to the assumption that R is the root of the ∆-system {supp(pα) : α ∈ S}. This shows supp(pα∗) ∩M = R. By (4.14) and (4.16), supp(p) ∈ M . It follows that supp(p) ∩ sup(pα∗) = supp(p) ∩ (sup(pα∗) ∩M) = supp(p) ∩ R = R. ⊣ (Claim 4.5.2) Since p ↾ R ≤P p ∗ ↾ R = p∗ = pα∗ ↾ R, q = p ∪ pα∗ ∈ P. We have q ≤P p. By pα∗ ‖–P “α ∗ ∈ Ṡ ” and q ≤P pα∗ , we have q ‖–P “α ∗ ∈ Ċ ∩ Ṡ ”. In particular q ‖– P “ Ċ ∩ Ṡ 6= ∅ ”. (Lemma 4.5) The arguments of the following two lemmas are also well-known. For Lemma 4.6 see e.g. [12]. Lemma 4.6. Suppose that P = Pα is an I-support product and G is a (V,P)- generic filter. For X, Y ⊆ δ, let Z = X ∩Y . Then, in V [G], for any κ ∈ CardV [G], we have [On]<κ ∩ (V [GX ] \ V [GZ ]) ∩ (V [GY ] \ V [GZ ]) = ∅. Lemma 4.7. Suppose that κ ≤ δ and P = Pα is a κ-c.c. homogeneous I- support product, p ∈ P, ȧ0, ..., ȧn−1 are P-names with (4.18) supp(ȧ0), ..., supp(ȧn−1) ⊆ X for some X ⊆ δ and ϕ = ϕ(x0, ...,xn−1) is a formula in LZF (possibly with some parameters from V ). (a) If (4.19) p ‖–P “ϕ(ȧ0, ..., ȧn−1) ” and (4.20) δ \X 6∈ I, then p ↾ X ‖–P “ϕ(ȧ0, ..., ȧn−1) ”. (b) If (4.21) p ‖–P “ (∃x ∈ ωω) ϕ(x, ȧ1, ..., ȧn−1) ”, (4.22) supp(p) ⊆ X and (4.23) |X \ (supp(p) ∪ supp(ȧ0) ∪ · · · ∪ supp(ȧn−1)) | ≥ κ, then there is a PX-name ȧ such that p ‖–P “ϕ(ȧ, ȧ1, ..., ȧn−1) ”. Proof. (a): Suppose that p ↾ X /‖–P “ϕ(ȧ0, ..., ȧn−1) ”. Then there is q ≤P p ↾ X such that q ‖–P “¬ϕ(ȧ0, ..., ȧn−1) ”. Let j : δ → δ be a bijection such that (4.24) j ↾ X = idX and (4.25) (j ′′ supp(q) \X) ∩ supp(p) = ∅. Note that the last condition is possible by (4.20). By (4.24) and (4.18), we have (4.26) j̃(q) ↾ X = j̃(q ↾ X) = q ↾ X and (4.27) j̃(ȧ0) = ȧ0,..., j̃(ȧn−1) = ȧn−1. By (4.27) and by the choice of q, we have j̃(q) ‖–P “¬ϕ(ȧ0, ..., ȧn−1) ”. On the other hand, by (4.26) p and j̃(q) are compatible. This is a contradiction to (4.19). (b): By maximal principle, there is a nice P-name ȧ′ of a real such that p ‖–P “ϕ(ȧ ′, ȧ1, ..., ȧn−1) ”. By the κ-c.c. of P, we have | supp(ȧ′) | < κ. By (4.18), (4.22) and (4.23), we can find a bijection j : δ → δ such that (4.28) j on supp(p) ∪ supp(ȧ1) ∪ · · · ∪ supp(ȧn−1) is the identity mapping, and (4.29) j ′′ supp(ȧ′) ⊆ X . By (4.28), j̃(p) = p and j̃(ȧ1) = ȧ1,..., j̃(ȧn−1) = ȧn−1. Let ȧ = j̃(ȧ ′). Then p ‖–P “ϕ(ȧ, ȧ1, ..., ȧn−1) ” and supp(ȧ) ⊆ X by (4.29). (Lemma 4.7) Proof of Theorem 4.3: By Proposition 2.5, we may assume that ‖– P “ κ ≤ 2 ℵ0 ”. In particular, by (4.11), (4.12) and (4.13), we may assume that δ ≥ κ. By the µ-c.c. of P, µ and κ remain regular cardinals in the generic extension by P. Let G be a (V,P)-generic filter. In V [G], let f : κ→ P(ω) and g : ((P(ω))) P(ω) be definable, say by a formula ϕ. We may assume that ϕ has a real a ∈ V [G] as its unique parameter. Let ḟ , ȧ and ġ be P-names of f , a and g respectively such that ‖–P “ ḟ : κ→ P(ω) ”, ‖– P “ ġ : ((P(ω))) → P(ω) ” and (4.30) ‖– P “ ∀x ∈ ((P(ω))) ∀x ∈ P(ω) ġ(x) = x↔ H(ℵ1) |= ϕ(x, x, ȧ) Suppose that, for a p ∈ G, (4.31) p ‖–P “ (i0) for IP(κ, µ) does not hold for ḟ and ġ ”. In particular, we have (4.32) p ‖–P “ ∀α < κ ({β ∈ κ : ḟ(β) = ḟ(α)} is non-stationary) ”. Claim 4.3.1. There is a stationary S ⊆ Eκ≥λ such that p ‖–P “ ḟ ↾ S is 1-1 ”. ⊢ By the κ-c.c. of P and by (4.32), there are club sets Cα ⊆ κ (in V ) for each α < κ such that p ‖–P “Cα ∩ {β ∈ κ : ḟ(β) = ḟ(α)} = ∅ ”. Then C = ∆α<κCα is club and S = E ≥λ ∩ C has the desired property. ⊣ (Claim 4.3.1) We show that p forces (i1) for ḟ and ġ. Let p′ ≤P p. It is enough to show that there is p∗ ≤P p ′ forcing (i1). By Lemma 4.4, Theorem 4.1, (4.10), (4.11) and (4.13), there are p′′ ≤P p slim P-name ȧ′ of a real, a stationary S∗ ⊆ S, a sequence 〈ẋ′α : α ∈ S ∗〉 of slim P-names and a sequence 〈pα : α ∈ S ∗〉 of conditions in P such that (4.33) (i) p′′ ‖– P “ ȧ = ȧ (ii) pα ≤P p ′′ and (iii) pα ‖– P “ ḟ(α) = ẋ α ” for every α ∈ S (4.34) dα = supp(pα)∪ supp(ȧ ′)∪ supp(ẋ′α), α ∈ S ∗ are all of the same cardinality and form a ∆-system with root R; (4.35) for each α, β ∈ S∗ there is a bijection jα,β : δ → δ such that (i) jα,β ↾ (δ \ (dα∆dβ)) = idδ\(dα∆dβ) , (ii) jα,β ′′dα = dβ, j̃α,β(pα) = pβ and (iii) j̃α,β(ẋ α) = ẋ β for every α, β ∈ S Note that, by (4.34), we have (4.36) supp(ȧ′) = supp(ȧ′) ∩ dα ⊆ R for every α ∈ S By (4.35), pα ↾ R for α ∈ S ∗ are all the same. Let q = pα ↾ R for some/any α ∈ S∗. q ≤P p ′′ by (4.33), (ii). Let Ṡ be a P-name such that (4.37) ‖– P “ Ṡ = {α ∈ S ∗ : pα ∈ Ġ} ”. By Lemma 4.5, q ‖–P “ Ṡ is stationary ”. Hence, by (4.31), (4.38) q ‖– P “ ∃n ∈ ω ∀α < κ | ġ ′′((ḟ ′′(Ṡ \ α))) | ≥ µ Let q′ ≤P q and n ∗ ∈ ω be such that (4.39) q′ ‖–P “ ∀α < κ (| ġ ′′((ḟ ′′(Ṡ \ α))) | ≥ µ) ”. (4.40) S∗∗ = {α ∈ S∗ : supp(q′) ∩ dα ⊆ R}. Since | supp(q′) | < λ by (4.11), we have (4.41) S∗ \ S∗∗ is of cardinality < λ. In particular S∗∗ is still stationary and (4.42) q′ ‖–P “ | ġ ′′((ḟ ′′(Ṡ ∩ S∗∗))) | ≥ µ ” by (4.39). Claim 4.3.2. There is 〈α0, ...,αn∗−1〉 ∈ ((S such that q′ ∪ pα0 ∪ · · · ∪ pαn∗−1 /‖– P “ ġ(〈ẋ , ..., ẋ′αn∗−1〉) ∈ V [ĠR] ”. ⊢ Otherwise, we would have q′ ∪ pβ0 ∪ · · · ∪ pβn∗−1 ‖–P “ ġ(〈ẋ , ..., ẋ′βn∗−1〉) ∈ V [ĠR] ” for all 〈β0, ..., βn∗−1〉 ∈ ((S Fix 〈α0, ...,αn∗−1〉 ∈ ((S and let D = {r ∈ P : r ≤P q ′ ∪ pα0 ∪ · · · ∪ pαn∗−1 , supp(r) ⊆ R ∪ {dαi : i < n ∗} ∪ supp(q′), r ‖– P “ ġ(ẋ , ..., ẋ′αn∗−1) = ẋ ” for some PR-name ẋ }. Let A be a maximal antichain in D. By the µ-c.c. of P, | A | < µ. For each r ∈ A, let ẋr be a PR-name such that r ‖– P “ ġ(ẋ , ..., ẋ′αn∗−1) = ẋr ” and Ẋ be a PR-name such that ‖– P “ Ẋ = {ẋr : r ∈ A} ”. Then, we have ‖–P “ | Ẋ | < µ ”. By Lemma 4.7, (a) q′ ∪ pα0 ∪ · · · ∪ pαn∗−1 ‖– P “ ġ(〈ẋ , ..., ẋ′αn∗−1〉) ∈ Ẋ ”. Hence by (4.35) and (4.9), we have q′ ∪ pβ0 ∪ · · · ∪ pβn∗−1 ‖–P “ ġ(〈ẋ , ..., ẋ′βn∗−1〉) ∈ Ẋ ” for all 〈β0, ..., βn∗−1〉 ∈ ((S . But this is a contradiction to (4.42). ⊣ (Claim 4.3.2) Let 〈α0, ...,αn∗−1〉 ∈ ((S be as in Claim 4.3.2 and (4.43) q′′ = q′ ∪ pα0 ∪ · · · ∪ pαn∗−1 . Note that (4.44) q′′ ‖–P “ f(αi) = ẋ ” for i < n∗ by (4.43). Let p∗ ≤P q ′′ be such that (4.45) p∗ ‖–P “ ġ(ẋ , ..., ẋ′αn∗−1) 6∈ V [ĠR] ”. By thinning out S∗∗ further, if necessary, we may assume that supp(p∗) ∩ supp(pα) ⊆ R for all α ∈ S ∗∗. For i < n∗, let Ṡi be a P-name such that (4.46) ‖– P “ Ṡi = {α ∈ S ∗∗ : j̃αi,α(p ∗) ∈ Ġ} ”. By Lemma 4.5, we have p∗ ‖–P “ Ṡi is a stationary subset of κ ” for all i < n ∗. Note that we have j̃αi,α(p ∗) ≤P pα by (4.43) and (4.35), (ii). Claim 4.3.3. p∗ ‖– P “ ∀β0 · · · ∀βn∗−1 〈β0, ..., βn∗−1〉 ∈ ((Ṡ0, ..., Ṡn∗−1)) → ġ(〈ḟ(β0), ..., ḟ(βn∗−1)〉) 6∈ V [ĠR] ⊢ Suppose that q ≤P p∗ and q ‖–P “ 〈β0, ..., βn∗−1〉 ∈ ((Ṡ0, ..., Ṡn∗−1)) ”. Then, by (4.46), q ‖–P “ j̃αi,βi(p ∗) ∈ Ġ ” for i < n∗. It follows that (4.47) q ‖– P “ j̃αi,βi(p ∗) ↾ dβi ∈ Ġ ” for i < n j̃ = j̃α0,β0 ◦ j̃α1,β1 ◦ · · · ◦ j̃αn∗−1,βn∗−1 . (4.48) j̃(p∗) = p∗ ↾ (δ \ dαi) ∪ j̃α0,β0(p ∗) ↾ dβ0 ∪ · · · ∪ j̃αn∗−1,βn∗−1(p ∗) ↾ dβn∗−1 by (4.35). Hence (4.49) q ‖– P “ j̃(p ∗) ∈ Ġ ” by q ≤P p ∗ and (4.47) and (4.48). By definition of j̃ and q′′, and by (4.35), we have (4.50) j̃(p∗) ≤P j̃(q ′′) ≤P j̃αi,βi(pαi) = pβi for i < n ∗ and (4.51) j̃(ẋ′αi) = ẋ for i < n∗ by (4.44). Hence by (4.45) q ‖–P “ ġ(〈ẋ , ..., ẋ′βn∗−1〉) 6∈ V [ĠR] ”. By (4.33), (4.49) and (4.50), it follows that q ‖–P “ ḟ(βi) = ẋ for i < n∗. Hence q ‖–P “ ġ(〈ḟ(β0), ..., ḟ(βn∗−1)〉) 6∈ V [GR] ”. ⊣ (Claim 4.3.3) To show that p∗ ‖–P “ (i1) holds ”, suppose that q ≤P p ∗ and 〈β0, ..., βn∗−1〉, 〈γ0, ..., γn∗−1〉 ∈ ((S are such that (4.52) {β0, ..., βn∗−1} ∩ {γ0, ..., γn∗−1} = ∅ and (4.53) q ‖– P “ 〈β0, ..., βn∗−1〉, 〈γ0, ..., γn∗−1〉 ∈ ((Ṡ0, ..., Ṡn∗−1)) ”. Note that it is enough to consider 〈β0, ...,βn∗−1〉, 〈γ0, ..., γn∗−1〉 ∈ ((S (4.52) since we can thin out ṠG0 ,..., Ṡ n∗−1 afterwards if necessary so that they are pairwise disjoint. By the remark after (4.46), we may assume that q ≤P p ∗ ∪ pβ0 ∪ · · · ∪ pβn∗−1 ∪ pγ0 ∪ · · · ∪ pγn∗−1 . By Lemma 4.7, (b), there are P-names ẏ, ż such that supp(ẏ) ∩ supp(ż) ⊆ R and p∗ ∪ pβ0 ∪ ··· ∪ pβn∗−1 ∪ pγ0 ∪ ··· ∪ pγn∗−1 ‖–P “ ġ(〈ḟ(β0), ..., ḟ(βn∗−1)〉) = ẏ ∧ ġ(〈ḟ(γ0), ..., ḟ(γn∗−1)〉) = ż ”. By Claim 4.3.3 and Lemma 4.6, it follows that q ≤P p ∗ ∪ pβ0 ∪ ··· ∪ pβn∗−1 ∪ pγ0 ∪ ··· ∪ pγn∗−1 ‖– P “ ġ(〈ḟ(β0〉, ..., ḟ(βn∗−1)) 6= ġ(〈ḟ(γ0), ..., ḟγn∗−1))〉 ”. Since q as above may be chosen below arbitrary r ≤P p ∗, it follows that p∗ ‖–P “ (i1) holds ”. (Theorem 4.3) Corollary 4.8. (a) Assume CH and P = Fn(µ, 2) for some cardinal µ. Then ‖–P “ IP(ℵ2,ℵ1) ” holds. (b) Assume GCH and P = Fn(µ, 2) for some cardinal µ. Then ‖–P “ IP(κ +,ℵ1) ” holds for every uncountable κ of uncountable cofinality and ‖–P “ IP(λ,ℵ1) ” for every inaccessible λ. (c) Assume CH and P is a finite support product of copies of a productively c.c.c. poset of cardinality ℵ1. Then ‖–P “ IP(ℵ2,ℵ1) ” holds. In particular, we have ‖–P “HP(ℵ2) ”. (d) Assume GCH and P is a finite support product of copies of a productively c.c.c. poset of cardinality ℵ1. Then ‖– P “ IP(κ +,ℵ1) ” holds for every uncountable κ of uncountable cofinality and ‖–P “ IP(λ,ℵ1) ” for every inaccessible λ. (e) Assume CH and P is a countable support product of copies of a proper poset of cardinality ℵ1 such that its product is also proper. Then ‖–P “ IP(ℵ2,ℵ2) ” holds. In particular, we have ‖–P “HP(ℵ2) ”. (f) Assume GCH and P is a countable support product of copies of a proper poset of cardinality ℵ1 such that its product is also proper. Then ‖–P “ IP(κ +,ℵ2) ” holds for every uncountable κ of uncountable cofinality and ‖–P “ IP(λ,ℵ2) ” for every inaccessible λ. Note that countable support products of Sacks or Prikry-Silver forcing are in- stances of (e) and (f) above. Proof. Under CH, ω1 = 2 <ω1 and ω2 is <ω1-inaccessible. In (a) and (b), P is forc- ing equivalent to a finite support product of copies of the countable poset Fn(ω, 2). Clearly P’s in all of (a) ∼ (f) are homogeneous; P’s in (a) ∼ (d) satisfy the c.c.c. and hence they are proper. Thus we can apply Theorem 4.3. The second parts of (c) and (e) follow from Theorem 2.9. (Corollary 4.8) Results similar to Theorem ?? and Corollary 4.8 also hold for partial orderings with product-like structure as those considered in [9]. Thus, we can prove e.g. that IP(ℵ2,ℵ2) together with clubsuit principle is consistent. In [7] it is shown that, if we start from a model V which is obtained by adding a dominating real to a model of GCH + Chang’s conjecture for ℵω, i.e. (ℵω+1,ℵω) →→ (ℵ1,ℵ0), then adding more than ℵω+1 Cohen reals forces ¬WFN. Since V satisfies GCH, IP(κ,ℵ1) is forced for every κ ≥ ℵ2 which is not a successor of a singular cardinal of cofinality ω by adding any number of Cohen reals by Corollary 4.8. In particular: Corollary 4.9. Suppose that Chang’s conjecture for ℵω is consistent. Then so is IP(ℵ2,ℵ1) ∧ b ∗ = ℵ1 ∧ ¬WFN. 5 Models of IP(ℵ2,ℵ2) ∧ ¬IP(ℵ2,ℵ1) Recall that Prikry-Silver forcing S is the forcing with partial functions with co- infinite domain, that is S = {f : f : D → 2, D ⊆ ω, |ω \D | = ℵ0} with the ordering f ≤S g ⇔ f ⊇ g for f , g ∈ S. A (V, S)-generic filter G gives rise to the function sG = G : ω → 2 which is often called a Prikry-Silver real. For f ∈ S let codom(f) = ω \ dom(f). It is easy to check that Prikry-Silver forcing S as well as its countable support products SI over any index set I satisfy the Axiom A. Hence they are all proper. Note that, by definition of ≤S, we have: (5.1) f , g ∈ S are incompatible if | codom(f) ∩ codom(g) | < ℵ0. (5.2) For any 〈f0, f1〉 ∈ S 2, there is 〈g0, g1〉 ≤S2 〈f0, f1〉 such that | codom(g0) ∩ codom(g1) | < ℵ0. Lemma 5.1. For any f ∈ S and 〈gn0 , g 1 〉 ∈ S 2, n ∈ ω such that | codom(gn0 ) ∩ codom(gn1 ) | < ℵ0 there is g ≤S f such that 〈g, g〉 is incompatible with all 〈g 0 , g n ∈ ω. Proof. Construct in ∈ 2, n ∈ ω and A ⊆ codom(f) recursively so that | codom(f) ∩ k≤n dom(g ) | = ℵ0 and |A ∩ codom(gnin) | < ℵ0 for all n ∈ ω. Then any extension g of f on ω \ A will do. (Lemma 5.1) Working in V = L, we can construct recursively a maximal antichain {〈gα0 , g 1 〉 : α < ω1} in S 2 such that (5.3) | codom(gα0 ) ∩ codom(g 1 ) | < ℵ0 for all α < ω1. Note that each step of the recursive construction is possible by (5.2) and (5.2). Furthermore by choosing 〈gα0 , g 1 〉 in each step of the construction according to the Σ12-well ordering of the reals (which exists because of V = L), we can make {〈gα0 , g 1 〉 : α < ω1} a Σ 2-set (actually we can even choose such a maximal antichain as a Π11-set arguing similarly to [16]). Let ϕ : S2 → ω2 be a Borel bijection and let g : ((ω2)) → ω2 be defined by (5.4) g(〈x0, ..., xn−1〉) = 0 , g 1 ) ; if n = 2, there is α < ω2 such that x0 ⊇ g 0 , x1 ⊇ g 1 and α ∗ is minimal among such α’s 0 ; otherwise. It is easy to check that g is a ∆13-set. Theorem 5.2. Assume V = L. Then we have ‖– Sω2 “ IP(ℵ2,ℵ2) and ¬IP(ℵ2,ℵ1) ”. Proof. ‖– Sω2 “ IP(ℵ2,ℵ2) ” follows from Corollary 4.8, (e). To show that ‖– Sω2 “¬IP(ℵ2,ℵ1) ”, let G be a (V, S ω2)-generic filter. Working in L[G], let sβ be the β’th Prikry-Silver real added by G for β < ω2. Let f : ω2 → be defined by (5.5) f(β) = sβ for β < ω2 and let g : ((ω2)) → ω2 be the mapping as in (5.4), or more precisely, let g be the mapping (in L[G]) defined by the ∆13 definition corresponding to (5.4). We show that f and g build a counter-example to IP(ℵ2,ℵ1). Since | rng(g) | ≤ ℵ1, (i1) clearly fails for these f and g. Hence we are done by showing that f and g do not satisfy (i0). Assume, for a contradiction, that f and g satisfy (i0). Returning to L, let ḟ , ġ, ṡβ, β < ω2 etc. be S ω2-names of f , g, sβ , β < ω2 etc. respectively. In particular, we can choose ḟ such that (5.6) ‖– Sω2 “ ḟ(β) = ṡβ ” for all β < ω2. Since Sω2 is proper, there are p ∈ G, Sω2-name Ṡ and a countable set Z (in L) such that (5.7) p ‖– Sω2 “ Ṡ ⊆ ω2 is stationary and ġ ′′((ḟ ′′Ṡ)) ⊆ Z ”. Let U = {β < ω2 : there is p ′ ≤Sω2 p such that p ′ ‖– Sω2 “ β ∈ Ṡ ”}. Then U is a stationary subset of ω2. For each β ∈ U , let pβ ≤Sω2 p be such that pβ ‖– Sω2 “β ∈ Ṡ ” and β ∈ supp(pβ). By ∆-System Lemma and CH, there is U∗ ∈ [U ]ℵ2 such that (5.8) supp(pβ), β ∈ U ∗ form a ∆-system with root R which is an initial segment of all of supp(pβ), β ∈ U (5.9) supR < minU∗; (5.10) pβ ↾ R, β ∈ U ∗ are all the same; and (5.11) pβ(β), β ∈ U ∗ are all the same, say h ∈ S. Note that pβ, β ∈ U ∗ are compatible by (5.8) and (5.10). (5.12) X = ϕ−1(Z). By Lemma 5.1, there is a k ≤S h such that 〈k, k〉 is incompatible with all 〈g 0 , g from the countable set X . Fix two distinct β, γ ∈ U∗ and let q ≤Sω2 pβ, pγ be defined by dom(q) = dom(pβ) ∪ dom(pγ) and (5.13) q(δ) = pβ(δ) ; if δ ∈ supp(pβ) \ {β} pγ(δ) ; else if δ ∈ supp(pγ) \ {γ} k ; else if δ = β or δ = γ for δ ∈ dom(q). By q ≤Sω2 pβ, pγ , we have q ‖– Sω2 “β, γ ∈ Ṡ ”. Thus the following claim yields a contradiction to (5.7): Claim 5.2.1. q ‖– Sω2 “ ġ(〈ḟ(β), ḟ(γ)〉) 6∈ Z ”. ⊢ By (5.6), we have to show q ‖– Sω2 “ ġ(〈ṡβ, ṡγ〉) 6∈ Z ”. First, we show that q ‖– Sω2 “ ġ(〈ṡβ, ṡγ〉) 6= 0 ”. Note that, by the complete embedding S2 ∋ 〈g0, g1〉 7→ {〈β, g0〉, 〈γ, g1〉} ∈ S {β,γ} ≤◦ Sω2 we have: {{〈β, gα0 〉, 〈γ, g 1 〉} : α < ω1} is a maximal antichain in S For any r ≤Sω2 q, let α ∗ < ω1 be such that r and {〈β, g 0 〉, 〈γ, g 1 〉} are com- patible. Let s ≤Sω2 r, {〈β, g 0 〉, 〈γ, g 1 〉}. Then we have s ‖– Sω2 “ ṡβ ⊇ g 0 , ṡγ ⊇ g Hence, by (5.4), it follows that s ‖–Sω2 “ ġ(〈ṡβ, ṡγ〉) 6= 0 ”. Now, suppose, for contradiction, that there is r ≤Qω2 q such that r ‖– Sω2 “ ġ(〈ṡβ, ṡγ〉) ∈ Z ”. Then, by the first part of the proof, there are s ≤Sω1 r and 〈g 0 , g 1 〉 ∈ X such that s ‖–Sω2 “ ṡβ ⊇ g 0 and ṡγ ⊇ g 1 ”. In particular s(β) and s(γ) are compatible with gα0 and g 1 , respectively. Since r ≤Sω2 q ≤Sω2 {〈β, k〉, 〈γ, k〉}, it follows that k is compatible with both of gα0 and g 1 . This is a contradiction to the choice of k. ⊣ (Claim 5.2.1) (Theorem 5.2) We can prove a Lemma similar to Lemma 5.1 for omega product of Sacks forcing. Thus, by a similar argument as above, we can also prove that IP(ℵ2,ℵ1) fails in a generic extension by countable support side-by-side product of Sacks forcing. 6 The Consistency of b∗ = ℵ2 ∧ do = ℵ1 In the following we shall refer by (A) the assertion that there is a structure 〈(ω2) 2, A,F〉 with the properties (6.1) ∼ (6.5) below. Recall that a mapping f : X → X is called an involution if it is a bijection exchanging (some) pairs of elements of X , that is, if f ◦ f = idX holds. (6.1) ω2 × ω2 ⊇ A ⊇ {〈α, β〉 ∈ ω2 × ω2 : β < α}; (6.2) For any C ∈ [ω2] ℵ0 there is an X ∈ [ω2] ℵ2 such that (C ×X) ∩ A = ∅; (6.3) For all 〈φ, ψ〉 ∈ F , φ and ψ are involutions on ω2; (6.4) For each 〈φ, ψ〉 ∈ F and for all 〈α, β〉 ∈ ω2 ×ω2, we have 〈α, β〉 ∈ A if and only if 〈φ(α), ψ(β)〉 ∈ A; (6.5) For any stationary S ⊆ Eω2ω1 and any Aζ , Bζ ∈ [ω2] ℵ0 for ζ ∈ S, there is a stationary T ⊆ S such that, for any n ∈ ω, if ζi, ηi ∈ T for i ∈ n are pairwise distinct (2n elements) then there is 〈φ, ψ〉 ∈ F such that φ ′′Aζi = Aηi , ψ ′′Bζi = Bηi ; and φ ↾ Aζi : Aζi → Aηi , ψ ↾ Bζi : Bζi → Bηi are order isomorphisms for all i ∈ n. The consistency of (A) together with CH over ZFC is proved in the next section. Below, we will prove the consistency of c = b∗ = ℵ2 ∧ do = ℵ1 ∧ ¬C s(ℵ2) by constructing a model of this combination of assertions starting from a model of (A) and CH. Let us begin with introducing some notation for the forcing construction we use in the proof. For a cardinal κ, a sequence f̄ = 〈fξ : ξ < κ〉 in ωω and X ⊆ κ, let Df̄ ,X be the canonical poset adding an element of ωω dominating {fξ : ξ ∈ X}. That is (6.6) Df̄ ,X = {〈s, F 〉 : s ∈ ω>ω, F ∈ [κ]<ℵ0} and, for 〈s, F 〉, 〈s′, F ′〉 ∈ Df̄ ,X , (6.7) 〈s′, F ′〉 ≤Df̄ ,X 〈s, F 〉 ⇔ s ′ ⊇ s, F ′ ⊇ F, ∀α ∈ F ∩X ∀n ∈ dom(s′) \ dom(s) (fα(n) ≤ s ′(n)). Since any 〈s, F 〉, 〈s′, F ′〉 ∈ Df̄ ,X with s = s ′ are compatible, we have: Lemma 6.1. Df̄ ,X is σ-centered. Note that the underlying set of Df̄ ,X does not depend on the sequence f̄ . So we shall denote this set with DX . Actually DX as a set does not depend on X either. Nevertheless we shall add the suffix X so that we can distinguish D’s by their intended function. Note also that, as a set, Df̄ ,Xα for any κ-sequence f̄ of reals is the same: we shall denote this set by If d ∈ DX and d = 〈s, F 〉 then we shall write s d and F d to denote these s and F respectively. In the following we assume that a sequence X̄ = 〈Xα : α < κ〉 of nonempty subsets of κ is fixed. Let (6.8) QX̄ = Cκ ∗ D ˙̄f,Xα where Cκ = Fn(κ × ω, ω) and ˙̄f denotes the Cκ-name of the sequence of Cohen reals (∈ ωω) of length κ added by Cκ. Thus, if G is a (V,Cκ)-generic set and cα is the α’th element of ˙̄fG, then cα(n) = m if and only if there is a condition c ∈ G such that 〈α, n〉 ∈ dom(c) and c(α, n) = m. (6.9) Q = {〈c, d〉 : c ∈ Cκ, d ∈ α∈κDXα , ξ∈dom(d) F d(ξ) × dom(sd(ξ)) ⊆ dom(c)} For 〈c, d〉, 〈c′, d′〉 ∈ Q (6.10) 〈c′, d′〉 ≤ 〈c, d〉 ⇔ c′ ≤Cκ c, dom(d ′) ⊇ dom(d), ∀α ∈ dom(d) ′(α) ⊇ sd(α) ∧ F d ′(α) ⊇ F d(α)∧ ∀ξ ∈ F d(α) ∩Xα ∀n ∈ dom(s d′(α)) \ dom(sd(α)) c′(ξ, n) ≤ sd ′(α)(n) The following can be shown easily by standard arguments: Lemma 6.2. Φ : Q → QX̄ ; 〈c, d〉 7→ 〈c, ď〉 is a dense embedding of Q into QX̄ . QX̄ and Q are thus forcing equivalent. For p ∈ Q with p = 〈c, d〉, let supp0(p) = {α < κ : 〈α, n〉 ∈ dom(c) for some n ∈ ω} and supp1(p) = dom(d). For a Q -name ȧ, supp0(ȧ) and supp1(ȧ) are also defined in analogy to (4.4). In Theorem 6.4, we assume CH + (A) and let, for a structure 〈(ω2) 2, A,F〉 as in (A), κ = ω2 and X̄ = 〈Xα : α < ω2〉 where Xα = {β ∈ ω2 : 〈α, β〉 ∈ A} for α < ω2. For such X̄ , the next lemma follows immediately from (6.3) and (6.4). Lemma 6.3. Suppose that 〈(ω2) 2, A,F〉 and X̄ are as above. If 〈φ, ψ〉 ∈ F , then the mapping j〈φ,ψ〉 : Q defined by j〈φ,ψ〉(〈c, d〉) = 〈c ′, d′〉 for 〈c, d〉 ∈ Q where c′ and d′ are such that dom(c′) = {〈φ(α), n〉 : 〈α, n〉 ∈ dom(c)}; c′(〈φ(α), n〉) = c(〈α, n〉) for 〈α, n〉 ∈ dom(c); dom(d′) = ψ ′′ dom(d); ′(ψ(ξ)) = F d(ξ) and sd ′(ψ(ξ)) = sd(ξ) for ξ ∈ dom(d) is an automorphism on the poset Q Similarly to Section 4, we shall also denote with j〈φ,ψ〉 the corresponding map- ping on Q -names. The following theorem together with the consistency result in Section 7 gives the consistency of the conjunction of the assertions c = b∗ = ℵ2, do = ℵ1 and ¬Cs(ℵ2) over ZFC. Theorem 6.4. Assume CH and (A). Let 〈(ω2) 2, A,F〉 be a structure satisfying (6.1) ∼ (6.5) and let X̄ = 〈Xα : α < ω2〉 where Xα = {β ∈ ω2 : 〈α, β〉 ∈ A}. Then ‖– “ c = b∗ = ℵ2 ∧ do = ℵ1 ∧ ¬C s(ℵ2) ”. Proof. First, we show that ‖– “ c = b∗ = ℵ2 ”. Let G be a (V,Q )-generic filter. Working in V [G], let f̄ = 〈cα : α < ω2〉 be the sequence of Cohen reals added by the Cω2 part of QX̄ and dα be the Hechler type real added by Df̄ ,Xα for α < ω2. By (6.1), {cα : α < γ} is bounded by dγ for all γ < ω2. On the other hand, {cα : α < ω2} is unbounded by (6.2) and the c.c.c. of Df̄ ,Xα (in V [f̄ ]). This shows that V [G] |= ℵ2 ≤ b ∗. Since |Q | = ℵ2 by CH, we have V [G] |= c ≤ ℵ2. To show that Q forces do = ℵ1, suppose that ḟα, α < ω2 are Q -names of elements of ωω, ϕ(x, y, z) a formula in LZF and ȧ a Q -name of an element of ωω such that (6.11) ‖– “H(ℵ1) |= ϕ(ḟα, ḟβ, ȧ) ” for all α < β < ω2. By Maximal Principle, it is enough to show that there are η1 < η0 < ω2 such that “H(ℵ1) |= ϕ(ḟη0 , ḟη1 , ȧ) ”. For ξ < ω2, let Aξ = supp1(ḟξ) ∪ supp1(ȧ) and Bξ = supp0(ḟξ) ∪ supp0(ȧ). By CH, ∆-System Lemma and (6.5), we can find a stationary S ⊆ Eω2ω1 such that (6.12) Aξ, ξ ∈ S form a ∆-system such that its root is an initial segment of each of Aξ, ξ ∈ S ; Bξ, ξ ∈ S form a ∆-system such that its root is an initial segment of each of Bξ, ξ ∈ S ; (6.13) for any distinct ζ0, ζ1, η0, η1 ∈ S, there is 〈φ, ψ〉 ∈ F such that (6.13a) φ ′′Aζi = Aηi , ψ ′′Bζi = Bηi ; and (6.13b) φ ↾ Aζi : Aζi → Aηi , ψ ↾ Bζi : Bζi → Bηi are order isomorphisms for i ∈ 2; (6.14) j〈φ,ψ〉(ḟζ) = ḟη for any distinct ζ , η ∈ S and 〈φ, ψ〉 ∈ F as in (6.13) with ζ0 = ζ and η0 = η. Note that, by (6.12) and (6.13b), we have (6.15) j〈φ,ψ〉(ȧ) = ȧ for any 〈φ, ψ〉 as in (6.13). Now, let ζ0, ζ1, η0, η1 ∈ S be four distinct elements of S such that ζ0 < ζ1 and η1 < η0. By (6.11), we have “H(ℵ1) |= ϕ(ḟζ0 , ḟζ1, ȧ) ” . Hence, by mapping this situation by j〈φ,ψ〉 for 〈φ, ψ〉 ∈ F as in (6.13) for these ζ0, ζ1, η0, η1 , we obtain “H(ℵ1) |= ϕ(ḟη0 , ḟη1 , ȧ) ” . Thus, η0, η1 above are as desired. Finally, we show that Q forces the negation of Cs(ℵ2). Let 〈r0n, s n〉, n ∈ ω list all quadruples of finite sequences r 0, s0, r1, s1 ∈ ω>ω such that (6.16) | r0 | = | s0 | = | r1 | = | s1 | and (6.17) 〈r0, s0〉 6= 〈r1, s1〉 if | r0 | > 0. We further assume that the enumeration 〈〈r0n, s n〉 : n ∈ ω〉 is arranged so (6.18) | r0n | ≤ n for all n ∈ ω. Now, working in V [G], let aα, α < ω2 be the subsets of ω defined by n ∈ aα ⇔ one of the following (6.19) and (6.20) holds: (6.19) r0n ⊆ cα, s n ⊆ dα+1, cα(n) = 0, dα+1(n) = 1, cα(n+ 1) = 2 and dα+1(n+ 1) = 3 ; (6.20) r1n ⊆ cα, s n ⊆ dα+1, cα(n) = 2, dα+1(n) = 3, cα(n+ 1) = 0 and dα+1(n+ 1) = 1 . (6.21) aα,n = aα \ {k : | r k | < n} for α < ω2 and n ∈ ω. We show that the matrix 〈aα,n : α < ω2, n ∈ ω〉 together with T = 2ω is a counter-example to Cs(ℵ2). For this, it is enough to prove the following: Claim 6.4.1. If S0, S1 are cofinal subsets of ω2, then (1) there exist n < ω, α ∈ S0 and β ∈ S1 such that aα,n ∩ aβ,n = ∅; and (2) for any t ∈ 2ω, there are α ∈ S0 and β ∈ S1 such that aα,t(0) ∩ aβ,t(1) 6= ∅. ⊢ Working in the ground model, let Ṡ0 and Ṡ1 beQ†X̄ -names for the cofinal subsets of ω2. Let p ∈ Q . For α < ω2, let pα ∈ Q and γα, δα ∈ ω2 be such that (6.22) γα < δα < γβ < δβ for all α < β < ω2 ; (6.23) pα ≤Q† p , pα = 〈c α, dα〉 for all α < ω2 ; and (6.24) pα ‖–Q† “ γα ∈ Ṡ0 , δα ∈ Ṡ1 ”. By ∆-System Lemma, we find a stationary U ⊆ Eω2ω1 and Aα, Bα ∈ [ω2] <ℵ0 for α ∈ U such that (6.25) supp0(pα) ⊆ Aα, supp1(pα) ⊆ Bα ; (6.26) Aα, α ∈ U form a ∆-system with root A; Bα, α ∈ U form a ∆-system with root B; (6.27) γα, γα + 1, δα, δα + 1 ∈ (Aα ∩Bα) \ (A ∪B) . By thinning out U further, if necessary, we may also assume that there are some k∗, n∗ ∈ ω such that (6.28) dom(cα) = supp0(pα)× k ∗ and dom(sd α(ξ)) = k∗ for all ξ ∈ supp1(pα) ; (6.29) cα(γα, ·) = r n∗ , s dα(γα+1) = s0n∗ ; (6.30) cα(δα, ·) = r n∗ , s dα(δα+1) = s1n∗ . Without loss of generality, we may also assume that, for some fixed c∗, d∗, (6.31) cα ↾ A× k∗ = c∗ and 〈sd α(η) : η ∈ B〉 = d∗ for all α ∈ U . Note that pα, α ∈ U are compatible by (6.25), (6.26) and (6.31). Now, since Ṡ0, Ṡ1, p were arbitrary, Claim 6.4.1, (1) is proved by the following subclaim: Subclaim 6.4.1.1. For any α, β ∈ U with α < β, there is q ≤ p such that “ γα ∈ Ṡ0, δβ ∈ Ṡ1, ȧγα,n∗ ∩ ȧδβ ,n∗ = ∅ ”. ⊢ Let q = 〈cq, dq〉 be the common extension of pα and pβ such that (6.32) γα ∈ F dq(δβ+1) (6.33) dom(sd q(ξ)) = k∗ for all ξ ∈ dom(dq). Let G be a (V,Q )-generic filter with q ∈ G. In V [G], we have (6.34) cγα(m) ≤ dδβ+1(m) for all m ≥ k by (6.28), (6.32) and (6.33). Now, toward a contradiction, assume that aγα,n∗ ∩ aδβ ,n∗ 6= ∅ and let m ∈ aγα,n∗ ∩ aδβ ,n∗. By the definition of aα’s it follows that, for some i, j ∈ 2, we have rim ⊆ cγα , s m ⊆ dγα+1 ; rjm ⊆ cδβ , s m ⊆ dδβ+1. On the other hand, since q ∈ G, we have pα, pβ ∈ G. It follows that r0n∗ ⊆ cγα, s n∗ ⊆ dγα+1 ; r1n∗ ⊆ cδβ , s n∗ ⊆ dδβ+1 by (6.29) and (6.30). By the definition (6.21) of aγα,n’s, we have | r m | ≥ n ∗. Thus we have, either r0n∗ ⊆ r m ⊆ cγα , s n∗ ⊆ s m ⊆ dγα+1 ; r1n∗ ⊆ r m ⊆ cγβ , s n∗ ⊆ s m ⊆ dγβ+1 ; r0n∗ ⊆ r m ⊆ cγα , s n∗ ⊆ s m ⊆ dγα+1 ; r1n∗ ⊆ r m ⊆ cγβ , s n∗ ⊆ s m ⊆ dγβ+1 . In the first case, we must have cγα(m+ 1) = 2 and dδβ+1(m+ 1) = 1 by (6.19) and (6.20). This is a contradiction to (6.34). Similarly, in the second case, we have cγα(m) = 2 and dδβ+1(m) = 1. This is again a contradiction to (6.34). ⊣ (Subclaim 6.4.1.1) (2) of Claim 6.4.1 follows from the next subclaim: Subclaim 6.4.1.2. For any t ∈ 2ω and α, β ∈ U with α < β, there is q ≤ such that “ γα ∈ Ṡ0, δβ ∈ Ṡ1, ȧγ(α),t(0) ∩ ȧδ(β),t(1) 6= ∅ ”. ⊢ For each ξ ∈ {α, β}, let p̃ξ ≤Q† pξ with p̃ξ = 〈c̃ ξ, d̃ξ〉 and m ∈ ω be such that (6.35) c̃ξ(γξ, ·) = r m , s d̃ξ(γξ+1) = s0m ; c̃ ξ(δξ, ·) = r m , s d̃ξ(δξ+1) = s1m ; (6.36) | r0m | ≥ t(0), t(1) ; (6.37) supp0(p̃ξ) = supp0(pξ) ; supp1(p̃ξ) = supp1(pξ) ; (6.38) c̃ξ ↾ A× ω = cξ ↾ A× ω and 〈sd̃ ξ(η) : η ∈ B〉 = 〈sd ξ(η) : η ∈ B〉. Let q0 = 〈cq 〉 be the maximal (with respect to ≤ ) common extension of p̃α and p̃β which exists because of (6.37) and (6.38). Extend q 0 further to q = 〈cq, dq〉 such that (6.39) | cq(γα, ·) | = | c q(δβ , ·) | = | s dq(γα+1) | = | sd q(δβ+1) | = m+ 2 ; (6.40) cq(γα, m) = 0, s dq(γα+1)(m) = 1, cq(γα, m+ 1) = 2, s dq(γα+1)(m+ 1) = 3 ; (6.41) cq(δβ, m) = 2, s dq(δβ+1)(m) = 3, cq(δβ, m+ 1) = 0, s dq(δβ+1)(m+ 1) = 1 . This is possible because γα 6∈ F (δβ+1) and δβ 6∈ F (γα+1) by the maximality of q0 and (6.37). By (6.35), (6.40), (6.41), by the definition (6.21) of aα,n’s, and since | r m | ≥ t(0), t(1), we have q ‖– “m ∈ ȧγα,t(0) ∩ ȧδβ ,t(1) ”. Since q ≤Q† pα, pβ, we also have “ γα ∈ Ṡ0, δβ ∈ Ṡ1 ”. Thus, q as above is as desired. ⊣ (Subclaim 6.4.1.2) ⊣ (Claim 6.4.1) (Theorem 6.4) Note that in the proof of ¬Cs(ℵ2) in Theorem 6.4, we used only (6.1) from the assumption (A). Note also that this proof actually shows that in the generic exten- sion the negation of C(ℵ2) from [13] holds which is a weakening of C s(ℵ2) obtained by replacing the condition “stationary” in the formulation of Cs(ℵ2) by “cofinal”. 7 Forcing CH + (A) In this section, we define under CH a σ-closed ℵ2-c.c. poset P0 which forces the combinatorial assertion (A) of the previous section. The poset P0 is defined as follows: p ∈ P0 ⇔ p = 〈X p, Y p, τ p, 〈φ : ξ ∈ Dp〉〉 where (7.1) Xp, Y p ∈ [ω2] (7.2) Dp ∈ [ω2] (7.3) for all ξ ∈ Dp, φ : Xp → Xp and ψ : Y p → Y p are involutions (that is, bijections φ such that φ−1 = φ); (7.4) for all ξ ∈ Dp, α ∈ Xp and β ∈ Y p, (7.4a) φ (α) < α + ξ + ω1 and (7.4b) ψ ξ (β) < β + ξ + ω1 ; Note that we have also α < φ (α) + ξ + ω1 and α < ψ (β) + ξ + ω1 for all ξ ∈ D α ∈ Xp and β ∈ Y p since φ ξ and ψ ξ are involutions by (7.3). (7.5) τ p : Xp × Y p → 2 ; (7.6) for all ξ ∈ Dp, α ∈ Xp and β ∈ Y p, we have τ p(α, β) = τ p(φ (α), ψ (β)) ; (7.7) τ p(α, β) = 1 for all 〈α, β〉 ∈ Xp × Y p with β < α. The ordering on P0 is defined by the following: For p, q ∈ P0 with p = 〈Xp, Y p, τ p, 〈φ ξ : ξ ∈ D p〉〉 and q = 〈Xq, Y q, τ q, 〈φ : ξ ∈ Dq〉〉, (7.8) p ≤P0 q ⇔ (7.8a) Xp ⊇ Xq, Y p ⊇ Y q ; (7.8b) Dp ⊇ Dq ; (7.8c) φ and ψ for all ξ ∈ Dq ; (7.8d) τ p ⊇ τ q and (7.8e) τ p ↾ (Xp \Xq)× Y q ≡ 1. For p ∈ P0 with p = 〈X p, Y p, τ p, 〈φ : ξ ∈ Dp〉〉, we intend to approximate the characteristic function of the set A in the assertion (A) by τ p. More precisely, in a generic extension V [G] for a (V,P0)-generic G, letting (7.9) τ = p∈G τ p ; φξ = p∈G φ and ψξ = p∈G ψ for ξ ∈ ω2 ; (7.10) A = τ−1 ′′{1} and F = {〈φξ, ψξ〉 : ξ ∈ ω2}, we are aiming to force 〈(ω2) 2, A,F〉 to satisfy (6.1) ∼ (6.5) in (A). Of the conditions in the definition of P0, (7.5) and (7.8d) force τ to be a function. Furthermore, τ : ω2×ω2 → 2 by density argument and the following Lemma 7.1, (a). (7.3) and (7.8c) make φξ and ψξ mappings for all ξ ∈ ω2; they are forced to be involutions on ω2 by (7.3) and the following Lemma 7.1, (a). Thus 〈(ω2) 2, A,F〉 is forced to satisfy (6.3). By (7.7) (and by the following Lemma 7.1, (a)), 〈(ω2) 2, A,F〉 is forced to satisfy the second inclusion of (6.1). By (7.6), 〈(ω2) 2, A,F〉 is forced to satisfy (6.4). (7.4) and (7.8e) are technical conditions whose role will be clear later in the course of the proof. By the definition of P0, it is clear that P0 is σ-closed. Thus, we are done by showing that P0 satisfies the ℵ2-c.c. and it forces that 〈(ω2) 2, A,F〉 as above satisfies the conditions (6.2) and (6.5). The next Lemma follows readily from the definition of P0. Lemma 7.1. (a) For any α, β < ω2, the set Dα,β = { p ∈ P0 : p = 〈X p, Y p, τ p, 〈φ : ξ ∈ Dp〉〉, α ∈ Xp and β ∈ Y p } is dense in P0. (b) For any C ∈ [ω2] ℵ0 and any β < ω2, EC,β = { p ∈ P0 : p = 〈X p, Y p, τ p, 〈φ : ξ ∈ Dp〉〉, C ⊆ Xp and for some δ ∈ Y p with δ ≥ β τ p(γ, δ) = 0 for all γ ∈ C } is dense in P0. In the rest of the section, we are going to work mainly in the ground model (where CH holds). Let τ̇ , φ̇ξ, ψ̇ξ for ξ ∈ ω2, Ȧ and Ḟ be P0-names of τ , φξ, ψξ for ξ ∈ ω2, A and F as above, respectively. Lemma 7.2. ‖– P0 “ 〈(ω2) 2, Ȧ, Ḟ〉 |= (6.2) ”. Proof. By density argument with Lemma 7.1, (b). (Lemma 7.2) For ξ < ω2, X , X ′, Y , Y ′ ∈ [ω2] ℵ0 with X ′ ⊆ X and Y ′ ⊆ Y , τ : X × Y → 2 and involutions φ′ : X ′ → X ′, ψ : Y ′ → Y ′, let us call the quintuple 〈X, Y, τ, φ′, ψ′〉 a ξ-extendable semi-condition if (7.11) φ′(α) < α + ξ + ω1 and ψ ′(β) < β + ξ + ω1 for all α ∈ X ′ and β ∈ Y ′ ; (7.12) τ(α, β) = τ(φ(α), ψ(β)) for all α ∈ X ′ and β ∈ Y ′ ; (7.13) τ(α, β) = 1 for all α ∈ X and β ∈ Y with β < α ; (7.14) τ ↾ ((X \X ′)× Y ′) ≡ 1. fig. 6 Note that the sets X ′ and Y ′, though not mentioned explicitly in the definition of ξ-extendable semi-condition, can be recovered from φ′ and ψ′. Note also that (7.14) holds vacuously if X = X ′. Hence, if p ∈ P0 with p = 〈Xp, Y p, τ p, 〈φ : ξ ∈ Dp〉〉, the quintuple 〈Xp, Y p, τ p, φξ, ψξ〉 is a ξ-extendable semi-condition for all ξ ∈ Dp. The following two lemmas explain the choice of the naming of ξ-extendable semi-conditions. Lemma 7.3. For any ξ < ω2, X, X ′, Y , Y ′ ∈ [ω2] ℵ0 with X ′ ⊆ X and Y ′ ⊆ Y , τ : X × Y → 2 as well as involutions φ′ : X ′ → X ′, ψ′ : Y ′ → Y ′, if 〈X, Y, τ, φ′, ψ′〉 is a ξ-extendable semi-condition then there are X̃ ⊇ X, Ỹ ⊇ Y , τ̃ : X̃ × Ỹ → 2 with τ̃ ⊇ τ and involutions φ̃ : X̃ → X̃, ψ̃ : Ỹ → Ỹ extending φ′ and ψ′ respectively such that 〈X̃, Ỹ , τ̃ , φ̃, ψ̃〉 is a ξ-extendable semi-condition and (7.15) τ̃ ↾ ((X̃ \X)× Y ) ≡ 1. fig. 7 Proof. Let X0 ∈ [ω2 \X ] ≤ℵ0 be such that (7.16) X0 is order-isomorphic to X \ X ′ and the order-isomorphism identifies points of distance less than ω1 (that is, if α ∈ X \ X ′ and α0 ∈ X0 are identified then we have α < α0 + ω1 and α0 < α+ ω1). Since X (and hence also X \X ′) is countable, we can easily choose the elements of X0 recursively in otp(X \ X ′) steps in accordance with (7.16). Put X̃ = X ∪̇ X0 and let φ̃ be the extension of φ which maps X0 order-isomorphically to X \X ′ and vice versa. Fix θ < ω1 such that (7.17) φ̃(α) ≤ α + ξ + θ for all α ∈ X̃ . There is such θ by (7.11), (7.16) and since | X̃ | ≤ ℵ0. Let Y0 ∈ [ω2 \ Y ] ≤ℵ0 be such that (7.18) Y0 is order-isomorphic to Y \Y ′ and the order-isomorphism identifies points of distance less than ξ + ω1; and (7.19) if β ∈ Y \ Y ′ and β0 ∈ Y0 are identified then β0 > β + ξ + θ. It is easy to see that the elements of such Y0 can be chosen recursively in otp(Y \Y steps. Now let Ỹ = Y ∪̇ Y0 and let ψ̃ be the extension of ψ ′ which maps Y0 order- isomorphically to Y \ Y ′ and vice versa. Finally define τ̃ : X̃ × Ỹ → 2 by (7.20) τ̃(α, β) = τ(α, β), if 〈α, β〉 ∈ X × Y τ(φ̃(α), ψ̃(β)), if 〈α, β〉 ∈ (X ′ × Y0) ∪ (X0 × Y ′) ∪ (X0 × Y0) 1, otherwise for every α ∈ X̃ and β ∈ Ỹ . X0 = X̃ \X fig. 8 〈X̃, Ỹ , τ̃ , φ̃, ψ̃〉 satisfies (7.11) by (7.17) and (7.18). It satisfies (7.12) by (7.20). Thus we are done by checking 〈X̃, Ỹ , τ̃ , φ̃, ψ̃〉 also satisfies (7.15) and (7.13). For (7.15), suppose that 〈α, β〉 ∈ (X̃ \X)× Y (= X0 × Y ). If 〈α, β〉 ∈ X0 × Y then τ̃ (α, β) = τ(φ̃(α), ψ̃(β)) by (7.20). But 〈φ̃(α), ψ̃(β)〉 ∈ (X \ X ′) × Y ′ by definition of φ̃ and ψ̃. Hence, by (7.14), we have τ̃ (α, β) = τ(φ̃(α), ψ̃(β)) = 1. If 〈α, β〉 ∈ X0 × (Y \ Y ′) then τ̃(α, β) = 1 by the “otherwise” clause of (7.20). For (7.13), it is enough to check that τ̃(α, β) = 1 for all 〈α, β〉 ∈ (X ′∪X0)×Y0 with β < α by (7.20) and (7.15). For such 〈α, β〉, we have τ̃ (α, β) = τ(φ̃(α), ψ̃(β)) by (7.20). Suppose that τ̃ (α, β) = 0. Then, since τ satisfies (7.13), we should have φ̃(α) ≤ ψ̃(β). By (7.19), we have β > ψ̃(β) + ξ + θ. On the other hand, by (7.17), we have α = φ̃2(α) ≤ φ̃(α) + ξ + θ. It follows that α ≤ φ̃(α) + ξ + θ ≤ ψ̃(β) + ξ + θ < β. This is a contradiction. (Lemma 7.3) A quartet p = 〈Xp, Y p, τ p, 〈φ : ξ ∈ Dp〉〉 (not necessarily an element of P0) with D p ∈ [ω2] ℵ0 is said to be an extendable condition if 〈Xp, Y p, τ p, φ ξ 〉 is a ξ-extendable semi-condition for all ξ ∈ Dp. For extendable conditions p, q with p = 〈Xp, Y p, τ p, 〈φ : ξ ∈ Dp〉〉, q = 〈Xq, Y q, τ q, 〈φ : ξ ∈ Dq〉〉, we denote p ≤1 q if (7.21) Xp ⊇ Xq, Y p ⊇ Y q, τ p ⊇ τ q, Dp ⊇ Dq, φ and ψ for all ξ ∈ Dp ; (7.22) τ p ↾ (Xp \Xq)× Y q ≡ 1. Note that for p, q ∈ P0, we have p ≤1 q if and only if p ≤P0 q. Lemma 7.4. (Extension Lemma) Suppose that p = 〈Xp, Y p, τ p, 〈φ : ξ ∈ Dp〉〉 is an extendable condition for some Dp ∈ [ω2] ℵ0. Then there is a q ∈ P0 with q = 〈Xq, Y q, τ q, 〈φ ξ : ξ ∈ D q〉〉 such that Dq = Dp and q ≤1 p. Furthermore , if p0 ∈ P0 is such that p ≤1 p0 then we have q ≤P0 p0. Proof. The second part of the lemma is clear once the condition q as in the claim of the lemma is found since (7.8e) holds for such q and p0 since the relation ≤1 is easily seen to be transitive. To construct the desired q ∈ P0, let 〈ξn : n ∈ ω〉 be an enumeration of D p such that each ξ ∈ Dp appears infinitely often in the enumeration. First, construct 〈Xn, Yn, τn, 〈φξ,n, ψξ,n : ξ ∈ D p〉〉, n ∈ ω recursively such that (7.23) 〈X0, Y0, τ0, 〈φξ,0, ψξ,0 : ξ ∈ D p〉〉 = p, (7.24) 〈Xn+1, Yn+1, τn+1, φξn,n+1, ψξn,n+1〉 is the ξn-extendable semi-condition which is constructed just as in Lemma 7.3 from the ξn-extendable semi-condition 〈Xn, Yn, τn, φξn,n, ψξn,n〉. (7.25) φξ,n+1 = φξ,n and ψξ,n+1 = ψξ,n for all ξ ∈ D p with ξ 6= ξn. Along with the recursive construction above, it can be shown easily that 〈Xn, Yn, τn, φξ,n, ψξ,n〉 is a ξ-extendable semi-condition for all n ∈ ω and ξ ∈ D p. Hence the construction in (7.24) is actually possible at each step. n∈ωXn , Y n∈ωXn , τ n∈ω τn , Dq = Dp and φ n∈ω φξ,n , ψ n∈ω ψξ,n for all ξ ∈ D For all ξ ∈ Dq, there are infinitely many n ∈ ω such that ξn = ξ. For such n, φξ,n is an involution on Xn and ψξ,n is an involution on Yn. It follows that φ ξ is an involution on Xq and ψ is an involution on Y q. Hence q = 〈Xq, Y q, τ q, 〈φ : ξ ∈ Dq〉〉 is a condition in P0. Also we have τ q ↾ (Xq \Xp)× Y p = n∈ω τ q ↾ (Xn+1 \Xn)× Y p ≡ 1. Thus, this q is as desired. (Lemma 7.4) Lemma 7.5. (CH) P0 satisfies the ℵ2-c.c. Proof. Actually we shall show that P0 satisfies a strong form of ℵ2-Knaster prop- erty. Suppose that pζ ∈ P0 with p ζ = 〈Xζ, Y ζ , τ ζ , 〈φ : ξ ∈ Dζ〉〉 for ζ ∈ ω2. By the ∆-System Lemma (Theorem 4.1) and the Pigeon Hole Principle, there are a stationary S ⊆ ω2, X , Y , D ∈ [ω2] ℵ0 , τ : X ×Y → 2 and φξ : X → X , ψξ : Y → Y for ξ ∈ D such that (7.26) Xζ, ζ ∈ S form a ∆-system with root X and Y ζ , ζ ∈ S form a ∆-system with root Y ; (7.27) τ ζ ↾ X × Y = τ for all ζ ∈ S ; (7.28) Dζ, ζ ∈ S form a ∆-system with root D ; (7.29) φ ↾ X = φξ and ψ ↾ Y = ψξ for all ζ ∈ S and ξ ∈ D ; (7.30) τ ζ ↾ (Xζ \X)× Y ≡ 1 for all ζ ∈ S . Note that (7.27) is possible since, by CH, there are at most |X×Y2 | ≤ 2ℵ0 = ℵ1 < ℵ2 many possible values of τ ζ ↾ X × Y . (7.29) is possible since, by CH, (7.4) and countability of D, there are at most ℵ1 possible values of 〈φ : ξ ∈ D〉 and : ξ ∈ D〉. (7.30) is possible by (7.7) and since we can choose S such that min(Xζ \X) > sup(Y ) for all ζ ∈ S. Now suppose ζ , η ∈ S with ζ < η. We show that pζ and pη are compatible. Let Xp = Xζ ∪Xη, Y p = Y ζ ∪ Y η and Dp = Dζ ∪Dη. For ξ ∈ Dp, let φ ξ : X p → Xp and ψ : Y p → Y p be defined by (7.31) φ , if ξ ∈ Dζ \D, , if ξ ∈ D, ξ , otherwise and ψ , if ξ ∈ Dζ \D, , if ξ ∈ D, ξ , otherwise. Finally let τ p : Xp × Y p → 2 be such that (7.32) τ p(α, β) = τ ζ(α, β), if 〈α, β〉 ∈ Xζ × Y ζ τ η(α, β), else if 〈α, β〉 ∈ Xη × Y η 1, otherwise. for all 〈α, β〉 ∈ Xp × Y p. It is easy to see that p = 〈Xp, Y p, τ p, 〈φ : ξ ∈ Dp〉〉 is an extendable condition and p ≤1 p ζ, pη. In particular, (7.22) for p ≤1 p ζ and p ≤1 p η holds because of (7.30) and “otherwise” clause of (7.32). By Extension Lemma (Lemma 7.4), there is a q ∈ P0 with q ≤1 p. Hence, by the second half of the lemma, it follows that q ≤P0 p ζ, pη. (Lemma 7.5) A modification of the ∆-system argument in the proof of Lemma 7.5 is also used to prove the following: Lemma 7.6. (CH) P0 forces (6.5). Proof. We show that P0 forces the following: (7.33) For any stationary S ⊆ Eω2ω1 and Aζ , Bζ ∈ [ω2] ℵ0 for ζ ∈ S, there is a stationary T ⊆ S such that for any n ∈ ω and pairwise distinct ζi, ηi ∈ T , i ∈ n, there is ξ < ω2 such that φ̇ξ ′′Aζi = Aηi and ψ̇ξ ′′Bζi = Bηi for all i ∈ n. Note that, by σ-closedness and ℵ2-c.c. of P0 (proved in Lemma 7.5), ω1 and ω2 in generic extensions by P0 remain ω1 and ω2. Suppose that Ṡ is a P0-name of a stationary subset of E . Let 〈Ȧζ : ζ ∈ Ṡ〉 and 〈Ḃζ : ζ ∈ Ṡ〉 be P0-names of sequences of countable subsets of ω2. Let S̃ = {ζ ∈ Eω2ω1 : /‖–P0 “ ζ 6∈ Ṡ ”}. Then we have ‖–P0 “ Ṡ ⊆ S̃ ” and hence S̃ is a stationary subset of E Since P0 is σ-closed, we can find pζ ∈ P0 and Aζ, Bζ ∈ [ω2] ℵ0 such that (7.34) pζ = 〈X ζ, Y ζ , τ ζ , 〈φ : ξ ∈ Dζ〉〉 and (7.35) pζ ‖– P0 “ ζ ∈ Ṡ , Ȧζ = Aζ and Ḃζ = Bζ ” for all ζ ∈ S̃. Without loss of generality, we may assume that (7.36) Aζ ⊆ X ζ and Bζ ⊆ Y By ∆-System Lemma (Theorem 4.1) and the Pigeon Hole Principle, there are a stationary S̃0 ⊆ S̃, X , Y , D ∈ [ω2] ℵ0 , τ : X × Y → 2 and φξ : X → X , ψξ : Y → Y for ξ ∈ D such that (7.37) Xζ, ζ ∈ S̃0 form a ∆-system with root X and Y ζ, ζ ∈ S̃0 form a ∆-system with root Y ; (7.38) sup(Y ) < min(Xζ \X) for all ζ ∈ S̃0 ; (7.39) τ ζ ↾ X × Y = τ for all ζ ∈ S̃0 ; (7.40) Dζ, ζ ∈ S̃0 form a ∆-system with root D ; (7.41) φ ↾ X = φξ and ψ ↾ Y = ψξ for all ζ ∈ S̃0 and ξ ∈ D ; (7.42) τ ζ ↾ (Xζ \X)× Y ≡ 1 for all ζ ∈ S̃0 (this follows from (7.38) and (7.7)) ; (7.43) Xζ, ζ ∈ S̃0 are order-isomorphic and Y ζ , ζ ∈ S̃0 are order-isomorphic; Note that the order-isomorphisms of Xζ ’s and Y ζ ’s do not move elements of X and Y , respectively. (7.44) the order-isomorphism sending Xζ to Xη sends τ ζ ↾ ((Xζ \ X) × Y ) to τ η ↾ ((Xη \X)× Y ) while the order-isomorphism sending Y ζ to Y η sends τ ζ ↾ (X × (Y ζ \ Y )) to τ η ↾ (X × (Y η \ Y )). These order-isomorphisms together send τ ζ ↾ ((Xζ \X)× (Y ζ \ Y )) to τ η ↾ ((Xη \X)× (Y η \ Y )); (7.45) the order-isomorphism sending Xζ to Xη sends Aζ to Aη, and the order- isomorphism sending Y ζ to Y η sends Bζ to Bη. Note that p̄ = 〈X, Y , τ , 〈φξ, ψξ : ξ ∈ D〉〉 is a condition in P0 and pζ ≤P0 p̄ for all ζ ∈ S̃0 (the condition (7.8e) for p̄ and pζ holds by (7.42)). Let Ṫ be a P0-name such that (7.46) ‖– P0 “ Ṫ = {ζ ∈ S̃0 : pζ ∈ Ġ } ” where Ġ is the standard P0-name of the generic set. Claim 7.6.1. p̄ ‖–P0 “ Ṫ is a stationary subset of ω2 ”. ⊢ Since P0 satisfies the ℵ2-c.c. by Lemma 7.5, for any P0-name Ċ of a club subset of ω2, there is a club subset C of ω2 (in the ground model) such that ‖–P0 “C ⊆ Ċ ”. Hence it is enough to show the following: (7.47) For any q ≤P0 p̄ and any club subset C of ω2, there are p ≤P0 q and ζ ∈ C ∩ S̃0 such that p ≤P0 pζ . To show (7.47), let q = 〈Xq, Y q, τ q, 〈φ ξ : ξ ∈ D q〉〉 and let ζ ∈ C ∩ S̃0 be such (7.48) (Xζ \X) ∩Xq = ∅ , (Y ζ \ Y ) ∩ Y q = ∅ and (Dζ \D) ∩Dq = ∅. This is possible by (7.37) and since C ∩ S̃0 is stationary. Let X∗ = Xq ∪Xζ, Y ∗ = Y q ∪ Y ζ , D∗ = Dq ∪Dζ. For ξ ∈ D∗, let φ∗ξ and ψ be partial functions from X∗ to X∗ and from Y ∗ to Y ∗ respectively defined by (7.49) φ∗ξ = , if ξ ∈ Dq \D, , if ξ ∈ D, , otherwise and ψ∗ξ = ψq, if ξ ∈ Dq \D, , if ξ ∈ D, , otherwise. Finally, let τ ∗ : X∗ × Y ∗ → 2 be defined by (7.50) τ ∗(α, β) = τ q(α, β), if 〈α, β〉 ∈ Xq × Y q, τ ζ(α, β), else if 〈α, β〉 ∈ Xζ × Y ζ , 1, otherwise for 〈α, β〉 ∈ X∗ × Y ∗. Then p∗ = 〈X∗, Y ∗, τ ∗, 〈φ∗ξ, ψ ξ : ξ ∈ D ∗〉〉 is an extendable condition and we have p∗ ≤1 q, pζ : (7.22) for p ∗ and pζ follows from τ ∗ ↾ ((X∗\Xζ)×Y ζ) = τ ∗ ↾ ((Xq \X)×(Y ζ \Y ))∪τ ∗ ↾ ((Xq \X)×Y ) ≡ 1 where we have τ ∗ ↾ ((Xq \X) × (Y ζ \ Y )) ≡ 1 by the definition (7.50) of τ ∗ and τ ∗ ↾ ((Xq \X) × Y ) ≡ 1 by q ≤P0 p̄ (in particular, by the condition (7.8e) in the definition of ≤P0). By Extension Lemma (Lemma 7.4) it follows that there is p ∈ P0 with p ≤1 p and hence p ≤P0 q, pζ . ⊣ (Claim 7.6.1) Claim 7.6.2. p̄ forces that Ṫ is as in (7.33) for 〈Ȧζ : ζ ∈ Ṡ〉 and 〈Ḃζ : ζ ∈ Ṡ〉. ⊢ By Claim 7.6.1 it is enough to prove the following: (7.51) For any q ≤P0 p̄ and n ∈ ω, if ζi, ηi ∈ S̃0 are pairwise distinct and q ‖– P0 “ ζi, ηi ∈ Ṫ for i ∈ n ”, then there is p ≤P0 q with p = 〈Xp, Y p, τ p, 〈φ : ξ ∈ Dp〉〉 and ξ0 ∈ D p such that φ ↾ Xζi : Xζi → Xηi and ψ ↾ Y ζi : Y ζi → Y ηi are order-isomorphisms for all i < n. Without loss of generality we may assume that (7.52) q ≤P0 pζi, pηi for all i < n. q = 〈Xq, Y q, τ q, 〈φ ξ : ξ ∈ D Take ξ0 ∈ ω2 \ (D q ∪ sup(Xq) ∪ sup(Y q)) and let D∗ = Dq ∪ {ξ0}. Let X ∗ = Xq, Y ∗ = Y q and τ ∗ = τ q. Let φ∗ξ0 : Xζi ∪ Xηi → Xζi ∪ Xηi be the involution sending Xζi order-isomorphically to Xηi and vice versa for all i < n and ψ∗ξ0 : Y ζi ∪ Y ηi → Y ζi ∪ Y ηi be the involution sending Y ζi order-isomorphically to Y ηi and vice versa for all i < n. Let φ∗ξ = φ and ψ∗ξ = ψ for ξ ∈ Dq. Then p∗ = 〈X∗, Y ∗, τ ∗, 〈φ∗ξ, ψ ξ : ξ ∈ D ∗〉〉 is an extendable condition with p∗ ≤1 q. To see this, we have to check 〈X ∗, Y ∗, τ ∗, φ∗ξ0, ψ 〉 satisfies (7.12) and (7.14). But this follows from (7.42), (7.52) and (7.45). By Extension Lemma (Lemma 7.4) there is p ∈ P0 with p ≤1 p ∗. Clearly p forces that ξ0 as above satisfies (7.33) together with Ṫ , 〈Ȧζ : ζ ∈ Ṡ〉 and 〈Ḃζ : ζ ∈ Ṡ〉. ⊣ (Claim 7.6.2) Since the argument above can be repeated below arbitrary element of P0, it follows that P0 forces (7.33). (Lemma 7.6) 8 A summary of consistency results and some open problems The following is a summary of consistency results in connection with the combina- torial principles in fig. 5 where (1) ∼ (7) below correspond to the separation lines (1) ∼ (7) drawn in fig. 9. (1): By adding random reals. More precisely, start from a model V of CH and force with (the positive elements of) the measure algebra B of, say, Maharam type ℵ2. B can be seen as a (measure theoretic) product of random forcing and inherits thus some of the homogeneity property of finite support product. This is used to prove do = ℵ1 in the generic extension. It is also well-known that the ground model functions from ω to ω dominate the functions in a generic extension by a measure algebra. Hence we have d = ℵ1 in the model. K. Kunen proved that there is a κ-Lusin gap for an uncountable κ in such a model. On the other hand, I. Juhász, L. Soukup and Z. Szentmiklóssy proved in [13] that there is no ℵ2-Lusin gap under Cs(ℵ2). This proves that C s(ℵ2) does not hold in the generic extension. This observation may be also interpreted as pinning down of the difference in the extent of homogeneity of product forcing and the forcing by measure theoretic products in terms of whether the principle Cs(ℵ2) holds. (2): A model constructed by J. Brendle and T. LaBerge in [1] realizes this separation. (3): By the model in Theorem 3.8 of I. Juhász and K. Kunen [12] in which Cs(ℵ2) and do > ℵ1 hold. The model is obtained by a finite support product of ℵ2 posets of cardinality ℵ1 starting from a model of CH. From this, it follows easily that b∗ = ℵ1 and d = ℵ2. (4): By adding Cohen reals. More exactly, start from a model V of CH and then add, say, ℵ2 Cohen reals (by Fn(ℵ2, 2)). Then by Corollary 4.8, (c) we have IP(ℵ2,ℵ1) in the generic extension. Just as in (3), we have d = ℵ2 in such a generic extension and it is shown in S. Fuchino, S. Koppelberg and S. Shelah [8] that WFN holds there. (5): By a model of Hechler. (6): By Theorem 6.4. (7): By Corollary 4.9. (8): By Theorem 5.2. See [11] for the proof of ‖– Sκ “¬WFN”. do = ℵ1 HP(ℵ2) h = ℵ1b ↑ = ℵ1b = ℵ1 b ∗ = ℵ1 Cs(ℵ2) d = ℵ1 (4)(5) IP(ℵ2,ℵ2) IP(ℵ2,ℵ1) fig. 9 Finally, we shall mention some open problems. In [5] it is shown that a = ℵ1 follows from WFN where a is the almost disjoint number. In [4], it is then shown that, under some additional assumptions, a = ℵ1 already follows from SEP which is a weakening of WFN. Therefore, it seems natural to ask the following question: Problem 1. Does a = ℵ1 follow from HP(ℵ2) or IP(ℵ2, λ) for λ = ℵ1,ℵ2 ? Problem 2. Does WFN imply HP(ℵ2) or do = ℵ1 ? The model of b∗ = ℵ2 and do = ℵ1 satisfies a strong form of negation of C s(ℵ2). This suggests the following problem: Problem 3. Does HP(ℵ2) (or even C s(ℵ2)) imply b ∗ = ℵ1 ? At the moment, we do not have any model separating HP(κ) and IP(κ, κ) for κ > κ1. Problem 4. Is HP(κ) + ¬IP(κ, κ) consistent for some (or any) κ > ℵ1 ? In Corollary 4.9 which realizes the separation (7) in fig.9, a very strong large cardinal property is assumed. Problem 5. Can we construct a model realizing (7) in fig.9 starting from ZFC without any large cardinal? The property (A) used in the proof of Theorem 6.4 and proved to be consistent with CH in Section 7 seems to be of its own interest. Problem 6. Is ¬(A) consistent with ZFC + CH (or even with ZFC + GCH) ? References [1] J. Brendle and T. LaBerge, Forcing tightness in products of fans, Fundamenta Mathematicae, 150, 211-226 (1996). [2] K. Ciesielski and J. Pawlikowski, The Covering Property Axiom, CPA, Cam- bridge University Press (2004). [3] K. Eda, M. Kada and Y. Yuasa, The tightness about sequential fans and combinatorial properties, Journal of Mathematical Society of Japan 49, 181– 197 (1997). [4] S. Fuchino and S. Geschke, Some combinatorial principles defined in terms of elementary submodels, Fundamenta Mathematicae 181, 233-255 (2004). [5] S. Fuchino, S. Geschke and L. Soukup, On the weak Freese-Nation property of P(ω), Archive for Mathematical Logic, Vol.40, 425-435 (2001). [6] S. Fuchino, S. Geschke and L. Soukup, Principles capturing features of generic extensions by almost side-by-side product, in preparation. [7] S. Fuchino, S. Geschke, S. Shelah and L. Soukup, On the weak Freese-Nation property of complete Boolean algebras, Annals of Pure and Applied Logic, 110 (1-3) (2001) 89-105. [8] S. Fuchino, S. Koppelberg and S. Shelah, Partial orderings with the weak Freese-Nation property, Annals of Pure and Applied Logic 80, 35–54 (1996). [9] S. Fuchino, S. Shelah and L. Soukup, Sticks and clubs, Annals of Pure and Applied Logic 90, no.1 (1997), 57-77. [10] S. Fuchino and S. Soukup, More set-theory around the weak Freese-Nation property, Fundamenta Matematicae 154, 159–176 (1997). [11] S. Geschke, On σ-Filtered Boolean Algebras, Dissertation, Freie Universität Berlin (1999). [12] I. Juhász and K. Kunen, The Power Set of ω, Elementary submodels and weakenings of CH, Fundamenta Mathematicae 170, 257–265 (2001). [13] I. Juhász, L. Soukup and Z. Szentmiklóssy, Combinatorial principles from adding Cohen reals, Proceedings of Logic Colloquium (1995). [14] M. Kada and Y. Yuasa, Cardinal invariants about shrinkability of unbounded sets, Topology and its Applications 74 (1996), 215–223. [15] K. Kunen: Set Theory, North-Holland (1980). [16] A. Miller, Infinite Combinatorics and Definability, Annals of Pure and Ap- plied Mathematical Logic, 41 (1989), 179-203. [17] S. Shelah, a(n unpublished?) note for I. Juhász (2002). Authors’ addresses Jörg Brendle Graduate School of Science and Technology Kobe University Rokko-dai 1-1, Nada KOBE 657-8501 Japan brendle@kurt.scitec.kobe-u.ac.jp Sakaé Fuchino Dept. of Natural Science and Mathematics College of Engineering, Chubu University, Kasugai AICHI 487-8501 Japan. fuchino@isc.chubu.ac.jp

We study combinatorial principles we call Homogeneity Principle HP(\kappa) and Injectivity Principle IP(\kappa,\lambda) for regular \kappa>\aleph_1 and \lambda\leq\kappa which are formulated in terms of coloring the ordinals <\kappa by reals.

Introduction The Cohen model which is obtained by adding at least ℵ2 Cohen reals over a model of GCH was the first and simplest model for the negation of CH, and it is still one of the most important. A plethora of statements have been shown to be consistent with ZFC by adjoining Cohen reals, and it is therefore natural to look for axioms which hold in the Cohen model and from which many such statements can be decided, that is, axioms which capture as much as possible of the combinatorial structure of the Cohen extension. Something similar has been done for the iterated Sacks model by Ciesielski and Pawlikowski who devised the Covering Property Axiom CPA [2]. For Cohen models, several such axioms have been proposed in the past. Some of them are homogeneity type statements, that is, they assert that given at least ω2 many reals, many of them “look similar”. Examples are the combinatorial principles Cs(κ), Ĉs(κ), and Fs(κ) introduced by I. Juhász, L. Soukup and Z. Szentmiklóssy [13] who showed they hold in Cohen models (see Section 2 below for definitions). On the other hand, rather different-looking statements have been also investi- gated in connection with Cohen models, for example, the axiom WFN asserting that 〈P(ω),⊆〉 has the weak Freese-Nation Property (see [8], [10] and [5]). Here a partial ordering 〈P,≤〉 has the weak Freese-Nation Property if there is a mapping f : P → [P ]ℵ0 such that for all p, q ∈ P , p ≤P q holds if and only if there is an r ∈ f(p) ∩ f(q) such that p ≤P r ≤P q. In [8], it is shown that WFN holds in a Cohen model for adding ℵn Cohen reals for any n < ω. If we start e.g. from V = L then WFN holds even after adding any number of Cohen reals ([10]). In [5], it was shown that WFN implies many of the known combinatorial properties of Cohen models and so it may be seen as an axiomatization of the combinatorial structure of the Cohen extension. Since WFN can be reformulated in terms of elementary submodels, WFN as well as some closely related statements have come to be known as elementary submodel type axioms (see [12] for this). At first glance it seemed that there would be no connection between these two types of axioms except that they both hold in a Cohen model. Surprisingly enough though S. Shelah [17] showed that Cs(ℵ2) follows from the combinatorial principle he called Princ, which is a consequence of WFN. The proof can be easily recast to show that WFN implies Cs(κ) for all regular κ > ℵ1 (see [6] for more details). In this paper, we introduce some new principles of the homogeneity type, namely, the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ, λ) which are formulated in terms of homogeneity of colorings of ordinals below the cardinal κ by reals. We establish that these axioms hold in Cohen models and address the question in which other models these axioms hold as well. It turns out that, in fact, these principles seem to capture a good deal of the combinatorial features of models of set theory obtained by forcing by the side-by-side (finite or countable support) product of copies of a fixed relatively small partial ordering (see Theorem 4.3 and Corollary 4.8). Though the relation of these principles to WFN is not yet completely clear, our principles imply the principles of I. Juhász, L. Soukup and Z. Szentmiklóssy (Theorem 2.7) and thus can be seen as natural strengthenings of these principles. Our paper is organized as follows. In Section 2, we review the principles Cs(κ), Ĉs(κ) and Fs(κ) of I. Juhász, L. Soukup and Z. Szentmiklóssy, and introduce our principles HP(κ) and IP(κ, λ). Some basic facts in ZFC concerning these principles are also proved. In particular, we show that Cs(κ) and Ĉs(κ) follow from HP(κ) (Theorem 2.7), Fs(κ) follows from IP(κ,ℵ1) (Theorem 2.8) and HP(κ) follows from IP(κ, κ) (Theorem 2.9). After reviewing some cardinal invariants introduced in [6] which are variants of the bounding number b and the shrinking number b∗ in [3], we study in Section 3 the effect of the combinatorial principles Cs(κ), Ĉs(κ) and HP(κ) on the values of these cardinal invariants. In Section 4 we give a forcing construction of models of IP(κ, λ) (Theorem 4.3) and its applications (Corollary 4.8). The results in this section improve consistency results in [13]. As a further application of Theorem 4.3 we show in Section 5 the consistency of ¬IP(ℵ2,ℵ1) and IP(ℵ2,ℵ2). One of the consequences of HP(ℵ2) discussed in Section 3 is that there is no definable well-ordering of length ω2 on any subset of ωω (or do = ℵ1 in our notation). Refining a forcing extension of Brendle and LaBerge [1], we prove in Section 6 the consistency of do = ℵ1 with b ∗ = ℵ2 (Theorem 6.4). We also show that the model of do = ℵ1 and b ∗ = ℵ2 we construct in this section satisfies a strong negation of Cs(ℵ2). Section 7 is devoted to the consistency proof of the combinatorial principle used in the proof of Theorem 6.4. In Section 8, we summarize the consistency results obtained in this paper to- gether with other consistency results established by some previously known con- structions. We discuss also some open problems at the end of the section. 2 Combinatorial principles formulated in terms of coloring of ordinals by reals For any set X , let (2.1) ((X)) = {~x ∈ Xn : ~x is injective} and (2.2) ((X)) ((X)) Likewise, for any sets X0,...,Xn−1, let (2.3) ((X0, ...,Xn−1)) = {~x ∈ X0 × · · · ×Xn−1 : ~x is injective}. For a cardinal κ, the following principle Cs(κ) was introduced by I. Juhász, L. Soukup and Z. Szentmiklóssy in [13]. Cs(κ): For any matrix 〈aα,n : α ∈ κ, n ∈ ω〉 of subsets of ω and T ⊆ ω>ω, at least one of the following holds: (c0) there is a stationary S ⊆ κ such that n<|t| aαn,t(n) 6= ∅ for all t ∈ T and 〈α0, ...,α|t|−1〉 ∈ ((S)) (c1) there exist t ∈ T and stationary S0,...,S|t|−1 ⊆ κ such that n<|t| aαn,t(n) = ∅ for all 〈α0, ...,α|t|−1〉 ∈ ((S0, ...,S|t|−1)). For any cardinal κ it is easy to see that Cs(κ) holds if and only if Cs(cf κ) holds. Thus it is enough to consider Cs(κ) for regular κ. The corresponding assertion is also true for other combinatorial principles we are going to introduce in this section. Hence, in the rest of this section, we shall assume that κ is a regular cardinal unless mentioned otherwise. The combinatorial principle Ĉs(κ), a sort of dual of the principle Cs(κ), is also considered in [13]: Ĉs(κ): For any T ⊆ ω<ω and any matrix 〈aα,n : α < κ, n ∈ ω〉 of subsets of ω, at least one of the following holds: (ĉ0) there is a stationary S ⊆ κ such that | n<|t| aαn,t(n) | < ℵ0 for every t ∈ T and 〈α0, ...,α| t |−1〉 ∈ ((S)) (ĉ1) there exist t ∈ T and stationary S0,...,S|t|−1 ⊆ κ such that | n<|t| aαn,t(n) | = ℵ0 for every 〈α0, ...,α| t |−1〉 ∈ ((S0, ...,S|t|−1)). The following is easily seen: Lemma 2.1. (I. Juhász, L. Soukup and Z. Szentmiklóssy [13]) (a) Neither of Cs(ℵ1) and Ĉ s(ℵ1) holds. (b) Cs(κ) and Ĉs(κ) hold for any regular κ > 2ℵ0. Let us call a subset A of H(ℵ1) definable if there are a formula ϕ and a ∈ H(ℵ1) such that A = {x ∈ H(ℵ1) : 〈H(ℵ1),∈〉 |= ϕ(x, a)}. Note that for any n ∈ ω, A ⊆ Rn is projective if and only if it is definable in our sense. Note also since all elements of H(ℵ1) can be coded by elements of ωω we may assume that a as above is an element of ωω. In Theorem 2.7, we show that the following Homogeneity Principle HP(κ) im- plies both of Cs(κ) and Ĉs(κ). HP(κ): For any f : κ→ P(ω) and any definable A ⊆ ((P(ω))) , at least one of the following holds: (h0) there is a stationary S ⊆ κ such that ((f ′′S)) \ {∅} ⊆ A; (h1) there are k ∈ ω \ 1 and stationary S0,...,Sk−1 ⊆ κ such that ((f ′′S0, ..., f ′′Sk−1)) ∩ A = ∅. Note that P(ω) in the definition of HP(κ) above can be replaced by R, ωω, (P(ω))n or (ωω)n etc. since these spaces can be coded as definable subsets of P(ω) and vice versa. As for Cs(κ) (and Ĉs(κ)), it is enough to consider HP(κ) for regular κ. Lemma 2.1 is also true for HP(κ): Lemma 2.2. (a) HP(ℵ1) does not hold. (b) HP(κ) holds for any regular κ > 2ℵ0. Proof. (a): This follows from Lemma 2.1 and Theorem 2.7. (b): Let κ > 2ℵ0 be a regular cardinal. Suppose that f : κ → P(ω) and A are as in the definition of HP(κ). Then there is a stationary S ⊆ κ such that f ↾ S is constant. If (h0) in the definition of HP(κ) does not hold then we must have ((f ′′S)) ∩A = ∅ since ((f ′′S)) = ∅ for all n > 1. Hence (h1) holds with n = 1 and S0 = S. (Lemma 2.2) The following combinatorial principle Fs(κ) is also introduced in [13]: Fs(κ): For any T ⊆ ω<ω and any matrix 〈aα,n : α < κ, n ∈ ω〉 of subsets of ω, at least one of the following holds: (f 0) there is a stationary S ⊆ κ such that n<|t|aαn,t(n) : t ∈ T and 〈α0, ...,α| t |−1〉 ∈ ((S)) } | ≤ ℵ0 ; (f 1) there are t ∈ T and stationary S0,...,S|t|−1 ⊆ κ such that for every 〈α0, ...,α| t |−1〉, 〈β0, ...,β| t |−1〉 ∈ ((S0, ...,S|t|−1)), if αn 6= βn for all n < |t|, then n<|t| aαn,t(n) 6= n<|t| aβn,t(n). Lemma 2.3. (I. Juhász, L. Soukup and Z. Szentmiklóssy [13]) (a) Fs(ℵ1) does not hold. (b) Fs(κ) holds for every regular κ > 2ℵ0. (c) Fs(κ) implies Ĉs(κ). A combinatorial principle in terms of coloring of ordinals by reals corresponding naturally to Fs(κ) might be the following Injectivity Principle IP(κ, λ) for cardinals κ and λ with λ ≤ κ: IP(κ, λ): For any f : κ→ P(ω) and definable g : ((P(ω))) → P(ω), at least one of the following holds: (i0) there is a stationary S ⊆ κ such that | g ′′((f ′′S)) | < λ for every n ∈ ω; (i1) there are k ∈ ω\1 and stationary S0,...,Sk−1 ⊆ κ such that for any 〈x0, ...,xk−1〉, 〈y0, ..., yk−1〉 ∈ ((f ′′S0, ..., f ′′Sk−1)), if xn 6= yn for all n < k, then we have g(x0, ..., xk−1) 6= g(y0, ..., yk−1). Again here, we may replace P(ω) in the definition of IP(κ, κ) above by R, ωω, (P(ω))n or (ωω)n etc. Lemma 2.4. (a) For λ ≤ λ′ ≤ κ, IP(κ, λ) implies IP(κ, λ′). (b) IP(ℵ1,ℵ1) does not hold. Proof. (a): Immediate from the definition. (b): By Lemma 2.2, (a) and Theorem 2.9. (Lemma 2.4) IP(κ,ℵ0) for a regular cardinal κ is equivalent to the cardinal inequality 2 ℵ0 < κ. Proposition 2.5. For a regular cardinal κ the following are equivalent: (a) IP(κ,ℵ0) holds; (b) 2 ℵ0 < κ; (c) IP(κ, 2) holds. Proof. (a) ⇒ (b): Suppose that 2ℵ0 ≥ κ. We show that IP(κ,ℵ0) does not hold. Let f : κ → P(ω) be any injective mapping and g : ((P(ω))) → P(ω) be defined by g(∅) = ∅, g(〈x〉) = ∅ for all x ∈ P(ω) and g(〈x0, ..., xn−1〉) = min{m ∈ ω : m ∈ x0 6↔ m ∈ x1} for 〈x0, ..., xn−1〉 ∈ ((P(ω))) with n ≥ 2. Let S be any stationary subset of κ. Then | g ′′((f ′′S)) | ≥ ℵ0: Suppose not and let k ∈ ω be such that g ′′((f ′′S)) Since P(k + 1) is finite, there are α, β ∈ S, α 6= β such that f(α) ∩ (k + 1) = f(β) ∩ (k + 1). But then, by definition of g, it follows that g(〈f(α), f(β)〉) > k. This is a contradiction. Thus (i0) does not hold for these f and g. On the other hand, for arbitrary stationary subsets S0,...,Sn−1 of κ, as there are only countably many values of g, we can find 〈x0, ..., xn−1〉, 〈y0, ..., yn−1〉 ∈ ((f ′′S0, ..., f ′′Sn−1)) such that xi 6= yi for all i < n and g(〈x0, ..., xn−1〉) = g(〈y0, ..., yn−1〉). Thus (i1) neither holds. (b) ⇒ (c): Suppose 2ℵ0 < κ. For f : κ → P(ω) and g : ((P(ω))) → P(ω) as in the definition of IP(κ, 2), there is a stationary S ⊆ κ such that f is constant on S. This S witnesses that (i0) holds. (c) ⇒ (a): This follows from Lemma 2.4, (a). (Proposition 2.5) Corollary 2.6. IP(ℵ2,ℵ0) is equivalent to CH. IP(ℵ2,ℵ1) and IP(ℵ2,ℵ2) are thus the first two non-trivial instances of IP(κ, λ). For κ ≥ ℵ2, the principles introduced in this section and some other principles discussed in [6] can be put together in the following diagram: IP(κ,ℵ1) Fs(κ) HP(κ) Ĉs(κ) Cs(κ) Princ(κ, κ) SEP(κ, κ) Theorem 2.8 Theorem 2.9 Theorem 2.7 [13] [17] (see also [6]) IP(κ, κ) Theorem 2.7 fig. 1 In the rest of the section, we shall prove the implications indicated by the thick arrows in fig.1. Theorem 2.7. For a regular cardinal κ, HP(κ) implies both Cs(κ) and Ĉs(κ). Proof. We prove that HP(κ) implies Cs(κ). The other implication can be proved similarly. By Lemma 2.1, (b), we may assume that κ ≤ 2ℵ0. Let 〈ti : i ∈ ω〉 be an enumeration of ω>ω such that (2.4) | ti | ≤ i for all i < ω and let ι : P(ω) → P(ω)ω be a definable bijection. For each x ∈ P(ω) and i < ω, let (x)i denote the i’th component of ι(x). Suppose that T ⊆ ω>ω and A = 〈aα,n : α < κ, n ∈ ω〉 is a matrix of subsets of ω. We show that either (c0) or (c1) holds for these A and T . Let g : κ → P(ω) be a fixed injective mapping which exists by κ ≤ 2ℵ0 . Let f : κ→ P(ω) be defined by (2.5) f(α) = ι−1(〈a′α,n : n ∈ ω〉) where (2.6) a′α,n = g(α), if n = 0, aα,n−1, otherwise. Note that f is injective by “if n = 0” clause of (2.6). For i < ω, let (2.7) A∗i = {〈x0, ..., xi−1〉 ∈ ((P(ω))) n<| ti | (xn)ti(n)+1 6= ∅}, if ti ∈ T, ((P(ω))) , otherwise (2.8) A = A∗i . It is easy to see that A is definable noting that T ∈ H(ℵ1) and hence T can be used as a parameter in the definition of A. By HP(κ), we have either (h0) or (h1) for these A and f . Assume first that (h0) holds. Then there is a stationary S ⊆ κ such that ((f ′′S)) \ {∅} ⊆ A. We show that this S witnesses (c0) for T and A: for t ∈ T , let i ∈ ω be such that t = ti. By (2.4), we have | t | ≤ i. For s ∈ ((S)) | t | , let s′ ∈ ((S)) be an end-extension of s. Then 〈f(s′(0)), ..., f(s′(i−1))〉 ∈ ((f ′′S)) since f is injective. Hence 〈f(s′(0)), ..., f(s′(i − 1))〉 ∈ A∗i by the assumption on S. By (2.7), we have n<| ti | (f(s′(n)))ti(n)+1 = n<| ti | s′(n),ti(n)+1 n<| t | as(n),t(n). Thus T and A satisfy (c0). Assume now that (h1) holds. In this case, there are i ∈ ω and stationary S0,...,Si−1 ⊆ κ such that (2.9) ((f ′′S0, ..., f ′′Si−1)) ⊆ ((P(ω))) \ A∗i . Let t = ti. Then t ∈ T by (2.9) and “otherwise” clause of (2.7). For s ∈ ((S0, ...,S|t|−1)), let s ′ ∈ ((S0, ...,Si−1)) be an end extension of s. Then we have 〈f(s′(0)), ..., f(s′(i− 1))〉 ∈ ((f ′′S0, ..., f ′′Si−1)). It follows that 〈f(s′(0)), ..., f(s′(i− 1))〉 ∈ ((P(ω))) \ A∗i by (2.9). Hence, by (2.7), we have n<| t |(f(s ′(n)))t(n)+1 = n<| ti | s′(n),t(n)+1 n<| t | as(n),t(n). Thus, T and A satisfy (c1) in this case. The proof of Ĉs(κ) from HP(κ) is exactly like the proof above with (2.7) replaced (2.7) A∗i = {〈x0, ..., xi−1〉 ∈ ((P(ω))) n<| ti | (xn)ti(n)+1 | < ℵ0}, if ti ∈ T, ((P(ω))) , otherwise. (Theorem 2.7) HP(κ) also imply other variants of Cs(κ). For example, let ∗Cs(κ): For any matrix 〈aα,n : α ∈ κ, n ∈ ω〉 of subsets of ω and T ⊆ at least one of the following holds: (∗c0) there is a stationary S ⊆ κ such that n<|t| aαn,t(n) is infinite for all t ∈ T and 〈α0, ...,α|t|−1〉 ∈ ((S)) (∗c1) there exist t ∈ T and stationary S0,...,S|t|−1 ⊆ κ such that n<|t| aαn,t(n) is finite for all 〈α0, ...,α|t|−1〉 ∈ ((S0, ...,S|t|−1)). It is easy to see by a proof similar to that of Theorem 2.7 that HP(κ) implies ∗Cs(κ) as well. The following can also be proved similarly to Theorem 2.7. Theorem 2.8. IP(κ,ℵ1) implies F s(κ). Theorem 2.9. IP(κ, κ) implies HP(κ). Proof. Suppose that A ⊆ ((P(ω))) is definable and f : κ → P(ω). If f−1[{x}] is stationary for some x ∈ P(ω), then, either (h0) holds for S = f−1[{x}] or (h1) holds for n = 1 and S0 = f −1[{x}] depending on whether x ∈ A or not. Otherwise let g : ((P(ω))) → P(ω) be defined by (2.10) g(∅) = ∅; (2.11) g(〈x〉) = ∅ for all x ∈ P(ω) and (2.12) g(〈x0, ..., xn−1, x〉) = ∅, if 〈x0, ..., xn−1〉 ∈ A, x otherwise for all 〈x0, ..., xn−1, x〉 ∈ ((P(ω))) . If (i0) holds for this g with S as in (i0), then, by (2.12), we must have g ′′((f ′′S)) = {∅}. Hence ((f ′′S)) \ {∅} ⊆ A. On the other hand, if (i1) holds for some n < ω and S0 ,...,Sn−1, then we should have n ≥ 2 by (2.11) and g(〈x0 , ..., xn−2, xn−1〉) = xn−1 for all xi ∈ f ′′Si, i < n by (2.12). It follows that ((f ′′S0, ..., f ′′Sn−2)) ⊆ ((P(ω))) \ A by (2.12). (Theorem 2.9) 3 The bounding number and its variations In this section, we show that the combinatorial principles introduced in the last section make some of the cardinal invariants from [6] small. Adopting the notation of [6], we consider the following spectra of cardinal num- bers in connection with a partial ordering 〈P,≤〉; unbounded spectrum, hereditary unbounded spectrum and the spectrum of length of P : S(P ) = {|X | : X ⊆ P,X is unbounded in P, ∀B ∈ [X ]<|X |(B is bounded in P )} , h(P ) = {|X | : X ⊆ P, ∀B ⊆ X (B is bounded in P ↔ |B | < |X |)} , ↑(P ) = {cf(C) : C ⊆ P, C is an unbounded chain} . Clearly, we have (3.1) S↑(P ) ⊆ Sh(P ) ⊆ S(P ). For P = 〈ωω,≤∗〉, we shall simply write S↑, Sh and S in place of S↑(〈ωω,≤∗〉), h(〈ωω,≤∗〉) and S(〈ωω,≤∗〉), respectively. Recall that the bounding number b is defined by b = min{|X | : X ⊆ ωω is unbounded with respect to ≤∗}. The variant b∗ of b was introduced and studied in [3] and [14] where ∗ = min{κ : ∀X ⊆ ωω X is unbounded → ∃X ′ ∈ [X ]≤κ(X ′ is unbounded) b and b∗ can be characterized in terms of S↑, Sh and S as follows: Lemma 3.1. (a) b = minS↑ = minSh = minS. (b) b∗ = supS. In analogy to Lemma 3.1, (b), let (3.2) b↑ = supS↑, bh = supSh. Recall also that the dominating number d is defined as: d = min{|X | : X ⊆ ωω, X dominates ωω}. By (3.1) and Lemma 3.1, we have Lemma 3.2. b ≤ b↑ ≤ bh ≤ b∗ ≤ d. fig. 2 DO = {cf(otp(〈X,R ↾ X〉)) : X ⊆ ωω, R is a definable binary relation and R ∩X2 well orders X} do = supDO. By definition, S↑ ⊆ DO. Hence Lemma 3.3. b↑ ≤ do. Lemma 3.2 and Lemma 3.3 may be put together into the following diagram: b ≤ b↑ ≤ bh ≤ b∗ ≤ d fig. 3 If S↑ has a maximal element then we have b↑ = maxS↑. In such case we shall say that b↑ is attained. Also we shall say that b∗, bh or do is attained if the corresponding set has a maximal element. In the following, Reg denotes the class of regular cardinals. The following lemma can be proved similarly to Lemma 3.7, (c). Lemma 3.4. ([6]) Sh ∩ Reg ⊆ DO. Corollary 3.5. If Sh ∩ Reg is cofinal in Sh then bh ≤ do. Note that the condition “Sh ∩ Reg is cofinal in Sh” holds if 2ℵ0 < ℵω or if b is regular and attained. Under this condition, we can thus improve the diagram in fig.3 to the following: b ≤ b↑ ≤ bh ≤ b∗ ≤ d fig. 4 For an ideal I over a set X , non(I) and cov(I) denote, as usual, the uniformity and the covering number of I, respectively. More exactly non(I) = min{|A | : A ∈ P(X) \ I} and cov(I) = min{|A | : A ⊆ I, A = X}. meager and null denote the ideal of meager sets and the ideal of null sets (over R) respectively. Lemma 3.6. Suppose that I is an ideal over R with Borel basis. Then we have min{non(I), cov(I)} ≤ do. In particular, we have min{non(meager), cov(meager)} ≤ do and min{non(null), cov(null)} ≤ do. Proof. Suppose that I ⊆ P(R) is an ideal with a Borel basis and κ = min{non(I), cov(I)}. We can construct inductively a sequence 〈〈fα, gα〉 : α < κ〉 such that (3.3) fα, gα ∈ ωω for all α < κ; (3.4) gα codes a Borel set Xα ⊆ ωω such that Xα ∈ I and {fβ : β < α} ⊆ Xα ; (3.5) fα 6∈ Xβ for all α < κ. Note that (3.4) is possible by κ ≤ non(I) and (3.5) by κ ≤ cov(I). The sequence 〈〈fα, gα〉 : α < κ〉 is well ordered in order type κ by the definable ordering: 〈f ′, g′〉 ≤ 〈f, g〉 ⇔ f ′ is an element of the Borel set coded by g. It follows that κ ≤ do. (Lemma 3.6) The following lemma shows the relations of cardinal numbers b, b↑, bh, do to the combinatorial principles introduced in Section 2. Lemma 3.7. (a) (I. Juhász, L. Soukup and Z. Szentmiklóssy [13]) If there is a ≤∗-chain of length κ then ¬Cs(κ) and ¬Ĉs(κ). In particular, κ ∈ S↑ implies ¬Cs(κ) and ¬Ĉs(κ). (b) Cs(κ) (or Ĉs(κ)) implies b↑ ≤ κ. If b↑ is attained then Cs(κ) (or Ĉs(κ)) implies b↑ < κ. (c) If κ ≤ λ for some λ ∈ Sh with cf λ ≥ κ then ¬Cs(κ) and ¬Ĉs(κ). (d) If Sh ∩ Reg is cofinal in Sh then Cs(κ) (or Ĉs(κ)) implies bh ≤ κ. If bh is regular and attained then Cs(κ) (or Ĉs(κ)) implies bh < κ. (e) κ ∈ DO implies ¬HP(κ). (f) HP(κ) implies do ≤ κ. If do is attained then HP(κ) implies do < κ. Proof. (a): See [13]. (b): This follows from (a). (c): Suppose that κ ≤ λ ∈ Sh and κ ≤ cf λ. We show ¬Cs(κ). ¬Ĉs(κ) can be proved similarly from these assumptions. Let X ⊆ ωω with |X | = λ be as in the definition of Sh. Then we can find fα ∈ X and gα ∈ ωω for α < κ such that (3.6) fα ≤ ∗ gβ for all α < β < κ; (3.7) fβ 6≤ ∗ g+α for all α ≤ β < κ where g α is defined by g α (k) = gα(k) + 1 for all k ∈ ω. Note that (3.7) is possible since cf(|X |) ≥ κ. For α < κ, let gα,n ∈ ωω, n ∈ ω be such that (3.8) {gα,n : n ∈ ω} = {g ∈ ωω : g =∗ gα}. (3.9) aα,0 = {〈k, ℓ〉 ∈ ω 2 : ℓ ≤ fα(k)} and (3.10) aα,n+1 = {〈k, ℓ〉 ∈ ω 2 : ℓ > gα,n(k)} for all n ∈ ω. We show that A = 〈aα,n : α ∈ κ, n ∈ ω〉 with T = {〈0, n〉 : n ∈ ω \ 1} is a counter-example to Cs(κ). Suppose first that S ⊆ κ is stationary. For any α ∈ S, let β ∈ S be such that α < β. Then we have fα ≤ ∗ gβ by (3.6). Hence there is n ∈ ω such that fα ≤ gβ,n. By (3.9) and (3.10), it follows that aα,0 ∩ aβ,n+1 = ∅. This shows that 〈A, T 〉 6|= (c0). Suppose now that S0, S1 ⊆ κ are stationary and 〈0, n〉 ∈ T . By the definition of T , it follows that n ∈ ω\1. Let α ∈ S0 and β ∈ S1 be such that β < α. Then, by (3.7), we have fα 6≤ . Thus, by (3.9) and (3.10), it follows that aα,0 ∩ aβ,n 6= ∅. This shows that 〈A, T 〉 6|= (c1). (d): This follows easily from (c). (e): Suppose that κ ∈ DO and let 〈X,R〉 be such that X ⊆ P(ω), R is a projective binary relation and otp(〈X,R ∩ X2〉) = κ. Let f : κ → P(ω) be the mapping sending α < κ to the α’th element of X with respect to R. Let A = R ∪ k∈ω\{2}((P(ω))) Then it is easily seen that 〈f, A〉 6|= (h0) and 〈f, A〉 6|= (h1). (f): This follows from (d) since DO is downward closed. (Lemma 3.7) Corollary 3.8. (a) HP(κ) implies min{non(I), cov(I)} ≤ κ for any ideal I over R with Borel basis. In particular, it implies min{non(meager), cov(meager)} ≤ κ and min{non(null), cov(null)} ≤ κ. (b) If do is attained then HP(κ) implies min{non(I), cov(I)} < κ for all any I over R with Borel basis. In particular, it implies min{non(meager), cov(meager)} < κ and min{non(null), cov(null)} < κ. Proof. By Lemma 3.6 and Lemma 3.7, (f). (Corollary 3.8) Corollary 3.9. (a) Cs(ℵ2) (or Ĉ s(ℵ2)) implies b h = ℵ1. (b) HP(ℵ2) implies do = min{non(meager), cov(meager)} = min{non(null), cov(null)} = ℵ1. Proof. (a): By Lemma 3.7, (d). (b): By Lemma 3.7, (f) and Corollary 3.8. (Corollary 3.9) do = ℵ1 HP(ℵ2) IP(ℵ2,ℵ1) h = ℵ1b ↑ = ℵ1b = ℵ1 b ∗ = ℵ1 Cs(ℵ2) Ĉs(ℵ2) Fs(ℵ2) Corollary 3.9,(b) Corollary 3.9,(a) fig. 5 4 A forcing construction of models of IP(κ, λ) In this section, we shall prove that IP(κ, λ) holds in a generic extension by a homogeneous product of copies of a relatively small partial ordering (Theorem 4.3). Let us begin with definition of some notions needed for precise formulation of the theorem. For cardinals κ and µ, κ is said to be µ-inaccessible if κ is regular and λµ < κ holds for all λ < κ. Similarly, we say that κ is <µ-inaccessible if κ is regular and λ<µ < κ holds for all λ < κ. Thus, if µ is a successor cardinal, say µ = µ+0 , then κ is <µ-inaccessible if and only if κ is µ0-inaccessible. In our context, <µ-inaccessibility is relevant because of the following variant of the ∆-System Lemma of Erdős and Rado. For cardinals µ < κ, let Eκ≥µ = {α < κ : cf(α) ≥ µ} and let Eκµ , E ≤µ etc. be defined analogously. Theorem 4.1. (P. Erdős and R. Rado, see [13]) Suppose that κ is <µ-inaccessible and S ⊆ Eκ≥µ is stationary in κ. For any sequence 〈xα : α ∈ S〉 of sets of cardinality < µ there is a stationary S∗ ⊆ S such that 〈xα : α ∈ S ∗〉 form a ∆-system. For a sequence Pα, α < δ of posets and an ideal I ⊆ P(δ), we consider the I-support product Pα of Pα, α < δ here as being defined as (4.1) Pα = {f : f : D → Pα for some D ∈ I and f(α) ∈ Pα \ {1lPα} for all α ∈ D } with the ordering (4.2) f ≤QI g ⇔ dom(f) ⊇ dom(g) and f(α) ≤Pα g(α) for all α ∈ dom(g) for all f , g ∈ Pα. In particular, 1lQI = ∅ is the largest element of with respect to ≤QI α<δ Pα Though this definition of product of posets is different from the standard one, it gives a poset forcing equivalent to the product given by the standard definition. The present definition is chosen here for the sake of smoother treatment of p ↾ X , P ↾ X , G ↾ X etc. (see (4.5), (4.7) etc.) As usual, the ideal [δ]<ℵ0 is denoted by fin and Pα is called the finite support product of Pα, α < δ. I. Juhász and K. Kunen [12] proved the following theorem for µ = ℵ1 and I = [δ]<ℵ0 . Their proof also applies to the following slight generalization. Theorem 4.2. (I. Juhász and K. Kunen [12]) Suppose that P = Pα for some ideal I ⊆ P(δ), P satisfies the µ-c.c. and |Pα | ≤ 2 <µ for all α < δ. Then, for all <µ-inaccessible κ we have ‖– P “C s(κ) ”. Suppose that I ⊆ P(δ) is an ideal and P = Pα is an I-support product of posets Pα, α < δ. For p ∈ P, the support supp(p) of p is defined by (4.3) supp(p) = dom(p). We assume in the following that P-names are constructed just as in [15]. For a P-name ȧ, the support supp(ȧ) is defined by (4.4) supp(ȧ) = {supp(p) : 〈ḃ, p〉 ∈ tcl(ȧ) for some P-name ḃ}. For X ∈ P(δ) (not necessarily in I), let (4.5) P ↾ X = {p ↾ X : p ∈ P}. By (4.1) and since I is an ideal, we have (4.6) P ↾ X = {p ∈ P : supp(p) ⊆ X}. In particular, (4.7) P ↾ X ⊆ P. Furthermore, it is easy to see that P ↾ X ≤◦ P. Thus, if G is a (V,P)-generic filter then G ∩ (P ↾ X) is a (V,P ↾ X)-generic filter. We shall denote the generic filter G ∩ (P ↾ X) by GX . Note that a P-name ȧ is a P ↾ X-name if and only if supp(ȧ) ⊆ X . We shall call an I-support product P = Pα homogeneous if Pα ∼= Pβ for all α, β < δ and I is translation invariant, that is, I = {j ′′x : x ∈ I} for all bijections j : δ → δ. Note that if I is translation invariant then I = [δ] < λ for some λ. For a homogeneous P = Pα, we shall always assume that a commutative system iα,β : Pα → Pβ, α, β < δ of isomorphisms is fixed. With such a fixed system of isomorphisms, every bijection j : δ → δ induces an isomorphism j̃ : P defined by (4.8) dom(j̃(p)) = j ′′ dom(p) ; for α ∈ dom(j̃(p)), j̃(p)(α) = ij−1(α),α ◦ p ◦ j −1(α) for all p ∈ P. For notational simplicity we shall denote the isomorphism on P-names induced from j̃ also by j̃. Note that for P and j as above, p ∈ P, P-names ȧ0,..., ȧn−1 and a formula ϕ in the language of set theory LZF, we have (4.9) p ‖–P “ϕ(ȧ0, ..., ȧn−1) ” if and only if j̃(p) ‖–P “ϕ(j̃(ȧ0), ..., j̃(ȧn−1)) ”. We are now ready to formulate the main result of the present section: Theorem 4.3. Suppose that (4.10) λ is a regular uncountable cardinal with 2<λ = λ, µ ∈ {λ, λ+} and κ is a <λ-inaccessible cardinal. Let P = Pα be a homogeneous I-support product such that (4.11) I ⊆ [δ]<λ; (4.12) |Pα | ≤ λ for all α < δ and P satisfies the µ-c.c.; (4.13) P is proper. Then ‖– P “ IP(κ, µ) ” holds. The proof of Theorem 4.3 will be given after the following Lemmas 4.4 to 4.7. As in [15], a P-name ẋ of a subset of ω for a poset P is called a nice P-name if there are antichains Aẋ,n, n ∈ ω in P such that ẋ = {〈ň, p〉 : p ∈ Aẋ,n}. Note that, for such a name ẋ, we have supp(ẋ) = n∈ω Aẋ,n. It is easy to see that, for all P-names ẋ of subsets of ω, there is a nice P-name ẋ′ such that ‖–P “ ẋ = ẋ We say that a nice P-name of a subset of ω with Aẋ,n, n ∈ ω as above is slim if Aẋ,n is countable for all n < ω. The following lemmas are well-known: Lemma 4.4. Suppose that P is a proper poset and p ∈ P. For any P-name ẋ of a subset of ω, there are q ≤P p and a slim P-name ẋ ′ such that q ‖–P “ ẋ = ẋ Proof. By the remark above, we may assume without loss of generality that ẋ is a nice P-name. Let Aẋ,n, n ∈ ω be as above and ẏ be a P-name such that ‖–P “ ẏ = {s ∈ P : s ∈ ( n∈ω Aẋ,n) ∩ Ġ} ”. Then we have ‖– P “ ẏ is a countable subset of P ”. As P is proper there exist q ≤P p and countable y ⊆ P such that q ‖– P “ ẏ ⊆ y ”. Let ẋ ′ = {〈ň, s〉 : n ∈ ω, s ∈ Aẋ,n ∩ y}. These q and ẋ ′ are as desired. (Lemma 4.4) Lemma 4.5. Suppose that P = Pα is a κ-c.c. I-support product for an ideal I ⊆ [δ]<λ and κ is <λ-inaccessible. If S ⊆ Eκ≥λ is stationary and pα ∈ P for α ∈ S are such that supp(pα), α ∈ S form a ∆-system with the root R and there is p∗ ∈ P ↾ R such that pα ↾ R = p ∗ for all α ∈ S, then p∗ ‖– P “ {α ∈ S : pα ∈ Ġ} is stationary ” where Ġ denotes the standard P-name of a (V,P)-generic filter. Proof. By κ-c.c. of P, κ remains a regular cardinal in P-generic extensions. Let Ṡ be a P-name of {α ∈ S : pα ∈ Ġ}. Suppose that Ċ is a P-name of a club subset of κ and p ≤P p ∗. It is enough to show that there is a q ≤P p such that q ‖– P “ Ċ ∩ Ṡ 6= ∅ ”. Let θ be sufficiently large and let M ≺ H(θ) be such that (4.14) I, P, κ, Ċ, 〈pα : α ∈ S〉, p ∈M ; (4.15) |M | < κ ∩M < κ; (4.16) [M ]<λ ⊆M and (4.17) α∗ ∈ S where α∗ = κ ∩M . (4.16) is possible since κ is <λ-inaccessible. (4.17) is possible since S ⊆ Eκ≥λ and S is stationary in κ. Claim 4.5.1. ‖–P “α ∗ ∈ Ċ ”. ⊢ Since P satisfies the κ-c.c., we have H(θ) |= ∀α < κ ∃β ∈ κ \ α ( ‖–P “β ∈ Ċ ”). By (4.14), and elementarity of M it follows that M |= ∀α < κ ∃β ∈ κ \ α ( ‖–P “β ∈ Ċ ”). Thus ‖–P “ Ċ ∩ α ∗ is unbounded in α∗ ”. Since ‖–P “ Ċ is a club in κ ”, it follows that ‖–P “α ∗ ∈ Ċ ”. ⊣ (Claim 4.5.1) Claim 4.5.2. supp(pα∗) ∩M = R and supp(p) ∩ supp(pα∗) = R . ⊢ Suppose u = (supp(pα∗)∩M)\R 6= ∅. By (4.16), u ∈M . Hence by elementarity M |= ∃α < κ (u ⊆ supp(pα)). Let α ∈ κ ∩M be such that u ⊆ supp(pα). Then α < α∗ and R ∪ u ⊆ pα ∩ pα∗ . This is a contradiction to the assumption that R is the root of the ∆-system {supp(pα) : α ∈ S}. This shows supp(pα∗) ∩M = R. By (4.14) and (4.16), supp(p) ∈ M . It follows that supp(p) ∩ sup(pα∗) = supp(p) ∩ (sup(pα∗) ∩M) = supp(p) ∩ R = R. ⊣ (Claim 4.5.2) Since p ↾ R ≤P p ∗ ↾ R = p∗ = pα∗ ↾ R, q = p ∪ pα∗ ∈ P. We have q ≤P p. By pα∗ ‖–P “α ∗ ∈ Ṡ ” and q ≤P pα∗ , we have q ‖–P “α ∗ ∈ Ċ ∩ Ṡ ”. In particular q ‖– P “ Ċ ∩ Ṡ 6= ∅ ”. (Lemma 4.5) The arguments of the following two lemmas are also well-known. For Lemma 4.6 see e.g. [12]. Lemma 4.6. Suppose that P = Pα is an I-support product and G is a (V,P)- generic filter. For X, Y ⊆ δ, let Z = X ∩Y . Then, in V [G], for any κ ∈ CardV [G], we have [On]<κ ∩ (V [GX ] \ V [GZ ]) ∩ (V [GY ] \ V [GZ ]) = ∅. Lemma 4.7. Suppose that κ ≤ δ and P = Pα is a κ-c.c. homogeneous I- support product, p ∈ P, ȧ0, ..., ȧn−1 are P-names with (4.18) supp(ȧ0), ..., supp(ȧn−1) ⊆ X for some X ⊆ δ and ϕ = ϕ(x0, ...,xn−1) is a formula in LZF (possibly with some parameters from V ). (a) If (4.19) p ‖–P “ϕ(ȧ0, ..., ȧn−1) ” and (4.20) δ \X 6∈ I, then p ↾ X ‖–P “ϕ(ȧ0, ..., ȧn−1) ”. (b) If (4.21) p ‖–P “ (∃x ∈ ωω) ϕ(x, ȧ1, ..., ȧn−1) ”, (4.22) supp(p) ⊆ X and (4.23) |X \ (supp(p) ∪ supp(ȧ0) ∪ · · · ∪ supp(ȧn−1)) | ≥ κ, then there is a PX-name ȧ such that p ‖–P “ϕ(ȧ, ȧ1, ..., ȧn−1) ”. Proof. (a): Suppose that p ↾ X /‖–P “ϕ(ȧ0, ..., ȧn−1) ”. Then there is q ≤P p ↾ X such that q ‖–P “¬ϕ(ȧ0, ..., ȧn−1) ”. Let j : δ → δ be a bijection such that (4.24) j ↾ X = idX and (4.25) (j ′′ supp(q) \X) ∩ supp(p) = ∅. Note that the last condition is possible by (4.20). By (4.24) and (4.18), we have (4.26) j̃(q) ↾ X = j̃(q ↾ X) = q ↾ X and (4.27) j̃(ȧ0) = ȧ0,..., j̃(ȧn−1) = ȧn−1. By (4.27) and by the choice of q, we have j̃(q) ‖–P “¬ϕ(ȧ0, ..., ȧn−1) ”. On the other hand, by (4.26) p and j̃(q) are compatible. This is a contradiction to (4.19). (b): By maximal principle, there is a nice P-name ȧ′ of a real such that p ‖–P “ϕ(ȧ ′, ȧ1, ..., ȧn−1) ”. By the κ-c.c. of P, we have | supp(ȧ′) | < κ. By (4.18), (4.22) and (4.23), we can find a bijection j : δ → δ such that (4.28) j on supp(p) ∪ supp(ȧ1) ∪ · · · ∪ supp(ȧn−1) is the identity mapping, and (4.29) j ′′ supp(ȧ′) ⊆ X . By (4.28), j̃(p) = p and j̃(ȧ1) = ȧ1,..., j̃(ȧn−1) = ȧn−1. Let ȧ = j̃(ȧ ′). Then p ‖–P “ϕ(ȧ, ȧ1, ..., ȧn−1) ” and supp(ȧ) ⊆ X by (4.29). (Lemma 4.7) Proof of Theorem 4.3: By Proposition 2.5, we may assume that ‖– P “ κ ≤ 2 ℵ0 ”. In particular, by (4.11), (4.12) and (4.13), we may assume that δ ≥ κ. By the µ-c.c. of P, µ and κ remain regular cardinals in the generic extension by P. Let G be a (V,P)-generic filter. In V [G], let f : κ→ P(ω) and g : ((P(ω))) P(ω) be definable, say by a formula ϕ. We may assume that ϕ has a real a ∈ V [G] as its unique parameter. Let ḟ , ȧ and ġ be P-names of f , a and g respectively such that ‖–P “ ḟ : κ→ P(ω) ”, ‖– P “ ġ : ((P(ω))) → P(ω) ” and (4.30) ‖– P “ ∀x ∈ ((P(ω))) ∀x ∈ P(ω) ġ(x) = x↔ H(ℵ1) |= ϕ(x, x, ȧ) Suppose that, for a p ∈ G, (4.31) p ‖–P “ (i0) for IP(κ, µ) does not hold for ḟ and ġ ”. In particular, we have (4.32) p ‖–P “ ∀α < κ ({β ∈ κ : ḟ(β) = ḟ(α)} is non-stationary) ”. Claim 4.3.1. There is a stationary S ⊆ Eκ≥λ such that p ‖–P “ ḟ ↾ S is 1-1 ”. ⊢ By the κ-c.c. of P and by (4.32), there are club sets Cα ⊆ κ (in V ) for each α < κ such that p ‖–P “Cα ∩ {β ∈ κ : ḟ(β) = ḟ(α)} = ∅ ”. Then C = ∆α<κCα is club and S = E ≥λ ∩ C has the desired property. ⊣ (Claim 4.3.1) We show that p forces (i1) for ḟ and ġ. Let p′ ≤P p. It is enough to show that there is p∗ ≤P p ′ forcing (i1). By Lemma 4.4, Theorem 4.1, (4.10), (4.11) and (4.13), there are p′′ ≤P p slim P-name ȧ′ of a real, a stationary S∗ ⊆ S, a sequence 〈ẋ′α : α ∈ S ∗〉 of slim P-names and a sequence 〈pα : α ∈ S ∗〉 of conditions in P such that (4.33) (i) p′′ ‖– P “ ȧ = ȧ (ii) pα ≤P p ′′ and (iii) pα ‖– P “ ḟ(α) = ẋ α ” for every α ∈ S (4.34) dα = supp(pα)∪ supp(ȧ ′)∪ supp(ẋ′α), α ∈ S ∗ are all of the same cardinality and form a ∆-system with root R; (4.35) for each α, β ∈ S∗ there is a bijection jα,β : δ → δ such that (i) jα,β ↾ (δ \ (dα∆dβ)) = idδ\(dα∆dβ) , (ii) jα,β ′′dα = dβ, j̃α,β(pα) = pβ and (iii) j̃α,β(ẋ α) = ẋ β for every α, β ∈ S Note that, by (4.34), we have (4.36) supp(ȧ′) = supp(ȧ′) ∩ dα ⊆ R for every α ∈ S By (4.35), pα ↾ R for α ∈ S ∗ are all the same. Let q = pα ↾ R for some/any α ∈ S∗. q ≤P p ′′ by (4.33), (ii). Let Ṡ be a P-name such that (4.37) ‖– P “ Ṡ = {α ∈ S ∗ : pα ∈ Ġ} ”. By Lemma 4.5, q ‖–P “ Ṡ is stationary ”. Hence, by (4.31), (4.38) q ‖– P “ ∃n ∈ ω ∀α < κ | ġ ′′((ḟ ′′(Ṡ \ α))) | ≥ µ Let q′ ≤P q and n ∗ ∈ ω be such that (4.39) q′ ‖–P “ ∀α < κ (| ġ ′′((ḟ ′′(Ṡ \ α))) | ≥ µ) ”. (4.40) S∗∗ = {α ∈ S∗ : supp(q′) ∩ dα ⊆ R}. Since | supp(q′) | < λ by (4.11), we have (4.41) S∗ \ S∗∗ is of cardinality < λ. In particular S∗∗ is still stationary and (4.42) q′ ‖–P “ | ġ ′′((ḟ ′′(Ṡ ∩ S∗∗))) | ≥ µ ” by (4.39). Claim 4.3.2. There is 〈α0, ...,αn∗−1〉 ∈ ((S such that q′ ∪ pα0 ∪ · · · ∪ pαn∗−1 /‖– P “ ġ(〈ẋ , ..., ẋ′αn∗−1〉) ∈ V [ĠR] ”. ⊢ Otherwise, we would have q′ ∪ pβ0 ∪ · · · ∪ pβn∗−1 ‖–P “ ġ(〈ẋ , ..., ẋ′βn∗−1〉) ∈ V [ĠR] ” for all 〈β0, ..., βn∗−1〉 ∈ ((S Fix 〈α0, ...,αn∗−1〉 ∈ ((S and let D = {r ∈ P : r ≤P q ′ ∪ pα0 ∪ · · · ∪ pαn∗−1 , supp(r) ⊆ R ∪ {dαi : i < n ∗} ∪ supp(q′), r ‖– P “ ġ(ẋ , ..., ẋ′αn∗−1) = ẋ ” for some PR-name ẋ }. Let A be a maximal antichain in D. By the µ-c.c. of P, | A | < µ. For each r ∈ A, let ẋr be a PR-name such that r ‖– P “ ġ(ẋ , ..., ẋ′αn∗−1) = ẋr ” and Ẋ be a PR-name such that ‖– P “ Ẋ = {ẋr : r ∈ A} ”. Then, we have ‖–P “ | Ẋ | < µ ”. By Lemma 4.7, (a) q′ ∪ pα0 ∪ · · · ∪ pαn∗−1 ‖– P “ ġ(〈ẋ , ..., ẋ′αn∗−1〉) ∈ Ẋ ”. Hence by (4.35) and (4.9), we have q′ ∪ pβ0 ∪ · · · ∪ pβn∗−1 ‖–P “ ġ(〈ẋ , ..., ẋ′βn∗−1〉) ∈ Ẋ ” for all 〈β0, ..., βn∗−1〉 ∈ ((S . But this is a contradiction to (4.42). ⊣ (Claim 4.3.2) Let 〈α0, ...,αn∗−1〉 ∈ ((S be as in Claim 4.3.2 and (4.43) q′′ = q′ ∪ pα0 ∪ · · · ∪ pαn∗−1 . Note that (4.44) q′′ ‖–P “ f(αi) = ẋ ” for i < n∗ by (4.43). Let p∗ ≤P q ′′ be such that (4.45) p∗ ‖–P “ ġ(ẋ , ..., ẋ′αn∗−1) 6∈ V [ĠR] ”. By thinning out S∗∗ further, if necessary, we may assume that supp(p∗) ∩ supp(pα) ⊆ R for all α ∈ S ∗∗. For i < n∗, let Ṡi be a P-name such that (4.46) ‖– P “ Ṡi = {α ∈ S ∗∗ : j̃αi,α(p ∗) ∈ Ġ} ”. By Lemma 4.5, we have p∗ ‖–P “ Ṡi is a stationary subset of κ ” for all i < n ∗. Note that we have j̃αi,α(p ∗) ≤P pα by (4.43) and (4.35), (ii). Claim 4.3.3. p∗ ‖– P “ ∀β0 · · · ∀βn∗−1 〈β0, ..., βn∗−1〉 ∈ ((Ṡ0, ..., Ṡn∗−1)) → ġ(〈ḟ(β0), ..., ḟ(βn∗−1)〉) 6∈ V [ĠR] ⊢ Suppose that q ≤P p∗ and q ‖–P “ 〈β0, ..., βn∗−1〉 ∈ ((Ṡ0, ..., Ṡn∗−1)) ”. Then, by (4.46), q ‖–P “ j̃αi,βi(p ∗) ∈ Ġ ” for i < n∗. It follows that (4.47) q ‖– P “ j̃αi,βi(p ∗) ↾ dβi ∈ Ġ ” for i < n j̃ = j̃α0,β0 ◦ j̃α1,β1 ◦ · · · ◦ j̃αn∗−1,βn∗−1 . (4.48) j̃(p∗) = p∗ ↾ (δ \ dαi) ∪ j̃α0,β0(p ∗) ↾ dβ0 ∪ · · · ∪ j̃αn∗−1,βn∗−1(p ∗) ↾ dβn∗−1 by (4.35). Hence (4.49) q ‖– P “ j̃(p ∗) ∈ Ġ ” by q ≤P p ∗ and (4.47) and (4.48). By definition of j̃ and q′′, and by (4.35), we have (4.50) j̃(p∗) ≤P j̃(q ′′) ≤P j̃αi,βi(pαi) = pβi for i < n ∗ and (4.51) j̃(ẋ′αi) = ẋ for i < n∗ by (4.44). Hence by (4.45) q ‖–P “ ġ(〈ẋ , ..., ẋ′βn∗−1〉) 6∈ V [ĠR] ”. By (4.33), (4.49) and (4.50), it follows that q ‖–P “ ḟ(βi) = ẋ for i < n∗. Hence q ‖–P “ ġ(〈ḟ(β0), ..., ḟ(βn∗−1)〉) 6∈ V [GR] ”. ⊣ (Claim 4.3.3) To show that p∗ ‖–P “ (i1) holds ”, suppose that q ≤P p ∗ and 〈β0, ..., βn∗−1〉, 〈γ0, ..., γn∗−1〉 ∈ ((S are such that (4.52) {β0, ..., βn∗−1} ∩ {γ0, ..., γn∗−1} = ∅ and (4.53) q ‖– P “ 〈β0, ..., βn∗−1〉, 〈γ0, ..., γn∗−1〉 ∈ ((Ṡ0, ..., Ṡn∗−1)) ”. Note that it is enough to consider 〈β0, ...,βn∗−1〉, 〈γ0, ..., γn∗−1〉 ∈ ((S (4.52) since we can thin out ṠG0 ,..., Ṡ n∗−1 afterwards if necessary so that they are pairwise disjoint. By the remark after (4.46), we may assume that q ≤P p ∗ ∪ pβ0 ∪ · · · ∪ pβn∗−1 ∪ pγ0 ∪ · · · ∪ pγn∗−1 . By Lemma 4.7, (b), there are P-names ẏ, ż such that supp(ẏ) ∩ supp(ż) ⊆ R and p∗ ∪ pβ0 ∪ ··· ∪ pβn∗−1 ∪ pγ0 ∪ ··· ∪ pγn∗−1 ‖–P “ ġ(〈ḟ(β0), ..., ḟ(βn∗−1)〉) = ẏ ∧ ġ(〈ḟ(γ0), ..., ḟ(γn∗−1)〉) = ż ”. By Claim 4.3.3 and Lemma 4.6, it follows that q ≤P p ∗ ∪ pβ0 ∪ ··· ∪ pβn∗−1 ∪ pγ0 ∪ ··· ∪ pγn∗−1 ‖– P “ ġ(〈ḟ(β0〉, ..., ḟ(βn∗−1)) 6= ġ(〈ḟ(γ0), ..., ḟγn∗−1))〉 ”. Since q as above may be chosen below arbitrary r ≤P p ∗, it follows that p∗ ‖–P “ (i1) holds ”. (Theorem 4.3) Corollary 4.8. (a) Assume CH and P = Fn(µ, 2) for some cardinal µ. Then ‖–P “ IP(ℵ2,ℵ1) ” holds. (b) Assume GCH and P = Fn(µ, 2) for some cardinal µ. Then ‖–P “ IP(κ +,ℵ1) ” holds for every uncountable κ of uncountable cofinality and ‖–P “ IP(λ,ℵ1) ” for every inaccessible λ. (c) Assume CH and P is a finite support product of copies of a productively c.c.c. poset of cardinality ℵ1. Then ‖–P “ IP(ℵ2,ℵ1) ” holds. In particular, we have ‖–P “HP(ℵ2) ”. (d) Assume GCH and P is a finite support product of copies of a productively c.c.c. poset of cardinality ℵ1. Then ‖– P “ IP(κ +,ℵ1) ” holds for every uncountable κ of uncountable cofinality and ‖–P “ IP(λ,ℵ1) ” for every inaccessible λ. (e) Assume CH and P is a countable support product of copies of a proper poset of cardinality ℵ1 such that its product is also proper. Then ‖–P “ IP(ℵ2,ℵ2) ” holds. In particular, we have ‖–P “HP(ℵ2) ”. (f) Assume GCH and P is a countable support product of copies of a proper poset of cardinality ℵ1 such that its product is also proper. Then ‖–P “ IP(κ +,ℵ2) ” holds for every uncountable κ of uncountable cofinality and ‖–P “ IP(λ,ℵ2) ” for every inaccessible λ. Note that countable support products of Sacks or Prikry-Silver forcing are in- stances of (e) and (f) above. Proof. Under CH, ω1 = 2 <ω1 and ω2 is <ω1-inaccessible. In (a) and (b), P is forc- ing equivalent to a finite support product of copies of the countable poset Fn(ω, 2). Clearly P’s in all of (a) ∼ (f) are homogeneous; P’s in (a) ∼ (d) satisfy the c.c.c. and hence they are proper. Thus we can apply Theorem 4.3. The second parts of (c) and (e) follow from Theorem 2.9. (Corollary 4.8) Results similar to Theorem ?? and Corollary 4.8 also hold for partial orderings with product-like structure as those considered in [9]. Thus, we can prove e.g. that IP(ℵ2,ℵ2) together with clubsuit principle is consistent. In [7] it is shown that, if we start from a model V which is obtained by adding a dominating real to a model of GCH + Chang’s conjecture for ℵω, i.e. (ℵω+1,ℵω) →→ (ℵ1,ℵ0), then adding more than ℵω+1 Cohen reals forces ¬WFN. Since V satisfies GCH, IP(κ,ℵ1) is forced for every κ ≥ ℵ2 which is not a successor of a singular cardinal of cofinality ω by adding any number of Cohen reals by Corollary 4.8. In particular: Corollary 4.9. Suppose that Chang’s conjecture for ℵω is consistent. Then so is IP(ℵ2,ℵ1) ∧ b ∗ = ℵ1 ∧ ¬WFN. 5 Models of IP(ℵ2,ℵ2) ∧ ¬IP(ℵ2,ℵ1) Recall that Prikry-Silver forcing S is the forcing with partial functions with co- infinite domain, that is S = {f : f : D → 2, D ⊆ ω, |ω \D | = ℵ0} with the ordering f ≤S g ⇔ f ⊇ g for f , g ∈ S. A (V, S)-generic filter G gives rise to the function sG = G : ω → 2 which is often called a Prikry-Silver real. For f ∈ S let codom(f) = ω \ dom(f). It is easy to check that Prikry-Silver forcing S as well as its countable support products SI over any index set I satisfy the Axiom A. Hence they are all proper. Note that, by definition of ≤S, we have: (5.1) f , g ∈ S are incompatible if | codom(f) ∩ codom(g) | < ℵ0. (5.2) For any 〈f0, f1〉 ∈ S 2, there is 〈g0, g1〉 ≤S2 〈f0, f1〉 such that | codom(g0) ∩ codom(g1) | < ℵ0. Lemma 5.1. For any f ∈ S and 〈gn0 , g 1 〉 ∈ S 2, n ∈ ω such that | codom(gn0 ) ∩ codom(gn1 ) | < ℵ0 there is g ≤S f such that 〈g, g〉 is incompatible with all 〈g 0 , g n ∈ ω. Proof. Construct in ∈ 2, n ∈ ω and A ⊆ codom(f) recursively so that | codom(f) ∩ k≤n dom(g ) | = ℵ0 and |A ∩ codom(gnin) | < ℵ0 for all n ∈ ω. Then any extension g of f on ω \ A will do. (Lemma 5.1) Working in V = L, we can construct recursively a maximal antichain {〈gα0 , g 1 〉 : α < ω1} in S 2 such that (5.3) | codom(gα0 ) ∩ codom(g 1 ) | < ℵ0 for all α < ω1. Note that each step of the recursive construction is possible by (5.2) and (5.2). Furthermore by choosing 〈gα0 , g 1 〉 in each step of the construction according to the Σ12-well ordering of the reals (which exists because of V = L), we can make {〈gα0 , g 1 〉 : α < ω1} a Σ 2-set (actually we can even choose such a maximal antichain as a Π11-set arguing similarly to [16]). Let ϕ : S2 → ω2 be a Borel bijection and let g : ((ω2)) → ω2 be defined by (5.4) g(〈x0, ..., xn−1〉) = 0 , g 1 ) ; if n = 2, there is α < ω2 such that x0 ⊇ g 0 , x1 ⊇ g 1 and α ∗ is minimal among such α’s 0 ; otherwise. It is easy to check that g is a ∆13-set. Theorem 5.2. Assume V = L. Then we have ‖– Sω2 “ IP(ℵ2,ℵ2) and ¬IP(ℵ2,ℵ1) ”. Proof. ‖– Sω2 “ IP(ℵ2,ℵ2) ” follows from Corollary 4.8, (e). To show that ‖– Sω2 “¬IP(ℵ2,ℵ1) ”, let G be a (V, S ω2)-generic filter. Working in L[G], let sβ be the β’th Prikry-Silver real added by G for β < ω2. Let f : ω2 → be defined by (5.5) f(β) = sβ for β < ω2 and let g : ((ω2)) → ω2 be the mapping as in (5.4), or more precisely, let g be the mapping (in L[G]) defined by the ∆13 definition corresponding to (5.4). We show that f and g build a counter-example to IP(ℵ2,ℵ1). Since | rng(g) | ≤ ℵ1, (i1) clearly fails for these f and g. Hence we are done by showing that f and g do not satisfy (i0). Assume, for a contradiction, that f and g satisfy (i0). Returning to L, let ḟ , ġ, ṡβ, β < ω2 etc. be S ω2-names of f , g, sβ , β < ω2 etc. respectively. In particular, we can choose ḟ such that (5.6) ‖– Sω2 “ ḟ(β) = ṡβ ” for all β < ω2. Since Sω2 is proper, there are p ∈ G, Sω2-name Ṡ and a countable set Z (in L) such that (5.7) p ‖– Sω2 “ Ṡ ⊆ ω2 is stationary and ġ ′′((ḟ ′′Ṡ)) ⊆ Z ”. Let U = {β < ω2 : there is p ′ ≤Sω2 p such that p ′ ‖– Sω2 “ β ∈ Ṡ ”}. Then U is a stationary subset of ω2. For each β ∈ U , let pβ ≤Sω2 p be such that pβ ‖– Sω2 “β ∈ Ṡ ” and β ∈ supp(pβ). By ∆-System Lemma and CH, there is U∗ ∈ [U ]ℵ2 such that (5.8) supp(pβ), β ∈ U ∗ form a ∆-system with root R which is an initial segment of all of supp(pβ), β ∈ U (5.9) supR < minU∗; (5.10) pβ ↾ R, β ∈ U ∗ are all the same; and (5.11) pβ(β), β ∈ U ∗ are all the same, say h ∈ S. Note that pβ, β ∈ U ∗ are compatible by (5.8) and (5.10). (5.12) X = ϕ−1(Z). By Lemma 5.1, there is a k ≤S h such that 〈k, k〉 is incompatible with all 〈g 0 , g from the countable set X . Fix two distinct β, γ ∈ U∗ and let q ≤Sω2 pβ, pγ be defined by dom(q) = dom(pβ) ∪ dom(pγ) and (5.13) q(δ) = pβ(δ) ; if δ ∈ supp(pβ) \ {β} pγ(δ) ; else if δ ∈ supp(pγ) \ {γ} k ; else if δ = β or δ = γ for δ ∈ dom(q). By q ≤Sω2 pβ, pγ , we have q ‖– Sω2 “β, γ ∈ Ṡ ”. Thus the following claim yields a contradiction to (5.7): Claim 5.2.1. q ‖– Sω2 “ ġ(〈ḟ(β), ḟ(γ)〉) 6∈ Z ”. ⊢ By (5.6), we have to show q ‖– Sω2 “ ġ(〈ṡβ, ṡγ〉) 6∈ Z ”. First, we show that q ‖– Sω2 “ ġ(〈ṡβ, ṡγ〉) 6= 0 ”. Note that, by the complete embedding S2 ∋ 〈g0, g1〉 7→ {〈β, g0〉, 〈γ, g1〉} ∈ S {β,γ} ≤◦ Sω2 we have: {{〈β, gα0 〉, 〈γ, g 1 〉} : α < ω1} is a maximal antichain in S For any r ≤Sω2 q, let α ∗ < ω1 be such that r and {〈β, g 0 〉, 〈γ, g 1 〉} are com- patible. Let s ≤Sω2 r, {〈β, g 0 〉, 〈γ, g 1 〉}. Then we have s ‖– Sω2 “ ṡβ ⊇ g 0 , ṡγ ⊇ g Hence, by (5.4), it follows that s ‖–Sω2 “ ġ(〈ṡβ, ṡγ〉) 6= 0 ”. Now, suppose, for contradiction, that there is r ≤Qω2 q such that r ‖– Sω2 “ ġ(〈ṡβ, ṡγ〉) ∈ Z ”. Then, by the first part of the proof, there are s ≤Sω1 r and 〈g 0 , g 1 〉 ∈ X such that s ‖–Sω2 “ ṡβ ⊇ g 0 and ṡγ ⊇ g 1 ”. In particular s(β) and s(γ) are compatible with gα0 and g 1 , respectively. Since r ≤Sω2 q ≤Sω2 {〈β, k〉, 〈γ, k〉}, it follows that k is compatible with both of gα0 and g 1 . This is a contradiction to the choice of k. ⊣ (Claim 5.2.1) (Theorem 5.2) We can prove a Lemma similar to Lemma 5.1 for omega product of Sacks forcing. Thus, by a similar argument as above, we can also prove that IP(ℵ2,ℵ1) fails in a generic extension by countable support side-by-side product of Sacks forcing. 6 The Consistency of b∗ = ℵ2 ∧ do = ℵ1 In the following we shall refer by (A) the assertion that there is a structure 〈(ω2) 2, A,F〉 with the properties (6.1) ∼ (6.5) below. Recall that a mapping f : X → X is called an involution if it is a bijection exchanging (some) pairs of elements of X , that is, if f ◦ f = idX holds. (6.1) ω2 × ω2 ⊇ A ⊇ {〈α, β〉 ∈ ω2 × ω2 : β < α}; (6.2) For any C ∈ [ω2] ℵ0 there is an X ∈ [ω2] ℵ2 such that (C ×X) ∩ A = ∅; (6.3) For all 〈φ, ψ〉 ∈ F , φ and ψ are involutions on ω2; (6.4) For each 〈φ, ψ〉 ∈ F and for all 〈α, β〉 ∈ ω2 ×ω2, we have 〈α, β〉 ∈ A if and only if 〈φ(α), ψ(β)〉 ∈ A; (6.5) For any stationary S ⊆ Eω2ω1 and any Aζ , Bζ ∈ [ω2] ℵ0 for ζ ∈ S, there is a stationary T ⊆ S such that, for any n ∈ ω, if ζi, ηi ∈ T for i ∈ n are pairwise distinct (2n elements) then there is 〈φ, ψ〉 ∈ F such that φ ′′Aζi = Aηi , ψ ′′Bζi = Bηi ; and φ ↾ Aζi : Aζi → Aηi , ψ ↾ Bζi : Bζi → Bηi are order isomorphisms for all i ∈ n. The consistency of (A) together with CH over ZFC is proved in the next section. Below, we will prove the consistency of c = b∗ = ℵ2 ∧ do = ℵ1 ∧ ¬C s(ℵ2) by constructing a model of this combination of assertions starting from a model of (A) and CH. Let us begin with introducing some notation for the forcing construction we use in the proof. For a cardinal κ, a sequence f̄ = 〈fξ : ξ < κ〉 in ωω and X ⊆ κ, let Df̄ ,X be the canonical poset adding an element of ωω dominating {fξ : ξ ∈ X}. That is (6.6) Df̄ ,X = {〈s, F 〉 : s ∈ ω>ω, F ∈ [κ]<ℵ0} and, for 〈s, F 〉, 〈s′, F ′〉 ∈ Df̄ ,X , (6.7) 〈s′, F ′〉 ≤Df̄ ,X 〈s, F 〉 ⇔ s ′ ⊇ s, F ′ ⊇ F, ∀α ∈ F ∩X ∀n ∈ dom(s′) \ dom(s) (fα(n) ≤ s ′(n)). Since any 〈s, F 〉, 〈s′, F ′〉 ∈ Df̄ ,X with s = s ′ are compatible, we have: Lemma 6.1. Df̄ ,X is σ-centered. Note that the underlying set of Df̄ ,X does not depend on the sequence f̄ . So we shall denote this set with DX . Actually DX as a set does not depend on X either. Nevertheless we shall add the suffix X so that we can distinguish D’s by their intended function. Note also that, as a set, Df̄ ,Xα for any κ-sequence f̄ of reals is the same: we shall denote this set by If d ∈ DX and d = 〈s, F 〉 then we shall write s d and F d to denote these s and F respectively. In the following we assume that a sequence X̄ = 〈Xα : α < κ〉 of nonempty subsets of κ is fixed. Let (6.8) QX̄ = Cκ ∗ D ˙̄f,Xα where Cκ = Fn(κ × ω, ω) and ˙̄f denotes the Cκ-name of the sequence of Cohen reals (∈ ωω) of length κ added by Cκ. Thus, if G is a (V,Cκ)-generic set and cα is the α’th element of ˙̄fG, then cα(n) = m if and only if there is a condition c ∈ G such that 〈α, n〉 ∈ dom(c) and c(α, n) = m. (6.9) Q = {〈c, d〉 : c ∈ Cκ, d ∈ α∈κDXα , ξ∈dom(d) F d(ξ) × dom(sd(ξ)) ⊆ dom(c)} For 〈c, d〉, 〈c′, d′〉 ∈ Q (6.10) 〈c′, d′〉 ≤ 〈c, d〉 ⇔ c′ ≤Cκ c, dom(d ′) ⊇ dom(d), ∀α ∈ dom(d) ′(α) ⊇ sd(α) ∧ F d ′(α) ⊇ F d(α)∧ ∀ξ ∈ F d(α) ∩Xα ∀n ∈ dom(s d′(α)) \ dom(sd(α)) c′(ξ, n) ≤ sd ′(α)(n) The following can be shown easily by standard arguments: Lemma 6.2. Φ : Q → QX̄ ; 〈c, d〉 7→ 〈c, ď〉 is a dense embedding of Q into QX̄ . QX̄ and Q are thus forcing equivalent. For p ∈ Q with p = 〈c, d〉, let supp0(p) = {α < κ : 〈α, n〉 ∈ dom(c) for some n ∈ ω} and supp1(p) = dom(d). For a Q -name ȧ, supp0(ȧ) and supp1(ȧ) are also defined in analogy to (4.4). In Theorem 6.4, we assume CH + (A) and let, for a structure 〈(ω2) 2, A,F〉 as in (A), κ = ω2 and X̄ = 〈Xα : α < ω2〉 where Xα = {β ∈ ω2 : 〈α, β〉 ∈ A} for α < ω2. For such X̄ , the next lemma follows immediately from (6.3) and (6.4). Lemma 6.3. Suppose that 〈(ω2) 2, A,F〉 and X̄ are as above. If 〈φ, ψ〉 ∈ F , then the mapping j〈φ,ψ〉 : Q defined by j〈φ,ψ〉(〈c, d〉) = 〈c ′, d′〉 for 〈c, d〉 ∈ Q where c′ and d′ are such that dom(c′) = {〈φ(α), n〉 : 〈α, n〉 ∈ dom(c)}; c′(〈φ(α), n〉) = c(〈α, n〉) for 〈α, n〉 ∈ dom(c); dom(d′) = ψ ′′ dom(d); ′(ψ(ξ)) = F d(ξ) and sd ′(ψ(ξ)) = sd(ξ) for ξ ∈ dom(d) is an automorphism on the poset Q Similarly to Section 4, we shall also denote with j〈φ,ψ〉 the corresponding map- ping on Q -names. The following theorem together with the consistency result in Section 7 gives the consistency of the conjunction of the assertions c = b∗ = ℵ2, do = ℵ1 and ¬Cs(ℵ2) over ZFC. Theorem 6.4. Assume CH and (A). Let 〈(ω2) 2, A,F〉 be a structure satisfying (6.1) ∼ (6.5) and let X̄ = 〈Xα : α < ω2〉 where Xα = {β ∈ ω2 : 〈α, β〉 ∈ A}. Then ‖– “ c = b∗ = ℵ2 ∧ do = ℵ1 ∧ ¬C s(ℵ2) ”. Proof. First, we show that ‖– “ c = b∗ = ℵ2 ”. Let G be a (V,Q )-generic filter. Working in V [G], let f̄ = 〈cα : α < ω2〉 be the sequence of Cohen reals added by the Cω2 part of QX̄ and dα be the Hechler type real added by Df̄ ,Xα for α < ω2. By (6.1), {cα : α < γ} is bounded by dγ for all γ < ω2. On the other hand, {cα : α < ω2} is unbounded by (6.2) and the c.c.c. of Df̄ ,Xα (in V [f̄ ]). This shows that V [G] |= ℵ2 ≤ b ∗. Since |Q | = ℵ2 by CH, we have V [G] |= c ≤ ℵ2. To show that Q forces do = ℵ1, suppose that ḟα, α < ω2 are Q -names of elements of ωω, ϕ(x, y, z) a formula in LZF and ȧ a Q -name of an element of ωω such that (6.11) ‖– “H(ℵ1) |= ϕ(ḟα, ḟβ, ȧ) ” for all α < β < ω2. By Maximal Principle, it is enough to show that there are η1 < η0 < ω2 such that “H(ℵ1) |= ϕ(ḟη0 , ḟη1 , ȧ) ”. For ξ < ω2, let Aξ = supp1(ḟξ) ∪ supp1(ȧ) and Bξ = supp0(ḟξ) ∪ supp0(ȧ). By CH, ∆-System Lemma and (6.5), we can find a stationary S ⊆ Eω2ω1 such that (6.12) Aξ, ξ ∈ S form a ∆-system such that its root is an initial segment of each of Aξ, ξ ∈ S ; Bξ, ξ ∈ S form a ∆-system such that its root is an initial segment of each of Bξ, ξ ∈ S ; (6.13) for any distinct ζ0, ζ1, η0, η1 ∈ S, there is 〈φ, ψ〉 ∈ F such that (6.13a) φ ′′Aζi = Aηi , ψ ′′Bζi = Bηi ; and (6.13b) φ ↾ Aζi : Aζi → Aηi , ψ ↾ Bζi : Bζi → Bηi are order isomorphisms for i ∈ 2; (6.14) j〈φ,ψ〉(ḟζ) = ḟη for any distinct ζ , η ∈ S and 〈φ, ψ〉 ∈ F as in (6.13) with ζ0 = ζ and η0 = η. Note that, by (6.12) and (6.13b), we have (6.15) j〈φ,ψ〉(ȧ) = ȧ for any 〈φ, ψ〉 as in (6.13). Now, let ζ0, ζ1, η0, η1 ∈ S be four distinct elements of S such that ζ0 < ζ1 and η1 < η0. By (6.11), we have “H(ℵ1) |= ϕ(ḟζ0 , ḟζ1, ȧ) ” . Hence, by mapping this situation by j〈φ,ψ〉 for 〈φ, ψ〉 ∈ F as in (6.13) for these ζ0, ζ1, η0, η1 , we obtain “H(ℵ1) |= ϕ(ḟη0 , ḟη1 , ȧ) ” . Thus, η0, η1 above are as desired. Finally, we show that Q forces the negation of Cs(ℵ2). Let 〈r0n, s n〉, n ∈ ω list all quadruples of finite sequences r 0, s0, r1, s1 ∈ ω>ω such that (6.16) | r0 | = | s0 | = | r1 | = | s1 | and (6.17) 〈r0, s0〉 6= 〈r1, s1〉 if | r0 | > 0. We further assume that the enumeration 〈〈r0n, s n〉 : n ∈ ω〉 is arranged so (6.18) | r0n | ≤ n for all n ∈ ω. Now, working in V [G], let aα, α < ω2 be the subsets of ω defined by n ∈ aα ⇔ one of the following (6.19) and (6.20) holds: (6.19) r0n ⊆ cα, s n ⊆ dα+1, cα(n) = 0, dα+1(n) = 1, cα(n+ 1) = 2 and dα+1(n+ 1) = 3 ; (6.20) r1n ⊆ cα, s n ⊆ dα+1, cα(n) = 2, dα+1(n) = 3, cα(n+ 1) = 0 and dα+1(n+ 1) = 1 . (6.21) aα,n = aα \ {k : | r k | < n} for α < ω2 and n ∈ ω. We show that the matrix 〈aα,n : α < ω2, n ∈ ω〉 together with T = 2ω is a counter-example to Cs(ℵ2). For this, it is enough to prove the following: Claim 6.4.1. If S0, S1 are cofinal subsets of ω2, then (1) there exist n < ω, α ∈ S0 and β ∈ S1 such that aα,n ∩ aβ,n = ∅; and (2) for any t ∈ 2ω, there are α ∈ S0 and β ∈ S1 such that aα,t(0) ∩ aβ,t(1) 6= ∅. ⊢ Working in the ground model, let Ṡ0 and Ṡ1 beQ†X̄ -names for the cofinal subsets of ω2. Let p ∈ Q . For α < ω2, let pα ∈ Q and γα, δα ∈ ω2 be such that (6.22) γα < δα < γβ < δβ for all α < β < ω2 ; (6.23) pα ≤Q† p , pα = 〈c α, dα〉 for all α < ω2 ; and (6.24) pα ‖–Q† “ γα ∈ Ṡ0 , δα ∈ Ṡ1 ”. By ∆-System Lemma, we find a stationary U ⊆ Eω2ω1 and Aα, Bα ∈ [ω2] <ℵ0 for α ∈ U such that (6.25) supp0(pα) ⊆ Aα, supp1(pα) ⊆ Bα ; (6.26) Aα, α ∈ U form a ∆-system with root A; Bα, α ∈ U form a ∆-system with root B; (6.27) γα, γα + 1, δα, δα + 1 ∈ (Aα ∩Bα) \ (A ∪B) . By thinning out U further, if necessary, we may also assume that there are some k∗, n∗ ∈ ω such that (6.28) dom(cα) = supp0(pα)× k ∗ and dom(sd α(ξ)) = k∗ for all ξ ∈ supp1(pα) ; (6.29) cα(γα, ·) = r n∗ , s dα(γα+1) = s0n∗ ; (6.30) cα(δα, ·) = r n∗ , s dα(δα+1) = s1n∗ . Without loss of generality, we may also assume that, for some fixed c∗, d∗, (6.31) cα ↾ A× k∗ = c∗ and 〈sd α(η) : η ∈ B〉 = d∗ for all α ∈ U . Note that pα, α ∈ U are compatible by (6.25), (6.26) and (6.31). Now, since Ṡ0, Ṡ1, p were arbitrary, Claim 6.4.1, (1) is proved by the following subclaim: Subclaim 6.4.1.1. For any α, β ∈ U with α < β, there is q ≤ p such that “ γα ∈ Ṡ0, δβ ∈ Ṡ1, ȧγα,n∗ ∩ ȧδβ ,n∗ = ∅ ”. ⊢ Let q = 〈cq, dq〉 be the common extension of pα and pβ such that (6.32) γα ∈ F dq(δβ+1) (6.33) dom(sd q(ξ)) = k∗ for all ξ ∈ dom(dq). Let G be a (V,Q )-generic filter with q ∈ G. In V [G], we have (6.34) cγα(m) ≤ dδβ+1(m) for all m ≥ k by (6.28), (6.32) and (6.33). Now, toward a contradiction, assume that aγα,n∗ ∩ aδβ ,n∗ 6= ∅ and let m ∈ aγα,n∗ ∩ aδβ ,n∗. By the definition of aα’s it follows that, for some i, j ∈ 2, we have rim ⊆ cγα , s m ⊆ dγα+1 ; rjm ⊆ cδβ , s m ⊆ dδβ+1. On the other hand, since q ∈ G, we have pα, pβ ∈ G. It follows that r0n∗ ⊆ cγα, s n∗ ⊆ dγα+1 ; r1n∗ ⊆ cδβ , s n∗ ⊆ dδβ+1 by (6.29) and (6.30). By the definition (6.21) of aγα,n’s, we have | r m | ≥ n ∗. Thus we have, either r0n∗ ⊆ r m ⊆ cγα , s n∗ ⊆ s m ⊆ dγα+1 ; r1n∗ ⊆ r m ⊆ cγβ , s n∗ ⊆ s m ⊆ dγβ+1 ; r0n∗ ⊆ r m ⊆ cγα , s n∗ ⊆ s m ⊆ dγα+1 ; r1n∗ ⊆ r m ⊆ cγβ , s n∗ ⊆ s m ⊆ dγβ+1 . In the first case, we must have cγα(m+ 1) = 2 and dδβ+1(m+ 1) = 1 by (6.19) and (6.20). This is a contradiction to (6.34). Similarly, in the second case, we have cγα(m) = 2 and dδβ+1(m) = 1. This is again a contradiction to (6.34). ⊣ (Subclaim 6.4.1.1) (2) of Claim 6.4.1 follows from the next subclaim: Subclaim 6.4.1.2. For any t ∈ 2ω and α, β ∈ U with α < β, there is q ≤ such that “ γα ∈ Ṡ0, δβ ∈ Ṡ1, ȧγ(α),t(0) ∩ ȧδ(β),t(1) 6= ∅ ”. ⊢ For each ξ ∈ {α, β}, let p̃ξ ≤Q† pξ with p̃ξ = 〈c̃ ξ, d̃ξ〉 and m ∈ ω be such that (6.35) c̃ξ(γξ, ·) = r m , s d̃ξ(γξ+1) = s0m ; c̃ ξ(δξ, ·) = r m , s d̃ξ(δξ+1) = s1m ; (6.36) | r0m | ≥ t(0), t(1) ; (6.37) supp0(p̃ξ) = supp0(pξ) ; supp1(p̃ξ) = supp1(pξ) ; (6.38) c̃ξ ↾ A× ω = cξ ↾ A× ω and 〈sd̃ ξ(η) : η ∈ B〉 = 〈sd ξ(η) : η ∈ B〉. Let q0 = 〈cq 〉 be the maximal (with respect to ≤ ) common extension of p̃α and p̃β which exists because of (6.37) and (6.38). Extend q 0 further to q = 〈cq, dq〉 such that (6.39) | cq(γα, ·) | = | c q(δβ , ·) | = | s dq(γα+1) | = | sd q(δβ+1) | = m+ 2 ; (6.40) cq(γα, m) = 0, s dq(γα+1)(m) = 1, cq(γα, m+ 1) = 2, s dq(γα+1)(m+ 1) = 3 ; (6.41) cq(δβ, m) = 2, s dq(δβ+1)(m) = 3, cq(δβ, m+ 1) = 0, s dq(δβ+1)(m+ 1) = 1 . This is possible because γα 6∈ F (δβ+1) and δβ 6∈ F (γα+1) by the maximality of q0 and (6.37). By (6.35), (6.40), (6.41), by the definition (6.21) of aα,n’s, and since | r m | ≥ t(0), t(1), we have q ‖– “m ∈ ȧγα,t(0) ∩ ȧδβ ,t(1) ”. Since q ≤Q† pα, pβ, we also have “ γα ∈ Ṡ0, δβ ∈ Ṡ1 ”. Thus, q as above is as desired. ⊣ (Subclaim 6.4.1.2) ⊣ (Claim 6.4.1) (Theorem 6.4) Note that in the proof of ¬Cs(ℵ2) in Theorem 6.4, we used only (6.1) from the assumption (A). Note also that this proof actually shows that in the generic exten- sion the negation of C(ℵ2) from [13] holds which is a weakening of C s(ℵ2) obtained by replacing the condition “stationary” in the formulation of Cs(ℵ2) by “cofinal”. 7 Forcing CH + (A) In this section, we define under CH a σ-closed ℵ2-c.c. poset P0 which forces the combinatorial assertion (A) of the previous section. The poset P0 is defined as follows: p ∈ P0 ⇔ p = 〈X p, Y p, τ p, 〈φ : ξ ∈ Dp〉〉 where (7.1) Xp, Y p ∈ [ω2] (7.2) Dp ∈ [ω2] (7.3) for all ξ ∈ Dp, φ : Xp → Xp and ψ : Y p → Y p are involutions (that is, bijections φ such that φ−1 = φ); (7.4) for all ξ ∈ Dp, α ∈ Xp and β ∈ Y p, (7.4a) φ (α) < α + ξ + ω1 and (7.4b) ψ ξ (β) < β + ξ + ω1 ; Note that we have also α < φ (α) + ξ + ω1 and α < ψ (β) + ξ + ω1 for all ξ ∈ D α ∈ Xp and β ∈ Y p since φ ξ and ψ ξ are involutions by (7.3). (7.5) τ p : Xp × Y p → 2 ; (7.6) for all ξ ∈ Dp, α ∈ Xp and β ∈ Y p, we have τ p(α, β) = τ p(φ (α), ψ (β)) ; (7.7) τ p(α, β) = 1 for all 〈α, β〉 ∈ Xp × Y p with β < α. The ordering on P0 is defined by the following: For p, q ∈ P0 with p = 〈Xp, Y p, τ p, 〈φ ξ : ξ ∈ D p〉〉 and q = 〈Xq, Y q, τ q, 〈φ : ξ ∈ Dq〉〉, (7.8) p ≤P0 q ⇔ (7.8a) Xp ⊇ Xq, Y p ⊇ Y q ; (7.8b) Dp ⊇ Dq ; (7.8c) φ and ψ for all ξ ∈ Dq ; (7.8d) τ p ⊇ τ q and (7.8e) τ p ↾ (Xp \Xq)× Y q ≡ 1. For p ∈ P0 with p = 〈X p, Y p, τ p, 〈φ : ξ ∈ Dp〉〉, we intend to approximate the characteristic function of the set A in the assertion (A) by τ p. More precisely, in a generic extension V [G] for a (V,P0)-generic G, letting (7.9) τ = p∈G τ p ; φξ = p∈G φ and ψξ = p∈G ψ for ξ ∈ ω2 ; (7.10) A = τ−1 ′′{1} and F = {〈φξ, ψξ〉 : ξ ∈ ω2}, we are aiming to force 〈(ω2) 2, A,F〉 to satisfy (6.1) ∼ (6.5) in (A). Of the conditions in the definition of P0, (7.5) and (7.8d) force τ to be a function. Furthermore, τ : ω2×ω2 → 2 by density argument and the following Lemma 7.1, (a). (7.3) and (7.8c) make φξ and ψξ mappings for all ξ ∈ ω2; they are forced to be involutions on ω2 by (7.3) and the following Lemma 7.1, (a). Thus 〈(ω2) 2, A,F〉 is forced to satisfy (6.3). By (7.7) (and by the following Lemma 7.1, (a)), 〈(ω2) 2, A,F〉 is forced to satisfy the second inclusion of (6.1). By (7.6), 〈(ω2) 2, A,F〉 is forced to satisfy (6.4). (7.4) and (7.8e) are technical conditions whose role will be clear later in the course of the proof. By the definition of P0, it is clear that P0 is σ-closed. Thus, we are done by showing that P0 satisfies the ℵ2-c.c. and it forces that 〈(ω2) 2, A,F〉 as above satisfies the conditions (6.2) and (6.5). The next Lemma follows readily from the definition of P0. Lemma 7.1. (a) For any α, β < ω2, the set Dα,β = { p ∈ P0 : p = 〈X p, Y p, τ p, 〈φ : ξ ∈ Dp〉〉, α ∈ Xp and β ∈ Y p } is dense in P0. (b) For any C ∈ [ω2] ℵ0 and any β < ω2, EC,β = { p ∈ P0 : p = 〈X p, Y p, τ p, 〈φ : ξ ∈ Dp〉〉, C ⊆ Xp and for some δ ∈ Y p with δ ≥ β τ p(γ, δ) = 0 for all γ ∈ C } is dense in P0. In the rest of the section, we are going to work mainly in the ground model (where CH holds). Let τ̇ , φ̇ξ, ψ̇ξ for ξ ∈ ω2, Ȧ and Ḟ be P0-names of τ , φξ, ψξ for ξ ∈ ω2, A and F as above, respectively. Lemma 7.2. ‖– P0 “ 〈(ω2) 2, Ȧ, Ḟ〉 |= (6.2) ”. Proof. By density argument with Lemma 7.1, (b). (Lemma 7.2) For ξ < ω2, X , X ′, Y , Y ′ ∈ [ω2] ℵ0 with X ′ ⊆ X and Y ′ ⊆ Y , τ : X × Y → 2 and involutions φ′ : X ′ → X ′, ψ : Y ′ → Y ′, let us call the quintuple 〈X, Y, τ, φ′, ψ′〉 a ξ-extendable semi-condition if (7.11) φ′(α) < α + ξ + ω1 and ψ ′(β) < β + ξ + ω1 for all α ∈ X ′ and β ∈ Y ′ ; (7.12) τ(α, β) = τ(φ(α), ψ(β)) for all α ∈ X ′ and β ∈ Y ′ ; (7.13) τ(α, β) = 1 for all α ∈ X and β ∈ Y with β < α ; (7.14) τ ↾ ((X \X ′)× Y ′) ≡ 1. fig. 6 Note that the sets X ′ and Y ′, though not mentioned explicitly in the definition of ξ-extendable semi-condition, can be recovered from φ′ and ψ′. Note also that (7.14) holds vacuously if X = X ′. Hence, if p ∈ P0 with p = 〈Xp, Y p, τ p, 〈φ : ξ ∈ Dp〉〉, the quintuple 〈Xp, Y p, τ p, φξ, ψξ〉 is a ξ-extendable semi-condition for all ξ ∈ Dp. The following two lemmas explain the choice of the naming of ξ-extendable semi-conditions. Lemma 7.3. For any ξ < ω2, X, X ′, Y , Y ′ ∈ [ω2] ℵ0 with X ′ ⊆ X and Y ′ ⊆ Y , τ : X × Y → 2 as well as involutions φ′ : X ′ → X ′, ψ′ : Y ′ → Y ′, if 〈X, Y, τ, φ′, ψ′〉 is a ξ-extendable semi-condition then there are X̃ ⊇ X, Ỹ ⊇ Y , τ̃ : X̃ × Ỹ → 2 with τ̃ ⊇ τ and involutions φ̃ : X̃ → X̃, ψ̃ : Ỹ → Ỹ extending φ′ and ψ′ respectively such that 〈X̃, Ỹ , τ̃ , φ̃, ψ̃〉 is a ξ-extendable semi-condition and (7.15) τ̃ ↾ ((X̃ \X)× Y ) ≡ 1. fig. 7 Proof. Let X0 ∈ [ω2 \X ] ≤ℵ0 be such that (7.16) X0 is order-isomorphic to X \ X ′ and the order-isomorphism identifies points of distance less than ω1 (that is, if α ∈ X \ X ′ and α0 ∈ X0 are identified then we have α < α0 + ω1 and α0 < α+ ω1). Since X (and hence also X \X ′) is countable, we can easily choose the elements of X0 recursively in otp(X \ X ′) steps in accordance with (7.16). Put X̃ = X ∪̇ X0 and let φ̃ be the extension of φ which maps X0 order-isomorphically to X \X ′ and vice versa. Fix θ < ω1 such that (7.17) φ̃(α) ≤ α + ξ + θ for all α ∈ X̃ . There is such θ by (7.11), (7.16) and since | X̃ | ≤ ℵ0. Let Y0 ∈ [ω2 \ Y ] ≤ℵ0 be such that (7.18) Y0 is order-isomorphic to Y \Y ′ and the order-isomorphism identifies points of distance less than ξ + ω1; and (7.19) if β ∈ Y \ Y ′ and β0 ∈ Y0 are identified then β0 > β + ξ + θ. It is easy to see that the elements of such Y0 can be chosen recursively in otp(Y \Y steps. Now let Ỹ = Y ∪̇ Y0 and let ψ̃ be the extension of ψ ′ which maps Y0 order- isomorphically to Y \ Y ′ and vice versa. Finally define τ̃ : X̃ × Ỹ → 2 by (7.20) τ̃(α, β) = τ(α, β), if 〈α, β〉 ∈ X × Y τ(φ̃(α), ψ̃(β)), if 〈α, β〉 ∈ (X ′ × Y0) ∪ (X0 × Y ′) ∪ (X0 × Y0) 1, otherwise for every α ∈ X̃ and β ∈ Ỹ . X0 = X̃ \X fig. 8 〈X̃, Ỹ , τ̃ , φ̃, ψ̃〉 satisfies (7.11) by (7.17) and (7.18). It satisfies (7.12) by (7.20). Thus we are done by checking 〈X̃, Ỹ , τ̃ , φ̃, ψ̃〉 also satisfies (7.15) and (7.13). For (7.15), suppose that 〈α, β〉 ∈ (X̃ \X)× Y (= X0 × Y ). If 〈α, β〉 ∈ X0 × Y then τ̃ (α, β) = τ(φ̃(α), ψ̃(β)) by (7.20). But 〈φ̃(α), ψ̃(β)〉 ∈ (X \ X ′) × Y ′ by definition of φ̃ and ψ̃. Hence, by (7.14), we have τ̃ (α, β) = τ(φ̃(α), ψ̃(β)) = 1. If 〈α, β〉 ∈ X0 × (Y \ Y ′) then τ̃(α, β) = 1 by the “otherwise” clause of (7.20). For (7.13), it is enough to check that τ̃(α, β) = 1 for all 〈α, β〉 ∈ (X ′∪X0)×Y0 with β < α by (7.20) and (7.15). For such 〈α, β〉, we have τ̃ (α, β) = τ(φ̃(α), ψ̃(β)) by (7.20). Suppose that τ̃ (α, β) = 0. Then, since τ satisfies (7.13), we should have φ̃(α) ≤ ψ̃(β). By (7.19), we have β > ψ̃(β) + ξ + θ. On the other hand, by (7.17), we have α = φ̃2(α) ≤ φ̃(α) + ξ + θ. It follows that α ≤ φ̃(α) + ξ + θ ≤ ψ̃(β) + ξ + θ < β. This is a contradiction. (Lemma 7.3) A quartet p = 〈Xp, Y p, τ p, 〈φ : ξ ∈ Dp〉〉 (not necessarily an element of P0) with D p ∈ [ω2] ℵ0 is said to be an extendable condition if 〈Xp, Y p, τ p, φ ξ 〉 is a ξ-extendable semi-condition for all ξ ∈ Dp. For extendable conditions p, q with p = 〈Xp, Y p, τ p, 〈φ : ξ ∈ Dp〉〉, q = 〈Xq, Y q, τ q, 〈φ : ξ ∈ Dq〉〉, we denote p ≤1 q if (7.21) Xp ⊇ Xq, Y p ⊇ Y q, τ p ⊇ τ q, Dp ⊇ Dq, φ and ψ for all ξ ∈ Dp ; (7.22) τ p ↾ (Xp \Xq)× Y q ≡ 1. Note that for p, q ∈ P0, we have p ≤1 q if and only if p ≤P0 q. Lemma 7.4. (Extension Lemma) Suppose that p = 〈Xp, Y p, τ p, 〈φ : ξ ∈ Dp〉〉 is an extendable condition for some Dp ∈ [ω2] ℵ0. Then there is a q ∈ P0 with q = 〈Xq, Y q, τ q, 〈φ ξ : ξ ∈ D q〉〉 such that Dq = Dp and q ≤1 p. Furthermore , if p0 ∈ P0 is such that p ≤1 p0 then we have q ≤P0 p0. Proof. The second part of the lemma is clear once the condition q as in the claim of the lemma is found since (7.8e) holds for such q and p0 since the relation ≤1 is easily seen to be transitive. To construct the desired q ∈ P0, let 〈ξn : n ∈ ω〉 be an enumeration of D p such that each ξ ∈ Dp appears infinitely often in the enumeration. First, construct 〈Xn, Yn, τn, 〈φξ,n, ψξ,n : ξ ∈ D p〉〉, n ∈ ω recursively such that (7.23) 〈X0, Y0, τ0, 〈φξ,0, ψξ,0 : ξ ∈ D p〉〉 = p, (7.24) 〈Xn+1, Yn+1, τn+1, φξn,n+1, ψξn,n+1〉 is the ξn-extendable semi-condition which is constructed just as in Lemma 7.3 from the ξn-extendable semi-condition 〈Xn, Yn, τn, φξn,n, ψξn,n〉. (7.25) φξ,n+1 = φξ,n and ψξ,n+1 = ψξ,n for all ξ ∈ D p with ξ 6= ξn. Along with the recursive construction above, it can be shown easily that 〈Xn, Yn, τn, φξ,n, ψξ,n〉 is a ξ-extendable semi-condition for all n ∈ ω and ξ ∈ D p. Hence the construction in (7.24) is actually possible at each step. n∈ωXn , Y n∈ωXn , τ n∈ω τn , Dq = Dp and φ n∈ω φξ,n , ψ n∈ω ψξ,n for all ξ ∈ D For all ξ ∈ Dq, there are infinitely many n ∈ ω such that ξn = ξ. For such n, φξ,n is an involution on Xn and ψξ,n is an involution on Yn. It follows that φ ξ is an involution on Xq and ψ is an involution on Y q. Hence q = 〈Xq, Y q, τ q, 〈φ : ξ ∈ Dq〉〉 is a condition in P0. Also we have τ q ↾ (Xq \Xp)× Y p = n∈ω τ q ↾ (Xn+1 \Xn)× Y p ≡ 1. Thus, this q is as desired. (Lemma 7.4) Lemma 7.5. (CH) P0 satisfies the ℵ2-c.c. Proof. Actually we shall show that P0 satisfies a strong form of ℵ2-Knaster prop- erty. Suppose that pζ ∈ P0 with p ζ = 〈Xζ, Y ζ , τ ζ , 〈φ : ξ ∈ Dζ〉〉 for ζ ∈ ω2. By the ∆-System Lemma (Theorem 4.1) and the Pigeon Hole Principle, there are a stationary S ⊆ ω2, X , Y , D ∈ [ω2] ℵ0 , τ : X ×Y → 2 and φξ : X → X , ψξ : Y → Y for ξ ∈ D such that (7.26) Xζ, ζ ∈ S form a ∆-system with root X and Y ζ , ζ ∈ S form a ∆-system with root Y ; (7.27) τ ζ ↾ X × Y = τ for all ζ ∈ S ; (7.28) Dζ, ζ ∈ S form a ∆-system with root D ; (7.29) φ ↾ X = φξ and ψ ↾ Y = ψξ for all ζ ∈ S and ξ ∈ D ; (7.30) τ ζ ↾ (Xζ \X)× Y ≡ 1 for all ζ ∈ S . Note that (7.27) is possible since, by CH, there are at most |X×Y2 | ≤ 2ℵ0 = ℵ1 < ℵ2 many possible values of τ ζ ↾ X × Y . (7.29) is possible since, by CH, (7.4) and countability of D, there are at most ℵ1 possible values of 〈φ : ξ ∈ D〉 and : ξ ∈ D〉. (7.30) is possible by (7.7) and since we can choose S such that min(Xζ \X) > sup(Y ) for all ζ ∈ S. Now suppose ζ , η ∈ S with ζ < η. We show that pζ and pη are compatible. Let Xp = Xζ ∪Xη, Y p = Y ζ ∪ Y η and Dp = Dζ ∪Dη. For ξ ∈ Dp, let φ ξ : X p → Xp and ψ : Y p → Y p be defined by (7.31) φ , if ξ ∈ Dζ \D, , if ξ ∈ D, ξ , otherwise and ψ , if ξ ∈ Dζ \D, , if ξ ∈ D, ξ , otherwise. Finally let τ p : Xp × Y p → 2 be such that (7.32) τ p(α, β) = τ ζ(α, β), if 〈α, β〉 ∈ Xζ × Y ζ τ η(α, β), else if 〈α, β〉 ∈ Xη × Y η 1, otherwise. for all 〈α, β〉 ∈ Xp × Y p. It is easy to see that p = 〈Xp, Y p, τ p, 〈φ : ξ ∈ Dp〉〉 is an extendable condition and p ≤1 p ζ, pη. In particular, (7.22) for p ≤1 p ζ and p ≤1 p η holds because of (7.30) and “otherwise” clause of (7.32). By Extension Lemma (Lemma 7.4), there is a q ∈ P0 with q ≤1 p. Hence, by the second half of the lemma, it follows that q ≤P0 p ζ, pη. (Lemma 7.5) A modification of the ∆-system argument in the proof of Lemma 7.5 is also used to prove the following: Lemma 7.6. (CH) P0 forces (6.5). Proof. We show that P0 forces the following: (7.33) For any stationary S ⊆ Eω2ω1 and Aζ , Bζ ∈ [ω2] ℵ0 for ζ ∈ S, there is a stationary T ⊆ S such that for any n ∈ ω and pairwise distinct ζi, ηi ∈ T , i ∈ n, there is ξ < ω2 such that φ̇ξ ′′Aζi = Aηi and ψ̇ξ ′′Bζi = Bηi for all i ∈ n. Note that, by σ-closedness and ℵ2-c.c. of P0 (proved in Lemma 7.5), ω1 and ω2 in generic extensions by P0 remain ω1 and ω2. Suppose that Ṡ is a P0-name of a stationary subset of E . Let 〈Ȧζ : ζ ∈ Ṡ〉 and 〈Ḃζ : ζ ∈ Ṡ〉 be P0-names of sequences of countable subsets of ω2. Let S̃ = {ζ ∈ Eω2ω1 : /‖–P0 “ ζ 6∈ Ṡ ”}. Then we have ‖–P0 “ Ṡ ⊆ S̃ ” and hence S̃ is a stationary subset of E Since P0 is σ-closed, we can find pζ ∈ P0 and Aζ, Bζ ∈ [ω2] ℵ0 such that (7.34) pζ = 〈X ζ, Y ζ , τ ζ , 〈φ : ξ ∈ Dζ〉〉 and (7.35) pζ ‖– P0 “ ζ ∈ Ṡ , Ȧζ = Aζ and Ḃζ = Bζ ” for all ζ ∈ S̃. Without loss of generality, we may assume that (7.36) Aζ ⊆ X ζ and Bζ ⊆ Y By ∆-System Lemma (Theorem 4.1) and the Pigeon Hole Principle, there are a stationary S̃0 ⊆ S̃, X , Y , D ∈ [ω2] ℵ0 , τ : X × Y → 2 and φξ : X → X , ψξ : Y → Y for ξ ∈ D such that (7.37) Xζ, ζ ∈ S̃0 form a ∆-system with root X and Y ζ, ζ ∈ S̃0 form a ∆-system with root Y ; (7.38) sup(Y ) < min(Xζ \X) for all ζ ∈ S̃0 ; (7.39) τ ζ ↾ X × Y = τ for all ζ ∈ S̃0 ; (7.40) Dζ, ζ ∈ S̃0 form a ∆-system with root D ; (7.41) φ ↾ X = φξ and ψ ↾ Y = ψξ for all ζ ∈ S̃0 and ξ ∈ D ; (7.42) τ ζ ↾ (Xζ \X)× Y ≡ 1 for all ζ ∈ S̃0 (this follows from (7.38) and (7.7)) ; (7.43) Xζ, ζ ∈ S̃0 are order-isomorphic and Y ζ , ζ ∈ S̃0 are order-isomorphic; Note that the order-isomorphisms of Xζ ’s and Y ζ ’s do not move elements of X and Y , respectively. (7.44) the order-isomorphism sending Xζ to Xη sends τ ζ ↾ ((Xζ \ X) × Y ) to τ η ↾ ((Xη \X)× Y ) while the order-isomorphism sending Y ζ to Y η sends τ ζ ↾ (X × (Y ζ \ Y )) to τ η ↾ (X × (Y η \ Y )). These order-isomorphisms together send τ ζ ↾ ((Xζ \X)× (Y ζ \ Y )) to τ η ↾ ((Xη \X)× (Y η \ Y )); (7.45) the order-isomorphism sending Xζ to Xη sends Aζ to Aη, and the order- isomorphism sending Y ζ to Y η sends Bζ to Bη. Note that p̄ = 〈X, Y , τ , 〈φξ, ψξ : ξ ∈ D〉〉 is a condition in P0 and pζ ≤P0 p̄ for all ζ ∈ S̃0 (the condition (7.8e) for p̄ and pζ holds by (7.42)). Let Ṫ be a P0-name such that (7.46) ‖– P0 “ Ṫ = {ζ ∈ S̃0 : pζ ∈ Ġ } ” where Ġ is the standard P0-name of the generic set. Claim 7.6.1. p̄ ‖–P0 “ Ṫ is a stationary subset of ω2 ”. ⊢ Since P0 satisfies the ℵ2-c.c. by Lemma 7.5, for any P0-name Ċ of a club subset of ω2, there is a club subset C of ω2 (in the ground model) such that ‖–P0 “C ⊆ Ċ ”. Hence it is enough to show the following: (7.47) For any q ≤P0 p̄ and any club subset C of ω2, there are p ≤P0 q and ζ ∈ C ∩ S̃0 such that p ≤P0 pζ . To show (7.47), let q = 〈Xq, Y q, τ q, 〈φ ξ : ξ ∈ D q〉〉 and let ζ ∈ C ∩ S̃0 be such (7.48) (Xζ \X) ∩Xq = ∅ , (Y ζ \ Y ) ∩ Y q = ∅ and (Dζ \D) ∩Dq = ∅. This is possible by (7.37) and since C ∩ S̃0 is stationary. Let X∗ = Xq ∪Xζ, Y ∗ = Y q ∪ Y ζ , D∗ = Dq ∪Dζ. For ξ ∈ D∗, let φ∗ξ and ψ be partial functions from X∗ to X∗ and from Y ∗ to Y ∗ respectively defined by (7.49) φ∗ξ = , if ξ ∈ Dq \D, , if ξ ∈ D, , otherwise and ψ∗ξ = ψq, if ξ ∈ Dq \D, , if ξ ∈ D, , otherwise. Finally, let τ ∗ : X∗ × Y ∗ → 2 be defined by (7.50) τ ∗(α, β) = τ q(α, β), if 〈α, β〉 ∈ Xq × Y q, τ ζ(α, β), else if 〈α, β〉 ∈ Xζ × Y ζ , 1, otherwise for 〈α, β〉 ∈ X∗ × Y ∗. Then p∗ = 〈X∗, Y ∗, τ ∗, 〈φ∗ξ, ψ ξ : ξ ∈ D ∗〉〉 is an extendable condition and we have p∗ ≤1 q, pζ : (7.22) for p ∗ and pζ follows from τ ∗ ↾ ((X∗\Xζ)×Y ζ) = τ ∗ ↾ ((Xq \X)×(Y ζ \Y ))∪τ ∗ ↾ ((Xq \X)×Y ) ≡ 1 where we have τ ∗ ↾ ((Xq \X) × (Y ζ \ Y )) ≡ 1 by the definition (7.50) of τ ∗ and τ ∗ ↾ ((Xq \X) × Y ) ≡ 1 by q ≤P0 p̄ (in particular, by the condition (7.8e) in the definition of ≤P0). By Extension Lemma (Lemma 7.4) it follows that there is p ∈ P0 with p ≤1 p and hence p ≤P0 q, pζ . ⊣ (Claim 7.6.1) Claim 7.6.2. p̄ forces that Ṫ is as in (7.33) for 〈Ȧζ : ζ ∈ Ṡ〉 and 〈Ḃζ : ζ ∈ Ṡ〉. ⊢ By Claim 7.6.1 it is enough to prove the following: (7.51) For any q ≤P0 p̄ and n ∈ ω, if ζi, ηi ∈ S̃0 are pairwise distinct and q ‖– P0 “ ζi, ηi ∈ Ṫ for i ∈ n ”, then there is p ≤P0 q with p = 〈Xp, Y p, τ p, 〈φ : ξ ∈ Dp〉〉 and ξ0 ∈ D p such that φ ↾ Xζi : Xζi → Xηi and ψ ↾ Y ζi : Y ζi → Y ηi are order-isomorphisms for all i < n. Without loss of generality we may assume that (7.52) q ≤P0 pζi, pηi for all i < n. q = 〈Xq, Y q, τ q, 〈φ ξ : ξ ∈ D Take ξ0 ∈ ω2 \ (D q ∪ sup(Xq) ∪ sup(Y q)) and let D∗ = Dq ∪ {ξ0}. Let X ∗ = Xq, Y ∗ = Y q and τ ∗ = τ q. Let φ∗ξ0 : Xζi ∪ Xηi → Xζi ∪ Xηi be the involution sending Xζi order-isomorphically to Xηi and vice versa for all i < n and ψ∗ξ0 : Y ζi ∪ Y ηi → Y ζi ∪ Y ηi be the involution sending Y ζi order-isomorphically to Y ηi and vice versa for all i < n. Let φ∗ξ = φ and ψ∗ξ = ψ for ξ ∈ Dq. Then p∗ = 〈X∗, Y ∗, τ ∗, 〈φ∗ξ, ψ ξ : ξ ∈ D ∗〉〉 is an extendable condition with p∗ ≤1 q. To see this, we have to check 〈X ∗, Y ∗, τ ∗, φ∗ξ0, ψ 〉 satisfies (7.12) and (7.14). But this follows from (7.42), (7.52) and (7.45). By Extension Lemma (Lemma 7.4) there is p ∈ P0 with p ≤1 p ∗. Clearly p forces that ξ0 as above satisfies (7.33) together with Ṫ , 〈Ȧζ : ζ ∈ Ṡ〉 and 〈Ḃζ : ζ ∈ Ṡ〉. ⊣ (Claim 7.6.2) Since the argument above can be repeated below arbitrary element of P0, it follows that P0 forces (7.33). (Lemma 7.6) 8 A summary of consistency results and some open problems The following is a summary of consistency results in connection with the combina- torial principles in fig. 5 where (1) ∼ (7) below correspond to the separation lines (1) ∼ (7) drawn in fig. 9. (1): By adding random reals. More precisely, start from a model V of CH and force with (the positive elements of) the measure algebra B of, say, Maharam type ℵ2. B can be seen as a (measure theoretic) product of random forcing and inherits thus some of the homogeneity property of finite support product. This is used to prove do = ℵ1 in the generic extension. It is also well-known that the ground model functions from ω to ω dominate the functions in a generic extension by a measure algebra. Hence we have d = ℵ1 in the model. K. Kunen proved that there is a κ-Lusin gap for an uncountable κ in such a model. On the other hand, I. Juhász, L. Soukup and Z. Szentmiklóssy proved in [13] that there is no ℵ2-Lusin gap under Cs(ℵ2). This proves that C s(ℵ2) does not hold in the generic extension. This observation may be also interpreted as pinning down of the difference in the extent of homogeneity of product forcing and the forcing by measure theoretic products in terms of whether the principle Cs(ℵ2) holds. (2): A model constructed by J. Brendle and T. LaBerge in [1] realizes this separation. (3): By the model in Theorem 3.8 of I. Juhász and K. Kunen [12] in which Cs(ℵ2) and do > ℵ1 hold. The model is obtained by a finite support product of ℵ2 posets of cardinality ℵ1 starting from a model of CH. From this, it follows easily that b∗ = ℵ1 and d = ℵ2. (4): By adding Cohen reals. More exactly, start from a model V of CH and then add, say, ℵ2 Cohen reals (by Fn(ℵ2, 2)). Then by Corollary 4.8, (c) we have IP(ℵ2,ℵ1) in the generic extension. Just as in (3), we have d = ℵ2 in such a generic extension and it is shown in S. Fuchino, S. Koppelberg and S. Shelah [8] that WFN holds there. (5): By a model of Hechler. (6): By Theorem 6.4. (7): By Corollary 4.9. (8): By Theorem 5.2. See [11] for the proof of ‖– Sκ “¬WFN”. do = ℵ1 HP(ℵ2) h = ℵ1b ↑ = ℵ1b = ℵ1 b ∗ = ℵ1 Cs(ℵ2) d = ℵ1 (4)(5) IP(ℵ2,ℵ2) IP(ℵ2,ℵ1) fig. 9 Finally, we shall mention some open problems. In [5] it is shown that a = ℵ1 follows from WFN where a is the almost disjoint number. In [4], it is then shown that, under some additional assumptions, a = ℵ1 already follows from SEP which is a weakening of WFN. Therefore, it seems natural to ask the following question: Problem 1. Does a = ℵ1 follow from HP(ℵ2) or IP(ℵ2, λ) for λ = ℵ1,ℵ2 ? Problem 2. Does WFN imply HP(ℵ2) or do = ℵ1 ? The model of b∗ = ℵ2 and do = ℵ1 satisfies a strong form of negation of C s(ℵ2). This suggests the following problem: Problem 3. Does HP(ℵ2) (or even C s(ℵ2)) imply b ∗ = ℵ1 ? At the moment, we do not have any model separating HP(κ) and IP(κ, κ) for κ > κ1. Problem 4. Is HP(κ) + ¬IP(κ, κ) consistent for some (or any) κ > ℵ1 ? In Corollary 4.9 which realizes the separation (7) in fig.9, a very strong large cardinal property is assumed. Problem 5. Can we construct a model realizing (7) in fig.9 starting from ZFC without any large cardinal? The property (A) used in the proof of Theorem 6.4 and proved to be consistent with CH in Section 7 seems to be of its own interest. Problem 6. Is ¬(A) consistent with ZFC + CH (or even with ZFC + GCH) ? References [1] J. Brendle and T. LaBerge, Forcing tightness in products of fans, Fundamenta Mathematicae, 150, 211-226 (1996). [2] K. Ciesielski and J. Pawlikowski, The Covering Property Axiom, CPA, Cam- bridge University Press (2004). [3] K. Eda, M. Kada and Y. Yuasa, The tightness about sequential fans and combinatorial properties, Journal of Mathematical Society of Japan 49, 181– 197 (1997). [4] S. Fuchino and S. Geschke, Some combinatorial principles defined in terms of elementary submodels, Fundamenta Mathematicae 181, 233-255 (2004). [5] S. Fuchino, S. Geschke and L. Soukup, On the weak Freese-Nation property of P(ω), Archive for Mathematical Logic, Vol.40, 425-435 (2001). [6] S. Fuchino, S. Geschke and L. Soukup, Principles capturing features of generic extensions by almost side-by-side product, in preparation. [7] S. Fuchino, S. Geschke, S. Shelah and L. Soukup, On the weak Freese-Nation property of complete Boolean algebras, Annals of Pure and Applied Logic, 110 (1-3) (2001) 89-105. [8] S. Fuchino, S. Koppelberg and S. Shelah, Partial orderings with the weak Freese-Nation property, Annals of Pure and Applied Logic 80, 35–54 (1996). [9] S. Fuchino, S. Shelah and L. Soukup, Sticks and clubs, Annals of Pure and Applied Logic 90, no.1 (1997), 57-77. [10] S. Fuchino and S. Soukup, More set-theory around the weak Freese-Nation property, Fundamenta Matematicae 154, 159–176 (1997). [11] S. Geschke, On σ-Filtered Boolean Algebras, Dissertation, Freie Universität Berlin (1999). [12] I. Juhász and K. Kunen, The Power Set of ω, Elementary submodels and weakenings of CH, Fundamenta Mathematicae 170, 257–265 (2001). [13] I. Juhász, L. Soukup and Z. Szentmiklóssy, Combinatorial principles from adding Cohen reals, Proceedings of Logic Colloquium (1995). [14] M. Kada and Y. Yuasa, Cardinal invariants about shrinkability of unbounded sets, Topology and its Applications 74 (1996), 215–223. [15] K. Kunen: Set Theory, North-Holland (1980). [16] A. Miller, Infinite Combinatorics and Definability, Annals of Pure and Ap- plied Mathematical Logic, 41 (1989), 179-203. [17] S. Shelah, a(n unpublished?) note for I. Juhász (2002). Authors’ addresses Jörg Brendle Graduate School of Science and Technology Kobe University Rokko-dai 1-1, Nada KOBE 657-8501 Japan brendle@kurt.scitec.kobe-u.ac.jp Sakaé Fuchino Dept. of Natural Science and Mathematics College of Engineering, Chubu University, Kasugai AICHI 487-8501 Japan. fuchino@isc.chubu.ac.jp

Evolution favors protein mutational robustness in sufficiently large populations Jesse D Bloom∗1 , Zhongyi Lu1 , David Chen1 , Alpan Raval2 , Ophelia S Venturelli1 and Frances H Arnold∗1 Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA Keck Graduate Institute of Applied Life Sciences and School of Mathematical Sciences, Claremont Graduate University, Claremont, CA 91711, USA Email: Jesse D Bloom∗- jesse.bloom@gmail.com; Zhongyi Lu - lu07@caltech.edu; David Chen - davidc@caltech.edu; Alpan Raval - alpan raval@kgi.edu; Ophelia S Venturelli - opheliav@stanford.edu; Frances H Arnold∗- frances@cheme.caltech.edu; Corresponding author Abstract Background: An important question is whether evolution favors properties such as mutational robustness or evolvability that do not directly benefit any individual, but can influence the course of future evolution. Functionally similar proteins can differ substantially in their robustness to mutations and capacity to evolve new functions, but it has remained unclear whether any of these differences might be due to evolutionary selection for these properties. Results: Here we use laboratory experiments to demonstrate that evolution favors protein mutational robustness if the evolving population is sufficiently large. We neutrally evolve cytochrome P450 proteins under identical selection pressures and mutation rates in populations of different sizes, and show that proteins from the larger and thus more polymorphic population tend towards higher mutational robust- ness. Proteins from the larger population also evolve greater stability, a biophysical property that is known to enhance both mutational robustness and evolvability. The excess mutational robustness and stability is well described by existing mathematical theories, and can be quantitatively related to the way that the proteins occupy their neutral network. Conclusions: Our work is the first experimental demonstration of the general tendency of evolution to favor mutational robustness and protein stability in highly polymorphic populations. We suggest that this phenomenon may contribute to the mutational robustness and evolvability of viruses and bacteria that exist in large populations. http://arxiv.org/abs/0704.1885v1 Background Proteins are quite tolerant of mutations, al- lowing evolution to produce highly diverged sequences that fold to similar structures and perform conserved biochemical functions [1, 2]. However, proteins with nearly identical struc- tures and functions may differ in their robust- ness to mutation [3–5], as well as in their ca- pacity to acquire new functions [5]. The fact that mutational robustness and evolvability can vary among the functionally equivalent proteins produced by natural sequence divergence makes these properties important hidden dimensions in evolution — direct selection for protein func- tion is blind to them, yet they can play a cru- cial role in enabling future evolution. Whether the evolutionary process somehow promotes the acquisition of mutational robustness and evolv- ability therefore remains a major question [6–8]. Previous experiments have identified several specific evolutionary conditions that can affect mutational robustness. For example, genetic complementation decreases the mutational ro- bustness of viruses [9], while high mutation rates favor mutational robustness in simulated digi- tal organisms [10]. However, theory [11] makes the much broader — and heretofore experimen- tally untested — prediction that extra muta- tional robustness will arise quite generally in sufficiently large populations. This prediction cannot be understood in the standard frame- work of Kimura’s neutral theory [12], since one of the usual assumptions of the neutral theory is that mutational robustness is constant. (Al- though Takahata [13] treated the consequences of stochastically fluctuating neutrality on the molecular clock, he did not describe how muta- tional robustness might change systematically during evolution.) However, changes in mu- tational robustness can be described by envi- sioning evolution as occurring on neutral net- works, or sets of functionally equivalent pro- teins that are connected by single mutational steps [14–17]. In a seminal theoretical analysis of evolution on neutral networks, van Nimwe- gen and coworkers [11] predicted that the extent of mutational robustness should depend on the degree of population polymorphism. Here we briefly summarize their reasoning, since it moti- vates our experimental work. We also refer the reader to chapter 16 of [8], which contains an excellent explanation of the densely mathemat- ical work of van Nimwegen and coworkers [11]. If an evolving population is mostly monomorphic, then each mutation is either lost or goes to fixation before another muta- tion occurs. The population is therefore usually clustered at a single genotype and rarely expe- riences mutations, meaning that selection does not distinguish between genotypes of different mutational robustness. All nodes of the neutral network are thus equivalent and will be occupied by the population with equal probability [11]. On the other hand, a highly polymorphic pop- ulation is always spread across many nodes of the neutral network. When mutations occur, the members of the population at highly con- nected nodes have a better chance of surviving, causing them to be favored by evolution and increasing the average mutational robustness [11, 17–20]. Specifically, a highly polymorphic population occupies each node with a proba- bility proportional to its eigenvector central- ity [11, 17], a measure of how connected it is to other connected nodes (a variant of eigen- vector centrality is used by Google’s PageRank algorithm to rank a webpage’s importance in the network of internet links [21]). Figure 1A illustrates how mostly monomorphic and highly polymorphic populations are predicted to oc- cupy a neutral network. For proteins, changes in neutral network occupancy should be man- ifested by changes in thermodynamic stabil- ity [22], with proteins from highly polymorphic populations predicted to be more stable than their counterparts from mostly monomorphic populations (Figure 1B). Note that the extent of polymorphism depends on the product of the mutation rate and population size, meaning that protein populations of different sizes are predicted to evolve to different levels of mu- tational robustness and stability even if they experience the same mutation rate. Results and Discussion Design of neutral evolution experiment To test whether high population polymorphism drives an increase in mutational robustness and protein stability, we performed laboratory evo- lution experiments on cytochrome P450 pro- teins. The basic idea was to neutrally evolve P450s under a constant selection pressure in populations that were either monomorphic or highly polymorphic, and observe whether the proteins evolved to different levels of mutational robustness and stability. The evolution experi- ments started with a P450 BM3 heme domain that had been engineered to hydroxylate 12-p- nitrophenoxydodecanoic acid (12-pNCA) [23]. We imposed the selection criterion that Es- cherichia coli cells expressing the P450 had to yield lysate with enough active enzyme to hy- droxylate a specified amount of 12-pNCA in 40 minutes. This criterion roughly corresponds to the case in which an enzyme must catalyze a biochemically relevant reaction at some mini- mal level in order for its host to survive. Note that other properties such as stability and ex- pression level can vary freely, provided that the criterion for total activity is met. The properties of a neutrally evolving pro- tein eventually “equilibrate,” much as the prop- erties of an isolated physical system under some macroscopic constraint tend towards the values that maximize the system’s internal entropy. For proteins, this usually means that stability, expression, and activity drift towards their low- est tolerable values, since the vast majority of random sequences do not encode stable, well- expressed enzymes (that is, natural selection must work against sequence entropy to maintain a functional protein) [22, 24]. The initial P450 had been engineered for maximal activity [23], meaning that it was not equilibrated to the more mild selection criterion of the experiments. We therefore neutrally evolved this initial P450 for 16 generations, introducing random mutations with error-prone PCR and retaining all mu- tants that met the selection criterion for total activity on 12-pNCA. The procedure used for this equilibration evolution was similar to that for the polymorphic neutral evolution described below. As expected, expression, stability, and activity all dropped during the equilibration evolution. At the end of the equilibration evo- lution, we chose a single sequence as the parent for the neutral evolution experiments. The gene encoding this parent sequence contained 29 nu- cleotide mutations and 13 amino acid mutations relative to the initial P450 (Additional File 1). We used this parent gene to begin three parallel sets of neutral evolution experiments, which we named “monomorphic,” “polymor- phic,” and “unselected” (Figure 2). The monomorphic experiments capture the case where the population moves as a single entity, the polymorphic experiment captures the case where the population spreads across many se- quences, and the unselected experiments show how the gene evolves in the absence of selection for protein function. In all experiments, at each generation we used error-prone PCR to intro- duce an average of 1.4 nucleotide mutations per P450 gene (Table 1). The mutant genes were ligated into a plasmid and transformed into E. coli [25], and transformants were selected using the plasmid’s antibiotic resistance marker. For the unselected case, we randomly picked one of the mutants, recovered the mutant gene with a plasmid mini-prep, and used this mutant as the template for the next generation of error- prone PCR. We performed four independent replicates of unselected evolution, evolving each for 12 generations. For the monomorphic and polymorphic pop- ulations, we imposed the selection criterion that the P450s hydroxylate 12-pNCA with at least 75% of the total activity of the original par- ent gene. We expressed the P450s in E. coli, and then assayed the cell lysates for activity in a high-throughput 96-well plate format. The total amount of product produced by 80 µl of clarified lysate in 40 minutes was compared to the median of four control wells containing the original parent P450 to determine if the mutant met the selection criterion. The only differ- ence between the monomorphic and polymor- phic experiments was the size of the evolving populations. In the monomorphic limit, each mutation is either lost or goes to fixation before the next occurs. We enforced this evolutionary dynamic by holding the population size to a single protein sequence, similar to the “blind ant” random walk of [11]. At each generation, we assayed a single mutant. If this mutant met the selection criterion, then it was carried over to the next generation, corresponding to a neu- tral mutation going to fixation. If the mutant failed the selection criterion, then the popu- lation stayed at the previous sequence for the next generation, corresponding to a mutation lost to selection. If all of the mutants assayed had zero or one mutations, then this proto- col would correspond exactly to the equations of [11, 22]. However, in order to achieve appre- ciable sequence evolution on a laboratory time scale, we used a mutation rate that sometimes produced multiple mutations in a generation. We mathematically describe this situation in the Mathematical Appendix; here we simply note that it is possible to think of each gener- ation as introducing a single mutational event rather than a single mutation. We performed 22 independent replicates of monomorphic evo- lution, evolving each for 25 generations. In the polymorphic limit, the population spreads across many sequences. To implement this experimentally, we assayed 435 mutants at each generation. The selection criterion was used to classify each mutant as functional or nonfunctional. In neutral evolution, all func- tional mutants reproduce with equal proba- bility. We therefore pooled equal volumes of stationary-phase cultures of each functional mu- tant and recovered the pooled genes with a mini- prep. The polymorphic evolution experiment therefore approaches the equations of [11, 22], again with the exception that a sequence may undergo multiple mutations at a single gener- ation. We give the equations describing this situation in the Mathematical Appendix. Since the population evolves deterministically in the polymorphic limit [11, 22], a single replicate was performed. Because mutations accumu- late more rapidly in the polymorphic experi- ments than the monomorphic ones, we evolved the polymorphic population for 15 generations rather than 25. Mutations and mutational robustness Figure 3 shows how mutations accumulated dur- ing the course of the neutral evolution experi- ments (full data are in Table 2 and Additional File 2). Since the unselected protein popula- tions evolve without constraint, mutations ac- cumulate at the same rate at which they are introduced by error-prone PCR, 1.4 nucleotide mutations per generation. Because selection eliminates mutations that disrupt P450 activ- ity, mutations accumulate more slowly in the monomorphic and polymorphic populations. Mutations accumulate more rapidly in the poly- morphic population than in the monomorphic populations. This difference in rates is pre- dicted by the equations in the Mathematical Appendix to be a consequence of the fact that the polymorphic population is more mutation- ally robust, and so can tolerate more of the possible mutations. To test directly whether the polymorphic population evolves higher average mutational robustness, we measured the fraction of 435 random mutants that met the selection crite- rion. Figure 4 shows that the polymorphic pop- ulation neutrally evolved to a markedly higher mutational robustness than the monomorphic populations, with 50± 2% of the final polymor- phic population mutants continuing to function versus 39 ± 2% for the final monomorphic pop- ulations (Chi-square P -value of 10−3 that these values are significantly different). The only dif- ference between the two types of populations was their size, so evolution has clearly favored mutational robustness in the larger and thus more polymorphic population. This finding represents the first experimental support for the prediction that highly polymorphic popu- lations evolve excess mutational robustness [11]. Theory predicts that the excess mutational robustness of a highly polymorphic protein population comes from increased protein sta- bility [22]. Because the P450 variants unfold irreversibly, an equilibrium thermodynamic sta- bility ∆Gf cannot be measured. We therefore determined stability to irreversible thermal and chemical denaturation, two highly correlated measures of P450 stability that have previously been shown to contribute to mutational ro- bustness [5] (see Additional Files 3, 4, and 5). Figure 5 shows that proteins from the polymor- phic population were in fact more stable than their counterparts from the monomorphic pop- ulation. We also observed that proteins in the polymorphic population tended to accumulate to higher levels in E. coli (Figure 5). Elevated expression could be a byproduct of increased stability, or it could independently increase mu- tational robustness by allowing the proteins to better tolerate mutations that decrease codon adaptation or reduce folding efficiency. It is pos- sible that additional unrecognized biophysical factors also contributed to the excess mutational robustness of the polymorphic population, but no such factors were immediately obvious. Interpretation in terms of the P450 neutral network The higher mutational robustness of the poly- morphic population is due to the fact that it occupies the P450 gene neutral network dif- ferently than the monomorphic populations. Measurements from the evolution experiments can therefore be used to infer basic proper- ties of the underlying neutral network of P450 genes, as originally noted by van Nimwegen and coworkers [11]. In the Mathematical Appendix, we derive approximations for the normalized principal eigenvalue 〈ν〉∞ and the normalized average connectivity 〈ν〉o of the neutral net- work, where in both cases the normalization is obtained by dividing by the network coordi- nation number. We obtain 〈ν〉∞ = 0.51 and 〈ν〉o = 0.35 for the P450 gene neutral net- work. Our ability to consistently estimate these two parameters from four different experimental measurements supports the idea that the the- ory that we elaborate in the Mathematical Ap- pendix appropriately describes the experiments. The difference between 〈ν〉∞ and 〈ν〉o is a mea- sure of the extent to which some P450 neutral network nodes have more connections than oth- ers. We note that 〈ν〉∞ is approximately equal to the exponential decline parameter for the asymptotic decline in the fraction of functional mutants with increasing numbers of random nu- cleotide mutations [3, 26,27] (see Mathematical Appendix). Previous studies looking at this ex- ponential decline have reported 〈ν〉∞ = 0.7 for subtilisin [26], 〈ν〉∞ = 0.7 for 3-methyladenine DNA glycosylase [27], and 〈ν〉∞ = 0.7 - 0.8 for TEM1 β-lactamase [3]. These comparisons suggest that P450 has a sparser neutral net- work (smaller 〈ν〉∞) than these other proteins. We suspect, however, that these earlier studies (one of which is our own) overestimate 〈ν〉∞ due to insufficient equilibration of the starting sequence. We believe that the approach of the current work is more accurate for determining 〈ν〉∞ because the measurements are made after many mutations have equilibrated the initial se- quence. This approach could be used in future experiments to compare the neutral network connectivities of proteins from different fami- lies. Conclusions We have demonstrated that neutral evolution favors more mutationally robust proteins when the evolving population is highly polymorphic. Strikingly, the excess mutational robustness is due only to population polymorphism, and so will arise in any population of sufficiently large size. Our work is the first experimental demon- stration of this phenomenon, which is predicted to occur quite generally in neutrally evolving proteins and nucleic acids [11]. Furthermore, we were able to identify one of the biophysical factors underlying the increase in mutational robustness by showing that proteins from the highly polymorphic population are more sta- ble. We recognize that evolution in a biological context will be more complex. In our exper- iments, fitness was the P450’s ability to be expressed in active form by bacteria grown to saturation in an environment with plentiful nu- trients. Biological fitness, however, depends on numerous additional and subtle effects such as the metabolic costs of synthesis or the burdens imposed by misfolded molecules. Some muta- tions that are neutral in the experiments may therefore have deleterious effects in a biologi- cal setting [28]. The experiments nonetheless capture the overriding constraint that proteins retain their biochemical functions. Our success in quantitatively explaining the results supports the notion that important aspects of protein evolution can be described simply in terms of mutational effects on stability [22,28]. An obvious question is whether evolution in nature favors mutational robustness by the process we have demonstrated. Whether natu- ral populations will neutrally evolve mutational robustness depends on whether they are suffi- ciently polymorphic, which will be the case if the product of their effective population size N and per protein per generation mutation rate µ is much greater than one [11, 12]. Accurately estimating Nµ, which is closely related to the widely used parameter θ in population genet- ics, for natural populations is difficult [29, 30] (note that since mutational robustness is a protein-wide property, the relevant mutation rate is per protein, which is ≈ 102 to 103 larger than the per codon mutation rate). For hu- mans and other multicellular organisms, Nµ is probably too small [31] for their proteins to neutrally evolve mutational robustness. But estimates [31, 32] place Nµ ≈ 10 to 100 for typical-length proteins in bacteria, and it is probably much higher for many viruses [33,34]. It is therefore likely that many viral and some bacterial proteins have neutrally evolved extra mutational robustness. The neutral evolution of protein mutational robustness may also contribute to adaptive evo- lution. Experiments have shown that extra sta- bility increases a protein’s evolvability by allow- ing it to tolerate a wider range of functionally beneficial but destabilizing mutations [5]. A similar phenomenon seems to occur in natural evolution, where functionally neutral but stabi- lizing mutations can play a key role in adaptive evolution by counterbalancing the destabilizing effects of other functionally beneficial muta- tions [35]. Viruses and perhaps bacteria may thus benefit from large population sizes and high mutation rates that drive an increase in the mutational robustness and stability of their proteins, which in turn enhances the capacity of these proteins to rapidly change their sequences and evolve new functions. Methods Equilibration evolution of the P450 protein We began with a 21B3 P450 peroxygenase that had been engineered for highly efficient hydrox- ylation of 12-pNCA [23] (sequence shown in Ad- ditional File 6). This P450 was not well equi- librated to the constant selection criterion that we planned to impose, since it had substantially higher total activity. We therefore neutrally evolved it for 16 generations in order to cre- ate P450s that were better equilibrated to the selection criterion. We evolved two parallel pop- ulations, which we named R1 and R2. The pro- cedure was exactly identical to that described below for the polymorphic evolution with the following exceptions: • Starting sequence: the starting sequence for the equilibration evolution was the 21B3 sequence. • Population size: each of the two equili- bration evolution populations had a size of 174 sequences rather than the 435 used for the polymorphic evolution. • Selection criterion: the sequences were re- quired to have at least 75% of the total activity of the 21B3 P450. • Mutation rate: the mutation rate for the equilibration evolution was much higher than for the polymorphic evolution. The error-prone PCR protocol used 200 µM manganese chloride (MnCl2), rather than the 25 µM used for the polymorphic evo- lution. We estimate that this error-prone PCR protocol introduced ≈ 4 nucleotide mutations per P450 gene at each genera- tion during the equilibration evolution. We performed 16 generations of equilibration evolution, and then randomly selected 23 func- tional mutants from each of the R1 and R2 populations (Additional File 7). We picked one of these mutants, R1-11, for use as the parent for the neutral evolution experiments. Detailed protocol for evolution experiments We began with the R1-11 P450 BM3 heme domain variant (sequence in Additional File 1) cloned into the pCWori [25] plasmid with a 5’ BamH1 and 3’ EcoR1 site as described in [5]. The cloning primers were pCWori for (5’- GAAACAGGATCCATCGATGCTTAGGAGGTCAT- 3’ and pCWori rev clone (5’-GCTCATGTTTGACAGCTTATCATCG- 3’). We used error-prone PCR to generate mutants, taking great care to make the error- prone PCR protocol repeatable by using a rela- tively small number of thermal cycles. This was both to control the mutation rate by ensuring that the reaction did not saturate the reagents (which would cause the number of doublings to become sensitive to the initial template concen- tration), and to avoid the PCR-based recombi- nation events which can occur during with the last few thermal cycles of PCR reactions [36,37]. The PCR reactions were 100 µl in volume, and contained ≈ 13 ng of plasmid template (cor- responding to ≈ 3 ng of template gene), 7 mM magnesium chloride MgCl2, 1 × Applied BioSystems PCR Buffer II without MgCl2, 25 µM MnCl2, 0.5 µM pCWori for primer, 0.5 µM pCWori rev primer, 200 µM of dATP and dGTP, 500 µm of dTTP and dCTP, and 5 units of Applied Biosystems AmpliTaq polymerase. The reactions were run on the BLOCK setting of a MJ Research PCR machine with a program of 95oC for 2 minutes, then 15 cycles of (95oC for 30 seconds, 57oC for 30 seconds, 72oC for 90 seconds), and then cooling to 4oC. This proto- col yielded roughly 1-1.5 µg of product gene (as quantified by gel electrophoresis versus a known standard), for a PCR efficiency of ≈ 0.5. Se- quencing the unselected populations at the end of the experiment indicated that this protocol introduced an average of 1.4 ± 0.2 nucleotide mutations, with the nucleotide error-spectrum shown in Table 1. Because the number of PCR doublings is large compared the average muta- tion rate, the distribution of mutations among sequences should be well-described by the Pois- son distribution [38,39]. The mutant genes from the error-prone PCR were purified over a ZymoResearch DNA clean and concentrator column, and digested at 37oC with EcoR1 and BamH1. The digested genes were then purified from an agarose gel with ZymoResearch DNA gel extraction columns, and ligated into pCWori plasmid that had been digested with BamH1 and EcoR1 and dephos- phorylated. The ligations were transformed into electro-competent catalase-free Escherichia coli [25] (the catalase is removed because it breaks down the hydrogen peroxide utilized by the P450 peroxygenase), plated on Luria Broth (LB) plates containing 100 µg/ml of ampicillin to select for the plasmid’s antibiotic resistance marker, and grown at 37oC. Transformation of a control ligation reaction without any digested gene yielded at least 100-fold fewer colonies, indicating that the rate of plasmid self-ligation was less than one percent. Individual mutant colonies from the plates were picked into 96-well 2 ml deep-well plates containing 400 µl of LB supplemented with 100 µg/ml ampicillin. Each plate contained four parental control wells with cells carrying the parent R1-11 gene, four null control wells with cells carrying the pCWori plasmid without a P450 gene, and a non-inoculated well to check for contamination. For the polymorphic pop- ulation, we picked five such plates with all 87 other wells containing different mutants for a total population size of 5 × 87 = 435 mutants. For the 22 monomorphic populations (we be- gan with 24 populations but two had to be discarded due to contamination), we picked a single colony for growth and screening. For the unselected populations we picked a single colony for growth without screening. The LB deep-well plates were grown for 16-20 hours at 30oC, 210 revolutions per minute (rpm), and 80% relative humidity in a Kuhner humidified shaker. To ex- press the P450 mutants, we prepared 2 ml deep well plates containing 400 µl per well of terrific broth (TB) supplemented with 200 µM iso- propyl β-D-thiogalactoside (IPTG), 100 µg/ml ampicillin, and 500 µM of δ-aminolevulinic acid. We used a pipetting robot inoculated these TB plates with 100 µl from the LB plates. We stored the LB deep-well plates at 4oC, and grew the TB deep-well plates in the humidi- fied shaker at 30oC, 210 rpm, and 80% relative humidity for 22-24 hours. After this growth, the cells were harvested by centrifuging the TB deep-well plates at 4000×g for 5 minutes and discarding the liquid. The cell pellets were flash-frozen in liquid nitrogen to aid in cell lysis. To lyse the cells for the assays, we resus- pended the cell pellets in 300 µl of 100 mM [4- (2-hydroxyethyl)-1-piperazinepropanesulfonic acid] (EPPS) (pH 8.2) with 0.5 mg/ml lysozyme and 4 units/ml of deoxyribonuclease by pipet- ting 40 times with the pipetting robot. We then incubated the plates at 37oC for 30 minutes, again resuspended with the pipetting robot, and put back at 37oC for an addition 30 minutes. We then pelleted the cell debris by centrifu- gation at 6000×g for 5 minutes at 4oC. The pipetting robot was used to dispense 80 µl of the clarified lysate into 96-well microtiter plates (Rainin). We prepared a 6× stock of 1.5 mM 12-pNCA in 36% dimethyl sulfoxide (DMSO) and the EPPS buffer (the 12-pNCA was stored in the DMSO solution and combined with the buffer immediately before use). We used a mul- tichannel pipette to add 20 µl of this substrate stock to each well of the microtiter plate. We briefly mixed the plates with “shake” setting of a 96-well plate spectrophotometer, and read an absorbance baseline at 398 nm. We then immediately added 20 µl of a freshly prepared solution of 24 mM hydrogen peroxide in the EPPS buffer to initiate the reaction, and mixed again. The final reaction conditions were there- fore the EPPS buffer with 6% DMSO, 4 mM hydrogen peroxide, and 250 µM 12-pNCA. After 40 minutes we quantified the amount of enzy- matic product by the increase in absorbance at 398 nm. This absorbance increase is due to the 4-nitrophenolate molecule released after the P450 hydroxylates the twelfth carbon of the 12-pNCA molecule [23]. To score the mutants as functional or nonfunctional, we compared their gain in absorbance minus the median null control reading to that of the median parental control reading minus the median null control reading. All mutants that had at least 75% of the parental gain were scored as functional, all other mutants were scored as nonfunctional. We used the information from these assays to select the parents for the next generation. For the unselected population we did not require the mutants to be functional, so the selected mutant was used to start a 4 ml culture of LB with 100 µg/ml ampicillin, and the plasmid DNA was harvested with a mini-prep. This plasmid DNA was used as the template for the next round of error-prone PCR. Therefore, after the first gen- eration the four unselected replicates diverged into four separate error-prone PCR reactions. These unselected replicates were evolved for a total of twelve generations, and were sequenced at every third generation. For the polymorphic population, all mu- tants that were functional contributed an equal amount of plasmid DNA as template for the next generation. In order to do this, we col- lected 50 µl of the culture from the LB deep-well plate for each mutant that was scored as func- tional. All of these LB aliquots were pooled, and then the plasmid DNA was collected with a mini-prep. The pool of plasmid DNA was used as template for the next generation’s error- prone PCR reactions. We performed 15 gener- ations of evolution for this polymorphic pop- ulation. Note that at each generation we are assaying 435 mutants as part of the evolution- ary procedure, so this provides information on mutational robustness. At every third gen- eration, we also selected a random sample of functional mutants for sequencing. After 15 generations, we randomly selected 22 mutants for stability measurements and sequencing anal- ysis. The random selections were made from all functional mutants with the Python computer language random number generator. For the monomorphic populations, at each generation we assayed just a single mutant. If that mutant was nonfunctional, then at that generation the population stayed at its original sequence. In that case, for the next generation we simply picked a new mutant from the previ- ous generation’s plate of transformed mutants. If the mutant we screened was functional, then that mutant represented the new population. We therefore grew a 4 ml LB culture with 100 µg/ml of ampicillin, and collected the plasmid DNA with a miniprep. That plasmid DNA was then used as the template for the next genera- tion’s error-prone PCR reaction. We thus had 22 (actually 24 before 2 were contaminated) in- dependent monomorphic populations that were being evolved in parallel. Each was evolved for 25 generations, and at the end of these 25 gen- erations we measured the stability of the final sequence of each population. Each time an as- sayed mutant was functional, we sequenced the new P450 gene. We also measured the average mutational robustness of the monomorphic pop- ulations at every fifth generation. To do this, we did a pooled mini-prep of equal volumes of LB cultures of all 22 replicates to obtain a equal mix of plasmid DNA. We then performed error-prone PCR on this mix, and assayed 435 mutants to measure the fraction functional. Full neutral evolution data are in Additional File 2. Test for recombination during error-prone During the polymorphic population evolution, we performed error-prone PCR on a mix of different plasmids. It is common for PCR on mixed templates to lead to recombination events during the reaction [36, 37]. We at- tempted to reduce this recombination by using a small number of thermal cycles. However, in order to test for recombination, we analyzed the sequences of the final 22 selected members of the polymorphic population. There are a variety of statistical tests to detect recombi- nation in a set of sequences. A comparison of these tests by Posada [40] found that the Max-Chi2 method developed by John Maynard Smith [41] performs well. A publicly avail- able implementation of this method [42] is at http://www.lifesci.sussex.ac.uk/CSE/test/maxchi.php. We used this implementation to analyze the 22 final polymorphic sequences, and the resulting P -value was 0.29 after 100 random permuta- tions, indicating that there is not significant recombination. Measurement of P450 stabilities We measured the stabilities to both irreversible thermal and irreversible urea denaturation of the final (generation 25) member of each monomorphic population, as well as of the 22 randomly selected members of the polymorphic population. As discussed in the Supplemen- tary Information of [5], cytochrome P450 BM3 heme domains (and indeed most P450s) dena- ture irreversibly, forcing us to use resistance to irreversible denaturation to quantify protein stability. The first stability measure is the T50, defined as the temperature at which half of the protein is denatured after a 10 minute incu- bation. The second stability measure is the [urea]50, defined as the urea concentration at which half of the protein denatures after a 4 hour room-temperature incubation. Each set of measurements (those of T50 and [urea]50) was performed on all of the mutants in the same day, and each mutant was treated identi- cally. Therefore, it is possible to make accurate comparisons of the relative values of the mea- surements within the data set. However, the absolute values of the T50 and [urea]50 values may be less accurate. Therefore, care should be taken in comparing the absolute value of these measurements to those of other studies (such as [5]). Both the T50 and [urea]50 measurements were performed in clarified cell lysate. The protein was expressed using catalase-free E. coli [25] containing the encoding gene on the IPTG inducible pCWori [25] plasmid. We used freshly streaked cells to inoculate 2 ml cultures of LB supplemented with 100 µg/ml of ampi- cillin, and grew these starter cultures overnight with shaking at 37oC. We then used 0.5 ml http://www.lifesci.sussex.ac.uk/CSE/test/maxchi.php from these starter cultures to inoculate 1 L flasks containing 200 ml of TB supplemented with 100 µg/ml of ampicillin. The TB cultures were grown at 30oC and 210 rpm until they reached an optical density at 600 nm of ≈0.9, at which point IPTG and δ-aminolevulinic acid were added to a final concentration of 0.5 mM each. The cultures were grown for an additional 19 hours, then the cells were harvested by pel- letting 50 ml aliquots at 5,500 g and 4oC for 10 min, and stored at -20oC. To obtain clarified lysate, each pellet was resuspended in 8 ml of 100 mM EPPS (pH 8.2) and lysed by sonica- tion, while being kept on ice. The cell debris was pelleted by centrifugation at 8,000 g and 4oC for 10 minutes, and the clarified lysate was decanted and kept on ice. For the T50 measurements, 125 µl of clar- ified lysate from a single mutant was added to all 12 wells in a row of a 96-well hard-shell thin-wall microplate (MJ Research). The plate was heated for 10 minutes using the gradient method of an Eppendorf Mastercycler gradient PCR machine, with the gradient set at either 33oC-45oC or 46oC-58oC (each mutant was ex- posed to both of these gradients), the machine on the BLOCK setting, and the heated lid set to 75oC with the lid WAIT option. The plate was then cooled to 4oC, removed from the PCR ma- chine, and centrifuged at 5,500 g and 4oC for 5 minutes to pellet any debris. A pipetting robot was used to dispense 80 µl of the supernatent into a 96-well microtiter plate (Rainin), and the amount of remaining properly folded P450 was quantified from the carbon monoxide difference spectrum as described below. The T50 values were determined by fitting sigmoidal curves the percent of remaining protein as shown in Addi- tional File 3. Our ability to accurately compare T50 values within the data set requires that each well in a given column of the gradient PCR ma- chine be at the same temperature. We used a thermocouple to measure the temperature of the wells with the machine lid open, and con- firmed that the wells were within a few tenths of a degree of the same temperature. Further evidence for the consistency of our T50 val- ues comes from the fact that two independent measurements of the T50 for our R1-11 par- ent yielded values that differed by only 0.1oC. However, the absolute values of the measured temperatures are less accurate. Thermocouple measurements indicated that, with the machine lid open, the wells were ≈ 1oC cooler than the indicated temperature. We were unable to as- certain the temperatures with the heated lid closed, but based on comparisons water bath measurements, the temperatures with the lid closed slightly exceeded the indicated tempera- tures. For the [urea]50 measurements, 125 µl of the clarified lysate from a single mutant was added to all 12 wells in a row of a 96-well microtiter plate. A pipetting robot was then used to add and mix 125 µl of a 2X solution of urea in 100 mM EPPS (pH 8.2) so that each subsequent column had a higher concentration of urea, and so that the final urea concentrations were those shown in Additional File 4. The plates were left on the bench at room temperature for 4 hours, and the amount of remaining properly folded P450 was quantified from the carbon monoxide difference spectrum as described be- low. The [urea]50 values were determined by fitting sigmoidal curves to the percent of re- maining protein. Evidence for the consistency of the [urea]50 measurements comes from the fact that two independent measurements of the [urea]50 for our R1-11 parent yielded values that differed by only 0.01 M. In addition, the [urea]50 and T50 values are highly correlated (Additional File 5), indicating that they provide consistent measures of stability. For both the T50 and [urea]50 measurements, the folded P450 was quantified from the car- bon monoxide difference spectrum [43]. The microtiter plates containing the P450 samples were first used to read blank spectra at 450 and 490 nm using a Tecan Safire 2 plate reader. The plates were then incubated for 10 minutes in an airtight oven with carbon monoxide. The plates were removed form the oven and 10 µl of 0.1 M sodium hydrosulfite in 1.3 M potassium phos- phate (pH 8.0) was immediately added to each well. After 5-10 minutes, spectra were again read at 450 and 490 nm. The amount of P450 is proportional to the increase in the signal at 450 nm after this procedure minus the change in the signal at 490 nm. Mathematical Appendix Contents A.1 Mathematical background 12 A.2 Monomorphic limit 14 A.3 Polymorphic limit 15 A.4 Approximations for polymorphic limit 16 A.5 Approximations for monomorphic limit 18 A.6 Interpretation in terms of neutral networks 20 A.7 Detailed justification for approximating pM by po 22 A.1 Mathematical background The first purpose of this appendix is to provide mathematical equations that describe the experi- ments. The second is to show how four measurements from the experiments can be used to calculate two quantities that describe the topology of the underlying protein neutral network. We will derive two equations for both quantitites, each in terms of a different measurement. The fact that the four equations will be seen to yield consistent results provides evidence for the accuracy of the following calculations. Our calculations are based on a view of neutral protein evolution as a process constrained by a stability threshold, a view that we originally introduced to explain experimental protein mutagen- esis results [3]. The calculations closely parallel our earlier work [22], which is in turn based on a general theoretical treatment of evolution on neutral networks by van Nimwegen and coworkers [11]. These calculations will probably be most thoroughly understood by first reading those works. The primary difference between the current calculations and [22] is that previously we assumed that the per generation per protein mutation rate µ was ≪ 1, so that at each generation a protein was either unmutated (with probability 1−µ) or experienced a single mutation (with probability µ). In contrast, here we allow the mutation rate to be arbitrarily large, so that a protein may experience multiple mutations in a single generation (in this sense the calculations resemble the generalization by Wilke [18] of [11]). Specifically, let fm be the probability that a protein experiences m mutations in a single generation. Here we derive results for arbitrary fm, and then approximations relevant to the form of fm in the experiments. In the limiting case of small mutation rate (where f0 = 1− µ, f1 = µ, and fm = 0 for m > 1), the calculations here reduce to those in [22]. Proteins evolving in nature typically experience very low mutation rates, so [22] probably offers the best description of natural protein evolution. The calculations presented here are designed to specifically treat the evolutionary dynamics of the experiments. A protein’s thermodynamic stability is described by its free energy of folding, ∆Gf , with more negative values indicating more stable proteins. As described in several previous papers [3, 5, 22], we assume that selection requires a protein to fold with some minimal stability ∆Gminf , so that a protein adequately folds if and only if ∆Gf ≤ ∆G f . The amount of extra stability a protein possesses relative to the stability threshold is given by ∆Gextraf = ∆Gf − ∆G f ; note that all folded proteins will have ∆Gextraf ≤ 0. We further assume that as long as ∆G extra f ≤ 0, selection is indifferent to the exact amount of extra stability that a protein possesses (see [22] for a discussion of the limitations of this assumption). We conceptually divide the continuous variable of protein stability into small discrete bins of width b. Specifically, a protein is in bin i if it has ∆Gextraf be- tween (1− i) b and −ib, where i = 1, 2, . . .. Mutating a protein changes its stability by an amount ∆∆G (defined as the stability of the mutant protein minus the stability of the initial protein), and so may move it to a new stability bin. In [22], we defined a matrix W with elements Wij giving the transition probabilities that a single mutation changes a protein’s stability from bin j to bin i. We noted that W could be computed from the distribution of ∆∆G values for all single mutations, and argued that W remains fairly constant during neutral evolution since the distribution of ∆∆G values remains relatively unchanged. However, we emphasize that (as discussed in detail in [22]) the constancy of the ∆∆G distribution remains an assumption, albeit one that has now been shown to be quite accurate for lattice proteins [3, 22,44] and provide a consistent theoretical explanation for a growing body of experimental results (the current work as well as [3]). Since we are allowing for larger mutation rates, and we must consider the possibility that a protein’s stability might change due to multiple mutations at a single generation. Therefore, we make a more general definition of Wij,m as the probability that m random mutations to a protein in stability bin j change its stability to bin i, and let Wm be the matrix with elements Wij,m. Note that Wm only describes mutations that cause transitions from one folded protein to another, since the stability bins i = 1, 2, . . . all correspond to folded proteins. As before [22], we assume that Wm is roughly constant during evolution, meaning that the distribution of ∆∆G values for multiple mutations is roughly constant during neutral evolution. Note that if m = 1, then Wm is just the matrix W that can be computed from the distribution of single-mutant ∆∆G values [22]. We will now use the matrices Wm to calculate the following characteristics of a population that has evolved to equilibrium: the distribution of stabilities, the average number of mutations 〈m〉T accumulated after T generations, and the average fraction 〈F〉 of stably folded proteins in the population. We then introduce a few approximations (that should be quite accurate for the experimental work in this paper) that greatly simplify these calculations. Finally, we relate the calculations to properties of the underlying protein neutral network. As described generally by van Nimwegen and coworkers [11], the evolutionary dynamics depend on whether the evolving population tends to be monomorphic or highly polymorphic. When the per sequence per generation mutation rate µ is ≪ 1, whether the population is mostly monomorphic or highly polymorphic is determined by the product of the population size N and µ: when Nµ ≪ 1 the population is mostly monomorphic, and when Nµ ≫ 1 the population is highly polymorphic. However, with multiple mutations per generation, Nµ is no longer an appropriate parameter to distinguish between mono- and polymorphism, since if the population size is sufficiently small the population can still be monomorphic even if there are multiple mutations per generation. Specifi- cally, in one set of experiments we constrained the population to be monomorphic (by maintaining a population size of one), but still allowed the single protein in this population to experience more than one mutation at a generation. So we instead denote the populations as either monomorphic or polymorphic. We indicate quantities calculated for the monomorphic population by the subscript M (i.e. 〈F〉M ) and those calculated for the polymorphic population by the subscript P (i.e. 〈F〉P ). A.2 Monomorphic limit In the limit of a completely monomorphic population, all of the proteins are in a single stability bin. Let pi (t) be the probability that the population is in stability bin i at time t, and let p (t) be the column vector with elements pi (t). At each generation there is a probability f0 that there is no mutation that becomes fixed in the population, a probability of Wij,mpj that the population experiences a mutational event (which could be a single mutation or several simultaneous mutations) that moves it into bin i, and a probability Wji,m that the population is in bin i and experiences one or more mutations that move it to another bin of stably folded proteins. Define νi,m = Wji,m to be the fraction of m-mutants of a protein in bin i that still fold, and let Vm be the matrix with diagonal elements given by Vii,m = νi,m and all other elements zero. The time evolution of p is p (t+ 1) = fm (Wm −Vm) p (t) (1) where I is the identity matrix. Note that mutations that destabilize a protein beyond the stabil- ity threshold are immediately lost to natural selection, and so leave the population in its original stability bin. This describes the experiments for the monomorphic populations, where we retain the parental sequence if the single mutant we generate is nonfunctional. Equation 1 corresponds to Equation (1) of [22], and the blind ant random walk described by van Nimwegen and coworkers [11]. Equation 1 describes a Markov process with a non-negative, irreducible, and acyclic transition matrix, and so p approaches a unique stationary distribution (equilibrium value) of pM given by the eigenvector equation fm (Wm −Vm) pM. (2) Once p has reached equilibrium, the average fraction of proteins that still stably fold at each generation is 〈F〉M = e pM (3) where e = (1, . . . , 1) is the unit row vector. To calculate 〈m〉T,M , the average number of mutations accumulated after T generations once the population has equilibrated, we note that at each generation there is a probability of fmpj Wij,m that a randomly chosen protein is in bin j, experiences m mutations, and still stably folds. The average number of mutations accumulated in a single generation is simply the average ofm weighted over this probability. So summing over all values of m and j, we see that 〈m〉T,M = Te mfmWmpM. (4) This equation corresponds to Equation (6) of [22], which was derived using an embedded Markov process formalism. Here we have foregone this formalism for the more intuitive argument presented above, since we do not attempt to calculate higher moments of the number of mutations. A.3 Polymorphic limit In the limit when the population is highly polymorphic, at each generation there are sequences in many different stability bins. In this case, we describe the distribution of stabilities by the column vector x (t), with element xi (t) giving the fraction of proteins in stability bin i at time t. At generation t, the fraction of mutants that continue to fold is 〈F〉t = e x (t) . (5) Therefore, in order to maintain a constant population size, each remaining protein must produce an average of αt = 〈F〉t offspring. The population therefore evolves according to x (t+ 1) = αt x (t) . (6) After the population evolves for a sufficiently long period of time, x will approach an equilibrium value of xP. At this equilibrium, the average fraction of mutants that fold at each generation is 〈F〉P = e xP, (7) and the equilibrium reproduction rate is α = 〈F〉P . Therefore, xP = α xP. (8) Equations 7 and 8 can be combined to show that xP and 〈F〉P can be calculated from the eigenvector equation (〈F〉P − f0)xP = fmWmxP, (9) with (〈F〉P − f0) the principal eigenvalue of the nonnegative and irreducible matrix fmWm. Equation 9 corresponds to Equation (14) of [22], Equation (6) of the work by van Nimwegen and coworkers [11], and Equation (13) of the work by Wilke [18]. We now calculate 〈m〉T,P , the average number of mutations accumulated in T generations after the population has equilibrated. At equilibrium, there is a probability of fmxj Wij,m that a protein is in bin j, experiences m mutations, and still stably folds. Subsequently, all of these folded proteins produce an average of α offspring. The average number of mutations accumulated in a single generation is simply the average of m weighted over this probability, and then multiplied by the average reproduction rate. So summing over all values of m and j, we obtain 〈m〉T,P = αTe mfmWmxP = mfmWmxP. (10) This equation is the counterpart of Equation (18) of [22], where we have again foregone the em- bedded Markov process formalism for a more intuitive derivation. A.4 Approximations for polymorphic limit We can dramatically simplify the results from the previous sections with several reasonable ap- proximations. The first approximation is that the ∆∆G values for random mutations are roughly additive, and is supported by a number of experimental studies of the thermodynamic effects of mutations [45–47]. We have previously shown that this approximation can be used to accurately describe experimental protein mutagenesis data with a simple stability threshold model [3]. Under this approximation, the distribution of net ∆∆G values for multiple mutations can be computed from the distribution of ∆∆G values for single mutations by performing convolutions of the single- mutation ∆∆G distribution [3], meaning that Wm for arbitrary m can be computed solely from the distribution of ∆∆G values for single mutations. However, to simplify the equations from previous sections, we need to express Wm for arbitrary m only in terms of W (recall that W = W1). Since W only contains information about stability transitions from folded proteins to other folded pro- teins, if we make the second approximation that a protein that is destabilized beyond the minimal stability threshold by one mutation is not re-stabilized to a folded protein by a subsequent muta- tion, then Wm = W m. This approximation that unfolded proteins are not re-stabilized should be quite accurate since stabilizing mutations tend to be relatively rare and small in magnitude [48–51] (this is the underlying idea behind the Markov chain approximation that was shown to be highly accurate for lattice proteins [44]). To summarize, if ∆∆G values are roughly additive and stabilizing mutations are rare, we have the approximation Wm ≈ W m. (11) Simplifying the equations of the previous sections also requires assigning a specific functional form to fm, the probability that a sequence undergoesmmutations. Here we assume that mutations are Poisson distributed among sequences, so that e−µµm where µ = mfm is the average number of mutations per protein per generation. When the mu- tations are introduced by error-prone PCR, the Poisson distribution is an excellent approximation to the true theoretical distribution of mutations created by error-prone PCR [38,39] provided that µ is much less than the number of PCR doublings, as is the case in all of the experiments in the current work. We now use the approximations of Equations 11 and 12 to simplify the results given above for the highly polymorphic limit. We begin by using these approximations to rewrite Equation 9 as 〈F〉P − e xP = e WmxP. (13) This equation makes clear that xP is the principal eigenvector of the matrix Wm, therefore xP must also be the principal eigenvector of W. Now in our earlier work [22], we defined the principal eigenvector of W as x∞, called the corresponding eigenvalue 〈ν〉∞, and showed that this eigenvalue is shown the average fraction of single mutations that are neutral in a population that is evolving with Nµ ≫ 1 and µ ≪ 1. Therefore, with the approximation of Equation 11, xP and x∞ are equal, and are both defined by the same eigenvector equation, 〈ν〉∞xP = WxP = Wx∞ = 〈ν〉∞x∞. (14) Combining Equations 13 and 14 we have, 〈F〉PxP = e (µ〈ν〉∞) 1−〈ν〉∞ xP (15) Equation 15 can be solved to yield 〈ν〉∞ = 1 + ln 〈F〉P . (16) Similarly, we can simplify Equation 10, 〈m〉T,P = mfmWmxP 1−〈ν〉∞ eWmxP = Te−µ〈ν〉∞ (µ〈ν〉∞) = Tµ〈ν〉∞e −µ〈ν〉∞ (µ〈ν〉∞) = Tµ〈ν〉∞. (17) Solving this equation for 〈ν〉∞ yields 〈ν〉∞ = 〈m〉T,P . (18) A.5 Approximations for monomorphic limit We now simplify the equations for the monomorphic limit. This requires several further approxima- tions. We begin by approximating that the stability probability distribution pM given by Equation 2 by the distribution po defined in [22] as satisfying 0 = (W −V)po. (19) The basic rationale behind approximating pM with po is that Equation 2 can be viewed as a per- turbation to Equation 19 [52]. Essentially, po is an eigenvector of the matrix W −V while pM is the corresponding eigenvector of the matrix W − V + (Wm −Vm). The latter matrix can be viewed as a perturbation to the first, since the sum (Wm −Vm) is small. This smallness is due to the fact that Wm tends to zero with large m, causing Vm to tend towards the identity matrix. In addition, the µm/m! terms tend to zero with large m. Therefore, the terms in the summation are all simply either a perturbation to W − V or involve subtracting terms that are fractions of the identity matrix. The perturbations lead to bounded changes in the eigenvec- tors [52], while the identity matrix terms do not change the eigenvectors. Below we give a more rigorous justification of the assumption that pM is approximately equal to po. We need one additional approximation to make further progress. Both Equations 3 and 4 con- tain terms of the form Wmpo, and even if we use Equation 11 to rewrite these terms as W there are no further clear simplifications. However, any probability vector that is multiplied re- peatedly by W and normalized will eventually converge to x∞ = xP (since this is the principal eigenvector of W). We make the approximation that this convergence is sufficiently rapid to be essentially complete after a single multiplication. This approximation is supported by both protein mutagenesis studies [3,26,27] that indicate that proteins rapidly converge to an exponential decline in the fraction folded (indicating the stability distribution has equilibrated, as discussed below, and by lattice protein studies showing the same [3, 44]. Therefore, we make the approximation that eWmpo = 〈ν〉oeW m−1x∞ = 〈ν〉o〈ν〉∞ where 〈ν〉o = eWpo has the same definition as in [22], where it was defined as the average fraction of functional single mutants of a population evolving with µ ≪ 1 and Nµ ≪ 1. We use these approximations to simplify Equation 3 as 〈F〉M = e = e−µ eWmpo = e−µ 1 + µ〈ν〉o (µ〈ν〉∞) = e−µ (µ〈ν〉∞) = e−µ eµ〈ν〉∞ − 1 . (20) Solving this equation for 〈ν〉o, we find 〈ν〉o = 〈ν〉∞ (〈F〉Me µ − 1) eµ〈ν〉∞ − 1 . (21) We now use the approximations to simplify Equation 4 as 〈m〉T,M = Te mfmWmpM = Te−µ eWmpo = Te−µ〈ν〉o = µTe−µ〈ν〉o (µ〈ν〉∞) = µT 〈ν〉oe 〈ν〉∞−1 . (22) Solving this equation for 〈ν〉o yields 〈ν〉o = 〈m〉T,Me 1−〈ν〉∞ . (23) To recap, we now have equations to calculate 〈ν〉∞ and 〈ν〉o from experimentally measurable quantities. Equations 16 and 18 allow us to calculate 〈ν〉∞ from 〈F〉P and 〈m〉T,P , respectively. Given this calculated value of 〈ν〉∞, Equations 21 and 23 then allow us to calculate 〈ν〉o from 〈F〉M and 〈m〉T,M , respectively. The fact that we have two equations each for 〈ν〉∞ and 〈ν〉o allows us to assess the self-consistency of the approach. A.6 Interpretation in terms of neutral networks Throughout the preceding calculations, we have referred to 〈ν〉∞ and 〈ν〉o as we defined them in [22]: namely, as the average neutrality of protein populations evolving with µ ≪ 1 and Nµ either ≫ 1 or ≪ 1, respectively. However, van Nimwegen and coworkers [11] have shown that they can also be interpreted in terms of the underlying neutral network. In the experiments we make mutations at the nucleotide (rather than amino acid) level, so each point in our sequence space corresponds to a different gene. Every gene that yields an amount of protein sufficient to hydroxylate the twelfth carbon of 12-p-nitrophenoxydodecanoic acid with at least 75% of the total activity conferred by the original R1-11 parent gene represents a node on this neutral network. We note that in the experiments (and also usually in natural evolution), the edges on the neutral network are not all completely equivalent or fully undirected, since some mutations are more likely to occur than others (for example, error-prone PCR with Taq polymerase is more likely to cause an A→G mutation than an A→C mutation). In the analysis that follows, we ignore this complication and assume all neutral network edges are equivalent. In an extremely insightful analysis, van Nimwegen and coworkers [11] have shown that impor- tant characteristics of a neutral network can be inferred from evolutionary quantities. Specifically, they have shown that if a population is evolving with µ ≪ 1 and Nµ ≫ 1, then the average neutrality (which we have denoted by 〈ν〉∞) is equal to the principal eigenvalue of the adjacency matrix of the neutral network, normalized by the network coordination number (number of possi- ble connections per node). In addition, they pointed out that a population evolving with µ ≪ 1 and Nµ ≪ 1 moves like a blind ant random walk, meaning that the average neutrality (which we have denoted by 〈ν〉o) is equal to the average connectivity of a neutral network node divided by the network coordination number. In our P450 experiments, we have measured the values needed to estimate 〈ν〉∞ and 〈ν〉o using Equations 16, 18, 21, and 23. Using the final values listed in Table 2, 〈F〉P = 0.50 and 〈F〉M = 0.39. Taking the final nucleotide mutation values from Table 2, 〈m〉T,P /T = 0.69 and 〈m〉T,M/T = 0.31. The average mutation rate, computed from the un- selected population, is µ = 1.40. So using Equation 16, 〈ν〉∞ = 0.53, while using Equation 18, 〈ν〉∞ = 0.49. The consistency of these two values supports the idea that the calculations above accurately describe the evolutionary process. Taking the average value of these two measurement as 〈ν〉∞ = 0.51, we can then use Equations 21 and 23 to calculate 〈ν〉o. We calculate values of 0.28 and 0.43, respectively. These estimates differ by more than those for 〈ν〉∞, perhaps because additional approximations have gone into the derivation of the relevant equations (in addition, we have made no attempt to carry out the rather complex propagation of the sampling errors of Table 2). However, the values are still in a similar range. Taking the average of these two values, we estimate that 〈ν〉o = 0.35. So overall, we predict that each functional P450 gene should have an average fraction of 0.35 of its sequence nearest neighbors also encoding a functional gene, for an average of about 1,500 neighbor genes. We predict that the principal eigenvalue of the neutral network adjacency matrix is 0.51 ×3L. The fact that principal eigenvalue exceeds the average connectivity indicates that the neutral network is not a regular graph, but instead has some nodes more highly connected than others. The value for 〈ν〉∞ calculated above can also be related to measurements from protein mu- tagenesis experiments. A number of studies [3, 26, 27] have observed that the probability that a protein remains functional after m mutations falls off exponentially with the number of mutations. In fact, the decline is not always exponential for the first few mutations if the starting protein has especially high or low stability [3] or activity [53], but will still converge to this exponential form after a few mutations [3,44,54]. The stability threshold model can be used to relate this decline to 〈ν〉∞, as is done indirectly in the Markov chain approximation of [44]. Here we make that connec- tion explicit. The initial protein has a stability that falls into some stability bin i. Therefore, its stability can be described by the column vector y0, which has element i equal to one and all other elements equal to zero. Now imagine constructing all single mutants of this protein. The fraction of these single mutants that still fold is just eWy0, and the distribution of stabilities among the single mutants is y1 = Wy0 (note that the elements of y1 no longer sum to one). Similarly, after m mutations, the fraction of mutants that still fold is eWmy0, and the distribution of stabilities among the m-mutants is ym = Wmy0. With the approximation of Equation 11, ym = W This makes it clear that ym will converge to a vector proportional to x∞, the principal eigenvector of W. Once this convergence is complete, each new mutation simply reduces the fraction of mu- tants that fold by a factor of 〈ν〉∞, the principal eigenvalue of W (and the spectral radius of the neutral network normalized by the coordination number). Therefore, what we have called 〈ν〉∞ in the present work and [22] is equal to what is called x in [27], q in [26], and 〈ν〉 in [3]. The major difficulty that is faced in extracting 〈ν〉∞ by the method of those three studies [3, 26,27] is that it is not possible to directly assay mutants with m mutations, but instead only possible to assay a set of mutants with a distribution of m. All three studies use different (and valid) methods to account for this distribution, but this accounting is still difficult because most of the functional mutants come from the low m end of the distribution. This makes it hard to get accurate values for the fraction functional after large numbers of mutations, since most of the functional mutants in the set come from sequences with few mutations. For this reason, we believe the current method of measuring 〈ν〉∞ is more accurate. A second caution about comparing values of 〈ν〉∞ from different studies is that its value depends on the nucleotide error-spectrum of the experiment, since differ- ent mutagenesis methods create different distributions of nucleotide and amino acid mutation types. We also briefly mention how we arrived at an estimate of 〈ν〉∞ for 3-methyladenine DNA glycosylase from the data of [27]. This paper reports that a fraction x = 0.34 of amino acid mutations inactivate the protein. We would like to determine the fraction 〈ν〉∞ of nucleotide mu- tations that do not inactivate the protein. Roughly 75% of random mutations to a gene will be synonymous. Therefore, m amino acid mutations should cause about 4m/3 nucleotide mutations. The study of [27] measures that after m mutations, a fraction (1− x) of the mutants are func- tional. That means that 〈ν〉∞ fraction should be functional. Equating these expressions yields 〈ν〉∞ = exp log (1− x) . So using x = 0.34, we arrive at 〈ν〉∞ = 0.73. A.7 Detailed justification for approximating pM by po Here we provide a detailed justification for the approximation that pM is about equal to po. In the monomorphic limit, the time evolution of p is given by Equation 1, and the stationary distribution pM is given by Equation 2. We assume the approximations of Equations 11 and 12 and show that we can approximate pM by po, where po is given by Equation 19. To justify this approximation, we insert po into the right hand side of Equation 1 and ask to what extent po is left unaltered by the dynamics. If po is found to be stationary to good approximation then, by uniqueness of the stationary distribution of an ergodic process, po would be a good approximation to pM. We therefore suppose that at some time t the distribution is given by po and compute, using Equation 1, the change in po after one generation. The new distribution at time t+ 1 is given by p (t+ 1) = fm (W m −Vm) po. (24) Using (V −W)po = 0, and taking components of the above equation, we obtain pi (t+ 1) = p0i + fm [(W m −Vm)po]i . (25) Thus po would be an approximately stationary distribution of the dynamics if | fm [(W m −Vm)po]i| ≪ p0i. We now proceed to show that this will be the case in most situations of interest by deriving upper and lower bounds on the second term of the right hand side of Equation 25. Consider first the term (Wmpo)i, which can be written as (Wmpo)i = k1,...,km Wik1Wk1k2 · · ·Wkm−1kmp0km k1,...,km−1 Wik1Wk1k2 · · ·Wkm−2km−1νkm−1p0km−1 , (26) where we have used Wpo = Vpo in the second equality. We now note that νk ≤ νmax for all k, where νmax is the maximum neutrality, maximized over all bins. This leads to the successive inequalities (Wmpo)i ≤ νmax k1,...,km−1 Wik1Wk1k2 · · ·Wkm−2km−1p0km−1 = νmax k1,...,km−2 Wik1Wk1k2 · · ·Wkm−3km−2νkm−2p0km−2 k1,...,km−2 Wik1Wk1k2 · · ·Wkm−3km−2p0km−2 ≤ νm−1 Wik1p0k1 , (27) yielding the upper bound (Wmpo)i ≤ ν νip0i. (28) In an identical manner, we obtain the lower bound (Wmpo)i ≥ ν νip0i, (29) where νmin is the smallest neutrality, minimized over all bins. Note that both inequalities above become exact equalities when all bins have the same neutrality ν, which could be interpreted as either νmin or νmax. Having obtained inequality constraints on (Wmpo)i, we now consider the term (Vmpo)i, which can be written as (Vmpo)i = p0iνi,m = p0i (Wm)ji = p0i j,k1,...,km−1 Wjk1Wk1k2 · · ·Wkm−1i = p0i k1,...,km−1 νk1Wk1k2 · · ·Wkm−1i ≤ p0iνmax k1,...,km−1 Wk1k2 · · ·Wkm−1i ≤ p0iν Wkm−1i, (30) which yields an identical upper bound to that on (Wmpo)i, namely, (Vmpo)i ≤ ν νip0i, (31) and similarly (Vmpo)i ≥ ν νip0i. (32) It should again be noted that both the above inequalities become exact equalities when all bins have a common neutrality ν. We are now in a position to estimate bounds on the magnitude of the second term of Equation 25. Using the four inequalities of Equations 28, 29, 31, and 32 above, we have − νm−1 νip0i ≤ [(W m −Vm)po]i ≤ − νm−1 νip0i, (33) or equivalently, |[(Wm −Vm)po]i| ≤ − νm−1 νip0i, (34) where the inequality above becomes an exact equality when all bins have the same neutrality. How- ever, in this limit, the right hand side of the above equation vanishes, and therefore the second term of Equation 25 is identically zero in this case, giving the result that pM is exactly equal to po when all bins have the same neutrality, even if µ is arbitrarily large. We now carry out the sum over m to obtain an upper bound on the second term of Equation 25 in the more general and realistic case of unequal neutrality bins. Using Equation 34 and the specific Poisson form of fm, we obtain an upper bound on the fractional change in p0i in one generation: pi(t+ 1)− p0i ≤ νie − νm−1 = νie eµνmax − 1 eµνmin − 1 . (35) The above bound vanishes for small µ, is an increasing function of νmax − νmin, and is typically much smaller than 1. An extreme estimate of the size of the fractional change can be made when νmax = 1 and νmin = 0. In this case, using µ = 1.4 (the value in our experiments), the above inequality simplifies to pi(t+ 1)− p0i 1− e−µ − µe−µ ≃ 0.41νi. (36) Noting that νi < 1, the fractional change in p0i is therefore reasonably controlled even in the most extreme case. For realistic situations, the fractional change in p0i is expected to be much lower, thus justifying the use of po as the stationary distribution of the dynamics of Equation 1. Author Contributions JDB and FHA designed the project and wrote the paper. JDB and ZL performed the bulk of the experiments; OSV assisted with the experiments. JDB and DC analyzed the data. JDB and AR performed the theoretical work. Acknowledgments We thank Claus O Wilke for helpful advice and comments. 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(B) Proteins evolving in a highly polymorphic population are predicted to be more stable than their counterparts in a mostly monomorphic population [22]. The histograms illustrate the distributions of stabilities for the two cases. The increased stability is a biophysical manifestation of excess mutational robustness, since more stable proteins are more mutationally robust [3–5]. polymorphic monomorphic unselected Figure 2 - Outline of the neutral evolution experimental procedure. For the polymorphic population, error-prone PCR was used to generate mutant P450 genes. These genes were ligated into a plasmid and transformed into E. coli. Individual mutants (435) were picked, expressed in E. coli, and assayed for enzymatic activity. All mutants that met the selection criterion contributed an equal amount of plasmid DNA as template for the next generation of error-prone PCR. The monomorphic populations were treated similarly, except only a single mutant was assayed at each generation. If this mutant met the selection criterion then it became the template for the next generation of error-prone PCR; otherwise at the next generation another colony was picked from the same plate. In the unselected populations a single mutant was picked and used as the template for the next generation of error-prone PCR. PSfrag replacements generation 〈mnt〉 unselected polymorphic monomorphic PSfrag replacements 0 10 20 generation 〈maa〉 〈 unselected polymorphic monomorphic Figure 3 - Accumulation of nucleotide (〈mnt〉) and nonsynonymous (〈maa〉) mutations in the experimentally evolved P450 populations. For the unselected and monomorphic populations, numbers are the average over all replicates at the indicated generation; for the polymorphic population they are from a random sample, with sampling standard error shown. PSfrag replacements 0 10 20 generation polymorphic monomorphic Figure 4 - The polymorphic population neutrally evolved a higher average mutational robustness than the monomorphic populations. The fraction functional was determined by assaying 435 mutants (average of 1.5 nucleotide mutations per gene). Error bars show binomial standard error. For the monomorphic population, numbers are the average over all replicates. PSfrag replacements polymorphic monomorphic T50 ( [urea]50 (M) percent parental expression 38 40 42 44 PSfrag replacements polymorphic monomorphic T50 ( [urea]50 (M) percent parental expression number 0.4 0.6 0.8 1.0 PSfrag replacements polymorphic monomorphic T50 ( [urea]50 (M) percent parental expression number 20 60 100 140 Figure 5 - The more mutationally robust proteins are more stable. The P450s from the polymorphic population neutrally evolved higher stability and expression levels than their counterparts from the monomorphic populations. The histograms show the distributions for the final protein from all monomorphic replicates and for the same number of randomly chosen proteins from the final polymorphic population. The plots show (left to right) the temperature at which half the protein irreversibly denatured after 10 minutes (T50), the urea concentration at which half the protein denatured after 4 hours ([urea]50), and the expression level relative to that of the original parental P450. The means are significantly different, with unequal variance t-test P -values of 0.02, 0.005, and 0.04, respectively. Tables Table 1 - Error-prone PCR nucleotide mutation spectrum. Spectrum of nucleotide mutations introduced by the error-prone PCR procedure used in the neutral evolution experiments. The spectrum was determined by sequencing the four final (generation 12) sequences from the unselected population, since in these sequences the mutations accumulate without constraint. As has been previously noted for error-prone PCR with Taq polymerase [3,5,26], the nucleotide error spectrum is biased towards certain types of mutations. Total nucleotide mutations 67 % synonymous mutations 25 Mutation types (%) A →T, T →A 19.4 A →C, T →G 1.5 A →G, T →C 64.2 G →A, C →T 4.5 G →C, C →G 0.0 G →T, C →A 1.5 frameshift 9.0 Table 2 - Neutral evolution robustness and mutation data. Each row is for a different generation, T . Entries of NA indicate that no measurement was made. The 〈mnt〉 and 〈maa〉 are the average number of nucleotide mutations and nonsynonymous mutations, respectively. Numbers in parentheses are total counts over the total samples. Subscripts indicate the population type: U for unselected, P for polymorphic, and M for monomorphic. For the unselected and monomorphic populations, numbers represent averages of all replicates. For the polymorphic population, numbers are for a random sample of functional mutants. 〈F〉P and 〈F〉M are the fraction of functional mutants out of 435 assayed. T 〈mnt〉U 〈maa〉U 〈mnt〉P 〈maa〉P 〈mnt〉M 〈maa〉M 〈F〉P 〈F〉M 0 0 0 0 0 0 0 0.48 (210 / 435) 0.48 (210 / 435) 1 NA NA NA NA 0.1 (3 / 22) 0.3 (6 / 22) 0.48 (208 / 435) NA 2 NA NA NA NA 0.4 (9 / 22) 0.8 (17 / 22) 0.49 (215 / 435) NA 3 5.0 (20 / 4) 3.5 (14 / 4) 2.7 (27 / 10) 1.4 (14 / 10) 1.0 (23 / 22) 0.4 (9 / 22) 0.49 (215 / 435) NA 4 NA NA NA NA 1.5 (32 / 22) 0.7 (15 / 22) 0.48 (208 / 435) NA 5 NA NA NA NA 2.2 (48 / 22) 1.1 (25 / 22) 0.45 (197 / 435) 0.43 (185 / 435) 6 9.8 (39 / 4) 7.5 (30 / 4) 5.5 (55 / 10) 2.1 (21 / 10) 2.6 (58 / 22) 1.4 (31 / 22) 0.46 (198 / 435) NA 7 NA NA NA NA 3.1 (69 / 22) 1.8 (39 / 22) 0.52 (227 / 435) NA 8 NA NA NA NA 3.4 (74 / 22) 1.8 (40 / 22) 0.46 (200 / 435) NA 9 13.0 (52 / 4) 10.3 (41 / 4) 6.7 (61 / 9) 3.1 (28 / 9) 3.7 (82 / 22) 2.1 (46 / 22) 0.47 (203 / 435) NA 10 NA NA NA NA 4.2 (92 / 22) 2.4 (52 / 22) 0.46 (199 / 435) 0.40 (175 / 435) 11 NA NA NA NA 4.6 (102 / 22) 2.5 (56 / 22) 0.48 (207 / 435) NA 12 16.8 (67 / 4) 12.5 (50 / 4) 7.8 (70 / 9) 3.3 (30 / 9) 4.9 (107 / 22) 2.6 (58 / 22) 0.52 (228 / 435) NA 13 NA NA NA NA 5.0 (110 / 22) 2.7 (60 / 22) 0.52 (227 / 435) NA 14 NA NA NA NA 5.3 (116 / 22) 2.9 (64 / 22) 0.50 (216 / 435) NA 15 NA NA 10.3 (227 / 22) 3.8 (83 / 22) 5.6 (123 / 22) 3.0 (67 / 22) 0.50 (219 / 435) 0.39 (171 / 435) 16 NA NA NA NA 5.8 (127 / 22) 3.0 (67 / 22) NA NA 17 NA NA NA NA 6.0 (133 / 22) 3.1 (69 / 22) NA NA 18 NA NA NA NA 6.3 (137 / 22) 3.2 (71 / 22) NA NA 19 NA NA NA NA 6.3 (138 / 22) 3.3 (72 / 22) NA NA 20 NA NA NA NA 6.6 (145 / 22) 3.4 (75 / 22) NA 0.37 (160 / 435) 21 NA NA NA NA 6.9 (152 / 22) 3.6 (79 / 22) NA NA 22 NA NA NA NA 7.1 (156 / 22) 3.7 (81 / 22) NA NA 23 NA NA NA NA 7.2 (158 / 22) 3.7 (81 / 22) NA NA 24 NA NA NA NA 7.3 (161 / 22) 3.8 (83 / 22) NA NA 25 NA NA NA NA 7.7 (169 / 22) 4.0 (87 / 22) NA 0.39 (169 / 435) Additional material Additional file 1 - Sequence of the parent P450 used start neutral evolution. FASTA file with sequence of the R1-11 P450 BME used as the neutral evolution parent. This sequence was isolated after the equilibration evolution. Additional file 2 - Information about sequences from neutral evolution experiments. The entries give the name of the mutant, the number of nonsynonymous and nucleotide mutations relative to the R1-11 parent, the [urea]50 value if measured, the T50 value if measured, the percent of the parental expression level if measured, and then a list of all of the mutations. Amino acid mutations are numbered in the standard P450 numbering scheme. The names of the mutants indicate their origin. Names beginning with “P-G3” are randomly chosen functional mutants from generation 3 of the polymorphic population, etc. Names of the form “P1,” “P2,”, etc. are the 22 functional mutants that were randomly chosen from the final (generation 15) polymorphic population. Numbers P5 and P12 are missing because two of the original 24 randomly selected polymorphic population members were randomly chosen to be discarded after it was discovered that two of the 24 monomorphic replicates were contaminated. Names beginning with “U1” indicate that sequences are from the first unselected replicate, etc. Names beginning “M1” indicate sequences are from the first monomorphic replicate, etc. Replicates “M9” and “M10” were discarded due to contamination during the experiment. For each replicate, we sequenced each new functional mutant. The last functional mutant after 25 generations represents the final sequence for that replicate, and is given an abbreviated name without the generation suffix. Additional file 3 - Thermostability measurements. Raw data from the T50 thermostability measurements. Additional file 4 - Urea stability measurements. Raw data from the [urea]50 thermostability measurements. Additional file 5 - Correlation of thermal and urea stabilities. The T50 and [urea]50 values are highly correlated. Additional file 6 - Sequence of initial P450 used to start equilibration evolution. FASTA file with sequence of the 21B3 P450 BM3 heme domain described in [23]. This P450 was used as the initial parent to start the equilibration evolution. Additional file 7 - Mutations accumulated during equilibration evolution. The file lists the mutations in the 46 P450 variants selected at the end of the equilibration evolution. Each line gives the name of the variant, with the prefix indicating whether it came from the R1 or R2 population. The next entries give the number of nucleotide and nonsynonymous mutations. All of the individual mutations relative to 21B3 are then listed. Amino acid mutations are numbered in the standard P450 numbering scheme, with the threonine after the N-terminal methionine given the number one. A.1 Mathematical background A.2 Monomorphic limit A.3 Polymorphic limit A.4 Approximations for polymorphic limit A.5 Approximations for monomorphic limit A.6 Interpretation in terms of neutral networks A.7 Detailed justification for approximating pM by po

BACKGROUND: An important question is whether evolution favors properties such as mutational robustness or evolvability that do not directly benefit any individual, but can influence the course of future evolution. Functionally similar proteins can differ substantially in their robustness to mutations and capacity to evolve new functions, but it has remained unclear whether any of these differences might be due to evolutionary selection for these properties. RESULTS: Here we use laboratory experiments to demonstrate that evolution favors protein mutational robustness if the evolving population is sufficiently large. We neutrally evolve cytochrome P450 proteins under identical selection pressures and mutation rates in populations of different sizes, and show that proteins from the larger and thus more polymorphic population tend towards higher mutational robustness. Proteins from the larger population also evolve greater stability, a biophysical property that is known to enhance both mutational robustness and evolvability. The excess mutational robustness and stability is well described by existing mathematical theories, and can be quantitatively related to the way that the proteins occupy their neutral network. CONCLUSIONS: Our work is the first experimental demonstration of the general tendency of evolution to favor mutational robustness and protein stability in highly polymorphic populations. We suggest that this phenomenon may contribute to the mutational robustness and evolvability of viruses and bacteria that exist in large populations.

Evolution favors protein mutational robustness in sufficiently large populations Jesse D Bloom∗1 , Zhongyi Lu1 , David Chen1 , Alpan Raval2 , Ophelia S Venturelli1 and Frances H Arnold∗1 Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA Keck Graduate Institute of Applied Life Sciences and School of Mathematical Sciences, Claremont Graduate University, Claremont, CA 91711, USA Email: Jesse D Bloom∗- jesse.bloom@gmail.com; Zhongyi Lu - lu07@caltech.edu; David Chen - davidc@caltech.edu; Alpan Raval - alpan raval@kgi.edu; Ophelia S Venturelli - opheliav@stanford.edu; Frances H Arnold∗- frances@cheme.caltech.edu; Corresponding author Abstract Background: An important question is whether evolution favors properties such as mutational robustness or evolvability that do not directly benefit any individual, but can influence the course of future evolution. Functionally similar proteins can differ substantially in their robustness to mutations and capacity to evolve new functions, but it has remained unclear whether any of these differences might be due to evolutionary selection for these properties. Results: Here we use laboratory experiments to demonstrate that evolution favors protein mutational robustness if the evolving population is sufficiently large. We neutrally evolve cytochrome P450 proteins under identical selection pressures and mutation rates in populations of different sizes, and show that proteins from the larger and thus more polymorphic population tend towards higher mutational robust- ness. Proteins from the larger population also evolve greater stability, a biophysical property that is known to enhance both mutational robustness and evolvability. The excess mutational robustness and stability is well described by existing mathematical theories, and can be quantitatively related to the way that the proteins occupy their neutral network. Conclusions: Our work is the first experimental demonstration of the general tendency of evolution to favor mutational robustness and protein stability in highly polymorphic populations. We suggest that this phenomenon may contribute to the mutational robustness and evolvability of viruses and bacteria that exist in large populations. http://arxiv.org/abs/0704.1885v1 Background Proteins are quite tolerant of mutations, al- lowing evolution to produce highly diverged sequences that fold to similar structures and perform conserved biochemical functions [1, 2]. However, proteins with nearly identical struc- tures and functions may differ in their robust- ness to mutation [3–5], as well as in their ca- pacity to acquire new functions [5]. The fact that mutational robustness and evolvability can vary among the functionally equivalent proteins produced by natural sequence divergence makes these properties important hidden dimensions in evolution — direct selection for protein func- tion is blind to them, yet they can play a cru- cial role in enabling future evolution. Whether the evolutionary process somehow promotes the acquisition of mutational robustness and evolv- ability therefore remains a major question [6–8]. Previous experiments have identified several specific evolutionary conditions that can affect mutational robustness. For example, genetic complementation decreases the mutational ro- bustness of viruses [9], while high mutation rates favor mutational robustness in simulated digi- tal organisms [10]. However, theory [11] makes the much broader — and heretofore experimen- tally untested — prediction that extra muta- tional robustness will arise quite generally in sufficiently large populations. This prediction cannot be understood in the standard frame- work of Kimura’s neutral theory [12], since one of the usual assumptions of the neutral theory is that mutational robustness is constant. (Al- though Takahata [13] treated the consequences of stochastically fluctuating neutrality on the molecular clock, he did not describe how muta- tional robustness might change systematically during evolution.) However, changes in mu- tational robustness can be described by envi- sioning evolution as occurring on neutral net- works, or sets of functionally equivalent pro- teins that are connected by single mutational steps [14–17]. In a seminal theoretical analysis of evolution on neutral networks, van Nimwe- gen and coworkers [11] predicted that the extent of mutational robustness should depend on the degree of population polymorphism. Here we briefly summarize their reasoning, since it moti- vates our experimental work. We also refer the reader to chapter 16 of [8], which contains an excellent explanation of the densely mathemat- ical work of van Nimwegen and coworkers [11]. If an evolving population is mostly monomorphic, then each mutation is either lost or goes to fixation before another muta- tion occurs. The population is therefore usually clustered at a single genotype and rarely expe- riences mutations, meaning that selection does not distinguish between genotypes of different mutational robustness. All nodes of the neutral network are thus equivalent and will be occupied by the population with equal probability [11]. On the other hand, a highly polymorphic pop- ulation is always spread across many nodes of the neutral network. When mutations occur, the members of the population at highly con- nected nodes have a better chance of surviving, causing them to be favored by evolution and increasing the average mutational robustness [11, 17–20]. Specifically, a highly polymorphic population occupies each node with a proba- bility proportional to its eigenvector central- ity [11, 17], a measure of how connected it is to other connected nodes (a variant of eigen- vector centrality is used by Google’s PageRank algorithm to rank a webpage’s importance in the network of internet links [21]). Figure 1A illustrates how mostly monomorphic and highly polymorphic populations are predicted to oc- cupy a neutral network. For proteins, changes in neutral network occupancy should be man- ifested by changes in thermodynamic stabil- ity [22], with proteins from highly polymorphic populations predicted to be more stable than their counterparts from mostly monomorphic populations (Figure 1B). Note that the extent of polymorphism depends on the product of the mutation rate and population size, meaning that protein populations of different sizes are predicted to evolve to different levels of mu- tational robustness and stability even if they experience the same mutation rate. Results and Discussion Design of neutral evolution experiment To test whether high population polymorphism drives an increase in mutational robustness and protein stability, we performed laboratory evo- lution experiments on cytochrome P450 pro- teins. The basic idea was to neutrally evolve P450s under a constant selection pressure in populations that were either monomorphic or highly polymorphic, and observe whether the proteins evolved to different levels of mutational robustness and stability. The evolution experi- ments started with a P450 BM3 heme domain that had been engineered to hydroxylate 12-p- nitrophenoxydodecanoic acid (12-pNCA) [23]. We imposed the selection criterion that Es- cherichia coli cells expressing the P450 had to yield lysate with enough active enzyme to hy- droxylate a specified amount of 12-pNCA in 40 minutes. This criterion roughly corresponds to the case in which an enzyme must catalyze a biochemically relevant reaction at some mini- mal level in order for its host to survive. Note that other properties such as stability and ex- pression level can vary freely, provided that the criterion for total activity is met. The properties of a neutrally evolving pro- tein eventually “equilibrate,” much as the prop- erties of an isolated physical system under some macroscopic constraint tend towards the values that maximize the system’s internal entropy. For proteins, this usually means that stability, expression, and activity drift towards their low- est tolerable values, since the vast majority of random sequences do not encode stable, well- expressed enzymes (that is, natural selection must work against sequence entropy to maintain a functional protein) [22, 24]. The initial P450 had been engineered for maximal activity [23], meaning that it was not equilibrated to the more mild selection criterion of the experiments. We therefore neutrally evolved this initial P450 for 16 generations, introducing random mutations with error-prone PCR and retaining all mu- tants that met the selection criterion for total activity on 12-pNCA. The procedure used for this equilibration evolution was similar to that for the polymorphic neutral evolution described below. As expected, expression, stability, and activity all dropped during the equilibration evolution. At the end of the equilibration evo- lution, we chose a single sequence as the parent for the neutral evolution experiments. The gene encoding this parent sequence contained 29 nu- cleotide mutations and 13 amino acid mutations relative to the initial P450 (Additional File 1). We used this parent gene to begin three parallel sets of neutral evolution experiments, which we named “monomorphic,” “polymor- phic,” and “unselected” (Figure 2). The monomorphic experiments capture the case where the population moves as a single entity, the polymorphic experiment captures the case where the population spreads across many se- quences, and the unselected experiments show how the gene evolves in the absence of selection for protein function. In all experiments, at each generation we used error-prone PCR to intro- duce an average of 1.4 nucleotide mutations per P450 gene (Table 1). The mutant genes were ligated into a plasmid and transformed into E. coli [25], and transformants were selected using the plasmid’s antibiotic resistance marker. For the unselected case, we randomly picked one of the mutants, recovered the mutant gene with a plasmid mini-prep, and used this mutant as the template for the next generation of error- prone PCR. We performed four independent replicates of unselected evolution, evolving each for 12 generations. For the monomorphic and polymorphic pop- ulations, we imposed the selection criterion that the P450s hydroxylate 12-pNCA with at least 75% of the total activity of the original par- ent gene. We expressed the P450s in E. coli, and then assayed the cell lysates for activity in a high-throughput 96-well plate format. The total amount of product produced by 80 µl of clarified lysate in 40 minutes was compared to the median of four control wells containing the original parent P450 to determine if the mutant met the selection criterion. The only differ- ence between the monomorphic and polymor- phic experiments was the size of the evolving populations. In the monomorphic limit, each mutation is either lost or goes to fixation before the next occurs. We enforced this evolutionary dynamic by holding the population size to a single protein sequence, similar to the “blind ant” random walk of [11]. At each generation, we assayed a single mutant. If this mutant met the selection criterion, then it was carried over to the next generation, corresponding to a neu- tral mutation going to fixation. If the mutant failed the selection criterion, then the popu- lation stayed at the previous sequence for the next generation, corresponding to a mutation lost to selection. If all of the mutants assayed had zero or one mutations, then this proto- col would correspond exactly to the equations of [11, 22]. However, in order to achieve appre- ciable sequence evolution on a laboratory time scale, we used a mutation rate that sometimes produced multiple mutations in a generation. We mathematically describe this situation in the Mathematical Appendix; here we simply note that it is possible to think of each gener- ation as introducing a single mutational event rather than a single mutation. We performed 22 independent replicates of monomorphic evo- lution, evolving each for 25 generations. In the polymorphic limit, the population spreads across many sequences. To implement this experimentally, we assayed 435 mutants at each generation. The selection criterion was used to classify each mutant as functional or nonfunctional. In neutral evolution, all func- tional mutants reproduce with equal proba- bility. We therefore pooled equal volumes of stationary-phase cultures of each functional mu- tant and recovered the pooled genes with a mini- prep. The polymorphic evolution experiment therefore approaches the equations of [11, 22], again with the exception that a sequence may undergo multiple mutations at a single gener- ation. We give the equations describing this situation in the Mathematical Appendix. Since the population evolves deterministically in the polymorphic limit [11, 22], a single replicate was performed. Because mutations accumu- late more rapidly in the polymorphic experi- ments than the monomorphic ones, we evolved the polymorphic population for 15 generations rather than 25. Mutations and mutational robustness Figure 3 shows how mutations accumulated dur- ing the course of the neutral evolution experi- ments (full data are in Table 2 and Additional File 2). Since the unselected protein popula- tions evolve without constraint, mutations ac- cumulate at the same rate at which they are introduced by error-prone PCR, 1.4 nucleotide mutations per generation. Because selection eliminates mutations that disrupt P450 activ- ity, mutations accumulate more slowly in the monomorphic and polymorphic populations. Mutations accumulate more rapidly in the poly- morphic population than in the monomorphic populations. This difference in rates is pre- dicted by the equations in the Mathematical Appendix to be a consequence of the fact that the polymorphic population is more mutation- ally robust, and so can tolerate more of the possible mutations. To test directly whether the polymorphic population evolves higher average mutational robustness, we measured the fraction of 435 random mutants that met the selection crite- rion. Figure 4 shows that the polymorphic pop- ulation neutrally evolved to a markedly higher mutational robustness than the monomorphic populations, with 50± 2% of the final polymor- phic population mutants continuing to function versus 39 ± 2% for the final monomorphic pop- ulations (Chi-square P -value of 10−3 that these values are significantly different). The only dif- ference between the two types of populations was their size, so evolution has clearly favored mutational robustness in the larger and thus more polymorphic population. This finding represents the first experimental support for the prediction that highly polymorphic popu- lations evolve excess mutational robustness [11]. Theory predicts that the excess mutational robustness of a highly polymorphic protein population comes from increased protein sta- bility [22]. Because the P450 variants unfold irreversibly, an equilibrium thermodynamic sta- bility ∆Gf cannot be measured. We therefore determined stability to irreversible thermal and chemical denaturation, two highly correlated measures of P450 stability that have previously been shown to contribute to mutational ro- bustness [5] (see Additional Files 3, 4, and 5). Figure 5 shows that proteins from the polymor- phic population were in fact more stable than their counterparts from the monomorphic pop- ulation. We also observed that proteins in the polymorphic population tended to accumulate to higher levels in E. coli (Figure 5). Elevated expression could be a byproduct of increased stability, or it could independently increase mu- tational robustness by allowing the proteins to better tolerate mutations that decrease codon adaptation or reduce folding efficiency. It is pos- sible that additional unrecognized biophysical factors also contributed to the excess mutational robustness of the polymorphic population, but no such factors were immediately obvious. Interpretation in terms of the P450 neutral network The higher mutational robustness of the poly- morphic population is due to the fact that it occupies the P450 gene neutral network dif- ferently than the monomorphic populations. Measurements from the evolution experiments can therefore be used to infer basic proper- ties of the underlying neutral network of P450 genes, as originally noted by van Nimwegen and coworkers [11]. In the Mathematical Appendix, we derive approximations for the normalized principal eigenvalue 〈ν〉∞ and the normalized average connectivity 〈ν〉o of the neutral net- work, where in both cases the normalization is obtained by dividing by the network coordi- nation number. We obtain 〈ν〉∞ = 0.51 and 〈ν〉o = 0.35 for the P450 gene neutral net- work. Our ability to consistently estimate these two parameters from four different experimental measurements supports the idea that the the- ory that we elaborate in the Mathematical Ap- pendix appropriately describes the experiments. The difference between 〈ν〉∞ and 〈ν〉o is a mea- sure of the extent to which some P450 neutral network nodes have more connections than oth- ers. We note that 〈ν〉∞ is approximately equal to the exponential decline parameter for the asymptotic decline in the fraction of functional mutants with increasing numbers of random nu- cleotide mutations [3, 26,27] (see Mathematical Appendix). Previous studies looking at this ex- ponential decline have reported 〈ν〉∞ = 0.7 for subtilisin [26], 〈ν〉∞ = 0.7 for 3-methyladenine DNA glycosylase [27], and 〈ν〉∞ = 0.7 - 0.8 for TEM1 β-lactamase [3]. These comparisons suggest that P450 has a sparser neutral net- work (smaller 〈ν〉∞) than these other proteins. We suspect, however, that these earlier studies (one of which is our own) overestimate 〈ν〉∞ due to insufficient equilibration of the starting sequence. We believe that the approach of the current work is more accurate for determining 〈ν〉∞ because the measurements are made after many mutations have equilibrated the initial se- quence. This approach could be used in future experiments to compare the neutral network connectivities of proteins from different fami- lies. Conclusions We have demonstrated that neutral evolution favors more mutationally robust proteins when the evolving population is highly polymorphic. Strikingly, the excess mutational robustness is due only to population polymorphism, and so will arise in any population of sufficiently large size. Our work is the first experimental demon- stration of this phenomenon, which is predicted to occur quite generally in neutrally evolving proteins and nucleic acids [11]. Furthermore, we were able to identify one of the biophysical factors underlying the increase in mutational robustness by showing that proteins from the highly polymorphic population are more sta- ble. We recognize that evolution in a biological context will be more complex. In our exper- iments, fitness was the P450’s ability to be expressed in active form by bacteria grown to saturation in an environment with plentiful nu- trients. Biological fitness, however, depends on numerous additional and subtle effects such as the metabolic costs of synthesis or the burdens imposed by misfolded molecules. Some muta- tions that are neutral in the experiments may therefore have deleterious effects in a biologi- cal setting [28]. The experiments nonetheless capture the overriding constraint that proteins retain their biochemical functions. Our success in quantitatively explaining the results supports the notion that important aspects of protein evolution can be described simply in terms of mutational effects on stability [22,28]. An obvious question is whether evolution in nature favors mutational robustness by the process we have demonstrated. Whether natu- ral populations will neutrally evolve mutational robustness depends on whether they are suffi- ciently polymorphic, which will be the case if the product of their effective population size N and per protein per generation mutation rate µ is much greater than one [11, 12]. Accurately estimating Nµ, which is closely related to the widely used parameter θ in population genet- ics, for natural populations is difficult [29, 30] (note that since mutational robustness is a protein-wide property, the relevant mutation rate is per protein, which is ≈ 102 to 103 larger than the per codon mutation rate). For hu- mans and other multicellular organisms, Nµ is probably too small [31] for their proteins to neutrally evolve mutational robustness. But estimates [31, 32] place Nµ ≈ 10 to 100 for typical-length proteins in bacteria, and it is probably much higher for many viruses [33,34]. It is therefore likely that many viral and some bacterial proteins have neutrally evolved extra mutational robustness. The neutral evolution of protein mutational robustness may also contribute to adaptive evo- lution. Experiments have shown that extra sta- bility increases a protein’s evolvability by allow- ing it to tolerate a wider range of functionally beneficial but destabilizing mutations [5]. A similar phenomenon seems to occur in natural evolution, where functionally neutral but stabi- lizing mutations can play a key role in adaptive evolution by counterbalancing the destabilizing effects of other functionally beneficial muta- tions [35]. Viruses and perhaps bacteria may thus benefit from large population sizes and high mutation rates that drive an increase in the mutational robustness and stability of their proteins, which in turn enhances the capacity of these proteins to rapidly change their sequences and evolve new functions. Methods Equilibration evolution of the P450 protein We began with a 21B3 P450 peroxygenase that had been engineered for highly efficient hydrox- ylation of 12-pNCA [23] (sequence shown in Ad- ditional File 6). This P450 was not well equi- librated to the constant selection criterion that we planned to impose, since it had substantially higher total activity. We therefore neutrally evolved it for 16 generations in order to cre- ate P450s that were better equilibrated to the selection criterion. We evolved two parallel pop- ulations, which we named R1 and R2. The pro- cedure was exactly identical to that described below for the polymorphic evolution with the following exceptions: • Starting sequence: the starting sequence for the equilibration evolution was the 21B3 sequence. • Population size: each of the two equili- bration evolution populations had a size of 174 sequences rather than the 435 used for the polymorphic evolution. • Selection criterion: the sequences were re- quired to have at least 75% of the total activity of the 21B3 P450. • Mutation rate: the mutation rate for the equilibration evolution was much higher than for the polymorphic evolution. The error-prone PCR protocol used 200 µM manganese chloride (MnCl2), rather than the 25 µM used for the polymorphic evo- lution. We estimate that this error-prone PCR protocol introduced ≈ 4 nucleotide mutations per P450 gene at each genera- tion during the equilibration evolution. We performed 16 generations of equilibration evolution, and then randomly selected 23 func- tional mutants from each of the R1 and R2 populations (Additional File 7). We picked one of these mutants, R1-11, for use as the parent for the neutral evolution experiments. Detailed protocol for evolution experiments We began with the R1-11 P450 BM3 heme domain variant (sequence in Additional File 1) cloned into the pCWori [25] plasmid with a 5’ BamH1 and 3’ EcoR1 site as described in [5]. The cloning primers were pCWori for (5’- GAAACAGGATCCATCGATGCTTAGGAGGTCAT- 3’ and pCWori rev clone (5’-GCTCATGTTTGACAGCTTATCATCG- 3’). We used error-prone PCR to generate mutants, taking great care to make the error- prone PCR protocol repeatable by using a rela- tively small number of thermal cycles. This was both to control the mutation rate by ensuring that the reaction did not saturate the reagents (which would cause the number of doublings to become sensitive to the initial template concen- tration), and to avoid the PCR-based recombi- nation events which can occur during with the last few thermal cycles of PCR reactions [36,37]. The PCR reactions were 100 µl in volume, and contained ≈ 13 ng of plasmid template (cor- responding to ≈ 3 ng of template gene), 7 mM magnesium chloride MgCl2, 1 × Applied BioSystems PCR Buffer II without MgCl2, 25 µM MnCl2, 0.5 µM pCWori for primer, 0.5 µM pCWori rev primer, 200 µM of dATP and dGTP, 500 µm of dTTP and dCTP, and 5 units of Applied Biosystems AmpliTaq polymerase. The reactions were run on the BLOCK setting of a MJ Research PCR machine with a program of 95oC for 2 minutes, then 15 cycles of (95oC for 30 seconds, 57oC for 30 seconds, 72oC for 90 seconds), and then cooling to 4oC. This proto- col yielded roughly 1-1.5 µg of product gene (as quantified by gel electrophoresis versus a known standard), for a PCR efficiency of ≈ 0.5. Se- quencing the unselected populations at the end of the experiment indicated that this protocol introduced an average of 1.4 ± 0.2 nucleotide mutations, with the nucleotide error-spectrum shown in Table 1. Because the number of PCR doublings is large compared the average muta- tion rate, the distribution of mutations among sequences should be well-described by the Pois- son distribution [38,39]. The mutant genes from the error-prone PCR were purified over a ZymoResearch DNA clean and concentrator column, and digested at 37oC with EcoR1 and BamH1. The digested genes were then purified from an agarose gel with ZymoResearch DNA gel extraction columns, and ligated into pCWori plasmid that had been digested with BamH1 and EcoR1 and dephos- phorylated. The ligations were transformed into electro-competent catalase-free Escherichia coli [25] (the catalase is removed because it breaks down the hydrogen peroxide utilized by the P450 peroxygenase), plated on Luria Broth (LB) plates containing 100 µg/ml of ampicillin to select for the plasmid’s antibiotic resistance marker, and grown at 37oC. Transformation of a control ligation reaction without any digested gene yielded at least 100-fold fewer colonies, indicating that the rate of plasmid self-ligation was less than one percent. Individual mutant colonies from the plates were picked into 96-well 2 ml deep-well plates containing 400 µl of LB supplemented with 100 µg/ml ampicillin. Each plate contained four parental control wells with cells carrying the parent R1-11 gene, four null control wells with cells carrying the pCWori plasmid without a P450 gene, and a non-inoculated well to check for contamination. For the polymorphic pop- ulation, we picked five such plates with all 87 other wells containing different mutants for a total population size of 5 × 87 = 435 mutants. For the 22 monomorphic populations (we be- gan with 24 populations but two had to be discarded due to contamination), we picked a single colony for growth and screening. For the unselected populations we picked a single colony for growth without screening. The LB deep-well plates were grown for 16-20 hours at 30oC, 210 revolutions per minute (rpm), and 80% relative humidity in a Kuhner humidified shaker. To ex- press the P450 mutants, we prepared 2 ml deep well plates containing 400 µl per well of terrific broth (TB) supplemented with 200 µM iso- propyl β-D-thiogalactoside (IPTG), 100 µg/ml ampicillin, and 500 µM of δ-aminolevulinic acid. We used a pipetting robot inoculated these TB plates with 100 µl from the LB plates. We stored the LB deep-well plates at 4oC, and grew the TB deep-well plates in the humidi- fied shaker at 30oC, 210 rpm, and 80% relative humidity for 22-24 hours. After this growth, the cells were harvested by centrifuging the TB deep-well plates at 4000×g for 5 minutes and discarding the liquid. The cell pellets were flash-frozen in liquid nitrogen to aid in cell lysis. To lyse the cells for the assays, we resus- pended the cell pellets in 300 µl of 100 mM [4- (2-hydroxyethyl)-1-piperazinepropanesulfonic acid] (EPPS) (pH 8.2) with 0.5 mg/ml lysozyme and 4 units/ml of deoxyribonuclease by pipet- ting 40 times with the pipetting robot. We then incubated the plates at 37oC for 30 minutes, again resuspended with the pipetting robot, and put back at 37oC for an addition 30 minutes. We then pelleted the cell debris by centrifu- gation at 6000×g for 5 minutes at 4oC. The pipetting robot was used to dispense 80 µl of the clarified lysate into 96-well microtiter plates (Rainin). We prepared a 6× stock of 1.5 mM 12-pNCA in 36% dimethyl sulfoxide (DMSO) and the EPPS buffer (the 12-pNCA was stored in the DMSO solution and combined with the buffer immediately before use). We used a mul- tichannel pipette to add 20 µl of this substrate stock to each well of the microtiter plate. We briefly mixed the plates with “shake” setting of a 96-well plate spectrophotometer, and read an absorbance baseline at 398 nm. We then immediately added 20 µl of a freshly prepared solution of 24 mM hydrogen peroxide in the EPPS buffer to initiate the reaction, and mixed again. The final reaction conditions were there- fore the EPPS buffer with 6% DMSO, 4 mM hydrogen peroxide, and 250 µM 12-pNCA. After 40 minutes we quantified the amount of enzy- matic product by the increase in absorbance at 398 nm. This absorbance increase is due to the 4-nitrophenolate molecule released after the P450 hydroxylates the twelfth carbon of the 12-pNCA molecule [23]. To score the mutants as functional or nonfunctional, we compared their gain in absorbance minus the median null control reading to that of the median parental control reading minus the median null control reading. All mutants that had at least 75% of the parental gain were scored as functional, all other mutants were scored as nonfunctional. We used the information from these assays to select the parents for the next generation. For the unselected population we did not require the mutants to be functional, so the selected mutant was used to start a 4 ml culture of LB with 100 µg/ml ampicillin, and the plasmid DNA was harvested with a mini-prep. This plasmid DNA was used as the template for the next round of error-prone PCR. Therefore, after the first gen- eration the four unselected replicates diverged into four separate error-prone PCR reactions. These unselected replicates were evolved for a total of twelve generations, and were sequenced at every third generation. For the polymorphic population, all mu- tants that were functional contributed an equal amount of plasmid DNA as template for the next generation. In order to do this, we col- lected 50 µl of the culture from the LB deep-well plate for each mutant that was scored as func- tional. All of these LB aliquots were pooled, and then the plasmid DNA was collected with a mini-prep. The pool of plasmid DNA was used as template for the next generation’s error- prone PCR reactions. We performed 15 gener- ations of evolution for this polymorphic pop- ulation. Note that at each generation we are assaying 435 mutants as part of the evolution- ary procedure, so this provides information on mutational robustness. At every third gen- eration, we also selected a random sample of functional mutants for sequencing. After 15 generations, we randomly selected 22 mutants for stability measurements and sequencing anal- ysis. The random selections were made from all functional mutants with the Python computer language random number generator. For the monomorphic populations, at each generation we assayed just a single mutant. If that mutant was nonfunctional, then at that generation the population stayed at its original sequence. In that case, for the next generation we simply picked a new mutant from the previ- ous generation’s plate of transformed mutants. If the mutant we screened was functional, then that mutant represented the new population. We therefore grew a 4 ml LB culture with 100 µg/ml of ampicillin, and collected the plasmid DNA with a miniprep. That plasmid DNA was then used as the template for the next genera- tion’s error-prone PCR reaction. We thus had 22 (actually 24 before 2 were contaminated) in- dependent monomorphic populations that were being evolved in parallel. Each was evolved for 25 generations, and at the end of these 25 gen- erations we measured the stability of the final sequence of each population. Each time an as- sayed mutant was functional, we sequenced the new P450 gene. We also measured the average mutational robustness of the monomorphic pop- ulations at every fifth generation. To do this, we did a pooled mini-prep of equal volumes of LB cultures of all 22 replicates to obtain a equal mix of plasmid DNA. We then performed error-prone PCR on this mix, and assayed 435 mutants to measure the fraction functional. Full neutral evolution data are in Additional File 2. Test for recombination during error-prone During the polymorphic population evolution, we performed error-prone PCR on a mix of different plasmids. It is common for PCR on mixed templates to lead to recombination events during the reaction [36, 37]. We at- tempted to reduce this recombination by using a small number of thermal cycles. However, in order to test for recombination, we analyzed the sequences of the final 22 selected members of the polymorphic population. There are a variety of statistical tests to detect recombi- nation in a set of sequences. A comparison of these tests by Posada [40] found that the Max-Chi2 method developed by John Maynard Smith [41] performs well. A publicly avail- able implementation of this method [42] is at http://www.lifesci.sussex.ac.uk/CSE/test/maxchi.php. We used this implementation to analyze the 22 final polymorphic sequences, and the resulting P -value was 0.29 after 100 random permuta- tions, indicating that there is not significant recombination. Measurement of P450 stabilities We measured the stabilities to both irreversible thermal and irreversible urea denaturation of the final (generation 25) member of each monomorphic population, as well as of the 22 randomly selected members of the polymorphic population. As discussed in the Supplemen- tary Information of [5], cytochrome P450 BM3 heme domains (and indeed most P450s) dena- ture irreversibly, forcing us to use resistance to irreversible denaturation to quantify protein stability. The first stability measure is the T50, defined as the temperature at which half of the protein is denatured after a 10 minute incu- bation. The second stability measure is the [urea]50, defined as the urea concentration at which half of the protein denatures after a 4 hour room-temperature incubation. Each set of measurements (those of T50 and [urea]50) was performed on all of the mutants in the same day, and each mutant was treated identi- cally. Therefore, it is possible to make accurate comparisons of the relative values of the mea- surements within the data set. However, the absolute values of the T50 and [urea]50 values may be less accurate. Therefore, care should be taken in comparing the absolute value of these measurements to those of other studies (such as [5]). Both the T50 and [urea]50 measurements were performed in clarified cell lysate. The protein was expressed using catalase-free E. coli [25] containing the encoding gene on the IPTG inducible pCWori [25] plasmid. We used freshly streaked cells to inoculate 2 ml cultures of LB supplemented with 100 µg/ml of ampi- cillin, and grew these starter cultures overnight with shaking at 37oC. We then used 0.5 ml http://www.lifesci.sussex.ac.uk/CSE/test/maxchi.php from these starter cultures to inoculate 1 L flasks containing 200 ml of TB supplemented with 100 µg/ml of ampicillin. The TB cultures were grown at 30oC and 210 rpm until they reached an optical density at 600 nm of ≈0.9, at which point IPTG and δ-aminolevulinic acid were added to a final concentration of 0.5 mM each. The cultures were grown for an additional 19 hours, then the cells were harvested by pel- letting 50 ml aliquots at 5,500 g and 4oC for 10 min, and stored at -20oC. To obtain clarified lysate, each pellet was resuspended in 8 ml of 100 mM EPPS (pH 8.2) and lysed by sonica- tion, while being kept on ice. The cell debris was pelleted by centrifugation at 8,000 g and 4oC for 10 minutes, and the clarified lysate was decanted and kept on ice. For the T50 measurements, 125 µl of clar- ified lysate from a single mutant was added to all 12 wells in a row of a 96-well hard-shell thin-wall microplate (MJ Research). The plate was heated for 10 minutes using the gradient method of an Eppendorf Mastercycler gradient PCR machine, with the gradient set at either 33oC-45oC or 46oC-58oC (each mutant was ex- posed to both of these gradients), the machine on the BLOCK setting, and the heated lid set to 75oC with the lid WAIT option. The plate was then cooled to 4oC, removed from the PCR ma- chine, and centrifuged at 5,500 g and 4oC for 5 minutes to pellet any debris. A pipetting robot was used to dispense 80 µl of the supernatent into a 96-well microtiter plate (Rainin), and the amount of remaining properly folded P450 was quantified from the carbon monoxide difference spectrum as described below. The T50 values were determined by fitting sigmoidal curves the percent of remaining protein as shown in Addi- tional File 3. Our ability to accurately compare T50 values within the data set requires that each well in a given column of the gradient PCR ma- chine be at the same temperature. We used a thermocouple to measure the temperature of the wells with the machine lid open, and con- firmed that the wells were within a few tenths of a degree of the same temperature. Further evidence for the consistency of our T50 val- ues comes from the fact that two independent measurements of the T50 for our R1-11 par- ent yielded values that differed by only 0.1oC. However, the absolute values of the measured temperatures are less accurate. Thermocouple measurements indicated that, with the machine lid open, the wells were ≈ 1oC cooler than the indicated temperature. We were unable to as- certain the temperatures with the heated lid closed, but based on comparisons water bath measurements, the temperatures with the lid closed slightly exceeded the indicated tempera- tures. For the [urea]50 measurements, 125 µl of the clarified lysate from a single mutant was added to all 12 wells in a row of a 96-well microtiter plate. A pipetting robot was then used to add and mix 125 µl of a 2X solution of urea in 100 mM EPPS (pH 8.2) so that each subsequent column had a higher concentration of urea, and so that the final urea concentrations were those shown in Additional File 4. The plates were left on the bench at room temperature for 4 hours, and the amount of remaining properly folded P450 was quantified from the carbon monoxide difference spectrum as described be- low. The [urea]50 values were determined by fitting sigmoidal curves to the percent of re- maining protein. Evidence for the consistency of the [urea]50 measurements comes from the fact that two independent measurements of the [urea]50 for our R1-11 parent yielded values that differed by only 0.01 M. In addition, the [urea]50 and T50 values are highly correlated (Additional File 5), indicating that they provide consistent measures of stability. For both the T50 and [urea]50 measurements, the folded P450 was quantified from the car- bon monoxide difference spectrum [43]. The microtiter plates containing the P450 samples were first used to read blank spectra at 450 and 490 nm using a Tecan Safire 2 plate reader. The plates were then incubated for 10 minutes in an airtight oven with carbon monoxide. The plates were removed form the oven and 10 µl of 0.1 M sodium hydrosulfite in 1.3 M potassium phos- phate (pH 8.0) was immediately added to each well. After 5-10 minutes, spectra were again read at 450 and 490 nm. The amount of P450 is proportional to the increase in the signal at 450 nm after this procedure minus the change in the signal at 490 nm. Mathematical Appendix Contents A.1 Mathematical background 12 A.2 Monomorphic limit 14 A.3 Polymorphic limit 15 A.4 Approximations for polymorphic limit 16 A.5 Approximations for monomorphic limit 18 A.6 Interpretation in terms of neutral networks 20 A.7 Detailed justification for approximating pM by po 22 A.1 Mathematical background The first purpose of this appendix is to provide mathematical equations that describe the experi- ments. The second is to show how four measurements from the experiments can be used to calculate two quantities that describe the topology of the underlying protein neutral network. We will derive two equations for both quantitites, each in terms of a different measurement. The fact that the four equations will be seen to yield consistent results provides evidence for the accuracy of the following calculations. Our calculations are based on a view of neutral protein evolution as a process constrained by a stability threshold, a view that we originally introduced to explain experimental protein mutagen- esis results [3]. The calculations closely parallel our earlier work [22], which is in turn based on a general theoretical treatment of evolution on neutral networks by van Nimwegen and coworkers [11]. These calculations will probably be most thoroughly understood by first reading those works. The primary difference between the current calculations and [22] is that previously we assumed that the per generation per protein mutation rate µ was ≪ 1, so that at each generation a protein was either unmutated (with probability 1−µ) or experienced a single mutation (with probability µ). In contrast, here we allow the mutation rate to be arbitrarily large, so that a protein may experience multiple mutations in a single generation (in this sense the calculations resemble the generalization by Wilke [18] of [11]). Specifically, let fm be the probability that a protein experiences m mutations in a single generation. Here we derive results for arbitrary fm, and then approximations relevant to the form of fm in the experiments. In the limiting case of small mutation rate (where f0 = 1− µ, f1 = µ, and fm = 0 for m > 1), the calculations here reduce to those in [22]. Proteins evolving in nature typically experience very low mutation rates, so [22] probably offers the best description of natural protein evolution. The calculations presented here are designed to specifically treat the evolutionary dynamics of the experiments. A protein’s thermodynamic stability is described by its free energy of folding, ∆Gf , with more negative values indicating more stable proteins. As described in several previous papers [3, 5, 22], we assume that selection requires a protein to fold with some minimal stability ∆Gminf , so that a protein adequately folds if and only if ∆Gf ≤ ∆G f . The amount of extra stability a protein possesses relative to the stability threshold is given by ∆Gextraf = ∆Gf − ∆G f ; note that all folded proteins will have ∆Gextraf ≤ 0. We further assume that as long as ∆G extra f ≤ 0, selection is indifferent to the exact amount of extra stability that a protein possesses (see [22] for a discussion of the limitations of this assumption). We conceptually divide the continuous variable of protein stability into small discrete bins of width b. Specifically, a protein is in bin i if it has ∆Gextraf be- tween (1− i) b and −ib, where i = 1, 2, . . .. Mutating a protein changes its stability by an amount ∆∆G (defined as the stability of the mutant protein minus the stability of the initial protein), and so may move it to a new stability bin. In [22], we defined a matrix W with elements Wij giving the transition probabilities that a single mutation changes a protein’s stability from bin j to bin i. We noted that W could be computed from the distribution of ∆∆G values for all single mutations, and argued that W remains fairly constant during neutral evolution since the distribution of ∆∆G values remains relatively unchanged. However, we emphasize that (as discussed in detail in [22]) the constancy of the ∆∆G distribution remains an assumption, albeit one that has now been shown to be quite accurate for lattice proteins [3, 22,44] and provide a consistent theoretical explanation for a growing body of experimental results (the current work as well as [3]). Since we are allowing for larger mutation rates, and we must consider the possibility that a protein’s stability might change due to multiple mutations at a single generation. Therefore, we make a more general definition of Wij,m as the probability that m random mutations to a protein in stability bin j change its stability to bin i, and let Wm be the matrix with elements Wij,m. Note that Wm only describes mutations that cause transitions from one folded protein to another, since the stability bins i = 1, 2, . . . all correspond to folded proteins. As before [22], we assume that Wm is roughly constant during evolution, meaning that the distribution of ∆∆G values for multiple mutations is roughly constant during neutral evolution. Note that if m = 1, then Wm is just the matrix W that can be computed from the distribution of single-mutant ∆∆G values [22]. We will now use the matrices Wm to calculate the following characteristics of a population that has evolved to equilibrium: the distribution of stabilities, the average number of mutations 〈m〉T accumulated after T generations, and the average fraction 〈F〉 of stably folded proteins in the population. We then introduce a few approximations (that should be quite accurate for the experimental work in this paper) that greatly simplify these calculations. Finally, we relate the calculations to properties of the underlying protein neutral network. As described generally by van Nimwegen and coworkers [11], the evolutionary dynamics depend on whether the evolving population tends to be monomorphic or highly polymorphic. When the per sequence per generation mutation rate µ is ≪ 1, whether the population is mostly monomorphic or highly polymorphic is determined by the product of the population size N and µ: when Nµ ≪ 1 the population is mostly monomorphic, and when Nµ ≫ 1 the population is highly polymorphic. However, with multiple mutations per generation, Nµ is no longer an appropriate parameter to distinguish between mono- and polymorphism, since if the population size is sufficiently small the population can still be monomorphic even if there are multiple mutations per generation. Specifi- cally, in one set of experiments we constrained the population to be monomorphic (by maintaining a population size of one), but still allowed the single protein in this population to experience more than one mutation at a generation. So we instead denote the populations as either monomorphic or polymorphic. We indicate quantities calculated for the monomorphic population by the subscript M (i.e. 〈F〉M ) and those calculated for the polymorphic population by the subscript P (i.e. 〈F〉P ). A.2 Monomorphic limit In the limit of a completely monomorphic population, all of the proteins are in a single stability bin. Let pi (t) be the probability that the population is in stability bin i at time t, and let p (t) be the column vector with elements pi (t). At each generation there is a probability f0 that there is no mutation that becomes fixed in the population, a probability of Wij,mpj that the population experiences a mutational event (which could be a single mutation or several simultaneous mutations) that moves it into bin i, and a probability Wji,m that the population is in bin i and experiences one or more mutations that move it to another bin of stably folded proteins. Define νi,m = Wji,m to be the fraction of m-mutants of a protein in bin i that still fold, and let Vm be the matrix with diagonal elements given by Vii,m = νi,m and all other elements zero. The time evolution of p is p (t+ 1) = fm (Wm −Vm) p (t) (1) where I is the identity matrix. Note that mutations that destabilize a protein beyond the stabil- ity threshold are immediately lost to natural selection, and so leave the population in its original stability bin. This describes the experiments for the monomorphic populations, where we retain the parental sequence if the single mutant we generate is nonfunctional. Equation 1 corresponds to Equation (1) of [22], and the blind ant random walk described by van Nimwegen and coworkers [11]. Equation 1 describes a Markov process with a non-negative, irreducible, and acyclic transition matrix, and so p approaches a unique stationary distribution (equilibrium value) of pM given by the eigenvector equation fm (Wm −Vm) pM. (2) Once p has reached equilibrium, the average fraction of proteins that still stably fold at each generation is 〈F〉M = e pM (3) where e = (1, . . . , 1) is the unit row vector. To calculate 〈m〉T,M , the average number of mutations accumulated after T generations once the population has equilibrated, we note that at each generation there is a probability of fmpj Wij,m that a randomly chosen protein is in bin j, experiences m mutations, and still stably folds. The average number of mutations accumulated in a single generation is simply the average ofm weighted over this probability. So summing over all values of m and j, we see that 〈m〉T,M = Te mfmWmpM. (4) This equation corresponds to Equation (6) of [22], which was derived using an embedded Markov process formalism. Here we have foregone this formalism for the more intuitive argument presented above, since we do not attempt to calculate higher moments of the number of mutations. A.3 Polymorphic limit In the limit when the population is highly polymorphic, at each generation there are sequences in many different stability bins. In this case, we describe the distribution of stabilities by the column vector x (t), with element xi (t) giving the fraction of proteins in stability bin i at time t. At generation t, the fraction of mutants that continue to fold is 〈F〉t = e x (t) . (5) Therefore, in order to maintain a constant population size, each remaining protein must produce an average of αt = 〈F〉t offspring. The population therefore evolves according to x (t+ 1) = αt x (t) . (6) After the population evolves for a sufficiently long period of time, x will approach an equilibrium value of xP. At this equilibrium, the average fraction of mutants that fold at each generation is 〈F〉P = e xP, (7) and the equilibrium reproduction rate is α = 〈F〉P . Therefore, xP = α xP. (8) Equations 7 and 8 can be combined to show that xP and 〈F〉P can be calculated from the eigenvector equation (〈F〉P − f0)xP = fmWmxP, (9) with (〈F〉P − f0) the principal eigenvalue of the nonnegative and irreducible matrix fmWm. Equation 9 corresponds to Equation (14) of [22], Equation (6) of the work by van Nimwegen and coworkers [11], and Equation (13) of the work by Wilke [18]. We now calculate 〈m〉T,P , the average number of mutations accumulated in T generations after the population has equilibrated. At equilibrium, there is a probability of fmxj Wij,m that a protein is in bin j, experiences m mutations, and still stably folds. Subsequently, all of these folded proteins produce an average of α offspring. The average number of mutations accumulated in a single generation is simply the average of m weighted over this probability, and then multiplied by the average reproduction rate. So summing over all values of m and j, we obtain 〈m〉T,P = αTe mfmWmxP = mfmWmxP. (10) This equation is the counterpart of Equation (18) of [22], where we have again foregone the em- bedded Markov process formalism for a more intuitive derivation. A.4 Approximations for polymorphic limit We can dramatically simplify the results from the previous sections with several reasonable ap- proximations. The first approximation is that the ∆∆G values for random mutations are roughly additive, and is supported by a number of experimental studies of the thermodynamic effects of mutations [45–47]. We have previously shown that this approximation can be used to accurately describe experimental protein mutagenesis data with a simple stability threshold model [3]. Under this approximation, the distribution of net ∆∆G values for multiple mutations can be computed from the distribution of ∆∆G values for single mutations by performing convolutions of the single- mutation ∆∆G distribution [3], meaning that Wm for arbitrary m can be computed solely from the distribution of ∆∆G values for single mutations. However, to simplify the equations from previous sections, we need to express Wm for arbitrary m only in terms of W (recall that W = W1). Since W only contains information about stability transitions from folded proteins to other folded pro- teins, if we make the second approximation that a protein that is destabilized beyond the minimal stability threshold by one mutation is not re-stabilized to a folded protein by a subsequent muta- tion, then Wm = W m. This approximation that unfolded proteins are not re-stabilized should be quite accurate since stabilizing mutations tend to be relatively rare and small in magnitude [48–51] (this is the underlying idea behind the Markov chain approximation that was shown to be highly accurate for lattice proteins [44]). To summarize, if ∆∆G values are roughly additive and stabilizing mutations are rare, we have the approximation Wm ≈ W m. (11) Simplifying the equations of the previous sections also requires assigning a specific functional form to fm, the probability that a sequence undergoesmmutations. Here we assume that mutations are Poisson distributed among sequences, so that e−µµm where µ = mfm is the average number of mutations per protein per generation. When the mu- tations are introduced by error-prone PCR, the Poisson distribution is an excellent approximation to the true theoretical distribution of mutations created by error-prone PCR [38,39] provided that µ is much less than the number of PCR doublings, as is the case in all of the experiments in the current work. We now use the approximations of Equations 11 and 12 to simplify the results given above for the highly polymorphic limit. We begin by using these approximations to rewrite Equation 9 as 〈F〉P − e xP = e WmxP. (13) This equation makes clear that xP is the principal eigenvector of the matrix Wm, therefore xP must also be the principal eigenvector of W. Now in our earlier work [22], we defined the principal eigenvector of W as x∞, called the corresponding eigenvalue 〈ν〉∞, and showed that this eigenvalue is shown the average fraction of single mutations that are neutral in a population that is evolving with Nµ ≫ 1 and µ ≪ 1. Therefore, with the approximation of Equation 11, xP and x∞ are equal, and are both defined by the same eigenvector equation, 〈ν〉∞xP = WxP = Wx∞ = 〈ν〉∞x∞. (14) Combining Equations 13 and 14 we have, 〈F〉PxP = e (µ〈ν〉∞) 1−〈ν〉∞ xP (15) Equation 15 can be solved to yield 〈ν〉∞ = 1 + ln 〈F〉P . (16) Similarly, we can simplify Equation 10, 〈m〉T,P = mfmWmxP 1−〈ν〉∞ eWmxP = Te−µ〈ν〉∞ (µ〈ν〉∞) = Tµ〈ν〉∞e −µ〈ν〉∞ (µ〈ν〉∞) = Tµ〈ν〉∞. (17) Solving this equation for 〈ν〉∞ yields 〈ν〉∞ = 〈m〉T,P . (18) A.5 Approximations for monomorphic limit We now simplify the equations for the monomorphic limit. This requires several further approxima- tions. We begin by approximating that the stability probability distribution pM given by Equation 2 by the distribution po defined in [22] as satisfying 0 = (W −V)po. (19) The basic rationale behind approximating pM with po is that Equation 2 can be viewed as a per- turbation to Equation 19 [52]. Essentially, po is an eigenvector of the matrix W −V while pM is the corresponding eigenvector of the matrix W − V + (Wm −Vm). The latter matrix can be viewed as a perturbation to the first, since the sum (Wm −Vm) is small. This smallness is due to the fact that Wm tends to zero with large m, causing Vm to tend towards the identity matrix. In addition, the µm/m! terms tend to zero with large m. Therefore, the terms in the summation are all simply either a perturbation to W − V or involve subtracting terms that are fractions of the identity matrix. The perturbations lead to bounded changes in the eigenvec- tors [52], while the identity matrix terms do not change the eigenvectors. Below we give a more rigorous justification of the assumption that pM is approximately equal to po. We need one additional approximation to make further progress. Both Equations 3 and 4 con- tain terms of the form Wmpo, and even if we use Equation 11 to rewrite these terms as W there are no further clear simplifications. However, any probability vector that is multiplied re- peatedly by W and normalized will eventually converge to x∞ = xP (since this is the principal eigenvector of W). We make the approximation that this convergence is sufficiently rapid to be essentially complete after a single multiplication. This approximation is supported by both protein mutagenesis studies [3,26,27] that indicate that proteins rapidly converge to an exponential decline in the fraction folded (indicating the stability distribution has equilibrated, as discussed below, and by lattice protein studies showing the same [3, 44]. Therefore, we make the approximation that eWmpo = 〈ν〉oeW m−1x∞ = 〈ν〉o〈ν〉∞ where 〈ν〉o = eWpo has the same definition as in [22], where it was defined as the average fraction of functional single mutants of a population evolving with µ ≪ 1 and Nµ ≪ 1. We use these approximations to simplify Equation 3 as 〈F〉M = e = e−µ eWmpo = e−µ 1 + µ〈ν〉o (µ〈ν〉∞) = e−µ (µ〈ν〉∞) = e−µ eµ〈ν〉∞ − 1 . (20) Solving this equation for 〈ν〉o, we find 〈ν〉o = 〈ν〉∞ (〈F〉Me µ − 1) eµ〈ν〉∞ − 1 . (21) We now use the approximations to simplify Equation 4 as 〈m〉T,M = Te mfmWmpM = Te−µ eWmpo = Te−µ〈ν〉o = µTe−µ〈ν〉o (µ〈ν〉∞) = µT 〈ν〉oe 〈ν〉∞−1 . (22) Solving this equation for 〈ν〉o yields 〈ν〉o = 〈m〉T,Me 1−〈ν〉∞ . (23) To recap, we now have equations to calculate 〈ν〉∞ and 〈ν〉o from experimentally measurable quantities. Equations 16 and 18 allow us to calculate 〈ν〉∞ from 〈F〉P and 〈m〉T,P , respectively. Given this calculated value of 〈ν〉∞, Equations 21 and 23 then allow us to calculate 〈ν〉o from 〈F〉M and 〈m〉T,M , respectively. The fact that we have two equations each for 〈ν〉∞ and 〈ν〉o allows us to assess the self-consistency of the approach. A.6 Interpretation in terms of neutral networks Throughout the preceding calculations, we have referred to 〈ν〉∞ and 〈ν〉o as we defined them in [22]: namely, as the average neutrality of protein populations evolving with µ ≪ 1 and Nµ either ≫ 1 or ≪ 1, respectively. However, van Nimwegen and coworkers [11] have shown that they can also be interpreted in terms of the underlying neutral network. In the experiments we make mutations at the nucleotide (rather than amino acid) level, so each point in our sequence space corresponds to a different gene. Every gene that yields an amount of protein sufficient to hydroxylate the twelfth carbon of 12-p-nitrophenoxydodecanoic acid with at least 75% of the total activity conferred by the original R1-11 parent gene represents a node on this neutral network. We note that in the experiments (and also usually in natural evolution), the edges on the neutral network are not all completely equivalent or fully undirected, since some mutations are more likely to occur than others (for example, error-prone PCR with Taq polymerase is more likely to cause an A→G mutation than an A→C mutation). In the analysis that follows, we ignore this complication and assume all neutral network edges are equivalent. In an extremely insightful analysis, van Nimwegen and coworkers [11] have shown that impor- tant characteristics of a neutral network can be inferred from evolutionary quantities. Specifically, they have shown that if a population is evolving with µ ≪ 1 and Nµ ≫ 1, then the average neutrality (which we have denoted by 〈ν〉∞) is equal to the principal eigenvalue of the adjacency matrix of the neutral network, normalized by the network coordination number (number of possi- ble connections per node). In addition, they pointed out that a population evolving with µ ≪ 1 and Nµ ≪ 1 moves like a blind ant random walk, meaning that the average neutrality (which we have denoted by 〈ν〉o) is equal to the average connectivity of a neutral network node divided by the network coordination number. In our P450 experiments, we have measured the values needed to estimate 〈ν〉∞ and 〈ν〉o using Equations 16, 18, 21, and 23. Using the final values listed in Table 2, 〈F〉P = 0.50 and 〈F〉M = 0.39. Taking the final nucleotide mutation values from Table 2, 〈m〉T,P /T = 0.69 and 〈m〉T,M/T = 0.31. The average mutation rate, computed from the un- selected population, is µ = 1.40. So using Equation 16, 〈ν〉∞ = 0.53, while using Equation 18, 〈ν〉∞ = 0.49. The consistency of these two values supports the idea that the calculations above accurately describe the evolutionary process. Taking the average value of these two measurement as 〈ν〉∞ = 0.51, we can then use Equations 21 and 23 to calculate 〈ν〉o. We calculate values of 0.28 and 0.43, respectively. These estimates differ by more than those for 〈ν〉∞, perhaps because additional approximations have gone into the derivation of the relevant equations (in addition, we have made no attempt to carry out the rather complex propagation of the sampling errors of Table 2). However, the values are still in a similar range. Taking the average of these two values, we estimate that 〈ν〉o = 0.35. So overall, we predict that each functional P450 gene should have an average fraction of 0.35 of its sequence nearest neighbors also encoding a functional gene, for an average of about 1,500 neighbor genes. We predict that the principal eigenvalue of the neutral network adjacency matrix is 0.51 ×3L. The fact that principal eigenvalue exceeds the average connectivity indicates that the neutral network is not a regular graph, but instead has some nodes more highly connected than others. The value for 〈ν〉∞ calculated above can also be related to measurements from protein mu- tagenesis experiments. A number of studies [3, 26, 27] have observed that the probability that a protein remains functional after m mutations falls off exponentially with the number of mutations. In fact, the decline is not always exponential for the first few mutations if the starting protein has especially high or low stability [3] or activity [53], but will still converge to this exponential form after a few mutations [3,44,54]. The stability threshold model can be used to relate this decline to 〈ν〉∞, as is done indirectly in the Markov chain approximation of [44]. Here we make that connec- tion explicit. The initial protein has a stability that falls into some stability bin i. Therefore, its stability can be described by the column vector y0, which has element i equal to one and all other elements equal to zero. Now imagine constructing all single mutants of this protein. The fraction of these single mutants that still fold is just eWy0, and the distribution of stabilities among the single mutants is y1 = Wy0 (note that the elements of y1 no longer sum to one). Similarly, after m mutations, the fraction of mutants that still fold is eWmy0, and the distribution of stabilities among the m-mutants is ym = Wmy0. With the approximation of Equation 11, ym = W This makes it clear that ym will converge to a vector proportional to x∞, the principal eigenvector of W. Once this convergence is complete, each new mutation simply reduces the fraction of mu- tants that fold by a factor of 〈ν〉∞, the principal eigenvalue of W (and the spectral radius of the neutral network normalized by the coordination number). Therefore, what we have called 〈ν〉∞ in the present work and [22] is equal to what is called x in [27], q in [26], and 〈ν〉 in [3]. The major difficulty that is faced in extracting 〈ν〉∞ by the method of those three studies [3, 26,27] is that it is not possible to directly assay mutants with m mutations, but instead only possible to assay a set of mutants with a distribution of m. All three studies use different (and valid) methods to account for this distribution, but this accounting is still difficult because most of the functional mutants come from the low m end of the distribution. This makes it hard to get accurate values for the fraction functional after large numbers of mutations, since most of the functional mutants in the set come from sequences with few mutations. For this reason, we believe the current method of measuring 〈ν〉∞ is more accurate. A second caution about comparing values of 〈ν〉∞ from different studies is that its value depends on the nucleotide error-spectrum of the experiment, since differ- ent mutagenesis methods create different distributions of nucleotide and amino acid mutation types. We also briefly mention how we arrived at an estimate of 〈ν〉∞ for 3-methyladenine DNA glycosylase from the data of [27]. This paper reports that a fraction x = 0.34 of amino acid mutations inactivate the protein. We would like to determine the fraction 〈ν〉∞ of nucleotide mu- tations that do not inactivate the protein. Roughly 75% of random mutations to a gene will be synonymous. Therefore, m amino acid mutations should cause about 4m/3 nucleotide mutations. The study of [27] measures that after m mutations, a fraction (1− x) of the mutants are func- tional. That means that 〈ν〉∞ fraction should be functional. Equating these expressions yields 〈ν〉∞ = exp log (1− x) . So using x = 0.34, we arrive at 〈ν〉∞ = 0.73. A.7 Detailed justification for approximating pM by po Here we provide a detailed justification for the approximation that pM is about equal to po. In the monomorphic limit, the time evolution of p is given by Equation 1, and the stationary distribution pM is given by Equation 2. We assume the approximations of Equations 11 and 12 and show that we can approximate pM by po, where po is given by Equation 19. To justify this approximation, we insert po into the right hand side of Equation 1 and ask to what extent po is left unaltered by the dynamics. If po is found to be stationary to good approximation then, by uniqueness of the stationary distribution of an ergodic process, po would be a good approximation to pM. We therefore suppose that at some time t the distribution is given by po and compute, using Equation 1, the change in po after one generation. The new distribution at time t+ 1 is given by p (t+ 1) = fm (W m −Vm) po. (24) Using (V −W)po = 0, and taking components of the above equation, we obtain pi (t+ 1) = p0i + fm [(W m −Vm)po]i . (25) Thus po would be an approximately stationary distribution of the dynamics if | fm [(W m −Vm)po]i| ≪ p0i. We now proceed to show that this will be the case in most situations of interest by deriving upper and lower bounds on the second term of the right hand side of Equation 25. Consider first the term (Wmpo)i, which can be written as (Wmpo)i = k1,...,km Wik1Wk1k2 · · ·Wkm−1kmp0km k1,...,km−1 Wik1Wk1k2 · · ·Wkm−2km−1νkm−1p0km−1 , (26) where we have used Wpo = Vpo in the second equality. We now note that νk ≤ νmax for all k, where νmax is the maximum neutrality, maximized over all bins. This leads to the successive inequalities (Wmpo)i ≤ νmax k1,...,km−1 Wik1Wk1k2 · · ·Wkm−2km−1p0km−1 = νmax k1,...,km−2 Wik1Wk1k2 · · ·Wkm−3km−2νkm−2p0km−2 k1,...,km−2 Wik1Wk1k2 · · ·Wkm−3km−2p0km−2 ≤ νm−1 Wik1p0k1 , (27) yielding the upper bound (Wmpo)i ≤ ν νip0i. (28) In an identical manner, we obtain the lower bound (Wmpo)i ≥ ν νip0i, (29) where νmin is the smallest neutrality, minimized over all bins. Note that both inequalities above become exact equalities when all bins have the same neutrality ν, which could be interpreted as either νmin or νmax. Having obtained inequality constraints on (Wmpo)i, we now consider the term (Vmpo)i, which can be written as (Vmpo)i = p0iνi,m = p0i (Wm)ji = p0i j,k1,...,km−1 Wjk1Wk1k2 · · ·Wkm−1i = p0i k1,...,km−1 νk1Wk1k2 · · ·Wkm−1i ≤ p0iνmax k1,...,km−1 Wk1k2 · · ·Wkm−1i ≤ p0iν Wkm−1i, (30) which yields an identical upper bound to that on (Wmpo)i, namely, (Vmpo)i ≤ ν νip0i, (31) and similarly (Vmpo)i ≥ ν νip0i. (32) It should again be noted that both the above inequalities become exact equalities when all bins have a common neutrality ν. We are now in a position to estimate bounds on the magnitude of the second term of Equation 25. Using the four inequalities of Equations 28, 29, 31, and 32 above, we have − νm−1 νip0i ≤ [(W m −Vm)po]i ≤ − νm−1 νip0i, (33) or equivalently, |[(Wm −Vm)po]i| ≤ − νm−1 νip0i, (34) where the inequality above becomes an exact equality when all bins have the same neutrality. How- ever, in this limit, the right hand side of the above equation vanishes, and therefore the second term of Equation 25 is identically zero in this case, giving the result that pM is exactly equal to po when all bins have the same neutrality, even if µ is arbitrarily large. We now carry out the sum over m to obtain an upper bound on the second term of Equation 25 in the more general and realistic case of unequal neutrality bins. Using Equation 34 and the specific Poisson form of fm, we obtain an upper bound on the fractional change in p0i in one generation: pi(t+ 1)− p0i ≤ νie − νm−1 = νie eµνmax − 1 eµνmin − 1 . (35) The above bound vanishes for small µ, is an increasing function of νmax − νmin, and is typically much smaller than 1. An extreme estimate of the size of the fractional change can be made when νmax = 1 and νmin = 0. In this case, using µ = 1.4 (the value in our experiments), the above inequality simplifies to pi(t+ 1)− p0i 1− e−µ − µe−µ ≃ 0.41νi. (36) Noting that νi < 1, the fractional change in p0i is therefore reasonably controlled even in the most extreme case. For realistic situations, the fractional change in p0i is expected to be much lower, thus justifying the use of po as the stationary distribution of the dynamics of Equation 1. Author Contributions JDB and FHA designed the project and wrote the paper. JDB and ZL performed the bulk of the experiments; OSV assisted with the experiments. JDB and DC analyzed the data. JDB and AR performed the theoretical work. Acknowledgments We thank Claus O Wilke for helpful advice and comments. 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(B) Proteins evolving in a highly polymorphic population are predicted to be more stable than their counterparts in a mostly monomorphic population [22]. The histograms illustrate the distributions of stabilities for the two cases. The increased stability is a biophysical manifestation of excess mutational robustness, since more stable proteins are more mutationally robust [3–5]. polymorphic monomorphic unselected Figure 2 - Outline of the neutral evolution experimental procedure. For the polymorphic population, error-prone PCR was used to generate mutant P450 genes. These genes were ligated into a plasmid and transformed into E. coli. Individual mutants (435) were picked, expressed in E. coli, and assayed for enzymatic activity. All mutants that met the selection criterion contributed an equal amount of plasmid DNA as template for the next generation of error-prone PCR. The monomorphic populations were treated similarly, except only a single mutant was assayed at each generation. If this mutant met the selection criterion then it became the template for the next generation of error-prone PCR; otherwise at the next generation another colony was picked from the same plate. In the unselected populations a single mutant was picked and used as the template for the next generation of error-prone PCR. PSfrag replacements generation 〈mnt〉 unselected polymorphic monomorphic PSfrag replacements 0 10 20 generation 〈maa〉 〈 unselected polymorphic monomorphic Figure 3 - Accumulation of nucleotide (〈mnt〉) and nonsynonymous (〈maa〉) mutations in the experimentally evolved P450 populations. For the unselected and monomorphic populations, numbers are the average over all replicates at the indicated generation; for the polymorphic population they are from a random sample, with sampling standard error shown. PSfrag replacements 0 10 20 generation polymorphic monomorphic Figure 4 - The polymorphic population neutrally evolved a higher average mutational robustness than the monomorphic populations. The fraction functional was determined by assaying 435 mutants (average of 1.5 nucleotide mutations per gene). Error bars show binomial standard error. For the monomorphic population, numbers are the average over all replicates. PSfrag replacements polymorphic monomorphic T50 ( [urea]50 (M) percent parental expression 38 40 42 44 PSfrag replacements polymorphic monomorphic T50 ( [urea]50 (M) percent parental expression number 0.4 0.6 0.8 1.0 PSfrag replacements polymorphic monomorphic T50 ( [urea]50 (M) percent parental expression number 20 60 100 140 Figure 5 - The more mutationally robust proteins are more stable. The P450s from the polymorphic population neutrally evolved higher stability and expression levels than their counterparts from the monomorphic populations. The histograms show the distributions for the final protein from all monomorphic replicates and for the same number of randomly chosen proteins from the final polymorphic population. The plots show (left to right) the temperature at which half the protein irreversibly denatured after 10 minutes (T50), the urea concentration at which half the protein denatured after 4 hours ([urea]50), and the expression level relative to that of the original parental P450. The means are significantly different, with unequal variance t-test P -values of 0.02, 0.005, and 0.04, respectively. Tables Table 1 - Error-prone PCR nucleotide mutation spectrum. Spectrum of nucleotide mutations introduced by the error-prone PCR procedure used in the neutral evolution experiments. The spectrum was determined by sequencing the four final (generation 12) sequences from the unselected population, since in these sequences the mutations accumulate without constraint. As has been previously noted for error-prone PCR with Taq polymerase [3,5,26], the nucleotide error spectrum is biased towards certain types of mutations. Total nucleotide mutations 67 % synonymous mutations 25 Mutation types (%) A →T, T →A 19.4 A →C, T →G 1.5 A →G, T →C 64.2 G →A, C →T 4.5 G →C, C →G 0.0 G →T, C →A 1.5 frameshift 9.0 Table 2 - Neutral evolution robustness and mutation data. Each row is for a different generation, T . Entries of NA indicate that no measurement was made. The 〈mnt〉 and 〈maa〉 are the average number of nucleotide mutations and nonsynonymous mutations, respectively. Numbers in parentheses are total counts over the total samples. Subscripts indicate the population type: U for unselected, P for polymorphic, and M for monomorphic. For the unselected and monomorphic populations, numbers represent averages of all replicates. For the polymorphic population, numbers are for a random sample of functional mutants. 〈F〉P and 〈F〉M are the fraction of functional mutants out of 435 assayed. T 〈mnt〉U 〈maa〉U 〈mnt〉P 〈maa〉P 〈mnt〉M 〈maa〉M 〈F〉P 〈F〉M 0 0 0 0 0 0 0 0.48 (210 / 435) 0.48 (210 / 435) 1 NA NA NA NA 0.1 (3 / 22) 0.3 (6 / 22) 0.48 (208 / 435) NA 2 NA NA NA NA 0.4 (9 / 22) 0.8 (17 / 22) 0.49 (215 / 435) NA 3 5.0 (20 / 4) 3.5 (14 / 4) 2.7 (27 / 10) 1.4 (14 / 10) 1.0 (23 / 22) 0.4 (9 / 22) 0.49 (215 / 435) NA 4 NA NA NA NA 1.5 (32 / 22) 0.7 (15 / 22) 0.48 (208 / 435) NA 5 NA NA NA NA 2.2 (48 / 22) 1.1 (25 / 22) 0.45 (197 / 435) 0.43 (185 / 435) 6 9.8 (39 / 4) 7.5 (30 / 4) 5.5 (55 / 10) 2.1 (21 / 10) 2.6 (58 / 22) 1.4 (31 / 22) 0.46 (198 / 435) NA 7 NA NA NA NA 3.1 (69 / 22) 1.8 (39 / 22) 0.52 (227 / 435) NA 8 NA NA NA NA 3.4 (74 / 22) 1.8 (40 / 22) 0.46 (200 / 435) NA 9 13.0 (52 / 4) 10.3 (41 / 4) 6.7 (61 / 9) 3.1 (28 / 9) 3.7 (82 / 22) 2.1 (46 / 22) 0.47 (203 / 435) NA 10 NA NA NA NA 4.2 (92 / 22) 2.4 (52 / 22) 0.46 (199 / 435) 0.40 (175 / 435) 11 NA NA NA NA 4.6 (102 / 22) 2.5 (56 / 22) 0.48 (207 / 435) NA 12 16.8 (67 / 4) 12.5 (50 / 4) 7.8 (70 / 9) 3.3 (30 / 9) 4.9 (107 / 22) 2.6 (58 / 22) 0.52 (228 / 435) NA 13 NA NA NA NA 5.0 (110 / 22) 2.7 (60 / 22) 0.52 (227 / 435) NA 14 NA NA NA NA 5.3 (116 / 22) 2.9 (64 / 22) 0.50 (216 / 435) NA 15 NA NA 10.3 (227 / 22) 3.8 (83 / 22) 5.6 (123 / 22) 3.0 (67 / 22) 0.50 (219 / 435) 0.39 (171 / 435) 16 NA NA NA NA 5.8 (127 / 22) 3.0 (67 / 22) NA NA 17 NA NA NA NA 6.0 (133 / 22) 3.1 (69 / 22) NA NA 18 NA NA NA NA 6.3 (137 / 22) 3.2 (71 / 22) NA NA 19 NA NA NA NA 6.3 (138 / 22) 3.3 (72 / 22) NA NA 20 NA NA NA NA 6.6 (145 / 22) 3.4 (75 / 22) NA 0.37 (160 / 435) 21 NA NA NA NA 6.9 (152 / 22) 3.6 (79 / 22) NA NA 22 NA NA NA NA 7.1 (156 / 22) 3.7 (81 / 22) NA NA 23 NA NA NA NA 7.2 (158 / 22) 3.7 (81 / 22) NA NA 24 NA NA NA NA 7.3 (161 / 22) 3.8 (83 / 22) NA NA 25 NA NA NA NA 7.7 (169 / 22) 4.0 (87 / 22) NA 0.39 (169 / 435) Additional material Additional file 1 - Sequence of the parent P450 used start neutral evolution. FASTA file with sequence of the R1-11 P450 BME used as the neutral evolution parent. This sequence was isolated after the equilibration evolution. Additional file 2 - Information about sequences from neutral evolution experiments. The entries give the name of the mutant, the number of nonsynonymous and nucleotide mutations relative to the R1-11 parent, the [urea]50 value if measured, the T50 value if measured, the percent of the parental expression level if measured, and then a list of all of the mutations. Amino acid mutations are numbered in the standard P450 numbering scheme. The names of the mutants indicate their origin. Names beginning with “P-G3” are randomly chosen functional mutants from generation 3 of the polymorphic population, etc. Names of the form “P1,” “P2,”, etc. are the 22 functional mutants that were randomly chosen from the final (generation 15) polymorphic population. Numbers P5 and P12 are missing because two of the original 24 randomly selected polymorphic population members were randomly chosen to be discarded after it was discovered that two of the 24 monomorphic replicates were contaminated. Names beginning with “U1” indicate that sequences are from the first unselected replicate, etc. Names beginning “M1” indicate sequences are from the first monomorphic replicate, etc. Replicates “M9” and “M10” were discarded due to contamination during the experiment. For each replicate, we sequenced each new functional mutant. The last functional mutant after 25 generations represents the final sequence for that replicate, and is given an abbreviated name without the generation suffix. Additional file 3 - Thermostability measurements. Raw data from the T50 thermostability measurements. Additional file 4 - Urea stability measurements. Raw data from the [urea]50 thermostability measurements. Additional file 5 - Correlation of thermal and urea stabilities. The T50 and [urea]50 values are highly correlated. Additional file 6 - Sequence of initial P450 used to start equilibration evolution. FASTA file with sequence of the 21B3 P450 BM3 heme domain described in [23]. This P450 was used as the initial parent to start the equilibration evolution. Additional file 7 - Mutations accumulated during equilibration evolution. The file lists the mutations in the 46 P450 variants selected at the end of the equilibration evolution. Each line gives the name of the variant, with the prefix indicating whether it came from the R1 or R2 population. The next entries give the number of nucleotide and nonsynonymous mutations. All of the individual mutations relative to 21B3 are then listed. Amino acid mutations are numbered in the standard P450 numbering scheme, with the threonine after the N-terminal methionine given the number one. A.1 Mathematical background A.2 Monomorphic limit A.3 Polymorphic limit A.4 Approximations for polymorphic limit A.5 Approximations for monomorphic limit A.6 Interpretation in terms of neutral networks A.7 Detailed justification for approximating pM by po

An algebraic generalization of Kripke structures∗ Sérgio Marcelino and Pedro Resende Abstract The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in ge- ometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study complete- ness properties of the quantale based semantics for the systems K, T, K4, S4, and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL, and the ramified temporal logic CTL. 1 Introduction It is well known that the set ℘(W × W ) of all the binary relations on a set W has the structure of a unital involutive quantale (see §2). Hence, a Kripke structure (W,R) as it appears in the semantics of propositional normal modal logic [3, 4], where W is the set of possible worlds and R ⊂W ×W is the accessibility relation, can be regarded as an example of a pointed unital involutive quantale (℘(W ×W ), R). This suggests a way of generalizing the notion of Kripke structure, namely in terms of a more general pointed unital involutive quantale (Q,α), and the purpose of this paper is to assess the usefulness of this idea with respect to the semantics of modal logic. ∗Research supported in part by Fundação para a Ciência e a Tecnologia through pro- gram POCI 2010/FEDER and project POCI/MAT/55958/2004. http://arxiv.org/abs/0704.1886v1 Not every unital involutive quantale is suitable for this purpose, and in this paper we restrict to the notion of stably supported quantale that has been introduced in [14]. In order to motivate this let us consider again the quantale ℘(W ×W ) of binary relations on W . Each relation R ⊂ W ×W has a domain dom(R) = {x ∈ W | ∃y (x, y) ∈ R} and, since the diagonal relation ∆W = {(x, y) ∈ W ×W | x = y} is of course isomorphic to W , we can equivalently replace the domain of R by the set ςR = {(x, x) ∈ ∆W | x ∈ dom(R)} , which we refer to as the support of R. This defines an operation ς : ℘(W ×W )→ ℘(W ×W ) that preserves unions and in addition satisfies the following properties, for all R, S ⊂W ×W : ςR ⊂ ∆W ςR ⊂ RR∗ R ⊂ ςRR ς(RS) ⊂ ςR A stably supported quantale, or simply ssq, is defined to be a unital involutive quantale Q equipped with a sup-lattice endomorphism ς : Q → Q which satisfies these properties; that is, for all a, b ∈ Q we have ςa ≤ e ςa ≤ aa∗ a ≤ ςa a ς(ab) ≤ ςa For each a ∈ Q the element ςa is called the support of a, the operation ς itself is referred to as the support of Q, and the set ςQ = {ςa | a ∈ Q} is necessarily a locale whose binary meet operation coincides with the multiplication of the quantale: a∧b = ab. For instance, if Q = ℘(W×W ) we have ςQ = ℘(∆W ) ∼= ℘(W ). The formulas of propositional modal logic can be easily interpreted on any pointed ssq (Q,α): there should be a valuation map v assigning to each formula ϕ an element v(ϕ) ∈ ςQ. The properties that such a map must satisfy are clear. For instance, conjunction of formulas should be interpreted as multiplication in Q: v(ϕ∧ψ) = v(ϕ)v(ψ); and the remaining propositional connectives are equally straightforward, both classically and intuitionistically (see §3). As regards the modal operator ♦ of possibility, we impose v(♦ϕ) = ς(αv(ϕ)) . This is easily seen to yield the usual interpretation on Kripke structures: v(♦ϕ) corresponds to the domain of the relation αv(ϕ), where α is the ac- cessibility relation. Algebraic semantics. If (Q,α) is a pointed ssq then ςQ becomes a locale equipped with unary operators in a natural way (§§3, 5), and it will be seen in this paper that from any such locale it is possible to obtain a pointed ssq, in fact giving us an adjunction between two categories, and generalizing the way in which a binary relation on W (i.e., an accessibility relation) corre- sponds classically to a unary operator ♦ : ℘(W ) → ℘(W ). This leads to a semantics that subsumes the classical algebraic semantics based on lattices with operators. The systems of modal logic that are characterized by special kinds of accessibility relations can be characterized by subcategories of the category StabQu∗ of pointed ssqs. For instance, the system S5, which is characterized by accessibility being an equivalence relation, will correspond to the full subcategory of StabQu∗ whose objects (Q,α) are those satisfying α 2 = α∗ = α ≥ e. Moreover, it is easy to address multi-modal logics, such as dynamic logic, logics of time and space, etc., in terms of the category whose objects are ssqs equipped with more than one point, the various points satisfying suitable relations. The category StabQu of ssqs has pleasant properties [14]. For instance, there is at most one support on any unital involutive quantale, and any homomorphism of unital involutive quantales h : Q → Q′ between ssqs Q and Q′ automatically preserves the support (hence, being stably supported is a property rather than extra structure). The category StabQu is therefore a full subcategory of the category of unital involutive quantales, and in fact it is reflective. There are presentations of ssqs by generators and relations, which play the role of “Lindenbaum algebras” in the context of the quantale- based semantics. For instance, the “Lindenbaum quantale” for S5 is the pointed ssq QS5 generated by the usual Lindenbaum algebra of S5 (with defining relations ensuring that it is a bounded sublattice of ς QS5) with the distinguished point α ∈ QS5 being subject to the relations α 2 = α∗ = α ≥ e. The usual Kripke structure based models for S5 can be identified with the relational representations of QS5, in other words the homomorphisms h : QS5 → ℘(W ×W ) of unital involutive quantales (see [10] for some properties of such representa- tions). The same identification of models with the relational representations of a quantale applies to many other well known examples of modal logics. Groupoids and inverse semigroups. An important aspect of the se- mantics described in this paper is that there are plenty of examples of ssqs besides the quantales of binary relations [14], arising from various geomet- ric or analytic structures, and thus we are provided with a uniform way of defining semantic interpretations of propositional normal modal logic based on such structures. More precisely, let G be a groupoid (i.e., a small category all of whose arrows are isomorphisms). Writing G also for the set of groupoid arrows, ℘(G) is an ssq; the quantales of binary relations ℘(W ×W ) are pre- cisely the quantales that arise from the so-called pair groupoid of W , which hasW as set of objects andW ×W as set of arrows, with the two projections W ×W → W being the domain and codomain maps of the groupoid. Even more generally, the topology Ω(G) of any topological étale groupoid G is a sub-ssq of ℘(G); and for localic étale groupoids, too, there is an ssq O(G) associated to each groupoid G. These facts are a part of the close relation [14] between the notions of étale groupoid (either topological or localic), inverse semigroup, and quantale, which can be summarized in the following diagram whose arrows denote bijections of objects up to isomorphism, or even, in the case of I and L∨, equivalences of categories: Inverse quantal frames Étale groupoids bisections Complete infinitely distributive inverse semigroups germspp The inverse quantal frames are the ssqs that arise from étale groupoids. It follows that the quantale semantics automatically provides a bridge between modal logic and those areas of mathematics where examples of étale groupoids and inverse semigroups occur, such as operator algebras and dif- ferential topology — see, e.g., [7, 8, 12]. As an example of the latter, foliated manifolds can be associated to dynamical systems, and from a foliation it is always possible to construct a topological étale groupoid [8]. We shall not deal with any such examples in this paper, but we mention that if we replace ℘(W ×W ) by a more general groupoid quantale, hence taking as models of propositional modal logic the homomorphisms Q → Ω(G) or Q → O(G) instead of Q → ℘(W ×W ) (where Q is a Lindenbaum quantale), we are led in a natural way to semantics which may be interesting, say, for applied logicians or computer scientists dealing with hybrid systems, logics of real time and space, etc. Overview. Let (Q,α) be a pointed ssq. Then the locale ςQ is canonically equipped with the two unary sup-latice endomorphisms ♦ and � defined by, for each x ∈ ςQ, ♦x = ς(αx) (1) �x = ς(α∗x) , (2) which are easily seen to satisfy the following conjugacy conditions (see §5): ♦x ∧ y ≤ ♦(x ∧ �y) �x ∧ y ≤ �(x ∧ ♦y) . Such a structure (L,♦,�), where L is a locale and ♦ and � satisfy the conjugacy conditions, will be called a bimodal frame, and there is a functor from the category of pointed ssqs to the obvious category of bimodal frames. A functor in the opposite direction can be easily obtained just from the knowledge that ssqs can be presented by generators and relations; given a bi- modal frame (L,♦,�) its associated quantale Q is generated by the elements of L plus an element α, with relations imposing both that L is a unital invo- lutive subquantale of Q and that (1) and (2) are satisfied. As we shall see, this defines a functor which is left adjoint to ς, and in fact it is a coreflection; hence, we always have an isomorphism of bimodal frames L ∼= ςQ. We can interpret this isomorphism in a logical sense as saying that no theorems are added in the process of interpreting ♦ and � in terms of the quantale operations. Hence, if we think of the addition of the quantale operations as a language extension then this extension is conservative — the conjugacy conditions are a complete axiomatization for the modal operators induced by α and α∗. This of course suggests looking at several systems of modal logic and their completeness theorems, which we shall do for K, T, K4, and S4, showing that the usual conditions on ♦ (and here also on �), as taken from the standard completeness theorems of modal logic, are precisely what is required for a coreflection to be obtained when the pointed ssqs (Q,α) under consideration satisfy the expected conditions, such as “reflexivity”, “transitivity”, etc., of the point α. Hence, the theorems which we prove in this paper can be re- garded as an algebraic generalization of the standard completeness theorems for these systems of propositional normal modal logic. It is worth noting that, as opposed to the classical theorems, these are now independent of the axiom of choice (which is required in the proof of the classical theorems, in the form of Zorn’s lemma). From our results we also obtain a complete axiomatization of system S5, where to the axioms of S4 one adds the axiom scheme ♦ϕ ∧ ψ → ♦(ϕ ∧ ♦ψ) . This is obtained from the conjugacy conditions by making ♦ = � and, as opposed to the usual axiom scheme ϕ→ �♦ϕ, it does not mention negation or the modal necessity operator. The remainder of this paper goes as follows. In §2 we provide some nec- essary background and preliminary results on quantales, presentations by generators and relations, etc. Then in §3 we describe in detail the quantale- based semantics of propositional normal modal logic, including the systems K, T, K4, S4, and S5. In order to illustrate the flexibility of this approach we provide additional examples, namely propositional intuitionistic logic, propo- sitional dynamic logic, and the ramified temporal logic CTL. Finally, after some technical results in §4 about graded quantales (the quantale analogue of graded rings), in §5 we address the adjunctions mentioned above. The mere existence of the adjunctions is a consequence of the existence of pre- sentations by generators and relations; in other words, it can be phrased in terms of the existence of the Lindenbaum quantales. However, in order to obtain additional information about the adjunctions and, in particular, in order to prove that they are coreflections, we shall need an actual construc- tion of the Lindenbaum quantales. This will be conveniently formulated in terms of “tensor algebras” over bimodal frames, and it will take up most of 2 Preliminaries Here we describe some background on sup-lattices, locales and quantales. General references are [5, 6, 9, 11, 15]. Sup-lattices. We shall denote the category of sup-lattices by SL. The objects are the complete lattices and the morphisms are the maps f : L→M that preserve arbitrary joins: for all X ⊂ L f(x) . We shall write 1L or simply 1 for the lattice unit (the greatest element) of a sup-lattice L, and 0L or simply 0 for the least element. Let (Li) be a family of sup-lattices. Their cartesian product Li is a sup-lattice with pointwise order and joins, and it is a product in the category SL. The products and the coproducts are isomorphic [6], similarly to abelian groups (but, contrary to the latter, also in the case of infinitary coproducts). We denote by Li the categorical coproduct of a set-indexed family of sup-lattices (Li), and call it direct sum. The tensor product of L and M is denoted by L ⊗M , and similarly to vector spaces, it is the image of a universal bi-morphism, where a sup-lattice bi-morphism f : L ×M → N is a map that preserves joins in each variable separately: f(x, y) f(x, y) . Concretely, L⊗M can be identified with the set of those subsets I ⊂ L×M such that ∈ I ⇐⇒ {x} × Y ⊂ I ∈ I ⇐⇒ X × {y} ⊂ I for all x ∈ L, y ∈ M , X ⊂ L, and Y ⊂ M . The universal bi-morphism L×M → L⊗M is defined by (x, y) 7→ x⊗ y, where x⊗ y is the least such set that contains the pair (x, y), which is called the pure tensor generated by (x, y). SL is a monoidal category with respect to this tensor product, with the powerset ℘( 1) of the singleton set as the tensor unit. Similarly to the category of abelian groups, the functor hom(N,−) has a left adjoint −⊗N for each N ; that is, we have the familiar isomorphism hom(M ⊗N,L) ∼= hom(M, hom(N,L)) , natural in the variables M and L, which in fact is an order isomorphism. As a consequence of colimit preservation by left adjoints, ⊗ distributes over Mi) ∼= (L⊗Mi) . Quotients of sup-lattices can be conveniently handled by means of closure operators (monotone endomaps j that satisfy a ≤ j(a) and j(j(a)) = j(a) for every element a). Let L be a sup-lattice and j a closure operator on L. The set of j-closed elements Lj = {x ∈ Q | x = j(x)} is a sup-lattice closed under the formation of meets in L, with joins given by (xi) = j( xi), and the map j : L → Lj is a (surjective) homomorphism of sup-lattices. Conversely, given a set S ⊂ L closed under meets in L, we obtain a closure operator jS : L→ L by jS(x) = {y ∈ S | x ≤ y} . These constructions are mutually inverse, jLj = j LjS = S , and every sup-lattice quotient arises in this way up to isomorphism. The relation to the usual description of quotients by means of congruence relations (i.e., equivalence relations on L which are sub-sup-latttices of L×L) is the following: from a closure operator j we obtain the congruence relation θj ⊂ L× L defined by (x, y) ∈ θj ⇐⇒ j(x) = j(y) [in particular (x, j(x)) ∈ θj ] and from a congruence relation θ ⊂ L × L we define a closure operator jθ by jθ(x) = where [x]θ is the congruence class of x; of course, we have jθj = j and θjθ = θ. Stably supported quantales. A quantale is a sup-lattice equipped with an associative multiplication, usually written (a, b) 7→ ab, which distributes over arbitrary joins: (abi) , (aib) . Hence, a quantale is a semigroup in SL. A quantale Q is unital if the multi- plication has a unit, which we denote by eQ, or simply e. Example 2.1 A locale, or frame, L is a sup-lattice satisfying the following distributivity property for all x ∈ L and Y ⊂ L: x ∧ y . Hence, a locale is a unital quantale whose multiplication is ∧ and whose unit e coincides with 1. In particular, it is a commutative and idempotent quantale. A quantale is a locale if and only if it is unital with e = 1 and it is idempotent [6]. An involutive quantale Q is a quantale equipped with an involution (−)∗ : Q→ Q , i.e., a join preserving operation that makes Q an involutive semigroup: (ab)∗ = b∗a∗, a∗∗ = a. Any involutive quantale satisfies 1∗ = 1 and, if it is unital, e∗ = e. Hence, a unital involutive quantale is an involutive monoid ℘( 1) ←− Q⊗Q in the monoidal category of sup-lattices, with ab = m(a⊗ b). Definition 2.2 Let Q be a unital involutive quantale. A support on Q is a sup-lattice endomorphism ς : Q→ Q satisfying, for all a ∈ Q: ςa ≤ e , (3) ςa ≤ aa∗ , (4) a ≤ ςaa . (5) A supported quantale is a unital involutive quantale equipped with a spec- ified support. On a supported quantale the set of supports ςQ coincides with ↓e = {x | x ≤ e} and it is a locale with ab = a ∧ b [14]. Definition 2.3 A support is stable if it satisfies ς(ab) = ς(aς(b)). A quantale equipped with a specified stable support is stably supported, or simply an ssq. Every homomorphism of unital involutive quantales preserves the support of an ssq, and thus the category of ssqs, StabQu, is defined to be the full subcategory of the category of unital involutive quantales Qu whose objects are the ssqs. Moreover, if a quantale is stably supported then it can have no other support, stable or not [14]. Hence, being stably supported should be regarded as a property of unital involutive quantales rather than extra structure. In [14] it has also been seen that the inclusion functor StabQu → Qu has a left adjoint (i.e., StabQu is a reflective subcategory of Qu). Any locale is an ssq with trivial involution and support: x∗ = x ςx = x . Nuclei and quotients. The quotients of ssqs are described in a similar way to those of sup-lattices. We give here an overview and refer to [14] for further details. Definition 2.4 A (quantic) nucleus on an ssq Q is a closure operator j : Q→ Q that satisfies, for all x, y ∈ Q, j(x)j(y) ≤ j(xy) , j(x)∗ ≤ j(x∗) , ς(j(x)) ≤ j(ς(x)) . We remark that the second condition is equivalent to j(x)∗ = j(x∗). The set of j-closed elements Qj = {x ∈ Q | x = j(x)} is an ssq with joins (xi) = j( xi), multiplication x · y = j(xy), with the same involution as Q, and support δx = j(ςx). The map j : Q → Qj is a (surjective) homomorphism of ssqs: j(xi) j(xy) = j(x) · j(y) j(x∗) = j(x)∗ j(ςx) = δx . Furthermore, every quotient arises in this way up to isomorphism. The set N(Q) of nuclei is a complete lattice under the pointwise order, with meets being calculate pointwise: j ≤ k ⇔ ∀x∈Q(j(x) ≤ k(x)), and jα(x) = (jα(x)). Furthermore, we have j ≤ k ⇔ Qk ⊂ Qj, and the join of nuclei corresponds to intersection of the respective sets of closed elements: j = jα if and only if Qj = Qjα . Definition 2.5 Let Q be an ssq, and R ⊂ Q×Q. The supported closure R of the binary relation R is the smallest relation that contains R and is closed for the quantale operations, i.e.: R ⊂ R ; (y, z) ∈ R ⇒ (ay, az) ∈ R, for all a ∈ Q ; (y, z) ∈ R ⇒ (ya, za) ∈ R, for all a ∈ Q ; (y, z) ∈ R ⇒ (ςy, ςz) ∈ R ; (y, z) ∈ R ⇒ (y∗, z∗) ∈ R . Contrarily to what is done in [13], we shall interpret each pair (y, z) ∈ R as an inequality y ≤ z, rather than an equation y = z. It is easy to see that there is a least quantic nucleus j such that j(y) ≤ j(z) for all (y, z) ∈ R: {j ∈ N(Q) | j(y) ≤ j(z) for all (y, z) ∈ R}. Analogously to the quotients of involutive quantales described in [13], the quantale QjR has a very simple description: QjR = {x ∈ Q | ∀(y,z)∈R (z ≤ x⇒ y ≤ x)}. We stress that nuclei and quotients of unital involutive quantales equipped with any sup-lattice endomorphism ς : Q → Q are handled in exactly the same way as described above for ssqs (we shall use this in §5 when dealing with quantales that are just “pre-supported”). The properties of ς pass to the quotients one by one: if j is a nucleus on Q and δ : Qj → Qj is the sup-lattice endomorphism defined by δ(a) = j(ςa), then if, say, the equation ς(ab) = ς(aςb) holds in Q then δ(ab) = δ(aδb) holds in Qj , etc. Generators and relations. Let G be a set (of “generators”). The con- struction of the unital involutive quantale Qu〈G〉 freely generated by G is described in [13]. Denoting by F : Qu → StabQu the left adjoint to the inclusion StabQu → Qu (cf. paragraph after ¶2.3), it follows that F (Qu〈G〉) is the free ssq generated by G, and we shall denote it by StabQu〈G〉. Definition 2.6 Let G and R ⊂ StabQu〈G〉 × StabQu〈G〉 be sets. The ssq presented by the generators in G and the relations in R is StabQu〈G | R〉 = StabQu〈G〉 If x ∈ G, one denotes by [x] the image of the generator x in the quantale being presented. This notation provides a useful way of describing the defin- ing relations of a quantale presentation: we just write the conditions with respect to which the injection of generators is universal, as in the following example for unital involutive quantales. Example 2.7 Let L be a sup-lattice. It follows from the universal proper- ties of the tensor product and the direct sum of sup-lattices that the unital involutive quantale freely generated by L with joins being preserved in the presentation, is isomorphic to the tensor quantale where I is the free involutive monoid on one generator, whose words can be concretely identified with the strings of symbols α and α∗ and whose unit we shall denote by ε, and L(d) = L⊗|d| = L⊗ · · · ⊗ L (|d| times) . Note that L(ε) = ℘( 1) is the neutral element of the tensor product. The multiplication is defined on pure tensors just by concatenation (x0 ⊗ . . .⊗ xn)(y1 ⊗ . . .⊗ ym) = x0 ⊗ . . .⊗ xn ⊗ y1 ⊗ . . .⊗ ym , where in the case of concatenation with elements of ℘( 1) we use the identi- fication ℘( 1) ⊗ L ∼= L ∼= L ⊗ ℘( 1), to produce the identity of the quantale e = 1℘( 1). The involution (−)∗ : TL→ TL is obtained from the isomorphisms L(w) → L(w ∗) that are given by x1 ⊗ · · · ⊗ xn 7→ xn ⊗ · · · ⊗ x1 . The injection of generators is the α-coprojection of the coproduct L = L(α) → L(d). 3 Quantale semantics of modal logic Propositional normal modal logic. In this section we describe the in- terpretations of the classical systems of modal logic K, T, K4, S4, and S5. For details on these we refer the reader to [3, 4]. We shall consider fixed a set Π of propositional symbols. The set Φ of propositional formulas is defined to be the least set containing Π such that for all ϕ, ψ ∈ Φ we have ¬ϕ ∈ Φ , ϕ ∨ ψ ∈ Φ , ♦ϕ ∈ Φ , where as usual we may define other connectives, for conjunction ∧, implica- tion →, and the necessity modal operator �, as abbreviations: ϕ ∧ ψ = ¬(¬ϕ ∨ ¬ψ) , ϕ→ ψ = ¬ϕ ∨ ψ , �ϕ = ¬♦¬ϕ . Definition 3.1 A generalized Kripke model consists of a triple (Q,α, v), where Q is an ssq, α ∈ Q is an accessibility element, and v : Φ → ςQ is an interpretation map satisfying the following properties for all ϕ, ψ ∈ Φ: v(ϕ ∨ ψ) = v(ϕ) ∨ v(ψ) v(ϕ)v(¬ϕ) = 0 v(ϕ) ∨ v(¬ϕ) = e v(♦ϕ) = ς(αv(ϕ)) . Remark 3.2 The above definition makes each element v(ϕ) be comple- mented in ςQ, with v(¬ϕ) being its (unique) complement, and it also follows that conjunction is interpreted as multiplication (equivalently, meet) in ςQ: v(ϕ ∧ ψ) = v(ϕ)v(ψ) . This means that we interpret the formulas inside a Boolean subalgebra of ςQ, hence obtaining a classical semantics of propositional modal logic, a fact that was already implicit in the definition of the conjunction and the implication as derived connectives. However, we point out that it is easy to define a (rather natural) semantics for intuitionistic modal logic. We shall describe this at the end of §3. As usual we say that a pointed ssq consists of an ssq Q together with a specified “point” α ∈ Q. A homomorphism of pointed ssqs is a homomor- phism of ssqs that preserves the point: h : (Q,α) → (R, β) α 7→ β. From now on we shall denote by BK the Lindenbaum algebra of system K (i.e., the set of formulas modulo equivalence, which is a Boolean algebra equipped with a finite join preserving endomorphism ♦). Definition 3.3 The Lindenbaum quantale for K is the pointed ssq QK which is presented by generators and relations with BK as the set of generators and with the following relations for all x, y ∈ BK, where we denote the selected point by α: [x ∨ y] = [x] ∨ [y] [¬x][x] = 0 [¬x] ∨ [x] = e [♦x] = ς(α[x]) . From the universal property of ssqs presented by generators and relations we immediately obtain: Theorem 3.4 There is a bijective correspondence between abstract Kripke models (Q,α, v) and homomorphims of unital involutive quantales QK −→ Q . In particular, if W is a set then a homomorphism ρ : QK −→ ℘(W ×W ) is the same as a model for system K with set of possible worlds W and accessibility relation ρ(α). In order to obtain similar facts for other systems, such as T, K4, S4, S5, one must define the appropriate Lindenbaum quantales. Definition 3.5 The Lindenbaum quantales for T, K4, S4, and S5, are the pointed ssqs QT, QK4, QS4, and QS5, respectively, which are presented by generators and relations similarly to QK, with the following additional rela- tions: QT: e ≤ α QK4: αα ≤ α QS4: e ≤ α ≥ αα QS5: e ≤ α = α ∗ ≥ αα Hence, QT is the quotient of QK by the least nucleus j such that j(e) ≤ j(α), and QK4 is the quotient ofQK by the least nucleus j such that j(αα) ≤ j(α). Then we have QS4 = QT ∩QK4 and QS5 is the quotient of QS4 by the least nucleus j such that j(α) = j(α∗). Notice that the relational representations of these quantales correspond to the expected classes of models: Theorem 3.6 The relational representations Q → ℘(W ×W ) of the Lin- denbaum quantales Q correspond bijectively to the Kripke models whose ac- cessibility relations are: • Reflexive, for Q = QT; • Transitive, for Q = QK4; • Preorders, for Q = QS4; • Equivalence relations, for Q = QS5. Propositional ramified temporal logic. Now we describe a similar se- mantics for the ramified temporal logic known as Computational Tree Logic (CTL), see [2]. As above, Π is a fixed set of propositional symbols. The set Φ of CTL formulas is defined to be the least set containing Π such that for all ϕ, ψ ∈ Φ we have ¬ϕ, ϕ ∨ ψ,EXϕ,EFϕ,EGϕ ∈ Φ , and we may define other modal operators as abbreviations: AXϕ ≡ ¬EX¬ϕ , AGϕ ≡ ¬EF¬ϕ , AFϕ ≡ ¬EG¬ϕ . Definition 3.7 The intuitive meaning of the various modalities is the fol- lowing: • EXϕ means that there is a possible future where ϕ will hold in the next time instant; • EFϕ means that there is a possible future where ϕ will eventually hold; • EGϕ means that there is a possible future where ϕ will always hold (including now); • AXϕ means that ϕ will certainly hold in the next time instant; • AGϕ means that ϕ will always hold (including now) in all possible future paths; • AFϕ means that in each possible future path ϕ will eventually hold. A generalized CTL model consists of a triple (Q,α, v), where Q is an ssq, α ∈ Q is an accessibility element that satisfies ς(α) = e (i.e., time never ends), and v : Φ → ςQ is an interpretation map satisfying the following properties for all ϕ, ψ ∈ Φ: v(ϕ ∨ ψ) = v(ϕ) ∨ v(ψ) v(ϕ)v(¬ϕ) = 0 v(ϕ) ∨ v(¬ϕ) = e v(EXϕ) = ς(αv(ϕ)) v(EFϕ) = ς v(EGϕ) = {a ∈ Q | a ≤ v(ϕ) ∧ ς(αa)} . It is easy to see that this interpretation conveys the intended meaning to the modal operators if we let Q = ℘(W ×W ) for some set W . Only the last condition, for EG, deserves an explanation. This says that EGϕ may be interpreted as the largest subset X ⊂ W such that every world x ∈ X satisfies the following two conditions: • x satisfies the formula ϕ; • there is a world y ∈ X such that (x, y) ∈ α. This guarantees that there is an infinite path (possibly with repetitions) x0, x1, x2, x3, . . . satisfying ϕ starting at any world x0 where EGϕ holds. Mathematically, the definition of v(EGϕ) is clarified by the Knaster–Tarski fixed point theorem: the set of all the lowerbounds of v(ϕ), S = {a ∈ ςQ | a ≤ v(ϕ)} , is a complete lattice and thus the set of pre-fixed points of the monotone operator f : S → S defined by f(a) = v(ϕ) ∧ ς(α a) has a join, which in fact is a fixed point; hence, v(EGϕ) is also the largest element a ∈ ςQ such that a = v(ϕ) ∧ ς(α a). Propositional dynamic logic. In the program logic of [1] there are modal- ities indexed by programs, which themselves form a set with some algebraic structure. Let Π be a fixed set of propositional symbols and Ξ a set of atomic programs. The sets F , of formulas, and P , of programs, give us the PDL language Φ = F ∪ P , and they are defined to be the least sets such that Π ⊂ F , Ξ ⊂ P , if ϕ, ψ ∈ F then ¬ϕ, ϕ ∨ ψ ∈ F , if p, q ∈ P then p ∪ q, p; q, p∗ ∈ P , if ϕ ∈ F and p ∈ P then 〈p〉ϕ ∈ F , if ϕ ∈ F then ϕ? ∈ P . Very briefly, the intuitive meaning of the program constructs is the following: • p∪ q is the program that behaves in a nondeterministic way either like p or like q (the choice is made at the beginning of the execution of the program, once and for all); • p; q is the program whose execution is that of p followed by q; • p∗ is the iteration of p, to be thought of as p executed sequentially zero or more times (not to be confused with the notation for the quantale involution); • ϕ? is the program that tests ϕ, succeeding if ϕ is found to be true, and failing otherwise. Definition 3.8 A generalized PDL model consists of a pair (Q, v), where Q is an ssq and v : Φ → Q is an interpretation map satisfying the following properties for all ϕ, ψ ∈ F and p, q ∈ P : v(ϕ ∨ ψ) = v(ϕ) ∨ v(ψ) v(ϕ)v(¬ϕ) = 0 v(ϕ) ∨ v(¬ϕ) = e v(〈p〉ϕ) = ς(v(p)v(ϕ)) v(p ∪ q) = v(p) ∨ v(q) v(p; q) = v(p)v(q) v(p∗) = v(p)n v(ϕ?) = v(ϕ) . This interpretation shows that to a large extent both the formulas and the programs are treated on an equal footing. In particular, p; q can be regarded as the (noncommutative) “conjunction” of p and q, and p ∪ q as their dis- junction, while a formula is just a particular kind of program (ϕ is identified with ϕ?). Intuitionistic modal logic. It is easy to define a semantics for intuitionis- tic propositional modal logic if we let ∧ and → be independent connectives. This is because the support ςQ of any ssq Q is a locale and therefore a Heyting algebra, and thus, denoting by \ the residuation operation of ςQ, b\a = {c ∈ ςQ | b ∧ c ≤ a} , the conditions on v can be simply replaced by the following: v(ϕ ∨ ψ) = v(ϕ) ∨ v(ψ) v(ϕ ∧ ψ) = v(ϕ)v(ψ) v(ϕ→ ψ) = v(ϕ)\v(ψ) v(¬ϕ) = v(ϕ)\0 v(♦ϕ) = ς(αv(ϕ)) . This would entirely define the intuitionistic semantics if we contented our- selves with defining � = ¬♦¬ as before. However, this is a bad interpretation of �, as for instance it usually does not satisfy the axiom of distributivity over meets �(ϕ ∧ ψ)↔ �ϕ ∧�ψ . Indeed, a much better interpretation, in particular one that satisfies distribu- tivity of � over (arbitrary) meets, is obtained if we let � be an independent connective interpreted as the right adjoint of a suitable sup-lattice endomor- phism, as we now describe. LetW be a set, let R ⊂W ×W be a binary relation, and let Q be the ssq ℘(W ×W ). Let ♦ and � be the sup-lattice endomorphisms of ςQ defined as in §1: ♦X = ς(RX) �X = ς(R∗X) . Equivalently, taking into account the isomorphism ςQ ∼= ℘(W ) we may consider ♦ and � to be endomorphisms of ℘(W ): ♦X = {y ∈ W | ∃x∈X (y, x) ∈ R} �X = {y ∈ W | ∃x∈X (x, y) ∈ R} . It is straightforward to verify that the usual necessity operator � : ℘(W )→ ℘(W ) , which is defined by �X = {y ∈ W | ∀x∈W (y, x) ∈ R⇒ x ∈ X} , is right adjoint to �, and we may take this as the natural definition of � when such a “possibility in the past” operator � is available — similarly, a “necessity in the past” operator � can be defined to be the right adjoint of �x ≤ y ⇐⇒ x ≤ �y ♦x ≤ y ⇐⇒ x ≤ �y . This leads to the following quantale-based intuitionistic semantics for propo- sitional modal logic, where we assume that ∧, →, and � are independent connectives: Definition 3.9 A generalized intuitionistic Kripke model consists of a triple (Q,α, v), where Q is an ssq, α ∈ Q is an accessibility element, and v : Φ→ ςQ is an interpretation map satisfying the following properties for all ϕ, ψ ∈ Φ: v(ϕ ∨ ψ) = v(ϕ) ∨ v(ψ) v(ϕ ∧ ψ) = v(ϕ)v(ψ) v(ϕ→ ψ) = v(ϕ)\v(ψ) v(¬ϕ) = v(ϕ)\0 v(♦ϕ) = ς(αv(ϕ)) v(�ϕ) = {x ∈ ςQ | ς(α∗x) ≤ v(ϕ)} . This definition illustrates a canonical way in which to define intuitionistic semantics for other propositional modal logics, including all the examples seen earlier in this section. We stress the fact that involutive quantales are essential for this, since without the involution we would neither have the � operator, nor a canonical definition of �. It is worth commenting on some aspects of the intuitionistic version of system S5, which similarly to its classical counterpart is based on imposing that the accessibility element α should be self-adjoint, and thus � = ♦. The unit of the adjunction relating � and � is the inequality x ≤ ��x , and thus for intuitionistic S5 we conclude that the axiom-scheme ϕ→ �♦ϕ (6) is always satisfied. It is well known that this scheme (together with those for S4) characterizes the classical system S5. Another axiom-scheme which is always satisfied by intuitionistic S5 is ♦�ϕ→ ϕ , which corresponds to the co-unit of the adjunction, and which classically (i.e., with � = ¬♦¬) is equivalent to (6). 4 Graded unital involutive quantales Basic definitions and properties. The usual notion of grading of a ring has a straightforward counterpart for quantales, which will be useful in §5. We shall study it now. Definition 4.1 LetM be an involutive monoid. A unital involutive quantale Q is graded over M if there is an M-indexed family of sub-sup-lattices Q(m) of Q satisfying the following two properties. 1. For each a ∈ Q there is one, and only one, element (am) ∈ such that 2. The following conditions hold for all m,n ∈ M : Q(m)Q(n) ⊂ Q(mn) 1Q(ε) = e ⊂ Q(m (The latter is equivalent to = Q(m For each m ∈M the set Q(m) is called the component of Q in degree m. Example 4.2 Recall the tensor quantale of ¶2.7: if L is a sup-lattice and I is the free involutive monoid on one generator α then the tensor quantale has an obvious grading over I such that L(ε) ∼= ℘( 1) and L(α) ∼= L. The following properties are immediate: Proposition 4.3 Let Q be a unital involutive quantale graded over an invo- lutive monoid M . 1. The map (am) 7→ m∈M am is an isomorphism of sup-lattices (m) → Q . 2. If m 6= n then Q(m) ∩Q(n) = {0}. 3. The union m∈M Q (m) is join-dense in Q. 4. ↓Q(m) = Q(m) for all m ∈M . There is a convenient alternative definition if the unital involutive quan- tale Q is also a locale (an example is the quantale TL of ¶4.2 if L is a locale, or the quantale T(L) of §5): Proposition 4.4 Let M be an involutive monoid, and let Q be a unital involutive quantale which is also a locale. Then Q is graded over M if and only if there is an M-indexed family (e(m)) of elements of Q satisfying the following properties: 1. 1Q = m∈M e (m) (i.e., covers Q); 2. e(m) ∧ e(n) = 0 if m 6= n; 3. e(m)e(n) ≤ e(mn); 4. e(ε) = e; ≤ e(m ∗) (equiv., = e(m Proof. It is clear that if Q is graded then it satisfies 1–5 if we let e(m) be 1Q(m) for each m ∈M . For the converse we define the component Q (m) to be ↓e(m): then any element a ∈ Q equals a ∧ 1 = a ∧ e(m) = a ∧ e(m) where a∧ e(m) ∈ Q(m) by definition of Q(m); and if bm then, for each n ∈M we have an = an ∧ am = an ∧ an ∧ bm = an ∧ bn , and in a similar way we obtain bn = bn ∧ an. Hence, we have an = bn, and thus each element a ∈ Q can be obtained uniquely as a join am. The rest is straightforward. Graded nuclei and quotients. The natural notion of quotient that re- spects the grading of a quantale is provided by the following definition: Definition 4.5 Let Q be a unital involutive quantale graded over an invo- lutive monoid M . A nucleus j : Q→ Q is graded if it satisfies the following two conditions for all (am) ∈ m∈M Q ⊂ Q(m); m∈M am m∈M j(am). Proposition 4.6 Let j be a graded nucleus as in the definition above. Then, 1. j(0) = 0 (the nucleus is “dense”); 2. Qj is graded, with each component being defined by (m) = j Proof. The first condition is obvious from the fact that 0 ∈ m∈M Q and thus j(0) ∈ m∈M Q (m) = {0}. For the second condition notice that if a ∈ Qj then on one hand we have a unique representation of a as a join and, on the other hand, a = j(a) = j j(am) , and thus am = j(am) for all m ∈M ; that is, the element am is necessarily in (m). The rest is straightforward. The nucleus induced by a binary relation is graded provided the relation respects the grading: Proposition 4.7 Let Q be a unital involutive quantale graded over an invo- lutive monoid M . Let also R ⊂ Q×Q be a binary relation that respects the grading in the sense that R ⊂ Q(m)×Q(m). Then jR is a graded nucleus. Proof. Let ⊏R ⊂ Q×Q be the preorder defined by a ⊏R b ⇐⇒ jR(a) ≤ jR(b) . Since this is also a sub-involutive-quantale of Q×Q, let us call it a congruence preorder. By a simple adaptation of the comments at the end of the sup- lattices section of §2, there is a bijection between congruence preorders and nuclei on Q, and ⊏R is the least congruence preorder on Q which contains R. It is clear that ⊏R respects the grading because R does, and thus if a ∈ Q then jR(a) ∈ Q (m), showing that jR satisfies the first of the properties of graded nuclei. In order to prove that it also satisfies the second property let k : Q→ Q be the map defined, for each (am) ∈ Q(m), as follows: jR(am) . Since jR is monotone we have k ≤ jR: jR(am) = k Now let us see that k is itself a nucleus. First, it is obvious that it is monotone and that it satisfies a ≤ k(a) for all a ∈ Q. It is also idempotent because jR(am) and the fact that jR(am) ∈ Q (m) allows us to conclude that the right hand side of the above equation equals jR(jR(am)) = jR(am) = k Now let us prove the condition relating k to the multiplication. For each pair (am), (bm) ∈ Q(m) we have jR(am) jR(bm) = jR(ap)jR(bq) jR(apbq) The condition relating k to the involution is equally simple and we omit it. Finally, it is obvious that for (a, b) ∈ R we have k(a) ≤ k(b), since jR(a) ≤ jR(b). But, by definition, jR is the least nucleus that satisfies this condition, and therefore we conclude that k = jR. Hence, jR is graded. 5 Construction of the Lindenbaum quantales The involutive tensor quantale of a frame. Let L be a frame, and denote by I the free involutive monoid on one generator α, whose words are the finite sequences of α and α∗, and whose unit we shall denote by ε. For each w ∈ I we shall denote by L(w) the sup-lattice L⊗(|w|+1), where |w| is the length of the word w (notice the difference with respect to ¶2.7): L(ε) = L , L(α) = L(α ∗) = L⊗ L , L(αα) = L(αα ∗) = L(α ∗α) = L(α ∗α∗) = L⊗ L⊗ L , For each w,w′ ∈ I we define a map ϕw,w′ : L (w) × L(w ′) → L(ww ϕw,w′(x1 ⊗ · · · ⊗ xn, y1 ⊗ · · · ⊗ ym) = x1 ⊗ · · · ⊗ (xn ∧ y1)⊗ · · · ⊗ ym . It is easy to see that this preserves joins in each variable, and thus it defines a sup-lattice homomorphism ϕw,w′ : L (w) ⊗ L(w ′) → L(ww Defining T(L) = (not the same as TL, cf. ¶¶2.7 and 4.2), and using the distributivity of⊗ over , we obtain the following sup-lattice homomorphism T(L)⊗T(L)→ T(L), T(L)⊗ T(L) ∼= // L(w) ⊗ L(w ϕw,w′ // // T(L) , where the rightmost homomorphism is the copairing of the family of sup- lattice embeddings L(ww ′) → T(L) which is given by the universal property of the sup-lattice coproduct ′). Hence, there is a bilinear multiplication T(L)× T(L)→ T(L). It is defined on pure tensors by (x1 ⊗ · · · ⊗ xn)(y1 ⊗ · · · ⊗ ym) = x1 ⊗ · · · ⊗ (xn ∧ y1)⊗ · · · ⊗ ym , and it is straightforward to see that it is associative, hence giving us a quan- tale multiplication on T(L). This multiplication has a unit, which coincides with 1L ∈ L (ε) = L, and an involution (−)∗ : T(L)→ T(L) is obtained from the isomorphisms L(w) → L(w ∗) that are given by x1 ⊗ · · · ⊗ xn 7→ xn ⊗ · · · ⊗ x1 . Hence, T(L) is a unital involutive quantale, and it is clearly graded over I, so that we can define: Definition 5.1 The tensor involutive quantale of L is the graded unital involutive quantale T(L). There is an obvious homomorphism of involutive monoids (−) : I → T(L) that sends each word w to 1L ⊗ · · · ⊗ 1L ∈ L (w). Hence, in particular, α = 1L ⊗ 1L ∈ L (α) and α∗ = 1L ⊗ 1L ∈ L (α∗). The quantale T(L) has the following universal property: Proposition 5.2 Let Q be a unital involutive quantale such that b = b2 = b∗ for all b ∈ ↓eQ (in particular, this implies that ↓eQ is a locale). Let also h : L→ ↓eQ be a homomorphism of locales, and let a ∈ Q. Then there is exactly one homomorphism of unital involutive quantales ϑ : T(L)→ Q such that: 1. ϑ(x) = h(x) for all x ∈ L(ε) = L; 2. ϑ(α) = a. Proof. By the universal property of the coproduct of sup-lattices, every sup-lattice homomorphism ϑ with domain T(L) is uniquely determined by its value on the pure tensors of T(L). Furthermore, every pure tensor x1 ⊗ · · · ⊗ xn ∈ L with wi = α, α ∗ can be written as a product of xi ∈ L, α ∈ L α and α∗ ∈ Lα and thus if ϑ is a homomorphism of unital involutive quantales that satisfies 1 and 2 then its value is uniquely determined on all the pure tensors. This proves that if ϑ exists then it is unique. In order to prove existence, assign to each pure tensor x1 ⊗ · · · ⊗ xn ∈ L the value h(x1)a1h(x2) . . . an−1h(xn) ∈ Q where ai = a or ai = a ∗ according to whether wi is α or α ∗, respectively. This assignment preserves joins in each variable xi separately and thus it defines a sup-lattice homomorphism ϑw : L (w) → Q. The copairing ϑ = [ϑw]w∈I : T(L)→ Q is easily seen to preserve the quantale multiplication, the unit and the invo- lution, and it satisfies conditions 1 and 2. For the following we denote by Frm the full subcategory of Qu whose objects are the locales (this is usually called the category of frames [5]). Corollary 5.3 Let Que be the full subcategory of Qu whose objects are those unital involutive quantales Q such that b = b2 = b∗ for all b ≤ e in Q. Let also Que∗ be the corresponding category of pointed quantales. There is an obvious functor Que∗ → Frm that to each quantale Q assigns ↓e, and such that h 7→ h|↓e for each homomorphism h. This functor has a left adjoint which to each locale L assigns the pointed quantale (T(L), α). Bimodal frames and pointed quantales. We have already mentioned that for an ssq Q, ςQ = ↓e is a locale. It is also clear that, for α in Q, the operators ♦x = ς(αx) and �x = ς(α∗x) preserve arbitrary joins, hence they are sup-lattice endomorphisms of ςQ. Definition 5.4 We say that two sup-lattice endomorphisms of L, ♦ and �, are conjugate modalities if for all x, y ∈ L we have ♦x ∧ y ≤ ♦(x ∧ �y) , �x ∧ y ≤ �(x ∧ ♦y) . A bimodal frame (L,♦,�) is a frame L equipped with two conjugate modal- ities ♦ and �. Proposition 5.5 Let Q be an ssq, and α ∈ Q. Then (ςQ, ς(α−), ς(α∗−)) is a bimodal frame. Proof. We only have to check that the two endomorphisms are conjugate modalities. From the fact that ς is a support (4) we have: ς(αx)y ≤ αxα∗y , and thus ♦x ∧ y = ς(αx)y = ς(ς(αx)y) ≤ ς(αxα∗y) = ς(αxς(α∗y)) = ♦(x ∧ �y) , using stability and the fact that x, y ∈ ςQ. The other conjugacy condition is obtained by interchanging α and α∗. From now on we shall refer to any sup-lattice endomorphism ς : Q→ Q on a unital involutive quantale Q such that ςa ≤ e for all a ∈ Q as a pre-support of Q. Given a bimodal frame (L,♦,�), a pre-support can be easily defined on pure tensors of T(L) (and then extended to joins of these in the obvious way): if x = x0 ⊗ · · · ⊗ xn is in degree w = w1 . . . wn (with wi ∈ {α, α ∗}) then ςx = x0 ∧ 〈w1〉(x1 ∧ 〈w2〉(. . .)) , where 〈wi〉 is ♦ or � according to whether wi = α or wi = α ∗, respectively. Recursively, we have: Definition 5.6 Let x = x0 ⊗ · · · ⊗ xn ∈ L (w1...wn). Then, • ςx = x, if n = 0; • ςx = x0 ∧ 〈w1〉(ςx ′), if n ≥ 1, where x′ = x1 ⊗ · · · ⊗ xn ∈ L (w2...wn). Lemma 5.7 The following properties hold for all a, b, c ∈ T(L): 1. ςe = e and ςa ≤ e (condition 3 in the definition of support); 2. ς(ςab) = ςaςb (in particular, ςςa = ςa); 3. ς(ab) = ς(aςb) (we say the pre-support ς is stable); 4. If ♦ and � are conjugate we have (a) ςa ≤ ς(aa∗); (b) ς(ςab) ≤ ς(aa∗b); (c) ς(cςab) ≤ ς(caa∗b). Proof. 1. ς T(L) = L and e is 1L. 2. It suffices to prove this for all the pure tensors b ∈ L(v), where v is an arbitrary degree. Let b = y0 ⊗ · · · ⊗ yp ∈ L (v1...vp) . We have ςa ∈ L(ε), and thus from ¶5.6 we obtain ς(ςab) = ς((ςa ∧ y0)⊗ · · · ⊗ yp) = (ςa ∧ y0) ∧ 〈v1〉(ς(y1 ⊗ · · · ⊗ yp)) = ςa ∧ (y0 ∧ 〈v1〉(ς(y1 ⊗ · · · ⊗ yp))) = ςaςb . 3. It suffices to prove this for all the pure tensors a ∈ L(w), where w is an arbitrary degree. Let then a = x0 ⊗ · · · ⊗ xn ∈ L (w1...wn) . The proof is done by induction on n. For the base case assume that n = 0; that is, we have a = ςa = x0 ∈ L (ε), and from 2 we obtain ς(ab) = ς(ςab) = ςaςb = aςb , whence ς(ab) = ςς(ab) = ς(aςb). Now for the induction step let n ≥ 1 and let a′ = x0 ⊗ · · · ⊗ xn−1 ∈ L (w1...wn−1) . We have a = a′wnxn and thus ς(ab) = ς(a′wnxnb) = ς(a′ς(wnxnb)) (Induction hyp.) = ς(a′〈wn〉(ς(xnb))) (Def. of ς) = ς(a′〈wn〉(xnςb)) (By 2) = ς(a′ς(wnxnςb))) (Def. of ς) = ς(a′wnxnςb) (Induction hyp.) = ς(aςb) . 4. First we remark that (4a) is an instance of (4b) (which in turn is an instance of (4c)). Moreover, (4b) implies (4c) due to stability: if we assume (4b) then ς(cςab) = ς(cς(ςab)) ≤ ς(cς(aa∗b)) = ς(caa∗b) . It now suffices to prove (4b). We shall prove ς(ςab) ≤ ς(aa∗b) for the particular case where a is a pure tensor a = x0 ⊗ · · · ⊗ xn in degree w = w1 . . . wn (n ≥ 1), which implies the general case. The proof is by induction on n. Base: for n = 0, we have a = ςa = aa∗, and thus ς(ςab) = ς(aa∗b). Step: for n ≥ 1, let r be x1 ⊗ · · · ⊗ xn ∈ L (w2...wn); hence, we have a = x0w1r, and thus using stability we obtain ς(aa∗b) = ς(x0w1rr ∗w∗1x0b) = ς(x0w1rr ∗ς(w∗1x0ςb)) . By the definition of ς (and making the symbol ∧ explicit for the mul- tiplication in ς T(L)) this equals x0 ∧ 〈w1〉ς(rr ∗〈w∗1〉(x0 ∧ ςb)) , which, by the induction hypothesis, is greater or equal to x0 ∧ 〈w1〉ς(ςr ∧ 〈w 1〉(x0 ∧ ςb)) , which in turn equals x0 ∧ 〈w1〉(ςr ∧ 〈w 1〉(x0 ∧ ςb)) (because the argument of the outermost occurrence of ς was in ς T(L)). Finally, by conjugacy of the operators 〈w1〉 and 〈w 1〉 the latter is greater or equal to x0 ∧ 〈w1〉(ςr) ∧ x0 ∧ ςb = x0 ∧ 〈w1〉(ςr) ∧ ςb = ς(x0w1r) ∧ ςb = ςa ∧ ςb = ς(ςab) . The supported quantale of a bimodal frame. So far we have obtained, from an arbitrary bimodal frame, a quantale with a stable pre-support. In order to obtain an actual supported quantale we shall impose the missing properties, namely ςa ≤ aa∗ (4) and a ≤ ςaa (5), by taking quotients of T(L). Definition 5.8 Let jς be the least nucleus j on T(L) such that j(a) = j(ςaa) . We define Tς(L,♦,�) to be T(L)jς . We also write Tς(L) if ♦ and � are clear from the context. From ¶4.7 it is clear that jς is a graded nucleus, and it is the identity on L(ε) = L because for all a ∈ L we have ςaa = a. Hence, we have concluded, just from the graded structure of T(L), that the injection of generators of L into Tς(L,♦,�) is 1–1. Proving a similar fact for the other axiom, ςa ≤ aa∗, is less easy, and we shall address this now. Let (L,♦,�) be a bimodal frame and let R = {(ςa, aa∗) | a ∈ Tς(L,♦,�)} . Definition 5.9 TK(L,♦,�) is Tς(L,♦,�)jR. As in ¶5.8 we may write TK(L). We shall denote the selected point jR(jς(α)) ∈ TK(L) by α. Lemma 5.10 Recall the definition (2.5) of R. If (y, z) ∈ R then ςy ≤ ςz in Tς(L,♦,�). Proof. We have ςy ≤ ςz for all (y, z) ∈ R if and only if the following two conditions hold for all (y, z) ∈ R and all a, b ∈ Tς(L): ς(ayb) ≤ ς(azb) (7) ς(ay∗b) ≤ ς(az∗b) . (8) In order to prove these two conditions we shall show that they hold for all (y, z) ∈ R and that they are preserved by the recursive rules of construction of R. Let (y, z) ∈ R; that is, let y and z be of the form y = ςt and z = tt∗. We ς(ayb) = ς(aςtb) ≤ ς(att∗b) = ς(azb) , from ¶5.7. We similarly have ς(ay∗b) ≤ ς(az∗b) because y and z are self- adjoint. Now assume that (7)–(8) hold for some pair (y, z) ∈ Tς(L) × Tς(L). We shall prove that they equally hold for the following pairs: (i) (ςy, ςz); (ii) (y∗, z∗); (iii) (qy, qz); and (iv) (yq, zq), for all q ∈ Tς(L). (i) Since (y, z) satisfies (7)–(8) we have ςy ≤ ςz (make a = b = 1L), and thus ς(aςyb) ≤ ς(aςzb) for all a, b ∈ Tς(L), proving (7) for the pair (ςy, ςt). Since ςy and ςz are self-adjoint, we also conclude (8) for the pair (ςy, ςt). (ii), (iii) and (iv) are obvious. Theorem 5.11 The unit of the adjunction L→ TK(L) is 1–1. Proof. Equivalently, we want to prove that ς Tς(L) ⊂ QjR, or, in other words, that for all (y, z) ∈ R and x ∈ L we have z ≤ x⇒ y ≤ x . (9) Let P ⊂ R be the subset of R consisting of all those (y, z) such that (9) holds for all x ∈ L. We shall prove that R ⊂ P and that P is closed under the recursive formation rules of R, hence showing that P = R because R is the least subset of Tς(L)× Tς(L) satisfying these conditions. Let (ςy, yy∗) ∈ R. If x ∈ L and yy∗ ≤ x we conclude that y = ςy = yy∗ ∈ L due to the grading of Tς(L) over the free involutive monoid I, and thus ςy ≤ x, showing that R ⊂ P . From now on let (y, z) be a fixed but arbitrary element of P . By ¶5.10 we conclude ςy ≤ ςz. Hence, (9) holds for the pair (ςy, ςz), and thus (ςy, ςz) ∈ P . Now let q ∈ Tς(L) and assume that qz ≤ x for some x ∈ L. Then z ≤ 1L (again due to the grading over I), and thus y ≤ 1L because (y, z) ∈ P (make x = 1L). Hence, again using the previous lemma we obtain qy = qςy ≤ qςz = qz ≤ x , showing that (qy, qz) ∈ P . In a similar way we conclude that (yq, zq) ∈ P . Finally, x ≤ 1L implies that x is self-adjoint, and thus the conditions y ∗ ≤ x and z∗ ≤ x are equivalent to y ≤ x and z ≤ x, respectively, showing that (y∗, z∗) ∈ P . T, K4, S4, S5. Now let us extend our results to the systems of modal logic T, K4, S4, and S5. As was explained in §3, we shall need to impose additional conditions on the selected element α ∈ TK, such as reflexivity (α ≥ 1L), transitivity (α 2 ≤ α), etc. In order to obtain again coreflections we shall also have to impose additional axioms on the modalities ♦ and � of L. As we mentioned in §1, we shall see that for T, K4 and S4 these coincide with the well known axioms for the corresponding systems of modal logic under consideration (in other words, the same axioms still produce complete axiomatizations for the new semantics), whereas for S5 a new axiomatization is defined just by imposing that ♦ should coincide with �. The proof techniques are very similar to those employed in the previous sections for the system K. In fact we could have already presented the theory for K in such a generality so as to be able to directly reuse the results now, but this would have obscured the main ideas, so for the sake of clarity we shall single out the general results only now. Lemma 5.12 Let ρ ⊂ Tς(L) × Tς(L) be any binary relation on Tς(L), and let ρ be the closure of ρ under the rules (y, z) ∈ ρ ⇒ (ςy, ςz) ∈ ρ (y, z) ∈ ρ ⇒ (ayb, azb) ∈ ρ for all a, b ∈ Tς(L) (y, z) ∈ ρ ⇒ (y∗, z∗) ∈ ρ . Assume that for all (y, z) ∈ ρ and all a, b ∈ Tς(L) we have ς(ayb) ≤ ς(azb) ς(ay∗b) ≤ ς(az∗b) . Then for all (y, z) ∈ ρ we have ςy ≤ ςz. Proof. This follows from a simple adaptation of the proof of ¶5.10. Lemma 5.13 Let P ⊂ Tς(L)×Tς(L) be the set of all those (y, z) satisfying the following two conditions: 1. ςy ≤ ςz; 2. z ≤ x⇒ y ≤ x for all x ∈ L. Then P is closed under the rules (y, z) ∈ P ⇒ (ςy, ςz) ∈ P (y, z) ∈ P ⇒ (ayb, azb) ∈ P for all a, b ∈ Tς(L) (y, z) ∈ P ⇒ (y∗, z∗) ∈ P . Proof. The proof of this lemma is contained in the proof of ¶5.11, where P was defined to be a subset of R, but in fact the only property of R used in order to prove the closure properties of P was the fact that for all (y, z) ∈ R we have ςy ≤ ςz. The other key ingredient is the fact that ab ∈ L implies a, b ∈ L for all a, b ∈ Tς(L), due to the grading of Tς(L) over I. Theorem 5.14 Let ρ ⊂ Tς(L)×Tς(L) be a binary relation such that for all (y, z) ∈ ρ and all a, b ∈ Tς(L) we have ς(ayb) ≤ ς(azb) (10) ς(ay∗b) ≤ ς(az∗b) , (11) and let Q be the (supported) quotient of Tς(L) generated by ρ. Then the injection of generators of L onto ςQ, η : L→ ς Tς(L)→ ςQ , is an isomorphism. Proof. This is an immediate consequence of the previous two lemmas, by a reasoning analogous to that of ¶5.11. Similarly to what we have done in §3 for the Lindenbaum quantales QT, QK4, etc. (see ¶3.5), we define TT(L), TK4(L),TS4(L) and TS5(L) to be quo- tients of TK(L) by analogous defining relations: TT(L): e ≤ α TK4(L): αα ≤ α TS4(L): e ≤ α ≥ αα TS5(L): e ≤ α = α ∗ ≥ αα Corollary 5.15 Let L be a bimodal frame such that for all x ∈ L the condi- tions x ≤ ♦x and x ≤ �x hold. Then the injection of generators L→ ς TT(L) is an isomorphism. Proof. The quantale TT(L) is the quotient of Tς(L) generated by the con- ditions ςa ≤ aa∗, for all a ∈ Tς(L) , α ≥ e . Define RT ⊂ Tς(L)× Tς(L) as follows: RT = {(ςy, yy ∗) | y ∈ Tς(L)}) ∪ {(1L,α)} . All we have to do is, by the previous theorem, prove that for all (y, z) ∈ RT the conditions (10)–(11) are satisfied for all a, b ∈ Tς(L). This has already been done for the pairs of the form (ςy, yy∗) in ¶5.10, so we only have to concern ourselves with the pair (1L,α). For a, b ∈ Q we have ς(ab) = ς(aς(b)) ≤ ς(a♦(ς(b))) = ς(aς(αb)) = ς(aαb) , where we have used stability of ς twice and the inequality follows from x ≤ ♦x and monotonicity of ς; this proves (10). Then (11) is proved in a similar way using the inequality x ≤ �x. Corollary 5.16 Let L be a bimodal frame such that for all x ∈ L the condi- tions ♦♦(x) ≤ ♦(x) and ��(x) ≤ �(x) hold. Then the injection of generators L→ TK4(L) is an isomorphism. Proof. It remains to prove that the pair (y, z) = (α2,α) satisfies (10)–(11). The first condition is proved as follows: ς(aαb) = ς(aς(αb)) = ς(a♦(ς(b))) ≥ ς(a♦2(ς(b))) = ς(aα2b) . (11) is proved in the same way once we replace α by α∗ and ♦ by � in the previous argument. Corollary 5.17 Let L be a bimodal frame such that for all x ∈ L the con- ditions x ≤ ♦x, x ≤ �x, ♦♦(x) ≤ ♦(x) and ��(x) ≤ �(x) hold. Then the injection of generators L→ TS4(L) is an isomorphism. Proof. Immediate from the previous two corollaries. Corollary 5.18 Let L be a bimodal frame such that for all x ∈ L the con- ditions of the previous corollary and ♦x = �x hold. Then the injection of generators L→ TS5(L) is an isomorphism. Proof. It remains to show that the pairs (α,α∗) and (α∗,α) satisfy (10) and (11), which is done as follows: ς(aαb) = ς(aς(αb)) = ς(a♦(ς(b))) = ς(a�(ς(b))) = ς(aα∗b) . The Lindenbaum quantales. Let BK be the Lindenbaum algebra for system K, as in §3, and let BT, BK4, BS4, and BS5 be the Lindenbaum algebras for systems T, K4, S4, and S5, respectively. These are modal lattices in the following sense: Definition 5.19 By a modal lattice is meant a bounded distributive lattice L equipped with an endomap ♦ that preserves finite joins. A bimodal lattice is a modal lattice equipped with another endomap � that preserves finite joins and in addition satisfies conjugacy relations similar to those of bimodal frames: ♦x ∧ y ≤ ♦(x ∧ �y) , �x ∧ y ≤ �(x ∧ ♦y) . The category Lat♦ of modal lattices has the modal lattices as objects and the homomorphisms of bounded lattices that preserve ♦ as morphisms. The category Lat♦� of bimodal lattices is defined analogously, with objects being the bimodal lattices and the morphisms being the homomorphisms of modal lattices that also preserve �. We shall also refer to a modal lattice as a • T-modal lattice if x ≤ ♦x for all x; • K4-modal lattice if ♦♦x ≤ ♦x for all x; • S4-modal lattice if it is both a T-modal lattice and a K4-modal lattice; • S5-modal lattice if it is an S4-modal lattice and ♦x∧ y ≤ ♦(x∧♦y) for all x and y (i.e., ♦ is conjugate to itself). The categories Lat♦T, Lat K4, Lat S4, and Lat S5 are, respectively, the full sub- categories of Lat♦ whose objects are the T-modal lattices, the K4-modal lattices, the S4-modal lattices, and the S5-modal lattices. For bimodal lattices we adopt a similar terminology: a bimodal lattice is referred to as a • T-bimodal lattice if x ≤ ♦x and x ≤ �x for all x; • K4-bimodal lattice if ♦♦x ≤ ♦x and ��x ≤ �x for all x; • S4-bimodal lattice if it is both a T-bimodal lattice and a K4-bimodal lattice; • S5-bimodal lattice if it is an S4-bimodal lattice and ♦x = �x for all x. The categories Lat♦�T , Lat K4, Lat S4 , and Lat S5 are, respectively, the full subcategories of Lat♦� whose objects are the T-bimodal lattices, the K4- bimodal lattices, the S4-bimodal lattices, and the S5-bimodal lattices. By standard universal algebra the forgetful functor Lat♦� → Lat♦ has a left adjoint which assigns to BK its “enveloping” bimodal lattice B K. Simi- larly, there are left adjoints Lat♦T → Lat Lat♦K4 → Lat Lat♦S4 → Lat and we write B′T, B K4, and B S4 for the respective images of BT, BK4, and BS4 under the left adjoints. For S5 the situation is simpler because the categories Lat♦S5 and Lat S5 are obviously isomorphic, since any S5-modal lattice becomes an S5-bimodal lattice just by defining � to coincide with ♦. We shall also write B′S5 for BS5 thus regarded as a bimodal lattice. Since ♦ and � preserve finite joins they can be extended canonically to sup-lattice endomorphisms of the ideal completion Idl(B′K), which is a frame because B′K is a distributive lattice. The conjugation relations are easily seen to be inherited from those of B′K, and thus Idl(B K) is a bimodal frame. Similar remarks apply to the other Lindenbaum algebras, and in addition Idl(B′T) satisfies the axioms of a T-bimodal lattice, Idl(B K4) satisfies the axioms of a K4-bimodal lattice, etc. (Hence, in particular, the propositions in ¶¶5.15–5.18 can be applied to Idl(B′T), Idl(B K4), Idl(B S4), and Idl(B respectively.) Summarizing, we have described a way of constructing functors from modal lattices to bimodal frames which are left adjoint to the obvious forget- ful functors. Composing these functors with the left adjoints from bimodal frames to pointed ssqs we obtain fromBK, BT, BK4, BS4, andBS5 pointed ssqs TK(Idl(B K)), TT(Idl(B T)), TK4(Idl(B K4)), TS4(Idl(B S4)), and TS5(Idl(B S5)), respectively. For each system S ∈ {K,T,K4, S4, S5} the map obtained by composing the following arrows (the leftmost one is just the natural quo- tient), BK → BS → B S → Idl(B S)→ TS(B has the same universal property as the injection of generators BK → QS. Hence, the quantale TS(B S) is a particular construction of the Lindenbaum quantale QS: Theorem 5.20 ∼= TK(Idl(B ∼= TT(Idl(B ∼= TK4(Idl(B ∼= TS4(Idl(B ∼= TS5(Idl(B Since, as we have remarked above, S5-modal lattices and S5-bimodal lattices are “the same”, our results immediately tell us that the unit of the adjunction between S5-modal lattices and pointed ssqs, BS5 → QS5 , is a monomorphism. In logical terms this means that a complete axiomati- zation for the system S5 (with the advantage of making no use of negation or the modal necessity operator) can be as follows: Theorem 5.21 S5 is complete for the following axiom schemata: ϕ → ♦ϕ ♦♦ϕ → ♦ϕ ♦ϕ ∧ ψ → ♦(ϕ ∧ ♦ψ) . We have not verified whether the remaining canonical mappings BK → B BT → B BK4 → B BS4 → B are monomorphisms (although we believe they are). This means that we have not verified completeness for the classical axiomatizations of K, T, K4, and S4. Of course, by this we mean we have not verified this in an arbitrary topos, for otherwise we know, from the classical completeness theorems of propositional normal modal logic, that the axiomatizations are complete: using Zorn’s Lemma we can find a Kripke structure (W,R) that gives us a monomorphism BK → B K → TK(Idl(B ∼= QK → ℘(W ×W ) implying that BK → B K is 1–1, and the same applies to T, K4, and S4. References [1] D. Harel, D. Kozen, J. Tiuryn, Dynamic Logic, MIT Press, 2000. [2] E.A. Emerson, Temporal and modal logic, in J. van Leeuwen (editor), Handbook of Theoretical Computer Science, vol. B, MIT Press, 1990, pp. 955–1072. [3] G.E. Hughes, M.J. Cresswell, An Introduction to Modal Logic, Methuen & Co. Ltd., London, 1968. [4] G.E. Hughes, M.J. Cresswell, A Companion to Modal Logic, Methuen & Co. Ltd., London, 1984. [5] P.T. Johnstone, Stone Spaces, Cambridge Stud. Adv. Math., vol. 3, Cambridge Univ. Press, 1982. [6] A. Joyal, M. Tierney, An Extension of the Galois Theory of Grothendieck, Mem. Amer. Math. Soc., vol. 309, American Mathemat- ical Society, 1984. [7] M.V. Lawson, Inverse Semigroups — The Theory of Partial Symmetries, World Scientific, 1998. [8] I. Moerdijk, J. Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge University Press, 2003. [9] C.J. Mulvey, Quantales, in M. Hazewinkel (editor), The Encyclopaedia of Mathematics, third supplement, Kluwer Academic Publishers, 2002, pp. 312–314. [10] C.J. Mulvey, P. Resende, A noncommutative theory of Penrose tilings, Internat. J. Theoret. Phys. 44 (2005) 655–689. [11] J. Paseka, J. Rosický, Quantales, in B. Coecke, D. Moore, A. Wilce, (editors), Current Research in Operational Quantum Logic: Algebras, Categories and Languages, Fund. Theories Phys., vol. 111, Kluwer Aca- demic Publishers, 2000, pp. 245–262. [12] A.L.T. Paterson, Groupoids, Inverse Semigroups, and Their Operator Algebras, Birkhäuser, 1999. [13] P. Resende, Tropological systems are points of quantales, J. Pure Appl. Algebra 173 (2002) 87–120. [14] P. Resende, Étale groupoids and their quantales, Adv. Math. 208 (2007) 147–209. [15] K. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Mathematics Series 234, Longman Scientific & Technical, 1990. Centro de Lógica e Computação Instituto Superior Técnico Universidade Técnica de Lisboa Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal E-mail: sergiortm@gmail.com Centro de Análise Matemática, Geometria e Sistemas Dinâmicos Departamento de Matemática do Instituto Superior Técnico Universidade Técnica de Lisboa Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal E-mail: pmr@math.ist.utl.pt Introduction Preliminaries Quantale semantics of modal logic Graded unital involutive quantales Construction of the Lindenbaum quantales

The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4, and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL, and the ramified temporal logic CTL.

Introduction It is well known that the set ℘(W × W ) of all the binary relations on a set W has the structure of a unital involutive quantale (see §2). Hence, a Kripke structure (W,R) as it appears in the semantics of propositional normal modal logic [3, 4], where W is the set of possible worlds and R ⊂W ×W is the accessibility relation, can be regarded as an example of a pointed unital involutive quantale (℘(W ×W ), R). This suggests a way of generalizing the notion of Kripke structure, namely in terms of a more general pointed unital involutive quantale (Q,α), and the purpose of this paper is to assess the usefulness of this idea with respect to the semantics of modal logic. ∗Research supported in part by Fundação para a Ciência e a Tecnologia through pro- gram POCI 2010/FEDER and project POCI/MAT/55958/2004. http://arxiv.org/abs/0704.1886v1 Not every unital involutive quantale is suitable for this purpose, and in this paper we restrict to the notion of stably supported quantale that has been introduced in [14]. In order to motivate this let us consider again the quantale ℘(W ×W ) of binary relations on W . Each relation R ⊂ W ×W has a domain dom(R) = {x ∈ W | ∃y (x, y) ∈ R} and, since the diagonal relation ∆W = {(x, y) ∈ W ×W | x = y} is of course isomorphic to W , we can equivalently replace the domain of R by the set ςR = {(x, x) ∈ ∆W | x ∈ dom(R)} , which we refer to as the support of R. This defines an operation ς : ℘(W ×W )→ ℘(W ×W ) that preserves unions and in addition satisfies the following properties, for all R, S ⊂W ×W : ςR ⊂ ∆W ςR ⊂ RR∗ R ⊂ ςRR ς(RS) ⊂ ςR A stably supported quantale, or simply ssq, is defined to be a unital involutive quantale Q equipped with a sup-lattice endomorphism ς : Q → Q which satisfies these properties; that is, for all a, b ∈ Q we have ςa ≤ e ςa ≤ aa∗ a ≤ ςa a ς(ab) ≤ ςa For each a ∈ Q the element ςa is called the support of a, the operation ς itself is referred to as the support of Q, and the set ςQ = {ςa | a ∈ Q} is necessarily a locale whose binary meet operation coincides with the multiplication of the quantale: a∧b = ab. For instance, if Q = ℘(W×W ) we have ςQ = ℘(∆W ) ∼= ℘(W ). The formulas of propositional modal logic can be easily interpreted on any pointed ssq (Q,α): there should be a valuation map v assigning to each formula ϕ an element v(ϕ) ∈ ςQ. The properties that such a map must satisfy are clear. For instance, conjunction of formulas should be interpreted as multiplication in Q: v(ϕ∧ψ) = v(ϕ)v(ψ); and the remaining propositional connectives are equally straightforward, both classically and intuitionistically (see §3). As regards the modal operator ♦ of possibility, we impose v(♦ϕ) = ς(αv(ϕ)) . This is easily seen to yield the usual interpretation on Kripke structures: v(♦ϕ) corresponds to the domain of the relation αv(ϕ), where α is the ac- cessibility relation. Algebraic semantics. If (Q,α) is a pointed ssq then ςQ becomes a locale equipped with unary operators in a natural way (§§3, 5), and it will be seen in this paper that from any such locale it is possible to obtain a pointed ssq, in fact giving us an adjunction between two categories, and generalizing the way in which a binary relation on W (i.e., an accessibility relation) corre- sponds classically to a unary operator ♦ : ℘(W ) → ℘(W ). This leads to a semantics that subsumes the classical algebraic semantics based on lattices with operators. The systems of modal logic that are characterized by special kinds of accessibility relations can be characterized by subcategories of the category StabQu∗ of pointed ssqs. For instance, the system S5, which is characterized by accessibility being an equivalence relation, will correspond to the full subcategory of StabQu∗ whose objects (Q,α) are those satisfying α 2 = α∗ = α ≥ e. Moreover, it is easy to address multi-modal logics, such as dynamic logic, logics of time and space, etc., in terms of the category whose objects are ssqs equipped with more than one point, the various points satisfying suitable relations. The category StabQu of ssqs has pleasant properties [14]. For instance, there is at most one support on any unital involutive quantale, and any homomorphism of unital involutive quantales h : Q → Q′ between ssqs Q and Q′ automatically preserves the support (hence, being stably supported is a property rather than extra structure). The category StabQu is therefore a full subcategory of the category of unital involutive quantales, and in fact it is reflective. There are presentations of ssqs by generators and relations, which play the role of “Lindenbaum algebras” in the context of the quantale- based semantics. For instance, the “Lindenbaum quantale” for S5 is the pointed ssq QS5 generated by the usual Lindenbaum algebra of S5 (with defining relations ensuring that it is a bounded sublattice of ς QS5) with the distinguished point α ∈ QS5 being subject to the relations α 2 = α∗ = α ≥ e. The usual Kripke structure based models for S5 can be identified with the relational representations of QS5, in other words the homomorphisms h : QS5 → ℘(W ×W ) of unital involutive quantales (see [10] for some properties of such representa- tions). The same identification of models with the relational representations of a quantale applies to many other well known examples of modal logics. Groupoids and inverse semigroups. An important aspect of the se- mantics described in this paper is that there are plenty of examples of ssqs besides the quantales of binary relations [14], arising from various geomet- ric or analytic structures, and thus we are provided with a uniform way of defining semantic interpretations of propositional normal modal logic based on such structures. More precisely, let G be a groupoid (i.e., a small category all of whose arrows are isomorphisms). Writing G also for the set of groupoid arrows, ℘(G) is an ssq; the quantales of binary relations ℘(W ×W ) are pre- cisely the quantales that arise from the so-called pair groupoid of W , which hasW as set of objects andW ×W as set of arrows, with the two projections W ×W → W being the domain and codomain maps of the groupoid. Even more generally, the topology Ω(G) of any topological étale groupoid G is a sub-ssq of ℘(G); and for localic étale groupoids, too, there is an ssq O(G) associated to each groupoid G. These facts are a part of the close relation [14] between the notions of étale groupoid (either topological or localic), inverse semigroup, and quantale, which can be summarized in the following diagram whose arrows denote bijections of objects up to isomorphism, or even, in the case of I and L∨, equivalences of categories: Inverse quantal frames Étale groupoids bisections Complete infinitely distributive inverse semigroups germspp The inverse quantal frames are the ssqs that arise from étale groupoids. It follows that the quantale semantics automatically provides a bridge between modal logic and those areas of mathematics where examples of étale groupoids and inverse semigroups occur, such as operator algebras and dif- ferential topology — see, e.g., [7, 8, 12]. As an example of the latter, foliated manifolds can be associated to dynamical systems, and from a foliation it is always possible to construct a topological étale groupoid [8]. We shall not deal with any such examples in this paper, but we mention that if we replace ℘(W ×W ) by a more general groupoid quantale, hence taking as models of propositional modal logic the homomorphisms Q → Ω(G) or Q → O(G) instead of Q → ℘(W ×W ) (where Q is a Lindenbaum quantale), we are led in a natural way to semantics which may be interesting, say, for applied logicians or computer scientists dealing with hybrid systems, logics of real time and space, etc. Overview. Let (Q,α) be a pointed ssq. Then the locale ςQ is canonically equipped with the two unary sup-latice endomorphisms ♦ and � defined by, for each x ∈ ςQ, ♦x = ς(αx) (1) �x = ς(α∗x) , (2) which are easily seen to satisfy the following conjugacy conditions (see §5): ♦x ∧ y ≤ ♦(x ∧ �y) �x ∧ y ≤ �(x ∧ ♦y) . Such a structure (L,♦,�), where L is a locale and ♦ and � satisfy the conjugacy conditions, will be called a bimodal frame, and there is a functor from the category of pointed ssqs to the obvious category of bimodal frames. A functor in the opposite direction can be easily obtained just from the knowledge that ssqs can be presented by generators and relations; given a bi- modal frame (L,♦,�) its associated quantale Q is generated by the elements of L plus an element α, with relations imposing both that L is a unital invo- lutive subquantale of Q and that (1) and (2) are satisfied. As we shall see, this defines a functor which is left adjoint to ς, and in fact it is a coreflection; hence, we always have an isomorphism of bimodal frames L ∼= ςQ. We can interpret this isomorphism in a logical sense as saying that no theorems are added in the process of interpreting ♦ and � in terms of the quantale operations. Hence, if we think of the addition of the quantale operations as a language extension then this extension is conservative — the conjugacy conditions are a complete axiomatization for the modal operators induced by α and α∗. This of course suggests looking at several systems of modal logic and their completeness theorems, which we shall do for K, T, K4, and S4, showing that the usual conditions on ♦ (and here also on �), as taken from the standard completeness theorems of modal logic, are precisely what is required for a coreflection to be obtained when the pointed ssqs (Q,α) under consideration satisfy the expected conditions, such as “reflexivity”, “transitivity”, etc., of the point α. Hence, the theorems which we prove in this paper can be re- garded as an algebraic generalization of the standard completeness theorems for these systems of propositional normal modal logic. It is worth noting that, as opposed to the classical theorems, these are now independent of the axiom of choice (which is required in the proof of the classical theorems, in the form of Zorn’s lemma). From our results we also obtain a complete axiomatization of system S5, where to the axioms of S4 one adds the axiom scheme ♦ϕ ∧ ψ → ♦(ϕ ∧ ♦ψ) . This is obtained from the conjugacy conditions by making ♦ = � and, as opposed to the usual axiom scheme ϕ→ �♦ϕ, it does not mention negation or the modal necessity operator. The remainder of this paper goes as follows. In §2 we provide some nec- essary background and preliminary results on quantales, presentations by generators and relations, etc. Then in §3 we describe in detail the quantale- based semantics of propositional normal modal logic, including the systems K, T, K4, S4, and S5. In order to illustrate the flexibility of this approach we provide additional examples, namely propositional intuitionistic logic, propo- sitional dynamic logic, and the ramified temporal logic CTL. Finally, after some technical results in §4 about graded quantales (the quantale analogue of graded rings), in §5 we address the adjunctions mentioned above. The mere existence of the adjunctions is a consequence of the existence of pre- sentations by generators and relations; in other words, it can be phrased in terms of the existence of the Lindenbaum quantales. However, in order to obtain additional information about the adjunctions and, in particular, in order to prove that they are coreflections, we shall need an actual construc- tion of the Lindenbaum quantales. This will be conveniently formulated in terms of “tensor algebras” over bimodal frames, and it will take up most of 2 Preliminaries Here we describe some background on sup-lattices, locales and quantales. General references are [5, 6, 9, 11, 15]. Sup-lattices. We shall denote the category of sup-lattices by SL. The objects are the complete lattices and the morphisms are the maps f : L→M that preserve arbitrary joins: for all X ⊂ L f(x) . We shall write 1L or simply 1 for the lattice unit (the greatest element) of a sup-lattice L, and 0L or simply 0 for the least element. Let (Li) be a family of sup-lattices. Their cartesian product Li is a sup-lattice with pointwise order and joins, and it is a product in the category SL. The products and the coproducts are isomorphic [6], similarly to abelian groups (but, contrary to the latter, also in the case of infinitary coproducts). We denote by Li the categorical coproduct of a set-indexed family of sup-lattices (Li), and call it direct sum. The tensor product of L and M is denoted by L ⊗M , and similarly to vector spaces, it is the image of a universal bi-morphism, where a sup-lattice bi-morphism f : L ×M → N is a map that preserves joins in each variable separately: f(x, y) f(x, y) . Concretely, L⊗M can be identified with the set of those subsets I ⊂ L×M such that ∈ I ⇐⇒ {x} × Y ⊂ I ∈ I ⇐⇒ X × {y} ⊂ I for all x ∈ L, y ∈ M , X ⊂ L, and Y ⊂ M . The universal bi-morphism L×M → L⊗M is defined by (x, y) 7→ x⊗ y, where x⊗ y is the least such set that contains the pair (x, y), which is called the pure tensor generated by (x, y). SL is a monoidal category with respect to this tensor product, with the powerset ℘( 1) of the singleton set as the tensor unit. Similarly to the category of abelian groups, the functor hom(N,−) has a left adjoint −⊗N for each N ; that is, we have the familiar isomorphism hom(M ⊗N,L) ∼= hom(M, hom(N,L)) , natural in the variables M and L, which in fact is an order isomorphism. As a consequence of colimit preservation by left adjoints, ⊗ distributes over Mi) ∼= (L⊗Mi) . Quotients of sup-lattices can be conveniently handled by means of closure operators (monotone endomaps j that satisfy a ≤ j(a) and j(j(a)) = j(a) for every element a). Let L be a sup-lattice and j a closure operator on L. The set of j-closed elements Lj = {x ∈ Q | x = j(x)} is a sup-lattice closed under the formation of meets in L, with joins given by (xi) = j( xi), and the map j : L → Lj is a (surjective) homomorphism of sup-lattices. Conversely, given a set S ⊂ L closed under meets in L, we obtain a closure operator jS : L→ L by jS(x) = {y ∈ S | x ≤ y} . These constructions are mutually inverse, jLj = j LjS = S , and every sup-lattice quotient arises in this way up to isomorphism. The relation to the usual description of quotients by means of congruence relations (i.e., equivalence relations on L which are sub-sup-latttices of L×L) is the following: from a closure operator j we obtain the congruence relation θj ⊂ L× L defined by (x, y) ∈ θj ⇐⇒ j(x) = j(y) [in particular (x, j(x)) ∈ θj ] and from a congruence relation θ ⊂ L × L we define a closure operator jθ by jθ(x) = where [x]θ is the congruence class of x; of course, we have jθj = j and θjθ = θ. Stably supported quantales. A quantale is a sup-lattice equipped with an associative multiplication, usually written (a, b) 7→ ab, which distributes over arbitrary joins: (abi) , (aib) . Hence, a quantale is a semigroup in SL. A quantale Q is unital if the multi- plication has a unit, which we denote by eQ, or simply e. Example 2.1 A locale, or frame, L is a sup-lattice satisfying the following distributivity property for all x ∈ L and Y ⊂ L: x ∧ y . Hence, a locale is a unital quantale whose multiplication is ∧ and whose unit e coincides with 1. In particular, it is a commutative and idempotent quantale. A quantale is a locale if and only if it is unital with e = 1 and it is idempotent [6]. An involutive quantale Q is a quantale equipped with an involution (−)∗ : Q→ Q , i.e., a join preserving operation that makes Q an involutive semigroup: (ab)∗ = b∗a∗, a∗∗ = a. Any involutive quantale satisfies 1∗ = 1 and, if it is unital, e∗ = e. Hence, a unital involutive quantale is an involutive monoid ℘( 1) ←− Q⊗Q in the monoidal category of sup-lattices, with ab = m(a⊗ b). Definition 2.2 Let Q be a unital involutive quantale. A support on Q is a sup-lattice endomorphism ς : Q→ Q satisfying, for all a ∈ Q: ςa ≤ e , (3) ςa ≤ aa∗ , (4) a ≤ ςaa . (5) A supported quantale is a unital involutive quantale equipped with a spec- ified support. On a supported quantale the set of supports ςQ coincides with ↓e = {x | x ≤ e} and it is a locale with ab = a ∧ b [14]. Definition 2.3 A support is stable if it satisfies ς(ab) = ς(aς(b)). A quantale equipped with a specified stable support is stably supported, or simply an ssq. Every homomorphism of unital involutive quantales preserves the support of an ssq, and thus the category of ssqs, StabQu, is defined to be the full subcategory of the category of unital involutive quantales Qu whose objects are the ssqs. Moreover, if a quantale is stably supported then it can have no other support, stable or not [14]. Hence, being stably supported should be regarded as a property of unital involutive quantales rather than extra structure. In [14] it has also been seen that the inclusion functor StabQu → Qu has a left adjoint (i.e., StabQu is a reflective subcategory of Qu). Any locale is an ssq with trivial involution and support: x∗ = x ςx = x . Nuclei and quotients. The quotients of ssqs are described in a similar way to those of sup-lattices. We give here an overview and refer to [14] for further details. Definition 2.4 A (quantic) nucleus on an ssq Q is a closure operator j : Q→ Q that satisfies, for all x, y ∈ Q, j(x)j(y) ≤ j(xy) , j(x)∗ ≤ j(x∗) , ς(j(x)) ≤ j(ς(x)) . We remark that the second condition is equivalent to j(x)∗ = j(x∗). The set of j-closed elements Qj = {x ∈ Q | x = j(x)} is an ssq with joins (xi) = j( xi), multiplication x · y = j(xy), with the same involution as Q, and support δx = j(ςx). The map j : Q → Qj is a (surjective) homomorphism of ssqs: j(xi) j(xy) = j(x) · j(y) j(x∗) = j(x)∗ j(ςx) = δx . Furthermore, every quotient arises in this way up to isomorphism. The set N(Q) of nuclei is a complete lattice under the pointwise order, with meets being calculate pointwise: j ≤ k ⇔ ∀x∈Q(j(x) ≤ k(x)), and jα(x) = (jα(x)). Furthermore, we have j ≤ k ⇔ Qk ⊂ Qj, and the join of nuclei corresponds to intersection of the respective sets of closed elements: j = jα if and only if Qj = Qjα . Definition 2.5 Let Q be an ssq, and R ⊂ Q×Q. The supported closure R of the binary relation R is the smallest relation that contains R and is closed for the quantale operations, i.e.: R ⊂ R ; (y, z) ∈ R ⇒ (ay, az) ∈ R, for all a ∈ Q ; (y, z) ∈ R ⇒ (ya, za) ∈ R, for all a ∈ Q ; (y, z) ∈ R ⇒ (ςy, ςz) ∈ R ; (y, z) ∈ R ⇒ (y∗, z∗) ∈ R . Contrarily to what is done in [13], we shall interpret each pair (y, z) ∈ R as an inequality y ≤ z, rather than an equation y = z. It is easy to see that there is a least quantic nucleus j such that j(y) ≤ j(z) for all (y, z) ∈ R: {j ∈ N(Q) | j(y) ≤ j(z) for all (y, z) ∈ R}. Analogously to the quotients of involutive quantales described in [13], the quantale QjR has a very simple description: QjR = {x ∈ Q | ∀(y,z)∈R (z ≤ x⇒ y ≤ x)}. We stress that nuclei and quotients of unital involutive quantales equipped with any sup-lattice endomorphism ς : Q → Q are handled in exactly the same way as described above for ssqs (we shall use this in §5 when dealing with quantales that are just “pre-supported”). The properties of ς pass to the quotients one by one: if j is a nucleus on Q and δ : Qj → Qj is the sup-lattice endomorphism defined by δ(a) = j(ςa), then if, say, the equation ς(ab) = ς(aςb) holds in Q then δ(ab) = δ(aδb) holds in Qj , etc. Generators and relations. Let G be a set (of “generators”). The con- struction of the unital involutive quantale Qu〈G〉 freely generated by G is described in [13]. Denoting by F : Qu → StabQu the left adjoint to the inclusion StabQu → Qu (cf. paragraph after ¶2.3), it follows that F (Qu〈G〉) is the free ssq generated by G, and we shall denote it by StabQu〈G〉. Definition 2.6 Let G and R ⊂ StabQu〈G〉 × StabQu〈G〉 be sets. The ssq presented by the generators in G and the relations in R is StabQu〈G | R〉 = StabQu〈G〉 If x ∈ G, one denotes by [x] the image of the generator x in the quantale being presented. This notation provides a useful way of describing the defin- ing relations of a quantale presentation: we just write the conditions with respect to which the injection of generators is universal, as in the following example for unital involutive quantales. Example 2.7 Let L be a sup-lattice. It follows from the universal proper- ties of the tensor product and the direct sum of sup-lattices that the unital involutive quantale freely generated by L with joins being preserved in the presentation, is isomorphic to the tensor quantale where I is the free involutive monoid on one generator, whose words can be concretely identified with the strings of symbols α and α∗ and whose unit we shall denote by ε, and L(d) = L⊗|d| = L⊗ · · · ⊗ L (|d| times) . Note that L(ε) = ℘( 1) is the neutral element of the tensor product. The multiplication is defined on pure tensors just by concatenation (x0 ⊗ . . .⊗ xn)(y1 ⊗ . . .⊗ ym) = x0 ⊗ . . .⊗ xn ⊗ y1 ⊗ . . .⊗ ym , where in the case of concatenation with elements of ℘( 1) we use the identi- fication ℘( 1) ⊗ L ∼= L ∼= L ⊗ ℘( 1), to produce the identity of the quantale e = 1℘( 1). The involution (−)∗ : TL→ TL is obtained from the isomorphisms L(w) → L(w ∗) that are given by x1 ⊗ · · · ⊗ xn 7→ xn ⊗ · · · ⊗ x1 . The injection of generators is the α-coprojection of the coproduct L = L(α) → L(d). 3 Quantale semantics of modal logic Propositional normal modal logic. In this section we describe the in- terpretations of the classical systems of modal logic K, T, K4, S4, and S5. For details on these we refer the reader to [3, 4]. We shall consider fixed a set Π of propositional symbols. The set Φ of propositional formulas is defined to be the least set containing Π such that for all ϕ, ψ ∈ Φ we have ¬ϕ ∈ Φ , ϕ ∨ ψ ∈ Φ , ♦ϕ ∈ Φ , where as usual we may define other connectives, for conjunction ∧, implica- tion →, and the necessity modal operator �, as abbreviations: ϕ ∧ ψ = ¬(¬ϕ ∨ ¬ψ) , ϕ→ ψ = ¬ϕ ∨ ψ , �ϕ = ¬♦¬ϕ . Definition 3.1 A generalized Kripke model consists of a triple (Q,α, v), where Q is an ssq, α ∈ Q is an accessibility element, and v : Φ → ςQ is an interpretation map satisfying the following properties for all ϕ, ψ ∈ Φ: v(ϕ ∨ ψ) = v(ϕ) ∨ v(ψ) v(ϕ)v(¬ϕ) = 0 v(ϕ) ∨ v(¬ϕ) = e v(♦ϕ) = ς(αv(ϕ)) . Remark 3.2 The above definition makes each element v(ϕ) be comple- mented in ςQ, with v(¬ϕ) being its (unique) complement, and it also follows that conjunction is interpreted as multiplication (equivalently, meet) in ςQ: v(ϕ ∧ ψ) = v(ϕ)v(ψ) . This means that we interpret the formulas inside a Boolean subalgebra of ςQ, hence obtaining a classical semantics of propositional modal logic, a fact that was already implicit in the definition of the conjunction and the implication as derived connectives. However, we point out that it is easy to define a (rather natural) semantics for intuitionistic modal logic. We shall describe this at the end of §3. As usual we say that a pointed ssq consists of an ssq Q together with a specified “point” α ∈ Q. A homomorphism of pointed ssqs is a homomor- phism of ssqs that preserves the point: h : (Q,α) → (R, β) α 7→ β. From now on we shall denote by BK the Lindenbaum algebra of system K (i.e., the set of formulas modulo equivalence, which is a Boolean algebra equipped with a finite join preserving endomorphism ♦). Definition 3.3 The Lindenbaum quantale for K is the pointed ssq QK which is presented by generators and relations with BK as the set of generators and with the following relations for all x, y ∈ BK, where we denote the selected point by α: [x ∨ y] = [x] ∨ [y] [¬x][x] = 0 [¬x] ∨ [x] = e [♦x] = ς(α[x]) . From the universal property of ssqs presented by generators and relations we immediately obtain: Theorem 3.4 There is a bijective correspondence between abstract Kripke models (Q,α, v) and homomorphims of unital involutive quantales QK −→ Q . In particular, if W is a set then a homomorphism ρ : QK −→ ℘(W ×W ) is the same as a model for system K with set of possible worlds W and accessibility relation ρ(α). In order to obtain similar facts for other systems, such as T, K4, S4, S5, one must define the appropriate Lindenbaum quantales. Definition 3.5 The Lindenbaum quantales for T, K4, S4, and S5, are the pointed ssqs QT, QK4, QS4, and QS5, respectively, which are presented by generators and relations similarly to QK, with the following additional rela- tions: QT: e ≤ α QK4: αα ≤ α QS4: e ≤ α ≥ αα QS5: e ≤ α = α ∗ ≥ αα Hence, QT is the quotient of QK by the least nucleus j such that j(e) ≤ j(α), and QK4 is the quotient ofQK by the least nucleus j such that j(αα) ≤ j(α). Then we have QS4 = QT ∩QK4 and QS5 is the quotient of QS4 by the least nucleus j such that j(α) = j(α∗). Notice that the relational representations of these quantales correspond to the expected classes of models: Theorem 3.6 The relational representations Q → ℘(W ×W ) of the Lin- denbaum quantales Q correspond bijectively to the Kripke models whose ac- cessibility relations are: • Reflexive, for Q = QT; • Transitive, for Q = QK4; • Preorders, for Q = QS4; • Equivalence relations, for Q = QS5. Propositional ramified temporal logic. Now we describe a similar se- mantics for the ramified temporal logic known as Computational Tree Logic (CTL), see [2]. As above, Π is a fixed set of propositional symbols. The set Φ of CTL formulas is defined to be the least set containing Π such that for all ϕ, ψ ∈ Φ we have ¬ϕ, ϕ ∨ ψ,EXϕ,EFϕ,EGϕ ∈ Φ , and we may define other modal operators as abbreviations: AXϕ ≡ ¬EX¬ϕ , AGϕ ≡ ¬EF¬ϕ , AFϕ ≡ ¬EG¬ϕ . Definition 3.7 The intuitive meaning of the various modalities is the fol- lowing: • EXϕ means that there is a possible future where ϕ will hold in the next time instant; • EFϕ means that there is a possible future where ϕ will eventually hold; • EGϕ means that there is a possible future where ϕ will always hold (including now); • AXϕ means that ϕ will certainly hold in the next time instant; • AGϕ means that ϕ will always hold (including now) in all possible future paths; • AFϕ means that in each possible future path ϕ will eventually hold. A generalized CTL model consists of a triple (Q,α, v), where Q is an ssq, α ∈ Q is an accessibility element that satisfies ς(α) = e (i.e., time never ends), and v : Φ → ςQ is an interpretation map satisfying the following properties for all ϕ, ψ ∈ Φ: v(ϕ ∨ ψ) = v(ϕ) ∨ v(ψ) v(ϕ)v(¬ϕ) = 0 v(ϕ) ∨ v(¬ϕ) = e v(EXϕ) = ς(αv(ϕ)) v(EFϕ) = ς v(EGϕ) = {a ∈ Q | a ≤ v(ϕ) ∧ ς(αa)} . It is easy to see that this interpretation conveys the intended meaning to the modal operators if we let Q = ℘(W ×W ) for some set W . Only the last condition, for EG, deserves an explanation. This says that EGϕ may be interpreted as the largest subset X ⊂ W such that every world x ∈ X satisfies the following two conditions: • x satisfies the formula ϕ; • there is a world y ∈ X such that (x, y) ∈ α. This guarantees that there is an infinite path (possibly with repetitions) x0, x1, x2, x3, . . . satisfying ϕ starting at any world x0 where EGϕ holds. Mathematically, the definition of v(EGϕ) is clarified by the Knaster–Tarski fixed point theorem: the set of all the lowerbounds of v(ϕ), S = {a ∈ ςQ | a ≤ v(ϕ)} , is a complete lattice and thus the set of pre-fixed points of the monotone operator f : S → S defined by f(a) = v(ϕ) ∧ ς(α a) has a join, which in fact is a fixed point; hence, v(EGϕ) is also the largest element a ∈ ςQ such that a = v(ϕ) ∧ ς(α a). Propositional dynamic logic. In the program logic of [1] there are modal- ities indexed by programs, which themselves form a set with some algebraic structure. Let Π be a fixed set of propositional symbols and Ξ a set of atomic programs. The sets F , of formulas, and P , of programs, give us the PDL language Φ = F ∪ P , and they are defined to be the least sets such that Π ⊂ F , Ξ ⊂ P , if ϕ, ψ ∈ F then ¬ϕ, ϕ ∨ ψ ∈ F , if p, q ∈ P then p ∪ q, p; q, p∗ ∈ P , if ϕ ∈ F and p ∈ P then 〈p〉ϕ ∈ F , if ϕ ∈ F then ϕ? ∈ P . Very briefly, the intuitive meaning of the program constructs is the following: • p∪ q is the program that behaves in a nondeterministic way either like p or like q (the choice is made at the beginning of the execution of the program, once and for all); • p; q is the program whose execution is that of p followed by q; • p∗ is the iteration of p, to be thought of as p executed sequentially zero or more times (not to be confused with the notation for the quantale involution); • ϕ? is the program that tests ϕ, succeeding if ϕ is found to be true, and failing otherwise. Definition 3.8 A generalized PDL model consists of a pair (Q, v), where Q is an ssq and v : Φ → Q is an interpretation map satisfying the following properties for all ϕ, ψ ∈ F and p, q ∈ P : v(ϕ ∨ ψ) = v(ϕ) ∨ v(ψ) v(ϕ)v(¬ϕ) = 0 v(ϕ) ∨ v(¬ϕ) = e v(〈p〉ϕ) = ς(v(p)v(ϕ)) v(p ∪ q) = v(p) ∨ v(q) v(p; q) = v(p)v(q) v(p∗) = v(p)n v(ϕ?) = v(ϕ) . This interpretation shows that to a large extent both the formulas and the programs are treated on an equal footing. In particular, p; q can be regarded as the (noncommutative) “conjunction” of p and q, and p ∪ q as their dis- junction, while a formula is just a particular kind of program (ϕ is identified with ϕ?). Intuitionistic modal logic. It is easy to define a semantics for intuitionis- tic propositional modal logic if we let ∧ and → be independent connectives. This is because the support ςQ of any ssq Q is a locale and therefore a Heyting algebra, and thus, denoting by \ the residuation operation of ςQ, b\a = {c ∈ ςQ | b ∧ c ≤ a} , the conditions on v can be simply replaced by the following: v(ϕ ∨ ψ) = v(ϕ) ∨ v(ψ) v(ϕ ∧ ψ) = v(ϕ)v(ψ) v(ϕ→ ψ) = v(ϕ)\v(ψ) v(¬ϕ) = v(ϕ)\0 v(♦ϕ) = ς(αv(ϕ)) . This would entirely define the intuitionistic semantics if we contented our- selves with defining � = ¬♦¬ as before. However, this is a bad interpretation of �, as for instance it usually does not satisfy the axiom of distributivity over meets �(ϕ ∧ ψ)↔ �ϕ ∧�ψ . Indeed, a much better interpretation, in particular one that satisfies distribu- tivity of � over (arbitrary) meets, is obtained if we let � be an independent connective interpreted as the right adjoint of a suitable sup-lattice endomor- phism, as we now describe. LetW be a set, let R ⊂W ×W be a binary relation, and let Q be the ssq ℘(W ×W ). Let ♦ and � be the sup-lattice endomorphisms of ςQ defined as in §1: ♦X = ς(RX) �X = ς(R∗X) . Equivalently, taking into account the isomorphism ςQ ∼= ℘(W ) we may consider ♦ and � to be endomorphisms of ℘(W ): ♦X = {y ∈ W | ∃x∈X (y, x) ∈ R} �X = {y ∈ W | ∃x∈X (x, y) ∈ R} . It is straightforward to verify that the usual necessity operator � : ℘(W )→ ℘(W ) , which is defined by �X = {y ∈ W | ∀x∈W (y, x) ∈ R⇒ x ∈ X} , is right adjoint to �, and we may take this as the natural definition of � when such a “possibility in the past” operator � is available — similarly, a “necessity in the past” operator � can be defined to be the right adjoint of �x ≤ y ⇐⇒ x ≤ �y ♦x ≤ y ⇐⇒ x ≤ �y . This leads to the following quantale-based intuitionistic semantics for propo- sitional modal logic, where we assume that ∧, →, and � are independent connectives: Definition 3.9 A generalized intuitionistic Kripke model consists of a triple (Q,α, v), where Q is an ssq, α ∈ Q is an accessibility element, and v : Φ→ ςQ is an interpretation map satisfying the following properties for all ϕ, ψ ∈ Φ: v(ϕ ∨ ψ) = v(ϕ) ∨ v(ψ) v(ϕ ∧ ψ) = v(ϕ)v(ψ) v(ϕ→ ψ) = v(ϕ)\v(ψ) v(¬ϕ) = v(ϕ)\0 v(♦ϕ) = ς(αv(ϕ)) v(�ϕ) = {x ∈ ςQ | ς(α∗x) ≤ v(ϕ)} . This definition illustrates a canonical way in which to define intuitionistic semantics for other propositional modal logics, including all the examples seen earlier in this section. We stress the fact that involutive quantales are essential for this, since without the involution we would neither have the � operator, nor a canonical definition of �. It is worth commenting on some aspects of the intuitionistic version of system S5, which similarly to its classical counterpart is based on imposing that the accessibility element α should be self-adjoint, and thus � = ♦. The unit of the adjunction relating � and � is the inequality x ≤ ��x , and thus for intuitionistic S5 we conclude that the axiom-scheme ϕ→ �♦ϕ (6) is always satisfied. It is well known that this scheme (together with those for S4) characterizes the classical system S5. Another axiom-scheme which is always satisfied by intuitionistic S5 is ♦�ϕ→ ϕ , which corresponds to the co-unit of the adjunction, and which classically (i.e., with � = ¬♦¬) is equivalent to (6). 4 Graded unital involutive quantales Basic definitions and properties. The usual notion of grading of a ring has a straightforward counterpart for quantales, which will be useful in §5. We shall study it now. Definition 4.1 LetM be an involutive monoid. A unital involutive quantale Q is graded over M if there is an M-indexed family of sub-sup-lattices Q(m) of Q satisfying the following two properties. 1. For each a ∈ Q there is one, and only one, element (am) ∈ such that 2. The following conditions hold for all m,n ∈ M : Q(m)Q(n) ⊂ Q(mn) 1Q(ε) = e ⊂ Q(m (The latter is equivalent to = Q(m For each m ∈M the set Q(m) is called the component of Q in degree m. Example 4.2 Recall the tensor quantale of ¶2.7: if L is a sup-lattice and I is the free involutive monoid on one generator α then the tensor quantale has an obvious grading over I such that L(ε) ∼= ℘( 1) and L(α) ∼= L. The following properties are immediate: Proposition 4.3 Let Q be a unital involutive quantale graded over an invo- lutive monoid M . 1. The map (am) 7→ m∈M am is an isomorphism of sup-lattices (m) → Q . 2. If m 6= n then Q(m) ∩Q(n) = {0}. 3. The union m∈M Q (m) is join-dense in Q. 4. ↓Q(m) = Q(m) for all m ∈M . There is a convenient alternative definition if the unital involutive quan- tale Q is also a locale (an example is the quantale TL of ¶4.2 if L is a locale, or the quantale T(L) of §5): Proposition 4.4 Let M be an involutive monoid, and let Q be a unital involutive quantale which is also a locale. Then Q is graded over M if and only if there is an M-indexed family (e(m)) of elements of Q satisfying the following properties: 1. 1Q = m∈M e (m) (i.e., covers Q); 2. e(m) ∧ e(n) = 0 if m 6= n; 3. e(m)e(n) ≤ e(mn); 4. e(ε) = e; ≤ e(m ∗) (equiv., = e(m Proof. It is clear that if Q is graded then it satisfies 1–5 if we let e(m) be 1Q(m) for each m ∈M . For the converse we define the component Q (m) to be ↓e(m): then any element a ∈ Q equals a ∧ 1 = a ∧ e(m) = a ∧ e(m) where a∧ e(m) ∈ Q(m) by definition of Q(m); and if bm then, for each n ∈M we have an = an ∧ am = an ∧ an ∧ bm = an ∧ bn , and in a similar way we obtain bn = bn ∧ an. Hence, we have an = bn, and thus each element a ∈ Q can be obtained uniquely as a join am. The rest is straightforward. Graded nuclei and quotients. The natural notion of quotient that re- spects the grading of a quantale is provided by the following definition: Definition 4.5 Let Q be a unital involutive quantale graded over an invo- lutive monoid M . A nucleus j : Q→ Q is graded if it satisfies the following two conditions for all (am) ∈ m∈M Q ⊂ Q(m); m∈M am m∈M j(am). Proposition 4.6 Let j be a graded nucleus as in the definition above. Then, 1. j(0) = 0 (the nucleus is “dense”); 2. Qj is graded, with each component being defined by (m) = j Proof. The first condition is obvious from the fact that 0 ∈ m∈M Q and thus j(0) ∈ m∈M Q (m) = {0}. For the second condition notice that if a ∈ Qj then on one hand we have a unique representation of a as a join and, on the other hand, a = j(a) = j j(am) , and thus am = j(am) for all m ∈M ; that is, the element am is necessarily in (m). The rest is straightforward. The nucleus induced by a binary relation is graded provided the relation respects the grading: Proposition 4.7 Let Q be a unital involutive quantale graded over an invo- lutive monoid M . Let also R ⊂ Q×Q be a binary relation that respects the grading in the sense that R ⊂ Q(m)×Q(m). Then jR is a graded nucleus. Proof. Let ⊏R ⊂ Q×Q be the preorder defined by a ⊏R b ⇐⇒ jR(a) ≤ jR(b) . Since this is also a sub-involutive-quantale of Q×Q, let us call it a congruence preorder. By a simple adaptation of the comments at the end of the sup- lattices section of §2, there is a bijection between congruence preorders and nuclei on Q, and ⊏R is the least congruence preorder on Q which contains R. It is clear that ⊏R respects the grading because R does, and thus if a ∈ Q then jR(a) ∈ Q (m), showing that jR satisfies the first of the properties of graded nuclei. In order to prove that it also satisfies the second property let k : Q→ Q be the map defined, for each (am) ∈ Q(m), as follows: jR(am) . Since jR is monotone we have k ≤ jR: jR(am) = k Now let us see that k is itself a nucleus. First, it is obvious that it is monotone and that it satisfies a ≤ k(a) for all a ∈ Q. It is also idempotent because jR(am) and the fact that jR(am) ∈ Q (m) allows us to conclude that the right hand side of the above equation equals jR(jR(am)) = jR(am) = k Now let us prove the condition relating k to the multiplication. For each pair (am), (bm) ∈ Q(m) we have jR(am) jR(bm) = jR(ap)jR(bq) jR(apbq) The condition relating k to the involution is equally simple and we omit it. Finally, it is obvious that for (a, b) ∈ R we have k(a) ≤ k(b), since jR(a) ≤ jR(b). But, by definition, jR is the least nucleus that satisfies this condition, and therefore we conclude that k = jR. Hence, jR is graded. 5 Construction of the Lindenbaum quantales The involutive tensor quantale of a frame. Let L be a frame, and denote by I the free involutive monoid on one generator α, whose words are the finite sequences of α and α∗, and whose unit we shall denote by ε. For each w ∈ I we shall denote by L(w) the sup-lattice L⊗(|w|+1), where |w| is the length of the word w (notice the difference with respect to ¶2.7): L(ε) = L , L(α) = L(α ∗) = L⊗ L , L(αα) = L(αα ∗) = L(α ∗α) = L(α ∗α∗) = L⊗ L⊗ L , For each w,w′ ∈ I we define a map ϕw,w′ : L (w) × L(w ′) → L(ww ϕw,w′(x1 ⊗ · · · ⊗ xn, y1 ⊗ · · · ⊗ ym) = x1 ⊗ · · · ⊗ (xn ∧ y1)⊗ · · · ⊗ ym . It is easy to see that this preserves joins in each variable, and thus it defines a sup-lattice homomorphism ϕw,w′ : L (w) ⊗ L(w ′) → L(ww Defining T(L) = (not the same as TL, cf. ¶¶2.7 and 4.2), and using the distributivity of⊗ over , we obtain the following sup-lattice homomorphism T(L)⊗T(L)→ T(L), T(L)⊗ T(L) ∼= // L(w) ⊗ L(w ϕw,w′ // // T(L) , where the rightmost homomorphism is the copairing of the family of sup- lattice embeddings L(ww ′) → T(L) which is given by the universal property of the sup-lattice coproduct ′). Hence, there is a bilinear multiplication T(L)× T(L)→ T(L). It is defined on pure tensors by (x1 ⊗ · · · ⊗ xn)(y1 ⊗ · · · ⊗ ym) = x1 ⊗ · · · ⊗ (xn ∧ y1)⊗ · · · ⊗ ym , and it is straightforward to see that it is associative, hence giving us a quan- tale multiplication on T(L). This multiplication has a unit, which coincides with 1L ∈ L (ε) = L, and an involution (−)∗ : T(L)→ T(L) is obtained from the isomorphisms L(w) → L(w ∗) that are given by x1 ⊗ · · · ⊗ xn 7→ xn ⊗ · · · ⊗ x1 . Hence, T(L) is a unital involutive quantale, and it is clearly graded over I, so that we can define: Definition 5.1 The tensor involutive quantale of L is the graded unital involutive quantale T(L). There is an obvious homomorphism of involutive monoids (−) : I → T(L) that sends each word w to 1L ⊗ · · · ⊗ 1L ∈ L (w). Hence, in particular, α = 1L ⊗ 1L ∈ L (α) and α∗ = 1L ⊗ 1L ∈ L (α∗). The quantale T(L) has the following universal property: Proposition 5.2 Let Q be a unital involutive quantale such that b = b2 = b∗ for all b ∈ ↓eQ (in particular, this implies that ↓eQ is a locale). Let also h : L→ ↓eQ be a homomorphism of locales, and let a ∈ Q. Then there is exactly one homomorphism of unital involutive quantales ϑ : T(L)→ Q such that: 1. ϑ(x) = h(x) for all x ∈ L(ε) = L; 2. ϑ(α) = a. Proof. By the universal property of the coproduct of sup-lattices, every sup-lattice homomorphism ϑ with domain T(L) is uniquely determined by its value on the pure tensors of T(L). Furthermore, every pure tensor x1 ⊗ · · · ⊗ xn ∈ L with wi = α, α ∗ can be written as a product of xi ∈ L, α ∈ L α and α∗ ∈ Lα and thus if ϑ is a homomorphism of unital involutive quantales that satisfies 1 and 2 then its value is uniquely determined on all the pure tensors. This proves that if ϑ exists then it is unique. In order to prove existence, assign to each pure tensor x1 ⊗ · · · ⊗ xn ∈ L the value h(x1)a1h(x2) . . . an−1h(xn) ∈ Q where ai = a or ai = a ∗ according to whether wi is α or α ∗, respectively. This assignment preserves joins in each variable xi separately and thus it defines a sup-lattice homomorphism ϑw : L (w) → Q. The copairing ϑ = [ϑw]w∈I : T(L)→ Q is easily seen to preserve the quantale multiplication, the unit and the invo- lution, and it satisfies conditions 1 and 2. For the following we denote by Frm the full subcategory of Qu whose objects are the locales (this is usually called the category of frames [5]). Corollary 5.3 Let Que be the full subcategory of Qu whose objects are those unital involutive quantales Q such that b = b2 = b∗ for all b ≤ e in Q. Let also Que∗ be the corresponding category of pointed quantales. There is an obvious functor Que∗ → Frm that to each quantale Q assigns ↓e, and such that h 7→ h|↓e for each homomorphism h. This functor has a left adjoint which to each locale L assigns the pointed quantale (T(L), α). Bimodal frames and pointed quantales. We have already mentioned that for an ssq Q, ςQ = ↓e is a locale. It is also clear that, for α in Q, the operators ♦x = ς(αx) and �x = ς(α∗x) preserve arbitrary joins, hence they are sup-lattice endomorphisms of ςQ. Definition 5.4 We say that two sup-lattice endomorphisms of L, ♦ and �, are conjugate modalities if for all x, y ∈ L we have ♦x ∧ y ≤ ♦(x ∧ �y) , �x ∧ y ≤ �(x ∧ ♦y) . A bimodal frame (L,♦,�) is a frame L equipped with two conjugate modal- ities ♦ and �. Proposition 5.5 Let Q be an ssq, and α ∈ Q. Then (ςQ, ς(α−), ς(α∗−)) is a bimodal frame. Proof. We only have to check that the two endomorphisms are conjugate modalities. From the fact that ς is a support (4) we have: ς(αx)y ≤ αxα∗y , and thus ♦x ∧ y = ς(αx)y = ς(ς(αx)y) ≤ ς(αxα∗y) = ς(αxς(α∗y)) = ♦(x ∧ �y) , using stability and the fact that x, y ∈ ςQ. The other conjugacy condition is obtained by interchanging α and α∗. From now on we shall refer to any sup-lattice endomorphism ς : Q→ Q on a unital involutive quantale Q such that ςa ≤ e for all a ∈ Q as a pre-support of Q. Given a bimodal frame (L,♦,�), a pre-support can be easily defined on pure tensors of T(L) (and then extended to joins of these in the obvious way): if x = x0 ⊗ · · · ⊗ xn is in degree w = w1 . . . wn (with wi ∈ {α, α ∗}) then ςx = x0 ∧ 〈w1〉(x1 ∧ 〈w2〉(. . .)) , where 〈wi〉 is ♦ or � according to whether wi = α or wi = α ∗, respectively. Recursively, we have: Definition 5.6 Let x = x0 ⊗ · · · ⊗ xn ∈ L (w1...wn). Then, • ςx = x, if n = 0; • ςx = x0 ∧ 〈w1〉(ςx ′), if n ≥ 1, where x′ = x1 ⊗ · · · ⊗ xn ∈ L (w2...wn). Lemma 5.7 The following properties hold for all a, b, c ∈ T(L): 1. ςe = e and ςa ≤ e (condition 3 in the definition of support); 2. ς(ςab) = ςaςb (in particular, ςςa = ςa); 3. ς(ab) = ς(aςb) (we say the pre-support ς is stable); 4. If ♦ and � are conjugate we have (a) ςa ≤ ς(aa∗); (b) ς(ςab) ≤ ς(aa∗b); (c) ς(cςab) ≤ ς(caa∗b). Proof. 1. ς T(L) = L and e is 1L. 2. It suffices to prove this for all the pure tensors b ∈ L(v), where v is an arbitrary degree. Let b = y0 ⊗ · · · ⊗ yp ∈ L (v1...vp) . We have ςa ∈ L(ε), and thus from ¶5.6 we obtain ς(ςab) = ς((ςa ∧ y0)⊗ · · · ⊗ yp) = (ςa ∧ y0) ∧ 〈v1〉(ς(y1 ⊗ · · · ⊗ yp)) = ςa ∧ (y0 ∧ 〈v1〉(ς(y1 ⊗ · · · ⊗ yp))) = ςaςb . 3. It suffices to prove this for all the pure tensors a ∈ L(w), where w is an arbitrary degree. Let then a = x0 ⊗ · · · ⊗ xn ∈ L (w1...wn) . The proof is done by induction on n. For the base case assume that n = 0; that is, we have a = ςa = x0 ∈ L (ε), and from 2 we obtain ς(ab) = ς(ςab) = ςaςb = aςb , whence ς(ab) = ςς(ab) = ς(aςb). Now for the induction step let n ≥ 1 and let a′ = x0 ⊗ · · · ⊗ xn−1 ∈ L (w1...wn−1) . We have a = a′wnxn and thus ς(ab) = ς(a′wnxnb) = ς(a′ς(wnxnb)) (Induction hyp.) = ς(a′〈wn〉(ς(xnb))) (Def. of ς) = ς(a′〈wn〉(xnςb)) (By 2) = ς(a′ς(wnxnςb))) (Def. of ς) = ς(a′wnxnςb) (Induction hyp.) = ς(aςb) . 4. First we remark that (4a) is an instance of (4b) (which in turn is an instance of (4c)). Moreover, (4b) implies (4c) due to stability: if we assume (4b) then ς(cςab) = ς(cς(ςab)) ≤ ς(cς(aa∗b)) = ς(caa∗b) . It now suffices to prove (4b). We shall prove ς(ςab) ≤ ς(aa∗b) for the particular case where a is a pure tensor a = x0 ⊗ · · · ⊗ xn in degree w = w1 . . . wn (n ≥ 1), which implies the general case. The proof is by induction on n. Base: for n = 0, we have a = ςa = aa∗, and thus ς(ςab) = ς(aa∗b). Step: for n ≥ 1, let r be x1 ⊗ · · · ⊗ xn ∈ L (w2...wn); hence, we have a = x0w1r, and thus using stability we obtain ς(aa∗b) = ς(x0w1rr ∗w∗1x0b) = ς(x0w1rr ∗ς(w∗1x0ςb)) . By the definition of ς (and making the symbol ∧ explicit for the mul- tiplication in ς T(L)) this equals x0 ∧ 〈w1〉ς(rr ∗〈w∗1〉(x0 ∧ ςb)) , which, by the induction hypothesis, is greater or equal to x0 ∧ 〈w1〉ς(ςr ∧ 〈w 1〉(x0 ∧ ςb)) , which in turn equals x0 ∧ 〈w1〉(ςr ∧ 〈w 1〉(x0 ∧ ςb)) (because the argument of the outermost occurrence of ς was in ς T(L)). Finally, by conjugacy of the operators 〈w1〉 and 〈w 1〉 the latter is greater or equal to x0 ∧ 〈w1〉(ςr) ∧ x0 ∧ ςb = x0 ∧ 〈w1〉(ςr) ∧ ςb = ς(x0w1r) ∧ ςb = ςa ∧ ςb = ς(ςab) . The supported quantale of a bimodal frame. So far we have obtained, from an arbitrary bimodal frame, a quantale with a stable pre-support. In order to obtain an actual supported quantale we shall impose the missing properties, namely ςa ≤ aa∗ (4) and a ≤ ςaa (5), by taking quotients of T(L). Definition 5.8 Let jς be the least nucleus j on T(L) such that j(a) = j(ςaa) . We define Tς(L,♦,�) to be T(L)jς . We also write Tς(L) if ♦ and � are clear from the context. From ¶4.7 it is clear that jς is a graded nucleus, and it is the identity on L(ε) = L because for all a ∈ L we have ςaa = a. Hence, we have concluded, just from the graded structure of T(L), that the injection of generators of L into Tς(L,♦,�) is 1–1. Proving a similar fact for the other axiom, ςa ≤ aa∗, is less easy, and we shall address this now. Let (L,♦,�) be a bimodal frame and let R = {(ςa, aa∗) | a ∈ Tς(L,♦,�)} . Definition 5.9 TK(L,♦,�) is Tς(L,♦,�)jR. As in ¶5.8 we may write TK(L). We shall denote the selected point jR(jς(α)) ∈ TK(L) by α. Lemma 5.10 Recall the definition (2.5) of R. If (y, z) ∈ R then ςy ≤ ςz in Tς(L,♦,�). Proof. We have ςy ≤ ςz for all (y, z) ∈ R if and only if the following two conditions hold for all (y, z) ∈ R and all a, b ∈ Tς(L): ς(ayb) ≤ ς(azb) (7) ς(ay∗b) ≤ ς(az∗b) . (8) In order to prove these two conditions we shall show that they hold for all (y, z) ∈ R and that they are preserved by the recursive rules of construction of R. Let (y, z) ∈ R; that is, let y and z be of the form y = ςt and z = tt∗. We ς(ayb) = ς(aςtb) ≤ ς(att∗b) = ς(azb) , from ¶5.7. We similarly have ς(ay∗b) ≤ ς(az∗b) because y and z are self- adjoint. Now assume that (7)–(8) hold for some pair (y, z) ∈ Tς(L) × Tς(L). We shall prove that they equally hold for the following pairs: (i) (ςy, ςz); (ii) (y∗, z∗); (iii) (qy, qz); and (iv) (yq, zq), for all q ∈ Tς(L). (i) Since (y, z) satisfies (7)–(8) we have ςy ≤ ςz (make a = b = 1L), and thus ς(aςyb) ≤ ς(aςzb) for all a, b ∈ Tς(L), proving (7) for the pair (ςy, ςt). Since ςy and ςz are self-adjoint, we also conclude (8) for the pair (ςy, ςt). (ii), (iii) and (iv) are obvious. Theorem 5.11 The unit of the adjunction L→ TK(L) is 1–1. Proof. Equivalently, we want to prove that ς Tς(L) ⊂ QjR, or, in other words, that for all (y, z) ∈ R and x ∈ L we have z ≤ x⇒ y ≤ x . (9) Let P ⊂ R be the subset of R consisting of all those (y, z) such that (9) holds for all x ∈ L. We shall prove that R ⊂ P and that P is closed under the recursive formation rules of R, hence showing that P = R because R is the least subset of Tς(L)× Tς(L) satisfying these conditions. Let (ςy, yy∗) ∈ R. If x ∈ L and yy∗ ≤ x we conclude that y = ςy = yy∗ ∈ L due to the grading of Tς(L) over the free involutive monoid I, and thus ςy ≤ x, showing that R ⊂ P . From now on let (y, z) be a fixed but arbitrary element of P . By ¶5.10 we conclude ςy ≤ ςz. Hence, (9) holds for the pair (ςy, ςz), and thus (ςy, ςz) ∈ P . Now let q ∈ Tς(L) and assume that qz ≤ x for some x ∈ L. Then z ≤ 1L (again due to the grading over I), and thus y ≤ 1L because (y, z) ∈ P (make x = 1L). Hence, again using the previous lemma we obtain qy = qςy ≤ qςz = qz ≤ x , showing that (qy, qz) ∈ P . In a similar way we conclude that (yq, zq) ∈ P . Finally, x ≤ 1L implies that x is self-adjoint, and thus the conditions y ∗ ≤ x and z∗ ≤ x are equivalent to y ≤ x and z ≤ x, respectively, showing that (y∗, z∗) ∈ P . T, K4, S4, S5. Now let us extend our results to the systems of modal logic T, K4, S4, and S5. As was explained in §3, we shall need to impose additional conditions on the selected element α ∈ TK, such as reflexivity (α ≥ 1L), transitivity (α 2 ≤ α), etc. In order to obtain again coreflections we shall also have to impose additional axioms on the modalities ♦ and � of L. As we mentioned in §1, we shall see that for T, K4 and S4 these coincide with the well known axioms for the corresponding systems of modal logic under consideration (in other words, the same axioms still produce complete axiomatizations for the new semantics), whereas for S5 a new axiomatization is defined just by imposing that ♦ should coincide with �. The proof techniques are very similar to those employed in the previous sections for the system K. In fact we could have already presented the theory for K in such a generality so as to be able to directly reuse the results now, but this would have obscured the main ideas, so for the sake of clarity we shall single out the general results only now. Lemma 5.12 Let ρ ⊂ Tς(L) × Tς(L) be any binary relation on Tς(L), and let ρ be the closure of ρ under the rules (y, z) ∈ ρ ⇒ (ςy, ςz) ∈ ρ (y, z) ∈ ρ ⇒ (ayb, azb) ∈ ρ for all a, b ∈ Tς(L) (y, z) ∈ ρ ⇒ (y∗, z∗) ∈ ρ . Assume that for all (y, z) ∈ ρ and all a, b ∈ Tς(L) we have ς(ayb) ≤ ς(azb) ς(ay∗b) ≤ ς(az∗b) . Then for all (y, z) ∈ ρ we have ςy ≤ ςz. Proof. This follows from a simple adaptation of the proof of ¶5.10. Lemma 5.13 Let P ⊂ Tς(L)×Tς(L) be the set of all those (y, z) satisfying the following two conditions: 1. ςy ≤ ςz; 2. z ≤ x⇒ y ≤ x for all x ∈ L. Then P is closed under the rules (y, z) ∈ P ⇒ (ςy, ςz) ∈ P (y, z) ∈ P ⇒ (ayb, azb) ∈ P for all a, b ∈ Tς(L) (y, z) ∈ P ⇒ (y∗, z∗) ∈ P . Proof. The proof of this lemma is contained in the proof of ¶5.11, where P was defined to be a subset of R, but in fact the only property of R used in order to prove the closure properties of P was the fact that for all (y, z) ∈ R we have ςy ≤ ςz. The other key ingredient is the fact that ab ∈ L implies a, b ∈ L for all a, b ∈ Tς(L), due to the grading of Tς(L) over I. Theorem 5.14 Let ρ ⊂ Tς(L)×Tς(L) be a binary relation such that for all (y, z) ∈ ρ and all a, b ∈ Tς(L) we have ς(ayb) ≤ ς(azb) (10) ς(ay∗b) ≤ ς(az∗b) , (11) and let Q be the (supported) quotient of Tς(L) generated by ρ. Then the injection of generators of L onto ςQ, η : L→ ς Tς(L)→ ςQ , is an isomorphism. Proof. This is an immediate consequence of the previous two lemmas, by a reasoning analogous to that of ¶5.11. Similarly to what we have done in §3 for the Lindenbaum quantales QT, QK4, etc. (see ¶3.5), we define TT(L), TK4(L),TS4(L) and TS5(L) to be quo- tients of TK(L) by analogous defining relations: TT(L): e ≤ α TK4(L): αα ≤ α TS4(L): e ≤ α ≥ αα TS5(L): e ≤ α = α ∗ ≥ αα Corollary 5.15 Let L be a bimodal frame such that for all x ∈ L the condi- tions x ≤ ♦x and x ≤ �x hold. Then the injection of generators L→ ς TT(L) is an isomorphism. Proof. The quantale TT(L) is the quotient of Tς(L) generated by the con- ditions ςa ≤ aa∗, for all a ∈ Tς(L) , α ≥ e . Define RT ⊂ Tς(L)× Tς(L) as follows: RT = {(ςy, yy ∗) | y ∈ Tς(L)}) ∪ {(1L,α)} . All we have to do is, by the previous theorem, prove that for all (y, z) ∈ RT the conditions (10)–(11) are satisfied for all a, b ∈ Tς(L). This has already been done for the pairs of the form (ςy, yy∗) in ¶5.10, so we only have to concern ourselves with the pair (1L,α). For a, b ∈ Q we have ς(ab) = ς(aς(b)) ≤ ς(a♦(ς(b))) = ς(aς(αb)) = ς(aαb) , where we have used stability of ς twice and the inequality follows from x ≤ ♦x and monotonicity of ς; this proves (10). Then (11) is proved in a similar way using the inequality x ≤ �x. Corollary 5.16 Let L be a bimodal frame such that for all x ∈ L the condi- tions ♦♦(x) ≤ ♦(x) and ��(x) ≤ �(x) hold. Then the injection of generators L→ TK4(L) is an isomorphism. Proof. It remains to prove that the pair (y, z) = (α2,α) satisfies (10)–(11). The first condition is proved as follows: ς(aαb) = ς(aς(αb)) = ς(a♦(ς(b))) ≥ ς(a♦2(ς(b))) = ς(aα2b) . (11) is proved in the same way once we replace α by α∗ and ♦ by � in the previous argument. Corollary 5.17 Let L be a bimodal frame such that for all x ∈ L the con- ditions x ≤ ♦x, x ≤ �x, ♦♦(x) ≤ ♦(x) and ��(x) ≤ �(x) hold. Then the injection of generators L→ TS4(L) is an isomorphism. Proof. Immediate from the previous two corollaries. Corollary 5.18 Let L be a bimodal frame such that for all x ∈ L the con- ditions of the previous corollary and ♦x = �x hold. Then the injection of generators L→ TS5(L) is an isomorphism. Proof. It remains to show that the pairs (α,α∗) and (α∗,α) satisfy (10) and (11), which is done as follows: ς(aαb) = ς(aς(αb)) = ς(a♦(ς(b))) = ς(a�(ς(b))) = ς(aα∗b) . The Lindenbaum quantales. Let BK be the Lindenbaum algebra for system K, as in §3, and let BT, BK4, BS4, and BS5 be the Lindenbaum algebras for systems T, K4, S4, and S5, respectively. These are modal lattices in the following sense: Definition 5.19 By a modal lattice is meant a bounded distributive lattice L equipped with an endomap ♦ that preserves finite joins. A bimodal lattice is a modal lattice equipped with another endomap � that preserves finite joins and in addition satisfies conjugacy relations similar to those of bimodal frames: ♦x ∧ y ≤ ♦(x ∧ �y) , �x ∧ y ≤ �(x ∧ ♦y) . The category Lat♦ of modal lattices has the modal lattices as objects and the homomorphisms of bounded lattices that preserve ♦ as morphisms. The category Lat♦� of bimodal lattices is defined analogously, with objects being the bimodal lattices and the morphisms being the homomorphisms of modal lattices that also preserve �. We shall also refer to a modal lattice as a • T-modal lattice if x ≤ ♦x for all x; • K4-modal lattice if ♦♦x ≤ ♦x for all x; • S4-modal lattice if it is both a T-modal lattice and a K4-modal lattice; • S5-modal lattice if it is an S4-modal lattice and ♦x∧ y ≤ ♦(x∧♦y) for all x and y (i.e., ♦ is conjugate to itself). The categories Lat♦T, Lat K4, Lat S4, and Lat S5 are, respectively, the full sub- categories of Lat♦ whose objects are the T-modal lattices, the K4-modal lattices, the S4-modal lattices, and the S5-modal lattices. For bimodal lattices we adopt a similar terminology: a bimodal lattice is referred to as a • T-bimodal lattice if x ≤ ♦x and x ≤ �x for all x; • K4-bimodal lattice if ♦♦x ≤ ♦x and ��x ≤ �x for all x; • S4-bimodal lattice if it is both a T-bimodal lattice and a K4-bimodal lattice; • S5-bimodal lattice if it is an S4-bimodal lattice and ♦x = �x for all x. The categories Lat♦�T , Lat K4, Lat S4 , and Lat S5 are, respectively, the full subcategories of Lat♦� whose objects are the T-bimodal lattices, the K4- bimodal lattices, the S4-bimodal lattices, and the S5-bimodal lattices. By standard universal algebra the forgetful functor Lat♦� → Lat♦ has a left adjoint which assigns to BK its “enveloping” bimodal lattice B K. Simi- larly, there are left adjoints Lat♦T → Lat Lat♦K4 → Lat Lat♦S4 → Lat and we write B′T, B K4, and B S4 for the respective images of BT, BK4, and BS4 under the left adjoints. For S5 the situation is simpler because the categories Lat♦S5 and Lat S5 are obviously isomorphic, since any S5-modal lattice becomes an S5-bimodal lattice just by defining � to coincide with ♦. We shall also write B′S5 for BS5 thus regarded as a bimodal lattice. Since ♦ and � preserve finite joins they can be extended canonically to sup-lattice endomorphisms of the ideal completion Idl(B′K), which is a frame because B′K is a distributive lattice. The conjugation relations are easily seen to be inherited from those of B′K, and thus Idl(B K) is a bimodal frame. Similar remarks apply to the other Lindenbaum algebras, and in addition Idl(B′T) satisfies the axioms of a T-bimodal lattice, Idl(B K4) satisfies the axioms of a K4-bimodal lattice, etc. (Hence, in particular, the propositions in ¶¶5.15–5.18 can be applied to Idl(B′T), Idl(B K4), Idl(B S4), and Idl(B respectively.) Summarizing, we have described a way of constructing functors from modal lattices to bimodal frames which are left adjoint to the obvious forget- ful functors. Composing these functors with the left adjoints from bimodal frames to pointed ssqs we obtain fromBK, BT, BK4, BS4, andBS5 pointed ssqs TK(Idl(B K)), TT(Idl(B T)), TK4(Idl(B K4)), TS4(Idl(B S4)), and TS5(Idl(B S5)), respectively. For each system S ∈ {K,T,K4, S4, S5} the map obtained by composing the following arrows (the leftmost one is just the natural quo- tient), BK → BS → B S → Idl(B S)→ TS(B has the same universal property as the injection of generators BK → QS. Hence, the quantale TS(B S) is a particular construction of the Lindenbaum quantale QS: Theorem 5.20 ∼= TK(Idl(B ∼= TT(Idl(B ∼= TK4(Idl(B ∼= TS4(Idl(B ∼= TS5(Idl(B Since, as we have remarked above, S5-modal lattices and S5-bimodal lattices are “the same”, our results immediately tell us that the unit of the adjunction between S5-modal lattices and pointed ssqs, BS5 → QS5 , is a monomorphism. In logical terms this means that a complete axiomati- zation for the system S5 (with the advantage of making no use of negation or the modal necessity operator) can be as follows: Theorem 5.21 S5 is complete for the following axiom schemata: ϕ → ♦ϕ ♦♦ϕ → ♦ϕ ♦ϕ ∧ ψ → ♦(ϕ ∧ ♦ψ) . We have not verified whether the remaining canonical mappings BK → B BT → B BK4 → B BS4 → B are monomorphisms (although we believe they are). This means that we have not verified completeness for the classical axiomatizations of K, T, K4, and S4. Of course, by this we mean we have not verified this in an arbitrary topos, for otherwise we know, from the classical completeness theorems of propositional normal modal logic, that the axiomatizations are complete: using Zorn’s Lemma we can find a Kripke structure (W,R) that gives us a monomorphism BK → B K → TK(Idl(B ∼= QK → ℘(W ×W ) implying that BK → B K is 1–1, and the same applies to T, K4, and S4. References [1] D. Harel, D. Kozen, J. Tiuryn, Dynamic Logic, MIT Press, 2000. [2] E.A. Emerson, Temporal and modal logic, in J. van Leeuwen (editor), Handbook of Theoretical Computer Science, vol. B, MIT Press, 1990, pp. 955–1072. [3] G.E. Hughes, M.J. Cresswell, An Introduction to Modal Logic, Methuen & Co. Ltd., London, 1968. [4] G.E. Hughes, M.J. Cresswell, A Companion to Modal Logic, Methuen & Co. Ltd., London, 1984. [5] P.T. Johnstone, Stone Spaces, Cambridge Stud. Adv. Math., vol. 3, Cambridge Univ. Press, 1982. [6] A. Joyal, M. Tierney, An Extension of the Galois Theory of Grothendieck, Mem. Amer. Math. Soc., vol. 309, American Mathemat- ical Society, 1984. [7] M.V. Lawson, Inverse Semigroups — The Theory of Partial Symmetries, World Scientific, 1998. [8] I. Moerdijk, J. Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge University Press, 2003. [9] C.J. Mulvey, Quantales, in M. Hazewinkel (editor), The Encyclopaedia of Mathematics, third supplement, Kluwer Academic Publishers, 2002, pp. 312–314. [10] C.J. Mulvey, P. Resende, A noncommutative theory of Penrose tilings, Internat. J. Theoret. Phys. 44 (2005) 655–689. [11] J. Paseka, J. Rosický, Quantales, in B. Coecke, D. Moore, A. Wilce, (editors), Current Research in Operational Quantum Logic: Algebras, Categories and Languages, Fund. Theories Phys., vol. 111, Kluwer Aca- demic Publishers, 2000, pp. 245–262. [12] A.L.T. Paterson, Groupoids, Inverse Semigroups, and Their Operator Algebras, Birkhäuser, 1999. [13] P. Resende, Tropological systems are points of quantales, J. Pure Appl. Algebra 173 (2002) 87–120. [14] P. Resende, Étale groupoids and their quantales, Adv. Math. 208 (2007) 147–209. [15] K. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Mathematics Series 234, Longman Scientific & Technical, 1990. Centro de Lógica e Computação Instituto Superior Técnico Universidade Técnica de Lisboa Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal E-mail: sergiortm@gmail.com Centro de Análise Matemática, Geometria e Sistemas Dinâmicos Departamento de Matemática do Instituto Superior Técnico Universidade Técnica de Lisboa Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal E-mail: pmr@math.ist.utl.pt Introduction Preliminaries Quantale semantics of modal logic Graded unital involutive quantales Construction of the Lindenbaum quantales

60K plateau of Tc (x) in YBa2Cu3O6+x, critical chain length Origin of the 60K plateau in YBa2Cu3O6+x V. M. Matic and N. Dj. Lazarov Laboratory of Theoretical Physics, Institute of Nuclear Sciences “Vinca”, 11001 Belgrade, Serbia Abstract A model for charge transfer mechanism in YBa2Cu3O6+x high-Tic cuprate based on critical chain length concept is proposed to account for 60K and 90K plateaus in Tc(x) dependence. It has been shown, when the statistics of CuO chain formation was described in terms of two dimensional asymmetric next-to-nearest neighbor Ising (ASYNNNI) model, that at any constant temperature below the top of OII phase there exists a uniquely defined value of critical chain length lcr(T) that yields a constant doping p(x)≈const over the regime of OII phase (related to 60K plateau of Tc(x)), while 90K plateau coincides with the monotonously increasing p(x) over optimal doping level p=0.16 in the regime of OI phase. Short length chains (l<lcr(T)), together with the first lcr(T)-2 holes in longer chains (l≥lcr(T)), are taken as not capable of attracting electrons from CuO2 planes. It is shown that only a part (≈41%) of the remaining l-lcr(T)+1 holes in the long chains can capture electrons. The results obtained indicate that the ASYNNNI model and two- plateaus-like behavior of Tc(x) in YBa2Cu3O6+x are closely connected. PACS: 74.72Bk, 64.60.Cn, 81.30Dz CuO2 layers are key ingredients in all high-Tc cuprates given the fact that superconductivity occurs in these materials when a part of 3d copper electrons, usually between 5% and 27%, is taken away from the layers. The missing electrons are commonly referred to as “holes” which can move throughout the layers and make the material superconducting if the temperature is low enough. The electrons are conventionally subtracted from the layers either by chemical substitution of interlayer metal atoms, as for example, substitution of La2+ by Sr3+ in La1-x(Sr)xCu2O4 and Y2+ by Ca3+ in Y1-b(Ca)bBa2Cu3O6+x, or by pumping oxygen into the material. Oxygen is introduced into separate layers in which it orders to form CuO chains and it is these chains that are known to act as efficient attractors of electrons from the CuO2 layers. The number of created holes per Cu atom is typically denoted as “doping” p and the notion that CuO2 layers have an unavoidable role in the onset of superconductivity is further corroborated by the fact that a number of important physical characteristics, as, for example, the pseudogap energy Eg and critical transition temperature Tc, are coupled to p by universal relations that are common to practically all high-Tc cuprates. Thus, it has been obtained empirically that the Tc is scaled with Tc,max (maximal transition temperature) through the following, approximately parabolic, function of the hole concentration [2] ( )[ ]2max, 16.06.821)( −−= pTpT cc , (1) which has its onset, maximum and termination at p=0.05, 0.16 (optimal doping) and 0.27, respectively. YBa2Cu3O6+x superconductor has probably been the most thoroughly studied compound of all high-Tc cuprates because it has a relatively simple synthesis route and it was the first superconductor discovered with Tc above the liquid nitrogen temperature. The Tc changes in nonlinear manner with oxygen composition revealing two well-known plateaus at 60K and 90K. These features are clearly associated with the Ortho-II and Ortho-I phases, respectively [3]. While the 90K plateau is in fact a broad maximum at x≈0.92 that is associated with transition from underdoped to overdoped regime, the origin of 60K plateau at 0.5<x<0.7 is not yet well understood [3-5] (although, in fact, some advance has recently been made along this line [6]). One popular explanation is that when the oxygen content is increased over x=0.5, where Ortho-II phase emerges in the form of alternating columns of fully occupied and empty oxygen sites (directed along b axes), additional oxygen fills the empty columns making a relatively small contribution to hole doping, inasmuch as they are far apart from each other and only a small fraction of them are able to form CuO chains that are long enough to initiate the charge transfer process. It has often been guessed that there might be a certain minimal (critical) chain length lcr defined so that only chains of length that is equal to, or greater than, lcr can effectively attract electrons from CuO2 layers [3-6]. However, even though the existence of Ortho-II phase was resolved a long ago in terms of the classical two dimensional asymmetric next nearest neighbor Ising (ASYNNNI) model [7], no convincing explanation has been provided over the last 20 years as to why exactly the concentration p(x) of holes induced in CuO2 sheets would remain constant when x changes within the Ortho-II phase regime, nor, if the critical chain length concept is presupposed, what the value of lcr would be equal to and how it can be determined. Although it is well known that high-Tc cuprates are complex quantum many-body systems with the pairing mechanism still remaining controversial, here we unambiguously grounds that it is the classical ASYNNNI model combined with the concept of minimal chain length (needed for charge transfer to take place) that accounts for constant doping at p≈0.094 in the region of 60K plateau, as well as for the broad maximum of Tc at optimal doping (p=0.16) at x≈0.92. There is a general agreement that copper in the chain (basal) plane can be either Cu1+, which is the case when it is not coupled to the in-plane oxygen (but connected only to two apical O(4) ions and therefore 2-fold coordinated), or Cu2+, when it is inserted within a CuO chain (4-fold coordinated), or at the chain end (3-fold coordinated). If isolated oxygen is introduced into the basal plane it then takes away two electrons from the two nearest neighbor Cu1+ transferring them into Cu2+ state. Thus, isolated oxygen does not have a tendency to attract an additional electron from other the parts of the system. When another oxygen is added to make a chain with two O atoms (chain of length l=2) there is only one electron available from its nearest neighbor copper coordination and the absence of another electron, needed for oxygen to become O2-, is usually referred to as a “hole”. In this way, a chain of length l is seen as to have created l- 1 chain-holes, which in principle can attract electrons from the other parts of the system, presumably from CuO2 layers. Since the state of quasi one dimensional electronic subsystem of a chain of a given length (say, l), which reflects the charge transfer effectiveness of the chain, is not expected to depend upon history of chain formation, one is free to assume that the chain has been formed by adding oxygen one by one as that would allow to shadow the evolution of charge transfer process as a function of l. Within the concept of critical chain length no charge transfer is supposed to occur unless l=lcr so that the first lcr-2 initially created holes will stay inactive in attracting electrons from the planes. As l further increases beyond lcr the transfer of charge is set on during which process the remaining l- lcr+1 holes are created. Therefore, our strategy for counting doping is, first of all, to discard contribution not only of all holes in short chains (l<lcr), but also of the first lcr-2 holes in long chains (l≥lcr), for they had been created before any charge transfer took place (we shortly denote these holes as passive holes). It then naturally evolves that the number of attracted electrons (transferred holes) from the chain of length l≥lcr should be as greater as more holes were created after the charge transfer process had been triggered (at l=lcr), i.e. it should be proportional to l-lcr+1 (we call these holes the active holes for their occurrence coincides with the development of charge transfer). In the case of Ortho- II phase, the concentration of active holes h (the number of active holes per Cu) is given +−++−= ∑ ∑ =cr crll ll crcr lfllnlfllnh )()1()()1(4 2211 αααα , (2) where nα1 and nα2 denote the fractions of 3-fold coordinated Cu ions on two different sublattices of oxygen sites (usually denoted as α1 and α2), and fα1(l) and fα2(l) are corresponding fractions of CuO chains of the same length l. If NCu and n=(nα1+nα2)/2 are total number and the fraction of 3-fold coordinated Cu in basal plane, then (n/2)NCu is the total number of CuO chains and the number of passive holes per Cu is surely not greater than (n/2)(lcr-2), given the fact that no one of chains with l<lcr has more than lcr-2 holes. At x=1 (OI stoichiometry) chains are very long, virtually infinite, and there are just a few chain ends in the system. This means that n tends to zero as x approaches to 1, so that the concentration of passive holes becomes negligible whatever the value of lcr. This in turn implies that practically each chain ordered oxygen has created one active hole, i.e. h≈1 at x≈1. Since, in the YBa2Cu3O6+x system, one chain plane supplies holes to two CuO2 layers, the doping would have been equal to p=0.5 if each active hole had succeeded to capture one electron. Experimental findings, however, clearly contradict such a scenario for it was found that p(x≈1) only slightly overshoots 19% [2]. Thus, at least at x≈1, it can be stated p=(χ/2)h where χ (≈40%), as defined by the number of effectively attracted electrons (transferred holes) per active hole, reflects the capability of an active hole to capture electron. On the other hand, long (infinite) chains also prevail at x=0.5 (OII stoichiometry) on every even column of oxygen sites. The concentrations of passive holes is also negligible here, bur h cannot be greater that 0.5; in fact, h can be only less than 0.5 due to thermally activated chain fragmentation (for example, one might expect that h≈0.48, as at x≈0.5, so in the region of the 60K plateau). Given the fact that 60K plateau of Tc, according to (1), corresponds to doping level p≈0.094 it appears that it is the same fraction (of nearly 40%) of active holes that is transferred to the layers not only at x≈1, but also at x≈0.5. The charge transfer model that we propose here assumes that the same percentage of active holes is transferred, not only at stoichiometries x=0.5 and x=1, at which long CuO chains are known to dominate, but as well at off-stoichiometry 0.5<x<1 (and also at x<0.5) where chain fragmentation is more intense. We therefore propose that doping is connected to the active hole concentration h, as given by (2), by p=(χ/2)h (χ≈40%) throughout the whole range of oxygen concentration 0<x<1. The quantity χ introduced in this way should be perceived as average capability of an active hole to capture an electron from the CuO2 planes that lie above, or below, the basal plane (averaging is done over all chains in the system, or equivalently, over the whole volume of the material). To calculate the hole concentration h (eq. (2)) and doping p=χh/2, at a given point of (x,T) space, it is necessary to determine the fractions of the 3-fold coordinated Cu and length distributions of CuO chains fα1(l) and fα2(l). We used Monte Carlo (MC) method applied to the ASYNNNI model to calculate these quantities for it is known that the model stabilizes both structures Ortho-II and Ortho-I that are responsible for 60K and 90K plateaus [9]. Although ASYNNNI model cannot stabilize other structures with longer periodicities along a axis, like Ortho-III, Ortho-IV and Ortho-V, their superstructure reflections have already been reported to be much weaker than those of the main phases [8,9], so that they are thought to appear only as small patches embedded in large domains of main phases [10]. Besides, since these structures were mainly observed at oxygen compositions that correspond to transition region between the two plateaus it implies that the ASYNNNI model alone should account for both plateaus of Tc(x), especially given the fact that none of these structures, except Ortho-II, was reported at x<0.62 [9]. The chain length distributions fα1(l) and fα2(l) were determined in the following way: In each MC step we counted the total numbers of chains Nα1 and Nα2, on sublattices α1 and α2 , respectively (Nα1 and Nα2 are in fact equal to one half of unlike V2 bonds on the corresponding α sublattices), as well as the numbers of chains of the same length, Nα1(l) and Nα2(l), for lengths ranging from l=1 to l=300. The ratios Nα1(l)/Nα1 and Nα2(l)/Nα2 were then equilibrated through the MC process and the so obtained values were finally assigned to fα1(l) and fα2(l). The MC calculations were performed using single-spin-flip Glauber dynamics, where the oxygen concentration x is a functions of temperature T and chemical potential μ. We have studied lattices with periodic boundary conditions that consisted of 400x400 oxygen chain sites (O(1) sites, that split into two nonequivalent sublattices α1 and α2, in OII phase), and as many sites on β sublattice (O(5) sites). One MC step included flipping of all 2X(400X400) lattice spins and one MC run (at a particular point (x, T)) typically consisted of 3·104 to 5·104 MC steps, where only every tenth was used to calculate chain length distributions fα1(l) and fα2(l), l=1,2, …,300, and other relevant quantities (oxygen sublattice occupancies x1 and x2, 3-fold Cu fractions n1 and n2, etc.). At a certain number of points we have even used a really large number of MC steps, ranging from 105 to 3·105. At all calculated points of (x,τ) space (τ is a quantity that scales with T according to τ=kBT/VB 1, where kBB is Boltzman constant and V1 the nearest neighbor O-O interaction of the ASYNNNI model) it was obtained [11] 2,1, iaviav α , (3) where lav,αi (i=1,2) denotes the average chain length on the corresponding sublattice (α1 or α2). Such a behavior of length distributions fα1(l) and fα2(l) was subsequently explained theoretically analyzing microscopic features of the ASYNNNI model lattice configurations [12]. In brief terms, the l dependence of probability of a chain to have particular length l can be derived in the following way: Consider a sequence of Ny oxygen chain sites that are aligned along b axis (Ny is a large number, and the sites are connected by copper mediated V2<0 bonds). Let xNy denotes the number of oxygen atoms on this column of α sites and let nNy stands for the number of unlike V2 bonds (n is therefore the fraction of 3-fold coordinated Cu along the column, and, consequently, 2x/n equals to the corresponding average chain length, lav). These xNy oxygen atoms are generally divided into (n/2)Ny groups (chains) that can have various lengths l=1,2,… . It is useful to recall that an each chain has two ends: one that is oriented towards positive side of b axes (the “positive” end) and the other one, oriented towards negative b axes (the “negative” end). Among these xNy oxygen atoms there are (n/2)Ny of them that are located at the positive chain end, and thus the probability for an oxygen to be lying at the positive chain end is equal to ω=n/2x={lav)-1. Consequently, 1-ω is probability for oxygen to be located either within the interior of the chain, or at the negative chain end. Assuming that chains are created by adding oxygen one by one, starting from the negative end, one arrives at the conclusion that the probability for obtaining chain of length l is equal to f(l)=ω(1-ω)l-1 [11,12] {such a form of f(l) dependence is known in the theory of probability as “geometric” probability distribution [13]). It should be noted, however, that a deeper analysis shows that the above reasoning applies only if fluctuations of energy of the ASYNNNI model are not too large [12]. Indeed, a certain deviation from linear behavior of ln[f(l)] versus l dependence has been found in the vicinity of the second order Ortho-I-to-Ortho-II phase transition curve (at x>0.5) [12] but, fortunately, such departures were observed only in a relatively narrow intervals Δx≈0.07 around critical points. Furthermore, our extended analysis (not shown here) shows that in the critical regime these deviations, of calculated fα1(l) and fα2(l) dependences from the expected behavior, were in a certain way compensated by summations in (2), so that the calculated h(x) dependences were obtained to vary smoothly over the transition region at all τ=const. The so obtained values of length distributions fα1(l) and fα2(l) for l=1,…,300 were inserted into (2) to calculate concentration h of active holes at different points (x,T). The geometric-like behavior of f(l) dependences ensures rather fast convergence of sums in (2). It should be mentioned, however, that the specific form of length distributions (3) makes it possible, instead of evaluating summations in (2) by first 300 terms, to transform each of sums into a closed analytical form, so that h would be connected through analytical expression with average oxygen occupancies, x1 and x2, the 3-fold Cu fractions n1 and n2, and the parameter lcr. Whatever the approach we used the calculated values of h were obtained to be practically indistinguishable one from the other (even in the critical region of the ortho-I/ortho-II transition), but we nevertheless gave advantage to calculating the first 300 terms in estimating sums in (2), as we wanted to keep under control the departures of length distributions from (3) that are known to occur in the critical regime [12]. In addition, at each point of (x,T) space h was calculated for the whole range of values of cutoff parameter lcr spanning from lcr=1 to lcr=50, so that h can be regarded as a function of three variables, i.e. h=h(x,τ,lcr) (aside from the fact that it also depends upon input parameters that define the ASYNNNI model, i.e. on O-O interactions V1>0 (nearest neighbor) and V2<0, V3>0 (next nearest neighbor)). There is one remarkable feature of the hole concentration h=h(x,τ,lcr), as defined by (2), that we have found while thoroughly analyzing its behavior: whatever the magnitudes of interactions V1, V2, and V3, when h is considered as a function of lcr at different points (xi,τ) that correspond to the same τ=const and oxygen concentrations xi, i=1,2,… spanning over the region of Ortho-II phase, all of these hxi(lcr), i=1,2,… functions intersect at a single, well defined value of lcr. This is shown in Figure 1 for interactions obtained by linear-muffin-tin orbital (LMTO) method [14] and at three different temperatures τ=0.45, 0.38 and 0.30, but the similar behavior we have also obtained (not shown here) for the so-called “canonical” interactions V2=-0.5V1, V3=0.5V1. From Figure 1 it can be clearly seen that at a given τ=const, the value of lcr at which all hxi(lcr) curves intersect depends on temperature in the way that it increases with the temperature decrease. Thus, at three temperatures τ=0.45, 0.38, 0.30, the intersection values were found to be lcr=4(5), 6(7), 11(12), respectively. At a given temperature, the so obtained intersection value of lcr we name “the optimal minimal (critical) length” (denoted by lcr,opt(τ)) for it is the value at which h(x) stays constant over the regime of Ortho-II phase. Such behavior of h(x) at τ=const, for the corresponding lcr,opt(τ), is shown in Figure 2 at two temperatures: τ=0.45 (Figures 2a and 2b, for lcr,opt(τ)=4 and 5, respectively) and τ=0.30 (Figure 2c, for lcr,opt(τ)=12). From these results it can be seen that indeed h(x) demonstrates a constant section at x>0.5 that is even more pronounced at lower temperatures. Calculated h(x) dependences were used to obtain doping versus x dependences, p(x)=χh(x)/2, that were then inserted into (1) to yield corresponding Tc(x)s (Figures 3a-c). The parameter χ was varied slightly around its expected value ≈40% [2] to achieve a better correlation between the so obtained Tc(x)’s and those from experiments [15] (shown by a solid line). We have indeed obtained h in the plateau regime to be slightly lower than 0.5: h=const=0.467 (0.450) at τ=0.45 for lcr=4 (lcr=5), h=const=0.483 at τ=0.38 for lcr=7, and h=const=0.495 at τ=0.30 for lcr=12. This gives χ=40.33%, 41.84%, 38.98%, 3804% for h=0.467, 0.450, 0.483, 0.495, respectively. Thus, not only experimental data on doping at x≈1 [2], but also the analysis at x≈0.5 (that falls into the regime of 60K plateau) both infer that χ is lying at some point between ≈38% and ≈42%. It should be noted that this result disagrees with estimations of Gawiec et al. [16,17] that long chains release up to 70% of their holes. Although we used here the ASYNNNI model interactions V1, V2, and V3 as obtained by Sterne and Wille [14] the issue of magnitudes of these interactions is still open so that new values were subsequently suggested [10]. Regardless of that, it seems very well established that nearest neighbor interaction V1 should be ranking around 6.9mRy [10,14], which fixes scaling between T and τ to Δτ≈0.1⇔ ΔT≈100K. On the other hand, as one of the most important features of YBa2Cu3O6+x phase diagram is the location of the top of Ortho-II phase along τ axis (at x≈0.5) and that it may well be affected by the magnitudes of V2 and V3 (which in fact are not known precisely), perhaps the best strategy to estimate the reduced temperature τ corresponding to room temperature is to determine the “distance” (in units of τ) between room temperature and the top of Ortho-II phase. According to experimental data the top of Ortho-II phase corresponds to ≈125-140ºC [5,9], while theoretically obtained phase diagram for the LMTO interactions [18] points at τ≈0.58, thus making τ≈0.45 a fairly reliable estimation of room temperature. The established correlation between room temperature and τ=0.45 renders lcr,opt(τ) (that plays the role of lcr in (2)) 4, or 5 (Figure 1a). A better estimation seems to be 5 for it yields a somewhat more pronounced 60K plateau than lcr=4 (Figures 3a and 3b), despite the fact that lcr=4 would be closer to lcr=3 that was proposed in some theoretical studies [16,17,19]. Besides, lcr=5 appears to be well correlated with χ≈2/5 and with the basic idea lying in the background of the minimal (critical) chain length concept suggesting that one isolated hole cannot efficiently attract an electron, but only a combined effect of several holes can achieve this goal. Both lcr=5 and χ≈2/5 imply that 3 chain-holes are still not enough to effectuate charge transfer, but that the joint impact of 5 holes suffices to attract two electrons (one per each CuO2 layer). In summary, despite the well known fact that in such highly correlated electron systems, as are the high-Tc cuprates, the nature of the controversial pairing mechanism is genuinely quantum mechanical, we have shown here that certain aspects of their behavior can be explained in terms of classical models. Such is the classical ASYNNNI model that successfully accounts for unusual two-plateaus-like behavior of Tc(x) in YBa2Cu3O6+x. The obtained Tc(x) dependence is in a remarkable correlation with experiment [15] for both lcr=4 and lcr=5, although we believe the later value is more realistic. It should be also pointed out that the presented results on Tc(x) dependence are in a qualitative agreement with some previous results on the same topic [20,21]. Generally, it can be expected that the capability of an active hole to attract an unpaired 3d electron, as expressed by the value of χ, should depend upon the density ρe of available electrons immediately above (below) the chain (aside from the fact that χ should also depend on a certain coupling between chains and planes). Our recently obtained results on Tc(x) in Y1- b(Ca)bBa2Cu3O6+x (b=0.2) system [22] seem to be lying along this line, since obtained χ≈33% can be understood in the light of the fact that introduction of ≈20% of Ca has additionally increased doping, and therefore reduced ρe, so that χ attained a lesser value than in the parent YBa2Cu3O6+x compound. Acknowledgements This work has been funded by the Serbian Ministry of Science and Technology through the Project 141014. References [1] G. V. M. Williams, J. L. Tallon, R. Michalak and R. Dupree, Phys. Rev. B 54, 6909 (1996). [2] J. L. Tallon, C. Bernhard, H. Shaked, R. L. Hitterman and J. D. Jorgensen, Phys. Rev. B 51, 12911 (1995). [3] Z. Yamani, W. A. MacFarlane, B. W. Statt, D. Bonn, R. Liang and W. N. Hardy, Physica C 405, 227 (2004). [4] R. Liang, D. A. Bonn and W. N. Hardy, Phys. Rev. B 73, 180505 (2006). [5] F. Yakhou, J. Y. Henry, P. Burlet, V. P. Plakhty, M. Vlasov and S. Moshkin, Physica C 333, 146 (2000). [6] V. M. Matic and N. Dj. Lazarov, Solid State Commun. 142, 165 (2007) (cond- mat/0611214). [7] L. T. Wille and D. de Fontaine, Phys. Rev. B 37, 2227 (1988). [8] Z. Islam, X. Liu, S. K. Sinha, J. C. Lang, S. C. Moss, D. Haskel, G. Srajer, P. Wochner, D. R. Lee, D. R. Haeffner and U. Welp, Phys. Rev. Lett. 93, 157008 (2004). [9] M.v. Zimmermann, J. R. Schneider, T. Frelo, N. H. Andersen, J. Madsen, J. Kall, H. F. Poulsen, R. Liang, P. Dosanjih and W. N. Hardy, Phys. Rev. B 68, 104515 (2003). [10] D. J. Liu, L. T. Einstein, P. A. Sterne and L. T. Wille, Phys. Rev. B 52, 9784 (1995). [11] V. M. Matic, N. Dj. Lazarov, V. Spasojevic, M. Milic and V. Kusigerski, Physica C 421, 49 (2005). [12] V. M. Matic and N. Dj. Lazarov, Physica C 443, 49 (2006). [13] M. J. Merkle and P. M. Vasic, “Probability and Statistics”, Edited by Elektrotechnic Faculty of Belgrade University (1998) (in Serbian). [14] P. A. Sterne and L. T. Wille, Physica C 162-164, 223 (1989). [15] J. D. Jorgensen, M. A. Beno, D. G. Hinks, L. Soderholm, K. J. Volin, R. L. Hitterman, J. D. Grace, J. K. Schulle, C. U. Segre, K. Zhang and M. S. Kleefisch, Phys. Rev. B 36, 3608 (1987). [16] P. Gawiec, D. R. Grempel, A. C. Riiser, H. Haugerud and G. Uimin, Phys. Rev. B 53, 5872 (1996). [17] P. Gawiec, D. R. Grempel, G. Uimin and J. Zittartz, Phys. Rev. B 53, 5880 (1996). [18] V. M. Matic, Physica A 184, 571 (1992). [19] K. Miitsen and O. M. Ivanenko, JETP Letters 82, 129 (2005). [20] N. Rezlesku, C. Buzea and C. Gh. Buzea, Physica C 247 (1995) 105. [21] C. Buzea and T. Yamashita, Physica C 357 (2001) 288. [22] V. M. Matic, N. Dj. Lazarov and I. M. Bradaric, arXiv:0705.0299 (to be submitted). Figure Captions Figure 1. Calculated h(lcr) dependences at τ=const for several values of oxygen composition x that span the range of OII phase: a) τ=0.45=const; 0.52<x<0.61 (shown at the top), b) τ=0.38=const; 0.52<x<0.62 (shown in the middle section), and c) τ=0.30=const; 0.52<x<0.67 (shown at the bottom). Figure 2. Calculated values of h as a function of x at τ=const: a) τ=0.45=const, for lcr=4 (shown at the top), b) τ=0.45=const, for lcr=5 (shown in the middle section), and c) τ=0.30=const, for lcr=12 (shown at the bottom). Figure 3, Calculated Tc(x) dependences, using h(x) dependences from the Figure 2 and thereafter obtained p(x)=χh(x)/2 dependences, that were then inserted into (1): a) τ=0.45=const, for lcr=4 and χ=39.4% (shown at the top), b) τ=0.45=const, for lcr=5 and χ=41% (shown in the middle section), and c) τ=0.30=const, for lcr=12 and χ=37.6% (shown at the bottom). -2 0 2 4 6 8 10 12 14 τ=0.30, 0.52<x<0.67 x=0.52 x=0.54 x=0.56 x=0.57 x=0.59 x=0.62 x=0.67 x=0.52 x=0.53 x=0.55 x=0.56 x=0.58 x=0.59 x=0.61 τ=0.45, 0.52<x<0.61 τ=0.38, 0.52<x<0.62 x=0.52 x=0.53 x=0.54 x=0.55 x=0.57 x=0.59 x=0.62 Figure 1 0.0 0.2 0.4 0.6 0.8 1.0 τ=0.45, l τ=0.45, l τ=0.30, l Figure 2 0.0 0.2 0.4 0.6 0.8 1.0 τ=0.45, l χ=0.394 τ=0.45, l χ=0.410 τ=0.45, l χ=0.376 Figure 3

A model for charge transfer mechanism in YBa2Cu3O6+x high-Tc cuprate based on critical chain length concept is proposed to account for 60K and 90K plateaus in Tc(x) dependence. It has been shown, when the statistics of CuO chain formation was described in terms of two dimensional asymmetric next-to-nearest neighbor Ising (ASYNNNI) model, that at any constant temperature below the top of OII phase there exists a uniquely defined value of critical chain length lcr(T) that yields a constant doping p(x)=const over the regime of OII phase (related to 60K plateau of Tc(x)), while 90K plateau coincides with the monotonously increasing p(x) over optimal doping level p=0.16 in the regime of OI phase. Short length chains (l<lcr(T)), together with the first lcr(T)-2 holes in longer chains (l>lcr(T)), are taken as not capable of attracting electrons from CuO2 planes.. It is shown that only a part (41%) of the remaining l-lcr(T)+1 holes in the long chains can capture electrons. The results obtained indicate that the ASYNNNI model and two-plateaus-like behavior of Tc(x) in YBa2Cu3O6+x are closely connected.

60K plateau of Tc (x) in YBa2Cu3O6+x, critical chain length Origin of the 60K plateau in YBa2Cu3O6+x V. M. Matic and N. Dj. Lazarov Laboratory of Theoretical Physics, Institute of Nuclear Sciences “Vinca”, 11001 Belgrade, Serbia Abstract A model for charge transfer mechanism in YBa2Cu3O6+x high-Tic cuprate based on critical chain length concept is proposed to account for 60K and 90K plateaus in Tc(x) dependence. It has been shown, when the statistics of CuO chain formation was described in terms of two dimensional asymmetric next-to-nearest neighbor Ising (ASYNNNI) model, that at any constant temperature below the top of OII phase there exists a uniquely defined value of critical chain length lcr(T) that yields a constant doping p(x)≈const over the regime of OII phase (related to 60K plateau of Tc(x)), while 90K plateau coincides with the monotonously increasing p(x) over optimal doping level p=0.16 in the regime of OI phase. Short length chains (l<lcr(T)), together with the first lcr(T)-2 holes in longer chains (l≥lcr(T)), are taken as not capable of attracting electrons from CuO2 planes. It is shown that only a part (≈41%) of the remaining l-lcr(T)+1 holes in the long chains can capture electrons. The results obtained indicate that the ASYNNNI model and two- plateaus-like behavior of Tc(x) in YBa2Cu3O6+x are closely connected. PACS: 74.72Bk, 64.60.Cn, 81.30Dz CuO2 layers are key ingredients in all high-Tc cuprates given the fact that superconductivity occurs in these materials when a part of 3d copper electrons, usually between 5% and 27%, is taken away from the layers. The missing electrons are commonly referred to as “holes” which can move throughout the layers and make the material superconducting if the temperature is low enough. The electrons are conventionally subtracted from the layers either by chemical substitution of interlayer metal atoms, as for example, substitution of La2+ by Sr3+ in La1-x(Sr)xCu2O4 and Y2+ by Ca3+ in Y1-b(Ca)bBa2Cu3O6+x, or by pumping oxygen into the material. Oxygen is introduced into separate layers in which it orders to form CuO chains and it is these chains that are known to act as efficient attractors of electrons from the CuO2 layers. The number of created holes per Cu atom is typically denoted as “doping” p and the notion that CuO2 layers have an unavoidable role in the onset of superconductivity is further corroborated by the fact that a number of important physical characteristics, as, for example, the pseudogap energy Eg and critical transition temperature Tc, are coupled to p by universal relations that are common to practically all high-Tc cuprates. Thus, it has been obtained empirically that the Tc is scaled with Tc,max (maximal transition temperature) through the following, approximately parabolic, function of the hole concentration [2] ( )[ ]2max, 16.06.821)( −−= pTpT cc , (1) which has its onset, maximum and termination at p=0.05, 0.16 (optimal doping) and 0.27, respectively. YBa2Cu3O6+x superconductor has probably been the most thoroughly studied compound of all high-Tc cuprates because it has a relatively simple synthesis route and it was the first superconductor discovered with Tc above the liquid nitrogen temperature. The Tc changes in nonlinear manner with oxygen composition revealing two well-known plateaus at 60K and 90K. These features are clearly associated with the Ortho-II and Ortho-I phases, respectively [3]. While the 90K plateau is in fact a broad maximum at x≈0.92 that is associated with transition from underdoped to overdoped regime, the origin of 60K plateau at 0.5<x<0.7 is not yet well understood [3-5] (although, in fact, some advance has recently been made along this line [6]). One popular explanation is that when the oxygen content is increased over x=0.5, where Ortho-II phase emerges in the form of alternating columns of fully occupied and empty oxygen sites (directed along b axes), additional oxygen fills the empty columns making a relatively small contribution to hole doping, inasmuch as they are far apart from each other and only a small fraction of them are able to form CuO chains that are long enough to initiate the charge transfer process. It has often been guessed that there might be a certain minimal (critical) chain length lcr defined so that only chains of length that is equal to, or greater than, lcr can effectively attract electrons from CuO2 layers [3-6]. However, even though the existence of Ortho-II phase was resolved a long ago in terms of the classical two dimensional asymmetric next nearest neighbor Ising (ASYNNNI) model [7], no convincing explanation has been provided over the last 20 years as to why exactly the concentration p(x) of holes induced in CuO2 sheets would remain constant when x changes within the Ortho-II phase regime, nor, if the critical chain length concept is presupposed, what the value of lcr would be equal to and how it can be determined. Although it is well known that high-Tc cuprates are complex quantum many-body systems with the pairing mechanism still remaining controversial, here we unambiguously grounds that it is the classical ASYNNNI model combined with the concept of minimal chain length (needed for charge transfer to take place) that accounts for constant doping at p≈0.094 in the region of 60K plateau, as well as for the broad maximum of Tc at optimal doping (p=0.16) at x≈0.92. There is a general agreement that copper in the chain (basal) plane can be either Cu1+, which is the case when it is not coupled to the in-plane oxygen (but connected only to two apical O(4) ions and therefore 2-fold coordinated), or Cu2+, when it is inserted within a CuO chain (4-fold coordinated), or at the chain end (3-fold coordinated). If isolated oxygen is introduced into the basal plane it then takes away two electrons from the two nearest neighbor Cu1+ transferring them into Cu2+ state. Thus, isolated oxygen does not have a tendency to attract an additional electron from other the parts of the system. When another oxygen is added to make a chain with two O atoms (chain of length l=2) there is only one electron available from its nearest neighbor copper coordination and the absence of another electron, needed for oxygen to become O2-, is usually referred to as a “hole”. In this way, a chain of length l is seen as to have created l- 1 chain-holes, which in principle can attract electrons from the other parts of the system, presumably from CuO2 layers. Since the state of quasi one dimensional electronic subsystem of a chain of a given length (say, l), which reflects the charge transfer effectiveness of the chain, is not expected to depend upon history of chain formation, one is free to assume that the chain has been formed by adding oxygen one by one as that would allow to shadow the evolution of charge transfer process as a function of l. Within the concept of critical chain length no charge transfer is supposed to occur unless l=lcr so that the first lcr-2 initially created holes will stay inactive in attracting electrons from the planes. As l further increases beyond lcr the transfer of charge is set on during which process the remaining l- lcr+1 holes are created. Therefore, our strategy for counting doping is, first of all, to discard contribution not only of all holes in short chains (l<lcr), but also of the first lcr-2 holes in long chains (l≥lcr), for they had been created before any charge transfer took place (we shortly denote these holes as passive holes). It then naturally evolves that the number of attracted electrons (transferred holes) from the chain of length l≥lcr should be as greater as more holes were created after the charge transfer process had been triggered (at l=lcr), i.e. it should be proportional to l-lcr+1 (we call these holes the active holes for their occurrence coincides with the development of charge transfer). In the case of Ortho- II phase, the concentration of active holes h (the number of active holes per Cu) is given +−++−= ∑ ∑ =cr crll ll crcr lfllnlfllnh )()1()()1(4 2211 αααα , (2) where nα1 and nα2 denote the fractions of 3-fold coordinated Cu ions on two different sublattices of oxygen sites (usually denoted as α1 and α2), and fα1(l) and fα2(l) are corresponding fractions of CuO chains of the same length l. If NCu and n=(nα1+nα2)/2 are total number and the fraction of 3-fold coordinated Cu in basal plane, then (n/2)NCu is the total number of CuO chains and the number of passive holes per Cu is surely not greater than (n/2)(lcr-2), given the fact that no one of chains with l<lcr has more than lcr-2 holes. At x=1 (OI stoichiometry) chains are very long, virtually infinite, and there are just a few chain ends in the system. This means that n tends to zero as x approaches to 1, so that the concentration of passive holes becomes negligible whatever the value of lcr. This in turn implies that practically each chain ordered oxygen has created one active hole, i.e. h≈1 at x≈1. Since, in the YBa2Cu3O6+x system, one chain plane supplies holes to two CuO2 layers, the doping would have been equal to p=0.5 if each active hole had succeeded to capture one electron. Experimental findings, however, clearly contradict such a scenario for it was found that p(x≈1) only slightly overshoots 19% [2]. Thus, at least at x≈1, it can be stated p=(χ/2)h where χ (≈40%), as defined by the number of effectively attracted electrons (transferred holes) per active hole, reflects the capability of an active hole to capture electron. On the other hand, long (infinite) chains also prevail at x=0.5 (OII stoichiometry) on every even column of oxygen sites. The concentrations of passive holes is also negligible here, bur h cannot be greater that 0.5; in fact, h can be only less than 0.5 due to thermally activated chain fragmentation (for example, one might expect that h≈0.48, as at x≈0.5, so in the region of the 60K plateau). Given the fact that 60K plateau of Tc, according to (1), corresponds to doping level p≈0.094 it appears that it is the same fraction (of nearly 40%) of active holes that is transferred to the layers not only at x≈1, but also at x≈0.5. The charge transfer model that we propose here assumes that the same percentage of active holes is transferred, not only at stoichiometries x=0.5 and x=1, at which long CuO chains are known to dominate, but as well at off-stoichiometry 0.5<x<1 (and also at x<0.5) where chain fragmentation is more intense. We therefore propose that doping is connected to the active hole concentration h, as given by (2), by p=(χ/2)h (χ≈40%) throughout the whole range of oxygen concentration 0<x<1. The quantity χ introduced in this way should be perceived as average capability of an active hole to capture an electron from the CuO2 planes that lie above, or below, the basal plane (averaging is done over all chains in the system, or equivalently, over the whole volume of the material). To calculate the hole concentration h (eq. (2)) and doping p=χh/2, at a given point of (x,T) space, it is necessary to determine the fractions of the 3-fold coordinated Cu and length distributions of CuO chains fα1(l) and fα2(l). We used Monte Carlo (MC) method applied to the ASYNNNI model to calculate these quantities for it is known that the model stabilizes both structures Ortho-II and Ortho-I that are responsible for 60K and 90K plateaus [9]. Although ASYNNNI model cannot stabilize other structures with longer periodicities along a axis, like Ortho-III, Ortho-IV and Ortho-V, their superstructure reflections have already been reported to be much weaker than those of the main phases [8,9], so that they are thought to appear only as small patches embedded in large domains of main phases [10]. Besides, since these structures were mainly observed at oxygen compositions that correspond to transition region between the two plateaus it implies that the ASYNNNI model alone should account for both plateaus of Tc(x), especially given the fact that none of these structures, except Ortho-II, was reported at x<0.62 [9]. The chain length distributions fα1(l) and fα2(l) were determined in the following way: In each MC step we counted the total numbers of chains Nα1 and Nα2, on sublattices α1 and α2 , respectively (Nα1 and Nα2 are in fact equal to one half of unlike V2 bonds on the corresponding α sublattices), as well as the numbers of chains of the same length, Nα1(l) and Nα2(l), for lengths ranging from l=1 to l=300. The ratios Nα1(l)/Nα1 and Nα2(l)/Nα2 were then equilibrated through the MC process and the so obtained values were finally assigned to fα1(l) and fα2(l). The MC calculations were performed using single-spin-flip Glauber dynamics, where the oxygen concentration x is a functions of temperature T and chemical potential μ. We have studied lattices with periodic boundary conditions that consisted of 400x400 oxygen chain sites (O(1) sites, that split into two nonequivalent sublattices α1 and α2, in OII phase), and as many sites on β sublattice (O(5) sites). One MC step included flipping of all 2X(400X400) lattice spins and one MC run (at a particular point (x, T)) typically consisted of 3·104 to 5·104 MC steps, where only every tenth was used to calculate chain length distributions fα1(l) and fα2(l), l=1,2, …,300, and other relevant quantities (oxygen sublattice occupancies x1 and x2, 3-fold Cu fractions n1 and n2, etc.). At a certain number of points we have even used a really large number of MC steps, ranging from 105 to 3·105. At all calculated points of (x,τ) space (τ is a quantity that scales with T according to τ=kBT/VB 1, where kBB is Boltzman constant and V1 the nearest neighbor O-O interaction of the ASYNNNI model) it was obtained [11] 2,1, iaviav α , (3) where lav,αi (i=1,2) denotes the average chain length on the corresponding sublattice (α1 or α2). Such a behavior of length distributions fα1(l) and fα2(l) was subsequently explained theoretically analyzing microscopic features of the ASYNNNI model lattice configurations [12]. In brief terms, the l dependence of probability of a chain to have particular length l can be derived in the following way: Consider a sequence of Ny oxygen chain sites that are aligned along b axis (Ny is a large number, and the sites are connected by copper mediated V2<0 bonds). Let xNy denotes the number of oxygen atoms on this column of α sites and let nNy stands for the number of unlike V2 bonds (n is therefore the fraction of 3-fold coordinated Cu along the column, and, consequently, 2x/n equals to the corresponding average chain length, lav). These xNy oxygen atoms are generally divided into (n/2)Ny groups (chains) that can have various lengths l=1,2,… . It is useful to recall that an each chain has two ends: one that is oriented towards positive side of b axes (the “positive” end) and the other one, oriented towards negative b axes (the “negative” end). Among these xNy oxygen atoms there are (n/2)Ny of them that are located at the positive chain end, and thus the probability for an oxygen to be lying at the positive chain end is equal to ω=n/2x={lav)-1. Consequently, 1-ω is probability for oxygen to be located either within the interior of the chain, or at the negative chain end. Assuming that chains are created by adding oxygen one by one, starting from the negative end, one arrives at the conclusion that the probability for obtaining chain of length l is equal to f(l)=ω(1-ω)l-1 [11,12] {such a form of f(l) dependence is known in the theory of probability as “geometric” probability distribution [13]). It should be noted, however, that a deeper analysis shows that the above reasoning applies only if fluctuations of energy of the ASYNNNI model are not too large [12]. Indeed, a certain deviation from linear behavior of ln[f(l)] versus l dependence has been found in the vicinity of the second order Ortho-I-to-Ortho-II phase transition curve (at x>0.5) [12] but, fortunately, such departures were observed only in a relatively narrow intervals Δx≈0.07 around critical points. Furthermore, our extended analysis (not shown here) shows that in the critical regime these deviations, of calculated fα1(l) and fα2(l) dependences from the expected behavior, were in a certain way compensated by summations in (2), so that the calculated h(x) dependences were obtained to vary smoothly over the transition region at all τ=const. The so obtained values of length distributions fα1(l) and fα2(l) for l=1,…,300 were inserted into (2) to calculate concentration h of active holes at different points (x,T). The geometric-like behavior of f(l) dependences ensures rather fast convergence of sums in (2). It should be mentioned, however, that the specific form of length distributions (3) makes it possible, instead of evaluating summations in (2) by first 300 terms, to transform each of sums into a closed analytical form, so that h would be connected through analytical expression with average oxygen occupancies, x1 and x2, the 3-fold Cu fractions n1 and n2, and the parameter lcr. Whatever the approach we used the calculated values of h were obtained to be practically indistinguishable one from the other (even in the critical region of the ortho-I/ortho-II transition), but we nevertheless gave advantage to calculating the first 300 terms in estimating sums in (2), as we wanted to keep under control the departures of length distributions from (3) that are known to occur in the critical regime [12]. In addition, at each point of (x,T) space h was calculated for the whole range of values of cutoff parameter lcr spanning from lcr=1 to lcr=50, so that h can be regarded as a function of three variables, i.e. h=h(x,τ,lcr) (aside from the fact that it also depends upon input parameters that define the ASYNNNI model, i.e. on O-O interactions V1>0 (nearest neighbor) and V2<0, V3>0 (next nearest neighbor)). There is one remarkable feature of the hole concentration h=h(x,τ,lcr), as defined by (2), that we have found while thoroughly analyzing its behavior: whatever the magnitudes of interactions V1, V2, and V3, when h is considered as a function of lcr at different points (xi,τ) that correspond to the same τ=const and oxygen concentrations xi, i=1,2,… spanning over the region of Ortho-II phase, all of these hxi(lcr), i=1,2,… functions intersect at a single, well defined value of lcr. This is shown in Figure 1 for interactions obtained by linear-muffin-tin orbital (LMTO) method [14] and at three different temperatures τ=0.45, 0.38 and 0.30, but the similar behavior we have also obtained (not shown here) for the so-called “canonical” interactions V2=-0.5V1, V3=0.5V1. From Figure 1 it can be clearly seen that at a given τ=const, the value of lcr at which all hxi(lcr) curves intersect depends on temperature in the way that it increases with the temperature decrease. Thus, at three temperatures τ=0.45, 0.38, 0.30, the intersection values were found to be lcr=4(5), 6(7), 11(12), respectively. At a given temperature, the so obtained intersection value of lcr we name “the optimal minimal (critical) length” (denoted by lcr,opt(τ)) for it is the value at which h(x) stays constant over the regime of Ortho-II phase. Such behavior of h(x) at τ=const, for the corresponding lcr,opt(τ), is shown in Figure 2 at two temperatures: τ=0.45 (Figures 2a and 2b, for lcr,opt(τ)=4 and 5, respectively) and τ=0.30 (Figure 2c, for lcr,opt(τ)=12). From these results it can be seen that indeed h(x) demonstrates a constant section at x>0.5 that is even more pronounced at lower temperatures. Calculated h(x) dependences were used to obtain doping versus x dependences, p(x)=χh(x)/2, that were then inserted into (1) to yield corresponding Tc(x)s (Figures 3a-c). The parameter χ was varied slightly around its expected value ≈40% [2] to achieve a better correlation between the so obtained Tc(x)’s and those from experiments [15] (shown by a solid line). We have indeed obtained h in the plateau regime to be slightly lower than 0.5: h=const=0.467 (0.450) at τ=0.45 for lcr=4 (lcr=5), h=const=0.483 at τ=0.38 for lcr=7, and h=const=0.495 at τ=0.30 for lcr=12. This gives χ=40.33%, 41.84%, 38.98%, 3804% for h=0.467, 0.450, 0.483, 0.495, respectively. Thus, not only experimental data on doping at x≈1 [2], but also the analysis at x≈0.5 (that falls into the regime of 60K plateau) both infer that χ is lying at some point between ≈38% and ≈42%. It should be noted that this result disagrees with estimations of Gawiec et al. [16,17] that long chains release up to 70% of their holes. Although we used here the ASYNNNI model interactions V1, V2, and V3 as obtained by Sterne and Wille [14] the issue of magnitudes of these interactions is still open so that new values were subsequently suggested [10]. Regardless of that, it seems very well established that nearest neighbor interaction V1 should be ranking around 6.9mRy [10,14], which fixes scaling between T and τ to Δτ≈0.1⇔ ΔT≈100K. On the other hand, as one of the most important features of YBa2Cu3O6+x phase diagram is the location of the top of Ortho-II phase along τ axis (at x≈0.5) and that it may well be affected by the magnitudes of V2 and V3 (which in fact are not known precisely), perhaps the best strategy to estimate the reduced temperature τ corresponding to room temperature is to determine the “distance” (in units of τ) between room temperature and the top of Ortho-II phase. According to experimental data the top of Ortho-II phase corresponds to ≈125-140ºC [5,9], while theoretically obtained phase diagram for the LMTO interactions [18] points at τ≈0.58, thus making τ≈0.45 a fairly reliable estimation of room temperature. The established correlation between room temperature and τ=0.45 renders lcr,opt(τ) (that plays the role of lcr in (2)) 4, or 5 (Figure 1a). A better estimation seems to be 5 for it yields a somewhat more pronounced 60K plateau than lcr=4 (Figures 3a and 3b), despite the fact that lcr=4 would be closer to lcr=3 that was proposed in some theoretical studies [16,17,19]. Besides, lcr=5 appears to be well correlated with χ≈2/5 and with the basic idea lying in the background of the minimal (critical) chain length concept suggesting that one isolated hole cannot efficiently attract an electron, but only a combined effect of several holes can achieve this goal. Both lcr=5 and χ≈2/5 imply that 3 chain-holes are still not enough to effectuate charge transfer, but that the joint impact of 5 holes suffices to attract two electrons (one per each CuO2 layer). In summary, despite the well known fact that in such highly correlated electron systems, as are the high-Tc cuprates, the nature of the controversial pairing mechanism is genuinely quantum mechanical, we have shown here that certain aspects of their behavior can be explained in terms of classical models. Such is the classical ASYNNNI model that successfully accounts for unusual two-plateaus-like behavior of Tc(x) in YBa2Cu3O6+x. The obtained Tc(x) dependence is in a remarkable correlation with experiment [15] for both lcr=4 and lcr=5, although we believe the later value is more realistic. It should be also pointed out that the presented results on Tc(x) dependence are in a qualitative agreement with some previous results on the same topic [20,21]. Generally, it can be expected that the capability of an active hole to attract an unpaired 3d electron, as expressed by the value of χ, should depend upon the density ρe of available electrons immediately above (below) the chain (aside from the fact that χ should also depend on a certain coupling between chains and planes). Our recently obtained results on Tc(x) in Y1- b(Ca)bBa2Cu3O6+x (b=0.2) system [22] seem to be lying along this line, since obtained χ≈33% can be understood in the light of the fact that introduction of ≈20% of Ca has additionally increased doping, and therefore reduced ρe, so that χ attained a lesser value than in the parent YBa2Cu3O6+x compound. Acknowledgements This work has been funded by the Serbian Ministry of Science and Technology through the Project 141014. References [1] G. V. M. Williams, J. L. Tallon, R. Michalak and R. Dupree, Phys. Rev. B 54, 6909 (1996). [2] J. L. Tallon, C. Bernhard, H. Shaked, R. L. Hitterman and J. D. Jorgensen, Phys. Rev. B 51, 12911 (1995). [3] Z. Yamani, W. A. MacFarlane, B. W. Statt, D. Bonn, R. Liang and W. N. Hardy, Physica C 405, 227 (2004). [4] R. Liang, D. A. Bonn and W. N. Hardy, Phys. Rev. B 73, 180505 (2006). [5] F. Yakhou, J. Y. Henry, P. Burlet, V. P. Plakhty, M. Vlasov and S. Moshkin, Physica C 333, 146 (2000). [6] V. M. Matic and N. Dj. Lazarov, Solid State Commun. 142, 165 (2007) (cond- mat/0611214). [7] L. T. Wille and D. de Fontaine, Phys. Rev. B 37, 2227 (1988). [8] Z. Islam, X. Liu, S. K. Sinha, J. C. Lang, S. C. Moss, D. Haskel, G. Srajer, P. Wochner, D. R. Lee, D. R. Haeffner and U. Welp, Phys. Rev. Lett. 93, 157008 (2004). [9] M.v. Zimmermann, J. R. Schneider, T. Frelo, N. H. Andersen, J. Madsen, J. Kall, H. F. Poulsen, R. Liang, P. Dosanjih and W. N. Hardy, Phys. Rev. B 68, 104515 (2003). [10] D. J. Liu, L. T. Einstein, P. A. Sterne and L. T. Wille, Phys. Rev. B 52, 9784 (1995). [11] V. M. Matic, N. Dj. Lazarov, V. Spasojevic, M. Milic and V. Kusigerski, Physica C 421, 49 (2005). [12] V. M. Matic and N. Dj. Lazarov, Physica C 443, 49 (2006). [13] M. J. Merkle and P. M. Vasic, “Probability and Statistics”, Edited by Elektrotechnic Faculty of Belgrade University (1998) (in Serbian). [14] P. A. Sterne and L. T. Wille, Physica C 162-164, 223 (1989). [15] J. D. Jorgensen, M. A. Beno, D. G. Hinks, L. Soderholm, K. J. Volin, R. L. Hitterman, J. D. Grace, J. K. Schulle, C. U. Segre, K. Zhang and M. S. Kleefisch, Phys. Rev. B 36, 3608 (1987). [16] P. Gawiec, D. R. Grempel, A. C. Riiser, H. Haugerud and G. Uimin, Phys. Rev. B 53, 5872 (1996). [17] P. Gawiec, D. R. Grempel, G. Uimin and J. Zittartz, Phys. Rev. B 53, 5880 (1996). [18] V. M. Matic, Physica A 184, 571 (1992). [19] K. Miitsen and O. M. Ivanenko, JETP Letters 82, 129 (2005). [20] N. Rezlesku, C. Buzea and C. Gh. Buzea, Physica C 247 (1995) 105. [21] C. Buzea and T. Yamashita, Physica C 357 (2001) 288. [22] V. M. Matic, N. Dj. Lazarov and I. M. Bradaric, arXiv:0705.0299 (to be submitted). Figure Captions Figure 1. Calculated h(lcr) dependences at τ=const for several values of oxygen composition x that span the range of OII phase: a) τ=0.45=const; 0.52<x<0.61 (shown at the top), b) τ=0.38=const; 0.52<x<0.62 (shown in the middle section), and c) τ=0.30=const; 0.52<x<0.67 (shown at the bottom). Figure 2. Calculated values of h as a function of x at τ=const: a) τ=0.45=const, for lcr=4 (shown at the top), b) τ=0.45=const, for lcr=5 (shown in the middle section), and c) τ=0.30=const, for lcr=12 (shown at the bottom). Figure 3, Calculated Tc(x) dependences, using h(x) dependences from the Figure 2 and thereafter obtained p(x)=χh(x)/2 dependences, that were then inserted into (1): a) τ=0.45=const, for lcr=4 and χ=39.4% (shown at the top), b) τ=0.45=const, for lcr=5 and χ=41% (shown in the middle section), and c) τ=0.30=const, for lcr=12 and χ=37.6% (shown at the bottom). -2 0 2 4 6 8 10 12 14 τ=0.30, 0.52<x<0.67 x=0.52 x=0.54 x=0.56 x=0.57 x=0.59 x=0.62 x=0.67 x=0.52 x=0.53 x=0.55 x=0.56 x=0.58 x=0.59 x=0.61 τ=0.45, 0.52<x<0.61 τ=0.38, 0.52<x<0.62 x=0.52 x=0.53 x=0.54 x=0.55 x=0.57 x=0.59 x=0.62 Figure 1 0.0 0.2 0.4 0.6 0.8 1.0 τ=0.45, l τ=0.45, l τ=0.30, l Figure 2 0.0 0.2 0.4 0.6 0.8 1.0 τ=0.45, l χ=0.394 τ=0.45, l χ=0.410 τ=0.45, l χ=0.376 Figure 3

N -HOMOGENEOUS SUPERALGEBRAS PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ ABSTRACT. We develop the theory of N -homogeneous algebras in a super setting, with par- ticular emphasis on the Koszul property. To any Hecke operator R on a vector superspace, we associate certain superalgebras SR,N and ΛR,N generalizing the ordinary symmetric and Grassmann algebra, respectively. We prove that these algebras are N -Koszul. For the special case where R is the ordinary supersymmetry, we derive an N -generalized super-version of MacMahon’s classical “master theorem”. CONTENTS Introduction 1 1. Review of linear superalgebra 5 2. The supercharacter 10 3. Homogeneous superalgebras 17 4. N -Koszul superalgebras 27 5. Koszul duality and master theorem 34 Appendix 38 References 41 INTRODUCTION 0.1. The theory of N -homogeneous algebras owes its existence primarily to the concerns of noncommutative geometry. In fact, as has been expounded by Manin in his landmark pub- lications [36], [37], quadratic algebras (the case N = 2) provide a convenient framework for the investigation of quantum group actions on noncommutative spaces. Moreover, certain Artin-Schelter regular algebras [1], natural noncommutative analogs of ordinary polynomial algebras, can be presented as associative algebras defined by cubic relations (N = 3). The latter algebras, as well as many of the quadratic algebras studied by Manin, enjoy the addi- tional “Koszul property” which will be of central importance in the present article; it will be reviewed in detail in 0.6 below. Motivated by these examples and others, Berger [5] initiated the systematic investigation of N -homogeneous algebras for all N ≥ 2, introducing in particular a natural extension of the notion of Koszul algebra from the familiar quadratic setting to general N -homogeneous 2000 Mathematics Subject Classification. Primary 16S37, 05A19. PHH is supported by the DFG through a Heisenberg-Fellowship. ML’s research is supported in part by NSA Grants H98230-05-1-0025 and H98230-07-1-0008 and by Lever- hulme Research Interchange Grant F/00158/X. http://arxiv.org/abs/0704.1888v2 2 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ algebras. Article [5] gives examples of N -Koszul algebras for all N ≥ 2; these are the so-called N -symmetric algebras, the special case N = 2 being the ordinary symmetric (poly- nomial) algebra. Following the general outline of Manin’s lecture notes [37] on the case of quadratic algebras, Berger, Dubois-Violette and Wambst developed the categorical aspects of N -homogeneous algebras in [7]. 0.2. Current interest in N -homogeneous algebras is fueled in part by the fact that they do occur naturally in mathematical physics and in combinatorics. Indeed, Connes and Dubois- Violette [10], [11] introduced a class of 3-homogeneous algebras, called Yang-Mills algebras, which are in fact 3-Koszul. There are two versions of Yang-Mills algebras: in the language of linear superalgebra, the first kind has even (parity 0̄) algebra generators while the second kind is generated by odd (parity 1̄) elements. Combinatorics enters the picture via MacMahon’s celebrated “master theorem” [35], specif- ically the recent quantum generalization of the master theorem due to Garoufalidis, Lê and Zeilberger [20]. As has been pointed out by two of the present authors in [28], the yoga of (quadratic) Koszul algebras leads to a rather effortless and conceptual proof of the quantum master theorem based on the fact that a certain quadratic algebra, known as quantum affine space, is Koszul. Further quantum generalizations and super versions of the master theorem have been obtained by several authors using a variety of approaches; see Foata and Han [17], [18], [19], Konvalinka and Pak [33], Etingof and Pak [16]. 0.3. From an algebraic point of view, MacMahon’s master theorem (MT) in its various incar- nations finds its most natural explanation by the phenomenon of “Koszul duality”. Indeed, all versions of MT can be expressed in the form that, for some algebra B, an equation Σ1 ·Σ2 = 1 holds for suitable power series Σ1,Σ2 ∈ BJtK. Here is a brief outline how one can arrive at such an equation starting with a given N -Koszul algebra A. Associated with A, there is a graded complex, K(A), which is exact in positive degrees, and a certain endomorphism bial- gebra, endA, which coacts on all components of K(A). These components therefore define elements of the representation ring RendA of endA, and exactness of K(A) in positive de- grees yields an equation in the power series ring RendAJtK. Due to the specific form of K(A), which is constructed from A together with its so-called dual algebra A!, the equation in ques- tion does indeed state that ρ1 · ρ2 = 1 holds for suitable ρ1, ρ2 ∈ RendAJtK. The last step in deriving a MT for A consists in using (super-)characters to transport the abstract duality equation ρ1 · ρ2 = 1 from RendAJtK to the power series ring over the algebra endA, where it takes a more explicit and useable form. Here then is the flow chart of our approach: N -Koszul algebra // exact Koszul complex // duality equation in RendAJtK // MT for A The actual labor involved in this process consists in the explicit evaluation of (super-)characters at the last arrow above. This step is often facilitated by specializing the bialgebra endA, which is highly noncommutative, to a more familiar algebra B via a homomorphism endA → B. For example: • MacMahon’s original MT [35] follows in the manner described above by starting with A = O(kd) = k[x1, . . . , xd], the ordinary polynomial algebra or “affine space”, and N -HOMOGENEOUS SUPERALGEBRAS 3 restricting the resulting MT over endO(kd) to the coordinate ring of d × d-matrices, O(Matd(k)) = k[x j | 1 ≤ i, j ≤ d]. • As was explained in [28], taking “quantum affine space” Oq(k d) as the point of de- parture one arrives at the quantum MT of Garoufalidis, Lê and Zeilberger [20] (and Konvalinka and Pak [33] in the multi-parameter case). The endomorphism bialgebra of Oq(k d) is exactly the algebra of right-quantum matrices as defined in [20]. • Berger’s N -symmetric algebra [5] leads to the N -generalization of the MT proved by Etingof and Pak [16] using the above approach, again after restricting to O(Matd(k)). 0.4. The present article aims to set forth an extension of the existing theory of N -homogeneous algebras to the category Vects of vector superspaces over some base field k. While this does not give rise to principal obstacles given that [37] and [7] are at hand as guiding references, the setting of superalgebra requires careful consideration of the order of terms and the so-called “rule of signs” will be ubiquitous in our formulæ. In view of the potential interdisciplinary interest of this material, we have opted to keep our presentation reasonably self-contained and complete. Therefore, in Sections 1 and 2, we deploy the requisite background material from superal- gebra in some detail before turning to N -homogeneous superalgebras in Section 3. The latter section, while following the general outline of [37] and [7] rather closely, also offers explicit discussions of a number of important examples. We interpolate the pure even and pure odd Yang-Mills algebras defined by Connes and Dubois-Violette [10], [11] by a family of super- algebras YMp|q and give a unified treatment of these algebras. (It turns out, however, that the mixed algebras YMp|q, with p and q both nonzero, are less well-behaved than the pure cases.) Moreover, we discuss a superized version of the N -symmetric algebras of Berger [5]. Finally, in Example 3.4, we introduce new N -homogeneous superversions of the symmetric algebra and the Grassmann algebra of a vector superspace V ; these are associated with any Hecke operator R : V ⊗2 → V ⊗2 and will be denoted by SR,N and ΛR,N , respectively. Sections 4 and 5 contain our main results: Theorem 4.5 shows that the superalgebras SR,N and ΛR,N are in fact N -Koszul, and Theorem 5.4 is superized version of the aforementioned N -generalized MT of Etingof and Pak [16, Theorem 2]. The special case N = 2 of The- orem 5.4 is a superization of the original master theorem of MacMahon [35]. The present article was motivated in part by a comment in Konvalinka and Pak [33, 13.4] asking for a “real” super-analog of the classical MT. 0.5. A considerable amount of research has been done by mathematical physicists on various quantum matrix identities. Some of these investigations have been carried out in a super set- ting; see, e.g., Gurevich, Pyatov and Saponov [23], [24] and the references therein. However, the techniques employed in these articles appear to be quite different from ours. After submitting this article, we also learned of recent work of Konvalinka [31], [32] which not only concerns MacMahon’s MT but also other matrix identities such as the determinantal identity of Sylvester. These identities are proved in [31], [32] by combinatorial means in various noncommutative settings including the right-quantum matrix algebra endOq(k 0.6. We conclude this Introduction by reviewing the precise definitions of N -homogeneous and N -Koszul algebras. Our basic reference is Berger [5]; see also [2], [7], [21]. 4 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Let A be a connected Z≥0-graded algebra over a field k; so A = n≥0An for k-subspaces An with A0 = k and AnAm ⊆ An+m. Choose a minimal generating set for the algebra A consisting of homogeneous elements of positive degree; this amounts to choosing a graded basis for a graded subspace V ⊆ A+ = n>0An such that A+ = A + ⊕ V . The grading of V imparts a grading to the tensor algebra T(V ) of the space V , and we have a graded presentation T(V )/I for some graded ideal I of T(V ), the ideal of relations of A. Recall that a graded vector space M = n∈ZMn is said to live in degrees ≥ n0 if Mn = 0 for all n < n0. Note that the relation ideal I lives in degrees ≥ 2, because T(V )0 ⊕T(V )1 ⊆ k⊕ V and k⊕ V injects into A. Fix an integer N ≥ 2 and define the jump function νN (i) = N if i is even N + 1 if i is odd (0.1) The following proposition is identical with [8, Proposition 2.1] except for the fact that we do not a priori assume A to be generated in degree 1. A proof is given in the Appendix. Proposition 0.1. The ideal I of relations of A lives in degrees ≥ N if and only if TorAi (k, k) lives in degrees ≥ νN (i) for all i ≥ 0. Following Berger [5], the graded algebra A is said to be N -Koszul if TorAi (k, k) is con- centrated in degree νN (i) for all i ≥ 0. This implies that the space of algebra generators V is concentrated in degree νN (1) = 1; so the algebra A is 1-generated. Moreover, choosing a minimal set of homogeneous ideal generators for the relation ideal I amounts to choosing a graded basis for a graded subspace R ⊆ I such that I = R⊕ (V ⊗ I + I ⊗ V ) (0.2) Then TorA2 (k, k) ∼= R and so R must be concentrated in degree νN (2) = N when A is N -Koszul. To summarize, all N -Koszul algebras are necessarily 1-generated and they have defining relations in degree N ; so there is a graded isomorphism A ∼= T(V )/(R) with R ⊆ V ⊗N Such algebras are called N -homogeneous. We remark that Green et al. [21] have studied N -Koszul algebras in the more general con- text where the grading A = n≥0An is not necessarily connected (A0 = k). In [21, The- orem 4.1], it is shown that an N -homogeneous algebra A with A0 split semisimple over k is N -Koszul if and only if the Yoneda Ext-algebra E(A) = n≥0 Ext A(A0,A0) is generated in degrees ≤ 2. Any N -homogeneous algebra A whose generating space V carries a Z2-grading and whose defining relations R are Z2-graded is naturally a k-superalgebra, that is, A has a Z2-grading (“parity”) besides the basic Z≥0-grading (“degree”). As will be reviewed below, this extra structure provides us with additional functions on Grothendieck rings, namely superdimension and supercharacters, which lead to natural formulations of the MT in a superized context. Note, however, that the defining property of N -Koszul algebras makes no reference to the Z2-grading of A. Thus, an N -homogeneous superalgebra is Koszul precisely if it is Koszul as an ordinary N -homogeneous algebra (forgetting the Z2-grading). N -HOMOGENEOUS SUPERALGEBRAS 5 0.7. Throughout k is a commutative field and ⊗ stands for ⊗ . Scalar multiplication in k-vector spaces will often, but not always, be written on the right while linear maps will act from the left. We tacitly assume throughout that char k 6= 2; further restrictions on the characteristic of k will be stated when required. 1. REVIEW OF LINEAR SUPERALGEBRA 1.1. Vector superspaces. A vector superspace over k is a k-vector space V equipped with a grading by the group Z2 = Z/2Z = {0̄, 1̄}. Thus, we have a decomposition V = V0̄ ⊕ V1̄ with k-subspaces V0̄ and V1̄ whose elements are called even and odd, respectively. In general, the Z2-degree of a homogeneous element a ∈ V is also called its parity; it will be denoted by â ∈ Z2. Vector superspaces over k form a category Vect whose morphisms are given by the linear maps preserving the Z2-grading; such maps are also called even linear maps. The dimension of an object V of Vects is the usual k-linear dimension. We shall use the notation d = dim V , p = dim V0̄ and q = dimk V1̄ So d = p+ q. The superdimension of a vector superspace V with d < ∞ is defined by sdimV = p− q ∈ Z When working with a fixed basis {xi} of a given V in Vect we shall assume that each xi is homogeneous; the parity of xi will be denoted by î. The basis x1, x2 . . . is called standard if î = 0̄ (i ≤ p) and î = 1̄ (i > p). 1.2. Tensors. The tensor product U ⊗V of vector superspaces U and V in Vects is the usual tensor product over k of the underlying vector spaces equipped with the natural Z2-grading: if a, b are homogeneous elements then the parity of a ⊗ b is â + b̂ ∈ Z2. Instead of the usual symmetry isomorphism U ⊗ V −→ V ⊗ U for interchanging terms in a tensor product we shall use the rule of signs, that is, the following functorial supersymmetry isomorphism in Vects cU,V : U ⊗ V −→ V ⊗ U , u⊗ v 7→ (−1)bubvv ⊗ u (1.1) for u, v homogeneous. (All formulas stated for homogeneous elements only are to be ex- tended to arbitrary elements by linearity.) The supersymmetry isomorphisms cU,V satisfy cV,U ◦ cU,V = IdU⊗V , and they are compatible with the usual associativity isomorphims aU,V,W : (U ⊗V )⊗W ∼= U ⊗ (V ⊗W ) in Vect , that is, they satisfy the “Hexagon Axiom”; see [29, Def. XIII.1.1]. Therefore, Vects is a symmetric tensor category; the unit object is the field k, with parity 0̄. See [29, Chap. XIII] or [12] for background on tensor categories. 1.3. Homomorphisms. The space Hom (V,U) of all k-linear maps between vector super- spaces V and U is again an object of Vects , with grading Hom (V,U)0̄ = Homk(V0̄, U0̄) ⊕ (V1̄, U1̄) and Homk(V,U)1̄ = Homk(V0̄, U1̄)⊕Homk(V1̄, U0̄); so (V,U)0̄ = HomVects (V,U) In particular, the linear dual V ∗ = Hom (V, k) belongs to Vects . Given homogeneous bases {xj} of V and {yi} of U we can describe any f ∈ Hom (V,U) by its matrix F = (F ij ): f(xj) = j (1.2) 6 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ When f is an even map then F ij = 0 unless î+ ĵ = 0̄. For finite-dimensional vector superspaces, we have the following functorial isomorphisms in Vects (see, e.g., [43, I.8]): U ⊗ V ∗ ∼= Hom (V,U) (1.3) via (u⊗ f)(v) = u〈f, v〉, and V ∗1 ⊗ . . . ⊗ V ∼= (Vm ⊗ . . . ⊗ V1) ∗ (1.4) via 〈f1 ⊗ . . .⊗ fm, vm ⊗ . . .⊗ v1〉 = i〈fi, vi〉. Here, we use the notation 〈f, v〉 = f(v) for the evaluation pairing evV = 〈 . , . 〉 : V ∗ ⊗ V → k in Vects . Similarly, we have a pairing V ⊗ V ∗ cV,V ∗ −→ V ∗ ⊗ V which yields an isomorphism −→ V ∗∗ (1.5) in Vects The isomorphism (1.3) (which is valid as long as one of U or V is finite-dimensional) has the following explicit description. Fix homogeneous bases {xj} of V and {yi} of U and let F = (F ij ) be the matrix of a given f ∈ Homk(V,U) with respect to these bases, as in (1.2). Let {xj} be the dual basis of V ∗, defined by 〈xj , xℓ〉 = δ (Kronecker delta). Then the image of f in U ⊗ V ∗ is given by i,j yi ⊗ x jF ij . Note also that xi and x i have the same parity. Finally, if U , V and W are vector superspaces, with U finite-dimensional, then the isomor- phism Id⊗cW,U∗ : V ⊗W ⊗U ∗ ∼−→V ⊗U∗ ⊗W together with (1.3) yields an isomorphism (U, V ⊗W ) −→ Hom (U, V )⊗W (1.6) in Vects which is explicitly given by (f ⊗ w)(u) = (−1)bwbuf(u) ⊗ w. Similarly, for vector superspaces U , U ′, V , V ′ with U , U ′ finite-dimensional, there is an isomorphism (U ⊗ U ′, V ⊗ V ′) −→ Hom (U, V )⊗Hom (U ′, V ′) (1.7) in Vects given by (f ⊗ g)(u ⊗ v) = (−1)bgbuf(u)⊗ g(v). 1.4. Supertrace. Let V be a finite-dimensional object of Vects . The supertrace is the map str : End (1.3) V ⊗ V ∗ −→ (1.3) k (1.8) in Vects . In order to describe the supertrace in terms of matrices, fix a basis {xi} of V consisting of homogeneous elements and let F = (F ij ) be the matrix of f ∈ Endk(V ) as in (1.2). Then str(f) = biF ii where î is the parity of xi (and of the dual basis vector x i ∈ V ∗) as in §1.1. Thus, str(IdV ) = sdimV.1 N -HOMOGENEOUS SUPERALGEBRAS 7 1.5. Action of the symmetric group. Given vector superspaces V1, . . . , Vn, we can consider the morphism ci : V1 ⊗ · · · ⊗ Vi ⊗ Vi+1 ⊗ · · · ⊗ Vn −→ V1 ⊗ · · · ⊗ Vi+1 ⊗ Vi ⊗ · · · ⊗ Vn in Vects which interchanges the factors Vi and Vi+1 via cVi,Vi+1 and is the identity on all other factors. More generally, for any σ ∈ Sn, the symmetric group consisting of all permutations of {1, 2, . . . , n}, one can define a morphism cσ : V1 ⊗ · · · ⊗ Vn −→ Vσ−1(1) ⊗ · · · ⊗ Vσ−1(n) in Vects as follows. Recall that Sn is generated by the transpositions σ1, . . . , σn−1 where σi interchanges i and i + 1 and leaves all other elements of {1, 2, . . . , n} fixed. The minimal length of a product in the σi’s which expresses a given element σ ∈ Sn is called the length of σ and denoted ℓ(σ); it is given by ℓ(σ) = #inv(σ) with inv(σ) = {(i, j) | i < j but σ(i) > σ(j)} Writing σ ∈ Sn as a product of certain σi, the analogous product of the maps ci yields a morphism cσ as above. This morphism is independent of the way σ is expressed in terms of the transpositions σi; see [43, I.4.13] or [29, Theorem XIII.1.3]. If all vi ∈ Vi are homogeneous cσ(v1 ⊗ · · · ⊗ vn) = (−1) (i,j)∈inv(σ) bvi bvjvσ−1(1) ⊗ · · · ⊗ vσ−1(n) (1.9) For example, if all vi are even then the ±-sign on the right is +, and if all vi are odd then it is sgn(σ), the signature of σ. Taking all Vi = V we obtain a representation c : Sn −→ AutVects (V ⊗n) where V ⊗n = V ⊗ · · · ⊗ V (n factors). Letting k[Sn] denote the group algebra of the symmetric group, this extends uniquely to an algebra map c : k[Sn] −→ EndVects (V ⊗n) (1.10) We will write ca := c(a) for a ∈ k[SN ]. For the dual superspace V ∗, besides the above representation c : k[Sn] −→ EndVects (V ∗⊗n), we also have the contragredient representation c∗ : k[Sn] −→ EndVects (V ∗⊗n) for the pairing 〈 . , . 〉 : V ∗⊗n ⊗ V ⊗n → k in (1.4). Explicitly, 〈c∗a(x), y〉 = 〈x, ca∗(y)〉 for all a ∈ k[Sn], x ∈ V ∗⊗n and y ∈ V ⊗n. Here, . ∗ : k[Sn] → k[Sn] is the involution sending σ ∈ Sn to σ −1. These two representations are related by c∗a = cτaτ (1.11) where τ = (1, n)(2, n−1) · · · ∈ Sn is the order reversal involution. One only needs to check (1.11) for the transpositions a = σi, which is straightforward. 8 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ 1.6. Hecke algebras. We recall some standard facts concerning Hecke algebras; these are suitable deformations of the group algebra k[Sn] considered above. For details, see [13], [14]. Fix 0 6= q ∈ k. The Hecke algebra Hn,q is generated as k-algebra by elements T1, . . . , Tn−1 subject to the relations (Ti + 1)(Ti − q) = 0 TiTi+1Ti = Ti+1TiTi+1 TiTj = TjTi if |i− j| ≥ 2 (1.12) When q = 1, one has an isomorphism Hn,1 −→ k[Sn], Ti 7→ σi where σi is the transposition (i, i + 1) as in §1.5. The algebra Hn,q has a k-basis {Tσ | σ ∈ Sn} so that (i) TId = 1 and Tσi = Ti; (ii) TσTσi = Tσσi if ℓ(σσi) = ℓ(σ) + 1; qTσσi + (q − 1)Tσ otherwise By k-linear extension of the rule T ∗σ := Tσ−1 (σ ∈ Sn) one obtains an involution . ∗ : Hn,q → Hn,q. Moreover, the elements T i := −qT q − 1− Ti also satisfy relations (1.12). Therefore, α(Ti) := −qT i (1.13) defines an algebra automorphism α : Hn,q → Hn,q of order 2. The Hecke algebra Hn,q is always a symmetric algebra, and Hn,q is a split semisimple k-algebra iff the following condition is satisfied: [n]q! := [i]q 6= 0 where [i]q := 1 + q + · · ·+ q i−1 (1.14) More precisely, if (1.14) holds then Matdλ×dλ(k) (1.15) where λ runs over all partitions of n and dλ denotes the number of standard λ-tableaux. The only partitions λ with dλ = 1 are λ = (n) and λ = (1 n). The central primitive idempotents of Hn,q for these partitions are given by Xn := [n]q! Tσ (1.16) Yn := [n]q−1 ! (−q)−ℓ(σ)Tσ (1.17) These idempotents are usually called the q-symmetrizer and the q-antisymmetrizer, respec- tively. One has XnTσ = TσXn = q ℓ(σ)Xn and YnTσ = TσYn = (−1) ℓ(σ)Yn (1.18) for σ ∈ Sn. Furthermore, α(Xn) = Yn. N -HOMOGENEOUS SUPERALGEBRAS 9 For later use, we note the following well-known consequence of (1.18). If M is any Hn,q- module, with corresponding representation µ : Hn,q → End (M), then Im(µ(Xn)) = Im(µ(Ti) + 1) (1.19) Indeed, (1.18) implies that Xn = [2] q (Ti + 1)Xn, which yields the inclusion ⊆. On the other hand, any m ∈ i=1 Im(µ(Ti) + 1) satisfies (µ(Ti) − q)(m) = 0 for all i, by (1.12). Therefore, µ(Tσ)(m) = q ℓ(σ)m holds for all σ ∈ Sn, and hence µ(Xn)(m) = [n]q! qℓ(σ)m = m. This proves ⊇. 1.7. Hecke operators. Again, let 0 6= q ∈ k. A Hecke operator (associated to q) on a vector superspace V is a morphism R : V ⊗2 → V ⊗2 in Vects satisfying the Hecke equation (R + 1)(R − q) = 0 and the Yang-Baxter equation R1R2R1 = R2R1R2 where R1 := R ⊗ IdV : V ⊗3 → V ⊗3 and similarly R2 := IdV ⊗R. The Hecke equation implies that R is invertible. Moreover, if R is a Hecke operator associated to q then so is −qR−1. Defining ρ(Ti) := Id V ⊗R ⊗ Id ⊗n−i−1 V , one obtains a representation ρ = ρn,R : Hn,q −→ EndVects (V ⊗n) (1.20) The representations ρn,R and ρn,−qR−1 are related by ρn,−qR−1 = ρn,R ◦ α, where α is the automorphism of Hn,q defined in (1.13). Example 1.1. The supersymmetry operator cV,V : V ⊗2 → V ⊗2 in (1.1) is a Hecke operator associated to q = 1, as is its negative, −cV,V . The representation ρcV,V of Hn,1 = k[Sn] in (1.20) is identical with (1.10). Example 1.2 (superized Drinfel’d-Jimbo [38], [27]). Let x1, . . . , xd be a standard basis of V as in §1.1. The super analog R = RDJ of the standard Drinfel’d-Jimbo Hecke operator is defined as follows. Writing R(xi ⊗ xj) = xk ⊗ xlR the matrix components Rk,li,j ∈ k are given by i,j = q2 − q2εi,j 1 + q2εi,j i,j + (−1) bibj q εi,j(q2 + 1) 1 + q2εi,j Here, εi,j = sgn(i− j). Thus, ii = q 2 if î = 0̄ ii = −1 if î = 1̄ ij = q 2 − 1 if i < j ij = (−1) bibjq if i 6= j (1.21) 10 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ and Rk,li,j = 0 in all other cases. One checks that R is a Hecke operator that is associated to 2. THE SUPERCHARACTER 2.1. Superalgebras, supercoalgebras etc. An algebra A in Vects is called a superalgebra over k; this is just an ordinary k-algebra such that the unit map k → A and the multiplication µ : A⊗A → A are morphisms in Vects . In other words, A is a Z2-graded k-algebra in the usual sense: A = A0̄ ⊕ A1̄ with k-subspaces A0̄ and A1̄ such that Ar̄As̄ ⊆ Ar+s. Homomorphisms of superalgebras, by definition, are algebra maps in Vects , that is, they preserve the Z2-grading. If V is a vector superspace in Vects then the tensor algebra T(V ) = n≥0 V ⊗n is a superalgebra via the Z2-grading of each V ⊗n as in §1.2. In general, if A is any superalgebra, then by selecting a Z2-graded subspace V ⊆ A which generates the algebra A, we obtain a canonical isomorphism of superalgebras T(V )/(R) −→ A (2.1) where (R) is the two-sided ideal of T(V ) that is generated by a Z2-graded linear subspace R ⊆ T(V ). Given superalgebras A and B, the tensor product A⊗ B is the superalgebra with the usual additive structure and grading and with multiplication µA⊗B defined by using the supersym- metry map (1.1): µA⊗B = (µA ⊗ µB) ◦ (IdA⊗cB,A ⊗ IdB) or, explicitly, (a⊗ b)(a′ ⊗ b′) = (−1) ba′bbaa′ ⊗ bb′ for homogeneous a′ ∈ A and b ∈ B. In other words, the canonical images of A and B in A⊗ B supercommute, in the sense that the supercommutator [a, b] = ab− (−1)ba bbba (2.2) vanishes for any pair of homogeneous elements a ∈ A and b ∈ B. Supercoalgebras, superbialgebras etc. are defined similarly as suitable objects of Vects such that all structure maps are maps in Vects . The compatibility between the comultiplica- tion ∆ and the multiplication of a superbialgebra B amounts to the following rule: ∆(ab) = (a),(b) (−1)ba(2) bb(1)a(1)b(1) ⊗ a(2)b(2) for homogeneous elements a, b ∈ B. Here we use the Sweedler notation ∆(a) = (a) a(1) ⊗ a(2) and a(1), a(2) are chosen homogeneous with â(1) + â(2) = â. Example 2.1 (Symmetric superalgebra [40, 3.2.5]). The symmetric superalgebra of a given V in Vects is defined by S(V ) = T(V )/ ([v,w]⊗ | v,w ∈ V ) where [v,w]⊗ is the supercommutator (2.2) in T(V ). Ignoring parity, S(V ) is isomorphic to S(V0̄)⊗Λ(V1̄), where S( . ) and Λ( . ) denote the ordinary symmetric and exterior (Grassmann) algebras, respectively. The symmetric superalgebra is a Hopf superalgebra: comultiplication ∆: S(V ) → S(V )⊗ S(V ) is given by ∆(v) = v ⊗ 1 + 1⊗ v for v ∈ V and extension to all N -HOMOGENEOUS SUPERALGEBRAS 11 of S(V ) by multiplicativity. Similarly, the counit ε : S(V ) → k is given by ε(v) = 0 and the antipode S : S(V ) → S(V ) by S(v) = −v for v ∈ V . 2.2. Comodules. We refer to [29, Chap. III] for background on comodules, comodule alge- bras etc. Given a superbialgebra B, we let ComodsB denote the category of all right B-comodules and B-comodule maps in Vects . Thus, for any object V in ComodsB, we have a “coaction” morphism δV : V → V ⊗B in Vects . If x1, . . . , xd is a fixed basis of V consisting of homogeneous elements, with î denoting the parity of xi as before, then we will write δV (xj) = xi ⊗ b j with b j ∈ Bbi+bj (2.3) The tensor product of vector superspaces makes ComodsB into a tensor category: if U and V are in ComodsB then B coacts on U ⊗ V by δU⊗V : U ⊗ V δU⊗δV −→ U ⊗ B ⊗ V ⊗ B −→ U ⊗ V ⊗ B ⊗ B Id⊗µB −→ U ⊗ V ⊗ B (2.4) If B is supercommutative as a superalgebra then the supersymmetry cU,V is a B-comodule morphism, i.e., δV⊗U ◦ cU,V = (cU,V ⊗ IdB) ◦ δU⊗V . Therefore Comod B is a symmetric tensor category in this case . 2.3. The supercharacter map. Let B denote a superbialgebra and let V be a finite dimen- sional object in ComodsB. The coaction δV is an even map in Homk(V, V ⊗B). Consider the following morphism in Vects χs : End δV ◦( . ) −→ Hom (V, V ⊗ B) (1.6) (V )⊗ B str⊗Id −→ k⊗ B = B (2.5) where str is the supertrace as in (1.8). This map will be called the supercharacter map of V . Forgetting parity and viewing all elements as even, the supertrace becomes the ordinary trace and the supercharacter becomes the usual character. These will be denoted by tr and χ, respectively. In particular, we have the element χsV := χ s(IdV ) ∈ B0̄ To obtain explicit formulas, fix a basis x1, . . . , xd of V consisting of homogeneous elements and let (F ij ) and (b j) be the matrices of f ∈ Endk(V ) and of δV with respect to this basis as in (1.2), (2.3). Then χs(f) = bibjbijF i (2.6) Let ε : B → k denote the counit of B. Then xj = i xiε(b j) holds in (2.3). Hence ε(bij) = δ j .1k and (2.6) gives ε(χs(f)) = str(f) (2.7) 12 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ When f is even formula (2.6) becomes χs(f) = i,j(−1) bibijF i , because F i = 0 unless î+ ĵ = 0̄. In particular, χsV = bibii (2.8) In the following, we let comodsB denote the full subcategory of Comod B consisting of all objects that are finite-dimensional over k. The supercharacter has the following properties analogous to standard properties of the ordinary character. Lemma 2.2. Let B denote a superbialgebra and let U , V and W be objects of comodsB . (a) If f : V → U and g : U → V are B-comodule maps (not necessarily even) then χs(f ◦ g) = (−1) bfbgχs(g ◦ f) (b) For f ∈ End (V ), g ∈ End (U) view f ⊗ g ∈ End (V ⊗ U) as in (1.7). Then χs(f ⊗ g) = χs(f)χs(g) (c) Given an exact sequence 0 → U −→ W → 0 in comodsB, let f ∈ Endk(V ) be such that f(µ(U)) ⊆ µ(U), and let g ∈ End (U), h ∈ End (W ) be the maps induced by f . Then χs(f) = χs(g) + χs(h) In particular, χsV = χ U + χ W . Moreover, if f ∈ EndcomodsB(V ) is a projection (i.e., f2 = f ) then χs(f) = χsIm f . Proof. (a) Let TV denote the map Hom (V, V ⊗ B) −→ B in (2.5); so χs(f) = TV (δV ◦ f). Since f and g are comodule maps, we have δU ◦ f = (f ⊗ IdB) ◦ δV and similarly for g. Putting h = δU ◦ f ∈ Hom (V,U ⊗ B) we obtain χs(f ◦ g) = TU (δU ◦ f ◦ g) = TU (h ◦ g) and χs(g ◦ f) = TV (δV ◦ g ◦ f) = TV ((g ⊗ IdB) ◦ h). Therefore, we must show that TU (h ◦ g) = (−1) bfbgTV ((g ⊗ IdB) ◦ h) Using the identification Hom (V,U⊗B) ∼= Hom (V,U)⊗B as in (1.6), write h = i fi⊗bi with fi ∈ Hom (V,U), bi ∈ B, and f̂i + b̂i = ĥ = f̂ . Then h ◦ g ∈ Hom (U,U ⊗ B) becomes the element ( i fi ⊗ bi) ◦ g = i(−1) bbibg(fi ◦ g) ⊗ bi ∈ End (U) ⊗ B, and (g ⊗ IdB) ◦ hHom (V, V ⊗B) becomes i(g ◦ fi)⊗ bi. The standard identity str(fi ◦ g) = bfibgstr(g ◦ fi) (cf., e.g., [40, p. 165 §3(b)]) now yields TU (h ◦ g) = bbibgstr(fi ◦ g)⊗ bi bbibg+bfibgstr(g ◦ fi)⊗ bi = (−1) bfbgTV ((g ⊗ IdB) ◦ h) as desired. (b) Fix homogeneous k-bases {xi} and {yℓ} of V and U , respectively, and write x̂i = î, ŷℓ = ℓ̂ as usual. Moreover, let (F j ) and (G m) be the matrices of f and g for these bases, as N -HOMOGENEOUS SUPERALGEBRAS 13 in (1.2). Then {xi ⊗ yℓ} is a basis of V ⊗ U , with xi ⊗ yℓ having parity î+ ℓ̂. Moreover, (f ⊗ g)(xj ⊗ ym) = (−1) bgbjf(xj)⊗ g(ym) = (−1)bg xi ⊗ yℓΦ j,m with Φ j,m = (−1) (bℓ+ bm)bjF ijG because Gmℓ = 0 unless ℓ̂+ m̂ = ĝ. Similarly, writing δV (xj) = i xi ⊗ b j with b j ∈ Bbi+bj and δU (ym) = ℓ yℓ ⊗ c m with c m ∈ Bbℓ+bm, one obtains using (2.4) δV⊗U (xj ⊗ ym) = xi ⊗ yℓ ⊗Ψ j,m with Ψ j,m = (−1) (bi+bj)bℓbijc Therefore, formula (2.6) becomes χs(f ⊗ g) = i,ℓ,j,m (−1)( bi+bℓ)(bj+ bm)Ψ i,ℓ,j,m bibj+bℓbmbijF = χs(f)χs(g) (c) Choose a basis {xi} of V consisting of homogeneous elements so that xi = µ(yi) for i ≤ dimU and let (F ij ) be the matrix of f for this basis. Then F j = 0 for i > dimU , j ≤ dimU . Moreover, the yi form a basis of U and the zi = π(xi) form a basis of W , and the matrices of g and h for these bases are (F ij )i,j≤dimU and (F j )i,j>dimU , respectively. Similarly, if (bij) is the matrix of δV with respect to the basis basis {xi} as in (2.3) then b j = 0 for i > dimU , j ≤ dimU , and the matrices of δU and δW for the given bases are (b j)i,j≤dimU and (bij)i,j>dimU , respectively. Therefore, χs(f) = bibjbijF i,j≤dimU bibjbijF i,j>dimU bibjbijF = χs(g) + χs(h) The remaining assertions are clear. � 2.4. The Grothendieck ring. Let B be a superbialgebra and let RB = K0(comod denote the Grothendieck group of the category comodsB. Thus, for each V in comod B, there is an element [V ] ∈ RB and each short exact sequence 0 → U → V → W → 0 in comod gives rise to an equation [V ] = [U ] + [W ] in RB. The group RB is in fact a ring with multiplication given by the tensor product of B-comodules. If B is supercommutative as a superalgebra then the ring RB is commutative; see §2.2. 14 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Both the ordinary dimension and the superdimension are additive on short exact sequences and multiplicative on tensor products. Hence they yield ring homomorphisms dim, sdim: RB → Z Parts (b) and (c) of Lemma 2.2 and formula (2.7) have the following immediate consequence: Corollary 2.3. The map [V ] 7→ χsV yields a well-defined ring homomorphism χ s : RB → B0̄. Furthermore, the following diagram commutes Z can. Forgetting the Z2-grading, the corollary also gives the more familiar version with χ and dim in place of χs and sdim, respectively. 2.5. General linear supergroup and Berezinian. Let V in Vects be finite-dimensional and fix a standard basis x1, . . . , xd with î = 0̄ (i ≤ p) and î = 1̄ (i > p). 2.5.1. For each supercommutative k-superalgebra R we denote by E(V )(R) the set of all R-linear maps V ⊗ R → V ⊗ R in Vects . Using the identification EndR(V ⊗ R) ∼= (V, V ⊗R) ∼= End (V )⊗R (see (1.6)), we may view E(V )(R) as the even subspace of End (V )⊗R: E(V )(R) = (End (V )⊗R)0̄ This defines a functor E(V ) from the category of supercommutative k-superalgebras to the category of semigroups. 2.5.2. Tensoring the supertrace str : End (V ) → k of (1.8) with IdR, we obtain an R-linear supertrace map str : End (V )⊗R → R in Vects which restricts to a map E(V )(R) → R0̄. The given standard basis x1, . . . , xd of V is an R-basis of V ⊗ R. In terms of this basis, an element φ ∈ E(V )(R) is given by φ(xj) = j with Φ j ∈ Rbi+bj (2.9) Thus φ is described by a supermatrix Φ = in standard form over R: (2.10) where A = i,j≤p and D = i,j>p are square matrices with entries in R0̄ while C,D are matrices over R1̄. The supertrace of φ is given by str(φ) = biΦii = tr(A)− tr(D) =: str(Φ) N -HOMOGENEOUS SUPERALGEBRAS 15 2.5.3. The functor E(V ) is represented by a supercommutative k-superbialgebra which coacts on V ; this algebra will be denoted by B = O(E(V )) Thus, there is a natural isomorphism of E(V ) with the functor Hom(B, ?) of parity preserving algebra homorphisms. In particular, the identity map on B corresponds to an element ξ ∈ E(V )(B). Let X = (xij)d×d be the matrix of ξ, as in (2.9). The elements x j have parity î + ĵ and they form a set of supercommuting algebraically independent generators of B. In fact, B is isomorphic to the symmetric superalgebra S(V ∗ ⊗ V ), with xij 7→ x i ⊗ xj , where {xi} ⊆ V ∗ is the dual basis for the given basis of V . We can think of X as the generic supermatrix with respect to the given basis of V : any supermatrix Φ = as in (2.9) comes from an algebra map B → R via xij 7→ Φ j . The canonical coaction δ : V → V ⊗B, the comultiplication ∆ and the counit ε of B are given by δ(xj) = xi ⊗ x ∆(xij) = xik ⊗ x ε(xij) = δ (2.11) These formulas can also be written as δ(x1, . . . , xd) = (x1, . . . , xd) ⊗X, ∆(X) = X ⊗X and ε(X) = 1. 2.5.4. Similarly, GL(V )(R) is defined, for any supercommutative k-superalgebra R, as the set of all invertible R-linear endomorphism of V ⊗R in Vects . The condition for a superma- trix Φ in standard form (as in (2.10)) to be invertible is that A and D are invertible as ordinary matrices over R0̄. In this case, the inverse of Φ is given by Φ−1 = (A−BD−1C)−1 −A−1B(D − CA−1B)−1 −D−1C(A−BD−1C)−1 (D − CA−1B)−1 See Berezin [3, Theorem 3.1 and Lemma 3.2]. The element ber(Φ) := det(A) det(D − CA−1B)−1 = det(D)−1 det(A−BD−1C) (2.12) is called the superdeterminant or Berezinian of Φ; it is an invertible element of R0̄. The functor GL(V ) is represented by a supercommutative Hopf superalgebra O(GL(V )) which is generated over B = O(E(V )) by det(X11)−1 and det(X22)−1, where X11 =( i,j≤p and X22 = i,j>p are the even blocks of the generic supermatrix X. By [3, Theorem 3.3], the Berezinian ber(X) is a group-like element in O(GL(V )). 2.6. Supersymmetric functions and exterior powers. Throughout this section, V will de- note a finite-dimensional vector superspace over k. We assume that the characteristic of k is zero. 16 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ 2.6.1. Let sgn(σ)σ ∈ k[Sn] be the antisymmetrizer idempotent of the group algebra k[Sn] and define nV := Im cYn ⊆ V ⊗n (2.13) where c : k[Sn] → EndVects (V ⊗n) is as in (1.10). Thus, ΛnV is the space of antisymmetric n-tensors, nV = {y ∈ V ⊗n | cσ(y) = sgn(σ)y for all σ ∈ Sn} For later use, we describe an explicit basis of ΛnV . To this end, fix a standard basis x1, . . . , xd of V , with î = 0̄ for i ≤ p and î = 1̄ for i > p. Then the products xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xin for sequences i = (i1, i2, . . . , in) ∈ {1, 2, . . . , d} n form a graded basis of V ⊗n that is permuted up to a ±-sign by the action of Sn on V ⊗n; see formula (1.9): cσ(xi) = sgni(σ)xσ(i) (2.14) sgni(σ) = (−1) (p,q)∈inv(σ) bip biq and σ(i) = (iσ−1(1), iσ−1(2), . . . , iσ−1(n)) Therefore, by elementary properties of monomial group representations, a k-basis of ΛnV is given by the nonzero elements cYn(xi) where i ranges over a transversal for the Sn- action on {1, 2, . . . , d}n. Such a transversal is provided by the weakly increasing sequences i ∈ {1, 2, . . . , d}n. Moreover, for a weakly increasing i, it is easily seen from (2.14) that cYn(xi) = 0 holds precisely if iℓ = iℓ+1 ≤ p for some ℓ. Therefore, a basis of Λ nV is given by the elements cYn(xi) with i = (i1 < i2 < · · · < im < im+1 ≤ · · · ≤ in) ∈ {1, 2, . . . , d} and im ≤ p < im+1. In particular, m+m′=n q +m′ − 1 (2.15) where p = dim V0̄ and q = dimk V1̄. Equivalently, the generating power series in ZJtK for the sequence dim nV is given by nV tn = (1 + t)p (1− t)q (2.16) When q > 0 then all ΛnV are nonzero. For additional details on exterior powers, see, e.g., [43, Sections I.5 and I.7]. 2.6.2. Consider the super bialgebra B = O(E(V )) as defined in §2.5.3 and recall that V is in comodsB. The representation c : k[Sn] → EndVects (V ⊗n) of (1.10) actually has image in EndcomodsB(V ⊗n), since B is supercommutative. Therefore, ΛnV also belongs to comodsB and we can define the nth elementary supersymmetric function by en := χ ΛnV = χ s(cYn) ∈ B0̄ Here, the equality χs ΛnV = χ s(cYn) holds by Lemma 2.2(c). Similarly, one defines the nth super power sum by pn := χ s(c(1,2,...,n)) ∈ B0̄ N -HOMOGENEOUS SUPERALGEBRAS 17 where (1, 2, . . . , n) ∈ Sn the cyclic permutation mapping 1 7→ 2 7→ 3 7→ . . . 7→ n 7→ 1. In terms of the generic supermatrix X from §2.5.3, one has pn = str(X Modulo the space spanned by the Lie commutators fg− gf with f, g ∈ k[Sn], the follow- ing relation is easily seen to hold in k[Sn]: nYn ≡ (−1)i−1(1, 2, . . . , i)Yn−i (with Y0 = 1). Applying the function χ s ◦ c : k[Sn] → B0̄ to this relation and using Lemma 2.2(a),(b), one obtains the Newton relations: nen = (−1)i−1pien−i Let t be a formal parameter (of parity 0̄) and consider the generating functions P (t) =∑ n≥1 pnt n−1 and E(t) = n≥0 ent n in B0̄JtK. The Newton relations can be written in the form P (−t) = d logE(t); see, e.g., [34, p. 23]. Combining this with the identity ber(exp(tX)) = exp(str(tX)) due to Berezin ([3, Chapter 3] or [40, p. 167]), one obtains the following expansion for the characteristic function ber(1 + tX) of generic supermatrix X: Proposition 2.4. ber(1 + tX) = n≥0 ent This proposition is known; see, e.g., Khudaverdian and Voronov [30, Prop. 1]. 3. HOMOGENEOUS SUPERALGEBRAS 3.1. N -homogeneous superalgebras. Let N be an integer with N ≥ 2. A homogeneous superalgebra of degree N or N -homogeneous superalgebra is an algebra A of the form (2.1) with V finite-dimensional and R ⊆ V ⊗N : A = A(V,R) ∼= T(V )/(R) The assumption R ⊆ V ⊗N implies that, besides the usual Z2-grading (“parity”), A also has a connected Z+-grading (“degree”), The algebra A is generated by A1 = V and all homogeneous components An are finite- dimensional objects of Vects . In fact, An ∼= V ⊗n/Rn with Rn := (R) ∩ V i+j+N=n V ⊗i ⊗R⊗ V ⊗j (3.1) Note that Rn = 0 for n < N ; so An ∼= V ⊗n if n < N . Morphisms of N -homogeneous superalgebras f : A = A(V,R) → A′ = A(V ′, R′) are morphism of superalgebras which also respect the Z+-grading. Equivalently, by restricting to degree 1, we have a morphism f1 : A1 = V → A 1 = V ′ in Vects whose N th tensor power sat- isfies f⊗N1 (R) ⊆ R ′. Thus, one has a category HNAlg of N -homogeneous k-superalgebras. 18 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Finally, N -homogeneous superalgebras with N = 2 are called quadratic superalgebras; for N = 3, they are called cubic, etc.. 3.2. Some examples. In order to explicitly describe a certain N -homogeneous superalgebra A = A(V,R), we will usually fix a Z2-graded k-basis x1, . . . , xd of V = A1 and denote the the parity of xi by î, as in §1.1. The xi form a set of algebra generators for A. Following Manin [38],[39], the d-tuple f = (1̂, . . . , d̂) ∈ Zd2 is called the format of the basis {xi}. Example 3.1 (Quantum superspace [39]). For a fixed family q of scalars 0 6= qij ∈ k (1 ≤ i < j ≤ d) and a given format f = (1̂, . . . , d̂) ∈ Zd2 of the basis x1, . . . , xd, the quadratic superalgebra A = Sfq is defined as the factor of T(V ) modulo the ideal generated by the elements ri := xi ⊗ xi ∈ (V ⊗2)0̄ (̂i = 1̄) (3.2) rij := xj ⊗ xi − qij(−1) bibjxi ⊗ xj ∈ (V bi+bj (i < j) (3.3) Thus, the algebra Sfq is generated by x1, . . . , xd subject to the defining relations xixi = 0 (̂i = 1̄) xjxi = qij(−1) bibjxixj (i < j). In the special case where all qij = 1, the algebra S q is the symmetric superalgebra S(V ) of V as in Example 2.1. The ordered monomials of the form xm11 x 2 . . . x , with i mi = n, mi ≥ 0 for all i and mi ≤ 1 if î = 1̄, form a k-basis of the n th homogeneous component of Sfq. Therefore, (Sfq)n = r+s=n r + p− 1 (3.4) where dimV0̄ = p and dimV1̄ = q as usual.Thus, the generating series of the dimensions is (Sfq)nt (1 + t)q (1− t)p Example 3.2 (Yang-Mills algebras [11],[10]). Fix a collection of elements x1, . . . , xd (d ≥ 2), numbered so as to have parity î = 0̄ for i ≤ p and î = 1̄ for i > p. Let G = (gij) ∈ GLd(k) be an invertible symmetric d × d-matrix satisfying gij = 0 if î 6= ĵ and consider the cubic superalgebra A that is generated by elements x1, . . . , xd subject to the relations gij [xi, [xj , xk]] = 0 (k = 1, . . . , d) (3.5) Here [ . , . ] is the supercommutator (2.2). The algebra A will be denoted by YMp|q (q = d−p). In particular, the pure even algebra YMd|0 is the ordinary Yang-Mills algebra introduced in [10] while YM0|d is the super Yang-Mills algebra as in [11]. As usual, put V = i kxi and let [ . , . ]⊗ denote the supercommutator in T(V ). Further- more, put rk = i,j gij [xi, [xj , xk]⊗]⊗ and R = k krk ⊆ V ⊗3; so YMp|q = T(V )/(R). Using the symmetry of G, we may replace the rk by simpler relations as follows. Choose an N -HOMOGENEOUS SUPERALGEBRAS 19 invertible d × d-matrix C = (cij) with cij = 0 if î 6= ĵ and such that C trGC is diagonal, i,j cirgijcjs = gsδ s . Replace the bases {xi} of V and {rk} of R by the new bases ijxj and sk = kℓrℓ where C −1 = (cij). Note that yi has parity î and sk has parity k̂, the parity of rk. A simple calculation shows that sk = i 6=k gi[yi, [yi, yk]⊗]⊗. Thus we obtain the following defining relations for the generators y1, . . . yd of YM i 6=k gi[yi, [yi, yk]] = 0 (k = 1, . . . , d) (3.6) The resulting algebras for d = 2 are as follows. Putting x = y1 and y = y2 we have two defining relations: [x, [x, y]] = 0 and [y, [y, x]] = 0. In the pure even case (x̂ = ŷ = 0̄), the supercommutators are the ordinary Lie commutators. So YM2|0 is the enveloping algebra of the Heisenberg Lie algebra; see [1, (0.4)]. In the pure odd case (x̂ = ŷ = 1̄), the two relations can be written as x2y = yx2 and yx2 = x2y. The resulting algebra YM0|2 is a cubic Artin-Schelter algebra of type S1 [1, (8.6)]. Thus, both unmixed algebras are Artin-Schelter regular of global dimension 3. In the mixed case, however (x̂ = 0̄, ŷ = 1̄), the relations say that x commutes with the Lie commutator [x, y] while y anticommutes: y[x, y] = −[x, y]y. Thus, [x, y] is a normal element of YM1|1 and YM1|1/([x, y]) is a polynomial algebra in two variables over k. Moreover, the calculation [x, y]2 = [x, [x, y]y] = −[x, y[x, y]] = −[x, y]2 shows that [x, y]2 = 0. Thus, the algebra YM1|1 is noetherian with Gelfand-Kirillov dimen- sion 2 and infinite global dimension. Returning to the case of general d ≥ 2, we now concentrate on the unmixed algebras intro- duced by Connes and Dubois-Violette. We will denote these algebras by YM+ = YMd|0 and YM− = YM0|d. In all formulas below, + applies to YM+ and − to YM−. The generators i 6=k gi[yi, [yi, yk]⊗]⊗ of the space of relations R can be written as sk = ℓ yℓ⊗mℓk = ℓmkℓ ⊗ yℓ with mℓk = gℓ (yℓ ⊗ yk − (1± 1)yk ⊗ yℓ) for ℓ 6= k i 6=k giyi ⊗ yi for ℓ = k Thus, putting Y = (y1, . . . , yd) and letting M denote the d × d-matrix over YM ± whose (ℓ, k)-entry is the image of mℓk, the defining relations (3.6) can be written as YM = 0 or MY tr = 0 (3.7) The defining relations (3.6) for A = YM− amount to the even element i giy i ∈ A2 being central in A. Example 3.3 (N -symmetric superalgebra; cf. [5]). Let N ≥ 2 be given and let V be a vector superspace V over a field k with char k = 0 or char k > N . Define SN (V ) = A(V,R) with R = Λ NV = cYN ⊆ V ⊗N where YN is the antisymmetrizer idempotent of the group algebra k[SN ]; see (2.13). This defines a functor SN ( . ) : Vect → HNAlg . Since 2cY2 is the supercommutator in T(V ), the algebra S2(V ) is just the symmetric superalgebra S(V ) of V ; see Example 3.1. The algebra SN (V ), for a pure even space V = V0̄ and general N ≥ 2, has been introduced in [5]. 20 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ If 2 ≤ M ≤ N then, viewing k[SM ] as a subalgebra of k[SN ] as usual, the antisymmetriz- ers of k[SN ] and k[SM ] satisfy YN = YMa for some a ∈ k[SN ]. Therefore, R = cYN ⊆ cYM = cYM ⊗ V ⊗(N−M) This shows that the identity map on V extends to an epimorphism of superalgebras SN(V ) ։ SM (V ). Now assume that dimk V = d and fix a standard basis x1, . . . , xd of V , with î = 0̄ for i ≤ p and î = 1̄ for i > p. From the basis for ΛNV exhibited in §2.6.1 we obtain that the algebra SN (V ) is generated by x1, . . . , xd subject to the relations (p,q)∈inv(σ) 1+ bip biqxi σ−1(1) σ−1(2) . . . xi σ−1(N) with 1 ≤ i1 < i2 < · · · < im ≤ p = dim V0̄ < im+1 ≤ · · · ≤ iN ≤ d = dimk V ; see formula (2.14). Example 3.4. The following construction generalizes Example 3.3. Fix N ≥ 2 and 0 6= q ∈ k and assume that condition (1.14) is satisfied. Given a Hecke operator R : V ⊗2 → V ⊗2 on a vector superspace V we define the N -homogeneous superalgebra ΛR,N := A(V,R) with R = Im ρR(XN ) ⊆ V ⊗N (3.8) where XN ∈ HN,q is the q-symmetrizer (1.16) and ρR is the representation (1.20) of HN,q. We also put SR,N := Λ−qR−1,N = A(V,R) with R = Im ρR(YN ) ⊆ V ⊗N (3.9) where YN ∈ HN,q is the antisymmetrizer (1.17). The algebra SN(V ) in Example 3.3 is identical with ScV,V ,N (q = 1). 3.3. The dual of a homogeneous superalgebra. Let A = A(V,R) be an N -homogeneous superalgebra. The dual A! of A is defined by A! = A(V ∗, R⊥) where, R⊥ ⊆ V ∗⊗N is the (homogeneous) subspace consisting of all elements that vanish on R ⊆ V ⊗N , using (1.4) in order to evaluate elements of V ∗⊗N on V ⊗N . Thus, (3.1) takes the A!n = V ∗⊗n/R⊥n with R i+j+N=n V ∗⊗i ⊗R⊥ ⊗ V ∗⊗j (3.10) Identifying V ∗⊗n with the linear dual of V ⊗n via (1.4), we have V ∗⊗i ⊗ R⊥ ⊗ V ∗⊗j =( V ⊗j ⊗R⊗ V ⊗i . Hence, R⊥n = i+j+N=n V ⊗j ⊗R⊗ V ⊗i (3.11) The canonical isomorphism V −→ V ∗∗ in (1.5) leads to an isomorphism V ⊗N V ∗∗⊗N which maps R onto R⊥⊥. Hence, A! ! ∼= A (3.12) N -HOMOGENEOUS SUPERALGEBRAS 21 Moreover, if f : A = A(V,R) → A′ = A(V ′, R′) is any morphism in HNAlg then the transpose of f1 : V → V ′ induces a morphism f ! : (A′)! → A! in HNAlg . Thus, we have a contravariant quasi-involutive dualization functor A 7→ A!, f 7→ f ! on HNAlg Example 3.5. The dual of A(V, 0) = T(V ) is A(V ∗, V ∗⊗N ); so T(V )! = T(V ∗)/ V ∗⊗N In particular, letting V = k be the unit object of Vects , we have A(k, 0) = k[t] (polynomial algebra) and A(k, 0)! = k[d]/(dN ), with t and d both having degree 1 and parity 0̄. Example 3.6 (Dual of quantum superspace). We will describe the dual A! of quantum su- perspace A = Sfq; see Example 3.1. Fix a homogeneous k-basis x1, . . . , xd with format f for V , and let x1, . . . , xd denote the dual basis of V ∗; this basis also has format f . Evaluat- ing an arbitrary element f = ℓ,m fℓmx ℓ ⊗ xm ∈ V ∗⊗2 on one of the generating relations ri, rij ∈ R in (3.2), (3.3) we obtain 〈f, ri〉 = fii and 〈f, rij〉 = fij − qij(−1) bibjfji. Therefore, the space R⊥ ⊆ V ∗⊗2 has a basis consisting of the elements sℓ := xℓ ⊗ xℓ (ℓ̂ = 0̄) and sℓ,k := xℓ ⊗ xk + qkℓ(−1) bkbℓxk ⊗ xℓ (k < ℓ). In summary, A! is generated by x1, . . . , xd subject to the defining relations xℓxℓ = 0 (ℓ̂ = 0̄) xℓxk = −qkℓ(−1) bkbℓxkxℓ (k < ℓ). Thus, A! is isomorphic to quantum superspace Sf with q′ij = (−1) bi+bjqij and f ′ = f + (1̄, . . . , 1̄) the format obtained from f by parity reversal in all components. Example 3.7 (Duals of the Yang-Mills algebras). Continuing with the notation of Exam- ple 3.2, we now desribe the algebra A! for A = YMp|q. We assume that char k = 0 and work with generators y1, . . . , yd of A satisfying (3.6). Let y1, . . . , yd denote the basis of V ∗ given by 〈yi, yj〉 = δ j and put γ = yi ∈ V ∗⊗2. Then, for the generators sk = i 6=k gi[yi, [yi, yk]⊗]⊗ of R as in Example 3.2, one computes 〈ya ⊗ yb ⊗ yc, sk〉 = gcδ k + (−1) bbgbδ k − (−1) babk(1 + (−1)ba)gaδ 〈yi ⊗ γ, sk〉 = δ (3.13) Therefore, the map ϕ 7→ ϕ− k〈ϕ, sk〉y k ⊗ γ is an epimorphism V ∗⊗3 ։ R⊥ ⊂ V ∗⊗3. We obtain that the algebra A! is generated by y1, . . . , yd subject to the relations yaybyc = (gcδ a + (−1) bbgbδ c − (−1)ba bb(1 + (−1)ba)gaδ b)g (3.14) where g = 1 iyi is the image of γ in A. Since A! is 3-homogeneous, we clearly have A!0 = k, A i = V ∗ and A!2 =⊕ i,j ky iyj ∼= V ∗⊗2. By (3.13), the elements yag form a k-basis of A!3 = V ∗⊗3/R⊥ ∼= R∗. Using the defining relations (3.14) it is not hard to see that A!4 = kg 2 and A!n = 0 for n ≥ 5. If A = YMp|q is of mixed type (i.e., p 6= 0 and q 6= 0) then g2 = 0. 22 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Example 3.8 (Dual of the N -symmetric superalgebra). Recall from Example 3.3 that SN (V ) = A(V,R) with R = cYN . Since YN is central in k[SN ] and stable under the inversion involution ∗ of k[SN ], it follows from (1.11) that 〈x, cYN (y)〉 = 〈cYN (x), y〉 holds for all x ∈ V ∗⊗N and y ∈ V ⊗N . Therefore, R⊥ = KerV ∗⊗N (cYN ) = (1− cYN ) V ∗⊗N and so SN (V ) ! = A V ∗, (1 − cYN )(V ∗⊗N ) Note that ⋂ i+j+N=n V ⊗i ⊗R⊗ V ⊗j = cYn (3.15) holds for all n ≥ N . This follows from (1.19). Alternatively, as has been noted in Exam- ple 3.3, we have cYn (V ⊗n) ⊆ R ⊗ V ⊗(n−N). In the same way, one sees that cYn (V ⊗n) ⊆ V ⊗i ⊗ R ⊗ V ⊗j whenever i + j + N = n. For the reverse inclusion, note that each x ∈ V ⊗i ⊗ R ⊗ V ⊗j satisfies cσℓ(x) = −x for all transpositions σℓ = (ℓ, ℓ + 1) ∈ Sn with i < ℓ < i+N . Hence, the left hand side of (3.15) is contained in the space of antisym- metric n-tensors, ΛnV = cYn (V ⊗n), thereby proving (3.15). We deduce from (3.10), (3.11) and (2.15) that SN (V ) dn if n < N r+s=n q+s−1 if n ≥ N (3.16) where d = dim V , p = dim V0̄ and q = dimk V1̄. 3.4. The operations ◦ and • on HNAlg . Let A = A(V,R) and A′ = A(V ′, R′) be N - homogeneous superalgebras. Following [37] and [7] we define the white and black products A ◦ A′ and A • A′ by A ◦A′ = A V ⊗ V ′, cπN R⊗ V ′⊗N + V ⊗N ⊗R′ A •A′ = A V ⊗ V ′, cπN where πN ∈ S2N is the inverse of the permutation (1, 2, . . . , 2N) 7→ (1, N + 1, 2, N + 2, . . . , k,N + k, . . . ,N, 2N) Explicitly, cπN : V ⊗N ⊗ V ′⊗N −→ (V ⊗ V ′)⊗N is the morphism in Vects that is given by v1 ⊗ . . . vN ⊗ v 1 ⊗ . . . v = (−1) bvj (v1 ⊗ v 1)⊗ . . . (vN ⊗ v N ) (3.17) Hence, cπN (R⊗R ′) and cπN R⊗ V ′⊗N + V ⊗N ⊗R′ are homogeneous subspaces of (V⊗ V ′)⊗N and so A ◦ A′ and A • A′ belong to HNAlg Under the isomorphism (V ′∗⊗V ∗)⊗N −→ (V ⊗V ′)∗⊗N which comes from (1.4), the rela- tions cπN R′⊥ ⊗R⊥ of A′! • A! map onto the relations R⊗ V ′⊗N + V ⊗N ⊗R′ of (A ◦ A′)!. In fact, by (1.11) we have c∗πN = cπN , because πN τ = τπN , and so 〈x, y〉 = 〈cπN (x), cπN (y)〉 holds for all x ∈ V ′∗ ⊗N ⊗ V ∗⊗N and y ∈ V ⊗N ⊗ V ′⊗N . Therefore, canonically, (A ◦A′)! ∼= A ′! • A! and (A • A′)! ∼= A′! ◦ A! (3.18) N -HOMOGENEOUS SUPERALGEBRAS 23 the two identities being equivalent by (3.12). By definition of ◦, the canonical isomorphisms k⊗V ∼= V ∼= V ⊗ k in Vects give isomor- phisms A(k, 0) ◦ A ∼= A ∼= A ◦ A(k, 0) in HNAlg , and (3.18) yields similar isomorphisms for •, with A(k, 0)! = k[d]/(dN ) replacing A(k, 0) = k[t]; see Example 3.5. The supersymmetry isomorphism cV,V ′ : V ⊗ V ′ ∼−→ V ′ ⊗ V in Vects (see (1.1)) yields isomorphisms A ◦ A′ ∼= A ′ ◦ A and A • A′ ∼= A′ • A (3.19) in HNAlg . To see this, note that the following diagram of isomorphisms in Vects commutes: V ⊗N ⊗ V ′⊗N cπN // V⊗N,V ′⊗N (V ⊗ V ′) V,V ′ V ′⊗N ⊗ V ⊗N cπN // (V ′ ⊗ V ) with v1⊗. . . vN⊗v 1⊗. . . v N 7→ (−1) bvj (v′1⊗v1)⊗. . . (v N⊗vN ) in both composites. Therefore, putting RA◦A′ = cπN R⊗ V ′⊗N + V ⊗N ⊗R′ and similarly for RA′◦A etc., we V,V ′ (RA◦A′) = cπN ◦ cV ⊗N ,V ′⊗N R⊗ V ′⊗N + V ⊗N ⊗R′ = cπN R′ ⊗ V ⊗N + V ′⊗N ⊗R = RA′◦A In the same way, one sees that c⊗N V,V ′ (RA•A′) = RA′•A. This proves (3.19). Similarly, the associativity isomorphism aV,V ′,V ′′ : (V ⊗ V ′) ⊗ V ′′ ∼= V ⊗ (V ′ ⊗ V ′′) in Vects leads to isomorphisms (A ◦ A′) ◦ A′′ ∼= A ◦ (A ′ ◦ A′′) and (A •A′) • A′′ ∼= A • (A′ • A′′) (3.20) in HNAlg . This is a consequence of the following commutative diagram of isomorphisms in Vects V ⊗N ⊗ V ′⊗N ⊗ V ′′⊗N cπN⊗Id V ⊗N,V ′⊗N,V ′′⊗N (V ⊗ V ′) ⊗ V ′′⊗N cπN // ((V ⊗ V ′)⊗ V ′′) V,V ′,V ′′ V ⊗N ⊗ V ′⊗N ⊗ V ′′⊗N Id⊗cπN // V ⊗N ⊗ (V ′ ⊗ V ′′)⊗N cπN // (V ⊗ (V ′ ⊗ V ′′)) Finally, the compatibility between the isomorphisms cV,V ′ and aV,V ′,V ′′ (see §1.2) is inher- ited by the isomorphisms (3.19) and (3.20) in HNAlg . To summarize: Proposition 3.9. The operations ◦ and • both make the category HNAlg of N -homogeneous k-superalgebras into a symmetric tensor category, with unit objects A(k, 0) = k[t] for ◦ and A(k, 0)! = k[d]/(dN ) for •. 24 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ 3.5. The superalgebra map i : A◦A′ → A⊗A′. Let A = A(V,R) and A′ = A(V ′, R′) be objects of HNAlg . The superalgebra A⊗A′ is generated by V ⊕ V ′ subject to the relations R+R′ ⊆ (V ⊕ V ′)⊗N and [V, V ′]⊗ ⊆ (V ⊕ V where [ . , . ]⊗ is the supercommutator (2.2) in the tensor algebra, as usual. Thus, A ⊗ A not N -homogeneous when N ≥ 3. Nonetheless, there always is an injective superalgebra homomorphism i : A ◦ A′ → A ⊗ A′ which is defined as follows. The linear embedding V ⊗ V ′ →֒ T(V )⊗ T(V ′) extends uniquely to a superalgebra map ι̃ : T(V ⊗ V ′) → T(V )⊗ T(V ′) (3.21) which doubles degrees: the restriction of ι̃ to degree n is the embedding T(V ⊗ V ′)n = (V ⊗ V −→ V ⊗n ⊗ V ′⊗n ⊆ (T(V )⊗ T(V ′))2n in Vects , where cπn is as in (3.17). Thus, ι̃ identifies the superalgebra T(V ⊗ V ′) with the (super) Segre product n≥0 V ⊗n ⊗ V ′⊗n of T(V ) and T(V ′). The map ι̃ sends RA◦A′ = cπN R⊗ V ′⊗N + V ⊗N ⊗R′ ⊆ (V ⊗V ′)⊗N to R⊗V ′⊗N + V ⊗N ⊗R′, the kernel of the canonical epimorphism V ⊗N ⊗ V ′⊗N ։ AN ⊗A N . Thus: Proposition 3.10. The algebra map ι̃ in (3.21) passes down to yield an injective homomor- phism k-superalgebras i : A ◦ A′ ֌ A ⊗ A′ which doubles degree. The image of i is the super Segre product n≥0An ⊗A n of A and A 3.6. Internal Hom. The isomorphisms (1.3) and (1.4) together with associativity lead to a functorial isomorphism (U ⊗ V,W ∗) ∼= Hom (U, (V ⊗W )∗) in Vects . Explicitly, if g ∈ Hom (U ⊗ V,W ∗) and g′ ∈ Hom (U, (V ⊗W )∗) correspond to each other under the above isomorphism then 〈g(u ⊗ v), w〉 = 〈g′(u), v ⊗ w〉 (3.22) holds for all u ∈ U , v ∈ V and w ∈ W . In particular, by restricting to 0̄-components, we have a k-linear isomorphism HomVects (U ⊗ V,W ∗) ∼= HomVects (U, (V ⊗W )∗) (3.23) This isomorphism leads to Proposition 3.11. There is a functorial isomorphism HomHNAlgs (A • B, C) ∼= HomHNAlgs (A, C ◦ B!) Proof. We follow Manin [37, 4.2]. Let A = A(U,R), B = A(V, S) and C = A(W,T ) be N -homogeneous superalgebras. We will prove the proposition in the following equivalent form; see (3.12) and (3.18): HomHNAlgs (A • B, C!) ∼= HomHNAlgs (A, (B • C)!) Recall that C! = A(W ∗, T⊥) and (B•C)! = A((V ⊗W )∗, (cπN (S⊗T )) ⊥). Let g : U⊗V → W ∗ be a morphism in Vects and let g′ : U → (V ⊗ W )∗ be the morphism in Vects N -HOMOGENEOUS SUPERALGEBRAS 25 corresponds to g under (3.23). We must show that, for homogeneous subspaces R ⊆ U⊗N , S ⊆ V ⊗N and T ⊆ W⊗N , g⊗N (cπN (R ⊗ S)) ⊆ T ⊥ ⇔ g′⊗N (R) (cπN (S ⊗ T )) Identifying T⊥⊥ with T as in §3.3, the first inclusion is equivalent to 〈g⊗N (cπN (R ⊗ S)) , T 〉 = 0 (3.24) while the second inclusion states that 〈g′⊗N (R), cπN (S ⊗ T )〉 = 0 (3.25) But (3.22) shows that (3.24) and (3.25) are equivalent, which proves the proposition. � Proposition 3.11 says that the tensor category (HNAlg , •) has an internal Hom which is given by Hom(A,B) = B ◦ A! Explicitly, Hom(A,B) is an object of HNAlg which represents the functor (HNAlg )op → Sets, X 7→ HomHNAlgs (X • A,B); so there is an isomorphism of functors HomHNAlgs (? • A,B) ∼= HomHNAlgs (?,Hom(A,B)) By general properties of Hom (see [12, Def. 1.6]), the morphism IdHom(A,B) corresponds to a morphism µ : Hom(A,B) • A → B (3.26) in HNAlg satisfying the following universal property: for any morphism f : X • A → B in HNAlg there exists a unique morphism g : X → Hom(A,B) such that the following diagram commutes: X • A g•IdA Hom(A,B) • A In degree 1, the map µ is simply IdV ⊗evU : V ⊗ U ∗ ⊗ U −→ V ⊗ k = V . From Hom(B, C) • Hom(A,B) • A Id •µ −→ Hom(B, C) • B −→ C one obtains in this way a composition morphism m : Hom(B, C) •Hom(A,B) → Hom(A, C) (3.27) in HNAlg . The morphisms µ and m satisfy the obvious associativity properties. 3.7. The superbialgebra endA. Following Manin [37, 4.2] we define hom(A,B) = Hom(A!,B!)! = A! • B for A, B in HNAlg . Applying the dualization functor to (3.26), (3.27) and recalling (3.18), we obtain morphisms δ◦ : A → B ◦ hom(B,A) ∆◦ : hom(A, C) → hom(A,B) ◦ hom(B, C) 26 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ in HNAlg . The associativity properties of µ and m translate into corresponding coassociativ- ity properties for δ◦ and ∆◦. Following δ◦ and ∆◦ by the algebra map i of Proposition 3.10, we obtain superalgebra maps δ : A → B ⊗ hom(B,A) (3.28) ∆: hom(A, C) → hom(A,B)⊗ hom(B, C) (3.29) Now take A = B = C = A(V,R) and put endA = hom(A,A); so endA = A! • A = A(V ∗ ⊗ V, cπN (R ⊥ ⊗R)) (3.30) Then (3.29) yields a coassociative superalgebra map ∆: endA → endA⊗ endA Moreover, by Proposition 3.11, the morphism A! −→ A! ∼= k[t] ◦ A! corresponds to a mor- phism endA = A! • A → k[t] in HNAlg . Following this morphism by the map t 7→ 1 we obtain a superalgebra map ε : endA → k which in degree 1 is the usual evaluation pairing evV : V ∗ ⊗ V → k in Vects . Finally, (3.28) provides us with a superalgebra map δA : A → A⊗ endA (3.31) Note that δA maps the degree n-component of A according to −→ (A ◦ endA)n −→ An ⊗ (endA)n →֒ An ⊗ endA (3.32) Fixing a graded k-basis x1, . . . , xd of V and denoting the dual basis of V ∗ by x1, . . . , xd as before, endA has algebra generators zij := x i ⊗ xj (3.33) of degree-1 and parity î+ ĵ. In terms of these generators, the maps ε, δA and ∆ are given by i ) = δ i or ε(Z) = 1 δA(xj) = xi ⊗ z j or δA(x1, . . . , xd) = (x1, . . . , xd)⊗ Z ∆(zij) = zik ⊗ z j or ∆(Z) = Z ⊗ Z (3.34) where Z = (zij)d×d. Proposition 3.12. Let A = A(V,R) be an N -homogeneous k-superalgebra. (a) With ∆ as comultiplication and ε as counit, the superalgebra endA becomes a super- bialgebra. Moreover, δA makes A into a graded right endA-comodule superalgebra. N -HOMOGENEOUS SUPERALGEBRAS 27 (b) Given any k-superalgebra B and a morphism of superalgebras δ : A → A⊗ B satis- fying δ(V ) ⊆ V ⊗ B, there is a unique morphism of superalgebras ϕ : endA → B such that the following diagram commutes: δA $$J A⊗ endA IdA ⊗ϕ The proposition is proved as in [37, §5] or [7, Theorem 3]. Example 3.13. When A = A(V, 0) = T(V ), we have endA = A(V ∗⊗V, 0) = T(V ∗⊗V ); endT(V ) = T(V ∗ ⊗ V ) the free superalgebra generated by the elements zij in (3.33). Example 3.14. By Examples 3.3 and 3.8, we have endSN (V ) = A V ∗ ⊗ V, cπN (1− cYN )(V ∗⊗N )⊗ cYN (V For example, the algebra end S2(V ) is generated by the elements z j with parity î+ ĵ subject to the relations ] + (−1) bi1 bi2+(bi1+bi2)bj1 [z ] = 0 where [ . , . ] is the supercommutator (2.2). This algebra is highly noncommutative, even for a pure even space V . Let O(E(V )) = S(V ∗ ⊗ V ) be the supercommutative superbialgebra as in §2.5.3, with generators xij . There is a map of superbialgebras ϕ : endSN (V ) → O(E(V )) , z j 7→ x j (3.35) Indeed, write B = O(E(V )) for brevity and recall the coaction δ : V → V ⊗ B, xj 7→∑ i xi ⊗ x j from (2.11). Since cYN ∈ EndcomodsB(V ⊗N ) (see §2.6.2), the map δ extends to a map of superalgebras δ : SN (V ) → SN (V )⊗ B Therefore, Proposition 3.12(b) yields the desired ϕ. Note that the coaction of end SN (V ) on V , when restricted along ϕ, becomes the canonical coaction of O(E(V )) on V ; see (2.11) and (3.34). 4. N -KOSZUL SUPERALGEBRAS Throughout this section, we fix an N -homogeneous superalgebra A = A(V,R). 28 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ 4.1. The graded dual A!∗. The graded dual A!∗ = A! ∗n of A! has a natural structure of a graded right endA-comodule. Indeed, the linear dual A! ∗n of the degree n-component of A! embeds into V ⊗n as follows. Recall from (3.11) that A! ∗n = V ⊗n if n < N i+j+N=n V ⊗i ⊗R⊗ V ⊗j if n ≥ N (4.1) This identification makes the graded dual A!∗ into a graded right endA-comodule. For, by (3.32) the coaction δA restricts in degree 1 to a map V → V ⊗endA which makes T(V ) into a graded right endA-comodule superalgebra. The structure map T(V ) → T(V )⊗endA sends R → R ⊗ endA. Therefore, each V ⊗i ⊗ R ⊗ V ⊗j is a endA-subcomodule of V ⊗(i+j+N), and hence A! ∗n is a endA-subcomodule of V ⊗n. Finally, for all n ≥ 0, A! ∗n+1 ⊆ V ⊗A n and A n+N ⊆ V ⊗N ⊗A! ∗n ∩R⊗ V ⊗n = R⊗A! ∗n (4.2) 4.2. The Koszul complex. The map A⊗ V ⊗(i+1) → A⊗ V ⊗i a⊗ (v1 ⊗ · · · ⊗ vi+1) 7→ av1 ⊗ (v2 ⊗ · · · ⊗ vi+1) is a morphism in the category ComodsendA of right endA-comodules, because the endA- coaction δA in (3.31) is a superalgebra map. Furthermore, this map is a left A-module map which preserves total degree, and it restricts to a map of endA-subcomodules d : A⊗A! ∗i+1 → AV ⊗A i →֒ A ⊗A which is the A-linear extension of the embedding (4.2). The map dN sends A! ∗i+N to AR ⊗ A! ∗i = 0; so d N = 0. In other words, we have an N -complex K(A) : . . . −→ A⊗A! ∗i+1 −→ A⊗A! ∗i −→ . . . −→ A −→ 0 (4.3) in ComodsendA consisting of graded-free left A-modules and A-module maps which pre- serve total degree. Therefore, K(A) splits into a direct sum of N -complexes K(A)n =⊕ i+j=nAi ⊗A j in comod endA. Following [7], the Koszul complex K(A) defined by Berger in [5] can be described as the following contraction of K(A): K(A) : . . . −→ A⊗A! ∗N+1 −→ A⊗A! ∗N −→ A⊗A! ∗1 −→ A −→ 0 (4.4) This is an ordinary complex in ComodsendA which splits into a direct sum of complexes K(A)n in comodsendA. The i th components of K(A) and of K(A)n are given by K(A)i = A⊗A ν(i) and K(A) i = An−ν(i) ⊗A with ν(i) = νN (i) as in (0.1). The differential on K(A) is δi : K(A)i → K(A)i−1 where δi = dN−1 for i even d for i odd N -HOMOGENEOUS SUPERALGEBRAS 29 Writing A+ = n>0 An = AV as usual, we have Ker δi ⊆ A+K(A)i for all i. Indeed, this is clear for odd i, since δi = d is injective on A . For even i, the restriction of δi = d N−1 to A! ∗ is given by dN−1 : A! ∗ = A! ∗ ν(i−1)+N−1 →֒ V ⊗(N−1) ⊗ ν(i−1) −→ AN−1 ⊗A ν(i−1) →֒ A ⊗A! ∗ ν(i−1) where the first embedding comes from (4.2). Since A! ∗ = A! ∗1 = V and A = A! ∗N = R by (4.1), the start of the Koszul complex, augmented by the canonical map A ։ k = A/A+ is as follows: −→ A⊗ V δ1=mult −→ A −→ k −→ 0 (4.5) This piece is easily seen to be exact: writing A = T(V )/I with I = (R) = I⊗V +T(V )⊗R as in (0.2), the map T(V )+ = T(V ) ⊗ V ։ A ⊗ V ։ A+ has kernel I . Thus, Ker δ1 = I/I ⊗ V = Im δ2. Hence (4.5) is the start of the minimal graded-free resolution of the left A-module k. 4.3. N -homogeneous Koszul superalgebras. Recall from the Introduction that an N -homoge- neous superalgebra A is called N -Koszul if TorAi (k, k) is concentrated in degree νN (i) for all i ≥ 0. By [5, Proposition 2.12] or [8, Theorem 2.4], this happens exactly if the Koszul complex K(A) is exact in degrees i > 0 and in view of (4.5), this amounts to exactness of K(A) in degrees i ≥ 2. In this case, K(A) −→ k −→ 0 is the minimal graded-free resolution of the trivial left A-module k. The Yoneda Ext-algebra E(A) = i≥0 Ext A(k, k) of an N -Koszul superalgebra A has the following description in terms of the dual algebra A!: ExtiA(k, k) ν(i) (i ≥ 0) Moreover, identifying ExtiA(k, k) and A , the Yoneda product f · g and the A!-product fg for f ∈ ExtiA(k, k) = A and g ∈ ExtjA(k, k) = A are related by f · g = (−1)ijfg when N = 2, and f · g = fg if i or j is even 0 if i and j are both odd for N > 2; see [21, Theorem 9.1], [8, Proposition 3.1]. Example 4.1. Quadratic algebras having a PBW-basis are 2-Koszul; see, e.g., [41, Chap. 4, Theorem 3.1]. This applies in particular to quantum superspace A = Afq; see Example 3.1. A PBW-basis in this case is given by the collection of ordered monomials xm11 x 2 . . . x d with mi ≥ 0 for all i and mi ≤ 1 if î = 1̄, as in Example 3.1. For a more general result, see [41, Chap. 4, Theorem 8.1]. 30 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Example 4.2. The unmixed Yang-Mills algebras A = YM± (see Example 3.2) were shown to be 3-Koszul in [10], [11]. Indeed, letting A[ℓ] denotes the shift of A that is defined by A[ℓ]n = Aℓ+n, the defining relations for A in the form (3.7) imply that the following complex of graded-free left A-modules is exact: 0 −→ A[−4] −→ A[−3]d −→ A[−1]d ·Y tr −→ A −→ k −→ 0 (4.6) The piece A[−3]d −→ A[−1]d ·Y tr −→ A −→ k −→ 0 is identical with (4.5). Therefore, (4.6) is the minimal graded-free resolution of k. The resolution shows that each TorAi (k, k) is con- centrated in degree ν3(i), and hence A is 3-Koszul. It also follows that (4.6) is isomorphic to K(A) → k → 0. In particular, (4.6) confirms the dimensions of the corresponding compo- nents A!n in Example 3.7. As has been pointed out in [10], [11], it follows from (4.6) that the Hilbert series HA(t) = n≥0 dimkAn t n of A = YM± has the form HA(t) = 1− dt+ dt3 − t4 (1− t2)(1− dt+ t2) If d > 2 then the series has a pole in the interval (0, 1), and hence dim An grows exponen- tially with n. Therefore, A is not noetherian in this case; see Stephenson and Zhang [42]. The mixed Yang-Mills algebras A = YMp|q with p 6= 0 and q 6= 0, on the other hand, are never 3-Koszul. For YM1|1 this follows from the description given in Example 3.2: this alge- bra has infinite global dimension. In general, one can check that the so-called extra condition (see (4.10) below) fails for A, and so A cannot be Koszul, by [5, Prop. 2.7]. Example 4.3. It has been shown in [5, Theorem 3.13] that the N -symmetric algebra SN(V ) of a pure even space V over a field of characteristic 0 is N -Koszul. An extension of this result will be offered in Theorem 4.5 below. 4.4. Confluence and Koszulity. For the convenience of the reader, we recall the notions of reduction operators and confluence and their relation to the Koszul property. Complete details can be found in Berger [4], [5]. Let V in Vects be given along with a graded basis X = {x1, . . . xd} that is ordered by x1 > x2 > · · · > xd. The tensors (“monomials”) xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xiN for i = (i1, i2, . . . , iN ) ∈ {1, 2, . . . , d} N form a basis of V ⊗N which will be given the lexicographical ordering. An X-reduction operator on V ⊗N is a projection S ∈ EndVects (V ⊗N ) such that either S(xi) = xi or S(xi) < xi holds for each i, where the latter inequality means that S(xi) is a linear combination (possibly 0) of monomials < xi. The monomials xi satisfying S(xi) = xi are called S-reduced, all other monomials are S-nonreduced. We denote by Red(S) and NRed(S)) the (super) subspaces of V ⊗N that are generated by the S-reduced monomials and the S-nonreduced monomials, respectively; so V ⊗N = Red(S) ⊕ NRed(S) and Im(S) = Red(S). Let LX(V ⊗N ) denote the collection of all X-reduction operators on V ⊗N . The proof of [4, Theorem 2.3] shows that the application S 7→ Ker(S) is a bijection between LX(V and the set of all super subspaces of V ⊗N . Hence LX(V ⊗N ) inherits a lattice structure: for S, S′ ∈ LX(V ⊗N ) one has X-reduction operators S ∧ S′ and S ∨ S′ on V ⊗N which are N -HOMOGENEOUS SUPERALGEBRAS 31 defined by Ker(S ∧ S′) = Ker(S) + Ker(S′) Ker(S ∨ S′) = Ker(S) ∩Ker(S′) A pair (S, S′) of X-reduction operators is said to be confluent if Red(S ∨ S′) = Red(S) + Red(S′) Since the inclusion ⊇ is always true, confluence of (S, S′) is equivalent to the inequality Im(S ∨ S′) ≤ dim (Im(S) + Im(S′)) (4.7) Let n ≥ N . Any X-reduction operator S on V ⊗N gives rise to X-reduction operators Sn,i on V ⊗n which are defined by Sn,i := IdV ⊗i ⊗S ⊗ IdV ⊗j (i+ j +N = n) A monomial xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xin of length n ≥ N is said to be S-reduced if xi is Sn,i-reduced for all i, that is, if every connected submonomial of xi of length N is S-reduced. Now let A = A(V,R) be an N -homogeneous superalgebra, and let S be the X-reduction operator on V ⊗N such that Ker(S) = R. The algebra A is said to be X-confluent if the pairs (SN+i,i, SN+i,0) of X-reduction operators on V ⊗N+i are confluent for i = 1, . . . , N − 1. By (4.7) this amounts to the inequalities Im(SN+i,i ∨ SN+i,0) ≤ dim (Im(SN+i,i) + Im(SN+i,0)) (4.8) being satisfied for i = 1, . . . , N − 1. Following Berger [5], we denote by Tn the lattice of super subspaces of V ⊗n that is gen- erated by the subspaces Rn,i := V ⊗i ⊗R⊗ V ⊗j = Ker(Sn,i) (i+ j +N = n) (4.9) The superalgebra A is said to be distributive if the lattices Tn are distributive for all n, that is, C ∩ (D + E) = (C ∩D) + (C ∩ E) holds for all C,D,E ∈ Tn. The following proposition states the operative facts concerning Koszulity for our purposes. Part (a) is identical with [5, Thm. 3.11] while (b) is [5, Prop. 3.4]. Proposition 4.4. Let A = A(V,R) be an N -homogeneous superalgebra. (a) If A is X-confluent for some totally ordered graded basis X of V then A is distributive. Moreover, letting S denote the X-reduction operator on V ⊗N such that Ker(S) = R, the classes in A of the S-reduced monomials xi1 ⊗ xi2 ⊗ · · · ⊗ xin with xij ∈ X form a k-basis of An for all n ≥ N . (b) Assume that A is distributive and the following “extra condition” is satisfied Rn+N,0 ∩Rn+N,n ⊆ Rn+N,n−1 (2 ≤ n ≤ N − 1) (4.10) Then A is N -Koszul. After these preparations, we are now ready to prove the following result. The quadratic case N = 2 is due to Gurevich [22]; see also Wambst [44]. Theorem 4.5. Let N ≥ 2 and 0 6= q ∈ k and assume that [n]q 6= 0 for all n ≥ 1. Then, for every Hecke operator R associated with q, the N -homogeneous superalgebra ΛR,N defined in (3.8) is N -Koszul. 32 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Proof. Put A = ΛR,N and recall that A = A(V,R) with R = Im ρN,R(XN ) ⊆ V The extra condition (4.10) is a consequence of equation (1.19). Indeed, (1.19) implies that the spaces Rn,i in (4.9) have the form Rn,i = i+N−1⋂ s=i+1 Im(ρn,R(Ts) + 1) ⊆ V ⊗n (4.11) Applying (4.11) with ρ = ρn+N,R we see that the left hand side of (4.10) is identical to Im(ρ(Ti) + 1) ∩ n+N−1⋂ i=n+1 Im(ρ(Ti) + 1) = n+N−1⋂ Im(ρ(Ti) + 1) where the equality holds because n + 1 ≤ N . The last expression is clearly contained in⋂n+N−2 i=n Im(ρ(Ti) + 1), which is identical to the right hand side of (4.10). This establishes the extra condition (4.10). In order to prove the distributivity of A, we follow the approach taken in [25]. We first prove the claim for the standard solution RDJ , i.e., the operator given in Example 1.2 with d = p and q = 0. As above, fix a basis X = {x1, . . . , xd} of V , ordered by x1 > x2 > · · · > xd, and consider the basis of V ⊗n consisting of the monomials xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xin for i = (i1, i2, . . . , in) ∈ {1, 2, . . . , d} n with the lexicographical ordering. By equation (1.21), the action of the generators Tj of the Hecke algebra H = Hn,q2 on this basis is given by Tj(xi) = q2xi if ij = ij+1 (q2 − 1)xi + qxσj(i) if ij < ij+1 qxσj(i) if ij > ij+1 (4.12) Here, σj = (j, j + 1) ∈ Sn and σ(i) = (iσ−1(1), iσ−1(2), . . . , iσ−1(n)) for σ ∈ Sn, as in Example 3.3. We claim that the H -submodule of V ⊗n that is generated by xi is given by H (xi) = i′∈Sn(i) kxi′ (4.13) where Sn(i) is the Sn-orbit of i. Indeed, (4.12) implies that each Tσ(xi) with σ ∈ Sn is a linear combination of basis vectors xi′ with i ′ ∈ Sn(i). Hence, ⊆ certainly holds in (4.13). For the reverse inclusion, let i∗ denote the unique non-decreasing sequence in Sn(i); so xi∗ = max{xi′ | i ′ ∈ Sn(i)}. The last formula in (4.12) implies that T (xi) = q r(i)xi∗ (4.14) where T is a suitable finite product of length r(i) ≥ 0 in the generators Tj . Since T is a unit in H , the inclusion ⊇ holds in (4.13), thereby proving the asserted equality. Furthermore, (4.14) and (1.18) (with q replaced by q2) give qr(i)Xn(xi) = Xn(xi∗). (4.15) N -HOMOGENEOUS SUPERALGEBRAS 33 These elements are nonzero. For, (4.15) implies that the elements Xn(xi∗) span the image of Xn on V ⊗n, and their number is d+n−1 which is equal to the rank of Xn (cf. [25, Eq. (5)]). It follows that Xn(V ⊗n) = Im ρn,RDJ (Xn) has a k-basis consisting of the elements Xn(xi∗) ∣∣ i∗ = (i1 ≤ i2 ≤ · · · ≤ in) ∈ {1, 2, . . . , d}n Next, writing Xn(xi) = i′∈Sn(i) λi′xi′ (4.16) with λi′ ∈ k, we claim that λσj(i′) = λi′ if i ′ = σj(i q±1λi′ otherwise To prove this, we may assume that i′ 6= σj(i ′). We compute the coefficient of xσj(i′) in TjXn(xi) in two ways: by (1.18) this coefficient is equal to q 2λσj(i′) while (4.12) yields the expression qλi′ + (q 2 − q1±1)λσj(i′). The claim follows from this. Writing an arbitrary σ ∈ Sn as a product of the inversions σj , we see that the coefficients λi′ in (4.16) only differ by a nonzero scalar, and hence they are all nonzero since Xn(xi) 6= 0. By Proposition 4.4, it suffices to check the X-confluence conditions (4.8) i = 1, . . . , N−1. So let S be the X-reduction operator on V ⊗N with Ker(S) = R. It is easy to see from the dis- cussion above (with n = N ) that S is given by S(xi∗) = (1−XN/λi∗)(xi∗) and S(xi) = xi for i 6= i∗. According to (4.11) and the discussion above, the dimension of (R⊗V ⊗i)∩(V ⊗i⊗ R) is d+N+i−1 . Thus, the dimension of the left hand side of (4.8) is dN+i− d+N+i−1 the other hand the monomials in V ⊗N+i that belong to NRed(SN+i,i) ∩ NRed(SN+i,0) are exactly those of the form xi∗ with i ∗ ∈ {1, . . . , d}N+i non-decreasing. Their number is pre- cisely d+N+i−1 . Therefore, the dimension of Im(SN+i,i) + Im(SN+i,0) = Red(SN+i,i) + Red(SN+i,0) is at least d N+i − d+N+i−1 . This proves the inequality in (4.8), thereby fin- ishing the proof of the theorem for the case R = RDJ . In order to deal with an arbitrary Hecke operator R, recall that Hn,q is split semisimple, having a representative set of simple modules Mλ indexed by the partitions λ ⊢ n; see (1.15). We denote the representation of Hn,q on Mλ by ρλ; it does not depend on the operator R but only on the partition λ. Let us fix a decomposition V ⊗n = into simple Hn,q-submodules Mt. Since all Mt are invariant under the operators ρn,R(Tj), formula (4.11) yields the decomposition Rn,i = i+N−1⋂ s=i+1 (ρn,R(Ts) + 1)(Mt) = Rn,i ∩Mt for all i. Therefore, by [25, Lemma 1.2], distributivity of the lattice Tn that is generated by the subspaces Rn,i of V ⊗n is equivalent to distributivity of the lattices Tn ∩Mt (t ∈ T ) that 34 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ are generated by the subspaces Rn,i ∩Mt = i+N−1⋂ s=i+1 (ρn,R(Ts) + 1)(Mt) of Mt. Now, each Mt is isomorphic to Mλ for some λ ⊢ n. Therefore, the lattice Tn ∩Mt is isomorphic to the lattice of subspaces of Mλ that is generated by the subspaces i+N−1⋂ s=i+1 (ρλ(Ts) + 1)(Mλ) with i + N ≤ n. Finally, when d = dimV > n, then all simple Hn,q-modules Mλ appear in V ⊗n; see [15, Proposition 5.1]. Thus, the distributivity of the lattice associated to RDJ , which we have already verified, implies the distributivity of the corresponding lattice for any Hecke operator R. This completes the proof. � 5. KOSZUL DUALITY AND MASTER THEOREM In this section, A = A(V,R) denotes an N -homogeneous superalgebra that is assumed to be N -Koszul (N ≥ 2). 5.1. By Koszulity, the complexes K(A)n : . . . → An−νN(i) ⊗A νN (i) → An−νN (i−1) ⊗A νN (i−1) → . . .→An → 0 are exact for n > 0. This yields equations in the Grothendieck ring RendA of the category comodsendA : ∑ (−1)i[An−νN (i)][A νN (i) ] = 0 (n > 0) (5.1) In the power series ring RendAJtK over the Grothendieck ring RendA, define the Poincaré series PA(t) = [An]t n and PA!∗(t) = [A! ∗n ]t For any power series P (t) = n ant n, we use the notation PN (t) := n≡0,1 mod N (−1)αN (n)ant where αN (n) = n − (n mod N) denotes the largest multiple of N less than or equal to n. Thus, P2(t) = P (t) and in general PN (−t) = n≡0,1 mod N (−1)n mod Nant (−1)iaνN (i)t νN (i) (5.2) In particular, PA!∗,N (−t) = (−1)i[A! ∗νN (i)]t νN (i) Equations (5.1) are equivalent to the following Koszul duality formula: N -HOMOGENEOUS SUPERALGEBRAS 35 Proposition 5.1. For any N -homogeneous Koszul superalgebra A, the identity PA(t)PA!∗,N (−t) = 1 holds in RendAJtK. Applying the ring homomorphism χsJtK : RendAJtK → (endA)0̄JtK, where χ s is the su- percharacter map as in Corollary 2.3, the formula in Proposition 5.1 takes the following form in (endA)0̄JtK: Corollary 5.2. χsAℓt m≡0,1 mod N (−1)m mod Nχs A! ∗m  = 1 Analogous formulas hold with the supercharacter χs replaced by the ordinary character χ or by one of the dimensions dim and sdim. By (3.32) the coaction of endA on A sends An to An⊗ (endA)n. A similar remark holds for the endA-coaction on A! ∗; see §4.1. Therefore, both factors in Corollary 5.2 actually belong to the Rees subring n≥0 Bnt n of BJtK, where we have put B = (endA)0̄. Example 5.3. As an application of the Hilbert series version of Corollary 5.2, we see that the duals A! of the Yang-Mills algebras A = YMp|q are never 3-Koszul. In fact, by Example 3.7, we have HA!(t) = 1+dt+d 2t2+dt3+t4 if p = 0 or q = 0 and HA!(t) = 1+dt+d 2t2+dt3 otherwise. In either case, HA!(t) −1 has a nonzero coefficient at t5, which rules out Koszulity. 5.2. A master theorem modeled on the N -symmetric superalgebra SN (V ). We put A = SN (V ) and use the notation of Examples 3.3 and 3.8. In particular, we assume that char k = 0 and work with a fixed basis x1, . . . , xd of V = A1 so that î = 0̄ for i ≤ p and î = 1̄ for i > p. From Example 3.3 (see also Proposition 4.4(a)), we know that a basis of Aℓ is given by the monomials xi = xi1xi2 . . . xiℓ for sequences i = (i1, . . . , iℓ) ∈ {1, . . . , d} ℓ such that i has no connected subsequence j = (j1, . . . , jN ) of length N satisfying 1 ≤ j1 < . . . < jm ≤ p < jm+1 ≤ . . . ≤ jN ≤ d = p+ q for some m. Adapting notation of Etingof and Pak [16] to our setting, we denote this set of sequences i by Λ(p|q,N)ℓ (5.3) For example, Λ(p|q, 2)ℓ consists of all weakly decreasing sequences i = (i1, . . . , iℓ) with entries from {1, . . . , d} and such that no repetition occurs in the range {p + 1, . . . , d}. In order to evaluate the character χsAℓ in Corollary 5.2, recall from (3.34) that the coaction δA : A → A⊗ endA is given on the generators xi of A by δA(xi) = xj ⊗ z i ∈ A⊗ endA where zji = x j ⊗ xi are the canonical generators of the algebra endA. For i = (i1, . . . , iℓ) ∈ Λ(p|q,N)ℓ, we have δA(xi) = δA(xi1)δA(xi2) . . . δA(xiℓ) ∈ Aℓ ⊗ endA Since Aℓ ⊗ endA = i∈Λ(p|q,N)ℓ xi ⊗ endA, we can define Z(i) ∈ (endA)0̄ by δA(xi) = xi ⊗ Z(i) + (terms supported on Λ(p|q,N)ℓ \ {i}) 36 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Then (2.8) becomes χsAℓ = i∈Λ(p|q,N)ℓ biZ(i) (5.4) with î = î1 + · · ·+ îℓ. Now consider the super bialgebra B = O(E(V )) = k[xij | 1 ≤ i, j ≤ d] defined in §2.5.3 and recall that the xij are supercommuting variables of parity î + ĵ over k. Restricting the comodule Aℓ to B along the map ϕ : endSN (V ) → B, z j 7→ x j in (3.35) we must replace Z(i) in (5.4) by X(i) := ϕ(Z(i)) ∈ B0̄. Thus, writing xj ⊗ x i ∈ A⊗ B and yi = yi1 . . . yiℓ ∈ Aℓ ⊗ B = j∈Λ(p|q,N)ℓ xj ⊗ B for i = (i1, . . . , iℓ), we have yi = xi ⊗X(i) + (terms supported on Λ(p|q,N)ℓ \ {i}) (5.5) As for the supercharacter of A! ∗m , recall from (4.1) and (3.15) that, for all n ≥ N , A! ∗n = i+j+N=n V ⊗i ⊗R⊗ V ⊗j = ΛnV Viewing A! ∗n = Λ nV as a comodule over B = O(E(V )), the supercharacter of A! ∗n is the nth elementary supersymmetric function en which we know, by Proposition 2.4, to be iden- tical to the coefficient at tn of the characteristic function ber(1 + tX) of the generic su- permatrix X = 1≤i,j≤d of type p|q; so the diagonal blocks X11 = 1≤i,j≤p X22 = p+1≤i,j≤p+q consist of even entries while all other entries are odd. To summarize, we obtain the following super-version of [16, Theorem 2]. Theorem 5.4. Let X = be the generic supermatrix of type p|q. Then i∈Λ(p|q,N)ℓ biX(i) tℓ m≡0,1 mod N (−1)m mod Nemt  = 1 holds in the power series ring k[xij | all i, j ]0̄JtK. Here Λ(p|q,N)ℓ and X(i) are defined by (5.3) and (5.5), respectively, and the em are the coefficients of the characteristic function ber(1 + tX) = n≥0 ent n of X. 5.3. As an application of Theorem 5.4 , we determine the superdimension Hilbert series HsA(t) = for the N -symmetric superalgebra A = SN (V ). For the pure even case, this was already done by Etingof and Pak [16] . The notations of §5.2 remain in effect. In view of Corollary 2.3, the superdimension Poincaré series follows by applying the counit ε : B → k to the equation in Theorem 5.4. Indeed, by (2.11), the counit ε sends X 7→ 1d×d, N -HOMOGENEOUS SUPERALGEBRAS 37 and hence the elements X(i) in (5.5) all map to 1. Therefore, the first factor in Theorem 5.4 becomes HsA(t) = i∈Λ(p|q,N)ℓ For the second factor, note that ber(1 + t 1d×d) = (1 + t) by (2.12). Thus, HsA(t) = i∈Λ(p|q,N)ℓ   m≡0,1 mod N (−1)m mod N if p ≥ q m≡0,1 mod N (−1)αN (m) m+ q − p− 1 q − p− 1 if p < q (5.6) where αN (m) = m− (m mod N) denotes the largest multiple of N less than or equal to m as in §5.1. 5.4. The ordinary Hilbert series HA(t) = ℓ≥0 dimkAℓ t ℓ of the N -symmetric superalge- bra A = SN(V ) is as follows. Recall from §5.2 that Aℓ = |Λ(p|q,N)ℓ| and from (3.16) that A!n = dn if n < N r+s=n q+s−1 if n ≥ N Therefore, the Hilbert series is HA(t) = |Λ(p|q,N)ℓ| t m≡0,1 mod N (−1)m mod N r+s=m q + s− 1 −1 (5.7) 5.5. Less is known about the Hilbert series of the N -homogeneous superalgebras A = ΛR,N associated to an arbitrary Hecke operator R : V ⊗2 → V ⊗2 on a vector superspace V ; see Example 3.4. Recall that A = A(V,R) with R = Im ρR(XN ) ⊆ V ⊗N . For any N - homogeneous algebra A = A(V,R), we have A!n = dimk i+j+N=n V ⊗j ⊗R⊗ V ⊗i 38 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ by (3.10) and (3.11). For R = Im ρR(XN ) in particular, (1.19) further implies that i+j+N=n V ⊗j ⊗R⊗ V ⊗i = ρR(Xn) holds for n ≥ N . Now [26, Theorem 3.5] implies that !(t) = ℓ=1(1 + aℓt)∏s m=1(1− bmt) where (r, s) is the birank of R and aℓ and bm are positive real numbers. For example, in the situation of 5.4, (r, s) = (p, q) and aℓ = bm = 1. For any complex power series P (t), the power series PN (−t) in (5.2) can be written as PN (−t) = (1− ζ−iN )P (ζ where ζN = e 2πi/N . In particular, HA!∗,N (−t) = (1− ζ−iN ) ℓ=1(1 + aℓζ N t)∏s m=1(1− bmζ QN,a,b(t)∏s m=1(1 + bmt+ . . .+ b m tN−1) for some real polynomial QN,a,b(t) with coefficients being polynomial in a = (aℓ) and b = (bm). Therefore, the Hilbert series of A has the form HA(t) = m=1(1 + bmt+ . . .+ b QN,a,b(t) (5.8) Notice that the fraction on the right-hand side is reduced. In particular, (5.7) has the form HA(t) = (1− tN )s (1− t)sQN,1,1(t) (5.9) APPENDIX For lack of a suitable reference, we include here a proof of Proposition 0.1 that was stated in the Introduction. Our proof is based on the proof of [8, Proposition 2.1] and on additional details that were communicated to us by Roland Berger. For the basics concerning graded algebras, we refer the reader to [9, Chap. II §11] or [6]. As in the Introduction, A = n≥0 An denotes an arbitrary connected Z≥0-graded k- algebra and V is a graded subspace of A+ = n>0An satisfying A+ = V ⊕ A +. Thus, T(V )/I −→ A for some graded ideal I of T(V ). For convenience, we state Proposition 0.1 again: Proposition. The relation ideal I of A lives in degrees ≥ N if and only if TorAi (k, k) lives in degrees ≥ νN (i) = N if i is even N + 1 if i is odd N -HOMOGENEOUS SUPERALGEBRAS 39 Proof. Let P : · · · → Pi −→ Pi−1 −→ · · · −→ P0 −→ k → 0 be a minimal graded-free resolution of the trivial left A-module k. Thus, all Pi have the form Pi = A⊗ Ei for some graded subspace Ei ⊆ Ker di−1 which is chosen so that Ker di−1 = Ei ⊕A+Ker di−1 (A.10) In particular, we may take E0 = k and E1 = V . The differential di : Pi → Pi−1 is the graded A-module map that is defined by the inclusion Ei →֒ Pi−1. By the graded Nakayama Lemma (e.g., [9, p. AII.171, Prop. 6]), our choice of Ei implies that Im di = AEi = Ker di−1 and Ker di ⊆ A+ ⊗ Ei = A+Pi (A.11) for all i. Consequently, the complex k⊗A P has zero differential, and hence TorAi (k, k) ∼= k⊗A Pi ∼= Ei In particular, TorA0 (k, k) ∼= k and TorA1 (k, k) ∼= V = A+/A live in degrees 0 = νN (0) and ≥ 1 = νN (1), respectively. Moreover, the kernel of d1 : P1 = (T(V )/I)⊗ V → P0 = A is exactly I/I ⊗ V , and so TorA2 (k, k) ∼= Ker d1/A+Ker d1 ∼= I/ (V ⊗ I + I ⊗ V ) Therefore, I lives in degrees ≥ N if and only if TorA2 (k, k) lives in degrees ≥ N = νN (2). For the remainder of the proof, assume that I lives in degrees ≥ N . We will show by induction on i that TorAi (k, k) = Ei lives in degrees ≥ νN (i) for all i. The cases i ≤ 2 have been checked above. Assume that Ei lives in degrees ≥ νN (i) and similarly for Ei−1. By (A.11), we know that Ei+1 ⊆ Ker di ⊆ A+ ⊗ Ei and so Ei+1 certainly lives in degrees ≥ νN (i)+1. Since νN (i)+1 = νN (i+1) when i is even (or when i is arbitrary and N = 2), we are done in these cases. From now on, we assume that i is odd. We must show that Ei+1 lives in degrees ≥ νN (i + 1) = N . Since Ei+1 ⊆ Ker di, it suffices to show that di is injective in degrees < i+1 N , and since Ei lives in degrees ≥ νN (i) = N+1, our goal is to show that di is injective on all homogeneous components Pi,n of Pi in degrees n = N + j with j = 1, . . . , N − 1. Put m = i−1 N for simplicity and note that Pi,m+j = Aj−ℓ ⊗ Ei,m+ℓ (A.12) Pi−1,m+j = Aj−k ⊗ Ei−1,m+k (A.13) since Ei−1 lives in degrees ≥ νN (i − 1) = m. The proposition will be a consequence of the following claims: (a) di is injective on all summands Aj−ℓ ⊗ Ei,m+ℓ in (A.12), and (b) the subspaces di (Aj−ℓ ⊗ Ei,m+ℓ) = Aj−ℓEi,m+ℓ for ℓ = 1, . . . , j form a direct sum inside Pi−1,m+j . 40 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ In order to prove (a), recall that the restriction of di to Ei,m+ℓ is the inclusion Ei,m+ℓ →֒ Pi−1,m+ℓ = Aℓ−k ⊗ Ei−1,m+k Hence, the effect of di on the ℓ th summand in (A.12) is the embedding Aj−ℓ ⊗ Ei,m+ℓ →֒ Aj−ℓ ⊗Aℓ−k ⊗ Ei−1,m+k followed by the map Aj−ℓ ⊗Aℓ−k ⊗ Ei−1,m+k −→ Aj−k ⊗ Ei−1,m+k ⊆ Pi−1,m+j which is given by the multiplication map Aj−ℓ ⊗ Aℓ−k → Aj−k. Since j − k < N , our hypothesis on I implies that Aj−k ∼= T(V )j−k, and similarly Aj−ℓ ∼= T(V )j−ℓ and Aℓ−k ∼= T(V )ℓ−k. Therefore, the above multiplication map is identical with the injection T(V )j−ℓ ⊗ T(V )ℓ−k →֒ T(V )j−k in T(V ). This proves (a). For (b), we proceed by induction on j. The case j = 1 being obvious, let 1 ≤ j ≤ N − 2 and assume that (ii) holds for 1, . . . , j. We wish to show that the subspaces Aj+1−ℓEi,m+ℓ (ℓ = 1, . . . , j + 1) of Pi−1,m+j+1 form a direct sum. First, by (A.10) we have Ei,m+j+1 ∩ A+Ker di−1 = 0 while ℓ=1Aj+1−ℓEi,m+ℓ ⊆ A+Ker di−1. Therefore, it suffices to show that the sum ℓ=1Aj+1−ℓEi,m+ℓ is direct. To this end, note that Aj+1−ℓ = d≥1 VdAj+1−d−ℓ holds for all ℓ ≤ j. Hence, Aj+1−ℓEi,m+ℓ = Aj+1−d−ℓEi,m+ℓ By induction, Aj+1−d−ℓEi,m+ℓ is a direct sum inside Pi−1,m+j+1−d. Thus, it suffices to show that the sum d≥1 VdPi−1,m+j+1−d ⊆ Pi−1,m+j+1 is direct. But (A.13) gives Pi−1,m+j+1 = Aj+1−k ⊗ Ei−1,m+k = T(V )j+1−k ⊗ Ei−1,m+k where the last equality holds since all j + 1− k < N . Therefore, VdPi−1,m+j+1−d = j+1−d⊕ T(V )j+1−d−k ⊗ Ei−1,m+k as desired. This proves (b), thereby completing the proof of the proposition. � Acknowledgement. The authors wish to thank Roland Berger for his helpful comments throughout the completion of this paper. 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Soc. 125 (1997), no. 6, 1593–1605. [43] Gijs M. Tuynman, Supermanifolds and supergroups, Mathematics and its Applications, vol. 570, Kluwer Academic Publishers, Dordrecht, 2004. [44] Marc Wambst, Complexes de Koszul quantiques, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 4, 1089–1156. MATHEMATIK, UNIVERSITY OF DUISBURG-ESSEN, GERMANY AND INSTITUTE OF MATHEMATICS, HANOI, VIETNAM E-mail address: hai.phung@uni-duisburg-essen.de LAMUSE, FACULTÉ DES SCIENCES ET TECHNIQUES, UNIVERSITÉ DE SAINT-ETIENNE, 23 RUE DU DOC- TEUR PAUL MICHELON, 42023 SAINT-ETIENNE CEDEX 2, FRANCE E-mail address: benoit.kriegk@univ-st-etienne.fr DEPARTMENT OF MATHEMATICS, TEMPLE UNIVERSITY, PHILADELPHIA, PA 19122-6094, USA E-mail address: lorenz@temple.edu http://arxiv.org/abs/math/0603169 http://arxiv.org/abs/math/0703203 http://arxiv.org/abs/math/0703213 http://arxiv.org/abs/math/0607737 Introduction 1. Review of linear superalgebra 2. The supercharacter 3. Homogeneous superalgebras 4. N-Koszul superalgebras 5. Koszul duality and master theorem Appendix References

We develop the theory of N-homogeneous algebras in a super setting, with particular emphasis on the Koszul property. To any Hecke operator on a vector superspace, we associate certain superalgebras and generalizing the ordinary symmetric and Grassmann algebra, respectively. We prove that these algebras are N-Koszul. For the special case where the Hecke operator is the ordinary supersymmetry, we derive an $N$-generalized super-version of MacMahon's classical "master theorem".

Introduction 1 1. Review of linear superalgebra 5 2. The supercharacter 10 3. Homogeneous superalgebras 17 4. N -Koszul superalgebras 27 5. Koszul duality and master theorem 34 Appendix 38 References 41 INTRODUCTION 0.1. The theory of N -homogeneous algebras owes its existence primarily to the concerns of noncommutative geometry. In fact, as has been expounded by Manin in his landmark pub- lications [36], [37], quadratic algebras (the case N = 2) provide a convenient framework for the investigation of quantum group actions on noncommutative spaces. Moreover, certain Artin-Schelter regular algebras [1], natural noncommutative analogs of ordinary polynomial algebras, can be presented as associative algebras defined by cubic relations (N = 3). The latter algebras, as well as many of the quadratic algebras studied by Manin, enjoy the addi- tional “Koszul property” which will be of central importance in the present article; it will be reviewed in detail in 0.6 below. Motivated by these examples and others, Berger [5] initiated the systematic investigation of N -homogeneous algebras for all N ≥ 2, introducing in particular a natural extension of the notion of Koszul algebra from the familiar quadratic setting to general N -homogeneous 2000 Mathematics Subject Classification. Primary 16S37, 05A19. PHH is supported by the DFG through a Heisenberg-Fellowship. ML’s research is supported in part by NSA Grants H98230-05-1-0025 and H98230-07-1-0008 and by Lever- hulme Research Interchange Grant F/00158/X. http://arxiv.org/abs/0704.1888v2 2 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ algebras. Article [5] gives examples of N -Koszul algebras for all N ≥ 2; these are the so-called N -symmetric algebras, the special case N = 2 being the ordinary symmetric (poly- nomial) algebra. Following the general outline of Manin’s lecture notes [37] on the case of quadratic algebras, Berger, Dubois-Violette and Wambst developed the categorical aspects of N -homogeneous algebras in [7]. 0.2. Current interest in N -homogeneous algebras is fueled in part by the fact that they do occur naturally in mathematical physics and in combinatorics. Indeed, Connes and Dubois- Violette [10], [11] introduced a class of 3-homogeneous algebras, called Yang-Mills algebras, which are in fact 3-Koszul. There are two versions of Yang-Mills algebras: in the language of linear superalgebra, the first kind has even (parity 0̄) algebra generators while the second kind is generated by odd (parity 1̄) elements. Combinatorics enters the picture via MacMahon’s celebrated “master theorem” [35], specif- ically the recent quantum generalization of the master theorem due to Garoufalidis, Lê and Zeilberger [20]. As has been pointed out by two of the present authors in [28], the yoga of (quadratic) Koszul algebras leads to a rather effortless and conceptual proof of the quantum master theorem based on the fact that a certain quadratic algebra, known as quantum affine space, is Koszul. Further quantum generalizations and super versions of the master theorem have been obtained by several authors using a variety of approaches; see Foata and Han [17], [18], [19], Konvalinka and Pak [33], Etingof and Pak [16]. 0.3. From an algebraic point of view, MacMahon’s master theorem (MT) in its various incar- nations finds its most natural explanation by the phenomenon of “Koszul duality”. Indeed, all versions of MT can be expressed in the form that, for some algebra B, an equation Σ1 ·Σ2 = 1 holds for suitable power series Σ1,Σ2 ∈ BJtK. Here is a brief outline how one can arrive at such an equation starting with a given N -Koszul algebra A. Associated with A, there is a graded complex, K(A), which is exact in positive degrees, and a certain endomorphism bial- gebra, endA, which coacts on all components of K(A). These components therefore define elements of the representation ring RendA of endA, and exactness of K(A) in positive de- grees yields an equation in the power series ring RendAJtK. Due to the specific form of K(A), which is constructed from A together with its so-called dual algebra A!, the equation in ques- tion does indeed state that ρ1 · ρ2 = 1 holds for suitable ρ1, ρ2 ∈ RendAJtK. The last step in deriving a MT for A consists in using (super-)characters to transport the abstract duality equation ρ1 · ρ2 = 1 from RendAJtK to the power series ring over the algebra endA, where it takes a more explicit and useable form. Here then is the flow chart of our approach: N -Koszul algebra // exact Koszul complex // duality equation in RendAJtK // MT for A The actual labor involved in this process consists in the explicit evaluation of (super-)characters at the last arrow above. This step is often facilitated by specializing the bialgebra endA, which is highly noncommutative, to a more familiar algebra B via a homomorphism endA → B. For example: • MacMahon’s original MT [35] follows in the manner described above by starting with A = O(kd) = k[x1, . . . , xd], the ordinary polynomial algebra or “affine space”, and N -HOMOGENEOUS SUPERALGEBRAS 3 restricting the resulting MT over endO(kd) to the coordinate ring of d × d-matrices, O(Matd(k)) = k[x j | 1 ≤ i, j ≤ d]. • As was explained in [28], taking “quantum affine space” Oq(k d) as the point of de- parture one arrives at the quantum MT of Garoufalidis, Lê and Zeilberger [20] (and Konvalinka and Pak [33] in the multi-parameter case). The endomorphism bialgebra of Oq(k d) is exactly the algebra of right-quantum matrices as defined in [20]. • Berger’s N -symmetric algebra [5] leads to the N -generalization of the MT proved by Etingof and Pak [16] using the above approach, again after restricting to O(Matd(k)). 0.4. The present article aims to set forth an extension of the existing theory of N -homogeneous algebras to the category Vects of vector superspaces over some base field k. While this does not give rise to principal obstacles given that [37] and [7] are at hand as guiding references, the setting of superalgebra requires careful consideration of the order of terms and the so-called “rule of signs” will be ubiquitous in our formulæ. In view of the potential interdisciplinary interest of this material, we have opted to keep our presentation reasonably self-contained and complete. Therefore, in Sections 1 and 2, we deploy the requisite background material from superal- gebra in some detail before turning to N -homogeneous superalgebras in Section 3. The latter section, while following the general outline of [37] and [7] rather closely, also offers explicit discussions of a number of important examples. We interpolate the pure even and pure odd Yang-Mills algebras defined by Connes and Dubois-Violette [10], [11] by a family of super- algebras YMp|q and give a unified treatment of these algebras. (It turns out, however, that the mixed algebras YMp|q, with p and q both nonzero, are less well-behaved than the pure cases.) Moreover, we discuss a superized version of the N -symmetric algebras of Berger [5]. Finally, in Example 3.4, we introduce new N -homogeneous superversions of the symmetric algebra and the Grassmann algebra of a vector superspace V ; these are associated with any Hecke operator R : V ⊗2 → V ⊗2 and will be denoted by SR,N and ΛR,N , respectively. Sections 4 and 5 contain our main results: Theorem 4.5 shows that the superalgebras SR,N and ΛR,N are in fact N -Koszul, and Theorem 5.4 is superized version of the aforementioned N -generalized MT of Etingof and Pak [16, Theorem 2]. The special case N = 2 of The- orem 5.4 is a superization of the original master theorem of MacMahon [35]. The present article was motivated in part by a comment in Konvalinka and Pak [33, 13.4] asking for a “real” super-analog of the classical MT. 0.5. A considerable amount of research has been done by mathematical physicists on various quantum matrix identities. Some of these investigations have been carried out in a super set- ting; see, e.g., Gurevich, Pyatov and Saponov [23], [24] and the references therein. However, the techniques employed in these articles appear to be quite different from ours. After submitting this article, we also learned of recent work of Konvalinka [31], [32] which not only concerns MacMahon’s MT but also other matrix identities such as the determinantal identity of Sylvester. These identities are proved in [31], [32] by combinatorial means in various noncommutative settings including the right-quantum matrix algebra endOq(k 0.6. We conclude this Introduction by reviewing the precise definitions of N -homogeneous and N -Koszul algebras. Our basic reference is Berger [5]; see also [2], [7], [21]. 4 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Let A be a connected Z≥0-graded algebra over a field k; so A = n≥0An for k-subspaces An with A0 = k and AnAm ⊆ An+m. Choose a minimal generating set for the algebra A consisting of homogeneous elements of positive degree; this amounts to choosing a graded basis for a graded subspace V ⊆ A+ = n>0An such that A+ = A + ⊕ V . The grading of V imparts a grading to the tensor algebra T(V ) of the space V , and we have a graded presentation T(V )/I for some graded ideal I of T(V ), the ideal of relations of A. Recall that a graded vector space M = n∈ZMn is said to live in degrees ≥ n0 if Mn = 0 for all n < n0. Note that the relation ideal I lives in degrees ≥ 2, because T(V )0 ⊕T(V )1 ⊆ k⊕ V and k⊕ V injects into A. Fix an integer N ≥ 2 and define the jump function νN (i) = N if i is even N + 1 if i is odd (0.1) The following proposition is identical with [8, Proposition 2.1] except for the fact that we do not a priori assume A to be generated in degree 1. A proof is given in the Appendix. Proposition 0.1. The ideal I of relations of A lives in degrees ≥ N if and only if TorAi (k, k) lives in degrees ≥ νN (i) for all i ≥ 0. Following Berger [5], the graded algebra A is said to be N -Koszul if TorAi (k, k) is con- centrated in degree νN (i) for all i ≥ 0. This implies that the space of algebra generators V is concentrated in degree νN (1) = 1; so the algebra A is 1-generated. Moreover, choosing a minimal set of homogeneous ideal generators for the relation ideal I amounts to choosing a graded basis for a graded subspace R ⊆ I such that I = R⊕ (V ⊗ I + I ⊗ V ) (0.2) Then TorA2 (k, k) ∼= R and so R must be concentrated in degree νN (2) = N when A is N -Koszul. To summarize, all N -Koszul algebras are necessarily 1-generated and they have defining relations in degree N ; so there is a graded isomorphism A ∼= T(V )/(R) with R ⊆ V ⊗N Such algebras are called N -homogeneous. We remark that Green et al. [21] have studied N -Koszul algebras in the more general con- text where the grading A = n≥0An is not necessarily connected (A0 = k). In [21, The- orem 4.1], it is shown that an N -homogeneous algebra A with A0 split semisimple over k is N -Koszul if and only if the Yoneda Ext-algebra E(A) = n≥0 Ext A(A0,A0) is generated in degrees ≤ 2. Any N -homogeneous algebra A whose generating space V carries a Z2-grading and whose defining relations R are Z2-graded is naturally a k-superalgebra, that is, A has a Z2-grading (“parity”) besides the basic Z≥0-grading (“degree”). As will be reviewed below, this extra structure provides us with additional functions on Grothendieck rings, namely superdimension and supercharacters, which lead to natural formulations of the MT in a superized context. Note, however, that the defining property of N -Koszul algebras makes no reference to the Z2-grading of A. Thus, an N -homogeneous superalgebra is Koszul precisely if it is Koszul as an ordinary N -homogeneous algebra (forgetting the Z2-grading). N -HOMOGENEOUS SUPERALGEBRAS 5 0.7. Throughout k is a commutative field and ⊗ stands for ⊗ . Scalar multiplication in k-vector spaces will often, but not always, be written on the right while linear maps will act from the left. We tacitly assume throughout that char k 6= 2; further restrictions on the characteristic of k will be stated when required. 1. REVIEW OF LINEAR SUPERALGEBRA 1.1. Vector superspaces. A vector superspace over k is a k-vector space V equipped with a grading by the group Z2 = Z/2Z = {0̄, 1̄}. Thus, we have a decomposition V = V0̄ ⊕ V1̄ with k-subspaces V0̄ and V1̄ whose elements are called even and odd, respectively. In general, the Z2-degree of a homogeneous element a ∈ V is also called its parity; it will be denoted by â ∈ Z2. Vector superspaces over k form a category Vect whose morphisms are given by the linear maps preserving the Z2-grading; such maps are also called even linear maps. The dimension of an object V of Vects is the usual k-linear dimension. We shall use the notation d = dim V , p = dim V0̄ and q = dimk V1̄ So d = p+ q. The superdimension of a vector superspace V with d < ∞ is defined by sdimV = p− q ∈ Z When working with a fixed basis {xi} of a given V in Vect we shall assume that each xi is homogeneous; the parity of xi will be denoted by î. The basis x1, x2 . . . is called standard if î = 0̄ (i ≤ p) and î = 1̄ (i > p). 1.2. Tensors. The tensor product U ⊗V of vector superspaces U and V in Vects is the usual tensor product over k of the underlying vector spaces equipped with the natural Z2-grading: if a, b are homogeneous elements then the parity of a ⊗ b is â + b̂ ∈ Z2. Instead of the usual symmetry isomorphism U ⊗ V −→ V ⊗ U for interchanging terms in a tensor product we shall use the rule of signs, that is, the following functorial supersymmetry isomorphism in Vects cU,V : U ⊗ V −→ V ⊗ U , u⊗ v 7→ (−1)bubvv ⊗ u (1.1) for u, v homogeneous. (All formulas stated for homogeneous elements only are to be ex- tended to arbitrary elements by linearity.) The supersymmetry isomorphisms cU,V satisfy cV,U ◦ cU,V = IdU⊗V , and they are compatible with the usual associativity isomorphims aU,V,W : (U ⊗V )⊗W ∼= U ⊗ (V ⊗W ) in Vect , that is, they satisfy the “Hexagon Axiom”; see [29, Def. XIII.1.1]. Therefore, Vects is a symmetric tensor category; the unit object is the field k, with parity 0̄. See [29, Chap. XIII] or [12] for background on tensor categories. 1.3. Homomorphisms. The space Hom (V,U) of all k-linear maps between vector super- spaces V and U is again an object of Vects , with grading Hom (V,U)0̄ = Homk(V0̄, U0̄) ⊕ (V1̄, U1̄) and Homk(V,U)1̄ = Homk(V0̄, U1̄)⊕Homk(V1̄, U0̄); so (V,U)0̄ = HomVects (V,U) In particular, the linear dual V ∗ = Hom (V, k) belongs to Vects . Given homogeneous bases {xj} of V and {yi} of U we can describe any f ∈ Hom (V,U) by its matrix F = (F ij ): f(xj) = j (1.2) 6 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ When f is an even map then F ij = 0 unless î+ ĵ = 0̄. For finite-dimensional vector superspaces, we have the following functorial isomorphisms in Vects (see, e.g., [43, I.8]): U ⊗ V ∗ ∼= Hom (V,U) (1.3) via (u⊗ f)(v) = u〈f, v〉, and V ∗1 ⊗ . . . ⊗ V ∼= (Vm ⊗ . . . ⊗ V1) ∗ (1.4) via 〈f1 ⊗ . . .⊗ fm, vm ⊗ . . .⊗ v1〉 = i〈fi, vi〉. Here, we use the notation 〈f, v〉 = f(v) for the evaluation pairing evV = 〈 . , . 〉 : V ∗ ⊗ V → k in Vects . Similarly, we have a pairing V ⊗ V ∗ cV,V ∗ −→ V ∗ ⊗ V which yields an isomorphism −→ V ∗∗ (1.5) in Vects The isomorphism (1.3) (which is valid as long as one of U or V is finite-dimensional) has the following explicit description. Fix homogeneous bases {xj} of V and {yi} of U and let F = (F ij ) be the matrix of a given f ∈ Homk(V,U) with respect to these bases, as in (1.2). Let {xj} be the dual basis of V ∗, defined by 〈xj , xℓ〉 = δ (Kronecker delta). Then the image of f in U ⊗ V ∗ is given by i,j yi ⊗ x jF ij . Note also that xi and x i have the same parity. Finally, if U , V and W are vector superspaces, with U finite-dimensional, then the isomor- phism Id⊗cW,U∗ : V ⊗W ⊗U ∗ ∼−→V ⊗U∗ ⊗W together with (1.3) yields an isomorphism (U, V ⊗W ) −→ Hom (U, V )⊗W (1.6) in Vects which is explicitly given by (f ⊗ w)(u) = (−1)bwbuf(u) ⊗ w. Similarly, for vector superspaces U , U ′, V , V ′ with U , U ′ finite-dimensional, there is an isomorphism (U ⊗ U ′, V ⊗ V ′) −→ Hom (U, V )⊗Hom (U ′, V ′) (1.7) in Vects given by (f ⊗ g)(u ⊗ v) = (−1)bgbuf(u)⊗ g(v). 1.4. Supertrace. Let V be a finite-dimensional object of Vects . The supertrace is the map str : End (1.3) V ⊗ V ∗ −→ (1.3) k (1.8) in Vects . In order to describe the supertrace in terms of matrices, fix a basis {xi} of V consisting of homogeneous elements and let F = (F ij ) be the matrix of f ∈ Endk(V ) as in (1.2). Then str(f) = biF ii where î is the parity of xi (and of the dual basis vector x i ∈ V ∗) as in §1.1. Thus, str(IdV ) = sdimV.1 N -HOMOGENEOUS SUPERALGEBRAS 7 1.5. Action of the symmetric group. Given vector superspaces V1, . . . , Vn, we can consider the morphism ci : V1 ⊗ · · · ⊗ Vi ⊗ Vi+1 ⊗ · · · ⊗ Vn −→ V1 ⊗ · · · ⊗ Vi+1 ⊗ Vi ⊗ · · · ⊗ Vn in Vects which interchanges the factors Vi and Vi+1 via cVi,Vi+1 and is the identity on all other factors. More generally, for any σ ∈ Sn, the symmetric group consisting of all permutations of {1, 2, . . . , n}, one can define a morphism cσ : V1 ⊗ · · · ⊗ Vn −→ Vσ−1(1) ⊗ · · · ⊗ Vσ−1(n) in Vects as follows. Recall that Sn is generated by the transpositions σ1, . . . , σn−1 where σi interchanges i and i + 1 and leaves all other elements of {1, 2, . . . , n} fixed. The minimal length of a product in the σi’s which expresses a given element σ ∈ Sn is called the length of σ and denoted ℓ(σ); it is given by ℓ(σ) = #inv(σ) with inv(σ) = {(i, j) | i < j but σ(i) > σ(j)} Writing σ ∈ Sn as a product of certain σi, the analogous product of the maps ci yields a morphism cσ as above. This morphism is independent of the way σ is expressed in terms of the transpositions σi; see [43, I.4.13] or [29, Theorem XIII.1.3]. If all vi ∈ Vi are homogeneous cσ(v1 ⊗ · · · ⊗ vn) = (−1) (i,j)∈inv(σ) bvi bvjvσ−1(1) ⊗ · · · ⊗ vσ−1(n) (1.9) For example, if all vi are even then the ±-sign on the right is +, and if all vi are odd then it is sgn(σ), the signature of σ. Taking all Vi = V we obtain a representation c : Sn −→ AutVects (V ⊗n) where V ⊗n = V ⊗ · · · ⊗ V (n factors). Letting k[Sn] denote the group algebra of the symmetric group, this extends uniquely to an algebra map c : k[Sn] −→ EndVects (V ⊗n) (1.10) We will write ca := c(a) for a ∈ k[SN ]. For the dual superspace V ∗, besides the above representation c : k[Sn] −→ EndVects (V ∗⊗n), we also have the contragredient representation c∗ : k[Sn] −→ EndVects (V ∗⊗n) for the pairing 〈 . , . 〉 : V ∗⊗n ⊗ V ⊗n → k in (1.4). Explicitly, 〈c∗a(x), y〉 = 〈x, ca∗(y)〉 for all a ∈ k[Sn], x ∈ V ∗⊗n and y ∈ V ⊗n. Here, . ∗ : k[Sn] → k[Sn] is the involution sending σ ∈ Sn to σ −1. These two representations are related by c∗a = cτaτ (1.11) where τ = (1, n)(2, n−1) · · · ∈ Sn is the order reversal involution. One only needs to check (1.11) for the transpositions a = σi, which is straightforward. 8 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ 1.6. Hecke algebras. We recall some standard facts concerning Hecke algebras; these are suitable deformations of the group algebra k[Sn] considered above. For details, see [13], [14]. Fix 0 6= q ∈ k. The Hecke algebra Hn,q is generated as k-algebra by elements T1, . . . , Tn−1 subject to the relations (Ti + 1)(Ti − q) = 0 TiTi+1Ti = Ti+1TiTi+1 TiTj = TjTi if |i− j| ≥ 2 (1.12) When q = 1, one has an isomorphism Hn,1 −→ k[Sn], Ti 7→ σi where σi is the transposition (i, i + 1) as in §1.5. The algebra Hn,q has a k-basis {Tσ | σ ∈ Sn} so that (i) TId = 1 and Tσi = Ti; (ii) TσTσi = Tσσi if ℓ(σσi) = ℓ(σ) + 1; qTσσi + (q − 1)Tσ otherwise By k-linear extension of the rule T ∗σ := Tσ−1 (σ ∈ Sn) one obtains an involution . ∗ : Hn,q → Hn,q. Moreover, the elements T i := −qT q − 1− Ti also satisfy relations (1.12). Therefore, α(Ti) := −qT i (1.13) defines an algebra automorphism α : Hn,q → Hn,q of order 2. The Hecke algebra Hn,q is always a symmetric algebra, and Hn,q is a split semisimple k-algebra iff the following condition is satisfied: [n]q! := [i]q 6= 0 where [i]q := 1 + q + · · ·+ q i−1 (1.14) More precisely, if (1.14) holds then Matdλ×dλ(k) (1.15) where λ runs over all partitions of n and dλ denotes the number of standard λ-tableaux. The only partitions λ with dλ = 1 are λ = (n) and λ = (1 n). The central primitive idempotents of Hn,q for these partitions are given by Xn := [n]q! Tσ (1.16) Yn := [n]q−1 ! (−q)−ℓ(σ)Tσ (1.17) These idempotents are usually called the q-symmetrizer and the q-antisymmetrizer, respec- tively. One has XnTσ = TσXn = q ℓ(σ)Xn and YnTσ = TσYn = (−1) ℓ(σ)Yn (1.18) for σ ∈ Sn. Furthermore, α(Xn) = Yn. N -HOMOGENEOUS SUPERALGEBRAS 9 For later use, we note the following well-known consequence of (1.18). If M is any Hn,q- module, with corresponding representation µ : Hn,q → End (M), then Im(µ(Xn)) = Im(µ(Ti) + 1) (1.19) Indeed, (1.18) implies that Xn = [2] q (Ti + 1)Xn, which yields the inclusion ⊆. On the other hand, any m ∈ i=1 Im(µ(Ti) + 1) satisfies (µ(Ti) − q)(m) = 0 for all i, by (1.12). Therefore, µ(Tσ)(m) = q ℓ(σ)m holds for all σ ∈ Sn, and hence µ(Xn)(m) = [n]q! qℓ(σ)m = m. This proves ⊇. 1.7. Hecke operators. Again, let 0 6= q ∈ k. A Hecke operator (associated to q) on a vector superspace V is a morphism R : V ⊗2 → V ⊗2 in Vects satisfying the Hecke equation (R + 1)(R − q) = 0 and the Yang-Baxter equation R1R2R1 = R2R1R2 where R1 := R ⊗ IdV : V ⊗3 → V ⊗3 and similarly R2 := IdV ⊗R. The Hecke equation implies that R is invertible. Moreover, if R is a Hecke operator associated to q then so is −qR−1. Defining ρ(Ti) := Id V ⊗R ⊗ Id ⊗n−i−1 V , one obtains a representation ρ = ρn,R : Hn,q −→ EndVects (V ⊗n) (1.20) The representations ρn,R and ρn,−qR−1 are related by ρn,−qR−1 = ρn,R ◦ α, where α is the automorphism of Hn,q defined in (1.13). Example 1.1. The supersymmetry operator cV,V : V ⊗2 → V ⊗2 in (1.1) is a Hecke operator associated to q = 1, as is its negative, −cV,V . The representation ρcV,V of Hn,1 = k[Sn] in (1.20) is identical with (1.10). Example 1.2 (superized Drinfel’d-Jimbo [38], [27]). Let x1, . . . , xd be a standard basis of V as in §1.1. The super analog R = RDJ of the standard Drinfel’d-Jimbo Hecke operator is defined as follows. Writing R(xi ⊗ xj) = xk ⊗ xlR the matrix components Rk,li,j ∈ k are given by i,j = q2 − q2εi,j 1 + q2εi,j i,j + (−1) bibj q εi,j(q2 + 1) 1 + q2εi,j Here, εi,j = sgn(i− j). Thus, ii = q 2 if î = 0̄ ii = −1 if î = 1̄ ij = q 2 − 1 if i < j ij = (−1) bibjq if i 6= j (1.21) 10 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ and Rk,li,j = 0 in all other cases. One checks that R is a Hecke operator that is associated to 2. THE SUPERCHARACTER 2.1. Superalgebras, supercoalgebras etc. An algebra A in Vects is called a superalgebra over k; this is just an ordinary k-algebra such that the unit map k → A and the multiplication µ : A⊗A → A are morphisms in Vects . In other words, A is a Z2-graded k-algebra in the usual sense: A = A0̄ ⊕ A1̄ with k-subspaces A0̄ and A1̄ such that Ar̄As̄ ⊆ Ar+s. Homomorphisms of superalgebras, by definition, are algebra maps in Vects , that is, they preserve the Z2-grading. If V is a vector superspace in Vects then the tensor algebra T(V ) = n≥0 V ⊗n is a superalgebra via the Z2-grading of each V ⊗n as in §1.2. In general, if A is any superalgebra, then by selecting a Z2-graded subspace V ⊆ A which generates the algebra A, we obtain a canonical isomorphism of superalgebras T(V )/(R) −→ A (2.1) where (R) is the two-sided ideal of T(V ) that is generated by a Z2-graded linear subspace R ⊆ T(V ). Given superalgebras A and B, the tensor product A⊗ B is the superalgebra with the usual additive structure and grading and with multiplication µA⊗B defined by using the supersym- metry map (1.1): µA⊗B = (µA ⊗ µB) ◦ (IdA⊗cB,A ⊗ IdB) or, explicitly, (a⊗ b)(a′ ⊗ b′) = (−1) ba′bbaa′ ⊗ bb′ for homogeneous a′ ∈ A and b ∈ B. In other words, the canonical images of A and B in A⊗ B supercommute, in the sense that the supercommutator [a, b] = ab− (−1)ba bbba (2.2) vanishes for any pair of homogeneous elements a ∈ A and b ∈ B. Supercoalgebras, superbialgebras etc. are defined similarly as suitable objects of Vects such that all structure maps are maps in Vects . The compatibility between the comultiplica- tion ∆ and the multiplication of a superbialgebra B amounts to the following rule: ∆(ab) = (a),(b) (−1)ba(2) bb(1)a(1)b(1) ⊗ a(2)b(2) for homogeneous elements a, b ∈ B. Here we use the Sweedler notation ∆(a) = (a) a(1) ⊗ a(2) and a(1), a(2) are chosen homogeneous with â(1) + â(2) = â. Example 2.1 (Symmetric superalgebra [40, 3.2.5]). The symmetric superalgebra of a given V in Vects is defined by S(V ) = T(V )/ ([v,w]⊗ | v,w ∈ V ) where [v,w]⊗ is the supercommutator (2.2) in T(V ). Ignoring parity, S(V ) is isomorphic to S(V0̄)⊗Λ(V1̄), where S( . ) and Λ( . ) denote the ordinary symmetric and exterior (Grassmann) algebras, respectively. The symmetric superalgebra is a Hopf superalgebra: comultiplication ∆: S(V ) → S(V )⊗ S(V ) is given by ∆(v) = v ⊗ 1 + 1⊗ v for v ∈ V and extension to all N -HOMOGENEOUS SUPERALGEBRAS 11 of S(V ) by multiplicativity. Similarly, the counit ε : S(V ) → k is given by ε(v) = 0 and the antipode S : S(V ) → S(V ) by S(v) = −v for v ∈ V . 2.2. Comodules. We refer to [29, Chap. III] for background on comodules, comodule alge- bras etc. Given a superbialgebra B, we let ComodsB denote the category of all right B-comodules and B-comodule maps in Vects . Thus, for any object V in ComodsB, we have a “coaction” morphism δV : V → V ⊗B in Vects . If x1, . . . , xd is a fixed basis of V consisting of homogeneous elements, with î denoting the parity of xi as before, then we will write δV (xj) = xi ⊗ b j with b j ∈ Bbi+bj (2.3) The tensor product of vector superspaces makes ComodsB into a tensor category: if U and V are in ComodsB then B coacts on U ⊗ V by δU⊗V : U ⊗ V δU⊗δV −→ U ⊗ B ⊗ V ⊗ B −→ U ⊗ V ⊗ B ⊗ B Id⊗µB −→ U ⊗ V ⊗ B (2.4) If B is supercommutative as a superalgebra then the supersymmetry cU,V is a B-comodule morphism, i.e., δV⊗U ◦ cU,V = (cU,V ⊗ IdB) ◦ δU⊗V . Therefore Comod B is a symmetric tensor category in this case . 2.3. The supercharacter map. Let B denote a superbialgebra and let V be a finite dimen- sional object in ComodsB. The coaction δV is an even map in Homk(V, V ⊗B). Consider the following morphism in Vects χs : End δV ◦( . ) −→ Hom (V, V ⊗ B) (1.6) (V )⊗ B str⊗Id −→ k⊗ B = B (2.5) where str is the supertrace as in (1.8). This map will be called the supercharacter map of V . Forgetting parity and viewing all elements as even, the supertrace becomes the ordinary trace and the supercharacter becomes the usual character. These will be denoted by tr and χ, respectively. In particular, we have the element χsV := χ s(IdV ) ∈ B0̄ To obtain explicit formulas, fix a basis x1, . . . , xd of V consisting of homogeneous elements and let (F ij ) and (b j) be the matrices of f ∈ Endk(V ) and of δV with respect to this basis as in (1.2), (2.3). Then χs(f) = bibjbijF i (2.6) Let ε : B → k denote the counit of B. Then xj = i xiε(b j) holds in (2.3). Hence ε(bij) = δ j .1k and (2.6) gives ε(χs(f)) = str(f) (2.7) 12 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ When f is even formula (2.6) becomes χs(f) = i,j(−1) bibijF i , because F i = 0 unless î+ ĵ = 0̄. In particular, χsV = bibii (2.8) In the following, we let comodsB denote the full subcategory of Comod B consisting of all objects that are finite-dimensional over k. The supercharacter has the following properties analogous to standard properties of the ordinary character. Lemma 2.2. Let B denote a superbialgebra and let U , V and W be objects of comodsB . (a) If f : V → U and g : U → V are B-comodule maps (not necessarily even) then χs(f ◦ g) = (−1) bfbgχs(g ◦ f) (b) For f ∈ End (V ), g ∈ End (U) view f ⊗ g ∈ End (V ⊗ U) as in (1.7). Then χs(f ⊗ g) = χs(f)χs(g) (c) Given an exact sequence 0 → U −→ W → 0 in comodsB, let f ∈ Endk(V ) be such that f(µ(U)) ⊆ µ(U), and let g ∈ End (U), h ∈ End (W ) be the maps induced by f . Then χs(f) = χs(g) + χs(h) In particular, χsV = χ U + χ W . Moreover, if f ∈ EndcomodsB(V ) is a projection (i.e., f2 = f ) then χs(f) = χsIm f . Proof. (a) Let TV denote the map Hom (V, V ⊗ B) −→ B in (2.5); so χs(f) = TV (δV ◦ f). Since f and g are comodule maps, we have δU ◦ f = (f ⊗ IdB) ◦ δV and similarly for g. Putting h = δU ◦ f ∈ Hom (V,U ⊗ B) we obtain χs(f ◦ g) = TU (δU ◦ f ◦ g) = TU (h ◦ g) and χs(g ◦ f) = TV (δV ◦ g ◦ f) = TV ((g ⊗ IdB) ◦ h). Therefore, we must show that TU (h ◦ g) = (−1) bfbgTV ((g ⊗ IdB) ◦ h) Using the identification Hom (V,U⊗B) ∼= Hom (V,U)⊗B as in (1.6), write h = i fi⊗bi with fi ∈ Hom (V,U), bi ∈ B, and f̂i + b̂i = ĥ = f̂ . Then h ◦ g ∈ Hom (U,U ⊗ B) becomes the element ( i fi ⊗ bi) ◦ g = i(−1) bbibg(fi ◦ g) ⊗ bi ∈ End (U) ⊗ B, and (g ⊗ IdB) ◦ hHom (V, V ⊗B) becomes i(g ◦ fi)⊗ bi. The standard identity str(fi ◦ g) = bfibgstr(g ◦ fi) (cf., e.g., [40, p. 165 §3(b)]) now yields TU (h ◦ g) = bbibgstr(fi ◦ g)⊗ bi bbibg+bfibgstr(g ◦ fi)⊗ bi = (−1) bfbgTV ((g ⊗ IdB) ◦ h) as desired. (b) Fix homogeneous k-bases {xi} and {yℓ} of V and U , respectively, and write x̂i = î, ŷℓ = ℓ̂ as usual. Moreover, let (F j ) and (G m) be the matrices of f and g for these bases, as N -HOMOGENEOUS SUPERALGEBRAS 13 in (1.2). Then {xi ⊗ yℓ} is a basis of V ⊗ U , with xi ⊗ yℓ having parity î+ ℓ̂. Moreover, (f ⊗ g)(xj ⊗ ym) = (−1) bgbjf(xj)⊗ g(ym) = (−1)bg xi ⊗ yℓΦ j,m with Φ j,m = (−1) (bℓ+ bm)bjF ijG because Gmℓ = 0 unless ℓ̂+ m̂ = ĝ. Similarly, writing δV (xj) = i xi ⊗ b j with b j ∈ Bbi+bj and δU (ym) = ℓ yℓ ⊗ c m with c m ∈ Bbℓ+bm, one obtains using (2.4) δV⊗U (xj ⊗ ym) = xi ⊗ yℓ ⊗Ψ j,m with Ψ j,m = (−1) (bi+bj)bℓbijc Therefore, formula (2.6) becomes χs(f ⊗ g) = i,ℓ,j,m (−1)( bi+bℓ)(bj+ bm)Ψ i,ℓ,j,m bibj+bℓbmbijF = χs(f)χs(g) (c) Choose a basis {xi} of V consisting of homogeneous elements so that xi = µ(yi) for i ≤ dimU and let (F ij ) be the matrix of f for this basis. Then F j = 0 for i > dimU , j ≤ dimU . Moreover, the yi form a basis of U and the zi = π(xi) form a basis of W , and the matrices of g and h for these bases are (F ij )i,j≤dimU and (F j )i,j>dimU , respectively. Similarly, if (bij) is the matrix of δV with respect to the basis basis {xi} as in (2.3) then b j = 0 for i > dimU , j ≤ dimU , and the matrices of δU and δW for the given bases are (b j)i,j≤dimU and (bij)i,j>dimU , respectively. Therefore, χs(f) = bibjbijF i,j≤dimU bibjbijF i,j>dimU bibjbijF = χs(g) + χs(h) The remaining assertions are clear. � 2.4. The Grothendieck ring. Let B be a superbialgebra and let RB = K0(comod denote the Grothendieck group of the category comodsB. Thus, for each V in comod B, there is an element [V ] ∈ RB and each short exact sequence 0 → U → V → W → 0 in comod gives rise to an equation [V ] = [U ] + [W ] in RB. The group RB is in fact a ring with multiplication given by the tensor product of B-comodules. If B is supercommutative as a superalgebra then the ring RB is commutative; see §2.2. 14 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Both the ordinary dimension and the superdimension are additive on short exact sequences and multiplicative on tensor products. Hence they yield ring homomorphisms dim, sdim: RB → Z Parts (b) and (c) of Lemma 2.2 and formula (2.7) have the following immediate consequence: Corollary 2.3. The map [V ] 7→ χsV yields a well-defined ring homomorphism χ s : RB → B0̄. Furthermore, the following diagram commutes Z can. Forgetting the Z2-grading, the corollary also gives the more familiar version with χ and dim in place of χs and sdim, respectively. 2.5. General linear supergroup and Berezinian. Let V in Vects be finite-dimensional and fix a standard basis x1, . . . , xd with î = 0̄ (i ≤ p) and î = 1̄ (i > p). 2.5.1. For each supercommutative k-superalgebra R we denote by E(V )(R) the set of all R-linear maps V ⊗ R → V ⊗ R in Vects . Using the identification EndR(V ⊗ R) ∼= (V, V ⊗R) ∼= End (V )⊗R (see (1.6)), we may view E(V )(R) as the even subspace of End (V )⊗R: E(V )(R) = (End (V )⊗R)0̄ This defines a functor E(V ) from the category of supercommutative k-superalgebras to the category of semigroups. 2.5.2. Tensoring the supertrace str : End (V ) → k of (1.8) with IdR, we obtain an R-linear supertrace map str : End (V )⊗R → R in Vects which restricts to a map E(V )(R) → R0̄. The given standard basis x1, . . . , xd of V is an R-basis of V ⊗ R. In terms of this basis, an element φ ∈ E(V )(R) is given by φ(xj) = j with Φ j ∈ Rbi+bj (2.9) Thus φ is described by a supermatrix Φ = in standard form over R: (2.10) where A = i,j≤p and D = i,j>p are square matrices with entries in R0̄ while C,D are matrices over R1̄. The supertrace of φ is given by str(φ) = biΦii = tr(A)− tr(D) =: str(Φ) N -HOMOGENEOUS SUPERALGEBRAS 15 2.5.3. The functor E(V ) is represented by a supercommutative k-superbialgebra which coacts on V ; this algebra will be denoted by B = O(E(V )) Thus, there is a natural isomorphism of E(V ) with the functor Hom(B, ?) of parity preserving algebra homorphisms. In particular, the identity map on B corresponds to an element ξ ∈ E(V )(B). Let X = (xij)d×d be the matrix of ξ, as in (2.9). The elements x j have parity î + ĵ and they form a set of supercommuting algebraically independent generators of B. In fact, B is isomorphic to the symmetric superalgebra S(V ∗ ⊗ V ), with xij 7→ x i ⊗ xj , where {xi} ⊆ V ∗ is the dual basis for the given basis of V . We can think of X as the generic supermatrix with respect to the given basis of V : any supermatrix Φ = as in (2.9) comes from an algebra map B → R via xij 7→ Φ j . The canonical coaction δ : V → V ⊗B, the comultiplication ∆ and the counit ε of B are given by δ(xj) = xi ⊗ x ∆(xij) = xik ⊗ x ε(xij) = δ (2.11) These formulas can also be written as δ(x1, . . . , xd) = (x1, . . . , xd) ⊗X, ∆(X) = X ⊗X and ε(X) = 1. 2.5.4. Similarly, GL(V )(R) is defined, for any supercommutative k-superalgebra R, as the set of all invertible R-linear endomorphism of V ⊗R in Vects . The condition for a superma- trix Φ in standard form (as in (2.10)) to be invertible is that A and D are invertible as ordinary matrices over R0̄. In this case, the inverse of Φ is given by Φ−1 = (A−BD−1C)−1 −A−1B(D − CA−1B)−1 −D−1C(A−BD−1C)−1 (D − CA−1B)−1 See Berezin [3, Theorem 3.1 and Lemma 3.2]. The element ber(Φ) := det(A) det(D − CA−1B)−1 = det(D)−1 det(A−BD−1C) (2.12) is called the superdeterminant or Berezinian of Φ; it is an invertible element of R0̄. The functor GL(V ) is represented by a supercommutative Hopf superalgebra O(GL(V )) which is generated over B = O(E(V )) by det(X11)−1 and det(X22)−1, where X11 =( i,j≤p and X22 = i,j>p are the even blocks of the generic supermatrix X. By [3, Theorem 3.3], the Berezinian ber(X) is a group-like element in O(GL(V )). 2.6. Supersymmetric functions and exterior powers. Throughout this section, V will de- note a finite-dimensional vector superspace over k. We assume that the characteristic of k is zero. 16 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ 2.6.1. Let sgn(σ)σ ∈ k[Sn] be the antisymmetrizer idempotent of the group algebra k[Sn] and define nV := Im cYn ⊆ V ⊗n (2.13) where c : k[Sn] → EndVects (V ⊗n) is as in (1.10). Thus, ΛnV is the space of antisymmetric n-tensors, nV = {y ∈ V ⊗n | cσ(y) = sgn(σ)y for all σ ∈ Sn} For later use, we describe an explicit basis of ΛnV . To this end, fix a standard basis x1, . . . , xd of V , with î = 0̄ for i ≤ p and î = 1̄ for i > p. Then the products xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xin for sequences i = (i1, i2, . . . , in) ∈ {1, 2, . . . , d} n form a graded basis of V ⊗n that is permuted up to a ±-sign by the action of Sn on V ⊗n; see formula (1.9): cσ(xi) = sgni(σ)xσ(i) (2.14) sgni(σ) = (−1) (p,q)∈inv(σ) bip biq and σ(i) = (iσ−1(1), iσ−1(2), . . . , iσ−1(n)) Therefore, by elementary properties of monomial group representations, a k-basis of ΛnV is given by the nonzero elements cYn(xi) where i ranges over a transversal for the Sn- action on {1, 2, . . . , d}n. Such a transversal is provided by the weakly increasing sequences i ∈ {1, 2, . . . , d}n. Moreover, for a weakly increasing i, it is easily seen from (2.14) that cYn(xi) = 0 holds precisely if iℓ = iℓ+1 ≤ p for some ℓ. Therefore, a basis of Λ nV is given by the elements cYn(xi) with i = (i1 < i2 < · · · < im < im+1 ≤ · · · ≤ in) ∈ {1, 2, . . . , d} and im ≤ p < im+1. In particular, m+m′=n q +m′ − 1 (2.15) where p = dim V0̄ and q = dimk V1̄. Equivalently, the generating power series in ZJtK for the sequence dim nV is given by nV tn = (1 + t)p (1− t)q (2.16) When q > 0 then all ΛnV are nonzero. For additional details on exterior powers, see, e.g., [43, Sections I.5 and I.7]. 2.6.2. Consider the super bialgebra B = O(E(V )) as defined in §2.5.3 and recall that V is in comodsB. The representation c : k[Sn] → EndVects (V ⊗n) of (1.10) actually has image in EndcomodsB(V ⊗n), since B is supercommutative. Therefore, ΛnV also belongs to comodsB and we can define the nth elementary supersymmetric function by en := χ ΛnV = χ s(cYn) ∈ B0̄ Here, the equality χs ΛnV = χ s(cYn) holds by Lemma 2.2(c). Similarly, one defines the nth super power sum by pn := χ s(c(1,2,...,n)) ∈ B0̄ N -HOMOGENEOUS SUPERALGEBRAS 17 where (1, 2, . . . , n) ∈ Sn the cyclic permutation mapping 1 7→ 2 7→ 3 7→ . . . 7→ n 7→ 1. In terms of the generic supermatrix X from §2.5.3, one has pn = str(X Modulo the space spanned by the Lie commutators fg− gf with f, g ∈ k[Sn], the follow- ing relation is easily seen to hold in k[Sn]: nYn ≡ (−1)i−1(1, 2, . . . , i)Yn−i (with Y0 = 1). Applying the function χ s ◦ c : k[Sn] → B0̄ to this relation and using Lemma 2.2(a),(b), one obtains the Newton relations: nen = (−1)i−1pien−i Let t be a formal parameter (of parity 0̄) and consider the generating functions P (t) =∑ n≥1 pnt n−1 and E(t) = n≥0 ent n in B0̄JtK. The Newton relations can be written in the form P (−t) = d logE(t); see, e.g., [34, p. 23]. Combining this with the identity ber(exp(tX)) = exp(str(tX)) due to Berezin ([3, Chapter 3] or [40, p. 167]), one obtains the following expansion for the characteristic function ber(1 + tX) of generic supermatrix X: Proposition 2.4. ber(1 + tX) = n≥0 ent This proposition is known; see, e.g., Khudaverdian and Voronov [30, Prop. 1]. 3. HOMOGENEOUS SUPERALGEBRAS 3.1. N -homogeneous superalgebras. Let N be an integer with N ≥ 2. A homogeneous superalgebra of degree N or N -homogeneous superalgebra is an algebra A of the form (2.1) with V finite-dimensional and R ⊆ V ⊗N : A = A(V,R) ∼= T(V )/(R) The assumption R ⊆ V ⊗N implies that, besides the usual Z2-grading (“parity”), A also has a connected Z+-grading (“degree”), The algebra A is generated by A1 = V and all homogeneous components An are finite- dimensional objects of Vects . In fact, An ∼= V ⊗n/Rn with Rn := (R) ∩ V i+j+N=n V ⊗i ⊗R⊗ V ⊗j (3.1) Note that Rn = 0 for n < N ; so An ∼= V ⊗n if n < N . Morphisms of N -homogeneous superalgebras f : A = A(V,R) → A′ = A(V ′, R′) are morphism of superalgebras which also respect the Z+-grading. Equivalently, by restricting to degree 1, we have a morphism f1 : A1 = V → A 1 = V ′ in Vects whose N th tensor power sat- isfies f⊗N1 (R) ⊆ R ′. Thus, one has a category HNAlg of N -homogeneous k-superalgebras. 18 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Finally, N -homogeneous superalgebras with N = 2 are called quadratic superalgebras; for N = 3, they are called cubic, etc.. 3.2. Some examples. In order to explicitly describe a certain N -homogeneous superalgebra A = A(V,R), we will usually fix a Z2-graded k-basis x1, . . . , xd of V = A1 and denote the the parity of xi by î, as in §1.1. The xi form a set of algebra generators for A. Following Manin [38],[39], the d-tuple f = (1̂, . . . , d̂) ∈ Zd2 is called the format of the basis {xi}. Example 3.1 (Quantum superspace [39]). For a fixed family q of scalars 0 6= qij ∈ k (1 ≤ i < j ≤ d) and a given format f = (1̂, . . . , d̂) ∈ Zd2 of the basis x1, . . . , xd, the quadratic superalgebra A = Sfq is defined as the factor of T(V ) modulo the ideal generated by the elements ri := xi ⊗ xi ∈ (V ⊗2)0̄ (̂i = 1̄) (3.2) rij := xj ⊗ xi − qij(−1) bibjxi ⊗ xj ∈ (V bi+bj (i < j) (3.3) Thus, the algebra Sfq is generated by x1, . . . , xd subject to the defining relations xixi = 0 (̂i = 1̄) xjxi = qij(−1) bibjxixj (i < j). In the special case where all qij = 1, the algebra S q is the symmetric superalgebra S(V ) of V as in Example 2.1. The ordered monomials of the form xm11 x 2 . . . x , with i mi = n, mi ≥ 0 for all i and mi ≤ 1 if î = 1̄, form a k-basis of the n th homogeneous component of Sfq. Therefore, (Sfq)n = r+s=n r + p− 1 (3.4) where dimV0̄ = p and dimV1̄ = q as usual.Thus, the generating series of the dimensions is (Sfq)nt (1 + t)q (1− t)p Example 3.2 (Yang-Mills algebras [11],[10]). Fix a collection of elements x1, . . . , xd (d ≥ 2), numbered so as to have parity î = 0̄ for i ≤ p and î = 1̄ for i > p. Let G = (gij) ∈ GLd(k) be an invertible symmetric d × d-matrix satisfying gij = 0 if î 6= ĵ and consider the cubic superalgebra A that is generated by elements x1, . . . , xd subject to the relations gij [xi, [xj , xk]] = 0 (k = 1, . . . , d) (3.5) Here [ . , . ] is the supercommutator (2.2). The algebra A will be denoted by YMp|q (q = d−p). In particular, the pure even algebra YMd|0 is the ordinary Yang-Mills algebra introduced in [10] while YM0|d is the super Yang-Mills algebra as in [11]. As usual, put V = i kxi and let [ . , . ]⊗ denote the supercommutator in T(V ). Further- more, put rk = i,j gij [xi, [xj , xk]⊗]⊗ and R = k krk ⊆ V ⊗3; so YMp|q = T(V )/(R). Using the symmetry of G, we may replace the rk by simpler relations as follows. Choose an N -HOMOGENEOUS SUPERALGEBRAS 19 invertible d × d-matrix C = (cij) with cij = 0 if î 6= ĵ and such that C trGC is diagonal, i,j cirgijcjs = gsδ s . Replace the bases {xi} of V and {rk} of R by the new bases ijxj and sk = kℓrℓ where C −1 = (cij). Note that yi has parity î and sk has parity k̂, the parity of rk. A simple calculation shows that sk = i 6=k gi[yi, [yi, yk]⊗]⊗. Thus we obtain the following defining relations for the generators y1, . . . yd of YM i 6=k gi[yi, [yi, yk]] = 0 (k = 1, . . . , d) (3.6) The resulting algebras for d = 2 are as follows. Putting x = y1 and y = y2 we have two defining relations: [x, [x, y]] = 0 and [y, [y, x]] = 0. In the pure even case (x̂ = ŷ = 0̄), the supercommutators are the ordinary Lie commutators. So YM2|0 is the enveloping algebra of the Heisenberg Lie algebra; see [1, (0.4)]. In the pure odd case (x̂ = ŷ = 1̄), the two relations can be written as x2y = yx2 and yx2 = x2y. The resulting algebra YM0|2 is a cubic Artin-Schelter algebra of type S1 [1, (8.6)]. Thus, both unmixed algebras are Artin-Schelter regular of global dimension 3. In the mixed case, however (x̂ = 0̄, ŷ = 1̄), the relations say that x commutes with the Lie commutator [x, y] while y anticommutes: y[x, y] = −[x, y]y. Thus, [x, y] is a normal element of YM1|1 and YM1|1/([x, y]) is a polynomial algebra in two variables over k. Moreover, the calculation [x, y]2 = [x, [x, y]y] = −[x, y[x, y]] = −[x, y]2 shows that [x, y]2 = 0. Thus, the algebra YM1|1 is noetherian with Gelfand-Kirillov dimen- sion 2 and infinite global dimension. Returning to the case of general d ≥ 2, we now concentrate on the unmixed algebras intro- duced by Connes and Dubois-Violette. We will denote these algebras by YM+ = YMd|0 and YM− = YM0|d. In all formulas below, + applies to YM+ and − to YM−. The generators i 6=k gi[yi, [yi, yk]⊗]⊗ of the space of relations R can be written as sk = ℓ yℓ⊗mℓk = ℓmkℓ ⊗ yℓ with mℓk = gℓ (yℓ ⊗ yk − (1± 1)yk ⊗ yℓ) for ℓ 6= k i 6=k giyi ⊗ yi for ℓ = k Thus, putting Y = (y1, . . . , yd) and letting M denote the d × d-matrix over YM ± whose (ℓ, k)-entry is the image of mℓk, the defining relations (3.6) can be written as YM = 0 or MY tr = 0 (3.7) The defining relations (3.6) for A = YM− amount to the even element i giy i ∈ A2 being central in A. Example 3.3 (N -symmetric superalgebra; cf. [5]). Let N ≥ 2 be given and let V be a vector superspace V over a field k with char k = 0 or char k > N . Define SN (V ) = A(V,R) with R = Λ NV = cYN ⊆ V ⊗N where YN is the antisymmetrizer idempotent of the group algebra k[SN ]; see (2.13). This defines a functor SN ( . ) : Vect → HNAlg . Since 2cY2 is the supercommutator in T(V ), the algebra S2(V ) is just the symmetric superalgebra S(V ) of V ; see Example 3.1. The algebra SN (V ), for a pure even space V = V0̄ and general N ≥ 2, has been introduced in [5]. 20 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ If 2 ≤ M ≤ N then, viewing k[SM ] as a subalgebra of k[SN ] as usual, the antisymmetriz- ers of k[SN ] and k[SM ] satisfy YN = YMa for some a ∈ k[SN ]. Therefore, R = cYN ⊆ cYM = cYM ⊗ V ⊗(N−M) This shows that the identity map on V extends to an epimorphism of superalgebras SN(V ) ։ SM (V ). Now assume that dimk V = d and fix a standard basis x1, . . . , xd of V , with î = 0̄ for i ≤ p and î = 1̄ for i > p. From the basis for ΛNV exhibited in §2.6.1 we obtain that the algebra SN (V ) is generated by x1, . . . , xd subject to the relations (p,q)∈inv(σ) 1+ bip biqxi σ−1(1) σ−1(2) . . . xi σ−1(N) with 1 ≤ i1 < i2 < · · · < im ≤ p = dim V0̄ < im+1 ≤ · · · ≤ iN ≤ d = dimk V ; see formula (2.14). Example 3.4. The following construction generalizes Example 3.3. Fix N ≥ 2 and 0 6= q ∈ k and assume that condition (1.14) is satisfied. Given a Hecke operator R : V ⊗2 → V ⊗2 on a vector superspace V we define the N -homogeneous superalgebra ΛR,N := A(V,R) with R = Im ρR(XN ) ⊆ V ⊗N (3.8) where XN ∈ HN,q is the q-symmetrizer (1.16) and ρR is the representation (1.20) of HN,q. We also put SR,N := Λ−qR−1,N = A(V,R) with R = Im ρR(YN ) ⊆ V ⊗N (3.9) where YN ∈ HN,q is the antisymmetrizer (1.17). The algebra SN(V ) in Example 3.3 is identical with ScV,V ,N (q = 1). 3.3. The dual of a homogeneous superalgebra. Let A = A(V,R) be an N -homogeneous superalgebra. The dual A! of A is defined by A! = A(V ∗, R⊥) where, R⊥ ⊆ V ∗⊗N is the (homogeneous) subspace consisting of all elements that vanish on R ⊆ V ⊗N , using (1.4) in order to evaluate elements of V ∗⊗N on V ⊗N . Thus, (3.1) takes the A!n = V ∗⊗n/R⊥n with R i+j+N=n V ∗⊗i ⊗R⊥ ⊗ V ∗⊗j (3.10) Identifying V ∗⊗n with the linear dual of V ⊗n via (1.4), we have V ∗⊗i ⊗ R⊥ ⊗ V ∗⊗j =( V ⊗j ⊗R⊗ V ⊗i . Hence, R⊥n = i+j+N=n V ⊗j ⊗R⊗ V ⊗i (3.11) The canonical isomorphism V −→ V ∗∗ in (1.5) leads to an isomorphism V ⊗N V ∗∗⊗N which maps R onto R⊥⊥. Hence, A! ! ∼= A (3.12) N -HOMOGENEOUS SUPERALGEBRAS 21 Moreover, if f : A = A(V,R) → A′ = A(V ′, R′) is any morphism in HNAlg then the transpose of f1 : V → V ′ induces a morphism f ! : (A′)! → A! in HNAlg . Thus, we have a contravariant quasi-involutive dualization functor A 7→ A!, f 7→ f ! on HNAlg Example 3.5. The dual of A(V, 0) = T(V ) is A(V ∗, V ∗⊗N ); so T(V )! = T(V ∗)/ V ∗⊗N In particular, letting V = k be the unit object of Vects , we have A(k, 0) = k[t] (polynomial algebra) and A(k, 0)! = k[d]/(dN ), with t and d both having degree 1 and parity 0̄. Example 3.6 (Dual of quantum superspace). We will describe the dual A! of quantum su- perspace A = Sfq; see Example 3.1. Fix a homogeneous k-basis x1, . . . , xd with format f for V , and let x1, . . . , xd denote the dual basis of V ∗; this basis also has format f . Evaluat- ing an arbitrary element f = ℓ,m fℓmx ℓ ⊗ xm ∈ V ∗⊗2 on one of the generating relations ri, rij ∈ R in (3.2), (3.3) we obtain 〈f, ri〉 = fii and 〈f, rij〉 = fij − qij(−1) bibjfji. Therefore, the space R⊥ ⊆ V ∗⊗2 has a basis consisting of the elements sℓ := xℓ ⊗ xℓ (ℓ̂ = 0̄) and sℓ,k := xℓ ⊗ xk + qkℓ(−1) bkbℓxk ⊗ xℓ (k < ℓ). In summary, A! is generated by x1, . . . , xd subject to the defining relations xℓxℓ = 0 (ℓ̂ = 0̄) xℓxk = −qkℓ(−1) bkbℓxkxℓ (k < ℓ). Thus, A! is isomorphic to quantum superspace Sf with q′ij = (−1) bi+bjqij and f ′ = f + (1̄, . . . , 1̄) the format obtained from f by parity reversal in all components. Example 3.7 (Duals of the Yang-Mills algebras). Continuing with the notation of Exam- ple 3.2, we now desribe the algebra A! for A = YMp|q. We assume that char k = 0 and work with generators y1, . . . , yd of A satisfying (3.6). Let y1, . . . , yd denote the basis of V ∗ given by 〈yi, yj〉 = δ j and put γ = yi ∈ V ∗⊗2. Then, for the generators sk = i 6=k gi[yi, [yi, yk]⊗]⊗ of R as in Example 3.2, one computes 〈ya ⊗ yb ⊗ yc, sk〉 = gcδ k + (−1) bbgbδ k − (−1) babk(1 + (−1)ba)gaδ 〈yi ⊗ γ, sk〉 = δ (3.13) Therefore, the map ϕ 7→ ϕ− k〈ϕ, sk〉y k ⊗ γ is an epimorphism V ∗⊗3 ։ R⊥ ⊂ V ∗⊗3. We obtain that the algebra A! is generated by y1, . . . , yd subject to the relations yaybyc = (gcδ a + (−1) bbgbδ c − (−1)ba bb(1 + (−1)ba)gaδ b)g (3.14) where g = 1 iyi is the image of γ in A. Since A! is 3-homogeneous, we clearly have A!0 = k, A i = V ∗ and A!2 =⊕ i,j ky iyj ∼= V ∗⊗2. By (3.13), the elements yag form a k-basis of A!3 = V ∗⊗3/R⊥ ∼= R∗. Using the defining relations (3.14) it is not hard to see that A!4 = kg 2 and A!n = 0 for n ≥ 5. If A = YMp|q is of mixed type (i.e., p 6= 0 and q 6= 0) then g2 = 0. 22 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Example 3.8 (Dual of the N -symmetric superalgebra). Recall from Example 3.3 that SN (V ) = A(V,R) with R = cYN . Since YN is central in k[SN ] and stable under the inversion involution ∗ of k[SN ], it follows from (1.11) that 〈x, cYN (y)〉 = 〈cYN (x), y〉 holds for all x ∈ V ∗⊗N and y ∈ V ⊗N . Therefore, R⊥ = KerV ∗⊗N (cYN ) = (1− cYN ) V ∗⊗N and so SN (V ) ! = A V ∗, (1 − cYN )(V ∗⊗N ) Note that ⋂ i+j+N=n V ⊗i ⊗R⊗ V ⊗j = cYn (3.15) holds for all n ≥ N . This follows from (1.19). Alternatively, as has been noted in Exam- ple 3.3, we have cYn (V ⊗n) ⊆ R ⊗ V ⊗(n−N). In the same way, one sees that cYn (V ⊗n) ⊆ V ⊗i ⊗ R ⊗ V ⊗j whenever i + j + N = n. For the reverse inclusion, note that each x ∈ V ⊗i ⊗ R ⊗ V ⊗j satisfies cσℓ(x) = −x for all transpositions σℓ = (ℓ, ℓ + 1) ∈ Sn with i < ℓ < i+N . Hence, the left hand side of (3.15) is contained in the space of antisym- metric n-tensors, ΛnV = cYn (V ⊗n), thereby proving (3.15). We deduce from (3.10), (3.11) and (2.15) that SN (V ) dn if n < N r+s=n q+s−1 if n ≥ N (3.16) where d = dim V , p = dim V0̄ and q = dimk V1̄. 3.4. The operations ◦ and • on HNAlg . Let A = A(V,R) and A′ = A(V ′, R′) be N - homogeneous superalgebras. Following [37] and [7] we define the white and black products A ◦ A′ and A • A′ by A ◦A′ = A V ⊗ V ′, cπN R⊗ V ′⊗N + V ⊗N ⊗R′ A •A′ = A V ⊗ V ′, cπN where πN ∈ S2N is the inverse of the permutation (1, 2, . . . , 2N) 7→ (1, N + 1, 2, N + 2, . . . , k,N + k, . . . ,N, 2N) Explicitly, cπN : V ⊗N ⊗ V ′⊗N −→ (V ⊗ V ′)⊗N is the morphism in Vects that is given by v1 ⊗ . . . vN ⊗ v 1 ⊗ . . . v = (−1) bvj (v1 ⊗ v 1)⊗ . . . (vN ⊗ v N ) (3.17) Hence, cπN (R⊗R ′) and cπN R⊗ V ′⊗N + V ⊗N ⊗R′ are homogeneous subspaces of (V⊗ V ′)⊗N and so A ◦ A′ and A • A′ belong to HNAlg Under the isomorphism (V ′∗⊗V ∗)⊗N −→ (V ⊗V ′)∗⊗N which comes from (1.4), the rela- tions cπN R′⊥ ⊗R⊥ of A′! • A! map onto the relations R⊗ V ′⊗N + V ⊗N ⊗R′ of (A ◦ A′)!. In fact, by (1.11) we have c∗πN = cπN , because πN τ = τπN , and so 〈x, y〉 = 〈cπN (x), cπN (y)〉 holds for all x ∈ V ′∗ ⊗N ⊗ V ∗⊗N and y ∈ V ⊗N ⊗ V ′⊗N . Therefore, canonically, (A ◦A′)! ∼= A ′! • A! and (A • A′)! ∼= A′! ◦ A! (3.18) N -HOMOGENEOUS SUPERALGEBRAS 23 the two identities being equivalent by (3.12). By definition of ◦, the canonical isomorphisms k⊗V ∼= V ∼= V ⊗ k in Vects give isomor- phisms A(k, 0) ◦ A ∼= A ∼= A ◦ A(k, 0) in HNAlg , and (3.18) yields similar isomorphisms for •, with A(k, 0)! = k[d]/(dN ) replacing A(k, 0) = k[t]; see Example 3.5. The supersymmetry isomorphism cV,V ′ : V ⊗ V ′ ∼−→ V ′ ⊗ V in Vects (see (1.1)) yields isomorphisms A ◦ A′ ∼= A ′ ◦ A and A • A′ ∼= A′ • A (3.19) in HNAlg . To see this, note that the following diagram of isomorphisms in Vects commutes: V ⊗N ⊗ V ′⊗N cπN // V⊗N,V ′⊗N (V ⊗ V ′) V,V ′ V ′⊗N ⊗ V ⊗N cπN // (V ′ ⊗ V ) with v1⊗. . . vN⊗v 1⊗. . . v N 7→ (−1) bvj (v′1⊗v1)⊗. . . (v N⊗vN ) in both composites. Therefore, putting RA◦A′ = cπN R⊗ V ′⊗N + V ⊗N ⊗R′ and similarly for RA′◦A etc., we V,V ′ (RA◦A′) = cπN ◦ cV ⊗N ,V ′⊗N R⊗ V ′⊗N + V ⊗N ⊗R′ = cπN R′ ⊗ V ⊗N + V ′⊗N ⊗R = RA′◦A In the same way, one sees that c⊗N V,V ′ (RA•A′) = RA′•A. This proves (3.19). Similarly, the associativity isomorphism aV,V ′,V ′′ : (V ⊗ V ′) ⊗ V ′′ ∼= V ⊗ (V ′ ⊗ V ′′) in Vects leads to isomorphisms (A ◦ A′) ◦ A′′ ∼= A ◦ (A ′ ◦ A′′) and (A •A′) • A′′ ∼= A • (A′ • A′′) (3.20) in HNAlg . This is a consequence of the following commutative diagram of isomorphisms in Vects V ⊗N ⊗ V ′⊗N ⊗ V ′′⊗N cπN⊗Id V ⊗N,V ′⊗N,V ′′⊗N (V ⊗ V ′) ⊗ V ′′⊗N cπN // ((V ⊗ V ′)⊗ V ′′) V,V ′,V ′′ V ⊗N ⊗ V ′⊗N ⊗ V ′′⊗N Id⊗cπN // V ⊗N ⊗ (V ′ ⊗ V ′′)⊗N cπN // (V ⊗ (V ′ ⊗ V ′′)) Finally, the compatibility between the isomorphisms cV,V ′ and aV,V ′,V ′′ (see §1.2) is inher- ited by the isomorphisms (3.19) and (3.20) in HNAlg . To summarize: Proposition 3.9. The operations ◦ and • both make the category HNAlg of N -homogeneous k-superalgebras into a symmetric tensor category, with unit objects A(k, 0) = k[t] for ◦ and A(k, 0)! = k[d]/(dN ) for •. 24 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ 3.5. The superalgebra map i : A◦A′ → A⊗A′. Let A = A(V,R) and A′ = A(V ′, R′) be objects of HNAlg . The superalgebra A⊗A′ is generated by V ⊕ V ′ subject to the relations R+R′ ⊆ (V ⊕ V ′)⊗N and [V, V ′]⊗ ⊆ (V ⊕ V where [ . , . ]⊗ is the supercommutator (2.2) in the tensor algebra, as usual. Thus, A ⊗ A not N -homogeneous when N ≥ 3. Nonetheless, there always is an injective superalgebra homomorphism i : A ◦ A′ → A ⊗ A′ which is defined as follows. The linear embedding V ⊗ V ′ →֒ T(V )⊗ T(V ′) extends uniquely to a superalgebra map ι̃ : T(V ⊗ V ′) → T(V )⊗ T(V ′) (3.21) which doubles degrees: the restriction of ι̃ to degree n is the embedding T(V ⊗ V ′)n = (V ⊗ V −→ V ⊗n ⊗ V ′⊗n ⊆ (T(V )⊗ T(V ′))2n in Vects , where cπn is as in (3.17). Thus, ι̃ identifies the superalgebra T(V ⊗ V ′) with the (super) Segre product n≥0 V ⊗n ⊗ V ′⊗n of T(V ) and T(V ′). The map ι̃ sends RA◦A′ = cπN R⊗ V ′⊗N + V ⊗N ⊗R′ ⊆ (V ⊗V ′)⊗N to R⊗V ′⊗N + V ⊗N ⊗R′, the kernel of the canonical epimorphism V ⊗N ⊗ V ′⊗N ։ AN ⊗A N . Thus: Proposition 3.10. The algebra map ι̃ in (3.21) passes down to yield an injective homomor- phism k-superalgebras i : A ◦ A′ ֌ A ⊗ A′ which doubles degree. The image of i is the super Segre product n≥0An ⊗A n of A and A 3.6. Internal Hom. The isomorphisms (1.3) and (1.4) together with associativity lead to a functorial isomorphism (U ⊗ V,W ∗) ∼= Hom (U, (V ⊗W )∗) in Vects . Explicitly, if g ∈ Hom (U ⊗ V,W ∗) and g′ ∈ Hom (U, (V ⊗W )∗) correspond to each other under the above isomorphism then 〈g(u ⊗ v), w〉 = 〈g′(u), v ⊗ w〉 (3.22) holds for all u ∈ U , v ∈ V and w ∈ W . In particular, by restricting to 0̄-components, we have a k-linear isomorphism HomVects (U ⊗ V,W ∗) ∼= HomVects (U, (V ⊗W )∗) (3.23) This isomorphism leads to Proposition 3.11. There is a functorial isomorphism HomHNAlgs (A • B, C) ∼= HomHNAlgs (A, C ◦ B!) Proof. We follow Manin [37, 4.2]. Let A = A(U,R), B = A(V, S) and C = A(W,T ) be N -homogeneous superalgebras. We will prove the proposition in the following equivalent form; see (3.12) and (3.18): HomHNAlgs (A • B, C!) ∼= HomHNAlgs (A, (B • C)!) Recall that C! = A(W ∗, T⊥) and (B•C)! = A((V ⊗W )∗, (cπN (S⊗T )) ⊥). Let g : U⊗V → W ∗ be a morphism in Vects and let g′ : U → (V ⊗ W )∗ be the morphism in Vects N -HOMOGENEOUS SUPERALGEBRAS 25 corresponds to g under (3.23). We must show that, for homogeneous subspaces R ⊆ U⊗N , S ⊆ V ⊗N and T ⊆ W⊗N , g⊗N (cπN (R ⊗ S)) ⊆ T ⊥ ⇔ g′⊗N (R) (cπN (S ⊗ T )) Identifying T⊥⊥ with T as in §3.3, the first inclusion is equivalent to 〈g⊗N (cπN (R ⊗ S)) , T 〉 = 0 (3.24) while the second inclusion states that 〈g′⊗N (R), cπN (S ⊗ T )〉 = 0 (3.25) But (3.22) shows that (3.24) and (3.25) are equivalent, which proves the proposition. � Proposition 3.11 says that the tensor category (HNAlg , •) has an internal Hom which is given by Hom(A,B) = B ◦ A! Explicitly, Hom(A,B) is an object of HNAlg which represents the functor (HNAlg )op → Sets, X 7→ HomHNAlgs (X • A,B); so there is an isomorphism of functors HomHNAlgs (? • A,B) ∼= HomHNAlgs (?,Hom(A,B)) By general properties of Hom (see [12, Def. 1.6]), the morphism IdHom(A,B) corresponds to a morphism µ : Hom(A,B) • A → B (3.26) in HNAlg satisfying the following universal property: for any morphism f : X • A → B in HNAlg there exists a unique morphism g : X → Hom(A,B) such that the following diagram commutes: X • A g•IdA Hom(A,B) • A In degree 1, the map µ is simply IdV ⊗evU : V ⊗ U ∗ ⊗ U −→ V ⊗ k = V . From Hom(B, C) • Hom(A,B) • A Id •µ −→ Hom(B, C) • B −→ C one obtains in this way a composition morphism m : Hom(B, C) •Hom(A,B) → Hom(A, C) (3.27) in HNAlg . The morphisms µ and m satisfy the obvious associativity properties. 3.7. The superbialgebra endA. Following Manin [37, 4.2] we define hom(A,B) = Hom(A!,B!)! = A! • B for A, B in HNAlg . Applying the dualization functor to (3.26), (3.27) and recalling (3.18), we obtain morphisms δ◦ : A → B ◦ hom(B,A) ∆◦ : hom(A, C) → hom(A,B) ◦ hom(B, C) 26 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ in HNAlg . The associativity properties of µ and m translate into corresponding coassociativ- ity properties for δ◦ and ∆◦. Following δ◦ and ∆◦ by the algebra map i of Proposition 3.10, we obtain superalgebra maps δ : A → B ⊗ hom(B,A) (3.28) ∆: hom(A, C) → hom(A,B)⊗ hom(B, C) (3.29) Now take A = B = C = A(V,R) and put endA = hom(A,A); so endA = A! • A = A(V ∗ ⊗ V, cπN (R ⊥ ⊗R)) (3.30) Then (3.29) yields a coassociative superalgebra map ∆: endA → endA⊗ endA Moreover, by Proposition 3.11, the morphism A! −→ A! ∼= k[t] ◦ A! corresponds to a mor- phism endA = A! • A → k[t] in HNAlg . Following this morphism by the map t 7→ 1 we obtain a superalgebra map ε : endA → k which in degree 1 is the usual evaluation pairing evV : V ∗ ⊗ V → k in Vects . Finally, (3.28) provides us with a superalgebra map δA : A → A⊗ endA (3.31) Note that δA maps the degree n-component of A according to −→ (A ◦ endA)n −→ An ⊗ (endA)n →֒ An ⊗ endA (3.32) Fixing a graded k-basis x1, . . . , xd of V and denoting the dual basis of V ∗ by x1, . . . , xd as before, endA has algebra generators zij := x i ⊗ xj (3.33) of degree-1 and parity î+ ĵ. In terms of these generators, the maps ε, δA and ∆ are given by i ) = δ i or ε(Z) = 1 δA(xj) = xi ⊗ z j or δA(x1, . . . , xd) = (x1, . . . , xd)⊗ Z ∆(zij) = zik ⊗ z j or ∆(Z) = Z ⊗ Z (3.34) where Z = (zij)d×d. Proposition 3.12. Let A = A(V,R) be an N -homogeneous k-superalgebra. (a) With ∆ as comultiplication and ε as counit, the superalgebra endA becomes a super- bialgebra. Moreover, δA makes A into a graded right endA-comodule superalgebra. N -HOMOGENEOUS SUPERALGEBRAS 27 (b) Given any k-superalgebra B and a morphism of superalgebras δ : A → A⊗ B satis- fying δ(V ) ⊆ V ⊗ B, there is a unique morphism of superalgebras ϕ : endA → B such that the following diagram commutes: δA $$J A⊗ endA IdA ⊗ϕ The proposition is proved as in [37, §5] or [7, Theorem 3]. Example 3.13. When A = A(V, 0) = T(V ), we have endA = A(V ∗⊗V, 0) = T(V ∗⊗V ); endT(V ) = T(V ∗ ⊗ V ) the free superalgebra generated by the elements zij in (3.33). Example 3.14. By Examples 3.3 and 3.8, we have endSN (V ) = A V ∗ ⊗ V, cπN (1− cYN )(V ∗⊗N )⊗ cYN (V For example, the algebra end S2(V ) is generated by the elements z j with parity î+ ĵ subject to the relations ] + (−1) bi1 bi2+(bi1+bi2)bj1 [z ] = 0 where [ . , . ] is the supercommutator (2.2). This algebra is highly noncommutative, even for a pure even space V . Let O(E(V )) = S(V ∗ ⊗ V ) be the supercommutative superbialgebra as in §2.5.3, with generators xij . There is a map of superbialgebras ϕ : endSN (V ) → O(E(V )) , z j 7→ x j (3.35) Indeed, write B = O(E(V )) for brevity and recall the coaction δ : V → V ⊗ B, xj 7→∑ i xi ⊗ x j from (2.11). Since cYN ∈ EndcomodsB(V ⊗N ) (see §2.6.2), the map δ extends to a map of superalgebras δ : SN (V ) → SN (V )⊗ B Therefore, Proposition 3.12(b) yields the desired ϕ. Note that the coaction of end SN (V ) on V , when restricted along ϕ, becomes the canonical coaction of O(E(V )) on V ; see (2.11) and (3.34). 4. N -KOSZUL SUPERALGEBRAS Throughout this section, we fix an N -homogeneous superalgebra A = A(V,R). 28 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ 4.1. The graded dual A!∗. The graded dual A!∗ = A! ∗n of A! has a natural structure of a graded right endA-comodule. Indeed, the linear dual A! ∗n of the degree n-component of A! embeds into V ⊗n as follows. Recall from (3.11) that A! ∗n = V ⊗n if n < N i+j+N=n V ⊗i ⊗R⊗ V ⊗j if n ≥ N (4.1) This identification makes the graded dual A!∗ into a graded right endA-comodule. For, by (3.32) the coaction δA restricts in degree 1 to a map V → V ⊗endA which makes T(V ) into a graded right endA-comodule superalgebra. The structure map T(V ) → T(V )⊗endA sends R → R ⊗ endA. Therefore, each V ⊗i ⊗ R ⊗ V ⊗j is a endA-subcomodule of V ⊗(i+j+N), and hence A! ∗n is a endA-subcomodule of V ⊗n. Finally, for all n ≥ 0, A! ∗n+1 ⊆ V ⊗A n and A n+N ⊆ V ⊗N ⊗A! ∗n ∩R⊗ V ⊗n = R⊗A! ∗n (4.2) 4.2. The Koszul complex. The map A⊗ V ⊗(i+1) → A⊗ V ⊗i a⊗ (v1 ⊗ · · · ⊗ vi+1) 7→ av1 ⊗ (v2 ⊗ · · · ⊗ vi+1) is a morphism in the category ComodsendA of right endA-comodules, because the endA- coaction δA in (3.31) is a superalgebra map. Furthermore, this map is a left A-module map which preserves total degree, and it restricts to a map of endA-subcomodules d : A⊗A! ∗i+1 → AV ⊗A i →֒ A ⊗A which is the A-linear extension of the embedding (4.2). The map dN sends A! ∗i+N to AR ⊗ A! ∗i = 0; so d N = 0. In other words, we have an N -complex K(A) : . . . −→ A⊗A! ∗i+1 −→ A⊗A! ∗i −→ . . . −→ A −→ 0 (4.3) in ComodsendA consisting of graded-free left A-modules and A-module maps which pre- serve total degree. Therefore, K(A) splits into a direct sum of N -complexes K(A)n =⊕ i+j=nAi ⊗A j in comod endA. Following [7], the Koszul complex K(A) defined by Berger in [5] can be described as the following contraction of K(A): K(A) : . . . −→ A⊗A! ∗N+1 −→ A⊗A! ∗N −→ A⊗A! ∗1 −→ A −→ 0 (4.4) This is an ordinary complex in ComodsendA which splits into a direct sum of complexes K(A)n in comodsendA. The i th components of K(A) and of K(A)n are given by K(A)i = A⊗A ν(i) and K(A) i = An−ν(i) ⊗A with ν(i) = νN (i) as in (0.1). The differential on K(A) is δi : K(A)i → K(A)i−1 where δi = dN−1 for i even d for i odd N -HOMOGENEOUS SUPERALGEBRAS 29 Writing A+ = n>0 An = AV as usual, we have Ker δi ⊆ A+K(A)i for all i. Indeed, this is clear for odd i, since δi = d is injective on A . For even i, the restriction of δi = d N−1 to A! ∗ is given by dN−1 : A! ∗ = A! ∗ ν(i−1)+N−1 →֒ V ⊗(N−1) ⊗ ν(i−1) −→ AN−1 ⊗A ν(i−1) →֒ A ⊗A! ∗ ν(i−1) where the first embedding comes from (4.2). Since A! ∗ = A! ∗1 = V and A = A! ∗N = R by (4.1), the start of the Koszul complex, augmented by the canonical map A ։ k = A/A+ is as follows: −→ A⊗ V δ1=mult −→ A −→ k −→ 0 (4.5) This piece is easily seen to be exact: writing A = T(V )/I with I = (R) = I⊗V +T(V )⊗R as in (0.2), the map T(V )+ = T(V ) ⊗ V ։ A ⊗ V ։ A+ has kernel I . Thus, Ker δ1 = I/I ⊗ V = Im δ2. Hence (4.5) is the start of the minimal graded-free resolution of the left A-module k. 4.3. N -homogeneous Koszul superalgebras. Recall from the Introduction that an N -homoge- neous superalgebra A is called N -Koszul if TorAi (k, k) is concentrated in degree νN (i) for all i ≥ 0. By [5, Proposition 2.12] or [8, Theorem 2.4], this happens exactly if the Koszul complex K(A) is exact in degrees i > 0 and in view of (4.5), this amounts to exactness of K(A) in degrees i ≥ 2. In this case, K(A) −→ k −→ 0 is the minimal graded-free resolution of the trivial left A-module k. The Yoneda Ext-algebra E(A) = i≥0 Ext A(k, k) of an N -Koszul superalgebra A has the following description in terms of the dual algebra A!: ExtiA(k, k) ν(i) (i ≥ 0) Moreover, identifying ExtiA(k, k) and A , the Yoneda product f · g and the A!-product fg for f ∈ ExtiA(k, k) = A and g ∈ ExtjA(k, k) = A are related by f · g = (−1)ijfg when N = 2, and f · g = fg if i or j is even 0 if i and j are both odd for N > 2; see [21, Theorem 9.1], [8, Proposition 3.1]. Example 4.1. Quadratic algebras having a PBW-basis are 2-Koszul; see, e.g., [41, Chap. 4, Theorem 3.1]. This applies in particular to quantum superspace A = Afq; see Example 3.1. A PBW-basis in this case is given by the collection of ordered monomials xm11 x 2 . . . x d with mi ≥ 0 for all i and mi ≤ 1 if î = 1̄, as in Example 3.1. For a more general result, see [41, Chap. 4, Theorem 8.1]. 30 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Example 4.2. The unmixed Yang-Mills algebras A = YM± (see Example 3.2) were shown to be 3-Koszul in [10], [11]. Indeed, letting A[ℓ] denotes the shift of A that is defined by A[ℓ]n = Aℓ+n, the defining relations for A in the form (3.7) imply that the following complex of graded-free left A-modules is exact: 0 −→ A[−4] −→ A[−3]d −→ A[−1]d ·Y tr −→ A −→ k −→ 0 (4.6) The piece A[−3]d −→ A[−1]d ·Y tr −→ A −→ k −→ 0 is identical with (4.5). Therefore, (4.6) is the minimal graded-free resolution of k. The resolution shows that each TorAi (k, k) is con- centrated in degree ν3(i), and hence A is 3-Koszul. It also follows that (4.6) is isomorphic to K(A) → k → 0. In particular, (4.6) confirms the dimensions of the corresponding compo- nents A!n in Example 3.7. As has been pointed out in [10], [11], it follows from (4.6) that the Hilbert series HA(t) = n≥0 dimkAn t n of A = YM± has the form HA(t) = 1− dt+ dt3 − t4 (1− t2)(1− dt+ t2) If d > 2 then the series has a pole in the interval (0, 1), and hence dim An grows exponen- tially with n. Therefore, A is not noetherian in this case; see Stephenson and Zhang [42]. The mixed Yang-Mills algebras A = YMp|q with p 6= 0 and q 6= 0, on the other hand, are never 3-Koszul. For YM1|1 this follows from the description given in Example 3.2: this alge- bra has infinite global dimension. In general, one can check that the so-called extra condition (see (4.10) below) fails for A, and so A cannot be Koszul, by [5, Prop. 2.7]. Example 4.3. It has been shown in [5, Theorem 3.13] that the N -symmetric algebra SN(V ) of a pure even space V over a field of characteristic 0 is N -Koszul. An extension of this result will be offered in Theorem 4.5 below. 4.4. Confluence and Koszulity. For the convenience of the reader, we recall the notions of reduction operators and confluence and their relation to the Koszul property. Complete details can be found in Berger [4], [5]. Let V in Vects be given along with a graded basis X = {x1, . . . xd} that is ordered by x1 > x2 > · · · > xd. The tensors (“monomials”) xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xiN for i = (i1, i2, . . . , iN ) ∈ {1, 2, . . . , d} N form a basis of V ⊗N which will be given the lexicographical ordering. An X-reduction operator on V ⊗N is a projection S ∈ EndVects (V ⊗N ) such that either S(xi) = xi or S(xi) < xi holds for each i, where the latter inequality means that S(xi) is a linear combination (possibly 0) of monomials < xi. The monomials xi satisfying S(xi) = xi are called S-reduced, all other monomials are S-nonreduced. We denote by Red(S) and NRed(S)) the (super) subspaces of V ⊗N that are generated by the S-reduced monomials and the S-nonreduced monomials, respectively; so V ⊗N = Red(S) ⊕ NRed(S) and Im(S) = Red(S). Let LX(V ⊗N ) denote the collection of all X-reduction operators on V ⊗N . The proof of [4, Theorem 2.3] shows that the application S 7→ Ker(S) is a bijection between LX(V and the set of all super subspaces of V ⊗N . Hence LX(V ⊗N ) inherits a lattice structure: for S, S′ ∈ LX(V ⊗N ) one has X-reduction operators S ∧ S′ and S ∨ S′ on V ⊗N which are N -HOMOGENEOUS SUPERALGEBRAS 31 defined by Ker(S ∧ S′) = Ker(S) + Ker(S′) Ker(S ∨ S′) = Ker(S) ∩Ker(S′) A pair (S, S′) of X-reduction operators is said to be confluent if Red(S ∨ S′) = Red(S) + Red(S′) Since the inclusion ⊇ is always true, confluence of (S, S′) is equivalent to the inequality Im(S ∨ S′) ≤ dim (Im(S) + Im(S′)) (4.7) Let n ≥ N . Any X-reduction operator S on V ⊗N gives rise to X-reduction operators Sn,i on V ⊗n which are defined by Sn,i := IdV ⊗i ⊗S ⊗ IdV ⊗j (i+ j +N = n) A monomial xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xin of length n ≥ N is said to be S-reduced if xi is Sn,i-reduced for all i, that is, if every connected submonomial of xi of length N is S-reduced. Now let A = A(V,R) be an N -homogeneous superalgebra, and let S be the X-reduction operator on V ⊗N such that Ker(S) = R. The algebra A is said to be X-confluent if the pairs (SN+i,i, SN+i,0) of X-reduction operators on V ⊗N+i are confluent for i = 1, . . . , N − 1. By (4.7) this amounts to the inequalities Im(SN+i,i ∨ SN+i,0) ≤ dim (Im(SN+i,i) + Im(SN+i,0)) (4.8) being satisfied for i = 1, . . . , N − 1. Following Berger [5], we denote by Tn the lattice of super subspaces of V ⊗n that is gen- erated by the subspaces Rn,i := V ⊗i ⊗R⊗ V ⊗j = Ker(Sn,i) (i+ j +N = n) (4.9) The superalgebra A is said to be distributive if the lattices Tn are distributive for all n, that is, C ∩ (D + E) = (C ∩D) + (C ∩ E) holds for all C,D,E ∈ Tn. The following proposition states the operative facts concerning Koszulity for our purposes. Part (a) is identical with [5, Thm. 3.11] while (b) is [5, Prop. 3.4]. Proposition 4.4. Let A = A(V,R) be an N -homogeneous superalgebra. (a) If A is X-confluent for some totally ordered graded basis X of V then A is distributive. Moreover, letting S denote the X-reduction operator on V ⊗N such that Ker(S) = R, the classes in A of the S-reduced monomials xi1 ⊗ xi2 ⊗ · · · ⊗ xin with xij ∈ X form a k-basis of An for all n ≥ N . (b) Assume that A is distributive and the following “extra condition” is satisfied Rn+N,0 ∩Rn+N,n ⊆ Rn+N,n−1 (2 ≤ n ≤ N − 1) (4.10) Then A is N -Koszul. After these preparations, we are now ready to prove the following result. The quadratic case N = 2 is due to Gurevich [22]; see also Wambst [44]. Theorem 4.5. Let N ≥ 2 and 0 6= q ∈ k and assume that [n]q 6= 0 for all n ≥ 1. Then, for every Hecke operator R associated with q, the N -homogeneous superalgebra ΛR,N defined in (3.8) is N -Koszul. 32 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Proof. Put A = ΛR,N and recall that A = A(V,R) with R = Im ρN,R(XN ) ⊆ V The extra condition (4.10) is a consequence of equation (1.19). Indeed, (1.19) implies that the spaces Rn,i in (4.9) have the form Rn,i = i+N−1⋂ s=i+1 Im(ρn,R(Ts) + 1) ⊆ V ⊗n (4.11) Applying (4.11) with ρ = ρn+N,R we see that the left hand side of (4.10) is identical to Im(ρ(Ti) + 1) ∩ n+N−1⋂ i=n+1 Im(ρ(Ti) + 1) = n+N−1⋂ Im(ρ(Ti) + 1) where the equality holds because n + 1 ≤ N . The last expression is clearly contained in⋂n+N−2 i=n Im(ρ(Ti) + 1), which is identical to the right hand side of (4.10). This establishes the extra condition (4.10). In order to prove the distributivity of A, we follow the approach taken in [25]. We first prove the claim for the standard solution RDJ , i.e., the operator given in Example 1.2 with d = p and q = 0. As above, fix a basis X = {x1, . . . , xd} of V , ordered by x1 > x2 > · · · > xd, and consider the basis of V ⊗n consisting of the monomials xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xin for i = (i1, i2, . . . , in) ∈ {1, 2, . . . , d} n with the lexicographical ordering. By equation (1.21), the action of the generators Tj of the Hecke algebra H = Hn,q2 on this basis is given by Tj(xi) = q2xi if ij = ij+1 (q2 − 1)xi + qxσj(i) if ij < ij+1 qxσj(i) if ij > ij+1 (4.12) Here, σj = (j, j + 1) ∈ Sn and σ(i) = (iσ−1(1), iσ−1(2), . . . , iσ−1(n)) for σ ∈ Sn, as in Example 3.3. We claim that the H -submodule of V ⊗n that is generated by xi is given by H (xi) = i′∈Sn(i) kxi′ (4.13) where Sn(i) is the Sn-orbit of i. Indeed, (4.12) implies that each Tσ(xi) with σ ∈ Sn is a linear combination of basis vectors xi′ with i ′ ∈ Sn(i). Hence, ⊆ certainly holds in (4.13). For the reverse inclusion, let i∗ denote the unique non-decreasing sequence in Sn(i); so xi∗ = max{xi′ | i ′ ∈ Sn(i)}. The last formula in (4.12) implies that T (xi) = q r(i)xi∗ (4.14) where T is a suitable finite product of length r(i) ≥ 0 in the generators Tj . Since T is a unit in H , the inclusion ⊇ holds in (4.13), thereby proving the asserted equality. Furthermore, (4.14) and (1.18) (with q replaced by q2) give qr(i)Xn(xi) = Xn(xi∗). (4.15) N -HOMOGENEOUS SUPERALGEBRAS 33 These elements are nonzero. For, (4.15) implies that the elements Xn(xi∗) span the image of Xn on V ⊗n, and their number is d+n−1 which is equal to the rank of Xn (cf. [25, Eq. (5)]). It follows that Xn(V ⊗n) = Im ρn,RDJ (Xn) has a k-basis consisting of the elements Xn(xi∗) ∣∣ i∗ = (i1 ≤ i2 ≤ · · · ≤ in) ∈ {1, 2, . . . , d}n Next, writing Xn(xi) = i′∈Sn(i) λi′xi′ (4.16) with λi′ ∈ k, we claim that λσj(i′) = λi′ if i ′ = σj(i q±1λi′ otherwise To prove this, we may assume that i′ 6= σj(i ′). We compute the coefficient of xσj(i′) in TjXn(xi) in two ways: by (1.18) this coefficient is equal to q 2λσj(i′) while (4.12) yields the expression qλi′ + (q 2 − q1±1)λσj(i′). The claim follows from this. Writing an arbitrary σ ∈ Sn as a product of the inversions σj , we see that the coefficients λi′ in (4.16) only differ by a nonzero scalar, and hence they are all nonzero since Xn(xi) 6= 0. By Proposition 4.4, it suffices to check the X-confluence conditions (4.8) i = 1, . . . , N−1. So let S be the X-reduction operator on V ⊗N with Ker(S) = R. It is easy to see from the dis- cussion above (with n = N ) that S is given by S(xi∗) = (1−XN/λi∗)(xi∗) and S(xi) = xi for i 6= i∗. According to (4.11) and the discussion above, the dimension of (R⊗V ⊗i)∩(V ⊗i⊗ R) is d+N+i−1 . Thus, the dimension of the left hand side of (4.8) is dN+i− d+N+i−1 the other hand the monomials in V ⊗N+i that belong to NRed(SN+i,i) ∩ NRed(SN+i,0) are exactly those of the form xi∗ with i ∗ ∈ {1, . . . , d}N+i non-decreasing. Their number is pre- cisely d+N+i−1 . Therefore, the dimension of Im(SN+i,i) + Im(SN+i,0) = Red(SN+i,i) + Red(SN+i,0) is at least d N+i − d+N+i−1 . This proves the inequality in (4.8), thereby fin- ishing the proof of the theorem for the case R = RDJ . In order to deal with an arbitrary Hecke operator R, recall that Hn,q is split semisimple, having a representative set of simple modules Mλ indexed by the partitions λ ⊢ n; see (1.15). We denote the representation of Hn,q on Mλ by ρλ; it does not depend on the operator R but only on the partition λ. Let us fix a decomposition V ⊗n = into simple Hn,q-submodules Mt. Since all Mt are invariant under the operators ρn,R(Tj), formula (4.11) yields the decomposition Rn,i = i+N−1⋂ s=i+1 (ρn,R(Ts) + 1)(Mt) = Rn,i ∩Mt for all i. Therefore, by [25, Lemma 1.2], distributivity of the lattice Tn that is generated by the subspaces Rn,i of V ⊗n is equivalent to distributivity of the lattices Tn ∩Mt (t ∈ T ) that 34 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ are generated by the subspaces Rn,i ∩Mt = i+N−1⋂ s=i+1 (ρn,R(Ts) + 1)(Mt) of Mt. Now, each Mt is isomorphic to Mλ for some λ ⊢ n. Therefore, the lattice Tn ∩Mt is isomorphic to the lattice of subspaces of Mλ that is generated by the subspaces i+N−1⋂ s=i+1 (ρλ(Ts) + 1)(Mλ) with i + N ≤ n. Finally, when d = dimV > n, then all simple Hn,q-modules Mλ appear in V ⊗n; see [15, Proposition 5.1]. Thus, the distributivity of the lattice associated to RDJ , which we have already verified, implies the distributivity of the corresponding lattice for any Hecke operator R. This completes the proof. � 5. KOSZUL DUALITY AND MASTER THEOREM In this section, A = A(V,R) denotes an N -homogeneous superalgebra that is assumed to be N -Koszul (N ≥ 2). 5.1. By Koszulity, the complexes K(A)n : . . . → An−νN(i) ⊗A νN (i) → An−νN (i−1) ⊗A νN (i−1) → . . .→An → 0 are exact for n > 0. This yields equations in the Grothendieck ring RendA of the category comodsendA : ∑ (−1)i[An−νN (i)][A νN (i) ] = 0 (n > 0) (5.1) In the power series ring RendAJtK over the Grothendieck ring RendA, define the Poincaré series PA(t) = [An]t n and PA!∗(t) = [A! ∗n ]t For any power series P (t) = n ant n, we use the notation PN (t) := n≡0,1 mod N (−1)αN (n)ant where αN (n) = n − (n mod N) denotes the largest multiple of N less than or equal to n. Thus, P2(t) = P (t) and in general PN (−t) = n≡0,1 mod N (−1)n mod Nant (−1)iaνN (i)t νN (i) (5.2) In particular, PA!∗,N (−t) = (−1)i[A! ∗νN (i)]t νN (i) Equations (5.1) are equivalent to the following Koszul duality formula: N -HOMOGENEOUS SUPERALGEBRAS 35 Proposition 5.1. For any N -homogeneous Koszul superalgebra A, the identity PA(t)PA!∗,N (−t) = 1 holds in RendAJtK. Applying the ring homomorphism χsJtK : RendAJtK → (endA)0̄JtK, where χ s is the su- percharacter map as in Corollary 2.3, the formula in Proposition 5.1 takes the following form in (endA)0̄JtK: Corollary 5.2. χsAℓt m≡0,1 mod N (−1)m mod Nχs A! ∗m  = 1 Analogous formulas hold with the supercharacter χs replaced by the ordinary character χ or by one of the dimensions dim and sdim. By (3.32) the coaction of endA on A sends An to An⊗ (endA)n. A similar remark holds for the endA-coaction on A! ∗; see §4.1. Therefore, both factors in Corollary 5.2 actually belong to the Rees subring n≥0 Bnt n of BJtK, where we have put B = (endA)0̄. Example 5.3. As an application of the Hilbert series version of Corollary 5.2, we see that the duals A! of the Yang-Mills algebras A = YMp|q are never 3-Koszul. In fact, by Example 3.7, we have HA!(t) = 1+dt+d 2t2+dt3+t4 if p = 0 or q = 0 and HA!(t) = 1+dt+d 2t2+dt3 otherwise. In either case, HA!(t) −1 has a nonzero coefficient at t5, which rules out Koszulity. 5.2. A master theorem modeled on the N -symmetric superalgebra SN (V ). We put A = SN (V ) and use the notation of Examples 3.3 and 3.8. In particular, we assume that char k = 0 and work with a fixed basis x1, . . . , xd of V = A1 so that î = 0̄ for i ≤ p and î = 1̄ for i > p. From Example 3.3 (see also Proposition 4.4(a)), we know that a basis of Aℓ is given by the monomials xi = xi1xi2 . . . xiℓ for sequences i = (i1, . . . , iℓ) ∈ {1, . . . , d} ℓ such that i has no connected subsequence j = (j1, . . . , jN ) of length N satisfying 1 ≤ j1 < . . . < jm ≤ p < jm+1 ≤ . . . ≤ jN ≤ d = p+ q for some m. Adapting notation of Etingof and Pak [16] to our setting, we denote this set of sequences i by Λ(p|q,N)ℓ (5.3) For example, Λ(p|q, 2)ℓ consists of all weakly decreasing sequences i = (i1, . . . , iℓ) with entries from {1, . . . , d} and such that no repetition occurs in the range {p + 1, . . . , d}. In order to evaluate the character χsAℓ in Corollary 5.2, recall from (3.34) that the coaction δA : A → A⊗ endA is given on the generators xi of A by δA(xi) = xj ⊗ z i ∈ A⊗ endA where zji = x j ⊗ xi are the canonical generators of the algebra endA. For i = (i1, . . . , iℓ) ∈ Λ(p|q,N)ℓ, we have δA(xi) = δA(xi1)δA(xi2) . . . δA(xiℓ) ∈ Aℓ ⊗ endA Since Aℓ ⊗ endA = i∈Λ(p|q,N)ℓ xi ⊗ endA, we can define Z(i) ∈ (endA)0̄ by δA(xi) = xi ⊗ Z(i) + (terms supported on Λ(p|q,N)ℓ \ {i}) 36 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ Then (2.8) becomes χsAℓ = i∈Λ(p|q,N)ℓ biZ(i) (5.4) with î = î1 + · · ·+ îℓ. Now consider the super bialgebra B = O(E(V )) = k[xij | 1 ≤ i, j ≤ d] defined in §2.5.3 and recall that the xij are supercommuting variables of parity î + ĵ over k. Restricting the comodule Aℓ to B along the map ϕ : endSN (V ) → B, z j 7→ x j in (3.35) we must replace Z(i) in (5.4) by X(i) := ϕ(Z(i)) ∈ B0̄. Thus, writing xj ⊗ x i ∈ A⊗ B and yi = yi1 . . . yiℓ ∈ Aℓ ⊗ B = j∈Λ(p|q,N)ℓ xj ⊗ B for i = (i1, . . . , iℓ), we have yi = xi ⊗X(i) + (terms supported on Λ(p|q,N)ℓ \ {i}) (5.5) As for the supercharacter of A! ∗m , recall from (4.1) and (3.15) that, for all n ≥ N , A! ∗n = i+j+N=n V ⊗i ⊗R⊗ V ⊗j = ΛnV Viewing A! ∗n = Λ nV as a comodule over B = O(E(V )), the supercharacter of A! ∗n is the nth elementary supersymmetric function en which we know, by Proposition 2.4, to be iden- tical to the coefficient at tn of the characteristic function ber(1 + tX) of the generic su- permatrix X = 1≤i,j≤d of type p|q; so the diagonal blocks X11 = 1≤i,j≤p X22 = p+1≤i,j≤p+q consist of even entries while all other entries are odd. To summarize, we obtain the following super-version of [16, Theorem 2]. Theorem 5.4. Let X = be the generic supermatrix of type p|q. Then i∈Λ(p|q,N)ℓ biX(i) tℓ m≡0,1 mod N (−1)m mod Nemt  = 1 holds in the power series ring k[xij | all i, j ]0̄JtK. Here Λ(p|q,N)ℓ and X(i) are defined by (5.3) and (5.5), respectively, and the em are the coefficients of the characteristic function ber(1 + tX) = n≥0 ent n of X. 5.3. As an application of Theorem 5.4 , we determine the superdimension Hilbert series HsA(t) = for the N -symmetric superalgebra A = SN (V ). For the pure even case, this was already done by Etingof and Pak [16] . The notations of §5.2 remain in effect. In view of Corollary 2.3, the superdimension Poincaré series follows by applying the counit ε : B → k to the equation in Theorem 5.4. Indeed, by (2.11), the counit ε sends X 7→ 1d×d, N -HOMOGENEOUS SUPERALGEBRAS 37 and hence the elements X(i) in (5.5) all map to 1. Therefore, the first factor in Theorem 5.4 becomes HsA(t) = i∈Λ(p|q,N)ℓ For the second factor, note that ber(1 + t 1d×d) = (1 + t) by (2.12). Thus, HsA(t) = i∈Λ(p|q,N)ℓ   m≡0,1 mod N (−1)m mod N if p ≥ q m≡0,1 mod N (−1)αN (m) m+ q − p− 1 q − p− 1 if p < q (5.6) where αN (m) = m− (m mod N) denotes the largest multiple of N less than or equal to m as in §5.1. 5.4. The ordinary Hilbert series HA(t) = ℓ≥0 dimkAℓ t ℓ of the N -symmetric superalge- bra A = SN(V ) is as follows. Recall from §5.2 that Aℓ = |Λ(p|q,N)ℓ| and from (3.16) that A!n = dn if n < N r+s=n q+s−1 if n ≥ N Therefore, the Hilbert series is HA(t) = |Λ(p|q,N)ℓ| t m≡0,1 mod N (−1)m mod N r+s=m q + s− 1 −1 (5.7) 5.5. Less is known about the Hilbert series of the N -homogeneous superalgebras A = ΛR,N associated to an arbitrary Hecke operator R : V ⊗2 → V ⊗2 on a vector superspace V ; see Example 3.4. Recall that A = A(V,R) with R = Im ρR(XN ) ⊆ V ⊗N . For any N - homogeneous algebra A = A(V,R), we have A!n = dimk i+j+N=n V ⊗j ⊗R⊗ V ⊗i 38 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ by (3.10) and (3.11). For R = Im ρR(XN ) in particular, (1.19) further implies that i+j+N=n V ⊗j ⊗R⊗ V ⊗i = ρR(Xn) holds for n ≥ N . Now [26, Theorem 3.5] implies that !(t) = ℓ=1(1 + aℓt)∏s m=1(1− bmt) where (r, s) is the birank of R and aℓ and bm are positive real numbers. For example, in the situation of 5.4, (r, s) = (p, q) and aℓ = bm = 1. For any complex power series P (t), the power series PN (−t) in (5.2) can be written as PN (−t) = (1− ζ−iN )P (ζ where ζN = e 2πi/N . In particular, HA!∗,N (−t) = (1− ζ−iN ) ℓ=1(1 + aℓζ N t)∏s m=1(1− bmζ QN,a,b(t)∏s m=1(1 + bmt+ . . .+ b m tN−1) for some real polynomial QN,a,b(t) with coefficients being polynomial in a = (aℓ) and b = (bm). Therefore, the Hilbert series of A has the form HA(t) = m=1(1 + bmt+ . . .+ b QN,a,b(t) (5.8) Notice that the fraction on the right-hand side is reduced. In particular, (5.7) has the form HA(t) = (1− tN )s (1− t)sQN,1,1(t) (5.9) APPENDIX For lack of a suitable reference, we include here a proof of Proposition 0.1 that was stated in the Introduction. Our proof is based on the proof of [8, Proposition 2.1] and on additional details that were communicated to us by Roland Berger. For the basics concerning graded algebras, we refer the reader to [9, Chap. II §11] or [6]. As in the Introduction, A = n≥0 An denotes an arbitrary connected Z≥0-graded k- algebra and V is a graded subspace of A+ = n>0An satisfying A+ = V ⊕ A +. Thus, T(V )/I −→ A for some graded ideal I of T(V ). For convenience, we state Proposition 0.1 again: Proposition. The relation ideal I of A lives in degrees ≥ N if and only if TorAi (k, k) lives in degrees ≥ νN (i) = N if i is even N + 1 if i is odd N -HOMOGENEOUS SUPERALGEBRAS 39 Proof. Let P : · · · → Pi −→ Pi−1 −→ · · · −→ P0 −→ k → 0 be a minimal graded-free resolution of the trivial left A-module k. Thus, all Pi have the form Pi = A⊗ Ei for some graded subspace Ei ⊆ Ker di−1 which is chosen so that Ker di−1 = Ei ⊕A+Ker di−1 (A.10) In particular, we may take E0 = k and E1 = V . The differential di : Pi → Pi−1 is the graded A-module map that is defined by the inclusion Ei →֒ Pi−1. By the graded Nakayama Lemma (e.g., [9, p. AII.171, Prop. 6]), our choice of Ei implies that Im di = AEi = Ker di−1 and Ker di ⊆ A+ ⊗ Ei = A+Pi (A.11) for all i. Consequently, the complex k⊗A P has zero differential, and hence TorAi (k, k) ∼= k⊗A Pi ∼= Ei In particular, TorA0 (k, k) ∼= k and TorA1 (k, k) ∼= V = A+/A live in degrees 0 = νN (0) and ≥ 1 = νN (1), respectively. Moreover, the kernel of d1 : P1 = (T(V )/I)⊗ V → P0 = A is exactly I/I ⊗ V , and so TorA2 (k, k) ∼= Ker d1/A+Ker d1 ∼= I/ (V ⊗ I + I ⊗ V ) Therefore, I lives in degrees ≥ N if and only if TorA2 (k, k) lives in degrees ≥ N = νN (2). For the remainder of the proof, assume that I lives in degrees ≥ N . We will show by induction on i that TorAi (k, k) = Ei lives in degrees ≥ νN (i) for all i. The cases i ≤ 2 have been checked above. Assume that Ei lives in degrees ≥ νN (i) and similarly for Ei−1. By (A.11), we know that Ei+1 ⊆ Ker di ⊆ A+ ⊗ Ei and so Ei+1 certainly lives in degrees ≥ νN (i)+1. Since νN (i)+1 = νN (i+1) when i is even (or when i is arbitrary and N = 2), we are done in these cases. From now on, we assume that i is odd. We must show that Ei+1 lives in degrees ≥ νN (i + 1) = N . Since Ei+1 ⊆ Ker di, it suffices to show that di is injective in degrees < i+1 N , and since Ei lives in degrees ≥ νN (i) = N+1, our goal is to show that di is injective on all homogeneous components Pi,n of Pi in degrees n = N + j with j = 1, . . . , N − 1. Put m = i−1 N for simplicity and note that Pi,m+j = Aj−ℓ ⊗ Ei,m+ℓ (A.12) Pi−1,m+j = Aj−k ⊗ Ei−1,m+k (A.13) since Ei−1 lives in degrees ≥ νN (i − 1) = m. The proposition will be a consequence of the following claims: (a) di is injective on all summands Aj−ℓ ⊗ Ei,m+ℓ in (A.12), and (b) the subspaces di (Aj−ℓ ⊗ Ei,m+ℓ) = Aj−ℓEi,m+ℓ for ℓ = 1, . . . , j form a direct sum inside Pi−1,m+j . 40 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ In order to prove (a), recall that the restriction of di to Ei,m+ℓ is the inclusion Ei,m+ℓ →֒ Pi−1,m+ℓ = Aℓ−k ⊗ Ei−1,m+k Hence, the effect of di on the ℓ th summand in (A.12) is the embedding Aj−ℓ ⊗ Ei,m+ℓ →֒ Aj−ℓ ⊗Aℓ−k ⊗ Ei−1,m+k followed by the map Aj−ℓ ⊗Aℓ−k ⊗ Ei−1,m+k −→ Aj−k ⊗ Ei−1,m+k ⊆ Pi−1,m+j which is given by the multiplication map Aj−ℓ ⊗ Aℓ−k → Aj−k. Since j − k < N , our hypothesis on I implies that Aj−k ∼= T(V )j−k, and similarly Aj−ℓ ∼= T(V )j−ℓ and Aℓ−k ∼= T(V )ℓ−k. Therefore, the above multiplication map is identical with the injection T(V )j−ℓ ⊗ T(V )ℓ−k →֒ T(V )j−k in T(V ). This proves (a). For (b), we proceed by induction on j. The case j = 1 being obvious, let 1 ≤ j ≤ N − 2 and assume that (ii) holds for 1, . . . , j. We wish to show that the subspaces Aj+1−ℓEi,m+ℓ (ℓ = 1, . . . , j + 1) of Pi−1,m+j+1 form a direct sum. First, by (A.10) we have Ei,m+j+1 ∩ A+Ker di−1 = 0 while ℓ=1Aj+1−ℓEi,m+ℓ ⊆ A+Ker di−1. Therefore, it suffices to show that the sum ℓ=1Aj+1−ℓEi,m+ℓ is direct. To this end, note that Aj+1−ℓ = d≥1 VdAj+1−d−ℓ holds for all ℓ ≤ j. Hence, Aj+1−ℓEi,m+ℓ = Aj+1−d−ℓEi,m+ℓ By induction, Aj+1−d−ℓEi,m+ℓ is a direct sum inside Pi−1,m+j+1−d. Thus, it suffices to show that the sum d≥1 VdPi−1,m+j+1−d ⊆ Pi−1,m+j+1 is direct. But (A.13) gives Pi−1,m+j+1 = Aj+1−k ⊗ Ei−1,m+k = T(V )j+1−k ⊗ Ei−1,m+k where the last equality holds since all j + 1− k < N . Therefore, VdPi−1,m+j+1−d = j+1−d⊕ T(V )j+1−d−k ⊗ Ei−1,m+k as desired. This proves (b), thereby completing the proof of the proposition. � Acknowledgement. The authors wish to thank Roland Berger for his helpful comments throughout the completion of this paper. 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Koszul duality and master theorem Appendix References

First and second sound modes at finite temperature in trapped Fermi gases from BCS to BEC Yan He, Qijin Chen, Chih-Chun Chien, and K. Levin James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637 (Dated: August 11, 2021) We determine the temperature T dependence of first and second sound mode frequencies for trapped Fermi gases undergoing BCS to Bose Einstein condensation (BEC) crossover. Our results are based on the two fluid equations in conjunction with a microscopic calculation of thermodynamical variables. As in experiment and at unitarity, we show that the lowest radial breathing mode is T independent. At finite T , higher order breathing modes strongly mix with second sound. Their complex T dependence should provide an alternative way of measuring the transition temperature, Tc. PACS numbers: 03.75.Hh, 03.75.Ss, 74.20.-z arXiv:0704.1889 The recent discovery of the superfluid phases of trapped Fermi gases has led to considerable interest in their collective mode spectrum [1, 2, 3, 4, 5, 6, 7]. Among the modes of ex- perimental interest are breathing modes as well as propagating first sound. While originally theoretical attention [8, 9, 10] was focused on ground state properties experimental mea- surements are naturally not restricted to temperature T = 0. Indeed there is an interesting body of information which is emerging in these Fermi gases about the finite temperature behavior [4, 5, 6] of the breathing modes and, more recently about the propagating sound velocity [7]. The purpose of this paper is to compute sound mode fre- quencies in spherically trapped Fermi gases undergoing BCS to Bose-Einstein condensation (BEC) crossover, at general T . We present a solution of the linearized two fluid equations and compare with recent experiments. We focus on the radial breathing modes and present predictions for second sound, as well. The structure of the two fluid equations for Bose [11] and Fermi gases [12] has been rather extensively discussed. In the crossover regime, the normal fluid is novel [13, 14], containing both fermions and noncondensed pairs, which have not been systematically included in previous collective mode literature. Of great importance to the field as a whole is the future possibility of second sound observations, in large part because this may ultimately help assign more precise experimental val- ues to the transition temperature Tc. While existing experi- mental techniques such as vortex observation [15] and den- sity profile features [16] help establish superfluidity they pro- vide lower bounds on Tc or determine its value for the special case of a population imbalanced system. Thermodynamical experiments measure Tc more directly [17, 18] but have been confined to unitarity. Thus other techniques, such as second sound observation will be of great value. One of the princi- pal results of the present paper is an analysis of how second sound is coupled to the breathing modes. We demonstrate that higher order breathing modes will reveal Tc through this cou- pling, and therefore are an alternative to direct second sound measurements. However, the lowest breathing mode appears to be remarkably T independent at unitarity. This has been ob- served experimentally [4, 5] and argued to follow from isen- tropic considerations [6]. Here we show, that even when we treat the full coupling between first and second sound, we ob- tain similar T independent behavior at unitarity. At the core of the two fluid theory is the assumption that hydrodynamics is valid and that there are frequent collisions which produce a state of local thermodynamical equilibrium. Although there are some exceptions [19], reaching the two fluid regime has not been easy for atomic Bose gases. Two fluid dynamics are more readily achieved for Fermi gases, principally because in the crossover regime the large scatter- ing lengths produce sufficient collisions. Nevertheless there has been considerable theoretical interest in setting up [12] and solving [20] the two fluid equations for Bose conden- sates. Indeed, hydrodynamical approaches have successfully addressed both the T = 0 and normal state regimes of the Bose gases [21] and demonstrated that the breathing mode frequencies are the same. Here, by contrast we address the Fermi gas case in a trap. Because they interact more strongly near unitarity, hydrodynamical descriptions have been argued quite convincingly [6, 9, 22] to be valid. Previous theoretical work has been confined to T = 0 treat- ments of a harmonic trap, or to finite T theories [22] of a uniform gas. Our work is most similar in spirit to an earlier Bose gas study [20] although we introduce different numeri- cal techniques as well as address fermions rather than bosons. We note that the input thermodynamics of systems undergo- ing the BCS-BEC crossover which is used in the present pa- per, has been rather well calibrated against experimental mea- surements in Ref. [17] and is based on a finite temperature extension of the simplest (BCS-Leggett) ground state. In the absence of a trap our results are for the most part similar to those in Ref. [22]. We begin with the two fluid equations which describe the dynamical coupling of the superfluid velocity vs and the nor- mal velocity vn. Just as in the spirit of the original Landau two fluid equations we ignore viscosity terms. In the presence of a trap potential Vext = mωhor 2, the two fluid equations are given by m∂vs +∇(µ+Vext+m ) = 0, ∂j +∇·Π = −n∇Vext, + ∇ · j = 0 and ∂(ns) + ∇ · (nsvn) = 0, with Πij = pδij + nsv s + nnv n, n = ns + nn and http://arxiv.org/abs/0704.1889v2 j = nsvs + nnvn. Here µ is the chemical potential, p the pressure, and s the entropy per particle. Moreover, we have nnvn + nsvs = nv. Here ns (vs) and nn (vn) denote the superfluid and normal densities (velocities), respectively. We use the subscript “0” to denote equilibrium quantities such that vs0 = vn0 = 0, ∇(µ + Vext) = 0, ∇p0 = −n0∇Vext, and n0, s0, µ0, p0 are independent of time t. Combining this with the thermodynamic relation dµ = −s dT + dp/n, we have ∇T0 = 0, implying that temperature T0 is constant in the trap. It then follows that in equilibrium µ = µ0 − Vext, consistent with the Thomas-Fermi approximation. For small deviations from equilibrium, we may linearize the two fluid equations. Eliminating the velocities vs and vn, one finds [12, 20] = ∇ · −∇ · (δT n0∇s0) (1) ns0n0s − (∇s0) 2δT +∇s0 · ∇ We will focus on δµ(r) and δT (r) as the principal variables. This choice, which is different from that in Ref. [20], is made because both variables are non vanishing at the trap edge so that in a basis set expansion they will satisfy the same bound- ary conditions. Moreover, the two fluid equations are simplest in this form. Expressing δs, δp and δn in terms of δµ, δT , the two fluid equations can be rewritten as , (3) , (4) A = ∇ · (n∇δµ) +∇ · (ns∇δT ) , s2∇δT +∇s · ∇δµ+ s∇s · ∇δT . It is understood that all coefficients of δµ and δT are calcu- lated in equilibrium so that we drop the subscript “0”. The thermodynamical quantities in equilibrium can be calculated following Ref. [23], based on the standard local density ap- proximation (LDA), µ(r) = µ0 − Vext(r). Their derivatives with respect to T and µ can be calculated analytically, and their gradients can be obtained via ∇f = − ∇Vext, where f denotes any of the variables (n, ns, nn, s). To solve the two coupled differential equations (3) and (4), we assume a simple harmonic time dependence δµ, δT ∝ e−iωt. We cast the differential two fluid equations into an eigenfunction problem with ω2 playing the role of eigenvalue and the eigenfunctions given by the amplitudes of δµ and δT . Since neither T nor µ depends on the density, they will not vanish at the trap edge. Our boundary conditions require that all thermodynamic variables be smooth (but not necessarily zero) at the trap edge. At finite T , the density in the trap de- creases exponentially when the local chemical potential be- comes negative at large radius. We choose, thus, to expand δµ and δT in terms of Jacobi polynomials. For our numerics we choose the matrix dimension to be 300; we have similarly investigated matrices of dimension 200 up to 900, and found little change in our principal findings. We now turn to an important aspect of our numerics. Be- cause we generate some 300 frequencies in our numerical ap- proach it is essential to establish a mechanism for systemati- cally identifying first and second sound modes. To help find such a “fingerprint”, we introduce a “decoupling approxima- tion” based on reducing Eqs. (3) and (4) to δ̈µ = g1(δµ, δT ), δ̈T = g2(δµ, δT ), where g1,2 are known functions. We elim- inate cross terms by setting δT = 0 in g1 and δµ = 0 in g2. With these two decoupled equations it is then relatively straightforward to associate a profile plot of the numerically calculated δp(r) and nδs(r) vs r within a trap with first or sec- ond sound-like modes. Here we convert δµ and δT to δp and δs via δp = nδµ + nsδT and δs = We stress that this procedure differs somewhat from that in Ref. [20] where it was argued that one could associate second sound with a mode in which the oscillations in the thermody- namical variable δT (r) were constrained to be in the conden- sate region of the trap. As explained below, we do not find this to be the case. An important check on our procedure is that we find that there is no sign of second sound above Tc. Up to this point everything is general applying to both Fermi and Bose superfluids. All that is needed is a micro- scopic theory for thermodynamical variables. Here we use a calculational framework we have developed for treating BCS- BEC crossover in trapped Fermi gases [23, 24] which em- phasizes the importance of pseudogap effects or finite mo- mentum pairs. The local thermodynamical potential (density) Ω = Ωf + Ωb is associated with a contribution from gapped fermionic excitationsΩf as well as from non-condensed pairs, called Ωb. We have Ωf = − [(ξk − Ek)− 2T ln(1 + e −Ek/T )], Ωb = −Z∆ 2µpair + T ln(1 − e−Ωq/T ) . (5) Here µpair is the chemical potential of the pairs which is zero below Tc and the pair dispersion Ωq, along with the (in- verse) residue Z , can be derived from a microscopic T -matrix theory, described elsewhere [13, 14]. Using Ω one then ar- rives at thermodynamical properties such as the entropy den- sity ns = −∂Ω as well as self consistent equations for the total excitation gap ∆, the contribution to ∆ from noncon- densed pairs (called the pseudogap) and the number equa- tions. These self-consistent (local) equations are simply given 0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 1 r/Rmax T = 0.1 n = 1 T = 0.2 n = 1 (a) (b) (c) (d) Figure 1: (Color online) Behavior of the first (upper row) and second sound (lower row) modes for a spherical trap at unitarity within the “decoupling approximation” (see text). The left column shows the T -dependence of the frequencies, while the right shows correspond- ing typical spatial oscillation profiles for δp [(black) solid lines] and nδs [(red) dashed lines] at the lowest frequencies, which provide fin- gerprints of first and second sound. For the second sound, nδs dom- inates and changes sign within the condensate. Here Tc ≈ 0.27TF , and the arrows indicate the condensate edge. by ∂Ω = 0, ∂Ω ∂µpair = 0 and n = −∂Ω , subject to the total number constraint N = d3r n(r). When T < Tc, we can use the gap equation and chain rule to eliminate the variable ∆, which is a function of µ and T , via ∂∆ and ∂∆ . Similarly, when T > Tc we use the gap and pseudogap equations to eliminate the variables ∆ and µpair to arrive at thermodynamical quantities. Figure 1 shows the lowest two collective modes at unitar- ity in a spherical trap, obtained by solving the chemical po- tential or temperature fluctuation equation in the decoupling approximation scheme described earlier. It is evident that our approximated breathing mode frequency [Fig. 1(a)] is inde- pendent of temperature. We understand this result by noting that the decoupled equation for the breathing mode is given by −ω2δµ = Cµµ1∇ 2δµ+Cµµ2 · ∇δµ where Cµµ1 = n and Cµµ2 = ∇µ. At unitarity, n µ. The only T -dependence contained in µ0 ≡ µ(r = 0) can be elimi- nated via a simple re-scaling of r → r 2µ0/mω ho, yielding a T -independent breathing mode frequency. These arguments can be shown to be equivalent to the isentropic assumption of Ref. [6], where vs = vn is assumed in the two fluid model, leading to two simpler coupled equations associated with the Euler and the continuity equations. By contrast, the second sound mode frequency we obtain increases rapidly with temperature. Some typical oscillation profiles of δp(r) and nδs(r) are shown in the right two pan- els of Figure 1. Although the “entropy density oscillations” nδs(r) fall off at large r, the entropy per particle δs(r) os- cillations (not shown) increase very rapidly upon entering the normal region. Consequently temperature fluctuations δT be- 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 T/TF = 0.01 T/TF = 0.2 1/kFa = 0 T/TF = 0.15 1/kFa = 0 T/TF = 0.08 1/kFa = 1 r/Rmax 1/kFa = 0 (a) (b) (c) (d) Figure 2: (Color online) Typical spatial oscillation profiles for δp [(black) solid lines] and nδs [(red) dashed lines] obtained from the fully coupled equations for a spherical trap at (a-c) unitarity and (d) 1/kF a = 1 at different T , for the first (top row) and second (bottom row) sound modes. Also shown in (a) is the T = 0 analytical result [(green) dot-dashed curve]. The arrows indicate condensate edge. come large at the trap edge. Our identification of first sound for the decoupled case leads us to associate this mode in a coupled situation with a profile for which at the trap center δp has a large amplitude, while nδs is almost zero (with a small peak near the trap edge) as in Fig. 1(b). By contrast, in the trap center the second sound mode has large entropy fluctuations, while the pressure fluctu- ations are almost zero [Fig. 1(d)]. These features will serve as fingerprints for distinguishing (lower order) first and second sound modes at finite temperatures. Figure 2 shows some typical eigenfunction profiles of the lowest modes obtained in the spherical trap upon solution of the fully coupled two fluid equations. The first row corre- sponds to the breathing mode in the unitary case. The good agreement between the very low T result for δp and the T = 0 analytical solution (green dashed line) in Fig. 2(a) helps vali- date our numerical scheme. Figure 2(b) corresponds to a high temperature breathing mode. In this regime the pseudogap re- gion outside the superfluid core is relatively large and the peak in nδs(r) is accordingly very broad. The lower two panels correspond to second sound modes for the unitary and BEC cases. By contrast with the breathing modes, here nδs has a larger amplitude than δp with an opposite sign. Clearly this is very similar to what we observed in the decoupled mode analysis of Fig. 1. Figures 3(a-c) address the fully coupled equations and show the behavior of the lowest breathing (upper branch) and sec- ond sound mode (lower branch) frequencies as a function of temperature in a spherical trap for 1/kFa = 1, 0, and −0.5, respectively. For all three values of 1/kFa we find very little sign of Tc in the lowest breathing mode frequency. In con- trast, the second sound mode frequencies increase with T and disappear above Tc. Figure 3(d) presents a more complete series of modes for the unitary case. Here the lines serve as guides to the eye for the lowest (blue) and higher order (green) 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 1/kFa = 1 1/kFa = 0 1/kFa = -0.5 1/kFa = 0 (a) (b) (c) (d) Figure 3: (Color online) Temperature dependence of breathing mode and second sound frequencies in a spherical trap. Panels (a-c) are for the near-BEC, unitary, and near-BCS cases, respectively. The upper and lower branches in (a-c) represent the lowest breathing mode and second sound frequencies, respectively. In (d), more complete results (open circles) are shown at unitarity, where the lines serve as guides to the eye for the breathing mode (nearly horizontal blue and green lines) and second sound (increasing red curves) frequencies. breathing mode, and second sound (red) frequencies. At unitarity [Fig. 3(d)] one can identify a sequence of higher order breathing modes which precisely overlap with analytical calculations for T = 0. Importantly, only the low- est of these is found to be a constant in temperature; the others are found to mix with second sound modes, as indicated in the solid and dashed (red) lines in Fig. 3(d). The behavior of the lowest mode helps to justify the isentropic assumption made in Ref. [6]. We understand this by referring back to the de- coupled profiles in Figs. 1(b) and 1(c), which are seen to be quite distinct. By contrast the profiles of the decoupled first and second sound modes at higher order (not shown) appear more similar to each other than their lowest order counter- parts. Indeed, the behavior of the profiles at higher order is associated with an increasing number of nodes in the curves of δp and nδs (not shown), which leads to a greater similarity between first and second sound profiles and helps explain why the higher order modes are more readily mixed [25]. In summary, we have presented predictions for future ex- periments on higher breathing modes and second sound in a trap. We find that only the lowest breathing mode frequency has very weak T dependence. For the unitary case this tem- perature insensitivity was clearly observed by Thomas and co- workers [6]. As a result of this experiment it should not be surprising that we find relatively weak dependencies on ei- ther side of the Feshbach resonance for this breathing mode frequency. In the literature there are experimental claims (at 1/kFa = 1.0) which are consistent with a decrease [4] in the radial breathing mode frequency, as indeed we find here, al- though ours is probably too weak to reconcile the different findings in Refs. [4] and [2, 3]. Finally, our more complete studies at unitarity show that if higher order radial breathing modes can be accessed, because of their strong hybridization with second sound, it may be possible to use these breathing modes, rather than direct second sound to determine the tran- sition temperatures Tc. This should be of value since there are currently few experiments which can assign a value to Tc over the wider crossover regime. This work was supported by Grant Nos. NSF PHY- 0555325 and NSF-MRSEC DMR-0213745; we thank John Thomas for suggesting this problem and Cheng Chin for use- ful conversations. [1] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. Hecker-Denschlag, and R. Grimm, Science 305, 1128 (2004). [2] J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov, and J. E. Thomas, Phys. Rev. Lett. 92, 150402 (2004). [3] J. Kinast, A. Turlapov, and J. E. Thomas, Phys. Rev. A 70, 051401(R) (2004). [4] A. Altmeyer, S. Riedl, C. Kohstall, M. J. Wright, R. Geursen, M. Bartenstein, C. Chin, J. H. Denschlag, and R. Grimm, Phys. Rev. Lett. 98, 040401 (2007). [5] J. Kinast, A. Turlapov, and J. E. Thomas, Phys. Rev. Lett. 94, 170404 (2005). [6] J. E. Thomas, J. Kinast, and A. Turlapov, Phys. Rev. Lett. 95, 120402 (2005). [7] J. Joseph, B. Clancy, L. Luo, J. Kinast, A. Turlapov, and J. E. Thomas (2006), cond-mat/0612567. [8] H. Hu, A. Minguzzi, X.-J. Liu, and M. P. Tosi, Phys. Rev. Lett. 93, 190403 (2004). [9] H. Heiselberg, Phys. Rev. Lett. 93, 040402 (2004). [10] G. E. Astrakharchik, R. Combescot, X. Leyronas, and S. Stringari, Phys. Rev. Lett. 95, 030404 (2005). [11] A. Griffin and E. Zaremba, Phys. Rev. A 56, 4839 (1997). [12] E. Taylor and A. Griffin, Phys. Rev. A 72, 053630 (2005). [13] Q. J. Chen, I. Kosztin, B. Jankó, and K. Levin, Phys. Rev. Lett. 81, 4708 (1998). [14] Q. J. Chen, J. Stajic, S. N. Tan, and K. Levin, Phys. Rep. 412, 1 (2005). [15] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, and W. Ket- terle, Nature 435, 170404 (2005). [16] M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ket- terle, Science 311, 492 (2006). [17] J. Kinast, A. Turlapov, J. E. Thomas, Q. J. Chen, J. Stajic, and K. Levin, Science 307, 1296 (2005), published online 27 Jan- uary 2005; doi:10.1126/science.1109220. [18] L. Luo, B. Clancy, J. Joseph, J. Kinast, and J. E. Thomas, Phys. Rev. Lett. 98, 080402 (2007). [19] D. M. Stamper-Kurn, H.-J. Miesner, S. Inouye, M. R. Andrews, and W. Ketterle, Phys. Rev. Lett. 81, 500 (1998). [20] V. B. Shenoy and T.-L. Ho, Phys. Rev. Lett. 80, 3895 (1998). [21] A. Griffin, W.-C. Wu, and S. Stringari, Phys. Rev. Lett. 78, 1838 (1997). [22] H. Heiselberg, Phys. Rev. A 73, 013607 (2006). [23] Q. J. Chen, J. Stajic, and K. Levin, Phys. Rev. Lett. 95, 260405 (2005). [24] J. Stajic, Q. J. Chen, and K. Levin, Phys. Rev. Lett. 94, 060401 (2005). [25] It is interesting to note that there exists a lower frequency first- sound like mode at finite T , which we also find to be present in the “free” (but hydrodynamic) Fermi gas limit at T 6= 0.

We determine the temperature $T$ dependence of first and second sound mode frequencies for trapped Fermi gases undergoing BCS to Bose Einstein condensation (BEC) crossover. Our results are based on the two fluid equations in conjunction with a microscopic calculation of thermodynamical variables. As in experiment and at unitarity, we show that the lowest radial breathing mode is $T$ independent. At finite $T$, higher order breathing modes strongly mix with second sound. Their complex $T$ dependence should provide an alternative way of measuring the transition temperature, $T_c$.

First and second sound modes at finite temperature in trapped Fermi gases from BCS to BEC Yan He, Qijin Chen, Chih-Chun Chien, and K. Levin James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637 (Dated: August 11, 2021) We determine the temperature T dependence of first and second sound mode frequencies for trapped Fermi gases undergoing BCS to Bose Einstein condensation (BEC) crossover. Our results are based on the two fluid equations in conjunction with a microscopic calculation of thermodynamical variables. As in experiment and at unitarity, we show that the lowest radial breathing mode is T independent. At finite T , higher order breathing modes strongly mix with second sound. Their complex T dependence should provide an alternative way of measuring the transition temperature, Tc. PACS numbers: 03.75.Hh, 03.75.Ss, 74.20.-z arXiv:0704.1889 The recent discovery of the superfluid phases of trapped Fermi gases has led to considerable interest in their collective mode spectrum [1, 2, 3, 4, 5, 6, 7]. Among the modes of ex- perimental interest are breathing modes as well as propagating first sound. While originally theoretical attention [8, 9, 10] was focused on ground state properties experimental mea- surements are naturally not restricted to temperature T = 0. Indeed there is an interesting body of information which is emerging in these Fermi gases about the finite temperature behavior [4, 5, 6] of the breathing modes and, more recently about the propagating sound velocity [7]. The purpose of this paper is to compute sound mode fre- quencies in spherically trapped Fermi gases undergoing BCS to Bose-Einstein condensation (BEC) crossover, at general T . We present a solution of the linearized two fluid equations and compare with recent experiments. We focus on the radial breathing modes and present predictions for second sound, as well. The structure of the two fluid equations for Bose [11] and Fermi gases [12] has been rather extensively discussed. In the crossover regime, the normal fluid is novel [13, 14], containing both fermions and noncondensed pairs, which have not been systematically included in previous collective mode literature. Of great importance to the field as a whole is the future possibility of second sound observations, in large part because this may ultimately help assign more precise experimental val- ues to the transition temperature Tc. While existing experi- mental techniques such as vortex observation [15] and den- sity profile features [16] help establish superfluidity they pro- vide lower bounds on Tc or determine its value for the special case of a population imbalanced system. Thermodynamical experiments measure Tc more directly [17, 18] but have been confined to unitarity. Thus other techniques, such as second sound observation will be of great value. One of the princi- pal results of the present paper is an analysis of how second sound is coupled to the breathing modes. We demonstrate that higher order breathing modes will reveal Tc through this cou- pling, and therefore are an alternative to direct second sound measurements. However, the lowest breathing mode appears to be remarkably T independent at unitarity. This has been ob- served experimentally [4, 5] and argued to follow from isen- tropic considerations [6]. Here we show, that even when we treat the full coupling between first and second sound, we ob- tain similar T independent behavior at unitarity. At the core of the two fluid theory is the assumption that hydrodynamics is valid and that there are frequent collisions which produce a state of local thermodynamical equilibrium. Although there are some exceptions [19], reaching the two fluid regime has not been easy for atomic Bose gases. Two fluid dynamics are more readily achieved for Fermi gases, principally because in the crossover regime the large scatter- ing lengths produce sufficient collisions. Nevertheless there has been considerable theoretical interest in setting up [12] and solving [20] the two fluid equations for Bose conden- sates. Indeed, hydrodynamical approaches have successfully addressed both the T = 0 and normal state regimes of the Bose gases [21] and demonstrated that the breathing mode frequencies are the same. Here, by contrast we address the Fermi gas case in a trap. Because they interact more strongly near unitarity, hydrodynamical descriptions have been argued quite convincingly [6, 9, 22] to be valid. Previous theoretical work has been confined to T = 0 treat- ments of a harmonic trap, or to finite T theories [22] of a uniform gas. Our work is most similar in spirit to an earlier Bose gas study [20] although we introduce different numeri- cal techniques as well as address fermions rather than bosons. We note that the input thermodynamics of systems undergo- ing the BCS-BEC crossover which is used in the present pa- per, has been rather well calibrated against experimental mea- surements in Ref. [17] and is based on a finite temperature extension of the simplest (BCS-Leggett) ground state. In the absence of a trap our results are for the most part similar to those in Ref. [22]. We begin with the two fluid equations which describe the dynamical coupling of the superfluid velocity vs and the nor- mal velocity vn. Just as in the spirit of the original Landau two fluid equations we ignore viscosity terms. In the presence of a trap potential Vext = mωhor 2, the two fluid equations are given by m∂vs +∇(µ+Vext+m ) = 0, ∂j +∇·Π = −n∇Vext, + ∇ · j = 0 and ∂(ns) + ∇ · (nsvn) = 0, with Πij = pδij + nsv s + nnv n, n = ns + nn and http://arxiv.org/abs/0704.1889v2 j = nsvs + nnvn. Here µ is the chemical potential, p the pressure, and s the entropy per particle. Moreover, we have nnvn + nsvs = nv. Here ns (vs) and nn (vn) denote the superfluid and normal densities (velocities), respectively. We use the subscript “0” to denote equilibrium quantities such that vs0 = vn0 = 0, ∇(µ + Vext) = 0, ∇p0 = −n0∇Vext, and n0, s0, µ0, p0 are independent of time t. Combining this with the thermodynamic relation dµ = −s dT + dp/n, we have ∇T0 = 0, implying that temperature T0 is constant in the trap. It then follows that in equilibrium µ = µ0 − Vext, consistent with the Thomas-Fermi approximation. For small deviations from equilibrium, we may linearize the two fluid equations. Eliminating the velocities vs and vn, one finds [12, 20] = ∇ · −∇ · (δT n0∇s0) (1) ns0n0s − (∇s0) 2δT +∇s0 · ∇ We will focus on δµ(r) and δT (r) as the principal variables. This choice, which is different from that in Ref. [20], is made because both variables are non vanishing at the trap edge so that in a basis set expansion they will satisfy the same bound- ary conditions. Moreover, the two fluid equations are simplest in this form. Expressing δs, δp and δn in terms of δµ, δT , the two fluid equations can be rewritten as , (3) , (4) A = ∇ · (n∇δµ) +∇ · (ns∇δT ) , s2∇δT +∇s · ∇δµ+ s∇s · ∇δT . It is understood that all coefficients of δµ and δT are calcu- lated in equilibrium so that we drop the subscript “0”. The thermodynamical quantities in equilibrium can be calculated following Ref. [23], based on the standard local density ap- proximation (LDA), µ(r) = µ0 − Vext(r). Their derivatives with respect to T and µ can be calculated analytically, and their gradients can be obtained via ∇f = − ∇Vext, where f denotes any of the variables (n, ns, nn, s). To solve the two coupled differential equations (3) and (4), we assume a simple harmonic time dependence δµ, δT ∝ e−iωt. We cast the differential two fluid equations into an eigenfunction problem with ω2 playing the role of eigenvalue and the eigenfunctions given by the amplitudes of δµ and δT . Since neither T nor µ depends on the density, they will not vanish at the trap edge. Our boundary conditions require that all thermodynamic variables be smooth (but not necessarily zero) at the trap edge. At finite T , the density in the trap de- creases exponentially when the local chemical potential be- comes negative at large radius. We choose, thus, to expand δµ and δT in terms of Jacobi polynomials. For our numerics we choose the matrix dimension to be 300; we have similarly investigated matrices of dimension 200 up to 900, and found little change in our principal findings. We now turn to an important aspect of our numerics. Be- cause we generate some 300 frequencies in our numerical ap- proach it is essential to establish a mechanism for systemati- cally identifying first and second sound modes. To help find such a “fingerprint”, we introduce a “decoupling approxima- tion” based on reducing Eqs. (3) and (4) to δ̈µ = g1(δµ, δT ), δ̈T = g2(δµ, δT ), where g1,2 are known functions. We elim- inate cross terms by setting δT = 0 in g1 and δµ = 0 in g2. With these two decoupled equations it is then relatively straightforward to associate a profile plot of the numerically calculated δp(r) and nδs(r) vs r within a trap with first or sec- ond sound-like modes. Here we convert δµ and δT to δp and δs via δp = nδµ + nsδT and δs = We stress that this procedure differs somewhat from that in Ref. [20] where it was argued that one could associate second sound with a mode in which the oscillations in the thermody- namical variable δT (r) were constrained to be in the conden- sate region of the trap. As explained below, we do not find this to be the case. An important check on our procedure is that we find that there is no sign of second sound above Tc. Up to this point everything is general applying to both Fermi and Bose superfluids. All that is needed is a micro- scopic theory for thermodynamical variables. Here we use a calculational framework we have developed for treating BCS- BEC crossover in trapped Fermi gases [23, 24] which em- phasizes the importance of pseudogap effects or finite mo- mentum pairs. The local thermodynamical potential (density) Ω = Ωf + Ωb is associated with a contribution from gapped fermionic excitationsΩf as well as from non-condensed pairs, called Ωb. We have Ωf = − [(ξk − Ek)− 2T ln(1 + e −Ek/T )], Ωb = −Z∆ 2µpair + T ln(1 − e−Ωq/T ) . (5) Here µpair is the chemical potential of the pairs which is zero below Tc and the pair dispersion Ωq, along with the (in- verse) residue Z , can be derived from a microscopic T -matrix theory, described elsewhere [13, 14]. Using Ω one then ar- rives at thermodynamical properties such as the entropy den- sity ns = −∂Ω as well as self consistent equations for the total excitation gap ∆, the contribution to ∆ from noncon- densed pairs (called the pseudogap) and the number equa- tions. These self-consistent (local) equations are simply given 0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 1 r/Rmax T = 0.1 n = 1 T = 0.2 n = 1 (a) (b) (c) (d) Figure 1: (Color online) Behavior of the first (upper row) and second sound (lower row) modes for a spherical trap at unitarity within the “decoupling approximation” (see text). The left column shows the T -dependence of the frequencies, while the right shows correspond- ing typical spatial oscillation profiles for δp [(black) solid lines] and nδs [(red) dashed lines] at the lowest frequencies, which provide fin- gerprints of first and second sound. For the second sound, nδs dom- inates and changes sign within the condensate. Here Tc ≈ 0.27TF , and the arrows indicate the condensate edge. by ∂Ω = 0, ∂Ω ∂µpair = 0 and n = −∂Ω , subject to the total number constraint N = d3r n(r). When T < Tc, we can use the gap equation and chain rule to eliminate the variable ∆, which is a function of µ and T , via ∂∆ and ∂∆ . Similarly, when T > Tc we use the gap and pseudogap equations to eliminate the variables ∆ and µpair to arrive at thermodynamical quantities. Figure 1 shows the lowest two collective modes at unitar- ity in a spherical trap, obtained by solving the chemical po- tential or temperature fluctuation equation in the decoupling approximation scheme described earlier. It is evident that our approximated breathing mode frequency [Fig. 1(a)] is inde- pendent of temperature. We understand this result by noting that the decoupled equation for the breathing mode is given by −ω2δµ = Cµµ1∇ 2δµ+Cµµ2 · ∇δµ where Cµµ1 = n and Cµµ2 = ∇µ. At unitarity, n µ. The only T -dependence contained in µ0 ≡ µ(r = 0) can be elimi- nated via a simple re-scaling of r → r 2µ0/mω ho, yielding a T -independent breathing mode frequency. These arguments can be shown to be equivalent to the isentropic assumption of Ref. [6], where vs = vn is assumed in the two fluid model, leading to two simpler coupled equations associated with the Euler and the continuity equations. By contrast, the second sound mode frequency we obtain increases rapidly with temperature. Some typical oscillation profiles of δp(r) and nδs(r) are shown in the right two pan- els of Figure 1. Although the “entropy density oscillations” nδs(r) fall off at large r, the entropy per particle δs(r) os- cillations (not shown) increase very rapidly upon entering the normal region. Consequently temperature fluctuations δT be- 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 T/TF = 0.01 T/TF = 0.2 1/kFa = 0 T/TF = 0.15 1/kFa = 0 T/TF = 0.08 1/kFa = 1 r/Rmax 1/kFa = 0 (a) (b) (c) (d) Figure 2: (Color online) Typical spatial oscillation profiles for δp [(black) solid lines] and nδs [(red) dashed lines] obtained from the fully coupled equations for a spherical trap at (a-c) unitarity and (d) 1/kF a = 1 at different T , for the first (top row) and second (bottom row) sound modes. Also shown in (a) is the T = 0 analytical result [(green) dot-dashed curve]. The arrows indicate condensate edge. come large at the trap edge. Our identification of first sound for the decoupled case leads us to associate this mode in a coupled situation with a profile for which at the trap center δp has a large amplitude, while nδs is almost zero (with a small peak near the trap edge) as in Fig. 1(b). By contrast, in the trap center the second sound mode has large entropy fluctuations, while the pressure fluctu- ations are almost zero [Fig. 1(d)]. These features will serve as fingerprints for distinguishing (lower order) first and second sound modes at finite temperatures. Figure 2 shows some typical eigenfunction profiles of the lowest modes obtained in the spherical trap upon solution of the fully coupled two fluid equations. The first row corre- sponds to the breathing mode in the unitary case. The good agreement between the very low T result for δp and the T = 0 analytical solution (green dashed line) in Fig. 2(a) helps vali- date our numerical scheme. Figure 2(b) corresponds to a high temperature breathing mode. In this regime the pseudogap re- gion outside the superfluid core is relatively large and the peak in nδs(r) is accordingly very broad. The lower two panels correspond to second sound modes for the unitary and BEC cases. By contrast with the breathing modes, here nδs has a larger amplitude than δp with an opposite sign. Clearly this is very similar to what we observed in the decoupled mode analysis of Fig. 1. Figures 3(a-c) address the fully coupled equations and show the behavior of the lowest breathing (upper branch) and sec- ond sound mode (lower branch) frequencies as a function of temperature in a spherical trap for 1/kFa = 1, 0, and −0.5, respectively. For all three values of 1/kFa we find very little sign of Tc in the lowest breathing mode frequency. In con- trast, the second sound mode frequencies increase with T and disappear above Tc. Figure 3(d) presents a more complete series of modes for the unitary case. Here the lines serve as guides to the eye for the lowest (blue) and higher order (green) 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 1/kFa = 1 1/kFa = 0 1/kFa = -0.5 1/kFa = 0 (a) (b) (c) (d) Figure 3: (Color online) Temperature dependence of breathing mode and second sound frequencies in a spherical trap. Panels (a-c) are for the near-BEC, unitary, and near-BCS cases, respectively. The upper and lower branches in (a-c) represent the lowest breathing mode and second sound frequencies, respectively. In (d), more complete results (open circles) are shown at unitarity, where the lines serve as guides to the eye for the breathing mode (nearly horizontal blue and green lines) and second sound (increasing red curves) frequencies. breathing mode, and second sound (red) frequencies. At unitarity [Fig. 3(d)] one can identify a sequence of higher order breathing modes which precisely overlap with analytical calculations for T = 0. Importantly, only the low- est of these is found to be a constant in temperature; the others are found to mix with second sound modes, as indicated in the solid and dashed (red) lines in Fig. 3(d). The behavior of the lowest mode helps to justify the isentropic assumption made in Ref. [6]. We understand this by referring back to the de- coupled profiles in Figs. 1(b) and 1(c), which are seen to be quite distinct. By contrast the profiles of the decoupled first and second sound modes at higher order (not shown) appear more similar to each other than their lowest order counter- parts. Indeed, the behavior of the profiles at higher order is associated with an increasing number of nodes in the curves of δp and nδs (not shown), which leads to a greater similarity between first and second sound profiles and helps explain why the higher order modes are more readily mixed [25]. In summary, we have presented predictions for future ex- periments on higher breathing modes and second sound in a trap. We find that only the lowest breathing mode frequency has very weak T dependence. For the unitary case this tem- perature insensitivity was clearly observed by Thomas and co- workers [6]. As a result of this experiment it should not be surprising that we find relatively weak dependencies on ei- ther side of the Feshbach resonance for this breathing mode frequency. In the literature there are experimental claims (at 1/kFa = 1.0) which are consistent with a decrease [4] in the radial breathing mode frequency, as indeed we find here, al- though ours is probably too weak to reconcile the different findings in Refs. [4] and [2, 3]. Finally, our more complete studies at unitarity show that if higher order radial breathing modes can be accessed, because of their strong hybridization with second sound, it may be possible to use these breathing modes, rather than direct second sound to determine the tran- sition temperatures Tc. This should be of value since there are currently few experiments which can assign a value to Tc over the wider crossover regime. This work was supported by Grant Nos. NSF PHY- 0555325 and NSF-MRSEC DMR-0213745; we thank John Thomas for suggesting this problem and Cheng Chin for use- ful conversations. [1] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. Hecker-Denschlag, and R. Grimm, Science 305, 1128 (2004). [2] J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov, and J. E. Thomas, Phys. Rev. Lett. 92, 150402 (2004). [3] J. Kinast, A. Turlapov, and J. E. Thomas, Phys. Rev. A 70, 051401(R) (2004). [4] A. Altmeyer, S. Riedl, C. Kohstall, M. J. Wright, R. Geursen, M. 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Computation and visualization of Casimir forces in arbitrary geometries: non-monotonic lateral-wall forces and failure of proximity-force approximations Alejandro Rodriguez,1 Mihai Ibanescu,1 Davide Iannuzzi,2 Federico Capasso,3 J. D. Joannopoulos,1 and Steven G. Johnson1 1Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 2Faculty of Sciences, Department of Physics and Astronomy, Vrije Universiteit Amsterdam, The Netherlands 3Department of Physics, Harvard University, Cambridge, MA 02139 We present a method of computing Casimir forces for arbitrary geometries, with any desired ac- curacy, that can directly exploit the efficiency of standard numerical-electromagnetism techniques. Using the simplest possible finite-difference implementation of this approach, we obtain both agree- ment with past results for cylinder-plate geometries, and also present results for new geometries. In particular, we examine a piston-like problem involving two dielectric and metallic squares sliding between two metallic walls, in two and three dimensions, respectively, and demonstrate non-additive and non-monotonic changes in the force due to these lateral walls. PACS numbers: Casimir forces arise between macroscopic objects due to changes in the zero-point energy associated with quan- tum fluctuations of the electromagnetic field [1]. This spectacular effect has been subject to many experimental validations, as reviewed in Ref. 2. All of the experiments reported so far have been based on simple geometries (parallel plates, crossed cylinders, or spheres and plates). For more complex geometries, calculations become ex- tremely cumbersome and often require drastic approx- imations, a limitation that has hampered experimental and theoretical work beyond the standard geometries. In this letter, we present a method to compute Casimir forces in arbitrary geometries and materials, with no un- controlled approximations, that can exploit the efficient solution of well-studied problems in classical computa- tional electromagnetism. Using this method, which we first test for geometries with known solutions, we predict a non-monotonic change in the force arising from lateral side walls in a less-familiar piston-like geometry (Fig. 2). Such a lateral-wall force cannot be predicted by “addi- tive” methods based on proximity-force or other purely two-body–interaction approximations, due to symmetry, and it is difficult to find a simple correction to give a non-monotonic force. We are able to compute forces for both perfect metals and arbitrary dispersive dielectrics, and we also obtain a visual map of the stress tensor that directly depicts the interaction forces between objects. The Casimir force was originally predicted for parallel metal plates, and the theory was subsequently extended to straighforward formulas for any planar-multilayer di- electric distribution ε(x, ω) via the generalized Lifshitz formula [3]. In order to handle more arbitrary geome- tries, two avenues have been pursued. First, one can em- ploy approximations derived from limits such as that of parallel plates; these methods include the proximity-force approximation (PFA) and its refinements [4], renormal- ized Casimir-Polder [5] or semi-classical interactions [6], multiple-scattering expansions [7], classical ray optics [8], and various perturbative techniques [9, 10]. Such meth- ods, however, involve uncontrolled approximations when applied to arbitrary geometries outside their range of ap- plicability, and have even been observed to give quali- tatively incorrect results [11, 12]. Therefore, researchers have instead sought numerical methods applicable to ar- bitrary geometries that converge to the exact result given sufficient computational resources. One such method uses a path-integral representation for the effective ac- tion [13], and has predicted the force between a cylin- der and a plane or between corrugated surfaces. Ref. 13 uses a surface parameterization of the fields coupled via vacuum Green’s functions, requiring O(N2) storage and O(N3) time for N degrees of freedom, making scaling to three dimensions (3d) problematic. Another exact method is the “world-line approach” [12, 14, 15], based on Monte-Carlo path-integral calculations. (The scaling of the world-line method involves a statistical analysis, determined by the relative feature sizes in the geome- try, that is beyond the scope of this Letter.) Further- more, the methods of Ref. 13 and Ref. 14 have currently only been demonstrated for perfect-metallic z-invariant structures—in this case, the vector unknowns can be de- composed into TE (E · ẑ = 0) and TM (H · ẑ = 0) scalar fields with Dirichlet or Neumann boundary conditions— although generalizations have been proposed [16]. Here, we propose a method based on evaluation of the mean stress tensor via the fluctuation-dissipation theorem, which only involves repeated evaluation of the electro- magnetic imaginary-frequency Green’s function. For a volume discretization with N degrees of freedom and an efficient iterative solver, this requires O(N) stor- age and O(N2−1/d) time in d dimensions. Further- more, because evaluation of the Green’s function is such a standard problem in classical computational electro- magnetism, it will be possible to exploit many develop- ments in fast solvers, such as finite-element, or boundary- element methods [17]. To illustrate the method, our initial implementation is based on the simplest-possible finite-difference frequency-domain (FDFD) method, as described below. As derived by Dzyaloshinskĭı et al. [1], the net Casimir force on a body can be expressed as an integral over any closed surface around the body of the mean electromag- netic stress tensor 〈Tij〉, integrated over imaginary fre- quencies ω = iw: surface 〈Tij(r, iw)〉 dSj . (1) For a 3d z-invariant structure, the z integral is replaced by an integral over the corresponding wavevector, result- ing in a net force per unit length. The stress tensor is defined as usual by: 〈Tij(r, iw)〉 = 〈Hi(r)Hj(r)〉 − 〈Hk(r)Hk(r)〉 + ε(r, iw) 〈Ei(r)Ej(r)〉 − 〈Ek(r)Ek(r)〉 The connection to quantum mechanics arises from the correlation functions of the fluctuating fields, such as 〈EiEj〉, given via the fluctuation-dissipation theorem in terms of the imaginary-ω Green’s function Gij(iw; r−r′): 〈Ei(r)Ej(r′)〉 = Gij(iw; r− r′) (3) 〈Hi(r)Hj(r′)〉 = −(∇×)i`(∇′×)jmG`m(iw; r− r′) , (4) where the Green’s function Gij solves the equation:[ ∇×∇×+ ε(r, iw) Gj(iw; r−r′) = êjδ(r−r′) (5) for a unit vector êj in the j direction, and obeys the usual boundary conditions on the electric field from classical electromagnetism. (The above expressions are at zero temperature; the nonzero-temperature force is found by changing dw in Eq. 1 into a discrete summation [1].) Although the Green’s function (and thus Tij) is formally infinite at r = r′, this divergence is conventionally re- moved by subtracting the vacuum Green’s function; in a numerical method with discretized space, as below, there is no divergence and no additional regularization is required. (The vacuum Green’s function gives zero net contribution to the dS integral, and therefore need not be removed as long as the integrand is finite.) Historically, this stress-tensor expression was used to derive the standard Lifshitz formula for parallel plates, where Gij is known analytically. However, it also forms an ideal starting point for a computational method, be- cause the Green’s function for arbitrary geometries is routinely computed numerically by a variety of tech- niques [17]. Furthermore, the problem actually becomes easier for an imaginary ω. First, for an imaginary ω, the linear operator in Eq. 5 is real-symmetric and positive- definite for w 6= 0, since the dielectric function ε(ω) is purely real and positive along the imaginary-ω axis for physical materials without gain, due to causality. Second, the imaginary-ω Green’s function is exponentially decay- ing rather than oscillating, leading to a well-behaved non- oscillatory integrand in Eq. 1. To illustrate this method, we employed the simplest possible computational technique: we perform a FDFD discretization of Eq. 5 with a staggered Yee grid [18] and periodic boundaries, inverting the linear operator by a conjugate-gradient method. The presence of discon- tinuous material interfaces degrades second-order finite- difference methods to only first-order accuracy, and the uniform spatial resolution is also suboptimal, but we found FDFD to be nevertheless adequate for small 2d geometries. The periodicity leads to artificial “wrap- around” forces that decay rapidly with cell size L (at least as 1/L3 in 2d and 1/L4 in 3d); we chose cell sizes large enough to make these contributions negligible (< 1%). The computational process is as follows: pick some sur- face/contour around a given body, evaluate the Green’s function for every grid point on this surface in order to compute the surface integral of the stress tensor, which is then integrated over w by adaptive quadrature. 1 10 FIG. 1: Casimir force between a radius-R cylinder and a plate (inset), relative to the proximity-force approximation FPFA, vs. normalized separation a/R. The solid lines are the Casimir force computed in Ref. 19 for TE (gray) and TM (blue) polarizations, along with results computed by our method with a simple finite-difference discretization (gray squares). Error bars were estimated for some data points by using computations at multiple spatial resolutions. Inset shows interaction stress tensor ∆〈Txx〉 at a typical imaginary frequency w = 2πc/a, where red indicates attractive stress. Before we attempt to study new geometries with our method, it is important to check it against known re- sults. The simplest cases, of parallel metallic or dielec- tric plates, of course match the known result from the Liftshitz formula and are not reproduced here. A more complicated geometry, consisting of a perfect metallic cylinder adjacent to a perfect metallic plate in 3d, was solved numerically by Ref. 19, to which our results are compared in Fig. 1. Ref. 19 used a specialized Fourier- Bessel basis specific to this cylindrical geometry, which should have exponential (spectral) convergence. Our use of a simple uniform grid was necessarily much less effi- cient, especially with the first-order accuracy, but was able to match the Ref. 19 results within ∼ 3% using rea- sonable computational resources. A simple grid has the advantage of being very general, as illustrated below, but other general bases with much greater efficiency are pos- sible using finite-element or boundary-element methods; the latter, in particular, could use a spectral Fourier ba- sis analogous to Ref. 19 and exploit a fast-multipole or similar O(N logN) solver technique [17]. Also shown, in the inset of Fig. 1, is a plot of the interaction stress-tensor component ∆〈Txx〉 at a typical imaginary frequency w = 2πc/a. By “interaction” stress- tensor ∆〈Tij〉, we mean the total 〈Tij〉 of the full geome- try minus the sum of the 〈Tij〉’s computed for each body in isolation. Here, the stress tensors of the isolated cylin- der and plate have been subtracted, giving us a way to visualize the force due to the interaction. As described below, such stress plots reveal the regions in which two objects most strongly affect one another, and therefore reveal where a change of the geometry would have the most impact. (In contrast, Ref. 12 plots an interaction- energy density that does not directly reveal the force, since the force requires the energy to be differentiated with respect to a. For example, Ref. 12’s subtracted en- ergy density apparently goes nearly to zero as a metallic surface is approached, whereas the stress tensor cannot since the stress integration surface is arbitrary.) We now consider a more complicated geometry in which there are interactions between multiple bodies: a 3d “piston”-like structure, shown in the inset of Fig. 2, consisting of two z-invariant metal s × s squares sepa- rated by a distance a from one another (here, s = a) and separated by a distance h from infinite metal plates on either side. We then compute the Casimir force per unit z between the two squares as a function of the sep- aration h. The result for perfect conductors is shown in Fig. 2, plotted for the TE and TM polarizations and also showing the total force. (Error bars are not shown because the estimated error is < 1%.) In the limit of h → 0, this structure approaches the “Casimir piston,” which has been solved analytically [20, 21] (and also in 2d for the TM polarization [22]). Our results, extrapo- lated to h = 0, agree with these results to within ≈ 2% (although we have computational difficulties for small h due to the high resolution required to resolve a small 0 0.2 0.4 0.6 0.8 1 total FIG. 2: Casimir force per unit length between metal squares F/FPFA, vs. distance from metal plate h (inset), normalized by the total TE+TM force per unit length obtained using the PFA, FPFA = ~csζ(3)/480πa4. The total force is plotted (black squares) along with the TE (red dots) and TM (blue circles) contributions. feature in FDFD). For h > 0, however, the result is sur- prising in at least two ways. First, the total force is non-monotonic in h, due to a competition between the TE and TM contributions to the forces. Second, the h dependence of the force is a lateral effect of the parallel plates on the squares, which would be zero by symmetry in PFA or any other two-body–interaction approxima- tion. Although lateral-wall effects can clearly arise qual- itatively in various approximations, such as in ray optics or in PFA restricted to “line-of-sight” interactions, non- monotonicity is more surprising[24]. Also, in the large-h limit, the force remains different from PFA due to finite-s “edge” effects [12], which are captured by our method. Our method is also capable, without modification, of handling dielectric materials. This is demonstrated in Fig. 3, where the Casimir force is shown for the case where the squares are made of gold, using the experimen- tal Drude ε(ω) from Ref. 23 for a separation a = 1µm. Here, our calculation is for a purely 2d geometry (equiv- alently, 3d restricted to z-invariant fields/currents), and for comparison we also plot the corresponding 2d force for perfect-metal squares (although the two cases are normalized differently as described in the caption). As might be expected, the dielectric squares have a weaker interaction than the perfect-metal squares, but are still non-monotonic. Note also the qualitative similarity be- tween the perfect-metal results of Figs. 2 and 3, reflecting the fact that the force contributions are dominated by the zero-wavevector (z-invariant) fields. Extrapolated to h = 0, the perfect-metal TM force agrees with the known analytical result [22] to within ≈ 3%. 0 0.2 0.4 0.6 0.8 1 1.2 sΑu a x FIG. 3: Solid lines: Casimir force between 2d gold squares F/FPFA, vs. distance from metal plate h (inset), using ex- perimental ε(ω) [23], normalized by the total force obtained using the PFA. (Here, the PFA force is computed for x-infinite gold slabs). The total force is plotted (black squares) along with the TE (red dots) and TM (blue circles) contributions. Dashed lines: force for 2d perfect-metal squares, normalized by the perfect-metal PFA force FPFA = ~csζ(3)/8πa3. To further explore the source of the h-dependence, we plot the TM interaction-stress maps ∆〈Txx〉 and ∆〈Txy〉 in Fig. 4, for the 2d perfect-metal squares from Fig. 3. The stress plots of Fig. 4 are computed at a typical fre- quency w = 2πc/a, and for varying distances from the metal plates (h = 0.5, 1.0, 2.0). As shown, the mag- nitudes of both the xx (a–c) and xy (d–f) components of the stress tensor change dramatically as the metal plates are brought closer to the squares. For example, one change in the force integral comes from Txy, which for isolated squares has an asymmetric pattern at the four corners that will contribute to the attractive force, whereas the presence of the plates induces a more sym- metric pattern of stresses at the four corners that will have nearly zero integral. This results in a decreasing TM force with decreasing h. Because stress maps indi- cate where bodies interact and with what signs, it may be useful in future work to explore whether they can be used to design unusual behaviors such as non-additive, non-monotonic, or even repulsive forces. This work was supported in part by the Nanoscale Sci- ence and Engineering Center (NSEC) under NSF con- tract PHY-0117795, by the Materials Research Science and Engineering Center program of the NSF under award DMR-9400334, and by a DOE Comp. Science Grad. Fel- lowship under grant DE–FG02-97ER25308. D. I. grate- fully acknowledges support from the Netherlands Organ- isation for Scientific Research (NWO), under the IRI Scheme Vernieuwingsimpuls VIDI-680-47-209. Txx Txy FIG. 4: (a–f): TM stress map of the 2d-analogous geome- try of Fig. 2 for various h. The intearaction stress tensors 〈Txx〉 (left) and 〈Txy〉 (right) for: (a),(d): h = 0.5a; (b),(e): h = a; and (c),(f): h = 2a, where blue/white/red = repul- sive/zero/attractive. [1] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics: Part 2 (Pergamon, Oxford, 1980). [2] R. Onofrio, New J. Phys. 8, 237 (2006). [3] M. S. Tomaš, Phys. Rev. A 66, 052103 (2002). [4] M. Bordag, Phys. Rev. D 73, 125018 (2006). [5] M. Tajmar, Intl. J. Mod. Phys. C 15, 1387 (2004). [6] M. Schaden and L. Spruch, Phys. Rev. A 58, 935 (1998). [7] R. Balian and B. Duplantier, Ann. Phys. 112, 165 (1978). [8] R. L. Jaffe and A. Scardicchio, Phys. Rev. Lett. 92, 070402 (2004). [9] T. Emig, A. Hanke, R. Golestanian, and M. Kardar, Phys. Rev. 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We present a method of computing Casimir forces for arbitrary geometries, with any desired accuracy, that can directly exploit the efficiency of standard numerical-electromagnetism techniques. Using the simplest possible finite-difference implementation of this approach, we obtain both agreement with past results for cylinder-plate geometries, and also present results for new geometries. In particular, we examine a piston-like problem involving two dielectric and metallic squares sliding between two metallic walls, in two and three dimensions, respectively, and demonstrate non-additive and non-monotonic changes in the force due to these lateral walls.

Computation and visualization of Casimir forces in arbitrary geometries: non-monotonic lateral-wall forces and failure of proximity-force approximations Alejandro Rodriguez,1 Mihai Ibanescu,1 Davide Iannuzzi,2 Federico Capasso,3 J. D. Joannopoulos,1 and Steven G. Johnson1 1Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 2Faculty of Sciences, Department of Physics and Astronomy, Vrije Universiteit Amsterdam, The Netherlands 3Department of Physics, Harvard University, Cambridge, MA 02139 We present a method of computing Casimir forces for arbitrary geometries, with any desired ac- curacy, that can directly exploit the efficiency of standard numerical-electromagnetism techniques. Using the simplest possible finite-difference implementation of this approach, we obtain both agree- ment with past results for cylinder-plate geometries, and also present results for new geometries. In particular, we examine a piston-like problem involving two dielectric and metallic squares sliding between two metallic walls, in two and three dimensions, respectively, and demonstrate non-additive and non-monotonic changes in the force due to these lateral walls. PACS numbers: Casimir forces arise between macroscopic objects due to changes in the zero-point energy associated with quan- tum fluctuations of the electromagnetic field [1]. This spectacular effect has been subject to many experimental validations, as reviewed in Ref. 2. All of the experiments reported so far have been based on simple geometries (parallel plates, crossed cylinders, or spheres and plates). For more complex geometries, calculations become ex- tremely cumbersome and often require drastic approx- imations, a limitation that has hampered experimental and theoretical work beyond the standard geometries. In this letter, we present a method to compute Casimir forces in arbitrary geometries and materials, with no un- controlled approximations, that can exploit the efficient solution of well-studied problems in classical computa- tional electromagnetism. Using this method, which we first test for geometries with known solutions, we predict a non-monotonic change in the force arising from lateral side walls in a less-familiar piston-like geometry (Fig. 2). Such a lateral-wall force cannot be predicted by “addi- tive” methods based on proximity-force or other purely two-body–interaction approximations, due to symmetry, and it is difficult to find a simple correction to give a non-monotonic force. We are able to compute forces for both perfect metals and arbitrary dispersive dielectrics, and we also obtain a visual map of the stress tensor that directly depicts the interaction forces between objects. The Casimir force was originally predicted for parallel metal plates, and the theory was subsequently extended to straighforward formulas for any planar-multilayer di- electric distribution ε(x, ω) via the generalized Lifshitz formula [3]. In order to handle more arbitrary geome- tries, two avenues have been pursued. First, one can em- ploy approximations derived from limits such as that of parallel plates; these methods include the proximity-force approximation (PFA) and its refinements [4], renormal- ized Casimir-Polder [5] or semi-classical interactions [6], multiple-scattering expansions [7], classical ray optics [8], and various perturbative techniques [9, 10]. Such meth- ods, however, involve uncontrolled approximations when applied to arbitrary geometries outside their range of ap- plicability, and have even been observed to give quali- tatively incorrect results [11, 12]. Therefore, researchers have instead sought numerical methods applicable to ar- bitrary geometries that converge to the exact result given sufficient computational resources. One such method uses a path-integral representation for the effective ac- tion [13], and has predicted the force between a cylin- der and a plane or between corrugated surfaces. Ref. 13 uses a surface parameterization of the fields coupled via vacuum Green’s functions, requiring O(N2) storage and O(N3) time for N degrees of freedom, making scaling to three dimensions (3d) problematic. Another exact method is the “world-line approach” [12, 14, 15], based on Monte-Carlo path-integral calculations. (The scaling of the world-line method involves a statistical analysis, determined by the relative feature sizes in the geome- try, that is beyond the scope of this Letter.) Further- more, the methods of Ref. 13 and Ref. 14 have currently only been demonstrated for perfect-metallic z-invariant structures—in this case, the vector unknowns can be de- composed into TE (E · ẑ = 0) and TM (H · ẑ = 0) scalar fields with Dirichlet or Neumann boundary conditions— although generalizations have been proposed [16]. Here, we propose a method based on evaluation of the mean stress tensor via the fluctuation-dissipation theorem, which only involves repeated evaluation of the electro- magnetic imaginary-frequency Green’s function. For a volume discretization with N degrees of freedom and an efficient iterative solver, this requires O(N) stor- age and O(N2−1/d) time in d dimensions. Further- more, because evaluation of the Green’s function is such a standard problem in classical computational electro- magnetism, it will be possible to exploit many develop- ments in fast solvers, such as finite-element, or boundary- element methods [17]. To illustrate the method, our initial implementation is based on the simplest-possible finite-difference frequency-domain (FDFD) method, as described below. As derived by Dzyaloshinskĭı et al. [1], the net Casimir force on a body can be expressed as an integral over any closed surface around the body of the mean electromag- netic stress tensor 〈Tij〉, integrated over imaginary fre- quencies ω = iw: surface 〈Tij(r, iw)〉 dSj . (1) For a 3d z-invariant structure, the z integral is replaced by an integral over the corresponding wavevector, result- ing in a net force per unit length. The stress tensor is defined as usual by: 〈Tij(r, iw)〉 = 〈Hi(r)Hj(r)〉 − 〈Hk(r)Hk(r)〉 + ε(r, iw) 〈Ei(r)Ej(r)〉 − 〈Ek(r)Ek(r)〉 The connection to quantum mechanics arises from the correlation functions of the fluctuating fields, such as 〈EiEj〉, given via the fluctuation-dissipation theorem in terms of the imaginary-ω Green’s function Gij(iw; r−r′): 〈Ei(r)Ej(r′)〉 = Gij(iw; r− r′) (3) 〈Hi(r)Hj(r′)〉 = −(∇×)i`(∇′×)jmG`m(iw; r− r′) , (4) where the Green’s function Gij solves the equation:[ ∇×∇×+ ε(r, iw) Gj(iw; r−r′) = êjδ(r−r′) (5) for a unit vector êj in the j direction, and obeys the usual boundary conditions on the electric field from classical electromagnetism. (The above expressions are at zero temperature; the nonzero-temperature force is found by changing dw in Eq. 1 into a discrete summation [1].) Although the Green’s function (and thus Tij) is formally infinite at r = r′, this divergence is conventionally re- moved by subtracting the vacuum Green’s function; in a numerical method with discretized space, as below, there is no divergence and no additional regularization is required. (The vacuum Green’s function gives zero net contribution to the dS integral, and therefore need not be removed as long as the integrand is finite.) Historically, this stress-tensor expression was used to derive the standard Lifshitz formula for parallel plates, where Gij is known analytically. However, it also forms an ideal starting point for a computational method, be- cause the Green’s function for arbitrary geometries is routinely computed numerically by a variety of tech- niques [17]. Furthermore, the problem actually becomes easier for an imaginary ω. First, for an imaginary ω, the linear operator in Eq. 5 is real-symmetric and positive- definite for w 6= 0, since the dielectric function ε(ω) is purely real and positive along the imaginary-ω axis for physical materials without gain, due to causality. Second, the imaginary-ω Green’s function is exponentially decay- ing rather than oscillating, leading to a well-behaved non- oscillatory integrand in Eq. 1. To illustrate this method, we employed the simplest possible computational technique: we perform a FDFD discretization of Eq. 5 with a staggered Yee grid [18] and periodic boundaries, inverting the linear operator by a conjugate-gradient method. The presence of discon- tinuous material interfaces degrades second-order finite- difference methods to only first-order accuracy, and the uniform spatial resolution is also suboptimal, but we found FDFD to be nevertheless adequate for small 2d geometries. The periodicity leads to artificial “wrap- around” forces that decay rapidly with cell size L (at least as 1/L3 in 2d and 1/L4 in 3d); we chose cell sizes large enough to make these contributions negligible (< 1%). The computational process is as follows: pick some sur- face/contour around a given body, evaluate the Green’s function for every grid point on this surface in order to compute the surface integral of the stress tensor, which is then integrated over w by adaptive quadrature. 1 10 FIG. 1: Casimir force between a radius-R cylinder and a plate (inset), relative to the proximity-force approximation FPFA, vs. normalized separation a/R. The solid lines are the Casimir force computed in Ref. 19 for TE (gray) and TM (blue) polarizations, along with results computed by our method with a simple finite-difference discretization (gray squares). Error bars were estimated for some data points by using computations at multiple spatial resolutions. Inset shows interaction stress tensor ∆〈Txx〉 at a typical imaginary frequency w = 2πc/a, where red indicates attractive stress. Before we attempt to study new geometries with our method, it is important to check it against known re- sults. The simplest cases, of parallel metallic or dielec- tric plates, of course match the known result from the Liftshitz formula and are not reproduced here. A more complicated geometry, consisting of a perfect metallic cylinder adjacent to a perfect metallic plate in 3d, was solved numerically by Ref. 19, to which our results are compared in Fig. 1. Ref. 19 used a specialized Fourier- Bessel basis specific to this cylindrical geometry, which should have exponential (spectral) convergence. Our use of a simple uniform grid was necessarily much less effi- cient, especially with the first-order accuracy, but was able to match the Ref. 19 results within ∼ 3% using rea- sonable computational resources. A simple grid has the advantage of being very general, as illustrated below, but other general bases with much greater efficiency are pos- sible using finite-element or boundary-element methods; the latter, in particular, could use a spectral Fourier ba- sis analogous to Ref. 19 and exploit a fast-multipole or similar O(N logN) solver technique [17]. Also shown, in the inset of Fig. 1, is a plot of the interaction stress-tensor component ∆〈Txx〉 at a typical imaginary frequency w = 2πc/a. By “interaction” stress- tensor ∆〈Tij〉, we mean the total 〈Tij〉 of the full geome- try minus the sum of the 〈Tij〉’s computed for each body in isolation. Here, the stress tensors of the isolated cylin- der and plate have been subtracted, giving us a way to visualize the force due to the interaction. As described below, such stress plots reveal the regions in which two objects most strongly affect one another, and therefore reveal where a change of the geometry would have the most impact. (In contrast, Ref. 12 plots an interaction- energy density that does not directly reveal the force, since the force requires the energy to be differentiated with respect to a. For example, Ref. 12’s subtracted en- ergy density apparently goes nearly to zero as a metallic surface is approached, whereas the stress tensor cannot since the stress integration surface is arbitrary.) We now consider a more complicated geometry in which there are interactions between multiple bodies: a 3d “piston”-like structure, shown in the inset of Fig. 2, consisting of two z-invariant metal s × s squares sepa- rated by a distance a from one another (here, s = a) and separated by a distance h from infinite metal plates on either side. We then compute the Casimir force per unit z between the two squares as a function of the sep- aration h. The result for perfect conductors is shown in Fig. 2, plotted for the TE and TM polarizations and also showing the total force. (Error bars are not shown because the estimated error is < 1%.) In the limit of h → 0, this structure approaches the “Casimir piston,” which has been solved analytically [20, 21] (and also in 2d for the TM polarization [22]). Our results, extrapo- lated to h = 0, agree with these results to within ≈ 2% (although we have computational difficulties for small h due to the high resolution required to resolve a small 0 0.2 0.4 0.6 0.8 1 total FIG. 2: Casimir force per unit length between metal squares F/FPFA, vs. distance from metal plate h (inset), normalized by the total TE+TM force per unit length obtained using the PFA, FPFA = ~csζ(3)/480πa4. The total force is plotted (black squares) along with the TE (red dots) and TM (blue circles) contributions. feature in FDFD). For h > 0, however, the result is sur- prising in at least two ways. First, the total force is non-monotonic in h, due to a competition between the TE and TM contributions to the forces. Second, the h dependence of the force is a lateral effect of the parallel plates on the squares, which would be zero by symmetry in PFA or any other two-body–interaction approxima- tion. Although lateral-wall effects can clearly arise qual- itatively in various approximations, such as in ray optics or in PFA restricted to “line-of-sight” interactions, non- monotonicity is more surprising[24]. Also, in the large-h limit, the force remains different from PFA due to finite-s “edge” effects [12], which are captured by our method. Our method is also capable, without modification, of handling dielectric materials. This is demonstrated in Fig. 3, where the Casimir force is shown for the case where the squares are made of gold, using the experimen- tal Drude ε(ω) from Ref. 23 for a separation a = 1µm. Here, our calculation is for a purely 2d geometry (equiv- alently, 3d restricted to z-invariant fields/currents), and for comparison we also plot the corresponding 2d force for perfect-metal squares (although the two cases are normalized differently as described in the caption). As might be expected, the dielectric squares have a weaker interaction than the perfect-metal squares, but are still non-monotonic. Note also the qualitative similarity be- tween the perfect-metal results of Figs. 2 and 3, reflecting the fact that the force contributions are dominated by the zero-wavevector (z-invariant) fields. Extrapolated to h = 0, the perfect-metal TM force agrees with the known analytical result [22] to within ≈ 3%. 0 0.2 0.4 0.6 0.8 1 1.2 sΑu a x FIG. 3: Solid lines: Casimir force between 2d gold squares F/FPFA, vs. distance from metal plate h (inset), using ex- perimental ε(ω) [23], normalized by the total force obtained using the PFA. (Here, the PFA force is computed for x-infinite gold slabs). The total force is plotted (black squares) along with the TE (red dots) and TM (blue circles) contributions. Dashed lines: force for 2d perfect-metal squares, normalized by the perfect-metal PFA force FPFA = ~csζ(3)/8πa3. To further explore the source of the h-dependence, we plot the TM interaction-stress maps ∆〈Txx〉 and ∆〈Txy〉 in Fig. 4, for the 2d perfect-metal squares from Fig. 3. The stress plots of Fig. 4 are computed at a typical fre- quency w = 2πc/a, and for varying distances from the metal plates (h = 0.5, 1.0, 2.0). As shown, the mag- nitudes of both the xx (a–c) and xy (d–f) components of the stress tensor change dramatically as the metal plates are brought closer to the squares. For example, one change in the force integral comes from Txy, which for isolated squares has an asymmetric pattern at the four corners that will contribute to the attractive force, whereas the presence of the plates induces a more sym- metric pattern of stresses at the four corners that will have nearly zero integral. This results in a decreasing TM force with decreasing h. Because stress maps indi- cate where bodies interact and with what signs, it may be useful in future work to explore whether they can be used to design unusual behaviors such as non-additive, non-monotonic, or even repulsive forces. This work was supported in part by the Nanoscale Sci- ence and Engineering Center (NSEC) under NSF con- tract PHY-0117795, by the Materials Research Science and Engineering Center program of the NSF under award DMR-9400334, and by a DOE Comp. Science Grad. Fel- lowship under grant DE–FG02-97ER25308. D. I. grate- fully acknowledges support from the Netherlands Organ- isation for Scientific Research (NWO), under the IRI Scheme Vernieuwingsimpuls VIDI-680-47-209. Txx Txy FIG. 4: (a–f): TM stress map of the 2d-analogous geome- try of Fig. 2 for various h. The intearaction stress tensors 〈Txx〉 (left) and 〈Txy〉 (right) for: (a),(d): h = 0.5a; (b),(e): h = a; and (c),(f): h = 2a, where blue/white/red = repul- sive/zero/attractive. [1] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics: Part 2 (Pergamon, Oxford, 1980). [2] R. Onofrio, New J. Phys. 8, 237 (2006). [3] M. S. Tomaš, Phys. Rev. A 66, 052103 (2002). [4] M. Bordag, Phys. Rev. D 73, 125018 (2006). [5] M. Tajmar, Intl. J. Mod. Phys. C 15, 1387 (2004). [6] M. Schaden and L. Spruch, Phys. Rev. A 58, 935 (1998). [7] R. Balian and B. Duplantier, Ann. Phys. 112, 165 (1978). [8] R. L. Jaffe and A. Scardicchio, Phys. Rev. Lett. 92, 070402 (2004). [9] T. Emig, A. Hanke, R. Golestanian, and M. Kardar, Phys. Rev. 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Axioms for a local Reidemeister trace in fixed point and coincidence theory on differentiable manifolds P. Christopher Staecker August 24, 2021 Abstract We give axioms which characterize the local Reidemeister trace for orientable differentiable manifolds. The local Reidemeister trace in fixed point theory is already known, and we provide both uniqueness and exis- tence results for the local Reidemeister trace in coincidence theory. 1 Introduction The Reidemeister trace is a fundamental invariant in topological fixed point theory, generalizing both the Lefschetz and Nielsen numbers. It was originally defined by Reidemeister in [11]. A more modern treatment, under the name “generalized Lefschetz number,” was given by Husseini in [9]. If X is a finite connected CW-complex with universal covering space X̃ and fundamental group π, then the cellular chain complex Cq(X̃) is a free Zπ- module. If f : X → X is a cellular map and f̃ : X̃ → X̃ is a lift of f , then the induced map f̃q : Cq(X̃) → Cq(X̃) can be viewed as a matrix with entries in Zπ (with respect to some chosen Zπ basis for Cq(X̃)). We then define RT (f, f̃) = (−1)qρ(tr(f̃q)), where tr is the sum of the diagonal entries of the matrix, and ρ is the projection into the “Reidemeister classes” of π. The Reidemeister trace, then, is an element of ZR, where R is the set of Reidemeister classes. Wecken, in [13], proved what we will refer to as the Wecken Trace Theorem, that RT (f, f̃) = [α]∈R ind([α]) [α], where ind([α]) is the index of the Nielsen fixed point class associated to [α] (see e.g. [10]). Thus the number of terms appearing in the Reidemeister trace with http://arxiv.org/abs/0704.1891v2 nonzero coefficient is equal to the Nielsen number of f , and by the Lefschetz- Hopf Theorem, the sum of the coefficients is equal to the Lefschetz number of Recent work of Furi, Pera, and Spadini in [6] has given a new proof of the uniqueness of the fixed point index on orientable manifolds with respect to three natural axioms. In [12] their approach was extended to the coincidence index. The result is the following theorem: Theorem 1. Let X and Y be oriented differentiable manifolds of the same dimension. The coincidence index ind(f, g, U) of two mappings f, g : X → Y over some open set U ⊂ X is the unique integer-valued function satisfying the following axioms: • (Additivity) If U1 and U2 are disjoint open subsets of U whose union con- tains all coincidence points of f and g on U , then ind(f, g, U) = ind(f, g, U1) + ind(f, g, U2). • (Homotopy) If f and g are “admissably homotopic” to f ′ and g′, then ind(f, g, U) = ind(f ′, g′, U) • (Normalization) If L(f, g) denotes the coincidence Lefschetz number of f and g, then ind(f, g,X) = L(f, g). In the spirit of the above theorem, we demonstrate the existence and unique- ness of a local Reidemeister trace in coincidence theory subject to five axioms. A local Reidemeister trace for fixed point theory was given by Fares and Hart in [5], but no Reidemeister trace (local or otherwise) has appeared in the literature for coincidence theory. We note that recent work by Gonçalves and Weber in [8] gives axioms for the Reidemeister trace in fixed point theory using entirely different methods. Their work uses no locality properties, and is based on axioms for the Lefschetz number by Arkowitz and Brown in [1]. In Section 2 we present our axiom set, and we prove the uniqueness in coincidence theory in Section 3. In the special case of local fixed point theory, we can obtain a slightly stronger uniqueness result which we discuss in Section 4. Section 5 is a demonstration of the existence in the setting of coincidence theory. This paper contains pieces of the author’s doctoral dissertation. The author would like to thank his dissertation advisor Robert F. Brown for assistance with both the dissertation work and with this paper. The author would also like to thank Peter Wong, who guided the early dissertation work and interested him in the coincidence Reidemeister trace. 2 The Axioms Throughout the paper, unless otherwise stated, let X and Y denote connected orientable differentiable manifolds of the same dimension. All maps f, g : X → Y will be assumed to be continuous. The universal covering spaces of X and Y will be denoted X̃ and Ỹ with projection maps pX : X̃ → X and pY : Ỹ → Y . A lift of some map f : X → Y is a map f̃ : X̃ → Ỹ with pY ◦ f̃ = f ◦ pX . Let f, g : X → Y be maps, with induced homomorphisms φ, ψ : π1(X) → π1(Y ) respectively. We will view elements of π1(X) and π1(Y ) as covering transformations, so that for any x̃ ∈ X̃ and σ ∈ π1(X), we have f̃(σx̃) = φ(σ)f̃ (x̃) and g̃(σx̃) = ψ(σ)g̃(x̃). We will partition the elements of π1(Y ) into equivalence classes defined by the “doubly twisted conjugacy” relation: α ∼ β ⇐⇒ α = ψ(σ)−1βφ(σ). The equivalence classes with respect to this relation (denoted e.g. [α]) are called Reidemeister classes. The set of Reidemeister classes is denoted R[f, g]. For any set S, let ZS denote the free abelian group generated by S, whose elements we write as sums of elements of S with integer coefficients. For any such abelian group, there is a homomorphism c : ZS → Z defined as the sum of the coefficients: for si ∈ S and ki ∈ Z, and i ranging over a finite set. For some maps f, g : X → Y and an open subset U ⊂ X , let Coin(f, g, U) = {x ∈ U | f(x) = g(x)}. We say that the triple (f, g, U) is admissable if Coin(f, g, U) is compact. Two triples (f, g, U) and (f ′, g′, U) are admissably homotopic if there is some pair of homotopies Ft, Gt : X × [0, 1] → X of f, g to f ′, g′ with {(x, t) ∈ U × [0, 1] | Ft(x) = Gt(x)} compact. Let C(X,Y ) be the set of admissable tuples, all tuples of the form (f, f̃ , g, g̃, U) where f, g : X → Y are maps, (f, g, U) is an admissable triple, and f̃ and g̃ are lifts of f and g. Let (f, f̃ , g, g̃, U), (f ′, f̃ ′, g′, g̃′, U) ∈ C(X,Y ) with (f, g, U) admissably ho- motopic to (f ′, g′, U) by homotopies Ft, Gt. By the homotopy lifting property, there are unique lifted homotopies F̃t, G̃t : X̃ × [0, 1] → Ỹ with F̃0 = f̃ and G̃0 = g̃. If we additionaly have F̃1 = f̃ ′ and G̃1 = g̃ ′, then we say that the tuples (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U) are admisssably homotopic. Throughout the following, let RT be any function which to an admissable tuple (f, f̃ , g, g̃, U) ∈ C(X,Y ) associates an element of ZR[f, g]. Our first three axioms for the local Reidemeister trace are modeled after the axioms of Theorem Axiom 1 (Additivity). Given (f, f̃ , g, g̃, U) ∈ C(X,Y ), if U1 and U2 are disjoint open subsets of U with Coin(f, g, U) ⊂ U1 ∪ U2, then RT(f, f̃ , g, g̃, U) = RT(f, f̃ , g, g̃, U1) + RT(f, f̃ , g, g̃, U2). Axiom 2 (Homotopy). If (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U) are admissably ho- motopic admissable tuples, then RT(f, f̃ , g, g̃, U) = RT(f ′, f̃ ′, g′, g̃′, U). Axiom 3 (Normalization). If (f, f̃ , g, g̃, X) ∈ C(X,Y ), then c(RT(f, f̃ , g, g̃, X)) = L(f, g), where L(f, g) is the Lefschetz number of f and g. We will require one additional axiom to make some connections with Nielsen theory, based on a well-known property of the Reidemeister trace: Axiom 4 (Lift invariance). For any (f, f̃ , g, g̃, U) ∈ C(X,Y ), and any α, β ∈ π1(Y ) we have c(RT(f, f̃ , g, g̃, U)) = c(RT(f, αf̃ , g, βg̃, U)). The four axioms above are enough to demonstrate some relationships be- tween RT and the coincidence index. Proposition 1. If RT satisfies the homotopy, additivity, normalization, and lift invariance axioms, then c(RT(f, f̃ , g, g̃, U)) = ind(f, g, U) for any (f, f̃ , g, g̃, U) ∈ C(X,Y ), where ind denotes the coincidence index (see [7]). Proof. Let ω = c ◦ RT : C(X,Y ) → Z. By the lift invariance axiom, ω is independent of the choice of lifts. Thus ω can be viewed as a function from the set of all admissable triples to Z. It is clear that ω satisfies the three axioms of Theorem 1, since they are implied by our additivity, homotopy, and normalization axioms for RT (disregarding the lift parameters). Thus ω is the coincidence index. Proposition 2. If RT satisfies the additivity, homotopy, normalization, and lift invariance axioms and c(RT(f, f̃ , g, g̃, U)) 6= 0, then there is some σ ∈ π1(Y ) such that σf̃ and g̃ have a coincidence on p−1X (U). Proof. By Proposition 1, if c(RT (f, f̃ , g, g̃, U)) 6= 0 then ind(f, g, U) 6= 0, and so f and g have a coincidence on U . Let x ∈ U be this coincidence point, and choose x̃ ∈ p−1X (x). Then since f̃ and g̃ are lifts, the points f̃(x̃) and g̃(x̃) will project to the same point of Y by pY . Thus there is some covering transformation σ with σf̃(x̃) = g̃(x̃). The four axioms given above are not sufficient to uniquely characterize the Reidemeister trace in fixed point or coincidence theory. For instance, the func- tion defined by T (f, f̃ , g, g̃, U) = ind(f, g, U)[1], where [1] is the Reidemeister class of the trivial element 1 ∈ π1(Y ), satisfies all of the axioms above, but provides none of the expected data concerning R[f, g], and so that function cannot be the Reidemeister trace. An additional axiom is needed, one which somehow indicates the elements of R[f, g] which are to appear in the Reidemeister trace. Our final axiom is a sort of strengthening of Proposition 2, which specifies the Reidemeister data associated to the coincidence points. Axiom 5 (Coincidence of lifts). If [α] appears with nonzero coefficient in RT(f, f̃ , g, g̃, U), then αf̃ and g̃ have a coincidence on p−1 Any function RT which to a tuple (f, f̃ , g, g̃, U) ∈ C(X,Y ) associates an element of ZR[f, g], and satisfies the additivity, homotopy, normalization, lift invariance, and coincidence of lifts axioms we will call a local Reidemeister trace. Our main result (Theorem 3) states that there is a unique such function. 3 Uniqueness Let (f, f̃ , g, g̃, U) ∈ C(X,Y ), let Ũ = p−1 (U), and let C(f̃ , g̃, Ũ , [α]) = pX(Coin(αf̃ , g̃, Ũ)). For each α we have C(f̃ , g̃, Ũ , [α]) ⊂ Coin(f, g, U), and such coincidence sets are called coincidence classes. That these classes are well defined is a consequence of the following lemma, which appears in slightly different language as Lemma 2.3 of [4]. Lemma 1. Let α, β ∈ π1(Y ), maps f, g : X → Y , and an open subset U ⊂ X be given. Then: • [α] = [β] if and only if pX Coin(αf̃ , g̃, Ũ) = pX Coin(βf̃ , g̃, Ũ) for any lifts f̃ , g̃. • If [α] 6= [β], then pX Coin(αf̃ , g̃, Ũ) and pX Coin(αf̃ , g̃, Ũ) are disjoint for any lifts f̃ , g̃. Given the above notation, the coincidence of lifts axiom could be restated as follows: If [α] appears with nonzero coefficient in RT (f, f̃ , g, g̃, U), then C(f̃ , g̃, Ũ , [α]) is nonempty. For each coincidence point x in U , define [xef,eg] ∈ R[f, g] as that class [α] for which x ∈ C(f̃ , g̃, Ũ , [α]). Theorem 2. If RT is a local Reidemeister trace and Coin(f, g, U) is a set of isolated points, then RT(f, f̃ , g, g̃, U) = x∈Coin(f,g,U) ind(f, g, Ux)[xef,eg], where Ux is an isolating neighborhood for the coincidence point x. Proof. By the additivity property, we need only show that RT (f, f̃ , g, g̃, Ux) = ind(f, g, Ux)[xef,eg]. First, we observe that no element of R[f, g] other than [xef,eg] appears as a term with nonzero coefficient in RT (f, f̃ , g, g̃, Ux): If some [β] does appear with nonzero coefficient, then we know by the coincidence of lifts axiom that βf̃ and g̃ have a coincidence on Ũx = p (Ux). Projection of this coincidence point gives a coincidence point in Ux which necessarily must be x, since x is the only coincidence point in Ux. Thus x ∈ pX Coin(βf̃ , g̃, Ũx), which means that [β] = [xef,eg]. Since [xef,eg] is the only element of R[f, g] appearing in RT(f, f̃ , g, g̃, U), we RT(f, f̃ , g, g̃, Ux) = k[xef,eg] for some k ∈ Z (possibly k = 0). Proposition 1 says that the coefficient sum must equal the index, and so k = ind(f, g, Ux) as desired. The above is a strong result for maps whose coincidence sets are isolated. In order to leverage this result for arbitrary maps, we will make use of a technical lemma, a combination of Lemmas 13 and 15 from [12]. Lemma 2. Let (f, g, U) be an admissable triple, and let V ⊂ U be an open subset containing Coin(f, g, U) with compact closure V̄ ⊂ U . Then (f, g, V ) is admissably homotopic to an admissable triple (f ′, g′, V ), where f ′ and g′ have isolated coincidence points in V . The above lemma is used to approximate any maps by maps having isolated coincidence points, and we obtain our uniqueness theorem: Theorem 3. There is at most one local Reidemeister trace defined on C(X,Y ). Proof. Let RT be local Reidemeister trace, and take (f, f̃ , g, g̃, U) ∈ C(X,Y ). Then by Lemma 2 there is an open subset V ⊂ U with Coin(f, U) ⊂ V and maps f ′, g′ with isolated coincidence points with (f, g, V ) admissably homotopic to (f ′, g′, V ). Then by the homotopy axiom there are lifts f̃ ′, g̃′ of f and g with RT (f, f̃ , g, g̃, U) = RT (f ′, f̃ ′, g′, g̃′, V ). The coincidence points of f ′ and g′ in V are isolated, so we have RT (f, f̃ , g, g̃, U) = x∈Coin(f ′,g′,V ) ind(f ′, g′, Vx)[xef ′,eg′ ], where Vx is an isolating neighborhood of the coincidence point x. This gives an explicit formula for the computation of RT (f, f̃ , g, g̃, U). The only choice made in the computation is of the admissable homotopy to (f ′, g′, V ), but any alternative choice must give the same local Reidemeister trace by the homotopy axiom. Thus all local Reidemeister traces must be computed in the same way, giving the same result, which means that there can be only one. 4 Uniqueness in fixed point theory In the special case where Y = X and g is taken to be the identity map id : X → X , the above method can be used with slight modifications to prove a uniqueness result for the local Reidemeister trace in the fixed point theory of possibly nonorientable manifolds. We have not in this paper made explicit use of the orientability hypothesis, but it is a necessary hypothesis for the uniqueness of the coincidence index in Theorem 1, which was used in Proposition 1. An accounting of orientations is needed in coincidence theory to distinguish between points of index +1 and index−1 (though see [4] for an approach to an index for nonorientable manifolds, which does not always give an integer). Orientability is not needed in local fixed point theory, since the notion of an orientation preserving selfmap is well-defined locally, even on a manifold with no global orientation. Thus the uniqueness of the fixed point index in [6] does not require orientability, and we will not require it here. Let C(X) be the set of all tuples of the form (f, f̃ , ı̃, U), where f : X → X is a selfmap, f̃ : X̃ → X̃ is a lift of f , the map ı̃ : X̃ → X̃ is a lift of the identity map, and U is an open subset of X with compact fixed point set Fix(f, U) = Coin(f, id, U). Let R[f ] = R[f, id]. Two tuples (f, f̃ , ı̃, U) and (f ′, f̃ ′, ı̃, U) are said to be admissably homotopic if there is some homotopy Ft of f to f ′ with {(x, t) | Ft(x) = x} compact, and Ft lifts to a homotopy of f̃ to f̃ ′. Our uniqueness theorem is then: Theorem 4. If X is a (possibly nonorientable) differentiable manifold, then there is a unique function taking an admissable tuple (f, f̃ , ı̃, U) to an element of ZR[f ] satisfying the following axioms: • (Additivity) If U1 and U2 are disjoint open subsets of U with Fix(f, U) ⊂ U1 ∪ U2, then RT(f, f̃ , ı̃, U) = RT(f, f̃ , ı̃, U1) + RT(f, f̃ , ı̃, U2) • (Homotopy) If (f, f̃ , ı̃, U) is admissably homotopic to (f ′, f̃ ′, ı̃, U), then RT(f, f̃ , ı̃, U) = RT(f ′, f̃ ′, ı̃, U) • (Weak normalization) If f is a constant map, then c(RT(f, f̃ , ı̃, U)) = 1 • (Lift invariance) For any α, β ∈ π1(X), we have c(RT(f, f̃ , ı̃, U)) = c(RT(f, αf̃ , βı̃, U)) • (Coincidence of lifts) If [α] appears with nonzero coefficient in RT(f, f̃ , ı̃, U), then αf̃ and ı̃ have a coincidence point on p−1X (U). Proof. First we note that a result analagous to Proposition 1 can be obtained in the fixed point setting using only the weak normalization axiom: Using the three axioms of [6], which make use of an appropriately weakened normalization axiom, we see that c ◦ RT is the fixed point index. Then letting g = id in the proof of Theorem 2, we have that, if f has isolated fixed points, RT(f, f̃ , ı̃, U) = x∈Fix(f,Ux) ind(f, Ux)[xef,eı], where ind denotes the fixed point index, and Ux is an isolating neighborhood for the fixed point x. A fixed point version of Lemma 2 can be found in Lemmas 4.1 and 3.3 of [6], and the proof of Theorem 3 can be mimicked to obtain our uniqueness result. Note that the uniqueness in fixed point theory requires only a weakened version of the normalization axiom. A uniqueness result for coincidence theory using only the weak normalization axiom can be obtained if we restrict ourselves to self-maps of a particular (not necessarily orientable) manifold. This would use a proof similar to the above, using results from Section 5 of [12]. 5 Existence The existence of a local Reidemeister trace in fixed point theory for connected finite dimensional locally compact polyhedra is established by Fares and Hart in [5]. There, the slightly more general localH-Reidemeister trace is defined, called “the local generalized H-Lefschetz number”. An extension of this paper to the mod H theory would not be difficult. The fact that the mod H Reidemeister classes are unions of ordinary Reidemeister classes allows the same results to be obtained without substantial modifications. In [5], the additivity and homotopy axioms are proved in Proposition 3.2.9 and Proposition 3.2.8, respectively. A strong version of the lift invariance ax- iom (see our Theorem 6) is proved in Proposition 3.2.4. The coincidence of lifts axiom is not stated explicitly by Fares and Hart, but is a straightforward con- sequence of their trace-like definition (if some [α] has nonzero coefficient in the Reidemeister trace, it neccesarily comes from some simplex in the covering space containing a fixed point of αf̃). A result analogous to the Wecken Trace Theo- rem (which trivially implies the normalization and weak normalization axioms) is given in Theorem 3.3.1. No Reidemeister trace for coincidence theory, either local or global, has appeared previously in the literature. The proof of Theorem 3 furnishes the appropriate definition, as follows: Given an admissable tuple (f, f̃ , g, g̃, U), we find (by Lemma 2) an admissably homotopic tuple (f ′, f̃ ′, g′, g̃′, V ) with isolated coincidence points, and we define RT (f, f̃ , g, g̃, U) = x∈Coin(f ′,g′,V ) ind(f ′, g′, Vx)[xef ′,eg′ ], where Vx is an isolating neighborhood for the coincidence point x. The above is well defined provided that it is independent of the choice of the admissably homotopic tuple. This is ensured by the following lemma: Lemma 3. If (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U) are admissably homotopic tu- ples with isolated coincidence points, then x∈Coin(f,g,U) ind(f, g, Ux)[xef,eg] = x′∈Coin(f ′,g′,U) ind(f ′, g′, Ux′)[x ef ′,eg′ where Ux is an isolating neighborhood for the coincidence point x ∈ Coin(f, g, U), and Ux′ is an isolating neighborhood of the coincidence point x ′ ∈ Coin(f ′, g′, U). Proof. We define the index of a coincidence class C of f and g as follows: indC = ind(f, g, Ux). A class is called essential if its index is nonzero. Since f and g are homotopic to f ′ and g′, we have R[f, g] = R[f ′, g′]. Call this common set of Reidemeister classes R. Letting Ũ = p−1X (U), the statement of the Lemma is equivalent to [α]∈R indC(f̃ , g̃, Ũ , [α])[α] = [α]∈R indC(f̃ ′, g̃′, Ũ , [α])[α], and we need only show that indC(f̃ , g̃, Ũ , [α]) = indC(f̃ ′, g̃′, Ũ , [α]) for any [α]. We will prove this using Brooks’s notion of homotopy-relatedness of coincidence classes, exposited in detail in [2] and briefly in [3]. Let Ft, F̃t, Gt, G̃t be homotopies realizing the admissable homotopy of (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U). Two coincidence points x ∈ Coin(f, g, U) and x′ ∈ Coin(f ′, g′, U) are (Ft, Gt)–related if there is some path γ(t) in X connecting x to x ′ such that the paths Ft(γ(t)) and Gt(γ(t)) are homotopic in Y as paths with fixed end- points. Two coincidence classes are related if at least one point of one is related to at least one point of the other. Theorem II.22 of [2] shows that the notion of (Ft, Gt)-relatedness gives a bijective correspondence between the essential coin- cidence classes of (f, g) and those of (f ′, g′). Theorem IV.24 of [2] further shows that any two such related classes will have the same index. What remains is an elementary argument using covering-space theory. Let C = C(f̃ , g̃, Ũ , [α]), and let C′ be the unique coincidence class of (f ′, g′) which is (Ft, Gt)-related to C. We need only show that C ′ = C(f̃ ′, g̃′, Ũ , [α]), and thus (since homotopy-relatedness preserves the index) that indC(f̃ , g̃, Ũ , [α]) = indC(f̃ ′, g̃′, Ũ , [α]). Choose a point x ∈ C, and let x′ be a point in C′ which is (Ft, Gt) related to x. Then there is some path γ in X from x to x′ with Ft(γ(t)) homotopic to Gt(γ(t)). Let x̃ be some point with pX(x̃) = x and αf̃(x̃) = g̃(x̃). We can lift γ to a path γ̃ in X̃ starting at x̃. Since Ft(γ(t)) is homotopic to Gt(γ(t)), we will have F̃t(γ̃(t)) homotopic to G̃t(γ̃(t)), which in particular means that they will have the same endpoint. This common endpoint is αf̃ ′(γ̃(1)) = g̃′(γ̃(1)), which must project by pX to the point x ′. Thus x′ ∈ pX(Coin(αf̃ ′, g̃′, Ũ)), and so C′ = C(f̃ ′, g̃′, Ũ , [α]), as desired. We have thus produced a meaningful definition of a local coincidence Reide- meister trace on orientable differentiable manifolds of the same dimension, and the proof above suffices to give: Theorem 5 (Wecken Coincidence Trace Theorem). Let RT be the unique lo- cal coincidence Reidemeister trace satisfying our five axioms. Then for any (f, f̃ , g, g̃, U) ∈ C(X,Y ) with Ũ = p−1X (U), we have RT(f, f̃ , g, g̃, U) = [α]∈R[f,g] indC(f̃ , g̃, Ũ , [α])[α]. In conclusion we prove a stronger form of the lift invariance axiom, a coin- cidence version of a well-known property of the Reidemeister trace. Theorem 6. Let RT be the unique local coincidence Reidemeister trace satis- fying our five axioms. If RT(f, f̃ , g, g̃, U) = [σ]∈R[f,g] k[σ][σ] for k[σ] ∈ Z, then for any α, β ∈ π1(Y ), we have RT(f, αf̃ , g, βg̃, U) = [σ]∈R[f,g] k[σ][βσα Proof. Letting Ũ = p−1 (U), by Theorem 5 we know that k[σ] = indC(f̃ , g̃, Ũ , [σ]). Then we have C(αf̃ , βg̃, [σ]) = pX Coin(σαf̃ , βg̃, Ũ) = pX Coin(β −1σαf̃ , g̃, Ũ) = C(f̃ , g̃, [β−1σα]), and thus indC(αf̃ , βg̃, Ũ , [σ]) = k[β−1σα]. Now by Theorem 5 again, we have RT(f, αf̃ , g, βg̃, Ũ) = [σ]∈R[f,g] indC(αf̃ , βg̃, [σ])[σ] = [σ]∈R[f,g] k[β−1σα][σ] [γ]∈R[f,g] k[γ][βγα as desired. References [1] M. Arkowitz and R. Brown. The Lefschetz-Hopf theorem and axioms for the Lefschetz number. Fixed Point Theory and Applications, 1:1–11, 2004. [2] R. Brooks. Coincidences, Roots, and Fixed Points. Doctoral dissertation, University of California, Los Angeles, 1967. [3] R. Brooks and R. Brown. A lower bound on the ∆-Nielsen number. Trans- actions of the American Mathematical Society, 143:555–564, 1969. [4] R. Dobreńko and J. Jezierski. The coincidence Nielsen number on nonori- entable manifolds. The Rocky Mountain Journal of Mathematics, 23:67–85, 1993. [5] J. Fares and E. Hart. A generalized Lefschetz number for local Nielsen fixed point theory. Topology and Its Applications, 59:1–23, 1994. [6] M. Furi, M. P. Pera, and M. Spadini. On the uniqueness of the fixed point index on differentiable manifolds. Fixed Point Theory and Applications, 4:251–259, 2004. [7] D. L. Gonçalves. Coincidence theory. In R.F. Brown, editor, The Handbook of Topological Fixed Point Theory, pages 3–42. Springer, 2005. [8] D. L. Gonçalves and J. Weber. Axioms for the equivariant Lefschetz num- ber and for the Reidemeister trace. Journal of Fixed Point Theory and Applications, 2:55–72, 2007. [9] S. Husseini. Generalized Lefschetz numbers. Transactions of the American Mathematical Society, 272:247–274, 1982. [10] B. Jiang. Lectures on Nielsen fixed point theory. Contemporary Mathemat- ics 14, American Mathematical Society, 1983. [11] K. Reidemeister. Automorphismen von Homotopiekettenringen. Mathema- tische Annalen, 112:586–593, 1936. [12] P. C. Staecker. On the uniqueness of the coincidence index on orientable dif- ferentiable manifolds. Topology and Its Applications, 154:1961–1970, 2007, arxiv eprint math.GN/0607751. http://arxiv.org/abs/math/0607751 [13] F. Wecken. Fixpunktklassen I, II, III. Mathematische Annalen, 117, 118:659–671, 216–234, 544–577, 1941, 1942. Introduction The Axioms Uniqueness Uniqueness in fixed point theory Existence

We give axioms which characterize the local Reidemeister trace for orientable differentiable manifolds. The local Reidemeister trace in fixed point theory is already known, and we provide both uniqueness and existence results for the local Reidemeister trace in coincidence theory.

Introduction The Reidemeister trace is a fundamental invariant in topological fixed point theory, generalizing both the Lefschetz and Nielsen numbers. It was originally defined by Reidemeister in [11]. A more modern treatment, under the name “generalized Lefschetz number,” was given by Husseini in [9]. If X is a finite connected CW-complex with universal covering space X̃ and fundamental group π, then the cellular chain complex Cq(X̃) is a free Zπ- module. If f : X → X is a cellular map and f̃ : X̃ → X̃ is a lift of f , then the induced map f̃q : Cq(X̃) → Cq(X̃) can be viewed as a matrix with entries in Zπ (with respect to some chosen Zπ basis for Cq(X̃)). We then define RT (f, f̃) = (−1)qρ(tr(f̃q)), where tr is the sum of the diagonal entries of the matrix, and ρ is the projection into the “Reidemeister classes” of π. The Reidemeister trace, then, is an element of ZR, where R is the set of Reidemeister classes. Wecken, in [13], proved what we will refer to as the Wecken Trace Theorem, that RT (f, f̃) = [α]∈R ind([α]) [α], where ind([α]) is the index of the Nielsen fixed point class associated to [α] (see e.g. [10]). Thus the number of terms appearing in the Reidemeister trace with http://arxiv.org/abs/0704.1891v2 nonzero coefficient is equal to the Nielsen number of f , and by the Lefschetz- Hopf Theorem, the sum of the coefficients is equal to the Lefschetz number of Recent work of Furi, Pera, and Spadini in [6] has given a new proof of the uniqueness of the fixed point index on orientable manifolds with respect to three natural axioms. In [12] their approach was extended to the coincidence index. The result is the following theorem: Theorem 1. Let X and Y be oriented differentiable manifolds of the same dimension. The coincidence index ind(f, g, U) of two mappings f, g : X → Y over some open set U ⊂ X is the unique integer-valued function satisfying the following axioms: • (Additivity) If U1 and U2 are disjoint open subsets of U whose union con- tains all coincidence points of f and g on U , then ind(f, g, U) = ind(f, g, U1) + ind(f, g, U2). • (Homotopy) If f and g are “admissably homotopic” to f ′ and g′, then ind(f, g, U) = ind(f ′, g′, U) • (Normalization) If L(f, g) denotes the coincidence Lefschetz number of f and g, then ind(f, g,X) = L(f, g). In the spirit of the above theorem, we demonstrate the existence and unique- ness of a local Reidemeister trace in coincidence theory subject to five axioms. A local Reidemeister trace for fixed point theory was given by Fares and Hart in [5], but no Reidemeister trace (local or otherwise) has appeared in the literature for coincidence theory. We note that recent work by Gonçalves and Weber in [8] gives axioms for the Reidemeister trace in fixed point theory using entirely different methods. Their work uses no locality properties, and is based on axioms for the Lefschetz number by Arkowitz and Brown in [1]. In Section 2 we present our axiom set, and we prove the uniqueness in coincidence theory in Section 3. In the special case of local fixed point theory, we can obtain a slightly stronger uniqueness result which we discuss in Section 4. Section 5 is a demonstration of the existence in the setting of coincidence theory. This paper contains pieces of the author’s doctoral dissertation. The author would like to thank his dissertation advisor Robert F. Brown for assistance with both the dissertation work and with this paper. The author would also like to thank Peter Wong, who guided the early dissertation work and interested him in the coincidence Reidemeister trace. 2 The Axioms Throughout the paper, unless otherwise stated, let X and Y denote connected orientable differentiable manifolds of the same dimension. All maps f, g : X → Y will be assumed to be continuous. The universal covering spaces of X and Y will be denoted X̃ and Ỹ with projection maps pX : X̃ → X and pY : Ỹ → Y . A lift of some map f : X → Y is a map f̃ : X̃ → Ỹ with pY ◦ f̃ = f ◦ pX . Let f, g : X → Y be maps, with induced homomorphisms φ, ψ : π1(X) → π1(Y ) respectively. We will view elements of π1(X) and π1(Y ) as covering transformations, so that for any x̃ ∈ X̃ and σ ∈ π1(X), we have f̃(σx̃) = φ(σ)f̃ (x̃) and g̃(σx̃) = ψ(σ)g̃(x̃). We will partition the elements of π1(Y ) into equivalence classes defined by the “doubly twisted conjugacy” relation: α ∼ β ⇐⇒ α = ψ(σ)−1βφ(σ). The equivalence classes with respect to this relation (denoted e.g. [α]) are called Reidemeister classes. The set of Reidemeister classes is denoted R[f, g]. For any set S, let ZS denote the free abelian group generated by S, whose elements we write as sums of elements of S with integer coefficients. For any such abelian group, there is a homomorphism c : ZS → Z defined as the sum of the coefficients: for si ∈ S and ki ∈ Z, and i ranging over a finite set. For some maps f, g : X → Y and an open subset U ⊂ X , let Coin(f, g, U) = {x ∈ U | f(x) = g(x)}. We say that the triple (f, g, U) is admissable if Coin(f, g, U) is compact. Two triples (f, g, U) and (f ′, g′, U) are admissably homotopic if there is some pair of homotopies Ft, Gt : X × [0, 1] → X of f, g to f ′, g′ with {(x, t) ∈ U × [0, 1] | Ft(x) = Gt(x)} compact. Let C(X,Y ) be the set of admissable tuples, all tuples of the form (f, f̃ , g, g̃, U) where f, g : X → Y are maps, (f, g, U) is an admissable triple, and f̃ and g̃ are lifts of f and g. Let (f, f̃ , g, g̃, U), (f ′, f̃ ′, g′, g̃′, U) ∈ C(X,Y ) with (f, g, U) admissably ho- motopic to (f ′, g′, U) by homotopies Ft, Gt. By the homotopy lifting property, there are unique lifted homotopies F̃t, G̃t : X̃ × [0, 1] → Ỹ with F̃0 = f̃ and G̃0 = g̃. If we additionaly have F̃1 = f̃ ′ and G̃1 = g̃ ′, then we say that the tuples (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U) are admisssably homotopic. Throughout the following, let RT be any function which to an admissable tuple (f, f̃ , g, g̃, U) ∈ C(X,Y ) associates an element of ZR[f, g]. Our first three axioms for the local Reidemeister trace are modeled after the axioms of Theorem Axiom 1 (Additivity). Given (f, f̃ , g, g̃, U) ∈ C(X,Y ), if U1 and U2 are disjoint open subsets of U with Coin(f, g, U) ⊂ U1 ∪ U2, then RT(f, f̃ , g, g̃, U) = RT(f, f̃ , g, g̃, U1) + RT(f, f̃ , g, g̃, U2). Axiom 2 (Homotopy). If (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U) are admissably ho- motopic admissable tuples, then RT(f, f̃ , g, g̃, U) = RT(f ′, f̃ ′, g′, g̃′, U). Axiom 3 (Normalization). If (f, f̃ , g, g̃, X) ∈ C(X,Y ), then c(RT(f, f̃ , g, g̃, X)) = L(f, g), where L(f, g) is the Lefschetz number of f and g. We will require one additional axiom to make some connections with Nielsen theory, based on a well-known property of the Reidemeister trace: Axiom 4 (Lift invariance). For any (f, f̃ , g, g̃, U) ∈ C(X,Y ), and any α, β ∈ π1(Y ) we have c(RT(f, f̃ , g, g̃, U)) = c(RT(f, αf̃ , g, βg̃, U)). The four axioms above are enough to demonstrate some relationships be- tween RT and the coincidence index. Proposition 1. If RT satisfies the homotopy, additivity, normalization, and lift invariance axioms, then c(RT(f, f̃ , g, g̃, U)) = ind(f, g, U) for any (f, f̃ , g, g̃, U) ∈ C(X,Y ), where ind denotes the coincidence index (see [7]). Proof. Let ω = c ◦ RT : C(X,Y ) → Z. By the lift invariance axiom, ω is independent of the choice of lifts. Thus ω can be viewed as a function from the set of all admissable triples to Z. It is clear that ω satisfies the three axioms of Theorem 1, since they are implied by our additivity, homotopy, and normalization axioms for RT (disregarding the lift parameters). Thus ω is the coincidence index. Proposition 2. If RT satisfies the additivity, homotopy, normalization, and lift invariance axioms and c(RT(f, f̃ , g, g̃, U)) 6= 0, then there is some σ ∈ π1(Y ) such that σf̃ and g̃ have a coincidence on p−1X (U). Proof. By Proposition 1, if c(RT (f, f̃ , g, g̃, U)) 6= 0 then ind(f, g, U) 6= 0, and so f and g have a coincidence on U . Let x ∈ U be this coincidence point, and choose x̃ ∈ p−1X (x). Then since f̃ and g̃ are lifts, the points f̃(x̃) and g̃(x̃) will project to the same point of Y by pY . Thus there is some covering transformation σ with σf̃(x̃) = g̃(x̃). The four axioms given above are not sufficient to uniquely characterize the Reidemeister trace in fixed point or coincidence theory. For instance, the func- tion defined by T (f, f̃ , g, g̃, U) = ind(f, g, U)[1], where [1] is the Reidemeister class of the trivial element 1 ∈ π1(Y ), satisfies all of the axioms above, but provides none of the expected data concerning R[f, g], and so that function cannot be the Reidemeister trace. An additional axiom is needed, one which somehow indicates the elements of R[f, g] which are to appear in the Reidemeister trace. Our final axiom is a sort of strengthening of Proposition 2, which specifies the Reidemeister data associated to the coincidence points. Axiom 5 (Coincidence of lifts). If [α] appears with nonzero coefficient in RT(f, f̃ , g, g̃, U), then αf̃ and g̃ have a coincidence on p−1 Any function RT which to a tuple (f, f̃ , g, g̃, U) ∈ C(X,Y ) associates an element of ZR[f, g], and satisfies the additivity, homotopy, normalization, lift invariance, and coincidence of lifts axioms we will call a local Reidemeister trace. Our main result (Theorem 3) states that there is a unique such function. 3 Uniqueness Let (f, f̃ , g, g̃, U) ∈ C(X,Y ), let Ũ = p−1 (U), and let C(f̃ , g̃, Ũ , [α]) = pX(Coin(αf̃ , g̃, Ũ)). For each α we have C(f̃ , g̃, Ũ , [α]) ⊂ Coin(f, g, U), and such coincidence sets are called coincidence classes. That these classes are well defined is a consequence of the following lemma, which appears in slightly different language as Lemma 2.3 of [4]. Lemma 1. Let α, β ∈ π1(Y ), maps f, g : X → Y , and an open subset U ⊂ X be given. Then: • [α] = [β] if and only if pX Coin(αf̃ , g̃, Ũ) = pX Coin(βf̃ , g̃, Ũ) for any lifts f̃ , g̃. • If [α] 6= [β], then pX Coin(αf̃ , g̃, Ũ) and pX Coin(αf̃ , g̃, Ũ) are disjoint for any lifts f̃ , g̃. Given the above notation, the coincidence of lifts axiom could be restated as follows: If [α] appears with nonzero coefficient in RT (f, f̃ , g, g̃, U), then C(f̃ , g̃, Ũ , [α]) is nonempty. For each coincidence point x in U , define [xef,eg] ∈ R[f, g] as that class [α] for which x ∈ C(f̃ , g̃, Ũ , [α]). Theorem 2. If RT is a local Reidemeister trace and Coin(f, g, U) is a set of isolated points, then RT(f, f̃ , g, g̃, U) = x∈Coin(f,g,U) ind(f, g, Ux)[xef,eg], where Ux is an isolating neighborhood for the coincidence point x. Proof. By the additivity property, we need only show that RT (f, f̃ , g, g̃, Ux) = ind(f, g, Ux)[xef,eg]. First, we observe that no element of R[f, g] other than [xef,eg] appears as a term with nonzero coefficient in RT (f, f̃ , g, g̃, Ux): If some [β] does appear with nonzero coefficient, then we know by the coincidence of lifts axiom that βf̃ and g̃ have a coincidence on Ũx = p (Ux). Projection of this coincidence point gives a coincidence point in Ux which necessarily must be x, since x is the only coincidence point in Ux. Thus x ∈ pX Coin(βf̃ , g̃, Ũx), which means that [β] = [xef,eg]. Since [xef,eg] is the only element of R[f, g] appearing in RT(f, f̃ , g, g̃, U), we RT(f, f̃ , g, g̃, Ux) = k[xef,eg] for some k ∈ Z (possibly k = 0). Proposition 1 says that the coefficient sum must equal the index, and so k = ind(f, g, Ux) as desired. The above is a strong result for maps whose coincidence sets are isolated. In order to leverage this result for arbitrary maps, we will make use of a technical lemma, a combination of Lemmas 13 and 15 from [12]. Lemma 2. Let (f, g, U) be an admissable triple, and let V ⊂ U be an open subset containing Coin(f, g, U) with compact closure V̄ ⊂ U . Then (f, g, V ) is admissably homotopic to an admissable triple (f ′, g′, V ), where f ′ and g′ have isolated coincidence points in V . The above lemma is used to approximate any maps by maps having isolated coincidence points, and we obtain our uniqueness theorem: Theorem 3. There is at most one local Reidemeister trace defined on C(X,Y ). Proof. Let RT be local Reidemeister trace, and take (f, f̃ , g, g̃, U) ∈ C(X,Y ). Then by Lemma 2 there is an open subset V ⊂ U with Coin(f, U) ⊂ V and maps f ′, g′ with isolated coincidence points with (f, g, V ) admissably homotopic to (f ′, g′, V ). Then by the homotopy axiom there are lifts f̃ ′, g̃′ of f and g with RT (f, f̃ , g, g̃, U) = RT (f ′, f̃ ′, g′, g̃′, V ). The coincidence points of f ′ and g′ in V are isolated, so we have RT (f, f̃ , g, g̃, U) = x∈Coin(f ′,g′,V ) ind(f ′, g′, Vx)[xef ′,eg′ ], where Vx is an isolating neighborhood of the coincidence point x. This gives an explicit formula for the computation of RT (f, f̃ , g, g̃, U). The only choice made in the computation is of the admissable homotopy to (f ′, g′, V ), but any alternative choice must give the same local Reidemeister trace by the homotopy axiom. Thus all local Reidemeister traces must be computed in the same way, giving the same result, which means that there can be only one. 4 Uniqueness in fixed point theory In the special case where Y = X and g is taken to be the identity map id : X → X , the above method can be used with slight modifications to prove a uniqueness result for the local Reidemeister trace in the fixed point theory of possibly nonorientable manifolds. We have not in this paper made explicit use of the orientability hypothesis, but it is a necessary hypothesis for the uniqueness of the coincidence index in Theorem 1, which was used in Proposition 1. An accounting of orientations is needed in coincidence theory to distinguish between points of index +1 and index−1 (though see [4] for an approach to an index for nonorientable manifolds, which does not always give an integer). Orientability is not needed in local fixed point theory, since the notion of an orientation preserving selfmap is well-defined locally, even on a manifold with no global orientation. Thus the uniqueness of the fixed point index in [6] does not require orientability, and we will not require it here. Let C(X) be the set of all tuples of the form (f, f̃ , ı̃, U), where f : X → X is a selfmap, f̃ : X̃ → X̃ is a lift of f , the map ı̃ : X̃ → X̃ is a lift of the identity map, and U is an open subset of X with compact fixed point set Fix(f, U) = Coin(f, id, U). Let R[f ] = R[f, id]. Two tuples (f, f̃ , ı̃, U) and (f ′, f̃ ′, ı̃, U) are said to be admissably homotopic if there is some homotopy Ft of f to f ′ with {(x, t) | Ft(x) = x} compact, and Ft lifts to a homotopy of f̃ to f̃ ′. Our uniqueness theorem is then: Theorem 4. If X is a (possibly nonorientable) differentiable manifold, then there is a unique function taking an admissable tuple (f, f̃ , ı̃, U) to an element of ZR[f ] satisfying the following axioms: • (Additivity) If U1 and U2 are disjoint open subsets of U with Fix(f, U) ⊂ U1 ∪ U2, then RT(f, f̃ , ı̃, U) = RT(f, f̃ , ı̃, U1) + RT(f, f̃ , ı̃, U2) • (Homotopy) If (f, f̃ , ı̃, U) is admissably homotopic to (f ′, f̃ ′, ı̃, U), then RT(f, f̃ , ı̃, U) = RT(f ′, f̃ ′, ı̃, U) • (Weak normalization) If f is a constant map, then c(RT(f, f̃ , ı̃, U)) = 1 • (Lift invariance) For any α, β ∈ π1(X), we have c(RT(f, f̃ , ı̃, U)) = c(RT(f, αf̃ , βı̃, U)) • (Coincidence of lifts) If [α] appears with nonzero coefficient in RT(f, f̃ , ı̃, U), then αf̃ and ı̃ have a coincidence point on p−1X (U). Proof. First we note that a result analagous to Proposition 1 can be obtained in the fixed point setting using only the weak normalization axiom: Using the three axioms of [6], which make use of an appropriately weakened normalization axiom, we see that c ◦ RT is the fixed point index. Then letting g = id in the proof of Theorem 2, we have that, if f has isolated fixed points, RT(f, f̃ , ı̃, U) = x∈Fix(f,Ux) ind(f, Ux)[xef,eı], where ind denotes the fixed point index, and Ux is an isolating neighborhood for the fixed point x. A fixed point version of Lemma 2 can be found in Lemmas 4.1 and 3.3 of [6], and the proof of Theorem 3 can be mimicked to obtain our uniqueness result. Note that the uniqueness in fixed point theory requires only a weakened version of the normalization axiom. A uniqueness result for coincidence theory using only the weak normalization axiom can be obtained if we restrict ourselves to self-maps of a particular (not necessarily orientable) manifold. This would use a proof similar to the above, using results from Section 5 of [12]. 5 Existence The existence of a local Reidemeister trace in fixed point theory for connected finite dimensional locally compact polyhedra is established by Fares and Hart in [5]. There, the slightly more general localH-Reidemeister trace is defined, called “the local generalized H-Lefschetz number”. An extension of this paper to the mod H theory would not be difficult. The fact that the mod H Reidemeister classes are unions of ordinary Reidemeister classes allows the same results to be obtained without substantial modifications. In [5], the additivity and homotopy axioms are proved in Proposition 3.2.9 and Proposition 3.2.8, respectively. A strong version of the lift invariance ax- iom (see our Theorem 6) is proved in Proposition 3.2.4. The coincidence of lifts axiom is not stated explicitly by Fares and Hart, but is a straightforward con- sequence of their trace-like definition (if some [α] has nonzero coefficient in the Reidemeister trace, it neccesarily comes from some simplex in the covering space containing a fixed point of αf̃). A result analogous to the Wecken Trace Theo- rem (which trivially implies the normalization and weak normalization axioms) is given in Theorem 3.3.1. No Reidemeister trace for coincidence theory, either local or global, has appeared previously in the literature. The proof of Theorem 3 furnishes the appropriate definition, as follows: Given an admissable tuple (f, f̃ , g, g̃, U), we find (by Lemma 2) an admissably homotopic tuple (f ′, f̃ ′, g′, g̃′, V ) with isolated coincidence points, and we define RT (f, f̃ , g, g̃, U) = x∈Coin(f ′,g′,V ) ind(f ′, g′, Vx)[xef ′,eg′ ], where Vx is an isolating neighborhood for the coincidence point x. The above is well defined provided that it is independent of the choice of the admissably homotopic tuple. This is ensured by the following lemma: Lemma 3. If (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U) are admissably homotopic tu- ples with isolated coincidence points, then x∈Coin(f,g,U) ind(f, g, Ux)[xef,eg] = x′∈Coin(f ′,g′,U) ind(f ′, g′, Ux′)[x ef ′,eg′ where Ux is an isolating neighborhood for the coincidence point x ∈ Coin(f, g, U), and Ux′ is an isolating neighborhood of the coincidence point x ′ ∈ Coin(f ′, g′, U). Proof. We define the index of a coincidence class C of f and g as follows: indC = ind(f, g, Ux). A class is called essential if its index is nonzero. Since f and g are homotopic to f ′ and g′, we have R[f, g] = R[f ′, g′]. Call this common set of Reidemeister classes R. Letting Ũ = p−1X (U), the statement of the Lemma is equivalent to [α]∈R indC(f̃ , g̃, Ũ , [α])[α] = [α]∈R indC(f̃ ′, g̃′, Ũ , [α])[α], and we need only show that indC(f̃ , g̃, Ũ , [α]) = indC(f̃ ′, g̃′, Ũ , [α]) for any [α]. We will prove this using Brooks’s notion of homotopy-relatedness of coincidence classes, exposited in detail in [2] and briefly in [3]. Let Ft, F̃t, Gt, G̃t be homotopies realizing the admissable homotopy of (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U). Two coincidence points x ∈ Coin(f, g, U) and x′ ∈ Coin(f ′, g′, U) are (Ft, Gt)–related if there is some path γ(t) in X connecting x to x ′ such that the paths Ft(γ(t)) and Gt(γ(t)) are homotopic in Y as paths with fixed end- points. Two coincidence classes are related if at least one point of one is related to at least one point of the other. Theorem II.22 of [2] shows that the notion of (Ft, Gt)-relatedness gives a bijective correspondence between the essential coin- cidence classes of (f, g) and those of (f ′, g′). Theorem IV.24 of [2] further shows that any two such related classes will have the same index. What remains is an elementary argument using covering-space theory. Let C = C(f̃ , g̃, Ũ , [α]), and let C′ be the unique coincidence class of (f ′, g′) which is (Ft, Gt)-related to C. We need only show that C ′ = C(f̃ ′, g̃′, Ũ , [α]), and thus (since homotopy-relatedness preserves the index) that indC(f̃ , g̃, Ũ , [α]) = indC(f̃ ′, g̃′, Ũ , [α]). Choose a point x ∈ C, and let x′ be a point in C′ which is (Ft, Gt) related to x. Then there is some path γ in X from x to x′ with Ft(γ(t)) homotopic to Gt(γ(t)). Let x̃ be some point with pX(x̃) = x and αf̃(x̃) = g̃(x̃). We can lift γ to a path γ̃ in X̃ starting at x̃. Since Ft(γ(t)) is homotopic to Gt(γ(t)), we will have F̃t(γ̃(t)) homotopic to G̃t(γ̃(t)), which in particular means that they will have the same endpoint. This common endpoint is αf̃ ′(γ̃(1)) = g̃′(γ̃(1)), which must project by pX to the point x ′. Thus x′ ∈ pX(Coin(αf̃ ′, g̃′, Ũ)), and so C′ = C(f̃ ′, g̃′, Ũ , [α]), as desired. We have thus produced a meaningful definition of a local coincidence Reide- meister trace on orientable differentiable manifolds of the same dimension, and the proof above suffices to give: Theorem 5 (Wecken Coincidence Trace Theorem). Let RT be the unique lo- cal coincidence Reidemeister trace satisfying our five axioms. Then for any (f, f̃ , g, g̃, U) ∈ C(X,Y ) with Ũ = p−1X (U), we have RT(f, f̃ , g, g̃, U) = [α]∈R[f,g] indC(f̃ , g̃, Ũ , [α])[α]. In conclusion we prove a stronger form of the lift invariance axiom, a coin- cidence version of a well-known property of the Reidemeister trace. Theorem 6. Let RT be the unique local coincidence Reidemeister trace satis- fying our five axioms. If RT(f, f̃ , g, g̃, U) = [σ]∈R[f,g] k[σ][σ] for k[σ] ∈ Z, then for any α, β ∈ π1(Y ), we have RT(f, αf̃ , g, βg̃, U) = [σ]∈R[f,g] k[σ][βσα Proof. Letting Ũ = p−1 (U), by Theorem 5 we know that k[σ] = indC(f̃ , g̃, Ũ , [σ]). Then we have C(αf̃ , βg̃, [σ]) = pX Coin(σαf̃ , βg̃, Ũ) = pX Coin(β −1σαf̃ , g̃, Ũ) = C(f̃ , g̃, [β−1σα]), and thus indC(αf̃ , βg̃, Ũ , [σ]) = k[β−1σα]. Now by Theorem 5 again, we have RT(f, αf̃ , g, βg̃, Ũ) = [σ]∈R[f,g] indC(αf̃ , βg̃, [σ])[σ] = [σ]∈R[f,g] k[β−1σα][σ] [γ]∈R[f,g] k[γ][βγα as desired. References [1] M. Arkowitz and R. Brown. The Lefschetz-Hopf theorem and axioms for the Lefschetz number. Fixed Point Theory and Applications, 1:1–11, 2004. [2] R. Brooks. Coincidences, Roots, and Fixed Points. Doctoral dissertation, University of California, Los Angeles, 1967. [3] R. Brooks and R. Brown. A lower bound on the ∆-Nielsen number. 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ON (n+ 2)-DIMENSIONAL n-LIE ALGEBRAS DONALD W. BARNES Abstract. I show that an (n + 2)-dimensional n-Lie algebra over an alge- braically closed field must have a subalgebra of codimension 1. R. Bai, X. Wang, H. An andW. Xiao [1] have been working on the classification of the 5-dimensional 3-Lie algebras over an algebraically closed field of characteristic 2. To complete their classification, they ask if such an algebra must have a subalgebra of dimension 4. The following theorem answers that question. Theorem 0.1. Let A be an (n+2)-dimensional n-Lie algebra over the algebraically closed field F . Then A has a subalgebra of codimension 1. Proof. We denote the derived algebra [A,A, . . . , A] of A by A(1). If A(1) < A, then A has a subalgebra of codimension 1 since any subspace containing A(1) is a subalgebra. Hence we may assume A(1) = A and so, that A is not nilpotent. Let H be a minimal Engel subalgebra. Then n−1 ≤ dim(H) ≤ n+1. As F is infinite, H is a Cartan subalgebra by Barnes [2, Theorem 4.3]. If dim(H) = n+1, the result holds, so we may assume dim(H) ≤ n. This implies thatH is abelian and is represented on A by commuting linear transformations. Since F is algebraically closed, they have a common eigenvector u. Thus we have [h1, h2, . . . , hn−1, u] = α(h1, h2, . . . , hn−1)u for all h1, h2, . . . , hn−1 ∈ H , where α is a linear map H ∧(n−1) → F . If dim(H) = n, then 〈H,u〉 is an (n+ 1)-dimensional subalgebra. Suppose dim(H) = n − 1, H = 〈a1, . . . , an−1〉. Let d be the inner derivation d(a1, . . . , an−1) of A. For each eigenvalue λ of d, we have the λ-component Aλ = {a ∈ A | (d− λI)n+2a = 0} of A, where I denotes the identity transformation. We have A0 = H and A is the direct sum of the components for the eigenvalues of d. Let λ1, . . . , λn be (not necessarily distinct) eigenvalues of d. Then [Aλ1 , . . . , Aλn ] ⊆ Aλ1+···+λn . Since H ⊂ A(1), we either have two eigenvalues, say α, β with sum 0 or we have α + β + γ = 0. Suppose first that α + β = 0. Suppose char(F ) 6= 2. We have eigenvectors u, v for α, β. Then 〈H,u, v〉 is an (n+1)-dimensional subalgebra. Suppose char(F ) = 2. Then we can choose u, v such that [a1, . . . , an−1, u] = αu and [a1, . . . , an−1, v] = αv + θu for some θ. Again we have that 〈H,u, v〉 is an (n+ 1)-dimensional subalgebra. Now suppose α + β + γ = 0. Suppose char(F ) 6= 2. Then α + β is not an eigenvalue of d, so [h1, . . . , hn−2, u, v] = 0 for all h1, . . . , hn−2 ∈ H . Thus 〈H,u, v〉 is an (n + 1)-dimensional subalgebra of A. Now suppose char(F ) = 2. We have 2000 Mathematics Subject Classification. Primary 17B05, 17B30. Key words and phrases. n-Lie algebras, Engel subalgebras. This work was done while the author was an Honorary Associate of the School of Mathematics and Statistics, University of Sydney. http://arxiv.org/abs/0704.1892v1 2 DONALD W. BARNES the distinct non-zero eigenvalues α, β, γ = α + β and corresponding eigenvectors u, v, w. If n = 2, 3, then H 6⊆ A(1), so we may suppose n ≥ 4. For some re-ordering of the basis ofH , we have [a1, . . . , an−3, u, v, w] 6= 0. Denote the string a1, . . . , an−3 by a . We apply the Jacobi identity to the product P = [−→a , an−2, u, [ a , an−1, v, w]]. Since [ a , an−1, v, w] ∈ 〈u〉, P = 0. But P = [−→a , [−→a , an−2, u, an−1], v, w] + [ a , an−1, [ a , an−2, u, v], w] + [−→a , an−1, v, [ a , an−2, u, w]] = [−→a , αu, v, w] + 0 + 0 since [−→a , an−2, u, v] ∈ 〈w〉 and [ a , an−2, u, w] ∈ 〈v〉. Therefore α = 0 contrary to the definition of α. Thus this case cannot arise. � References 1. R. Bai, X. Wang, H. An and W. Xiao, The classification of 5-dimensional 3-Lie algebras over the field of characteristic 2, private communication to the author. 2. D. W. Barnes, Engel subalgebras of n-Lie algebra, Acta Math. Sinica, Series B, to appear. arXiv:math.RA/0610347. 1 Little Wonga Rd, Cremorne NSW 2090 Australia E-mail address: donwb@iprimus.com.au http://arxiv.org/abs/math/0610347 References

I show that an (n+2)-dimensional n-Lie algebra over an algebraically closed field must have a subalgeba of codimension 1.

ON (n+ 2)-DIMENSIONAL n-LIE ALGEBRAS DONALD W. BARNES Abstract. I show that an (n + 2)-dimensional n-Lie algebra over an alge- braically closed field must have a subalgebra of codimension 1. R. Bai, X. Wang, H. An andW. Xiao [1] have been working on the classification of the 5-dimensional 3-Lie algebras over an algebraically closed field of characteristic 2. To complete their classification, they ask if such an algebra must have a subalgebra of dimension 4. The following theorem answers that question. Theorem 0.1. Let A be an (n+2)-dimensional n-Lie algebra over the algebraically closed field F . Then A has a subalgebra of codimension 1. Proof. We denote the derived algebra [A,A, . . . , A] of A by A(1). If A(1) < A, then A has a subalgebra of codimension 1 since any subspace containing A(1) is a subalgebra. Hence we may assume A(1) = A and so, that A is not nilpotent. Let H be a minimal Engel subalgebra. Then n−1 ≤ dim(H) ≤ n+1. As F is infinite, H is a Cartan subalgebra by Barnes [2, Theorem 4.3]. If dim(H) = n+1, the result holds, so we may assume dim(H) ≤ n. This implies thatH is abelian and is represented on A by commuting linear transformations. Since F is algebraically closed, they have a common eigenvector u. Thus we have [h1, h2, . . . , hn−1, u] = α(h1, h2, . . . , hn−1)u for all h1, h2, . . . , hn−1 ∈ H , where α is a linear map H ∧(n−1) → F . If dim(H) = n, then 〈H,u〉 is an (n+ 1)-dimensional subalgebra. Suppose dim(H) = n − 1, H = 〈a1, . . . , an−1〉. Let d be the inner derivation d(a1, . . . , an−1) of A. For each eigenvalue λ of d, we have the λ-component Aλ = {a ∈ A | (d− λI)n+2a = 0} of A, where I denotes the identity transformation. We have A0 = H and A is the direct sum of the components for the eigenvalues of d. Let λ1, . . . , λn be (not necessarily distinct) eigenvalues of d. Then [Aλ1 , . . . , Aλn ] ⊆ Aλ1+···+λn . Since H ⊂ A(1), we either have two eigenvalues, say α, β with sum 0 or we have α + β + γ = 0. Suppose first that α + β = 0. Suppose char(F ) 6= 2. We have eigenvectors u, v for α, β. Then 〈H,u, v〉 is an (n+1)-dimensional subalgebra. Suppose char(F ) = 2. Then we can choose u, v such that [a1, . . . , an−1, u] = αu and [a1, . . . , an−1, v] = αv + θu for some θ. Again we have that 〈H,u, v〉 is an (n+ 1)-dimensional subalgebra. Now suppose α + β + γ = 0. Suppose char(F ) 6= 2. Then α + β is not an eigenvalue of d, so [h1, . . . , hn−2, u, v] = 0 for all h1, . . . , hn−2 ∈ H . Thus 〈H,u, v〉 is an (n + 1)-dimensional subalgebra of A. Now suppose char(F ) = 2. We have 2000 Mathematics Subject Classification. Primary 17B05, 17B30. Key words and phrases. n-Lie algebras, Engel subalgebras. This work was done while the author was an Honorary Associate of the School of Mathematics and Statistics, University of Sydney. http://arxiv.org/abs/0704.1892v1 2 DONALD W. BARNES the distinct non-zero eigenvalues α, β, γ = α + β and corresponding eigenvectors u, v, w. If n = 2, 3, then H 6⊆ A(1), so we may suppose n ≥ 4. For some re-ordering of the basis ofH , we have [a1, . . . , an−3, u, v, w] 6= 0. Denote the string a1, . . . , an−3 by a . We apply the Jacobi identity to the product P = [−→a , an−2, u, [ a , an−1, v, w]]. Since [ a , an−1, v, w] ∈ 〈u〉, P = 0. But P = [−→a , [−→a , an−2, u, an−1], v, w] + [ a , an−1, [ a , an−2, u, v], w] + [−→a , an−1, v, [ a , an−2, u, w]] = [−→a , αu, v, w] + 0 + 0 since [−→a , an−2, u, v] ∈ 〈w〉 and [ a , an−2, u, w] ∈ 〈v〉. Therefore α = 0 contrary to the definition of α. Thus this case cannot arise. � References 1. R. Bai, X. Wang, H. An and W. Xiao, The classification of 5-dimensional 3-Lie algebras over the field of characteristic 2, private communication to the author. 2. D. W. Barnes, Engel subalgebras of n-Lie algebra, Acta Math. Sinica, Series B, to appear. arXiv:math.RA/0610347. 1 Little Wonga Rd, Cremorne NSW 2090 Australia E-mail address: donwb@iprimus.com.au http://arxiv.org/abs/math/0610347 References

Elastic properties of vanadium pentoxide aggregates and topological defects L. V. Elnikova A. I. Alikhanov Institute for Theoretical and Experimental Physics, 25, B. Cheremushkinskaya st., 117218 Moscow, Russia (Dated: September 12, 2021) We study the aqueous solution of vanadium pentoxide by using topology methods. The exper- iments by Zocher, Kaznacheev, and Dogic exhibited, that in the sol phases of V2O5 − H2O, the tactoid droplets of V2O5 can coalesce. In the magnetic field, this effect is associated with a gauge field action, viz. we consider coalescence (in the topologically more convenient term, ”junction”) of droplets as annihilation of topological defects, concerning with the tactoid geometry. We have shown, that in the magnetic field, the tactoid junction is mainly caused by non-Abelian monopoles (vortons), whereas the Abelian defects almost do not annihilate. Taking into account this annihila- tion mechanism, the estimations of time-aging of the V2O5 −H2O sols may be specified. PACS numbers: I. INTRODUCTION The tactoid sol phase of the V2O5 −H2O system has been discovered at the 20-th years of the last century by Zocher (see references in [1, 2]). At the beginning of our century, the tactoid drops (tactoids) have been in- vestigated on the optical experiments by Kaznacheev [2], Lavrentovich [3], Dogic (see [4] and references in [5]), and their coworkers. The tactoid phase is chemically classi- fied as the lyotropic inorganic nematic [1]. The tactoids coexist with the isotropic liquid phase at the mass con- centration of V2O5, amounting 0.3-2.1 percents, and un- der other standard conditions [2]. The thermodynamic parameters and pHcause the dy- namics of their formation, in particular, the junction. The tactoid geometry is evolved complicatedly (and mutually inversely) in depending on time-aging of the sols [2]. Due to the de Gennes’s theory [6], the tactoid shape stabilization is defined by competition between the elastic energy of the nematic phase, the surface energy, and the anchoring energy [2]. The minimum of the tactoid free energy provides an equilibrium shape of a droplet. The measured macroscopic elastic moduli are in a very large ratio (K3 > 100), that distinguishes V2O5 − H2O from other lyotropic liquid crystals (LC), whose typical values of K3 are in order of ten. In the magnetic field, the prolate droplets are aligned by their long axes parallel to the field. Then the special case of the junction of tactoid poles may be observed [1, Remarkably, that the sol phases of V2O5 −H2O were conditionally sorted on a shape polarity and a nematic director field [5] as of a homogeneous and a non-uniform field, and of the spherical and the bispherical [2] drops with boojums. Strikingly simultaneously, these phases have been parsed (see [5, 7, 8] and references therein) basing on the experiments by Dogic (references in [5]), performed independently of Kaznacheev. In this paper, we study the mesomorphism of the V2O5−H2O system during the tactoid junction and spec- ify the character of the mesomorphic consequence there. Our goal is to define the influence of junction onto dy- namic parameters of the sol system, including time-aging of the sols. In addition, aging of these sols in water is an applied problem of ecology, since V2O5 contains in coal impurities, generated in result of work of thermal power stations. From a topological standpoint, poles of a tactoid are the point defects, boojums. As will readily be observed, we have to do with a quantum phase transition, the anal- ogous topological singularities of two poles (each admit- ting a flux) were announced by Haldane [9] for the quan- tum Hall semiconductors. Also, there is a convenient analogy with the boojum formalism for the superfluid phases of 3He and 4He [10], however their varied topol- ogy descriptions does not allow to explain the case of the tactoid coalescence. II. FORMALISM Geometry of the droplets obeys the local nematic order parameter n, which is oriented relatively to a droplet surface (Fig. 1). The free energy functional of a tactoid in the magnetic field is summed up from the Frank elastic energy Fel and the magnetic energy Fm [2, 5]: F = Fel + Fm, (1) Fel = (∇·n)2+K2 (n·∇×n)2+K3 [n×(∇×n)]2−K24∇·[n·∇·n+n×(∇×n)]2]. The magnetic energy density has the form −χa (n ·H)2, (where χa is the anisotropy of magnetic susceptibility, and H is the magnetic field). The terms at K1, K2, and K3 elastic constants in 2 mean splay, twist, and bend deformations of a bulk ne- matic respectively, n is the coordinate dependent nematic director. The term at K24 relates to saddle-splay defor- mation mode [5]. In this continuum, the tactoid boojums http://arxiv.org/abs/0704.1893v7 FIG. 1: The director field on the tactoid surface, taken over [12]. Ri and α are the geometric parameters, γ = ( tan(α1/2 tan(α/2) 0 ≤ γ ≤ 1, the vectors ei i = ϕkazn, ξ, ηkazn denote the bispherical coordinates. were revealed by Kaznacheev [2] and by van der Schoot [5] practically identically, independently of one another. The final result of tactoid classification is the existence of four regimes of form is possible, which depend on anchor- ing between the local director and the tactoid surface, and also on the total tactoid volume [11]. Only at the week-coupled limit (γ = 0), Kaznacheev found an equi- librium shape of a tactoid [2], [13], a fortiori at Fm = 0 and without the terms of K24-s in (2). At the limit (Fig. 1), the free energy (1) is the almost non-analytical func- tion on f(α, γ) [2, 13]: 4π(sinα− α cosα) + π(3 sinα− 3α sinα− α2 sinα) + π sin3 α sin θ (cosh ηkazn + cosα) dηkazn + [sinα(20 + cosα)− 3α cosα(7 + 2 sin2 α)] (3) here θ is the parameter with the too long dependence of α, γ, ηkazn [13], the last term of (4) corresponds to the magnetic energy at γ → 1. For γ, see Fig.1. Nematic surface defects of the tactoids [12] are of the homotopic group π2(R, R̃) = P ×Q, the defects of the P group are living only at the surface (P group is the kernel of the homomorphism π1(R̃) → π1(R) and consists of integers [3]), andQ’s defects are arrived from the interior. (Here R and R̃ denote the space of degenerate states in the volume and the non-vanishing states on the surface, which are arrived from the interior, respectively). The interior may be inhabited by hedgehogs. All of these point defects keep within the exact homotopic sequence [12]: π2(R̃) −→ π2(R) −→ π1(R̃) −→ π1(R). (4) Boojums are characterized by topological charges m and n [3], which depend on a configuration of a nematic di- rector’s field. Annihilation of the boojums of the ad- jacent tactoids does not mean an influence of the rais- ing hedgehog’s (in topology, they are not arbitrary float- ing to the tactoid surface). Kurik and Lavrentovich [14] have mentioned about some strings, connecting opposite boojums via a hedgehog in nematic droplets, however, non-triviality of π1 group hampered the revealing of the droplet junction without the disclination concept. How- ever, in our case we reasonably ignore lacking disclina- tions (see the conclusions by Balachandran et al. [15]). Interaction scales are the ’dipole length’ Ldip, and the ’correlation length’ Lξ [16], which are characterized an action of the group of the order parameter. We assume Ldip is in connection with a long-axis of a tactoid. In the Cartesian coordinates (x, y, z), the director field has the configuration n = n(0, 0, 1−cosh ηkazn cos ξ cosh ηkazn−cos ξ ), where ηkazn, ξ are the bispherical coordinates [2]. Quite evidently, that tactoid system is provided by a gauge field [17] (and a field with SU(2) symmetry). Concerning an universality class of the system, take the V2O5 droplet surface as belonging to SO(3) group of ro- tations of the two-dimensional sphere (here ’tactoid’) S2 [16]. U(1) will a group of rotations around a droplet axis, which is agree closely with the magnetic phase group of 3He − A [18]. U(1)’s winding is realized of non-trivial topology of tactoids. The SO(3) and SU(2) groups are locally isomorphic (as their Lie algebras) and are connected by the homo- morphism, SO(3) ∼ SU(2)/Z2, where our Z2 is the boo- jum’s boundary condition. In our standpoint, at the bulk junction, the group SO(3)n ×U(1)2n broken down to SO(3)n−1 ×U(1)2n−1, where n is a number of tactoids. A model of the sol should involve the monopole solu- tions, according to the theorem [16] about requirement of their existence (π2(G/H) −→ π1(H)). On the other hand, inasmuch as π1(H) = Z ⊗ Z ⊗ Z ⊗ ...⊗ Z︸ ︷︷ ︸ , the V2O5 − H2O sols are of the group G. The tactoid annihilation may be described ei- ther by non-Abelian or Abelian theory in depending on the global field SU(2). Besides, we have to expect ap- pearance of a compensative vector field [19]. Here, an each tactoid, in correspondence to two poles (boojums) on a tactoid surface, may contain two vortons with their tails (the wide and ”over-Witten’s” definition for vortons see in [20, 21, 22], this is a kind of monopoles with the definite pair of topological charges, vortex and azimuthal windings). Just as vortices, they appear, if the order parameter has extra degrees of freedom besides of the overall phase [23, 24]. In the tactoid free energy, the terms of twisted deformations [8] may play a role in these excitations. By introducing a necessary parametrization, the free energy equation, analogous to [2], was proposed in [8], where the free parameters permit to be the non- commutative relations in the droplet symmetry. Let us note, that we use the factor-space CP 1 in accordance to a chiral (gauge) field (2) [17]. Though, due to the electromagnetic (no topological) reasons, the sol tactoids can survive coalescence owing to the Coulomb attraction in water. But from topology [14], we do not yet know about appearance of a physical field from the configuration of defects. We have to note, that because of in-homogeneity of a system, we have a wide class of string models for a prototype. III. ANNIHILATION OF TOPOLOGICAL DEFECTS So, a junction of droplets means, that the surface point defect (boojum) configuration may be unstable (γ 6= 0). We discuss the Abelian and non-Abelian string config- urations [18, 25, 26, 28, 29], which support the sols of tactoid nematics. Their combinations and interactions are expected to define of the junction of tactoids. A. Abelian space The Abelian character of pair boojums and monopoles, and also their integer charge were proven [10]. Boojums of charge N = ±1 live at Lξ ≪ L ≪ Ldip [16, 20]. But from the surface field phenomenology [2] of a solitary tac- toid, one can not define a flux number k [16], concerning an each boojum, only what k = 1 is preferable for their pairing configuration, and k = 2 for a unit singularity. In this scenario, annihilation of charge-opposite (topo- logical) ’particles’ is possible. Abelian monopoles may be associated with locations of boojums, but, due to the topological properties of our G, we ignore them. Let us consider only vortons of the Abelian gauge. They are unstable [26], and appear to- gether with the neutral strings. The open question is which velocity will greater: of the tactoid coalescence or the vorton decay. In the U(1) gauge, the loop-radius dependent criterion of the vorton stability was found and analyzed numer- ically in the case of the potential expressed in the el- liptic ansatz [27], as well as in the well-known Witten’s U(1) × U(1) case (see review [21), that is an analogous phase transition from U(1)×U(1) to U(1) for two neigh- bouring randomly oriented tactoids, in absent of mag- netic field. B. Non-Abelian space Usual Lagrangians of non-Abelian theories are often linearized into the Bogomolny-Prasad-Sommerfeld (BPS) equations [30]. A number of applications corresponding to similar strings were considered, for example, in [15, 21, 22, 24, 25, 28, 29, 31, 32]. In the phase diagram [8], the regions of twist states were indicated. If the tactoid junction carry out there, for spherical and prolate droplets, one may make an analogy between the non-Abelian vortons and ”rotation” of the nematic order parameter, in spite of the ansatz (α(η) = α0 sin η [8]) condition, labeled one of the topo- logical invariants. Let us formulate the string model with the boson La- grangian density (due to [21, 22]) L = −1 µν − 1 µν −Dµ~φ† ·Dµ~φ− V (~φ). (5) Fµν = ∂µAν − ∂νAµ (6) are the Abelian field strengths. The global curvature is Gµν = ∂µHν − ∂νHµ + gHµ ×Hν . (7) The gauge covariant derivatives of vacuums are: Dµ(~φ) = ∂µ~φ− ieAµ~φ+ gHµ × ~φ, (8) In the formulas (5) - (8), µ and ν are indices of the gauge field A and of the metrics g. Hµ and φ are the three- dimensional vectors in the SU(2) Lie algebra. The field potential V (φ) is expressing from (2). Due to [13] x = a sin ξ cosϕ cosh η − cos ξ , y = a sin ξ sinϕ cosh η − cos ξ , z = a sinh η coshη − cos ξ the bulk elastic energy [13] of a tactoid equals to dηkazn 4K1 sinh 2 ηkazn sin ξ +K3 sin (cosh ηkazn − cos ξ)3 The corresponding vector potential is V (~φ) = λ+ (~φ+ · ~φ− 1 ζ2)2 + k|~φ · ~φ|2. (11) At the parameter k > 0, the vacuum is characterize by ~φ · ~φ, ~φ† · ~φ = 1 − sin ξ sinh ηkazn cosϕkazn cosh ηkazn−cos ξ − sin ξ sinh ηkazn sinϕkazn cosh ηkazn−cos ξ 1−coshkazn cos ξ cosh ηkazn−cos ξ  . (12) The generators of SU(2) are denoted as Ti=1,2,3. T0 is the generator of U(1). −iT1(~φj) = −ǫijk~φj , −iT0(~φ)j = −~φj . Q = T2 + T0 is the annihilation condition. The string generator (TS = T3) does not commutate with the charge generator: [TS , Q] = [T3, T2] = −iT1 [22]. Here ~φ(α) = e−iαT3 ~φ are also meaning the generators. Between the vacuums, the angular dependence is estab- lished Q(θ) = e−iθTsQeiθTs [29]. Tactoid vortices revolve SU(2). R, α are introduced to describe the tactoid ge- ometry (Fig.1). Further, we need to solve the next equations of motion: gFµν = jα = je[~φ† ×Dν ~φ− ~φ · (Dν ~φ)†], (13) gGµν = Jα = g[~φ† ×Dν ~φ+ ~φ× (Dν ~φ)†], (14) g~φ = . (15) To confirm the existence of vortons, labeled by vac- uum, and estimate the energy T2, the first-order Bogo- mol’ny’s equations are usually applied. the first-order Bogomolny’s equations are usually applied. For example, in the sigma-model limit of the Lagrangian of the type (5), the non-Abelian votrons with the (1, 1)-, (1, 2)- and other pairs of winding numbers in SU(2) were numeri- cally revealed by Radu and Volkov [21] just lately; to be solvable, their model has included four free parameters in the potential (Fig. 2). There was numerically proven with help of Gauss- Tschebuchev algorithm, that in the U(1) gauge, the sta- ble vortons may appear [27], whereas in SO(3) it is not so [26]. The stability criterion includes the radius R of the vortex loop, which may be compared with the Kaznacheev-van der Schoot theoretical analysis [2], [7], and with the lattice Monte Carlo simulations, performed by Bates [11]. IV. DYNAMICS AND ESTIMATIONS FOR TIME-AGING OF THE TACTOID SOLS Along with these assertions on the configurations sup- plied with Non-Abelian gauge fields, the approximate methods of analysis exist for quite attainable numer- ical simulations of vorton states. One of there is so called Abelian projection [33]. So, following the Maxi- mal Abelian (MaA) projection approach, we fix SU(2) gauge and leave the winding group U(1) unfixed. In ap- plied numerical tasks, Abelian approximations of (11) are yet acceptable. For example, whether is an analogous Abelian projec- tion of the V2O5 −H2O tactoid configuration realized in the 2D ferromagnetic systems and thin films [34], if there FIG. 2: The energy density of n=m=1 vortons [21], plotted by Radu and Volkov numerically at four free parameters, where z and ? are the polar coordinates. FIG. 3: Competition between the magnetic Φm and the elastic and the surface energies Φel +Φs of a tactoid, divided by σ = 10−3 erg/cm, errors are not indicated; as this is a qualitative view of (3) and experiments [2], [12]; the data at γ → 1, from which α ≈ 32o. are defined the same topological invariants? This simpli- fication is useful to estimate the case of annihilating par- ticles with whole unit opposite charges [3]. One may ex- press the vorton dynamics by the Landau-Lifshitz equa- tion (LLE), including dissipation (labeled by the constant ad [34]), and write: = m× f− αdm× (m× f), f ≡ △m−Qmzêz, (16) where m is the magnetization vector, Q is the free pa- rameter, êz is the unit vector in the z magnetization direction. According to the definition [3], the topologi- cal invariant N connected with the topological density n is N = 1 nǫµνr 3, ǫµν is the asymmetric tensor with (µ, ν) = (1, 2), V is a tactoid volume, and the vector r denotes its space. The magnetic field stretches large tactoids (a in- creases), whereas to annihilate, the tactoid shape should become more oblate [12], Fig. 3. Therefore, the equilib- rium angle a, corresponding to the large tactoid shape, exists also for coalescence in the magnetic field. To de- fine a is not difficult from the next simple algebra with (1) and (2), by using the definitions [3, 10-11]. From (1) - (3) and (12), dynamics characteristics of a solitary tactoid may be expressed as: (χH)4 )2 ∼ α2d( 1− cosh ηkazn cos ξ cosh ηkazn − cos ξ )2. (17) E. g. stretch of a tactoid in z-axis direction increases its magnetic energy, and the magnetic field is precipitating for annihilation of droplets, as a free volume decreases. On the experiment [14], the next parameters are mea- sured: Ci = , i = 1, 3, Ki are modulii of (2), and σ is the surface tension. C3-s order is hundreds micrometers. For C1-s, these are about unit. Both of they are drop- down with time, but according to (1)-(2), have not affect on the magnetic term. V. CONCLUSIONS We composed the topological classification of sols V2O5 − H2O, owing to which, the qualitative practical predictions for thermodynamic states of these sols may be performed. The cosmological theory of superconductive strings supposes that the nematic tactoids in V2O5−H2O annihilate in accordance with non-Abelian statistics. This process, carried out in magnetic field, increases a time-aging of the sols, but does not yield to direct exact estimations, since its nature is principally Non-Abelian. One may connect an actual electromagnetic interaction in the V2O5 − H2O solution via pH value and discuss questions on the tactoid junction in frames of chemistry, which we have wittingly ignored in favor of the impor- tant topological role. The process of tactoid junction in magnetic field leads to rise of the additional electromag- netic field changing pH of water around tactoids and, for one’s part, time-aging [2, 13]. These observations may be important for ecology, as long as vanadium pentox- ide is contained in impurities of coal soles, which are the components of wastes of thermoelectric power stations and are included in the impurity parameters at the back- ground control for radiation. [1] A. S. Sonin, J. Mater. Chem. 8, 2557 (1998). [2] A. V. Kaznacheev, M. M. Bogdanov, and S. A. Taraskin JETP, 95, 57 (2002). [3] G. E. Volovik and O. D. Lavrentovich, Sov. Phys. JETP 58, 1159(1983). [4] E. Barry, Z. Hensel, and Z. Dogic, Phys. Rev. L. 96, 018305 (2006). [5] P. Prinsen and P. van der Schoot, Phys. Rev. E. 68, 021701 (2003). [6] P. G. de Gennes, The physics of liquid crystals, Caledon Press, Oxford (1974). [7] P. Prinsen and P. van der Schoot, Eur. Phys. J. E, 13, 35 (2004). [8] P. Prinsen and P. van der Schoot, J. Phys.: Condens. Matter, 16, 8835 (2004). [9] F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983). [10] G. E. Volovik and V. P. Mineev, Zh. Eksp. Teor. Fiz. 83, 1025 (1982) (in Russian). [11] M. A. Bates, Chem. Phys. Lett. 368, 87 (2003). [12] G. E. Volovik, Pis’ma v Zh. Eksp. Teor. Fiz. 28, 65 (1978) (in Russian). [13] A. V. Kaznacheev, M. M. Bogdanov, and A. S. Sonin, JETP, 97, 1159 (2003). [14] M. V. Kurik and O. D. Lavrentovich, UFN, 154 381 (1988) (in Russian). [15] A. P. Balachandran, F. Lizzi, and V. G. J. Rodgers, Phys. Rev. Lett. 52, 1818 (1994). [16] M. Monastyrsky, Topology of Gauge Fields and Con- densed Matter, Springer (1993). [17] A. M. Polyakov, Gauge Fields and Strings, Harwood Aca- demic Publishers, Chur, Switzerald (1987). [18] F. A. Bais and A. M. J. Schakel, J. Phys.: Condens. Matter 2, 5053-5064 (1990). [19] N. N. Bogolubov and D. V. Shirkov, Quantum Fields, Moscow, Nauka (1993). [20] S. Blaha, Phys. Rev. Lett. 36, 873 (1976); A. Saupe, Mol. Cryst. 21, 211 (1973). [21] E. Radu, and M. S. Volkov, E-print ArXiv, hep- th/0804.1357. [22] T. W. B. Kibble, G. Lozano, and A. J. Yates, Phys. Rev. D. 56, 1204 (1997). [23] D. Ivanov, Phys. Rev. Lett, 86, 268 (2001). [24] P. Mc Graw, Phys. Rev. D. 50, 952 (1994). [25] F. A. Bais and C. J. M. Mathy, Annals of Physics 322, 709 (2007). [26] J. R. Morris, Il Nuovo Cimento A 106, 355 (1993). [27] Y. Lemperiere and E. P. S. Shellard, Phys. Rev. Lett. 91, 141601-1 (2003). [28] M. Shifman and A. Yung, Phys. Rev. D. 70, 045004 (2004). [29] M. Oshikawa, Y. B. Kim, K. Shtengel, C. Nayak, and S. Tewari, Ann. Phys. 322, 1477 (2007). [30] E. B. Bogomolny, Sov. J. Nucl. Phys. 24 449 (1976). [31] K. Janich, H. R. Trebin, 1981 in Physics of Defects (Am- sterdam: North Holland Publ.) [32] H.-K. Lo and J. Preskill, Phys. Rev. D. 48, 4821 (1993). [33] M. N. Chernodub and F. V. Gubarev, JETP Lett. 62 (1995) 100; B. L. G. Bakker, M. N. Chernodub, and M. I. Polikarpov, Phys. Rev. Lett. 80 (1998) 30. [34] S. Komineas, Phys. Rev. Lett. 99 (2007) 117202.

We study the aqueous solution of vanadium pentoxide by using topology methods. The experiments by Zocher, Kaznacheev, and Dogic exhibited, that in the sol phases of $V_2O_5-H_2O$, the tactoid droplets of $V_2O_5$ can coalesce. In the magnetic field, this effect is associated with a gauge field action, viz. we consider coalescence (in the topologically more convenient term, "junction") of droplets as annihilation of topological defects, concerning with the tactoid geometry. We have shown, that in the magnetic field, the tactoid junction is mainly caused by non-Abelian monopoles (vortons), whereas the Abelian defects almost do not annihilate. Taking into account this annihilation mechanism, the estimations of time-aging of the $V_2O_5-H_2O$ sols may be specified

Elastic properties of vanadium pentoxide aggregates and topological defects L. V. Elnikova A. I. Alikhanov Institute for Theoretical and Experimental Physics, 25, B. Cheremushkinskaya st., 117218 Moscow, Russia (Dated: September 12, 2021) We study the aqueous solution of vanadium pentoxide by using topology methods. The exper- iments by Zocher, Kaznacheev, and Dogic exhibited, that in the sol phases of V2O5 − H2O, the tactoid droplets of V2O5 can coalesce. In the magnetic field, this effect is associated with a gauge field action, viz. we consider coalescence (in the topologically more convenient term, ”junction”) of droplets as annihilation of topological defects, concerning with the tactoid geometry. We have shown, that in the magnetic field, the tactoid junction is mainly caused by non-Abelian monopoles (vortons), whereas the Abelian defects almost do not annihilate. Taking into account this annihila- tion mechanism, the estimations of time-aging of the V2O5 −H2O sols may be specified. PACS numbers: I. INTRODUCTION The tactoid sol phase of the V2O5 −H2O system has been discovered at the 20-th years of the last century by Zocher (see references in [1, 2]). At the beginning of our century, the tactoid drops (tactoids) have been in- vestigated on the optical experiments by Kaznacheev [2], Lavrentovich [3], Dogic (see [4] and references in [5]), and their coworkers. The tactoid phase is chemically classi- fied as the lyotropic inorganic nematic [1]. The tactoids coexist with the isotropic liquid phase at the mass con- centration of V2O5, amounting 0.3-2.1 percents, and un- der other standard conditions [2]. The thermodynamic parameters and pHcause the dy- namics of their formation, in particular, the junction. The tactoid geometry is evolved complicatedly (and mutually inversely) in depending on time-aging of the sols [2]. Due to the de Gennes’s theory [6], the tactoid shape stabilization is defined by competition between the elastic energy of the nematic phase, the surface energy, and the anchoring energy [2]. The minimum of the tactoid free energy provides an equilibrium shape of a droplet. The measured macroscopic elastic moduli are in a very large ratio (K3 > 100), that distinguishes V2O5 − H2O from other lyotropic liquid crystals (LC), whose typical values of K3 are in order of ten. In the magnetic field, the prolate droplets are aligned by their long axes parallel to the field. Then the special case of the junction of tactoid poles may be observed [1, Remarkably, that the sol phases of V2O5 −H2O were conditionally sorted on a shape polarity and a nematic director field [5] as of a homogeneous and a non-uniform field, and of the spherical and the bispherical [2] drops with boojums. Strikingly simultaneously, these phases have been parsed (see [5, 7, 8] and references therein) basing on the experiments by Dogic (references in [5]), performed independently of Kaznacheev. In this paper, we study the mesomorphism of the V2O5−H2O system during the tactoid junction and spec- ify the character of the mesomorphic consequence there. Our goal is to define the influence of junction onto dy- namic parameters of the sol system, including time-aging of the sols. In addition, aging of these sols in water is an applied problem of ecology, since V2O5 contains in coal impurities, generated in result of work of thermal power stations. From a topological standpoint, poles of a tactoid are the point defects, boojums. As will readily be observed, we have to do with a quantum phase transition, the anal- ogous topological singularities of two poles (each admit- ting a flux) were announced by Haldane [9] for the quan- tum Hall semiconductors. Also, there is a convenient analogy with the boojum formalism for the superfluid phases of 3He and 4He [10], however their varied topol- ogy descriptions does not allow to explain the case of the tactoid coalescence. II. FORMALISM Geometry of the droplets obeys the local nematic order parameter n, which is oriented relatively to a droplet surface (Fig. 1). The free energy functional of a tactoid in the magnetic field is summed up from the Frank elastic energy Fel and the magnetic energy Fm [2, 5]: F = Fel + Fm, (1) Fel = (∇·n)2+K2 (n·∇×n)2+K3 [n×(∇×n)]2−K24∇·[n·∇·n+n×(∇×n)]2]. The magnetic energy density has the form −χa (n ·H)2, (where χa is the anisotropy of magnetic susceptibility, and H is the magnetic field). The terms at K1, K2, and K3 elastic constants in 2 mean splay, twist, and bend deformations of a bulk ne- matic respectively, n is the coordinate dependent nematic director. The term at K24 relates to saddle-splay defor- mation mode [5]. In this continuum, the tactoid boojums http://arxiv.org/abs/0704.1893v7 FIG. 1: The director field on the tactoid surface, taken over [12]. Ri and α are the geometric parameters, γ = ( tan(α1/2 tan(α/2) 0 ≤ γ ≤ 1, the vectors ei i = ϕkazn, ξ, ηkazn denote the bispherical coordinates. were revealed by Kaznacheev [2] and by van der Schoot [5] practically identically, independently of one another. The final result of tactoid classification is the existence of four regimes of form is possible, which depend on anchor- ing between the local director and the tactoid surface, and also on the total tactoid volume [11]. Only at the week-coupled limit (γ = 0), Kaznacheev found an equi- librium shape of a tactoid [2], [13], a fortiori at Fm = 0 and without the terms of K24-s in (2). At the limit (Fig. 1), the free energy (1) is the almost non-analytical func- tion on f(α, γ) [2, 13]: 4π(sinα− α cosα) + π(3 sinα− 3α sinα− α2 sinα) + π sin3 α sin θ (cosh ηkazn + cosα) dηkazn + [sinα(20 + cosα)− 3α cosα(7 + 2 sin2 α)] (3) here θ is the parameter with the too long dependence of α, γ, ηkazn [13], the last term of (4) corresponds to the magnetic energy at γ → 1. For γ, see Fig.1. Nematic surface defects of the tactoids [12] are of the homotopic group π2(R, R̃) = P ×Q, the defects of the P group are living only at the surface (P group is the kernel of the homomorphism π1(R̃) → π1(R) and consists of integers [3]), andQ’s defects are arrived from the interior. (Here R and R̃ denote the space of degenerate states in the volume and the non-vanishing states on the surface, which are arrived from the interior, respectively). The interior may be inhabited by hedgehogs. All of these point defects keep within the exact homotopic sequence [12]: π2(R̃) −→ π2(R) −→ π1(R̃) −→ π1(R). (4) Boojums are characterized by topological charges m and n [3], which depend on a configuration of a nematic di- rector’s field. Annihilation of the boojums of the ad- jacent tactoids does not mean an influence of the rais- ing hedgehog’s (in topology, they are not arbitrary float- ing to the tactoid surface). Kurik and Lavrentovich [14] have mentioned about some strings, connecting opposite boojums via a hedgehog in nematic droplets, however, non-triviality of π1 group hampered the revealing of the droplet junction without the disclination concept. How- ever, in our case we reasonably ignore lacking disclina- tions (see the conclusions by Balachandran et al. [15]). Interaction scales are the ’dipole length’ Ldip, and the ’correlation length’ Lξ [16], which are characterized an action of the group of the order parameter. We assume Ldip is in connection with a long-axis of a tactoid. In the Cartesian coordinates (x, y, z), the director field has the configuration n = n(0, 0, 1−cosh ηkazn cos ξ cosh ηkazn−cos ξ ), where ηkazn, ξ are the bispherical coordinates [2]. Quite evidently, that tactoid system is provided by a gauge field [17] (and a field with SU(2) symmetry). Concerning an universality class of the system, take the V2O5 droplet surface as belonging to SO(3) group of ro- tations of the two-dimensional sphere (here ’tactoid’) S2 [16]. U(1) will a group of rotations around a droplet axis, which is agree closely with the magnetic phase group of 3He − A [18]. U(1)’s winding is realized of non-trivial topology of tactoids. The SO(3) and SU(2) groups are locally isomorphic (as their Lie algebras) and are connected by the homo- morphism, SO(3) ∼ SU(2)/Z2, where our Z2 is the boo- jum’s boundary condition. In our standpoint, at the bulk junction, the group SO(3)n ×U(1)2n broken down to SO(3)n−1 ×U(1)2n−1, where n is a number of tactoids. A model of the sol should involve the monopole solu- tions, according to the theorem [16] about requirement of their existence (π2(G/H) −→ π1(H)). On the other hand, inasmuch as π1(H) = Z ⊗ Z ⊗ Z ⊗ ...⊗ Z︸ ︷︷ ︸ , the V2O5 − H2O sols are of the group G. The tactoid annihilation may be described ei- ther by non-Abelian or Abelian theory in depending on the global field SU(2). Besides, we have to expect ap- pearance of a compensative vector field [19]. Here, an each tactoid, in correspondence to two poles (boojums) on a tactoid surface, may contain two vortons with their tails (the wide and ”over-Witten’s” definition for vortons see in [20, 21, 22], this is a kind of monopoles with the definite pair of topological charges, vortex and azimuthal windings). Just as vortices, they appear, if the order parameter has extra degrees of freedom besides of the overall phase [23, 24]. In the tactoid free energy, the terms of twisted deformations [8] may play a role in these excitations. By introducing a necessary parametrization, the free energy equation, analogous to [2], was proposed in [8], where the free parameters permit to be the non- commutative relations in the droplet symmetry. Let us note, that we use the factor-space CP 1 in accordance to a chiral (gauge) field (2) [17]. Though, due to the electromagnetic (no topological) reasons, the sol tactoids can survive coalescence owing to the Coulomb attraction in water. But from topology [14], we do not yet know about appearance of a physical field from the configuration of defects. We have to note, that because of in-homogeneity of a system, we have a wide class of string models for a prototype. III. ANNIHILATION OF TOPOLOGICAL DEFECTS So, a junction of droplets means, that the surface point defect (boojum) configuration may be unstable (γ 6= 0). We discuss the Abelian and non-Abelian string config- urations [18, 25, 26, 28, 29], which support the sols of tactoid nematics. Their combinations and interactions are expected to define of the junction of tactoids. A. Abelian space The Abelian character of pair boojums and monopoles, and also their integer charge were proven [10]. Boojums of charge N = ±1 live at Lξ ≪ L ≪ Ldip [16, 20]. But from the surface field phenomenology [2] of a solitary tac- toid, one can not define a flux number k [16], concerning an each boojum, only what k = 1 is preferable for their pairing configuration, and k = 2 for a unit singularity. In this scenario, annihilation of charge-opposite (topo- logical) ’particles’ is possible. Abelian monopoles may be associated with locations of boojums, but, due to the topological properties of our G, we ignore them. Let us consider only vortons of the Abelian gauge. They are unstable [26], and appear to- gether with the neutral strings. The open question is which velocity will greater: of the tactoid coalescence or the vorton decay. In the U(1) gauge, the loop-radius dependent criterion of the vorton stability was found and analyzed numer- ically in the case of the potential expressed in the el- liptic ansatz [27], as well as in the well-known Witten’s U(1) × U(1) case (see review [21), that is an analogous phase transition from U(1)×U(1) to U(1) for two neigh- bouring randomly oriented tactoids, in absent of mag- netic field. B. Non-Abelian space Usual Lagrangians of non-Abelian theories are often linearized into the Bogomolny-Prasad-Sommerfeld (BPS) equations [30]. A number of applications corresponding to similar strings were considered, for example, in [15, 21, 22, 24, 25, 28, 29, 31, 32]. In the phase diagram [8], the regions of twist states were indicated. If the tactoid junction carry out there, for spherical and prolate droplets, one may make an analogy between the non-Abelian vortons and ”rotation” of the nematic order parameter, in spite of the ansatz (α(η) = α0 sin η [8]) condition, labeled one of the topo- logical invariants. Let us formulate the string model with the boson La- grangian density (due to [21, 22]) L = −1 µν − 1 µν −Dµ~φ† ·Dµ~φ− V (~φ). (5) Fµν = ∂µAν − ∂νAµ (6) are the Abelian field strengths. The global curvature is Gµν = ∂µHν − ∂νHµ + gHµ ×Hν . (7) The gauge covariant derivatives of vacuums are: Dµ(~φ) = ∂µ~φ− ieAµ~φ+ gHµ × ~φ, (8) In the formulas (5) - (8), µ and ν are indices of the gauge field A and of the metrics g. Hµ and φ are the three- dimensional vectors in the SU(2) Lie algebra. The field potential V (φ) is expressing from (2). Due to [13] x = a sin ξ cosϕ cosh η − cos ξ , y = a sin ξ sinϕ cosh η − cos ξ , z = a sinh η coshη − cos ξ the bulk elastic energy [13] of a tactoid equals to dηkazn 4K1 sinh 2 ηkazn sin ξ +K3 sin (cosh ηkazn − cos ξ)3 The corresponding vector potential is V (~φ) = λ+ (~φ+ · ~φ− 1 ζ2)2 + k|~φ · ~φ|2. (11) At the parameter k > 0, the vacuum is characterize by ~φ · ~φ, ~φ† · ~φ = 1 − sin ξ sinh ηkazn cosϕkazn cosh ηkazn−cos ξ − sin ξ sinh ηkazn sinϕkazn cosh ηkazn−cos ξ 1−coshkazn cos ξ cosh ηkazn−cos ξ  . (12) The generators of SU(2) are denoted as Ti=1,2,3. T0 is the generator of U(1). −iT1(~φj) = −ǫijk~φj , −iT0(~φ)j = −~φj . Q = T2 + T0 is the annihilation condition. The string generator (TS = T3) does not commutate with the charge generator: [TS , Q] = [T3, T2] = −iT1 [22]. Here ~φ(α) = e−iαT3 ~φ are also meaning the generators. Between the vacuums, the angular dependence is estab- lished Q(θ) = e−iθTsQeiθTs [29]. Tactoid vortices revolve SU(2). R, α are introduced to describe the tactoid ge- ometry (Fig.1). Further, we need to solve the next equations of motion: gFµν = jα = je[~φ† ×Dν ~φ− ~φ · (Dν ~φ)†], (13) gGµν = Jα = g[~φ† ×Dν ~φ+ ~φ× (Dν ~φ)†], (14) g~φ = . (15) To confirm the existence of vortons, labeled by vac- uum, and estimate the energy T2, the first-order Bogo- mol’ny’s equations are usually applied. the first-order Bogomolny’s equations are usually applied. For example, in the sigma-model limit of the Lagrangian of the type (5), the non-Abelian votrons with the (1, 1)-, (1, 2)- and other pairs of winding numbers in SU(2) were numeri- cally revealed by Radu and Volkov [21] just lately; to be solvable, their model has included four free parameters in the potential (Fig. 2). There was numerically proven with help of Gauss- Tschebuchev algorithm, that in the U(1) gauge, the sta- ble vortons may appear [27], whereas in SO(3) it is not so [26]. The stability criterion includes the radius R of the vortex loop, which may be compared with the Kaznacheev-van der Schoot theoretical analysis [2], [7], and with the lattice Monte Carlo simulations, performed by Bates [11]. IV. DYNAMICS AND ESTIMATIONS FOR TIME-AGING OF THE TACTOID SOLS Along with these assertions on the configurations sup- plied with Non-Abelian gauge fields, the approximate methods of analysis exist for quite attainable numer- ical simulations of vorton states. One of there is so called Abelian projection [33]. So, following the Maxi- mal Abelian (MaA) projection approach, we fix SU(2) gauge and leave the winding group U(1) unfixed. In ap- plied numerical tasks, Abelian approximations of (11) are yet acceptable. For example, whether is an analogous Abelian projec- tion of the V2O5 −H2O tactoid configuration realized in the 2D ferromagnetic systems and thin films [34], if there FIG. 2: The energy density of n=m=1 vortons [21], plotted by Radu and Volkov numerically at four free parameters, where z and ? are the polar coordinates. FIG. 3: Competition between the magnetic Φm and the elastic and the surface energies Φel +Φs of a tactoid, divided by σ = 10−3 erg/cm, errors are not indicated; as this is a qualitative view of (3) and experiments [2], [12]; the data at γ → 1, from which α ≈ 32o. are defined the same topological invariants? This simpli- fication is useful to estimate the case of annihilating par- ticles with whole unit opposite charges [3]. One may ex- press the vorton dynamics by the Landau-Lifshitz equa- tion (LLE), including dissipation (labeled by the constant ad [34]), and write: = m× f− αdm× (m× f), f ≡ △m−Qmzêz, (16) where m is the magnetization vector, Q is the free pa- rameter, êz is the unit vector in the z magnetization direction. According to the definition [3], the topologi- cal invariant N connected with the topological density n is N = 1 nǫµνr 3, ǫµν is the asymmetric tensor with (µ, ν) = (1, 2), V is a tactoid volume, and the vector r denotes its space. The magnetic field stretches large tactoids (a in- creases), whereas to annihilate, the tactoid shape should become more oblate [12], Fig. 3. Therefore, the equilib- rium angle a, corresponding to the large tactoid shape, exists also for coalescence in the magnetic field. To de- fine a is not difficult from the next simple algebra with (1) and (2), by using the definitions [3, 10-11]. From (1) - (3) and (12), dynamics characteristics of a solitary tactoid may be expressed as: (χH)4 )2 ∼ α2d( 1− cosh ηkazn cos ξ cosh ηkazn − cos ξ )2. (17) E. g. stretch of a tactoid in z-axis direction increases its magnetic energy, and the magnetic field is precipitating for annihilation of droplets, as a free volume decreases. On the experiment [14], the next parameters are mea- sured: Ci = , i = 1, 3, Ki are modulii of (2), and σ is the surface tension. C3-s order is hundreds micrometers. For C1-s, these are about unit. Both of they are drop- down with time, but according to (1)-(2), have not affect on the magnetic term. V. CONCLUSIONS We composed the topological classification of sols V2O5 − H2O, owing to which, the qualitative practical predictions for thermodynamic states of these sols may be performed. The cosmological theory of superconductive strings supposes that the nematic tactoids in V2O5−H2O annihilate in accordance with non-Abelian statistics. This process, carried out in magnetic field, increases a time-aging of the sols, but does not yield to direct exact estimations, since its nature is principally Non-Abelian. One may connect an actual electromagnetic interaction in the V2O5 − H2O solution via pH value and discuss questions on the tactoid junction in frames of chemistry, which we have wittingly ignored in favor of the impor- tant topological role. The process of tactoid junction in magnetic field leads to rise of the additional electromag- netic field changing pH of water around tactoids and, for one’s part, time-aging [2, 13]. These observations may be important for ecology, as long as vanadium pentox- ide is contained in impurities of coal soles, which are the components of wastes of thermoelectric power stations and are included in the impurity parameters at the back- ground control for radiation. [1] A. S. Sonin, J. Mater. Chem. 8, 2557 (1998). [2] A. V. Kaznacheev, M. M. Bogdanov, and S. A. Taraskin JETP, 95, 57 (2002). [3] G. E. Volovik and O. D. Lavrentovich, Sov. Phys. JETP 58, 1159(1983). [4] E. Barry, Z. Hensel, and Z. Dogic, Phys. Rev. L. 96, 018305 (2006). [5] P. Prinsen and P. van der Schoot, Phys. Rev. E. 68, 021701 (2003). [6] P. G. de Gennes, The physics of liquid crystals, Caledon Press, Oxford (1974). [7] P. Prinsen and P. van der Schoot, Eur. Phys. J. E, 13, 35 (2004). [8] P. Prinsen and P. van der Schoot, J. Phys.: Condens. Matter, 16, 8835 (2004). [9] F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983). [10] G. E. Volovik and V. P. Mineev, Zh. Eksp. Teor. Fiz. 83, 1025 (1982) (in Russian). [11] M. A. Bates, Chem. Phys. Lett. 368, 87 (2003). [12] G. E. Volovik, Pis’ma v Zh. Eksp. Teor. Fiz. 28, 65 (1978) (in Russian). [13] A. V. Kaznacheev, M. M. Bogdanov, and A. S. Sonin, JETP, 97, 1159 (2003). [14] M. V. Kurik and O. D. Lavrentovich, UFN, 154 381 (1988) (in Russian). [15] A. P. Balachandran, F. Lizzi, and V. G. J. Rodgers, Phys. Rev. Lett. 52, 1818 (1994). 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LORENTZ INVARIANCE AND SPIN PROPERTIES OF REFLECTION SYMMETRIC SUM OF VECTORS Non-Associativity of Lorentz Transformation and Associative Reflection Symmetric Transformation Mushfiq Ahmad Department of Physics, Rajshahi University, Rajshahi, Bangladesh. E-mail: mushfiqahmad@ru.ac.bd M. Shah Alam Department of Physics, Shah Jalal University of Science and Technology, Sylhet, Bangladesh. E-mail: salam@sust.edu Abstract Each of the two moving observers observes the relative velocity of the other. The two velocities should be equal and opposite. We have shown that this relativistic requirement is not fulfilled by Lorentz transformation. We have also shown that the reason is that Lorentz transformation is not associative. Reciprocal symmetric transformation is associative and fulfills relativistic requirements. 1. Introduction Two observers are moving with velocities U and V. Their relativity velocity according to one observer should be W. According to the other observer it should be –W. Arrows give the positive directions of the vectors. Negative vectors have the opposite directions. According to triangular law of addition of vectors, we must have andUVW +−= ˆ)( VUW +−=− ˆ)( . Therefore, U}V{ +−− ˆ)( VU +−= ˆ)( (1.1) If is non-commutative +̂ U}V{ +−− ˆ)( )(ˆ UV −+≠ (1.2) 2. Lorentz-Einstein Transformation mailto:mushfiqahmad@ http://ru.ac.bd:20000/ym/Compose?To=salam@sust.edu If a body is traveling with velocity U and an observer is traveling with velocity V, the relative velocity, , according to Lorentz-Einstein transformation is given by. EL−W −−−+− =+−=− 1)/(11)/(1 (2.1) We have used the symbol to represent Lorentz-Einstein addition of velocities. If the object and the observer swap positions so that U is the velocity of the observer and V is the velocity of the object, the relative velocity, , is given by )(U/11V)(U/1 V(-U)W −−−+− (2.2) In general they are not equal and opposite. ELEL −− −≠ WW (2.3) An exception is when U and V are parallel. In that case they are equal. (2.1) and (2.2) show that ELEl −− ≠−=−+ WWUV (2.4) 3. Non-Associativity of Lorentz Transformation Consider 3 vectors U, V and W using definition (2.1) of+ , we can calculate and and show that in general WV)(U ++ W)(VU ++ ≠++ WV)(U W)(VU ++ (3.1) We can by pass the cumbersome calculation and prove (3.1) by using the following theorem. 4. Theorem: If operation is associative +̂ )(ˆ)()ˆ( UVVU −+−=+− (4.1a) And if operation is also non-commutative +̂ )(ˆ)()ˆ( VUVU −+−≠+− (4.1b) (4.1a) is a necessary condition for associativity. (4.1b) is a necessary condition for associativity and non-commutativity. Proof: Consider )(ˆ)(ˆ}ˆ{ UVVU −+−++ (4.2) Using associativity and (4.2) becomes 0)(ˆ =−+ VV =−+−++ )(ˆ)(ˆ}ˆ{ UVVU 0)(ˆ)}(ˆ{ˆ =−+−++ UVVU (4.3) Therefore, 0)(ˆ)(ˆˆ =−+−++ }UV{V}{U (4.5) Comparing (4.5) with 0ˆˆˆ =+−++ V)(UV)(U (4.6) We find that )(ˆ)()ˆ( UVVU −+−=+− (4.7) If is also non-commutative +̂ U}{V +− ˆ )(ˆ)( UV −+−≠ (4.8) 5. Non-Associativity of Lorentz Transformation (2.4) contradicts (4.7) and (4.8). Therefore, Lorentz transformation is not associative. 6. Associativity of Reciprocal Symmetric Transformation Relative velocity according to reciprocal transformation is defined as VxUUV U(-V)W ˆ (6.1) A direct calculation will show that for any U, V and W =++ WV)(U ˆˆ W)(VU ++ ˆˆ (6.2) (6.2) can also be seen from the fact that (6.1) is built from Pauli quaternion1 and Pauli quaternion is associative. Definition (6.1) fulfills requirement (4.1) )(ˆ)()ˆ( VUUV −+−=+− (6.3) 7. Comparison between Lorentz and Reciprocal Symmetric Transformations Lorentz transformation and reciprocal symmetric transformations have the same magnitude. |)||ˆ| VUUV +=+ (7.1) Reciprocal symmetric transformation has a complex x-product term. This x-product term ensures (6.3). In Lorentz transformation (2.1) the product term comes as a dot-product, which does not change sing when order is changed. We have already seen2 the complex part explicitly shows the rotation hidden in Lorentz transformation. We have also seen3 that the complex part explicitly shows the origin of spin. 8. Conclusion We have shown that Lorentz Transformation is not associative and that Reciprocal Symmetric transformation is associative. Reciprocal Symmetric transformation fulfills requirement of Lorentz invariance4. It is complex. This complex part gives rise to spin5. 1 Mushfiq Ahmad, M. Shah Alam, M.O.G. Talukder. Comparison between Spin and Rotation Properties Of Lorentz Einstein and Reflection Symmetric Transformations.http://www.arxiv.org/abs/math-ph/0701067 2 Mushfiq Ahmad, M. Shah Alam, M.O.G. Talukder. Comparison between Spin and Rotation Properties Of Lorentz Einstein and Reflection Symmetric Transformations.http://www.arxiv.org/abs/math-ph/0701067 3 Mushfiq Ahmad. Reciprocal Symmetry and the Origin of Spin. http://www.arxiv.org/abs/math-ph/0702043 4 Mushfiq AhmadReciprocal Symmetry and Equivalence between Relativistic and Quantum Mechanical Concepts. http://www.arxiv.org/abs/math-ph/0611024. 5 Mushfiq Ahmad. Reciprocal Symmetry and the Origin of Spin. http://www.arxiv.org/abs/math-ph/0702043 http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Alam_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Talukder_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Alam_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Talukder_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1 Abstract (1.1) (1.2) If a body is traveling with velocity U and an observer is traveling with velocity V, the relative velocity,�, according to Lorentz-Einstein transformation is given by. (2.1) We have used the symbol � to represent Lorentz-Einstein addition of velocities. If the object and the observer swap positions so that U is the velocity of the observer and V is the velocity of the object, the relative velocity,�, is given by (2.2) (2.3) (2.4) (3.1) And if operation � is also non-commutative (4.1a) is a necessary condition for associativity. (4.1b) is a necessary condition for associativity and non-commutativity. Proof: Consider (4.2) (4.3) (4.5) (6.2) (6.3) (7.1)

Each of the two moving observers observes the relative velocity of the other. The two velocities should be equal and opposite. We have shown that this relativistic requirement is not fulfilled by Lorentz transformation. We have also shown that the reason is that Lorentz transformation is not associative. Reciprocal symmetric transformation is associative and fulfills relativistic requirements.

Introduction Two observers are moving with velocities U and V. Their relativity velocity according to one observer should be W. According to the other observer it should be –W. Arrows give the positive directions of the vectors. Negative vectors have the opposite directions. According to triangular law of addition of vectors, we must have andUVW +−= ˆ)( VUW +−=− ˆ)( . Therefore, U}V{ +−− ˆ)( VU +−= ˆ)( (1.1) If is non-commutative +̂ U}V{ +−− ˆ)( )(ˆ UV −+≠ (1.2) 2. Lorentz-Einstein Transformation mailto:mushfiqahmad@ http://ru.ac.bd:20000/ym/Compose?To=salam@sust.edu If a body is traveling with velocity U and an observer is traveling with velocity V, the relative velocity, , according to Lorentz-Einstein transformation is given by. EL−W −−−+− =+−=− 1)/(11)/(1 (2.1) We have used the symbol to represent Lorentz-Einstein addition of velocities. If the object and the observer swap positions so that U is the velocity of the observer and V is the velocity of the object, the relative velocity, , is given by )(U/11V)(U/1 V(-U)W −−−+− (2.2) In general they are not equal and opposite. ELEL −− −≠ WW (2.3) An exception is when U and V are parallel. In that case they are equal. (2.1) and (2.2) show that ELEl −− ≠−=−+ WWUV (2.4) 3. Non-Associativity of Lorentz Transformation Consider 3 vectors U, V and W using definition (2.1) of+ , we can calculate and and show that in general WV)(U ++ W)(VU ++ ≠++ WV)(U W)(VU ++ (3.1) We can by pass the cumbersome calculation and prove (3.1) by using the following theorem. 4. Theorem: If operation is associative +̂ )(ˆ)()ˆ( UVVU −+−=+− (4.1a) And if operation is also non-commutative +̂ )(ˆ)()ˆ( VUVU −+−≠+− (4.1b) (4.1a) is a necessary condition for associativity. (4.1b) is a necessary condition for associativity and non-commutativity. Proof: Consider )(ˆ)(ˆ}ˆ{ UVVU −+−++ (4.2) Using associativity and (4.2) becomes 0)(ˆ =−+ VV =−+−++ )(ˆ)(ˆ}ˆ{ UVVU 0)(ˆ)}(ˆ{ˆ =−+−++ UVVU (4.3) Therefore, 0)(ˆ)(ˆˆ =−+−++ }UV{V}{U (4.5) Comparing (4.5) with 0ˆˆˆ =+−++ V)(UV)(U (4.6) We find that )(ˆ)()ˆ( UVVU −+−=+− (4.7) If is also non-commutative +̂ U}{V +− ˆ )(ˆ)( UV −+−≠ (4.8) 5. Non-Associativity of Lorentz Transformation (2.4) contradicts (4.7) and (4.8). Therefore, Lorentz transformation is not associative. 6. Associativity of Reciprocal Symmetric Transformation Relative velocity according to reciprocal transformation is defined as VxUUV U(-V)W ˆ (6.1) A direct calculation will show that for any U, V and W =++ WV)(U ˆˆ W)(VU ++ ˆˆ (6.2) (6.2) can also be seen from the fact that (6.1) is built from Pauli quaternion1 and Pauli quaternion is associative. Definition (6.1) fulfills requirement (4.1) )(ˆ)()ˆ( VUUV −+−=+− (6.3) 7. Comparison between Lorentz and Reciprocal Symmetric Transformations Lorentz transformation and reciprocal symmetric transformations have the same magnitude. |)||ˆ| VUUV +=+ (7.1) Reciprocal symmetric transformation has a complex x-product term. This x-product term ensures (6.3). In Lorentz transformation (2.1) the product term comes as a dot-product, which does not change sing when order is changed. We have already seen2 the complex part explicitly shows the rotation hidden in Lorentz transformation. We have also seen3 that the complex part explicitly shows the origin of spin. 8. Conclusion We have shown that Lorentz Transformation is not associative and that Reciprocal Symmetric transformation is associative. Reciprocal Symmetric transformation fulfills requirement of Lorentz invariance4. It is complex. This complex part gives rise to spin5. 1 Mushfiq Ahmad, M. Shah Alam, M.O.G. Talukder. Comparison between Spin and Rotation Properties Of Lorentz Einstein and Reflection Symmetric Transformations.http://www.arxiv.org/abs/math-ph/0701067 2 Mushfiq Ahmad, M. Shah Alam, M.O.G. Talukder. Comparison between Spin and Rotation Properties Of Lorentz Einstein and Reflection Symmetric Transformations.http://www.arxiv.org/abs/math-ph/0701067 3 Mushfiq Ahmad. Reciprocal Symmetry and the Origin of Spin. http://www.arxiv.org/abs/math-ph/0702043 4 Mushfiq AhmadReciprocal Symmetry and Equivalence between Relativistic and Quantum Mechanical Concepts. http://www.arxiv.org/abs/math-ph/0611024. 5 Mushfiq Ahmad. Reciprocal Symmetry and the Origin of Spin. http://www.arxiv.org/abs/math-ph/0702043 http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Alam_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Talukder_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Alam_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Talukder_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1 http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1 Abstract (1.1) (1.2) If a body is traveling with velocity U and an observer is traveling with velocity V, the relative velocity,�, according to Lorentz-Einstein transformation is given by. (2.1) We have used the symbol � to represent Lorentz-Einstein addition of velocities. If the object and the observer swap positions so that U is the velocity of the observer and V is the velocity of the object, the relative velocity,�, is given by (2.2) (2.3) (2.4) (3.1) And if operation � is also non-commutative (4.1a) is a necessary condition for associativity. (4.1b) is a necessary condition for associativity and non-commutativity. Proof: Consider (4.2) (4.3) (4.5) (6.2) (6.3) (7.1)

Jamming dynamics in grain mixtures : An extended hydrodynamic approach Supurna Sinha Raman Research Institute, Bangalore 560 080,India (Dated: November 16, 2018) Abstract We study jamming in granular mixtures from the novel point of view of extended hydrodynamics. Using a hard sphere binary mixture model we predict that a few large grains are expected to get caged more effectively in a matrix of small grains compared to a few small grains in a matrix of larger ones. A similar effect has been experimentally seen in the context of colloidal mixtures. PACS numbers: PACS numbers: 87.15.-v,05.40.-a,36.20.-r http://arxiv.org/abs/0704.1895v1 I. INTRODUCTION In recent years granular matter has emerged as an active area of research[1]. Interest in this field has grown as a result of observations coming from a large number of interesting and relatively low-tech experiments [2]. In particular, a vibrated granular system[3] consisting of a large number of macroscopic grains in motion, provides us with a “scaled up” fluid system where we can explore the similarities and differences between such a large scale system and a microscopic atomic fluid. Recent studies[1, 4, 5] in dense granular systems indicate that there is a striking similar- ity between the dynamics of granular materials and glassy dynamics in atomic fluids and colloids[6]. In this Letter, we exploit this analogy to understand some facets of jamming in granular matter. In particular, we study the role of compositional disorder in form- ing jammed configurations which slow down the dynamics and eventually result in a state of structural arrest characteristic of a glass[7, 8, 9, 10]. The observations made here are analogous to jamming effects studied in atomic liquid mixtures and colloids [7, 11, 12]. The behavior of a fluid at large length and time scales is well described by hydrodynamics. The hydrodynamic description has been extended to molecular length scales in an extended or generalized hydrodynamic description [7, 13]. In a generalized hydrodynamic framework the basic structure of the hydrodynamic equations is retained and the static susceptibili- ties and transport coefficients are wave-vector dependent to account for the nontrivial static correlations that come into play on molecular scales in a dense liquid. The generalized hydro- dynamic description has proved to be very successful in describing the dramatic narrowing of the central diffusive peak in a neutron scattering spectrum S(k, ω) of a dense liquid[14]. Such a slow decay of density fluctuations is a precursor to glassy dynamics in dense liquids. A binary liquid mixture turns out to be more effective in forming glasses compared to a one component liquid since compositional disorder leads to jammed configurations which prevent the system from reaching its global equilibrium crystalline configuration[15]. Gen- eralized hydrodynamic studies of dense binary hard sphere mixtures in the context of glass transition suggest that at intermediate wavevectors (i.e. equivalently on length scales of the order of the average of the molecular diameters of the two species) the density fluctuations of the two species emerge as the slowest decaying fluctuations and therefore dominate the slow dynamics of the system [7, 12]. In a self-consistent mode-coupling theory (MCT), nonlinear couplings of these slowly decaying modes of density fluctuations lead to a glassy state where structural relaxation is frozen[8, 9]. In this Letter we use, for the first time, generalized hydrodynamic techniques developed in the realm of atomic liquids to understand the behavior of jammed configurations in granular mixtures. Such an approach to granular matter enables us to take into consideration non- trivial static correlations stemming from the granularity or finite sizes of the particles, which in turn, influences the high density jamming dynamics of the system. We thus gain new insight into granular mixture dynamics and make predictions for future experiments in such systems. II. EXTENDED HYDRODYNAMIC APPROACH TO A BINARY MIXTURE OF GRAINS Consider a binary granular mixture of hard spheres of diameters σ1 and σ2 (σ2>σ1), masses m1 and m2, number densities n1 and n2 and of total packing fraction η = 1 + n2σ 2 ]. An analysis of the extended hydrodynamic equations of such a binary hard sphere mixture suggests that on length scales of the order of the size of the hard sphere particles [7] momentum and temperature fluctuations decay very fast and the slow dynamics of the system can be well described in terms of the modes of density fluctuations of the two species[12, 16]. This enables us to confine to a two mode description of the system on length scales of the order of the diameters of the hard sphere granular particles at high densities. It is convenient to describe the system in terms of the following linear combinations of the density fluctuations of the two species- the total mass density fluctuation at wavevector ~k ρ~k = ρ1~k + ρ2~k and the concentration fluctuation at wavevector ~k c~k = where ρ and ρ are the mass density fluctuations of species 1 and 2, ρ1 = m1n1 and ρ2 = m2n2 are the equilibrium mass densities of species 1 and 2 and ρ = ρ1 + ρ2 is the total equilibrium mass density. The set of Laplace transformed coupled extended hydrodynamic equations of this system is given by: ρχT (k)γL(k) ρ~k(z) + βγL(k) f1(k) f2(k) c~k(z) = ρ~k(t = 0) (1) z + k2D(k) f1(k) + f2(k) c~k(z) n1n2 [m1f2(k)−m2f1(k)] ρ~k(z) = c~k(t = 0) (2) where χT (k) is the generalized isothermal compressibility, defined in terms of the partial static structure factors Sij(k) with i = 1, 2 and j = 1, 2: χT (k) = χ S11(k)S22(k)− S 12(k) x2S11(k) + x1S22(k)− 2 x1x2S12(k) Here χ0T is the compressibility of the granular gas in the dilute limit. γL(k) is the generalized longitudinal viscosity[7]. f1(k) = S22(k) + S12(k) S11(k)S22(k)− S 12(k) f2(k) = S11(k) + S12(k) S11(k)S22(k)− S 12(k) are combinations of partial static structure factors and D(k) is the coefficient of mutual diffusion[7, 17]. This set of equations leads to two extended hydrodynamic diffusive modes. Since we are interested in exploring the packing aspects of jamming in a binary granular mixture which is controlled by the sizes rather than the masses of the particles, we confine ourselves to the case of equal masses m1 = m2 for the two species[15]. Here we analyze two illuminating special cases to bring out the role of size difference and packing in the jamming process which triggers the transition to a glassy state: (i) a system composed of a few large spheres in a matrix of small spheres and (ii) a system composed of a few small spheres in a matrix of large spheres. In both these extreme limits (x2 = n2/n << 1 and x1 = n1/n << 1) the cross terms in the expressions representing the eigenvalues for the extended diffusive modes are negligibly small and the mode structure reduces to: (k) ≃ −k2 ρχT (k)γL(k) z+(k) ≃ −k 2D(k)[ f1(k) + f2(k)] (4) Thus, there are two relevant modes: z (k), which governs the relaxation of total mass density fluctuations and z+(k), which governs the relaxation of concentration fluctuations. Let us analyze these modes for case (i). In this case, since the mixture consists of a large number of small spheres, the static compressibility χT (k) which is the main determinant of the dynamics of a dense liquid, is given by χT (k) ≃ χ TS11(k), i.e. it is dominated by the static structure factor of the small spheres. Thus, z (k) ≃ −k2 1 S11(k)γL(k) . Consequently there is a significant slowing down of the dynamics of density fluctuations of the background matrix at the location of the first peak of S11(k)[18]. In this case the static structure of large (type 2) spheres is flat and is given by S22(k) ≃ 1. Thus, z+(k) ≃ −k 2 D02 S22(k) ≃ −k2D02, where D02 is the dilute gas limit of the diffusion coefficient of large spheres[17]. In case (ii), the large (type 2) and small (type 1) spheres switch roles. The final picture that emerges is the following. Caging and slowing down of dynamics is more effective for x2 << 1 firstly because the ratio D02/D01 = σ1/σ2 < 1 [7, 19] and a few large spheres diffuse slower in the background of small spheres compared to a few small spheres in the background of large spheres. In addition, in both cases there is also a significant slowing down of structural relaxation due to softening of the mode of total density fluctuations z (k) around the location of the peak of the static structure factor of the majority particles. In other words, caging is more efficient in a mixture dominated by small spheres compared to one dominated by large spheres. This is the main prediction made in this Letter for future experiments on dense granular mixtures designed to probe the efficiency of caging in such systems. The main point that we emphasize in this Letter is that by exploiting the analogy be- tween glass forming atomic liquids and granular matter we can draw some definite testable conclusions about caging dynamics in grains. To summarize, we have, for the first time applied extended or generalized hydrodynamic techniques to dynamics of granular matter. In particular, we consider a dense binary mixture of hard spheres and analyze the modes of density fluctuations which dominate the slow dynamics on length scales of the order of the sizes of the granular particles constituting the system. Our analysis points to some differences in jamming behavior between a mixture dominated by small spheres and one dominated by large spheres. Effects similar to the ones predicted here have been observed in confocal microscopy studies in colloidal mixtures [11]. The predictions made here can be tested against experiments in vibrated dense granular mixtures. We expect our analysis to be valid for a granular mixture consisting of nearly elastic spheres of comparable masses [20]. While the present analysis captures the onset of glassy behavior in granular mixtures, it would be worthwhile to do a mode-coupling study for such a system using the extended hydrodynamic modes that stem out of our analysis as an input to understand glass transition in granular mixtures. In future, one can do a more complete analysis where the effects of momentum and temperature fluctuations and inelastic collisions [21] are taken into consideration and check if the high density particle- scale dynamics presented here survives the inclusion of these effects. [1] See for instance, H. A. Maske, J. Brujic and S. F. Edwards, The physics of Granular Media, Wiley-VCH, (2004). [2] See for instance, K. Feitosa and N. Menon, Physical Review Letters 88 , 198301 (2002). [3] X. Yang, C. Huan, D. Candela, R. W. Mair and R. L. Walsworth Phys. Rev. Lett. 88, 044301 (2002); X. Yang, C. Huan, D. Candela, R. W. Mair and R. L. Walsworth Physical Review E 69, 041302 (2004); [4] A. J. Liu and S. R. Nagel, Nature 396, 21 (1998). [5] See for instance M. Nicodemi et al, “Statistical Mechanics of jamming and segregation in granular media” in “Unifying Concepts in Granular Media and Glasses” edts. A. Coniglio, A. Fierro, H. J. Herrmann and M. Nicodemi. [6] W. Gotze and L. Sjorgen, Rep. Prog. Phys 55, 241 (1992). [7] M. C. Marchetti and Supurna Sinha, Physical Review A 41 3214 (1990); Supurna Sinha and M. C. Marchetti, Physical Review A 46 4942 (1992). [8] U. Harbola and S. P. Das Physical Review E 65, 036138 (2002). [9] J. Bosse and J. S. Thakur, Physical Review Letters 59, 998 (1987). [10] J. Bosse and Y. Kaneko, Physical Review Letters 74, 4023 (1995). [11] A. D. Dinsmore, E. R. Weeks, V. Prasad, A. Levitt and D. A. Weitz, Applied Optics 40, 4152 (2001). [12] Supurna Sinha Physical Review E 49 3504 (1994). [13] T. R. Kirkpatrick, Physical Review A 32, 3130 (1985). [14] P. G. de Gennes, Physica 25, 825 (1959). [15] J. N. Roux, J. L. Barrat and J. P. Hansen Journal Of Physics : Condensed Matter 1, 7171 (1989). [16] A more familiar example is the phenomenon of de Gennes narrowing which happens in the context of dense simple liquids. [17] J. Ferziger and H. Kaper, Mathematical Theory Of Transport Processes in Gases, North- Holland, Amsterdam, (1972). [18] Also, we are focussing on a high viscosity (i.e. large γL(k)[[7, 13]]) regime which contributes to the process of structural slowing down. [19] See for instance, G. V. Vijayadamodar and B. Bagchi Journal Of Chemical Physics 93, 689 (1990). [20] In such a domain the mixture can be described fairly well as one sharing a common temperature T . See for instance P. Zamankhan Physical Review E 52, 4877 (1995) and V. Garzo and J. W. Dufty Physical Review E 60, 5706 (1999). [21] See, for instance, J. W. Dufty and J. J. Brey Physical Review E 68, 030302 (2003); J. J. Brey and J. W. Dufty Physical Review E 72, 011303 (2005); Introduction Extended Hydrodynamic Approach To A Binary Mixture Of Grains References

We study jamming in granular mixtures from the novel point of view of extended hydrodynamics. Using a hard sphere binary mixture model we predict that a few large grains are expected to get caged more effectively in a matrix of small grains compared to a few small grains in a matrix of larger ones. A similar effect has been experimentally seen in the context of colloidal mixtures.

Introduction Extended Hydrodynamic Approach To A Binary Mixture Of Grains References

Path Distinguishability in Double Scattering of Light by Atoms Christian Miniatura,1, 2 Cord A. Müller,3 Yin Lu,1 Guangquan Wang,1 and Berthold-Georg Englert1 Department of Physics, Faculty of Science, National University of Singapore, Singapore 117542, Singapore Institut Non Linéaire de Nice, UMR 6618 du CNRS, UNSA, 1361 route des Lucioles, 06560 Valbonne, France Physikalisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany (Dated: 15 April 2007) Wave-particle duality finds a natural application for electrons or light propagating in disordered media where coherent corrections to transport are given by two-wave interference. For scatterers with internal degrees of freedom, these corrections are observed to be much smaller than would be expected for structureless scatterers. By examining the basic example of the scattering of one photon by two spin- 1 atoms—a case-study for coherent backscattering—we demonstrate that the loss of interference strength is associated with which-path information stored by the scattering atoms. PACS numbers: 42.25.Hz, 03.65.Nk Einstein’s and de Broglie’s wave-particle duality (WPD)—the ability of a quantum system to display the seemingly contradictory attributes that one would have regarded as wave-like and particle-like and, therefore, as mutually exclusive in pre-quantum physics—is arguably the most important phenomenological consequence of Bohr’s principle of complementarity. The quantitative aspects of WPD are particularly well understood [1] in the context of two-paths interferometers where a definite path is particle-like and the interference between the am- plitudes of the two paths is wave-like. The wave-like interference strength is quantified by the familiar Michelson’s fringe visibility V , and the particle- like path knowledge is measured by the path distinguisha- bility D, which is perhaps less familiar and has this op- erational meaning: The odds for guessing the path right are (1+D)/2. The extreme situations of no path knowl- edge and high fringe visibility (ideally D = 0, V = 1) or full path knowledge and no fringes (D = 1, V = 0) are standard textbook fare. A rather recent experiment with a bearing on the matter discussed below is the one carried out by Eichmann et al. in 1993 [2]. The compromises allowed by the laws of physics in in- termediary situations are restricted by the duality rela- tion [3, 4] D2 + V2 ≤ 1 . (1) It is worth noting that well pronounced wave-like and particle-like aspects can coexist: With odds for guessing the path right of 99% (D = 0.98), we can have well visible fringes of 20% visibility (V = 0.2). The two pioneering experiments that tested the dual- ity relation employed two-path interferometers for atoms [5] and photons [6], whereby internal degrees of freedom of the interfering objects themselves were used to pro- vide the path information. By contrast, in the situation that we examine here—coherent double scattering—the which-path information is stored in the deflecting ele- ments of the interferometer (atoms) and not carried by the interfering objects (photons). We wish to show how these notions of path knowledge and interference strength are naturally applied to coher- ent wave transport. In the semi-classical regime of weak localization, most of the coherent effects in wave trans- port can be explained by two-wave interference between amplitudes propagating in opposite direction along loop- like scattering paths [7, 8]. These interference corrections to transport can dramatically alter the diffusion process and even suppress it [9, 10]. They are sensitive to sev- eral “dephasing” processes but most of them are circum- vented at sufficiently low temperatures [11, 12, 13]. Here we address an intrinsic dephasing mechanism which survives at zero temperature: the path knowledge stored in the internal degrees of freedom of the scatterers. Indeed, when the scatterers have an internal structure, the interference corrections to transport are observed to be rather small, for example in the scattering of electrons by magnetic impurities at very low temperatures [12]. The same effect has been observed in the coherent backscattering (CBS) of light by cold rubidium atoms [14, 15, 16]. This coherent multiple scattering effect arises when an optically thick sample of scatterers is il- luminated by coherent light. It, too, results from inter- ference of light amplitudes, here of the two amplitudes associated with traversing the same path in opposite di- rection. The endpoints of each scattering path thus play the role of Young slits and give rise to an angular fringe pattern in the far-field. Owing to the varying separa- tion between the endpoints, these patterns have differ- ent fringe spacings but they all display a bright fringe at backscattering. Thus, the sum of all fringe patterns displays an angular peak around the backscattering di- rection [17]. We quantify the strength of this interference in a nat- ural manner by the relative excess of the peak intensity over the background, the analog of the fringe visibility V in this context. For atoms with a spin-0 ground state, this CBS peak-to-background ratio reaches its maximal possible value of 2 in the helicity-preserving polarization channel [18], corresponding to V = 1, whereas it is very http://arxiv.org/abs/0704.1896v2 ✓✏ 2✒✑ ............. . ............. ........................ ........................ ....................... ....................... ....................... ...................... .............. ..................... . ............ ........................ ........................ ....................... ....................... ....................... ...................... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ............. ........................ ..... ..... ..... ........................ ...... ...... ..... ......................... ........................................................................................... ......... ......... ........................ ............ ............ ........................ ....................... ....................... ...................... −1/2 1/2 . ......................................................................................... .......................................................................................... . ......................................................................................... .......................................................................................... ........................................................................................................... ........... .......... ........................................................................................................... ........... .......... ......................................................................................................................................................................................................... ..... ..... .......... .......... .......... .......... ..... ..... FIG. 1: (a) The two paths in coherent backscattering. Along path A the photon is first scattered by atom 1, then by atom 2; along path B the order is reversed. (b) Level scheme of the transition. Both the ground state and the excited state are doublets with total angular momentum j = 1 , and the magnetic quantum numbers m = ± 1 label the sublevels. small for atoms with a degenerate ground state [14], as is the situation with a Zeeman fine structure or a hyperfine structure. Our point is that these results can be recast and understood, both qualitatively and quantitatively, in terms of wave-particle duality. Indeed, when scatter- ing the photon the atoms may undergo a change in their ground state—a circumstance equally crucial in the single-scattering situation of the Eichmann et al. exper- iment [2]. This is to say that the atoms can store which- path information so that the experimenter can find out, in principle if not in practice, which of the two atoms scattered first and which second. As a consequence, the strength of the coherent corrections to transport is bounded by the distinguishability of the paths inside the sample, and the height of the CBS peak is limited by the amount of path knowledge available. We consider the simplest possible scenario that ex- hibits the effect: double scattering off two identical spin- atoms (atom 1 and atom 2), with the photon resonant with a 1 dipole transition. In fact, this is the situ- ation of the Eichmann et al. experiment where the scat- terers are Hg+ ions. This geometry is simplest for mul- tiple scattering to set in and is at the heart of the CBS phenomenon. Since our focus is on the influence of the internal atomic structure, we assume that the atoms are so stiffly trapped that there is no relevant contribution from the atomic recoil (the storage of CBS which-path information in the center-of-mass degrees of freedom of mobile atoms is studied in Ref. [19]). Put differently, we take for granted that the atomic center-of-mass degrees of freedom do not store which-path information. To simplify the problem further, we assume that the distance between the atoms is sufficiently large for the double scattering contribution to dominate over all other multiple scattering processes (triple, quadruple, . . . ). As illustrated in Fig. 1, path A is the case when atom 1 scatters first and atom 2 second (sequence 1 → 2); path B is the sequence 2 → 1. The paths are geometrically identical but traversed in opposite directions. The two atoms together compose the path detector: the change of their internal states bears witness of the actual path. Since the ground states are spin- 1 states, the path de- tector is a qubit pair, which is a 4-state system. During the scattering process, however, the excited states of the atoms are involved as well, and the details of the scatter- ing interaction determine the over-all effect on the atoms. This net before-to-after change in the combined ground states of both atoms is given by a completely positive two-qubit map. We establish this map by first recalling that the atom- photon interaction is described by quasi-resonant point- dipole elastic scattering. The corresponding transition operator is proportional to T = (dd) ⊗ |r〉〈r| where r is the atom’s position vector and d is the dipole vector operator of the atomic transition. The omitted propor- tionality factor depends on the oscillator strength of the transition. It determines the probability of the double scattering event and is, therefore, crucial for an actual experiment. But in the present context this probability is irrelevant because the final two-atom state is conditioned on successful double scattering. Bearing this conditioning in mind, we consistently leave all further proportionality factors implicit. For an incoming photon with wave vector k and trans- verse polarization ǫ, the matrix elements of T are 〈m′,k′ǫ′|T |m,kǫ〉 = 〈m′, ǫ′|(dd)|m, ǫ〉 ei(k−k ′)·r , (2) where k′ and ǫ′ are the wave vector and polarization of the outgoing photon, and m and m′ are the magnetic quantum numbers of the initial and final ground state, respectively [16]. Since the scattering is elastic, we have |k| = |k′|. The dyadic operator T = (dd) acts on the internal degrees of freedom of the photon (the polarization states) and of the atom (the magnetic sublevels of the angular momentum multiplets). Its matrix elements read 〈m′, ǫ′|(dd)|m, ǫ〉 = 〈m′|(ǫ′∗ · d)(d · ǫ)|m〉 . (3) The matrix elements of the vector operator d are the Clebsch–Gordan coefficients that characterize the cou- pling of spin-1 (photon) with spin- 1 (ground state) to give spin- 1 (excited state); all the coefficients have equal magnitude for such a 1 transition. As a conse- quence, we have effectively T = (σ σ) for initial and final ground states, where σ is the Pauli vector operator for the spin- 1 ground state. [20] We consider the exact backscattering geometry where k = −k′, which we choose parallel to the z axis of the coordinate system; the magnetic quantum numbers ±1/2 in Fig. 1(b) also refer to the z direction. For path A, the double scattering operator that acts on the two atomic ground-state qubits is TA = ǫ ′∗ · (σ2σ2) · (1− nn) · (σ1σ1) · ǫ = −ǫ′∗ · σ2 (σ2 × n) · (n× σ1)σ1 · ǫ , (4) where 1− nn is the dyadic projector onto the plane or- thogonal to the unit vector n that points from one scat- terer to the other. The double-scattering operator TB for path B is obtained by interchanging 1 ↔ 2 in (4). With ρin denoting the initial two-qubit state of the two scattering atoms, the final states are then given by ρA,B = TA,B ρin TA,B with wA,B = tr TA,B ρin TA,B , (5) where the normalizing denominators take care of all the proportionality factors that we left implicit. The weights of the two paths are wA/(wA +wB) and wB/(wA +wB), respectively. In addition to the initial two-atom state ρin, these weights and the final states depend on the pre- selected polarization ǫ of the incoming photon and the post-selected polarization ǫ′ of the outgoing photon, on which the ensemble of events is conditioned. Since the final states of the atoms are different for the two paths, there is which-path information stored in the atoms, which—in principle—can be extracted by a suit- able measurement, although in practice it could be very difficult to implement such a measurement. The optimal measurement would provide as much path knowledge as is available, quantified by the distinguishability of the paths, which is given by [1, 3, 4] ∣wAρA − wBρB wA + wB ∣TA ρin TA † − TB ρin TB wA + wB . (6) This is supplemented by the visibility TA ρin TB wA + wB , (7) the quantitative measure for the interference strength of the two paths. Irrespective of the detailed form of ρin and the operators TA, TB, the duality relation (1) is obeyed by this distinguishability and visibility [1]. We now restrict the discussion to symmetric initial two-qubit states of the form ρin = (11 − pσ1 · σ2) with − 1 ≤ p ≤ 1 as required by the positivity of ρin. This one-parameter family of initial states encompasses some cases of particular physical interest: the completely mixed state (p = 0); the projector on the singlet state of vanishing total angular momentum (p = 1); the projec- tor on the triplet sector of unit total angular momentum (p = − 1 ). It is worth recalling that two-qubit states of this form are separable for p ≤ 1 and entangled for p > 1 but, as illustrated by Eqs. (9) below, nothing remarkable happens to D and V at the transition from p < 1 p > 1 For all values of p, the initial state ρin is invariant under the interchange 1 ↔ 2 and, therefore, the interferometer is symmetric in the sense that both paths occur with equal a priori probability (wA = wB). As a consequence, the difference of operators in (6) is antisymmetric under 1 ↔ 2 and thus of the form TA ρin TA † − TB ρin TB wA + wB = a·(σ1−σ2)+b·(σ1×σ2) (8) with two numerical vectors a and b that depend on the photon polarizations ǫ, ǫ′, the unit vector n, and the initial-state parameter p. The right-hand side in (8) is a rank-2 operator with its nonzero eigenvalues given by a2 + b2, and so we get D = 4 a2 + b2 for the dis- tinguishability of the paths. In this manner we arrive at explicit expressions for D and V [21, 22]. We will report the full technical details elsewhere and focus here on the particular situation in which the line connecting the two atoms in Fig. 1(a) is perpendicular to the incoming and outgoing propagation directions, that is: we choose the unit vector n along the x axis. For this perpendicular geometry, one has 1 + p+ 2pu 2(1 + pu) 1− u′2 , ∣(1 + p)(1 + uu′)− 2p(1− u′) 2(1 + pu) , (9) where u and u′ are the x components of the Stokes vec- tors associated with the incoming and outgoing photon polarizations [23]. These are such that (1− V)V for p ≤ 0 , (1− V) 2p/(1 + p) + V for p ≥ 0 . Clearly, the duality relation (1) is obeyed for all p values. The relation is only saturated for p = 1, in which case the initial atomic state is pure and the equal sign is expected to hold in (1) on general grounds [4]. For the completely-mixed initial state (p = 0) we have 1− u′2 , V = (1 + uu′) . (11) Here, the distinguishability does not depend at all on the initial polarization—a surprising feature that is particu- lar to the 1 transition in the perpendicular geometry and is not generic. This observation about the perpen- dicular geometry can be understood as follows. Since all Clebsch–Gordan coefficients are of equal size, the first scatterer has uniform a priori probability of reaching either one of its ground states, irrespective of the polarization of the incoming photon. Yet, when con- ditioned on the direction into which the photon is re- emitted, the probability is not uniform as a rule, but it is for the perpendicular geometry. Therefore, there is no which-path information stored in the final completely- mixed state of the atom that scatters first. When the observed polarization of the outgoing photon is an equal-weight superposition of the in-plane and out- of-plane linear polarizations (u′ = 0), the final state of the second scatterer is a corresponding pure state. So, when finding only one atom in this pure state, we can infer the path with certainty, but if both atoms are found in this state, we know nothing about the path and will guess wrong half of the time. Both cases are equally probable, so that our betting odds are 75%, which is consistent with D = 1 for u′ = 0 in (11), as it should be. In this case the visibility is V = 1 irrespective of the incoming polarization. The distinguishability is zero for an outgoing photon linearly polarized in the plane of the drawing in Fig. 1(a), when u′ = 1, or perpendicular to this plane (u′ = −1). The corresponding visibility takes on any value between 0 and 1. The case V = 0 happens for photons with per- pendicular polarizations, one with in-plane polarization, the other perpendicular (uu′ = −1). The case V = 1 oc- curs for photons with parallel linear polarizations, both in the plane or both perpendicular to it (uu′ = 1). Let us now turn to the situation of an initial singlet state (p = 1). As noted above, the duality relation (1) is then saturated and we have 1− u′2 , V = ∣ . (12) The fact that the distinguishability does not depend on the initial polarization can be understood by an argu- ment similar to the one given above for the p = 0 case. Irrespective of the incoming photon polarization and for, say, a left-circular outgoing photon, the final states of the atoms have (m1,m2) = ( ) for path A and ) for path B whereas the reversed situation occurs for a right-circular outgoing photon. This means that if the outgoing light is analyzed in the circular channels (u′ = 0), perfect path knowledge is available (D = 1) and no interference will be observed (V = 0). Conversely, if the outgoing light is analyzed in the channels of linear in- plane and out-of-plane polarization ( ∣ = 1), no path knowledge is available (D = 0) and one recovers full in- terference strength (V = 1). Finally, one can think of mimicking the physics of the CBS phenomenon by an angular average over the direc- tion n. We first calculate the average of the difference operator in (8) and then compute the resulting distin- guishability as the trace of its modulus. The correspond- ing visibility is obtained as the angular average of the n dependent visibility (7). For p = 0, which applies to most of the available experimental data, the largest averaged distinguishability is D = 1 ; it is found in the helicity- preserving polarization channel. The smallest average visibility is also found in this channel, its value is V = 2 Even if a direct quantitative comparison with the real CBS situation cannot be made at this stage, this result is nevertheless consistent with the experimental observa- tion that the lowest CBS peaks are actually found in this detection channel [14]. In summary, we have demonstrated that the concept of wave-particle duality proves relevant and useful for our understanding of some aspects of the interference effects in multiple scattering. To make solid quantitative con- tact with actual CBS experiments, the analysis must be extended to account for scattering by three and more atoms. The stage for this future research is now set. Ch. M. and C. M. wish to thank the Science Faculty and the Physics Department of NUS for their kind hospi- tality and financial support. This work was supported by by A∗Star Grant No. 012-104-0040, and by NUS Grant WBS: R-144-000-179-112. [1] See, e.g., B.-G. Englert and J.A. Bergou, Opt. Commun. 179, 337 (2000), and the references therein, in particular [3] and [4]. [2] U. Eichmann et al., Phys. Rev. Lett. 70, 2359 (1993); W.M. Itano et al., Phys. Rev. A 57, 4176 (1998). [3] G. Jaeger, A. Shimony, and L. Vaidman, Phys. Rev. A 51, 54 (1995). [4] B.-G. Englert, Phys. Rev. Lett. 77, 2154 (1996). [5] S. Dürr, T. Nonn, and G. Rempe, Nature (London) 395, 33 (1998); Phys. Rev. Lett. 81, 5705 (1998). [6] P.D.D. Schwindt, P.G. Kwiat, and B.-G. Englert, Phys. Rev. A 60, 4285 (1999). [7] E. Akkermans et al. (eds.), Mesoscopic Quantum Physics (North-Holland, Amsterdam 1995). [8] G. Bergmann, Phys. Rep. 107, 1 (1984). [9] D.S. Wiersma et al., Nature (London) 390, 671 (1997). [10] C.M. Aegerter, M. Störzer, and G. Maret, Europhys. Lett. 75, 562 (2006). [11] Y. Imry, arXiv:cond-mat/0202044 (2002). [12] F. Pierre et al., Phys. Rev. B 68, 085413 (2003). [13] G. Labeyrie et al., Phys. Rev. Lett. 97, 013004 (2006). [14] G. Labeyrie et al., Phys. Rev. Lett. 83, 5266 (1999). [15] T. Jonckheere et al., Phys. Rev. Lett. 85, 4269 (2000). [16] C. Müller et al., Phys. Rev. A 64, 053804 (2001). [17] G. Labeyrie et al., J. Opt. B: Quantum Semiclass. Opt. 2, 672 (2000). [18] Y. Bidel et al., Phys. Rev. Lett. 88, 203902 (2002). [19] C. Wickles and C. Müller, Europhys. Lett. 74, 240 (2006). [20] In the case of a 1 transition, T = (σ σ)− 31 has an admixture of the unit dyadic [22]. [21] Y. Lu, Wave-Particle Duality in Coherent Multiple Scat- tering , unpublished (2006). [22] G. Wang, Quantitative Wave-Particle Duality in Double Scattering Effect , unpublished (2007). [23] M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford 1980). http://arxiv.org/abs/cond-mat/0202044

Wave-particle duality finds a natural application for electrons or light propagating in disordered media where coherent corrections to transport are given by two-wave interference. For scatterers with internal degrees of freedom, these corrections are observed to be much smaller than would be expected for structureless scatterers. By examining the basic example of the scattering of one photon by two spin-1/2 atoms--a case-study for coherent backscattering--we demonstrate that the loss of interference strength is associated with which-path information stored by the scattering atoms.

Path Distinguishability in Double Scattering of Light by Atoms Christian Miniatura,1, 2 Cord A. Müller,3 Yin Lu,1 Guangquan Wang,1 and Berthold-Georg Englert1 Department of Physics, Faculty of Science, National University of Singapore, Singapore 117542, Singapore Institut Non Linéaire de Nice, UMR 6618 du CNRS, UNSA, 1361 route des Lucioles, 06560 Valbonne, France Physikalisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany (Dated: 15 April 2007) Wave-particle duality finds a natural application for electrons or light propagating in disordered media where coherent corrections to transport are given by two-wave interference. For scatterers with internal degrees of freedom, these corrections are observed to be much smaller than would be expected for structureless scatterers. By examining the basic example of the scattering of one photon by two spin- 1 atoms—a case-study for coherent backscattering—we demonstrate that the loss of interference strength is associated with which-path information stored by the scattering atoms. PACS numbers: 42.25.Hz, 03.65.Nk Einstein’s and de Broglie’s wave-particle duality (WPD)—the ability of a quantum system to display the seemingly contradictory attributes that one would have regarded as wave-like and particle-like and, therefore, as mutually exclusive in pre-quantum physics—is arguably the most important phenomenological consequence of Bohr’s principle of complementarity. The quantitative aspects of WPD are particularly well understood [1] in the context of two-paths interferometers where a definite path is particle-like and the interference between the am- plitudes of the two paths is wave-like. The wave-like interference strength is quantified by the familiar Michelson’s fringe visibility V , and the particle- like path knowledge is measured by the path distinguisha- bility D, which is perhaps less familiar and has this op- erational meaning: The odds for guessing the path right are (1+D)/2. The extreme situations of no path knowl- edge and high fringe visibility (ideally D = 0, V = 1) or full path knowledge and no fringes (D = 1, V = 0) are standard textbook fare. A rather recent experiment with a bearing on the matter discussed below is the one carried out by Eichmann et al. in 1993 [2]. The compromises allowed by the laws of physics in in- termediary situations are restricted by the duality rela- tion [3, 4] D2 + V2 ≤ 1 . (1) It is worth noting that well pronounced wave-like and particle-like aspects can coexist: With odds for guessing the path right of 99% (D = 0.98), we can have well visible fringes of 20% visibility (V = 0.2). The two pioneering experiments that tested the dual- ity relation employed two-path interferometers for atoms [5] and photons [6], whereby internal degrees of freedom of the interfering objects themselves were used to pro- vide the path information. By contrast, in the situation that we examine here—coherent double scattering—the which-path information is stored in the deflecting ele- ments of the interferometer (atoms) and not carried by the interfering objects (photons). We wish to show how these notions of path knowledge and interference strength are naturally applied to coher- ent wave transport. In the semi-classical regime of weak localization, most of the coherent effects in wave trans- port can be explained by two-wave interference between amplitudes propagating in opposite direction along loop- like scattering paths [7, 8]. These interference corrections to transport can dramatically alter the diffusion process and even suppress it [9, 10]. They are sensitive to sev- eral “dephasing” processes but most of them are circum- vented at sufficiently low temperatures [11, 12, 13]. Here we address an intrinsic dephasing mechanism which survives at zero temperature: the path knowledge stored in the internal degrees of freedom of the scatterers. Indeed, when the scatterers have an internal structure, the interference corrections to transport are observed to be rather small, for example in the scattering of electrons by magnetic impurities at very low temperatures [12]. The same effect has been observed in the coherent backscattering (CBS) of light by cold rubidium atoms [14, 15, 16]. This coherent multiple scattering effect arises when an optically thick sample of scatterers is il- luminated by coherent light. It, too, results from inter- ference of light amplitudes, here of the two amplitudes associated with traversing the same path in opposite di- rection. The endpoints of each scattering path thus play the role of Young slits and give rise to an angular fringe pattern in the far-field. Owing to the varying separa- tion between the endpoints, these patterns have differ- ent fringe spacings but they all display a bright fringe at backscattering. Thus, the sum of all fringe patterns displays an angular peak around the backscattering di- rection [17]. We quantify the strength of this interference in a nat- ural manner by the relative excess of the peak intensity over the background, the analog of the fringe visibility V in this context. For atoms with a spin-0 ground state, this CBS peak-to-background ratio reaches its maximal possible value of 2 in the helicity-preserving polarization channel [18], corresponding to V = 1, whereas it is very http://arxiv.org/abs/0704.1896v2 ✓✏ 2✒✑ ............. . ............. ........................ ........................ ....................... ....................... ....................... ...................... .............. ..................... . ............ ........................ ........................ ....................... ....................... ....................... ...................... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ............. ........................ ..... ..... ..... ........................ ...... ...... ..... ......................... ........................................................................................... ......... ......... ........................ ............ ............ ........................ ....................... ....................... ...................... −1/2 1/2 . ......................................................................................... .......................................................................................... . ......................................................................................... .......................................................................................... ........................................................................................................... ........... .......... ........................................................................................................... ........... .......... ......................................................................................................................................................................................................... ..... ..... .......... .......... .......... .......... ..... ..... FIG. 1: (a) The two paths in coherent backscattering. Along path A the photon is first scattered by atom 1, then by atom 2; along path B the order is reversed. (b) Level scheme of the transition. Both the ground state and the excited state are doublets with total angular momentum j = 1 , and the magnetic quantum numbers m = ± 1 label the sublevels. small for atoms with a degenerate ground state [14], as is the situation with a Zeeman fine structure or a hyperfine structure. Our point is that these results can be recast and understood, both qualitatively and quantitatively, in terms of wave-particle duality. Indeed, when scatter- ing the photon the atoms may undergo a change in their ground state—a circumstance equally crucial in the single-scattering situation of the Eichmann et al. exper- iment [2]. This is to say that the atoms can store which- path information so that the experimenter can find out, in principle if not in practice, which of the two atoms scattered first and which second. As a consequence, the strength of the coherent corrections to transport is bounded by the distinguishability of the paths inside the sample, and the height of the CBS peak is limited by the amount of path knowledge available. We consider the simplest possible scenario that ex- hibits the effect: double scattering off two identical spin- atoms (atom 1 and atom 2), with the photon resonant with a 1 dipole transition. In fact, this is the situ- ation of the Eichmann et al. experiment where the scat- terers are Hg+ ions. This geometry is simplest for mul- tiple scattering to set in and is at the heart of the CBS phenomenon. Since our focus is on the influence of the internal atomic structure, we assume that the atoms are so stiffly trapped that there is no relevant contribution from the atomic recoil (the storage of CBS which-path information in the center-of-mass degrees of freedom of mobile atoms is studied in Ref. [19]). Put differently, we take for granted that the atomic center-of-mass degrees of freedom do not store which-path information. To simplify the problem further, we assume that the distance between the atoms is sufficiently large for the double scattering contribution to dominate over all other multiple scattering processes (triple, quadruple, . . . ). As illustrated in Fig. 1, path A is the case when atom 1 scatters first and atom 2 second (sequence 1 → 2); path B is the sequence 2 → 1. The paths are geometrically identical but traversed in opposite directions. The two atoms together compose the path detector: the change of their internal states bears witness of the actual path. Since the ground states are spin- 1 states, the path de- tector is a qubit pair, which is a 4-state system. During the scattering process, however, the excited states of the atoms are involved as well, and the details of the scatter- ing interaction determine the over-all effect on the atoms. This net before-to-after change in the combined ground states of both atoms is given by a completely positive two-qubit map. We establish this map by first recalling that the atom- photon interaction is described by quasi-resonant point- dipole elastic scattering. The corresponding transition operator is proportional to T = (dd) ⊗ |r〉〈r| where r is the atom’s position vector and d is the dipole vector operator of the atomic transition. The omitted propor- tionality factor depends on the oscillator strength of the transition. It determines the probability of the double scattering event and is, therefore, crucial for an actual experiment. But in the present context this probability is irrelevant because the final two-atom state is conditioned on successful double scattering. Bearing this conditioning in mind, we consistently leave all further proportionality factors implicit. For an incoming photon with wave vector k and trans- verse polarization ǫ, the matrix elements of T are 〈m′,k′ǫ′|T |m,kǫ〉 = 〈m′, ǫ′|(dd)|m, ǫ〉 ei(k−k ′)·r , (2) where k′ and ǫ′ are the wave vector and polarization of the outgoing photon, and m and m′ are the magnetic quantum numbers of the initial and final ground state, respectively [16]. Since the scattering is elastic, we have |k| = |k′|. The dyadic operator T = (dd) acts on the internal degrees of freedom of the photon (the polarization states) and of the atom (the magnetic sublevels of the angular momentum multiplets). Its matrix elements read 〈m′, ǫ′|(dd)|m, ǫ〉 = 〈m′|(ǫ′∗ · d)(d · ǫ)|m〉 . (3) The matrix elements of the vector operator d are the Clebsch–Gordan coefficients that characterize the cou- pling of spin-1 (photon) with spin- 1 (ground state) to give spin- 1 (excited state); all the coefficients have equal magnitude for such a 1 transition. As a conse- quence, we have effectively T = (σ σ) for initial and final ground states, where σ is the Pauli vector operator for the spin- 1 ground state. [20] We consider the exact backscattering geometry where k = −k′, which we choose parallel to the z axis of the coordinate system; the magnetic quantum numbers ±1/2 in Fig. 1(b) also refer to the z direction. For path A, the double scattering operator that acts on the two atomic ground-state qubits is TA = ǫ ′∗ · (σ2σ2) · (1− nn) · (σ1σ1) · ǫ = −ǫ′∗ · σ2 (σ2 × n) · (n× σ1)σ1 · ǫ , (4) where 1− nn is the dyadic projector onto the plane or- thogonal to the unit vector n that points from one scat- terer to the other. The double-scattering operator TB for path B is obtained by interchanging 1 ↔ 2 in (4). With ρin denoting the initial two-qubit state of the two scattering atoms, the final states are then given by ρA,B = TA,B ρin TA,B with wA,B = tr TA,B ρin TA,B , (5) where the normalizing denominators take care of all the proportionality factors that we left implicit. The weights of the two paths are wA/(wA +wB) and wB/(wA +wB), respectively. In addition to the initial two-atom state ρin, these weights and the final states depend on the pre- selected polarization ǫ of the incoming photon and the post-selected polarization ǫ′ of the outgoing photon, on which the ensemble of events is conditioned. Since the final states of the atoms are different for the two paths, there is which-path information stored in the atoms, which—in principle—can be extracted by a suit- able measurement, although in practice it could be very difficult to implement such a measurement. The optimal measurement would provide as much path knowledge as is available, quantified by the distinguishability of the paths, which is given by [1, 3, 4] ∣wAρA − wBρB wA + wB ∣TA ρin TA † − TB ρin TB wA + wB . (6) This is supplemented by the visibility TA ρin TB wA + wB , (7) the quantitative measure for the interference strength of the two paths. Irrespective of the detailed form of ρin and the operators TA, TB, the duality relation (1) is obeyed by this distinguishability and visibility [1]. We now restrict the discussion to symmetric initial two-qubit states of the form ρin = (11 − pσ1 · σ2) with − 1 ≤ p ≤ 1 as required by the positivity of ρin. This one-parameter family of initial states encompasses some cases of particular physical interest: the completely mixed state (p = 0); the projector on the singlet state of vanishing total angular momentum (p = 1); the projec- tor on the triplet sector of unit total angular momentum (p = − 1 ). It is worth recalling that two-qubit states of this form are separable for p ≤ 1 and entangled for p > 1 but, as illustrated by Eqs. (9) below, nothing remarkable happens to D and V at the transition from p < 1 p > 1 For all values of p, the initial state ρin is invariant under the interchange 1 ↔ 2 and, therefore, the interferometer is symmetric in the sense that both paths occur with equal a priori probability (wA = wB). As a consequence, the difference of operators in (6) is antisymmetric under 1 ↔ 2 and thus of the form TA ρin TA † − TB ρin TB wA + wB = a·(σ1−σ2)+b·(σ1×σ2) (8) with two numerical vectors a and b that depend on the photon polarizations ǫ, ǫ′, the unit vector n, and the initial-state parameter p. The right-hand side in (8) is a rank-2 operator with its nonzero eigenvalues given by a2 + b2, and so we get D = 4 a2 + b2 for the dis- tinguishability of the paths. In this manner we arrive at explicit expressions for D and V [21, 22]. We will report the full technical details elsewhere and focus here on the particular situation in which the line connecting the two atoms in Fig. 1(a) is perpendicular to the incoming and outgoing propagation directions, that is: we choose the unit vector n along the x axis. For this perpendicular geometry, one has 1 + p+ 2pu 2(1 + pu) 1− u′2 , ∣(1 + p)(1 + uu′)− 2p(1− u′) 2(1 + pu) , (9) where u and u′ are the x components of the Stokes vec- tors associated with the incoming and outgoing photon polarizations [23]. These are such that (1− V)V for p ≤ 0 , (1− V) 2p/(1 + p) + V for p ≥ 0 . Clearly, the duality relation (1) is obeyed for all p values. The relation is only saturated for p = 1, in which case the initial atomic state is pure and the equal sign is expected to hold in (1) on general grounds [4]. For the completely-mixed initial state (p = 0) we have 1− u′2 , V = (1 + uu′) . (11) Here, the distinguishability does not depend at all on the initial polarization—a surprising feature that is particu- lar to the 1 transition in the perpendicular geometry and is not generic. This observation about the perpen- dicular geometry can be understood as follows. Since all Clebsch–Gordan coefficients are of equal size, the first scatterer has uniform a priori probability of reaching either one of its ground states, irrespective of the polarization of the incoming photon. Yet, when con- ditioned on the direction into which the photon is re- emitted, the probability is not uniform as a rule, but it is for the perpendicular geometry. Therefore, there is no which-path information stored in the final completely- mixed state of the atom that scatters first. When the observed polarization of the outgoing photon is an equal-weight superposition of the in-plane and out- of-plane linear polarizations (u′ = 0), the final state of the second scatterer is a corresponding pure state. So, when finding only one atom in this pure state, we can infer the path with certainty, but if both atoms are found in this state, we know nothing about the path and will guess wrong half of the time. Both cases are equally probable, so that our betting odds are 75%, which is consistent with D = 1 for u′ = 0 in (11), as it should be. In this case the visibility is V = 1 irrespective of the incoming polarization. The distinguishability is zero for an outgoing photon linearly polarized in the plane of the drawing in Fig. 1(a), when u′ = 1, or perpendicular to this plane (u′ = −1). The corresponding visibility takes on any value between 0 and 1. The case V = 0 happens for photons with per- pendicular polarizations, one with in-plane polarization, the other perpendicular (uu′ = −1). The case V = 1 oc- curs for photons with parallel linear polarizations, both in the plane or both perpendicular to it (uu′ = 1). Let us now turn to the situation of an initial singlet state (p = 1). As noted above, the duality relation (1) is then saturated and we have 1− u′2 , V = ∣ . (12) The fact that the distinguishability does not depend on the initial polarization can be understood by an argu- ment similar to the one given above for the p = 0 case. Irrespective of the incoming photon polarization and for, say, a left-circular outgoing photon, the final states of the atoms have (m1,m2) = ( ) for path A and ) for path B whereas the reversed situation occurs for a right-circular outgoing photon. This means that if the outgoing light is analyzed in the circular channels (u′ = 0), perfect path knowledge is available (D = 1) and no interference will be observed (V = 0). Conversely, if the outgoing light is analyzed in the channels of linear in- plane and out-of-plane polarization ( ∣ = 1), no path knowledge is available (D = 0) and one recovers full in- terference strength (V = 1). Finally, one can think of mimicking the physics of the CBS phenomenon by an angular average over the direc- tion n. We first calculate the average of the difference operator in (8) and then compute the resulting distin- guishability as the trace of its modulus. The correspond- ing visibility is obtained as the angular average of the n dependent visibility (7). For p = 0, which applies to most of the available experimental data, the largest averaged distinguishability is D = 1 ; it is found in the helicity- preserving polarization channel. The smallest average visibility is also found in this channel, its value is V = 2 Even if a direct quantitative comparison with the real CBS situation cannot be made at this stage, this result is nevertheless consistent with the experimental observa- tion that the lowest CBS peaks are actually found in this detection channel [14]. In summary, we have demonstrated that the concept of wave-particle duality proves relevant and useful for our understanding of some aspects of the interference effects in multiple scattering. To make solid quantitative con- tact with actual CBS experiments, the analysis must be extended to account for scattering by three and more atoms. The stage for this future research is now set. Ch. M. and C. M. wish to thank the Science Faculty and the Physics Department of NUS for their kind hospi- tality and financial support. This work was supported by by A∗Star Grant No. 012-104-0040, and by NUS Grant WBS: R-144-000-179-112. [1] See, e.g., B.-G. Englert and J.A. Bergou, Opt. Commun. 179, 337 (2000), and the references therein, in particular [3] and [4]. [2] U. Eichmann et al., Phys. Rev. Lett. 70, 2359 (1993); W.M. Itano et al., Phys. Rev. A 57, 4176 (1998). [3] G. Jaeger, A. Shimony, and L. Vaidman, Phys. Rev. A 51, 54 (1995). [4] B.-G. Englert, Phys. Rev. Lett. 77, 2154 (1996). [5] S. Dürr, T. Nonn, and G. Rempe, Nature (London) 395, 33 (1998); Phys. Rev. Lett. 81, 5705 (1998). [6] P.D.D. Schwindt, P.G. Kwiat, and B.-G. Englert, Phys. Rev. A 60, 4285 (1999). [7] E. Akkermans et al. (eds.), Mesoscopic Quantum Physics (North-Holland, Amsterdam 1995). [8] G. Bergmann, Phys. Rep. 107, 1 (1984). [9] D.S. Wiersma et al., Nature (London) 390, 671 (1997). [10] C.M. Aegerter, M. Störzer, and G. Maret, Europhys. Lett. 75, 562 (2006). [11] Y. Imry, arXiv:cond-mat/0202044 (2002). [12] F. Pierre et al., Phys. Rev. B 68, 085413 (2003). [13] G. Labeyrie et al., Phys. Rev. Lett. 97, 013004 (2006). [14] G. Labeyrie et al., Phys. Rev. Lett. 83, 5266 (1999). [15] T. Jonckheere et al., Phys. Rev. Lett. 85, 4269 (2000). [16] C. Müller et al., Phys. Rev. A 64, 053804 (2001). [17] G. Labeyrie et al., J. Opt. B: Quantum Semiclass. Opt. 2, 672 (2000). [18] Y. Bidel et al., Phys. Rev. Lett. 88, 203902 (2002). [19] C. Wickles and C. Müller, Europhys. Lett. 74, 240 (2006). [20] In the case of a 1 transition, T = (σ σ)− 31 has an admixture of the unit dyadic [22]. [21] Y. Lu, Wave-Particle Duality in Coherent Multiple Scat- tering , unpublished (2006). [22] G. Wang, Quantitative Wave-Particle Duality in Double Scattering Effect , unpublished (2007). [23] M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford 1980). http://arxiv.org/abs/cond-mat/0202044

Constraints on transmission, dispersion, and density of states in dielectric multilayers and stepwise potential barriers with arbitrary layer arrangement S. V. Zhukovsky1,2, ∗ and S. V. Gaponenko1, † 1Institute of Molecular and Atomic Physics, National Academy of Belarus, Nezavisimosti Ave. 70, 220072 Minsk, Belarus 2Physilakisches Institut, Universität Bonn, Nussallee 12, D-53115 Bonn, Germany Normal-incidence transmission and dispersion properties of optical multilayers and one-dimensional stepwise potential barriers in the non-tunneling regime are analytically investigated. The optical paths of every constituent layer in a multilayer structure, as well as the parameters of every step of the stepwise potential barrier, are constrained by a generalized quarter-wave condition. No other restrictions on the structure geometry is imposed, i.e., the layers are arranged arbitrarily. We show that the density of states (DOS) spectra of the multilayer or barrier in question are subject to integral conservation rules similar to the Barnett-Loudon sum rule but ocurring within a finite frequency or energy interval. In the optical case, these frequency intervals are regular. For the potential barriers, only non-periodic energy intervals can be present in the spectrum of any given structure, and only if the parameters of constituent potential steps are properly chosen. The integral conservation relations derived analytically have also been verified numerically. The relations can be used in dispersion-engineered multilayer-based devices, e.g., ultrashort pulse compressors or ultracompact optical delay lines, as well as to design multiple-quantum-well electronic heterostructures with engineered DOS. I. INTRODUCTION Over the centuries, the concept of homogeneity has played a major part in both mathematics and physics. The very name of a fundamental monograph on electrodynamics [1], Elec- trodynamics of Continuous Media, suggests that there should also exist electrodynamics of discontinuous media, quite dis- tinct and yet unexplored. Indeed, most real-world physical phenomena and processes are usually neither continuous nor homogeneous, and all seemingly homogeneous substances are in fact discontinuous on the molecular and atomic level. The reason why the concept of homogeneous media is applicable and produces good results in electrodynamics is that, in the first place, the microscopic structure is so much smaller than typical electromagnetic wavelengths that an effective-medium approximation is valid. Secondly, many macroscopic systems can be broken up into several homogeneous parts, the rela- tively large size of which making the studies of the whole sys- tem comparatively simple. The intermediate case of mesoscopic structures where in- homogeneities appear on the scale not minuscule enough to use an effective-medium approach but not too large to allow finite-size effects to be neglected has appeared more or less recently. This was largely motivated by the advancement of technology, allowing such structures to be fabricated and char- acterized. Even the first steps in this direction have already caused major advancements. The onset of semiconductor het- erostructures was a breakthrough in electronics, the pioneers in the area awarded the Nobel Prize in 2000 [2]. The intro- duction of quantum mesoscopic systems such as nanocrystals and quantum dots opens new horizons in many areas, includ- ing biological sensor design and solid-state quantum compu- tation (see, e.g., [3] and references therein). The introduction ∗Electronic address: sergei@th.physik.uni-bonn.de †Electronic address: s.gaponenko@dragon.bas-net.by of micro- and nanostructured optical materials has opened up whole new areas of photonic crystal research, integrated op- tics, and the newly-emerging metamaterial physics (see [4]), with innumerable applications in telecommunication. It appears that by arranging the matter in a mesoscopically structured fashion, one can engineer its properties (e.g., elec- tronic and/or optical) with considerable freedom. One can achieve as rich a variety as seen among natural substances due to a known diversity in their molecular-sized chemical compo- sition. This freedom is especially increased when the concept of structuring is extended beyond periodicity (see the recent review [5]), such as in quasiperiodic [6] or fractal media [7]. Often it is even possible to design a structured medium in or- der to achieve the chosen desired properties [8, 9]. Structured media can even exhibit optical properties beyond what occurs in natural materials, e.g., negative refraction [10] and the abil- ity to slow down or stop light pulses [11]. All physical phenomena that involve interaction between light and matter appear to be altered in inhomogeneous me- dia. This alteration is believed to be a fundamental physical principle involving modification of the properties of the vac- uum (electromagnetic or electronic) in the vicinity of inhomo- geneities. Such modification is generally described using the concept of the density of states (DOS) [12]; however, this con- cept is not without discussion points as regards definition of the DOS in finite vs. infinite media [13]. Despite those diffi- culties, the DOS concept appears a promising candidate for a universal approach towards consistent description of physical processes in arbitrarily inhomogeneous media. As an example, it should be clear that the above men- tioned modification of the vacuum cannot be totally arbitrary. Causality had been shown to restrict the modification of spon- taneous emission rate by spectral redistribution with the total emission rate remaining unaffected (the Barnett-Loudon sum rule [14]). Since spontaneous emission is related to the DOS, this rule can be expressed as impossibility to change the total “number” of states, but only to redistribute them spectrally, which appears to be intuitively clear and heuristically potent. In this paper we report on another, related limitation con- http://arxiv.org/abs/0704.1897v2 mailto:sergei@th.physik.uni-bonn.de mailto:s.gaponenko@dragon.bas-net.by cerning the modification of transmission and dispersion prop- erties of optical dielectric multilayers as well as of electronic heterostructures consisting of stepwise potential wells and barriers in the non-tunneling regime. For the optical case, making all constituent layers commen- surate in optical path produces a set of equidistant single- layer reflection-free (SLRF) points 2mω0 where (and only where) the dispersion relation of the structure coincides with that of a homogeneous medium (i.e., k(2mω0) ∝ 2mω0). We have found that the optical DOS integrated between these points does not depend on the structure geometry and does not change if the constituent layers (whose optical paths are commensurate with respect to each other) are rearranged. The degree of modification to the optical properties as due to inho- mogeneity of the structure is thus shown to be limited not only in its amount but also in its extent. This means that not only is a DOS enhancement in one spectral region compensated for in some other region, but also the compensation must occur within the distance 2ω0 between the SLRF points, which is a spectral interval preset by the structural parameters of the constituent element. For the stepwise potential, similar relations have been shown to exist. The single-layer reflection-free points do oc- cur but are no longer equidistant. For any given values of pa- rameters for constituent elements, there can be either none or a multitude of aperiodically located SLRF points for all struc- tures. In the latter case, the integral constraints on the DOS can still be obtained, but they are more complex. In both op- tical and quantum case, the analytical relations obtained have been confirmed in numerical calculations. The paper is organized as follows. In Sec. II we introduce the reader to the structures under study and provide the neces- sary basic notation. In Sec. III, we discuss the concept of the DOS and its relation to the spectral properties of the structure. In Secs. IV and V we derive the constraints on the DOS for optical multilayers and for binary stepwise potential barriers, respectively. In Sec. VI the results obtained in the previous sections are compared and discussed. Finally, Sec. VII sum- marizes the paper. II. OPTICAL AND ELECTRONIC HETEROSTRUCTURES We start by considering a one-dimensional dielectric mul- tilayer nanostructure of N layers, each layer having a thick- ness dj and a refractive index nj , infinite in the transverse directions and surrounded on both sides by free space (n0 = 1). Consider a normally incident plane monochromatic wave propagating through such a structure. This problem is one- dimensional, and unless the multilayer structure contains op- tically anisotropic materials, it can be described using scalar electric field governed by the scalar Helmholtz equation [4] E(x) + ε(x) E(x) = 0. (1) LetR(ω) and T (ω) denote the complex (i.e., taking into ac- count the phase shift) reflection and transmission coefficients of the multilayer structure, respectively. Let us now assume that all the layers have such parameters that the optical path njdj is the same for any j, so that n1d1 = n2d2 = · · · = njdj = · · · = nNdN ≡ πc/2ω0. (2) where ω0 is defined as the central frequency. We call any structure that conforms to Eq. (2) the quarter wave (QW) mul- tilayer structure. It can be shown that for any even multiple of ω0 the prop- agating wave passes each constituent layer without reflection (no internal reflections at the layer interfaces), and thus gains the phase shift ∆ϕ = ω njdj , which is the same for all lay- ers in view of the QW condition as expressed with Eq. (2). (See also [15] for more detail on phase relations in Fresnel reflection from one layer.) As a result, the structure becomes fully transparent (|T (2mω0)| = 1) regardless of the number or arrangement of constituent layers, and the total phase shift becomes a simple sum of the shifts for all the layers: T (ωm = 2mω0) = exp  = exp (iNmπ) . Eq. (3) essentially provides a set of equidistant frequency points [we will call them single-layer reflection-free (SLRF) points] where the propagation phase (and hence, the wave number) is linearly dependent on frequency regardless of the structure. Indeed, the dispersion relation at these points j=1 dj) km(ωm) = k(2mω0) = linearly relates km and ωm, as is the case for a homogeneous medium. This linear dependence occurs only at the set of SLRF points ωm = 2mω0, and Eqs. (3)–(4) are not true any- where between these points. Note that among all the transmis- sion resonances present in a given multilayer’s spectrum, the SLRF points represent stationary waves without any correla- tions on a length scale greater than the optical path of one con- stituent layer. As such, they are naturally the least localized non-evanescent eigenstates possible in any given QW multi- layer. Moreover, QW structures are known to possess spectral pe- riodicity in transmittance [16] |T (ω + 2mω0)| = |T (ω)| (5) and mirror symmetry within each period [16] |T ((2m+ 1)ω0 + ω)| = |T ((2m+ 1)ω0 − ω)| , 0 < ω < ω0. Now let us note that Eq. (1) is isomorphic to the Schrödinger equation governing a quantum particle with mass mp and energy Ep in a stepwise potential u(x) ψ(x) + [Ep − u(x)]ψ(x) = 0 (7) Figure 1: A binary optical multilayer (a) together with its quantum mechanical counterpart: a particle with energy Ep = ~ω in 1D periodic (b) and non-periodic (c) binary stepwise potential. If the potential function is a constant (u(x) = −u0), the solution of Eq. (7) is a plane-wave function ψ0(x) = exp ikx = exp Ep + u0 , (8) which is analogous to a plane optical wave with the wave vec- tor k = 2mp/~ Ep + u0. If Ep > u(x) for any x, then k is real and the particle interacts with a potential barrier in the non-tunneling (Ramsauer) regime. This is a quantum mechanical analogy to electromagnetic wave propagation in a dielectric structure. Similar to the optical case, one can con- sider a stepwise potential barrier consisting of N “elementary wells” (Fig. 1b,c). The role of refractive index is taken by the potential energy uj in every step of the whole potential func- tion. The frequency is replaced by the particle energy Ep, which can be expressed in terms of de Broglie frequency [17] as Ep = ~ω. The case Ep < u(x), which causes imaginary wave vector in Eq. (8), is the tunneling case analogous to light propagation in absorbing media (e.g., metals). It is outside the scope of the present paper. To assure Ep > u(x), let us assume Ep ≥ 0 and u(x) < 0 from now on. Furthermore, it is commonly known that if the potential represents a single step (u(x) = −u0 for, say, x < x0 and u(x) = −u1 otherwise), one can introduce the coefficients r01 = k0 − k1 k0 + k1 , t01 = k0 + k1 , (9) which, when squared, denote the probability of finding the impinging quantum particle reflected or transmitted, respec- tively [17]. One can name them the reflection and transmis- sion coefficient for matter waves, a potential step correspond- ing to a single interface between dielectric media. Since the expressions for r and t are the same (the wave vector k taking the role of refractive index), one can use the same transfer- matrix formalism for determining both the stationary electro- magnetic wave distribution in a multilayer [18] and the steady- state wave function for quantum particles travelling through a complex stepwise potential [19]. Note that Eq. (8) indicates that the “refractive index” introduced in this way possesses an inherent quadratic dispersion. Finally, we call a multilayer structure binary if it can be rep- resented as consisting of two types of constituent layers (de- noted as binary digits 0 and 1, following the notation in [20]), to which two combinations of refractive index and thickness (n0; d0) and (n1; d1) are attributed. By arranging the 0’s and 1’s in different sequences, it is possible to vary the geometry of the structure very widely, making it periodic (if 0 and 1 alternate, as in 10101010101), disordered (if the sequence is randomly determined), or deterministically aperiodic (e.g., quasiperiodic [6] or fractal [7]). A binary potential barrier, with constituent elementary wells associated with (u0; d0) and (u1; d1), can be introduced likewise. For brevity, we will oc- casionally use the term “layers” for both types of constituent elements. Note that whenever the sequence contains two identical lay- ers (e.g., “00” in 1010100101) , it will of course mean in prac- tice that the corresponding structure will contain a single layer with thickness 2d0. However, for the purpose of this work we will regard such combinations as two separate constituent lay- ers. The number of layers of both typesN0 andN1, as well as their total number N = N0 + N1, will then remain the same regardless of layer rearrangement, indicative of the transmis- sion coefficient phase at the SLRF points [see Eq. (3)]. III. OPTICAL AND QUANTUM DENSITY OF STATES As mentioned in Section I, any inhomogeneity present in space is known to modify the properties of the quantum or electromagnetic vacuum in its vicinity. This modification takes the form of the change in the local DOS. It is believed to affect all phenomena that involve light-matter interaction, such as spontaneous emission or Raman scattering (see, e.g., [12] and references therein). Physically, the local DOS N (r, E) is directly related to the trace of Green’s function for the system in question: N (r, E) ∝ Im Tr G(r, r, E). By taking the integral Green’s functionG(E) in place of the local one, one obtains the value of the DOS N (E) that is characteristic to the whole system for a given value of energy. In a spatially finite system (a potential well with infinite walls or a closed resonator), only the states with a discrete set of energy (or frequency) eigenvalues are allowed. Supposing that these eigenvalues are dense enough, this integral DOS has a rather loose mathematically but very intuitive meaning of the number of these discrete states per unit energy. In an open resonator, any value of energy corresponds to an eigenstate, and the DOS transforms into a continuous spec- trum N (E), indicative of spectral characteristics for the over- Figure 2: (Color online) Transmittance |T (ω)|2 (dashed line) and normalized averaged local DOS ρ(ω)v0 [as of Eqs. (15) and (18), solid line] for multilayer structures of different geometry: (a) single 9d1-thick layer; (b) 3-layer structure; (c) 9-layer periodic structure; (d) Fabry-Pérot- like periodic structure with a half-wave defect; (e) coupled-defect structure; (f) fractal Cantor-like structure. All multilayers have N = 9. The vertical scale is chosen alike for all plots for the ease of comparison, the insets showing the scale of clipped peaks. The area [0, 1] is shaded to show the allowed region for transmittance, as well as to provide a visual guide for estimating the integral of ρ(ω) [see Eq. (19)]. all vacuum modification. It had been shown [21, 22, 23] that a finite-sized inhomogeneous potential located in an infinite 1D space causes the local DOS integrated over the whole space to undergo an overall finite modification ∆N (E) ≡ [N (x,E) −N0(x,E)] dx. Note that we are consider- ing a finite-sized inhomogeneity in an infinite space (an open resonator), as opposed to a finite system isolated from the out- side space (a closed resonator). In the former, ∆N was found to be proportional to the derivative of the total phase accumu- lated by the wave packet during its transmission through the inhomogeneity: ∆N (E) = (1/π) (dϕ/dE) , (10) which, as seen from Eq. (8), becomes asymptotically zero for very large energies compared to the potential (i.e., if nothing gets in the particle’s way). A similar expression can be used to determine the local density of electromagnetic states, also called the optical DOS (for details on its definition in 2D and 3D case, see Refs. [13, 24]). The transition from local to in- tegral DOS can be made in a similar manner to the quantum system. In the 1D case (the wave propagation in a multilayer is a 1D problem when only normal-incidence states are taken into account) the modification to the optical DOS ∆N (ω) is also likewise related to the derivative of transmission phase [25]: ∆N (ω) = (1/π) (d(ϕ− ϕ0)/dω) . (11) The subtraction of the free-space phase shift ϕ0 ensures that ∆N (ω) = 0 in free space. In the work by Barnett and Loudon [14] it has been shown that the modification of spon- taneous emission rate Γ by inhomogeneous medium (as com- pared to the free-space emission rate Γ0) integrated over the whole spectrum must be zero (the Barnett-Loudon sum rule): Γ(r, ω) Γ0(ω) dω = 0. (12) The emission rate Γ is proportional to the local DOS N , and the local optical DOS is frequency independent in free space (see [12]). Spectral integration in Eq. (12) can be interchanged with spatial integration over Γ(r, ω). Hence, a similar relation holds for the integral DOS: ∆N (ω)dω = 0. (13) The transition from local to integral DOS, as well as the transition from∆N to N , involves renormalization and there- fore may be ambiguous. A simple way to counter the diver- gence is to accompany the transition from ϕ − ϕ0 to ϕ with normalizing the DOS by the total thickness of the inhomoge- neous medium D (see [26]): ρ(ω) ≡ π∆N (ω) , (14) The authors in [26] simply define ρ as the optical DOS without going into further details. We can see that it is in fact neither local nor integral, but rather has the meaning of local DOS modified by a finite inhomogeneity, averaged in infinite 1D space. In the absence of any inhomogeneity, Eq. (14) gives ρ(ω) = 1/c, a known value for the DOS in 1D free space. Further, Eq. (14) can be used to calculate ρ(ω) from the complex transmission coefficient T (ω) as ρ(ω) = [Im T (ω)] Re T (ω)− Im T (ω) [Re T (ω)]′ D |T (ω)|2 , (15) the derivation taken with respect to ω. One must realize that the concept of the DOS introduced as in Eqs. (10)–(11), and especially, as in Eq. (14), is not without controversy. Questions arise already as to the physical mean- ing of the quantities involved. For example, one can define a “wave number” k a posteriori from the phase of the transmit- ted wave k(ω) = Arg T (ω) , (16) which would equal the actual wave number in a homogeneous medium, or the Bloch wave number in a periodic multilayer at transmission resonances [26]. In such special cases, ρ would equal the inverse group velocity (ρ(ω) = (dω/dk)−1), the latter also equal to the energy velocity. In the general case of non-periodic structures, however, the concepts of phase, group, and energy velocity, as well as their mutual correspondence, need to be re-examined. For instance, the “phase time” defined as dϕ/dω is, in general, not equal to the pulse’s actual “dwell time” (see [25, 27]), although, ad- mittedly, both have a similar frequency dependence and in some cases the phase time is a very good approximation for the dwell time [28]. That said, it is safer not to assign any direct physical meaning to k defined in Eq. (16) in the general case. We will thus treat it like a parameter within the scope of the present paper. Another point is that the applicability of 1D models for electromagnetic problems is in general of limited value. The reason is that reduction of Maxwell’s equations to the scalar wave equation (1), e.g., for multilayered media does not re- ally make the problem entirely one-dimensional. In reality one deals with finite-width beams rather than plane waves and with excited atoms that can emit in any direction. The lat- eral width of the multilayers is finite, too. As pointed out in Ref. [29], the optical DOS reduces to the expression (14) only if off-axis wave propagation is totally left out. That kept in mind, the function ρ(ω) nevertheless under- goes the same dramatic variation as does the transmittance itself when the structure geometry is varied (see Fig. 2). The peaks in |T (ω)|2 and ρ(ω) obviously correspond to each other. Note that this correspondence is a physical property of multi- layers rather than just a mathematical property of Eqs. (14)– (15). Indeed, one can show analytically that ρ(ω) ∝ |T (ω)|2 for any single layer. This proportionality is due to the phase structure of the Airy formulas, and is obviously not there for arbitrary ϕ(ω) in Eq. (14). Numerical calculations confirm that spectral features in |T (ω)|2 and ρ(ω) also correspond for an N -layer structure, e.g., as seen in Fig. 2. It should be pos- sible to show this analytically by induction but it is outside the scope of the present paper. We note instead that the same correspondence was observed in higher-dimensional systems (e.g., in slab photonic crystals [30]). Besides, one can observe that the sharper is the transmis- sion resonance around some frequency ωr, the larger is the value of ρ(ωr). Sharper transmission resonances correspond to stationary waves with greater energy localization, and it takes longer for greater energies to build up inside the struc- ture. Hence, it takes longer for resonant transmission to mani- fest in such cases. Therefore the maxima of ρ(ω) are just those points where delayed light propagation is likely to be experi- enced. The DOS spectrum is thus valuable as a quick visual guide for determining the resonant behavior of any multilayer structure, as employed earlier [31]. Finally, let us note that although k in Eq. (16) cannot be as- signed a direct physical meaning in the general case, it can be used as a parameter, which can provide some heuristic guid- ance in experiments on the group velocity dispersion-related effects (e.g., pertaining to propagation, compression, delay, and chirp compensation of ultrashort laser pulses). For some examples involving non-periodic structures, the reader is re- ferred to Refs. [32, 33, 34]. IV. CONSTRAINTS IN MULTILAYERS In the previous section, the use of ρ(ω) defined by Eq. (15) as a meaningful characteristic of the structure’s optical prop- erties has been motivated. It was demonstrated that ρ(ω) can be strongly modified by altering the geometry of the structure (Fig. 2). We proceed to show that the degree of geometry- induced modification imposed on ρ has fundamental limita- tions. One of these is the Barnett-Loudon sum rule – Eq. (13) holds both for the quantum mechanical and for the electro- magnetic case when the corresponding expression for ∆N (ω) is used. If the medium is a QW multilayer, the constraint becomes stricter and involves integration over finite rather than infi- nite frequency intervals. As the transmission properties in the SLRF points are given by Eq. (3), the integral of ρ(ω) be- tween those points can be evaluated explicitly using Eqs. (16) Figure 3: (Color online) The schematic frequency dependence of propagation phase for (a) optical waves [Eq. (23)] and (b) quantum wave function [Eq. (22)] in a slab of homogeneous dielectric and in a length of constant potential, respectively. and (3)–(4) (see also [16]): 2(m+1)ω0 ρ(ω)dω = k[2(m+1)ω0] k[2mω0] , (17) which holds regardless of the geometrical arrangement of the constituent layers in the structure, provided that the layers obey the relation (2). One can further introduce the “bulk velocity prameter” from the minimum time it takes light to traverse the multilayer, internal reflections neglected, as (djnj/c) . (18) This is a parameter independent either of ω or of the layer arrangement of the structure. Making a transition to the di- mensionless frequency η ≡ ω/ω0 and taking into account the symmetry condition (6), we arrive at v0ρ(η)dη = 1 (19) and, further, since v0 = c and ρ0 = 1/c in free space, ∆ [v0ρ(η)] dη = (v0ρ(η)− cρ0) dη = 0 (20) for any integer m ≥ 0. The conditions (19)–(20) have no less a universal charac- ter than Eq. (13). They physically mean that the modification of the transmission or dispersion properties due to layer rear- rangement in QW multilayers is only possible within a finite frequency range ω0. One can see in Fig. 2 that despite appar- ently dramatic modification of ρ(ω), the enhancement in one portion of the spectrum appears compensated by a gap in an- other portion, so that the overall DOS, integrated between the SLRF points, remains unaffected. It is also important to real- ize that within ω0, one can achieve any desired spectral shape, given the sufficient number of layers and sufficient freedom in their arrangement. For example, a heuristic optimization algorithm was recently used to demonstrate that certain ape- riodic sequences can be employed to fabricate structures with desired spectral properties [9]. Note, too, the inverse proportionality between ω0 and the optical path of the constituent layers. It follows that if the QW condition (2) is broken but the quantities njdj all remain commensurate, the same reasoning can be applied. Eqs. (17) and (19) can then be obtained by subdivision of the constituent layers, accompanied by the according increase in the central frequency (ω0 → Nω0). In the limiting case of mathemati- cally incommensurate layers,N goes to infinity, and the struc- ture appears to possess the same freedom as a continuously inhomogeneous medium would, retaining only asymptotic re- lation ρ(ω)dω = , (21) associated with increasing ω0 to infinity in Eq. (20), and con- sequently, representing the sum rule (13). V. CONSTRAINTS IN POTENTIAL BARRIERS The concept of optical DOS appeared in electrodynamics largely by the influence of the quantum DOS. Such transfer of concepts makes use of the analogy between the Helmholtz (1) and Schrödinger (7) equations, as well as between a mul- tilayer and a potential barrier, as outlined in Sec. II. In this section, we will attempt to make these analogies work back- wards and determine if, and to what extent, the relation (17) can be generalized to the quantum mechanical case. Consider a binary stepwise potential and the particle with mass mp and energy Ep ≡ ~ω interacting with it in the non- tunneling regime (Fig. 1b,c). Making use of the expression (8) for k, we can derive the frequency dependence for prop- agation phase of the particle’s wave function for constant po- tential corresponding to one elementary potential well. Com- pared to the same dependence for an optical wave in a homo- geneous slab corresponding to one constituent layer, it has the form (α ≡ 2mp/~) ϕ(qm)(ω) ≡ k(qm)(ω)d = dα ~ω + u; (22) ϕ(opt)(ω) = ndω/c. (23) Fig. 3 schematically shows both these dependencies. In the optical case (Fig. 3a) the only variable parameter is the slope given by nd. Hence, with the aid of Eq. (2) it becomes possi- ble to achieve exactly the same dispersion relation, and hence the same set of SLRF points, for both constituent layers when n0 6= n1. This is what forms the foundation for reasoning pre- sented in the previous section. In the quantum case (Fig. 3b) u and d are seen to contribute in an essentially different way. Is is thus not possible to arrive at the same dispersion relation for two different potentials (u0 6= u1). However, one can still define a set of frequency points (though no longer equidistant) where ϕ(qm)(ωj) = jπ. In these points, as can be seen from Eqs. (8) and (9), the whole structure would be totally “transparent” for incoming quan- tum particles (the Ramsauer effect). If the structure is binary, the frequencies for both kinds of elementary wells are given (0,1) = j20,1π 2/d20,1α 2 − u0,1. (24) Since two different parabolic curves can still have intersec- tion points, one can manage to achieve ω for two pairs of j0 and j1. The reasoning presented in the previous section can then be reproduced involving the quantity ρ de- fined exactly as in Eqs. (14) and (17) and having the same physical background. The dependence on ω, however, will be more complex due to inherent dispersion as seen in Eq. (8). For simplicity and for the sake of further analogy between optical and quantum systems, let us require one of the equal frequency pairs in Eq. (24) to correspond to ω = 0 (which is always true for optical waves where all dispersion curves pass trough the origin, see Fig. 3a). In this case we arrive at α2d20u0 = α 2d21u1 = j 2, (25) which can be seen as a quantum analogy to the condition (2). The second pair (j0, j1) can then be found as an integer solu- tion of the equation (first suggested in our earlier work [35]) j20 − 1− β/j2qw j21 = β, j0,1 > jqw; β ≡ (u0 − u1)α2d20/π2. It can be seen that for any integer j0,1 > jqw there is a ratio- nal β that solves Eq. (26). But β is related to the parameters of the constituent potential wells. So, the inverse problem, i.e, finding suitable j0,1 for a given β, is far more interesting from a physical point of view. However, is not so straightforward and is mathematically related to finding Pythagorean triples in integer numbers. One can confirm numerically that there are a multitude of solutions for many rational values of β (see Table I). Some of them can be represented via recurrent rela- tions, e.g., for β = 1 some of the solutions represent a series: (i+1) 1 = j 1 + 6 + 4i, j 0 = j 1 − 1, (27) where j 1 = j (min) qw = 2. Other cases are more complex, but they, too, can be seen to form distinct solution branches (Fig. 4). Once j0 and j1 have been found, an analogous relation to Eq. (17) can be formulated as 0=~ωjqw ρ(E)dE = [N0(j0 − jqw) +N1(j1 − jqw)] . Note that Eq. (28) is more complicated than its optical counterpart (17), and becomes, in general, dependent on the number of constituent potential wells N1,2. This dependence cannot be eliminated because one sees from Eq. (26) that it is impossible to have j0 = j1 without violating the assertion that j0,1 > jqw. It is still, however, completely independent of layer rearrangement. In this sense, Eq. (26) represents a uni- versal quantum mechanical conservation relation for the DOS over a finite energy interval. To demonstrate the results numerically, we have considered a stepwise AlAs/GaAs quantum well (∆u = 1000 meV). To aim at β = 4/5, we have taken d0 = 21.2 Å, d1 = 47.4 Å, according to Eqs. (25)–(26). One possible solution of Eq. (26) would then be jqw = 1, j0 = 2, j1 = 4 (see Table I). The structures made of N = 9 elementary wells were used, and the numbersN0 andN1 were fixed, too, at the values 4 and 5, respectively. We see in Fig. 5 that both ρ(E) and the transmission T (E) are subject to quite a strong modification. It resembles the modification seen in dielectric multilayers (compare, e.g., Fig. 5a,b with Fig. 2a,c). Two differences are the presence of a decaying background due to the inherent dispersion [see Eq. (22)] and the lack of periodicity because Eq. (2) can no longer be satisfied. However, if we integrate ρ(E) between the SLRF points (~ωjqw = 0 and ~ωj0 = ~ωj1 = 3.75 eV) as provided by Eq. (26), we can see that the integral does not change when the layers are rearranged. Table II provides the results for nine different structures and for several upper integration limits. It can be seen that both below and above 3.75 eV the integrals vary from structure to structure. When, however, the correct integration limits are chosen, the difference vanishes and all integrals equal 19, which is the right-hand side of Eq. (28) for the chosen values of parameters. VI. DISCUSSION The equations (17)–(20) and (28) constitute the main result of this paper pertaining to optical and electronic heterostruc- tures, respectively. In both cases, we are dealing with conser- vation of the DOS ρ integrated across a finite energy or fre- quency region. As discussed in Sec. III, ρ represents the av- eraged local DOS as modified by the presence of finite-sized inhomogeneous structure in an infinite 1D free space. It is re- lated to the dispersion and the transmission properties of the heterostructures in question [see Eqs. (14)–(15)]. Table I: Some values of β that allow integer solutions of Eq. (26), along with some of such solutions obtained numerically. β < 1 (jqw : j0, j1) β ≥ 1 (jqw : j0, j1) 1/4 (1 : 13, 15); (1 : 181, 209); (2 : 122, 126) 1 (2 : 7, 8); (2 : 26, 30); (3 : 17, 18); (3 : 99, 105); (4 : 31, 32) 1/3 (1 : 9, 11); (1 : 89, 109); (2 : 90, 94) 3/2 (2 : 8, 10); (2 : 68, 86); (3 : 63, 69) 1/2 (1 : 5, 7); (1 : 29, 41); (2 : 58, 62); (3 : 207, 213) 2 (2 : 10, 14); (2 : 58, 82); (3 : 45, 51) 2/5 (1 : 7, 9); (1 : 55, 71); (2 : 74, 78) 4 (3 : 7, 9); (3 : 18, 24); (4 : 14, 16); (4 : 52, 60); (5 : 23, 25) 2/3 (1 : 3, 5); (1 : 11, 19); (2 : 42, 46); (3 : 153, 159) 6 (3 : 9, 15); (3 : 33, 57); (4 : 16, 20) 4/5 (1 : 2, 4); (1 : 5, 11); (1 : 13, 19); (2 : 34, 38) 10 (4 : 8, 12); (4 : 32, 52); (5 : 35, 45) Figure 4: The distribution of integer solutions j0,1 of Eq. (26) for (a) β = 1, (b) β = 4, and (c) β = 4/5. Distinct solution groups (“branches”) can be seen. These properties, as well as the DOS, can undergo dramatic modification as compared to those of homogeneous media (see Figs. 2 and 5) because a multilayer structure or a step- wise potential barrier can be very complex. Nevertheless, the modification appears to be limited both in its amount [see the right hand side of Eqs. (17) and (28)] and in its extent (by the finite integration limits in those equations). There is an elegant physical explanation for the relations obtained. By engineering the geometrical properties of an in- homogeneous structure, it is only posible to redistribute the Table II: Numerically evaluated integrals (D/~π) ρ(E)dE [as in Eq. (28)] from 0 to several upper energy values for nine structures with N = 9, N0 = 4, and N1 = 5 (same as in Fig. 5). Standard deviation of the values across all structures for each upper integra- tion limit is provided in the lowest row. The limit of 3.75 eV (the obtained value of the SLRF point) is accompanied by a drop in stan- dard deviation down to 10−8, which falls within accuracy limits for numerical integration. Structure 0. . . 1 eV 3 eV 3.75 eV . . . 5 eV 001111100 7.5531 16.4036 19.0000 22.8568 010111100 7.5526 16.3991 19.0000 22.8593 100111100 7.5481 16.3982 19.0000 22.8582 110011100 7.5207 16.4016 19.0000 22.8574 010111010 7.5890 16.3991 19.0000 22.8600 100110011 7.5177 16.4048 19.0000 22.8592 100111001 7.5198 16.4017 19.0000 22.8512 110010011 7.5373 16.3996 19.0000 22.8603 101010101 7.5880 16.3982 19.0000 22.8657 Std. deviation 0.027 0.0045 8× 10−9 0.0038 available electromagnetic or quantum states across the spec- trum, but impossible to alter the “total number” of the states. The latter turns out to be related to the size or “1D volume” of the structure [as seen by the presence of N at the right-hand side of Eqs. (17) and (28)] and represents integrated character- istics of the structure-affected vacuum. So, an enhancement of the DOS in some pars of the spectrum (like the band edge res- onances for a periodic structure in Fig. 2c) giving rise, e.g., to the spontaneous emission enhancement, is inevitably accom- panied by a suppression of the DOS in other spectral region (like the band gap in the same figure), leading to the inhibition of light propagation and all phenomena involving light-matter interaction [4]. In this sense, the results obtained resemble already known constraints on the DOS like the Barnett-Loudon sum rule (13). However, in the relations obtained in this work the integra- tion involved is finite rather than infinite. For the optical case, this means a tighter restriction on the spectral redistribution of the DOS. The compensation of suppression snd enhancement must occur within the frequency interval ω0. This interval is determined solely by the elementary constituent element of the structure in question [see Eq. (2)]. It is totally indepen- dent of geometrical arrangement of these elements. In other words, the QW condition (2) enforces the existence of certain points in the spectrum (the SLRF points) that cannot be “tran- scended” by electromagnetic states that are “pushed around” the spectrum by layer rearrangement. On the other hand, the spectral properties of the structure can be arbitrary everywhere between the SLRF points (3). It should also be noted that the increase of N causes the details in the spectra to become finer, and the variation of T (ω) and ρ(ω) to get more rapid. These results can help to understand the underlying physics of complex media. Similar conclusions can be formulated for a quantum par- Figure 5: The quantum averaged DOS ρ(E) (top) and transmittance ˛T (E)2 ˛ (bottom) for an AlAs/GaAs quantum well and β = 4/5 in three stepwise potential wells differing only by the elementary well rearrangement: (a) single-layer structure; (b) periodic structure; (c) non-periodic structure. The portion between the SLRF points [0. . . 3.75 eV] is depicted, which corresponds to a solution of Eq. (26) for jqw = 1, j0 = 2, j1 = 4. ticle in a stepwise potential barrier. However, the inherent quadratic dispersion as seen in Eqs. (8) and (22) results in many differences. First and foremost, the SLRF points are no longer guaranteed. Not only a relative restriction on con- stituent elements (25) analogous to the QW condition (2) is required, but also individual constraints on u0,1 and d0,1 are necessary, so as to provide special values of β as determined by Eq. (26). This makes the binarity of the structure an impor- tant requirement in contrast to the optical case where Eqs. (2) and (23) can be extended to as many kinds of constituent ele- ments as needed. Because the equation (26) is quadratic rather than linear, the SLRF points occur far more seldom than in the optical case and are no longer equidistant. However, they still do occur on a regular basis if they occur at all for a given choice of parameters, as seen in Fig, 4. In this way, we have provided a way for engineering an electronic heterostructure where the DOS modification due to structure complexity is confined in a finite spectral region. The structure itself can be arbitrarily complex because Eqs. (24)–(26) do not depend on N in any way. To conclude this section, let us note that the structures in question appear to possess other conservation relations. As can be seen, e.g., in Fig. 2a–c, the transmission spectra contain the same number of resonance peaks in the interval [0; 2ω0], namely, nine, which equals the number of constituent layers. Bearing a loose resemblance to the energy level splitting in solids if one regards the layers as “atoms”, this was found to be a general property of such multilayers [16]. However, additional restrictions on the structures seem to be necessary, such as the outermost layers of the structure being 1 rather than 0 (compare, e.g., Figs. 2d, e). This requires additional investigations and remains a subject for further studies. VII. CONCLUSIONS AND OUTLOOK To summarize, we have investigated the possible degree of modification to transmission and dispersion properties, as well as the averaged local DOS, in discretely inhomogeneous me- dia. Both electromagnetic waves propagating in a dielectric multilayer structure and a quantum particle propagating over a stepwise, multiple-well potential barrier, have been consid- ered (Fig. 1). In both cases, certain constraints on the con- stituent elements of the structure [Eqs. (2) and (25)] allow to derive the conservation relations over finite frequency or en- ergy regions [Eqs. (17)–(20) and (28), respectively]. Both re- lations hold regardless of the structure geometry (at least in the sense of rearrangement of constituent elements) and are, in this sense, universal, despite the fact that the spectral prop- erties themselves can be strongly geometry-dependent. The quantum case appears to be more complicated than the opti- cal one and requires more conditions to be fulfilled, as im- plied by a quadratic character of Eq. (26). The analytical re- sults obtained have been verified by numerical calculations (see Figs. 2, 5, and Table II). The results obtained contribute to understanding the physics of complex inhomogeneous media. They can be ap- plied in the design of heterostructures with engineered disper- sion, such as chirp compensation, pulse compression or delay line devices. A more detailed studies of the relations obtained would also be useful. It would be of interest to find out if, and to what extent, the results can be applied to the case of optical multilayers made of dispersive and/or absorptive materials, as well as for potential barriers in the tunneling regime. Acknowledgments The authors are thankful to J. P. Dowling, H. V. Demir, D. N. Chigrin, E. P. Petrov, and A. V. Lavrinenko for stimulat- ing discussions, as well as to C. Kremers for helpful advice. 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Normal-incidence transmission and dispersion properties of optical multilayers and one-dimensional stepwise potential barriers in the non-tunneling regime are analytically investigated. The optical paths of every constituent layer in a multilayer structure, as well as the parameters of every step of the stepwise potential barrier, are constrained by a generalized quarter-wave condition. No other restrictions on the structure geometry is imposed, i.e., the layers are arranged arbitrarily. We show that the density of states (DOS) spectra of the multilayer or barrier in question are subject to integral conservation rules similar to the Barnett-Loudon sum rule but ocurring within a finite frequency or energy interval. In the optical case, these frequency intervals are regular. For the potential barriers, only non-periodic energy intervals can be present in the spectrum of any given structure, and only if the parameters of constituent potential steps are properly chosen. Abstract The integral conservation relations derived analytically have also been verified numerically. The relations can be used in dispersion-engineered multilayer-based devices, e.g., ultrashort pulse compressors or ultracompact optical delay lines, as well as to design multiple-quantum-well electronic heterostructures with engineered DOS.

Constraints on transmission, dispersion, and density of states in dielectric multilayers and stepwise potential barriers with arbitrary layer arrangement S. V. Zhukovsky1,2, ∗ and S. V. Gaponenko1, † 1Institute of Molecular and Atomic Physics, National Academy of Belarus, Nezavisimosti Ave. 70, 220072 Minsk, Belarus 2Physilakisches Institut, Universität Bonn, Nussallee 12, D-53115 Bonn, Germany Normal-incidence transmission and dispersion properties of optical multilayers and one-dimensional stepwise potential barriers in the non-tunneling regime are analytically investigated. The optical paths of every constituent layer in a multilayer structure, as well as the parameters of every step of the stepwise potential barrier, are constrained by a generalized quarter-wave condition. No other restrictions on the structure geometry is imposed, i.e., the layers are arranged arbitrarily. We show that the density of states (DOS) spectra of the multilayer or barrier in question are subject to integral conservation rules similar to the Barnett-Loudon sum rule but ocurring within a finite frequency or energy interval. In the optical case, these frequency intervals are regular. For the potential barriers, only non-periodic energy intervals can be present in the spectrum of any given structure, and only if the parameters of constituent potential steps are properly chosen. The integral conservation relations derived analytically have also been verified numerically. The relations can be used in dispersion-engineered multilayer-based devices, e.g., ultrashort pulse compressors or ultracompact optical delay lines, as well as to design multiple-quantum-well electronic heterostructures with engineered DOS. I. INTRODUCTION Over the centuries, the concept of homogeneity has played a major part in both mathematics and physics. The very name of a fundamental monograph on electrodynamics [1], Elec- trodynamics of Continuous Media, suggests that there should also exist electrodynamics of discontinuous media, quite dis- tinct and yet unexplored. Indeed, most real-world physical phenomena and processes are usually neither continuous nor homogeneous, and all seemingly homogeneous substances are in fact discontinuous on the molecular and atomic level. The reason why the concept of homogeneous media is applicable and produces good results in electrodynamics is that, in the first place, the microscopic structure is so much smaller than typical electromagnetic wavelengths that an effective-medium approximation is valid. Secondly, many macroscopic systems can be broken up into several homogeneous parts, the rela- tively large size of which making the studies of the whole sys- tem comparatively simple. The intermediate case of mesoscopic structures where in- homogeneities appear on the scale not minuscule enough to use an effective-medium approach but not too large to allow finite-size effects to be neglected has appeared more or less recently. This was largely motivated by the advancement of technology, allowing such structures to be fabricated and char- acterized. Even the first steps in this direction have already caused major advancements. The onset of semiconductor het- erostructures was a breakthrough in electronics, the pioneers in the area awarded the Nobel Prize in 2000 [2]. The intro- duction of quantum mesoscopic systems such as nanocrystals and quantum dots opens new horizons in many areas, includ- ing biological sensor design and solid-state quantum compu- tation (see, e.g., [3] and references therein). The introduction ∗Electronic address: sergei@th.physik.uni-bonn.de †Electronic address: s.gaponenko@dragon.bas-net.by of micro- and nanostructured optical materials has opened up whole new areas of photonic crystal research, integrated op- tics, and the newly-emerging metamaterial physics (see [4]), with innumerable applications in telecommunication. It appears that by arranging the matter in a mesoscopically structured fashion, one can engineer its properties (e.g., elec- tronic and/or optical) with considerable freedom. One can achieve as rich a variety as seen among natural substances due to a known diversity in their molecular-sized chemical compo- sition. This freedom is especially increased when the concept of structuring is extended beyond periodicity (see the recent review [5]), such as in quasiperiodic [6] or fractal media [7]. Often it is even possible to design a structured medium in or- der to achieve the chosen desired properties [8, 9]. Structured media can even exhibit optical properties beyond what occurs in natural materials, e.g., negative refraction [10] and the abil- ity to slow down or stop light pulses [11]. All physical phenomena that involve interaction between light and matter appear to be altered in inhomogeneous me- dia. This alteration is believed to be a fundamental physical principle involving modification of the properties of the vac- uum (electromagnetic or electronic) in the vicinity of inhomo- geneities. Such modification is generally described using the concept of the density of states (DOS) [12]; however, this con- cept is not without discussion points as regards definition of the DOS in finite vs. infinite media [13]. Despite those diffi- culties, the DOS concept appears a promising candidate for a universal approach towards consistent description of physical processes in arbitrarily inhomogeneous media. As an example, it should be clear that the above men- tioned modification of the vacuum cannot be totally arbitrary. Causality had been shown to restrict the modification of spon- taneous emission rate by spectral redistribution with the total emission rate remaining unaffected (the Barnett-Loudon sum rule [14]). Since spontaneous emission is related to the DOS, this rule can be expressed as impossibility to change the total “number” of states, but only to redistribute them spectrally, which appears to be intuitively clear and heuristically potent. In this paper we report on another, related limitation con- http://arxiv.org/abs/0704.1897v2 mailto:sergei@th.physik.uni-bonn.de mailto:s.gaponenko@dragon.bas-net.by cerning the modification of transmission and dispersion prop- erties of optical dielectric multilayers as well as of electronic heterostructures consisting of stepwise potential wells and barriers in the non-tunneling regime. For the optical case, making all constituent layers commen- surate in optical path produces a set of equidistant single- layer reflection-free (SLRF) points 2mω0 where (and only where) the dispersion relation of the structure coincides with that of a homogeneous medium (i.e., k(2mω0) ∝ 2mω0). We have found that the optical DOS integrated between these points does not depend on the structure geometry and does not change if the constituent layers (whose optical paths are commensurate with respect to each other) are rearranged. The degree of modification to the optical properties as due to inho- mogeneity of the structure is thus shown to be limited not only in its amount but also in its extent. This means that not only is a DOS enhancement in one spectral region compensated for in some other region, but also the compensation must occur within the distance 2ω0 between the SLRF points, which is a spectral interval preset by the structural parameters of the constituent element. For the stepwise potential, similar relations have been shown to exist. The single-layer reflection-free points do oc- cur but are no longer equidistant. For any given values of pa- rameters for constituent elements, there can be either none or a multitude of aperiodically located SLRF points for all struc- tures. In the latter case, the integral constraints on the DOS can still be obtained, but they are more complex. In both op- tical and quantum case, the analytical relations obtained have been confirmed in numerical calculations. The paper is organized as follows. In Sec. II we introduce the reader to the structures under study and provide the neces- sary basic notation. In Sec. III, we discuss the concept of the DOS and its relation to the spectral properties of the structure. In Secs. IV and V we derive the constraints on the DOS for optical multilayers and for binary stepwise potential barriers, respectively. In Sec. VI the results obtained in the previous sections are compared and discussed. Finally, Sec. VII sum- marizes the paper. II. OPTICAL AND ELECTRONIC HETEROSTRUCTURES We start by considering a one-dimensional dielectric mul- tilayer nanostructure of N layers, each layer having a thick- ness dj and a refractive index nj , infinite in the transverse directions and surrounded on both sides by free space (n0 = 1). Consider a normally incident plane monochromatic wave propagating through such a structure. This problem is one- dimensional, and unless the multilayer structure contains op- tically anisotropic materials, it can be described using scalar electric field governed by the scalar Helmholtz equation [4] E(x) + ε(x) E(x) = 0. (1) LetR(ω) and T (ω) denote the complex (i.e., taking into ac- count the phase shift) reflection and transmission coefficients of the multilayer structure, respectively. Let us now assume that all the layers have such parameters that the optical path njdj is the same for any j, so that n1d1 = n2d2 = · · · = njdj = · · · = nNdN ≡ πc/2ω0. (2) where ω0 is defined as the central frequency. We call any structure that conforms to Eq. (2) the quarter wave (QW) mul- tilayer structure. It can be shown that for any even multiple of ω0 the prop- agating wave passes each constituent layer without reflection (no internal reflections at the layer interfaces), and thus gains the phase shift ∆ϕ = ω njdj , which is the same for all lay- ers in view of the QW condition as expressed with Eq. (2). (See also [15] for more detail on phase relations in Fresnel reflection from one layer.) As a result, the structure becomes fully transparent (|T (2mω0)| = 1) regardless of the number or arrangement of constituent layers, and the total phase shift becomes a simple sum of the shifts for all the layers: T (ωm = 2mω0) = exp  = exp (iNmπ) . Eq. (3) essentially provides a set of equidistant frequency points [we will call them single-layer reflection-free (SLRF) points] where the propagation phase (and hence, the wave number) is linearly dependent on frequency regardless of the structure. Indeed, the dispersion relation at these points j=1 dj) km(ωm) = k(2mω0) = linearly relates km and ωm, as is the case for a homogeneous medium. This linear dependence occurs only at the set of SLRF points ωm = 2mω0, and Eqs. (3)–(4) are not true any- where between these points. Note that among all the transmis- sion resonances present in a given multilayer’s spectrum, the SLRF points represent stationary waves without any correla- tions on a length scale greater than the optical path of one con- stituent layer. As such, they are naturally the least localized non-evanescent eigenstates possible in any given QW multi- layer. Moreover, QW structures are known to possess spectral pe- riodicity in transmittance [16] |T (ω + 2mω0)| = |T (ω)| (5) and mirror symmetry within each period [16] |T ((2m+ 1)ω0 + ω)| = |T ((2m+ 1)ω0 − ω)| , 0 < ω < ω0. Now let us note that Eq. (1) is isomorphic to the Schrödinger equation governing a quantum particle with mass mp and energy Ep in a stepwise potential u(x) ψ(x) + [Ep − u(x)]ψ(x) = 0 (7) Figure 1: A binary optical multilayer (a) together with its quantum mechanical counterpart: a particle with energy Ep = ~ω in 1D periodic (b) and non-periodic (c) binary stepwise potential. If the potential function is a constant (u(x) = −u0), the solution of Eq. (7) is a plane-wave function ψ0(x) = exp ikx = exp Ep + u0 , (8) which is analogous to a plane optical wave with the wave vec- tor k = 2mp/~ Ep + u0. If Ep > u(x) for any x, then k is real and the particle interacts with a potential barrier in the non-tunneling (Ramsauer) regime. This is a quantum mechanical analogy to electromagnetic wave propagation in a dielectric structure. Similar to the optical case, one can con- sider a stepwise potential barrier consisting of N “elementary wells” (Fig. 1b,c). The role of refractive index is taken by the potential energy uj in every step of the whole potential func- tion. The frequency is replaced by the particle energy Ep, which can be expressed in terms of de Broglie frequency [17] as Ep = ~ω. The case Ep < u(x), which causes imaginary wave vector in Eq. (8), is the tunneling case analogous to light propagation in absorbing media (e.g., metals). It is outside the scope of the present paper. To assure Ep > u(x), let us assume Ep ≥ 0 and u(x) < 0 from now on. Furthermore, it is commonly known that if the potential represents a single step (u(x) = −u0 for, say, x < x0 and u(x) = −u1 otherwise), one can introduce the coefficients r01 = k0 − k1 k0 + k1 , t01 = k0 + k1 , (9) which, when squared, denote the probability of finding the impinging quantum particle reflected or transmitted, respec- tively [17]. One can name them the reflection and transmis- sion coefficient for matter waves, a potential step correspond- ing to a single interface between dielectric media. Since the expressions for r and t are the same (the wave vector k taking the role of refractive index), one can use the same transfer- matrix formalism for determining both the stationary electro- magnetic wave distribution in a multilayer [18] and the steady- state wave function for quantum particles travelling through a complex stepwise potential [19]. Note that Eq. (8) indicates that the “refractive index” introduced in this way possesses an inherent quadratic dispersion. Finally, we call a multilayer structure binary if it can be rep- resented as consisting of two types of constituent layers (de- noted as binary digits 0 and 1, following the notation in [20]), to which two combinations of refractive index and thickness (n0; d0) and (n1; d1) are attributed. By arranging the 0’s and 1’s in different sequences, it is possible to vary the geometry of the structure very widely, making it periodic (if 0 and 1 alternate, as in 10101010101), disordered (if the sequence is randomly determined), or deterministically aperiodic (e.g., quasiperiodic [6] or fractal [7]). A binary potential barrier, with constituent elementary wells associated with (u0; d0) and (u1; d1), can be introduced likewise. For brevity, we will oc- casionally use the term “layers” for both types of constituent elements. Note that whenever the sequence contains two identical lay- ers (e.g., “00” in 1010100101) , it will of course mean in prac- tice that the corresponding structure will contain a single layer with thickness 2d0. However, for the purpose of this work we will regard such combinations as two separate constituent lay- ers. The number of layers of both typesN0 andN1, as well as their total number N = N0 + N1, will then remain the same regardless of layer rearrangement, indicative of the transmis- sion coefficient phase at the SLRF points [see Eq. (3)]. III. OPTICAL AND QUANTUM DENSITY OF STATES As mentioned in Section I, any inhomogeneity present in space is known to modify the properties of the quantum or electromagnetic vacuum in its vicinity. This modification takes the form of the change in the local DOS. It is believed to affect all phenomena that involve light-matter interaction, such as spontaneous emission or Raman scattering (see, e.g., [12] and references therein). Physically, the local DOS N (r, E) is directly related to the trace of Green’s function for the system in question: N (r, E) ∝ Im Tr G(r, r, E). By taking the integral Green’s functionG(E) in place of the local one, one obtains the value of the DOS N (E) that is characteristic to the whole system for a given value of energy. In a spatially finite system (a potential well with infinite walls or a closed resonator), only the states with a discrete set of energy (or frequency) eigenvalues are allowed. Supposing that these eigenvalues are dense enough, this integral DOS has a rather loose mathematically but very intuitive meaning of the number of these discrete states per unit energy. In an open resonator, any value of energy corresponds to an eigenstate, and the DOS transforms into a continuous spec- trum N (E), indicative of spectral characteristics for the over- Figure 2: (Color online) Transmittance |T (ω)|2 (dashed line) and normalized averaged local DOS ρ(ω)v0 [as of Eqs. (15) and (18), solid line] for multilayer structures of different geometry: (a) single 9d1-thick layer; (b) 3-layer structure; (c) 9-layer periodic structure; (d) Fabry-Pérot- like periodic structure with a half-wave defect; (e) coupled-defect structure; (f) fractal Cantor-like structure. All multilayers have N = 9. The vertical scale is chosen alike for all plots for the ease of comparison, the insets showing the scale of clipped peaks. The area [0, 1] is shaded to show the allowed region for transmittance, as well as to provide a visual guide for estimating the integral of ρ(ω) [see Eq. (19)]. all vacuum modification. It had been shown [21, 22, 23] that a finite-sized inhomogeneous potential located in an infinite 1D space causes the local DOS integrated over the whole space to undergo an overall finite modification ∆N (E) ≡ [N (x,E) −N0(x,E)] dx. Note that we are consider- ing a finite-sized inhomogeneity in an infinite space (an open resonator), as opposed to a finite system isolated from the out- side space (a closed resonator). In the former, ∆N was found to be proportional to the derivative of the total phase accumu- lated by the wave packet during its transmission through the inhomogeneity: ∆N (E) = (1/π) (dϕ/dE) , (10) which, as seen from Eq. (8), becomes asymptotically zero for very large energies compared to the potential (i.e., if nothing gets in the particle’s way). A similar expression can be used to determine the local density of electromagnetic states, also called the optical DOS (for details on its definition in 2D and 3D case, see Refs. [13, 24]). The transition from local to in- tegral DOS can be made in a similar manner to the quantum system. In the 1D case (the wave propagation in a multilayer is a 1D problem when only normal-incidence states are taken into account) the modification to the optical DOS ∆N (ω) is also likewise related to the derivative of transmission phase [25]: ∆N (ω) = (1/π) (d(ϕ− ϕ0)/dω) . (11) The subtraction of the free-space phase shift ϕ0 ensures that ∆N (ω) = 0 in free space. In the work by Barnett and Loudon [14] it has been shown that the modification of spon- taneous emission rate Γ by inhomogeneous medium (as com- pared to the free-space emission rate Γ0) integrated over the whole spectrum must be zero (the Barnett-Loudon sum rule): Γ(r, ω) Γ0(ω) dω = 0. (12) The emission rate Γ is proportional to the local DOS N , and the local optical DOS is frequency independent in free space (see [12]). Spectral integration in Eq. (12) can be interchanged with spatial integration over Γ(r, ω). Hence, a similar relation holds for the integral DOS: ∆N (ω)dω = 0. (13) The transition from local to integral DOS, as well as the transition from∆N to N , involves renormalization and there- fore may be ambiguous. A simple way to counter the diver- gence is to accompany the transition from ϕ − ϕ0 to ϕ with normalizing the DOS by the total thickness of the inhomoge- neous medium D (see [26]): ρ(ω) ≡ π∆N (ω) , (14) The authors in [26] simply define ρ as the optical DOS without going into further details. We can see that it is in fact neither local nor integral, but rather has the meaning of local DOS modified by a finite inhomogeneity, averaged in infinite 1D space. In the absence of any inhomogeneity, Eq. (14) gives ρ(ω) = 1/c, a known value for the DOS in 1D free space. Further, Eq. (14) can be used to calculate ρ(ω) from the complex transmission coefficient T (ω) as ρ(ω) = [Im T (ω)] Re T (ω)− Im T (ω) [Re T (ω)]′ D |T (ω)|2 , (15) the derivation taken with respect to ω. One must realize that the concept of the DOS introduced as in Eqs. (10)–(11), and especially, as in Eq. (14), is not without controversy. Questions arise already as to the physical mean- ing of the quantities involved. For example, one can define a “wave number” k a posteriori from the phase of the transmit- ted wave k(ω) = Arg T (ω) , (16) which would equal the actual wave number in a homogeneous medium, or the Bloch wave number in a periodic multilayer at transmission resonances [26]. In such special cases, ρ would equal the inverse group velocity (ρ(ω) = (dω/dk)−1), the latter also equal to the energy velocity. In the general case of non-periodic structures, however, the concepts of phase, group, and energy velocity, as well as their mutual correspondence, need to be re-examined. For instance, the “phase time” defined as dϕ/dω is, in general, not equal to the pulse’s actual “dwell time” (see [25, 27]), although, ad- mittedly, both have a similar frequency dependence and in some cases the phase time is a very good approximation for the dwell time [28]. That said, it is safer not to assign any direct physical meaning to k defined in Eq. (16) in the general case. We will thus treat it like a parameter within the scope of the present paper. Another point is that the applicability of 1D models for electromagnetic problems is in general of limited value. The reason is that reduction of Maxwell’s equations to the scalar wave equation (1), e.g., for multilayered media does not re- ally make the problem entirely one-dimensional. In reality one deals with finite-width beams rather than plane waves and with excited atoms that can emit in any direction. The lat- eral width of the multilayers is finite, too. As pointed out in Ref. [29], the optical DOS reduces to the expression (14) only if off-axis wave propagation is totally left out. That kept in mind, the function ρ(ω) nevertheless under- goes the same dramatic variation as does the transmittance itself when the structure geometry is varied (see Fig. 2). The peaks in |T (ω)|2 and ρ(ω) obviously correspond to each other. Note that this correspondence is a physical property of multi- layers rather than just a mathematical property of Eqs. (14)– (15). Indeed, one can show analytically that ρ(ω) ∝ |T (ω)|2 for any single layer. This proportionality is due to the phase structure of the Airy formulas, and is obviously not there for arbitrary ϕ(ω) in Eq. (14). Numerical calculations confirm that spectral features in |T (ω)|2 and ρ(ω) also correspond for an N -layer structure, e.g., as seen in Fig. 2. It should be pos- sible to show this analytically by induction but it is outside the scope of the present paper. We note instead that the same correspondence was observed in higher-dimensional systems (e.g., in slab photonic crystals [30]). Besides, one can observe that the sharper is the transmis- sion resonance around some frequency ωr, the larger is the value of ρ(ωr). Sharper transmission resonances correspond to stationary waves with greater energy localization, and it takes longer for greater energies to build up inside the struc- ture. Hence, it takes longer for resonant transmission to mani- fest in such cases. Therefore the maxima of ρ(ω) are just those points where delayed light propagation is likely to be experi- enced. The DOS spectrum is thus valuable as a quick visual guide for determining the resonant behavior of any multilayer structure, as employed earlier [31]. Finally, let us note that although k in Eq. (16) cannot be as- signed a direct physical meaning in the general case, it can be used as a parameter, which can provide some heuristic guid- ance in experiments on the group velocity dispersion-related effects (e.g., pertaining to propagation, compression, delay, and chirp compensation of ultrashort laser pulses). For some examples involving non-periodic structures, the reader is re- ferred to Refs. [32, 33, 34]. IV. CONSTRAINTS IN MULTILAYERS In the previous section, the use of ρ(ω) defined by Eq. (15) as a meaningful characteristic of the structure’s optical prop- erties has been motivated. It was demonstrated that ρ(ω) can be strongly modified by altering the geometry of the structure (Fig. 2). We proceed to show that the degree of geometry- induced modification imposed on ρ has fundamental limita- tions. One of these is the Barnett-Loudon sum rule – Eq. (13) holds both for the quantum mechanical and for the electro- magnetic case when the corresponding expression for ∆N (ω) is used. If the medium is a QW multilayer, the constraint becomes stricter and involves integration over finite rather than infi- nite frequency intervals. As the transmission properties in the SLRF points are given by Eq. (3), the integral of ρ(ω) be- tween those points can be evaluated explicitly using Eqs. (16) Figure 3: (Color online) The schematic frequency dependence of propagation phase for (a) optical waves [Eq. (23)] and (b) quantum wave function [Eq. (22)] in a slab of homogeneous dielectric and in a length of constant potential, respectively. and (3)–(4) (see also [16]): 2(m+1)ω0 ρ(ω)dω = k[2(m+1)ω0] k[2mω0] , (17) which holds regardless of the geometrical arrangement of the constituent layers in the structure, provided that the layers obey the relation (2). One can further introduce the “bulk velocity prameter” from the minimum time it takes light to traverse the multilayer, internal reflections neglected, as (djnj/c) . (18) This is a parameter independent either of ω or of the layer arrangement of the structure. Making a transition to the di- mensionless frequency η ≡ ω/ω0 and taking into account the symmetry condition (6), we arrive at v0ρ(η)dη = 1 (19) and, further, since v0 = c and ρ0 = 1/c in free space, ∆ [v0ρ(η)] dη = (v0ρ(η)− cρ0) dη = 0 (20) for any integer m ≥ 0. The conditions (19)–(20) have no less a universal charac- ter than Eq. (13). They physically mean that the modification of the transmission or dispersion properties due to layer rear- rangement in QW multilayers is only possible within a finite frequency range ω0. One can see in Fig. 2 that despite appar- ently dramatic modification of ρ(ω), the enhancement in one portion of the spectrum appears compensated by a gap in an- other portion, so that the overall DOS, integrated between the SLRF points, remains unaffected. It is also important to real- ize that within ω0, one can achieve any desired spectral shape, given the sufficient number of layers and sufficient freedom in their arrangement. For example, a heuristic optimization algorithm was recently used to demonstrate that certain ape- riodic sequences can be employed to fabricate structures with desired spectral properties [9]. Note, too, the inverse proportionality between ω0 and the optical path of the constituent layers. It follows that if the QW condition (2) is broken but the quantities njdj all remain commensurate, the same reasoning can be applied. Eqs. (17) and (19) can then be obtained by subdivision of the constituent layers, accompanied by the according increase in the central frequency (ω0 → Nω0). In the limiting case of mathemati- cally incommensurate layers,N goes to infinity, and the struc- ture appears to possess the same freedom as a continuously inhomogeneous medium would, retaining only asymptotic re- lation ρ(ω)dω = , (21) associated with increasing ω0 to infinity in Eq. (20), and con- sequently, representing the sum rule (13). V. CONSTRAINTS IN POTENTIAL BARRIERS The concept of optical DOS appeared in electrodynamics largely by the influence of the quantum DOS. Such transfer of concepts makes use of the analogy between the Helmholtz (1) and Schrödinger (7) equations, as well as between a mul- tilayer and a potential barrier, as outlined in Sec. II. In this section, we will attempt to make these analogies work back- wards and determine if, and to what extent, the relation (17) can be generalized to the quantum mechanical case. Consider a binary stepwise potential and the particle with mass mp and energy Ep ≡ ~ω interacting with it in the non- tunneling regime (Fig. 1b,c). Making use of the expression (8) for k, we can derive the frequency dependence for prop- agation phase of the particle’s wave function for constant po- tential corresponding to one elementary potential well. Com- pared to the same dependence for an optical wave in a homo- geneous slab corresponding to one constituent layer, it has the form (α ≡ 2mp/~) ϕ(qm)(ω) ≡ k(qm)(ω)d = dα ~ω + u; (22) ϕ(opt)(ω) = ndω/c. (23) Fig. 3 schematically shows both these dependencies. In the optical case (Fig. 3a) the only variable parameter is the slope given by nd. Hence, with the aid of Eq. (2) it becomes possi- ble to achieve exactly the same dispersion relation, and hence the same set of SLRF points, for both constituent layers when n0 6= n1. This is what forms the foundation for reasoning pre- sented in the previous section. In the quantum case (Fig. 3b) u and d are seen to contribute in an essentially different way. Is is thus not possible to arrive at the same dispersion relation for two different potentials (u0 6= u1). However, one can still define a set of frequency points (though no longer equidistant) where ϕ(qm)(ωj) = jπ. In these points, as can be seen from Eqs. (8) and (9), the whole structure would be totally “transparent” for incoming quan- tum particles (the Ramsauer effect). If the structure is binary, the frequencies for both kinds of elementary wells are given (0,1) = j20,1π 2/d20,1α 2 − u0,1. (24) Since two different parabolic curves can still have intersec- tion points, one can manage to achieve ω for two pairs of j0 and j1. The reasoning presented in the previous section can then be reproduced involving the quantity ρ de- fined exactly as in Eqs. (14) and (17) and having the same physical background. The dependence on ω, however, will be more complex due to inherent dispersion as seen in Eq. (8). For simplicity and for the sake of further analogy between optical and quantum systems, let us require one of the equal frequency pairs in Eq. (24) to correspond to ω = 0 (which is always true for optical waves where all dispersion curves pass trough the origin, see Fig. 3a). In this case we arrive at α2d20u0 = α 2d21u1 = j 2, (25) which can be seen as a quantum analogy to the condition (2). The second pair (j0, j1) can then be found as an integer solu- tion of the equation (first suggested in our earlier work [35]) j20 − 1− β/j2qw j21 = β, j0,1 > jqw; β ≡ (u0 − u1)α2d20/π2. It can be seen that for any integer j0,1 > jqw there is a ratio- nal β that solves Eq. (26). But β is related to the parameters of the constituent potential wells. So, the inverse problem, i.e, finding suitable j0,1 for a given β, is far more interesting from a physical point of view. However, is not so straightforward and is mathematically related to finding Pythagorean triples in integer numbers. One can confirm numerically that there are a multitude of solutions for many rational values of β (see Table I). Some of them can be represented via recurrent rela- tions, e.g., for β = 1 some of the solutions represent a series: (i+1) 1 = j 1 + 6 + 4i, j 0 = j 1 − 1, (27) where j 1 = j (min) qw = 2. Other cases are more complex, but they, too, can be seen to form distinct solution branches (Fig. 4). Once j0 and j1 have been found, an analogous relation to Eq. (17) can be formulated as 0=~ωjqw ρ(E)dE = [N0(j0 − jqw) +N1(j1 − jqw)] . Note that Eq. (28) is more complicated than its optical counterpart (17), and becomes, in general, dependent on the number of constituent potential wells N1,2. This dependence cannot be eliminated because one sees from Eq. (26) that it is impossible to have j0 = j1 without violating the assertion that j0,1 > jqw. It is still, however, completely independent of layer rearrangement. In this sense, Eq. (26) represents a uni- versal quantum mechanical conservation relation for the DOS over a finite energy interval. To demonstrate the results numerically, we have considered a stepwise AlAs/GaAs quantum well (∆u = 1000 meV). To aim at β = 4/5, we have taken d0 = 21.2 Å, d1 = 47.4 Å, according to Eqs. (25)–(26). One possible solution of Eq. (26) would then be jqw = 1, j0 = 2, j1 = 4 (see Table I). The structures made of N = 9 elementary wells were used, and the numbersN0 andN1 were fixed, too, at the values 4 and 5, respectively. We see in Fig. 5 that both ρ(E) and the transmission T (E) are subject to quite a strong modification. It resembles the modification seen in dielectric multilayers (compare, e.g., Fig. 5a,b with Fig. 2a,c). Two differences are the presence of a decaying background due to the inherent dispersion [see Eq. (22)] and the lack of periodicity because Eq. (2) can no longer be satisfied. However, if we integrate ρ(E) between the SLRF points (~ωjqw = 0 and ~ωj0 = ~ωj1 = 3.75 eV) as provided by Eq. (26), we can see that the integral does not change when the layers are rearranged. Table II provides the results for nine different structures and for several upper integration limits. It can be seen that both below and above 3.75 eV the integrals vary from structure to structure. When, however, the correct integration limits are chosen, the difference vanishes and all integrals equal 19, which is the right-hand side of Eq. (28) for the chosen values of parameters. VI. DISCUSSION The equations (17)–(20) and (28) constitute the main result of this paper pertaining to optical and electronic heterostruc- tures, respectively. In both cases, we are dealing with conser- vation of the DOS ρ integrated across a finite energy or fre- quency region. As discussed in Sec. III, ρ represents the av- eraged local DOS as modified by the presence of finite-sized inhomogeneous structure in an infinite 1D free space. It is re- lated to the dispersion and the transmission properties of the heterostructures in question [see Eqs. (14)–(15)]. Table I: Some values of β that allow integer solutions of Eq. (26), along with some of such solutions obtained numerically. β < 1 (jqw : j0, j1) β ≥ 1 (jqw : j0, j1) 1/4 (1 : 13, 15); (1 : 181, 209); (2 : 122, 126) 1 (2 : 7, 8); (2 : 26, 30); (3 : 17, 18); (3 : 99, 105); (4 : 31, 32) 1/3 (1 : 9, 11); (1 : 89, 109); (2 : 90, 94) 3/2 (2 : 8, 10); (2 : 68, 86); (3 : 63, 69) 1/2 (1 : 5, 7); (1 : 29, 41); (2 : 58, 62); (3 : 207, 213) 2 (2 : 10, 14); (2 : 58, 82); (3 : 45, 51) 2/5 (1 : 7, 9); (1 : 55, 71); (2 : 74, 78) 4 (3 : 7, 9); (3 : 18, 24); (4 : 14, 16); (4 : 52, 60); (5 : 23, 25) 2/3 (1 : 3, 5); (1 : 11, 19); (2 : 42, 46); (3 : 153, 159) 6 (3 : 9, 15); (3 : 33, 57); (4 : 16, 20) 4/5 (1 : 2, 4); (1 : 5, 11); (1 : 13, 19); (2 : 34, 38) 10 (4 : 8, 12); (4 : 32, 52); (5 : 35, 45) Figure 4: The distribution of integer solutions j0,1 of Eq. (26) for (a) β = 1, (b) β = 4, and (c) β = 4/5. Distinct solution groups (“branches”) can be seen. These properties, as well as the DOS, can undergo dramatic modification as compared to those of homogeneous media (see Figs. 2 and 5) because a multilayer structure or a step- wise potential barrier can be very complex. Nevertheless, the modification appears to be limited both in its amount [see the right hand side of Eqs. (17) and (28)] and in its extent (by the finite integration limits in those equations). There is an elegant physical explanation for the relations obtained. By engineering the geometrical properties of an in- homogeneous structure, it is only posible to redistribute the Table II: Numerically evaluated integrals (D/~π) ρ(E)dE [as in Eq. (28)] from 0 to several upper energy values for nine structures with N = 9, N0 = 4, and N1 = 5 (same as in Fig. 5). Standard deviation of the values across all structures for each upper integra- tion limit is provided in the lowest row. The limit of 3.75 eV (the obtained value of the SLRF point) is accompanied by a drop in stan- dard deviation down to 10−8, which falls within accuracy limits for numerical integration. Structure 0. . . 1 eV 3 eV 3.75 eV . . . 5 eV 001111100 7.5531 16.4036 19.0000 22.8568 010111100 7.5526 16.3991 19.0000 22.8593 100111100 7.5481 16.3982 19.0000 22.8582 110011100 7.5207 16.4016 19.0000 22.8574 010111010 7.5890 16.3991 19.0000 22.8600 100110011 7.5177 16.4048 19.0000 22.8592 100111001 7.5198 16.4017 19.0000 22.8512 110010011 7.5373 16.3996 19.0000 22.8603 101010101 7.5880 16.3982 19.0000 22.8657 Std. deviation 0.027 0.0045 8× 10−9 0.0038 available electromagnetic or quantum states across the spec- trum, but impossible to alter the “total number” of the states. The latter turns out to be related to the size or “1D volume” of the structure [as seen by the presence of N at the right-hand side of Eqs. (17) and (28)] and represents integrated character- istics of the structure-affected vacuum. So, an enhancement of the DOS in some pars of the spectrum (like the band edge res- onances for a periodic structure in Fig. 2c) giving rise, e.g., to the spontaneous emission enhancement, is inevitably accom- panied by a suppression of the DOS in other spectral region (like the band gap in the same figure), leading to the inhibition of light propagation and all phenomena involving light-matter interaction [4]. In this sense, the results obtained resemble already known constraints on the DOS like the Barnett-Loudon sum rule (13). However, in the relations obtained in this work the integra- tion involved is finite rather than infinite. For the optical case, this means a tighter restriction on the spectral redistribution of the DOS. The compensation of suppression snd enhancement must occur within the frequency interval ω0. This interval is determined solely by the elementary constituent element of the structure in question [see Eq. (2)]. It is totally indepen- dent of geometrical arrangement of these elements. In other words, the QW condition (2) enforces the existence of certain points in the spectrum (the SLRF points) that cannot be “tran- scended” by electromagnetic states that are “pushed around” the spectrum by layer rearrangement. On the other hand, the spectral properties of the structure can be arbitrary everywhere between the SLRF points (3). It should also be noted that the increase of N causes the details in the spectra to become finer, and the variation of T (ω) and ρ(ω) to get more rapid. These results can help to understand the underlying physics of complex media. Similar conclusions can be formulated for a quantum par- Figure 5: The quantum averaged DOS ρ(E) (top) and transmittance ˛T (E)2 ˛ (bottom) for an AlAs/GaAs quantum well and β = 4/5 in three stepwise potential wells differing only by the elementary well rearrangement: (a) single-layer structure; (b) periodic structure; (c) non-periodic structure. The portion between the SLRF points [0. . . 3.75 eV] is depicted, which corresponds to a solution of Eq. (26) for jqw = 1, j0 = 2, j1 = 4. ticle in a stepwise potential barrier. However, the inherent quadratic dispersion as seen in Eqs. (8) and (22) results in many differences. First and foremost, the SLRF points are no longer guaranteed. Not only a relative restriction on con- stituent elements (25) analogous to the QW condition (2) is required, but also individual constraints on u0,1 and d0,1 are necessary, so as to provide special values of β as determined by Eq. (26). This makes the binarity of the structure an impor- tant requirement in contrast to the optical case where Eqs. (2) and (23) can be extended to as many kinds of constituent ele- ments as needed. Because the equation (26) is quadratic rather than linear, the SLRF points occur far more seldom than in the optical case and are no longer equidistant. However, they still do occur on a regular basis if they occur at all for a given choice of parameters, as seen in Fig, 4. In this way, we have provided a way for engineering an electronic heterostructure where the DOS modification due to structure complexity is confined in a finite spectral region. The structure itself can be arbitrarily complex because Eqs. (24)–(26) do not depend on N in any way. To conclude this section, let us note that the structures in question appear to possess other conservation relations. As can be seen, e.g., in Fig. 2a–c, the transmission spectra contain the same number of resonance peaks in the interval [0; 2ω0], namely, nine, which equals the number of constituent layers. Bearing a loose resemblance to the energy level splitting in solids if one regards the layers as “atoms”, this was found to be a general property of such multilayers [16]. However, additional restrictions on the structures seem to be necessary, such as the outermost layers of the structure being 1 rather than 0 (compare, e.g., Figs. 2d, e). This requires additional investigations and remains a subject for further studies. VII. CONCLUSIONS AND OUTLOOK To summarize, we have investigated the possible degree of modification to transmission and dispersion properties, as well as the averaged local DOS, in discretely inhomogeneous me- dia. Both electromagnetic waves propagating in a dielectric multilayer structure and a quantum particle propagating over a stepwise, multiple-well potential barrier, have been consid- ered (Fig. 1). In both cases, certain constraints on the con- stituent elements of the structure [Eqs. (2) and (25)] allow to derive the conservation relations over finite frequency or en- ergy regions [Eqs. (17)–(20) and (28), respectively]. Both re- lations hold regardless of the structure geometry (at least in the sense of rearrangement of constituent elements) and are, in this sense, universal, despite the fact that the spectral prop- erties themselves can be strongly geometry-dependent. The quantum case appears to be more complicated than the opti- cal one and requires more conditions to be fulfilled, as im- plied by a quadratic character of Eq. (26). The analytical re- sults obtained have been verified by numerical calculations (see Figs. 2, 5, and Table II). The results obtained contribute to understanding the physics of complex inhomogeneous media. They can be ap- plied in the design of heterostructures with engineered disper- sion, such as chirp compensation, pulse compression or delay line devices. A more detailed studies of the relations obtained would also be useful. It would be of interest to find out if, and to what extent, the results can be applied to the case of optical multilayers made of dispersive and/or absorptive materials, as well as for potential barriers in the tunneling regime. Acknowledgments The authors are thankful to J. P. Dowling, H. V. Demir, D. N. Chigrin, E. P. Petrov, and A. V. Lavrinenko for stimulat- ing discussions, as well as to C. Kremers for helpful advice. 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Electromagnetic Casimir densities for a wedge with a coaxial cylindrical shell A. A. Saharian∗ Department of Physics, Yerevan State University, 1 Alex Manoogian Street, 375025 Yerevan, Armenia October 29, 2018 Abstract Vacuum expectation values of the field square and the energy-momentum tensor for the electromagnetic field are investigated for the geometry of a wedge with a coaxal cylindrical boundary. All boundaries are assumed to be perfectly conducting and both regions inside and outside the shell are considered. By using the generalized Abel-Plana formula, the vacuum expectation values are presented in the form of the sum of two terms. The first one corresponds to the geometry of the wedge without the cylindrical shell and the second term is induced by the presence of the shell. The vacuum energy density induced by the shell is negative for the interior region and is positive for the exterior region. The asymptotic behavior of the vacuum expectation values are investigated in various limiting cases. It is shown that the vacuum forces acting on the wedge sides due to the presence of the cylindrical boundary are always attractive. PACS numbers: 03.70.+k 1 Introduction The Casimir effect is among the most interesting macroscopic manifestations of vacuum fluctu- ations. It have important implications on all scales, from cosmological to subnuclear, and has become in recent decades an increasingly popular topic in quantum field theory. In addition to its fundamental interest the Casimir effect also plays an important role in the fabrication and operation of nano- and micro-scale mechanical systems. The imposition of boundary conditions on a quantum field leads to the modification of the spectrum for the zero-point fluctuations and results in the shift in the vacuum expectation values for physical quantities such as the energy density and stresses. In particular, the confinement of quantum fluctuations causes forces that act on constraining boundaries. The particular features of the resulting vacuum forces depend on the nature of the quantum field, the type of spacetime manifold, the boundary geometries and the specific boundary conditions imposed on the field. Since the original work by Casimir [1] many theoretical and experimental works have been done on this problem (see, e.g., [2, 3, 4, 5] and references therein). Many different approaches have been used: mode summation method E-mail: saharian@ictp.it http://arxiv.org/abs/0704.1898v1 with combination of the zeta function regularization technique, Green function formalism, mul- tiple scattering expansions, heat-kernel series, etc. Advanced field-theoretical methods have been developed for Casimir calculations during the past years [6, 7, 8]. However, there are still difficulties in both interpretation and renormalization of the Casimir effect. Straightforward computations of geometry dependencies are conceptually complicated, since relevant informa- tion is subtly encoded in the fluctuations spectrum [8]. Analytic solutions can usually be found only for highly symmetric geometries including planar, spherically and cylindrically symmetric boundaries. Recently the Casimir energy has been evaluated exactly for several less symmetric configurations of experimental interest. These include a sphere in front of a plane and a cylinder in front of a plane [9]. A great deal of attention received the investigations of quantum effects for cylindrical bound- aries. In addition to traditional problems of quantum electrodynamics under the presence of material boundaries, the Casimir effect for cylindrical geometries can also be important to the flux tube models of confinement [10, 11] and for determining the structure of the vacuum state in interacting field theories [12]. The calculation of the vacuum energy of electromagnetic field with boundary conditions defined on a cylinder turned out to be technically a more involved problem than the analogous one for a sphere. First the Casimir energy of an infinite perfectly conducting cylindrical shell has been calculated in Ref. [13] by introducing ultraviolet cutoff and later the corresponding result was derived by zeta function technique [14, 15, 16]. The local characteristics of the corresponding electromagnetic vacuum such as energy density and vacuum stresses are considered in [17] for the interior and exterior regions of a conducting cylindrical shell, and in [18] for the region between two coaxial shells (see also [19]). The vacuum forces acting on the boundaries in the geometry of two cylinders are also considered in Refs. [20]. The scalar Casimir densities for a single and two coaxial cylindrical shells with Robin boundary conditions are investigated in Refs. [21, 22]. Less symmetric configuration of two eccentric per- fectly conducting cylinders is considered in Ref. [23]. Vacuum energy for a perfectly conducting cylinder of elliptical section is evaluated in Ref. [24] by the mode summation method, using the ellipticity as a perturbation parameter. The Casimir forces acting on two parallel plates inside a conducting cylindrical shell are investigated in Ref. [25]. Aside from their own theoretical and experimental interest, the exactly solvable problems with this type of boundaries are useful for testing the validity of various approximations used to deal with more complicated geometries. From this point of view the wedge with a coaxial cylindrical boundary is an interesting system, since the geometry is nontrivial and it includes two dynamical parameters, radius of the cylindrical shell and opening angle of the wedge. This geometry is also interesting from the point of view of general analysis for surface divergences in the expectation values of local physical observables for boundaries with discontinuities. The nonsmoothness of the boundary generates additional contributions to the heat kernel coefficients (see, for instance, the discussion in [26, 27, 28] and references therein). The present paper is concerned with local analysis of the vacuum of the electromagnetic field constrained to satisfy perfectly conducting boundary conditions on boundary surfaces of a wedge with a coaxial cylin- drical boundary. Namely, we will study the vacuum expectation values of the field squares and the energy-momentum tensor for the electromagnetic field for both regions inside and outside the cylindrical shell. In addition to describing the physical structure of the quantum field at a given point, the energy-momentum tensor acts as the source of gravity in the Einstein equa- tions. It therefore plays an important role in modelling a self-consistent dynamics involving the gravitational field. The vacuum expectation value of the square of the electric field determines the electromagnetic force on a neutral polarizable particle. Some most relevant investigations to the present paper are contained in Refs. [2, 29, 30, 31, 32, 33], where the geometry of a wedge without a cylindrical boundary is considered for a conformally coupled scalar and electromag- netic fields in a four dimensional spacetime. The total Casimir energy of a semi-circular infinite cylindrical shell with perfectly conducting walls is considered in [34] by using the zeta function technique. For a scalar field with an arbitrary curvature coupling parameter the Wightman function, the vacuum expectation values of the field square and the energy-momentum tensor in the geometry of a wedge with an arbitrary opening angle and with a cylindrical boundary are evaluated in [35, 36]. Note that, unlike the case of conformally coupled fields, for a general coupling the vacuum energy-momentum tensor is angle-dependent and diverges on the wedge sides. Our method here employs the mode summation and is based on a variant of the gen- eralized Abel-Plana formula [37] (see also Refs. [19, 38]). This enables us to extract from the vacuum expectation values the parts due to a wedge without the cylindrical shell and to present the parts induced by the shell in terms of strongly convergent integrals. Note that the closely related problem of the vacuum densities induced by a cylindrical boundary in the geometry of a cosmic string is investigated in Refs. [39, 40] for both scalar and electromagnetic fields. We have organized the paper as follows. In the next section we describe the structure of the modes for a wedge with a cylindrical shell in the region inside the shell. By applying to the corresponding mode sums the generalized Abel-Plana formula, we evaluate the vacuum expectation values of the electric and magnetic field square. Various limiting cases of the general formulae are discussed. Section 3 is devoted to the investigation of the vacuum expectation values for the energy-momentum tensor of the electromagnetic field in the region inside the shell. The additional vacuum forces acting on the wedge sides due to the presence of the cylindrical boundary are evaluated. In section 4 we consider the vacuum densities for a wedge with the cylindrical shell in the exterior region with respect to the shell. Formulae for the shell contributions are derived and the corresponding surface divergences are investigated. The vacuum forces acting on the wedge sides are discussed. The main results are summarized and discussed in section 5. 2 Vacuum expectation values of the field square inside a cylin- drical shell Consider a wedge with the opening angle φ0 and with a coaxial cylindrical boundary of radius a (see figure 1) assuming that all boundaries are perfectly conducting. In accordance with the problem symmetry, in the discussion below the cylindrical coordinates (r, φ, z) will be used. We are interested in the vacuum expectation values (VEVs) of the field square and the energy- momentum tensor for the electromagnetic field. Expanding the field operator in terms of the creation and annihilation operators and using the commutation relations, the VEV for a quantity F {Ai, Ak} bilinear in the field can be presented in the form of the mode-sum 〈0|F {Ai, Ak} |0〉 = F {Aαi, A∗αk} , (1) where {Aαi, A∗αk} is a complete set of solutions of the classical field equations satisfying the boundary conditions on the bounding surfaces and specified by a set of quantum numbers α. In accordance with formula (1), for the evaluation of the VEVs for the square of the electric and magnetic fields and the energy-momentum tensor, the corresponding eigenfunctions are needed. In this section we consider the region inside the cylindrical shell (region I in figure 1). For the geometry under consideration there are two different types of the eigenfunctions corresponding to the transverse magnetic (TM) and transverse electric (TE) waves. In the discussion below we will specify these modes by the index λ = 0 and λ = 1 for the TM and TE waves respectively. In the Coulomb gauge, the vector potentials for the TM and TE modes are Figure 1: Geometry of a wedge with a coaxial cylindrical boundary with radius a. given by the formulae Aα = βα (1/iω) γ2e3 + ik∇t Jq|m|(γr) sin(qmφ) exp [i (kz − ωt)] , λ = 0 −e3 ×∇t Jq|m|(γr) cos(qmφ) exp [i (kz − ωt)] , λ = 1 , (2) where e3 is the unit vector along the axis of the wedge, ∇t is the part of the nabla operator transverse to this axis, Jν(x) is the Bessel function of the first kind, and ω2 = γ2 + k2, q = π/φ0. (3) In Eq. (2), m = 1, 2, . . . for λ = 0 and m = 0, 1, 2, . . . for λ = 1. The normalization coefficient βα is found from the orthonormalization condition for the vector potential: dV Aα ·A∗α′ = δαα′ , (4) where the integration goes over the region inside the shell. From this condition, by using the standard integral involving the square of the Bessel function, one finds β2α = 4qTqm(γa) δm, δm = 1/2, m = 0 1, m 6= 0 , (5) where we have introduced the notation Tν(x) = x ν (x) + (1− ν2/x2)J2ν (x) . (6) Eigenfunctions (2) satisfy the standard boundary conditions for the electric and magnetic fields, n×E = 0 and n ·B = 0, on the wedge sides corresponding to φ = 0 and φ = φ0, with n being the normal to the boundary. The eigenvalues for the quantum number γ are determined by the boundary conditions on the cylindrical shell. From the latter it follows that these eigenvalues are solutions of the equation J (λ)qm (γa) = 0, λ = 0, 1, (7) where we use the notations J ν (x) = Jν(x) and J ν (x) = J ν(x). We will denote the corre- sponding eigenmodes by γa = j m,n, n = 1, 2, . . ., assuming that the zeros j m,n are arranged in ascending order. Consequently, the eigenfunctions are specified by the set of quantum numbers α = (k,m, λ, n). First we consider the VEVs of the squares of the electric and magnetic fields inside the shell. Substituting the eigenfunctions (2) into the corresponding mode-sum formula, we find 〈0|F 2|0〉 = 4q λ=0,1 m,n Tqm(j m,n + k2a2 ×g(ηFλ)[Φ(λ)qm(φ), Jqm(j(λ)m,nr/a)], (8) where F = E,B with ηEλ = λ, ηBλ = 1−λ, and the prime in the summation over m means that the term m = 0 should be halved. In formula (8), for a given function f(x), we have introduced the notations g(0)[Φ(φ), f(x)] = (k2r2/x2) Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2 +Φ2(φ)f2(x), (9) g(1)[Φ(φ), f(x)] = (1 + k2r2/x2) Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2 , (10) Φ(λ)ν (φ) = sin(νφ), λ = 0 cos(νφ), λ = 1 . (11) The expressions (8) corresponding to the electric and magnetic fields are divergent. They may be regularized introducing a cutoff function ψµ(ω) with the cutting parameter µ which makes the divergent expressions finite and satisfies the condition ψµ(ω) → 1 for µ → 0. After the renormalization the cutoff function is removed by taking the limit µ → 0. An alternative way is to consider the product of the fields at different spacetime points and to take the coincidence limit after the subtraction of the corresponding Minkowskian part. Our approach here follows the first method. As we do not know the explicit expressions for the zeros j m,n as functions on m and n, and the summand in formula (8) is strongly oscillating function for large values of m and n, this formula is not convenient for the further evaluation of the VEVs of the field square. In order to obtain an alternative representation, we apply to the series over n the generalized Abel-Plana summation formula [37] (see also [38]) Tqm(j m,n)f(j m,n) = dx f(x) + qm (z) qm (z) qm (x) qm (x) e−qmπif(eπi/2x) + eqmπif(e−πi/2x) , (12) where Yν(z) is the Neumann function and Iν(z), Kν(z) are the modified Bessel functions. As it can be seen, for points away from the shell the contribution to the VEVs coming from the second integral term on the right-hand side of (12) is finite in the limit µ → 0 and, hence, the cutoff function in this term can be safely removed. As a result the VEVs can be written in the 〈0|F 2|0〉 = 〈0w|F 2|0w〉+ , (13) where 〈0w|F 2|0w〉 = γ3ψµ(ω) γ2 + k2 J ′2qm(γr) + J2qm(γr) + J2qm(γr) −(−1)ηF1 cos(2qmφ) J ′2qm(γr)− J2qm(γr) , (14) 〈F 2〉cyl = λ=0,1 qm (xa) qm (xa) G(ηFλ)[k,Φ qm(φ), Iqm(xr)]√ x2 − k2 . (15) In formula Eq. (15) we have introduced the notations G(0)[k,Φ(φ), f(x)] = (k2r2/x2) Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2 +Φ2(φ)f2(x), (16) G(1)[k,Φ(φ), f(x)] = (k2r2/x2 − 1) Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2 . (17) The second term on the right-hand side of Eq. (13) vanishes in the limit a → ∞ and the first one does not depend on a. Thus, we can conclude that the term 〈0w|F 2|0w〉 corresponds to the part in the VEVs when the cylindrical shell is absent with the corresponding vacuum state |0w〉. Hence, the application of the generalized Abel-Plana formula enables us to extract from the VEVs the parts induced by the cylindrical shell without specifying the cutoff function. In addition, these parts are presented in terms of the exponentially convergent integrals. First, let us concentrate on the part corresponding to the wedge without a cylindrical shell. First of all we note that in Eq. (14) the part which does not depend on the angular coordinate φ is the same as in the corresponding problem of the cosmic string geometry with the angle deficit 2π − φ0 (see Ref. [40]), which we will denote by 〈0s|F 2|0s〉. For this part we have 〈0s|F 2|0s〉 = γ3ψµ(ω) γ2 + k2 J ′2qm(γr) + J2qm(γr) + J2qm(γr) = 〈0M|F 2|0M〉 − (q2 − 1)(q2 + 11) 180πr4 , (18) where 〈0M|F 2|0M〉 is the part corresponding to the Minkowskian spacetime without boundaries and in the last expression we have removed the cutoff. To evaluate the part in (14) which depends on φ, we firstly consider the case when the parameter q is an integer. In this case the summation over m can be done by using the formula [41, 42] cos(2qmφ)J2qm(y) = J0(2y sin(φ+ φ0l)). (19) The formulae for the other series entering in Eq. (14) are obtained from (19) taking the deriva- tives with respect to φ and y. In particular, for the combination appearing in the angle-dependent part we obtain cos(2qmφ) J ′2qm(y)− J2qm(y) J ′1(2y sin(φ+ φ0l)). (20) Substituting this in formula (14), the integrals remained are evaluated by introducing polar coordinates in the (k, γ)-plane. In this way one finds 〈0w|F 2|0w〉 = 〈0s|F 2|0s〉 − 3(−1)ηF1 sin−4(φ+ lπ/q). (21) The sum on the right hand-side of this formula is evaluated by the double differentiation of the relation [41] cos−2(x+ lπ/q) = q2 sin−2(qx+ qπ/2). (22) Finally, for the renormalised VEVs of the field square in the geometry of a wedge without a cylindrical boundary we find 〈F 2〉w,ren = − (q2 − 1)(q2 + 11) 180πr4 (−1)ηF1q2 2πr4 sin2(qφ) 2 sin2(qφ) + 1− q2 , (23) with ηE1 = 1 and ηB1 = 0. Though we have derived this formula for integer values of the parameter q, by the analytic continuation it is valid for non-integer values of this parameter as well. The expression on the right of formula (23) is invariant under the replacement φ→ φ0 −φ and, as we could expect, the VEVs are symmetric with respect to the half-plane φ = φ0/2. Formula (23) for F = E was derived in Ref. [32] within the framework of Schwinger’s source theory. For q = 1 from formula (23) as a special case we obtain the renormalized VEVs of the field square for a conducting plate. In this case x = r sinφ is the distance from the plate and one has 〈F 2〉pl,ren = − 3(−1)ηF1 . (24) Another special case q = 1/2 corresponds to the geometry of a half-plane. In (23) taking the limit r → ∞, with x0 = rφ0 being fixed, we obtain the corresponding results in the region between two parallel plates located at the points x = 0 and x = x0: 〈F 2〉2pl,ren = − 180x40 − (−1) ηF1π3 2x40 sin 2(πx/x0) 2 sin2(πx/x0) . (25) Now, we turn to the investigation of the parts in the VEVs of the field square induced by the cylindrical boundary and given by formula (15). By using the formula dk km xf(x)√ x2 − k2 dxxm+1f(x), (26) these parts are presented in the form 〈F 2〉cyl = λ=0,1 qm (xa) qm (xa) G(ηFλ)[Φ(λ)qm(φ), Iqm(xr)]. (27) Here, for given functions f(x) and Φ(φ), we have introduced the notations G(0)[Φ(φ), f(x)] = Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2 + 2Φ2(φ)f2(x), (28) G(1)[Φ(φ), f(x)] = −Φ2(φ)f ′2(x)− Φ′2(φ)f2(x)/x2. (29) As we see the parts in the VEVs induced by the cylindrical shell are symmetric with respect to the half-plane φ = φ0/2. The expression in the right-hand side of (27) is finite for 0 < r < a including the points on the wedge sides, and diverges on the shell. To find the leading term in the corresponding asymptotic expansion, we note that near the shell the main contribution comes from large values of m. By using the uniform asymptotic expansions of the modified Bessel functions (see, for instance, [43]) for large values of the order, up to the leading order, for the points a−r ≪ a| sinφ|, a| sin(φ0−φ)| we find 〈F 2〉cyl ≈ − 3(−1)ηF1 4π(a− r)4 . (30) For the points near the edges (r = a, φ = 0, φ0) the leading terms in the corresponding asymp- totic expansions are the same as for the geometry of a wedge with the opening angle φ0 = π/2. The leading terms given by formula (30) are the same as for the geometry of a single plate (see (24)). They do not depend on φ0 and have opposite signs for the electric and magnetic fields. In particular, the leading terms are cancelled in the evaluation of the vacuum energy density. Surface divergences originate in the unphysical nature of perfect conductor boundary conditions and are well-known in quantum field theory with boundaries. In reality the expectation values will attain a limiting value on the conductor surface, which will depend on the molecular de- tails of the conductor. From the formulae given above it follows that the main contribution to 〈F 2〉cyl are due to the frequencies ω . (a − r)−1. Hence, we expect that formula (27) is valid for real conductors up to distances r for which (a− r)−1 ≪ ω0, with ω0 being the characteristic frequency, such that for ω > ω0 the conditions for perfect conductivity fail. Near the edge r = 0, assuming that r/a≪ 1, the asymptotic behavior of the part induced in the VEVs of the field square by the cylindrical shell depends on the parameter q. For q > 1+ηF1, the dominant contribution comes from the lowest mode m = 0 and to the leading order one has 〈F 2〉cyl ≈ −(−1)ηF1 21−ηF1q )2ηF1 K1(x) I1(x) . (31) In this case the quantity 〈B2〉cyl takes a finite limiting value on the edge r = 0, whereas 〈E2〉cyl vanishes as r2. For q < 1 + ηF1 the main contribution comes form the mode with m = 1 and the shell-induced parts diverge on the edge r = 0. The leading terms are given by the formula 〈F 2〉cyl ≈ − (−1)ηF1q(r/a)2(q−1) 22q−1πΓ2(q)a4 dxx2q+1 Kq(x) Iq(x) K ′q(x) I ′q(x) . (32) As for the points near the shell, here the leading divergences in the VEVs of the electric and magnetic fields are cancelled in the evaluation of the vacuum energy density. For q = 1+ηF1 the main contribution comes from the modes m = 0, 1 and the corresponding asymptotic behavior is obtained by summing the right-hand sides of Eqs. (31) and (32). In accordance with (23), near the edge r = 0 the total VEV is dominated by the part coming from the wedge without the cylindrical shell. Here we have considered the VEVs for the field square. The VEVs for the bilinear products of the fields at different spacetime points may be evaluated in a similar way. Now, we turn to the investigation of the behavior of the VEVs induced by the cylindrical boundary in the limit q ≫ 1. In this limit the order of the modified Bessel functions is large for m 6= 0. By using the corresponding asymptotic formulae it can be seen that the contribution of these terms is suppressed by the factor exp[−2qm ln(a/r)]. As a result, the main contribution comes from the lowest mode m = 0 and the VEVs induced by the cylindrical shell are propor- tional to q. Note that in this limit the part corresponding to the wedge without the cylindrical shell behaves as q4. 3 Vacuum energy-momentum tensor inside the cylindrical shell Now let us consider the VEV of the energy-momentum tensor in the region inside the cylindrical shell. Substituting the eigenfunctions (2) into the corresponding mode-sum formula, for the non- zero components we obtain (no summation over i) 〈0|T ii |0〉 = 2π2a3 λ=0,1 m,n Tqm(j m,n + k2a2 f (i)[Φ(λ)qm(φ), Jqm(j m,nr/a)], (33) 〈0|T 12 |0〉 = − m sin(2qmφ) λ=0,1 (−1)λ m,nTqm(j m,n + k2a2 J2qm(j m,nr/a), (34) where we have introduced the notations f (j)[Φ(φ), f(x)] = (−1)i 2k2/γ2 + 1 Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/y2 +Φ2(φ)f2(x), (35) f (l)[Φ(φ), f(x)] = (−1)lΦ2(φ)f ′2(x)− Φ2(φ) + (−1)lΦ′2(φ)/x2 f2(x), (36) with j = 0, 3 and l = 1, 2. As in the case of the field square, in formulae (33) and (34) we introduce a cutoff function and apply formula (12) for the summation over n. This enables us to present the vacuum energy-momentum tensor in the form of the sum 〈0|T ki |0〉 = 〈0w|T ki |0w〉+ 〈T ki 〉cyl, (37) where 〈0w|T ki |0w〉 is the part corresponding to the geometry of a wedge without a cylindrical boundary and 〈T ki 〉cyl is induced by the cylindrical shell. By taking into account (26), the latter may be written in the form (no summation over i) 〈T ii 〉cyl = λ=0,1 qm (xa) qm (xa) F (i)[Φ(λ)qm(φ), Iqm(xr)], (38) 〈T 12 〉cyl = m sin(2qmφ) λ=0,1 (−1)λ qm (xa) qm (xa) I2qm(xr), (39) with the notations F (i)[Φ(φ), f(y)] = Φ2(φ)f2(y), i = 0, 3, (40) F (i)[Φ(φ), f(y)] = −(−1)iΦ2(φ)f ′2(y)− Φ2(φ)− (−1)iΦ′2(φ)/y2 f2(y), i = 1, 2. (41) The diagonal components are symmetric with respect to the half-plane φ = φ0/2, whereas the off-diagonal component is an odd function under the replacement φ → φ0 − φ. As it can be easily checked, the tensor 〈T ki 〉cyl is traceless and satisfies the covariant continuity equation ∇k〈T ki 〉cyl = 0. For the geometry under consideration the latter leads to the relations r〈T 12 〉cyl 〈T 22 〉cyl = 0, (42) r〈T 11 〉cyl 〈T 21 〉cyl = 〈T 22 〉cyl. (43) As it is seen from formula (39), the off-diagonal component 〈T 12 〉cyl vanishes at the wedge sides and for these points the VEV of the energy-momentum tensor is diagonal. By using the in- equalities I ′ν(x) < 1 + ν2/x2Iν(x) and −K ′ν(x) > 1 + ν2/x2Kν(x) for the modified Bessel functions, it can be seen that K ′ν(x)/I ν(x) + Kν(x)/Iν(x) < 0. From this relation it follows that the vacuum energy density induced by the cylindrical shell in the interior region is always negative. The renormalized VEV of the energy-momentum tensor for the geometry without the cylin- drical shell is obtained by using the corresponding formulae for the field square. For the corre- sponding energy density one finds 〈T 00 〉w,ren = 〈E2〉w,ren + 〈B2〉w,ren = −(q 2 − 1)(q2 + 11) 720π2r4 . (44) As we see the parts in the VEVs of the field square which diverge on the wedge sides cancel out and the corresponding energy density is finite everywhere except the edge. Formula (44) coincides with the corresponding result for the geometry of the cosmic string (see [44, 45]) with the angle deficit 2π − φ0 and in the corresponding formula q = 2π/φ0. Other components are found from the tracelessness condition and the continuity equation and one has [29, 30] (see also 〈T ki 〉w,ren = − (q2 − 1)(q2 + 11) 720π2r4 diag(1, 1,−3, 1). (45) As we could expect this VEV vanishes for the geometry of a single plate corresponding to q = 1. In the limit r → ∞, for fixed values x0 = rφ0, from (45) the standard result for the geometry of two parallel conducting plates is obtained. The force acting on the wedge sides is determined by the component 〈T 22 〉ren of the vacuum energy-momentum tensor evaluated for φ = 0 and φ = φ0. On the base of formula (37) for the corresponding effective pressure one has p2 = −〈T 22 〉ren|φ=0,φ0 = p2w + p2cyl, (46) where p2w = − (q2 − 1)(q2 + 11) 240π2r4 , (47) is the normal force acting per unit surface of the wedge for the case without a cylindrical boundary and the additional term p2cyl = −〈T 22 〉cyl|φ=0,φ0 = − λ=0,1 qm (xa) qm (xa) F (λ)qm [Iqm(xr)], (48) with the notation F (λ)ν [f(y)] = ν2f2(y)/y2, λ = 0 −f ′2(y)− f2(y), λ = 1 , (49) is induced by the cylindrical shell. From formula (47) we see that the corresponding vacuum forces are attractive for q > 1 and repulsive for q < 1. In particular, the equilibrium position corresponding to the geometry of a single plate (q = 1) is unstable. As regards to the part induced by the cylindrical shell, from (48) it follows that p2cyl < 0 and, hence, the corresponding forces are always attractive. Now, let us discuss the behavior of the boundary-induced part in the VEV of the energy- momentum tensor in the asymptotic region of the parameters. Near the cylindrical shell the main contribution comes from large values of m. Thus, using the uniform asymptotic expansions for the modified Bessel functions for large values of the order, up to the leading order, for the points a− r ≪ a| sinφ|, a| sin(φ0 − φ)| we find 〈T 00 〉cyl ≈ − 〈T 22 〉cyl ≈ − (a− r)−3 60π2a , 〈T 11 〉cyl ≈ (a− r)−2 60π2a2 . (50) These leading terms are the same as those for a cylindrical shell when the wedge is absent. For the points near the edges (r = a, φ = 0, φ0) the leading terms in the corresponding asymptotic expansions are the same as for the geometry of a wedge with the opening angle φ0 = π/2. The latter are given by (45) with q = 2. Near the edge, r → 0, for the components (no summation over i) 〈T ii 〉cyl, i = 0, 3, the main contribution comes from the mode m = 0 and we find 〈T ii 〉cyl ≈ 4π2a4 K ′0(x) I ′0(x) = −0.0590 , i = 0, 3. (51) For the components (no summation over i) 〈T ii 〉cyl, i = 1, 2, when q > 1 the main contribution again comes form m = 0 term and one has 〈T ii 〉cyl ≈ −〈T 00 〉cyl, i = 1, 2. For q < 1 the main contribution into the components 〈T ii 〉cyl, i = 1, 2, comes from the term m = 1 and we have (no summation over i) 〈T ii 〉cyl ≈ (−1)iq cos(2qφ) 22q+1π2Γ2(q)a4 )2(q−1) dxx2q+1 Kq(x) Iq(x) K ′q(x) I ′q(x) , i = 1, 2. (52) In this case the radial and azimuthal stresses induced by the cylindrical shell diverge on the edge r = 0. In the case q = 1 the sum of the contributions of the terms with m = 0 and m = 1 given by formulae (51) and (52) should be taken. For the off-diagonal component the main contribution comes from the m = 1 mode with the leading term 〈T 12 〉cyl ≈ q sin(2qφ) 22q+1π2Γ2(q)a3 )2q−1 dxx2q+1 Kq(x) Iq(x) K ′q(x) I ′q(x) , (53) and this component vanishes on the edge for q > 1/2. In the limit q ≫ 1, the contribution of the modes with m > 1 is suppressed by the factor exp[−2qm ln(a/r)] and the main contribution comes from the m = 0 mode. The leading terms are given by the formulae (no summation over i) 〈T ii 〉cyl ≈ 4π2a4 K ′0(x) I ′0(x) I20 (xr/a), i = 0, 3, (54) 〈T ii 〉cyl ≈ − 4π2a4 K ′0(x) I ′0(x) I20 (xr/a) + (−1)iI21 (xr/a) , i = 1, 2. (55) Though in this limit the vacuum densities are large, due to the factor 1/q in the spatial volume, the corresponding global quantities tend to finite value. In particular, as it follows from Eq. (55), in the limit under consideration one has 〈T ii 〉cyl > 0. Note that in the same limit the parts corresponding to the wedge without the cylindrical shell behave as q4 and, hence, for points not too close to the shell these parts dominate in the VEVs. In figures 2-5 we have plotted the parts in the VEVs of the energy-momentum tensor induced by the cylindrical shell, a4〈T ki 〉cyl, as functions of x = (r/a) cos φ and y = (r/a) sinφ, for a wedge with the opening angle φ0 = π/2. In figure 6 we have presented the dependence of the effective azimuthal pressure induced by the cylindrical shell on the wedge sides, a4p2cyl, as a function of r/a for different values of the parameter q. Figure 2: The part in the VEV of the energy density, a4〈T 00 〉cyl, induced by the cylindrical boundary as a function on x = (r/a) cos φ and y = (r/a) sin φ for a wedge with φ0 = π/2. Figure 3: The part in the VEV of the radial stress, a4〈T 11 〉cyl, induced by the cylindrical boundary as a function on x = (r/a) cos φ and y = (r/a) sinφ for a wedge with φ0 = π/2. Figure 4: The part in the VEV of the azimuthal stress, a4〈T 22 〉cyl, induced by the cylindrical boundary as a function on x = (r/a) cos φ and y = (r/a) sin φ for a wedge with φ0 = π/2. Figure 5: The part in the VEV of the off-diagonal component, a4〈T 21 〉cyl, induced by the cylin- drical boundary as a function on x = (r/a) cos φ and y = (r/a) sinφ for a wedge with φ0 = π/2. 0.1 0.2 0.3 0.4 0.5 Figure 6: The effective azimuthal pressure induced by the cylindrical shell on the wedge sides, a4p2cyl, as a function of r/a. The numbers near the curves correspond to the values of the parameter q. There are several special cases of interest for the geometry of boundaries we have consid- ered. The case φ0 = π corresponds to the semi-circular cylinder. The Casimir energy for the corresponding interior region is evaluated in Ref. [34] by using the zeta function technique. The case φ0 = 2π corresponds to the geometry of a cylindrical shell with a coaxial half-plane. And finally, the limit φ0 → 0, r, a → ∞, assuming that a − r and aφ0 ≡ b are fixed, corresponds to the geometry of two parallel plates separated by a distance b, perpendicularly intersected by the third plate. In the latter case it is convenient to introduce rectangular coordinates (x′1, x′2, x′3) = (x, y, z) with the relations x = a − r, y = rφ. The components of the tensors in these coordinates we will denote by primes. The corresponding vacuum energy-momentum tensor is presented in the form 〈0|T ′ik |0〉 = 〈T ′ik 〉(0) + 〈T ′ik 〉(1), (56) where 〈T ′ik 〉(0) is the vacuum expectation value in the region between two parallel plates located at y = 0 and y = a and 〈T ′ik 〉(1) is induced by the intersecting plate at x = 0. The latter is related to the quantities investigated above by formulae 〈T ′ii 〉(1) = lim 〈T ii 〉cyl, 〈T ′12 〉(1) = − lim 〈T 12 〉cyl, (57) with lim corresponding to the limit a → ∞, φ0 → 0 for fixed a − r and aφ0. Taking this limit in the term with m = 0 of formula (38) we replace the modified Bessel functions by the leading terms of the corresponding asymptotic formulae for large values of the argument and the integral is taken elementary. For the terms with m 6= 0 in formulae (38), (39) we note that in the limit under consideration one has q = π/φ0 → ∞, and the order of the Bessel modified functions tends to infinity. Introducing a new integration variable x→ qmx, we can replace these functions by their uniform asymptotic expansions for large values of the order. After these replacements the integration and the further summation over m are done by using the formulae from [46]. 4 Vacuum densities in the exterior region In this section we consider the VEVs for the field square and the energy-momentum tensor in the region outside the cylindrical boundary (region II in figure 1). The corresponding eigenfunctions for the vector potential are obtained from formulae (2) by the replacement Jqm(γr) → g(λ)qm(γa, γr) = Jqm(γr)Y (λ)qm (γa)− Yqm(γr)J (λ)qm (γa), (58) where, as before, λ = 0, 1 correspond to the waves of the electric and magnetic types, respec- tively. Now, the eigenvalues for γ are continuous and in the normalization condition (4) the corresponding part on the right is presented by the delta function. As the normalization inte- gral diverges for γ′ = γ, the main contribution into the integral comes from large values of r and we can replace the cylindrical functions with the argument γr by their asymptotic expressions for large values of the argument. By this way it can be seen that the normalization coefficient in the exterior region is determined by the relation β−2α = J (λ)2qm (γa) + Y qm (γa) . (59) Substituting the eigenfunctions into the corresponding mode-sum formula, for the VEV of the field square one finds 〈0|F 2|0〉 = λ=0,1 k2 + γ2 g(ηFλ)[Φ qm(φ), g qm(γa, γr)] qm (γa) + Y qm (γa) , (60) where the functions g(ηFλ) qm(φ), g qm(γa, γr) are defined by relations (9), (10) with the function f(x) = g qm(γa, x). To extract from this VEV the part induced by the cylindrical shell, we subtract from the right-hand side the corresponding expression for the wedge without the cylindrical boundary. The latter is given by formula (14). The corresponding difference can be further evaluated by using the identity g(ηFλ)[Φ qm(φ), g qm(γa, γr)] qm (γa) + Y qm (γa) = g(ηFλ)[Φ(λ)qm(φ), Jqm(γr)] qm (γa) (s)(λ) qm (γa) g(ηFλ)[Φ(λ)qm(φ),H qm(γr)], (61) where H (1,2) qm (z) are the Hankel functions. In order to transform the integral over γ with the last term on the right of (61), in the complex plane γ we rotate the integration contour by the angle π/2 for the term with s = 1 and by the angle −π/2 for the term with s = 2. Due to the well-known properties of the Hankel functions the integrals over the corresponding parts of the circles of large radius in the upper and lower half-planes vanish. After introducing the modified Bessel functions and integrating over k with the help of formula (26), we can write the VEVs of the field square in the form (13), where the part induced by the cylindrical shell is given by the formula 〈F 2〉cyl = λ=0,1 qm (xa) qm(xa) G(ηFλ)[Φ(λ)qm(φ),Kqm(xr)]. (62) In this formula the functions G(ηFλ) [Φ(φ), f(x)] are defined by formulae (28), (29). Comparing this result with formula (27), we see that the expressions for the shell-induced parts in the interior and exterior regions are related by the interchange Iqm ⇄ Kqm. The VEV (62) diverges on the cylindrical shell with the leading term being the same as that for the interior region. At large distances from the cylindrical shell we introduce a new integration variable y = xr and expand the integrand over a/r. For q > 1 the main contribution comes from the lowest mode m = 0 and up to the leading order we have 〈E2〉cyl ≈ , 〈B2〉cyl ≈ − 15πr4 . (63) For q < 1 the dominant contribution into the VEVs at large distances is due to the mode m = 1 with the leading term 〈F 2〉cyl ≈ − 4q2(q + 1) cos(2qφ) 2q + 3 + (−1)ηF1 q + 1 2q + 1 . (64) For the case q = 1 the contributions of the modes m = 0 and m = 1 are of the same order and the corresponding leading terms are obtained by summing these contributions. The latter are given by the right-hand sides of formulae (63) and (64). As we see, at large distances the part induced by the cylindrical shell is suppressed with respect to the part corresponding to the wedge without the shell by the factor (a/r)2β with β = min(1, q). Now we turn to the VEVs of the energy-momentum tensor in the exterior region. Substi- tuting the eigenfunctions into the corresponding mode-sum formula, one finds (no summation over i) 〈0|T ii |0〉 = λ=0,1 k2 + γ2 f (i)[Φ qm(φ), g qm(γa, γr)] qm (γa) + Y qm (γa) , (65) 〈0|T 12 |0〉 = − m sin(2qmφ) λ=0,1 (−1)λ qm (γa, γr) k2 + γ2 . (66) Subtracting from these VEVs the corresponding expression for the wedge without the cylindrical boundary, analogously to the case of the field square, it can be seen that the VEVs are presented in the form (37), with the parts induced by the cylindrical shell given by the formulae (no summation over i) 〈T ii 〉cyl = λ=0,1 qm (xa) qm (xa) F (i)[Φ(λ)qm(φ),Kqm(xr)], (67) 〈T 12 〉cyl = m sin(2qmφ) λ=0,1 (−1)λ qm (xa) qm (xa) K2qm(xr). (68) Here the functions F (i) [Φ(φ), f(y)] are defined by formulae (40), (41). By using the inequality given in the paragraph after formula (43), we can show that the vacuum energy density induced by the cylindrical shell in the exterior region is positive. In the way similar to that for the interior region, for the force acting on the wedge sides is presented in the form of the sum (46), where the part corresponding to the wedge without a cylindrical shell is determined by formula (47) and for the part due to the presence of the cylindrical shell we have p2cyl = −〈T 22 〉cyl|φ=0,φ0 = − λ=0,1 qm (xa) qm(xa) F (λ)qm [Kqm(xr)]. (69) In this formula, the function F ν [f(y)] is defined by relations (49) and the corresponding forces are always attractive. The leading divergence in the boundary induced part (67) on the cylindrical surface is given by the same formulae as for the interior region. For large distances from the shell and for q > 1 the main contribution into the VEVs of the diagonal components comes from the m = 0, λ = 1 term and one has (no summation over i) 〈T ii 〉cyl ≈ − 15π2r4 , c0 = c3 = 2, c1 = 1, c2 = −5. (70) In the case q < 1 the main contribution into the VEVs of the diagonal components at large distances from the cylindrical shell comes from the m = 1 mode. The leading terms in the corresponding asymptotic expansions are given by the formulae 〈T ii 〉cyl ≈ −q2(q + 1)ci(q) cos(2qφ) , (71) with the notations c0(q) = c3(q) = 2q + 3 , c1(q) = 2q2 + q + 1 (2q + 1)(2q + 3) , c2(q) = − q + 1 2q + 1 . (72) In the case q = 1 the asymptotic terms are determined by the sum of the contributions coming from the modes m = 0 and m = 1. The latter are given by formulae (70), (71). For the off- diagonal component, for all values q the main contribution at large distances comes from the m = 1 mode with the leading term 〈T 12 〉cyl ≈ − q3(q + 1) 2q + 1 sin(2qφ) . (73) For large values of q, q ≫ 1, the contribution of the terms with m > 0 is suppressed by the factor exp[−2qm ln(r/a)] and the main contribution comes form the m = 0 term with the behavior 〈F 2〉cyl ∝ q and 〈T ki 〉cyl ∝ q. In figure 7 we have plotted the dependence of the effective azimuthal pressure induced by the cylindrical shell on the wedge sides, a4p2cyl, as a function of r/a for q = 1. 1.4 1.5 1.6 1.7 1.8 1.9 2 Figure 7: The effective azimuthal pressure induced by the cylindrical shell on the wedge sides, a4p2cyl, as a function of r/a in the exterior region for q = 1. The curves for the values q = 0.5, 3 are close to the plotted one. 5 Conclusion In this paper we have investigated the polarization of the electromagnetic vacuum by a wedge with a coaxial cylindrical boundary, assuming that all boundaries are perfectly conducting. Both regions inside and outside of the cylindrical shell (regions I and II in figure 1) are considered. In section 2 we have evaluated the VEVs of the field square in the interior region. The corresponding mode-sums contain series over the zeros of the Bessel function for TM modes and its derivative for TE modes. For the summation of these series we used a variant of the generalized Abel- Plana formula. The latter enables us to extract from the VEVs the parts corresponding to the geometry of a wedge without a cylindrical shell and to present the parts induced by the shell in terms of integrals which are exponentially convergent for points away from the boundaries. For the wedge without the cylindrical shell the VEVs of the field square are presented in the form (23). The first term on the right of this formula corresponds to the VEVs for the geometry of a cosmic string with the angle deficit 2π − φ0. The angle-dependent parts in the VEVs of the electric and magnetic fields have opposite signs and are cancelled in the evaluation of the vacuum energy density. The parts induced by the cylindrical shell are presented in the form (27). We have discussed this general formula in various asymptotic regions of the parameters including the points near the edges and near the shell. In section 3 we consider the VEV of the energy-momentum tensor in the region inside the shell. As for the field square, the application of the Abel-Plana formula allows us to present this VEV in the form of the sum of purely wedge and shell-induced parts, formula (37). For the geometry of a wedge without the cylindrical boundary the vacuum energy-momentum tensor does not depend on the angle φ and is the same as in the geometry of the cosmic string and is given by formula (45). The corresponding vacuum forces acting on the wedge sides are attractive for φ0 < π and repulsive for φ0 > π. In particular, the equilibrium position corresponding to the geometry of a single plate (φ0 = π) is unstable. For the region inside the shell the part in the VEV of the energy-momentum tensor induced by the presence of the cylindrical shell is non-diagonal and the corresponding components are given by formulae (38), (39). The vacuum energy density induced by the cylindrical shell in the interior region is negative. We have investigated the vacuum densities induced by the cylindrical shell in various asymptotic regions of the parameters. For points near the cylindrical shell the leading terms in the asymptotic expansions over the distance from the shell are given by formulae (50). These terms are the same as those for a cylindrical shell when the wedge is absent. For a wedge with φ0 < π the part in the vacuum energy-momentum tensor induced by the shell is finite on the edge r = 0. For φ0 > π the shell-induced parts in the energy density and the axial stress remain finite, whereas the radial and azimuthal stresses diverge as r2(π/φ0−1). The corresponding off-diagonal component behaves like r2π/φ0−1 for all values φ0. For the points near the edges (r = a, φ = 0, φ0) the leading terms in the corresponding asymptotic expansions are the same as for the geometry of a wedge with the opening angle φ0 = π/2. In the limit of small opening angles, φ0 ≪ π, the shell-induced parts behave like 1/φ0. In the same limit the parts corresponding to the wedge without the shell behave as 1/φ40, and for points not too close to the shell these parts dominate in the VEV of the energy-momentum tensor. The presence of the shell leads to additional forces acting on the wedge sides. The corresponding effective azimuthal pressure is given by formula (48) and these forces are always attractive. The VEVs of the field square and the energy-momentum tensor in the region outside the cylindrical shell are investigated in section 4. As in the case of the interior region, these VEVs are presented as sums of the parts corresponding to the wedge without the cylindrical shell and the parts induced by the shell. The latter are given by formula (62) for the field square and by formulae (67), (66) for the components of the energy-momentum tensor. In the exterior region the vacuum energy density induced by the cylindrical shell is always positive. Additional forces acting on the wedge sides due to the presence of the shell are given by formula (69). As in the case of the interior region these forces are attractive. For large values of the parameter q, the contribution into the parts induced by the cylindrical shell coming from the modes with m 6= 0 is exponentially suppressed, whereas the contribution of the lowest mode m = 0 is proportional to q. Though in this limit the vacuum densities are large, due to the factor 1/q in the spatial volume element, the corresponding global quantities tend to finite limiting values. 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Tarloyan, Phys. Rev. D 74, 025017 (2006). [40] E.R. Bezerra de Mello, V.B. Bezerra, and A.A. Saharian, Phys. Lett. B 645, 245 (2007). [41] A.P. Prudnikov, Yu.A. Brychkov, and O.I. Marichev, Integrals and Series (Gordon and Breach, New York, 1986), Vol. 2. [42] P.C.W. Davies and V. Sahni, Class. Quantum Grav. 5, 1 (1988). [43] Handbook of Mathematical Functions, edited by M. Abramowitz and I.A. Stegun, (Dover, New York, 1972). [44] V.P. Frolov and E.M. Serebriany, Phys. Rev. D 35, 3779 (1987). [45] J.S. Dowker, Phys. Rev. D 36, 3742 (1987). [46] A.P. Prudnikov, Yu.A. Brychkov, and O.I. Marichev, Integrals and Series (Gordon and Breach, New York, 1986), Vol. 1. http://arxiv.org/abs/hep-th/0609093 Introduction Vacuum expectation values of the field square inside a cylindrical shell Vacuum energy-momentum tensor inside the cylindrical shell Vacuum densities in the exterior region Conclusion

Vacuum expectation values of the field square and the energy-momentum tensor for the electromagnetic field are investigated for the geometry of a wedge with a coaxal cylindrical boundary. All boundaries are assumed to be perfectly conducting and both regions inside and outside the shell are considered. By using the generalized Abel-Plana formula, the vacuum expectation values are presented in the form of the sum of two terms. The first one corresponds to the geometry of the wedge without the cylindrical shell and the second term is induced by the presence of the shell. The vacuum energy density induced by the shell is negative for the interior region and is positive for the exterior region. The asymptotic behavior of the vacuum expectation values are investigated in various limiting cases. It is shown that the vacuum forces acting on the wedge sides due to the presence of the cylindrical boundary are always attractive.

Introduction The Casimir effect is among the most interesting macroscopic manifestations of vacuum fluctu- ations. It have important implications on all scales, from cosmological to subnuclear, and has become in recent decades an increasingly popular topic in quantum field theory. In addition to its fundamental interest the Casimir effect also plays an important role in the fabrication and operation of nano- and micro-scale mechanical systems. The imposition of boundary conditions on a quantum field leads to the modification of the spectrum for the zero-point fluctuations and results in the shift in the vacuum expectation values for physical quantities such as the energy density and stresses. In particular, the confinement of quantum fluctuations causes forces that act on constraining boundaries. The particular features of the resulting vacuum forces depend on the nature of the quantum field, the type of spacetime manifold, the boundary geometries and the specific boundary conditions imposed on the field. Since the original work by Casimir [1] many theoretical and experimental works have been done on this problem (see, e.g., [2, 3, 4, 5] and references therein). Many different approaches have been used: mode summation method E-mail: saharian@ictp.it http://arxiv.org/abs/0704.1898v1 with combination of the zeta function regularization technique, Green function formalism, mul- tiple scattering expansions, heat-kernel series, etc. Advanced field-theoretical methods have been developed for Casimir calculations during the past years [6, 7, 8]. However, there are still difficulties in both interpretation and renormalization of the Casimir effect. Straightforward computations of geometry dependencies are conceptually complicated, since relevant informa- tion is subtly encoded in the fluctuations spectrum [8]. Analytic solutions can usually be found only for highly symmetric geometries including planar, spherically and cylindrically symmetric boundaries. Recently the Casimir energy has been evaluated exactly for several less symmetric configurations of experimental interest. These include a sphere in front of a plane and a cylinder in front of a plane [9]. A great deal of attention received the investigations of quantum effects for cylindrical bound- aries. In addition to traditional problems of quantum electrodynamics under the presence of material boundaries, the Casimir effect for cylindrical geometries can also be important to the flux tube models of confinement [10, 11] and for determining the structure of the vacuum state in interacting field theories [12]. The calculation of the vacuum energy of electromagnetic field with boundary conditions defined on a cylinder turned out to be technically a more involved problem than the analogous one for a sphere. First the Casimir energy of an infinite perfectly conducting cylindrical shell has been calculated in Ref. [13] by introducing ultraviolet cutoff and later the corresponding result was derived by zeta function technique [14, 15, 16]. The local characteristics of the corresponding electromagnetic vacuum such as energy density and vacuum stresses are considered in [17] for the interior and exterior regions of a conducting cylindrical shell, and in [18] for the region between two coaxial shells (see also [19]). The vacuum forces acting on the boundaries in the geometry of two cylinders are also considered in Refs. [20]. The scalar Casimir densities for a single and two coaxial cylindrical shells with Robin boundary conditions are investigated in Refs. [21, 22]. Less symmetric configuration of two eccentric per- fectly conducting cylinders is considered in Ref. [23]. Vacuum energy for a perfectly conducting cylinder of elliptical section is evaluated in Ref. [24] by the mode summation method, using the ellipticity as a perturbation parameter. The Casimir forces acting on two parallel plates inside a conducting cylindrical shell are investigated in Ref. [25]. Aside from their own theoretical and experimental interest, the exactly solvable problems with this type of boundaries are useful for testing the validity of various approximations used to deal with more complicated geometries. From this point of view the wedge with a coaxial cylindrical boundary is an interesting system, since the geometry is nontrivial and it includes two dynamical parameters, radius of the cylindrical shell and opening angle of the wedge. This geometry is also interesting from the point of view of general analysis for surface divergences in the expectation values of local physical observables for boundaries with discontinuities. The nonsmoothness of the boundary generates additional contributions to the heat kernel coefficients (see, for instance, the discussion in [26, 27, 28] and references therein). The present paper is concerned with local analysis of the vacuum of the electromagnetic field constrained to satisfy perfectly conducting boundary conditions on boundary surfaces of a wedge with a coaxial cylin- drical boundary. Namely, we will study the vacuum expectation values of the field squares and the energy-momentum tensor for the electromagnetic field for both regions inside and outside the cylindrical shell. In addition to describing the physical structure of the quantum field at a given point, the energy-momentum tensor acts as the source of gravity in the Einstein equa- tions. It therefore plays an important role in modelling a self-consistent dynamics involving the gravitational field. The vacuum expectation value of the square of the electric field determines the electromagnetic force on a neutral polarizable particle. Some most relevant investigations to the present paper are contained in Refs. [2, 29, 30, 31, 32, 33], where the geometry of a wedge without a cylindrical boundary is considered for a conformally coupled scalar and electromag- netic fields in a four dimensional spacetime. The total Casimir energy of a semi-circular infinite cylindrical shell with perfectly conducting walls is considered in [34] by using the zeta function technique. For a scalar field with an arbitrary curvature coupling parameter the Wightman function, the vacuum expectation values of the field square and the energy-momentum tensor in the geometry of a wedge with an arbitrary opening angle and with a cylindrical boundary are evaluated in [35, 36]. Note that, unlike the case of conformally coupled fields, for a general coupling the vacuum energy-momentum tensor is angle-dependent and diverges on the wedge sides. Our method here employs the mode summation and is based on a variant of the gen- eralized Abel-Plana formula [37] (see also Refs. [19, 38]). This enables us to extract from the vacuum expectation values the parts due to a wedge without the cylindrical shell and to present the parts induced by the shell in terms of strongly convergent integrals. Note that the closely related problem of the vacuum densities induced by a cylindrical boundary in the geometry of a cosmic string is investigated in Refs. [39, 40] for both scalar and electromagnetic fields. We have organized the paper as follows. In the next section we describe the structure of the modes for a wedge with a cylindrical shell in the region inside the shell. By applying to the corresponding mode sums the generalized Abel-Plana formula, we evaluate the vacuum expectation values of the electric and magnetic field square. Various limiting cases of the general formulae are discussed. Section 3 is devoted to the investigation of the vacuum expectation values for the energy-momentum tensor of the electromagnetic field in the region inside the shell. The additional vacuum forces acting on the wedge sides due to the presence of the cylindrical boundary are evaluated. In section 4 we consider the vacuum densities for a wedge with the cylindrical shell in the exterior region with respect to the shell. Formulae for the shell contributions are derived and the corresponding surface divergences are investigated. The vacuum forces acting on the wedge sides are discussed. The main results are summarized and discussed in section 5. 2 Vacuum expectation values of the field square inside a cylin- drical shell Consider a wedge with the opening angle φ0 and with a coaxial cylindrical boundary of radius a (see figure 1) assuming that all boundaries are perfectly conducting. In accordance with the problem symmetry, in the discussion below the cylindrical coordinates (r, φ, z) will be used. We are interested in the vacuum expectation values (VEVs) of the field square and the energy- momentum tensor for the electromagnetic field. Expanding the field operator in terms of the creation and annihilation operators and using the commutation relations, the VEV for a quantity F {Ai, Ak} bilinear in the field can be presented in the form of the mode-sum 〈0|F {Ai, Ak} |0〉 = F {Aαi, A∗αk} , (1) where {Aαi, A∗αk} is a complete set of solutions of the classical field equations satisfying the boundary conditions on the bounding surfaces and specified by a set of quantum numbers α. In accordance with formula (1), for the evaluation of the VEVs for the square of the electric and magnetic fields and the energy-momentum tensor, the corresponding eigenfunctions are needed. In this section we consider the region inside the cylindrical shell (region I in figure 1). For the geometry under consideration there are two different types of the eigenfunctions corresponding to the transverse magnetic (TM) and transverse electric (TE) waves. In the discussion below we will specify these modes by the index λ = 0 and λ = 1 for the TM and TE waves respectively. In the Coulomb gauge, the vector potentials for the TM and TE modes are Figure 1: Geometry of a wedge with a coaxial cylindrical boundary with radius a. given by the formulae Aα = βα (1/iω) γ2e3 + ik∇t Jq|m|(γr) sin(qmφ) exp [i (kz − ωt)] , λ = 0 −e3 ×∇t Jq|m|(γr) cos(qmφ) exp [i (kz − ωt)] , λ = 1 , (2) where e3 is the unit vector along the axis of the wedge, ∇t is the part of the nabla operator transverse to this axis, Jν(x) is the Bessel function of the first kind, and ω2 = γ2 + k2, q = π/φ0. (3) In Eq. (2), m = 1, 2, . . . for λ = 0 and m = 0, 1, 2, . . . for λ = 1. The normalization coefficient βα is found from the orthonormalization condition for the vector potential: dV Aα ·A∗α′ = δαα′ , (4) where the integration goes over the region inside the shell. From this condition, by using the standard integral involving the square of the Bessel function, one finds β2α = 4qTqm(γa) δm, δm = 1/2, m = 0 1, m 6= 0 , (5) where we have introduced the notation Tν(x) = x ν (x) + (1− ν2/x2)J2ν (x) . (6) Eigenfunctions (2) satisfy the standard boundary conditions for the electric and magnetic fields, n×E = 0 and n ·B = 0, on the wedge sides corresponding to φ = 0 and φ = φ0, with n being the normal to the boundary. The eigenvalues for the quantum number γ are determined by the boundary conditions on the cylindrical shell. From the latter it follows that these eigenvalues are solutions of the equation J (λ)qm (γa) = 0, λ = 0, 1, (7) where we use the notations J ν (x) = Jν(x) and J ν (x) = J ν(x). We will denote the corre- sponding eigenmodes by γa = j m,n, n = 1, 2, . . ., assuming that the zeros j m,n are arranged in ascending order. Consequently, the eigenfunctions are specified by the set of quantum numbers α = (k,m, λ, n). First we consider the VEVs of the squares of the electric and magnetic fields inside the shell. Substituting the eigenfunctions (2) into the corresponding mode-sum formula, we find 〈0|F 2|0〉 = 4q λ=0,1 m,n Tqm(j m,n + k2a2 ×g(ηFλ)[Φ(λ)qm(φ), Jqm(j(λ)m,nr/a)], (8) where F = E,B with ηEλ = λ, ηBλ = 1−λ, and the prime in the summation over m means that the term m = 0 should be halved. In formula (8), for a given function f(x), we have introduced the notations g(0)[Φ(φ), f(x)] = (k2r2/x2) Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2 +Φ2(φ)f2(x), (9) g(1)[Φ(φ), f(x)] = (1 + k2r2/x2) Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2 , (10) Φ(λ)ν (φ) = sin(νφ), λ = 0 cos(νφ), λ = 1 . (11) The expressions (8) corresponding to the electric and magnetic fields are divergent. They may be regularized introducing a cutoff function ψµ(ω) with the cutting parameter µ which makes the divergent expressions finite and satisfies the condition ψµ(ω) → 1 for µ → 0. After the renormalization the cutoff function is removed by taking the limit µ → 0. An alternative way is to consider the product of the fields at different spacetime points and to take the coincidence limit after the subtraction of the corresponding Minkowskian part. Our approach here follows the first method. As we do not know the explicit expressions for the zeros j m,n as functions on m and n, and the summand in formula (8) is strongly oscillating function for large values of m and n, this formula is not convenient for the further evaluation of the VEVs of the field square. In order to obtain an alternative representation, we apply to the series over n the generalized Abel-Plana summation formula [37] (see also [38]) Tqm(j m,n)f(j m,n) = dx f(x) + qm (z) qm (z) qm (x) qm (x) e−qmπif(eπi/2x) + eqmπif(e−πi/2x) , (12) where Yν(z) is the Neumann function and Iν(z), Kν(z) are the modified Bessel functions. As it can be seen, for points away from the shell the contribution to the VEVs coming from the second integral term on the right-hand side of (12) is finite in the limit µ → 0 and, hence, the cutoff function in this term can be safely removed. As a result the VEVs can be written in the 〈0|F 2|0〉 = 〈0w|F 2|0w〉+ , (13) where 〈0w|F 2|0w〉 = γ3ψµ(ω) γ2 + k2 J ′2qm(γr) + J2qm(γr) + J2qm(γr) −(−1)ηF1 cos(2qmφ) J ′2qm(γr)− J2qm(γr) , (14) 〈F 2〉cyl = λ=0,1 qm (xa) qm (xa) G(ηFλ)[k,Φ qm(φ), Iqm(xr)]√ x2 − k2 . (15) In formula Eq. (15) we have introduced the notations G(0)[k,Φ(φ), f(x)] = (k2r2/x2) Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2 +Φ2(φ)f2(x), (16) G(1)[k,Φ(φ), f(x)] = (k2r2/x2 − 1) Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2 . (17) The second term on the right-hand side of Eq. (13) vanishes in the limit a → ∞ and the first one does not depend on a. Thus, we can conclude that the term 〈0w|F 2|0w〉 corresponds to the part in the VEVs when the cylindrical shell is absent with the corresponding vacuum state |0w〉. Hence, the application of the generalized Abel-Plana formula enables us to extract from the VEVs the parts induced by the cylindrical shell without specifying the cutoff function. In addition, these parts are presented in terms of the exponentially convergent integrals. First, let us concentrate on the part corresponding to the wedge without a cylindrical shell. First of all we note that in Eq. (14) the part which does not depend on the angular coordinate φ is the same as in the corresponding problem of the cosmic string geometry with the angle deficit 2π − φ0 (see Ref. [40]), which we will denote by 〈0s|F 2|0s〉. For this part we have 〈0s|F 2|0s〉 = γ3ψµ(ω) γ2 + k2 J ′2qm(γr) + J2qm(γr) + J2qm(γr) = 〈0M|F 2|0M〉 − (q2 − 1)(q2 + 11) 180πr4 , (18) where 〈0M|F 2|0M〉 is the part corresponding to the Minkowskian spacetime without boundaries and in the last expression we have removed the cutoff. To evaluate the part in (14) which depends on φ, we firstly consider the case when the parameter q is an integer. In this case the summation over m can be done by using the formula [41, 42] cos(2qmφ)J2qm(y) = J0(2y sin(φ+ φ0l)). (19) The formulae for the other series entering in Eq. (14) are obtained from (19) taking the deriva- tives with respect to φ and y. In particular, for the combination appearing in the angle-dependent part we obtain cos(2qmφ) J ′2qm(y)− J2qm(y) J ′1(2y sin(φ+ φ0l)). (20) Substituting this in formula (14), the integrals remained are evaluated by introducing polar coordinates in the (k, γ)-plane. In this way one finds 〈0w|F 2|0w〉 = 〈0s|F 2|0s〉 − 3(−1)ηF1 sin−4(φ+ lπ/q). (21) The sum on the right hand-side of this formula is evaluated by the double differentiation of the relation [41] cos−2(x+ lπ/q) = q2 sin−2(qx+ qπ/2). (22) Finally, for the renormalised VEVs of the field square in the geometry of a wedge without a cylindrical boundary we find 〈F 2〉w,ren = − (q2 − 1)(q2 + 11) 180πr4 (−1)ηF1q2 2πr4 sin2(qφ) 2 sin2(qφ) + 1− q2 , (23) with ηE1 = 1 and ηB1 = 0. Though we have derived this formula for integer values of the parameter q, by the analytic continuation it is valid for non-integer values of this parameter as well. The expression on the right of formula (23) is invariant under the replacement φ→ φ0 −φ and, as we could expect, the VEVs are symmetric with respect to the half-plane φ = φ0/2. Formula (23) for F = E was derived in Ref. [32] within the framework of Schwinger’s source theory. For q = 1 from formula (23) as a special case we obtain the renormalized VEVs of the field square for a conducting plate. In this case x = r sinφ is the distance from the plate and one has 〈F 2〉pl,ren = − 3(−1)ηF1 . (24) Another special case q = 1/2 corresponds to the geometry of a half-plane. In (23) taking the limit r → ∞, with x0 = rφ0 being fixed, we obtain the corresponding results in the region between two parallel plates located at the points x = 0 and x = x0: 〈F 2〉2pl,ren = − 180x40 − (−1) ηF1π3 2x40 sin 2(πx/x0) 2 sin2(πx/x0) . (25) Now, we turn to the investigation of the parts in the VEVs of the field square induced by the cylindrical boundary and given by formula (15). By using the formula dk km xf(x)√ x2 − k2 dxxm+1f(x), (26) these parts are presented in the form 〈F 2〉cyl = λ=0,1 qm (xa) qm (xa) G(ηFλ)[Φ(λ)qm(φ), Iqm(xr)]. (27) Here, for given functions f(x) and Φ(φ), we have introduced the notations G(0)[Φ(φ), f(x)] = Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/x2 + 2Φ2(φ)f2(x), (28) G(1)[Φ(φ), f(x)] = −Φ2(φ)f ′2(x)− Φ′2(φ)f2(x)/x2. (29) As we see the parts in the VEVs induced by the cylindrical shell are symmetric with respect to the half-plane φ = φ0/2. The expression in the right-hand side of (27) is finite for 0 < r < a including the points on the wedge sides, and diverges on the shell. To find the leading term in the corresponding asymptotic expansion, we note that near the shell the main contribution comes from large values of m. By using the uniform asymptotic expansions of the modified Bessel functions (see, for instance, [43]) for large values of the order, up to the leading order, for the points a−r ≪ a| sinφ|, a| sin(φ0−φ)| we find 〈F 2〉cyl ≈ − 3(−1)ηF1 4π(a− r)4 . (30) For the points near the edges (r = a, φ = 0, φ0) the leading terms in the corresponding asymp- totic expansions are the same as for the geometry of a wedge with the opening angle φ0 = π/2. The leading terms given by formula (30) are the same as for the geometry of a single plate (see (24)). They do not depend on φ0 and have opposite signs for the electric and magnetic fields. In particular, the leading terms are cancelled in the evaluation of the vacuum energy density. Surface divergences originate in the unphysical nature of perfect conductor boundary conditions and are well-known in quantum field theory with boundaries. In reality the expectation values will attain a limiting value on the conductor surface, which will depend on the molecular de- tails of the conductor. From the formulae given above it follows that the main contribution to 〈F 2〉cyl are due to the frequencies ω . (a − r)−1. Hence, we expect that formula (27) is valid for real conductors up to distances r for which (a− r)−1 ≪ ω0, with ω0 being the characteristic frequency, such that for ω > ω0 the conditions for perfect conductivity fail. Near the edge r = 0, assuming that r/a≪ 1, the asymptotic behavior of the part induced in the VEVs of the field square by the cylindrical shell depends on the parameter q. For q > 1+ηF1, the dominant contribution comes from the lowest mode m = 0 and to the leading order one has 〈F 2〉cyl ≈ −(−1)ηF1 21−ηF1q )2ηF1 K1(x) I1(x) . (31) In this case the quantity 〈B2〉cyl takes a finite limiting value on the edge r = 0, whereas 〈E2〉cyl vanishes as r2. For q < 1 + ηF1 the main contribution comes form the mode with m = 1 and the shell-induced parts diverge on the edge r = 0. The leading terms are given by the formula 〈F 2〉cyl ≈ − (−1)ηF1q(r/a)2(q−1) 22q−1πΓ2(q)a4 dxx2q+1 Kq(x) Iq(x) K ′q(x) I ′q(x) . (32) As for the points near the shell, here the leading divergences in the VEVs of the electric and magnetic fields are cancelled in the evaluation of the vacuum energy density. For q = 1+ηF1 the main contribution comes from the modes m = 0, 1 and the corresponding asymptotic behavior is obtained by summing the right-hand sides of Eqs. (31) and (32). In accordance with (23), near the edge r = 0 the total VEV is dominated by the part coming from the wedge without the cylindrical shell. Here we have considered the VEVs for the field square. The VEVs for the bilinear products of the fields at different spacetime points may be evaluated in a similar way. Now, we turn to the investigation of the behavior of the VEVs induced by the cylindrical boundary in the limit q ≫ 1. In this limit the order of the modified Bessel functions is large for m 6= 0. By using the corresponding asymptotic formulae it can be seen that the contribution of these terms is suppressed by the factor exp[−2qm ln(a/r)]. As a result, the main contribution comes from the lowest mode m = 0 and the VEVs induced by the cylindrical shell are propor- tional to q. Note that in this limit the part corresponding to the wedge without the cylindrical shell behaves as q4. 3 Vacuum energy-momentum tensor inside the cylindrical shell Now let us consider the VEV of the energy-momentum tensor in the region inside the cylindrical shell. Substituting the eigenfunctions (2) into the corresponding mode-sum formula, for the non- zero components we obtain (no summation over i) 〈0|T ii |0〉 = 2π2a3 λ=0,1 m,n Tqm(j m,n + k2a2 f (i)[Φ(λ)qm(φ), Jqm(j m,nr/a)], (33) 〈0|T 12 |0〉 = − m sin(2qmφ) λ=0,1 (−1)λ m,nTqm(j m,n + k2a2 J2qm(j m,nr/a), (34) where we have introduced the notations f (j)[Φ(φ), f(x)] = (−1)i 2k2/γ2 + 1 Φ2(φ)f ′2(x) + Φ′2(φ)f2(x)/y2 +Φ2(φ)f2(x), (35) f (l)[Φ(φ), f(x)] = (−1)lΦ2(φ)f ′2(x)− Φ2(φ) + (−1)lΦ′2(φ)/x2 f2(x), (36) with j = 0, 3 and l = 1, 2. As in the case of the field square, in formulae (33) and (34) we introduce a cutoff function and apply formula (12) for the summation over n. This enables us to present the vacuum energy-momentum tensor in the form of the sum 〈0|T ki |0〉 = 〈0w|T ki |0w〉+ 〈T ki 〉cyl, (37) where 〈0w|T ki |0w〉 is the part corresponding to the geometry of a wedge without a cylindrical boundary and 〈T ki 〉cyl is induced by the cylindrical shell. By taking into account (26), the latter may be written in the form (no summation over i) 〈T ii 〉cyl = λ=0,1 qm (xa) qm (xa) F (i)[Φ(λ)qm(φ), Iqm(xr)], (38) 〈T 12 〉cyl = m sin(2qmφ) λ=0,1 (−1)λ qm (xa) qm (xa) I2qm(xr), (39) with the notations F (i)[Φ(φ), f(y)] = Φ2(φ)f2(y), i = 0, 3, (40) F (i)[Φ(φ), f(y)] = −(−1)iΦ2(φ)f ′2(y)− Φ2(φ)− (−1)iΦ′2(φ)/y2 f2(y), i = 1, 2. (41) The diagonal components are symmetric with respect to the half-plane φ = φ0/2, whereas the off-diagonal component is an odd function under the replacement φ → φ0 − φ. As it can be easily checked, the tensor 〈T ki 〉cyl is traceless and satisfies the covariant continuity equation ∇k〈T ki 〉cyl = 0. For the geometry under consideration the latter leads to the relations r〈T 12 〉cyl 〈T 22 〉cyl = 0, (42) r〈T 11 〉cyl 〈T 21 〉cyl = 〈T 22 〉cyl. (43) As it is seen from formula (39), the off-diagonal component 〈T 12 〉cyl vanishes at the wedge sides and for these points the VEV of the energy-momentum tensor is diagonal. By using the in- equalities I ′ν(x) < 1 + ν2/x2Iν(x) and −K ′ν(x) > 1 + ν2/x2Kν(x) for the modified Bessel functions, it can be seen that K ′ν(x)/I ν(x) + Kν(x)/Iν(x) < 0. From this relation it follows that the vacuum energy density induced by the cylindrical shell in the interior region is always negative. The renormalized VEV of the energy-momentum tensor for the geometry without the cylin- drical shell is obtained by using the corresponding formulae for the field square. For the corre- sponding energy density one finds 〈T 00 〉w,ren = 〈E2〉w,ren + 〈B2〉w,ren = −(q 2 − 1)(q2 + 11) 720π2r4 . (44) As we see the parts in the VEVs of the field square which diverge on the wedge sides cancel out and the corresponding energy density is finite everywhere except the edge. Formula (44) coincides with the corresponding result for the geometry of the cosmic string (see [44, 45]) with the angle deficit 2π − φ0 and in the corresponding formula q = 2π/φ0. Other components are found from the tracelessness condition and the continuity equation and one has [29, 30] (see also 〈T ki 〉w,ren = − (q2 − 1)(q2 + 11) 720π2r4 diag(1, 1,−3, 1). (45) As we could expect this VEV vanishes for the geometry of a single plate corresponding to q = 1. In the limit r → ∞, for fixed values x0 = rφ0, from (45) the standard result for the geometry of two parallel conducting plates is obtained. The force acting on the wedge sides is determined by the component 〈T 22 〉ren of the vacuum energy-momentum tensor evaluated for φ = 0 and φ = φ0. On the base of formula (37) for the corresponding effective pressure one has p2 = −〈T 22 〉ren|φ=0,φ0 = p2w + p2cyl, (46) where p2w = − (q2 − 1)(q2 + 11) 240π2r4 , (47) is the normal force acting per unit surface of the wedge for the case without a cylindrical boundary and the additional term p2cyl = −〈T 22 〉cyl|φ=0,φ0 = − λ=0,1 qm (xa) qm (xa) F (λ)qm [Iqm(xr)], (48) with the notation F (λ)ν [f(y)] = ν2f2(y)/y2, λ = 0 −f ′2(y)− f2(y), λ = 1 , (49) is induced by the cylindrical shell. From formula (47) we see that the corresponding vacuum forces are attractive for q > 1 and repulsive for q < 1. In particular, the equilibrium position corresponding to the geometry of a single plate (q = 1) is unstable. As regards to the part induced by the cylindrical shell, from (48) it follows that p2cyl < 0 and, hence, the corresponding forces are always attractive. Now, let us discuss the behavior of the boundary-induced part in the VEV of the energy- momentum tensor in the asymptotic region of the parameters. Near the cylindrical shell the main contribution comes from large values of m. Thus, using the uniform asymptotic expansions for the modified Bessel functions for large values of the order, up to the leading order, for the points a− r ≪ a| sinφ|, a| sin(φ0 − φ)| we find 〈T 00 〉cyl ≈ − 〈T 22 〉cyl ≈ − (a− r)−3 60π2a , 〈T 11 〉cyl ≈ (a− r)−2 60π2a2 . (50) These leading terms are the same as those for a cylindrical shell when the wedge is absent. For the points near the edges (r = a, φ = 0, φ0) the leading terms in the corresponding asymptotic expansions are the same as for the geometry of a wedge with the opening angle φ0 = π/2. The latter are given by (45) with q = 2. Near the edge, r → 0, for the components (no summation over i) 〈T ii 〉cyl, i = 0, 3, the main contribution comes from the mode m = 0 and we find 〈T ii 〉cyl ≈ 4π2a4 K ′0(x) I ′0(x) = −0.0590 , i = 0, 3. (51) For the components (no summation over i) 〈T ii 〉cyl, i = 1, 2, when q > 1 the main contribution again comes form m = 0 term and one has 〈T ii 〉cyl ≈ −〈T 00 〉cyl, i = 1, 2. For q < 1 the main contribution into the components 〈T ii 〉cyl, i = 1, 2, comes from the term m = 1 and we have (no summation over i) 〈T ii 〉cyl ≈ (−1)iq cos(2qφ) 22q+1π2Γ2(q)a4 )2(q−1) dxx2q+1 Kq(x) Iq(x) K ′q(x) I ′q(x) , i = 1, 2. (52) In this case the radial and azimuthal stresses induced by the cylindrical shell diverge on the edge r = 0. In the case q = 1 the sum of the contributions of the terms with m = 0 and m = 1 given by formulae (51) and (52) should be taken. For the off-diagonal component the main contribution comes from the m = 1 mode with the leading term 〈T 12 〉cyl ≈ q sin(2qφ) 22q+1π2Γ2(q)a3 )2q−1 dxx2q+1 Kq(x) Iq(x) K ′q(x) I ′q(x) , (53) and this component vanishes on the edge for q > 1/2. In the limit q ≫ 1, the contribution of the modes with m > 1 is suppressed by the factor exp[−2qm ln(a/r)] and the main contribution comes from the m = 0 mode. The leading terms are given by the formulae (no summation over i) 〈T ii 〉cyl ≈ 4π2a4 K ′0(x) I ′0(x) I20 (xr/a), i = 0, 3, (54) 〈T ii 〉cyl ≈ − 4π2a4 K ′0(x) I ′0(x) I20 (xr/a) + (−1)iI21 (xr/a) , i = 1, 2. (55) Though in this limit the vacuum densities are large, due to the factor 1/q in the spatial volume, the corresponding global quantities tend to finite value. In particular, as it follows from Eq. (55), in the limit under consideration one has 〈T ii 〉cyl > 0. Note that in the same limit the parts corresponding to the wedge without the cylindrical shell behave as q4 and, hence, for points not too close to the shell these parts dominate in the VEVs. In figures 2-5 we have plotted the parts in the VEVs of the energy-momentum tensor induced by the cylindrical shell, a4〈T ki 〉cyl, as functions of x = (r/a) cos φ and y = (r/a) sinφ, for a wedge with the opening angle φ0 = π/2. In figure 6 we have presented the dependence of the effective azimuthal pressure induced by the cylindrical shell on the wedge sides, a4p2cyl, as a function of r/a for different values of the parameter q. Figure 2: The part in the VEV of the energy density, a4〈T 00 〉cyl, induced by the cylindrical boundary as a function on x = (r/a) cos φ and y = (r/a) sin φ for a wedge with φ0 = π/2. Figure 3: The part in the VEV of the radial stress, a4〈T 11 〉cyl, induced by the cylindrical boundary as a function on x = (r/a) cos φ and y = (r/a) sinφ for a wedge with φ0 = π/2. Figure 4: The part in the VEV of the azimuthal stress, a4〈T 22 〉cyl, induced by the cylindrical boundary as a function on x = (r/a) cos φ and y = (r/a) sin φ for a wedge with φ0 = π/2. Figure 5: The part in the VEV of the off-diagonal component, a4〈T 21 〉cyl, induced by the cylin- drical boundary as a function on x = (r/a) cos φ and y = (r/a) sinφ for a wedge with φ0 = π/2. 0.1 0.2 0.3 0.4 0.5 Figure 6: The effective azimuthal pressure induced by the cylindrical shell on the wedge sides, a4p2cyl, as a function of r/a. The numbers near the curves correspond to the values of the parameter q. There are several special cases of interest for the geometry of boundaries we have consid- ered. The case φ0 = π corresponds to the semi-circular cylinder. The Casimir energy for the corresponding interior region is evaluated in Ref. [34] by using the zeta function technique. The case φ0 = 2π corresponds to the geometry of a cylindrical shell with a coaxial half-plane. And finally, the limit φ0 → 0, r, a → ∞, assuming that a − r and aφ0 ≡ b are fixed, corresponds to the geometry of two parallel plates separated by a distance b, perpendicularly intersected by the third plate. In the latter case it is convenient to introduce rectangular coordinates (x′1, x′2, x′3) = (x, y, z) with the relations x = a − r, y = rφ. The components of the tensors in these coordinates we will denote by primes. The corresponding vacuum energy-momentum tensor is presented in the form 〈0|T ′ik |0〉 = 〈T ′ik 〉(0) + 〈T ′ik 〉(1), (56) where 〈T ′ik 〉(0) is the vacuum expectation value in the region between two parallel plates located at y = 0 and y = a and 〈T ′ik 〉(1) is induced by the intersecting plate at x = 0. The latter is related to the quantities investigated above by formulae 〈T ′ii 〉(1) = lim 〈T ii 〉cyl, 〈T ′12 〉(1) = − lim 〈T 12 〉cyl, (57) with lim corresponding to the limit a → ∞, φ0 → 0 for fixed a − r and aφ0. Taking this limit in the term with m = 0 of formula (38) we replace the modified Bessel functions by the leading terms of the corresponding asymptotic formulae for large values of the argument and the integral is taken elementary. For the terms with m 6= 0 in formulae (38), (39) we note that in the limit under consideration one has q = π/φ0 → ∞, and the order of the Bessel modified functions tends to infinity. Introducing a new integration variable x→ qmx, we can replace these functions by their uniform asymptotic expansions for large values of the order. After these replacements the integration and the further summation over m are done by using the formulae from [46]. 4 Vacuum densities in the exterior region In this section we consider the VEVs for the field square and the energy-momentum tensor in the region outside the cylindrical boundary (region II in figure 1). The corresponding eigenfunctions for the vector potential are obtained from formulae (2) by the replacement Jqm(γr) → g(λ)qm(γa, γr) = Jqm(γr)Y (λ)qm (γa)− Yqm(γr)J (λ)qm (γa), (58) where, as before, λ = 0, 1 correspond to the waves of the electric and magnetic types, respec- tively. Now, the eigenvalues for γ are continuous and in the normalization condition (4) the corresponding part on the right is presented by the delta function. As the normalization inte- gral diverges for γ′ = γ, the main contribution into the integral comes from large values of r and we can replace the cylindrical functions with the argument γr by their asymptotic expressions for large values of the argument. By this way it can be seen that the normalization coefficient in the exterior region is determined by the relation β−2α = J (λ)2qm (γa) + Y qm (γa) . (59) Substituting the eigenfunctions into the corresponding mode-sum formula, for the VEV of the field square one finds 〈0|F 2|0〉 = λ=0,1 k2 + γ2 g(ηFλ)[Φ qm(φ), g qm(γa, γr)] qm (γa) + Y qm (γa) , (60) where the functions g(ηFλ) qm(φ), g qm(γa, γr) are defined by relations (9), (10) with the function f(x) = g qm(γa, x). To extract from this VEV the part induced by the cylindrical shell, we subtract from the right-hand side the corresponding expression for the wedge without the cylindrical boundary. The latter is given by formula (14). The corresponding difference can be further evaluated by using the identity g(ηFλ)[Φ qm(φ), g qm(γa, γr)] qm (γa) + Y qm (γa) = g(ηFλ)[Φ(λ)qm(φ), Jqm(γr)] qm (γa) (s)(λ) qm (γa) g(ηFλ)[Φ(λ)qm(φ),H qm(γr)], (61) where H (1,2) qm (z) are the Hankel functions. In order to transform the integral over γ with the last term on the right of (61), in the complex plane γ we rotate the integration contour by the angle π/2 for the term with s = 1 and by the angle −π/2 for the term with s = 2. Due to the well-known properties of the Hankel functions the integrals over the corresponding parts of the circles of large radius in the upper and lower half-planes vanish. After introducing the modified Bessel functions and integrating over k with the help of formula (26), we can write the VEVs of the field square in the form (13), where the part induced by the cylindrical shell is given by the formula 〈F 2〉cyl = λ=0,1 qm (xa) qm(xa) G(ηFλ)[Φ(λ)qm(φ),Kqm(xr)]. (62) In this formula the functions G(ηFλ) [Φ(φ), f(x)] are defined by formulae (28), (29). Comparing this result with formula (27), we see that the expressions for the shell-induced parts in the interior and exterior regions are related by the interchange Iqm ⇄ Kqm. The VEV (62) diverges on the cylindrical shell with the leading term being the same as that for the interior region. At large distances from the cylindrical shell we introduce a new integration variable y = xr and expand the integrand over a/r. For q > 1 the main contribution comes from the lowest mode m = 0 and up to the leading order we have 〈E2〉cyl ≈ , 〈B2〉cyl ≈ − 15πr4 . (63) For q < 1 the dominant contribution into the VEVs at large distances is due to the mode m = 1 with the leading term 〈F 2〉cyl ≈ − 4q2(q + 1) cos(2qφ) 2q + 3 + (−1)ηF1 q + 1 2q + 1 . (64) For the case q = 1 the contributions of the modes m = 0 and m = 1 are of the same order and the corresponding leading terms are obtained by summing these contributions. The latter are given by the right-hand sides of formulae (63) and (64). As we see, at large distances the part induced by the cylindrical shell is suppressed with respect to the part corresponding to the wedge without the shell by the factor (a/r)2β with β = min(1, q). Now we turn to the VEVs of the energy-momentum tensor in the exterior region. Substi- tuting the eigenfunctions into the corresponding mode-sum formula, one finds (no summation over i) 〈0|T ii |0〉 = λ=0,1 k2 + γ2 f (i)[Φ qm(φ), g qm(γa, γr)] qm (γa) + Y qm (γa) , (65) 〈0|T 12 |0〉 = − m sin(2qmφ) λ=0,1 (−1)λ qm (γa, γr) k2 + γ2 . (66) Subtracting from these VEVs the corresponding expression for the wedge without the cylindrical boundary, analogously to the case of the field square, it can be seen that the VEVs are presented in the form (37), with the parts induced by the cylindrical shell given by the formulae (no summation over i) 〈T ii 〉cyl = λ=0,1 qm (xa) qm (xa) F (i)[Φ(λ)qm(φ),Kqm(xr)], (67) 〈T 12 〉cyl = m sin(2qmφ) λ=0,1 (−1)λ qm (xa) qm (xa) K2qm(xr). (68) Here the functions F (i) [Φ(φ), f(y)] are defined by formulae (40), (41). By using the inequality given in the paragraph after formula (43), we can show that the vacuum energy density induced by the cylindrical shell in the exterior region is positive. In the way similar to that for the interior region, for the force acting on the wedge sides is presented in the form of the sum (46), where the part corresponding to the wedge without a cylindrical shell is determined by formula (47) and for the part due to the presence of the cylindrical shell we have p2cyl = −〈T 22 〉cyl|φ=0,φ0 = − λ=0,1 qm (xa) qm(xa) F (λ)qm [Kqm(xr)]. (69) In this formula, the function F ν [f(y)] is defined by relations (49) and the corresponding forces are always attractive. The leading divergence in the boundary induced part (67) on the cylindrical surface is given by the same formulae as for the interior region. For large distances from the shell and for q > 1 the main contribution into the VEVs of the diagonal components comes from the m = 0, λ = 1 term and one has (no summation over i) 〈T ii 〉cyl ≈ − 15π2r4 , c0 = c3 = 2, c1 = 1, c2 = −5. (70) In the case q < 1 the main contribution into the VEVs of the diagonal components at large distances from the cylindrical shell comes from the m = 1 mode. The leading terms in the corresponding asymptotic expansions are given by the formulae 〈T ii 〉cyl ≈ −q2(q + 1)ci(q) cos(2qφ) , (71) with the notations c0(q) = c3(q) = 2q + 3 , c1(q) = 2q2 + q + 1 (2q + 1)(2q + 3) , c2(q) = − q + 1 2q + 1 . (72) In the case q = 1 the asymptotic terms are determined by the sum of the contributions coming from the modes m = 0 and m = 1. The latter are given by formulae (70), (71). For the off- diagonal component, for all values q the main contribution at large distances comes from the m = 1 mode with the leading term 〈T 12 〉cyl ≈ − q3(q + 1) 2q + 1 sin(2qφ) . (73) For large values of q, q ≫ 1, the contribution of the terms with m > 0 is suppressed by the factor exp[−2qm ln(r/a)] and the main contribution comes form the m = 0 term with the behavior 〈F 2〉cyl ∝ q and 〈T ki 〉cyl ∝ q. In figure 7 we have plotted the dependence of the effective azimuthal pressure induced by the cylindrical shell on the wedge sides, a4p2cyl, as a function of r/a for q = 1. 1.4 1.5 1.6 1.7 1.8 1.9 2 Figure 7: The effective azimuthal pressure induced by the cylindrical shell on the wedge sides, a4p2cyl, as a function of r/a in the exterior region for q = 1. The curves for the values q = 0.5, 3 are close to the plotted one. 5 Conclusion In this paper we have investigated the polarization of the electromagnetic vacuum by a wedge with a coaxial cylindrical boundary, assuming that all boundaries are perfectly conducting. Both regions inside and outside of the cylindrical shell (regions I and II in figure 1) are considered. In section 2 we have evaluated the VEVs of the field square in the interior region. The corresponding mode-sums contain series over the zeros of the Bessel function for TM modes and its derivative for TE modes. For the summation of these series we used a variant of the generalized Abel- Plana formula. The latter enables us to extract from the VEVs the parts corresponding to the geometry of a wedge without a cylindrical shell and to present the parts induced by the shell in terms of integrals which are exponentially convergent for points away from the boundaries. For the wedge without the cylindrical shell the VEVs of the field square are presented in the form (23). The first term on the right of this formula corresponds to the VEVs for the geometry of a cosmic string with the angle deficit 2π − φ0. The angle-dependent parts in the VEVs of the electric and magnetic fields have opposite signs and are cancelled in the evaluation of the vacuum energy density. The parts induced by the cylindrical shell are presented in the form (27). We have discussed this general formula in various asymptotic regions of the parameters including the points near the edges and near the shell. In section 3 we consider the VEV of the energy-momentum tensor in the region inside the shell. As for the field square, the application of the Abel-Plana formula allows us to present this VEV in the form of the sum of purely wedge and shell-induced parts, formula (37). For the geometry of a wedge without the cylindrical boundary the vacuum energy-momentum tensor does not depend on the angle φ and is the same as in the geometry of the cosmic string and is given by formula (45). The corresponding vacuum forces acting on the wedge sides are attractive for φ0 < π and repulsive for φ0 > π. In particular, the equilibrium position corresponding to the geometry of a single plate (φ0 = π) is unstable. For the region inside the shell the part in the VEV of the energy-momentum tensor induced by the presence of the cylindrical shell is non-diagonal and the corresponding components are given by formulae (38), (39). The vacuum energy density induced by the cylindrical shell in the interior region is negative. We have investigated the vacuum densities induced by the cylindrical shell in various asymptotic regions of the parameters. For points near the cylindrical shell the leading terms in the asymptotic expansions over the distance from the shell are given by formulae (50). These terms are the same as those for a cylindrical shell when the wedge is absent. For a wedge with φ0 < π the part in the vacuum energy-momentum tensor induced by the shell is finite on the edge r = 0. For φ0 > π the shell-induced parts in the energy density and the axial stress remain finite, whereas the radial and azimuthal stresses diverge as r2(π/φ0−1). The corresponding off-diagonal component behaves like r2π/φ0−1 for all values φ0. For the points near the edges (r = a, φ = 0, φ0) the leading terms in the corresponding asymptotic expansions are the same as for the geometry of a wedge with the opening angle φ0 = π/2. In the limit of small opening angles, φ0 ≪ π, the shell-induced parts behave like 1/φ0. In the same limit the parts corresponding to the wedge without the shell behave as 1/φ40, and for points not too close to the shell these parts dominate in the VEV of the energy-momentum tensor. The presence of the shell leads to additional forces acting on the wedge sides. The corresponding effective azimuthal pressure is given by formula (48) and these forces are always attractive. The VEVs of the field square and the energy-momentum tensor in the region outside the cylindrical shell are investigated in section 4. As in the case of the interior region, these VEVs are presented as sums of the parts corresponding to the wedge without the cylindrical shell and the parts induced by the shell. The latter are given by formula (62) for the field square and by formulae (67), (66) for the components of the energy-momentum tensor. In the exterior region the vacuum energy density induced by the cylindrical shell is always positive. Additional forces acting on the wedge sides due to the presence of the shell are given by formula (69). As in the case of the interior region these forces are attractive. For large values of the parameter q, the contribution into the parts induced by the cylindrical shell coming from the modes with m 6= 0 is exponentially suppressed, whereas the contribution of the lowest mode m = 0 is proportional to q. Though in this limit the vacuum densities are large, due to the factor 1/q in the spatial volume element, the corresponding global quantities tend to finite limiting values. 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arXiv:0704.1899v1 [astro-ph] 15 Apr 2007 Progenitors of Long Gamma-ray Bursts Melvyn B. Davies∗, Andrew J. Levan†,∗, Josefin Larsson∗∗,∗, Andrew R. King‡ and Andrew S. Fruchter§ ∗Lund Observatory, Box 43, SE-221 00 Lund, Sweden †Department of Physics, University of Warwick, Coventry, CV4 7AL ∗∗Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK ‡Department of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK §Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA Abstract. Pinpointing the progenitors of long duration gamma-ray bursts (LGRBs) remains an extremely important question, although it is now clear that at least a fraction of LGRBs originate in the core collapse of massive stars in type Ic supernovae, the pathways to the production of these stars, and their initial masses, remain uncertain. Rotation is thought to be vital in the creation of LGRBs, and it is likely that black hole creation is also necessary. We suggest that these two constraints can be met if the GRB progenitors are very massive stars > 20 M⊙ and are formed in tight binary systems. Using simple models we compare the predictions of this scenario with observations and find that the location of GRBs on their host galaxies are suggestive of main- sequence masses in excess of 20 M⊙, while 50% of the known compact binary systems may have been sufficiently close to have had the necessary rotation rates for GRB creation. Thus, massive stars in compact binaries are a likely channel for at least some fraction of LGRBs. Keywords: Gamma-ray bursts, Supernovae, Neutron Stars, Black holes PACS: 98.70.Rz, 97.60.Bw, 97.60.Jd, 97.60.Lf INTRODUCTION The link between some long-duration gamma-ray bursts (LGRBs) and stellar collapse is now firmly established [1, 2, 3] In particular LGRBs appear to originate in type Ic supernovae and frequently in hypernovae. There is significant diversity in the LGRB population, including the highly luminous (“classical") burst population, originating from a mean redshift of 2.8 [4], a nearby lower luminosity sub-class, which has a much higher space density (e.g. [5]), and possibly some LGRBs which are not associated with supernovae. This suggests there may be some variety in the progenitors of the bursts, and that they may not come from systems undergoing identical evolution. Nonetheless, there remains a strong drive to identify the progenitor stars themselves. In the standard picture, core-collapse supernovae lead to LGRBs when the stellar core has a rapid rotation just prior to collapse. Some of the infalling material then forms a torus around the newly-formed central compact object (often assumed to be a black hole) and subsequent accretion of the material in the torus then fuels the LGRB. We are naturally left with two questions: 1) What leads to the high rotation rates required to form a torus? 2) What is the minimum progenitor mass required (or, in other words is the formation of a black hole (BH) rather than a neutron star (NS) necessary)? For massive single stars, it is not clear whether sufficiently high central rotation rates may be maintained to produce a torus on core collapse (e.g. [6, 7]), although it http://arxiv.org/abs/0704.1899v1 has been suggested that rapidly rotating metal–poor stars can retain sufficient angular momentum [8, 9] Alternatively, binary scenarios suggest a way of removing the envelope and providing a source of angular momentum (e.g. [10, 11]). We have explored this idea further [12] and present a summary of the key results here. In addition, we have explored whether the observation of Fruchter et al. [13] that LGRBs are more concentrated on their host galaxy light than SNe may be due to the greater mass of a LGRB progenitor than a typical SNe progenitor [14]. A review of the key results of this work is also presented here. NEUTRON STAR BINARIES AND LONG-DURATION GAMMA-RAY BURSTS Evolutionary pathways to compact object binaries Binaries containing two compact objects (ie white dwarfs, neutron stars or black holes) can be formed through several channels (see e.g. [15]). The basic scheme is shown in Fig. 1, and is essentially the same as that described in Bhattacharya and van den Heuvel [16]. The primary evolves first to produce either a black hole or neutron star, possibly following a phase of mass transfer to the secondary. When the secondary evolves, and fills its Roche lobe, the formation of a common envelope phase is likely owing to the mass ratio of the system. The black hole/neutron star and the helium-star core of the secondary will then spiral towards each other as the enshrouding gaseos envelope is ejected. The post-common envelope system will be a compact binary. A key point is that the helium-star core will be rotating rapidly because of tidal locking. In at least some systems, the rotation rate will be sufficiently large that when the secondary explodes as a core-collapse supernova, some of the infalling material will form a torus around the newly-formed neutron star or black hole. Discs around neutron stars and black holes In a core-collapse supernova, the material within the stellar core collapses to form a black hole or neutron star. If a core of a massive star is rotating sufficiently rapidly, some of the infalling material will be centrifugally supported (as it will carry its angular momentum with it) and form a torus around the central black hole or neutron star. The torus will have a radius of DGMc/c 2 (where Mc is the mass of the black hole or neutron star) if the specific angular momentum of the infalling material is given by DGMc/c. In order to have a stable orbit around a (non-rotating) black hole, we require D ≥ 6 (or a radius of around 12 km for a 1.4 M⊙ black hole). Forming a disc around a proto-neutron star requires more angular momentum as a disc would have to be formed with a radius somewhat larger than 10 km. If we assume that tidal locking occurs at the beginning of the helium main sequence and that for the late stages of evolution of the helium star the core decouples from the MS MS NS/BH NS/BH NS/BH He SN/GRB FIGURE 1. An evolutionary pathway to the creation of a binary containing a rapidly-rotating core- collapse supernova in a tight orbit. The primary evolves first, possibly transferring material to the sec- ondary (stage 1). It then produces a neutron star (NS) or black hole (BH), when it explodes as a core- collapse supernova (stage 2). The secondary then evolves, filling its Roche lobe (stage 3) and transferring material to the NS/BH producing a common envelope phase (stage 4). The NS/BH and He core of the secondary spiral together ejecting the surrounding envelope producing a very compact binary (stage 5). Tidal locking produces a rapidly-rotating He star such that the rotation is significant when the secondary explodes as a core-collapse supernova, with a torus being formed around the central compact object by infalling material. envelope. In this case, following Podsiadlowski et al. [11] we can compare the required angular momentum at the time of collapse with that at the edge of the iron core at the start of the helium main sequence, and, assuming tidal locking equate this to an orbital frequency (i.e. ω = DGMc/R cc, where Rc is the radius of the iron core). Assuming synchronous rotation, this gives a critical orbital separation [12] a < (4Mtotc 2R4c/9DGM 1/3, (1) )4/3(Mtot )1/3( Mc )−2/3 , (2) where Mc and Rc are the core mass and radius, and Mtot is the total mass of the binary. We use here the models of Heger et al. [17], the key parameters are Mc = 1.7 M⊙, Rc ∼ 0.1R⊙, for a He star of mass 7.71 M⊙, this yields a critical separation of ∼ 3 R⊙ for the formation of a disc at the innermost stable orbit of a 1.7 M⊙ BH. Do systems exist which are sufficiently tight to form discs when the secondary un- dergoes a core-collapse supernova? In order to answer this question, one can consider the observed compact binaries containing black holes or neutron stars which would be formed after the second supernova explosion (assuming the binary remained bound). There are currently no known black hole-neutron star binaries. However there are cur- rently eight NS-NS binaries and three WD-NS binaries. The semi-major axis and ec- centricity of a binary provides limits to the separation of the binary at the moment of the second supernova. In Fig. 2 we plot the semi-major axes and eccentricities of the observed systems. The plot also illustrates the minimum separation required to form a disc around the second compact object for discs of various radii. From this figure, we can see that discs are likely to have formed during the second supernova in about half of all osberved binaries. Energy released from such discs may power some of the long gamma-ray bursts, as discussed below. The contribution of compact binaries to long gamma-ray bursts Long-duration gamma-ray bursts are commonly thought to originate from discs around black holes located at the centre of a supernova. However, one can also con- sider whether discs around newly-formed neutron stars may also produce some form of gamma-ray burst. The maximum energy released in the accretion of the torus is given by E = GMns/bhMacc/Rns/bh, or Ens = 3.6×1053(Macc/M⊙) ergs, for a 1.4 M⊙ NS and Ebh = 1×1054(Macc/M⊙) ergs, for any mass BH (since Mbh ∝ Rbh). The extrapolation from disc accretion to gamma-ray luminosity is far from trivial since it requires an assumption about the conversion of accretion luminosity into γ-ray energy. One plausible mechanism of providing this energy is via neutrino-antineutrino annihilation. If this is assumed to be the energy source then the accretion energy can be related to the observed gamma-ray energy via several efficiency factors which account for the conversion of accretion energy to neutrinos, the cross section for neutrino - antineutrino annihilation, the subsequent fraction of energy that is transferred into a FIGURE 2. The eccentricity and semi-major axis of observed systems containing either two neutron stars, or a neutron star and white dwarf. The error bars give an indication of the range of separations between the two stars during their orbit. Neglecting the inspiral the separation at the time of the supernova must be taken from this range (see Table 1 for the calculations including the effects of inspiral via gravitational radiation). The three vertical lines represent the critical separations necessary at the time of core collapse for a centrifugally supported disc to have formed at a distance of 6M, 20M and 50M from the newly formed compact object, based on the models of Heger et al. [17] described in section 3 (where M = GMns/bh/c 2 and is ∼ 12,40 and 100 km for D=6,20 and 50 for a 1.4 M⊙ NS). Lower total masses, as were likely the case of J0737-3039 require slightly tighter orbits, although the orbit is only a weak function of the total mass (M tot ). If the binary separation is less than this (i.e. to the left of the line) then disc formation is favoured. The data are from Champion et al. [18] and Lorimer [19]. In cases where the masses of the two components have not be measured 1.4 M⊙ has been assumed. Known NS-NS binaries are indicated with circles, while NS-WD or those with uncertain companions are marked with open circles baryon free jet, and finally the fraction of this energy which is emitted as gamma- rays [20]. Following Oechslin and Janka [20] we assume that the product of the these efficiencies is ∼ 10−3. Thus the observed luminosities of low luminosity GRBs of 1048 − 1050 ergs can be explained by the accretion of 0.01 < (Macc/M⊙) < 0.3 of material from the disc. More-massive versions of the systems which formed the observed NS-NS binaries are expected to form BH-BH binaries and are thus candidates for classical long gamma-ray bursts. Could both supernovae in such systems produce a gamma-ray burst? One can consider the separation required in order to give the core of the primary a sufficiently- high rotation rate. It is found that the size of the stellar is restricted to less than a few solar radii. In other words, the primary of a binary containing two massive stars is unlikely to be tidally spun-up sufficiently to produce a gamma-ray burst. However the black-hole it produces will work effectively to tidally spin-up the core of the secondary star following a pathway similar to that shown in Fig. 1. A NEW CONSTRAINT FOR GAMMA-RAY BURST PROGENITOR The basic idea Recently Fruchter et al. [13] have conducted a survey of the galactic environments of both long duration GRBs and core collapse SNe (i.e. all types of core collapse events, including SN II, Ib and Ic). These results demonstrate that GRBs are highly concentrated on their host light, significantly more so than the core collapse supernova population. Fruchter et al. [13] further suggest that this can be explained as being due to the GRBs originating in the most massive stars, which, upon core collapse form black holes rather than neutron stars. Here we further explore this possibility and attempt to derive plausible limits on the progenitor lifetime and mass based on the observed distributions of core collapse SNe and GRBs upon their host light. Using a simple model, motivated by the distributions of young star clusters in a local starburst galaxy, we explore the expected distributions of stars of different masses upon their host galaxies and compare these to the observed distributions from Fruchter et al. [13]. Our results demonstrate that for plausible models more massive stars are always more concentrated on their host light than lower mass stars. Further, given that supernovae originate from stars with initial masses > 8 M⊙, we find that the observed distributions of GRBs on their host galaxies can naturally be explained by progenitors with initial masses in excess of 20 The model GRB host galaxies at high redshift are typically starburst galaxies. The Antennae (NGC 4038/4039) are a natural, local, analogue and were used as a template for con- structing a simple model for GRB host galaxies. In this model we consider the popula- tion of young stellar clusters, as well as the stellar population contributing to the back- ground light within the galaxy. The galaxy model consists of several key parameters: the surface density of clusters; the distribution of cluster masses; the distribution of cluster ages; the distribution of background light; and the distribution of clusters on the back- ground light. The first two properties were taken from observations of NGC 4038/4039 when viewed at z ∼ 1. The surface density of clusters expressed in terms of number of clusters per pixel is ∼ 0.15, although this is far from uniform across the galaxy. We use the observed distribution of cluster masses, which follows dN/dMcl ∝ M−2cl , with Mcl,min = 4 ·104 M⊙ and Mcl,max = 106 M⊙, where Mcl is the cluster mass. The age dis- FIGURE 3. Fraction of objects plotted against fraction of light for observed SNe (red) and GRBs (blue) together with the results from our model (black lines). Black lines from top to bottom correspond to minimum progenitor masses of 8, 20, 40, 60, and 80 M⊙. tribution of clusters was taken from Fall et al. [21] and follows dN/dτ ∝ τ−1, where τ is the cluster age. In order to mimic an (almost) instantaneous burst of star forma- tion, clusters are created in the model according to this distribution over a period of 107 years. The distribution of the background light adopted lies in the middle of the distri- butions observed for galaxies hosting supernovae and gamma-ray bursts and is given by dN/dLpix ∝ L−1.5pix with Lpix,max/Lpix,min = 20, where Lpix is the luminosity of a pixel. We distribute the stellar clusters such that there is a correlation between background brightness and the number of clusters present in a pixel. The degree of correlation was chosen to give a good match to the cluster distribution observed in NGC 4038/39. Results Using the parameters described in the previous section we performed runs for min- imum progenitor masses of 8, 20, 40, 60, and 80 M⊙. The results are shown as black lines in Fig. 3 together with the observed distributions of SNe (in red) and GRBs (in blue) from Fruchter et al. [13]. FIGURE 4. KS-probabilities of our model results following the observed SN (red) and GRB (blue) distributions. The probabilities are plotted as a function of the minimum progenitor mass and were calculated for masses of 8, 20, 40, 60, and 80 M⊙. A spline has been fitted through the data points. The model distributions for all masses were KS-tested against the observed SN and GRB distributions and the resulting probabilities are shown as a function of mass in Fig. 4. While the probability of following the SN distribution decreases with increasing mass, the likelihood of following the observed GRB distribution increases rapidly from 8 to 40 M⊙ and then flattens out, reaching a weak maximum around 60 M⊙. The shapes of the two probability functions look the same for all realisations of the model, although the peak probabilities can change by about 0.1 between different runs. These results strongly suggest that GRB progenitors are significantly more massive than SN progenitors. SUMMARY We have suggested that GRBs originate from the most massive stars [14] and that many of these stars exist within tight binary systems [12]. After the GRB a NS-NS, NS-BH or BH-BH binary may remain, as the separation at the time of the GRB is moderately small many of the systems will merge within a Hubble time and might be observable as a short GRB. These systems are spectacular, at some point in their evolution they will be observed as two supernova, two GRBs as well as X-ray binary (and possibly even ultraluminous X-ray source) phases. 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Pinpointing the progenitors of long duration gamma-ray bursts (LGRBs) remains an extremely important question, although it is now clear that at least a fraction of LGRBs originate in the core collapse of massive stars in type Ic supernovae, the pathways to the production of these stars, and their initial masses, remain uncertain. Rotation is thought to be vital in the creation of LGRBs, and it is likely that black hole creation is also necessary. We suggest that these two constraints can be met if the GRB progenitors are very massive stars (>20 solar masses) and are formed in tight binary systems. Using simple models we compare the predictions of this scenario with observations and find that the location of GRBs on their host galaxies are suggestive of main-sequence masses in excess of 20 solar masses, while 50% of the known compact binary systems may have been sufficiently close to have had the necessary rotation rates for GRB creation. Thus, massive stars in compact binaries are a likely channel for at least some fraction of LGRBs.

arXiv:0704.1899v1 [astro-ph] 15 Apr 2007 Progenitors of Long Gamma-ray Bursts Melvyn B. Davies∗, Andrew J. Levan†,∗, Josefin Larsson∗∗,∗, Andrew R. King‡ and Andrew S. Fruchter§ ∗Lund Observatory, Box 43, SE-221 00 Lund, Sweden †Department of Physics, University of Warwick, Coventry, CV4 7AL ∗∗Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK ‡Department of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK §Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA Abstract. Pinpointing the progenitors of long duration gamma-ray bursts (LGRBs) remains an extremely important question, although it is now clear that at least a fraction of LGRBs originate in the core collapse of massive stars in type Ic supernovae, the pathways to the production of these stars, and their initial masses, remain uncertain. Rotation is thought to be vital in the creation of LGRBs, and it is likely that black hole creation is also necessary. We suggest that these two constraints can be met if the GRB progenitors are very massive stars > 20 M⊙ and are formed in tight binary systems. Using simple models we compare the predictions of this scenario with observations and find that the location of GRBs on their host galaxies are suggestive of main- sequence masses in excess of 20 M⊙, while 50% of the known compact binary systems may have been sufficiently close to have had the necessary rotation rates for GRB creation. Thus, massive stars in compact binaries are a likely channel for at least some fraction of LGRBs. Keywords: Gamma-ray bursts, Supernovae, Neutron Stars, Black holes PACS: 98.70.Rz, 97.60.Bw, 97.60.Jd, 97.60.Lf INTRODUCTION The link between some long-duration gamma-ray bursts (LGRBs) and stellar collapse is now firmly established [1, 2, 3] In particular LGRBs appear to originate in type Ic supernovae and frequently in hypernovae. There is significant diversity in the LGRB population, including the highly luminous (“classical") burst population, originating from a mean redshift of 2.8 [4], a nearby lower luminosity sub-class, which has a much higher space density (e.g. [5]), and possibly some LGRBs which are not associated with supernovae. This suggests there may be some variety in the progenitors of the bursts, and that they may not come from systems undergoing identical evolution. Nonetheless, there remains a strong drive to identify the progenitor stars themselves. In the standard picture, core-collapse supernovae lead to LGRBs when the stellar core has a rapid rotation just prior to collapse. Some of the infalling material then forms a torus around the newly-formed central compact object (often assumed to be a black hole) and subsequent accretion of the material in the torus then fuels the LGRB. We are naturally left with two questions: 1) What leads to the high rotation rates required to form a torus? 2) What is the minimum progenitor mass required (or, in other words is the formation of a black hole (BH) rather than a neutron star (NS) necessary)? For massive single stars, it is not clear whether sufficiently high central rotation rates may be maintained to produce a torus on core collapse (e.g. [6, 7]), although it http://arxiv.org/abs/0704.1899v1 has been suggested that rapidly rotating metal–poor stars can retain sufficient angular momentum [8, 9] Alternatively, binary scenarios suggest a way of removing the envelope and providing a source of angular momentum (e.g. [10, 11]). We have explored this idea further [12] and present a summary of the key results here. In addition, we have explored whether the observation of Fruchter et al. [13] that LGRBs are more concentrated on their host galaxy light than SNe may be due to the greater mass of a LGRB progenitor than a typical SNe progenitor [14]. A review of the key results of this work is also presented here. NEUTRON STAR BINARIES AND LONG-DURATION GAMMA-RAY BURSTS Evolutionary pathways to compact object binaries Binaries containing two compact objects (ie white dwarfs, neutron stars or black holes) can be formed through several channels (see e.g. [15]). The basic scheme is shown in Fig. 1, and is essentially the same as that described in Bhattacharya and van den Heuvel [16]. The primary evolves first to produce either a black hole or neutron star, possibly following a phase of mass transfer to the secondary. When the secondary evolves, and fills its Roche lobe, the formation of a common envelope phase is likely owing to the mass ratio of the system. The black hole/neutron star and the helium-star core of the secondary will then spiral towards each other as the enshrouding gaseos envelope is ejected. The post-common envelope system will be a compact binary. A key point is that the helium-star core will be rotating rapidly because of tidal locking. In at least some systems, the rotation rate will be sufficiently large that when the secondary explodes as a core-collapse supernova, some of the infalling material will form a torus around the newly-formed neutron star or black hole. Discs around neutron stars and black holes In a core-collapse supernova, the material within the stellar core collapses to form a black hole or neutron star. If a core of a massive star is rotating sufficiently rapidly, some of the infalling material will be centrifugally supported (as it will carry its angular momentum with it) and form a torus around the central black hole or neutron star. The torus will have a radius of DGMc/c 2 (where Mc is the mass of the black hole or neutron star) if the specific angular momentum of the infalling material is given by DGMc/c. In order to have a stable orbit around a (non-rotating) black hole, we require D ≥ 6 (or a radius of around 12 km for a 1.4 M⊙ black hole). Forming a disc around a proto-neutron star requires more angular momentum as a disc would have to be formed with a radius somewhat larger than 10 km. If we assume that tidal locking occurs at the beginning of the helium main sequence and that for the late stages of evolution of the helium star the core decouples from the MS MS NS/BH NS/BH NS/BH He SN/GRB FIGURE 1. An evolutionary pathway to the creation of a binary containing a rapidly-rotating core- collapse supernova in a tight orbit. The primary evolves first, possibly transferring material to the sec- ondary (stage 1). It then produces a neutron star (NS) or black hole (BH), when it explodes as a core- collapse supernova (stage 2). The secondary then evolves, filling its Roche lobe (stage 3) and transferring material to the NS/BH producing a common envelope phase (stage 4). The NS/BH and He core of the secondary spiral together ejecting the surrounding envelope producing a very compact binary (stage 5). Tidal locking produces a rapidly-rotating He star such that the rotation is significant when the secondary explodes as a core-collapse supernova, with a torus being formed around the central compact object by infalling material. envelope. In this case, following Podsiadlowski et al. [11] we can compare the required angular momentum at the time of collapse with that at the edge of the iron core at the start of the helium main sequence, and, assuming tidal locking equate this to an orbital frequency (i.e. ω = DGMc/R cc, where Rc is the radius of the iron core). Assuming synchronous rotation, this gives a critical orbital separation [12] a < (4Mtotc 2R4c/9DGM 1/3, (1) )4/3(Mtot )1/3( Mc )−2/3 , (2) where Mc and Rc are the core mass and radius, and Mtot is the total mass of the binary. We use here the models of Heger et al. [17], the key parameters are Mc = 1.7 M⊙, Rc ∼ 0.1R⊙, for a He star of mass 7.71 M⊙, this yields a critical separation of ∼ 3 R⊙ for the formation of a disc at the innermost stable orbit of a 1.7 M⊙ BH. Do systems exist which are sufficiently tight to form discs when the secondary un- dergoes a core-collapse supernova? In order to answer this question, one can consider the observed compact binaries containing black holes or neutron stars which would be formed after the second supernova explosion (assuming the binary remained bound). There are currently no known black hole-neutron star binaries. However there are cur- rently eight NS-NS binaries and three WD-NS binaries. The semi-major axis and ec- centricity of a binary provides limits to the separation of the binary at the moment of the second supernova. In Fig. 2 we plot the semi-major axes and eccentricities of the observed systems. The plot also illustrates the minimum separation required to form a disc around the second compact object for discs of various radii. From this figure, we can see that discs are likely to have formed during the second supernova in about half of all osberved binaries. Energy released from such discs may power some of the long gamma-ray bursts, as discussed below. The contribution of compact binaries to long gamma-ray bursts Long-duration gamma-ray bursts are commonly thought to originate from discs around black holes located at the centre of a supernova. However, one can also con- sider whether discs around newly-formed neutron stars may also produce some form of gamma-ray burst. The maximum energy released in the accretion of the torus is given by E = GMns/bhMacc/Rns/bh, or Ens = 3.6×1053(Macc/M⊙) ergs, for a 1.4 M⊙ NS and Ebh = 1×1054(Macc/M⊙) ergs, for any mass BH (since Mbh ∝ Rbh). The extrapolation from disc accretion to gamma-ray luminosity is far from trivial since it requires an assumption about the conversion of accretion luminosity into γ-ray energy. One plausible mechanism of providing this energy is via neutrino-antineutrino annihilation. If this is assumed to be the energy source then the accretion energy can be related to the observed gamma-ray energy via several efficiency factors which account for the conversion of accretion energy to neutrinos, the cross section for neutrino - antineutrino annihilation, the subsequent fraction of energy that is transferred into a FIGURE 2. The eccentricity and semi-major axis of observed systems containing either two neutron stars, or a neutron star and white dwarf. The error bars give an indication of the range of separations between the two stars during their orbit. Neglecting the inspiral the separation at the time of the supernova must be taken from this range (see Table 1 for the calculations including the effects of inspiral via gravitational radiation). The three vertical lines represent the critical separations necessary at the time of core collapse for a centrifugally supported disc to have formed at a distance of 6M, 20M and 50M from the newly formed compact object, based on the models of Heger et al. [17] described in section 3 (where M = GMns/bh/c 2 and is ∼ 12,40 and 100 km for D=6,20 and 50 for a 1.4 M⊙ NS). Lower total masses, as were likely the case of J0737-3039 require slightly tighter orbits, although the orbit is only a weak function of the total mass (M tot ). If the binary separation is less than this (i.e. to the left of the line) then disc formation is favoured. The data are from Champion et al. [18] and Lorimer [19]. In cases where the masses of the two components have not be measured 1.4 M⊙ has been assumed. Known NS-NS binaries are indicated with circles, while NS-WD or those with uncertain companions are marked with open circles baryon free jet, and finally the fraction of this energy which is emitted as gamma- rays [20]. Following Oechslin and Janka [20] we assume that the product of the these efficiencies is ∼ 10−3. Thus the observed luminosities of low luminosity GRBs of 1048 − 1050 ergs can be explained by the accretion of 0.01 < (Macc/M⊙) < 0.3 of material from the disc. More-massive versions of the systems which formed the observed NS-NS binaries are expected to form BH-BH binaries and are thus candidates for classical long gamma-ray bursts. Could both supernovae in such systems produce a gamma-ray burst? One can consider the separation required in order to give the core of the primary a sufficiently- high rotation rate. It is found that the size of the stellar is restricted to less than a few solar radii. In other words, the primary of a binary containing two massive stars is unlikely to be tidally spun-up sufficiently to produce a gamma-ray burst. However the black-hole it produces will work effectively to tidally spin-up the core of the secondary star following a pathway similar to that shown in Fig. 1. A NEW CONSTRAINT FOR GAMMA-RAY BURST PROGENITOR The basic idea Recently Fruchter et al. [13] have conducted a survey of the galactic environments of both long duration GRBs and core collapse SNe (i.e. all types of core collapse events, including SN II, Ib and Ic). These results demonstrate that GRBs are highly concentrated on their host light, significantly more so than the core collapse supernova population. Fruchter et al. [13] further suggest that this can be explained as being due to the GRBs originating in the most massive stars, which, upon core collapse form black holes rather than neutron stars. Here we further explore this possibility and attempt to derive plausible limits on the progenitor lifetime and mass based on the observed distributions of core collapse SNe and GRBs upon their host light. Using a simple model, motivated by the distributions of young star clusters in a local starburst galaxy, we explore the expected distributions of stars of different masses upon their host galaxies and compare these to the observed distributions from Fruchter et al. [13]. Our results demonstrate that for plausible models more massive stars are always more concentrated on their host light than lower mass stars. Further, given that supernovae originate from stars with initial masses > 8 M⊙, we find that the observed distributions of GRBs on their host galaxies can naturally be explained by progenitors with initial masses in excess of 20 The model GRB host galaxies at high redshift are typically starburst galaxies. The Antennae (NGC 4038/4039) are a natural, local, analogue and were used as a template for con- structing a simple model for GRB host galaxies. In this model we consider the popula- tion of young stellar clusters, as well as the stellar population contributing to the back- ground light within the galaxy. The galaxy model consists of several key parameters: the surface density of clusters; the distribution of cluster masses; the distribution of cluster ages; the distribution of background light; and the distribution of clusters on the back- ground light. The first two properties were taken from observations of NGC 4038/4039 when viewed at z ∼ 1. The surface density of clusters expressed in terms of number of clusters per pixel is ∼ 0.15, although this is far from uniform across the galaxy. We use the observed distribution of cluster masses, which follows dN/dMcl ∝ M−2cl , with Mcl,min = 4 ·104 M⊙ and Mcl,max = 106 M⊙, where Mcl is the cluster mass. The age dis- FIGURE 3. Fraction of objects plotted against fraction of light for observed SNe (red) and GRBs (blue) together with the results from our model (black lines). Black lines from top to bottom correspond to minimum progenitor masses of 8, 20, 40, 60, and 80 M⊙. tribution of clusters was taken from Fall et al. [21] and follows dN/dτ ∝ τ−1, where τ is the cluster age. In order to mimic an (almost) instantaneous burst of star forma- tion, clusters are created in the model according to this distribution over a period of 107 years. The distribution of the background light adopted lies in the middle of the distri- butions observed for galaxies hosting supernovae and gamma-ray bursts and is given by dN/dLpix ∝ L−1.5pix with Lpix,max/Lpix,min = 20, where Lpix is the luminosity of a pixel. We distribute the stellar clusters such that there is a correlation between background brightness and the number of clusters present in a pixel. The degree of correlation was chosen to give a good match to the cluster distribution observed in NGC 4038/39. Results Using the parameters described in the previous section we performed runs for min- imum progenitor masses of 8, 20, 40, 60, and 80 M⊙. The results are shown as black lines in Fig. 3 together with the observed distributions of SNe (in red) and GRBs (in blue) from Fruchter et al. [13]. FIGURE 4. KS-probabilities of our model results following the observed SN (red) and GRB (blue) distributions. The probabilities are plotted as a function of the minimum progenitor mass and were calculated for masses of 8, 20, 40, 60, and 80 M⊙. A spline has been fitted through the data points. The model distributions for all masses were KS-tested against the observed SN and GRB distributions and the resulting probabilities are shown as a function of mass in Fig. 4. While the probability of following the SN distribution decreases with increasing mass, the likelihood of following the observed GRB distribution increases rapidly from 8 to 40 M⊙ and then flattens out, reaching a weak maximum around 60 M⊙. The shapes of the two probability functions look the same for all realisations of the model, although the peak probabilities can change by about 0.1 between different runs. These results strongly suggest that GRB progenitors are significantly more massive than SN progenitors. SUMMARY We have suggested that GRBs originate from the most massive stars [14] and that many of these stars exist within tight binary systems [12]. After the GRB a NS-NS, NS-BH or BH-BH binary may remain, as the separation at the time of the GRB is moderately small many of the systems will merge within a Hubble time and might be observable as a short GRB. These systems are spectacular, at some point in their evolution they will be observed as two supernova, two GRBs as well as X-ray binary (and possibly even ultraluminous X-ray source) phases. 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arXiv:0704.1900v3 [hep-ph] 15 Nov 2007 Preprint typeset in JHEP style - HYPER VERSION Causal vs. Analytic constraints on anomalous quartic gauge couplings L. Vecchi INFN, Sezione di Trieste and Scuola Internazionale Superiore di Studi Avanzati via Beirut 4, I-34014 Trieste, Italy Abstract: We derive one loop constraints on the anomalous quartic gauge couplings using a general non-forward dispersion relation for the elastic scattering amplitude of two longitudinally polarized vector bosons. We show that for exactly chiral theories more stringent bounds can be obtained by the assumption that the underlying theory satisfies the causality principle of Special Relativity. Keywords: Spontaneous Symmetry Breaking, Chiral Lagrangians, Beyond the Standard Model. http://arxiv.org/abs/0704.1900v3 http://jhep.sissa.it/stdsearch http://jhep.sissa.it/stdsearch Contents 1. Introduction 1 2. Analytical bounds 2 2.1 Application to the gauged chiral lagrangian 3 2.2 Derivation of the analytical bounds 3 3. Causal bounds 5 4. Conclusions 7 1. Introduction The general structure of an effective lagrangian is dictated by the interplay between quan- tum mechanics, Poincaré invariance, and internal symmetries. Its coefficients are not con- strained by the symmetries and must be determined by experiments. Unitarity usually sets an upper bound on the energy scale below which a perturbative effective approach is reliable. We can interpret the standard model (SM) as an effective theory extending its la- grangian to include new non-renormalizable operators with unknown coefficients. Some of them enter the scattering amplitudes of longitudinally polarized vector bosons. These are called anomalous quartic gauge couplings since they measure the deviation from the SM predictions. These coefficients are necessarily connected with the not yet observed Higgs sector. In the case the Higgs boson is not a fundamental state, or even no Higgs boson will be observed, they provide important informations on the nature of the electro-weak symmetry breaking sector. Whereas there are no significant experimental bounds on them at the moment [1], theoretical arguments can reduce significantly their allowed range and can serve as a guide for future experiments. The authors of [2] have noticed that the coefficients of a general effective lagrangian may be constrained by requiring the S-matrix of the full theory respects some desirable property such as analyticity, crossing symmetry, Lorentz invariance and unitarity. We follow these authors and consider the SM SU(2)× U(1) → U(1) breaking pattern in the case there exists a light Higgs-like boson as well as in the case no Higgs boson can propagate under the cut off of the effective theory. We show that a general non- forward dispersion relation leads to a less constraining bound than the one derived by the request the UV completion respects the causality principle of Special Relativity. This is not surprising because it is commonly believed that the analytical properties of the S-matrix are a consequence of its causal nature. – 1 – 2. Analytical bounds We briefly review an analytic tool which has been used in the context of the chiral la- grangian of QCD to constrain some effective coefficients. Consider a multiplet of scalar particles, which to be definite we call pions πa, having mass m. Assume they are lighter than any other quanta and that they have appropriate quantum numbers to forbid the transition 2π → π. The other states can be general unstable quanta of complex masses M much greater than 2m. The S-matrix element for a general transition 2π → 2π is a Lorentz scalar function of the Mandelstam variables s, t, u and of the mass m2. We study the amplitude for the elastic scattering πaπb → πaπb and assume it can be analytically continued to the complex variables s, t. We denote this analytical function by F (s, t) and require that its domain of analyticity be dictated entirely by the optical theorem and the crossing symmetry. More precisely, we assume that the singularities come from simple poles in the correspondence of the physical masses of the quantum states which can be produced in the reaction, and branch cuts in the real axis starting at the threshold of multi-particle production. Since no mass-less particle exchange is included in F (s, t), the analytical amplitude satisfies a twice subtracted dispersion relation for a variety of complex t [3]. For any non singular complex point s, t we can write: d2F (s, t) + P = ImF (x+ iε, t) (x− s)3 ImFu(x+ iε, t) (x− u)3 (2.1) where we defined u = 4m2 − s− t and used the crossing symmetry to write the amplitude in the u-channel as Fu(x, t) = F (4m 2 − x− t, t). The P on the left hand side of (2.1) denotes the second derivative of the residues. By the analyticity assumption this term comes entirely from the complex simple poles produced by the exchange of unstable states. In our discussion the pole term can be neglected since its contribution turns out not to be relevant . In the case of forward scattering (t = 0) the imaginary part ImF (x, 0) is proportional to the total cross section of the transition 2π →’everything’ and is therefore non negative. The crossing symmetry leads to a similar result for the u-channel. We conclude that F ′′(s, 0) is a strictly positive function for any real center of mass energy s in the range 0 ≤ s ≤ 4m2. The analyticity assumption can be used to generalize the domain of positivity of the imaginary part of the amplitude. This can be seen by expanding ImF (x+ iε, t) in partial waves in the physical region and observing that, due to the optical theorem and the prop- erties of the Legendre polynomials, any derivative with respect to t at the point x ≥ 4m2, t = 0 is non negative. The Taylor series of ImF (x+ iε, t) for t ≥ 0 is therefore greater than zero. Since an analog result holds for the u-channel, we conclude that the second derivative F ′′(s, t) is strictly positive (and analytical) for any real kinematical invariant belonging to the triangle ∆ = s, t, u| 0 ≤ s, t, u ≤ 4m2 In QCD, the scattering of pions at a scale comparable with their masses is very well described by the chiral lagrangian. The 4 pion operators produce order s2 corrections to – 2 – the scattering amplitude and eq. (2.1) implies positive bounds on some combination of their coefficients (see [4], for example). 2.1 Application to the gauged chiral lagrangian We can think of the SM as an effective theory and extend its action to include non renor- malizable operators in the standard way [5]. The anomalous quartic gauge couplings enter the scattering amplitude of two longi- tudinally polarized gauge bosons at order s2. We expect that the method outlined in the previous section may be used to bound these coefficients. There exists, however, a fundamental difference from the QCD case. The assumptions made to derive the relation (2.1) are the analytic, Lorentz and crossing symmetric nature together with the asymptotic behavior of the amplitude F (s, t). A sufficient condition for the latter hypothesis to hold is that no massless particle exchange contribute to F (Froissart bound). In the electroweak case this latter assumption is not natural because of the presence of the electromagnetic interactions. Although we may consider only amplitudes with no single photon exchange (like W±Z0 → W±Z0 for example), there is still an operative difficulty due to the fact that the amplitude F is generally dominated by the SM graphs at low energy scales. These latter give rise to positive contributions to F (s, t), since the SM is well defined even for vanishing coefficients, and one is lead to conclude that eq. (2.1) implies that the effective operators involved cannot produce a ”too large and negative” contribution to the amplitude F (s, t) and that, as a consequence, no significant bound can be derived in the gauged theory. Notice that this is also true in the absence of a light Higgs boson as far as the CM energy is of the order of the Z0 mass. One way to overcome these apparent complications is considering amplitudes with no single photon exchange and evaluating them at a high scale s ≫ m2Z with the equivalence theorem (ET). In this case one has to prove the positivity of the second derivative of the amplitude is guaranteed in the energy regime in which the approach is defined [6]. Another way, which we decide to follow, is working in the global limit. The crucial observation in order to justify this assumption is that in the matching between the effective lagrangian and the UV theory the transverse gauge bosons contribute, because of their weak coupling, in a subdominant way to the effective coefficients of our interest. An accurate estimate of them, and the respective bounds, can therefore be obtained neglecting completely the gauge structure and studying the coefficients of the global theory. Using this conceptually different (though operationally equivalent) perspective we can study any two by two elastic scattering amplitude and generalize the analysis of [6] to non-forward scattering. 2.2 Derivation of the analytical bounds We first specialize to the case there appears no Higgs-like boson under a cut off Λ. In this context the basic tool is a non linearly realized effective lagrangian for the break- ing pattern SU(2)× U(1) → U(1) written in terms of a SU(2) matrix U = exp(iπaσa/v), where σa are the three Pauli matrices with a = 1, 2, 3 and v ≃ 250 GeV is the EW vacuum. – 3 – As usual, under a global SU(2)L × U(1)Y transformation U → LUR†, where L ∈ SU(2)L and R ∈ U(1)Y ⊂ SU(2)R. Assuming m2Z ≪ Λ2 and working at energies comparable with the Z0 mass, the most general lagrangian respecting the above symmetries and up to O(s2) is given in reference [7]. The globally symmetric version is: LEWChL = − Tr (VµV 2v2[Tr(TVµ)] + α4[Tr(VµVν)] 2 + α5[Tr(VµV µ)]2 + α6Tr(VµVν)Tr(TV µ)Tr(TV ν) + α7Tr(VµV µ)Tr(TVν)Tr(TV α10[Tr(TVµ)Tr(TVν)] 2, (2.2) where Vµ = (∂µU)U † and T = Uσ3U †. We stress that in this idealized scenario the πa are exact Goldstone bosons. To avoid any complication with the asymptotic behavior of the amplitude we can introduce by hand a πa mass and proceed as in QCD. This mass is actually the consequence of an explicit symmetry breaking term in the UV theory. Being interested in constraining the underlying symmetric theory we are forced to take m2 ≪ m2Z , s. The bounds we derive differ from the QCD ones for this very reason. Although no mass gap is present in this context, an approximate positive constraint for F ′′(s, t) can be derived. This we do by noticing that a general dispersion relation like (2.1) can be used to bound the anomalous quartic couplings only if the O(s3) contribution to F (s, t) is negligible. In this regime the second derivative F ′′(s, t) is dominantly s indepen- dent and, for a small non vanishing imaginary part for s, the dispersion relation can be approximated as: d2F (s, t) ImF (x+ iε, t) ImFu(x+ iε, t) (2.3) where the limit m2/s → 0 was assumed and the resonant pole term has been neglected. Eq. (2.3) shows that, as far as O(s3) are negligible compared to O(s2), the second derivative of the amplitude is strictly positive. Before evaluating the bounds we notice that the smallness of the EW precision tests T parameter [8] is conveniently achieved by assuming the existence of an approximate global SU(2)C custodial symmetry under which the Goldstone boson matrix transforms as the adjoint representation. The dominant coefficients associated to anomalous quartic gauge operators are α4 and α5 and any π aπb → πcπd scattering amplitude can be written in terms of a function A(s, t, u). The relevant processes turn out to be: A(π0π0 −→ π0π0) = A(s, t, u) +A(t, s, u) +A(u, t, s) A(π±π0 −→ π±π0) = A(t, s, u), (2.4) where, at one loop level and in the limit m2/s → 0, we have [9] A(s, t, u) = 2α5(µ)s 2 + α4(µ)(t 2 + u2) + (4π)2 10s2 + 13(t2 + u2) 96π2v4 t(t− u) log + u(u− t) log + 3s2 log .(2.5) – 4 – Notice that we have chosen to work with the renormalized coefficients α4,5(µ) as defined by the modified minimal subtraction scheme, rather than using the non standard normalization of [9]. We can now derive (2.4) twice with respect to s and evaluate the result at s + iε, t, where 0 < s, t ≪ Λ2. It is convenient to choose a different representation for the kinematical invariants in order to eliminate the logarithms in the final result. We define a scale w = s(s+ t) = −su > s and obtain: α4(w) + α5(w) > − (4π)2 α4(w) > (4π)2 . (2.6) For t = 0 we have α4 + α5 & −0.40 × 10−3 and α4 & −0.35 × 10−3 at an arbitrary scale w = s ≪ Λ2. This result coincides with the one obtained in [6], as expected. In the case of non-forward scattering, the bound on α4(w) cannot get arbitrarily large (large w or, equivalently, large t) because at some unknown scale, much smaller than Λ2, the O(s3) corrections become relevant in the determination of the amplitude and the bound would not apply. Without a detailed knowledge of the perturbative expansion in the weak coupling s/Λ2, (that is, of the full theory!) we cannot realistically tell which is the strongest bound derived by this analysis. What we can certainly do is to compare (2.6) with the well known constraints on the corresponding parameters l1 = 4α5 and l2 = 4α4 of QCD. Strong bounds on these coefficients have been evaluated in the triangle ∆ [10]. We may interpret our analysis as a study of the axiomatic constraints on the two pion amplitudes in the complementary region m2 ≪ s ≪ Λ2. Using the notation introduced in [9] we translate (2.6) into 2l̄1 + 4l̄4 & 3 and l̄2 & 0.3. These constraints are compatible with the experimental observations [11] but are less stringent than those obtained in [10]. We conclude that our analysis does not lead to an improvement of the bounds on l̄1,2. If the chiral symmetry is exact, on the other hand, eqs. (2.6) represent stringent bounds on the anomalous quartic couplings implied by the assumptions of analyticity, crossing symmetry, unitarity and Lorentz invariance of the S-matrix. Eq. (2.3) is not rigorous if a light state enters the processes under consideration and therefore (2.6) are not valid if a Higgs-like scalar propagates under the cutoff. In the next paragraph we discuss an approach which works in this context as well, provided the chiral symmetry is exact. 3. Causal bounds Given a general solution of the equations of motion derived from (2.2) we can study the oscillations around it. Consistency with Special Relativity requires the oscillations to prop- agate sub-luminally. This request may be expressed as a constraint on the same coefficients which enter the elastic scattering of two Goldstone bosons because the dynamics of the os- cillation on the background can be interpreted as a scattering process on a macroscopic – 5 – ‘object‘. If the background has a constant gradient, the presence of super-luminal propa- gations sum up and can in principle become manifest in the low energy regime [2]. A constant gradient solutions admitted by the lagrangian (2.2) is defined by π0 = µ, where σ is a generic isospin direction and the constant vector Cµ is fine-tuned in order to satisfy C2 ≪ v4. The quadratic lagrangian for the oscillations δπ = π−π0 around the background have the general form: L = δπ δπ, (3.1) with α = α4, α4 + α5. In the evaluation of (3.1) we neglected O(Cx/v) terms. We can imagine in fact the non trivial background to be switched on in a finite space-time domain so that the latter approximation is seen as a consequence of the fine-tuning of the parameter A perturbative study of the interacting field δπ is in principle possible for energies under a certain scale (to be definite we call this scale the cut-off of the effective theory). By assumption, this cut off is arbitrarily close to Λ as C2/v4 goes to zero and, having this fact in mind, we simply denote it as Λ. A necessary condition for such a perturbative study to make any sense is that the quadratic lagrangian be well defined. This is the case for (3.1) only if α ≥ 0. In fact, the field δπ has velocity dE/dp = E/p (where pµ = (E, p̄) and |p̄| = p) and for α < 0 its quanta propagate super-luminally. It is important to notice that the presence of super-luminal modes is not the conse- quence of a bad choice of the vacuum. The quadratic hamiltonian is stable in any vacuum (parametrized by Cµ) if α is ’sufficiently small’ but generally leads to violations of the causality principle of Special Relativity when α < 0. In the latter hypothesis then different inertial frames may not agree on the physical observations and, for example, the quadratic hamiltonian may appear unbounded from below to a general Lorentzian frame boosted with a sufficiently high velocity. We finally interpret the constraint α ≥ 0 as a causal bound. The effective coefficients α which appear in the perturbative analysis are actually the renormalized couplings so that the above bound can be extended to all energy scales w < Λ2, where the perturbative study is assumed to be meaningful, after taking into account the running effect: α4(w) + α5(w) ≥ (4π)2 α4(w) ≥ (4π)2 . (3.2) This approach may be applied even to scenarios in which a scalar Higgs, composite or fundamental, can propagate under the cut off. In this latter case the causal constraints read α4 ≥ 0 and α4 + α5 ≥ 0 but now the coefficients do not have any scale dependence because the theory has no extra-SM divergences at order s2. Therefore, the possibility α4 = α5 = 0 can not and must not be excluded (consider the particular example of the – 6 – SM). The analytical bounds, which would imply a strict inequality, do not apply as already noticed. The bounds 3.2 cannot be compared to the QCD ones because π0 does not solve the equations of motion when m 6= 0. 4. Conclusions We have derived general bounds on the anomalous quartic gauge couplings using two distinct approaches. The causal one relies on the absence of superluminal propagations. The analytical one relies on the assumption of analyticity, crossing and Lorentz symmetry together with a good behavior at infinity of the scattering amplitude F (s, t). The latter method works in the context of a strongly coupled theory with no Higgs propagating at low energy only. In this scenario (2.6) can be compared to (3.2). We see that the bound on α4 +α5 is clearly dominated by the causal result and that this is also the case for α4 if, roughly, the ratio (w/s)2 does not exceed 16 log(Λ/ w). We cannot tell if the analytical bound still apply up to this scale More importantly, if the fermionic effects are considered separately from α4,5, a realistic estimate of the constraints should take the fermions couplings to the Goldstone bosons into account. It is easy to see that the one loop effect induced by the SM fermions gives rise to a positive contribution to the second derivative of the amplitude. This of course lowers the analytical bounds while the causal argument remains valid and (3.2) is not altered. The bound (3.2) for the higgsless scenario, together with the constraint α4 ≥ 0 and α4+α5 ≥ 0 for the light Higgs-like scenario provide the most stringent and reliable bounds on the effective coefficients α4,5. In order to have a rough estimate of (3.2) we assume Λ ∼ 1 TeV and get α4 + α5 & 3.8×10−3, α4 & 2.5×10−3 at the Z0 pole. These values lie inside the very wide experimental bounds −0.1 . α4,5 . 0.1. Eqs. (3.2) significantly reduce the allowed range. The experimental constraints are extremely weak since they have been derived by estimating the loop corrections induced by α4,5 on the electroweak precision parameters [1]. A direct measurement of the anomalous gauge couplings turns out to be of fundamental importance in order to have some insight on the actual nature of the electroweak breaking sector [13]. LHC may improve the bounds [1] by an order of magnitude but the linear collider seems far more appropriate to resolve the coefficients [12]. The measurement of a negative value of α4 and α4+α5 at the next linear collider would therefore signal a breaking of causality, irrespective of the presence of a light scalar state like the Higgs boson. This seems a rather unlikely possibility because it would require too drastic a modification of our physical understanding. A more conservative point of view consists in interpreting the bounds (3.2) as theoretical constraints on the full theory. Acknowledgments This work is partially supported by MIUR and the RTN European Program MRTN-CT- 2004-503369. – 7 – References [1] O. J. P. Eboli, M. C. Gonzalez-Garcia and J. K. Mizukoshi, Phys. Rev. D 74, 073005 (2006) [arXiv:hep-ph/0606118]. H. J. He, Y. P. Kuang and C. P. Yuan, Phys. Rev. D 55, 3038 (1997) [arXiv:hep-ph/9611316]; A. S. Belyaev, O. J. P. Eboli, M. C. Gonzalez-Garcia, J. K. Mizukoshi, S. F. Novaes and I. Zacharov, Phys. Rev. D 59, 015022 (1999) [arXiv:hep-ph/9805229]. [2] A. Adams, N. Arkani-Hamed, S. Dubovsky, N. Nicolis, R. Rattazzi, JHEP 0610:014,(2006) [3] A. Martin, Nuovo Cim. A 42, 930 (1966) [4] T. N. Pham and T. N. Truong, Phys. Rev. D 31, 3027 (1985). [5] C. G. . Callan, S. R. Coleman, J. Wess and B. Zumino, Phys. Rev. 177, 2247 (1969). [6] J. Distler, B. Grinstein, R. A. Porto and I. Z. Rothstein, Phys. Rev. Lett. 98, 041601 (2007) [arXiv:hep-ph/0604255]. [7] T. Appelquist and G. H. Wu, Phys. Rev. D 48, 3235 (1993) [arXiv:hep-ph/9304240]. [8] M. E. Peskin and T. Takeuchi, Phys. Rev. D 46, 381 (1992). [9] J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984). [10] B. Ananthanarayan, D. Toublan and G. Wanders, Phys. Rev. D 51, 1093 (1995) [arXiv:hep-ph/9410302]. [11] J. Bijnens, Prog. Part. Nucl. Phys. 58, 521 (2007) [arXiv:hep-ph/0604043]. [12] E. Boos, H. J. He, W. Kilian, A. Pukhov, C. P. Yuan and P. M. Zerwas, Phys. Rev. D 61, 077901 (2000) [arXiv:hep-ph/9908409]. [13] M. Fabbrichesi and L. Vecchi, Phys. Rev. D 76, 056002 (2007) [arXiv:hep-ph/0703236]. – 8 –

We derive one loop constraints on the anomalous quartic gauge couplings using a general non-forward dispersion relation for the elastic scattering amplitude of two longitudinally polarized vector bosons. We compare this result with another one derived by the assumption that the underlying theory satisfies the causality principle of Special Relativity and show that this latter is more constraining.

Introduction 1 2. Analytical bounds 2 2.1 Application to the gauged chiral lagrangian 3 2.2 Derivation of the analytical bounds 3 3. Causal bounds 5 4. Conclusions 7 1. Introduction The general structure of an effective lagrangian is dictated by the interplay between quan- tum mechanics, Poincaré invariance, and internal symmetries. Its coefficients are not con- strained by the symmetries and must be determined by experiments. Unitarity usually sets an upper bound on the energy scale below which a perturbative effective approach is reliable. We can interpret the standard model (SM) as an effective theory extending its la- grangian to include new non-renormalizable operators with unknown coefficients. Some of them enter the scattering amplitudes of longitudinally polarized vector bosons. These are called anomalous quartic gauge couplings since they measure the deviation from the SM predictions. These coefficients are necessarily connected with the not yet observed Higgs sector. In the case the Higgs boson is not a fundamental state, or even no Higgs boson will be observed, they provide important informations on the nature of the electro-weak symmetry breaking sector. Whereas there are no significant experimental bounds on them at the moment [1], theoretical arguments can reduce significantly their allowed range and can serve as a guide for future experiments. The authors of [2] have noticed that the coefficients of a general effective lagrangian may be constrained by requiring the S-matrix of the full theory respects some desirable property such as analyticity, crossing symmetry, Lorentz invariance and unitarity. We follow these authors and consider the SM SU(2)× U(1) → U(1) breaking pattern in the case there exists a light Higgs-like boson as well as in the case no Higgs boson can propagate under the cut off of the effective theory. We show that a general non- forward dispersion relation leads to a less constraining bound than the one derived by the request the UV completion respects the causality principle of Special Relativity. This is not surprising because it is commonly believed that the analytical properties of the S-matrix are a consequence of its causal nature. – 1 – 2. Analytical bounds We briefly review an analytic tool which has been used in the context of the chiral la- grangian of QCD to constrain some effective coefficients. Consider a multiplet of scalar particles, which to be definite we call pions πa, having mass m. Assume they are lighter than any other quanta and that they have appropriate quantum numbers to forbid the transition 2π → π. The other states can be general unstable quanta of complex masses M much greater than 2m. The S-matrix element for a general transition 2π → 2π is a Lorentz scalar function of the Mandelstam variables s, t, u and of the mass m2. We study the amplitude for the elastic scattering πaπb → πaπb and assume it can be analytically continued to the complex variables s, t. We denote this analytical function by F (s, t) and require that its domain of analyticity be dictated entirely by the optical theorem and the crossing symmetry. More precisely, we assume that the singularities come from simple poles in the correspondence of the physical masses of the quantum states which can be produced in the reaction, and branch cuts in the real axis starting at the threshold of multi-particle production. Since no mass-less particle exchange is included in F (s, t), the analytical amplitude satisfies a twice subtracted dispersion relation for a variety of complex t [3]. For any non singular complex point s, t we can write: d2F (s, t) + P = ImF (x+ iε, t) (x− s)3 ImFu(x+ iε, t) (x− u)3 (2.1) where we defined u = 4m2 − s− t and used the crossing symmetry to write the amplitude in the u-channel as Fu(x, t) = F (4m 2 − x− t, t). The P on the left hand side of (2.1) denotes the second derivative of the residues. By the analyticity assumption this term comes entirely from the complex simple poles produced by the exchange of unstable states. In our discussion the pole term can be neglected since its contribution turns out not to be relevant . In the case of forward scattering (t = 0) the imaginary part ImF (x, 0) is proportional to the total cross section of the transition 2π →’everything’ and is therefore non negative. The crossing symmetry leads to a similar result for the u-channel. We conclude that F ′′(s, 0) is a strictly positive function for any real center of mass energy s in the range 0 ≤ s ≤ 4m2. The analyticity assumption can be used to generalize the domain of positivity of the imaginary part of the amplitude. This can be seen by expanding ImF (x+ iε, t) in partial waves in the physical region and observing that, due to the optical theorem and the prop- erties of the Legendre polynomials, any derivative with respect to t at the point x ≥ 4m2, t = 0 is non negative. The Taylor series of ImF (x+ iε, t) for t ≥ 0 is therefore greater than zero. Since an analog result holds for the u-channel, we conclude that the second derivative F ′′(s, t) is strictly positive (and analytical) for any real kinematical invariant belonging to the triangle ∆ = s, t, u| 0 ≤ s, t, u ≤ 4m2 In QCD, the scattering of pions at a scale comparable with their masses is very well described by the chiral lagrangian. The 4 pion operators produce order s2 corrections to – 2 – the scattering amplitude and eq. (2.1) implies positive bounds on some combination of their coefficients (see [4], for example). 2.1 Application to the gauged chiral lagrangian We can think of the SM as an effective theory and extend its action to include non renor- malizable operators in the standard way [5]. The anomalous quartic gauge couplings enter the scattering amplitude of two longi- tudinally polarized gauge bosons at order s2. We expect that the method outlined in the previous section may be used to bound these coefficients. There exists, however, a fundamental difference from the QCD case. The assumptions made to derive the relation (2.1) are the analytic, Lorentz and crossing symmetric nature together with the asymptotic behavior of the amplitude F (s, t). A sufficient condition for the latter hypothesis to hold is that no massless particle exchange contribute to F (Froissart bound). In the electroweak case this latter assumption is not natural because of the presence of the electromagnetic interactions. Although we may consider only amplitudes with no single photon exchange (like W±Z0 → W±Z0 for example), there is still an operative difficulty due to the fact that the amplitude F is generally dominated by the SM graphs at low energy scales. These latter give rise to positive contributions to F (s, t), since the SM is well defined even for vanishing coefficients, and one is lead to conclude that eq. (2.1) implies that the effective operators involved cannot produce a ”too large and negative” contribution to the amplitude F (s, t) and that, as a consequence, no significant bound can be derived in the gauged theory. Notice that this is also true in the absence of a light Higgs boson as far as the CM energy is of the order of the Z0 mass. One way to overcome these apparent complications is considering amplitudes with no single photon exchange and evaluating them at a high scale s ≫ m2Z with the equivalence theorem (ET). In this case one has to prove the positivity of the second derivative of the amplitude is guaranteed in the energy regime in which the approach is defined [6]. Another way, which we decide to follow, is working in the global limit. The crucial observation in order to justify this assumption is that in the matching between the effective lagrangian and the UV theory the transverse gauge bosons contribute, because of their weak coupling, in a subdominant way to the effective coefficients of our interest. An accurate estimate of them, and the respective bounds, can therefore be obtained neglecting completely the gauge structure and studying the coefficients of the global theory. Using this conceptually different (though operationally equivalent) perspective we can study any two by two elastic scattering amplitude and generalize the analysis of [6] to non-forward scattering. 2.2 Derivation of the analytical bounds We first specialize to the case there appears no Higgs-like boson under a cut off Λ. In this context the basic tool is a non linearly realized effective lagrangian for the break- ing pattern SU(2)× U(1) → U(1) written in terms of a SU(2) matrix U = exp(iπaσa/v), where σa are the three Pauli matrices with a = 1, 2, 3 and v ≃ 250 GeV is the EW vacuum. – 3 – As usual, under a global SU(2)L × U(1)Y transformation U → LUR†, where L ∈ SU(2)L and R ∈ U(1)Y ⊂ SU(2)R. Assuming m2Z ≪ Λ2 and working at energies comparable with the Z0 mass, the most general lagrangian respecting the above symmetries and up to O(s2) is given in reference [7]. The globally symmetric version is: LEWChL = − Tr (VµV 2v2[Tr(TVµ)] + α4[Tr(VµVν)] 2 + α5[Tr(VµV µ)]2 + α6Tr(VµVν)Tr(TV µ)Tr(TV ν) + α7Tr(VµV µ)Tr(TVν)Tr(TV α10[Tr(TVµ)Tr(TVν)] 2, (2.2) where Vµ = (∂µU)U † and T = Uσ3U †. We stress that in this idealized scenario the πa are exact Goldstone bosons. To avoid any complication with the asymptotic behavior of the amplitude we can introduce by hand a πa mass and proceed as in QCD. This mass is actually the consequence of an explicit symmetry breaking term in the UV theory. Being interested in constraining the underlying symmetric theory we are forced to take m2 ≪ m2Z , s. The bounds we derive differ from the QCD ones for this very reason. Although no mass gap is present in this context, an approximate positive constraint for F ′′(s, t) can be derived. This we do by noticing that a general dispersion relation like (2.1) can be used to bound the anomalous quartic couplings only if the O(s3) contribution to F (s, t) is negligible. In this regime the second derivative F ′′(s, t) is dominantly s indepen- dent and, for a small non vanishing imaginary part for s, the dispersion relation can be approximated as: d2F (s, t) ImF (x+ iε, t) ImFu(x+ iε, t) (2.3) where the limit m2/s → 0 was assumed and the resonant pole term has been neglected. Eq. (2.3) shows that, as far as O(s3) are negligible compared to O(s2), the second derivative of the amplitude is strictly positive. Before evaluating the bounds we notice that the smallness of the EW precision tests T parameter [8] is conveniently achieved by assuming the existence of an approximate global SU(2)C custodial symmetry under which the Goldstone boson matrix transforms as the adjoint representation. The dominant coefficients associated to anomalous quartic gauge operators are α4 and α5 and any π aπb → πcπd scattering amplitude can be written in terms of a function A(s, t, u). The relevant processes turn out to be: A(π0π0 −→ π0π0) = A(s, t, u) +A(t, s, u) +A(u, t, s) A(π±π0 −→ π±π0) = A(t, s, u), (2.4) where, at one loop level and in the limit m2/s → 0, we have [9] A(s, t, u) = 2α5(µ)s 2 + α4(µ)(t 2 + u2) + (4π)2 10s2 + 13(t2 + u2) 96π2v4 t(t− u) log + u(u− t) log + 3s2 log .(2.5) – 4 – Notice that we have chosen to work with the renormalized coefficients α4,5(µ) as defined by the modified minimal subtraction scheme, rather than using the non standard normalization of [9]. We can now derive (2.4) twice with respect to s and evaluate the result at s + iε, t, where 0 < s, t ≪ Λ2. It is convenient to choose a different representation for the kinematical invariants in order to eliminate the logarithms in the final result. We define a scale w = s(s+ t) = −su > s and obtain: α4(w) + α5(w) > − (4π)2 α4(w) > (4π)2 . (2.6) For t = 0 we have α4 + α5 & −0.40 × 10−3 and α4 & −0.35 × 10−3 at an arbitrary scale w = s ≪ Λ2. This result coincides with the one obtained in [6], as expected. In the case of non-forward scattering, the bound on α4(w) cannot get arbitrarily large (large w or, equivalently, large t) because at some unknown scale, much smaller than Λ2, the O(s3) corrections become relevant in the determination of the amplitude and the bound would not apply. Without a detailed knowledge of the perturbative expansion in the weak coupling s/Λ2, (that is, of the full theory!) we cannot realistically tell which is the strongest bound derived by this analysis. What we can certainly do is to compare (2.6) with the well known constraints on the corresponding parameters l1 = 4α5 and l2 = 4α4 of QCD. Strong bounds on these coefficients have been evaluated in the triangle ∆ [10]. We may interpret our analysis as a study of the axiomatic constraints on the two pion amplitudes in the complementary region m2 ≪ s ≪ Λ2. Using the notation introduced in [9] we translate (2.6) into 2l̄1 + 4l̄4 & 3 and l̄2 & 0.3. These constraints are compatible with the experimental observations [11] but are less stringent than those obtained in [10]. We conclude that our analysis does not lead to an improvement of the bounds on l̄1,2. If the chiral symmetry is exact, on the other hand, eqs. (2.6) represent stringent bounds on the anomalous quartic couplings implied by the assumptions of analyticity, crossing symmetry, unitarity and Lorentz invariance of the S-matrix. Eq. (2.3) is not rigorous if a light state enters the processes under consideration and therefore (2.6) are not valid if a Higgs-like scalar propagates under the cutoff. In the next paragraph we discuss an approach which works in this context as well, provided the chiral symmetry is exact. 3. Causal bounds Given a general solution of the equations of motion derived from (2.2) we can study the oscillations around it. Consistency with Special Relativity requires the oscillations to prop- agate sub-luminally. This request may be expressed as a constraint on the same coefficients which enter the elastic scattering of two Goldstone bosons because the dynamics of the os- cillation on the background can be interpreted as a scattering process on a macroscopic – 5 – ‘object‘. If the background has a constant gradient, the presence of super-luminal propa- gations sum up and can in principle become manifest in the low energy regime [2]. A constant gradient solutions admitted by the lagrangian (2.2) is defined by π0 = µ, where σ is a generic isospin direction and the constant vector Cµ is fine-tuned in order to satisfy C2 ≪ v4. The quadratic lagrangian for the oscillations δπ = π−π0 around the background have the general form: L = δπ δπ, (3.1) with α = α4, α4 + α5. In the evaluation of (3.1) we neglected O(Cx/v) terms. We can imagine in fact the non trivial background to be switched on in a finite space-time domain so that the latter approximation is seen as a consequence of the fine-tuning of the parameter A perturbative study of the interacting field δπ is in principle possible for energies under a certain scale (to be definite we call this scale the cut-off of the effective theory). By assumption, this cut off is arbitrarily close to Λ as C2/v4 goes to zero and, having this fact in mind, we simply denote it as Λ. A necessary condition for such a perturbative study to make any sense is that the quadratic lagrangian be well defined. This is the case for (3.1) only if α ≥ 0. In fact, the field δπ has velocity dE/dp = E/p (where pµ = (E, p̄) and |p̄| = p) and for α < 0 its quanta propagate super-luminally. It is important to notice that the presence of super-luminal modes is not the conse- quence of a bad choice of the vacuum. The quadratic hamiltonian is stable in any vacuum (parametrized by Cµ) if α is ’sufficiently small’ but generally leads to violations of the causality principle of Special Relativity when α < 0. In the latter hypothesis then different inertial frames may not agree on the physical observations and, for example, the quadratic hamiltonian may appear unbounded from below to a general Lorentzian frame boosted with a sufficiently high velocity. We finally interpret the constraint α ≥ 0 as a causal bound. The effective coefficients α which appear in the perturbative analysis are actually the renormalized couplings so that the above bound can be extended to all energy scales w < Λ2, where the perturbative study is assumed to be meaningful, after taking into account the running effect: α4(w) + α5(w) ≥ (4π)2 α4(w) ≥ (4π)2 . (3.2) This approach may be applied even to scenarios in which a scalar Higgs, composite or fundamental, can propagate under the cut off. In this latter case the causal constraints read α4 ≥ 0 and α4 + α5 ≥ 0 but now the coefficients do not have any scale dependence because the theory has no extra-SM divergences at order s2. Therefore, the possibility α4 = α5 = 0 can not and must not be excluded (consider the particular example of the – 6 – SM). The analytical bounds, which would imply a strict inequality, do not apply as already noticed. The bounds 3.2 cannot be compared to the QCD ones because π0 does not solve the equations of motion when m 6= 0. 4. Conclusions We have derived general bounds on the anomalous quartic gauge couplings using two distinct approaches. The causal one relies on the absence of superluminal propagations. The analytical one relies on the assumption of analyticity, crossing and Lorentz symmetry together with a good behavior at infinity of the scattering amplitude F (s, t). The latter method works in the context of a strongly coupled theory with no Higgs propagating at low energy only. In this scenario (2.6) can be compared to (3.2). We see that the bound on α4 +α5 is clearly dominated by the causal result and that this is also the case for α4 if, roughly, the ratio (w/s)2 does not exceed 16 log(Λ/ w). We cannot tell if the analytical bound still apply up to this scale More importantly, if the fermionic effects are considered separately from α4,5, a realistic estimate of the constraints should take the fermions couplings to the Goldstone bosons into account. It is easy to see that the one loop effect induced by the SM fermions gives rise to a positive contribution to the second derivative of the amplitude. This of course lowers the analytical bounds while the causal argument remains valid and (3.2) is not altered. The bound (3.2) for the higgsless scenario, together with the constraint α4 ≥ 0 and α4+α5 ≥ 0 for the light Higgs-like scenario provide the most stringent and reliable bounds on the effective coefficients α4,5. In order to have a rough estimate of (3.2) we assume Λ ∼ 1 TeV and get α4 + α5 & 3.8×10−3, α4 & 2.5×10−3 at the Z0 pole. These values lie inside the very wide experimental bounds −0.1 . α4,5 . 0.1. Eqs. (3.2) significantly reduce the allowed range. The experimental constraints are extremely weak since they have been derived by estimating the loop corrections induced by α4,5 on the electroweak precision parameters [1]. A direct measurement of the anomalous gauge couplings turns out to be of fundamental importance in order to have some insight on the actual nature of the electroweak breaking sector [13]. LHC may improve the bounds [1] by an order of magnitude but the linear collider seems far more appropriate to resolve the coefficients [12]. The measurement of a negative value of α4 and α4+α5 at the next linear collider would therefore signal a breaking of causality, irrespective of the presence of a light scalar state like the Higgs boson. This seems a rather unlikely possibility because it would require too drastic a modification of our physical understanding. A more conservative point of view consists in interpreting the bounds (3.2) as theoretical constraints on the full theory. Acknowledgments This work is partially supported by MIUR and the RTN European Program MRTN-CT- 2004-503369. – 7 – References [1] O. J. P. Eboli, M. C. Gonzalez-Garcia and J. K. Mizukoshi, Phys. Rev. D 74, 073005 (2006) [arXiv:hep-ph/0606118]. H. J. He, Y. P. Kuang and C. P. Yuan, Phys. Rev. D 55, 3038 (1997) [arXiv:hep-ph/9611316]; A. S. Belyaev, O. J. P. Eboli, M. C. Gonzalez-Garcia, J. K. Mizukoshi, S. F. Novaes and I. Zacharov, Phys. Rev. D 59, 015022 (1999) [arXiv:hep-ph/9805229]. [2] A. Adams, N. Arkani-Hamed, S. Dubovsky, N. Nicolis, R. Rattazzi, JHEP 0610:014,(2006) [3] A. Martin, Nuovo Cim. A 42, 930 (1966) [4] T. N. Pham and T. N. Truong, Phys. Rev. D 31, 3027 (1985). [5] C. G. . Callan, S. R. Coleman, J. Wess and B. Zumino, Phys. Rev. 177, 2247 (1969). [6] J. Distler, B. Grinstein, R. A. Porto and I. Z. Rothstein, Phys. Rev. Lett. 98, 041601 (2007) [arXiv:hep-ph/0604255]. [7] T. Appelquist and G. H. Wu, Phys. Rev. D 48, 3235 (1993) [arXiv:hep-ph/9304240]. [8] M. E. Peskin and T. Takeuchi, Phys. Rev. D 46, 381 (1992). [9] J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984). [10] B. Ananthanarayan, D. Toublan and G. Wanders, Phys. Rev. D 51, 1093 (1995) [arXiv:hep-ph/9410302]. [11] J. Bijnens, Prog. Part. Nucl. Phys. 58, 521 (2007) [arXiv:hep-ph/0604043]. [12] E. Boos, H. J. He, W. Kilian, A. Pukhov, C. P. Yuan and P. M. Zerwas, Phys. Rev. D 61, 077901 (2000) [arXiv:hep-ph/9908409]. [13] M. Fabbrichesi and L. Vecchi, Phys. Rev. D 76, 056002 (2007) [arXiv:hep-ph/0703236]. – 8 –

Classical Information Capacity of the Bosonic Broadcast Channel Saikat Guha Research Laboratory of Electronics Massachusetts Institute of Technology Cambridge, MA 02139 saikat@MIT.edu Jeffrey H. Shapiro Research Laboratory of Electronics Massachusetts Institute of Technology Cambridge, MA 02139 jhs@MIT.edu Abstract— We show that when coherent-state encoding is employed in conjunction with coherent detection, the Bosonic broadcast channel is equivalent to a classical degraded Gaussian broadcast channel whose capacity region is dual to that of the classical Gaussian multiple-access channel. We further show that if a minimum output-entropy conjecture holds true, then the ultimate classical information capacity of the Bosonic broadcast channel can be achieved by a coherent-state encoding. We provide some evidence in support of the conjecture. I. INTRODUCTION The past decade has seen several advances in evaluating classical information capacities of several important quantum communication channels [1]–[5]. Despite the theoretical ad- vances that have resulted [1], exact capacity results are not known for many important and practical quantum communi- cation channels. Here we extend the line of research aimed at evaluating capacities of Bosonic communication channels, which began with the capacity derivation for the input photon- number constrained lossless Bosonic channel [2], [3]. The capacity of the lossy Bosonic channel was found in [4], where it was shown that a modulation scheme using classical light (coherent states) suffices to achieve ultimate communication rates over this channel. Subsequent attempts to evaluate the capacity of the noisy Bosonic channel with additive Gaussian noise [5] led to a crucial conjecture on the minimum output entropy of a class of Bosonic channels [6]. Proving that conjecture would complete the capacity proof for the Bosonic channel with additive Gaussian noise, and it would show that this channel’s capacity is achievable with classical-light mod- ulation. More recent work that addressed Bosonic multiple- access communication channels [7] revealed that modulation of information using non-classical states of light is necessary to achieve ultimate single-user rates. In the present work, we study the classical information capacity of the Bosonic broadcast channel. A broadcast channel is the congregation of communication media connecting a single transmitter to two or more receivers. In general, the transmitter encodes and sends out independent information to each receiver in a way that each receiver can reliably decode its respective information. In Sec. II, we describe some recent work on the capacity region of the degraded quantum broadcast channel [8]. In Sec. III, we introduce the noiseless Bosonic broadcast chan- nel model, and derive its capacity region subject to a new minimum output entropy conjecture. In Sec. IV we show that a recent duality result between capacity regions of classical multiple-input, multiple-output Gaussian multiple-access and broadcast channels [9] does not hold for Bosonic channels. II. QUANTUM DEGRADED BROADCAST CHANNEL A quantum channel NA−B from Alice to Bob is a trace- preserving completely positive map that maps Alice’s single- use density operators ρ̂A to Bob’s, ρ̂B = NA−B(ρ̂A). The two- user quantum broadcast channel NA−BC is a quantum channel from sender Alice (A) to two independent receivers Bob (B) and Charlie (C). The quantum channel from Alice to Bob is obtained by tracing out C from the channel map, i.e.,NA−B ≡ TrC (NA−BC), with a similar definition for NA−C . We say that a broadcast channel NA−BC is degraded if there exists a degrading channel N degB−C from B to C satisfying NA−C = N degB−C ◦ NA−B . The degraded broadcast channel describes a physical scenario in which for each successive n uses of NA−BC Alice communicates a randomly generated classical message (m, k) ∈ (WB ,WC) to Bob and Charlie, where the message-sets WB and WC are sets of classical indices of sizes 2nRB and 2nRC respectively. The messages (m, k) are assumed to be uniformly distributed over (WB ,WC). Because of the degraded nature of the channel, Bob receives the entire message (m, k) whereas Charlie only receives the index k. To convey these message (m, k), Alice prepares n-channel use states that after transmission through the channel, result in bipartite conditional density matrices , ∀(m, k) ∈ (WB ,WC). The quantum states received by Bob and Charlie,{ respectively, can be found by tracing out the other receiver, viz., ρ̂B m,k ≡ TrCn , etc. A (2nRB , 2nRC , n, �) code for this channel consists of an encoder xn : (WB ,WC)→ An, (1) a positive operator-valued measure (POVM) {Λmk} on Bn and a POVM {Λ′k} on C n which satisfy1 ρ̂xn(m,k)(Λmk ⊗ Λ′k) ≥ 1− � (2) 1An, Bn, and Cn are the n channel use alphabets of Alice, Bob, and Charlie, with respective sizes |An|, |Bn|, and |Cn|. for every (m, k) ∈ (WB ,WC). A rate-pair (RB , RC) is achievable if there exists a sequence of (2nRB , 2nRC , n, �n) codes with �n → 0. The classical capacity region of the broadcast channel is defined as the convex hull of the closure of all achievable rate pairs (RB , RC). The classical capacity region of the two-user degraded quantum broadcast channel NA−BC was recently derived by Yard et. al. [8], and can be expressed in terms of the Holevo information [10], χ(pj , σ̂j) ≡ S pj σ̂j pjS(σ̂j), (3) where {pj} is a probability distribution associated with the density operators σ̂j , and S(ρ̂) ≡ −Tr(ρ̂ log ρ̂) is the von Neumann entropy of the quantum state ρ̂. Because χ may not be additive, the rate region (RB , RC) of the degraded broadcast channel must be computed by maximizing over successive uses of the channel, i.e., for n uses pj|i,N⊗nA−B(ρ̂ pj|iρ̂ pipj|iS ) , and (4) RC ≤ χ pi,∑ pj|iN⊗nA−C(ρ̂  /n, pipj|iρ̂ pj|iρ̂  , (5) where j ≡ (m, k) is a collective index and the states live in the Hilbert space H⊗n of n successive uses of the broadcast channel. The probabilities {pi} form a distribution over an auxiliary classical alphabet T , of size |T |, satisfying |T | ≤ min |An|, |Bn|2 + |Cn|2 + 1 . The ultimate rate- region is computed by maximizing the region specified by Eqs. (4) and (5), over {pi}, , and n, subject to the cardinality constraint on |T |. Fig. 1 illustrates the setup of the two-user degraded quantum channel. III. NOISELESS BOSONIC BROADCAST CHANNEL The two-user noiseless Bosonic broadcast channel NA−BC consists of a collection of spatial and temporal Bosonic modes at the transmitter (Alice), that interact with a minimal- quantum-noise environment and split into two sets of spatio- temporal modes en route to two independent receivers (Bob and Charlie). The multi-mode two-user Bosonic broadcast channelNA−BC is given by sNAs−BsCs , whereNAs−BsCs Fig. 1. Schematic diagram of the degraded single-mode Bosonic broadcast channel. The transmitter Alice (A) encodes her messages to Bob (B) and Charlie (C) in a classical index j, and over n successive uses of the channel, prepares a bipartite state ρ̂B j for them. is the broadcast-channel map for the sth mode, which can be obtained from the Heisenberg evolutions b̂s = ηs âs + 1− ηs ês, and (6) ĉs = 1− ηs âs − ηs ês, (7) where {âs} are Alice’s modal annihilation operators, and {b̂s}, {ĉs} are the corresponding modal annihilation operators for Bob and Charlie, respectively. The modal transmissivities {ηs} satisfy 0 ≤ ηs ≤ 1, ∀s, and the environment modes {ês} are in their vacuum states. We will limit our treatment here to the single-mode Bosonic broadcast channel, as the capacity of the multi-mode channel can in principle be obtained by summing up capacities of all spatio-temporal modes and maximizing the sum capacity region subject to an overall input-power budget using Lagrange multipliers, cf. [5], where this was done for the capacity of the multi-mode single-user lossy Bosonic channel. The principal result we have for the single-mode degraded Bosonic broadcast channel depends on a minimum output entropy conjecture (the strong form of Conjecture 2, see Appendix). Assuming this conjecture to be true, we have that the ultimate capacity region of the single-mode noiseless Bosonic broadcast channel (see Fig. 2) with a mean input photon-number constraint 〈â†â〉 ≤ N̄ is RB ≤ g(ηβN̄), and (8) RC ≤ g((1− η)N̄)− g((1− η)βN̄), (9) for 0 ≤ β ≤ 1, where g(x) = (1 + x) log(1 + x) − x log(x). This rate region is additive and achievable with single channel use coherent-state encoding with the distributions pT (t) = , and (10) pA|T (α|t) = 1− β t− α|2 . (11) Proof — It is straightforward to show that if η > 1/2, the Bosonic broadcast channel is a degraded quantum broadcast channel, in which Bob’s is the less-noisy receiver and Charlie’s is the more-noisy receiver. Yard et al.’s capacity region in Eqs. (4) and (5) requires finite-dimensional Hilbert spaces. Fig. 2. A single-mode noiseless Bosonic broadcast channel can be envisioned as a beam splitter with transmissivity η. With η > 1/2, the Bosonic broadcast channel reduces to a ‘degraded’ quantum broadcast channel, where Bob (B) is the less-noisy receiver and Charlie (C) is the more noisy (degraded) receiver. Nevertheless, we will use their result for the Bosonic broadcast channel, which has an infinite-dimensional state space, by extending it to infinite-dimensional state spaces through a limiting argument.2 The n = 1 rate-region for the Bosonic broadcast channel using a coherent-state encoding is thus: pT (t)S pA|T (α|t)| η α〉〈 η α| dα dt (12) RC ≤ S pT (t)pA|T (α|t)| 1− η α〉〈 1− η α| dα dt pT (t)S pA|T (α|t) 1− η α〉〈 1− η α| dα dt, (13) where we need to maximize the bounds for RB and RC over all joint distributions pT (t)pA|T (α|t) subject to 〈|α|2〉 ≤ N̄ . Note that A and T are complex-valued random variables, and the second term in the RB bound (4) vanishes, because the von Neumann entropy of a pure state is zero. Substituting Eqs. (10) and (11) into Eqs. (12) and (13), shows that the rate-region Eqs. (8) and (9) is achievable using single-use coherent state encoding. For the converse, assume that the rate pair (RB , RC) is achievable. Let {xn(m, k)}, and POVMs {Λmk} and {Λ′k} comprise any (2nRB , 2nRC , n, �) code in the achieving se- quence. Suppose that Bob and Charlie store their decoded messages in the classical registers ŴB and ŴC respectively. Let us use pWB ,WC (m, k) = pWB (m)pWC (k) to denote the joint probability mass function of the independent message registers WB and WC . As (RB , RC) is an achievable rate- 2When |T | and |A| are finite, and we are using coherent states, there will be a finite number of possible transmitted states, which leads to a finite number of possible states received by Bob and Charlie. Suppose we limit the auxiliary- input alphabet (T )—and hence the input (A) and the output alphabets (B and C)—to truncated coherent states within the finite-dimensional Hilbert space spanned by the Fock states {|0〉, |1〉, . . . , |K〉}, where K � N̄ . Applying Yard et al.’s theorem to the Hilbert space spanned by these truncated coherent states then gives us a broadcast channel capacity region that must be strictly an inner-bound of the rate-region given by unconditional equations (12) and (13). For made K sufficiently large, while maintaining the cardinality condition, the rate-region expressions given by Yard et. al.’s theorem will converge to Eqs (12) and (13). pair, there must exist �′n → 0, such that nRC = H(WC) ≤ I(WC ; ŴC) + n�′n ≤ χ(pWC (k), ρ̂ k ) + n� n, (14) where I(WC ; ŴC) ≡ H(ŴC) − H(ŴC |WC) is the Shan- non mutual information, and ρ̂C m pWB (m)ρ̂ m,k. The second line follows from Fano’s inequality and the third line follows from Holevo’s bound3. Similarly, for an �′′n → 0, we can bound nRB as nRB = H(WB) ≤ I(WB ; ŴB) + n�′′n ≤ χ(pWB (m), ρ̂ m ) + n� pWC (k)χ(pWB (m), ρ̂ m,k) + n� n, (15) where the three lines above follow from Fano’s inequality, Holevo’s bound and the concavity of Holevo information. In order to prove the converse, we now need to show that there exists a number β ∈ [0, 1], such that∑ pWC (k)χ(pWB (m), ρ̂ m,k) ≤ ng(ηβN̄), and χ(pWC (k), ρ̂ k ) ≤ ng((1− η)N̄)− ng((1− η)βN̄). From the non-negativity of the von Neumann entropy , it follows that k pWC (k)χ(pWB (m), ρ̂ m,k) ≤∑ k pWC (k)S m pWB (m)ρ̂ , as the second term of the Holevo information above is non-negative. Because the maximum von Neumann entropy of a single-mode Bosonic state with 〈â†â〉 ≤ N̄ is given by g(N̄), we have that 0 ≤ S ηN̄kj , (16) where, N̄k ≡ N̄kj , and N̄kj is the mean photon number of the jth symbol ρ̂ k of the n-symbol codeword k , for j ∈ {1, . . . , n}. Therefore, ∃βk ∈ [0, 1], ∀k ∈ WC , such that ηβkN̄k . (17) Because of the degraded nature of the channel, Charlie’s state can be obtained as the output of a beam splitter whose input states are Bob’s state (coupling coefficient η′ = (1 − η)/η to Charlie) and a vacuum state (coupling coefficient 1 − η′ to Charlie). It follows, from assuming the truth of Strong conjecture 2 (see Appendix), that (1− η)βkN̄k . (18) N̄ is the average number of photons per-use at the transmitter (Alice) averaged over the entire codebook. Thus, the mean 3Holevo’s bound [10]: Let X be the input alphabet for a channel, {pi, ρ̂i} the priors and modulating states, {Πj} be a POVM, and Y the resulting output (classical) alphabet. The Shannon mutual information I(X;Y ) is upper bounded by the Holevo information χ(pi, ρ̂i) photon-number of the n-use average codeword at Bob, ρ̂B k pWC (k)ρ̂ k , is ηN̄ . Hence, pWC (k)S ≤ S(ρ̂B ) ≤ ng , (19) where the second inequality follows from the convexity of von Neumann entropy. The monotonicity of g(x) then implies that there is a β ∈ [0, 1], such that k pWC (k)S ng(ηβN̄). Hence we have,∑ pWC (k)χ(pWB (m), ρ̂ m,k) ≤ ng(ηβN̄). (20) for some β ∈ [0, 1]. Equation (17), and the uniform distribu- tion pWC (k) = 1/2 nRC imply that∑ ηβkN̄k . (21) Using (21), the convexity of g(x), and η > 1/2, we have shown (proof omitted) that∑ (1− η)βkN̄k (1− η)βN̄ . (22) From Eq. (22), and Eq. (18) summed over k, we then obtain∑ pWC (k)S ≥ ng((1− η)βN̄). (23) Finally, writing Charlie’s Holevo information as χ(pWC (k), ρ̂ k ) = S pWC (k)ρ̂ pWC (k)S ≤ ng((1− η)N̄) pWC (k)S , (24) we can use Eq. (23) to get χ(pWC (k), ρ̂ k ) ≤ ng((1− η)N̄)− ng((1− η)βN̄), (25) which completes the proof. IV. DISCUSSION AND CONCLUSION Recently, Vishwanath et. al. [9] established a duality be- tween the dirty paper achievable region (recently proved to be the ultimate capacity region [11]) for the classical multiple- input, multiple-output (MIMO) Gaussian broadcast channel and the capacity region of the MIMO Gaussian multiple-access channel (MAC). The duality result states that if we evaluate the capacity regions of the MIMO Gaussian MAC—with fixed total received power P and channel-gain values—over all possible power-allocations between the users, the corners of those capacity regions trace out the capacity region of the MIMO Gaussian broadcast channel with transmitter power P and the same channel-gain values. Unlike this classical result, the capacity region of the Bosonic broadcast channel using coherent-state inputs is not equal to of the envelope of Fig. 3. Comparison of Bosonic broadcast and multiple-access channel capacity regions, in bits per channel use, for η = 0.8, and N̄ = 15. The red line is the conjectured ultimate broadcast capacity region, which lies below the green line—the envelope of the MAC capacity regions. the MAC capacity regions using coherent-state inputs. The capacity region of the Bosonic MAC using coherent-state inputs was first computed by Yen [7]. In Fig. 3 we compare the envelope of coherent-state MAC capacities to the capacity region of the coherent-state broadcast channel. This figure shows that with a fixed beam splitter and identical average photon number budgets, more collective classical information can be sent when the beam splitter is used as a multiple-access channel as opposed to when it is used as a broadcast channel. The broadcast channel capacity region that we have derived—modulo Strong conjecture 2—exceeds what can be accomplished with conventional optical receivers, as shown in Fig. 4. In this figure we compare the capacity regions attained by a coherent-state input alphabet using homodyne detection, heterodyne detection, and optimum reception. As is known for single-user Bosonic communications, homodyne detection per- forms better than heterodyne detection when the transmitters are starved for photons, because it has lower noise. Conversely, heterodyne detection outperforms homodyne detection when the transmitters are photon rich, because it has a factor-of-two bandwidth advantage. To bridge the gap between the coherent- detection capacity regions and the ultimate capacity region, one must use joint detection over long codewords. Future investigation will need to be done to realize better broadcast communication rates over the Bosonic broadcast channel. ACKNOWLEDGMENT This research was supported by the Defense Advanced Research Projects Agency. The authors thank Baris Erkmen for helpful discussions and for proving the Gaussian-state version of Strong conjecture 2. APPENDIX: MINIMUM OUTPUT ENTROPY CONJECTURES Let â and b̂ denote the two input modes of a lossless beam splitter of transmissivity η, to produce output modes η â + 1− η b̂ and d̂ = 1− η â − η b̂. In [6], we proposed the following minimum output entropy conjecture: Fig. 4. Comparison of Bosonic broadcast channel capacity regions, in bits per channel use, achieved by coherent-state encoding with homodyne detection [red, circles], heterodyne detection [blue, dashed], and optimum reception [black, solid], for η = 0.8, and N̄ = 1, 5, and 15. Conjecture 1 — Let the input b̂ be in a zero-mean thermal state with von Neumann entropy S(ρ̂B) = g(K). Then the von Neumann entropy of output ĉ is minimized when â is in the vacuum state, and the minimum output entropy is g((1−η)K). In this paper, we propose a new output entropy conjecture: Conjecture 2 — Let the input â be in its vacuum state, input b̂ in a zero-mean state with von Neumann entropy S(ρ̂B) = g(K). Then the von Neumann entropy of output ĉ is minimized when b̂ is in a thermal state with average photon number K, and the minimum output entropy is g((1− η)K). For the capacity proof of the Bosonic broadcast channel, we use Strong conjecture 2, which we now describe. Let the input modes {âi : 1 ≤ i ≤ n} be in a product state of n vacuum states, and let the von Neumann entropy of the joint state of the inputs {b̂i : 1 ≤ i ≤ n} be ng(K). Then, putting {b̂i : 1 ≤ i ≤ n} in a product state of mean-photon-number K thermal states minimizes the output von Neumann entropy of the joint state of {ĉi : 1 ≤ i ≤ n}. Moreover, this minimum output entropy is ng((1− η)K). Previous work has provided considerable evidence in sup- port of Conjecture 1 [6], [12]. In particular, we know that Conjecture 1 is true: when the state of â is Gaussian; when Wehrl entropy4 is considered instead of von Neumann entropy; and when Rényi entropy of integer order n ≥ 2 is considered instead of von Neumann entropy. Strong conjecture 1, i.e., the n-use version, has been proven: when the joint state of the {âi} is Gaussian [13]; and when Wehrl entropy is considered instead of von Neumann entropy. Other evidence in support of Conjecture 1 has been developed from entropy bounds [6], which show that the conjecture is asymptotically correct in the limit of weak and strong noise, and from simulated annealing starting with randomly selected initial states. In unpublished work, we have shown that Conjecture 2 is true: when the state of b̂ is Gaussian; when Wehrl entropy is considered instead of von Neumann entropy; and when the state of b̂ is mixed and diagonal in the Fock basis with a probability distribution that is either Poisson, Binomial, or Bose-Einstein. For Strong conjecture 2 we have shown that it is true: when the {b̂i} are in a Gaussian state; and when Wehrl entropy is considered instead of von Neumann entropy. REFERENCES [1] C. H. Bennett and P. W. 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We show that when coherent-state encoding is employed in conjunction with coherent detection, the Bosonic broadcast channel is equivalent to a classical degraded Gaussian broadcast channel whose capacity region is dual to that of the classical Gaussian multiple-access channel. We further show that if a minimum output-entropy conjecture holds true, then the ultimate classical information capacity of the Bosonic broadcast channel can be achieved by a coherent-state encoding. We provide some evidence in support of the conjecture.

Introduction Quantum Degraded Broadcast Channel Noiseless Bosonic Broadcast Channel Discussion and Conclusion References

Interplay of Anisotropy and Disorder in the Doping-Dependent Melting and Glass Transitions of Vortices in Bi2Sr2CaCu2O8+δ H. Beidenkopf,1, ∗ T. Verdene,1 Y. Myasoedov,1 H. Shtrikman,1 E. Zeldov,1 B. Rosenstein,2 D. Li,3 and T. Tamegai4 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel Electrophysics Department, National Chiao Tung University, Hsinchu 30050, Taiwan, Republic of China Department of Physics, Peking University, Beijing 100871, China Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan (Dated: November 11, 2018) We study the oxygen doping dependence of the equilibrium first-order melting and second-order glass transitions of vortices in Bi2Sr2CaCu2O8+δ. Doping affects both anisotropy and disorder. Anisotropy scaling is shown to collapse the melting lines only where thermal fluctuations are dom- inant. Yet, in the region where disorder breaks that scaling, the glass lines are still collapsed. A quantitative fit to melting and replica symmetry breaking lines of a 2D Ginzburg-Landau model further reveals that disorder amplitude weakens with doping, but to a lesser degree than thermal fluctuations, enhancing the relative role of disorder. PACS numbers: 74.25.Qt, 74.25.Dw, 74.72.Hs, 64.70.Pf Elasticity, thermal energy, disorder, and inter-layer coupling are some of the closely competing energy scales in the intricate H − T phase diagram of the vortex matter in the layered high temperature superconductor Bi2Sr2CaCu2O8+δ (BSCCO) [1, 2, 3, 4, 5, 6]. The low- temperature part of the equilibrium phase diagram of BSCCO was recently made accessible to experiment by vortex shaking [7, 8, 9]. It unveiled a first-order (FO) in- verse melting line, which continues the thermal melting line from high temperatures [8] separating low-field or- dered phases from high-field amorphous ones. A second- order (SO) transition line, at which low-temperature glassy phases get thermally depinned, was subsequently reported [9]. In this letter we study the oxygen doping dependence of these transition lines. We show that the SO line scales with material anisotropy even where the FO line does not, and that effective disorder weakens with doping, but gains relative dominance over thermal fluctuations. We present measurements of optimally doped (OPD), slightly over doped (SOD), over doped (OVD) and highly over doped (HOD) BSCCO crystals [10, 11] with critical temperatures Tc =92, 90, 88.5 and 86 K, respectively, corresponding to hole concentrations of 0.171, 0.180, 0.184 and 0.190 [12]. Various crystal geometries were studied with typical sizes of ∼300×300×40 µm3. They were mounted on 10 × 10 µm2 Hall sensor arrays, fabri- cated in a GaAs/AlGaAs heterostructure. At low tem- peratures we utilized a 350 Oe in-plane ac shaking field of 10 Hz to relax the pancake vortices towards their equi- librium configuration [7, 8, 9]. Conjugating local probes with shaking yielded the equilibrium reversible magneti- zation of the samples. Figure 1 shows the local induction, B(H,T ), measured in the SOD sample by sweeping the temperature, T , at a constant out-of-plane field, H , in presence of an in-plane shaking field. A linear term α T + β was subtracted FIG. 1: (color online) The local induction shows FO (dashed- dotted) and SO (dashed) transition lines in the H − T phase diagram. The induction in the high-temperature depinned liquid and solid phases was artificially flattened by a linear subtraction α T + β (hence the constant color). The low- temperature BrG and glassy phases have thus a finite positive slope. (a) Values of α and β used. (b) FO-SO-FO tran- sition sequence at a 380 Oe temperature sweep on the back- ground of the colorbar. (c) Discontinuous steps in dB/dT , manifesting the SO nature of the transition. from each temperature sweep (Fig. 1a) to flatten the orig- inally increasing (α > 0) local induction. As a result, the FO and SO singular behavior can be readily traced throughout the H − T phase diagram. The FOmelting transition is manifested by a discontin- uous step in the magnetization along the dashed-dotted line in Fig. 1. It separates the high-field amorphous glass and liquid phases from the low-field quasi-long-range- ordered Bragg glass (BrG) [5, 13]. The melting line be- comes nonmonotonic at a certain temperature [8, 9]. Its http://arxiv.org/abs/0704.1902v1 0.4 0.5 0.6 0.7 0.8 0.9 1 t=T/T 0.4 0.45 0.5 t=T/T bisection FO−SO maximum FIG. 2: (color online) FO and SO transition lines (solid and open symbols, respectively) measured with OPD (◦), SOD(�), OVD (♦) and HOD (△) samples. The inset shows the doping dependence of the maximum of the FO line and its bisection with the SO line. Dotted and dashed lines are second-degree polynomial fits. inverse-melting part is believed to be an order-disorder transition, induced mainly by quenched disorder and not by thermal fluctuations [1, 2, 3, 4, 5, 14, 15, 16]. The SO phase transition is manifested by a break in the slope of the magnetization at Tg, marked in Fig. 1 by the dashed line. It separates the fully flattened high- temperature magnetization (dB/dT |T>Tg − α ≈ 0) of constant color from a low-temperature region of nonzero slope (dB/dT |T<Tg − α > 0), whose color varies with temperature. The nonanaliticity at Tg is demonstrated in Fig. 1c, which shows ∼40 mG/K steps in dB/dT for temperature sweeps lying below as well as above the FO line. The location of the SO line above melting resembles earlier dynamic irreversibility measurements [17, 18, 19, 20, 21] and theoretical models of a glass transition [1, 2, 3, 4, 5, 22, 23, 24]. Few dynamic mea- surements found a similar line of thermal depinning be- low melting [9, 25, 26, 27, 28, 29, 30], which separates the BrG phase below it from a thermally depinned solid, whose detailed characteristics are not yet certain. We mapped three more samples of various doping lev- els that yielded the B−T phase diagram plotted in Fig. 2. The melting line shifts as a whole to higher fields with doping due to the decrease in anisotropy, which implies stronger inter-layer coupling and higher stiffness of the pancake vortex stacks [10, 30, 31, 32]. As the FO line shifts to higher fields, its maximum shifts towards in- creasingly higher temperatures (Fig. 2 inset). Identify- ing these maxima with the crossover from disorder to thermally dominated behavior [1, 2, 3, 4, 5] may sug- gest that doping enhances disorder. The SO transition line also shifts upwards, further signifying that disorder is enhanced with doping. However, we will show be- low that this conclusion is inaccurate. Moreover, arguing naively that the balance between characteristic energies constrains the SO line to bisect the FO one at its maxi- mum [1] is also too simplistic. The inset of Fig. 2 clearly shows that in over-doped samples the bisection point re- sides below the FO maximum temperature (the overlap of the two in the OPD sample is apparently accidental). Therefore, a naive description of thermal depinning can- not account for both the SO line and the FO maximum behavior (for instance, the latter is directly affected also by elasticity) [2, 3, 4]. We now present a quantitative scaling analysis of the doping dependence, parameterized by the sample anisotropy ratio ε2 = mab/mc (mi is the electronic effec- tive mass in the ith direction). For high-κ superconduc- tors, such as BSCCO, the Ginzburg-Lanadau (GL) free energy functional can be recast into an isotropic (ε = 1) form by rescaling its parameters [33]. We focus on one such transformation [34] that rescales space, magnetic induction, the penetration depth λ0 and the coherence length ξ0. Models for high-temperature melting are usu- ally independent of ξ0, and find λ0 to enter with some model-dependent power as a proportionality factor. For definiteness we resort to a specific model [6, 35] for a FO evaporation line with no disorder from vortex solid to pancake gas BE(t) ∝ (ε2/λ20d)(1− t2)/t, where t = T/Tc and d is the inter-plane separation. The scaling transfor- mation [34] indeed renders BE(t) isotropic. Figure 3 shows the rescaled FO and SO lines. The high-temperature parts of the FO lines are perfectly col- lapsed by dividing their induction axes by a constant [30]. BE(t) (dashed-dotted line) fits the collapsed melt- ing lines precisely, asserting that the multiplicands in this procedure are (ε◦/ε) 2, normalized by the OPD ε−1 ≈ 500 [36]. Anisotropy scaling collapses the data down to tth ≈ 0.58. Below tth, which appears to be independent of anisotropy, the rescaled FO lines disperse again, and the fit to BE(t) breaks. The flattening of the FO line to- wards an inverse-melting behavior results from quenched disorder, which gains dominance with decreasing tem- peratures [1]. Accordingly, above tth the FO transition is purely thermally-induced, and completely unaffected by disorder [14]. Just below tth disorder becomes a relevant, though not yet a dominant, energy scale. This counterparts the extremely low-temperature be- havior, where thermal energy becomes negligible relative to pinning, resulting in a flat temperature-independent behavior of the FO lines [1]. Indeed the FO lines in Fig. 3 tend to flatten towards their ends, below which 350 Oe - 10 Hz shaking is insufficient for detecting a reversible melting step. We thus conjectured a similar doping-independent threshold temperature td ≈ 0.25, be- low which thermal energy becomes irrelevant. We fitted the low-temperature order-disorder lines by a leading or- der expansion BOD(t & td) ∼ BOD(td)+Λ(t−td)2, which agrees very well with measurement. We can, therefore, estimate BOD(td), at which elasticity is balanced solely by the disordering potential. It increases monotonically with doping (inset of Fig. 3), stating that with reducing anisotropy elasticity gains dominance also over disorder. Yet, the most remarkable outcome of the anisotropy scaling shown in Fig. 3 is the simultaneous collapse of the SO transition lines (with zero freedom). The SO lines re- side in a temperature region where anisotropy scaling of the melting lines fails due to effects of disorder. Still, the same scaling transformation somehow succeeds in render- ing the glass-transition isotropic, even though this tran- sition is believed to be intimately related with the com- petition between disorder and thermal fluctuations. To gain deeper understanding of the low-temperature behavior we fit the measured transition lines to those predicted by a recent calculation [37], which gives access to the doping dependence of the model’s free parameters. It incorporates thermal, disordering and elastic energies to yield bisecting FO and SO lines. The pancake vortex system is modeled by the 2D GL theory. We interpret this single layer model as the outcome of an integration of all other layers out of a complete 3D theory, resulting in an effective 2D model whose renormalized coefficients may still depend on anisotropy ε of a 3D mass tensor. The free energy functional is averaged over gaussian disorder in the coefficients of the quadratic and quar- tic terms using the replica method. Taking the lowest- Landau-level (LLL) approximation yields for the repli- cated partition function Ψ1...Ψn G0(Ψa) + R̃|Ψa|2|Ψb|2 G0(Ψ) = aT |Ψ|2 + κ2(b − h)2 2Gibt where R̃ = a2T bR/32π 2, R is the disorder amplitude in the coefficient of the quadratic LLL term, b = B/H̃c2, aT = (b+t−1)/(2π2b2t2Gi)1/4 is the LLL parameter, Gi is the effective Ginzburg number that generally scales thermal fluctuations, and κ is the GL parameter. Randomness in the coefficient of the quartic LLL term, although crucial for obtaining the non-analyticity of SO, has negligible effect on the locus of the transition lines, and is therefore neglected in the present context. The FO line is calculated by equating the energies of the homogeneous and crystalline states under the influ- ence of disorder in gaussian approximation. This ex- tends beyond the earlier calculation, which treated dis- order perturbatively [38]. The glass line is found from the stability analysis of the replica symmetric solution. The replica symmetry breaking (RSB) is continuous. The corresponding Parisi function describing the hierarchial structure of the glassy state and its detailed derivation can be found in Ref. [37]. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t=T/T 1 1.3 1.6 1.9 ε / ε FIG. 3: (color online) Data collapse of the over-doped high- temperature (t > tth) FO lines (solid symbols) onto the OPD line, which also collapses the SO lines (open symbols). The fit to BE(t) (dashed-dotted) sets the anisotropy ratios ε/ε◦ to 1, 1.20, 1.54 and 1.85 for the OPD (◦), SOD (�), OVD (♦) and HOD (△) samples, respectively. The inset shows the linear dependence of the characteristic BOD(td) on ε/ε◦ (dotted line). The model’s free parameters are R, Gi, T̃c, and H̃c2 = . However, due to a hidden symmetry, given by Gi/H̃2c2 and Gi/R 2, the theory effectively has only three independent fitting parameters. We therefore fixed H̃c2 =100T, for simplicity, consistent with our limited range of hole concentrations [12]. Figure 4 presents two fitting strategies motivated by different physical behav- iors. The first (thick lines) is optimized to fit the col- lapsed parts of the phase diagram - SO lines and FO ones for t > tth. Interestingly, this set simultaneously provides a collapsed FO line also for t < tth. This results from the above symmetry of the model in which Gi and ε take the same role, defining a relation R ∝ Gi along which the calculated phase diagram remains unchanged under anisotropy rescaling. The thick lines in Fig.4 insets show the resulting parameters Gi = 3.91(ε0/ε − 0.37)2 and R = 1.54(ε0/ε − 0.37). The departure of the calculated FO line (thick dashed) from the measured ones below tth may indicate that theory lacks a symmetry-breaking mechanism once anisotropy scaling breaks in experiment. An alternative fit (thin lines), faithful to the com- plete FO lines, dictates the slightly different set of values Gi = 4.77(ε◦/ε − 0.38)2, R = 0.75(ε◦/ε − 0.2) (insets, thin lines). The excellent fit of the FO lines (dashed) suggests that the effective 2D model still captures the es- sential physics involved. Its validity should break down at very low temperatures, where indeed it misses the flat- tening of the FO lines as td is approached, and close to Tc due to critical fluctuations, which may explain the 10% overestimate in the fitted bare T̃c values [39]. Substitut- 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t=T/T 0.5 0.75 1 FIG. 4: (color online) Fits of FO (dashed) and SO (solid) lines of the LLL 2D GL model with H̃c2 =100 T, T̃c =103, 101, 98 and 95 K for the OPD (◦), SOD (�), OVD (♦) and HOD (△) samples, respectively. The values of Gi (a) and R (b) were fixed either by fitting the SO lines and FO ones above tth (thick lines) or by fitting the full FO lines (thin lines). ing this set of values in the calculation of the RSB lines (solid) with zero freedom also produces a good fit that improves with doping, but misses the anisotropy rescal- ing of the measured SO lines. Nevertheless, the discrep- ancy between the two sets of values of the different fitting strategies is small, and vanishes in the high doping limit. In Fig. 4a the reduced dominance of thermal fluctua- tions with doping is clearly captured by the decrease in Gi. Its dependence on anisotropy is closer to the 3D 1/ε2 than to the 2D ε-independent behavior. Remarkably, the disorder amplitude in Fig. 4b also decreases with doping. This contrasts the result of Ref. [40], which found that oxygen addition increases the defects concentration. The apparent contradiction is resolved by noting that the dis- order amplitude is affected also by the anisotropy depen- dence of the effective pinning potential, which decreases with doping as the vortex stacks get stiffer. The over- all decrease of R suggests that the latter mechanism is dominant, not the rising defect concentration. Last, while both Gi and R decrease with doping, Gi has a stronger dependence on anisotropy. This clarifies the true mechanism that shifts the maximum of the FO line and its bisection with the SO line towards higher temperatures with doping (Fig. 2). Although the magni- tude of the disorder amplitude decreases with increased doping, disorder still gains relative dominance over ther- mal fluctuations, which decrease faster with ε. In summary, ε2 scaling accounts for the reduced anisotropy of BSCCO samples with doping down to tth. Below tth, where disorder becomes relevant, ε 2 scaling still collapses the SO lines. From the quantitative agree- ment with the LLL 2D GL model we deduce that disor- der and thermal-fluctuations weaken relative to elasticity with increased doping, which shifts the FO line towards higher fields. Disorder still gains relative dominance over thermal fluctuations, which concurrently shifts the max- imum of the FO line and its bisection with the SO line towards higher temperatures. We thank V.M. Vinokur, and G.P. Mikitik for stim- ulating discussions. This work was supported by the Israel Science Foundation Center of Excellence, by the German-Israeli Foundation G.I.F., by Grant-in-aid for Scientific Research from the Ministry of Education, Cul- ture, Sports, Science, and Technology, Japan, by the Na- tional science foundation of China, and by the MOE ATU Program of R.O.C. NSC952112M009048. BR acknowl- edges the support of the Albert Einstein Minerva Center for Theoretical Physics and EZ the US-Israel Binational Science Foundation (BSF). ∗ Electronic address: haim.beidenkopf@weizmann.ac.il [1] D. Ertas and D.R. Nelson, Physica C 272, 79 (1996). [2] G.P. Mikitik and E.H. Brandt, Phys. Rev. B 68, 054509 (2003). [3] J. Kierfeld and V. Vinokur, Phys. Rev. B 69, 024501 (2004). [4] V. Vinokur et al., Physica C 295, 209 (1998). [5] T. Giamarchi and P. LeDoussal, Phys. Rev. B 55, 6577 (1997). [6] L.I. Glazman and A.E. Koshelev, Phys. Rev. B 43, 2835 (1991). [7] M. Willemin et al., Phys. Rev. Lett. 81, 4236 (1998). [8] N. 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We study the oxygen doping dependence of the equilibrium first-order melting and second-order glass transitions of vortices in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. Doping affects both anisotropy and disorder. Anisotropy scaling is shown to collapse the melting lines only where thermal fluctuations are dominant. Yet, in the region where disorder breaks that scaling, the glass lines are still collapsed. A quantitative fit to melting and replica symmetry breaking lines of a 2D Ginzburg-Landau model further reveals that disorder amplitude weakens with doping, but to a lesser degree than thermal fluctuations, enhancing the relative role of disorder.

Interplay of Anisotropy and Disorder in the Doping-Dependent Melting and Glass Transitions of Vortices in Bi2Sr2CaCu2O8+δ H. Beidenkopf,1, ∗ T. Verdene,1 Y. Myasoedov,1 H. Shtrikman,1 E. Zeldov,1 B. Rosenstein,2 D. Li,3 and T. Tamegai4 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel Electrophysics Department, National Chiao Tung University, Hsinchu 30050, Taiwan, Republic of China Department of Physics, Peking University, Beijing 100871, China Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan (Dated: November 11, 2018) We study the oxygen doping dependence of the equilibrium first-order melting and second-order glass transitions of vortices in Bi2Sr2CaCu2O8+δ. Doping affects both anisotropy and disorder. Anisotropy scaling is shown to collapse the melting lines only where thermal fluctuations are dom- inant. Yet, in the region where disorder breaks that scaling, the glass lines are still collapsed. A quantitative fit to melting and replica symmetry breaking lines of a 2D Ginzburg-Landau model further reveals that disorder amplitude weakens with doping, but to a lesser degree than thermal fluctuations, enhancing the relative role of disorder. PACS numbers: 74.25.Qt, 74.25.Dw, 74.72.Hs, 64.70.Pf Elasticity, thermal energy, disorder, and inter-layer coupling are some of the closely competing energy scales in the intricate H − T phase diagram of the vortex matter in the layered high temperature superconductor Bi2Sr2CaCu2O8+δ (BSCCO) [1, 2, 3, 4, 5, 6]. The low- temperature part of the equilibrium phase diagram of BSCCO was recently made accessible to experiment by vortex shaking [7, 8, 9]. It unveiled a first-order (FO) in- verse melting line, which continues the thermal melting line from high temperatures [8] separating low-field or- dered phases from high-field amorphous ones. A second- order (SO) transition line, at which low-temperature glassy phases get thermally depinned, was subsequently reported [9]. In this letter we study the oxygen doping dependence of these transition lines. We show that the SO line scales with material anisotropy even where the FO line does not, and that effective disorder weakens with doping, but gains relative dominance over thermal fluctuations. We present measurements of optimally doped (OPD), slightly over doped (SOD), over doped (OVD) and highly over doped (HOD) BSCCO crystals [10, 11] with critical temperatures Tc =92, 90, 88.5 and 86 K, respectively, corresponding to hole concentrations of 0.171, 0.180, 0.184 and 0.190 [12]. Various crystal geometries were studied with typical sizes of ∼300×300×40 µm3. They were mounted on 10 × 10 µm2 Hall sensor arrays, fabri- cated in a GaAs/AlGaAs heterostructure. At low tem- peratures we utilized a 350 Oe in-plane ac shaking field of 10 Hz to relax the pancake vortices towards their equi- librium configuration [7, 8, 9]. Conjugating local probes with shaking yielded the equilibrium reversible magneti- zation of the samples. Figure 1 shows the local induction, B(H,T ), measured in the SOD sample by sweeping the temperature, T , at a constant out-of-plane field, H , in presence of an in-plane shaking field. A linear term α T + β was subtracted FIG. 1: (color online) The local induction shows FO (dashed- dotted) and SO (dashed) transition lines in the H − T phase diagram. The induction in the high-temperature depinned liquid and solid phases was artificially flattened by a linear subtraction α T + β (hence the constant color). The low- temperature BrG and glassy phases have thus a finite positive slope. (a) Values of α and β used. (b) FO-SO-FO tran- sition sequence at a 380 Oe temperature sweep on the back- ground of the colorbar. (c) Discontinuous steps in dB/dT , manifesting the SO nature of the transition. from each temperature sweep (Fig. 1a) to flatten the orig- inally increasing (α > 0) local induction. As a result, the FO and SO singular behavior can be readily traced throughout the H − T phase diagram. The FOmelting transition is manifested by a discontin- uous step in the magnetization along the dashed-dotted line in Fig. 1. It separates the high-field amorphous glass and liquid phases from the low-field quasi-long-range- ordered Bragg glass (BrG) [5, 13]. The melting line be- comes nonmonotonic at a certain temperature [8, 9]. Its http://arxiv.org/abs/0704.1902v1 0.4 0.5 0.6 0.7 0.8 0.9 1 t=T/T 0.4 0.45 0.5 t=T/T bisection FO−SO maximum FIG. 2: (color online) FO and SO transition lines (solid and open symbols, respectively) measured with OPD (◦), SOD(�), OVD (♦) and HOD (△) samples. The inset shows the doping dependence of the maximum of the FO line and its bisection with the SO line. Dotted and dashed lines are second-degree polynomial fits. inverse-melting part is believed to be an order-disorder transition, induced mainly by quenched disorder and not by thermal fluctuations [1, 2, 3, 4, 5, 14, 15, 16]. The SO phase transition is manifested by a break in the slope of the magnetization at Tg, marked in Fig. 1 by the dashed line. It separates the fully flattened high- temperature magnetization (dB/dT |T>Tg − α ≈ 0) of constant color from a low-temperature region of nonzero slope (dB/dT |T<Tg − α > 0), whose color varies with temperature. The nonanaliticity at Tg is demonstrated in Fig. 1c, which shows ∼40 mG/K steps in dB/dT for temperature sweeps lying below as well as above the FO line. The location of the SO line above melting resembles earlier dynamic irreversibility measurements [17, 18, 19, 20, 21] and theoretical models of a glass transition [1, 2, 3, 4, 5, 22, 23, 24]. Few dynamic mea- surements found a similar line of thermal depinning be- low melting [9, 25, 26, 27, 28, 29, 30], which separates the BrG phase below it from a thermally depinned solid, whose detailed characteristics are not yet certain. We mapped three more samples of various doping lev- els that yielded the B−T phase diagram plotted in Fig. 2. The melting line shifts as a whole to higher fields with doping due to the decrease in anisotropy, which implies stronger inter-layer coupling and higher stiffness of the pancake vortex stacks [10, 30, 31, 32]. As the FO line shifts to higher fields, its maximum shifts towards in- creasingly higher temperatures (Fig. 2 inset). Identify- ing these maxima with the crossover from disorder to thermally dominated behavior [1, 2, 3, 4, 5] may sug- gest that doping enhances disorder. The SO transition line also shifts upwards, further signifying that disorder is enhanced with doping. However, we will show be- low that this conclusion is inaccurate. Moreover, arguing naively that the balance between characteristic energies constrains the SO line to bisect the FO one at its maxi- mum [1] is also too simplistic. The inset of Fig. 2 clearly shows that in over-doped samples the bisection point re- sides below the FO maximum temperature (the overlap of the two in the OPD sample is apparently accidental). Therefore, a naive description of thermal depinning can- not account for both the SO line and the FO maximum behavior (for instance, the latter is directly affected also by elasticity) [2, 3, 4]. We now present a quantitative scaling analysis of the doping dependence, parameterized by the sample anisotropy ratio ε2 = mab/mc (mi is the electronic effec- tive mass in the ith direction). For high-κ superconduc- tors, such as BSCCO, the Ginzburg-Lanadau (GL) free energy functional can be recast into an isotropic (ε = 1) form by rescaling its parameters [33]. We focus on one such transformation [34] that rescales space, magnetic induction, the penetration depth λ0 and the coherence length ξ0. Models for high-temperature melting are usu- ally independent of ξ0, and find λ0 to enter with some model-dependent power as a proportionality factor. For definiteness we resort to a specific model [6, 35] for a FO evaporation line with no disorder from vortex solid to pancake gas BE(t) ∝ (ε2/λ20d)(1− t2)/t, where t = T/Tc and d is the inter-plane separation. The scaling transfor- mation [34] indeed renders BE(t) isotropic. Figure 3 shows the rescaled FO and SO lines. The high-temperature parts of the FO lines are perfectly col- lapsed by dividing their induction axes by a constant [30]. BE(t) (dashed-dotted line) fits the collapsed melt- ing lines precisely, asserting that the multiplicands in this procedure are (ε◦/ε) 2, normalized by the OPD ε−1 ≈ 500 [36]. Anisotropy scaling collapses the data down to tth ≈ 0.58. Below tth, which appears to be independent of anisotropy, the rescaled FO lines disperse again, and the fit to BE(t) breaks. The flattening of the FO line to- wards an inverse-melting behavior results from quenched disorder, which gains dominance with decreasing tem- peratures [1]. Accordingly, above tth the FO transition is purely thermally-induced, and completely unaffected by disorder [14]. Just below tth disorder becomes a relevant, though not yet a dominant, energy scale. This counterparts the extremely low-temperature be- havior, where thermal energy becomes negligible relative to pinning, resulting in a flat temperature-independent behavior of the FO lines [1]. Indeed the FO lines in Fig. 3 tend to flatten towards their ends, below which 350 Oe - 10 Hz shaking is insufficient for detecting a reversible melting step. We thus conjectured a similar doping-independent threshold temperature td ≈ 0.25, be- low which thermal energy becomes irrelevant. We fitted the low-temperature order-disorder lines by a leading or- der expansion BOD(t & td) ∼ BOD(td)+Λ(t−td)2, which agrees very well with measurement. We can, therefore, estimate BOD(td), at which elasticity is balanced solely by the disordering potential. It increases monotonically with doping (inset of Fig. 3), stating that with reducing anisotropy elasticity gains dominance also over disorder. Yet, the most remarkable outcome of the anisotropy scaling shown in Fig. 3 is the simultaneous collapse of the SO transition lines (with zero freedom). The SO lines re- side in a temperature region where anisotropy scaling of the melting lines fails due to effects of disorder. Still, the same scaling transformation somehow succeeds in render- ing the glass-transition isotropic, even though this tran- sition is believed to be intimately related with the com- petition between disorder and thermal fluctuations. To gain deeper understanding of the low-temperature behavior we fit the measured transition lines to those predicted by a recent calculation [37], which gives access to the doping dependence of the model’s free parameters. It incorporates thermal, disordering and elastic energies to yield bisecting FO and SO lines. The pancake vortex system is modeled by the 2D GL theory. We interpret this single layer model as the outcome of an integration of all other layers out of a complete 3D theory, resulting in an effective 2D model whose renormalized coefficients may still depend on anisotropy ε of a 3D mass tensor. The free energy functional is averaged over gaussian disorder in the coefficients of the quadratic and quar- tic terms using the replica method. Taking the lowest- Landau-level (LLL) approximation yields for the repli- cated partition function Ψ1...Ψn G0(Ψa) + R̃|Ψa|2|Ψb|2 G0(Ψ) = aT |Ψ|2 + κ2(b − h)2 2Gibt where R̃ = a2T bR/32π 2, R is the disorder amplitude in the coefficient of the quadratic LLL term, b = B/H̃c2, aT = (b+t−1)/(2π2b2t2Gi)1/4 is the LLL parameter, Gi is the effective Ginzburg number that generally scales thermal fluctuations, and κ is the GL parameter. Randomness in the coefficient of the quartic LLL term, although crucial for obtaining the non-analyticity of SO, has negligible effect on the locus of the transition lines, and is therefore neglected in the present context. The FO line is calculated by equating the energies of the homogeneous and crystalline states under the influ- ence of disorder in gaussian approximation. This ex- tends beyond the earlier calculation, which treated dis- order perturbatively [38]. The glass line is found from the stability analysis of the replica symmetric solution. The replica symmetry breaking (RSB) is continuous. The corresponding Parisi function describing the hierarchial structure of the glassy state and its detailed derivation can be found in Ref. [37]. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t=T/T 1 1.3 1.6 1.9 ε / ε FIG. 3: (color online) Data collapse of the over-doped high- temperature (t > tth) FO lines (solid symbols) onto the OPD line, which also collapses the SO lines (open symbols). The fit to BE(t) (dashed-dotted) sets the anisotropy ratios ε/ε◦ to 1, 1.20, 1.54 and 1.85 for the OPD (◦), SOD (�), OVD (♦) and HOD (△) samples, respectively. The inset shows the linear dependence of the characteristic BOD(td) on ε/ε◦ (dotted line). The model’s free parameters are R, Gi, T̃c, and H̃c2 = . However, due to a hidden symmetry, given by Gi/H̃2c2 and Gi/R 2, the theory effectively has only three independent fitting parameters. We therefore fixed H̃c2 =100T, for simplicity, consistent with our limited range of hole concentrations [12]. Figure 4 presents two fitting strategies motivated by different physical behav- iors. The first (thick lines) is optimized to fit the col- lapsed parts of the phase diagram - SO lines and FO ones for t > tth. Interestingly, this set simultaneously provides a collapsed FO line also for t < tth. This results from the above symmetry of the model in which Gi and ε take the same role, defining a relation R ∝ Gi along which the calculated phase diagram remains unchanged under anisotropy rescaling. The thick lines in Fig.4 insets show the resulting parameters Gi = 3.91(ε0/ε − 0.37)2 and R = 1.54(ε0/ε − 0.37). The departure of the calculated FO line (thick dashed) from the measured ones below tth may indicate that theory lacks a symmetry-breaking mechanism once anisotropy scaling breaks in experiment. An alternative fit (thin lines), faithful to the com- plete FO lines, dictates the slightly different set of values Gi = 4.77(ε◦/ε − 0.38)2, R = 0.75(ε◦/ε − 0.2) (insets, thin lines). The excellent fit of the FO lines (dashed) suggests that the effective 2D model still captures the es- sential physics involved. Its validity should break down at very low temperatures, where indeed it misses the flat- tening of the FO lines as td is approached, and close to Tc due to critical fluctuations, which may explain the 10% overestimate in the fitted bare T̃c values [39]. Substitut- 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t=T/T 0.5 0.75 1 FIG. 4: (color online) Fits of FO (dashed) and SO (solid) lines of the LLL 2D GL model with H̃c2 =100 T, T̃c =103, 101, 98 and 95 K for the OPD (◦), SOD (�), OVD (♦) and HOD (△) samples, respectively. The values of Gi (a) and R (b) were fixed either by fitting the SO lines and FO ones above tth (thick lines) or by fitting the full FO lines (thin lines). ing this set of values in the calculation of the RSB lines (solid) with zero freedom also produces a good fit that improves with doping, but misses the anisotropy rescal- ing of the measured SO lines. Nevertheless, the discrep- ancy between the two sets of values of the different fitting strategies is small, and vanishes in the high doping limit. In Fig. 4a the reduced dominance of thermal fluctua- tions with doping is clearly captured by the decrease in Gi. Its dependence on anisotropy is closer to the 3D 1/ε2 than to the 2D ε-independent behavior. Remarkably, the disorder amplitude in Fig. 4b also decreases with doping. This contrasts the result of Ref. [40], which found that oxygen addition increases the defects concentration. The apparent contradiction is resolved by noting that the dis- order amplitude is affected also by the anisotropy depen- dence of the effective pinning potential, which decreases with doping as the vortex stacks get stiffer. The over- all decrease of R suggests that the latter mechanism is dominant, not the rising defect concentration. Last, while both Gi and R decrease with doping, Gi has a stronger dependence on anisotropy. This clarifies the true mechanism that shifts the maximum of the FO line and its bisection with the SO line towards higher temperatures with doping (Fig. 2). Although the magni- tude of the disorder amplitude decreases with increased doping, disorder still gains relative dominance over ther- mal fluctuations, which decrease faster with ε. In summary, ε2 scaling accounts for the reduced anisotropy of BSCCO samples with doping down to tth. Below tth, where disorder becomes relevant, ε 2 scaling still collapses the SO lines. From the quantitative agree- ment with the LLL 2D GL model we deduce that disor- der and thermal-fluctuations weaken relative to elasticity with increased doping, which shifts the FO line towards higher fields. Disorder still gains relative dominance over thermal fluctuations, which concurrently shifts the max- imum of the FO line and its bisection with the SO line towards higher temperatures. We thank V.M. Vinokur, and G.P. Mikitik for stim- ulating discussions. This work was supported by the Israel Science Foundation Center of Excellence, by the German-Israeli Foundation G.I.F., by Grant-in-aid for Scientific Research from the Ministry of Education, Cul- ture, Sports, Science, and Technology, Japan, by the Na- tional science foundation of China, and by the MOE ATU Program of R.O.C. NSC952112M009048. BR acknowl- edges the support of the Albert Einstein Minerva Center for Theoretical Physics and EZ the US-Israel Binational Science Foundation (BSF). ∗ Electronic address: haim.beidenkopf@weizmann.ac.il [1] D. Ertas and D.R. Nelson, Physica C 272, 79 (1996). [2] G.P. Mikitik and E.H. Brandt, Phys. Rev. B 68, 054509 (2003). [3] J. Kierfeld and V. Vinokur, Phys. Rev. B 69, 024501 (2004). [4] V. Vinokur et al., Physica C 295, 209 (1998). [5] T. Giamarchi and P. LeDoussal, Phys. Rev. B 55, 6577 (1997). [6] L.I. Glazman and A.E. Koshelev, Phys. Rev. B 43, 2835 (1991). [7] M. Willemin et al., Phys. Rev. Lett. 81, 4236 (1998). [8] N. 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Microsoft Word - NodaSuper2pdf.doc Growth window and possible mechanism of millimeter-thick single-walled carbon nanotube forests Kei Hasegawa1, Suguru Noda1,* , Hisashi Sugime1, Kazunori Kakehi1, Shigeo Maruyama2 and Yukio Yamaguchi1 1 Department of Chemical System Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2 Department of Mechanical Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Corresponding author. E-mail address: noda@chemsys.t.u-tokyo.ac.jp Our group recently reproduced the water-assisted growth method, so-called "super growth", of millimeter-thick single-walled carbon nanotube (SWNT) forests by using C2H4/ H2/ H2O/ Ar reactant gas and Fe/ Al2O3 catalyst. In this current work, a parametric study was carried out on both reaction and catalyst conditions. Results revealed that a thin Fe catalyst layer (about 0.5 nm) yielded rapid growth of SWNTs only when supported on Al2O3, and that Al2O3 support enhanced the activity of Fe, Co, and Ni catalysts. The growth window for the rapid SWNT growth was narrow, however. Optimum amount of added H2O increased the SWNT growth rate but further addition of H2O degraded both the SWNT growth rate and quality. Addition of H2 was also essential for rapid SWNT growth, but again, further addition decreased both the SWNT growth rate and quality. Because Al2O3 catalyzes hydrocarbon reforming, Al2O3 support possibly enhances the SWNT growth rate by supplying the carbon source to the catalyst nanoparticles. The origin of the narrow window for rapid SWNT growth will also be discussed. Keywords: Single-Walled Carbon Nanotubes, Vertically Aligned Nanotubes, Combinatorial Method, Growth Mechanism, Growth Window 1. INTRODUCTION Single-walled carbon nanotubes (SWNTs) have unique mechanical and electrical properties, and many applications for them have been proposed and researched. To realize practical applications, mass production of SWNTs is essential, and various catalytic chemical vapor deposition (CCVD) methods have been developed to achieve this mass production. There are two types of CCVD; one involving nanoparticle catalysts suspended in the gas phase and the other involving nanoparticle catalysts supported on substrates. A gas-phase production process, the so-called "HiPco process", is the first process to be used in the mass production of SWNTs.1 Recently, remarkable progress has been made in CCVD processes using supported catalysts. Submicrometer-thick films of randomly-oriented SWNTs have been the typical product when supported catalysts are used. In 2003, vertically aligned single-walled carbon nanotubes (VA-SWNTs) were realized2 by using alcohol catalytic CCVD (ACCVD).3 VA-SWNTs have now been achieved using several CVD methods and conditions.4-7 As a result, CCVD from substrates now has potential as a process in the mass production of SWNTs. Among those growth methods, the water-assisted method, so-called "super growth",4 realized an outstanding growth rate of a few micrometers per second, leading to millimeter-thick VA-SWNTs forests. However, many research groups have failed in reproducing "super growth", and the underlying mechanism of the growth rate enhancement by water remained unclear. By using our combinatorial method for catalyst optimization,8,9 we recently reproduced the "super growth" method and showed the important role of catalyst supports.10 In this current work, by doing a parametric study, we report in detail the effect of the catalyst and reaction conditions determined, and discuss the novel mechanism essential for rapid growth VA-SWNTs. 2. EXPERIMENTAL Catalysts were prepared on SiO2 substrates by sputter-deposition. An Al2O3 layer was formed by depositing 15-nm-thick Al on a substrate and then exposing the layer to ambient air. Fe was deposited on Al2O3 layers or directly on SiO2 substrates. For a separate experiment, gradient-thickness profiles of Fe were formed by using combinatorial masked deposition (CMD) method previously described.9 The catalysts were set in a tubular CVD reactor (22 mm in diameter and 300 mm in length), heated to a target temperature (typically 1093 K), and kept at that temperature for 10 min while being exposed to 27 kPa H2/75 kPa Ar at a flow rate of 500 sccm, to which H2O vapor was added at the same partial pressure as for the CVD condition (i.e., 0 to 0.03 kPa). During this heat treatment, Fe forms nanoparticles of a certain diameter and areal density depending on the initial Fe thickness.8 After the heat treatment, CVD was carried out by switching the gas to C2H4/ H2/ H2O / Ar. The standard condition was 8.0 kPa C2H4/ 27 kPa H2/ 0.01 kPa H2O/ 67 kPa Ar at 500 sccm at 1093 K for 10 min. The samples after CCVD were analyzed by using transmission electron microscopy (TEM) (JEOL JEM-2000EX), field emission scanning electron microscopy (FE-SEM) (Hitachi S-900), and micro-Raman scattering spectroscopy (Seki Technotron, STR-250) with an excitation wavelength at 488 nm. 3. RESULTS AND DISCUSSION 3.1. Standard condition of "super growth" Figure 1a shows a photograph of CNT samples grown by a combinatorial catalyst library under the standard condition. Millimeter-thick vertically aligned CNTs (VA-CNTs) were grown at regions where the Fe thickness was 0.4 nm or more. The maximum thickness of VA-CNTs was 1.2 mm at a Fe thickness of 0.5 nm. The thickness of VA-CNTs decreased when Fe thickness exceeded 0.5 nm. Figure 1b shows TEM images of CNTs grown under the same condition as Fig. 1a on substrates with uniform Fe thicknesses of 0.5 and 1.0 nm. SWNTs with a diameter around 4 nm mainly grew for 0.5-nm-thick Fe catalyst, whereas thicker CNTs grew for 1.0-nm-thick Fe catalyst. This difference in CNTs is because a thicker initial Fe layer yields larger Fe nanoparticles,8 indicating a narrow VA-SWNTs growth window for the initial Fe thickness. Figure 1c shows Raman spectra of the same CNT sample as Fig. 1a taken at Fe thicknesses of 0.5, 0.8, and 1.0 nm. Sharp, branched G-bands with a small D-band and peaks of radial breathing mode (RBM) were detected, indicating the existence of SWNTs. The G/D peak area ratios were smaller for thicker Fe layers (≥ 1 nm), because multi-walled CNTs (MWNTs) became the main product when a thicker Fe layer was used as catalyst. These figures show that "super growth" was achieved in this work. The growth temperature of the standard condition of this work is higher than that of the original "super growth" 4 because both the CNT thickness and the G/D ratio increased with increasing growth temperature. 3.2. Effects of catalyst metals and supports Effects of catalyst metals and their supports were investigated next. Figure 2a shows top-view photographs of CNT samples grown on the Fe/Al2O3, Co/Al2O3, and Ni/Al2O3 combinatorial catalyst libraries under the standard condition. The 0.5 0.8 1.0 Fe thickness / nm 2 mm(a) 0.5 0.8 1.0 Fe thickness / nm 2 mm(a) RBM ×10 Fe 0.5 nm Fe 0.8 nm Fe 1.0 nm 1200 1400 1600 Raman shift / cm-1 100 200 300 RBM ×10 Fe 0.5 nm Fe 0.8 nm Fe 1.0 nm 1200 1400 1600 Raman shift / cm-1 100 200 300 20 nm 20 nm20 nm Fig. 1 CNTs grown on the Fe/Al2O3 catalyst library under the standard condition (8.0 kPa C2H4/ 27 kPa H2/ 0.010 kPa H2O/ 67 kPa Ar at 1093 K for 10 min) . (a) Side view photograph of CNTs grown on the combinatorial catalyst library. (b) Raman spectra of the same sample at Fe thickness of 0.5, 0.8 and 1.0 nm. Intensity at the low wavenumber region (< 300 cm-1) was 10x magnified. (c) TEM images of CNTs grown on substrates with uniform Fe thickness of 0.5 nm (left) and 1.0 (right) nm. surfaces of both Fe/Al2O3 and Co/Al2O3 libraries became black at regions where the Fe thickness was 0.4 nm or more. On the other hand, the surface of Ni/Al2O3 was somewhat darkened only at the relatively thin Fe region around 0.5 nm. Figure 2b shows cross-sectional SEM images of these samples. About 200-μm-thick VA-CNTs grew for 0.5-nm-thick Co and about 0.4-μm-thick VA-CNTs grew for 0.5-nm-thick Ni.. Figure 2c shows top-view photographs of combinatorial catalyst libraries of Fe/SiO2, Co/SiO2, and Ni/SiO2 after CVD under the standard condition. The results were completely different from those libraries with Al2O3 support layer (i.e., Fe/Al2O3, Co/Al2O3, and Ni/Al2O3); VA-CNTs did not form on any library and only part of the surface of Fe/SiO2 for the thickness range from 0.4 to 0.6 nm darkened slightly compared with the library before CVD. As for Co/SiO2 and Ni/SiO2, negligible change appeared in color. Figure 2d shows the Raman spectra for the Fe/ SiO2 sample in Fig. 2b at Fe thickness of 0.5 nm, indicating growth of SWNTs. Fe catalyst supported on SiO2 can actually grow CNTs, including SWNTs, although the CNT yield is much smaller (submicrometer thickness or less) than that grown using catalysts supported on Al2O3 (up to millimeter thickness), regardless of the thickness of the catalyst. Next, we examined if an Al2O3 catalyst support layer actually acts as Al2O3 rather than metallic Al. Another catalyst library was prepared by first depositing 15-nm-thick Al on SiO2, then exposing the layer to ambient air, then oxidizing the layer under air at 973 K for 5 min, and finally depositing a gradient thickness profile of Fe on the layer at ambient temperature. Figure 3a shows photographs of the Fe/ Al2O3 catalyst libraries before CVD. In the thick Fe region (≥ 2 nm), both libraries were metallic silver in color. In the thinner region (< 2 nm), the Al layer only exposed to air was gray metallic in color, indicating the existence of metallic Al, whereas the Al layer oxidized at 973 K was completely transparent, indicating the complete oxidation of the Al layer. Then, CVD was carried out on these libraries. Figure 3b shows the photograph after CVD of the library with the Al layer oxidized at high temperature. The growth of the VA-CNT films was similar to that without high-temperature oxidation of the Al layer (Fig. 1a), indicating that the surface of the Al layer was oxidized by just being exposed to air, and that the layer acted as an Al2O3 support. Fe nanoparticles can grow SWNTs on either SiO2 or Al2O3 supports, but they need to be 0.5 0.8 1.0 Metal thickness / nm 0.5 0.8 1.0 Metal thickness / nm 0.5 0.8 1.0 Metal thickness / nm 0.5 0.8 1.0 Metal thickness / nm RBM ×10 1200 1400 1600 Raman shift / cm-1 100 200 300 RBM ×10 1200 1400 1600 Raman shift / cm-1 100 200 300 Fig. 2 Effects of catalyst metals and supports on CNT growth under the standard condition. (a,c) Top-view photographs of CNTs samples grown by combinatorial catalyst libraries prepared on Al2O3 (a) and SiO2 (c). All of the catalyst metals (i.e. Fe, Co, Ni) had the same thickness profiles between 0.2 nm (left) and 3 nm (right). (b) Cross-sectional FE-SEM images of the CNTs samples grown by 0.5-nm-thick Co and Ni catalysts prepared on Al2O3. (d) Raman spectrum of the CNTs samples grown by 0.5-nm-thick Fe catalyst prepared on SiO2 supports. Intensity at the low wavenumber region (<300 cm-1) was 10x magnified. supported on Al2O3 to achieve rapid growth of VA-CNTs at a rate of micrometers per second. In addition, Al2O3 support enhances the CNT growth by other catalysts (i.e., Co and Ni). Al2O3 support layer should have an essential role in growing CNTs. 3.3. Effects of H2O and H2 Figure 4a shows the thickness profiles of CNTs grown on Fe/Al2O3 catalyst libraries under the standard condition, except that the amount of added H2O (i.e. partial pressure) was varied. Without H2O, CNTs grew only at the relatively thin Fe region (0.3- 1.0 nm) and the maximum thickness of VA-CNTs was 0.7 mm at the 0.5 nm-thick Fe region. Addition of 0.010 kPa H2O enhanced the growth, especially at the thick Fe region (> 0.7 nm), and the maximum VA-CNT thickness increased to 1.0 mm at the 0.5-nm-thick Fe region. Further H2O addition (0.030 kPa) inhibited CNT growth at the thin Fe region (0.3- 0.6 nm) where SWNTs grew at the lower H2O partial pressures (≤ 0.010 kPa). Figure 4b shows the G/D ratios for each CVD condition at Fe thickness of 0.5, 0.8, and 1.0 nm. Slight addition of H2O (0.010 kPa) did not affect the G/D ratio at the thin Fe region (0.5 nm), but decreased the G/D ratio at the thicker regions (0.8 and 1.0 nm). Further addition of H2O (0.030 kPa) significantly decreased the G/D ratio at all regions. These results show that proper amount of H2O enhances the growth rate of CNTs, but excessive H2O decreases the growth rate of CNTs from small nanoparticles (i.e. small Fe thickness) and degrades the quality of CNTs possibly by oxidation. In conclusion, excess H2O totally inhibits "super growth" of SWNTs. Figure 5a shows thickness profiles of CNTs grown under the standard condition except that the amount of added H2 (i.e. partial pressure) was varied. When a lower amount of H2 was added (2.7 kPa), VA-CNTs became much thinner at Fe thickness of 0.7 nm or less. When a large amount of H2 was added (54 kPa), CNTs grew at any Fe thickness at a reduced VA-CNT thickness (around 0.4 mm). Figure 5b shows the G/D ratios for each CVD condition at Fe thickness of 0.5, 0.8, and 1.0 0.5 0.8 1.0 Metal thickness / nm 0.5 0.8 1.0 Metal thickness / nm 0.5 0.8 1.0 Fe thickness / nm 2 mm(b) 0.5 0.8 1.0 Fe thickness / nm 2 mm(b) Fig. 3 Effect of the preparation process of Al2O3 layer. (a) Photographs of Fe/Al2O3 catalyst libraries before CVD. Al2O3 layer was formed by oxidizing 15-nm-thick Al layer only by exposure to ambient air (top) and then by oxidizing in air at 973 K (bottom). (b) Photograph of CNT grown on the Fe/Al2O3 library with Al2O3 layer oxidized at 973 K (same library as Fig. 3a, bottom). 0.2 0.4 0.6 0.8 1T Fe thickness / nm 0.4 0.6 0.8 1.0 Fe thickness / nm 0 kPa 0.010 kPa 0.030 kPa 0.4 0.6 0.8 1.0 Fe thickness / nm 0 kPa 0.010 kPa 0.030 kPa Fig. 4 Effect of H2O on CNT growth. CVD was carried out by using Fe/Al2O3 combinatorial catalyst libraries under the standard condition except the amount of added H2O (partial pressure) was varied. (a) Relationship between thickness of VA-CNTs and Fe thickness at different H2O pressures. (b) Relationship between G/D ratio of Raman spectra and Fe thickness at different H2O pressures. nm. The G/D ratios decreased in either case of lower (2.7 kPa) and higher (54 kPa) H2 was added. These results indicate that an optimum amount of H2 is needed for rapid growth of VA-SWNTs of relatively good quality. 3.4. Possible mechanism of rapid growth of VA-SWNTs Based on the results discussed above, we propose three necessary conditions for "super growth" of SWNTs. The first condition is that the Fe catalyst needs to be thin enough, about 0.5 nm, so that the catalyst nanoparticles are small enough to grow SWNTs with small diameters. The second condition is that the catalyst nanoparticles need to be supported on Al2O3. The third is that the partial pressures of both H2O and H2 need to be carefully adjusted; these gases are essential but excessive amounts degrade the growth rate and/or the quality of SWNTs. Al2O3 and its related materials have long been used as catalysts for decomposition and dehydrogenation of hydrocarbons.11,12 Figure 6 shows a schematic of our proposed mechanism explaining the enhancement effect of Al2O3 on CNT growth. In this mechanism, C2H4 or its derivatives adsorb on Al2O3, diffuse over Al2O3 surface to catalyst nanoparticles, and then are incorporated in Fe nanoparticles. In contrast, when Fe is deposited directly on SiO2, C2H4 can be decomposed only on Fe nanoparticles whose surface is largely covered with growing CNTs. As a result, the growth rate of CNTs on Al2O3 is much faster than that on SiO2. In addition, the catalytic activity of Al2O3 strongly depends on its crystalline structure.11 α-Al2O3, the most stable phase, has low activity,11 and thus, rapid CNT growth does not occur when catalysts are supported on sapphire (i.e. monocrystalline α-Al2O3). Next, we discuss why the window of "super growth" for SWNTs is narrow. One reason might be due to the catalyst deactivation mechanism, i.e. coking of either Al2O3 surface or Fe nanoparticles. In the dehydrogenation process of hydrocarbons,11 Al2O3 easily loses its catalytic activity due to carbon deposition, and thus, H2O vapor has long been used to remove carbon byproducts. H2 also keeps the Al2O3 surface reactive by controlling the balance between dehydrogenation and hydrogenation of carbon surface species, as is known in the hydrocracking process.13 In contrast, excessive H2 inhibits the growth of SWNTs by hydrogenating C2H4-derivatives adsorbed on Al2O3. Concerning Fe nanoparticles, when the incoming flux of carbon into Fe nanoparticles increases, carbon in Fe nanoparticles will be highly supersaturated, resulting not only in increased SWNT growth rate but also in graphite formation on the surface of the nanoparticles. The fewer walls and larger free energy of SWNTs than MWNTs may 0.2 0.4 0.6 0.8 1 Fe thickness / nm 27 kPa 2.7 kPa 54 kPa 0.2 0.4 0.6 0.8 1 Fe thickness / nm 27 kPa 2.7 kPa 54 kPa 27 kPa 2.7 kPa 54 kPa 0.4 0.6 0.8 1 Fe Thickness / nm 27 kPa 2.7 kPa 54 kPa 0.4 0.6 0.8 1 Fe Thickness / nm Fig. 5 Effect of H2 on CNT growth. CVD was carried out by using Fe/Al2O3 combinatorial catalyst libraries under the standard condition except the amount of added H2 (partial pressure) was varied. (a) Relationship between thickness of VA-CNTs and Fe thickness at different H2 pressures. (b) Relationship between G/D ratio of Raman spectra and Fe thickness at different H2 pressures. SiO2 Al2O3 C2H4 C2H4 SiO2 Al2O3 C2H4 C2H4 Fig. 6 Schematic of enhancement mechanism of SWNT catalytic growth by Al2O3 support. make the degree of supersaturation larger for nanoparticles growing SWNTs than those growing MWNTs. This may be the reason why the "super growth" window for SWNTs is such narrow compared with MWNTs. (Note that MWNTs grow rapidly under a wide window of reaction conditions if catalyst nanoparticles are supported on Al2O3). In conclusive, two conditions are needed to sustain the rapid growth of SWNTs: first, the partial pressures of C2H4, H2, and H2O need to be balanced to suppress coking of Al2O3, and second, the incoming flux of carbon into Fe nanoparticles must not be too large, that means nanotubes must not grow too rapid, to prevent carbonization of Fe nanoparticles. 4. CONCLUSION Rapid growth of VA-SWNTs, namely, a few micrometers per second, or so-called "super growth", was reproduced in this study, and the growth window was clarified. The standard condition of this work was 8.0 kPa C2H4/ 27 kPa H2/ 0.01 kPa H2O/ 67 kPa Ar at 500 sccm at 1093 K for 10 min using a tubular CVD reactor (22 mm in diameter and 300 mm in length). Results showed that for the rapid growth of SWNTs, small nanoparticles formed from a thin Fe layer (about 0.5 nm) need to be supported on Al2O3 and that only the optimal amounts of H2O and H2 should be used. We proposed a novel mechanism by which Al2O3 enhances the growth rate of SWNTs, and offered a simple explanation of the effect of H2O and H2 on the growth rate. Namely, Al2O3 supports supply a carbon source to Fe nanoparticles, and H2O and H2 prevent catalyst deactivation by keeping the Al2O3 surface reactive and by balancing the carbon fluxes among the gas-phase, Al2O3 support, Fe nanoparticles, and growing SWNTs. Acknowledgements: The authors thank Z. Zhang for her help in the Raman measurements. This work is financially supported in part by the Grant-in-Aid for Young Scientists (A), 18686062, 2006, from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. References and Notes: 1. P. Nikolaev, M. J. Bronikowski, R. K. Bradley, F. Rohmund, D. T. Colbert, K. A. Smith, and R. E. Smalley, Chem. Phys. Lett. 313, 91 (1999). 2. S. Maruyama, R. Kojima, Y. Miyauchi, S. Chiashi, and M. Kohno, Chem. Phys. Lett. 360, 229 (2002). 3. Y. Murakami, S. Chiashi, Y. Miyauchi, M. Hu, M. Ogura, T. Okubo, and S. Maruyama, Chem. Phys. Lett. 385, 298 (2004). 4. K. Hata, D. N. Futaba, K. Mizuno, T. Nanami, M. Yumura, and S. Iijima, Science 306, 1362 (2004). 5. G. Zhong, T. Iwasaki, K. Honda, Y. Furukawa, I. Ohdomari, and H. Kawarada, Jpn. J. Appl. Phys. 44, 1558 (2004). 6. L. Zhang, Y. Tan, and D. E. Resasco, Chem. Phys. Lett. 422, 198 (2006). 7. G. Zhang, D. Mann, L. Zhang, A. Jabey, Y. Li, E. Yenilmez, Q. Wang, J. P. McVittie, Y. Nishi, J. Gibbons, and H. Dai, Proc. Nat. Acad. Sci. 102, 16141 (2005). 8. S. Noda, Y. Tsuji, Y. Murakami, and S. Maruyama, Appl. Phys. Lett. 86, 173106 (2005). 9. S. Noda, H. Sugime, T. Osawa, Y. Tsuji, S. Chiashi, Y. Murakami, and S. Maruyama, Carbon 44, 1414 (2006). 10. S. Noda, K. Hasegawa, H. Sugime, K. Kakehi, Z. Zhang, S. Maruyama, and Y. Yamaguchi, Jpn. J. Appl. Phys., in press. 11. D. S. MacIver, W. H. Wilmot, and J. M. Bridges, J. Catal. 3, 502 (1964). 12. F. J. Dumez and G. F. Froment, Ind. Eng. Chem., Process Des. Dev. 15, 291 (1976). 13. A. I. L. Cava, C. A. Bernardo, and D. L. Trimm, Carbon 20, 219 (1982).

Our group recently reproduced the water-assisted growth method, so-called "super growth", of millimeter-thick single-walled carbon nanotube (SWNT) forests by using C2H4/ H2/ H2O/ Ar reactant gas and Fe/ Al2O3 catalyst. In this current work, a parametric study was carried out on both reaction and catalyst conditions. Results revealed that a thin Fe catalyst layer (about 0.5 nm) yielded rapid growth of SWNTs only when supported on Al2O3, and that Al2O3 support enhanced the activity of Fe, Co, and Ni catalysts. The growth window for the rapid SWNT growth was narrow, however. Optimum amount of added H2O increased the SWNT growth rate but further addition of H2O degraded both the SWNT growth rate and quality. Addition of H2 was also essential for rapid SWNT growth, but again, further addition decreased both the SWNT growth rate and quality. Because Al2O3 catalyzes hydrocarbon reforming, Al2O3 support possibly enhances the SWNT growth rate by supplying the carbon source to the catalyst nanoparticles. The origin of the narrow window for rapid SWNT growth will also be discussed.

Microsoft Word - NodaSuper2pdf.doc Growth window and possible mechanism of millimeter-thick single-walled carbon nanotube forests Kei Hasegawa1, Suguru Noda1,* , Hisashi Sugime1, Kazunori Kakehi1, Shigeo Maruyama2 and Yukio Yamaguchi1 1 Department of Chemical System Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2 Department of Mechanical Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Corresponding author. E-mail address: noda@chemsys.t.u-tokyo.ac.jp Our group recently reproduced the water-assisted growth method, so-called "super growth", of millimeter-thick single-walled carbon nanotube (SWNT) forests by using C2H4/ H2/ H2O/ Ar reactant gas and Fe/ Al2O3 catalyst. In this current work, a parametric study was carried out on both reaction and catalyst conditions. Results revealed that a thin Fe catalyst layer (about 0.5 nm) yielded rapid growth of SWNTs only when supported on Al2O3, and that Al2O3 support enhanced the activity of Fe, Co, and Ni catalysts. The growth window for the rapid SWNT growth was narrow, however. Optimum amount of added H2O increased the SWNT growth rate but further addition of H2O degraded both the SWNT growth rate and quality. Addition of H2 was also essential for rapid SWNT growth, but again, further addition decreased both the SWNT growth rate and quality. Because Al2O3 catalyzes hydrocarbon reforming, Al2O3 support possibly enhances the SWNT growth rate by supplying the carbon source to the catalyst nanoparticles. The origin of the narrow window for rapid SWNT growth will also be discussed. Keywords: Single-Walled Carbon Nanotubes, Vertically Aligned Nanotubes, Combinatorial Method, Growth Mechanism, Growth Window 1. INTRODUCTION Single-walled carbon nanotubes (SWNTs) have unique mechanical and electrical properties, and many applications for them have been proposed and researched. To realize practical applications, mass production of SWNTs is essential, and various catalytic chemical vapor deposition (CCVD) methods have been developed to achieve this mass production. There are two types of CCVD; one involving nanoparticle catalysts suspended in the gas phase and the other involving nanoparticle catalysts supported on substrates. A gas-phase production process, the so-called "HiPco process", is the first process to be used in the mass production of SWNTs.1 Recently, remarkable progress has been made in CCVD processes using supported catalysts. Submicrometer-thick films of randomly-oriented SWNTs have been the typical product when supported catalysts are used. In 2003, vertically aligned single-walled carbon nanotubes (VA-SWNTs) were realized2 by using alcohol catalytic CCVD (ACCVD).3 VA-SWNTs have now been achieved using several CVD methods and conditions.4-7 As a result, CCVD from substrates now has potential as a process in the mass production of SWNTs. Among those growth methods, the water-assisted method, so-called "super growth",4 realized an outstanding growth rate of a few micrometers per second, leading to millimeter-thick VA-SWNTs forests. However, many research groups have failed in reproducing "super growth", and the underlying mechanism of the growth rate enhancement by water remained unclear. By using our combinatorial method for catalyst optimization,8,9 we recently reproduced the "super growth" method and showed the important role of catalyst supports.10 In this current work, by doing a parametric study, we report in detail the effect of the catalyst and reaction conditions determined, and discuss the novel mechanism essential for rapid growth VA-SWNTs. 2. EXPERIMENTAL Catalysts were prepared on SiO2 substrates by sputter-deposition. An Al2O3 layer was formed by depositing 15-nm-thick Al on a substrate and then exposing the layer to ambient air. Fe was deposited on Al2O3 layers or directly on SiO2 substrates. For a separate experiment, gradient-thickness profiles of Fe were formed by using combinatorial masked deposition (CMD) method previously described.9 The catalysts were set in a tubular CVD reactor (22 mm in diameter and 300 mm in length), heated to a target temperature (typically 1093 K), and kept at that temperature for 10 min while being exposed to 27 kPa H2/75 kPa Ar at a flow rate of 500 sccm, to which H2O vapor was added at the same partial pressure as for the CVD condition (i.e., 0 to 0.03 kPa). During this heat treatment, Fe forms nanoparticles of a certain diameter and areal density depending on the initial Fe thickness.8 After the heat treatment, CVD was carried out by switching the gas to C2H4/ H2/ H2O / Ar. The standard condition was 8.0 kPa C2H4/ 27 kPa H2/ 0.01 kPa H2O/ 67 kPa Ar at 500 sccm at 1093 K for 10 min. The samples after CCVD were analyzed by using transmission electron microscopy (TEM) (JEOL JEM-2000EX), field emission scanning electron microscopy (FE-SEM) (Hitachi S-900), and micro-Raman scattering spectroscopy (Seki Technotron, STR-250) with an excitation wavelength at 488 nm. 3. RESULTS AND DISCUSSION 3.1. Standard condition of "super growth" Figure 1a shows a photograph of CNT samples grown by a combinatorial catalyst library under the standard condition. Millimeter-thick vertically aligned CNTs (VA-CNTs) were grown at regions where the Fe thickness was 0.4 nm or more. The maximum thickness of VA-CNTs was 1.2 mm at a Fe thickness of 0.5 nm. The thickness of VA-CNTs decreased when Fe thickness exceeded 0.5 nm. Figure 1b shows TEM images of CNTs grown under the same condition as Fig. 1a on substrates with uniform Fe thicknesses of 0.5 and 1.0 nm. SWNTs with a diameter around 4 nm mainly grew for 0.5-nm-thick Fe catalyst, whereas thicker CNTs grew for 1.0-nm-thick Fe catalyst. This difference in CNTs is because a thicker initial Fe layer yields larger Fe nanoparticles,8 indicating a narrow VA-SWNTs growth window for the initial Fe thickness. Figure 1c shows Raman spectra of the same CNT sample as Fig. 1a taken at Fe thicknesses of 0.5, 0.8, and 1.0 nm. Sharp, branched G-bands with a small D-band and peaks of radial breathing mode (RBM) were detected, indicating the existence of SWNTs. The G/D peak area ratios were smaller for thicker Fe layers (≥ 1 nm), because multi-walled CNTs (MWNTs) became the main product when a thicker Fe layer was used as catalyst. These figures show that "super growth" was achieved in this work. The growth temperature of the standard condition of this work is higher than that of the original "super growth" 4 because both the CNT thickness and the G/D ratio increased with increasing growth temperature. 3.2. Effects of catalyst metals and supports Effects of catalyst metals and their supports were investigated next. Figure 2a shows top-view photographs of CNT samples grown on the Fe/Al2O3, Co/Al2O3, and Ni/Al2O3 combinatorial catalyst libraries under the standard condition. The 0.5 0.8 1.0 Fe thickness / nm 2 mm(a) 0.5 0.8 1.0 Fe thickness / nm 2 mm(a) RBM ×10 Fe 0.5 nm Fe 0.8 nm Fe 1.0 nm 1200 1400 1600 Raman shift / cm-1 100 200 300 RBM ×10 Fe 0.5 nm Fe 0.8 nm Fe 1.0 nm 1200 1400 1600 Raman shift / cm-1 100 200 300 20 nm 20 nm20 nm Fig. 1 CNTs grown on the Fe/Al2O3 catalyst library under the standard condition (8.0 kPa C2H4/ 27 kPa H2/ 0.010 kPa H2O/ 67 kPa Ar at 1093 K for 10 min) . (a) Side view photograph of CNTs grown on the combinatorial catalyst library. (b) Raman spectra of the same sample at Fe thickness of 0.5, 0.8 and 1.0 nm. Intensity at the low wavenumber region (< 300 cm-1) was 10x magnified. (c) TEM images of CNTs grown on substrates with uniform Fe thickness of 0.5 nm (left) and 1.0 (right) nm. surfaces of both Fe/Al2O3 and Co/Al2O3 libraries became black at regions where the Fe thickness was 0.4 nm or more. On the other hand, the surface of Ni/Al2O3 was somewhat darkened only at the relatively thin Fe region around 0.5 nm. Figure 2b shows cross-sectional SEM images of these samples. About 200-μm-thick VA-CNTs grew for 0.5-nm-thick Co and about 0.4-μm-thick VA-CNTs grew for 0.5-nm-thick Ni.. Figure 2c shows top-view photographs of combinatorial catalyst libraries of Fe/SiO2, Co/SiO2, and Ni/SiO2 after CVD under the standard condition. The results were completely different from those libraries with Al2O3 support layer (i.e., Fe/Al2O3, Co/Al2O3, and Ni/Al2O3); VA-CNTs did not form on any library and only part of the surface of Fe/SiO2 for the thickness range from 0.4 to 0.6 nm darkened slightly compared with the library before CVD. As for Co/SiO2 and Ni/SiO2, negligible change appeared in color. Figure 2d shows the Raman spectra for the Fe/ SiO2 sample in Fig. 2b at Fe thickness of 0.5 nm, indicating growth of SWNTs. Fe catalyst supported on SiO2 can actually grow CNTs, including SWNTs, although the CNT yield is much smaller (submicrometer thickness or less) than that grown using catalysts supported on Al2O3 (up to millimeter thickness), regardless of the thickness of the catalyst. Next, we examined if an Al2O3 catalyst support layer actually acts as Al2O3 rather than metallic Al. Another catalyst library was prepared by first depositing 15-nm-thick Al on SiO2, then exposing the layer to ambient air, then oxidizing the layer under air at 973 K for 5 min, and finally depositing a gradient thickness profile of Fe on the layer at ambient temperature. Figure 3a shows photographs of the Fe/ Al2O3 catalyst libraries before CVD. In the thick Fe region (≥ 2 nm), both libraries were metallic silver in color. In the thinner region (< 2 nm), the Al layer only exposed to air was gray metallic in color, indicating the existence of metallic Al, whereas the Al layer oxidized at 973 K was completely transparent, indicating the complete oxidation of the Al layer. Then, CVD was carried out on these libraries. Figure 3b shows the photograph after CVD of the library with the Al layer oxidized at high temperature. The growth of the VA-CNT films was similar to that without high-temperature oxidation of the Al layer (Fig. 1a), indicating that the surface of the Al layer was oxidized by just being exposed to air, and that the layer acted as an Al2O3 support. Fe nanoparticles can grow SWNTs on either SiO2 or Al2O3 supports, but they need to be 0.5 0.8 1.0 Metal thickness / nm 0.5 0.8 1.0 Metal thickness / nm 0.5 0.8 1.0 Metal thickness / nm 0.5 0.8 1.0 Metal thickness / nm RBM ×10 1200 1400 1600 Raman shift / cm-1 100 200 300 RBM ×10 1200 1400 1600 Raman shift / cm-1 100 200 300 Fig. 2 Effects of catalyst metals and supports on CNT growth under the standard condition. (a,c) Top-view photographs of CNTs samples grown by combinatorial catalyst libraries prepared on Al2O3 (a) and SiO2 (c). All of the catalyst metals (i.e. Fe, Co, Ni) had the same thickness profiles between 0.2 nm (left) and 3 nm (right). (b) Cross-sectional FE-SEM images of the CNTs samples grown by 0.5-nm-thick Co and Ni catalysts prepared on Al2O3. (d) Raman spectrum of the CNTs samples grown by 0.5-nm-thick Fe catalyst prepared on SiO2 supports. Intensity at the low wavenumber region (<300 cm-1) was 10x magnified. supported on Al2O3 to achieve rapid growth of VA-CNTs at a rate of micrometers per second. In addition, Al2O3 support enhances the CNT growth by other catalysts (i.e., Co and Ni). Al2O3 support layer should have an essential role in growing CNTs. 3.3. Effects of H2O and H2 Figure 4a shows the thickness profiles of CNTs grown on Fe/Al2O3 catalyst libraries under the standard condition, except that the amount of added H2O (i.e. partial pressure) was varied. Without H2O, CNTs grew only at the relatively thin Fe region (0.3- 1.0 nm) and the maximum thickness of VA-CNTs was 0.7 mm at the 0.5 nm-thick Fe region. Addition of 0.010 kPa H2O enhanced the growth, especially at the thick Fe region (> 0.7 nm), and the maximum VA-CNT thickness increased to 1.0 mm at the 0.5-nm-thick Fe region. Further H2O addition (0.030 kPa) inhibited CNT growth at the thin Fe region (0.3- 0.6 nm) where SWNTs grew at the lower H2O partial pressures (≤ 0.010 kPa). Figure 4b shows the G/D ratios for each CVD condition at Fe thickness of 0.5, 0.8, and 1.0 nm. Slight addition of H2O (0.010 kPa) did not affect the G/D ratio at the thin Fe region (0.5 nm), but decreased the G/D ratio at the thicker regions (0.8 and 1.0 nm). Further addition of H2O (0.030 kPa) significantly decreased the G/D ratio at all regions. These results show that proper amount of H2O enhances the growth rate of CNTs, but excessive H2O decreases the growth rate of CNTs from small nanoparticles (i.e. small Fe thickness) and degrades the quality of CNTs possibly by oxidation. In conclusion, excess H2O totally inhibits "super growth" of SWNTs. Figure 5a shows thickness profiles of CNTs grown under the standard condition except that the amount of added H2 (i.e. partial pressure) was varied. When a lower amount of H2 was added (2.7 kPa), VA-CNTs became much thinner at Fe thickness of 0.7 nm or less. When a large amount of H2 was added (54 kPa), CNTs grew at any Fe thickness at a reduced VA-CNT thickness (around 0.4 mm). Figure 5b shows the G/D ratios for each CVD condition at Fe thickness of 0.5, 0.8, and 1.0 0.5 0.8 1.0 Metal thickness / nm 0.5 0.8 1.0 Metal thickness / nm 0.5 0.8 1.0 Fe thickness / nm 2 mm(b) 0.5 0.8 1.0 Fe thickness / nm 2 mm(b) Fig. 3 Effect of the preparation process of Al2O3 layer. (a) Photographs of Fe/Al2O3 catalyst libraries before CVD. Al2O3 layer was formed by oxidizing 15-nm-thick Al layer only by exposure to ambient air (top) and then by oxidizing in air at 973 K (bottom). (b) Photograph of CNT grown on the Fe/Al2O3 library with Al2O3 layer oxidized at 973 K (same library as Fig. 3a, bottom). 0.2 0.4 0.6 0.8 1T Fe thickness / nm 0.4 0.6 0.8 1.0 Fe thickness / nm 0 kPa 0.010 kPa 0.030 kPa 0.4 0.6 0.8 1.0 Fe thickness / nm 0 kPa 0.010 kPa 0.030 kPa Fig. 4 Effect of H2O on CNT growth. CVD was carried out by using Fe/Al2O3 combinatorial catalyst libraries under the standard condition except the amount of added H2O (partial pressure) was varied. (a) Relationship between thickness of VA-CNTs and Fe thickness at different H2O pressures. (b) Relationship between G/D ratio of Raman spectra and Fe thickness at different H2O pressures. nm. The G/D ratios decreased in either case of lower (2.7 kPa) and higher (54 kPa) H2 was added. These results indicate that an optimum amount of H2 is needed for rapid growth of VA-SWNTs of relatively good quality. 3.4. Possible mechanism of rapid growth of VA-SWNTs Based on the results discussed above, we propose three necessary conditions for "super growth" of SWNTs. The first condition is that the Fe catalyst needs to be thin enough, about 0.5 nm, so that the catalyst nanoparticles are small enough to grow SWNTs with small diameters. The second condition is that the catalyst nanoparticles need to be supported on Al2O3. The third is that the partial pressures of both H2O and H2 need to be carefully adjusted; these gases are essential but excessive amounts degrade the growth rate and/or the quality of SWNTs. Al2O3 and its related materials have long been used as catalysts for decomposition and dehydrogenation of hydrocarbons.11,12 Figure 6 shows a schematic of our proposed mechanism explaining the enhancement effect of Al2O3 on CNT growth. In this mechanism, C2H4 or its derivatives adsorb on Al2O3, diffuse over Al2O3 surface to catalyst nanoparticles, and then are incorporated in Fe nanoparticles. In contrast, when Fe is deposited directly on SiO2, C2H4 can be decomposed only on Fe nanoparticles whose surface is largely covered with growing CNTs. As a result, the growth rate of CNTs on Al2O3 is much faster than that on SiO2. In addition, the catalytic activity of Al2O3 strongly depends on its crystalline structure.11 α-Al2O3, the most stable phase, has low activity,11 and thus, rapid CNT growth does not occur when catalysts are supported on sapphire (i.e. monocrystalline α-Al2O3). Next, we discuss why the window of "super growth" for SWNTs is narrow. One reason might be due to the catalyst deactivation mechanism, i.e. coking of either Al2O3 surface or Fe nanoparticles. In the dehydrogenation process of hydrocarbons,11 Al2O3 easily loses its catalytic activity due to carbon deposition, and thus, H2O vapor has long been used to remove carbon byproducts. H2 also keeps the Al2O3 surface reactive by controlling the balance between dehydrogenation and hydrogenation of carbon surface species, as is known in the hydrocracking process.13 In contrast, excessive H2 inhibits the growth of SWNTs by hydrogenating C2H4-derivatives adsorbed on Al2O3. Concerning Fe nanoparticles, when the incoming flux of carbon into Fe nanoparticles increases, carbon in Fe nanoparticles will be highly supersaturated, resulting not only in increased SWNT growth rate but also in graphite formation on the surface of the nanoparticles. The fewer walls and larger free energy of SWNTs than MWNTs may 0.2 0.4 0.6 0.8 1 Fe thickness / nm 27 kPa 2.7 kPa 54 kPa 0.2 0.4 0.6 0.8 1 Fe thickness / nm 27 kPa 2.7 kPa 54 kPa 27 kPa 2.7 kPa 54 kPa 0.4 0.6 0.8 1 Fe Thickness / nm 27 kPa 2.7 kPa 54 kPa 0.4 0.6 0.8 1 Fe Thickness / nm Fig. 5 Effect of H2 on CNT growth. CVD was carried out by using Fe/Al2O3 combinatorial catalyst libraries under the standard condition except the amount of added H2 (partial pressure) was varied. (a) Relationship between thickness of VA-CNTs and Fe thickness at different H2 pressures. (b) Relationship between G/D ratio of Raman spectra and Fe thickness at different H2 pressures. SiO2 Al2O3 C2H4 C2H4 SiO2 Al2O3 C2H4 C2H4 Fig. 6 Schematic of enhancement mechanism of SWNT catalytic growth by Al2O3 support. make the degree of supersaturation larger for nanoparticles growing SWNTs than those growing MWNTs. This may be the reason why the "super growth" window for SWNTs is such narrow compared with MWNTs. (Note that MWNTs grow rapidly under a wide window of reaction conditions if catalyst nanoparticles are supported on Al2O3). In conclusive, two conditions are needed to sustain the rapid growth of SWNTs: first, the partial pressures of C2H4, H2, and H2O need to be balanced to suppress coking of Al2O3, and second, the incoming flux of carbon into Fe nanoparticles must not be too large, that means nanotubes must not grow too rapid, to prevent carbonization of Fe nanoparticles. 4. CONCLUSION Rapid growth of VA-SWNTs, namely, a few micrometers per second, or so-called "super growth", was reproduced in this study, and the growth window was clarified. The standard condition of this work was 8.0 kPa C2H4/ 27 kPa H2/ 0.01 kPa H2O/ 67 kPa Ar at 500 sccm at 1093 K for 10 min using a tubular CVD reactor (22 mm in diameter and 300 mm in length). Results showed that for the rapid growth of SWNTs, small nanoparticles formed from a thin Fe layer (about 0.5 nm) need to be supported on Al2O3 and that only the optimal amounts of H2O and H2 should be used. We proposed a novel mechanism by which Al2O3 enhances the growth rate of SWNTs, and offered a simple explanation of the effect of H2O and H2 on the growth rate. Namely, Al2O3 supports supply a carbon source to Fe nanoparticles, and H2O and H2 prevent catalyst deactivation by keeping the Al2O3 surface reactive and by balancing the carbon fluxes among the gas-phase, Al2O3 support, Fe nanoparticles, and growing SWNTs. Acknowledgements: The authors thank Z. Zhang for her help in the Raman measurements. This work is financially supported in part by the Grant-in-Aid for Young Scientists (A), 18686062, 2006, from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. References and Notes: 1. P. Nikolaev, M. J. Bronikowski, R. K. Bradley, F. Rohmund, D. T. Colbert, K. A. Smith, and R. E. Smalley, Chem. Phys. Lett. 313, 91 (1999). 2. S. Maruyama, R. Kojima, Y. Miyauchi, S. Chiashi, and M. Kohno, Chem. Phys. Lett. 360, 229 (2002). 3. Y. Murakami, S. Chiashi, Y. Miyauchi, M. Hu, M. Ogura, T. Okubo, and S. Maruyama, Chem. Phys. Lett. 385, 298 (2004). 4. K. Hata, D. N. Futaba, K. Mizuno, T. Nanami, M. Yumura, and S. Iijima, Science 306, 1362 (2004). 5. G. Zhong, T. Iwasaki, K. Honda, Y. Furukawa, I. Ohdomari, and H. Kawarada, Jpn. J. Appl. Phys. 44, 1558 (2004). 6. L. Zhang, Y. Tan, and D. E. Resasco, Chem. Phys. Lett. 422, 198 (2006). 7. G. Zhang, D. Mann, L. Zhang, A. Jabey, Y. Li, E. Yenilmez, Q. Wang, J. P. McVittie, Y. Nishi, J. Gibbons, and H. Dai, Proc. Nat. Acad. Sci. 102, 16141 (2005). 8. S. Noda, Y. Tsuji, Y. Murakami, and S. Maruyama, Appl. Phys. Lett. 86, 173106 (2005). 9. S. Noda, H. Sugime, T. Osawa, Y. Tsuji, S. Chiashi, Y. Murakami, and S. Maruyama, Carbon 44, 1414 (2006). 10. S. Noda, K. Hasegawa, H. Sugime, K. Kakehi, Z. Zhang, S. Maruyama, and Y. Yamaguchi, Jpn. J. Appl. Phys., in press. 11. D. S. MacIver, W. H. Wilmot, and J. M. Bridges, J. Catal. 3, 502 (1964). 12. F. J. Dumez and G. F. Froment, Ind. Eng. Chem., Process Des. Dev. 15, 291 (1976). 13. A. I. L. Cava, C. A. Bernardo, and D. L. Trimm, Carbon 20, 219 (1982).

Nonlocal interactions versus viscosity in turbulence A. Bershadskii ICAR, P.O. Box 31155, Jerusalem 91000, Israel It is shown that nonlocal interactions determine energy spectrum in isotropic turbulence at small Reynolds numbers. It is also shown that for moderate Reynolds numbers the bottleneck effect is determined by the same nonlocal interactions. Role of the large and small scales covariance at the nonlocal interactions and in energy balance has been investigated. A possible hydrodynamic mech- anism of the nonlocal solution instability at large scales has been briefly discussed. A quantitative relationship between effective strain of the nonlocal interactions and viscosity has been found. All results are supported by comparison with the data of experiments and numerical simulations. PACS numbers: 47.27.-i, 47.27.Gs I. INTRODUCTION Classic wind-tunnel experiments [1] showed that there is no scaling behavior in a presumably isotropic turbulence at low Reynolds numbers. Moreover, recent high-resolution numerical simulations [2] performed for low-Reynolds- number (Rλ ≈ 10− 60) show no hint of scaling-like behavior of the velocity increments even when ESS [3] is applied. For moderate Reynolds numbers numerical simulations show ’excess’ power just before the dissipation (Kolmogorov’s) wavenumber kd (a hump in the compensated energy spectra), see for instance [4]. This non-scaling effect (bottleneck effect [5]) is usually related with reducing efficiency of the energy cascade toward kd [5],[6],[7]. For the small Reynolds numbers applicability of the energy cascade idea is problematic for entire range of scales. It is shown in recent paper [8] that nonlocal interactions become dominating in comparison with the local ones just in the near-dissipation range of scales (cf. also [9]). In this range the viscous effects cannot be neglected and scaling asymptote corresponding to the nonlocal regime cannot be observed [8] (in the inertial range the local interactions are presumably dominating ones and the nonlocal scaling asymptote also cannot be observed). However, the nonlocal scaling asymptote can be used as a zeroth term in a perturbation approach taking into account the viscosity effects. There are many ways to develop such perturbation approach. For instance, in the paper [8] a perturbation approach giving logarithmic corrections to the scaling was developed, provided by significant role of the kink instabilities of the vortex filaments at moderate and large values of Reynolds number [10]. This ’logarithmic’-perturbation approach is shown to be effective in a vicinity of the crossover scale rc, where exchange of stability between local and nonlocal regimes takes place [8]. This vicinity is rather wide at moderate values of Reynolds number, when an overlap between these regimes is a strong phenomenon [8]. For small Reynolds numbers, however, the kink instabilities of the small vortex tubes is suppressed by the strong viscosity (see, for instance [11] and references therein). Therefore, for the small Reynolds numbers another, adequate just for this case, perturbation approach should be developed. An approach of such type is suggested in present paper. Starting from this approach and using comparison with results of numerical simulations [12] the laboratory experiments [1] it is shown that energy spectrum for small Reynolds numbers is determined by the nonlocal interactions even in isotropic turbulence. The same nonlocal interactions provide a hydrodynamic mechanism for the so-called bottleneck effect for moderate Reynolds numbers. It is also shown that large and small scales covariance at the nonlocal interactions plays a significant role in these phenomena. A quantitative relationship between effective strain of the nonlocal interactions and viscosity has been found using dynamical equations. II. PERTURBATIONS TO SCALING Let us following to the paper [8] consider a dimensional function E(k) of a dimensional argument k. And let us construct a dimensionless function of the same argument α(k) = E−1dE k−1dk . (1) If for kd ≫ k we have no relevant fixed scale (scaling situation), then for these values of k the function α(k) must be independent on k, i.e. α(k) ≃ const for kd ≫ k. For turbulence k could be a wavenumber and kd could be a dissipation wavenumber (kd = 1/η, where η = (ν 3/〈ε〉)1/4 is so-called viscous scale [24]). Solution of equation (1) with constant α can be readily found as E(k) ∼= ck α (2) http://arxiv.org/abs/0704.1904v4 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 FIG. 1: One-dimensional spectra measured in nearly isotropic turbulence downstream of a grid at small Reynolds numbers (the data are reported in [1]). The solid curves are drawn in the figure to indicate correspondence of the data to the equation (10) (non-local regime). where c is a dimension constant. This is the well-known power law corresponding to the scaling situations. Let us now consider an analytic approach, which allows us to find corrections of all orders to the approximate power law, related to the fixed scale kd. In the non-scaling situation let us denote f ≡ ln(E/A), x ≡ ln(k/kd) (3) where A and kd are dimensional constants used for normalization. In these variables, equation (1) can be rewritten as = α(x) (4). In the non-scaling situation x is a dimensionless variable, hence the dimensionless function α(x) can be non-constant. Since the ’pure’ scaling corresponds to k/kd ≪ 1 we will use an analytic expansion in power series α(x) = α0 + α1(k/kd) + ...+ αn(k/kd) n + ... (5) where αn are dimensionless constants. Choice of the small parameter for analytic perturbation approach is determined by physical situation, which one intends to consider. For instance, formoderate Reynolds numbers it would be generally preferable to consider the x−1 as a small parameter [8]. In this paper, however, we intend to consider energy spectra in isotropic turbulence at small Reynolds numbers and corresponding phenomena in a relatively close vicinity of the dissipation scale at moderate Reynolds numbers. The kink instabilities of the vortex filaments, which are a significant factor near rc (see above and [8]) are presumably not significant in situations with strong viscous effects [11]. Therefore 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 rc≅ηs FIG. 2: Three-dimensional spectra from a DNS performed in [12] for different Reynolds numbers up to Rλ = 125. The solid curves in this figure corresponds to the best fit by equation (10) (nonlocal interactions). The value of α1 = 6.0 ± 0.1 is the same in all cases. the choice of x−1 does not seem to be relevant here. On the other hand, parameter k/kd seems to be less related to the specific hydrodynamic structures dominating processes in turbulence and more relevant to a sheer taking into account corrections to the scaling providing by viscosity. After substitution of the analytic expansion (5) into Eq. (4) the zeroth order approximation gives the power law (2) with α ∼= α0. First order analytic approximation, when one takes only the two first terms in the analytic expansion (5), gives E(k) ∼= ck α0eα1k/kd . (6) (cf , for instance, [13]-[16]). Corrections of the higher orders can be readily found in this perturbation approach. III. NONLOCAL INTERACTIONS Let us recall that in isotropic turbulence a complete separation of local and non-local interactions is possible in principle. It was shown by Kadomtsev [17] that this separation plays a crucial role for the local Kolmogorov’s cascade regime with scaling energy spectrum E(k) ≃ K 〈ε〉2/3k−5/3 (7) where 〈ε〉 is the average of the energy dissipation rate, ε, k = 1/r is the wave-number, and K is the so-called Kolmogorov constant. This separation should be effective for the both ends. That is, if there exists a solution with the local scaling (7) as an asymptote, then there should also exist a solution with the non-local scaling asymptote. 0.001 0.01 0.1 1 0.001 0.01 0.1 1 rc ηs FIG. 3: As in Fig. 2 but for Rλ = 460. The upper part of the figure shows the same energy spectrum as the lower one but in the compensate (according to the Kolmogorov’s scaling) form. The solid curves are the best fit corresponding to the nonlocal approximation (10) with the same α1 = 6.0± 0.1 as in Figs 2. Of course, the two solutions with these asymptotes should be alternatively stable (unstable) in different regions of scales. It is expected, that the local (Kolmogorov’s) solution is stable (i.e. statistically dominating) in inertial range (that means instability of the non-local solution in this range of scales). Roughly speaking, in non-local solution for small scales r only non-local interactions with large scales L (1 ≫ r/L) are dynamically significant (the interaction among the small scales is negligible compared with interaction via large scales) and the non-local interactions is determined by large scale strain/shear. This means that one should add to the energy flux 〈ε〉-parameter (which is a governing parameter for the both solutions) an additional parameter such as the strain s for the non-local solution. As far as we know it was noted for the first time by Nazarenko and Laval [18] that dimensional considerations applied to the non-local asymptote result in the power-law energy spectrum E(k) ≃ c k−1 (8) both for two- and three-dimensional cases. Linear dependence of the spectrum (8) on 〈ε〉 is determined by the linear nature of equations corresponding to the non-local asymptote that together with the dimensional considerations results in (8) [18]. Interesting numerical simulations were performed in [19]. In these simulations local and non-local interactions have been alternatively removed. For the first case a tendency toward a spectrum flatter than ’-5/3’ is observed near and beyond the separating scale (beyond which local interactions are ignored), that supports Eq. (8). Following to the perturbation approach suggested above both local and non-local regimes can be corrected. The first order correction is E(k) ≃ K〈ε〉2/3k−5/3e−β(k/kd) (9) E(k) ≃ c k−1e−α1(k/kd) (10) for the local and non-local regimes respectively (K, c, α1 and β are dimensionless constants). Figure 1 shows one-dimensional spectra measured in nearly isotropic turbulence downstream of a grid at small Reynolds numbers (the data are reported in [1]). The solid curves are drawn in the figure to indicate correspondence of the data to the equation (10) (non-local regime). The experimental data, however, is not controlled enough to study fine properties of the isotropic turbulence. Therefore below we will mainly use the data obtained in numerical simulations. It is shown in [8] that there is an ’exchange of stability’ phenomenon at certain kc = 1/rc. That is, for k < kc the Kolmogorov’s regime is stable and the non-local regime is unstable, whereas for k > kc the Kolmogorov’s regime is unstable and the non-local regime is stable. For this scenario, at k = kc the Kolmogorov’s regime is still asymptotically scale-invariant (i.e. Eq. (7) gives an adequate approximation for this regime), while for the non-local regime the first order correction is substantial (i.e. Eq. (10) should be used at k > kc for the non-local regime). In this scenario the Kolmogorov’s regime plays significant role in the viscous stabilization of the non-local regime for k > kc (see [19]), but for these k the non-local regime becomes statistically dominating instead of the Kolmogorov’s one (it is well known that only for energy spectra steeper than k−3 it can be rigorously proved that the dominant interactions are nonlocal (cf Eq. (10) that provides such steepness). It should be noted that in [20] a second scaling solution E(k) ∝ Pk−1 was obtained in addition to the Kolmogorov scaling using the Clebsch formulation of hydrodynamics. The prefactor P = (〈ε〉ν)1/2 denotes a certain flux in the wavenumber space [20]. It will be shown below that the prefactor in Eqs. (8),(10) can be transformed into the one formally equal to P (see Eq. (15) and Fig. 4). This could mean that the studied nonlocal regime (more precisely: its scaling asymptote) is closely related to the Yakhot-Zakharov scaling solution introduced in the Ref. [20]. Such identification (if valid) allows us understand instability of the nonlocal regime for large scales (small k). Indeed, it is shown in [21] that reconnection process breaks conservation of the integral determining the flux P for the large scales (small k) and , therefore, there is no possibility for realization of the scaling asymptote k−1 of this solution. Following to Newell [21][22] the nonconservation of the integral in large scales generally follows from nonlocality of the viscosity term in the equations formulated for the Clebsch variables. Since for the large scales the nonlocal regime should be represented by its asymptote E ∝ k−1 (as an intermediate asymptote [23] the nonconservation of the integral makes this regime unstable for the large scales (small k). Figure 2 shows three-dimensional energy spectra calculated using data from a high-resolution direct numerical simulation of homogeneous steady three-dimensional turbulence [12] for different Reynold numbers up to Rλ = 125. The solid curves in this figure corresponds to the best fit by equation (10) (nonlocal interactions). The exponent α1 ≃ 6.0±0.1 for all considered values of Rλ. The arrows indicate the rc = 1/kc scales calculated using corresponding DLLL(r) (see [8]). In figure 3 we show the data obtained in the same DNS as those shown in Fig. 2 but for Rλ = 460. In the upper part of this figure we show the energy spectrum in the compensate (according to Kolmogorov’s scaling Eq. (7)) form. One can clear see the hump corresponding to the bottleneck effect. The solid curves are the best fit to the the same nonlocal spectrum (10) with the same (universal) value of α1 ≃ 6.0 ± 0.1 (see also below). One can see that the bottleneck effect is determined by just the same nonlocal interactions as the above considered energy spectra for small Reynolds numbers. IV. STRAIN AND VISCOSITY It is shown in Ref. [8] that for sufficiently large Reynolds numbers, providing a visible inertial interval, there is an overlapping between the two regimes: non-local and local (Kolmogorov). This overlapping is based on the very nature of the stability exchange between the two statistical regimes. However, in analogy with the viscous scale η = (ν3/〈ε〉)1/4 [24] there should be a scale ηs such that for scales r < ηs contribution of the local interactions will be drastically decreased in comparison with the nonlocal ones. In the analogy with η one can calculate ηs using the dimensional considerations as (cf for shear flows [25]). The dynamical equations provide us with a relationship [24] 〈ε〉 = 2ν k2E(k)dk (12) For E(k) given by Eq. (10) the dissipation function k2E(k) has its maximum at k = kd/6. If kd ≫ ks = 1/ηs (see below) the maximum of the dissipation function k2E(k) is located just between ks and kd. Therefore, we can estimate 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 (<ε>ν)1/2 38 - 460 FIG. 4: The prefactor in the nonlocal approximation to the energy spectra (10): c〈ε〉/s - circles, versus (〈ε〉ν)1/2. The straight line with the slope equals to 18 indicates agreement with Eq. (15). (12) as 〈ε〉 ≃ 2ν k2E(k)dk (13) Substituting (10) (with α1 = 6) into (13) we obtain relationship 〈ε〉1/2ν−1/2 (14) Using the relationship (14) one can estimate the prefactor in the nonlocal approximation to the energy spectra (10) ≃ 18(〈ε〉ν)1/2 (15) Now using the data of the DNS [12] (cf Figs. 2,3) let us calculate the prefactor c〈ε〉/s. Results of these calculations are shown in figure 4 as circles (Reλ = 38, 54, 70, 125, 284, 380, 460). This figure shows the prefactor c 〈ε〉/s against (〈ε〉ν)1/2. The straight line with the slope equals to 18 indicates agreement with Eq. (15). It should be noted that in the DNS [12] 〈ε〉 ≃ const for Rλ ≥ 70, in agreement with the well known Kolmogov’s hypothesis [24]. Therefore, for Rλ ≥ 70 (when 〈ε〉 ≃ const [12]) it follows from Eq. (14) that s ∝ ν −1/2. This relationship provides us also with dependence of the strain s on Rλ. Due to the nonlinear character of the Navier-Stokes equations the so-called triadic type of interactions is dominating mechanism of the dynamical interactions in turbulence (see, for instance, [9]). A triad corresponding to the nonlocal interactions involves two short-wave-number modes and one long-wave-number mode (see a sketch in figure 5). Ac- cordingly, at the nonlocal interactions two characteristic space scales are actively involved: large-scale characteristic scale ηs (11) and small-scale (viscous or Kolmogorov) characteristic space scale η = (ν 3/〈ε〉)1/4. It is naturally that the large scales should be normalized by ηs while the small scales should be normalized by η. This could cause an obvious problem at the nonlocal interactions. However, the large and small scales covariance relationship at the nonlocal interactions follows directly from the relationship (14) ≃ const (16) (i.e. the relation ηs/η is independent on Rλ). And vice versa, the large and small scales covariance (15) results in the relationship of the type (14). The covariance at the nonlocal interactions supports right balance between the energy flux to the small scales and their dissipative capacity (cf, for instance, [26],[27]). FIG. 5: Sketch of the nonlocal triadic interaction in the physical and in the Fourier space Actually, the scale ηs = 1/ks should provide an edge of applicability of the approximation (10) to the real energy spectra. If one compares the ηs determined by this way from the Figs. 2,3,6 one can see that indeed ηs/η ≃ const in agreement with Eq. (16) (we have also checked this with the data for Rλ = 284, 380 [12] and the data reported in [29]). Moreover, taking into account a continuity condition of the energy spectrum at the point k = ks (and using Eqs. (7) and (10)): K 〈ε〉2/3k−5/3s ≃ c k−1s e −α1(ks/kd) (17) we obtain equation K ≃ c exp[−(2c)3/2/α21] (18) Substituting the Kolmogorov constant K ≃ 1.6 (see [12],[28]) and α1 ≃ 6 into this equation we obtain c ≃ 2. Then, substituting this value of c into Eq. (16) we obtain ks/kd ≃ 0.037. The last value is in agreement with the available data (see Figs. 2,3,6 and [29]). Together with the universality of α1 ≃ 6.0 (which also is a consequence of the scale covariance) Eq. (16) determines the well known from DNSs [12],[29] universality (independence on Rλ) of the position of the ’hump’ in the axes kη for the bottleneck effect. V. PASSIVE SCALAR For passive scalar θ in the isotropic turbulence the equations (9),(10) should be replaced [8] by the equations Eθ(k) ∝ 〈ε〉−1/3〈εθ〉 k −5/3e−γ(k/kd) (19) Eθ(k) ∝ k−1e−δ(k/kd) (20) for the local and non-local regimes respectively (γ and δ are dimensionless constants, and 〈εθ〉 is the average value of dissipation rate of scalar variance). Figure 6 shows three-dimensional passive scalar spectrum from a DNS performed in [30] for Péclet number Pλ = 427 (the Schmidt number is unity, i.e Pλ = Rλ). The solid curves in this figure correspond to the best fit by equation (20) (nonlocal interactions). In the upper part of this figure we show the spectrum in the compensate (according to the Corrsin-Obukhov scaling [30]) form. One can clear see the hump corresponding to the bottleneck effect. The arrows show position of the ηs scale. Comparing with Figs. 2,3 one can see that for the passive scalar ηs/η takes the same universal value as for the velocity field (the large and small scales covariance (15)). 0.001 0.01 0.1 1 0.001 0.01 0.1 1 FIG. 6: The passive scalar DNS data (circles) for homogeneous isotropic turbulence described in [30], Péclet number Pλ = 427 and the Schmidt number is unity (i.e Pλ = Rλ). The solid curves in this figure correspond to the best fit by equation (20) (nonlocal interactions). Acknowledgments I thank K.R. Sreenivasan for inspiring cooperation. I also thank T. Nakano, D. Fukayama and T. Gotoh for sharing their data and discussions. [1] G. Comte-Bellot and S. Corrsin, J. Fluid Mech., 48, 273 (1971). [2] J. Schumacher, K. R. Sreenivasan, and V. Yakhot, New Journal of Physics 9, 89 (2007). [3] R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioloi, and S. Succi, Phys. Rev. E., 48, R29 (1993). [4] T. Gotoh and D. Fukayama, Phys. Rev. Lett. 86, 3775 (2001). [5] G. Falkovich, Phys. Fluids 6, 1411 (1994). [6] D. Lohse and A. Muller-Groeling, Phys. Rev. Lett., 74, 1747 (1995); Phys. Rev. E 54, 395 (1996). [7] M.K. Verma and D. Donzis, J. Phys. A, 40, 4401 (2007). [8] A. Bershadskii, J. Stat. Phys., online first: DOI 10.1007/s10955-007-9322-0 (see also arXiv:nlin.CD/0603070). [9] J.A. Domaradzki, Phys. Fluids A, 4, 2037 (1992). [10] K.R. Sreenivasan and A. Bershadskii, J. Fluid. Mech. 554, 477 (2006). [11] P.R. Woodward, D.H. Porter, B. K. Edgar, S. E. Anderson, and G. Basset, Comput. Appl. Math. 14, 97 (1995). [12] T. Gotoh, D. Fukayama and T. Nakano, Phys. Fluids, 14, 1065 (2002). [13] K.R. Sreenivasan, J. Fluid Mech., 151, 81 (1985). [14] C. Foias, O. Manley, and L. Sirovich, Phys. Fluids, A 2, 464 (1990). [15] Z-S. She and E. Jackson, Phys. Fluids A, 5 1526 (1993). [16] M. Nelkin, Adv. Phys. 43, 143 (1994). [17] B.B. Kadomtsev, Plasma Turbulence (Academic Press, New York, 1965). [18] S. Nazarenko and J.-P. Laval, J. Fluid Mech., 408, 301 (2000). [19] J-P. Laval, B. Dubrulle and S. Nazarenko, Phys. Fluids, 13: 1995 (2001). [20] V. Yakhot and V. Zakharov, Physica D, 64 379 (1993). [21] S.V. Nazarenko, Physica D, 102 343 (1997). http://arxiv.org/abs/nlin/0603070 [22] A.C. Newell, private communication. [23] G.I. Barrenblatt, Scaling, self-similarity, and intermediate asymptotics (Plenum Press, New York/London, 218 p., 1979). [24] A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics, Vol. 2, (MIT Press, Cambridge 1975). [25] F. Toschi, G. Amati, S. Succi, R. Benzi, and R. Piva, Phys. Rev. Lett., 82, 5044 (1999). [26] B. Dubrulle and J. Graner, J. Phys. II, 6, 797 (1996). [27] A. Pocheau, Europhys. Lett, 35, 183 (1996). [28] K. R. Sreenivasan, Phys. Fluids 7, 2778 (1995). [29] T. Ishihara, Y. Kaneda, M. Yokokawa, K. Itakura and A. Uno, J. Phys. Soc. of Japan, 74, 1464 (2005). [30] T. Watanabe and T. Gotoh, New J. Phys. 6: Art. No. 40. (2004) Introduction Perturbations to scaling Nonlocal interactions Strain and viscosity Passive scalar Acknowledgments References

It is shown that nonlocal interactions determine energy spectrum in isotropic turbulence at small Reynolds numbers. It is also shown that for moderate Reynolds numbers the bottleneck effect is determined by the same nonlocal interactions. Role of the large and small scales covariance at the nonlocal interactions and in energy balance has been investigated. A possible hydrodynamic mechanism of the nonlocal solution instability at large scales has been briefly discussed. A quantitative relationship between effective strain of the nonlocal interactions and viscosity has been found. All results are supported by comparison with the data of experiments and numerical simulations.

Introduction Perturbations to scaling Nonlocal interactions Strain and viscosity Passive scalar Acknowledgments References

NIKHEF-2007-09, SPIN-07-11, ITP-UU-07-18 Quantum radiative corrections to slow-roll inflation Ante Bilandžić1, 2, ∗ and Tomislav Prokopec1, † 1Institute for Theoretical Physics (ITP) & Spinoza Institute, Utrecht University, Leuvenlaan 4, Postbus 80.195, 3508 TD Utrecht, The Netherlands 2NIKHEF, Kruislaan 409, 1098 SJ Amsterdam, The Netherlands We consider the nonminimally coupled λϕ4 scalar field theory in de Sitter space and construct the renormalization group improved renormalized effective theory at the one-loop level. Based on the corresponding quantum Friedmann equation and the scalar field equation of motion, we calculate the quantum radiative corrections to the scalar spectral index ns, gravitational wave spectral index ng and the ratio r of tensor to scalar perturbations. When compared with the standard (tree-level) values, we find that the quantum contributions are suppressed by λN2 where N denotes the number of e-foldings. Hence there is an N2 enhancement with respect to the näıve expectation, which is due to the infrared enhancement of scalar vacuum fluctuations characterising de Sitter space. Since observations constrain λ to be very small λ ∼ 10−12 and N ∼ 50−60, the quantum corrections in this inflationary model are unobservably small. PACS numbers: 98.80.-k, 98.80.Cq 1. INTRODUCTION Given the fact that we live in the era of precision cosmology, it is important to establish a framework within which the quantum radiative corrections for observables induced by the vacuum fluctuations of matter fields can be calculated. Such radiative corrections may be important in some inflationary models. In this paper we consider the nonminimal λϕ4 inflationary model which includes a nonvanishing coupling ξ to the curvature scalar. We consider this model for simplicity; once the framework for calculating the quantum (radiative) corrections is established, it can be ∗A.Bilandzic@students.uu.nl, anteb@nikhef.nl †T.Prokopec@phys.uu.nl http://arxiv.org/abs/0704.1905v2 mailto:A.Bilandzic@students.uu.nl, anteb@nikhef.nl mailto:T.Prokopec@phys.uu.nl quite easily applied to other inflationary models [1]. The minimally coupled λϕ4 inflationary model (with ξ = 0) is already more than two standard deviations disfavored by cosmological observations [2]. However, for a certain choice of the coupling ξ to the background, the resulting nonminimally coupled scalar field model can still match the experimental data [3, 4, 5, 6], basically because the model then produces the spectral index of the massive chaotic inflaton model. In this work we show that this is indeed the case, but only for a rather limited values of ξ, namely for ξ which satisfies ≪ |ξ| ≪ 1 , (ξ < 0) , (1) where Ñ ≃ N + 1 and N is the number of e-foldings. If the condition |dH/dt| ≪ H2 is not fulfilled, then our framework is not applicable, because we have constructed the de Sitter invariant scalar field propagator with the assumption that the Ricci scalar R = 6(2H2 + dH/dt) ≃ 12H2, where H = H(t) denotes the Hubble parameter. This condition is fulfilled in most of inflationary models and thus does not present a significant constraint to our model. During inflation the amplitude of field correlators at the classical level is suppressed by powers of the Hubble parameter, but the quantum corrections to field correlators can depend on the whole history of inflation, leaving hence the possibility that in a cumulative manner quantum corrections can become important and even detectable by future experiments [7, 8, 9, 10, 11]. Such cumulative effects are claimed to be present in the analysis done recently in Refs. [12, 13]. In our analysis no such cumulative effects are present. The quantum radiative corrections to slow-roll parameters in inflation have been firstly calculated in Refs. [7, 8, 9]. The authors begin by considering single field inflationary models and subsequently generalize their analysis to include the inflaton coupling to a light scalar and light fermionic field. While Refs. [7, 8] consider quantum corrections to the equation of motion in momentum space, in this work we make use of the effective action techniques. Our results are in a qualitative disagree- ment with those of Refs. [7, 8, 9]. One important difference is in that in their analysis the authors of [7, 8, 9] neglect the inflaton coupling to the background curvature (Ricci scalar), which within our framework yields the dominant contribution to the quantum radiative corrections during slow-roll inflation. A second important difference is that we made our analysis by using the de Sitter invari- ant propagator, while the proper analysis should be conducted by making use of a scalar propagator suitable for quasi-de Sitter spaces. Our method is based on the effective action approach. We arrive at our one-loop effective action by making use of the position space propagator at coincidence. Within this method we are able to use the well established machinery of dimensional regularization, renormalization and renormalization group improvement of our resulting effective field theory. The authors of Ref. [9] advocate the use of the dynamical renormalisation group method (DRG) [14]. In that novel method the secular terms, which induce a logarithmic growth (with conformal time) of the mode functions, are resummed to yield the renormalisation group improved mode functions. These improved mode functions exhibit regulated late time infrared divergences, rendering the mode functions infrared finite. More specifically, within our framework we obtain a quantum infrared enhancement to slow-roll parameters which is, when compared to the classical values, proportional to the number of e-foldings squared N2. This enhancement is due to the scalar field mass generated by the coupling to the background curvature scalar. The authors of [7, 8, 9] obtain a quadratic enhancement but for the λϕ4 model without including the inflaton coupling to the Ricci scalar. On the other hand, when compared to the classical value, the quantum corrections generated by the λϕ4 interaction term are enhanced in our framework only linearly by N . It is by now a well established fact that quantum effects can have quite a dramatic impact during inflation. An example is the breakdown of conformal invariance for the photons of scalar electrodynamics, which has as a consequence a photon mass generation during inflation and a generation of cosmological scale magnetic fields [15]. Similarly the quantum radiative effects break conformal invariance of the fermions of the Yukawa theory in de Sitter space [16]. As a result fermions acquire a mass during inflation [16, 17], having as a consequence a production of fermions during inflation and possibly inflationary baryogenesis. Finally, the canonical coupling of gravitons to fermions enhances the production of fermions during inflation [18]. The main result of our work is the quantum correction to the scalar spectral index (107–108). The leading order contribution reads (ns − 1)Q = λÑ(ξ − 1/6) (1− κ)2(1− 2 +O(λ ln(Ñ)) , (κ = 8Ñξ, Ñ ≈ N + 1− ξ/2) , (2) which is to be compared with the classical contribution, (ns − 1)C = − . (3) The leading contribution (2) originates solely from the resummation of the mass insertions generated by the coupling to the background curvature. Since λ ∼ 10−12 the quantum contribution (2) is indeed too small to be observable. Note that the condition (98) implies −Ñ/3 ≪ κ < 1, such that κ can be large and negative. If this is the case the classical spectral index (3) becomes consistent with the current CMB data [2]. We futhermore calculate the quantum corrections to the spectrum of curvature perturbation, to the tensor spectral index, and to the ratio of the tensor-to-scalar spectrum. The present work is organized as follows. In Section 2 we first recall the basics of de Sitter space and then sketch the derivation of the de Sitter invariant Chernikov-Tagirov propagator. In Section 3 we use this propagator and the techniques of dimensional regularization and renormalization to derive the one-loop improved effective potential for our theory. This procedure requires one counterterm for the quartic self-coupling constant λ and one for the coupling to the background ξ, which are defined at an arbitrary scale ϕ0. In Section 4 we use the standard renormalization group (RG) techniques to improve our effective potential. Having obtained the RG improved effective potential, in Section 5 we calculate the corresponding quantum scalar field stress-energy tensor in the slow-roll approximation. By making use of the quantum Friedmann equation and of the scalar field equation of motion, we then develop our framework within which the quantum radiative corrections from the vacuum matter fluctuations to slow-roll parameters can be calculated. In particular, we organize the quantum radiative corrections to slow-roll parameters ǫ and η into two distinct parts. The first part arises from the one-loop resummation of the mass insertions generated by the quartic self-coupling in the presence of a scalar (inflaton) condensate, while the second part arises from the resummation of the scalar mass insertions generated by the coupling to the background. Both of these quantum corrections are suppressed by the coupling constant λ but they are enhanced by the number of e-foldings squared. Based on these results we then calculate the quantum radiative corrections to the observables: the spectrum of curvature perturbation and its spectral index, the tensor spectral index and the ratio of tensor-to-scalar spectra. Finally, in Section 6 we summarize our results and discuss their physical implications. 2. PROPAGATOR IN DE SITTER SPACE 2.1. de Sitter space A four dimensional de Sitter space is perhaps best viewed as a 4-dimensional hyperboloid em- bedded into the 5-dimensional Minkowski space-time with the line element, ds25 = −dX20 + dX21 + dX22 + dX23 + dX24 . (4) FIG. 1: The embedding of de Sitter space into a five dimensional flat space-time. The vertical line corresponds to the time coordinate, X0 = T , and the radial coordinate R = X21 +X 4 . At each point (T,R) there is a unit 3-sphere S3, which is for the sake of clarity represented by a circle S1 erected at each point (T,R). The Hubble radius RH = 1/H is the coordinate distance R of the hyperboloid from the origin at T = 0. The embedded hyperboloid of de Sitter space is shown in Figure 1, and it is determined by −X20 +X21 +X22 +X23 +X24 = , (5) where H denotes the Hubble parameter. The symmetry group of de Sitter space, SO(1, 4), is manifest by this embedding. One defines the de Sitter invariant distance functions as, Z(X ;X ′) = H2 A,B=0 ηABXAX B = 1− Y (X ;X ′) , ηAB = diag(−1, 1, 1, 1, 1) . (6) We shall use the following flat 4-dimensional coordinates (which cover 1/2 of the de Sitter manifold), sinh(Ht) + eHt‖~x‖2 , (−∞ < t <∞) , Xi = e Htxi , (−∞ < xi <∞, i = 1, 2, 3) , cosh(Ht)− H eHt‖~x‖2 , (7) in which the metric tensor reduces to the form ds2 = −dt2 + a2d~x 2 , (8) with the scale factor a = eHt. When written in terms of conformal time η, defined as adη = dt, the metric tensor acquires the conformal form, gµν = a 2(η)ηµν , a = − (η < 0) , ηµν = diag(−1, 1, 1, 1) . (9) The invariant distance functions Z(X ;X ′) ≡ z(x; x′) and Y (X ;X ′) ≡ y(x; x′) reduce in these coordinates to the simple form, z(x; x′) = 1− 1 y(x; x′) , y(x; x′) = aa′H2∆x2 , (10) with a = a(η) = −1/(Hη), a′ = a(η′) = −1/(Hη′), and ∆x2(x; x′) = −(|η − η′| − iε)2 + ‖~x− ~x ′‖2 , (11) where (for a later use) we introduced the infinitesimal parameter ε > 0, which defines how the poles of the propagator (discussed in the next section) contribute. In these coordinates the curvature of spatial sections vanishes, and thus they are also known as flat (Euclidean) coordinates, in which de Sitter space appears as uniformly expanding. By solving the relevant geodesic equations for x0 and xi (i = 1, 2, 3) one can show that the de Sitter invariant distance function y = y(x; x′ ) is related to the geodesic distance ℓ = ℓ(x; x′ ) by the following simple relation, y(x; x′ ) = 4 sin2 Hℓ(x; x′ ) . (12) 2.2. Scalar propagator in de Sitter space The dynamics of the scalar field are specified by the following tree-level action, −gLϕ , (13) with the Lagrangean, −gLϕ = gµν(∂µϕ)(∂νϕ)− Vb(ϕ) , (14) where Vb(ϕ) = ξbRϕ2 + ϕ4 . (15) In the above expression λb and ξb are the bare values of the quartic self-coupling and the nonminimal coupling to the Ricci curvature scalar R, respectively, and g = det(gµν). For simplicity we set the bare scalar mass mb = 0. The theory (13–15) is a simplified version of the Yukawa theory in de Sitter background studied in Refs. [20? ]. An early related work can be found in Ref. [19]. The scalar propagator in a curved background space can be defined as the expectation value, i∆(x; x′) = 〈x| −g(�−m2ϕ − ξbRD) |x′〉 , m2ϕ(ϕ) ≡ , (16) where |x〉 is the eigenstate of the position operator x̂ (i.e. x̂|x〉 = x|x〉), � = (−g)−1/2∂µ(−g)1/2gµν∂ν denotes the d’Alambertian and g = det[gµν ]. This Feynman propagator solves the following equation in de Sitter space in general D space-time dimensions (needed for dimensional regularization and renormalization), −g(�−m2ϕ − ξbRD)i∆(x; x′) = iδD(x− x′) , (17) where δD(x− x′) is the D-dimensional Dirac δ-distribution, RD = D(D − 1)H2 is the Ricci scalar in a D dimensional de Sitter space, and H is the Hubble parameter. The de Sitter invariant form of (17) is [22, 23, 24, 25] (1− z2) d −Dz d m2ϕ + ξbRD iG(y) = iδD(x− x′) , (18) where the invariant propagator is defined as, iG(y) = i∆(x; x′). Here we made use of Eqs. (10–11) and of ∂µ ≡ (∂µz) δ 0µ y + 2a ′H∆xµ . (19) The properly normalized de Sitter invariant solution of Eq. (18), which near the light-cone and in the massless limit reduces to the Hadamard form, iG(y) y→0−→ H (2π)D/2 y2−D/2, y0 , (20) is unique, iG(y) = ) 2F1 (D − 1 + νD, D − 1 − νD; , (21) where ((D − 1)2 m2ϕ + ξbRD . (22) This is the Chernikov-Tagirov propagator for de Sitter space [22, 23] generalized to D space-time dimensions. The pole prescription defined by the iε-prescription in Eq. (11) implies that the prop- agator (21) corresponds to the time-ordered (Feynman) propagator. For a discussion of other propagators relevant for expanding space-times in the Schwinger-Keldysh in-in formalism we refer to [10, 25]. 3. EFFECTIVE POTENTIAL The one-loop effective action for a real scalar field reads, Γϕ[gµν , ϕ] ≡ −gLϕ = SHE[gµν ] + Sϕ[gµν , ϕ] + Tr ln −g(�−m2ϕ − ξbRD) , (23) FIG. 2: The one-loop diagram (vacuum bubble) contributing to the scalar effective theory (23) in a curved background. where Tr refers to the space-time integration dDx, Sϕ is the tree-level scalar field action (13) and SHE denotes the Hilbert-Einstein action, SHE = − 16πGN −gRD . (24) The last term in Eq. (23) represents the one-loop contribution to the effective action δ1Lϕ, whose graphical representation is shown in Figure 2. We shall now evaluate the general expression (23) in de Sitter space. It is convenient to differentiate the one-loop contribution δ1Lϕ with respect to the scalar mass, ∂δ1Lϕ −g(�−m2ϕ − ξbRD) i∆(x; x) . (25) Now making use of Eq. (21) one obtains, ∂δ1Lϕ iG(y)|y→0 = − (4π)D/2 ) . (26) Separating the divergent and finite contributions in (26) yields ∂δ1Lϕ (4π)D/2 )(m2ϕ − (D−2) + ξbD(D−1) − 2(1− 6ξb) , (27) where we made use of, D − 4 − (1− γE) +O(D − 4) , (28) γE ≃ 0.577 is the Euler constant, RD=4 ≡ R = 12H2, and + 2(1−6ξb) , (29) where m2ϕ = λϕ 2/2 is defined in Eq. (16). When integrated, Eq. (27) gives the following contribution to the effective Lagrangean, δ1Lϕ = − 2(4π)D/2 HD−4m4ϕ − (D−2)− ξbD(D−1) HD−2m2ϕ , w = where the integral is an indefinite integral. In order to renormalize our Lagrangean Lϕ we will add to it the counterterms λ0 and ξ0 and apply the renormalization conditions which will determine the finite parts of those counterterms, Lϕ, ren = − gµν(∂µϕ)(∂νϕ)− Vren(ϕ) , (31) where now Vren(ϕ) = ξbRϕ2 + ξ0Rϕ2 − δ1Lϕ, (32) and δ1Lϕ is given by Eq. (30). We renormalize our Lagrangean at an arbitrary scale ϕ0, = λb = λb + λ(ϕ0, H 2)− δ 4(δ1L) δ(H2)δϕ2 = D(D−1)ξb = D(D −1)ξb +D(D −1)ξ(ϕ0, H2)− δ3(δ1L) δ(H2)δϕ2 , (33) which yields λ(ϕ0, H 2) = − 3 2(4π)D/2 Γ(1− D )µD−4λ2b 0 + 24H 2(ξb − 1/6) 0 + 24H 2(ξb − 1/6) ξ(ϕ0, H 2(4π)D/2 Γ(1− D D − 2 D(D − 1) 0 + 24H 2(ξb − 1/6) . (34) From (33) it follows that the counterterms λ0 and ξ0 are given by λ0 ≡ λ(ϕ0, 0) = − 2(4π)D/2 µD−4λ2b − ξ0 ≡ ξ(ϕ0, 0) = 2(4π)D/2 )[ D−2 D(D − 1) . (35) Given thatH and ϕ are dynamical fields, the counterterm parameters λ0 and ξ0 must be independent of H and ϕ, which is indeed satisfied by (35). Now making use of Eqs. (30), (32) and (35) we can calculate the renormalized Lagrangean (31). The result is, Lϕ, ren = − gµν∂µϕ∂νϕ− ( 2H2 ( 2H2 . (36) This is the fully renormalized effective Lagrangean. We now consider the two asymptotic forms of (36), first the ultraviolet (UV) limit. The following asymptotic expansion of the di-gamma function, ψ(z) = (d/dz)[ln(Γ(z))], is then useful (cf. Eq. (8.344) in [26]), ψ(z) = ln(z)− 1 120z4 +O(z−6) , (37) such that = ln(w)− +O(1/w3) , ν2 = − w . (38) Upon evaluating the integral in (27) one obtains ln(w)− ln(w) +O(1/w) . (39) Taking account of this, we can recast Eq. (36) to the form, Lϕ = − gµν∂µϕ∂νϕ− 2 + 24H2 ξb − 16 2 + 24H2 ξb − 16 +O(R3). (40) This is the UV form of (36). In the limit H2 → 0 to flat Minkowski spacetime, the first line in (40) reproduces the classical Coleman-Weinberg result [27]. To complete the analysis of the effective Lagrangean (36) we still need to consider the small field limit of the integral in (36). Let us first consider the expansion which is applicable around the poles of the di-gamma function, which are located at + n , n = 0, 1, 2, 3, . . . . (41) This implies that the poles are located at wn = −n(n + 1) (42) and in the vicinity of the poles we can write w = wn + δw . (43) With the above definitions, a conformally coupled scalar field with ξ = 1/6 corresponds to n = 0 and a minimally coupled scalar field with ξ = 0 corresponds to n = 1. More generally we have, (1− n)(2 + n) , (n ≥ 0) , (44) such that for n > 1 all ξn < 0. In particular, ξ2 = −1/3, ξ3 = −5/6, etc. Now we can expand ν as ν = νn − (δw)2 and by making use of + ℓ− ν we can finally write + bn + cnδw +O((δw)2) . (47) In the above expressions an = n(n+ 1)(2n+ 1) , bn = −2n(n + 1)ψ(n+ 1)− (2n+ 1)− n(n+ 1) 2n+ 1 2n+ 1 + 2ψ(n+ 1) + 2n(n+ 1) 2n+ 1 −ψ′(1) + ψ′(n+ 1)− 2(2n+ 1)2 , (48) and we made use of = ψ(n + 1)− ψ(1) , = −ψ′(n + 1) + ψ′(1) , (49) where ψ(1) = −γE = 0.577215... is the Euler constant, ψ′(1) = π2/6 and ψ′(z + 1) = ψ ′(z)− 1/z2. We can now write the infrared limit of the renormalized Lagrangean (36). From Eqs. (47–48) it follows that the integral in the last line of Eq. (36) has the infrared limit, an ln(δwn) + bn(δwn) + cn(δwn) (δwn) , (50) where δwn = + 12δξn , δξn = ξb + (n− 1)(n+ 2) ≪ 1 (51) and an, bn and cn are defined in Eq. (48). 4. RENORMALIZATION GROUP ANALYSIS In deriving expression (36) we have introduced an arbitrary renormalization scale ϕ0 by definin- ing the renormalization conditions (33). From this definition an arbitrary scale ϕ0 enters into the expressions for the counterterms (35) and hence into the renormalized Lagrangean (36). How- ever, as it was stressed in the classic paper of Coleman and Weinberg in 1973, the change of the renormalization scale can only change the definitions of coupling constants, not the physics [27]. By applying the same reasoning in our case, we arrive at the following conclusion: a small change in ϕ0 in the expression for the physical quantity of interest can always be compensated for by an appropriate small change in λ and ξ. The convenient way of expressing this statement is Veff(ϕ0, λ, ξ, ϕ) = 0, (52) which is the standard Callan-Symanzik (CS) equation written for the theory at hand. The renor- malization group functions βλ and βξ are given by the following relations, βλ = −ϕ0 βξ = −ϕ0 . (53) Within the one-loop approximation the renormalization group functions βλ and βξ are uniquely determined as the coefficients of the divergent logarithmic terms appearing in the counterterms λ0 and ξ0 (35). It follows (writing λb and ξb from now on as λ and ξ, respectively): . (54) These expressions for βλ and βξ we will use to determine the running of λ and ξ with the scale ϕ0. We shall now solve the Callan-Symanzik equation (52). From the theory of partial differential equations we can make use of the method of characteristics [29, 30, 31, 32]. Applying this method to (52) we can write down the solution to the Callan-Symanzik equation (52) as Veff(ϕ0, λ, ξ, ϕ) = Veff(ϕ0(t), λ(t), ξ(t), ϕ(t)), (55) where ϕ0(t), λ(t), ξ(t), ϕ(t) are the running parameters. The t-dependence of the running parameters is given (to the order we are working in) by the following differential equations: dϕ0(t) = ϕ0(t), dϕ(t) dλ(t) = βλ(λ(t)), dξ(t) = βξ(ξ(t), λ(t)). (56) The boundary conditions (at t = 0) are, ϕ0(0) = ϕ0, ϕ(0) = ϕ, λ(0) = λ, ξ(0) = ξ. (57) The solutions of the first two differential equations in (56) are trivial and read, ϕ20(t) = ϕ 2t, ϕ(t) = ϕ . (58) When combined with the previous results (54), the last two differential equations in (56) are solved λ(t) = 1− 3λ , (59) ξ(t) = + (ξ − , (60) where here λ = λ(0) and ξ = ξ(0). These solutions imply that our model exhibits an infrared fixed point, where the coupling constants are λFP = 0 and ξFP = 1/6. In this limit the theory possesses an enhanced symmetry (conformal symmetry) and it can be reduced to a pure metric theory (see Ref. [33] for a nonperturbative proof of this statement). We stress that the parameter t in the above relations is completely arbitrary. The basic idea of the renormalization group (RG) improvement of an effective potential is that we can choose t in such a way that the perturbation series for the effective potential converges more rapidly. Indeed by suitably choosing t one can extend the range of validity of the effective theory to a larger range of the dynamical quantities H and ϕ by replacing the perturbative expression on the left-hand side of (55) by its right-hand side. This is intimately related to the fact that the unimproved expression for the effective potential is actually valid only for ϕ’s sufficiently close to ϕ0. Since the change of the arbitrary scale ϕ0 corresponds just to a reparametrization of the coupling constants within our theory, the unimproved effective potential is valid only near ϕ ∼ ϕ0 and H ∼ 0 and thus not a very useful quantity. If we, on the other hand, require that the effective theory does not depend on ϕ0, then the improved effective potential (which solves the Callan-Symanzik equation) remains valid whenever the coupling constants are small. Since the perturbation series for the effective potential is characterized by the occurrence of powers of logarithmic terms, we choose1 t = ln . (61) The improved expression for the renormalized effective potential now becomes VRG(ϕ) = +6H2ϕ2 where λ and ξ are now t-dependent according to (59) and (60). We remark that even after the improvement logarithmic terms still appear in the expression for the effective potential. As it can be seen from Eq. (40), these logarithmic terms vanish in the limit when H → 0. Had we introduced the renormalization scale H0 for the Hubble parameter and then solved the CS equation for this case, we could in principle get rid off all logarithmic terms. However, as we shall see later, for the calculation of the quantum radiative corrections to slow-roll parameters the expression (62) suffices because our final results do not depend on H . With this in mind we use Eq. (62) in the next section. For an alternative approach to the renormalisation group improved scalar effective theories in de Sitter space see Refs. [34, 35]. 1 The choice (61) is not unique. However, this is the unique choice for which ϕ0∂ϕ0 = −∂t when H2 → 0 in the CS equation (52). For any other choice of t there is an additional prefactor in front of ∂t, and after dividing the CS equation with that prefactor one can solve it as if the β functions are modified, which in turn will modify the functions λ(t) and ξ(t). This in principle leads to a different effective RG improved theory, which however differs from the one we use here only at higher orders in the coupling constants. 5. SLOW-ROLL PARAMETERS In this section we calculate the quantum2 one-loop corrections to the slow-roll parameters ǫ and η arising from the scalar matter vacuum fluctuations in inflation. Within the slow-roll approxi- mation we can drop the kinetic term in the action because it is formally second order in slow-roll parameters [28]. That implies that – within the slow-roll approximation – the leading contribution to the stress-energy tensor is given by3 Tµν = −gµν VRG(ϕ)− δVRG(ϕ) δ lnH , (64) where VRG(ϕ) is the improved renormalized effective potential (62). A straightforward calculation yields, Tµν = −gµν A+ 3H2ϕ2B , (65) where we have introduced A ≡ λ+ 3λ X − 25 B ≡ ξ + λ(ξ − 1 X − 4 + 1 36(ξ − 1 , (66) X ≡ ln . (67) The above result for the stress-energy tensor we insert into the Einstein equation Rµν − gµνR = 8πGNTµν (68) to obtain the following quantum Friedmann equation 3M2Pl A + 3H2ϕ2B , (69) 2 In this work when we refer to the ‘classical’ value of a parameter we mean its tree-level value. When we refer to the ‘quantum correction’ we mean the one-loop contribution to the corresponding parameter. 3 In Eq. (64) we have neglected the tree-level contributions to the stress-energy tensor, which are proportional to ξ(d/dt)2(ϕ2) and ξH(d/dt)(ϕ2). These terms can be neglected based on the observation that the condition Ḣ ≪ H2 together with the slow-roll approximation imply 3ξHϕϕ̇ ≪ 1 V (ϕ) . (63) For a derivation of this condition we refer to Ref. [36]. where M2Pl = 1/(8πGN). If we take from A and B, defined in (66), the leading (classical) contribu- tions, then we can from (69) extract the classical Friedmann equation in the form, H2C = 72M2Pl 1− ξ ϕ , (70) which in the limit ξ → 0 reduces to the well-known result. We shall use equation (70) to calculate the number of e-foldings N in the next section. In order to determine the slow-roll parameters we still need an expression for ϕ̇. From Eq. (62) and the slow-roll form4 of the scalar field equation, 3Hϕ̇+ = 0 , (71) we obtain ϕ̇ = −W , (72) where W ≡ ϕ C + 12H2ϕD + 72 E , (73) C ≡ λ+ (4π)4 D ≡ ξ + λ(ξ − 1 X − 3 + 36(ξ − 1 λ2(ξ − 1 (4π)4 48(ξ − 1 λ(ξ − 1 (4π)4 . (74) Upon inserting the leading contributions from Eq. (74) into Eq. (72), the classical expression for ϕ̇ follows immediately, ϕ̇C = − + 12ξH2Cϕ . (75) It is important to note that with the above definitions C = A+ βλ +O(λ3) , D = B + βξ +O(λ2) . (76) We keep the parameter E in (73) for completeness, although it yields only higher order contributions comparing to other parameters defined by (66) and (74). 4 Within the slow-roll approximation we can drop the ϕ̈ term in Eq. (71) because that term is second order in slow-roll parameters. We now proceed by making use of the standard result for the spectrum of primordial curvature perturbation [28] PR(k) = . (77) In deriving this expression the canonical quantization of the inflaton field has been performed in the standard way, by studying the evolution of small perturbations around the inflaton condensate. Since in our approach the quantum corrections are calculated at the level of the effective potential, which changes the on-shell structure of the theory but does not change the structure of Eq. (77), we conjecture that Eq. (77) can be used without any further generalization for the calculation of the one-loop quantum corrections to the spectrum of curvature perturbation and the implied slow-roll parameters that arise from the matter vacuum fluctuations within the framework proposed in this work. A proof of this conjecture is nevertheless desirable. The right-hand side of (77) is evaluated at the horizon exit, at which k = aH , because during slow-roll inflation the Hubble parameter does not change significantly over a few Hubble times [28]. The scalar spectral index ns is defined as ns − 1 = d lnPR d ln k , (78) which after some algebra, by making use of (77) and (72), yields ns − 1 = − . (79) By analogy with the standard result ns − 1 = −6ǫ+ 2η , (80) which is valid at the classical level for various inflationary models, we define ǫ ≡ W η ≡ 1 , (81) such that equation (80) still holds for the quantum case. On the other hand, for the gravitational wave spectrum we use the result . (82) The gravitational wave spectral index ng is defined as d lnPg d ln k , (83) from which it follows that within our framework, ng = − . (84) After taking into account the definition of ǫ from (81) we obtain that the standard result, ng = −2ǫ , (85) remains valid for the quantum case as well. It is also convenient to introduce the ratio r between the gravitational wave spectrum and the spectrum of primordial curvature perturbation, r ≡ Pg , (86) which here turns into 9M2Pl . (87) With the above definitions the standard relation, r = 16ǫ is not any more satisfied at the quantum level. However, we still expect to reproduce it at the classical limit (but see the discussion below). The final result for ǫ and η, which includes both the classical and quantum contributions, we write in the form ǫ = ǫC + ǫQ , η = ηC + ηQ, (88) and we separate the quantum contributions into the following two characteristic parts, Qǫλ + βξ Qǫξ , Qηλ + βξ Qηξ . (89) Although the two contributions are formally of the same order of magnitude, they have a different origin. The former contribution in Eq. (89) arises as a result of the resummation of the mass insertions m2ϕ = λϕ 2/2 generated by the quartic self-interaction in the presence of an inflaton condensate. The latter contribution is a consequence of the resummation induced by the effective mass parameter 12ξH2 generated by the inflaton field coupled to the background curvature. Now we shall present our results, first the classical part. 5.1. Classical contributions ǫC and ηC After a straightforward calculation, we arrive at , (90) where z and κ are defined by z ≡ ϕ κ ≡ ξz = ξ . (91) It is clear that in the limit when ξ → 0, i.e. when κ → 0, we recover the standard expressions for the slow-roll parameters in the λϕ4 inflationary model; namely ǫ = 8M2Pl/ϕ 2 and η = 12M2Pl/ϕ This is not surprising since in this limit our theory reduces precisely to that inflationary model. We now introduce the number of e-foldings N = − Hdt , (92) which somewhat surprisingly, when calculated classically (H → HC), gives the same result as the λϕ4 inflationary model ϕ2 − ϕ2end . (93) However, a mild ξ-dependence does enter the expression for N through the value of the inflaton field at the end of inflation, ϕend, which is determined from the condition ǫC(ϕend) = 1. From (90) it follows5 ϕ2end ≃ 4M2Pl(2− ξ), (94) and finally = 8Ñ , Ñ ≃ N + 1− ξ . (95) We shall use the above notation when writing the quantum contributions to slow-roll parameters. 5 The result (94) is valid to the leading order in ξ. 5.2. Quantum contributions ǫQ and ηQ In calculating the quantum contributions to slow-roll parameters we must take into account the observational constraint required by the near scale invariance of the spectrum λϕ2 + 12ξH2 ≪ H2 . (96) In view of equations (41)-(48) the observational constraint (96) implies that, in order to study the quantum radiative corrections to slow-roll parameters, we need the infrared limit of the RG improved theory (62). This is the opposite limit from the ultraviolet limit in which our effective theory reduces to the Coleman-Weinberg form (40) [27]. That means that in order to study the quantum radiative corrections to slow-roll parameters, one needs to focus on the infrared radiative corrections which are specific for (quasi-)de Sitter spaces, and completely absent in Minkowski space (which is related by a conformal rescaling to our n = 0 case), and hence has a very different infrared structure. In particular, the most singular term an/δw in (48) is absent in the conformal n = 0 case (a0 = 0). In summary, that means that the infrared sector plays a crucial role in determining the quantum corrections to slow-roll parameters. After taking into account the constraint (96) and the expression (70) for the classical Friedmann equation, we arrive at the condition − 24ξ ≪ 1 , (97) Together with the condition 8Ñξ < 1, this equation then gives, ≪ ξ < . (98) Recall that typically N (and hence also Ñ) is between 50 and 60 such that the term 9/(2Ñ) ∼ 10−1 ≪ 1 in Eq. (98) can be to a good approximation neglected. We proceed by writing approximately the relation (67) as (δw ≪ 1) X = ln − 2γE + +O(δw) , (99) from which it follows = − 1 ζ , (100) where we have introduced ζ ≡ − . (101) Now the calculation of the slow-roll parameters ǫ and η is straightforward; here we present only our final results. Some intermediate steps and results can be found in Appendix A. For the quantum contribution (89) to the slow-roll parameter ǫ we obtain Qǫλ = 1− 29 κ+ 17 (1− κ)2(1− 2 6− 5κ+ κ2 Qǫξ = − (1− κ)2(1− 2 (1− κ)2 2− σ − 2κ+ 1 (1− κ)2 , (102) and for the η slow-roll parameter defined in (89) Qηλ = 1− 29 κ+ 17 (1− κ)2(1− 2 11− 10κ+ 3κ2 = 2Qǫλ − 1 + κ Qηξ = − (1− κ)2(1− 2 (1− κ)2 18− 8σ − 20κ+ 6κ2 (1− κ)2 = 2Qǫξ + 2 . (103) In Eqs. (102) and (103) we abbreviated z ≡ ϕ = 8Ñ , κ ≡ zξ , + γE + ln 6 . (104) Since typically the number of e-foldings required during inflationary epoch ranges between 50 and 60, it is evident from (104) that z is of the order 5×102, which justifies the ordering of the quantum corrections in powers of z. The leading contributions are the terms linear in z, and they are present only in Qǫξ and Qηξ in (102) and (103). Both Qǫξ and Qηξ contain also the next-to-leading terms of the order ln(z). The (subleading) terms of the order z0 are in fact the leading contributions to Qǫλ and Qηλ in (102) and (103). 5.3. Tensor and scalar spectral indices ng and ns We can now easily calculate the tensor and scalar spectral indices from the results for the slow-roll parameters ǫ and η (90), (102–103). Note first that the tensor spectral index ng (85) can be trivially obtained by summing the classical (90) and quantum (102) contributions for ǫ, since from our definition (81) it follows that Eq. (85) is valid also at the quantum level. Next we consider the scalar spectral index ns. By making use of Eqs. (90) and (102–103) and separating again the classical and quantum contributions as, ns − 1 = (ns − 1)C + (ns − 1)Q , (105) where (ns − 1)Q = Q(ns−1)λ + βξ Q(ns−1)ξ , (106) we arrive at the classical scalar spectral index, (ns − 1)C = − , z = 8Ñ , κ = 8Ñξ . (107) The quantum contributions are given by, Q(ns−1)λ = − 1− 29 κ+ 17 (1− κ)2(1− 2 14− 10κ , (108) Q(ns−1)ξ = (1− κ)2(1− 2 (1− κ)2 − 12− 8σ − 8κ (1− κ)2 Notice that the quantum contributions (βλ/λ)Q(ns−1)λ and βξQ(ns−1)ξ are both much smaller than the classical contribution (ns−1)C due to the fact that both βλ/λ and βξ are of the order of λ, which is constrained by experimental data to be of the order of λ ∼ 10−12 (we provide a more precise constraint for λ below). The quantum contributions can become significant only in an inflationary model in which the relevant coupling constant can be as large as the order of 10−3. It is important to notice that Q(ns−1)λ provides an infrared enhanced quantum correction proportional to the number of e-foldings N , while, on the other hand, Q(ns−1)ξ contains a correction which is enhanced by N when compared to the näıve expectation O(1/z) ∼ 1/N). This shows that in the de Sitter invariant limit the quantum corrections to slow-roll parameters accumulate only from the time the mode becomes super-Hubble until the end of inflation. For a mode which exits horizon N e-foldings before the end of inflation, the whole history of inflation before the Hubble exit (i.e. when the mode was sub-Hubble) is completely irrelevant and does not contribute in a cumulative manner to the quantum corrections. This disagrees with the result found in Refs. [12, 13], where it is claimed that the quantum loop corrections induce corrections to the slow-roll parameters and spectral indices which depend on the total duration of inflation and are thus enhanced when inflation lasts for a large number of e-foldings. This N2 enhancement is the main result of our work, and it resolves the Weinberg’s dilemma [10, 11]: how big can be the correction induced by the quantum fluctuations of light or massless scalar fields during inflation, given the fact that the (equal time and space) scalar field correlator for a massless minimally coupled scalar grows linearly with time during (de Sitter) inflation, 〈0|ϕ2(x) |0〉 = infinite + ln a , (109) 〈0|ϕ2(x) |0〉 N , (110) where here N denotes (minus) the number of e-foldings. By comparing Eqs. (85) (102) and (108) we observe that the following curious relation holds, (ng)Q = (ns − 1)Q + 2(1 + κ) − 4βξ , (111) such that the leading quantum contributions O(λÑ) and O(λ ln(Ñ)) to the tensor and scalar spectral indices are equal. This approximate equality can be traced back to the fact that ǫQ ≃ ηQ/2, from where it follows that [(W/H4)(dH2/dϕ)]Q ≃ [(1/H2)(dW/dϕ)]Q, or equiva- lently [d(W/H2)/dϕ]Q ≃ 0. The expression W/H2 is proportional to the square-root of the ratio r = Pg/PR, which according to Eq. (121) does not receive any quantum corrections that are am- plified by N . That means that, because the tensor and scalar spectra are identically affected by the quantum corrections enhanced by N , these leading corrections cancel in the ratio r. We do not have a deeper insight to why that is the case. Note that the approximate equality (111) does not hold for the corresponding classical parts (90) and (107). If inflationary models with large quantum corrections are found, Eq. (111) could be used to resolve the quantum from classical contributions to the tensor and scalar spectral indices. 5.4. The spectrum of curvature perturbation PR and the tensor-to-scalar ratio r Next we consider the spectrum of curvature perturbation (77). As usual we decompose the spectrum into the classical and quantum parts as, PR = (PR)C + (PR)Q , (112) where (PR)Q ≡ QPRλ + βξQPRξ . (113) Working within our framework we obtain, (PR)C = , (114) (PR)C 1− 25 (1− κ)(1− 2 (1− κ)− σ − (PR)C (1− κ)(1− 2 , (115) where Ñ = N +1− 1 ξ, z = 8Ñ and κ = 8ξÑ . This implies that, just like the spectral indices, when compared to the classical contribution, the quantum contribution to the spectrum is suppressed as λÑ2, and thus unobservably small for the model in consideration. The spectrum of curvature perturbation is an observable quantity and the three year WMAP data provide a strong constrain for it [2], PR ≈ 29.5× 10−10A , A = 0.801+0.043−0.054 . (116) In the case of a weak coupling to the background, i.e. when 8|ξ|Ñ ≪ 1 and Ñ ≈ N +1, Eqs. (114) and (116) imply λ50 ≈ 1.58× 10−12 , λ60 ≈ 9.25× 10−13 , (8|ξ|Ñ ≪ 1) , (117) where the subscripts on λ denote the number of e-foldings N . We stress that these values for λ are strictly speaking valid only for this particular model. Indeed, when we choose the coupling to the background in the range, ≫ −ξ > , (118) then from Eq. (114) we see that the value of λ can be up to one order of magnitude larger, λ50 ≃ 2.69× 10−11 × (−24ξ) , λ60 ≃ 1.88× 10−11 × (−24ξ) , (−24ξ ≪ 1) . (119) These larger values of λ are still too small however to render the quantum effects observable. In other inflationary models the relation for the spectrum of curvature perturbation (114) can in general be different, hence allowing for models in which couplings are larger. Furthermore, the quantum effects in some other models may be much stronger – of particular interest are hybrid inflationary models [1]. Let us now consider the ratio r of the gravitational wave and curvature spectrum. From Eq. (87) we obtain r = rC + rQ (120) (1− κ) + 128βξ , (121) where again z ≡ 8Ñ and κ ≡ zξ. We see that rC = 128/z, and after observing from (90) that ǫC = 8/z in the limit when ξ → 0 (i.e. when κ → 0), we obtain the standard result r = 16ǫ. This relation is violated both by the classical corrections from κ 6= 0 (ξ 6= 0) and by the quantum contributions. In particular, when κ = 8Ñξ ≪ −2, one gets r ≃ 32ǫ. When compared with the classical contribution (120), the quantum contribution to r (121) is as usual suppressed by λ, but – in contrast to the slow-roll parameters ǫQ and ηQ – it is not enhanced by powers of N . It is well known that the minimally coupled λϕ4 inflationary model is disfavored by observa- tions [2] by about 2 standard deviations. This is not however in general the case with the non- minimally coupled λϕ4 inflationary model. Indeed, from Eqs. (98) and (107) we infer that in the range ≫ −ξ > 1 . (122) the deviation of the classical spectral index of scalar curvature perturbation (107) from scale invari- ance is reduced approximately by a factor of 2/3, (ns − 1)C ≈ − , (−ξ ≫ (8Ñ)−1) , (123) while the value of rC (120) remains unchanged. This then implies that, similarly as in the minimally coupled massive inflationary model, the λϕ4 model with ξ in the range (122) falls roughly at the 1σ contour of Figure 14 in Ref. [2], rendering these nonminimally coupled λϕ4 inflationary models consistent with the three year WMAP data [2, 4, 5, 6]. We emphasize that, because ξ = ξ(ϕ0) runs logaritmically towards its infrared fixed point ξFP = 1/6, it is natural to assume that ξ deviates from zero. Indeed, even if we choose ξ = 0 (which corresponds to the coupling at some scale ϕ = ϕ0), the running of ξ will induce the dominant quantum contributions to slow-roll parameters. Thus for consistency it is necessary to consider the effects of nonminimal coupling, and choosing ξ in the range (122) is a priory as natural as any other choice (different, of course, from ξ = 1/6). In conclusion, we have found out that, even though the quantum effects to slow-roll parameters are enhanced by the number of e-foldings squared, they are suppressed by the small coupling constant λ. Due to the smallness of λ however, the quantum effects have a negligible impact on the plot presented for example in Figure 14 of [2]. 6. DISCUSSION In this work we develop a quantum field theoretic framework within which the quantum correc- tions to slow-roll parameters and observables from inflationary models can be calculated. Whe main purpose of this paper is methodical, and we postpone a detailed study of the quantum corrections to inflationary observables in various inflationary models to a future work [1]. We illustrate how our framework works by performing the relevant calculations in a concrete inflationary model chosen for its simplicity. More specifically, we consider a λϕ4 inflationary model (13–15) with a nonminimal coupling to the background curvature. Our formalism can be quite straightforwardly generalized to other inflationary models. Within our λϕ4 model we cal- culate the quantum corrections to the inflationary slow-roll parameters ǫ and η (89), (102–103), based on which we derive the scalar spetral index ns (105–108), the tensor spectral index ng (85), (103) and the spectrum of curvature perturbation PR (112–115). These corrections arise from the one-loop scalar vacuum fluctuations during de Sitter inflation. We find that the dominant quantum effects for the spectral indices (2), (85), (103) are suppressed as λN2 when compared to the classical (tree-level) results (3), (90). The dominant quantum contribution arises from the inflaton coupling to the background curvature. Our theoretic framework can be improved in several aspects. For example, one could generalize our calculation of the renormalization group improved effective action (62) to quasi-de Sitter spaces, which are important since these spaces comprise a large fraction of inflationary models. Next one should generalize our analysis to incorporate other matter degrees of freedom which would allow us to incorporate a broad spectrum of inflationary models. Even more importantly one should study the role of the interactions that couple matter and gravitational degrees of freedom. Appendix A Here we provide a more detailed derivation for the quantum contributions to slow-roll parameters given by the relations (102) and (103). We begin by abbreviating the equalities in Eq. (66) as A = λ + βλXA , B = ξ + βξXB , (124) where now X − 4 + ξ − 1 . (125) It is important to note that, in order to strictly follow our notation in Eq. (89), the last term in the definition of XB actually contributes to both Qǫλ and Qηλ. The reason is that, after taking into account the relations (54) for βλ and βξ, it follows immediately that, ξ − 1 . (126) The origin of this mixing is the peculiar 1/36 term in (62), which is suppressed by λ, but not by ξ − 1 It is convenient to introduce the parameter α as follows, , (127) which, from the definitions in (91), can also be expressed as . (128) Assuming that ξϕ2/M2Pl < 1 is far enough from 1, from the quantum Friedmann equation (69) we obtain the following expression, 1− βξαXB − . (129) Upon differentiating the quantum Friedmann equation (69) with respect to ϕ we get, λ+ βλζA + 36 (ξ + βξζB) 1 + βλ + βξα ) . (130) In deriving this expression we have used, βξ , (131) where dX/dϕ and ζ are given by Eqs. (100) and (101), respectively. In writing Eq. (130) we have also introduced ζA ≡ XA + (1 + ζ) , ζB ≡ XB + (1 + ζ) . (132) In order to evaluate ǫ from (81) we still need the expression for W . 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Odintsov, “Effective Lagrangian and the back reaction prob- lem in a selfinteracting O(N) scalar theory in curved space-time,” Phys. Rev. D 50 (1994) 5137 [arXiv:hep-th/9404084]. [36] A. Bilandžić, master’s thesis, “Quantum radiative corrections to slow roll inflation,” http://www1.phys.uu.nl/wwwitf/Teaching/Thesis.htm. http://arxiv.org/abs/hep-ph/9210229 http://arxiv.org/abs/hep-th/9401057 http://arxiv.org/abs/hep-th/9404084 http://www1.phys.uu.nl/wwwitf/Teaching/Thesis.htm Introduction Propagator in de Sitter space de Sitter space Scalar propagator in de Sitter space Effective potential Renormalization group analysis Slow-roll parameters Classical contributions C and C Quantum contributions Q and Q Tensor and scalar spectral indices ng and ns The spectrum of curvature perturbation PR and the tensor-to-scalar ratio r Discussion Appendix A References

We consider the nonminimally coupled lambda phi^4 scalar field theory in de Sitter space and construct the renormalization group improved renormalized effective theory at the one-loop level. Based on the corresponding quantum Friedmann equation and the scalar field equation of motion, we calculate the quantum radiative corrections to the scalar spectral index n_s, gravitational wave spectral index n_g and the ratio r of tensor to scalar perturbations. When compared with the standard (tree-level) values, we find that the quantum contributions are suppressed by lambda N^2 where N denotes the number of e-foldings. Hence there is an N^2 enhancement with respect to the naive expectation, which is due to the infrared enhancement of scalar vacuum fluctuations characterising de Sitter space. Since observations constrain lambda to be very small lambda ~ 10^(-12) and N ~ 50-60, the quantum corrections in this inflationary model are unobservably small.

Introduction Propagator in de Sitter space de Sitter space Scalar propagator in de Sitter space Effective potential Renormalization group analysis Slow-roll parameters Classical contributions C and C Quantum contributions Q and Q Tensor and scalar spectral indices ng and ns The spectrum of curvature perturbation PR and the tensor-to-scalar ratio r Discussion Appendix A References

Ab initio wavefunction based methods for excited states in solids: correlation corrections to the band structure of ionic oxides L. Hozoi, U. Birkenheuer, and P. Fulde Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany A. Mitrushchenkov and H. Stoll Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany (Dated: November 5, 2018) Ab initio wavefunction based methods are applied to the study of electron correlation effects on the band structure of oxide systems. We choose MgO as a prototype closed-shell ionic oxide. Our analysis is based on a local Hamiltonian approach and performed on finite fragments cut from the infinite solid. Localized Wannier functions and embedding potentials are obtained from prior peri- odic Hartree-Fock (HF) calculations. We investigate the role of various electron correlation effects in reducing the HF band gap and modifying the band widths. On-site and nearest-neighbor charge relaxation as well as long-range polarization effects are calculated. Whereas correlation effects are essential for computing accurate band gaps, we found that they produce smaller changes on the HF band widths, at least for this material. Surprisingly, a broadening effect is obtained for the O 2p valence bands. The ab initio data are in good agreement with the energy gap and band width derived from thermoreflectance and x-ray photoemission experiments. The results show that the wavefunction based approach applied here allows for well controlled approximations and a transpar- ent identification of the microscopic processes which determine the electronic band structure. I. INTRODUCTION The proper treatment of electron correlation effects in molecules and solids stands at the heart of modern elec- tronic structure theory1,2,3,4,5. In the study of molecular systems at least, wavefunction based quantum chemistry is known to provide a rigorous theoretical framework for addressing the electron correlation problem. The stan- dard quantum chemical methods4,5 make possible the construction of approximate wavefunctions at levels of increasing sophistication and accuracy and offer thus a systematic route to converged results. Advanced wave- function based calculations can be routinely performed nowadays for small and medium size molecules. However, algorithms able to treat electron correlation effects in pe- riodic systems are still at their infancy. The simplest cor- relation method is based on the Møller-Plesset equations and second-order Møller-Plesset (MP2) perturbational schemes for solids were actually implemented by several groups6,7,8,9,10,11,12,13,14. As an extension of MP2, ab ini- tio many-body Green’s function techniques were devel- oped too15,16,17,18,19. In addition, investigations based on coupled-cluster (CC) theory were initiated13,20,21,22,23. The main point when applying quantum chemical methods to correlation calculations is to make use of the local character of the correlation hole which is surround- ing an electron. The latter optimizes the Coulomb re- pulsion between electrons and its accurate description is the essence of the correlation treatment. Starting point is a Hartree-Fock (HF) calculation on top of which the correlation calculations are implemented. For insulators and semiconductors, the development of efficient compu- tational tools is facilitated by the use of optimally local- ized Wannier functions. Several orbital localization pro- cedures were proposed in the context of periodic HF cal- culations. They normally rest on the a posteriori trans- formation of the crystal Bloch orbitals, see for example Refs. 6,14,20,21,22,24,25. Alternative approaches were elaborated by Shukla et al.26,27,28,29 and in the Toulouse group11,30, where the self-consistent-field (SCF) equa- tions are solved directly in the Wannier representation. Local correlation methods such as the incremental scheme of Stoll31,32 and the local Hamiltonian formal- ism of Horsch et al.33 and Gräfenstein et al.34,35,36 were applied before to the rigorous determination of both ground-state13,26,27,28 and excited-state18,37,38,39,40 properties of infinite systems. Materials under inves- tigation were crystalline LiH, LiF, and LiCl18,26,27,28, diamond38,40, silicon38, beryllium39, carbon, and boron- nitrogen chains13, and trans-polyacetylene37. In the present paper we extend these studies to the case of ionic oxide compounds and choose MgO as a prototype and, at the same time, relatively simple insulating oxide. We compute correlation induced corrections to the valence and conduction HF energy bands and find that our final estimate for the fundamental gap is in rather good agree- ment with the experimental data. Our analysis provides also a clear picture of the major correlation effects that are responsible for the reduction of the HF band gap. The present study on magnesium oxide should open the way to similar calculations for more complex materials such as the 3d transition-metal oxides. II. COMPUTATIONAL APPROACH Periodic HF calculations were carried out for crys- talline MgO with the crystal package41. Wannier functions associated with the HF valence and low-lying conduction bands were determined with the Wannier- http://arxiv.org/abs/0704.1906v1 Boys localization module24 of the same program. For conduction-band states, this may be a somewhat tedious task. However, localized Wannier orbitals can also be de- rived in more difficult cases with band disentanglement techniques25. For computing the correlation induced corrections to the valence and conduction bands we adopt the same quasiparticle picture and local Hamiltonian formalism as employed in Refs. 33,34,35,36,37,38,39,40. The physical reasoning is as follows. When an electron (or hole) is added to the N -particle system, its surroundings relax and polarize due to the additional charge. This response of the system lowers the energy it takes to add the ex- tra electron (or hole). The particle plus the modified surroundings move together through the system in the form of a Bloch wave and define a quasiparticle. At the same time, some of the correlation contributions which are present in the ground-state of the N -particle system are no longer operative in the (N+1) [or (N−1)] -particle system, because some excitations are now blocked. This effect is called loss of ground-state correlation and must be accounted for too33,38. A basic feature of our approach is that the correlation treatment is performed on a finite cluster C cut from the extended periodic system. With the cluster-in-solid embedding technique elaborated by Birkenheuer et al.38,42,43, the cluster C is divided into an active region CA that supports the local occupied and virtual orbitals entering the post-HF calculations and a spatial “buffer” domain CB including a number of atomic sites (and basis functions) whose role is to provide a good representation for the tails of the Wannier-like orbitals of the active region CA. Thus C = CA + CB. All Wannier orbitals centered in CB (and also the environment) will be held frozen in the correlation calculations. The orbital set associated with the finite-size clus- ter is obtained from the data supplied by the Wannier- Boys localization module of crystal24. The interface program44 written for this purpose yields in addition an embedding potential corresponding to the frozen environ- ment, i.e., the surrounding HF electron sea. The detailed procedure for constructing the embedded cluster and the associated orbital set is described elsewhere42,43. Here we merely give an outline of this. The core, valence, and low-lying conduction-band cluster orbitals are generated by projecting the crystal Wannier functions onto the set of atomic basis functions attached to the cluster region |w′n(R)〉 = β,α∈C |β〉S−1 〈α|wn(R)〉 , (1) where α and β are Gaussian basis functions centered within the region C, S−1 represents the inverse overlap matrix for the basis set attached to C, |wn(R)〉 is a Wan- nier orbital with index n and centered in the unit cell corresponding to the lattice vector R, and |w′n(R)〉 is its projected counterpart expressed exclusively in terms of basis functions centered in C. The |w′n(R)〉 func- tions are neither normalized nor are they orthogonal to each other because of the projection procedure mentioned above. Therefore they are group-wise orthonormalized in the following order: active core, active occupied, active low-lying conduction-band orbitals, buffer core, buffer oc- cupied, buffer low-lying conduction-band orbitals. This way the contamination of the most important types of or- bitals is minimized. Group-wise orthonormalization ac- tually means to use Schmidt orthogonalization for the inter-group orthogonalization and Löwdin orthonormal- ization inside the groups. This set of orthonormal or- bitals will be denoted by |w̃′n(R)〉. For the construction of the variational space to be used in the subsequent cor- relation calculations we follow the prescription suggested by Pulay and Saebø45. So-called virtual projected atomic orbitals (PAO’s) are generated from the Gaussian basis functions associated with the active region CA by pro- jecting out the occupied and the low-lying conduction- band orbitals |w̃′n(R)〉 via a Schmidt orthogonalization scheme. Thereafter, the PAO’s are Löwdin orthonormal- ized among themselves to facilitate their subsequent use in the correlation calculations. By making the buffer region CB sufficiently large the original Wannier orbitals of CA are well represented. For example, for the Wannier orbitals corresponding to the lowest four conduction bands of MgO and centered at a Mg site, the six ligands surrounding the given Mg ion should be included in CA because large contributions arise not only from the Mg 3s and 3p basis functions but also from the nearest-neighbor (NN) oxygen 2s,3s and 2p,3p components. Farther neighbors need not be included in the central region CA but may be put into the buffer zone CB because the weight of the longer-range tails is small. Whereas a high-quality description can be easily achieved for the Wannier orbitals centered in the active region, the representation of the Wannier functions in the buffer zone is less accurate. The impact of this deficiency on the correlation calculations, however, is completely compensated by an appropriate choice of the embedding potential. Actually, the Gaussian orbital representation V embαβ of the embedding potential is constructed by V embαβ = F − F [P C ]αβ , α, β ∈ C (2) where F crys is the self-consistent Fock operator from the periodic HF calculation and F [P C ] is the Fock operator associated with the density operator P C = 2 |w̃′ν〉〈w̃′ν | (3) arising from all occupied orbitals |w̃′ν〉 which enter the subsequent correlation calculations explicitly. This way, the correlation calculations are effectively performed in an infinite frozen HF environment. The data concerning the occupied and virtual orbitals of the cluster is transferred via the crystal-molpro interface44 to the quantum chemistry program mol- 46. The same holds for the matrix representation of the self-consitent Fock operator of the periodic host system. The embedding potential itself is con- structed according to Eqs. (2) and (3) using the MA- TROP module of the molpro program package. Local electron removal and electron addition one- particle configurations can be defined in terms of the set of occupied and virtual orbitals localized within the spa- tial domain CA: |ΦN−1 〉 = cRpσ|Φ〉 and |ΦN+1Rqσ 〉 = c |Φ〉 , (4) where cRpσ and c are annihilation and creation op- erators for the valence and conduction band σ-spin or- bitals |w̃′p(R)〉 and |w̃′q(R)〉, respectively, and |Φ〉 is the single-determinant ground-state wavefunction of the N - electron system. For clusters which are large enough, the Hartree-Fock valence and conduction energy bands of the periodic crystal can be recovered by diagonalizing k-dependent matrices of the following form: HHFnn′(k) = eikR〈ΦN∓1 0nσ |H − E 0 |ΦN∓1Rn′σ〉 . (5) Here 0 stands for the reference unit cell and EHF0 denotes the ground-state HF energy of the neutral N -electron system. The diagonal terms R=0, n′=n in this expres- sion are directly related to the on-site excitation energies within the Koopmans approximation4,5, i.e., ionization potentials IPHFpp (0) = 〈ΦN−10pσ |H |Φ 0pσ 〉 − E 0 = −ǫHF0p > 0 (6) and electron affinities EAHFqq (0) = E 0 − 〈ΦN+10qσ |H |Φ 0qσ 〉 = −ǫ 0q , (7) with the latter being negative in the case of MgO. The off-diagonal terms are the hopping matrix elements in a tight-binding representation, tHFnn′(R) = 〈ΦN∓10nσ |H − EHF0 |Φ 〉 . (8) Starting from the Hartree-Fock energy bands we in- clude next the effects of electron correlations. Within the quasiparticle approximation we may introduce for the (N+1)-particle system a wavefunction of the following form: |ΨN+1 〉 = exp (S) c† |Ψ〉 , (9) where |Ψ〉 is the ground-state of the N electron system. The operator S does not commute with c and de- scribes the correlation hole. For a more detailed discus- sion we refer to Ref. 1. Alternatively, we may write |ΨN+1 〉 = Ω c† |Φ〉, where |Φ〉 is the SCF ground-state and Ω acts like a wave- or Moeller operator3. Therefore, when correlations are taken into account, the analogue of (5) becomes Hnn′(k) = eikR〈ΨN∓1 0nσ |H − E0|Ψ 〉 , (10) where E0 is the energy of the N -particle correlated ground-state |Ψ〉 (see also Refs. 34,35,36,37,38). The cor- relation hole around an added electron (or hole) consists of a short-range and a long-range part. The short-range part originates from intra-atomic and short-range inter- atomic relaxation and polarization effects. We construct the short-range part of the correlation hole by separate orbital optimizations. Thereby the Wannier orbital to which the extra electron (hole) is attached is kept frozen and the changes within a finite region around it are deter- mined by performing an additional SCF calculation38,39. This is sufficient if electron correlations are weak or mod- erate. If they are strong we would have to take into account the fact that changes in the nearby surround- ings are decreased when correlation effects among the electrons in that neighborhood are accounted for. The long-range part of the correlation hole consists of long range polarization of the environment. The effect of long- range polarization on the diagonal Hamiltonian matrix elements is estimated in this paper by applying the ap- proximation of a dielectric continuum. With the known dielectric constant of MgO, ǫ0=9.7, we can directly de- termine the polarization energy of a charge ±e outside a sphere of radiusR. In addition, differential correlation ef- fects related to the existence of a different number of elec- trons in the system’s ground-state and in the (N±1) ex- cited states are investigated by subsequent configuration- interaction (CI) calculations. III. CORRELATION CORRECTIONS TO THE BAND STRUCTURE OF MGO Magnesium oxide is a prototype closed-shell ionic ma- terial that crystallizes under normal conditions in the rocksalt structure. It is extensively used in materials sci- ence as substrate for the epitaxial growth of films of other compounds. In recent years, it has attracted renewed in- FIG. 1: Hartree-Fock band structure of bulk MgO. The core O 1s and Mg 1s− 2p bands are not shown in the figure. FIG. 2: Plot of the Mg 3s-like conduction-band Wannier or- bital after projection onto a [Mg19O14] cluster. There is sub- stantial weight at the NN oxygen sites. terest because of its use as tunnel barrier in magnetic tun- nel junctions47. A different area where MgO attracted at- tention is optoelectronics. It turns out that when alloyed with ZnO, depending on the precise chemical composi- tion, the band gap of the system can be tuned along an interval ranging from 3.3 to 7.8 eV48. The latter number represents the fundamental gap of MgO49. The valence bands of MgO have oxygen 2p character, whereas the lowest conduction bands are mainly related to the Mg 3s and 3p orbitals, with some admixture from the O 2s,3s and 2p,3p functions. The Hartree-Fock en- ergy bands are shown in Fig.1. We employed for our calculations the lattice constant reported by Sasaki et al., a=4.217 Å50, and Gaussian-type basis sets from the standard crystal library. Basis sets of triple-zeta qual- ity, Mg 8-51151, were used for the cations and triple-zeta basis sets suplemented with polarization functions, O 8- 411∗51, were applied for the more polarizable O ions. At the Hartree-Fock level and with this choice of the basis functions, the fundamental gap of the system is 16.20 eV, i.e., 8.4 eV larger than the experimental value. As illustrated in Fig.1, there is a separation of about 16 eV between the O 2s and O 2p bands. There is also a clear separation between the low-lying Mg 3s, 3p complex and the other conduction bands. Density-functional calcula- tions within the local density approximation and using the same Gaussian basis sets as for the HF calculations give a gap of 5.0 eV between the valence and conduction bands, nearly 3 eV lower than observed in experiment. Projected Wannier orbitals associated with the Mg 3s, 3p conduction-band complex are plotted in Fig.2 and Fig.3. We note that the Wannier-Boys localization mod- ule of the crystal program yields a set of Mg sp3 hybrids for the lowest four conduction-band states. In order to arrive at a set of s and x, y, z -like functions, we applied to those projected hybrids a Pipek-Mezey localization procedure52, as implemented in the molpro package46. In this localization scheme, the number of atomic or- bitals contributing to a given molecular-like composite is minimized. Whereas the Mg 3s and 3p Wannier func- tions have substantial weight at the NN ligand sites, the valence-band O 2p (and 2s) Wannier orbitals are much more compact and contributions from neighboring ions are not visible in plots like those shown in Figs. 2 and 3. In the calculations reported here the norm of the pro- jected active orbitals is typically larger than 99.0% and never below 98.5% of the original crystal Wannier func- tions. A. Correlation induced corrections to the diagonal matrix elements Relaxation and polarization effects in the immedi- ate neighborhood of an oxygen hole were computed by separate restricted open-shell HF (ROHF) calculations on [O39Mg30] clusters. The multiconfiguration MCSCF module of the molpro package was employed for this purpose. The active region CA of the [O39Mg30] clus- ter contains a central 2p5 (or 2s1) O− ion and four of the twelve nearest O2− ligands. These four oxygen neighbors are all chosen to be in the same plane and we denote them as O1xy,...O xy (see Fig.4). In addition, we include in the cluster C the NN cations and all the O’s in the next co- ordination shell of each of the Oixy sites, as sketched in Fig.4. These additional Mg and O neighbors represent the so-called buffer region CB and ensure an accurate de- scription of the tails of the orbitals centered in the active region CA. When performing the ROHF calculations, the oxygen hole orbital is kept frozen39. We also freeze in our calcu- lations the core-like 1s, 2s, and 2p shells of all Mg ions. Orbital relaxation effects are listed in Table I for both 2s FIG. 3: Mg p-like conduction-band Wannier orbital after pro- jection onto a [Mg19O14] cluster. and 2p oxygen holes. The on-site relaxation effect is quite large, more than 2 eV. There is also a substantial relax- ation/polarization effect associated with the first oxygen neighbors. For the four Oixy ligands included in the ac- tive region CA of the cluster, this effect amounts to about 0.41 eV in the presence of an O 2s hole and to 0.40 eV in the presence of a 2p hole. In Table I, we multiplied these numbers by three because there is a total of twelve oxygens in that coordination shell. We obtain thus a good estimate for the relaxation and polarization effects up to the nearest O neighbors. The fact that these relax- ation/polarization effects are additive was checked by ex- tra calculations with smaller, double-zeta basis sets51 on two different clusters: a [O39Mg30] cluster including only the four Oixy sites in the active region and a [O55Mg38] cluster where all twelve first oxygen neighbors were al- lowed to polarize. The four lowest-energy (N+1) conduction-band states imply Mg 3s1 and 3p1 electron configurations. ROHF calculations were performed for such configurations on a [Mg19O38] cluster with a [MgO6] kernel as active region, CA. Beyond the [MgO6] kernel, this cluster incorporates again all Mg and O ions in the first two coordination shells of the active ligands. The on-site relaxation effects associated with the addition of an electron in a localized Mg 3s or 3p Wannier orbital are vanishingly small. The relaxation effects at the adjacent O sites induce energy shifts of 0.80–0.85 eV, see Table I. In these calculations the open-shell active orbitals (Mg 3s1 or 3p1) were again kept frozen39. We note that the energetic effect is nearly the same for the 3s1 and 3p1 conduction-band states. At the scale of Fig.1 at least, it induces an uniform down- FIG. 4: Sketch of the [O39Mg30] cluster employed for the calculation of short-range relaxation effects on the O valence- band states. The shortest line segments are Mg–O “bonds”. The so-called active region CA includes a central O ion and four nearest oxygen neighbors Oixy in the “horizontal” plane, see text. These active ions are shown as small black spheres. TABLE I: Correlation induced corrections to the diagonal Hamiltonian matrix elements for the valence-band O 2s, 2p and conduction-band Mg 3s, 3p states. All numbers are in eV. Negative corrections induce upwards shifts of the valence bands and shifts to lower energies for the conduction bands. ∆Hnn(0) O 2s O 2p Mg 3s Mg 3p On-site orb. relaxation −2.64 −2.04 — — NN orb. relaxation −1.23 −1.20 −0.81 −0.84 Long-range polarization −1.80 −1.80 −2.25 −2.25 Total −5.67 −5.04 −3.06 −3.09 wards shift of the center of gravity of the 3s−3p band complex. The data listed in Table I indicate that the on-site orbital relaxation and relaxation and polarization effects at the nearest oxygen sites in the presence of an extra electron or extra electron hole results in a reduction of the HF band gap by about 4.05 eV, that is, more than 45% of the difference between the HF and experimental values. Large corrections are also expected to arise from long- range polarization effects. The long-range polarization energy of a dielectric due to the presence of an extra charge±e can be expressed as ∆E(∞) = ∆E(R)−C/R1, where ǫ0 − 1 e2 , (11) ǫ0 is the static dielectric constant of the material, and R defines a sphere around the extra charge beyond which the dielectric response reaches its asymptotic value ǫ0. The energy increment ∆E(R) denotes the relaxation and polarization energy up to the radius R around the added particle. The constantC can be obtained by choosing two different radii R1 and R2 where the quantities ∆E(R1) and ∆E(R2) are computed, see for example Ref. 35. However, we adopt here a simpler approach. We cal- culate the corrections due to long-range polarization by using the experimental value for the static dielectric con- stant, ǫ0 = 9.7. Since relaxation and polarization effects related to the nearest oxygen neighbors were already ac- counted for, both, for the valence-band hole states and the conduction-band electrons (see Table I) and since the core-like electrons of the Mg2+ ions can be ignored in these calculations, we set R as the average of the radii of the first and second oxygen coordination shells around a localized 2p hole or 3s(3p) electron: R = (a 2/2 + a)/2 for the O 2p valence-band states, where a = 4.217 Å is the lattice constant50, and R = (a/2 + a 3/2)/2 for the conduction-band states. The corrections to the diagonal matrix elements of the Hamiltonian are then ∆Hnn(0) = − ǫ0 − 1 , (12) about −1.80 eV for the O 2s/2p bands and −2.25 eV for the Mg 3s/3p bands. These numbers are also included in Table I. Before we discuss band narrowing (or broadening) due to correlation effects which also affect the band gap, we consider the loss of ground-state correlations. The latter leads again to a shift of the center of gravity of the bands. As pointed out in the previous section, some of the config- urations that are present in the N -particle ground-state are blocked when an electron is added or removed. We investigated such correlations by CI calculations with sin- gle and double excitations (CISD) and discuss first dif- ferential effects for the N and (N−1) states. Since the oxygen valence-band Wannier orbitals are rather local- ized, we designed a cluster with a single O ion in the active region. Around this central O site we added one shell of Mg ions (6 Mg’s) and two shells of anions (12+6 O’s) to build the buffer region CB. In the CISD calcula- tions for the N and (N−1) configurations we correlate the 2s and 2p orbitals of the central O ion. Thereby the occupancy of the hole orbital is kept frozen in the calculations for the (N−1) states, which is referred to as the frozen hole approximation38,39. Sets of separately optimized orbitals were used for the hole states of the (N−1)-particle system, as discussed above. We found that for a 2p hole the correction to the on-site matrix element of the Hamiltonian is ∆Hnn(0) = 0.85 eV, i.e., the 2p valence bands are downshifted by 0.85 eV. For the O 2s hole states, this correction amounts to 0.99 eV. One would expect similarly an upwards shift of the conduction bands. However, the situation is somewhat different here. When an extra electron is attached to the Mg2+ ion, it polarizes the closed shells of the core. This is the domi- nant effect now because Mg2+ has no valence electrons. We may employ for our analysis the high-quality results obtained for a free Mg ion by Doll et al.53. The correc- tion to the ROHF Mg+→Mg2+ ionization potential was found to be 0.27 eV in Ref. 53. A similar differential cor- relation effect is occuring for the conduction-band states in bulk MgO. Therefore the conduction bands are shifted downwards instead of upwards and a partial cancellation between loss of ground-state correlation effects for the valence and conduction bands is taking place. The net result is a slight increase of the gap between the valence and conduction bands, in the range of 0.5 eV. Among the different contributions discussed here, this appears to produce the smallest corrections to the gap. More advanced calculations for studying such differential cor- relation effects are left for future work. To summarize the results listed in Table I, relaxation and polarization effects in bulk MgO are responsible for a reduction of the Hartree-Fock gap by 8.1 eV, which represents about 95% of the difference between the HF and experimental values, 16.2 and 7.8, respectively. Im- proved agreement is expected between our results and the experimental data when applying higher-quality ba- sis sets. We mention in this context that a reduction of 3.8 eV is obtained for the HF gap when going from va- lence double-zeta to triple-zeta basis sets. Nevertheless, this large energy difference is mainly related to the very poor representation of the conduction-band states in the TABLE II: Nearest–neighbor (NN), RNN = (1, 1, 0)a/2, and next-nearest-neighbor (NNN), RNNN = (1, 0, 0)a, hopping matrix elements for the conduction-band Mg 3s and 3p or- bitals, see text. Results of frozen-orbital CI (FO-CI) are listed in the second column; NOCI results in terms of separately optimized, relaxed orbitals (RO-NOCI) are given in the third column. All numbers are in eV. tnn′(R) FO-CI RO-NOCI tNN : 3s − 3s 0.41 0.42 3px(y) − 3px(y) 0.66 0.69 3px(y) − 3py(x) 0.72 0.77 3pz − 3pz 0.13 0.13 tNNN : 3s − 3s 0.36 0.37 3px − 3px 0.77 0.74 3py(z) − 3py(z) 0.13 0.12 calculations with double-zeta basis functions and such ef- fects will be less substantial by further extension of the basis sets. We also keep in mind that differential correla- tion effects due to the existence of a different number of electrons in the system’s ground-state and in the (N±1) excited states determine a small correction in the oppo- site direction, i.e., a slight increase of the fundamental B. Off-diagonal matrix elements We discuss next the effect of correlations on the widths of the different bands. For that purpose the off-diagonal matrix elements of the effective Hamiltonian (10) have to be determined, i.e., the so-called hopping terms. At the Hartree-Fock level, these matrix elements are obtained by solving 2×2 secular equations where both wavefunc- tions are expressed in terms of localized HF orbitals, see Eqs. (4), (5), and (8). Relaxation and polarization ef- fects in the nearby surroundings of the added electron (or hole) are obtained by separate SCF optimizations for the (N±1) states. The separate optimization of the (N±1) wavefunctions leads to sets of non-orthogonal or- bitals. There will be thus both Hamiltonian and over- lap matrix elements between the extra electron (extra hole) wavefunctions ΨN±1 0nσ and Ψ . The calculation of such matrix elements has been recently implemented in molpro54. It is based on the transformation of the CI vectors to bi-orthogonal orbitals and follows an idea suggested by Malmqvist55. A similar approach was devel- oped by Broer et al. in Groningen56. Instead of putting these matrix elements directly into the eigenvalue equa- tions that determine the Bloch energies ǫn(k) we extract from these data effective hopping parameters associated with various pairs of orbitals. Comparison between such effective hopping terms and the HF off-diagonal Hamil- tonian matrix elements offers an insightful picture of how correlation effects modify the inter-site interactions and FIG. 5: Sketch of the [Mg28O36] cluster employed for the calculation of the NN conduction-band hopppings. The active region CA includes two NN Mg sites, small grey spheres, plus the bridging and apical ligands, large black spheres. consequently the widths of the bands. For energetically degenerate states, the effective hopping is defined as tnn′ = (Hnn′ − Snn′Hnn)/(1− S2nn′) , (13) where Hnn′ and Snn′ are the Hamiltonian and over- lap matrix elements between the (N ± 1) states n and n′. Since the separately optimized wavefunctions are ex- pressed in terms of sets of non-orthogonal orbitals, this type of secular problem is usually referred to as non- orthogonal CI (NOCI). In the case of mutually orthogo- nal states Snn′ is zero and tnn′(R) = Hnn′(R). Nearest-neighbor and next-nearest-neighbor (NNN) hopping matrix elements for the more diffuse conduction- band Mg 3s and 3p orbitals are listed in Table II. In the rocksalt structure, there are two O ions bridging two NN cations, whereas two NNN Mg’s share a single oxygen. We designed a [Mg28O36] cluster for calculating the NN hoppings and a [Mg36O47] cluster for the NNN matrix elements. As active regions, we employed [Mg2O6] and [Mg2O9] kernels, respectively, see Fig.5 and Fig.6. As in the calculations for the on-site matrix elements, for each of these clusters we included in the buffer region CB all metal and O ions in the first two coordination shells of the active oxygens. Since the second cluster consists of 83 atoms, we reduced the computational effort by re- moving the polarization functions at the oxygen sites for this particular cluster. Two different values are given in Table II for each hopping matrix element. Results ex- tracted from 2×2 CI calculations in terms of frozen HF Wannier orbitals are listed in the second column. In the third column, we allowed for full relaxation of the 2s and 2p orbitals of the O ions included in the active region CA, for each of the (N+1) configuration state functions en- tering the 2×2 CI. As illustrated in Figs. 5 and 6, those active anions are the ligand(s) bridging two Mg sites and the ligands which are nearest oxygen neighbors of one of the active Mg sites and also of the bridging ligand(s). The results show that the separate optimization of the (N+1) wavefunctions induces only minor changes on the electron hoppings, in the range of few percent. An inter- esting feature is that variations occur in both directions, i.e., some of the effective hoppings are slightly enlarged by taking into account short-range relaxation and polar- ization effects and some are reduced. Since these changes are quite small, the width (and the overall structure) of the lower conduction bands will change very little. Re- garding the longer-range polarization effects, we expect that their influence on the hopping terms is negligible. In the case of non-equivalent orbitals, neighboring Mg 3s and Mg 3p, it is more difficult to define some NOCI ef- fective hoppings and the corresponding data is not shown in Table II. Nevertheless, the effect of short-range relax- ation and polarization is also small for these interactions, with reductions of the CI splittings of few meV. The same type of analysis was applied to the (N−1) O 2s1/2p5 hole states. The spatial extent of the oxygen or- bitals and the inter-site matrix elements are significantly smaller. Since the largest relaxation effects concern or- bitals at the same site, see Table I, we included in a first step only two ligands in the active regions of our clusters. Nevertheless, few other atomic shells were added around these active sites in the buffer region CB. We considered a [O28Mg10] cluster for computing the relaxation effects on the NN hoppings and a [O32Mg11] cluster for the NNN terms. The results are collected in Table III. The largest corrections arise for the NN matrix elements, with abso- lute values that are similar to the corrections obtained for the NN conduction-band hoppings. In relative numbers, these corrections are somewhat larger, by 10% to 20% for the NN hoppings. A very different situation occurs in strongly correlated oxides such as the layered cuprates, where the existence of an antiferromagnetic spin back- ground determines a reduction of the effective quasipar- ticle hoppings by a factor of four57. FIG. 6: Sketch of the [Mg36O47] cluster employed for the cal- culation of the NNN conduction-band hopppings. The active region CA includes two Mg sites, small grey spheres, and nine O neighbors, large black spheres. TABLE III: NN, RNN = (1, 1, 0)a/2, and NNN, RNNN = (1, 0, 0)a, hopping matrix elements for the valence-band states. Results of frozen-orbital CI (FO-CI) are listed in the second column; NOCI results in terms of separately op- timized, relaxed orbitals (RO-NOCI) are given in the third column. All numbers are in eV. Data for the oxygen 2s or- bitals are also included, although the O 2s band is much be- low the O 2p bands. The values in parentheses include orbital relaxation effects at four additional O sites, see text. tnn′(R) FO-CI RO-NOCI tNN : 2s− 2s 0.10 0.12 (0.11) 2px(y) − 2px(y) 0.32 0.37 (0.36) 2px(y) − 2py(x) 0.42 0.49 (0.47) 2pz − 2pz 0.12 0.14 (0.13) tNNN : 2s− 2s 0.01 0.01 2px − 2px 0.06 0.06 The trends displayed in Table III are confirmed by multi-reference CISD [MRCI(SD)] calculations. We cor- related in these calculations the eight 2s and 2p orbitals at the two NN oxygen sites and the reference active space included those two orbitals involved in the hopping pro- cess. The hopping matrix element is half of the en- ergy separation between the lowest two eigenstates. The MRCI values for the 2px−2px and 2px−2py hoppings, for example, are 0.33 and 0.43 eV. These numbers are again larger than the HF results, although the magnitude of the effect is smaller as compared to the NOCI calculations. Having the data of the SCF calculations for the (N−1) oxygen hole states at hand, we performed an analysis of the changes produced in the composition of the (relaxed) orbitals in the immediate vicinity of an O− 2s1 or 2p5 anion. We found that an oxygen hole is causing polariza- tion and “bending” of the 2p orbitals at the NN ligand sites. The “bending” of the NN 2p orbitals towards the O− site takes place through both p−s and pi−pj mixing. These effects result in stronger inter-site orbital overlap and explain the fact that the effective NOCI hoppings are larger than the corresponding HF values. A similar analysis for the conduction-band (N+1) states is compli- cated by the presence of several sets of “active” orbitals, Mg 3s, 3p and bridging O 2p, and we could draw no clear conclusions in that case. Extra calculations were performed for the NN hop- ping matrix elements with four additional oxygen sites included in the active region of the cluster, CA. Those are the four ligands which are nearest neighbors to both O ions involved in the hopping process. They are situated in the median plane of the segment RNN = (1, 1, 0)a/2. A cluster composed of 62 sites, [O40Mg22], was employed for these calculations. The results are given in Table III in parentheses. The corrections due to relaxation and po- larization at the four nearest O neighbors are very small, 0.01 to 0.02 eV. The effect of these corrections is to re- duce somewhat the absolute values of the NN hoppings. Experimental studies for characterizing the valence electronic structure of MgO have been carried out using x-ray photoelectron spectroscopy (XPS)58 and angle-resolved ultraviolet photoelectron spectroscopy (ARUPS)59. The measured width of the O 2p bands is about 6.5 eV58,59. The HF valence-band width is 5.50 eV and inclusion of local correlations leads to a slight broadening of the O 2p bands, which brings the ab initio result in good agreement with the experiment. For com- parison, density-functional calculations within the local density approximation and using the same Gaussian ba- sis sets as in the HF calculations predict a width of 4.68 eV for the O 2p bands. The correlation induced corrections to the widths of the bands also influence the band gap. In the fcc lat- tice the dispersion of p bands at the Γ point depends on two of the nearest-neighbor hoppings, ǫx(Γ) = const. + (110) x,x − 4t(011)x,x + ...60,61. With the notations from Ta- ble III, t (011) x,x = t (110) z,z . Corrections of 0.04 eV for t (110) and 0.01 eV for t (011) x,x (or t (110) z,z ), see Table III, imply an upwards shift of the O 2p bands at the Γ point and a narrowing of the fundamental gap by about 0.30 eV. For the Mg 3s−3p conduction-band complex such changes at the Γ point are smaller because the correlation induced corrections to the 3s−3s and 3s−3p inter-site matrix elements are lower. IV. SUMMARY AND CONCLUSIONS We have analyzed the different correlation contribu- tions to the energy gap of MgO and to the widths of the conduction and valence bands. This was done within the quasiparticle description. As regards correlation ef- fects we have distinguished between relaxation and po- larization around an electron or hole added to the (neu- tral) ground-state. The net result is a reduction of the Hartree-Fock gap from 16.2 eV to a value of 8.1 eV. This has to be compared with a measured energy gap of 7.8 eV. Within the local density approximation (LDA) to density functional theory a gap of 5.0 eV is found. This is not surprising since LDA is known to produce too small gaps for insulators. The calculations were performed with triple-zeta basis sets. Since on the Hartree-Fock level the calculated gap differs for double- and triple-zeta basis sets by 3.8 eV, one may consider the good agreement with the experimental gap as somewhat fortuitous. Triple-zeta basis sets are known, however, to produce reliable results in quantum chemistry and therefore a further extension of the basis set should keep the corrections small. Presently we are not able to work with larger basis sets. It was shown that a large contribution to the correla- tion induced corrections to the fundamental gap comes from on-site and nearest-neighbor relaxation, i.e., from the immediate neighborhood of the added particle. But also the long-range part of the polarization generated by the extra particle contributes substantially to the reduc- tion of the HF gap. This long-range part can be treated in a continuum approximation thereby using the known dielectric constant of MgO. The so-called loss of ground- state correlations makes a small contribution in MgO. The reason is that the conduction-band Wannier orbitals have predominant Mg 3s or 3p character. The added electron will essentially go thus to a Mg site where it polarizes the closed 1s2, 2s2, and 2p6 shells. This effect reduces the gap and counteracts the loss of ground-state correlations which occurs when an electron is removed (hole state) and therefore can no longer contribute to the correlations of the remaining ones. Finally, also changes in the widths of the bands influence the energy gap. One surprising effect which we found is an enhance- ment of the width of the valence bands when local corre- lations are taken into account. 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Ab initio wavefunction based methods are applied to the study of electron correlation effects on the band structure of oxide systems. We choose MgO as a prototype closed-shell ionic oxide. Our analysis is based on a local Hamiltonian approach and performed on finite fragments cut from the infinite solid. Localized Wannier functions and embedding potentials are obtained from prior periodic Hartree-Fock (HF) calculations. We investigate the role of various electron correlation effects in reducing the HF band gap and modifying the band widths. On-site and nearest-neighbor charge relaxation as well as long-range polarization effects are calculated. Whereas correlation effects are essential for computing accurate band gaps, we found that they produce smaller changes on the HF band widths, at least for this material. Surprisingly, a broadening effect is obtained for the O 2p valence bands. The ab initio data are in good agreement with the energy gap and band width derived from thermoreflectance and x-ray photoemission experiments. The results show that the wavefunction based approach applied here allows for well controlled approximations and a transparent identification of the microscopic processes which determine the electronic band structure.

Ab initio wavefunction based methods for excited states in solids: correlation corrections to the band structure of ionic oxides L. Hozoi, U. Birkenheuer, and P. Fulde Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany A. Mitrushchenkov and H. Stoll Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany (Dated: November 5, 2018) Ab initio wavefunction based methods are applied to the study of electron correlation effects on the band structure of oxide systems. We choose MgO as a prototype closed-shell ionic oxide. Our analysis is based on a local Hamiltonian approach and performed on finite fragments cut from the infinite solid. Localized Wannier functions and embedding potentials are obtained from prior peri- odic Hartree-Fock (HF) calculations. We investigate the role of various electron correlation effects in reducing the HF band gap and modifying the band widths. On-site and nearest-neighbor charge relaxation as well as long-range polarization effects are calculated. Whereas correlation effects are essential for computing accurate band gaps, we found that they produce smaller changes on the HF band widths, at least for this material. Surprisingly, a broadening effect is obtained for the O 2p valence bands. The ab initio data are in good agreement with the energy gap and band width derived from thermoreflectance and x-ray photoemission experiments. The results show that the wavefunction based approach applied here allows for well controlled approximations and a transpar- ent identification of the microscopic processes which determine the electronic band structure. I. INTRODUCTION The proper treatment of electron correlation effects in molecules and solids stands at the heart of modern elec- tronic structure theory1,2,3,4,5. In the study of molecular systems at least, wavefunction based quantum chemistry is known to provide a rigorous theoretical framework for addressing the electron correlation problem. The stan- dard quantum chemical methods4,5 make possible the construction of approximate wavefunctions at levels of increasing sophistication and accuracy and offer thus a systematic route to converged results. Advanced wave- function based calculations can be routinely performed nowadays for small and medium size molecules. However, algorithms able to treat electron correlation effects in pe- riodic systems are still at their infancy. The simplest cor- relation method is based on the Møller-Plesset equations and second-order Møller-Plesset (MP2) perturbational schemes for solids were actually implemented by several groups6,7,8,9,10,11,12,13,14. As an extension of MP2, ab ini- tio many-body Green’s function techniques were devel- oped too15,16,17,18,19. In addition, investigations based on coupled-cluster (CC) theory were initiated13,20,21,22,23. The main point when applying quantum chemical methods to correlation calculations is to make use of the local character of the correlation hole which is surround- ing an electron. The latter optimizes the Coulomb re- pulsion between electrons and its accurate description is the essence of the correlation treatment. Starting point is a Hartree-Fock (HF) calculation on top of which the correlation calculations are implemented. For insulators and semiconductors, the development of efficient compu- tational tools is facilitated by the use of optimally local- ized Wannier functions. Several orbital localization pro- cedures were proposed in the context of periodic HF cal- culations. They normally rest on the a posteriori trans- formation of the crystal Bloch orbitals, see for example Refs. 6,14,20,21,22,24,25. Alternative approaches were elaborated by Shukla et al.26,27,28,29 and in the Toulouse group11,30, where the self-consistent-field (SCF) equa- tions are solved directly in the Wannier representation. Local correlation methods such as the incremental scheme of Stoll31,32 and the local Hamiltonian formal- ism of Horsch et al.33 and Gräfenstein et al.34,35,36 were applied before to the rigorous determination of both ground-state13,26,27,28 and excited-state18,37,38,39,40 properties of infinite systems. Materials under inves- tigation were crystalline LiH, LiF, and LiCl18,26,27,28, diamond38,40, silicon38, beryllium39, carbon, and boron- nitrogen chains13, and trans-polyacetylene37. In the present paper we extend these studies to the case of ionic oxide compounds and choose MgO as a prototype and, at the same time, relatively simple insulating oxide. We compute correlation induced corrections to the valence and conduction HF energy bands and find that our final estimate for the fundamental gap is in rather good agree- ment with the experimental data. Our analysis provides also a clear picture of the major correlation effects that are responsible for the reduction of the HF band gap. The present study on magnesium oxide should open the way to similar calculations for more complex materials such as the 3d transition-metal oxides. II. COMPUTATIONAL APPROACH Periodic HF calculations were carried out for crys- talline MgO with the crystal package41. Wannier functions associated with the HF valence and low-lying conduction bands were determined with the Wannier- http://arxiv.org/abs/0704.1906v1 Boys localization module24 of the same program. For conduction-band states, this may be a somewhat tedious task. However, localized Wannier orbitals can also be de- rived in more difficult cases with band disentanglement techniques25. For computing the correlation induced corrections to the valence and conduction bands we adopt the same quasiparticle picture and local Hamiltonian formalism as employed in Refs. 33,34,35,36,37,38,39,40. The physical reasoning is as follows. When an electron (or hole) is added to the N -particle system, its surroundings relax and polarize due to the additional charge. This response of the system lowers the energy it takes to add the ex- tra electron (or hole). The particle plus the modified surroundings move together through the system in the form of a Bloch wave and define a quasiparticle. At the same time, some of the correlation contributions which are present in the ground-state of the N -particle system are no longer operative in the (N+1) [or (N−1)] -particle system, because some excitations are now blocked. This effect is called loss of ground-state correlation and must be accounted for too33,38. A basic feature of our approach is that the correlation treatment is performed on a finite cluster C cut from the extended periodic system. With the cluster-in-solid embedding technique elaborated by Birkenheuer et al.38,42,43, the cluster C is divided into an active region CA that supports the local occupied and virtual orbitals entering the post-HF calculations and a spatial “buffer” domain CB including a number of atomic sites (and basis functions) whose role is to provide a good representation for the tails of the Wannier-like orbitals of the active region CA. Thus C = CA + CB. All Wannier orbitals centered in CB (and also the environment) will be held frozen in the correlation calculations. The orbital set associated with the finite-size clus- ter is obtained from the data supplied by the Wannier- Boys localization module of crystal24. The interface program44 written for this purpose yields in addition an embedding potential corresponding to the frozen environ- ment, i.e., the surrounding HF electron sea. The detailed procedure for constructing the embedded cluster and the associated orbital set is described elsewhere42,43. Here we merely give an outline of this. The core, valence, and low-lying conduction-band cluster orbitals are generated by projecting the crystal Wannier functions onto the set of atomic basis functions attached to the cluster region |w′n(R)〉 = β,α∈C |β〉S−1 〈α|wn(R)〉 , (1) where α and β are Gaussian basis functions centered within the region C, S−1 represents the inverse overlap matrix for the basis set attached to C, |wn(R)〉 is a Wan- nier orbital with index n and centered in the unit cell corresponding to the lattice vector R, and |w′n(R)〉 is its projected counterpart expressed exclusively in terms of basis functions centered in C. The |w′n(R)〉 func- tions are neither normalized nor are they orthogonal to each other because of the projection procedure mentioned above. Therefore they are group-wise orthonormalized in the following order: active core, active occupied, active low-lying conduction-band orbitals, buffer core, buffer oc- cupied, buffer low-lying conduction-band orbitals. This way the contamination of the most important types of or- bitals is minimized. Group-wise orthonormalization ac- tually means to use Schmidt orthogonalization for the inter-group orthogonalization and Löwdin orthonormal- ization inside the groups. This set of orthonormal or- bitals will be denoted by |w̃′n(R)〉. For the construction of the variational space to be used in the subsequent cor- relation calculations we follow the prescription suggested by Pulay and Saebø45. So-called virtual projected atomic orbitals (PAO’s) are generated from the Gaussian basis functions associated with the active region CA by pro- jecting out the occupied and the low-lying conduction- band orbitals |w̃′n(R)〉 via a Schmidt orthogonalization scheme. Thereafter, the PAO’s are Löwdin orthonormal- ized among themselves to facilitate their subsequent use in the correlation calculations. By making the buffer region CB sufficiently large the original Wannier orbitals of CA are well represented. For example, for the Wannier orbitals corresponding to the lowest four conduction bands of MgO and centered at a Mg site, the six ligands surrounding the given Mg ion should be included in CA because large contributions arise not only from the Mg 3s and 3p basis functions but also from the nearest-neighbor (NN) oxygen 2s,3s and 2p,3p components. Farther neighbors need not be included in the central region CA but may be put into the buffer zone CB because the weight of the longer-range tails is small. Whereas a high-quality description can be easily achieved for the Wannier orbitals centered in the active region, the representation of the Wannier functions in the buffer zone is less accurate. The impact of this deficiency on the correlation calculations, however, is completely compensated by an appropriate choice of the embedding potential. Actually, the Gaussian orbital representation V embαβ of the embedding potential is constructed by V embαβ = F − F [P C ]αβ , α, β ∈ C (2) where F crys is the self-consistent Fock operator from the periodic HF calculation and F [P C ] is the Fock operator associated with the density operator P C = 2 |w̃′ν〉〈w̃′ν | (3) arising from all occupied orbitals |w̃′ν〉 which enter the subsequent correlation calculations explicitly. This way, the correlation calculations are effectively performed in an infinite frozen HF environment. The data concerning the occupied and virtual orbitals of the cluster is transferred via the crystal-molpro interface44 to the quantum chemistry program mol- 46. The same holds for the matrix representation of the self-consitent Fock operator of the periodic host system. The embedding potential itself is con- structed according to Eqs. (2) and (3) using the MA- TROP module of the molpro program package. Local electron removal and electron addition one- particle configurations can be defined in terms of the set of occupied and virtual orbitals localized within the spa- tial domain CA: |ΦN−1 〉 = cRpσ|Φ〉 and |ΦN+1Rqσ 〉 = c |Φ〉 , (4) where cRpσ and c are annihilation and creation op- erators for the valence and conduction band σ-spin or- bitals |w̃′p(R)〉 and |w̃′q(R)〉, respectively, and |Φ〉 is the single-determinant ground-state wavefunction of the N - electron system. For clusters which are large enough, the Hartree-Fock valence and conduction energy bands of the periodic crystal can be recovered by diagonalizing k-dependent matrices of the following form: HHFnn′(k) = eikR〈ΦN∓1 0nσ |H − E 0 |ΦN∓1Rn′σ〉 . (5) Here 0 stands for the reference unit cell and EHF0 denotes the ground-state HF energy of the neutral N -electron system. The diagonal terms R=0, n′=n in this expres- sion are directly related to the on-site excitation energies within the Koopmans approximation4,5, i.e., ionization potentials IPHFpp (0) = 〈ΦN−10pσ |H |Φ 0pσ 〉 − E 0 = −ǫHF0p > 0 (6) and electron affinities EAHFqq (0) = E 0 − 〈ΦN+10qσ |H |Φ 0qσ 〉 = −ǫ 0q , (7) with the latter being negative in the case of MgO. The off-diagonal terms are the hopping matrix elements in a tight-binding representation, tHFnn′(R) = 〈ΦN∓10nσ |H − EHF0 |Φ 〉 . (8) Starting from the Hartree-Fock energy bands we in- clude next the effects of electron correlations. Within the quasiparticle approximation we may introduce for the (N+1)-particle system a wavefunction of the following form: |ΨN+1 〉 = exp (S) c† |Ψ〉 , (9) where |Ψ〉 is the ground-state of the N electron system. The operator S does not commute with c and de- scribes the correlation hole. For a more detailed discus- sion we refer to Ref. 1. Alternatively, we may write |ΨN+1 〉 = Ω c† |Φ〉, where |Φ〉 is the SCF ground-state and Ω acts like a wave- or Moeller operator3. Therefore, when correlations are taken into account, the analogue of (5) becomes Hnn′(k) = eikR〈ΨN∓1 0nσ |H − E0|Ψ 〉 , (10) where E0 is the energy of the N -particle correlated ground-state |Ψ〉 (see also Refs. 34,35,36,37,38). The cor- relation hole around an added electron (or hole) consists of a short-range and a long-range part. The short-range part originates from intra-atomic and short-range inter- atomic relaxation and polarization effects. We construct the short-range part of the correlation hole by separate orbital optimizations. Thereby the Wannier orbital to which the extra electron (hole) is attached is kept frozen and the changes within a finite region around it are deter- mined by performing an additional SCF calculation38,39. This is sufficient if electron correlations are weak or mod- erate. If they are strong we would have to take into account the fact that changes in the nearby surround- ings are decreased when correlation effects among the electrons in that neighborhood are accounted for. The long-range part of the correlation hole consists of long range polarization of the environment. The effect of long- range polarization on the diagonal Hamiltonian matrix elements is estimated in this paper by applying the ap- proximation of a dielectric continuum. With the known dielectric constant of MgO, ǫ0=9.7, we can directly de- termine the polarization energy of a charge ±e outside a sphere of radiusR. In addition, differential correlation ef- fects related to the existence of a different number of elec- trons in the system’s ground-state and in the (N±1) ex- cited states are investigated by subsequent configuration- interaction (CI) calculations. III. CORRELATION CORRECTIONS TO THE BAND STRUCTURE OF MGO Magnesium oxide is a prototype closed-shell ionic ma- terial that crystallizes under normal conditions in the rocksalt structure. It is extensively used in materials sci- ence as substrate for the epitaxial growth of films of other compounds. In recent years, it has attracted renewed in- FIG. 1: Hartree-Fock band structure of bulk MgO. The core O 1s and Mg 1s− 2p bands are not shown in the figure. FIG. 2: Plot of the Mg 3s-like conduction-band Wannier or- bital after projection onto a [Mg19O14] cluster. There is sub- stantial weight at the NN oxygen sites. terest because of its use as tunnel barrier in magnetic tun- nel junctions47. A different area where MgO attracted at- tention is optoelectronics. It turns out that when alloyed with ZnO, depending on the precise chemical composi- tion, the band gap of the system can be tuned along an interval ranging from 3.3 to 7.8 eV48. The latter number represents the fundamental gap of MgO49. The valence bands of MgO have oxygen 2p character, whereas the lowest conduction bands are mainly related to the Mg 3s and 3p orbitals, with some admixture from the O 2s,3s and 2p,3p functions. The Hartree-Fock en- ergy bands are shown in Fig.1. We employed for our calculations the lattice constant reported by Sasaki et al., a=4.217 Å50, and Gaussian-type basis sets from the standard crystal library. Basis sets of triple-zeta qual- ity, Mg 8-51151, were used for the cations and triple-zeta basis sets suplemented with polarization functions, O 8- 411∗51, were applied for the more polarizable O ions. At the Hartree-Fock level and with this choice of the basis functions, the fundamental gap of the system is 16.20 eV, i.e., 8.4 eV larger than the experimental value. As illustrated in Fig.1, there is a separation of about 16 eV between the O 2s and O 2p bands. There is also a clear separation between the low-lying Mg 3s, 3p complex and the other conduction bands. Density-functional calcula- tions within the local density approximation and using the same Gaussian basis sets as for the HF calculations give a gap of 5.0 eV between the valence and conduction bands, nearly 3 eV lower than observed in experiment. Projected Wannier orbitals associated with the Mg 3s, 3p conduction-band complex are plotted in Fig.2 and Fig.3. We note that the Wannier-Boys localization mod- ule of the crystal program yields a set of Mg sp3 hybrids for the lowest four conduction-band states. In order to arrive at a set of s and x, y, z -like functions, we applied to those projected hybrids a Pipek-Mezey localization procedure52, as implemented in the molpro package46. In this localization scheme, the number of atomic or- bitals contributing to a given molecular-like composite is minimized. Whereas the Mg 3s and 3p Wannier func- tions have substantial weight at the NN ligand sites, the valence-band O 2p (and 2s) Wannier orbitals are much more compact and contributions from neighboring ions are not visible in plots like those shown in Figs. 2 and 3. In the calculations reported here the norm of the pro- jected active orbitals is typically larger than 99.0% and never below 98.5% of the original crystal Wannier func- tions. A. Correlation induced corrections to the diagonal matrix elements Relaxation and polarization effects in the immedi- ate neighborhood of an oxygen hole were computed by separate restricted open-shell HF (ROHF) calculations on [O39Mg30] clusters. The multiconfiguration MCSCF module of the molpro package was employed for this purpose. The active region CA of the [O39Mg30] clus- ter contains a central 2p5 (or 2s1) O− ion and four of the twelve nearest O2− ligands. These four oxygen neighbors are all chosen to be in the same plane and we denote them as O1xy,...O xy (see Fig.4). In addition, we include in the cluster C the NN cations and all the O’s in the next co- ordination shell of each of the Oixy sites, as sketched in Fig.4. These additional Mg and O neighbors represent the so-called buffer region CB and ensure an accurate de- scription of the tails of the orbitals centered in the active region CA. When performing the ROHF calculations, the oxygen hole orbital is kept frozen39. We also freeze in our calcu- lations the core-like 1s, 2s, and 2p shells of all Mg ions. Orbital relaxation effects are listed in Table I for both 2s FIG. 3: Mg p-like conduction-band Wannier orbital after pro- jection onto a [Mg19O14] cluster. and 2p oxygen holes. The on-site relaxation effect is quite large, more than 2 eV. There is also a substantial relax- ation/polarization effect associated with the first oxygen neighbors. For the four Oixy ligands included in the ac- tive region CA of the cluster, this effect amounts to about 0.41 eV in the presence of an O 2s hole and to 0.40 eV in the presence of a 2p hole. In Table I, we multiplied these numbers by three because there is a total of twelve oxygens in that coordination shell. We obtain thus a good estimate for the relaxation and polarization effects up to the nearest O neighbors. The fact that these relax- ation/polarization effects are additive was checked by ex- tra calculations with smaller, double-zeta basis sets51 on two different clusters: a [O39Mg30] cluster including only the four Oixy sites in the active region and a [O55Mg38] cluster where all twelve first oxygen neighbors were al- lowed to polarize. The four lowest-energy (N+1) conduction-band states imply Mg 3s1 and 3p1 electron configurations. ROHF calculations were performed for such configurations on a [Mg19O38] cluster with a [MgO6] kernel as active region, CA. Beyond the [MgO6] kernel, this cluster incorporates again all Mg and O ions in the first two coordination shells of the active ligands. The on-site relaxation effects associated with the addition of an electron in a localized Mg 3s or 3p Wannier orbital are vanishingly small. The relaxation effects at the adjacent O sites induce energy shifts of 0.80–0.85 eV, see Table I. In these calculations the open-shell active orbitals (Mg 3s1 or 3p1) were again kept frozen39. We note that the energetic effect is nearly the same for the 3s1 and 3p1 conduction-band states. At the scale of Fig.1 at least, it induces an uniform down- FIG. 4: Sketch of the [O39Mg30] cluster employed for the calculation of short-range relaxation effects on the O valence- band states. The shortest line segments are Mg–O “bonds”. The so-called active region CA includes a central O ion and four nearest oxygen neighbors Oixy in the “horizontal” plane, see text. These active ions are shown as small black spheres. TABLE I: Correlation induced corrections to the diagonal Hamiltonian matrix elements for the valence-band O 2s, 2p and conduction-band Mg 3s, 3p states. All numbers are in eV. Negative corrections induce upwards shifts of the valence bands and shifts to lower energies for the conduction bands. ∆Hnn(0) O 2s O 2p Mg 3s Mg 3p On-site orb. relaxation −2.64 −2.04 — — NN orb. relaxation −1.23 −1.20 −0.81 −0.84 Long-range polarization −1.80 −1.80 −2.25 −2.25 Total −5.67 −5.04 −3.06 −3.09 wards shift of the center of gravity of the 3s−3p band complex. The data listed in Table I indicate that the on-site orbital relaxation and relaxation and polarization effects at the nearest oxygen sites in the presence of an extra electron or extra electron hole results in a reduction of the HF band gap by about 4.05 eV, that is, more than 45% of the difference between the HF and experimental values. Large corrections are also expected to arise from long- range polarization effects. The long-range polarization energy of a dielectric due to the presence of an extra charge±e can be expressed as ∆E(∞) = ∆E(R)−C/R1, where ǫ0 − 1 e2 , (11) ǫ0 is the static dielectric constant of the material, and R defines a sphere around the extra charge beyond which the dielectric response reaches its asymptotic value ǫ0. The energy increment ∆E(R) denotes the relaxation and polarization energy up to the radius R around the added particle. The constantC can be obtained by choosing two different radii R1 and R2 where the quantities ∆E(R1) and ∆E(R2) are computed, see for example Ref. 35. However, we adopt here a simpler approach. We cal- culate the corrections due to long-range polarization by using the experimental value for the static dielectric con- stant, ǫ0 = 9.7. Since relaxation and polarization effects related to the nearest oxygen neighbors were already ac- counted for, both, for the valence-band hole states and the conduction-band electrons (see Table I) and since the core-like electrons of the Mg2+ ions can be ignored in these calculations, we set R as the average of the radii of the first and second oxygen coordination shells around a localized 2p hole or 3s(3p) electron: R = (a 2/2 + a)/2 for the O 2p valence-band states, where a = 4.217 Å is the lattice constant50, and R = (a/2 + a 3/2)/2 for the conduction-band states. The corrections to the diagonal matrix elements of the Hamiltonian are then ∆Hnn(0) = − ǫ0 − 1 , (12) about −1.80 eV for the O 2s/2p bands and −2.25 eV for the Mg 3s/3p bands. These numbers are also included in Table I. Before we discuss band narrowing (or broadening) due to correlation effects which also affect the band gap, we consider the loss of ground-state correlations. The latter leads again to a shift of the center of gravity of the bands. As pointed out in the previous section, some of the config- urations that are present in the N -particle ground-state are blocked when an electron is added or removed. We investigated such correlations by CI calculations with sin- gle and double excitations (CISD) and discuss first dif- ferential effects for the N and (N−1) states. Since the oxygen valence-band Wannier orbitals are rather local- ized, we designed a cluster with a single O ion in the active region. Around this central O site we added one shell of Mg ions (6 Mg’s) and two shells of anions (12+6 O’s) to build the buffer region CB. In the CISD calcula- tions for the N and (N−1) configurations we correlate the 2s and 2p orbitals of the central O ion. Thereby the occupancy of the hole orbital is kept frozen in the calculations for the (N−1) states, which is referred to as the frozen hole approximation38,39. Sets of separately optimized orbitals were used for the hole states of the (N−1)-particle system, as discussed above. We found that for a 2p hole the correction to the on-site matrix element of the Hamiltonian is ∆Hnn(0) = 0.85 eV, i.e., the 2p valence bands are downshifted by 0.85 eV. For the O 2s hole states, this correction amounts to 0.99 eV. One would expect similarly an upwards shift of the conduction bands. However, the situation is somewhat different here. When an extra electron is attached to the Mg2+ ion, it polarizes the closed shells of the core. This is the domi- nant effect now because Mg2+ has no valence electrons. We may employ for our analysis the high-quality results obtained for a free Mg ion by Doll et al.53. The correc- tion to the ROHF Mg+→Mg2+ ionization potential was found to be 0.27 eV in Ref. 53. A similar differential cor- relation effect is occuring for the conduction-band states in bulk MgO. Therefore the conduction bands are shifted downwards instead of upwards and a partial cancellation between loss of ground-state correlation effects for the valence and conduction bands is taking place. The net result is a slight increase of the gap between the valence and conduction bands, in the range of 0.5 eV. Among the different contributions discussed here, this appears to produce the smallest corrections to the gap. More advanced calculations for studying such differential cor- relation effects are left for future work. To summarize the results listed in Table I, relaxation and polarization effects in bulk MgO are responsible for a reduction of the Hartree-Fock gap by 8.1 eV, which represents about 95% of the difference between the HF and experimental values, 16.2 and 7.8, respectively. Im- proved agreement is expected between our results and the experimental data when applying higher-quality ba- sis sets. We mention in this context that a reduction of 3.8 eV is obtained for the HF gap when going from va- lence double-zeta to triple-zeta basis sets. Nevertheless, this large energy difference is mainly related to the very poor representation of the conduction-band states in the TABLE II: Nearest–neighbor (NN), RNN = (1, 1, 0)a/2, and next-nearest-neighbor (NNN), RNNN = (1, 0, 0)a, hopping matrix elements for the conduction-band Mg 3s and 3p or- bitals, see text. Results of frozen-orbital CI (FO-CI) are listed in the second column; NOCI results in terms of separately optimized, relaxed orbitals (RO-NOCI) are given in the third column. All numbers are in eV. tnn′(R) FO-CI RO-NOCI tNN : 3s − 3s 0.41 0.42 3px(y) − 3px(y) 0.66 0.69 3px(y) − 3py(x) 0.72 0.77 3pz − 3pz 0.13 0.13 tNNN : 3s − 3s 0.36 0.37 3px − 3px 0.77 0.74 3py(z) − 3py(z) 0.13 0.12 calculations with double-zeta basis functions and such ef- fects will be less substantial by further extension of the basis sets. We also keep in mind that differential correla- tion effects due to the existence of a different number of electrons in the system’s ground-state and in the (N±1) excited states determine a small correction in the oppo- site direction, i.e., a slight increase of the fundamental B. Off-diagonal matrix elements We discuss next the effect of correlations on the widths of the different bands. For that purpose the off-diagonal matrix elements of the effective Hamiltonian (10) have to be determined, i.e., the so-called hopping terms. At the Hartree-Fock level, these matrix elements are obtained by solving 2×2 secular equations where both wavefunc- tions are expressed in terms of localized HF orbitals, see Eqs. (4), (5), and (8). Relaxation and polarization ef- fects in the nearby surroundings of the added electron (or hole) are obtained by separate SCF optimizations for the (N±1) states. The separate optimization of the (N±1) wavefunctions leads to sets of non-orthogonal or- bitals. There will be thus both Hamiltonian and over- lap matrix elements between the extra electron (extra hole) wavefunctions ΨN±1 0nσ and Ψ . The calculation of such matrix elements has been recently implemented in molpro54. It is based on the transformation of the CI vectors to bi-orthogonal orbitals and follows an idea suggested by Malmqvist55. A similar approach was devel- oped by Broer et al. in Groningen56. Instead of putting these matrix elements directly into the eigenvalue equa- tions that determine the Bloch energies ǫn(k) we extract from these data effective hopping parameters associated with various pairs of orbitals. Comparison between such effective hopping terms and the HF off-diagonal Hamil- tonian matrix elements offers an insightful picture of how correlation effects modify the inter-site interactions and FIG. 5: Sketch of the [Mg28O36] cluster employed for the calculation of the NN conduction-band hopppings. The active region CA includes two NN Mg sites, small grey spheres, plus the bridging and apical ligands, large black spheres. consequently the widths of the bands. For energetically degenerate states, the effective hopping is defined as tnn′ = (Hnn′ − Snn′Hnn)/(1− S2nn′) , (13) where Hnn′ and Snn′ are the Hamiltonian and over- lap matrix elements between the (N ± 1) states n and n′. Since the separately optimized wavefunctions are ex- pressed in terms of sets of non-orthogonal orbitals, this type of secular problem is usually referred to as non- orthogonal CI (NOCI). In the case of mutually orthogo- nal states Snn′ is zero and tnn′(R) = Hnn′(R). Nearest-neighbor and next-nearest-neighbor (NNN) hopping matrix elements for the more diffuse conduction- band Mg 3s and 3p orbitals are listed in Table II. In the rocksalt structure, there are two O ions bridging two NN cations, whereas two NNN Mg’s share a single oxygen. We designed a [Mg28O36] cluster for calculating the NN hoppings and a [Mg36O47] cluster for the NNN matrix elements. As active regions, we employed [Mg2O6] and [Mg2O9] kernels, respectively, see Fig.5 and Fig.6. As in the calculations for the on-site matrix elements, for each of these clusters we included in the buffer region CB all metal and O ions in the first two coordination shells of the active oxygens. Since the second cluster consists of 83 atoms, we reduced the computational effort by re- moving the polarization functions at the oxygen sites for this particular cluster. Two different values are given in Table II for each hopping matrix element. Results ex- tracted from 2×2 CI calculations in terms of frozen HF Wannier orbitals are listed in the second column. In the third column, we allowed for full relaxation of the 2s and 2p orbitals of the O ions included in the active region CA, for each of the (N+1) configuration state functions en- tering the 2×2 CI. As illustrated in Figs. 5 and 6, those active anions are the ligand(s) bridging two Mg sites and the ligands which are nearest oxygen neighbors of one of the active Mg sites and also of the bridging ligand(s). The results show that the separate optimization of the (N+1) wavefunctions induces only minor changes on the electron hoppings, in the range of few percent. An inter- esting feature is that variations occur in both directions, i.e., some of the effective hoppings are slightly enlarged by taking into account short-range relaxation and polar- ization effects and some are reduced. Since these changes are quite small, the width (and the overall structure) of the lower conduction bands will change very little. Re- garding the longer-range polarization effects, we expect that their influence on the hopping terms is negligible. In the case of non-equivalent orbitals, neighboring Mg 3s and Mg 3p, it is more difficult to define some NOCI ef- fective hoppings and the corresponding data is not shown in Table II. Nevertheless, the effect of short-range relax- ation and polarization is also small for these interactions, with reductions of the CI splittings of few meV. The same type of analysis was applied to the (N−1) O 2s1/2p5 hole states. The spatial extent of the oxygen or- bitals and the inter-site matrix elements are significantly smaller. Since the largest relaxation effects concern or- bitals at the same site, see Table I, we included in a first step only two ligands in the active regions of our clusters. Nevertheless, few other atomic shells were added around these active sites in the buffer region CB. We considered a [O28Mg10] cluster for computing the relaxation effects on the NN hoppings and a [O32Mg11] cluster for the NNN terms. The results are collected in Table III. The largest corrections arise for the NN matrix elements, with abso- lute values that are similar to the corrections obtained for the NN conduction-band hoppings. In relative numbers, these corrections are somewhat larger, by 10% to 20% for the NN hoppings. A very different situation occurs in strongly correlated oxides such as the layered cuprates, where the existence of an antiferromagnetic spin back- ground determines a reduction of the effective quasipar- ticle hoppings by a factor of four57. FIG. 6: Sketch of the [Mg36O47] cluster employed for the cal- culation of the NNN conduction-band hopppings. The active region CA includes two Mg sites, small grey spheres, and nine O neighbors, large black spheres. TABLE III: NN, RNN = (1, 1, 0)a/2, and NNN, RNNN = (1, 0, 0)a, hopping matrix elements for the valence-band states. Results of frozen-orbital CI (FO-CI) are listed in the second column; NOCI results in terms of separately op- timized, relaxed orbitals (RO-NOCI) are given in the third column. All numbers are in eV. Data for the oxygen 2s or- bitals are also included, although the O 2s band is much be- low the O 2p bands. The values in parentheses include orbital relaxation effects at four additional O sites, see text. tnn′(R) FO-CI RO-NOCI tNN : 2s− 2s 0.10 0.12 (0.11) 2px(y) − 2px(y) 0.32 0.37 (0.36) 2px(y) − 2py(x) 0.42 0.49 (0.47) 2pz − 2pz 0.12 0.14 (0.13) tNNN : 2s− 2s 0.01 0.01 2px − 2px 0.06 0.06 The trends displayed in Table III are confirmed by multi-reference CISD [MRCI(SD)] calculations. We cor- related in these calculations the eight 2s and 2p orbitals at the two NN oxygen sites and the reference active space included those two orbitals involved in the hopping pro- cess. The hopping matrix element is half of the en- ergy separation between the lowest two eigenstates. The MRCI values for the 2px−2px and 2px−2py hoppings, for example, are 0.33 and 0.43 eV. These numbers are again larger than the HF results, although the magnitude of the effect is smaller as compared to the NOCI calculations. Having the data of the SCF calculations for the (N−1) oxygen hole states at hand, we performed an analysis of the changes produced in the composition of the (relaxed) orbitals in the immediate vicinity of an O− 2s1 or 2p5 anion. We found that an oxygen hole is causing polariza- tion and “bending” of the 2p orbitals at the NN ligand sites. The “bending” of the NN 2p orbitals towards the O− site takes place through both p−s and pi−pj mixing. These effects result in stronger inter-site orbital overlap and explain the fact that the effective NOCI hoppings are larger than the corresponding HF values. A similar analysis for the conduction-band (N+1) states is compli- cated by the presence of several sets of “active” orbitals, Mg 3s, 3p and bridging O 2p, and we could draw no clear conclusions in that case. Extra calculations were performed for the NN hop- ping matrix elements with four additional oxygen sites included in the active region of the cluster, CA. Those are the four ligands which are nearest neighbors to both O ions involved in the hopping process. They are situated in the median plane of the segment RNN = (1, 1, 0)a/2. A cluster composed of 62 sites, [O40Mg22], was employed for these calculations. The results are given in Table III in parentheses. The corrections due to relaxation and po- larization at the four nearest O neighbors are very small, 0.01 to 0.02 eV. The effect of these corrections is to re- duce somewhat the absolute values of the NN hoppings. Experimental studies for characterizing the valence electronic structure of MgO have been carried out using x-ray photoelectron spectroscopy (XPS)58 and angle-resolved ultraviolet photoelectron spectroscopy (ARUPS)59. The measured width of the O 2p bands is about 6.5 eV58,59. The HF valence-band width is 5.50 eV and inclusion of local correlations leads to a slight broadening of the O 2p bands, which brings the ab initio result in good agreement with the experiment. For com- parison, density-functional calculations within the local density approximation and using the same Gaussian ba- sis sets as in the HF calculations predict a width of 4.68 eV for the O 2p bands. The correlation induced corrections to the widths of the bands also influence the band gap. In the fcc lat- tice the dispersion of p bands at the Γ point depends on two of the nearest-neighbor hoppings, ǫx(Γ) = const. + (110) x,x − 4t(011)x,x + ...60,61. With the notations from Ta- ble III, t (011) x,x = t (110) z,z . Corrections of 0.04 eV for t (110) and 0.01 eV for t (011) x,x (or t (110) z,z ), see Table III, imply an upwards shift of the O 2p bands at the Γ point and a narrowing of the fundamental gap by about 0.30 eV. For the Mg 3s−3p conduction-band complex such changes at the Γ point are smaller because the correlation induced corrections to the 3s−3s and 3s−3p inter-site matrix elements are lower. IV. SUMMARY AND CONCLUSIONS We have analyzed the different correlation contribu- tions to the energy gap of MgO and to the widths of the conduction and valence bands. This was done within the quasiparticle description. As regards correlation ef- fects we have distinguished between relaxation and po- larization around an electron or hole added to the (neu- tral) ground-state. The net result is a reduction of the Hartree-Fock gap from 16.2 eV to a value of 8.1 eV. This has to be compared with a measured energy gap of 7.8 eV. Within the local density approximation (LDA) to density functional theory a gap of 5.0 eV is found. This is not surprising since LDA is known to produce too small gaps for insulators. The calculations were performed with triple-zeta basis sets. Since on the Hartree-Fock level the calculated gap differs for double- and triple-zeta basis sets by 3.8 eV, one may consider the good agreement with the experimental gap as somewhat fortuitous. Triple-zeta basis sets are known, however, to produce reliable results in quantum chemistry and therefore a further extension of the basis set should keep the corrections small. Presently we are not able to work with larger basis sets. It was shown that a large contribution to the correla- tion induced corrections to the fundamental gap comes from on-site and nearest-neighbor relaxation, i.e., from the immediate neighborhood of the added particle. But also the long-range part of the polarization generated by the extra particle contributes substantially to the reduc- tion of the HF gap. This long-range part can be treated in a continuum approximation thereby using the known dielectric constant of MgO. The so-called loss of ground- state correlations makes a small contribution in MgO. The reason is that the conduction-band Wannier orbitals have predominant Mg 3s or 3p character. The added electron will essentially go thus to a Mg site where it polarizes the closed 1s2, 2s2, and 2p6 shells. This effect reduces the gap and counteracts the loss of ground-state correlations which occurs when an electron is removed (hole state) and therefore can no longer contribute to the correlations of the remaining ones. Finally, also changes in the widths of the bands influence the energy gap. One surprising effect which we found is an enhance- ment of the width of the valence bands when local corre- lations are taken into account. 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Mermin, Solid State Physics (Thomson Brooks/Cole, 1976). http://arxiv.org/abs/cond-mat/0611720

Mon. Not. R. Astron. Soc. 000, 1–14 (2007) Printed 7 September 2021 (MN LATEX style file v2.2) Abundances in intermediate-mass AGB stars undergoing third dredge-up and hot-bottom burning J.A. McSaveney1,2, P.R. Wood1, M. Scholz3, J.C. Lattanzio2 and K.H. Hinkle4 1Mount Stromlo Observatory, Research School of Astronomy and Astrophysics, Australian National University, Cotter Road, Weston Creek ACT 2611, Australia. 2Centre for Stellar and Planetary Astrophysics, Department of Mathematical Sciences, Monash University, Melbourne, Australia. 3 Institut f. Theoretische Astrophysik d. Univ. Heidelberg, Albert-Ueberle-Strasse 2, 69120 Heidelberg, Germany, and School of Physics, University of Sydney, NSW 2600, Australia. 4 National Optical Astronomy Observatory, PO Box 26732, Tucson AZ 85726-6732, USA. Accepted 2007 ?. Received 2007 ?; in original form 2007 ? ABSTRACT High dispersion near-infrared spectra have been taken of seven highly-evolved, variable, intermediate-mass (4-6 M⊙) AGB stars in the LMC and SMC in order to look for C, N and O variations that are expected to arise from third dredge-up and hot-bottom burning. The pulsation of the objects has been modelled, yielding stellar masses, and spectral synthesis calculations have been performed in order to derive abundances from the observed spectra. For two stars, abundances of C, N, O, Na, Al, Ti, Sc and Fe were derived and compared with the abundances predicted by detailed AGB models. Both stars show very large N enhancements and C deficiencies. These results provide the first observational confirmation of the long-predicted production of primary nitrogen by the combination of third dredge-up and hot-bottom burning in intermediate-mass AGB stars. It was not possible to derive abundances for the remaining five stars: three were too cool to model, while another two had strong shocks in their atmospheres which caused strong emission to fill the line cores and made abundance determination impossible. The latter occurrence allows us to predict the pulsation phase interval during which observations should be made if successful abundance analysis is to be possible. Key words: stars: AGB and post-AGB – stars: abundances – stars: oscillations. 1 INTRODUCTION AGB stars are predicted to be major contributors to the enrichment of the interstellar medium in carbon, nitro- gen and s-process elements (e.g. Iben and Truran 1978; Dray et al. 2003). The 12C and s-process elements are brought to the surface of AGB stars during the third dredge- up (Iben and Renzini 1983) and they are then ejected into the interstellar medium by the extensive stellar winds that occur during the AGB phase of evolution. The 14N produc- tion is mainly from the envelopes of the more massive AGB stars where the hot-bottom burning process (Scalo et al. 1975) can cycle the entire envelope through the outer parts of the hydrogen-burning shell, converting 12C to 14N. In the case where the 12C in the envelope has a component that comes from third dredge-up, the 14N resulting from the dredged-up 12C is of primary origin. There is strong evidence that some of the 14N in the universe is of pri- mary origin (van Zee et al. 1998), with the primary source dominating for metal abundances less than one third so- lar. Hot-bottom burning in intermediate-mass (M & 3M⊙) AGB stars is a possible source for this primary nitrogen. Detailed stellar evolution and nucleosynthesis calculations suggest that the hot-bottom burning in AGB stars may also involve the Ne-Na and Mg-Al chains, leading to changes in the surface abundances of Al amd Mg as well as ni- trogen (Karakas and Lattanzio 2003; Lattanzio and Wood 2004). Unfortunately, all the evolution and nucleosynthe- sis calculations are very uncertain due to the problem of treating convection, especially the critical process of convec- tive overshoot. Observational constraints from intermediate- mass AGB stars are required to constrain the convection parameters such as mixing length and overshoot distance, which are critical to the nucleosynthesis results. This study was initiated to provide such constraints. Considerable numbers of intermediate-mass AGB stars are known in the Magellanic Clouds (Wood, Bessell & Fox 1983), and these stars can be used to provide the obser- vational tests required for the evolution and nucleosynthe- sis calculations. Some work has been done in this area by Smith et al. (2002) who examined the C, N and O abun- c© 2007 RAS http://arxiv.org/abs/0704.1907v1 2 J.A. McSaveney et al. dances in a sample of luminous AGB stars in the LMC. They found that 14N was enhanced in a way consistent with first dredge-up only, without the need for hot-bottom burn- ing. However, the stars studied by Smith et al. (2002) were not known variable stars, meaning that they were not near the end of their AGB evolution where the effects of dredge- up and hot-bottom burning should be most pronounced. In addition, the lack of pulsation means that there was no way to estimate the mass of the stars involved. In this study, we have aimed to get surface abundances for a small sample of intermediate-mass, pulsating AGB stars. These stars are near the end of their AGB evolution and are about to enter their final superwind stage where most of the material currently in their envelopes will be ejected back into the interstellar medium (in fact, two of the stars are IRAS sources already exhibiting strong super- winds). Since they are pulsating, we can also estimate their current masses from pulsation theory. On the other hand, because of their low effective temperatures and their pulsa- tion, the model atmospheres required for the interpretation of their spectra are complicated and difficult, and must in- volve the dynamics of the atmosphere. In Section 2, we describe the sample of stars and the photometry and spectra obtained for them. In Section 3, we describe the pulsation models, in Section 4 the model atmospheres are described and the derived abundances are given. The results are discussed in the final section. 2 OBSERVATIONS 2.1 Near-infrared photometry The sample of objects observed in this study is given in Table 1. All these stars are oxygen-rich, pulsating, lumi- nous AGB stars. All the stars have been monitored in J and K with the ANU 2.3m telescope at Siding Spring Ob- servatory, using either a single channel photometer (prior to March 1994) or the infrared array CASPIR (McGregor et al. 1994).The mean bolometric luminosity Mbol of each star was derived from the JK photometry and monitoring, using the bolometric corrections from Houdashelt et al. (2000a,b) with [Fe/H ] = −0.3 and -0.6 for the LMC and SMC, respec- tively. Distance modulii of 18.54 and 18.93 and reddenings E(B-V) of 0.08 and 0.12 were assumed for the LMC and SMC, respectively (Keller and Wood 2006). The least evolved of the stars in Table 1, with lowest luminosity, smallest amplitude and shortest pulsation pe- riod, is the most luminous red giant in the LMC cluster NGC1866. At the other extreme are two IRAS sources with very long pulsation periods and large amplitudes. The lu- minosities and periods of the stars in the sample mean that their current masses lie in the range 3–8 M⊙ (Wood et al. 1983). 2.2 The high-resolution near-infrared spectra Spectra were taken of these stars using the Phoenix spectro- graph (Hinkle et al. 2003) on the Gemini South telescope. The observations were done in service mode, and the obser- vation dates are listed in Table 1. For each star, observations were made in 3 separate bands, as shown in Figures 1, 2 and Table 1. The intermediate-mass AGB star sample Star Mbol P 1 ∆K MC Date2 NGC1866 #4 -6.00 158 0.10 LMC 030210 HV 2576 -6.61 530 0.25 LMC 030209 HV 11303 -5.59 534 0.70 SMC 030728 HV 12149 -6.84 742 0.70 SMC 020920 GM 103 -6.80 1070 1.30 SMC 030728 IRAS 04516-6902 -6.80 1090 1.30 LMC 030211 IRAS 04509-6922 -7.33 1290 1.50 LMC 021202 1 Pulsation period in days. 2 Date of Gemini observation in the form yymmdd. For GM 103 and HV 11303 the observations were spread over 6 days and the given date is the mean value. Table 2. The modelled stars Star L/L⊙ Teff M/M⊙ ℓ/Hp α NGC1866#4 20000 3490 4.0 2.57 0.05 HV 2576 35000 3350 6.0 2.21 0.51 HV 11303 34000 3490 4.6 2.21 0.58 GM 103 41840 3040 6.0 1.40 1.30 1 Turbulent viscosity parameter. 3. Firstly, an observation centred near 1.554 µm was made so that the OH lines could be measured for the derivation of an oxygen abundance; lines of CN also occur in this piece of spectrum. Secondly, an observation near 2.340 µm was made to measure 12CO (and possible 13CO) lines and hence the 12C abundance; given the C abundance, the CN lines near 1.554 µm then yield a 14N abundance. The Na abun- dance was also obtained from this spectral region. Thirdly, an observation at 2.112 µm was made to include lines of Al (and possible Mg). Lines of Fe, Sc and Ti also occur in the spectra, giving an estimate of the metal abundance. 3 PULSATION MODELS Pulsation models and model atmospheres were made for four stars in Table 2: NGC1866 #4, HV2576, HV11303 and GM103. The spectra of the remaining stars in Table 1 (HV12149, IRAS04516-6902 and IRAS04509-6922) were considered too difficult to model at this stage because of the extreme appearance of the spectra, with no clear con- tinuum remaining (see Figs. 1, 2 and 3). This is presumably a result of the very large amplitude and very low effective temperatures of these stars. The J and K photometry for the four stars for which modelling was attempted is given in Table 3. Typical photometric errors are less than 0.03 mag- nitudes. The J and K magnitudes have been converted to those on the AAO system of Allen and Cragg (1983) using the conversions in McGregor (1994). For each observed star, given estimates of the mean lu- minosity L and the effective temperature Teff from the J and K photometry, the stellar radius was computed from the definition L = 4πσR2T 4eff . Then, given the radius and the known pulsation period, the current stellar mass was com- puted from the P -M -R relation of stellar pulsation. These values of mass and luminosity were then used in the nonlin- ear pulsation calculations for the star. c© 2007 RAS, MNRAS 000, 1–14 Abundances in intermediate-mass AGB stars 3 1.548 1.55 1.552 1.554 1.556 1.558 CNOH Ti GM 103 HV11303 HV 2576 NGC 1866 #4 HV 12149 IRAS04509 IRAS04516 Wavelength (µ m) Figure 1. The spectra of all objects in the region near 1.554 µm. The most important lines used for abundance determination are marked. 2.104 2.106 2.108 2.11 2.112 2.114 2.116 2.118 Al Fe GM 103 HV11303 HV 2576 NGC 1866 #4 HV 12149 IRAS04509 IRAS04516 Wavelength (µ m) Figure 2. The same as Figure 1 but for the region near 2.112 µm. c© 2007 RAS, MNRAS 000, 1–14 4 J.A. McSaveney et al. 2.332 2.334 2.336 2.338 2.34 2.342 2.344 2.346 GM 103 HV11303 HV 2576 NGC 1866 #4 HV 12149 IRAS04509 IRAS04516 Na Sc CO Wavelength (µ m) Figure 3. The same as Figure 1 but for the region near 2.340 µm. Table 3. JK photometry JD241 J-K K JD24 J-K K NGC1866#4 HV11303 (continued) 50731 1.23 9.71 52804 1.29 10.02 50772 1.27 9.69 52924 1.26 9.97 50885 1.24 9.68 53516 1.20 9.71 51157 1.30 9.67 53574 1.20 9.38 52564 1.29 9.71 52714 1.33 9.64 GM103 52803 1.25 9.69 45980 1.66 9.55 52924 1.28 9.69 46341 1.49 8.65 46645 1.54 8.57 HV25762 48142 1.63 9.31 52563 1.39 9.00 48168 1.70 9.23 52715 1.43 9.13 48640 1.64 8.36 52804 1.29 8.97 48851 1.62 8.78 52924 1.36 8.87 48934 1.63 9.13 48992 1.67 9.33 HV113032,3 49259 1.89 9.80 49259 1.19 9.81 49317 1.87 9.82 49291 - 9.61 49374 1.68 9.45 49317 1.17 9.47 52564 1.65 9.33 49374 1.19 9.32 52714 1.59 8.74 52564 1.21 9.27 52803 1.61 8.61 52715 1.39 9.56 52924 1.64 8.40 1 JD24 is Julian Date - 2400000. 2 Extra photometry is given in Wood et al. (1983). 3 Extra photometry is given in Catchpole and Feast (1981). The nonlinear pulsation code described in Keller and Wood (2006), and references therein, was used in this study. The turbulent viscosity parameter αν was adjusted to give the observed K light curve amplitude. The K and V mag- nitudes were computed using L and Teff from the pulsa- tion models together with the bolometric corrections from Houdashelt et al. (2000a,b). For three of the four modelled stars, MACHO light curves are available: the V magnitude was computed from the MACHO magnitude using the transforms in Bessell and Germany (1999). For the fourth star, HV11303, an Eros2 light curve was kindly provided by Patrick Tisserand (pri- vate communication), who also provided a conversion from the two Eros2 bands to the V band. The combined V and K light curves give a V -K colour curve. The effective tem- peratures of the models were adjusted (by altering the ra- tio of mixing-length to pressure scale height ℓ/Hp to repro- duce the observed V -K colour when available, or the J-K in the one remaining case of HV11303 (the Eros2 light curve was obtained after the modelling was completed). The final model parameters are given in Table 2. In all cases, a helium abundance Y = 0.30 was assumed, while a metal abundance Z=0.01 was assumed for the LMC stars and Z=0.004 was assumed for the SMC stars. In order to construct a model atmosphere taking account of the dynamical processes in the outer lay- ers, procedures similar to those described in Bessell et al. (1996), Hofmann et al. (1998), Scholz and Wood (2000) and Ireland et al. (2004a,b) were used, with updates to the molecular opacity from Ireland and Scholz (2006). Briefly, c© 2007 RAS, MNRAS 000, 1–14 Abundances in intermediate-mass AGB stars 5 a pulsation model at the same pulsation phase as the phase of the Gemini observation was extracted from each pulsa- tion series. The radial profiles of gas pressure and velocity for this model were transferred to a non-grey, spherical at- mosphere code, and the temperature structure was iterated to a new state, now consistent with the atmosphere code. Once the temperature, gas pressure, density and velocity structure were all consistent in the atmosphere code, the model atmosphere structure was transferred to a separate spectrum synthesis code which was used to compute line profiles (see Section 4). In the atmosphere code, half- and quarter-solar metal abundances from Grevesse et al. (1996) were adopted for LMC and SMC stars, respectively. Radiative and local ther- modynamic equilibrium was adopted and, in particular, the shock-heated region behind the shock front was assumed to have negligible width and to have no effects on the temper- ature stratification. The pulsation series and the extracted dynamical model used as input to the atmosphere code is now described for each star. 3.1 NGC1866 #4 NGC1866 #4 is the most luminous red giant in the LMC cluster NGC1866 (the designation #4 is that given by Fro- gel, Mould and Blanco 1990: note that Maceroni et al. 2002 have mislabelled #4 as #2 in their paper). The MACHO light curve of this star shows a period of 158 days. Mod- elling the cluster HR-diagram, Brocato et al. (2003) derive masses for the He-burning giants of 3.8–4.15 M⊙. Our de- rived mass for NGC1866 #4 is 4 M⊙. With this mass and the computed mean luminosity and Teff , this star must be pulsating in the first overtone mode, a result also consistent with the small amplitude. Fig. 4 shows the observed and computed K, V and Mbol light curves. The period and amplitude of the pulsation and the V -K colour are reproduced well. The Gemini observa- tion was made at a phase of ∼0.1 after visual maximum. The structure of the pulsation model at this time is shown in Fig. 5. The pulsation amplitude of this star is quite small so there are no shock waves present in the atmosphere. The temperature solution from the model atmosphere program produces a cooling of the outer layers and a warming at moderate optical depths compared to the modified Edding- ton approximation used in the pulsation models. 3.2 HV2576 HV2576 is an LMC star which is more luminous than NGC1866 #4. It has a much longer period, a larger ampli- tude, and it pulsates in the fundamental mode. The derived mass for HV2576 is 6 M⊙. Fig. 6 shows the observed and computed K, V and Mbol light curves. Like many of the intermediate-mass, large- amplitude AGB pulsators in the Magellanic Clouds, this star shows a double-humped optical light curve. The pulsation models do not accurately reproduce the double hump but they do show some evidence for it. The K light curve does not show evidence for a prominent hump, but the points are too sparse in the light curve to be definitive. Overall, the Figure 4. The K, V and Mbol light curves of NGC1866 #4. The solid points are model values (several cycles are overplotted), while the crosses are observations or Mbol values computed from the J and K photometry in Table 3. The V values are computed from the MACHO MB and MR photometry using the transforms of Bessell and Germany (1999). The arrows show the phase of the Gemini observations and the phase of the pulsation model extracted for model atmosphere computation. period and amplitude of the pulsation and the large V -K colour (V -K ∼ 7) are reproduced well. The Gemini observa- tion was made near minimum light of the K light curve, or between the double-humped maximum of the optical light curve. The structure of the pulsation model at this time is shown in Fig. 7. The velocity gradient through the atmo- sphere at the time of observation was small and there was no shock wave present in the atmosphere. As with NGC1866 #4, the temperature solution from the model atmosphere program produces a cooling of the outer layers and a warm- ing at moderate optical depths compared to the modified Eddington approximation used in the pulsation models. 3.3 HV11303 HV11303 is an SMC star which is almost a twin of HV2576 in the LMC in terms of its pulsation period and luminosity. However, it has a much larger pulsation amplitude. It pul- sates in the fundamental mode and the derived mass is 4.6 M⊙. This mass is smaller than that of HV2576, which may explain the larger pulsation amplitude. Fig. 8 shows the observed and computedK, V and Mbol light curves. The pulsation models reproduce the observed K light curve reasonably well. The Gemini observation was made near minimum light of theK light curve. The structure of the pulsation model at this time is shown in Fig. 9. Note that there is an uncertainty of about 0.1 in the phasing of c© 2007 RAS, MNRAS 000, 1–14 6 J.A. McSaveney et al. Figure 5. The density ρ, temperature T and velocity v plotted against radius R in a pulsation model of NGC1866 #4 at the same phase as the phase of Gemini observation of the star. The dashed temperature profile is that obtained from the non-grey model atmosphere code. Figure 6. The same as Fig. 4 but for HV2576. Figure 7. The same as Fig. 5 but for HV2576. Figure 8. The same as Fig. 4 but for HV11303. c© 2007 RAS, MNRAS 000, 1–14 Abundances in intermediate-mass AGB stars 7 Figure 9. The same as Fig. 5 but for HV11303. the models relative to the K light curve. With the adopted phasing, a strong shock was just emerging into the atmo- sphere at the time of Gemini observation. In addition, there was a large velocity gradient through the atmosphere above the shock. The temperature solution from the model atmo- sphere program shows a large temperature drop in the outer layers. This behaviour is typical for extended atmospheres in a certain parameter range where strong water absorption abruptly dominates in the high layers (Scholz 1985). The warmer, grey pulsation model used as input to the atmo- sphere code does not have water-forming low temperatures. 3.4 GM103 GM103 (Groenewegen et al. 1995) is also an SMC star but of greater luminosity, amplitude and period than HV11303. It pulsates in the fundamental mode and the derived mass is 6 M⊙. Fig. 10 shows the observed and computed K, V and Mbol light curves. The pulsation models reproduce the K light curve and the large V -K colour (V -K ∼ 8) reasonably well. However, the V light curve of the pulsation models seems to rise towards maximum earlier than the observed light curve. The Gemini observation was made near maxi- mum visible light, or about phase 0.1 before maximum of the K light curve. The structure of the pulsation model at this time is shown in Fig. 11. This star has a strong shock situated in the middle part of the atmosphere at the time of Gemini observation, with moderate velocity gradients above and below the shock. The temperature solution for GM103 from the model atmosphere program shows a larger devi- ation from the pulsation solution than in any other star. This is due to the strong density drop at the shock, with Figure 10. The same as Fig. 4 but for GM103. Figure 11. The same as Fig. 5 but for GM103. c© 2007 RAS, MNRAS 000, 1–14 8 J.A. McSaveney et al. the model atmosphere’s water-dominated temperature drop- ping rapidly near what is the effective edge of the star at the shock front. 4 SPECTRAL SYNTHESIS AND ABUNDANCE DERIVATION 4.1 Line synthesis calculations The spherical model atmospheres described in Section 3 were put into a compatible spectrum synthesis program (us- ing spherical geometry) to generate line spectra. The spec- trum synthesis program was that used by Scholz (1992), Bessell et al. (1996) and Scholz and Wood (2000), except that partial pressures of a larger number of molecules were calculated using a computer program supplied by K. Ohnaka (private communication). This program uses an equation of state based on molecular constants of Tsuji (1973), updated for CN using data from Costes et al. (1990) and for TiO from Tsuji (1978). Lines are assumed to be formed in local thermodynamic equilibrium. As shock-heating behind the shock front is neglected in these models, emission compo- nents of synthetic line profiles may only occur as a conse- quence of large atmospheric extension (P-Cygni-like emis- sion; see Scholz 1992). The overall spectrum in each of the three filter re- gions was computed using a reasonably comprehensive set of atomic and diatomic molecular lines, although only specific lines were used in the actual abundance deriva- tions. For atomic lines, wavelengths, excitation potentials and log gf values were taken from the Vienna Atomic Line Database (VALD, Kupka et al. 1999), with param- eters of lines selected for abundance determination be- ing checked against those of Smith et al. (2002). Molecular line data proved more problematic as no single source ex- ists. CO lines were taken from Goorvitch (1994) and CN lines from Aoki and Tsuji (1997) and the SCAN database (Jorgensen and Larsson 1990). CH lines were also obtained from the SCAN database (Jorgensen et al. 1996). In each of these cases, lines were cross-checked against those from Smith et al. (2002). The OH lines proved most problem- atic. An incomplete selection of lines was assembled from Smith et al. (2002) and Melendez et al. (2001): several ad- ditional lines listed in the source Abrams et al. (1994) could not be not included due to a lack of gf values. 4.2 The derivation of abundances Starting abundances were adopted by approximating LMC and SMC metal abundances as half and quarter solar, re- spectively, with solar defined from Asplund et al. (2005) (see Table 5). The CO and OH lines computed with these abun- dances were examined to establish the microturbulent ve- locity. As these relatively massive stars are expected to have altered CNO abundances as a result of nucleosynthetic pro- cessing, a series of CNO values based on the AGB evolution calculations of Karakas (2003) and Karakas and Lattanzio (2003) was adopted, and the resultant spectra were then calculated. The CNO combination giving the best overall fit to the observed spectra was used as the primary starting 1.5533 1.5534 1.5535 1.5536 1.5537 1.5538 Wavelength (µ m) A(O) = 8.13; 8.28; 8.43 OH OH Figure 12. Fitted OH lines for HV2576. In this and subsequent figures, fits are shown for the three abundance values shown on the figure (where A(X) = log[n(X)/n(H)]+12). The dashed lines correspond to changes of ±0.15 dex from the adopted abundance (solid line). 1.5533 1.5534 1.5535 1.5536 1.5537 1.5538 Wavelength (µ m) A(O) = 8.65; 8.80; 8.95 OH OH Figure 13. Fitted OH lines for NGC1866#4. point for subsequent refinement of the individual CNO abun- dances. Once the best CNO abundances had been found, the metallic abundances were refined to give a best fit. Lines used for final abundance derivations are listed in Table 4, and a selection of the fitted lines are shown in Figs. 12 to The abundances derived from different lines of the same species show some scatter, with the adopted lines generally found to yield reasonably consistent values. Figs. 14 and 15 show examples of the fit errors for the CO lines. It is clear that neither CO line is fitted exactly by the final adopted abundance (that of the mean of the best fits to 6 lines), but both lines are fitted reasonably well, with one too strong and one too weak. Figs. 12 and 13, where Fe I lines are present on either side of the OH lines, show further examples of fit errors. The adopted Fe abundance is based on a fit to the 2.1095µm line in Figs. 24 and 25. However, in HV2576 (Fig. 12), this abundance is too high for the 1.553424µm Fe line to the blue and too low for the 1.553757µm line to the red (or, more likely, the models are missing an unidentified line at this wavelength). For NGC1866#4 (Fig. 13), the c© 2007 RAS, MNRAS 000, 1–14 Abundances in intermediate-mass AGB stars 9 2.342 2.3422 2.3424 2.3426 2.3428 2.343 Wavelength (µ m) A(C) = 6.86; 7.01; 7.16 CO CO Figure 14. Fitted CO lines for HV2576. 2.342 2.3422 2.3424 2.3426 2.3428 2.343 Wavelength (µ m) A(C) = 7.21; 7.36; 7.51 CO CO Figure 15. Fitted CO lines for NGC1866#4. 1.553424µm line fits well, while once again the 1.553757µm line is too weak (or an unidentified line is missing in the model). There were also cases where observed and computed line velocities do not match to better than a few km s−1, as can be seen in Figs. 14 to 17. The velocity mismatch could be due to a number of potential causes, including inexact adopted line wavelengths, inexact wavelength calibration, or the model velocities as a function of depth not being correct. The microturbulent velocity required for line broad- ening varied from star to star. The adopted value for NGC1866#4 (3 km s−1) is close to the value used by Smith et al. (2002). However, a larger value of 7 km s−1 was required for HV2576. This value was derived by fitting to the CO lines. Slightly different values would be obtained by fitting different lines. This is probably a reflection of different amounts of turbulence at different levels in the atmosphere: the CO lines tend to be broader than the metallic lines and they are, on average, formed further out in cooler layers. Our final derived abundances are shown in Table 5. For the elements Fe, Na, Al, Sc, Ti, and N, the error estimates in the table are those arising from the fit to the line only, as only one line was available. They are eye-estimates based on the fits 0.15 dex above and below the adopted optimum fit 1.556 1.5561 1.5562 1.5563 1.5564 1.5565 1.5566 1.5567 Wavelength (µ m) A(N) = 8.35; 8.50; 8.65 Figure 16. Fitted CN line for HV2576. 1.556 1.5561 1.5562 1.5563 1.5564 1.5565 1.5566 1.5567 Wavelength (µ m) A(N) = 8.40; 8,55; 8.70 Figure 17. Fitted CN line for NGC1866#4. 2.34 2.3402 2.3404 2.3406 2.3408 2.341 Wavelength (µ m) A(Sc) = 2.75; 2.90; 3.05 Figure 18. Fitted Sc I line for HV2576. c© 2007 RAS, MNRAS 000, 1–14 10 J.A. McSaveney et al. 2.34 2.3402 2.3404 2.3406 2.3408 2.341 Wavelength (µ m) A(Sc) = 3.25; 3.40; 3.55 Figure 19. Fitted Sc I line for NGC1866#4. 1.554 1.5541 1.5542 1.5543 1.5544 1.5545 1.5546 1.5547 1.5548 Wavelength (µ m) A(Ti) = 3.45; 3.60; 3.75 Figure 20. Fitted Ti I line for HV2576. 1.554 1.5541 1.5542 1.5543 1.5544 1.5545 1.5546 1.5547 1.5548 Wavelength (µ m) A(Ti) = 4.55; 4.70; 4.85 Figure 21. Fitted Ti I line for NGC1866#4. 2.3375 2.3376 2.3377 2.3378 2.3379 2.338 2.3381 2.3382 2.3383 Wavelength (µ m) A(Na) = 5.20; 5.35; 5.50 Figure 22. Fitted Na I line for HV2576. 2.3375 2.3376 2.3377 2.3378 2.3379 2.338 2.3381 2.3382 2.3383 Wavelength (µ m) A(Na) = 5.60; 5.75; 5.90 Figure 23. Fitted Na I line for NGC1866#4. 2.1088 2.109 2.1092 2.1094 2.1096 2.1098 Wavelength (µ m) A(Al) = 5.35; 5.50; 5.65 A(Fe) = 7.05; 7.20; 7.35 Al I Fe I Figure 24. Fitted Fe I and Al lines for HV2576. c© 2007 RAS, MNRAS 000, 1–14 Abundances in intermediate-mass AGB stars 11 2.1088 2.109 2.1092 2.1094 2.1096 2.1098 Wavelength (µ m) A(Al) = 5.60; 5.75; 5.90 A(Fe) = 6.90; 7.05; 7.20 Al I Fe I Figure 25. Fitted Fe I and Al lines for NGC1866#4. Table 4. Spectral line data. λ χ Source (Å) (eV) log(gf) 21095.446 6.20 -0.69 VALD 23378.945 3.75 -0.420 Smith et al. (2002) 23404.756 1.44 -1.278 Smith et al. (2002) 15543.720 1.88 -1.481 Smith et al. (2002) 21093.029 4.09 -0.31 VALD 15535.489 0.51 -5.23 Smith et al. (2002) 15536.707 0.51 -5.23 Melendez et al. (2001) 15560.271 0.30 -5.31 Smith et al. (2002) 15565.815 0.90 -5.00 Melendez et al. (2001) 23396.260 0.37 -5.17 Smith et al. (2002) 23398.224 1.73 -4.44 Smith et al. (2002) 23406.345 0.00 -6.57 Smith et al. (2002) 23408.509 0.36 -5.19 Smith et al. (2002) 23424.328 1.80 -4.427 Smith et al. (2002) 23426.322 0.00 -6.565 Smith et al. (2002) 15563.367 1.15 -1.141 Smith et al. (2002) abundance. The C and O values are based on averages of the fits to 6 and 4 lines, respectively, with the stated errors being based on the standard deviation. The lines used are given in Table 4. Systematic errors, such as errors in the pulsation model and its phase, will undoubtedly add significantly to the fit error, but such additional errors are hard to estimate. 4.3 HV11303 and GM103 As noted in Section 3, the SMC stars HV11303 and GM103 both had strong shock fronts in their atmospheres at the time of observation with Gemini South. This has made abundance derivation with our atmosphere code impossible. Fig. 26 shows a sample of the spectrum for each star, along with an attempt at synthesisizing the spectrum using the pulsation models from Section 3. In the case of GM103, the synthesized absorption lines are much stronger than the observed lines. A comparison of the observed and synthesisized spectra shows clearly that the observed line cores are filled in by strong emission in ev- ery case. An alternative explanation could be that each line is split into a pre- and post-shock absorption component, with no emission in the middle. However, the weakness of the observed absorption lines compared to the model lines suggests that there is indeed a strong emission component. In HV11303, the lines are much broader than in GM103, with evidence for an emission component as well as absorp- tion components corresponding to the pre- and post-shock velocities. As noted in Section 3, the spectral synthesis code ig- nores the immediate post-shock, high-temperature region so that there is no post-shock emission component in our synthesized spectra. When a strong shock is present in the atmosphere, this model deficiency prevents abundance de- termination from lines which have a post-shock emission contribution. These results show that observations useful for abun- dance analysis must be obtained at phases of the pulsation cycle when no strong shock wave is present in the atmo- sphere. In these relatively massive stars, the shock wave en- ters the lower part of the stellar atmosphere near minimum of the K light curve, or phase ∼0.2–0.3 past minimum vi- sual light. It has passed through the line forming part of the atmosphere half a pulsation cycle later, corresponding to ∼0.1–0.2 in phase past K maximum or after phase ∼0.4 of the visual light curve. 5 DISCUSSION The abundances obtained here for NGC1866#4 and HV2576 are shown with other comparable observations in Fig. 27. The Fe abundance is close to half solar, consistent with the values obtained by Smith et al. (2002), Hill et al. (2000) and other studies of young objects in the LMC. The other metal abundances are similarly near half-solar, except for Ti in HV2576 where our derived value seems anoma- lously low. The CNO abundances are the most interesting part of this study, since in a highly-evolved, intermediate-mass AGB star they will be altered by the 1st, 2nd and 3rd dredge-ups and by hot-bottom burning. Evidence for hot-bottom burning has been found in intermediate-mass AGB stars in the Magellanic Clouds by Plez et al. (1993) and Smith et al. (1995). Plez et al. (1993) examined spectra of 7 luminous AGB stars in the SMC and found them to be Li-rich, C-poor and with low 12C/13C ratios, consistent with hot-bottom burning. Smith et al. (1995) examined a large number of SMC and LMC stars and found many luminous AGB stars, including HV2576, showed strong Li lines that could be attributed to hot- bottom burning. In fact, HV2576 had the largest estimated Li abundance of all the stars examined. Maceroni et al. (2002) examined Li lines in the giant stars in NGC1866 and found that NGC1866#4 (which they mislabel as #2) c© 2007 RAS, MNRAS 000, 1–14 12 J.A. McSaveney et al. 2.335 2.336 2.337 2.338 2.339 2.34 Wavelength (µ m) 2.335 2.336 2.337 2.338 2.339 2.34 Wavelength (µ m) Figure 26. Top: A section of the 2.34 µm spectrum of GM103 (points), overlayed with a synthesised spectrum (continuous line). Bottom: The same piece of spectrum for HV11303. Table 5. Derived abundances Star A(Fe) A(12C) A(14N) A(16O) A(Na) A(Sc) A(Ti) A(Al) HV 2576 7.20 ± 0.10 7.01 ± 0.20 8.50 ± 0.05 8.28 ± 0.08 5.35 ± 0.10 2.90 ± 0.10 3.60 ± 0.05 5.50 ± 0.05 NGC 1866 #4 7.05 ± 0.15 7.36 ± 0.16 8.55 ± 0.15 8.80 ± 0.17 5.75 ± 0.15 3.40 ± 0.15 4.70 ± 0.15 5.75 ± 0.10 Smith et al. 2002 6.37 - 7.16 6.53 - 7.86 7.14 - 8.24 7.82 - 8.33 4.69 - 5.84 2.01 - 2.91 3.84 - 4.61 ... Half-solar 7.15 8.09 7.48 8.36 5.87 2.75 4.60 5.87 Notes: A(X) = log[n(X)/n(H)] + 12. Half-solar abundances are half the solar abundances in Asplund et al. (2005). The errors are estimates of the line fitting error alone - see Sect. 4.2. had enhanced Li compared to other stars in the cluster, in- dicating that hot-bottom burning is operating in this star also. There is evidence that the third dredge-up has been operating in HV2576, as well as hot-bottom burning. Wood et al. (1983) give a spectral type M5.5/S1 for HV2576, indicating the presence of excess ZrO in the spec- trum, which is a result of the dredge-up of s-process ele- ments during thermal pulses. Smith et al. (1995) noted that HV2576 had one of the strongest ZrO bands in their large sample of stars, indicating that a large amount of third dredge-up has occurred in this star. The combination of third dredge-up and hot-bottom burning is predicted to be a significant source of primary nitrogen in the universe. The 12C dredged up at a ther- mal pulse is converted into 14N during the next interpulse period when hot-bottom burning operates during the quies- cent H-shell burning phase. Lattanzio and Wood (2004) and Karakas (2003) (see their Figs. 2.37 and 5.6, respectively) show the effects of third dredge-up and hot-bottom burn- ing in intermediate-mass, luminous and sub-solar metallic- ity AGB stars such as those we are dealing with here. Ini- tially on the AGB, just at the onset of thermal pulses, hot- bottom burning by the CN cycle reduces the 12C abundance to ∼1/15 of its initial main-sequence value, and increases the 14N abundance to ∼5-6 times its initial value. The ratio 12C/14N at this time is ∼1/15. During subsequent evolution, the dredge-up of 12C by third dredge-up at thermal pulses, followed by hot-bottom burning, causes a steady rise over many pulsation cycles in both the C and N abundance, with the C/N ratio remaining almost constant at ∼1/15. Dur- ing this relatively long evolutionary process, there is a small decrease in the O abundance, but this effect would be too small for us to reliably detect. c© 2007 RAS, MNRAS 000, 1–14 Abundances in intermediate-mass AGB stars 13 Looking at Fig. 27 and Table 5, we see that the C abun- dance in both NGC1866#4 and HV2576 is about 0.1 times half-solar. At the same time, the N abundance is ∼10 times the half-solar value. Both these numbers are consistent with the strong operation of hot-bottom burning in the envelopes of these stars. Furthermore, the very large N abundances can only be explained by hot-bottom processing of dredged-up 12C into 14N i.e. a large fraction of the nitrogen in the en- velopes of these stars must be of primary origin. This is the first direct demonstration that this long-predicted source of primary nitrogen does exist. The O abundance in HV2576 is essentially unchanged from the half-solar value, as predicted by models. There is some evidence for an increase in the O abundance in NGC1866#4. If confirmed, this would lend support to con- vective theories such as that of Herwig et al. (1997) which lead to overshoot at convective boundaries, enrichment of the intershell region in both 12C and 16O at each helium shell flash, and subsequent dredge-up of both 12C and 16O in third dredge-up events. A strong disagreement between the observations and evolutionary models without convective overshoot relates to the total stellar mass required for the efficient opera- tion of hot-bottom burning. The mass of NGC1866#4 must be no more than ∼4 M⊙ because of cluster membership. This star shows evidence for efficient hot-bottom burning, while models without convective overshoot do not predict ef- ficient hot-bottom burning at masses this low. This suggests that overshoot (inwards) of convective material at convective boundaries is required to significantly extends the convec- tive envelope into nuclear-burning regions. The models of Ventura et al. (2002) use a convective theory that does pro- duce overshoot, and their 4 M⊙, half-solar metal abundance AGB models do have the efficient hot-bottom burning ob- served in NGC1866#4. At 6 M⊙, both the observations of HV2576 and models with or without convective overshoot show efficient hot-bottom burning. The elements Li, Na and Al can also be compared with evolutionary models. Smith et al. (1995) give A(Li) = 3.8 for HV2576 while Maceroni et al. (2002) give A(Li) = 1.5 (note that these authors used static model atmospheres, and observations at random phases where shocks may have been present in the stellar atmosphere). Models of AGB stars with efficient hot-bottom burning generally show an initial large increase in the surface Li abundance, after which it slowly decreases to low values (A(Li) . 1.0) while the CN cycle converts C into N - see Karakas (2003) and Ventura et al. (2002). The high Li value derived for HV2576, together with the observed high 14N and low 12C abundances, is consistent with model predictions for a 6 M⊙ star about one quarter of the way through its TPAGB phase and undergoing efficient hot-bottom burning - see Fig. C27 of Karakas (2003). The Na and Al abundances found in this study are marginally below the expected half-solar values and within the expected range of observed variation. Models of AGB stars (Karakas 2003) with efficient hot-bottom burning show a small increase in Al abundance and small changes in the total Mg abundance but these are too small to detect here. The 25Mg/24Mg ratio can increase greatly in such stars, but we are unable to detect isotopic changes. 0 1 2 3 4 5 6 7 8 9 C N O Fe Na Al Sc Ti Solar (Asplund et al 2005) Solar −0.3 (LMC 1st approximation) HV2576 NGC1866 4 NGC1866 (Hill et al 2000) Red Giant range (Smith et al 2002) Figure 27. Final abundances. 6 SUMMARY High-dispersion, near-infrared spectra have been used to de- rive abundances of luminous, intermediate-mass AGB stars in the Magellanic Clouds. The C and N abundances derived here provide the first observational demonstration that third dredge-up at thermal pulses followed by hot-bottom burn- ing can produce significant amounts of primary nitrogen in intermediate-mass AGB stars. The observed spectra and modelling show that only spectra taken during the part of the pulsation cycle near minimum visual light, equivalent to the latter part of the decline of the K light curve, are useful for abundance analysis. The results in this paper come from exploratory obser- vations and calculations and, in the end, only two stars were suitable for abundance analysis. It is clear that follow-up work with a larger sample of objects is required to confirm the results found here, and to seek C and N abundances as a function of stellar mass, luminosity and metal abundance. ACKNOWLEDGMENTS We would like to thank Amanda Karakas for useful dis- cussions about AGB nucleosynthesis models, Patrick Tis- serand for supplying the Eros2 light curve for HV 11303, and Keiichi Ohnaka for supplying his programs for partial pressure calculations. JAM is grateful for the discovery grant from the Australian Research Council (DP0343832) which supported her during the course of this work: PRW and JCL also received partial support from this grant. MS re- ceived support from a grant of the Deutsche Forschungs- gemeinschaft. This paper is based in part on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on be- half of the Gemini partnership: the National Science Foun- dation (United States), the Particle Physics and Astron- omy Research Council (United Kingdom), the National Re- search Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), CNPq (Brazil), and CONI- CRT (Argentina). The observations were obtained with the Phoenix infrared spectrograph, which was developed and c© 2007 RAS, MNRAS 000, 1–14 14 J.A. McSaveney et al. is operated by the National Optical Astronomy Observa- tory. The spectra were obtained as part of Gemini programs GS-2002A-Q-49, GS-2002B-Q-22 and GS-2003A-Q-25. We thank the observers at Gemini South for taking these spec- tra in service mode. REFERENCES Abrams M.C., Davis S.P., Rao M.L.P., Engleman R. 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High dispersion near-infrared spectra have been taken of seven highly-evolved, variable, intermediate-mass (4-6 Msun) AGB stars in the LMC and SMC in order to look for C, N and O variations that are expected to arise from third dredge-up and hot-bottom burning. The pulsation of the objects has been modelled, yielding stellar masses, and spectral synthesis calculations have been performed in order to derive abundances from the observed spectra. For two stars, abundances of C, N, O, Na, Al, Ti, Sc and Fe were derived and compared with the abundances predicted by detailed AGB models. Both stars show very large N enhancements and C deficiencies. These results provide the first observational confirmation of the long-predicted production of primary nitrogen by the combination of third dredge-up and hot-bottom burning in intermediate-mass AGB stars. It was not possible to derive abundances for the remaining five stars: three were too cool to model, while another two had strong shocks in their atmospheres which caused strong emission to fill the line cores and made abundance determination impossible. The latter occurrence allows us to predict the pulsation phase interval during which observations should be made if successful abundance analysis is to be possible.

Introduction Observations Near-infrared photometry The high-resolution near-infrared spectra Pulsation models NGC1866 #4 HV2576 HV11303 GM103 Spectral synthesis and abundance derivation Line synthesis calculations The derivation of abundances HV11303 and GM103 Discussion Summary

A PRACTICAL GUIDE TO STOCHASTIC SIMULATIONS OF REACTION-DIFFUSION PROCESSES RADEK ERBAN∗, S. JONATHAN CHAPMAN∗, AND PHILIP K. MAINI∗ Abstract. A practical introduction to stochastic modelling of reaction-diffusion processes is presented. No prior knowledge of stochastic simulations is assumed. The methods are explained using illustrative examples. The article starts with the classical Gillespie algorithm for the stochastic modelling of chemical reactions. Then stochastic algorithms for modelling molecular diffusion are given. Finally, basic stochastic reaction-diffusion methods are presented. The connections between stochastic simulations and deterministic models are explained and basic mathematical tools (e.g. chemical master equation) are presented. The article concludes with an overview of more advanced methods and problems. Key words. stochastic simulations, reaction-diffusion processes AMS subject classifications. 60G05, 92C40, 60J60, 92C15 1. Introduction. There are two fundamental approaches to the mathematical modelling of chemical reactions and diffusion: deterministic models which are based on differential equations; and stochastic simulations. Stochastic models provide a more detailed understanding of the reaction-diffusion processes. Such a description is often necessary for the modelling of biological systems where small molecular abundances of some chemical species make deterministic models inaccurate or even inapplicable. Stochastic models are also necessary when biologically observed phenomena depend on stochastic fluctuations (e.g. switching between two favourable states of the system). In this paper, we provide an accessible introduction for students to the stochastic modelling of the reaction-diffusion processes. We assume that students have a basic understanding of differential equations but we do not assume any prior knowledge of advanced probability theory or stochastic analysis. We explain stochastic simulation methods using illustrative examples. We also present basic theoretical tools which are used for analysis of stochastic methods. We start with a stochastic model of a single chemical reaction (degradation) in Section 2.1, introducing a basic stochastic simulation algorithm (SSA) and a mathematical equation suitable for its analysis (the so-called chemical master equation). Then we study systems of chemical reactions in the rest of Section 2, presenting the Gillespie SSA and some additional theoretical concepts. We introduce new theory whenever it provides more insights into the par- ticular example. We believe that such an example-based approach is more accessible for students than introducing the whole theory first. In Section 3, we study SSAs for modelling diffusion of molecules. We focus on models of diffusion which are later used for the stochastic modelling of reaction-diffusion processes. Such methods are presented in Section 4. We also introduce further theoretical concepts, including the reaction-diffusion master equation, the Smoluchowski equation and the Fokker-Planck equation. We conclude with Sections 5 and 6 where more advanced problems, methods and theory are discussed, giving references suitable for further reading. The stochastic methods and the corresponding theory are explained using several illustrative examples. We do not assume a prior knowledge of a particular computer language in this paper. A student might use any computer language to implement the examples from this paper. However, we believe that some students might benefit from ∗University of Oxford, Mathematical Institute, 24-29 St. Giles’, Oxford, OX1 3LB, United King- dom; e-mails: erban@maths.ox.ac.uk, chapman@maths.ox.ac.uk, maini@maths.ox.ac.uk. http://arXiv.org/abs/0704.1908v2 2 RADEK ERBAN ET AL. our computer codes which were used to compute the illustrative results in this paper. The computer codes (in Matlab or Fortran) can be downloaded from the website http://www.maths.ox.ac.uk/cmb/Education/ which is hosted by the Centre for Mathematical Biology in the Mathematical Institute, University of Oxford. 2. Stochastic simulation of chemical reactions. The goal of this section is to introduce stochastic methods for the modelling of (spatially homogeneous) systems of chemical reactions. We present the Gillespie SSA, the chemical master equation and its consequences [18, 19]. We start with the simplest case possible, that of modelling a single chemical reaction, in Section 2.1. We then study two simple systems of chemical reactions in Sections 2.2 and 2.3. 2.1. Stochastic simulation of degradation. Let us consider the single chem- ical reaction k−→ ∅ (2.1) where A is the chemical species of interest and k is the rate constant of the reaction. The symbol ∅ denotes chemical species which are of no further interest in what fol- lows. The rate constant k in (2.1) is defined so that k dt gives the probability that a randomly chosen molecule of chemical species A reacts (is degraded) during the time interval [t, t+ dt) where t is time and dt an (infinitesimally) small time step. In par- ticular, the probability that exactly one reaction (2.1) occurs during the infinitesimal time interval [t, t+ dt) is equal to A(t)k dt where we denote the number of molecules of chemical species A at time t simply as A(t). This notational convention will be used throughout the paper. Let us assume that we have n0 molecules of A in the system at time t = 0, i.e. A(0) = n0. Our first goal is to compute the number of molecules A(t) for times t > 0. To do that, we need a computer routine generating random numbers uniformly distributed in the interval (0, 1). Such a program is included in many modern programming languages (e.g. function rand in Matlab): It generates a number r ∈ (0, 1), so that the probability that r is in a subinterval (a, b) ⊂ (0, 1) is equal to b− a for any a, b ∈ (0, 1), a < b. The mathematical definition of the chemical reaction (2.1) can be directly used to design a “naive” SSA for simulating it. We choose a small time step ∆t. We compute the number of molecules A(t) at times t = i∆t, i = 1, 2, 3, . . . , as follows. Starting with t = 0 and A(0) = n0, we perform two steps at time t: (a1) Generate a random number r uniformly distributed in the interval (0, 1). (b1) If r < A(t)k∆t, then put A(t+∆t) = A(t)−1; otherwise, A(t+∆t) = A(t). Then continue with step (a1) for time t+ ∆t. Since r is a random number uniformly distributed in the interval (0, 1), the probability that r < A(t)k∆t is equal to A(t)k∆t. Consequently, step (b1) says that the proba- bility that the chemical reaction (2.1) occurs in the time interval [t, t+∆t) is equal to A(t)k∆t. Thus step (b1) correctly implements the definition of (2.1) provided that ∆t is small. The time evolution of A obtained by the “naive” SSA (a1)–(b1) is given in Figure 2.1(a) for k = 0.1 sec−1, A(0) = 20 and ∆t = 0.005 sec. We repeated the stochastic simulation twice and we plotted two realizations of SSA (a1)–(b1). We see in Figure 2.1(a) that two realizations of SSA (a1)–(b1) give two different results. Each time we run the algorithm, we obtain different results. This is generally true for any SSA. Therefore, one might ask what useful and reproducible information can http://www.maths.ox.ac.uk/cmb/Education/ STOCHASTIC REACTION-DIFFUSION PROCESSES 3 0 5 10 15 20 25 30 time [sec] first realization second realization 0 5 10 15 20 25 30 time [sec] Fig. 2.1. Stochastic simulation of chemical reaction (2.1) for k = 0.1 sec−1 and A(0) = 20. (a) Number of molecules of A as a function of time for two realizations of the “naive” SSA (a1)–(b1) for ∆t = 0.005 sec; (b) results of ten realizations of SSA (a2)–(c2)(solid lines; different colours show different realizations) and stochastic mean (2.8) plotted by the dashed line. be obtained from stochastic simulations? This question will be addressed later in this section. The probability that exactly one reaction (2.1) occurs during the infinitesimal time interval [t, t+dt) is equal to A(t)k dt. To design the SSA (a1)–(b1), we replaced dt by the finite time step ∆t. In order to get reasonably accurate results, we must ensure that A(t)k∆t≪ 1 during the simulation. We used k = 0.1 sec−1 and ∆t = 0.005 sec. Since A(t) ≤ A(0) = 20 for any t ≥ 0, we have that A(t)k∆t ∈ [0, 0.01] for any t ≥ 0. Consequently, the condition A(t)k∆t ≪ 1 is reasonably satisfied during the simulation. We might further increase the accuracy of the SSA (a1)–(b1) by decreasing ∆t. However, decreasing ∆t increases the computational intensity of the algorithm. The probability that the reaction (2.1) occurs during the time interval [t, t + ∆t) is less or equal to 1% for our parameter values. During most of the time steps, we generate a random number r in step (a1) to find out that no reaction occurs in step (b1). Hence, we need to generate a lot of random numbers before the reaction takes place. Our next task will be to design a more efficient method for the simulation of the chemical reaction (2.1). We will need only one random number to decide when the next reaction occurs. Moreover, the method will be exact. There will be no approximation in the derivation of the following SSA (a2)–(c2). Suppose that there are A(t) molecules at time t in the system. Our goal is to com- pute time t+τ when the next reaction (2.1) takes place. Let us denote by f(A(t), s) ds the probability that, given A(t) molecules at time t in the system, the next reaction occurs during the time interval [t+ s, t+ s+ds) where ds is an (infinitesimally) small time step. Let g(A(t), s) be the probability that no reaction occurs in interval [t, t+s). Then the probability f(A(t), s) ds can be computed as a product of g(A(t), s) and the probability that a reaction occurs in the time interval [t+ s, t+ s+ds) which is given according to the definition of (2.1) by A(t+ s)k ds. Thus we have f(A(t), s) ds = g(A(t), s)A(t + s)k ds. Since no reaction occurs in [t, t+ s), we have A(t+ s) = A(t). This implies f(A(t), s) ds = g(A(t), s)A(t)k ds. (2.2) 4 RADEK ERBAN ET AL. To compute the probability g(A(t), s), let us consider σ > 0. The probability that no reaction occurs in the interval [t, t+ σ + dσ) can be computed as the product of the probability that no reaction occurs in the interval [t, t + σ) and the probability that no reaction occurs in the interval [t+ σ, t+ σ + dσ). Hence g(A(t), σ + dσ) = g(A(t), σ)[1 −A(t+ σ)k dσ]. Since no reaction occurs in the interval [t, t + σ), we have A(t + σ) = A(t). Conse- quently, g(A(t), σ + dσ)− g(A(t), σ) = −A(t)k g(A(t), σ). Passing to the limit dσ → 0, we obtain the ordinary differential equation (in the σ variable) dg(A(t), σ) = −A(t)k g(A(t), σ). Solving this equation with initial condition g(A(t), 0) = 1, we obtain g(A(t), σ) = exp[−A(t)kσ]. Consequently, (2.2) can be written as f(A(t), s) ds = A(t)k exp[−A(t)ks] ds. (2.3) Our goal is to find τ such that t + τ is the time when the next reaction occurs, provided that there are A(t) molecules of A in the system at time t. Such τ ∈ (0,∞) is a random number which has to be generated consistently with the definition of the chemical reaction (2.1). To do that, we consider the function F (·) defined by F (τ) = exp[−A(t)kτ ]. (2.4) The function F (·) is monotone decreasing for A(t) > 0. If τ is a random number from the interval (0,∞), then F (τ) is a random number from the interval (0, 1). If τ is a random number chosen consistently with the reaction (2.1), then F (τ) is a random number uniformly distributed in the interval (0, 1) which can be shown as follows. Let a, b, a < b, be chosen arbitrarily in the interval (0, 1). The probability that F (τ) ∈ (a, b) is equal to the probability that τ ∈ (F−1(b), F−1(a)) which is given by the integral of f(A(t), s) over s in the interval (F−1(b), F−1(a)). Using (2.3) and (2.4), we obtain ∫ F−1(a) F−1(b) f(A(t), s) ds = ∫ F−1(a) F−1(b) A(t)k exp[−A(t)ks] ds ∫ F−1(a) F−1(b) ds = −F [F−1(a)] + F [F−1(b)] = b− a. Hence we have verifed that F (τ) is a random number uniformly distributed in (0, 1). Such a number can be obtained using the random number generator (e.g. function STOCHASTIC REACTION-DIFFUSION PROCESSES 5 rand in Matlab). Let us denote it by r. The previous observation implies that we can generate the time step τ by putting r = F (τ). Using (2.4), we obtain r = exp[−A(t)kτ ]. Solving for τ , we obtain the formula A(t)k . (2.5) Consequently, the SSA for the chemical reaction (2.1) can be written as follows. Starting with t = 0 and A(0) = n0, we perform three steps at time t: (a2) Generate a random number r uniformly distributed in the interval (0, 1). (b2) Compute the time when the next reaction (2.1) occurs as t + τ where τ is given by (2.5). (c2) Compute the number of molecules at time t+ τ by A(t+ τ) = A(t) − 1. Then continue with step (a2) for time t+ τ. Steps (a2)–(c2) are repeated until we reach the time when there is no molecule of A in the system, i.e. A = 0. SSA (a2)–(c2) computes the time of the next reaction t + τ using formula (2.5) in step (b2) with the help of one random number only. Then the reaction is performed in step (c2) by decreasing the number of molecules of chemical species A by 1. The time evolution of A obtained by SSA (a2)–(c2) is given in Figure 2.1(b). We plot ten realizations of SSA (a2)–(c2) for k = 0.1 sec−1 and A(0) = 20. Since the function A(t) has only integer values {0, 1, 2, . . . , 20}, it is not surprising that some of the computed curves A(t) partially overlap. On the other hand, all ten realizations yield different functions A(t). Even if we made millions of stochastic realizations, it would be very unlikely (with probability zero) that there would be two realizations giving exactly the same results. Therefore, the details of one realization A(t) are of no special interest (they depend on the sequence of random numbers obtained by the random number generator). However, averaging values of A at time t over many realizations (e.g. computing the stochastic mean of A), we obtain a reproducible characteristic of the system – see the dashed line in Figure 2.1(b). The stochastic mean of A(t) over (infinitely) many realizations can be also computed theoretically as follows. Let us denote by pn(t) the probability that there are n molecules of A at time t in the system, i.e. A(t) = n. Let us consider an (infinitesimally) small time step dt chosen such that the probability that two molecules are degraded during [t, t + dt) is negligible compared to the probability that only one molecule is degraded during [t, t+ dt). Then there are two possible ways for A(t + dt) to take the value n: either A(t) = n and no reaction occurred in [t, t+dt), or A(t) = n+1 and one molecule was degraded in [t, t+ dt). Hence pn(t+ dt) = pn(t)× (1− kn dt) + pn+1(t)× k(n+ 1) dt. A simple algebraic manipulation yields pn(t+ dt)− pn(t) = k(n+ 1) pn+1(t)− kn pn(t). Passing to the limit dt → 0, we obtain the so-called chemical master equation in the = k(n+ 1) pn+1 − kn pn. (2.6) 6 RADEK ERBAN ET AL. Equation (2.6) looks like an infinite system of ordinary differential equations (ODEs) for pn, n = 0, 1, 2, 3, . . . . Our initial condition A(0) = n0 implies that there are never more than n0 molecules in the system. Consequently, pn ≡ 0 for n > n0 and the system (2.6) reduces to a system of (n0 + 1) ODEs for pn, n ≤ n0. The equation for n = n0 reads as follows = −kn0 pn0 . Solving this equation with initial condition pn0(0) = 1, we get pn0(t) = exp[−kn0t]. Using this formula in the chemical master equation (2.6) for pn0−1(t), we obtain pn0−1(t) = kn0 exp[−kn0t]− k(n0 − 1) pn0−1(t). Solving this equation with initial condition pn0−1(0) = 0, we obtain pn0−1(t) = exp[−k(n0 − 1)t]n0(1 − exp[−kt]). Using mathematical induction, it is possible to pn(t) = exp[−knt] 1− exp[−kt] }n0−n . (2.7) The formula (2.7) provides all the information about the stochastic process which is defined by (2.1) and initial condition A(0) = n0. We can never say for sure that A(t) = n; we can only say that A(t) = n with probability pn(t). In particular, formula (2.7) can be used to derive a formula for the mean value of A(t) over (infinitely) many realizations, which is defined by M(t) = n pn(t). Using (2.7), we deduce M(t) = n pn(t) = n exp[−knt] 1− exp[−kt] }n0−n = n0 exp[−kt] n0 − 1 1− exp[−kt] }(n0−1)−(n−1){ exp[−kt] = n0 exp[−kt]. (2.8) The chemical master equation (2.6) and its solution (2.7) can be also used to quan- tify the stochastic fluctuations around the mean value (2.8), i.e. how much can an individual realization of SSA (a2)–(c2) differ from the mean value given by (2.8). We will present the corresponding theory and results on a more complicated illustrative example in Section 2.2. Finally, let us note that a classical deterministic description of the chemical reaction (2.1) is given by the ODE da/dt = −ka. Solving this equation with initial condition a(0) = n0, we obtain the function (2.8), i.e. the stochastic mean is equal to the solution of the corresponding deterministic ODE. However, we should emphasize that this is not true for general systems of chemical reactions, as we will see in Section 2.3 and Section 5.1. STOCHASTIC REACTION-DIFFUSION PROCESSES 7 2.2. Stochastic simulation of production and degradation. We consider a system of two chemical reactions k1−→ ∅, ∅ k2−→ A. (2.9) The first reaction describes the degradation of chemical A with the rate constant k1. It was already studied previously as reaction (2.1). We couple it with the second reaction which represents the production of chemical A with the rate constant k2. The exact meaning of the second chemical reaction in (2.9) is that one molecule of A is created during the time interval [t, t+ dt) with probability k2 dt. As before, the symbol ∅ denotes chemical species which are of no special interest to the modeller. The impact of other chemical species on the rate of production of A is assumed to be time independent and is already incorporated in the rate constant k2. To simulate the system of chemical reactions (2.9), we perform the following four steps at time t (starting with A(0) = n0 at time t = 0): (a3) Generate two random numbers r1, r2 uniformly distributed in (0, 1). (b3)Compute α0 = A(t)k1 + k2. (c3) Compute the time when the next chemical reaction takes place as t+τ where . (2.10) (d3) Compute the number of molecules at time t+ τ by A(t+ τ) = A(t) + 1 if r2 < k2/α0; A(t)− 1 if r2 ≥ k2/α0. (2.11) Then continue with step (a3) for time t+ τ. To justify that SSA (a3)–(d3) correctly simulates (2.9), let us note that the probability that any of the reactions in (2.9) takes place in the time interval [t, t+ dt) is equal to α0 dt. It is given as a sum of the probability that the first reaction occurs (A(t)k1dt) and the probability that the second reaction occurs (k2 dt). Formula (2.10) gives the time t + τ when the next reaction takes place. It can be justified using the same arguments as for formula (2.5). Once we know the time t+ τ , a molecule is produced with probability k2/α0, i.e. the second reaction in (2.9) takes place with probability k2/α0. Otherwise, a molecule is degraded, i.e. the first reaction in (2.9) occurs. The decision as to which reaction takes place is given in step (d3) with the help of the second uniformly distributed random number r2. Five realizations of SSA (a3)–(d3) are presented in Figure 2.2(a) as solid lines. We plot the number of molecules of A as a function of time for A(0) = 0, k1 = 0.1 sec and k2 = 1 sec −1. We see that, after an initial transient, the number of molecules A(t) fluctuates around its mean value. To compute the stochastic mean and quantify the stochastic fluctuations, we use the chemical master equation which can be written for the chemical system (2.9) in the following form = k1(n+ 1) pn+1 − k1n pn + k2 pn−1 − k2 pn (2.12) where pn(t) denotes the probability that A(t) = n for n = 0, 1, 2, 3, . . . . Let us note that the third term on the right hand side is missing in (2.12) for n = 0; i.e. we use 8 RADEK ERBAN ET AL. 0 20 40 60 80 100 time [sec] 0 2 4 6 8 10 12 14 16 18 20 22 number of molecules Gillespie SSA master equation Fig. 2.2. Stochastic simulation of the system of chemical reactions (2.9) for A(0) = 0, k1 = 0.1 sec−1 and k2 = 1 sec −1. (a) A(t) given by five realizations of SSA (a3)–(d3) (solid lines) and stochastic mean (dashed line). (b) Stationary distribution φ(n) obtained by long time simulation of SSA (a3)–(d3) (gray histogram) and by formulae (2.21)–(2.22) (red solid line). the convention that p −1 ≡ 0. The first two terms on the right hand side correspond to the first reaction in (2.9). They already appeared in equation (2.6). Production of A is described by the third and fourth term on the right hand side of (2.9). To derive the chemical master equation (2.12), we can use similar arguments as in the derivation of (2.6). The stochastic mean M(t) and variance V (t) are defined by M(t) = n pn(t), V (t) = n−M(t) pn(t). (2.13) The stochastic mean M(t) gives the average number of molecules of A at time t, while the variance V (t) describes the fluctuations. In Section 2.1, we first solved the chemical master equation (2.6) and then we used its solution (2.7) to compute M(t). Alternatively, we could use the chemical master equation to derive an evolution equation for M(t), i.e. we could find M(t) without solving the chemical master equation. Such an approach will be presented in this section. Multiplying (2.12) by n and summing over n, we obtain npn = k1 n(n+ 1) pn+1 − k1 n2 pn + k2 n pn−1 − k2 n pn. Using definition (2.13) on the left hand side and changing indices n + 1 → n (resp. n− 1→ n) in the first (resp. third) sum on the right hand side, we obtain (n− 1)n pn − k1 n2 pn + k2 (n+ 1) pn − k2 n pn. Adding the first and the second sum (resp. the third and the fourth sum) on the right hand side, we get = −k1 n pn + k2 pn. (2.14) STOCHASTIC REACTION-DIFFUSION PROCESSES 9 Since pn(t) is the probability that A(t) = n and A(t) is equal to a nonnegative integer with probability 1, we have pn(t) = 1. (2.15) Using this fact together with the definition of M(t), equation (2.14) implies the evo- lution equation for M(t) in the form = −k1M + k2. (2.16) The solution of (2.16) with initial condition M(0) = 0 is plotted as a dashed line in Figure 2.2(a). To derive the evolution equation for the variance V (t), let us first observe that definition (2.13) implies n2 pn(t) = V (t) +M(t) 2. (2.17) Multiplying (2.12) by n2 and summing over n, we obtain n2pn = k1 n2(n+ 1) pn+1 − k1 n3 pn + k2 n2 pn−1 − k2 n2 pn. Changing indices n + 1 → n (resp. n − 1 → n) in the first (resp. third) sum on the right hand side and adding the first and the second sum (resp. the third and the fourth sum) on the right hand side, we get n2pn = k1 (−2n2 + n) pn + k2 (2n+ 1) pn. Using (2.17), (2.15) and (2.13), we obtain = −2k1 V +M2 + k1M + 2k2M + k2. Substituting (2.16) for dM/dt, we derive the evolution equation for the variance V (t) in the following form = −2k1V + k1M + k2. (2.18) The time evolution of M(t) and V (t) is described by (2.16) and (2.18). Let us define the stationary values of M(t) and V (t) by Ms = lim M(t), Vs = lim V (t). (2.19) The values of Ms and Vs can be computed using the steady state equations corre- sponding to (2.16) and (2.18), namely by solving 0 = −k1Ms + k2, and 0 = −2k1Vs + k1Ms + k2. 10 RADEK ERBAN ET AL. Consequently, Ms = Vs = For our parameter values k1 = 0.1 sec −1 and k2 = 1 sec −1, we obtain Ms = Vs = 10. We see in Figure 2.2(a) that A(t) fluctuates after a sufficiently long time around the mean value Ms = 10. To quantify the fluctuations, one often uses the square root of Vs, the so-called mean standard deviation which is equal to More detailed information about the fluctuations is given by the so-called sta- tionary distribution φ(n), n = 0, 1, 2, 3, . . . , which is defined as φ(n) = lim pn(t). (2.20) This means that φ(n) is the probability that A(t) = n after an (infinitely) long time. One way to compute φ(n) is to run SSA (a3)–(d3) for a long time and create a histogram of values of A(t) at given time intervals. Using k1 = 0.1 sec −1 and k2 = 1 sec −1, the results of such a long time computation are presented in Figure 2.2(b) as a gray histogram. To compute it, we ran SSA (a3)–(d3) for 105 seconds, recording the value of A(t) every second and then dividing the whole histogram by the number of recordings, i.e. by 105. An alternative way to compute φ(n) is to use the steady state version of the chemical master equation (2.12), namely 0 = k1 φ(1)− k2 φ(0) 0 = k1(n+ 1)φ(n+ 1)− k1nφ(n) + k2 φ(n− 1)− k2 φ(n), for n ≥ 1, which implies φ(1) = φ(0), (2.21) φ(n+ 1) = k1(n+ 1) k1nφ(n) + k2 φ(n)− k2 φ(n− 1) , for n ≥ 1. (2.22) Consequently, we can express φ(n) for any n ≥ 1 in terms of φ(0). The formulae (2.21)–(2.22) yield an alternative way to compute φ(n). We put φ(0) = 1 and we compute φ(n), for sufficiently many n, by (2.21)–(2.22). Then we divide φ(n), n ≥ 0, φ(n). The results obtained by (2.21)–(2.22) are plotted in Figure 2.2(b) as a (red) solid line. As expected, the results compare well with the results obtained by the long time stochastic simulation. We can also find the formula for φ(n) directly. We let a reader to verify that the solution of the recurrence formula (2.21)–(2.22) can be written as φ(n) = (2.23) where C is a real constant. Using (2.15) and (2.20), we have φ(n) = 1. (2.24) Substituting (2.23) into the normalization condition (2.24), we get = C exp STOCHASTIC REACTION-DIFFUSION PROCESSES 11 where we used the Taylor series for the exponential function to get the last equal- ity. Consequently, C = exp[−k2/k1] which, together with (2.23), implies that the stationary distribution φ(n) is the Poisson distribution φ(n) = . (2.25) Thus the red solid line in Figure 2.2(b) which was obtained numerically by the re- currence formula (2.21)–(2.22) can be also viewed as the stationary distribution φ(n) given by the explicit exact formula (2.25). 2.3. Gillespie algorithm. SSAs (a2)–(c2) and (a3)–(d3) were special forms of the so-called Gillespie SSA. In this section, we present this algorithm for a more com- plicated illustrative example which will also involve second-order chemical reactions. Such chemical reactions are of the following form k1−→ C, A+B k2−→ D. (2.26) In the first equation, two molecules of A react with rate constant k1 to produce C. The probability that the reaction takes place in the time interval [t, t + dt) is equal to A(t)(A(t)− 1)k1dt. We define the propensity function of the first reaction as α1(t) = A(t)(A(t) − 1)k1. Then the probability that the first reaction occurs in the time interval [t, t+dt) is equal to α1(t) dt. The propensity function which corresponds to the second equation in (2.26) is defined as α2(t) = A(t)B(t)k1 and the probability that the second reaction occurs in the time interval [t, t+ dt) is equal to α2(t) dt. In such a case, one molecule of A and one molecule of B react to form a molecule of D. In general, the propensity function can be defined for any chemical reaction so that its product with dt gives the probability that the given reaction occurs in the infinitesimally small time interval [t, t+ dt). We consider that A and B can react according to (2.26). Moreover, we assume that they are also produced with constant rates, that is, we consider a system of four chemical equations: k1−→ ∅ A+B k2−→ ∅ (2.27) ∅ k3−→ A ∅ k4−→ B. (2.28) Let us note that we are not interested in chemical species C and D. Hence, we replaced them by ∅, consistent with our previous notation of unimportant chemical species. To simulate the system of chemical reactions (2.27)–(2.28), we perform the following four steps at time t (starting with A(0) = n0, B(0) = m0 at time t = 0): (a4) Generate two random numbers r1, r2 uniformly distributed in (0, 1). (b4) Compute the propensity functions of each reaction by α1 = A(t)(A(t)−1)k1, α2 = A(t)B(t)k2, α3 = k3 and α4 = k4. Compute α0 = α1 + α2 + α3 + α4. (c4) Compute the time when the next chemical reaction takes place as t+τ where . (2.29) 12 RADEK ERBAN ET AL. 0 20 40 60 80 100 time [sec] solution of ODEs 0 20 40 60 80 100 time [sec] solution of ODEs Fig. 2.3. Five realizations of SSA (a4)–(d4). Number of molecules of chemical species A (left panel) and B (right panel) are plotted as functions of time as solid lines. Different colours correspond to different realizations. The solution of (2.33)–(2.34) is given by the dashed line. We use A(0) = 0, B(0) = 0, k1 = 10 −3 sec−1, k2 = 10 −2 sec−1, k3 = 1.2 sec −1 and k4 = 1 sec (d4) Compute the number of molecules at time t+ τ by A(t+ τ) = A(t)− 2 if 0 ≤ r2 < α1/α0; A(t)− 1 if α1/α0 ≤ r2 < (α1 + α2)/α0; A(t) + 1 if (α1 + α2)/α0 ≤ r2 < (α1 + α2 + α3)/α0; A(t) if (α1 + α2 + α3)/α0 ≤ r2 < 1; (2.30) B(t+ τ) = B(t) if 0 ≤ r2 < α1/α0; B(t)− 1 if α1/α0 ≤ r2 < (α1 + α2)/α0; B(t) if (α1 + α2)/α0 ≤ r2 < (α1 + α2 + α3)/α0; B(t) + 1 if (α1 + α2 + α3)/α0 ≤ r2 < 1; (2.31) Then continue with step (a4) for time t+ τ. SSA (a4)–(d4) is a direct generalisation of SSA (a3)–(d3). At each time step, we first ask the question when will the next reaction occur? The answer is given by formula (2.29) which can be justified using the same arguments as formulae (2.5) or (2.10). Then we ask the question which reaction takes place. The probability that the i-th chemical reaction occurs is given by αi/α0. The decision which reaction takes place is given in step (d4) with the help of the second uniformly distributed random number r2. Knowing that the i-th reaction took place, we update the number of reactants and products accordingly. Results of five realizations of SSA (a4)–(d4) are plotted in Figure 2.3 as solid lines. We use A(0) = 0, B(0) = 0, k1 = 10 −3 sec−1, k2 = 10 −2 sec−1, k3 = 1.2 sec −1 and k4 = 1 sec −1. We plot the number of molecules of chemical species A and B as functions of time. We see that, after initial transients, A(t) and B(t) fluctuate around their average values. They can be estimated from long time stochastic simulations as 9.6 for A and 12.2 for B. Let pn,m(t) be the probability that A(t) = n and B(t) = m. The chemical master STOCHASTIC REACTION-DIFFUSION PROCESSES 13 equation can be written in the following form dpn,m = k1(n+ 2)(n+ 1) pn+2,m − k1n(n− 1) pn,m + k2(n+ 1)(m+ 1) pn+1,m+1 − k2nmpn,m + k3 pn−1,m − k3 pn,m + k4 pn,m−1 − k4 pn,m (2.32) for n, m ≥ 0, with the convention that pn,m ≡ 0 if n < 0 or m < 0. The first difference between (2.32) and the chemical master equations from the previous sections is that equation (2.32) is parametrised by two indices n and m. The second important difference is that (2.32) cannot be solved analytically as we did with (2.6). Moreover, it does not lead to closed evolution equations for stochastic means and variances; i.e. we cannot follow the same technique as in the case of equation (2.12). The approach from the previous section does not work. Let us note that the probability pn,m(t) is sometimes denoted by p(n,m, t); such a notational convention is often used when we consider systems of many chemical species. We will use it in the following sections to avoid long subscripts. The classical deterministic description of the chemical system (2.27)–(2.28) is given by the system of ODEs = −2k1a2 − k2 ab+ k3, (2.33) = −k2 ab+ k4. (2.34) The solution of (2.33)–(2.34) with initial conditions a(0) = 0 and b(0) = 0 is plotted as a dashed line in Figure 2.3. Let us note that the equations (2.33)–(2.34) do not describe the stochastic means of A(t) and B(t). For example, the steady state values of (2.33)–(2.34) are (for the parameter values of Figure 2.3) equal to as = bs = 10. On the other hand, the average values estimated from long time stochastic simulations are 9.6 for A and 12.2 for B. We will see later in Section 5.1 that the difference between the results of stochastic simulations and the corresponding ODEs can be even more significant. The stationary distribution is defined by φ(n,m) = lim pn,m(t). This can be computed by long time simulations of SSA (a4)–(d4) and is plotted in Figure 2.4(a). We see that there is a correlation between the values of A and B. This can also be observed in Figure 2.3. Looking at the blue realizations, we see that the values of A(t) are below the average and the values of B(t) are above the average, similarly for other realizations. One can also define the stationary distribution of A only by φ(n) = φ(n,m). (2.35) Summing the results of Figure 2.4(a) overm, we obtain φ(n) which is plotted in Figure 2.4(b) as a gray histogram. The red bar highlights the steady state value as = 10 of system (2.33)–(2.34). SSAs (a3)–(d3) and (a4)–(d4) were special forms of the so-called Gillespie SSA. To conclude this section, we formulate the Gillespie SSA in its full generality. Let us 14 RADEK ERBAN ET AL. number of A molecules 5 10 15 20 25 30 2 4 6 8 10 12 14 16 18 20 22 24 number of A molecules Fig. 2.4. (a) Stationary distribution φ(n,m) obtained by long time simulation of (a4)–(d4) for k1 = 10 −3 sec−1, k2 = 10 −2 sec−1, k3 = 1.2 sec −1 and k4 = 1 sec −1. (b) Stationary distribution of A obtained by (2.35). consider that we have a system of q chemical reactions. Let αi(t) be the propensity function of the i-th reaction, i = 1, 2, . . . , q, at time t, that is, αi(t) dt is the probability that the i-th reaction occurs in the time interval [t, t + dt). Then the Gillespie SSA consists of the following four steps at time t. (a5) Generate two random numbers r1, r2 uniformly distributed in (0, 1). (b5) Compute the propensity function αi(t) of each reaction. Compute αi(t). (2.36) (c5) Compute the time when the next chemical reaction takes place as t+τ where τ is given by (2.29). (d5) Compute which reaction occurs at time t+ τ . Find j such that αi and r2 < Then the j-th reaction takes place, i.e. update numbers of reactants and products of the j-th reaction. Continue with step (a5) for time t+ τ. The Gillespie SSA (a5)–(d5) provides an exact method for the stochastic simulation of systems of chemical reactions. It was applied previously as SSA (a2)–(c2) for the chemical reaction (2.1), as SSA (a3)–(d3) for the chemical system (2.9) and as SSA (a4)–(d4) for the chemical system (2.27)–(2.28). Our simple examples can be simulated quickly in Matlab (in less than a second on present-day computers). If one considers systems of many chemical reactions and many chemical species, then SSA (a5)–(d5) might be computationally intensive. In such a case, there are ways to make the Gillespie SSA more efficient. For example, it would be a waste of time to recompute all the propensity functions at each time step (step (b5)). We simu- late one reaction per one time step. Therefore, it makes sense to update only those STOCHASTIC REACTION-DIFFUSION PROCESSES 15 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 x [mm] x [mm] −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Fig. 3.1. (a) Six trajectories obtained by SSA (a6)–(b6) for D = 10−4 mm2 sec−1 and ∆t = 0.1 sec. Trajectories were started at the origin and followed for 10 minutes. (b) Probability distribution function ψ(x, y, t) given by (3.5) at time t = 10 min. propensity functions which are changed by the chemical reaction which was selected in step (d5) of SSA (a5)–(d5). A more detailed discussion about the efficient computer implementation of the Gillespie SSA can be found e.g. in [16]. 3. Diffusion. Diffusion is the random migration of molecules (or small particles) arising from motion due to thermal energy [3]. As shown by Einstein, the kinetic en- ergy of a molecule (e.g. protein) is proportional to the absolute temperature. In particular, the protein molecule has a non-zero instantaneous speed at, for exam- ple, room temperature or at the temperature of the human body. A typical protein molecule is immersed in the aqueous medium of a living cell. Consequently, it can- not travel too far before it bumps into other molecules (e.g. water molecules) in the solution. As a result, the trajectory of the molecule is not straight but it executes a random walk as shown in Figure 3.1(a). We plot six possible trajectories of the protein molecule with six different colours. All trajectories start at the origin and are followed for 10 minutes. We will provide more details about this figure together with the methods for simulating molecular diffusion in the rest of this section. Stochastic models of diffusion which are based on the Smoluchowski equation are introduced in Section 3.1. In Section 3.2, we introduce a model which is suitable for coupling with the Gillespie SSA. Both modelling approaches will be used later in Section 4 for the stochastic modelling of reaction-diffusion processes. Let us note that there exist other models of molecular diffusion – they will be discussed in Section 6. 3.1. Smoluchowski equation and diffusion. Let [X(t), Y (t), Z(t)] ∈ R3 be the position of a diffusing molecule at time t. Starting with [X(0), Y (0), Z(0)] = [x0, y0, z0], we want to compute the time evolution of [X(t), Y (t), Z(t)]. To do that, we make use of a generator of random numbers which are normally distributed with zero mean and unit variance. Such a generator is part of many modern computer languages (e.g. function randn in Matlab). Diffusive spreading of molecules is characterised by a single diffusion constant D which depends on the size of the molecule, absolute temperature and viscosity of the solution [3]. Choosing time step ∆t, we compute the time evolution of the position of the diffusing molecule by performing two steps at time t: 16 RADEK ERBAN ET AL. (a6) Generate three normally distributed (with zero mean and unit variance) random numbers ξx, ξy and ξz. (b6) Compute the position of the molecule at time t+ ∆t by X(t+ ∆t) = X(t) + 2D∆t ξx, (3.1) Y (t+ ∆t) = Y (t) + 2D∆t ξy , (3.2) Z(t+ ∆t) = Z(t) + 2D∆t ξz , (3.3) Then continue with step (a6) for time t+ ∆t. Choosing D = 10−4 mm2 sec−1 (diffusion constant of a typical protein molecule), [X(0), Y (0), Z(0)] = [0, 0, 0] and ∆t = 0.1 sec, we plot six realizations of SSA (a6)– (b6) in Figure 3.1(a). We plot only the x and y coordinates. We follow the diffusing molecule for 10 minutes. The position of the molecule at time t = 10 min is denoted as a black circle for each trajectory. Equations (3.1)–(3.3) are discretized versions of stochastic differential equations (SDEs) which are sometimes called Smoluchowski equations. Some basic facts about SDEs can be found e.g. in [2, 14]. A more accessible introduction to SDEs can be found in [23] which has a similar philosophy as our paper. Reference [23] is a nice algorithmic introduction to SDEs for students who do not have a prior knowledge of advanced probability theory or stochastic analysis. We will not go into details of SDEs in this paper, but only highlight some interesting facts which will be useful later. First, equations (3.1)–(3.3) are not coupled. To compute the time evolution of X(t), we do not need to know the time evolution of Y (t) or Z(t). We will later focus only on the time evolution of the x-th coordinate, effectively studying one-dimensional problems. Two-dimensional or three-dimensional problems can be treated similarly. Second, we see that different realizations of SSA (a6)–(b6) give different results. To get more reproducible quantities, we will shortly study the behaviour of several molecules. However, even in the case of a single diffusing molecule, there are still quantities whose evolution is deterministic. Let ϕ(x, y, t) dxdydz be the probability that X(t) ∈ [x, x + dx), Y (t) ∈ [y, y + dy) and Z(t) ∈ [z, z + dz) at time t. It can be shown that ϕ evolves according to the partial differential equation , (3.4) which is a special form of the so-called Fokker-Planck equation. Since our random walk starts at the origin, we can solve (3.4) with initial condition ϕ(x, y, z, 0) = δ(x, y, z) where δ is the Dirac distribution at the origin. We obtain ϕ(x, y, z, t) = (4Dπt)3/2 2 + y2 + z2 In order to visualise this probability distribution function, we integrate it over z to get probability distribution function ψ(x, y, t) = ϕ(x, y, z, t)dz = 2 + y2 . (3.5) This means that ψ(x, y, t) dxdy is the probability that X(t) ∈ [x, x+ dx) and Y (t) ∈ [y, y + dy) at time t. The function ψ(x, y, t) at time t = 10 min is plotted in Figure STOCHASTIC REACTION-DIFFUSION PROCESSES 17 0 0.2 0.4 0.6 0.8 1 x [mm] 0 0.2 0.4 0.6 0.8 1 time=4 min x [mm] Fig. 3.2. (a) Ten trajectories computed by SSA (a7)–(c7) for D = 10−4 mm2 sec−1, L = 1 mm, X(0) = 0.4 mm and ∆t = 0.1 sec. (b) Numbers of molecules in bins of length h = 25 µm at time t = 4 min. 3.1(b). It can be obtained also by computing many realizations of SSA (a6)–(b6) and plotting the histogram of positions of a molecule at time 10 min; such positions were denoted as black circles for the six illustrative trajectories in Figure 3.1(a). One important issue which was not addressed previously is that molecules diffuse in bounded volumes, i.e. the domain of interest has boundaries and suitable bound- ary conditions must be implemented. In the rest of this paper, we focus on one- dimensional problems to avoid technicalities. Hence, we effectively study diffusion of molecules in the one-dimensional interval [0, L]. Then the SSA can be formulated as follows: (a7) Generate a normally distributed (with zero mean and unit variance) random number ξ. (b7) Compute the position of the molecule at time t+ ∆t by X(t+ ∆t) = X(t) + 2D∆t ξ. (3.6) (c7) If X(t+ ∆t) computed by (3.6) is less than 0, then X(t+ ∆t) = −X(t)− 2D∆t ξ. If X(t+ ∆t) computed by (3.6) is greater than L, then X(t+ ∆t) = 2L−X(t)− 2D∆t ξ. Then continue with step (a7) for time t+ ∆t. The boundary condition implemented in step (c7) is the so-called reflective boundary condition or zero flux boundary condition. It can be used when there is no chem- ical interaction between the boundary and diffusing molecules. More complicated boundary conditions are discussed in [7, 8]. Choosing D = 10−4 mm2 sec−1, L = 1 mm, X(0) = 0.4 mm and ∆t = 0.1 sec, we plot ten realizations of SSA (a7)–(c7) in Figure 3.2(a). Let us assume that we have a system of 1000 molecules which are released at position x = 0.4 mm at time t = 0. Then Figure 3.2(a) can be viewed as a plot of the trajectories of ten representative molecules. Considering 1000 molecules, the trajectories of individual molecules are of no special interest. We are rather interested in spatial histograms (density of molecules). An example of such a plot is given in Figure 3.2(b). We simulate 1000 18 RADEK ERBAN ET AL. molecules, each following SSA (a7)–(c7). At time t = 4 min, we divided the domain of interest [0, L] into 40 bins of length h = L/40 = 25 µm. We calculated the number of molecules in each bin [(i− 1)h, ih), i = 1, 2, . . . , 40, at time t = 4 min and plotted them as a histogram. Let us note that the deterministic counterpart to the stochastic simulation is a solution of the corresponding Fokker-Planck equation (diffusion equation in our case) which, in one dimension with zero flux boundary conditions, reads as follows where (0) = (L) = 0. (3.7) The solution of (3.7) with the Dirac-like initial condition at x = 0.4 mm is plotted as a red solid line in Figure 3.2(b) for comparison. 3.2. Compartment-based approach to diffusion. In Section 3.1, we sim- ulated the behaviour of 1000 molecules by computing the individual trajectories of every molecule (using SSA (a7)–(c7)). At the end of the simulation, we divided the computational domain [0, L] into K = 40 compartments and we plotted numbers of molecules in each compartment in Figure 3.2(b). In particular, most of the com- puted information (1000 trajectories) was not used for the final result – the spatial histogram. We visualised only 40 numbers (numbers of molecules in compartments) instead of 1000 computed positions of molecules. In this section, we present a different SSA for the simulation of molecular diffusion. We redo the example from Section 3.1 but instead of simulating 1000 positions of the individual molecules, we are going to simulate directly the time evolution of 40 compartments. To do that, we divide the computational domain [0, L] into K = 40 compartments of length h = L/K = 25µm. We denote the number of molecules of chemical species A in the i-th compartment [(i− 1)h, ih) by Ai, i = 1, . . . ,K. We apply the Gillespie SSA to the following chain of “chemical reactions”: d−→←− d−→←− d−→←− . . . d−→←− AK (3.8) where d−→←− Ai+1 means that Ai d−→ Ai+1 and Ai+1 d−→ Ai. We will shortly show that the Gillespie SSA of (3.8) provides a correct model of diffusion provided that the rate constant d in (3.8) is chosen as d = D/h2 where D is the diffusion constant and h is the compartment length. The compartment- based SSA can be described as follows. Starting with initial condition Ai(t) = a0,i, i = 1, 2, . . . ,K, we perform six steps at time t: (a8) Generate two random numbers r1, r2 uniformly distributed in (0, 1). (b8) Compute propensity functions of reactions by αi = Ai(t)d for i = 1, 2, . . . ,K. Compute αi. (3.9) (c8) Compute the time at which the next chemical reaction takes place as t + τ where τ is given by (2.29). STOCHASTIC REACTION-DIFFUSION PROCESSES 19 (d8) If r2 < i=1 αi/α0, then find j ∈ {1, 2, . . . ,K − 1} such that αi and r2 < Then compute the number of molecules at time t+ τ by Aj(t+ τ) = Aj(t)− 1, (3.10) Aj+1(t+ τ) = Aj+1(t) + 1, (3.11) Ai(t+ τ) = Ai(t), for i 6= j, i 6= j + 1. (3.12) (e8) If r2 ≥ i=1 αi/α0, then find j ∈ {2, 3, . . . ,K} such that and r2 < Then compute the number of molecules at time t+ τ by Aj(t+ τ) = Aj(t)− 1, (3.13) Aj−1(t+ τ) = Aj−1(t) + 1, (3.14) Ai(t+ τ) = Ai(t), for i 6= j, i 6= j − 1. (3.15) (f8) Continue with step (a8) for time t+ τ. The first term on the right hand side of (3.9) corresponds to reactions Ai → Ai+1 (jumps to the right) and the second term corresponds to reactions Ai → Ai−1 (jumps to the left). The time of the next chemical reaction is computed in the step (c8) using formula (2.29) derived previously. The decision about which reaction takes place is done in steps (d8)–(e8) with the help of random number r2. Jumps to the right are implemented in step (d8) and jumps to the left in step (e8). We want to redo the example from Section 3.1, i.e. simulate 1000 molecules starting from position 0.4 mm in the interval [0, L] for L = 1 mm. We use K = 40. Since 0.4 mm is exactly a boundary between the 16th and 17th compartment, the initial condition is given by A16(0) = 500, A17(0) = 500 and Ai(0) = 0 for i 6= 16, i 6= 17. As D = 10−4 mm2 sec−1, we have d = D/h2 = 0.16 sec−1. The numbers Ai(t), i = 1, . . . ,K, at time t = 4 min, are plotted in Figure 3.3(a) as a histogram. This panel can be directly compared with Figure 3.2(b). The computational intensity of SSA (a8)–(f8) can be decreased using the appropriate way to implement it in the computer. For example, only one chemical reaction occurs per time step. Consequently, only two propensity functions change and need to be updated in step (b8). Moreover, the formula (3.9) can be simplifed as follows αi = 2 αi−α1−αK = 2d Ai(t)−α1−αK = 2dN−α1−αK , where N = 1000 is the total number of molecules in the simulation (this number is conserved because there is no creation or degradation of the molecules in the system). Hence, we need to recompute α0 only when there is a change in α1 or αK , i.e. whenever the boundary compartments were involved in the previous reaction. 20 RADEK ERBAN ET AL. 0 0.2 0.4 0.6 0.8 1 time=4 min x [mm] 0 0.2 0.4 0.6 0.8 1 x [mm] Fig. 3.3. Compartment-based SSA model of diffusion. (a) Numbers Ai(t), i = 1, 2, . . . ,K, at time t = 4 min obtained by SSA (a8)–(f8). We use d = D/h2 = 0.16 sec−1, K = 40 and initial condition A16(0) = 500, A17(0) = 500 and Ai(0) = 0 for i 6= 16, i 6= 17. (b) Ten realizations of the simulation of an individual molecule by SSA (a8)–(f8). SSA (a8)–(f8) does not compute the trajectories of individual molecules. However, we can still compute a plot comparable with Figure 3.2(a). To do that, we repeat the simulation with 1 molecule instead of 1000. Then, at given time t, exactly one of numbers Ai(t), i = 1, 2, . . . ,K, is non-zero and equal to 1. This is a position of the molecule at time t. Ten realizations of SSA (a8)–(f8) with one molecule released at 0.4 mm at t = 0 are plotted in Figure 3.3(b). This panel can be directly compared with Figure 3.2(a). Let p(n, t) be the joint probability that Ai(t) = ni, i = 1, . . . ,K, where we denoted n = [n1, n2, . . . , nK ]. Let us define operators Ri, Li : N K → NK by Ri : [n1, . . . , ni, ni+1, . . . , nK ]→ [n1, . . . , ni + 1, ni+1 − 1, . . . , nK ], i = 1, . . . ,K − 1, (3.16) Li : [n1, . . . , ni−1, ni, . . . , nK ]→ [n1, . . . , ni−1 − 1, ni + 1, . . . , nK ], i = 2, . . . ,K. (3.17) Then the chemical master equation, which corresponds to the system of chemical reactions given by (3.8), can be written as follows ∂P (n) (nj + 1)P (Rjn)− nj P (n) (nj + 1)P (Ljn)− nj P (n) (3.18) The stochastic mean is defined as the vector M(t) ≡ [M1,M2, . . . ,MK ] where Mi(t) = ni P (n, t) ≡ · · · ni P (n, t) (3.19) gives the mean number of molecules in the i-th compartment, i = 1, 2, . . . ,K. To derive an evolution equation for the stochastic mean vector M(t), we can follow the method from Section 2.2 – see derivation of (2.16) from chemical master equation (2.12). Multiplying (3.18) by ni and summing over n, we obtain (leaving the details STOCHASTIC REACTION-DIFFUSION PROCESSES 21 to the student) a system of equations for Mi of the form = d(Mi+1 +Mi−1 − 2Mi), i = 2, . . . ,K − 1, (3.20) = d(M2 −M1), = d(MK−1 −MK). (3.21) System (3.20)–(3.21) is equivalent to a discretization of (3.7) provided that d = D/h2. Hence, we have derived the relation between the rate constant d in (3.8), diffusion constant D and compartment length h. The solution of (3.7) with the Dirac-like initial condition at x = 0.4 mm is plotted for comparison as a red solid line in Figure 3.3(a). The noise is described by the variance vector V(t) ≡ [V1, V2, . . . , VK ] where Vi(t) = (ni −Mi(t))2 P (n, t) ≡ · · · (ni −Mi(t))2 P (n, t) (3.22) gives the variance of number of molecules in the i-th compartment, i = 1, 2, . . . ,K. To derive the evolution equation for the vector V(t), we define the matrix {Vi,j} by Vij = ninj P (n, t)−MiMj, for i, j = 1, 2, . . . ,K. Using (3.22), we obtain Vi = Vii for i = 1, 2, . . . ,K. Multiplying (3.18) by n i and summing over n, we obtain n2iP (n) = d n2i (nj + 1)P (Rjn)− n2inj P (n) n2i (nj + 1)P (Ljn)− n2inj P (n) . (3.23) Let us assume that i = 2, . . . ,K − 1. Let us consider the term corresponding to j = i in the first sum on the right hand side. We get n2i (ni + 1)P (Rin)− n2ini P (n) = (ni − 1)2ni P (n)− n2ini P (n) (−2n2i + ni)P (n) = −2Vi − 2M2i +Mi. First, we changed indices in the first sum Rin→ n and then we used definitions (3.19) and (3.22). Similarly, the term corresponding to j = i−1 in the first sum on the right hand side of (3.23) can be rewritten as n2i (ni−1 + 1)P (Ri−1n)− n2ini−1 P (n) = (2nini−1 + ni−1)P (n) = 2Vi,i−1 + 2MiMi−1 +Mi−1. 22 RADEK ERBAN ET AL. Other terms (corresponding to j 6= i, i− 1) in the first sum on the right hand side of (3.23) are equal to zero. The second sum can be handled analogously. We obtain n2iP (n) = d 2Vi,i−1 + 2MiMi−1 +Mi−1 − 2Vi − 2M2i +Mi 2Vi,i+1 + 2MiMi+1 +Mi+1 − 2Vi − 2M2i +Mi . (3.24) Using (3.22) and (3.20) on the left hand side of (3.24), we obtain n2iP (n) = + 2Mi + d(2MiMi+1 + 2MiMi−1 − 4M2i ). Substituting this into (3.24), we get Vi,i+1 + Vi,i−1 − 2Vi Mi+1 +Mi−1 + 2Mi (3.25) for i = 2, . . . ,K − 1. Similarly, we get V1,2 − V1 M2 +M1 , (3.26) VK,K−1 − VK MK−1 +MK . (3.27) We see that the evolution equation for the variance vector V(t) depends on the mean M, variance V and on non-diagonal terms of the matrix Vi,j . To get a closed system of equations, we have to derive evolution equations for Vi,j too. This can be done by multiplying (3.18) by ninj, summing over n and following the same arguments as before. We conclude this section with some consequences of (3.20)–(3.21) and (3.25)– (3.27). Looking at the steady states of equations (3.20)–(3.21), we obtain Mi = N/K, i = 1, 2, . . . ,K, where N is the total number of diffusing molecules. Moreover, the variance equations imply that Vi = N/K, i = 1, 2, . . . ,K, at the steady state. 4. Stochastic reaction-diffusion models. In this section, we add chemical reactions to both models of molecular diffusion which were presented in Section 3. We introduce two methods for the stochastic modelling of reaction-diffusion processes. The first one is based on the diffusion model from Section 3.2, the second one on the diffusion model from Section 3.1. We explain both methods using the same example. Namely, we consider molecules (e.g. protein) which diffuse in the domain [0, L] with diffusion constant D as we considered in Section 3. Moreover, we assume that protein molecules are degraded (in the whole domain) and produced in part of the domain, i.e. we consider the chemical reactions from Sections 2.1 and 2.2 in our illustrative reaction-diffusion model. The model has a realistic motivation which is discussed in more detail later in Section 5.2. In Section 4.3, we present another illustrative example of a reaction-diffusion process incorporating the nonlinear model (2.27)–(2.28). 4.1. Compartment-based reaction-diffusion SSA. We consider molecules of chemical species A which are diffusing in the domain [0, L], where L = 1 mm, with diffusion constant D = 10−4 mm2 sec−1. Initially, there are no molecules in the system. Molecules are produced in the part of the domain [0, L/5] with rate STOCHASTIC REACTION-DIFFUSION PROCESSES 23 0 0.2 0.4 0.6 0.8 1 x [mm] time=10 min 0 0.2 0.4 0.6 0.8 1 x [mm] time=30 min Fig. 4.1. One realization of the Gillespie SSA (a5)–(d5) for the system of chemical reactions (4.1)–(4.3). Gray histograms show numbers of molecules in compartments at time: (a) t = 10 min; (b) t = 30 min. Solution of (4.9)–(4.10) is plotted as the red solid line. kp = 0.012 µm −1 sec−1. This means that the probability that a molecule is created in the subinterval of the length 1 µm is equal to kp dt. Consequently, the probability that a molecule is created somewhere in the interval [0, L/5] is equal to kpL/5 dt. Molecules are degraded with rate k1 = 10 −3 sec−1 according to the chemical reaction (2.1). Following Section 3.2, we divide the computational domain [0, L] into K = 40 compartments of length h = L/K = 25µm. We denote the number of molecules of chemical species A in the i-th compartment [(i− 1)h, ih) by Ai, i = 1, . . . ,K. Then our reaction-diffusion process is described by the system of chemical reactions d−→←− d−→←− d−→←− . . . d−→←− AK , (4.1) k1−→ ∅, for i = 1, 2, . . . ,K, (4.2) ∅ k2−→ Ai, for i = 1, 2, . . . ,K/5. (4.3) Equation (4.1) describes diffusion and is identical to (3.8). In particular, the rate constant d is given by d = D/h2. Equation (4.2) describes the degradation of A and is, in fact, equation (2.1) applied to every compartment. Equation (4.3) describes the production of A in the first K/5 compartments (e.g. in part [0, L/5] of the computational domain). The rate constant k2 describes the rate of production per compartment. Since each compartment has length h, we have k2 = kph. The system of chemical reactions (4.1)–(4.3) is simulated using the Gillespie SSA (a5)–(d5). In our case, the propensity functions of reactions in (4.1) are given as Ai(t)d, the propensity functions of reactions in (4.2) are given as Ai(t)k1 and propen- sity functions of reactions in (4.3) are equal to k2. Starting with no molecules of A in the system, we compute one realization of SSA (a5)–(d5) for the system of reactions (4.1)–(4.3). We plot the numbers of molecules in compartments at two different times in Figure 4.1. 24 RADEK ERBAN ET AL. Let p(n, t) be the joint probability that Ai(t) = ni, i = 1, . . . ,K, where we use the notation n = [n1, n2, . . . , nK ]. Let us define operators Ri, Li : N K → NK by (3.16)–(3.17). Then the chemical master equation, which corresponds to the system of chemical reactions (4.1)–(4.3), can be written as follows ∂p(n) (ni + 1) p(Rin)− ni p(n) (ni + 1) p(Lin)− ni p(n) (ni + 1) p(n1, . . . , ni + 1, . . . , nK)− ni p(n) p(n1, . . . , ni − 1, . . . , nK)− p(n) . (4.4) The first two sums correspond to diffusion (4.1), the third sum to degradation (4.2) and the fourth sum to production (4.3). The stochastic mean is defined as the vector M(t) ≡ [M1,M2, . . . ,MK ] where Mi is given by (3.19). This gives the mean number of molecules in the i-th compartment, i = 1, 2, . . . ,K, at time t (averaged over many realizations of SSA (a5)–(d5)). To derive the evolution equation for the stochastic mean vector M(t), we can follow the method from Section 2.2 – see derivation of (2.16) from the chemical master equation (2.12). Multiplying (4.4) by ni and summing over all nj , j = 1, . . . ,K, we obtain (leaving the details to the student) a system of equations for Mi in the form = d(M2 −M1) + k2 − k1M1, (4.5) = d(Mi+1 +Mi−1 − 2Mi) + k2 − k1Mi, i = 2, . . . ,K/5, (4.6) = d(Mi+1 +Mi−1 − 2Mi)− k1Mi, i = K/5 + 1, . . . ,K − 1, (4.7) = d(MK−1 −MK)− k1MK . (4.8) System (4.5)–(4.8) is a discretized version of the reaction-diffusion equation + k2χ[0,L/5] − k1a (4.9) with zero-flux boundary conditions (0) = (L) = 0. (4.10) Here, χ[0,L/5] is the characteristic function of the interval [0, L/5]. Using initial con- dition a(·, 0) ≡ 0, we computed the solution of (4.9)–(4.10) numerically. It is plotted as a red solid line in Figure 4.1 for comparison. The concentration of molecules in the i-th compartment is defined as M i = Mi/h, i = 1, . . . ,K. Dividing (4.5)–(4.8) by h, we can write a system of ODEs for M i. It is a discretized version of the reaction-diffusion equation + kpχ[0,L/5] − k1a (4.11) STOCHASTIC REACTION-DIFFUSION PROCESSES 25 where a ≡ a(x, t) is the concentration of molecules of A at point x and time t. The equation (4.11) provides a classical deterministic description of the reaction-diffusion process. Its parametersD, kp and k1 are independent of h. Solving (4.11) is equivalent to solving (4.9). Consequently, the red solid line in Figure 4.1 can be also viewed as a plot of ah where a is the solution of the classical deterministic model (4.11) with the zero-flux boundary conditions. 4.2. Reaction-diffusion SSA based on the Smoluchowski equation. In this section, we present a SSA which implements the Smoluchowski model of diffusion from Section 3.1, that is, we follow the trajectories of individual molecules. Diffusion of each molecule is modelled according to the model (a7)–(c7). We explain the SSA using the reaction-diffusion example from Section 4.1. Choosing a small time step ∆t, we perform the following three steps at time t: (a9) For each molecule, compute its position at time t + ∆t according to steps (a7)–(c7). (b9) For each molecule, generate a random number r1 uniformly distributed in the interval (0, 1). If r1 < k1 ∆t, then remove the molecule from the system. (c9) Generate a random number r2 uniformly distributed in the interval (0, 1). If r2 < kpL/5 ∆t, then generate another random number r3 uniformly dis- tributed in the interval (0, 1) and introduce a new molecule at position r3L/5. Continue with step (a9) for time t+ ∆t. The degradation of molecules is modelled by step (b9). Equation (2.1) implies that k1 dt is the probability that a molecule is degraded in the time interval [t, t + dt) for infinitesimally small dt. SSA (a9)–(c9) replaces dt by the finite time step ∆t (compare with SSA (a1)–(b1)) which has to be chosen sufficiently small so that k1 ∆t ≪ 1. Similarly, the probability that a molecule is created in [0, L/5] in time interval [t, t + dt) is equal to kpL/5 dt. Consequently, we have to choose ∆t so small that kpL/5 ∆t is significantly less than 1. We choose ∆t = 10 −2 sec. Then k1 ∆t = 10 −5 and kpL/5 ∆t = 2.4 × 10−2 for our parameter values k1 = 10−3 sec−1, kp = 0.012 µm −1 sec−1 and L = 1 mm. Starting with no molecules of A in the sys- tem, we compute one realization of SSA (a9)–(c9). To visualise the results, we divide the interval [0, L] into 40 bins and we plot the numbers of molecules in bins at time 10 minutes in Figure 4.2(a). The same plot at time 30 minutes is given in Figure 4.2(b). We used the same number of bins to visualise the results of SSA (a9)–(c9) as we used previously in the compartment-based model. Thus Figure 4.2 is directly comparable with Figure 4.1. We also plot the solution of (4.9)–(4.10) as a red solid line for comparison. 4.3. Reaction-diffusion models of nonlinear chemical kinetics. In the previous sections, we studied an example of a reaction-diffusion model which did not include the second-order chemical reactions (2.26). We considered only production and degradation, i.e. we considered chemical reactions from Sections 2.1 and 2.2. In this section, we discuss generalisations of our approaches to models which involve second-order chemical reactions too. Our illustrative example is a reaction-diffusion process incorporating the nonlinear model (2.27)–(2.28). The second-order chemical reactions (2.26) require that two molecules collide (be close to each other) before the reaction can take place. The generalisation of SSA (a9)–(c9) to such a case is nontrivial and we will not present it in this paper (it can be found in [1]). Application of the Gillespie SSA (a5)–(d5) is more straightforward and is presented below. 26 RADEK ERBAN ET AL. 0 0.2 0.4 0.6 0.8 1 x [mm] time=10 min 0 0.2 0.4 0.6 0.8 1 x [mm] time=30 min Fig. 4.2. One realization of SSA (a9)–(c9). Dividing domain [0, L] into 40 bins, we plot the number of molecules in each bin at time: (a) t = 10 min; (b) t = 30 min. Solution of (4.9)–(4.10) is plotted as the red solid line. We consider that both chemical speciesA andB diffuse in the domain [0, L], where L = 1 mm, with diffusion constant D = 10−4 mm2 sec−1. Following the method of Section 4.1, we divide the computational domain [0, L] into K = 40 compartments of length h = L/K = 25µm. We denote the number of molecules of chemical species A (resp. B) in the i-th compartment [(i − 1)h, ih) by Ai (resp. Bi), i = 1, . . . ,K. Diffusion corresponds to two chains of “chemical reactions”: d−→←− d−→←− d−→←− . . . d−→←− AK (4.12) d−→←− d−→←− d−→←− . . . d−→←− BK (4.13) Molecules of A and B are assumed to react according to chemical reactions (2.27) in the whole domain with rate constants k1 = 10 −3 sec−1 and k2 = 10 −2 sec−1 per one compartment, that is, Ai +Ai k1−→ ∅, Ai +Bi k2−→ ∅, for i = 1, 2, . . . ,K. (4.14) Production of chemical species (2.28) is assumed to take place only in parts of the computational domain [0, L]. Molecules of chemical species A (resp. B) are assumed to be produced in subinterval [0, 9L/10] (resp. [2L/5, L]) with rate k3 = 1.2 sec (resp. k4 = 1 sec −1) per one compartment of length h, that is, ∅ k3−→ Ai, for i = 1, 2, . . . , 9K/10, (4.15) ∅ k4−→ Bi, for i = 2K/5, . . . ,K. (4.16) Starting with no molecules in the system at time t = 0, we present one realization of the Gillespie SSA (a5)–(d5) applied to the chemical system (4.12)–(4.16) in Figure 4.3. We plot the numbers of molecules of A and B at time 30 minutes. STOCHASTIC REACTION-DIFFUSION PROCESSES 27 0 0.2 0.4 0.6 0.8 1 x [mm] stochastic simulation solution of PDEs 0 0.2 0.4 0.6 0.8 1 x [mm] stochastic simulation solution of PDEs Fig. 4.3. One realization of the Gillespie SSA (a5)–(d5) for the system of chemical reactions (4.12)–(4.16). Numbers of molecules of chemical species A (left panel) and B (right panel) in compartments at time 30 minutes (gray histograms). Solution of (4.17)–(4.19) is plotted as the red solid line. We already observed in Section 2.3 that the analysis of the master equation for chemical systems involving the second order reactions is not trivial. It is not possible to derive the equation for stochastic means as was done in Section 4.1 for the linear model. Hence, we will not attempt such an approach here. We also observed in Section 4.1 that the equation for the mean vector (4.5)–(4.8) was actually equal to a discretized version of the reaction-diffusion equation (4.9)–(4.10) which would be used as a traditional deterministic description. When considering the nonlinear chemical model (4.14)–(4.16), we cannot derive the equation for the mean vector but we can still write a deterministic system of partial differential equations (PDEs). We simply add diffusion to the system of ODEs (2.33)–(2.34) to obtain − 2k1a2 − k2 ab+ k3χ[0,9L/10], (4.17) − k2 ab+ k4χ[2L/5,L], (4.18) and couple it with zero-flux boundary conditions (0) = (L) = (0) = (L) = 0. (4.19) Using initial condition a(·, 0) ≡ 0 and b(·, 0) ≡ 0, we can compute the solution of (4.17)–(4.19) numerically. It is plotted as a red solid line in Figure 4.3 for comparison. We see that (4.17)–(4.19) gives a reasonable description of the system when comparing with one realization of SSA (a5)–(d5). However, let us note that solution of (4.17)– (4.19) is not equal to the stochastic mean. The equations (4.17)–(4.19) can be also rewritten in terms of concentrations a = a/h and b = b/h as we did in the case of equations (4.9) and (4.11). Let us note that the rate constants scale with h as k1 = k1/h, k2 = k2/h, k3 = k3h, k4 = k4h where k1, k2, k3, k4 are independent of h. Consequently, the equations for concentrations a and b are independent of h. They can be written in terms of the parameters D, k1, k2, k3 and k4 only (compare with (4.11)). 28 RADEK ERBAN ET AL. Finally, let us discuss the choice of the compartment length h. In Sections 3.2 and 4.1, we considered linear models and we were able to derive the equations for the mean vectors (e.g. (4.5)–(4.8)). Dividing (4.5)–(4.8) by h and passing to the limit h → 0, we derive the corresponding deterministic reaction-diffusion PDE (4.11) which can be viewed (for linear models) as an equation for the probability distribution function of a single molecule (i.e. the exact description which we want to approximate by the compartment-based SSA). Consequently, we can increase the accuracy of the SSA by decreasing h. Considering the nonlinear model from this section, the continuum limit h→ 0 is not well-defined. The compartment-based SSA is generally considered valid only for a range of h values (i.e. the length h cannot be chosen arbitrarily small); conditions which the length h has to satisfy are subject of current research – see e.g. [24, Section 3.5]. 5. Two important remarks. We explained SSAs for chemical reactions, molec- ular diffusion and reaction-diffusion processes in the previous sections. This final section is devoted to two important questions: (a) Why do we care about stochastic modelling? The answer is given in Section 5.1 where we discuss connections between stochastic and deterministic modelling. In particular, we present examples where deterministic modelling fails and a stochastic approach is necessary. We start with a simple example of stochastic switching between favourable states of the system, a phenomenon which cannot be fully understood without stochastic modelling. Then we illustrate the fact that the stochastic model might have qualitatively different properties than its deterministic limit, i.e. the stochastic model is not just “equal” to the “noisy solution” of the corresponding deterministic equations. We present a simple system of chemical reactions for which the deterministic description converges to a steady state. On the other hand, the stochastic model of the same system of chemical reactions has oscillatory solutions. Finally, let us note that stochasticity plays important roles in biological applications, see e.g. [32, 6, 30]. (b) Why do we care about reaction-diffusion processes? The answer is given in Section 5.2 where we discuss biological pattern formation. Reaction-diffusion models are key components of models in developmental biology. We present stochastic ana- logues of two classical pattern forming models. The first one is the so-called French flag problem where we re-interpret the illustrative example from Sections 4.1 and 4.2. Then we present the reaction-diffusion pattern forming model based on the so-called Turing instability. 5.1. Deterministic vs. stochastic modelling. The models presented so far have one thing in common. One could use the deterministic description (given by ODEs or PDEs) and one would obtain a reasonable description of the system. In Sections 2.1, 2.2, 3.1, 3.2, 4.1 and 4.2, we studied linear models. We showed that the evolution equations for the stochastic mean are equal to (the discretized versions of) the corresponding deterministic differential equations. In Sections 2.3 and 4.3, we presented nonlinear models. We were not able to derive equations for the stochastic mean. However, we solved numerically the corresponding systems of deterministic equations (ODEs (2.33)–(2.34) in Section 2.3 and PDEs (4.17)–(4.19) in Section 4.3) and we obtained results comparable with the SSAs, i.e. results of the SSAs looked like “noisy solutions” of the corresponding differential equations. Here, we discuss exam- ples of problems when SSAs give results which cannot be obtained by corresponding deterministic models. Let us consider the model from Section 2.3. Its deterministic description is given STOCHASTIC REACTION-DIFFUSION PROCESSES 29 0 0.5 1 1.5 2 time [min] stochastic deterministic 0 20 40 60 80 100 time [min] Fig. 5.1. Simulation of (5.1). One realization of SSA (a5)–(d5) for the system of chemical reactions (5.1) (blue line) and the solution of the deterministic ODE (5.2) (red line). (a) The number of molecules of A as a function of time over the first two minutes of simulation. (b) Time evolution over 100 minutes. by the system of ODEs (2.33)–(2.34). Such a system has only one nonnegative (stable) steady state for our parameter values, namely as = bs = 10. It can be observed from Figure 2.3 that solutions of (2.33)–(2.34) converge to as and bs as t → ∞. This is true for any nonnegative initial condition. The results of SSAs show fluctuation about the means, which are roughly equal to as and bs (they are 9.6 for A and 12.2 for B). However, there are chemical systems which have two or more favourable states, so that the corresponding ODEs have more than one nonnegative stable steady state. For example, let us consider the system of chemical reactions for chemical A introduced by Schlögl [36] k1−→←− 3A, ∅ k3−→←− A. (5.1) The corresponding ODE is given as follows = − k2 a3 + k1 a2 − k4 a+ k3. (5.2) We choose the rate constants as follows: k1 = 0.18 min −1, k2 = 2.5 × 10−4 min−1, k3 = 2200 min −1 and k4 = 37.5 min −1. Then the ODE (5.2) has two stable steady states as1 = 100 and as2 = 400 and one unstable steady state au = 220. The solution of (5.2) converges to one of the steady states with the choice of the steady state dependent on the initial condition. Let us consider that there are initially no molecules of A in the system, i.e. A(0) = 0. The solution of (5.2) is plotted in Figure 5.1(a) as a red line. We see that the solution of (5.2) converges to the stable steady state as1 = 100. This is true for any initial condition A(0) ∈ [0, au). If A(0) > au, then the solution of (5.2) converges to the second stable steady state as2 = 400. Next, we use the Gillespie SSA (a5)–(d5) to simulate the chemical system (5.1). Starting with no molecules of A in the system, we plot one realization of SSA (a5)–(d5) in Figure 5.1(a) as a blue line. We see that the time evolution of A given by SSA (a5)–(d5) initially (over the first 2 minutes) looks like the noisy solution of (5.2). However, we can find significant differences between the stochastic and deterministic 30 RADEK ERBAN ET AL. 0 20 40 60 80 time [min] stochastic deterministic 10000 12000 number of A molecules stochastic deterministic Fig. 5.2. Self-induced stochastic resonance. (a) One realization of SSA (a5)–(d5) for the system of chemical reactions (5.3) (blue line) and solution of the deterministic ODEs (red line). (b) Comparison of the stochastic and deterministic trajectories in the (A,B)-plane. Nullclines of the deterministic ODEs are plotted as green lines. model if we observe both models over sufficiently large times – see Figure 5.1(b) where we plot the time evolution of A over the first 100 minutes. As expected, the solution of the deterministic model (5.2) stays forever close to the stable steady state as1 = 100. The number of molecules given by the stochastic model initially fluctuates around one of the favourable states of the system (which is close to as1 = 100). However, the fluctuations are sometimes so strong that the system spontaneously switches to another steady state (which is close to as2 = 400). This random switching is missed by the deterministic description. If one wants to find the mean switching time between favourable states of the system, then it is necessary to implement SSAs. Random switching between states has been found in gene regulatory networks [15, 21]. Theoretical or computational approaches for the analysis of suitable stochastic models are given in [25, 9]. Our next example is a nonlinear system of chemical equations for which the stochastic model has qualitatively different behaviour than its deterministic coun- terpart in some parameter regimes. The presented phenomenon is sometimes called self-induced stochastic resonance [27]. Following an example from [5], we consider the system of chemical reactions introduced by Schnakenberg [37] k1−→ 3A, ∅ k2−→←− A, ∅ k4−→ B, (5.3) where we choose the rate constants as k1 = 4×10−5 sec−1, k2 = 50 sec−1, k3 = 10 sec−1 and k4 = 25 sec −1. We use the Gillespie SSA (a5)–(d5) to simulate the time evolution of this system. To do that, let us note that the propensity function of the first reaction is equal to A(t)(A(t) − 1)B(t)k1. We also derive and solve the deter- ministic system of ODEs corresponding to (5.3). Using the same initial conditions [A,B] = [10, 10], we compare the results of the stochastic and deterministic models in Figure 5.2(a). We plot the time evolution of A(t). We see that the solution of the deterministic equations converges to a steady state while the stochastic model has os- cillatory solutions. Note that there is a log scale on the A-axis – numbers of A given by the (more precise) SSA vary between zero and ten thousand. If we use a linear STOCHASTIC REACTION-DIFFUSION PROCESSES 31 0 0.2 0.4 0.6 0.8 1 x [mm] N > 80 80 ≥ N > 30 30 ≥ N 0 0.2 0.4 0.6 0.8 1 x [mm] N > 80 80 ≥ N > 30 30 ≥ N Fig. 5.3. French flag problem. (a) Deterministic model. (b) Stochastic model. scale on the A-axis, then the low molecular fluctuations would be invisible and the solution of the SSAs would look as if there were “almost deterministic oscillations”, although it is the intrinsic noise which makes the oscillations possible. To understand this behaviour better, we plot the stochastic and deterministic trajectories in the (A,B)-plane in Figure 5.2(b). We include the nullclines of the deterministic system of ODEs (green lines). We see that the deterministic system follows a stable nullcline into the steady state (red circle). The stochastic model also initially “follows” this nullcline (with some noise) but it is the intrinsic noise which makes it possible for the stochastic model to leave the stable nullcline and oscillate (again we use a log scale on the A-axis). 5.2. Biological pattern formation. Reaction-diffusion processes are key el- ements of pattern forming mechanisms in developmental biology. The illustrative example from Sections 4.1 and 4.2 was a caricature of more complicated morphogen- esis applications [38, 33] where one assumes that some prepatterning in the domain exists and one wants to validate the reaction-diffusion mechanism of the next stage of the patterning of the embryo. In our example, we considered a chemical A which is produced in part [0, L/5] of domain [0, L]. Hence, the domain [0, L] was divided into two different regions (prepatterning) [0, L/5] and [L/5, L]. The simplest idea of further patterning is the so-called French flag problem [42]. We assume that the interval [0, L] describes a layer of cells which are sensitive to the concentration of chemical A. Let us assume that a cell can have three different fates (e.g. different genes are switched on or off) depending on the concentration of chemical A. Then the concentration gradient of A can help to distinguish three different regions in [0, L] – see Figure 5.3. If the concentration of A is high enough (above a certain threshold), a cell follows the first possible program (denoted blue in Figure 5.3). The “white program” (resp. “red program”) is followed for medium (resp. low) concentrations of A. The deterministic version of the French flag problem is presented in Figure 5.3(a). We consider a solution of (4.9)–(4.10) at time 30 minutes which is the red curve in Figure 4.1(b) or Figure 4.2(b). The solution of (4.9)–(4.10) is decreasing in space. Introducing two thresholds, we can clearly obtain three well-defined regions as seen in Figure 5.3(a). The stochastic version of the French flag problem is presented in Figure 5.3(b). We take the spatial histogram presented in Figure 4.2(b). We introduce two thresholds (80 and 30 molecules) as before and replot the histogram using the corre- 32 RADEK ERBAN ET AL. 0 0.2 0.4 0.6 0.8 1 x [mm] time=30 min 0 0.2 0.4 0.6 0.8 1 x [mm] time=30 min Fig. 5.4. Turing patterns. (a) Numbers of molecules of chemical species A in each compartment at time 30 minutes; (b) the same plot for chemical species B. sponding colours. Clearly, the resulting “French flag” is noisy. Different realizations of the SSA would lead to different noisy French flags. The same is true for the SSA from Figure 4.1(b). Our second example of patterning in developmental biology are the so-called Turing patterns [41, 17, 28, 39]. They do not require any prepatterning. Molecules are subject to the same chemical reactions in the whole domain of interest. For example, let us consider a system of two chemical species A and B which react according to the Schnakenberg system of chemical reactions (5.3). Let us choose the values of rate constants as k1 = 10 −6 sec−1, k2 = 1 sec −1, k3 = 0.02 sec −1 and k4 = 3 sec The corresponding deterministic system of ODEs for (5.3) has one nonnegative stable steady state equal to as = 200 and bs = 75 molecules. Introducing diffusion to the model, one steady state solution of the spatial problem is the constant one (as, bs) everywhere. However, such a solution might not be stable (i.e. might not exist in reality) if the diffusion constants of A and B differ significantly. We choose DA = 10−5 mm2 sec−1 and DB = 10 −3 mm2 sec−1, i.e. DB/DA = 100. To simulate the reaction-diffusion problem with the Schnakenberg system of chemical reactions (5.3), we follow the method of Section 4.1. We divide the computational domain [0, L] into K = 40 compartments of length h = L/K = 25µm. We denote the number of molecules of chemical species A (resp. B) in the i-th compartment [(i−1)h, ih) by Ai (resp. Bi), i = 1, . . . ,K. Diffusion is described by two chains of chemical reactions (4.12)–(4.13) where the rates of “chemical reactions” are equal to dA = DA/h 2 for chemical species A and dB = DB/h 2 for chemical species B. Chemical reactions (5.3) are considered in every compartment (the values of rate constants in (5.3) are already assumed to be expressed in units per compartment). Starting with a uniform distribution of chemicals Ai(0) = as = 200 and Bi(0) = bs = 75, i = 1, 2, . . . ,K, at time t = 0, we plot the numbers of molecules in each compartment at time t = 30 minutes computed by SSA (a5)–(d5) in Figure 5.4. To demonstrate the idea of patterning, compartments with many molecules (above steady state values as or bs) are plotted as blue; other compartments are plotted as red. We see in Figure 5.4(a) that chemical A can be clearly used to divide our computational domain into several regions. There are two and half blue peaks in this figure. The number of blue peaks depends on the size of the computational domain [0, L] and it is not a unique number STOCHASTIC REACTION-DIFFUSION PROCESSES 33 in general. The reaction-diffusion system has several favourable states with a different number of blue peaks. As discussed in Section 5.1, the solution of the corresponding deterministic model converges to one of the favourable (stable steady) states of the system. The stochastic model enables stochastic switching between the favourable states, i.e. between the states with a different number of blue peaks. 6. Discussion. We presented SSAs for systems of chemical reactions and molec- ular diffusion. Then we presented methods for simulating both reactions and diffusion at the same time. The algorithms for simulating (spatially homogeneous) systems of chemical reactions were based on the work of Gillespie [18]. We did not focus on the computer implementation of the algorithms. We chose simple examples which can be simulated quickly. If one considers systems of many equations, there are ways to make the Gillespie SSA more efficient [16]. For example, it would be a waste of time to recompute all the propensity functions at each time step. We simulate one reaction per one time step. Therefore, it makes sense to update only those propensity functions which are changed by the chemical reaction which was selected in step (d5) of SSA (a5)–(d5). We only briefly touched on the concept of the Fokker-Planck equation [34] when we discussed the Smoluchowski description of diffusion. It is worth noting that there are interesting connections between the chemical master equation (which is equivalent to the Gillespie SSA) and the Fokker-Planck equation which gives the time evolution of the probability distribution. Such connections are discussed (through the so-called chemical Langevin equation) in [20]. The Smoluchowski equation is actually the same mathematical object as the chemical Langevin equation, i.e. the stochastic differential equation [2]. An algorithmic introduction to stochastic differential equations can be found in [23]. We presented two models of diffusion in this paper. One was based on the chain of “chemical reactions” (3.8) computing the time evolution of the numbers of molecules in compartments. Coupling this model with the modelling of chemical reactions is straightforward and presented in Section 4.1; such a compartment-based approach is used e.g. in [40, 24, 22]. The second model for molecular diffusion was based on the Smoluchowski equation (3.6). It was an example of the so-called position jump process, that is, a molecule jumps to a different location at each time step. As a result, the trajectory of a molecule is discontinuous. The individual trajectories of diffusing molecules can be also modelled using the so-called velocity jump processes [29], that is, the position of a molecule x(t) follows the deterministic equation dx/dt = v where v(t), the velocity of the molecule, changes stochastically. Such stochastic processes can be used not only for the simulation of diffusing molecules but also for the description of movement of unicellular organisms like bacteria [10, 11] or amoeboid cells [12]. Velocity jump processes can be also described in terms of PDEs for the time evolution of the probability distributions to find a particle (molecule or cell) at a given place. Such equations are not exactly equal to the diffusion equation. However, they can be reduced in the appropriate limit to the diffusion equation [43, 7]. A classical review paper on diffusion and other stochastic processes was written by Chandrasekhar [4], a nice introduction to random walks in biology is the book by Berg [3]. In this paper, we used only reflective boundary conditions, that is, particles hitting the boundary were reflected back. Such boundary conditions are suitable whenever there is no chemical interaction between molecules in the solution and the boundary of the domain. Considering biological applications, it is often the case that molecules (e.g. proteins) react with the boundary (e.g. with receptors in the cellular mem- 34 RADEK ERBAN ET AL. brane). Then the boundary conditions have to be modified accordingly. It has to be assumed that some molecules which hit the boundary are reflected and some molecules are adsorbed by the boundary (e.g. become bound to the receptor or take part in membrane-based chemical reactions). The probability that a molecule is adsorbed rather than reflected depends on the chemical properties of the boundary and also on the SSA which is used for modelling (further details are given in [7, 8]). Our analysis of SSAs was based on the chemical master equation. We successfully derived equations for the means and variances in illustrative examples which did not include second-order reactions. Other first-order reaction networks can be also analysed using this framework [13]. The nonlinear chemical kinetics complicates the mathematical analysis significantly. We can write a deterministic description but it might be too far from the correct description of the system [35]. A review of more computational approaches for the analysis of SSAs can be found in [26]. Applications of such methods to stochastic reaction-diffusion processes is presented in [31]. Acknowledgements. This work was supported by the Biotechnology and Bi- ological Sciences Research Council (grant ref. 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A practical introduction to stochastic modelling of reaction-diffusion processes is presented. No prior knowledge of stochastic simulations is assumed. The methods are explained using illustrative examples. The article starts with the classical Gillespie algorithm for the stochastic modelling of chemical reactions. Then stochastic algorithms for modelling molecular diffusion are given. Finally, basic stochastic reaction-diffusion methods are presented. The connections between stochastic simulations and deterministic models are explained and basic mathematical tools (e.g. chemical master equation) are presented. The article concludes with an overview of more advanced methods and problems.

Introduction. There are two fundamental approaches to the mathematical modelling of chemical reactions and diffusion: deterministic models which are based on differential equations; and stochastic simulations. Stochastic models provide a more detailed understanding of the reaction-diffusion processes. Such a description is often necessary for the modelling of biological systems where small molecular abundances of some chemical species make deterministic models inaccurate or even inapplicable. Stochastic models are also necessary when biologically observed phenomena depend on stochastic fluctuations (e.g. switching between two favourable states of the system). In this paper, we provide an accessible introduction for students to the stochastic modelling of the reaction-diffusion processes. We assume that students have a basic understanding of differential equations but we do not assume any prior knowledge of advanced probability theory or stochastic analysis. We explain stochastic simulation methods using illustrative examples. We also present basic theoretical tools which are used for analysis of stochastic methods. We start with a stochastic model of a single chemical reaction (degradation) in Section 2.1, introducing a basic stochastic simulation algorithm (SSA) and a mathematical equation suitable for its analysis (the so-called chemical master equation). Then we study systems of chemical reactions in the rest of Section 2, presenting the Gillespie SSA and some additional theoretical concepts. We introduce new theory whenever it provides more insights into the par- ticular example. We believe that such an example-based approach is more accessible for students than introducing the whole theory first. In Section 3, we study SSAs for modelling diffusion of molecules. We focus on models of diffusion which are later used for the stochastic modelling of reaction-diffusion processes. Such methods are presented in Section 4. We also introduce further theoretical concepts, including the reaction-diffusion master equation, the Smoluchowski equation and the Fokker-Planck equation. We conclude with Sections 5 and 6 where more advanced problems, methods and theory are discussed, giving references suitable for further reading. The stochastic methods and the corresponding theory are explained using several illustrative examples. We do not assume a prior knowledge of a particular computer language in this paper. A student might use any computer language to implement the examples from this paper. However, we believe that some students might benefit from ∗University of Oxford, Mathematical Institute, 24-29 St. Giles’, Oxford, OX1 3LB, United King- dom; e-mails: erban@maths.ox.ac.uk, chapman@maths.ox.ac.uk, maini@maths.ox.ac.uk. http://arXiv.org/abs/0704.1908v2 2 RADEK ERBAN ET AL. our computer codes which were used to compute the illustrative results in this paper. The computer codes (in Matlab or Fortran) can be downloaded from the website http://www.maths.ox.ac.uk/cmb/Education/ which is hosted by the Centre for Mathematical Biology in the Mathematical Institute, University of Oxford. 2. Stochastic simulation of chemical reactions. The goal of this section is to introduce stochastic methods for the modelling of (spatially homogeneous) systems of chemical reactions. We present the Gillespie SSA, the chemical master equation and its consequences [18, 19]. We start with the simplest case possible, that of modelling a single chemical reaction, in Section 2.1. We then study two simple systems of chemical reactions in Sections 2.2 and 2.3. 2.1. Stochastic simulation of degradation. Let us consider the single chem- ical reaction k−→ ∅ (2.1) where A is the chemical species of interest and k is the rate constant of the reaction. The symbol ∅ denotes chemical species which are of no further interest in what fol- lows. The rate constant k in (2.1) is defined so that k dt gives the probability that a randomly chosen molecule of chemical species A reacts (is degraded) during the time interval [t, t+ dt) where t is time and dt an (infinitesimally) small time step. In par- ticular, the probability that exactly one reaction (2.1) occurs during the infinitesimal time interval [t, t+ dt) is equal to A(t)k dt where we denote the number of molecules of chemical species A at time t simply as A(t). This notational convention will be used throughout the paper. Let us assume that we have n0 molecules of A in the system at time t = 0, i.e. A(0) = n0. Our first goal is to compute the number of molecules A(t) for times t > 0. To do that, we need a computer routine generating random numbers uniformly distributed in the interval (0, 1). Such a program is included in many modern programming languages (e.g. function rand in Matlab): It generates a number r ∈ (0, 1), so that the probability that r is in a subinterval (a, b) ⊂ (0, 1) is equal to b− a for any a, b ∈ (0, 1), a < b. The mathematical definition of the chemical reaction (2.1) can be directly used to design a “naive” SSA for simulating it. We choose a small time step ∆t. We compute the number of molecules A(t) at times t = i∆t, i = 1, 2, 3, . . . , as follows. Starting with t = 0 and A(0) = n0, we perform two steps at time t: (a1) Generate a random number r uniformly distributed in the interval (0, 1). (b1) If r < A(t)k∆t, then put A(t+∆t) = A(t)−1; otherwise, A(t+∆t) = A(t). Then continue with step (a1) for time t+ ∆t. Since r is a random number uniformly distributed in the interval (0, 1), the probability that r < A(t)k∆t is equal to A(t)k∆t. Consequently, step (b1) says that the proba- bility that the chemical reaction (2.1) occurs in the time interval [t, t+∆t) is equal to A(t)k∆t. Thus step (b1) correctly implements the definition of (2.1) provided that ∆t is small. The time evolution of A obtained by the “naive” SSA (a1)–(b1) is given in Figure 2.1(a) for k = 0.1 sec−1, A(0) = 20 and ∆t = 0.005 sec. We repeated the stochastic simulation twice and we plotted two realizations of SSA (a1)–(b1). We see in Figure 2.1(a) that two realizations of SSA (a1)–(b1) give two different results. Each time we run the algorithm, we obtain different results. This is generally true for any SSA. Therefore, one might ask what useful and reproducible information can http://www.maths.ox.ac.uk/cmb/Education/ STOCHASTIC REACTION-DIFFUSION PROCESSES 3 0 5 10 15 20 25 30 time [sec] first realization second realization 0 5 10 15 20 25 30 time [sec] Fig. 2.1. Stochastic simulation of chemical reaction (2.1) for k = 0.1 sec−1 and A(0) = 20. (a) Number of molecules of A as a function of time for two realizations of the “naive” SSA (a1)–(b1) for ∆t = 0.005 sec; (b) results of ten realizations of SSA (a2)–(c2)(solid lines; different colours show different realizations) and stochastic mean (2.8) plotted by the dashed line. be obtained from stochastic simulations? This question will be addressed later in this section. The probability that exactly one reaction (2.1) occurs during the infinitesimal time interval [t, t+dt) is equal to A(t)k dt. To design the SSA (a1)–(b1), we replaced dt by the finite time step ∆t. In order to get reasonably accurate results, we must ensure that A(t)k∆t≪ 1 during the simulation. We used k = 0.1 sec−1 and ∆t = 0.005 sec. Since A(t) ≤ A(0) = 20 for any t ≥ 0, we have that A(t)k∆t ∈ [0, 0.01] for any t ≥ 0. Consequently, the condition A(t)k∆t ≪ 1 is reasonably satisfied during the simulation. We might further increase the accuracy of the SSA (a1)–(b1) by decreasing ∆t. However, decreasing ∆t increases the computational intensity of the algorithm. The probability that the reaction (2.1) occurs during the time interval [t, t + ∆t) is less or equal to 1% for our parameter values. During most of the time steps, we generate a random number r in step (a1) to find out that no reaction occurs in step (b1). Hence, we need to generate a lot of random numbers before the reaction takes place. Our next task will be to design a more efficient method for the simulation of the chemical reaction (2.1). We will need only one random number to decide when the next reaction occurs. Moreover, the method will be exact. There will be no approximation in the derivation of the following SSA (a2)–(c2). Suppose that there are A(t) molecules at time t in the system. Our goal is to com- pute time t+τ when the next reaction (2.1) takes place. Let us denote by f(A(t), s) ds the probability that, given A(t) molecules at time t in the system, the next reaction occurs during the time interval [t+ s, t+ s+ds) where ds is an (infinitesimally) small time step. Let g(A(t), s) be the probability that no reaction occurs in interval [t, t+s). Then the probability f(A(t), s) ds can be computed as a product of g(A(t), s) and the probability that a reaction occurs in the time interval [t+ s, t+ s+ds) which is given according to the definition of (2.1) by A(t+ s)k ds. Thus we have f(A(t), s) ds = g(A(t), s)A(t + s)k ds. Since no reaction occurs in [t, t+ s), we have A(t+ s) = A(t). This implies f(A(t), s) ds = g(A(t), s)A(t)k ds. (2.2) 4 RADEK ERBAN ET AL. To compute the probability g(A(t), s), let us consider σ > 0. The probability that no reaction occurs in the interval [t, t+ σ + dσ) can be computed as the product of the probability that no reaction occurs in the interval [t, t + σ) and the probability that no reaction occurs in the interval [t+ σ, t+ σ + dσ). Hence g(A(t), σ + dσ) = g(A(t), σ)[1 −A(t+ σ)k dσ]. Since no reaction occurs in the interval [t, t + σ), we have A(t + σ) = A(t). Conse- quently, g(A(t), σ + dσ)− g(A(t), σ) = −A(t)k g(A(t), σ). Passing to the limit dσ → 0, we obtain the ordinary differential equation (in the σ variable) dg(A(t), σ) = −A(t)k g(A(t), σ). Solving this equation with initial condition g(A(t), 0) = 1, we obtain g(A(t), σ) = exp[−A(t)kσ]. Consequently, (2.2) can be written as f(A(t), s) ds = A(t)k exp[−A(t)ks] ds. (2.3) Our goal is to find τ such that t + τ is the time when the next reaction occurs, provided that there are A(t) molecules of A in the system at time t. Such τ ∈ (0,∞) is a random number which has to be generated consistently with the definition of the chemical reaction (2.1). To do that, we consider the function F (·) defined by F (τ) = exp[−A(t)kτ ]. (2.4) The function F (·) is monotone decreasing for A(t) > 0. If τ is a random number from the interval (0,∞), then F (τ) is a random number from the interval (0, 1). If τ is a random number chosen consistently with the reaction (2.1), then F (τ) is a random number uniformly distributed in the interval (0, 1) which can be shown as follows. Let a, b, a < b, be chosen arbitrarily in the interval (0, 1). The probability that F (τ) ∈ (a, b) is equal to the probability that τ ∈ (F−1(b), F−1(a)) which is given by the integral of f(A(t), s) over s in the interval (F−1(b), F−1(a)). Using (2.3) and (2.4), we obtain ∫ F−1(a) F−1(b) f(A(t), s) ds = ∫ F−1(a) F−1(b) A(t)k exp[−A(t)ks] ds ∫ F−1(a) F−1(b) ds = −F [F−1(a)] + F [F−1(b)] = b− a. Hence we have verifed that F (τ) is a random number uniformly distributed in (0, 1). Such a number can be obtained using the random number generator (e.g. function STOCHASTIC REACTION-DIFFUSION PROCESSES 5 rand in Matlab). Let us denote it by r. The previous observation implies that we can generate the time step τ by putting r = F (τ). Using (2.4), we obtain r = exp[−A(t)kτ ]. Solving for τ , we obtain the formula A(t)k . (2.5) Consequently, the SSA for the chemical reaction (2.1) can be written as follows. Starting with t = 0 and A(0) = n0, we perform three steps at time t: (a2) Generate a random number r uniformly distributed in the interval (0, 1). (b2) Compute the time when the next reaction (2.1) occurs as t + τ where τ is given by (2.5). (c2) Compute the number of molecules at time t+ τ by A(t+ τ) = A(t) − 1. Then continue with step (a2) for time t+ τ. Steps (a2)–(c2) are repeated until we reach the time when there is no molecule of A in the system, i.e. A = 0. SSA (a2)–(c2) computes the time of the next reaction t + τ using formula (2.5) in step (b2) with the help of one random number only. Then the reaction is performed in step (c2) by decreasing the number of molecules of chemical species A by 1. The time evolution of A obtained by SSA (a2)–(c2) is given in Figure 2.1(b). We plot ten realizations of SSA (a2)–(c2) for k = 0.1 sec−1 and A(0) = 20. Since the function A(t) has only integer values {0, 1, 2, . . . , 20}, it is not surprising that some of the computed curves A(t) partially overlap. On the other hand, all ten realizations yield different functions A(t). Even if we made millions of stochastic realizations, it would be very unlikely (with probability zero) that there would be two realizations giving exactly the same results. Therefore, the details of one realization A(t) are of no special interest (they depend on the sequence of random numbers obtained by the random number generator). However, averaging values of A at time t over many realizations (e.g. computing the stochastic mean of A), we obtain a reproducible characteristic of the system – see the dashed line in Figure 2.1(b). The stochastic mean of A(t) over (infinitely) many realizations can be also computed theoretically as follows. Let us denote by pn(t) the probability that there are n molecules of A at time t in the system, i.e. A(t) = n. Let us consider an (infinitesimally) small time step dt chosen such that the probability that two molecules are degraded during [t, t + dt) is negligible compared to the probability that only one molecule is degraded during [t, t+ dt). Then there are two possible ways for A(t + dt) to take the value n: either A(t) = n and no reaction occurred in [t, t+dt), or A(t) = n+1 and one molecule was degraded in [t, t+ dt). Hence pn(t+ dt) = pn(t)× (1− kn dt) + pn+1(t)× k(n+ 1) dt. A simple algebraic manipulation yields pn(t+ dt)− pn(t) = k(n+ 1) pn+1(t)− kn pn(t). Passing to the limit dt → 0, we obtain the so-called chemical master equation in the = k(n+ 1) pn+1 − kn pn. (2.6) 6 RADEK ERBAN ET AL. Equation (2.6) looks like an infinite system of ordinary differential equations (ODEs) for pn, n = 0, 1, 2, 3, . . . . Our initial condition A(0) = n0 implies that there are never more than n0 molecules in the system. Consequently, pn ≡ 0 for n > n0 and the system (2.6) reduces to a system of (n0 + 1) ODEs for pn, n ≤ n0. The equation for n = n0 reads as follows = −kn0 pn0 . Solving this equation with initial condition pn0(0) = 1, we get pn0(t) = exp[−kn0t]. Using this formula in the chemical master equation (2.6) for pn0−1(t), we obtain pn0−1(t) = kn0 exp[−kn0t]− k(n0 − 1) pn0−1(t). Solving this equation with initial condition pn0−1(0) = 0, we obtain pn0−1(t) = exp[−k(n0 − 1)t]n0(1 − exp[−kt]). Using mathematical induction, it is possible to pn(t) = exp[−knt] 1− exp[−kt] }n0−n . (2.7) The formula (2.7) provides all the information about the stochastic process which is defined by (2.1) and initial condition A(0) = n0. We can never say for sure that A(t) = n; we can only say that A(t) = n with probability pn(t). In particular, formula (2.7) can be used to derive a formula for the mean value of A(t) over (infinitely) many realizations, which is defined by M(t) = n pn(t). Using (2.7), we deduce M(t) = n pn(t) = n exp[−knt] 1− exp[−kt] }n0−n = n0 exp[−kt] n0 − 1 1− exp[−kt] }(n0−1)−(n−1){ exp[−kt] = n0 exp[−kt]. (2.8) The chemical master equation (2.6) and its solution (2.7) can be also used to quan- tify the stochastic fluctuations around the mean value (2.8), i.e. how much can an individual realization of SSA (a2)–(c2) differ from the mean value given by (2.8). We will present the corresponding theory and results on a more complicated illustrative example in Section 2.2. Finally, let us note that a classical deterministic description of the chemical reaction (2.1) is given by the ODE da/dt = −ka. Solving this equation with initial condition a(0) = n0, we obtain the function (2.8), i.e. the stochastic mean is equal to the solution of the corresponding deterministic ODE. However, we should emphasize that this is not true for general systems of chemical reactions, as we will see in Section 2.3 and Section 5.1. STOCHASTIC REACTION-DIFFUSION PROCESSES 7 2.2. Stochastic simulation of production and degradation. We consider a system of two chemical reactions k1−→ ∅, ∅ k2−→ A. (2.9) The first reaction describes the degradation of chemical A with the rate constant k1. It was already studied previously as reaction (2.1). We couple it with the second reaction which represents the production of chemical A with the rate constant k2. The exact meaning of the second chemical reaction in (2.9) is that one molecule of A is created during the time interval [t, t+ dt) with probability k2 dt. As before, the symbol ∅ denotes chemical species which are of no special interest to the modeller. The impact of other chemical species on the rate of production of A is assumed to be time independent and is already incorporated in the rate constant k2. To simulate the system of chemical reactions (2.9), we perform the following four steps at time t (starting with A(0) = n0 at time t = 0): (a3) Generate two random numbers r1, r2 uniformly distributed in (0, 1). (b3)Compute α0 = A(t)k1 + k2. (c3) Compute the time when the next chemical reaction takes place as t+τ where . (2.10) (d3) Compute the number of molecules at time t+ τ by A(t+ τ) = A(t) + 1 if r2 < k2/α0; A(t)− 1 if r2 ≥ k2/α0. (2.11) Then continue with step (a3) for time t+ τ. To justify that SSA (a3)–(d3) correctly simulates (2.9), let us note that the probability that any of the reactions in (2.9) takes place in the time interval [t, t+ dt) is equal to α0 dt. It is given as a sum of the probability that the first reaction occurs (A(t)k1dt) and the probability that the second reaction occurs (k2 dt). Formula (2.10) gives the time t + τ when the next reaction takes place. It can be justified using the same arguments as for formula (2.5). Once we know the time t+ τ , a molecule is produced with probability k2/α0, i.e. the second reaction in (2.9) takes place with probability k2/α0. Otherwise, a molecule is degraded, i.e. the first reaction in (2.9) occurs. The decision as to which reaction takes place is given in step (d3) with the help of the second uniformly distributed random number r2. Five realizations of SSA (a3)–(d3) are presented in Figure 2.2(a) as solid lines. We plot the number of molecules of A as a function of time for A(0) = 0, k1 = 0.1 sec and k2 = 1 sec −1. We see that, after an initial transient, the number of molecules A(t) fluctuates around its mean value. To compute the stochastic mean and quantify the stochastic fluctuations, we use the chemical master equation which can be written for the chemical system (2.9) in the following form = k1(n+ 1) pn+1 − k1n pn + k2 pn−1 − k2 pn (2.12) where pn(t) denotes the probability that A(t) = n for n = 0, 1, 2, 3, . . . . Let us note that the third term on the right hand side is missing in (2.12) for n = 0; i.e. we use 8 RADEK ERBAN ET AL. 0 20 40 60 80 100 time [sec] 0 2 4 6 8 10 12 14 16 18 20 22 number of molecules Gillespie SSA master equation Fig. 2.2. Stochastic simulation of the system of chemical reactions (2.9) for A(0) = 0, k1 = 0.1 sec−1 and k2 = 1 sec −1. (a) A(t) given by five realizations of SSA (a3)–(d3) (solid lines) and stochastic mean (dashed line). (b) Stationary distribution φ(n) obtained by long time simulation of SSA (a3)–(d3) (gray histogram) and by formulae (2.21)–(2.22) (red solid line). the convention that p −1 ≡ 0. The first two terms on the right hand side correspond to the first reaction in (2.9). They already appeared in equation (2.6). Production of A is described by the third and fourth term on the right hand side of (2.9). To derive the chemical master equation (2.12), we can use similar arguments as in the derivation of (2.6). The stochastic mean M(t) and variance V (t) are defined by M(t) = n pn(t), V (t) = n−M(t) pn(t). (2.13) The stochastic mean M(t) gives the average number of molecules of A at time t, while the variance V (t) describes the fluctuations. In Section 2.1, we first solved the chemical master equation (2.6) and then we used its solution (2.7) to compute M(t). Alternatively, we could use the chemical master equation to derive an evolution equation for M(t), i.e. we could find M(t) without solving the chemical master equation. Such an approach will be presented in this section. Multiplying (2.12) by n and summing over n, we obtain npn = k1 n(n+ 1) pn+1 − k1 n2 pn + k2 n pn−1 − k2 n pn. Using definition (2.13) on the left hand side and changing indices n + 1 → n (resp. n− 1→ n) in the first (resp. third) sum on the right hand side, we obtain (n− 1)n pn − k1 n2 pn + k2 (n+ 1) pn − k2 n pn. Adding the first and the second sum (resp. the third and the fourth sum) on the right hand side, we get = −k1 n pn + k2 pn. (2.14) STOCHASTIC REACTION-DIFFUSION PROCESSES 9 Since pn(t) is the probability that A(t) = n and A(t) is equal to a nonnegative integer with probability 1, we have pn(t) = 1. (2.15) Using this fact together with the definition of M(t), equation (2.14) implies the evo- lution equation for M(t) in the form = −k1M + k2. (2.16) The solution of (2.16) with initial condition M(0) = 0 is plotted as a dashed line in Figure 2.2(a). To derive the evolution equation for the variance V (t), let us first observe that definition (2.13) implies n2 pn(t) = V (t) +M(t) 2. (2.17) Multiplying (2.12) by n2 and summing over n, we obtain n2pn = k1 n2(n+ 1) pn+1 − k1 n3 pn + k2 n2 pn−1 − k2 n2 pn. Changing indices n + 1 → n (resp. n − 1 → n) in the first (resp. third) sum on the right hand side and adding the first and the second sum (resp. the third and the fourth sum) on the right hand side, we get n2pn = k1 (−2n2 + n) pn + k2 (2n+ 1) pn. Using (2.17), (2.15) and (2.13), we obtain = −2k1 V +M2 + k1M + 2k2M + k2. Substituting (2.16) for dM/dt, we derive the evolution equation for the variance V (t) in the following form = −2k1V + k1M + k2. (2.18) The time evolution of M(t) and V (t) is described by (2.16) and (2.18). Let us define the stationary values of M(t) and V (t) by Ms = lim M(t), Vs = lim V (t). (2.19) The values of Ms and Vs can be computed using the steady state equations corre- sponding to (2.16) and (2.18), namely by solving 0 = −k1Ms + k2, and 0 = −2k1Vs + k1Ms + k2. 10 RADEK ERBAN ET AL. Consequently, Ms = Vs = For our parameter values k1 = 0.1 sec −1 and k2 = 1 sec −1, we obtain Ms = Vs = 10. We see in Figure 2.2(a) that A(t) fluctuates after a sufficiently long time around the mean value Ms = 10. To quantify the fluctuations, one often uses the square root of Vs, the so-called mean standard deviation which is equal to More detailed information about the fluctuations is given by the so-called sta- tionary distribution φ(n), n = 0, 1, 2, 3, . . . , which is defined as φ(n) = lim pn(t). (2.20) This means that φ(n) is the probability that A(t) = n after an (infinitely) long time. One way to compute φ(n) is to run SSA (a3)–(d3) for a long time and create a histogram of values of A(t) at given time intervals. Using k1 = 0.1 sec −1 and k2 = 1 sec −1, the results of such a long time computation are presented in Figure 2.2(b) as a gray histogram. To compute it, we ran SSA (a3)–(d3) for 105 seconds, recording the value of A(t) every second and then dividing the whole histogram by the number of recordings, i.e. by 105. An alternative way to compute φ(n) is to use the steady state version of the chemical master equation (2.12), namely 0 = k1 φ(1)− k2 φ(0) 0 = k1(n+ 1)φ(n+ 1)− k1nφ(n) + k2 φ(n− 1)− k2 φ(n), for n ≥ 1, which implies φ(1) = φ(0), (2.21) φ(n+ 1) = k1(n+ 1) k1nφ(n) + k2 φ(n)− k2 φ(n− 1) , for n ≥ 1. (2.22) Consequently, we can express φ(n) for any n ≥ 1 in terms of φ(0). The formulae (2.21)–(2.22) yield an alternative way to compute φ(n). We put φ(0) = 1 and we compute φ(n), for sufficiently many n, by (2.21)–(2.22). Then we divide φ(n), n ≥ 0, φ(n). The results obtained by (2.21)–(2.22) are plotted in Figure 2.2(b) as a (red) solid line. As expected, the results compare well with the results obtained by the long time stochastic simulation. We can also find the formula for φ(n) directly. We let a reader to verify that the solution of the recurrence formula (2.21)–(2.22) can be written as φ(n) = (2.23) where C is a real constant. Using (2.15) and (2.20), we have φ(n) = 1. (2.24) Substituting (2.23) into the normalization condition (2.24), we get = C exp STOCHASTIC REACTION-DIFFUSION PROCESSES 11 where we used the Taylor series for the exponential function to get the last equal- ity. Consequently, C = exp[−k2/k1] which, together with (2.23), implies that the stationary distribution φ(n) is the Poisson distribution φ(n) = . (2.25) Thus the red solid line in Figure 2.2(b) which was obtained numerically by the re- currence formula (2.21)–(2.22) can be also viewed as the stationary distribution φ(n) given by the explicit exact formula (2.25). 2.3. Gillespie algorithm. SSAs (a2)–(c2) and (a3)–(d3) were special forms of the so-called Gillespie SSA. In this section, we present this algorithm for a more com- plicated illustrative example which will also involve second-order chemical reactions. Such chemical reactions are of the following form k1−→ C, A+B k2−→ D. (2.26) In the first equation, two molecules of A react with rate constant k1 to produce C. The probability that the reaction takes place in the time interval [t, t + dt) is equal to A(t)(A(t)− 1)k1dt. We define the propensity function of the first reaction as α1(t) = A(t)(A(t) − 1)k1. Then the probability that the first reaction occurs in the time interval [t, t+dt) is equal to α1(t) dt. The propensity function which corresponds to the second equation in (2.26) is defined as α2(t) = A(t)B(t)k1 and the probability that the second reaction occurs in the time interval [t, t+ dt) is equal to α2(t) dt. In such a case, one molecule of A and one molecule of B react to form a molecule of D. In general, the propensity function can be defined for any chemical reaction so that its product with dt gives the probability that the given reaction occurs in the infinitesimally small time interval [t, t+ dt). We consider that A and B can react according to (2.26). Moreover, we assume that they are also produced with constant rates, that is, we consider a system of four chemical equations: k1−→ ∅ A+B k2−→ ∅ (2.27) ∅ k3−→ A ∅ k4−→ B. (2.28) Let us note that we are not interested in chemical species C and D. Hence, we replaced them by ∅, consistent with our previous notation of unimportant chemical species. To simulate the system of chemical reactions (2.27)–(2.28), we perform the following four steps at time t (starting with A(0) = n0, B(0) = m0 at time t = 0): (a4) Generate two random numbers r1, r2 uniformly distributed in (0, 1). (b4) Compute the propensity functions of each reaction by α1 = A(t)(A(t)−1)k1, α2 = A(t)B(t)k2, α3 = k3 and α4 = k4. Compute α0 = α1 + α2 + α3 + α4. (c4) Compute the time when the next chemical reaction takes place as t+τ where . (2.29) 12 RADEK ERBAN ET AL. 0 20 40 60 80 100 time [sec] solution of ODEs 0 20 40 60 80 100 time [sec] solution of ODEs Fig. 2.3. Five realizations of SSA (a4)–(d4). Number of molecules of chemical species A (left panel) and B (right panel) are plotted as functions of time as solid lines. Different colours correspond to different realizations. The solution of (2.33)–(2.34) is given by the dashed line. We use A(0) = 0, B(0) = 0, k1 = 10 −3 sec−1, k2 = 10 −2 sec−1, k3 = 1.2 sec −1 and k4 = 1 sec (d4) Compute the number of molecules at time t+ τ by A(t+ τ) = A(t)− 2 if 0 ≤ r2 < α1/α0; A(t)− 1 if α1/α0 ≤ r2 < (α1 + α2)/α0; A(t) + 1 if (α1 + α2)/α0 ≤ r2 < (α1 + α2 + α3)/α0; A(t) if (α1 + α2 + α3)/α0 ≤ r2 < 1; (2.30) B(t+ τ) = B(t) if 0 ≤ r2 < α1/α0; B(t)− 1 if α1/α0 ≤ r2 < (α1 + α2)/α0; B(t) if (α1 + α2)/α0 ≤ r2 < (α1 + α2 + α3)/α0; B(t) + 1 if (α1 + α2 + α3)/α0 ≤ r2 < 1; (2.31) Then continue with step (a4) for time t+ τ. SSA (a4)–(d4) is a direct generalisation of SSA (a3)–(d3). At each time step, we first ask the question when will the next reaction occur? The answer is given by formula (2.29) which can be justified using the same arguments as formulae (2.5) or (2.10). Then we ask the question which reaction takes place. The probability that the i-th chemical reaction occurs is given by αi/α0. The decision which reaction takes place is given in step (d4) with the help of the second uniformly distributed random number r2. Knowing that the i-th reaction took place, we update the number of reactants and products accordingly. Results of five realizations of SSA (a4)–(d4) are plotted in Figure 2.3 as solid lines. We use A(0) = 0, B(0) = 0, k1 = 10 −3 sec−1, k2 = 10 −2 sec−1, k3 = 1.2 sec −1 and k4 = 1 sec −1. We plot the number of molecules of chemical species A and B as functions of time. We see that, after initial transients, A(t) and B(t) fluctuate around their average values. They can be estimated from long time stochastic simulations as 9.6 for A and 12.2 for B. Let pn,m(t) be the probability that A(t) = n and B(t) = m. The chemical master STOCHASTIC REACTION-DIFFUSION PROCESSES 13 equation can be written in the following form dpn,m = k1(n+ 2)(n+ 1) pn+2,m − k1n(n− 1) pn,m + k2(n+ 1)(m+ 1) pn+1,m+1 − k2nmpn,m + k3 pn−1,m − k3 pn,m + k4 pn,m−1 − k4 pn,m (2.32) for n, m ≥ 0, with the convention that pn,m ≡ 0 if n < 0 or m < 0. The first difference between (2.32) and the chemical master equations from the previous sections is that equation (2.32) is parametrised by two indices n and m. The second important difference is that (2.32) cannot be solved analytically as we did with (2.6). Moreover, it does not lead to closed evolution equations for stochastic means and variances; i.e. we cannot follow the same technique as in the case of equation (2.12). The approach from the previous section does not work. Let us note that the probability pn,m(t) is sometimes denoted by p(n,m, t); such a notational convention is often used when we consider systems of many chemical species. We will use it in the following sections to avoid long subscripts. The classical deterministic description of the chemical system (2.27)–(2.28) is given by the system of ODEs = −2k1a2 − k2 ab+ k3, (2.33) = −k2 ab+ k4. (2.34) The solution of (2.33)–(2.34) with initial conditions a(0) = 0 and b(0) = 0 is plotted as a dashed line in Figure 2.3. Let us note that the equations (2.33)–(2.34) do not describe the stochastic means of A(t) and B(t). For example, the steady state values of (2.33)–(2.34) are (for the parameter values of Figure 2.3) equal to as = bs = 10. On the other hand, the average values estimated from long time stochastic simulations are 9.6 for A and 12.2 for B. We will see later in Section 5.1 that the difference between the results of stochastic simulations and the corresponding ODEs can be even more significant. The stationary distribution is defined by φ(n,m) = lim pn,m(t). This can be computed by long time simulations of SSA (a4)–(d4) and is plotted in Figure 2.4(a). We see that there is a correlation between the values of A and B. This can also be observed in Figure 2.3. Looking at the blue realizations, we see that the values of A(t) are below the average and the values of B(t) are above the average, similarly for other realizations. One can also define the stationary distribution of A only by φ(n) = φ(n,m). (2.35) Summing the results of Figure 2.4(a) overm, we obtain φ(n) which is plotted in Figure 2.4(b) as a gray histogram. The red bar highlights the steady state value as = 10 of system (2.33)–(2.34). SSAs (a3)–(d3) and (a4)–(d4) were special forms of the so-called Gillespie SSA. To conclude this section, we formulate the Gillespie SSA in its full generality. Let us 14 RADEK ERBAN ET AL. number of A molecules 5 10 15 20 25 30 2 4 6 8 10 12 14 16 18 20 22 24 number of A molecules Fig. 2.4. (a) Stationary distribution φ(n,m) obtained by long time simulation of (a4)–(d4) for k1 = 10 −3 sec−1, k2 = 10 −2 sec−1, k3 = 1.2 sec −1 and k4 = 1 sec −1. (b) Stationary distribution of A obtained by (2.35). consider that we have a system of q chemical reactions. Let αi(t) be the propensity function of the i-th reaction, i = 1, 2, . . . , q, at time t, that is, αi(t) dt is the probability that the i-th reaction occurs in the time interval [t, t + dt). Then the Gillespie SSA consists of the following four steps at time t. (a5) Generate two random numbers r1, r2 uniformly distributed in (0, 1). (b5) Compute the propensity function αi(t) of each reaction. Compute αi(t). (2.36) (c5) Compute the time when the next chemical reaction takes place as t+τ where τ is given by (2.29). (d5) Compute which reaction occurs at time t+ τ . Find j such that αi and r2 < Then the j-th reaction takes place, i.e. update numbers of reactants and products of the j-th reaction. Continue with step (a5) for time t+ τ. The Gillespie SSA (a5)–(d5) provides an exact method for the stochastic simulation of systems of chemical reactions. It was applied previously as SSA (a2)–(c2) for the chemical reaction (2.1), as SSA (a3)–(d3) for the chemical system (2.9) and as SSA (a4)–(d4) for the chemical system (2.27)–(2.28). Our simple examples can be simulated quickly in Matlab (in less than a second on present-day computers). If one considers systems of many chemical reactions and many chemical species, then SSA (a5)–(d5) might be computationally intensive. In such a case, there are ways to make the Gillespie SSA more efficient. For example, it would be a waste of time to recompute all the propensity functions at each time step (step (b5)). We simu- late one reaction per one time step. Therefore, it makes sense to update only those STOCHASTIC REACTION-DIFFUSION PROCESSES 15 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 x [mm] x [mm] −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Fig. 3.1. (a) Six trajectories obtained by SSA (a6)–(b6) for D = 10−4 mm2 sec−1 and ∆t = 0.1 sec. Trajectories were started at the origin and followed for 10 minutes. (b) Probability distribution function ψ(x, y, t) given by (3.5) at time t = 10 min. propensity functions which are changed by the chemical reaction which was selected in step (d5) of SSA (a5)–(d5). A more detailed discussion about the efficient computer implementation of the Gillespie SSA can be found e.g. in [16]. 3. Diffusion. Diffusion is the random migration of molecules (or small particles) arising from motion due to thermal energy [3]. As shown by Einstein, the kinetic en- ergy of a molecule (e.g. protein) is proportional to the absolute temperature. In particular, the protein molecule has a non-zero instantaneous speed at, for exam- ple, room temperature or at the temperature of the human body. A typical protein molecule is immersed in the aqueous medium of a living cell. Consequently, it can- not travel too far before it bumps into other molecules (e.g. water molecules) in the solution. As a result, the trajectory of the molecule is not straight but it executes a random walk as shown in Figure 3.1(a). We plot six possible trajectories of the protein molecule with six different colours. All trajectories start at the origin and are followed for 10 minutes. We will provide more details about this figure together with the methods for simulating molecular diffusion in the rest of this section. Stochastic models of diffusion which are based on the Smoluchowski equation are introduced in Section 3.1. In Section 3.2, we introduce a model which is suitable for coupling with the Gillespie SSA. Both modelling approaches will be used later in Section 4 for the stochastic modelling of reaction-diffusion processes. Let us note that there exist other models of molecular diffusion – they will be discussed in Section 6. 3.1. Smoluchowski equation and diffusion. Let [X(t), Y (t), Z(t)] ∈ R3 be the position of a diffusing molecule at time t. Starting with [X(0), Y (0), Z(0)] = [x0, y0, z0], we want to compute the time evolution of [X(t), Y (t), Z(t)]. To do that, we make use of a generator of random numbers which are normally distributed with zero mean and unit variance. Such a generator is part of many modern computer languages (e.g. function randn in Matlab). Diffusive spreading of molecules is characterised by a single diffusion constant D which depends on the size of the molecule, absolute temperature and viscosity of the solution [3]. Choosing time step ∆t, we compute the time evolution of the position of the diffusing molecule by performing two steps at time t: 16 RADEK ERBAN ET AL. (a6) Generate three normally distributed (with zero mean and unit variance) random numbers ξx, ξy and ξz. (b6) Compute the position of the molecule at time t+ ∆t by X(t+ ∆t) = X(t) + 2D∆t ξx, (3.1) Y (t+ ∆t) = Y (t) + 2D∆t ξy , (3.2) Z(t+ ∆t) = Z(t) + 2D∆t ξz , (3.3) Then continue with step (a6) for time t+ ∆t. Choosing D = 10−4 mm2 sec−1 (diffusion constant of a typical protein molecule), [X(0), Y (0), Z(0)] = [0, 0, 0] and ∆t = 0.1 sec, we plot six realizations of SSA (a6)– (b6) in Figure 3.1(a). We plot only the x and y coordinates. We follow the diffusing molecule for 10 minutes. The position of the molecule at time t = 10 min is denoted as a black circle for each trajectory. Equations (3.1)–(3.3) are discretized versions of stochastic differential equations (SDEs) which are sometimes called Smoluchowski equations. Some basic facts about SDEs can be found e.g. in [2, 14]. A more accessible introduction to SDEs can be found in [23] which has a similar philosophy as our paper. Reference [23] is a nice algorithmic introduction to SDEs for students who do not have a prior knowledge of advanced probability theory or stochastic analysis. We will not go into details of SDEs in this paper, but only highlight some interesting facts which will be useful later. First, equations (3.1)–(3.3) are not coupled. To compute the time evolution of X(t), we do not need to know the time evolution of Y (t) or Z(t). We will later focus only on the time evolution of the x-th coordinate, effectively studying one-dimensional problems. Two-dimensional or three-dimensional problems can be treated similarly. Second, we see that different realizations of SSA (a6)–(b6) give different results. To get more reproducible quantities, we will shortly study the behaviour of several molecules. However, even in the case of a single diffusing molecule, there are still quantities whose evolution is deterministic. Let ϕ(x, y, t) dxdydz be the probability that X(t) ∈ [x, x + dx), Y (t) ∈ [y, y + dy) and Z(t) ∈ [z, z + dz) at time t. It can be shown that ϕ evolves according to the partial differential equation , (3.4) which is a special form of the so-called Fokker-Planck equation. Since our random walk starts at the origin, we can solve (3.4) with initial condition ϕ(x, y, z, 0) = δ(x, y, z) where δ is the Dirac distribution at the origin. We obtain ϕ(x, y, z, t) = (4Dπt)3/2 2 + y2 + z2 In order to visualise this probability distribution function, we integrate it over z to get probability distribution function ψ(x, y, t) = ϕ(x, y, z, t)dz = 2 + y2 . (3.5) This means that ψ(x, y, t) dxdy is the probability that X(t) ∈ [x, x+ dx) and Y (t) ∈ [y, y + dy) at time t. The function ψ(x, y, t) at time t = 10 min is plotted in Figure STOCHASTIC REACTION-DIFFUSION PROCESSES 17 0 0.2 0.4 0.6 0.8 1 x [mm] 0 0.2 0.4 0.6 0.8 1 time=4 min x [mm] Fig. 3.2. (a) Ten trajectories computed by SSA (a7)–(c7) for D = 10−4 mm2 sec−1, L = 1 mm, X(0) = 0.4 mm and ∆t = 0.1 sec. (b) Numbers of molecules in bins of length h = 25 µm at time t = 4 min. 3.1(b). It can be obtained also by computing many realizations of SSA (a6)–(b6) and plotting the histogram of positions of a molecule at time 10 min; such positions were denoted as black circles for the six illustrative trajectories in Figure 3.1(a). One important issue which was not addressed previously is that molecules diffuse in bounded volumes, i.e. the domain of interest has boundaries and suitable bound- ary conditions must be implemented. In the rest of this paper, we focus on one- dimensional problems to avoid technicalities. Hence, we effectively study diffusion of molecules in the one-dimensional interval [0, L]. Then the SSA can be formulated as follows: (a7) Generate a normally distributed (with zero mean and unit variance) random number ξ. (b7) Compute the position of the molecule at time t+ ∆t by X(t+ ∆t) = X(t) + 2D∆t ξ. (3.6) (c7) If X(t+ ∆t) computed by (3.6) is less than 0, then X(t+ ∆t) = −X(t)− 2D∆t ξ. If X(t+ ∆t) computed by (3.6) is greater than L, then X(t+ ∆t) = 2L−X(t)− 2D∆t ξ. Then continue with step (a7) for time t+ ∆t. The boundary condition implemented in step (c7) is the so-called reflective boundary condition or zero flux boundary condition. It can be used when there is no chem- ical interaction between the boundary and diffusing molecules. More complicated boundary conditions are discussed in [7, 8]. Choosing D = 10−4 mm2 sec−1, L = 1 mm, X(0) = 0.4 mm and ∆t = 0.1 sec, we plot ten realizations of SSA (a7)–(c7) in Figure 3.2(a). Let us assume that we have a system of 1000 molecules which are released at position x = 0.4 mm at time t = 0. Then Figure 3.2(a) can be viewed as a plot of the trajectories of ten representative molecules. Considering 1000 molecules, the trajectories of individual molecules are of no special interest. We are rather interested in spatial histograms (density of molecules). An example of such a plot is given in Figure 3.2(b). We simulate 1000 18 RADEK ERBAN ET AL. molecules, each following SSA (a7)–(c7). At time t = 4 min, we divided the domain of interest [0, L] into 40 bins of length h = L/40 = 25 µm. We calculated the number of molecules in each bin [(i− 1)h, ih), i = 1, 2, . . . , 40, at time t = 4 min and plotted them as a histogram. Let us note that the deterministic counterpart to the stochastic simulation is a solution of the corresponding Fokker-Planck equation (diffusion equation in our case) which, in one dimension with zero flux boundary conditions, reads as follows where (0) = (L) = 0. (3.7) The solution of (3.7) with the Dirac-like initial condition at x = 0.4 mm is plotted as a red solid line in Figure 3.2(b) for comparison. 3.2. Compartment-based approach to diffusion. In Section 3.1, we sim- ulated the behaviour of 1000 molecules by computing the individual trajectories of every molecule (using SSA (a7)–(c7)). At the end of the simulation, we divided the computational domain [0, L] into K = 40 compartments and we plotted numbers of molecules in each compartment in Figure 3.2(b). In particular, most of the com- puted information (1000 trajectories) was not used for the final result – the spatial histogram. We visualised only 40 numbers (numbers of molecules in compartments) instead of 1000 computed positions of molecules. In this section, we present a different SSA for the simulation of molecular diffusion. We redo the example from Section 3.1 but instead of simulating 1000 positions of the individual molecules, we are going to simulate directly the time evolution of 40 compartments. To do that, we divide the computational domain [0, L] into K = 40 compartments of length h = L/K = 25µm. We denote the number of molecules of chemical species A in the i-th compartment [(i− 1)h, ih) by Ai, i = 1, . . . ,K. We apply the Gillespie SSA to the following chain of “chemical reactions”: d−→←− d−→←− d−→←− . . . d−→←− AK (3.8) where d−→←− Ai+1 means that Ai d−→ Ai+1 and Ai+1 d−→ Ai. We will shortly show that the Gillespie SSA of (3.8) provides a correct model of diffusion provided that the rate constant d in (3.8) is chosen as d = D/h2 where D is the diffusion constant and h is the compartment length. The compartment- based SSA can be described as follows. Starting with initial condition Ai(t) = a0,i, i = 1, 2, . . . ,K, we perform six steps at time t: (a8) Generate two random numbers r1, r2 uniformly distributed in (0, 1). (b8) Compute propensity functions of reactions by αi = Ai(t)d for i = 1, 2, . . . ,K. Compute αi. (3.9) (c8) Compute the time at which the next chemical reaction takes place as t + τ where τ is given by (2.29). STOCHASTIC REACTION-DIFFUSION PROCESSES 19 (d8) If r2 < i=1 αi/α0, then find j ∈ {1, 2, . . . ,K − 1} such that αi and r2 < Then compute the number of molecules at time t+ τ by Aj(t+ τ) = Aj(t)− 1, (3.10) Aj+1(t+ τ) = Aj+1(t) + 1, (3.11) Ai(t+ τ) = Ai(t), for i 6= j, i 6= j + 1. (3.12) (e8) If r2 ≥ i=1 αi/α0, then find j ∈ {2, 3, . . . ,K} such that and r2 < Then compute the number of molecules at time t+ τ by Aj(t+ τ) = Aj(t)− 1, (3.13) Aj−1(t+ τ) = Aj−1(t) + 1, (3.14) Ai(t+ τ) = Ai(t), for i 6= j, i 6= j − 1. (3.15) (f8) Continue with step (a8) for time t+ τ. The first term on the right hand side of (3.9) corresponds to reactions Ai → Ai+1 (jumps to the right) and the second term corresponds to reactions Ai → Ai−1 (jumps to the left). The time of the next chemical reaction is computed in the step (c8) using formula (2.29) derived previously. The decision about which reaction takes place is done in steps (d8)–(e8) with the help of random number r2. Jumps to the right are implemented in step (d8) and jumps to the left in step (e8). We want to redo the example from Section 3.1, i.e. simulate 1000 molecules starting from position 0.4 mm in the interval [0, L] for L = 1 mm. We use K = 40. Since 0.4 mm is exactly a boundary between the 16th and 17th compartment, the initial condition is given by A16(0) = 500, A17(0) = 500 and Ai(0) = 0 for i 6= 16, i 6= 17. As D = 10−4 mm2 sec−1, we have d = D/h2 = 0.16 sec−1. The numbers Ai(t), i = 1, . . . ,K, at time t = 4 min, are plotted in Figure 3.3(a) as a histogram. This panel can be directly compared with Figure 3.2(b). The computational intensity of SSA (a8)–(f8) can be decreased using the appropriate way to implement it in the computer. For example, only one chemical reaction occurs per time step. Consequently, only two propensity functions change and need to be updated in step (b8). Moreover, the formula (3.9) can be simplifed as follows αi = 2 αi−α1−αK = 2d Ai(t)−α1−αK = 2dN−α1−αK , where N = 1000 is the total number of molecules in the simulation (this number is conserved because there is no creation or degradation of the molecules in the system). Hence, we need to recompute α0 only when there is a change in α1 or αK , i.e. whenever the boundary compartments were involved in the previous reaction. 20 RADEK ERBAN ET AL. 0 0.2 0.4 0.6 0.8 1 time=4 min x [mm] 0 0.2 0.4 0.6 0.8 1 x [mm] Fig. 3.3. Compartment-based SSA model of diffusion. (a) Numbers Ai(t), i = 1, 2, . . . ,K, at time t = 4 min obtained by SSA (a8)–(f8). We use d = D/h2 = 0.16 sec−1, K = 40 and initial condition A16(0) = 500, A17(0) = 500 and Ai(0) = 0 for i 6= 16, i 6= 17. (b) Ten realizations of the simulation of an individual molecule by SSA (a8)–(f8). SSA (a8)–(f8) does not compute the trajectories of individual molecules. However, we can still compute a plot comparable with Figure 3.2(a). To do that, we repeat the simulation with 1 molecule instead of 1000. Then, at given time t, exactly one of numbers Ai(t), i = 1, 2, . . . ,K, is non-zero and equal to 1. This is a position of the molecule at time t. Ten realizations of SSA (a8)–(f8) with one molecule released at 0.4 mm at t = 0 are plotted in Figure 3.3(b). This panel can be directly compared with Figure 3.2(a). Let p(n, t) be the joint probability that Ai(t) = ni, i = 1, . . . ,K, where we denoted n = [n1, n2, . . . , nK ]. Let us define operators Ri, Li : N K → NK by Ri : [n1, . . . , ni, ni+1, . . . , nK ]→ [n1, . . . , ni + 1, ni+1 − 1, . . . , nK ], i = 1, . . . ,K − 1, (3.16) Li : [n1, . . . , ni−1, ni, . . . , nK ]→ [n1, . . . , ni−1 − 1, ni + 1, . . . , nK ], i = 2, . . . ,K. (3.17) Then the chemical master equation, which corresponds to the system of chemical reactions given by (3.8), can be written as follows ∂P (n) (nj + 1)P (Rjn)− nj P (n) (nj + 1)P (Ljn)− nj P (n) (3.18) The stochastic mean is defined as the vector M(t) ≡ [M1,M2, . . . ,MK ] where Mi(t) = ni P (n, t) ≡ · · · ni P (n, t) (3.19) gives the mean number of molecules in the i-th compartment, i = 1, 2, . . . ,K. To derive an evolution equation for the stochastic mean vector M(t), we can follow the method from Section 2.2 – see derivation of (2.16) from chemical master equation (2.12). Multiplying (3.18) by ni and summing over n, we obtain (leaving the details STOCHASTIC REACTION-DIFFUSION PROCESSES 21 to the student) a system of equations for Mi of the form = d(Mi+1 +Mi−1 − 2Mi), i = 2, . . . ,K − 1, (3.20) = d(M2 −M1), = d(MK−1 −MK). (3.21) System (3.20)–(3.21) is equivalent to a discretization of (3.7) provided that d = D/h2. Hence, we have derived the relation between the rate constant d in (3.8), diffusion constant D and compartment length h. The solution of (3.7) with the Dirac-like initial condition at x = 0.4 mm is plotted for comparison as a red solid line in Figure 3.3(a). The noise is described by the variance vector V(t) ≡ [V1, V2, . . . , VK ] where Vi(t) = (ni −Mi(t))2 P (n, t) ≡ · · · (ni −Mi(t))2 P (n, t) (3.22) gives the variance of number of molecules in the i-th compartment, i = 1, 2, . . . ,K. To derive the evolution equation for the vector V(t), we define the matrix {Vi,j} by Vij = ninj P (n, t)−MiMj, for i, j = 1, 2, . . . ,K. Using (3.22), we obtain Vi = Vii for i = 1, 2, . . . ,K. Multiplying (3.18) by n i and summing over n, we obtain n2iP (n) = d n2i (nj + 1)P (Rjn)− n2inj P (n) n2i (nj + 1)P (Ljn)− n2inj P (n) . (3.23) Let us assume that i = 2, . . . ,K − 1. Let us consider the term corresponding to j = i in the first sum on the right hand side. We get n2i (ni + 1)P (Rin)− n2ini P (n) = (ni − 1)2ni P (n)− n2ini P (n) (−2n2i + ni)P (n) = −2Vi − 2M2i +Mi. First, we changed indices in the first sum Rin→ n and then we used definitions (3.19) and (3.22). Similarly, the term corresponding to j = i−1 in the first sum on the right hand side of (3.23) can be rewritten as n2i (ni−1 + 1)P (Ri−1n)− n2ini−1 P (n) = (2nini−1 + ni−1)P (n) = 2Vi,i−1 + 2MiMi−1 +Mi−1. 22 RADEK ERBAN ET AL. Other terms (corresponding to j 6= i, i− 1) in the first sum on the right hand side of (3.23) are equal to zero. The second sum can be handled analogously. We obtain n2iP (n) = d 2Vi,i−1 + 2MiMi−1 +Mi−1 − 2Vi − 2M2i +Mi 2Vi,i+1 + 2MiMi+1 +Mi+1 − 2Vi − 2M2i +Mi . (3.24) Using (3.22) and (3.20) on the left hand side of (3.24), we obtain n2iP (n) = + 2Mi + d(2MiMi+1 + 2MiMi−1 − 4M2i ). Substituting this into (3.24), we get Vi,i+1 + Vi,i−1 − 2Vi Mi+1 +Mi−1 + 2Mi (3.25) for i = 2, . . . ,K − 1. Similarly, we get V1,2 − V1 M2 +M1 , (3.26) VK,K−1 − VK MK−1 +MK . (3.27) We see that the evolution equation for the variance vector V(t) depends on the mean M, variance V and on non-diagonal terms of the matrix Vi,j . To get a closed system of equations, we have to derive evolution equations for Vi,j too. This can be done by multiplying (3.18) by ninj, summing over n and following the same arguments as before. We conclude this section with some consequences of (3.20)–(3.21) and (3.25)– (3.27). Looking at the steady states of equations (3.20)–(3.21), we obtain Mi = N/K, i = 1, 2, . . . ,K, where N is the total number of diffusing molecules. Moreover, the variance equations imply that Vi = N/K, i = 1, 2, . . . ,K, at the steady state. 4. Stochastic reaction-diffusion models. In this section, we add chemical reactions to both models of molecular diffusion which were presented in Section 3. We introduce two methods for the stochastic modelling of reaction-diffusion processes. The first one is based on the diffusion model from Section 3.2, the second one on the diffusion model from Section 3.1. We explain both methods using the same example. Namely, we consider molecules (e.g. protein) which diffuse in the domain [0, L] with diffusion constant D as we considered in Section 3. Moreover, we assume that protein molecules are degraded (in the whole domain) and produced in part of the domain, i.e. we consider the chemical reactions from Sections 2.1 and 2.2 in our illustrative reaction-diffusion model. The model has a realistic motivation which is discussed in more detail later in Section 5.2. In Section 4.3, we present another illustrative example of a reaction-diffusion process incorporating the nonlinear model (2.27)–(2.28). 4.1. Compartment-based reaction-diffusion SSA. We consider molecules of chemical species A which are diffusing in the domain [0, L], where L = 1 mm, with diffusion constant D = 10−4 mm2 sec−1. Initially, there are no molecules in the system. Molecules are produced in the part of the domain [0, L/5] with rate STOCHASTIC REACTION-DIFFUSION PROCESSES 23 0 0.2 0.4 0.6 0.8 1 x [mm] time=10 min 0 0.2 0.4 0.6 0.8 1 x [mm] time=30 min Fig. 4.1. One realization of the Gillespie SSA (a5)–(d5) for the system of chemical reactions (4.1)–(4.3). Gray histograms show numbers of molecules in compartments at time: (a) t = 10 min; (b) t = 30 min. Solution of (4.9)–(4.10) is plotted as the red solid line. kp = 0.012 µm −1 sec−1. This means that the probability that a molecule is created in the subinterval of the length 1 µm is equal to kp dt. Consequently, the probability that a molecule is created somewhere in the interval [0, L/5] is equal to kpL/5 dt. Molecules are degraded with rate k1 = 10 −3 sec−1 according to the chemical reaction (2.1). Following Section 3.2, we divide the computational domain [0, L] into K = 40 compartments of length h = L/K = 25µm. We denote the number of molecules of chemical species A in the i-th compartment [(i− 1)h, ih) by Ai, i = 1, . . . ,K. Then our reaction-diffusion process is described by the system of chemical reactions d−→←− d−→←− d−→←− . . . d−→←− AK , (4.1) k1−→ ∅, for i = 1, 2, . . . ,K, (4.2) ∅ k2−→ Ai, for i = 1, 2, . . . ,K/5. (4.3) Equation (4.1) describes diffusion and is identical to (3.8). In particular, the rate constant d is given by d = D/h2. Equation (4.2) describes the degradation of A and is, in fact, equation (2.1) applied to every compartment. Equation (4.3) describes the production of A in the first K/5 compartments (e.g. in part [0, L/5] of the computational domain). The rate constant k2 describes the rate of production per compartment. Since each compartment has length h, we have k2 = kph. The system of chemical reactions (4.1)–(4.3) is simulated using the Gillespie SSA (a5)–(d5). In our case, the propensity functions of reactions in (4.1) are given as Ai(t)d, the propensity functions of reactions in (4.2) are given as Ai(t)k1 and propen- sity functions of reactions in (4.3) are equal to k2. Starting with no molecules of A in the system, we compute one realization of SSA (a5)–(d5) for the system of reactions (4.1)–(4.3). We plot the numbers of molecules in compartments at two different times in Figure 4.1. 24 RADEK ERBAN ET AL. Let p(n, t) be the joint probability that Ai(t) = ni, i = 1, . . . ,K, where we use the notation n = [n1, n2, . . . , nK ]. Let us define operators Ri, Li : N K → NK by (3.16)–(3.17). Then the chemical master equation, which corresponds to the system of chemical reactions (4.1)–(4.3), can be written as follows ∂p(n) (ni + 1) p(Rin)− ni p(n) (ni + 1) p(Lin)− ni p(n) (ni + 1) p(n1, . . . , ni + 1, . . . , nK)− ni p(n) p(n1, . . . , ni − 1, . . . , nK)− p(n) . (4.4) The first two sums correspond to diffusion (4.1), the third sum to degradation (4.2) and the fourth sum to production (4.3). The stochastic mean is defined as the vector M(t) ≡ [M1,M2, . . . ,MK ] where Mi is given by (3.19). This gives the mean number of molecules in the i-th compartment, i = 1, 2, . . . ,K, at time t (averaged over many realizations of SSA (a5)–(d5)). To derive the evolution equation for the stochastic mean vector M(t), we can follow the method from Section 2.2 – see derivation of (2.16) from the chemical master equation (2.12). Multiplying (4.4) by ni and summing over all nj , j = 1, . . . ,K, we obtain (leaving the details to the student) a system of equations for Mi in the form = d(M2 −M1) + k2 − k1M1, (4.5) = d(Mi+1 +Mi−1 − 2Mi) + k2 − k1Mi, i = 2, . . . ,K/5, (4.6) = d(Mi+1 +Mi−1 − 2Mi)− k1Mi, i = K/5 + 1, . . . ,K − 1, (4.7) = d(MK−1 −MK)− k1MK . (4.8) System (4.5)–(4.8) is a discretized version of the reaction-diffusion equation + k2χ[0,L/5] − k1a (4.9) with zero-flux boundary conditions (0) = (L) = 0. (4.10) Here, χ[0,L/5] is the characteristic function of the interval [0, L/5]. Using initial con- dition a(·, 0) ≡ 0, we computed the solution of (4.9)–(4.10) numerically. It is plotted as a red solid line in Figure 4.1 for comparison. The concentration of molecules in the i-th compartment is defined as M i = Mi/h, i = 1, . . . ,K. Dividing (4.5)–(4.8) by h, we can write a system of ODEs for M i. It is a discretized version of the reaction-diffusion equation + kpχ[0,L/5] − k1a (4.11) STOCHASTIC REACTION-DIFFUSION PROCESSES 25 where a ≡ a(x, t) is the concentration of molecules of A at point x and time t. The equation (4.11) provides a classical deterministic description of the reaction-diffusion process. Its parametersD, kp and k1 are independent of h. Solving (4.11) is equivalent to solving (4.9). Consequently, the red solid line in Figure 4.1 can be also viewed as a plot of ah where a is the solution of the classical deterministic model (4.11) with the zero-flux boundary conditions. 4.2. Reaction-diffusion SSA based on the Smoluchowski equation. In this section, we present a SSA which implements the Smoluchowski model of diffusion from Section 3.1, that is, we follow the trajectories of individual molecules. Diffusion of each molecule is modelled according to the model (a7)–(c7). We explain the SSA using the reaction-diffusion example from Section 4.1. Choosing a small time step ∆t, we perform the following three steps at time t: (a9) For each molecule, compute its position at time t + ∆t according to steps (a7)–(c7). (b9) For each molecule, generate a random number r1 uniformly distributed in the interval (0, 1). If r1 < k1 ∆t, then remove the molecule from the system. (c9) Generate a random number r2 uniformly distributed in the interval (0, 1). If r2 < kpL/5 ∆t, then generate another random number r3 uniformly dis- tributed in the interval (0, 1) and introduce a new molecule at position r3L/5. Continue with step (a9) for time t+ ∆t. The degradation of molecules is modelled by step (b9). Equation (2.1) implies that k1 dt is the probability that a molecule is degraded in the time interval [t, t + dt) for infinitesimally small dt. SSA (a9)–(c9) replaces dt by the finite time step ∆t (compare with SSA (a1)–(b1)) which has to be chosen sufficiently small so that k1 ∆t ≪ 1. Similarly, the probability that a molecule is created in [0, L/5] in time interval [t, t + dt) is equal to kpL/5 dt. Consequently, we have to choose ∆t so small that kpL/5 ∆t is significantly less than 1. We choose ∆t = 10 −2 sec. Then k1 ∆t = 10 −5 and kpL/5 ∆t = 2.4 × 10−2 for our parameter values k1 = 10−3 sec−1, kp = 0.012 µm −1 sec−1 and L = 1 mm. Starting with no molecules of A in the sys- tem, we compute one realization of SSA (a9)–(c9). To visualise the results, we divide the interval [0, L] into 40 bins and we plot the numbers of molecules in bins at time 10 minutes in Figure 4.2(a). The same plot at time 30 minutes is given in Figure 4.2(b). We used the same number of bins to visualise the results of SSA (a9)–(c9) as we used previously in the compartment-based model. Thus Figure 4.2 is directly comparable with Figure 4.1. We also plot the solution of (4.9)–(4.10) as a red solid line for comparison. 4.3. Reaction-diffusion models of nonlinear chemical kinetics. In the previous sections, we studied an example of a reaction-diffusion model which did not include the second-order chemical reactions (2.26). We considered only production and degradation, i.e. we considered chemical reactions from Sections 2.1 and 2.2. In this section, we discuss generalisations of our approaches to models which involve second-order chemical reactions too. Our illustrative example is a reaction-diffusion process incorporating the nonlinear model (2.27)–(2.28). The second-order chemical reactions (2.26) require that two molecules collide (be close to each other) before the reaction can take place. The generalisation of SSA (a9)–(c9) to such a case is nontrivial and we will not present it in this paper (it can be found in [1]). Application of the Gillespie SSA (a5)–(d5) is more straightforward and is presented below. 26 RADEK ERBAN ET AL. 0 0.2 0.4 0.6 0.8 1 x [mm] time=10 min 0 0.2 0.4 0.6 0.8 1 x [mm] time=30 min Fig. 4.2. One realization of SSA (a9)–(c9). Dividing domain [0, L] into 40 bins, we plot the number of molecules in each bin at time: (a) t = 10 min; (b) t = 30 min. Solution of (4.9)–(4.10) is plotted as the red solid line. We consider that both chemical speciesA andB diffuse in the domain [0, L], where L = 1 mm, with diffusion constant D = 10−4 mm2 sec−1. Following the method of Section 4.1, we divide the computational domain [0, L] into K = 40 compartments of length h = L/K = 25µm. We denote the number of molecules of chemical species A (resp. B) in the i-th compartment [(i − 1)h, ih) by Ai (resp. Bi), i = 1, . . . ,K. Diffusion corresponds to two chains of “chemical reactions”: d−→←− d−→←− d−→←− . . . d−→←− AK (4.12) d−→←− d−→←− d−→←− . . . d−→←− BK (4.13) Molecules of A and B are assumed to react according to chemical reactions (2.27) in the whole domain with rate constants k1 = 10 −3 sec−1 and k2 = 10 −2 sec−1 per one compartment, that is, Ai +Ai k1−→ ∅, Ai +Bi k2−→ ∅, for i = 1, 2, . . . ,K. (4.14) Production of chemical species (2.28) is assumed to take place only in parts of the computational domain [0, L]. Molecules of chemical species A (resp. B) are assumed to be produced in subinterval [0, 9L/10] (resp. [2L/5, L]) with rate k3 = 1.2 sec (resp. k4 = 1 sec −1) per one compartment of length h, that is, ∅ k3−→ Ai, for i = 1, 2, . . . , 9K/10, (4.15) ∅ k4−→ Bi, for i = 2K/5, . . . ,K. (4.16) Starting with no molecules in the system at time t = 0, we present one realization of the Gillespie SSA (a5)–(d5) applied to the chemical system (4.12)–(4.16) in Figure 4.3. We plot the numbers of molecules of A and B at time 30 minutes. STOCHASTIC REACTION-DIFFUSION PROCESSES 27 0 0.2 0.4 0.6 0.8 1 x [mm] stochastic simulation solution of PDEs 0 0.2 0.4 0.6 0.8 1 x [mm] stochastic simulation solution of PDEs Fig. 4.3. One realization of the Gillespie SSA (a5)–(d5) for the system of chemical reactions (4.12)–(4.16). Numbers of molecules of chemical species A (left panel) and B (right panel) in compartments at time 30 minutes (gray histograms). Solution of (4.17)–(4.19) is plotted as the red solid line. We already observed in Section 2.3 that the analysis of the master equation for chemical systems involving the second order reactions is not trivial. It is not possible to derive the equation for stochastic means as was done in Section 4.1 for the linear model. Hence, we will not attempt such an approach here. We also observed in Section 4.1 that the equation for the mean vector (4.5)–(4.8) was actually equal to a discretized version of the reaction-diffusion equation (4.9)–(4.10) which would be used as a traditional deterministic description. When considering the nonlinear chemical model (4.14)–(4.16), we cannot derive the equation for the mean vector but we can still write a deterministic system of partial differential equations (PDEs). We simply add diffusion to the system of ODEs (2.33)–(2.34) to obtain − 2k1a2 − k2 ab+ k3χ[0,9L/10], (4.17) − k2 ab+ k4χ[2L/5,L], (4.18) and couple it with zero-flux boundary conditions (0) = (L) = (0) = (L) = 0. (4.19) Using initial condition a(·, 0) ≡ 0 and b(·, 0) ≡ 0, we can compute the solution of (4.17)–(4.19) numerically. It is plotted as a red solid line in Figure 4.3 for comparison. We see that (4.17)–(4.19) gives a reasonable description of the system when comparing with one realization of SSA (a5)–(d5). However, let us note that solution of (4.17)– (4.19) is not equal to the stochastic mean. The equations (4.17)–(4.19) can be also rewritten in terms of concentrations a = a/h and b = b/h as we did in the case of equations (4.9) and (4.11). Let us note that the rate constants scale with h as k1 = k1/h, k2 = k2/h, k3 = k3h, k4 = k4h where k1, k2, k3, k4 are independent of h. Consequently, the equations for concentrations a and b are independent of h. They can be written in terms of the parameters D, k1, k2, k3 and k4 only (compare with (4.11)). 28 RADEK ERBAN ET AL. Finally, let us discuss the choice of the compartment length h. In Sections 3.2 and 4.1, we considered linear models and we were able to derive the equations for the mean vectors (e.g. (4.5)–(4.8)). Dividing (4.5)–(4.8) by h and passing to the limit h → 0, we derive the corresponding deterministic reaction-diffusion PDE (4.11) which can be viewed (for linear models) as an equation for the probability distribution function of a single molecule (i.e. the exact description which we want to approximate by the compartment-based SSA). Consequently, we can increase the accuracy of the SSA by decreasing h. Considering the nonlinear model from this section, the continuum limit h→ 0 is not well-defined. The compartment-based SSA is generally considered valid only for a range of h values (i.e. the length h cannot be chosen arbitrarily small); conditions which the length h has to satisfy are subject of current research – see e.g. [24, Section 3.5]. 5. Two important remarks. We explained SSAs for chemical reactions, molec- ular diffusion and reaction-diffusion processes in the previous sections. This final section is devoted to two important questions: (a) Why do we care about stochastic modelling? The answer is given in Section 5.1 where we discuss connections between stochastic and deterministic modelling. In particular, we present examples where deterministic modelling fails and a stochastic approach is necessary. We start with a simple example of stochastic switching between favourable states of the system, a phenomenon which cannot be fully understood without stochastic modelling. Then we illustrate the fact that the stochastic model might have qualitatively different properties than its deterministic limit, i.e. the stochastic model is not just “equal” to the “noisy solution” of the corresponding deterministic equations. We present a simple system of chemical reactions for which the deterministic description converges to a steady state. On the other hand, the stochastic model of the same system of chemical reactions has oscillatory solutions. Finally, let us note that stochasticity plays important roles in biological applications, see e.g. [32, 6, 30]. (b) Why do we care about reaction-diffusion processes? The answer is given in Section 5.2 where we discuss biological pattern formation. Reaction-diffusion models are key components of models in developmental biology. We present stochastic ana- logues of two classical pattern forming models. The first one is the so-called French flag problem where we re-interpret the illustrative example from Sections 4.1 and 4.2. Then we present the reaction-diffusion pattern forming model based on the so-called Turing instability. 5.1. Deterministic vs. stochastic modelling. The models presented so far have one thing in common. One could use the deterministic description (given by ODEs or PDEs) and one would obtain a reasonable description of the system. In Sections 2.1, 2.2, 3.1, 3.2, 4.1 and 4.2, we studied linear models. We showed that the evolution equations for the stochastic mean are equal to (the discretized versions of) the corresponding deterministic differential equations. In Sections 2.3 and 4.3, we presented nonlinear models. We were not able to derive equations for the stochastic mean. However, we solved numerically the corresponding systems of deterministic equations (ODEs (2.33)–(2.34) in Section 2.3 and PDEs (4.17)–(4.19) in Section 4.3) and we obtained results comparable with the SSAs, i.e. results of the SSAs looked like “noisy solutions” of the corresponding differential equations. Here, we discuss exam- ples of problems when SSAs give results which cannot be obtained by corresponding deterministic models. Let us consider the model from Section 2.3. Its deterministic description is given STOCHASTIC REACTION-DIFFUSION PROCESSES 29 0 0.5 1 1.5 2 time [min] stochastic deterministic 0 20 40 60 80 100 time [min] Fig. 5.1. Simulation of (5.1). One realization of SSA (a5)–(d5) for the system of chemical reactions (5.1) (blue line) and the solution of the deterministic ODE (5.2) (red line). (a) The number of molecules of A as a function of time over the first two minutes of simulation. (b) Time evolution over 100 minutes. by the system of ODEs (2.33)–(2.34). Such a system has only one nonnegative (stable) steady state for our parameter values, namely as = bs = 10. It can be observed from Figure 2.3 that solutions of (2.33)–(2.34) converge to as and bs as t → ∞. This is true for any nonnegative initial condition. The results of SSAs show fluctuation about the means, which are roughly equal to as and bs (they are 9.6 for A and 12.2 for B). However, there are chemical systems which have two or more favourable states, so that the corresponding ODEs have more than one nonnegative stable steady state. For example, let us consider the system of chemical reactions for chemical A introduced by Schlögl [36] k1−→←− 3A, ∅ k3−→←− A. (5.1) The corresponding ODE is given as follows = − k2 a3 + k1 a2 − k4 a+ k3. (5.2) We choose the rate constants as follows: k1 = 0.18 min −1, k2 = 2.5 × 10−4 min−1, k3 = 2200 min −1 and k4 = 37.5 min −1. Then the ODE (5.2) has two stable steady states as1 = 100 and as2 = 400 and one unstable steady state au = 220. The solution of (5.2) converges to one of the steady states with the choice of the steady state dependent on the initial condition. Let us consider that there are initially no molecules of A in the system, i.e. A(0) = 0. The solution of (5.2) is plotted in Figure 5.1(a) as a red line. We see that the solution of (5.2) converges to the stable steady state as1 = 100. This is true for any initial condition A(0) ∈ [0, au). If A(0) > au, then the solution of (5.2) converges to the second stable steady state as2 = 400. Next, we use the Gillespie SSA (a5)–(d5) to simulate the chemical system (5.1). Starting with no molecules of A in the system, we plot one realization of SSA (a5)–(d5) in Figure 5.1(a) as a blue line. We see that the time evolution of A given by SSA (a5)–(d5) initially (over the first 2 minutes) looks like the noisy solution of (5.2). However, we can find significant differences between the stochastic and deterministic 30 RADEK ERBAN ET AL. 0 20 40 60 80 time [min] stochastic deterministic 10000 12000 number of A molecules stochastic deterministic Fig. 5.2. Self-induced stochastic resonance. (a) One realization of SSA (a5)–(d5) for the system of chemical reactions (5.3) (blue line) and solution of the deterministic ODEs (red line). (b) Comparison of the stochastic and deterministic trajectories in the (A,B)-plane. Nullclines of the deterministic ODEs are plotted as green lines. model if we observe both models over sufficiently large times – see Figure 5.1(b) where we plot the time evolution of A over the first 100 minutes. As expected, the solution of the deterministic model (5.2) stays forever close to the stable steady state as1 = 100. The number of molecules given by the stochastic model initially fluctuates around one of the favourable states of the system (which is close to as1 = 100). However, the fluctuations are sometimes so strong that the system spontaneously switches to another steady state (which is close to as2 = 400). This random switching is missed by the deterministic description. If one wants to find the mean switching time between favourable states of the system, then it is necessary to implement SSAs. Random switching between states has been found in gene regulatory networks [15, 21]. Theoretical or computational approaches for the analysis of suitable stochastic models are given in [25, 9]. Our next example is a nonlinear system of chemical equations for which the stochastic model has qualitatively different behaviour than its deterministic coun- terpart in some parameter regimes. The presented phenomenon is sometimes called self-induced stochastic resonance [27]. Following an example from [5], we consider the system of chemical reactions introduced by Schnakenberg [37] k1−→ 3A, ∅ k2−→←− A, ∅ k4−→ B, (5.3) where we choose the rate constants as k1 = 4×10−5 sec−1, k2 = 50 sec−1, k3 = 10 sec−1 and k4 = 25 sec −1. We use the Gillespie SSA (a5)–(d5) to simulate the time evolution of this system. To do that, let us note that the propensity function of the first reaction is equal to A(t)(A(t) − 1)B(t)k1. We also derive and solve the deter- ministic system of ODEs corresponding to (5.3). Using the same initial conditions [A,B] = [10, 10], we compare the results of the stochastic and deterministic models in Figure 5.2(a). We plot the time evolution of A(t). We see that the solution of the deterministic equations converges to a steady state while the stochastic model has os- cillatory solutions. Note that there is a log scale on the A-axis – numbers of A given by the (more precise) SSA vary between zero and ten thousand. If we use a linear STOCHASTIC REACTION-DIFFUSION PROCESSES 31 0 0.2 0.4 0.6 0.8 1 x [mm] N > 80 80 ≥ N > 30 30 ≥ N 0 0.2 0.4 0.6 0.8 1 x [mm] N > 80 80 ≥ N > 30 30 ≥ N Fig. 5.3. French flag problem. (a) Deterministic model. (b) Stochastic model. scale on the A-axis, then the low molecular fluctuations would be invisible and the solution of the SSAs would look as if there were “almost deterministic oscillations”, although it is the intrinsic noise which makes the oscillations possible. To understand this behaviour better, we plot the stochastic and deterministic trajectories in the (A,B)-plane in Figure 5.2(b). We include the nullclines of the deterministic system of ODEs (green lines). We see that the deterministic system follows a stable nullcline into the steady state (red circle). The stochastic model also initially “follows” this nullcline (with some noise) but it is the intrinsic noise which makes it possible for the stochastic model to leave the stable nullcline and oscillate (again we use a log scale on the A-axis). 5.2. Biological pattern formation. Reaction-diffusion processes are key el- ements of pattern forming mechanisms in developmental biology. The illustrative example from Sections 4.1 and 4.2 was a caricature of more complicated morphogen- esis applications [38, 33] where one assumes that some prepatterning in the domain exists and one wants to validate the reaction-diffusion mechanism of the next stage of the patterning of the embryo. In our example, we considered a chemical A which is produced in part [0, L/5] of domain [0, L]. Hence, the domain [0, L] was divided into two different regions (prepatterning) [0, L/5] and [L/5, L]. The simplest idea of further patterning is the so-called French flag problem [42]. We assume that the interval [0, L] describes a layer of cells which are sensitive to the concentration of chemical A. Let us assume that a cell can have three different fates (e.g. different genes are switched on or off) depending on the concentration of chemical A. Then the concentration gradient of A can help to distinguish three different regions in [0, L] – see Figure 5.3. If the concentration of A is high enough (above a certain threshold), a cell follows the first possible program (denoted blue in Figure 5.3). The “white program” (resp. “red program”) is followed for medium (resp. low) concentrations of A. The deterministic version of the French flag problem is presented in Figure 5.3(a). We consider a solution of (4.9)–(4.10) at time 30 minutes which is the red curve in Figure 4.1(b) or Figure 4.2(b). The solution of (4.9)–(4.10) is decreasing in space. Introducing two thresholds, we can clearly obtain three well-defined regions as seen in Figure 5.3(a). The stochastic version of the French flag problem is presented in Figure 5.3(b). We take the spatial histogram presented in Figure 4.2(b). We introduce two thresholds (80 and 30 molecules) as before and replot the histogram using the corre- 32 RADEK ERBAN ET AL. 0 0.2 0.4 0.6 0.8 1 x [mm] time=30 min 0 0.2 0.4 0.6 0.8 1 x [mm] time=30 min Fig. 5.4. Turing patterns. (a) Numbers of molecules of chemical species A in each compartment at time 30 minutes; (b) the same plot for chemical species B. sponding colours. Clearly, the resulting “French flag” is noisy. Different realizations of the SSA would lead to different noisy French flags. The same is true for the SSA from Figure 4.1(b). Our second example of patterning in developmental biology are the so-called Turing patterns [41, 17, 28, 39]. They do not require any prepatterning. Molecules are subject to the same chemical reactions in the whole domain of interest. For example, let us consider a system of two chemical species A and B which react according to the Schnakenberg system of chemical reactions (5.3). Let us choose the values of rate constants as k1 = 10 −6 sec−1, k2 = 1 sec −1, k3 = 0.02 sec −1 and k4 = 3 sec The corresponding deterministic system of ODEs for (5.3) has one nonnegative stable steady state equal to as = 200 and bs = 75 molecules. Introducing diffusion to the model, one steady state solution of the spatial problem is the constant one (as, bs) everywhere. However, such a solution might not be stable (i.e. might not exist in reality) if the diffusion constants of A and B differ significantly. We choose DA = 10−5 mm2 sec−1 and DB = 10 −3 mm2 sec−1, i.e. DB/DA = 100. To simulate the reaction-diffusion problem with the Schnakenberg system of chemical reactions (5.3), we follow the method of Section 4.1. We divide the computational domain [0, L] into K = 40 compartments of length h = L/K = 25µm. We denote the number of molecules of chemical species A (resp. B) in the i-th compartment [(i−1)h, ih) by Ai (resp. Bi), i = 1, . . . ,K. Diffusion is described by two chains of chemical reactions (4.12)–(4.13) where the rates of “chemical reactions” are equal to dA = DA/h 2 for chemical species A and dB = DB/h 2 for chemical species B. Chemical reactions (5.3) are considered in every compartment (the values of rate constants in (5.3) are already assumed to be expressed in units per compartment). Starting with a uniform distribution of chemicals Ai(0) = as = 200 and Bi(0) = bs = 75, i = 1, 2, . . . ,K, at time t = 0, we plot the numbers of molecules in each compartment at time t = 30 minutes computed by SSA (a5)–(d5) in Figure 5.4. To demonstrate the idea of patterning, compartments with many molecules (above steady state values as or bs) are plotted as blue; other compartments are plotted as red. We see in Figure 5.4(a) that chemical A can be clearly used to divide our computational domain into several regions. There are two and half blue peaks in this figure. The number of blue peaks depends on the size of the computational domain [0, L] and it is not a unique number STOCHASTIC REACTION-DIFFUSION PROCESSES 33 in general. The reaction-diffusion system has several favourable states with a different number of blue peaks. As discussed in Section 5.1, the solution of the corresponding deterministic model converges to one of the favourable (stable steady) states of the system. The stochastic model enables stochastic switching between the favourable states, i.e. between the states with a different number of blue peaks. 6. Discussion. We presented SSAs for systems of chemical reactions and molec- ular diffusion. Then we presented methods for simulating both reactions and diffusion at the same time. The algorithms for simulating (spatially homogeneous) systems of chemical reactions were based on the work of Gillespie [18]. We did not focus on the computer implementation of the algorithms. We chose simple examples which can be simulated quickly. If one considers systems of many equations, there are ways to make the Gillespie SSA more efficient [16]. For example, it would be a waste of time to recompute all the propensity functions at each time step. We simulate one reaction per one time step. Therefore, it makes sense to update only those propensity functions which are changed by the chemical reaction which was selected in step (d5) of SSA (a5)–(d5). We only briefly touched on the concept of the Fokker-Planck equation [34] when we discussed the Smoluchowski description of diffusion. It is worth noting that there are interesting connections between the chemical master equation (which is equivalent to the Gillespie SSA) and the Fokker-Planck equation which gives the time evolution of the probability distribution. Such connections are discussed (through the so-called chemical Langevin equation) in [20]. The Smoluchowski equation is actually the same mathematical object as the chemical Langevin equation, i.e. the stochastic differential equation [2]. An algorithmic introduction to stochastic differential equations can be found in [23]. We presented two models of diffusion in this paper. One was based on the chain of “chemical reactions” (3.8) computing the time evolution of the numbers of molecules in compartments. Coupling this model with the modelling of chemical reactions is straightforward and presented in Section 4.1; such a compartment-based approach is used e.g. in [40, 24, 22]. The second model for molecular diffusion was based on the Smoluchowski equation (3.6). It was an example of the so-called position jump process, that is, a molecule jumps to a different location at each time step. As a result, the trajectory of a molecule is discontinuous. The individual trajectories of diffusing molecules can be also modelled using the so-called velocity jump processes [29], that is, the position of a molecule x(t) follows the deterministic equation dx/dt = v where v(t), the velocity of the molecule, changes stochastically. Such stochastic processes can be used not only for the simulation of diffusing molecules but also for the description of movement of unicellular organisms like bacteria [10, 11] or amoeboid cells [12]. Velocity jump processes can be also described in terms of PDEs for the time evolution of the probability distributions to find a particle (molecule or cell) at a given place. Such equations are not exactly equal to the diffusion equation. However, they can be reduced in the appropriate limit to the diffusion equation [43, 7]. A classical review paper on diffusion and other stochastic processes was written by Chandrasekhar [4], a nice introduction to random walks in biology is the book by Berg [3]. In this paper, we used only reflective boundary conditions, that is, particles hitting the boundary were reflected back. Such boundary conditions are suitable whenever there is no chemical interaction between molecules in the solution and the boundary of the domain. Considering biological applications, it is often the case that molecules (e.g. proteins) react with the boundary (e.g. with receptors in the cellular mem- 34 RADEK ERBAN ET AL. brane). Then the boundary conditions have to be modified accordingly. It has to be assumed that some molecules which hit the boundary are reflected and some molecules are adsorbed by the boundary (e.g. become bound to the receptor or take part in membrane-based chemical reactions). The probability that a molecule is adsorbed rather than reflected depends on the chemical properties of the boundary and also on the SSA which is used for modelling (further details are given in [7, 8]). Our analysis of SSAs was based on the chemical master equation. We successfully derived equations for the means and variances in illustrative examples which did not include second-order reactions. Other first-order reaction networks can be also analysed using this framework [13]. The nonlinear chemical kinetics complicates the mathematical analysis significantly. We can write a deterministic description but it might be too far from the correct description of the system [35]. A review of more computational approaches for the analysis of SSAs can be found in [26]. Applications of such methods to stochastic reaction-diffusion processes is presented in [31]. Acknowledgements. This work was supported by the Biotechnology and Bi- ological Sciences Research Council (grant ref. 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Non-linear electromagnetic response of graphene S. A. Mikhailov Institute for Theoretical Physics II, University of Augsburg, D-86135 Augsburg, Germany (Dated: October 22, 2018) It is shown that the massless energy spectrum of electrons and holes in graphene leads to the strongly non-linear electromagnetic response of this system. We predict that the graphene layer, irradiated by electromagnetic waves, emits radiation at higher frequency harmonics and can work as a frequency multiplier. The operating frequency of the graphene frequency multiplier can lie in a broad range from microwaves to the infrared. In the past two years a great deal of attention has been attracted by a recently discovered, new two-dimensional (2D) electronic system – graphene, built out of a single monolayer of carbon atoms with a honeycomb 2D crystal structure [1, 2]. The band structure of the charge carriers in this system consists of six Dirac cones at the corners of the hexagon- shaped Brillouin zone [3, 4], with the massless, linear electron/hole dispersion. The massless electron spectrum leads to unusual transport and electrodynamic properties, which have been intensively studied in the literature, see e.g. [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] and for review [32, 33]. Electrodynamic properties of graphene have been theoretically studied in Refs. [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. The frequency dependent conductivity[16, 17, 20, 21, 22], as well as plasmon [23, 25, 27, 29, 30], plasmon-polariton [24], and transverse electromagnetic wave spectra [31] have been investigated. In all these papers electrodynamic response of the system has been studied within the linear response theory (for instance, using the Kubo formalism, or the random phase approximation, or the self-consistent-field approach). In this Letter we show that, apart from all the fascinating and non-trivial properties of graphene predicted and observed so far, this material should also demonstrate strongly non-linear electrodynamic behavior. In particular, irradiation of the graphene sheet by a harmonic electromagnetic wave with the frequency Ω should lead to the emission of the higher harmonics with the frequencies mΩ, m = 3, 5, . . ., from the system. The operating frequency of such a frequency multiplier can vary from microwaves up to infrared, and the required ac electric field is rather low, especially at low carrier densities and low temperatures. The predicted non-linear electrodynamic properties of graphene may open up new exciting opportunities for building electronic and optoelectronic devices based on this material. To qualitatively demonstrate the non-linear behavior of graphene electrons consider a classical 2D particle with the charge −e and the energy spectrum ǫp = V p = V p2x + p y in the external electric field Ex(t) = E0 cosΩt. Here V is the velocity of 2D electrons in the energy band (in graphene V ≈ 108 cm/s [1, 2]). According to the classical equations of motion dpx/dt = −eEx(t) the momentum px will then be equal to px(t) = −(eE0/Ω) sinΩt, and the velocity vx = ∂ǫp/∂px is then vx(t) = −V sgn(sinΩt). If there are ns particles per unit area, the corresponding ac electric current jx(t) = ensV sgn(sinΩt) = ensV sinΩt+ sin 3Ωt+ sin 5Ωt+ . . . contains all odd Fourier harmonics, with the amplitudes jm, m = 1, 3, 5 . . ., falling down very slowly with the harmonics number, jm ∼ 1/m. Notice that at the density ns = 6 · 1012 cm−2 and at V ≃ 108 cm/s (parameters of Refs. [1, 2]) the current amplitude j0 = ensV in our simple estimate gives a giant value of j0 ≃ 100 A/cm. The above consideration does not take into account the Fermi distribution of charge carriers over the quantum states in the conduction and valence bands of graphene. To get a more accurate description of the non-linear phenomena in the considered system we use the kinetic Boltzmann theory, which allows one to get an exact response of the system not imposing any restrictions on the amplitude of the external ac electric field E(t). Using this quasi-classical approach we take into account the intra-band contribution to the ac electric current. The inter-band contribution to the electric current, due to the transitions between the hole and the electron bands, is ignored. This imposes certain restrictions on the frequency of radiation Ω, which will be discussed below. Consider a 2D electron/hole gas with the energy spectrum ǫp± = ±V p2x + p y under the action of the field E = (Ex, 0), where the sign + (−) corresponds to the electron (hole) band, Ex(t) = E0eαt cos(Ωt), and α → +0 describes an adiabatic switching on of the electric field. Assume that the Fermi energy ǫF lies in the electron (or the hole) band and that the temperature is small as compared to ǫF , T ≪ ǫF . The momentum distribution function of electrons fp+(t) ≡ fp(t) (we omit the sign + for brevity) in the collisionless approximation is described by the http://arxiv.org/abs/0704.1909v1 Boltzmann equation ∂fp(t) − ∂fp(t) αt cos(Ωt) = 0, (2) which has the exact solution fp(t) = F0 (px − p0(t), py) , (3) where F0(px, py) = 1 + exp p2x + p y − ǫF is the electronic Fermi function, and p0(t) = −(eE0/Ω)eαt sinΩt is the solution of the single particle equation of motion. The electric current j(t) = −egsgvS−1 vfp(t) then assumes the form jx(t) = − gsgveV (2πh̄)2 dpxdpy p2x + p F0 (px − p0(t), py) , (5) jy = 0, where gs = gv = 2 are the spin and valley degeneracies in graphene, and S is the sample area. After some lengthy but simple transformation, Eq. (5) can be rewritten as jx(t) = ensV 2QF (t) 1 +Q2F (t) ∫ π/2 cos2 xdx 2QF (t) cosx+ 1− 2QF (t) where ns ≡ ne = gsgvp 4πh̄2 gsgvǫ 4πh̄2V 2 is the density of electrons, pF = ǫF /V is the Fermi momentum, and QF (t) = − p0(t) sin(Ωt) ≡ QF0 sin(Ωt) (8) is the field parameter, proportional to the ac electric field E0. Figure 1a shows the current (6) as a function of time Ωt. One sees that in the low-field limit the response is linear. Expanding the current (6), we get at QF0 ≪ 1 jx(t) ≈ ensV QF0 sinΩt+ Q2F0 sin 3Ωt , (9) so that the linear response conductivity (in the collisionless approximation) is σǫF ,T=0(Ω) = . (10) The expression (10) coincides with the intra-band Drude conductivity, which can be obtained from the linear-response theory [16, 17, 20, 21, 22, 31]. As the inter-band conductivity is of order of e2/h̄ [16, 17, 20, 21, 22, 31], our quasi- classical approach is valid at h̄Ω <∼ ǫF . At the electron density ≃ 1011 − 1012 cm−2 this restricts the frequency by the value of 10− 30 THz. In the strong-field limit QF0 >∼ 1 Eq. (6) results in the formula (1). From the condition QF0 >∼ 1, rewritten as E0 >∼ , (11) −4 −2 0 2 4 −4 −2 0 2 4 FIG. 1: (Color online) The time dependence of the ac electric current, measured in units ensV , at harmonic excitation of the system at the frequency Ω. (a) The temperature is zero, T/ǫF = 0; the curves are labeled by the values of the electric field parameter QF0 = eE0V/ΩǫF . (b) The temperature is finite, the Fermi energy is zero, ǫF = 0; the curves are labeled by the values of the parameter QT0 = eE0V/ΩT . one sees that the required ac electric field grows linearly with the electromagnetic wave frequency and is proportional to the square root of the electron density. At f ≃ 50 GHz and ns ≃ 1011 cm−2, the inequality (11) is fulfilled at E0 >∼ 100 V/cm. This value can be reduced in systems with lower electron/hole density. Therefore, we consider now an opposite limiting case with ǫF = 0, but finite temperature T . At finite T and the vanishing ǫF = 0 both electrons and holes contribute to the charge carrier density ns = ne + nh = πgsgvT 12h̄2V 2 and to the current. Starting again from Eq. (5) but accounting for the hole contribution and putting ǫF = 0, we get jx(t) = ensV cos θ 1 + exp x2 +Q2T (t)− 2xQT (t) cos θ ) , (13) where QT (t) = − V p0(t) sinΩt ≡ QT0 sinΩt. (14) Figure 1b shows the current (13) as a function of time Ωt. In the low-field limit QT0 ≪ 1 we get from (13) the current jx(t) ≈ ensV QT (t) 6 ln 2 , (15) and the correct expression for the linear-response intra-band dynamic conductivity [17], σǫF=0,T (Ω) = 6 ln 2 gsgvT . (16) One sees that the quasi-classical approach is now valid at h̄Ω <∼ T . This restricts the frequency by the value of ≃ 200 GHz at T ∼ 10 K and ≃ 6 THz at room temperature. In the strong field regime QT0 >∼ 1 Eq. (13) is reduced, again, to (1). Figure 2 shows the Fourier components of the ac electric current, for m = 1, 3 and 5, as a function of the field parameter eE0V/ΩT at ǫF = 0. The strong-field condition now assumes the form E0 >∼ . (17) 0 2 4 6 8 10 eE0V/ΩT FIG. 2: (Color online) The Fourier components of the current (13), in arbitrary units, as a function of QT0 = eE0V/ΩT at ǫF /T = 0. At T ≃ 10 K and f ≃ 100 GHz this gives a moderate value of the required electric field E0 ≃ 5 V/cm. The efficiency of the predicted frequency multiplication effect can be increased further by using the resonance response of the system at the plasmon, the cyclotron, or the magnetoplasmon frequency. To summarize, we have investigated the non-linear electrodynamic response of 2D electrons and holes in graphene. We have shown that irradiation of graphene by the electromagnetic wave with the frequency Ω should lead to the higher harmonics generation at frequencies 3Ω, 5Ω, et cetera. The efficiency of the frequency up-conversion is rather high: the amplitudes of the higher harmonics of the ac electric current fall down slowly (as 1/m) with the harmonics index m. The presented quasi-classical theory is valid at h̄Ω <∼ max{ǫF , T }. This estimate shows that the effect works at frequencies up to 5− 10 THz, which opens up exciting opportunities for building new graphene devices for terahertz and sub-terahertz electronics. I wish to thank Klaus Ziegler for stimulating discussion. The work was partly supported by the Swedish Research Council and INTAS. [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005). [2] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 (2005). [3] P. R. Wallace, Phys. Rev. 71, 622 (1947). [4] G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984). [5] K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, Science online (February 15, 2007), cond-mat/0702408. [6] V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801 (2005). [7] K. Ziegler, Phys. Rev. Lett. 97, 266802 (2006). [8] M. I. Katsnelson, Europ. Phys. J. B 51, 157 (2006). [9] V. V. Cheianov and V. I. Fal’ko, Phys. Rev. Lett. 97, 226801 (2006). [10] J. Nilsson, A. H. Castro Neto, F. Guinea, and N. M. R. Peres, Phys. Rev. Lett. 97, 266801 (2006). [11] T. Y. Tudorovskiy and A. V. Chaplik, JETP Letters 84, 619 (2006). [12] K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 98, 076602 (2007). [13] J.-N. Fuchs and P. Lederer, Phys. Rev. Lett. 98, 016803 (2007). [14] A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg, Nature Physics 3, 36 (2007). [15] R. S. Deacon, K.-C. Chuang, R. J. Nicholas, K. S. Novoselov, and A. K. Geim (2007), arXiv:0704.0410. [16] V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73, 245411 (2006). [17] L. A. Falkovsky and A. A. Varlamov (2006), cond-mat/0606800. [18] M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A. de Heer, Phys. Rev. Lett. 97, 266405 (2006). [19] M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A. de Heer (2007), arXiv:0704.0585. [20] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. 96, 256802 (2006). [21] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, J. Phys. Condens. Matter 19, 026222 (2007). [22] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte (2007), arXiv:cond-mat/0701053. [23] E. H. Hwang and S. Das Sarma (2006), cond-mat/0610561. [24] O. Vafek, Phys. Rev. Lett. 97, 266406 (2006). [25] V. Apalkov, X.-F. Wang, and T. Chakraborty (2006), cond-mat/0611465. [26] D. S. L. Abergel and V. I. Fal’ko (2006), cond-mat/0610673. [27] V. Ryzhii, Jpn. J. Appl. Phys. 45, L923 (2006). [28] B. Trauzettel, Y. M. Blanter, and A. F. Morpurgo, Phys. Rev. B 75, 035305 (2007). [29] V. Ryzhii, A. Satou, and T. Otsuji, J. Appl. Phys. 101, 024509 (2007). [30] F. Rana and F. R. Ahmad (2007), arXiv:0704.0607v1 [cond-mat.mes-hall]. [31] S. A. Mikhailov and K. Ziegler (2007), arXiv:cond-mat/0703406. [32] M. I. Katsnelson, Materials Today 10, 20 (2007). [33] A. K. Geim and K. S. Novoselov, Nature Materials 6, 183 (2007). References

It is shown that the massless energy spectrum of electrons and holes in graphene leads to the strongly non-linear electromagnetic response of this system. We predict that the graphene layer, irradiated by electromagnetic waves, emits radiation at higher frequency harmonics and can work as a frequency multiplier. The operating frequency of the graphene frequency multiplier can lie in a broad range from microwaves to the infrared.

Non-linear electromagnetic response of graphene S. A. Mikhailov Institute for Theoretical Physics II, University of Augsburg, D-86135 Augsburg, Germany (Dated: October 22, 2018) It is shown that the massless energy spectrum of electrons and holes in graphene leads to the strongly non-linear electromagnetic response of this system. We predict that the graphene layer, irradiated by electromagnetic waves, emits radiation at higher frequency harmonics and can work as a frequency multiplier. The operating frequency of the graphene frequency multiplier can lie in a broad range from microwaves to the infrared. In the past two years a great deal of attention has been attracted by a recently discovered, new two-dimensional (2D) electronic system – graphene, built out of a single monolayer of carbon atoms with a honeycomb 2D crystal structure [1, 2]. The band structure of the charge carriers in this system consists of six Dirac cones at the corners of the hexagon- shaped Brillouin zone [3, 4], with the massless, linear electron/hole dispersion. The massless electron spectrum leads to unusual transport and electrodynamic properties, which have been intensively studied in the literature, see e.g. [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] and for review [32, 33]. Electrodynamic properties of graphene have been theoretically studied in Refs. [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. The frequency dependent conductivity[16, 17, 20, 21, 22], as well as plasmon [23, 25, 27, 29, 30], plasmon-polariton [24], and transverse electromagnetic wave spectra [31] have been investigated. In all these papers electrodynamic response of the system has been studied within the linear response theory (for instance, using the Kubo formalism, or the random phase approximation, or the self-consistent-field approach). In this Letter we show that, apart from all the fascinating and non-trivial properties of graphene predicted and observed so far, this material should also demonstrate strongly non-linear electrodynamic behavior. In particular, irradiation of the graphene sheet by a harmonic electromagnetic wave with the frequency Ω should lead to the emission of the higher harmonics with the frequencies mΩ, m = 3, 5, . . ., from the system. The operating frequency of such a frequency multiplier can vary from microwaves up to infrared, and the required ac electric field is rather low, especially at low carrier densities and low temperatures. The predicted non-linear electrodynamic properties of graphene may open up new exciting opportunities for building electronic and optoelectronic devices based on this material. To qualitatively demonstrate the non-linear behavior of graphene electrons consider a classical 2D particle with the charge −e and the energy spectrum ǫp = V p = V p2x + p y in the external electric field Ex(t) = E0 cosΩt. Here V is the velocity of 2D electrons in the energy band (in graphene V ≈ 108 cm/s [1, 2]). According to the classical equations of motion dpx/dt = −eEx(t) the momentum px will then be equal to px(t) = −(eE0/Ω) sinΩt, and the velocity vx = ∂ǫp/∂px is then vx(t) = −V sgn(sinΩt). If there are ns particles per unit area, the corresponding ac electric current jx(t) = ensV sgn(sinΩt) = ensV sinΩt+ sin 3Ωt+ sin 5Ωt+ . . . contains all odd Fourier harmonics, with the amplitudes jm, m = 1, 3, 5 . . ., falling down very slowly with the harmonics number, jm ∼ 1/m. Notice that at the density ns = 6 · 1012 cm−2 and at V ≃ 108 cm/s (parameters of Refs. [1, 2]) the current amplitude j0 = ensV in our simple estimate gives a giant value of j0 ≃ 100 A/cm. The above consideration does not take into account the Fermi distribution of charge carriers over the quantum states in the conduction and valence bands of graphene. To get a more accurate description of the non-linear phenomena in the considered system we use the kinetic Boltzmann theory, which allows one to get an exact response of the system not imposing any restrictions on the amplitude of the external ac electric field E(t). Using this quasi-classical approach we take into account the intra-band contribution to the ac electric current. The inter-band contribution to the electric current, due to the transitions between the hole and the electron bands, is ignored. This imposes certain restrictions on the frequency of radiation Ω, which will be discussed below. Consider a 2D electron/hole gas with the energy spectrum ǫp± = ±V p2x + p y under the action of the field E = (Ex, 0), where the sign + (−) corresponds to the electron (hole) band, Ex(t) = E0eαt cos(Ωt), and α → +0 describes an adiabatic switching on of the electric field. Assume that the Fermi energy ǫF lies in the electron (or the hole) band and that the temperature is small as compared to ǫF , T ≪ ǫF . The momentum distribution function of electrons fp+(t) ≡ fp(t) (we omit the sign + for brevity) in the collisionless approximation is described by the http://arxiv.org/abs/0704.1909v1 Boltzmann equation ∂fp(t) − ∂fp(t) αt cos(Ωt) = 0, (2) which has the exact solution fp(t) = F0 (px − p0(t), py) , (3) where F0(px, py) = 1 + exp p2x + p y − ǫF is the electronic Fermi function, and p0(t) = −(eE0/Ω)eαt sinΩt is the solution of the single particle equation of motion. The electric current j(t) = −egsgvS−1 vfp(t) then assumes the form jx(t) = − gsgveV (2πh̄)2 dpxdpy p2x + p F0 (px − p0(t), py) , (5) jy = 0, where gs = gv = 2 are the spin and valley degeneracies in graphene, and S is the sample area. After some lengthy but simple transformation, Eq. (5) can be rewritten as jx(t) = ensV 2QF (t) 1 +Q2F (t) ∫ π/2 cos2 xdx 2QF (t) cosx+ 1− 2QF (t) where ns ≡ ne = gsgvp 4πh̄2 gsgvǫ 4πh̄2V 2 is the density of electrons, pF = ǫF /V is the Fermi momentum, and QF (t) = − p0(t) sin(Ωt) ≡ QF0 sin(Ωt) (8) is the field parameter, proportional to the ac electric field E0. Figure 1a shows the current (6) as a function of time Ωt. One sees that in the low-field limit the response is linear. Expanding the current (6), we get at QF0 ≪ 1 jx(t) ≈ ensV QF0 sinΩt+ Q2F0 sin 3Ωt , (9) so that the linear response conductivity (in the collisionless approximation) is σǫF ,T=0(Ω) = . (10) The expression (10) coincides with the intra-band Drude conductivity, which can be obtained from the linear-response theory [16, 17, 20, 21, 22, 31]. As the inter-band conductivity is of order of e2/h̄ [16, 17, 20, 21, 22, 31], our quasi- classical approach is valid at h̄Ω <∼ ǫF . At the electron density ≃ 1011 − 1012 cm−2 this restricts the frequency by the value of 10− 30 THz. In the strong-field limit QF0 >∼ 1 Eq. (6) results in the formula (1). From the condition QF0 >∼ 1, rewritten as E0 >∼ , (11) −4 −2 0 2 4 −4 −2 0 2 4 FIG. 1: (Color online) The time dependence of the ac electric current, measured in units ensV , at harmonic excitation of the system at the frequency Ω. (a) The temperature is zero, T/ǫF = 0; the curves are labeled by the values of the electric field parameter QF0 = eE0V/ΩǫF . (b) The temperature is finite, the Fermi energy is zero, ǫF = 0; the curves are labeled by the values of the parameter QT0 = eE0V/ΩT . one sees that the required ac electric field grows linearly with the electromagnetic wave frequency and is proportional to the square root of the electron density. At f ≃ 50 GHz and ns ≃ 1011 cm−2, the inequality (11) is fulfilled at E0 >∼ 100 V/cm. This value can be reduced in systems with lower electron/hole density. Therefore, we consider now an opposite limiting case with ǫF = 0, but finite temperature T . At finite T and the vanishing ǫF = 0 both electrons and holes contribute to the charge carrier density ns = ne + nh = πgsgvT 12h̄2V 2 and to the current. Starting again from Eq. (5) but accounting for the hole contribution and putting ǫF = 0, we get jx(t) = ensV cos θ 1 + exp x2 +Q2T (t)− 2xQT (t) cos θ ) , (13) where QT (t) = − V p0(t) sinΩt ≡ QT0 sinΩt. (14) Figure 1b shows the current (13) as a function of time Ωt. In the low-field limit QT0 ≪ 1 we get from (13) the current jx(t) ≈ ensV QT (t) 6 ln 2 , (15) and the correct expression for the linear-response intra-band dynamic conductivity [17], σǫF=0,T (Ω) = 6 ln 2 gsgvT . (16) One sees that the quasi-classical approach is now valid at h̄Ω <∼ T . This restricts the frequency by the value of ≃ 200 GHz at T ∼ 10 K and ≃ 6 THz at room temperature. In the strong field regime QT0 >∼ 1 Eq. (13) is reduced, again, to (1). Figure 2 shows the Fourier components of the ac electric current, for m = 1, 3 and 5, as a function of the field parameter eE0V/ΩT at ǫF = 0. The strong-field condition now assumes the form E0 >∼ . (17) 0 2 4 6 8 10 eE0V/ΩT FIG. 2: (Color online) The Fourier components of the current (13), in arbitrary units, as a function of QT0 = eE0V/ΩT at ǫF /T = 0. At T ≃ 10 K and f ≃ 100 GHz this gives a moderate value of the required electric field E0 ≃ 5 V/cm. The efficiency of the predicted frequency multiplication effect can be increased further by using the resonance response of the system at the plasmon, the cyclotron, or the magnetoplasmon frequency. To summarize, we have investigated the non-linear electrodynamic response of 2D electrons and holes in graphene. We have shown that irradiation of graphene by the electromagnetic wave with the frequency Ω should lead to the higher harmonics generation at frequencies 3Ω, 5Ω, et cetera. The efficiency of the frequency up-conversion is rather high: the amplitudes of the higher harmonics of the ac electric current fall down slowly (as 1/m) with the harmonics index m. The presented quasi-classical theory is valid at h̄Ω <∼ max{ǫF , T }. This estimate shows that the effect works at frequencies up to 5− 10 THz, which opens up exciting opportunities for building new graphene devices for terahertz and sub-terahertz electronics. I wish to thank Klaus Ziegler for stimulating discussion. The work was partly supported by the Swedish Research Council and INTAS. [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005). [2] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 (2005). [3] P. R. Wallace, Phys. Rev. 71, 622 (1947). [4] G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984). [5] K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, Science online (February 15, 2007), cond-mat/0702408. [6] V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801 (2005). [7] K. Ziegler, Phys. Rev. Lett. 97, 266802 (2006). [8] M. I. Katsnelson, Europ. Phys. J. B 51, 157 (2006). [9] V. V. Cheianov and V. I. Fal’ko, Phys. Rev. Lett. 97, 226801 (2006). [10] J. Nilsson, A. H. Castro Neto, F. Guinea, and N. M. R. Peres, Phys. Rev. Lett. 97, 266801 (2006). [11] T. Y. Tudorovskiy and A. V. Chaplik, JETP Letters 84, 619 (2006). [12] K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 98, 076602 (2007). [13] J.-N. Fuchs and P. Lederer, Phys. Rev. Lett. 98, 016803 (2007). [14] A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg, Nature Physics 3, 36 (2007). [15] R. S. Deacon, K.-C. Chuang, R. J. Nicholas, K. S. Novoselov, and A. K. Geim (2007), arXiv:0704.0410. [16] V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73, 245411 (2006). [17] L. A. Falkovsky and A. A. Varlamov (2006), cond-mat/0606800. [18] M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A. de Heer, Phys. Rev. Lett. 97, 266405 (2006). [19] M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A. de Heer (2007), arXiv:0704.0585. [20] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. 96, 256802 (2006). [21] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, J. Phys. Condens. Matter 19, 026222 (2007). [22] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte (2007), arXiv:cond-mat/0701053. [23] E. H. Hwang and S. Das Sarma (2006), cond-mat/0610561. [24] O. Vafek, Phys. Rev. Lett. 97, 266406 (2006). [25] V. Apalkov, X.-F. Wang, and T. Chakraborty (2006), cond-mat/0611465. [26] D. S. L. Abergel and V. I. Fal’ko (2006), cond-mat/0610673. [27] V. Ryzhii, Jpn. J. Appl. Phys. 45, L923 (2006). [28] B. Trauzettel, Y. M. Blanter, and A. F. Morpurgo, Phys. Rev. B 75, 035305 (2007). [29] V. Ryzhii, A. Satou, and T. Otsuji, J. Appl. Phys. 101, 024509 (2007). [30] F. Rana and F. R. Ahmad (2007), arXiv:0704.0607v1 [cond-mat.mes-hall]. [31] S. A. Mikhailov and K. Ziegler (2007), arXiv:cond-mat/0703406. [32] M. I. Katsnelson, Materials Today 10, 20 (2007). [33] A. K. Geim and K. S. Novoselov, Nature Materials 6, 183 (2007). References

DRAFT VERSION NOVEMBER 1, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 ELLIPSOIDAL OSCILLATIONS INDUCED BY SUBSTELLAR COMPANIONS: A PROSPECT FOR THE KEPLER MISSION ERIC PFAHL1, PHIL ARRAS1,2 , AND BILL PAXTON1 Draft version November 1, 2018 ABSTRACT Hundreds of substellar companions to solar-type stars will be discovered with the Kepler satellite. Kepler’s extreme photometric precision gives access to low-amplitude stellar variability contributed by a variety of physical processes. We discuss in detail the periodic flux modulations arising from the tidal force on the star due to a substellar companion. An analytic expression for the variability is derived in the equilibrium-tide approximation. We demonstrate analytically and through numerical solutions of the linear, nonadiabatic stellar oscillation equations that the equilibrium-tide formula works extremely well for stars of mass <1.4M⊙ with thick surface convection zones. More massive stars with largely radiative envelopes do not conform to the equilibrium-tide approximation and can exhibit flux variations &10 times larger than naive estimates. Over the full range of stellar masses considered, we treat the oscillatory response of the convection zone by adapting a prescription that A. J. Brickhill developed for pulsating white dwarfs. Compared to other sources of periodic variability, the ellipsoidal lightcurve has a distinct dependence on time and system parameters. We suggest that ellipsoidal oscillations induced by giant planets may be detectable from as many as ∼100 of the 105 Kepler target stars. For the subset of these stars that show transits and have radial-velocity measurements, all system parameters are well constrained, and measurement of ellipsoidal variation provides a consistency check, as well as a test of the theory of forced stellar oscillations in a challenging regime. Subject headings: planetary systems — stars: oscillations — techniques: photometric 1. INTRODUCTION The upcoming Kepler3 satellite will continuously monitor ∼105 main-sequence stars of mass ≃0.5–1.5M⊙ over 4–6 years with fractional photometric precisions of ∼10−5. Such high sensitivity, which is unattainable from the ground, will allow for the robust detection of Earth-size planets that transit their host stars, and the measurement of asteroseismic oscilla- tions as a probe of stellar structure (e.g., Borucki et al. 2004; Basri et al. 2005). These missions will also discover hundreds of “hot Jupiters” with orbital periods of <10 days, revealed by their transits or reflected starlight (e.g., Jenkins & Doyle 2003). Continuous observations of these systems are likely to show a myriad of novel physical effects, including Doppler flux variability of the host stars (Loeb & Gaudi 2003), pho- tometric dips due to moons or rings around the planets (Sartoretti & Schneider 1999; Brown et al. 2001), and the impact of additional perturbing planets on transit timing (Miralda-Escudé 2002; Agol et al. 2005; Holman & Murray 2005). The same ideas apply if the companion is a more mas- sive brown dwarf, but these are rarely found in close orbits around solar-type stars (e.g., Grether & Lineweaver 2006). Here we scrutinize another mechanism for generating peri- odic variability of a star closely orbited by a giant planet or brown dwarf. A star subject to the tidal gravity of a binary companion has a nonspherical shape and surface-brightness distribution. In the simplest approximation, the stellar surface is a prolate ellipsoid with its long axis on the line connecting the two objects. As the tidal bulge tracks the orbital motion, differing amounts of light reach the observer. For a solar-type 1 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106; pfahl@kitp.ucsb.edu, paxton@kitp.ucsb.edu 2 Department of Astronomy, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904-4325; arras@virginia.edu 3 http://kepler.nasa.gov star orbited by a perturbing companion of mass Mp with pe- riod Porb, the expected fractional amplitude of this ellipsoidal variability is ∼10−2(Mp/M⊙)(1day/Porb) 2. This effect has a long history in the study of eclipsing binary stars (see the re- view by Wilson 1994), but was mentioned only recently in the exoplanet context. Udalski et al. (2002), Drake (2003), and Sirko & Paczyński (2003) noted that if ellipsoidal light variations are detected from the ground, where the fractional photometric precision is &10−3, then the perturber must be fairly massive (e.g., &0.1M⊙). They offered this idea as a test to distinguish be- tween planetary transits and eclipses by low-mass stars. The superior sensitivity of Kepler offers the possibility of mea- suring ellipsoidal variability induced by giant planets (Mp ∼ 10−3–10−2 M⊙) with orbital periods of .10 days. Loeb & Gaudi (2003) compare the ellipsoidal variability in- duced by a planetary companion to flux modulations arising from reflected starlight and the Doppler effect. The three am- plitudes are similar when the companion has an orbital period of .3 days and an optical albedo of .0.1. In a sufficiently long observation it should be possible to separately extract each of the signals, since their Fourier decompositions are dis- tinct. Precise physical modeling of the ellipsoidal lightcurve could provide an independent constraint on the mass of the companion, as well as important clues regarding stellar tidal interactions. Ellipsoidal variability is typically modeled under the as- sumption that the distorted star maintains hydrostatic balance and precisely fills a level surface of an appropriate poten- tial (e.g., the Roche potential). The measured flux is then just an integral of the intensity over the visible stellar sur- face, where the intensity includes the effects of limb darken- ing and gravity darkening (e.g., Kopal 1942). This approach is strictly valid only when the orbit is circular and the star rotates at the orbital frequency, so that a stationary configura- http://arxiv.org/abs/0704.1910v1 http://kepler.nasa.gov 2 PFAHL, ARRAS, & PAXTON tion exists in the coorbital frame. These conditions may not be satisfied when the companion has a low mass or long pe- riod, because of the weak tidal interaction. In fact, a state of tidal equilibrium may not be attainable in the case of a plan- etary companion (e.g., Rasio et al. 1996). Equilibrium mod- els of ellipsoidal lightcurves do have a realm of validity for noncircular orbits and asynchronously rototating stars, and have been applied successfully to somewhat eccentric bina- ries (e.g., Soszynski et al. 2004). However, by construction, such models ignore fluid inertia and the possibility exciting normal modes of oscillation, effects that may be of critical importance in a wide range of observationally relevant cir- cumstances. Here we apply the machinery of linear stellar oscillation theory to the weak tidal forcing of stars by sub- stellar companions. Conceptually, our investigation bridges Kepler’s planetary and astroseismology programs. Section 2 describes the geometry of the problem, provides quantitative measures for the strength of the tidal interaction, discusses our simplifying assumptions, and presents the math- ematical framework for calculating ellipsoidal variability. In § 3, we consider the equilibrium-tide approximation and de- rive an analytic expression for the ellipsoidal lightcurve. A brief review of von Zeipel’s theorem and its limitations is given in § 4. Tidally forced, nonadiabatic stellar oscillations are addressed in § 5, where we argue for a simple treatment of perturbed surface convection zones, use this prescription to calculate the ellipsoidal variability of deeply convective stars, estimate analytically the surface flux perturbation in mainly radiative stars, and show select numerical results. Our main conclusions are summarized in § 6. We conclude in § 7 with remarks on the measurement of ellipsoidal oscillations in the presence of other sources of periodic variability. 2. PRELIMINARIES Consider a star of mass M and radius R is orbited by a sub- stellar companion of mass Mp and radius Rp. We work in spherical coordinates (r,θ,φ) with the origin at the star’s cen- ter and the pole direction (θ = 0) parallel to the orbital angular momentum vector. The orbit is then described by (d,π/2,φp), where d and φp are, respectively, the time-dependent or- bital separation and true anomaly; φp = 0 marks the phase of periastron. We assume that the orbit is strictly Keple- rian with fixed semimajor axis a and eccentricity e, such that d = a(1 − e2)/(1 + ecosφp). The direction to the observer from the center of the star is (θo,φo), so that the conventional orbital inclination is I = π − θo. We imagine that the gravity of the companion raises nearly symmetrical tidal bulges on opposite sides of the star that ro- tate at the orbital frequency. A rough measure of both the height of the tides relative to the unperturbed stellar radius and the fractional amplitude of the ellipsoidal variability is given by the ratio of the tidal acceleration to the star’s surface gravity: ∼ 10−5 2.8 hr 1 day , (1) where MJ ≃ 10 −3 M⊙ is the mass of Jupiter, P∗ = 2π(R3/GM)1/2 = 2.8[(R/R⊙) 3(M⊙/M)] 1/2 hr is the dynami- cal time of the star. For main-sequence stars with R/R⊙ ≃ M/M⊙, we see that ε ∝ MpMP orb. The maximum value of ε is attained when the companion fills its Roche lobe at an orbital separation of a ≃ 2Rp(M/Mp) 1/3, which gives εmax ≃ ≃ 10−4 0.1R⊙ , (2) where we have applied a fixed value of Rp = 0.1R⊙, appro- priate for both giant planets and old brown dwarfs. Note that εmax ∼ 1 for massive brown dwarfs (Mp/MJ ∼ 80). Hereafter, we consider only cases with ε≪ 1. For orbital periods as short as ≃1 day, tidal torques on the star from a planetary companion are rather ineffective at alter- ing the stellar rotation rate (e.g., Rasio et al. 1996). Therefore, as already mentioned in § 1, we should not generally expect the star to rotate synchronously with the orbit, and so there is no frame in which the star appears static. This holds when the orbit is circular, and is obviously true when the there is a finite eccentricity. In fact, ≃30% of the known exoplanets4 with Porb < 10 days have eccentricities of >0.1. Small vari- able distortions of the star from its equilibrium state, due to a combination of asynchronous rotation and orbital eccentric- ity, should be viewed as waves excited by the tidal force of the companion. Our task is to study such tidally forced stel- lar oscillations in the linear domain in order to understand the corresponding lightcurves. In order to greatly simplify the mathematical description of the stellar oscillations, we assume that the star is nonrotating in the inertial frame. When the stellar rotation frequency is nonzero, but much smaller than the tidal forcing frequency, the effect of rotation is to introduce fine structure into the os- cillation frequency spectrum, and cause the oscillation eigen- functions to be slightly modified as a result of the Coriolis force (for a discussion, see Unno et al. 1989). Tidal pump- ing of a slowly rotating star by an orbiting companion has a dominant period of Porb/2—a few days in the cases of inter- est. By contrast, single solar-type stars with ages>1 Gyr tend to have rotation periods of >10 days (e.g., Skumanich 1972; Pace & Pasquini 2004); the Sun has an equatorial rotation pe- riod of ≃25 days. Slowly rotating stars with masses of ≃1M⊙ are prime targets for Kepler, since they exhibit low intrinsic variability. Based on this selection effect, and the inability of tidal torques to spin up the star, our assumption of vanishing stellar rotation seems generally justified. The general framework for calculating the measurable flux modulations associated with ellipsoidal stellar oscillations is as follows. We consider small perturbations to a spherical, nonrotating background stellar model, such that fluid ele- ments at equilibrium position x are displaced in a Lagrangian fashion to position x +ξ. Variations in the measured flux from an oscillating star arise from two physically distinct contribu- tions (e.g., Dziembowski 1977): (1) changes in the shape of the star due to radial fluid displacements ξr = ξ · er, where er is the radial unit vector, and (2) hot and cold spots generated by local Lagrangian perturbations ∆F to the heat flux. Our main task in §§ 3 and 5 is to compute ξr and ∆F according to the relevant physics. Given the dependences of ξr and ∆F on (r,θ,φ), it is straightforward to compute the time varying component of the measured flux. The flux5 received from a star at distance D is 4 http://vo.obspm.fr/exoplanetes/encyclo/encycl.html 5 Our calculations concern the bolometric flux, although is relatively http://vo.obspm.fr/exoplanetes/encyclo/encycl.html ELLIPSOIDAL OSCILLATIONS 3 (e.g., Robinson et al. 1982) dS n ·no F h(n ·no) (3) where dS is an area element at the stellar photosphere, F is the net flux of radiation out of the surface element, h is the limb-darkening function, n and no are unit vectors normal to the surface and toward the observer, respectively, and the in- tegration is over the visible stellar disk. Vertical displacement at the surface yields changes in F through changes in surface area and n · no. Following Dziembowski (1977), we expand ξr and ∆F in spherical harmonics, ξr(r,θ,φ, t) = ξr,ℓm(r, t)Yℓm(θ,φ) , (4) ∆F(r,θ,φ, t) = ∆Fℓm(r, t)Yℓm(θ,φ) , (5) and carry out the appropriate linear expansions to obtain the fractional variability (2bℓ − cℓ) ξor,ℓ . (6) Here ξor,ℓ and ∆F are components evaluated at the surface (r = R) and in the direction of the observer: ξor,ℓ ξr,ℓm(R, t) Yℓm(θo,φo) , (7) ∆Fℓm(R, t) Yℓm(θo,φo) . (8) The terms bℓ and cℓ are given by dµµPℓ h , cℓ = dµ(1 −µ2) where µ = n ·no, the Pℓ(µ) are ordinary Legendre polynomials, and h(µ) is normalized such that dµµh = 1. The linear limb-darkening function is h(µ) = (3 −γ) 1 −γ(1 −µ) ; (10) more general nonlinear functions of µ (e.g., Claret 2000) will not be considered here. The classical Eddington limb- darkening function is h = 1 + 3µ/2 (γ = 3/5; e.g., Mihalas 1970). Table 1 shows shows functional forms and particular values of bℓ and cℓ for ℓ = 2 and 3. 3. EQUILIBRIUM TIDE Vertical displacement of the stellar surface is often accu- rately modeled by assuming that the tidally perturbed fluid remains in hydrostatic balance. The cause and magnitude of the surface flux perturbation is a more complicated affair. In this section, we apply a simple parameterization of the flux perturbation and obtain a complete set of formulae for com- puting the ellipsoidal lightcurve. Subsequent sections provide more detailed calculations. In particular, we show in § 5.2 straightforward to modify the analysis for narrow-band measurements. TABLE 1 LIMB DARKENING PARAMETERS bℓ cℓ ℓ General γ γ = 3/5 General γ γ = 3/5 2 (1 +γ)/[20(3 −γ)] 13/40 3(1 + 3γ)[10(3 −γ)] 39/20 3 γ/[4(3 −γ)] 1/16 3γ/(3 −γ) 3/4 that stars with deep convective envelopes (the majority of Ke- pler targets) have surface flux variations that conform to the equilibrium-tide approximation. When the tidal forces on the stellar fluid change sufficiently slowly, the star can stay very nearly in hydrostatic equilib- rium. If the net acceleration required to balance the pressure gradient is derivable from a potential, then equilibrium im- plies that a fluid element remains on an equipotential surface. Since we neglect stellar rotation, there is no centrifugal force, and the total potential is the sum of the gravitational potential ϕ from the spherical background stellar model and the per- turbing tidal potential U ∼ εϕ≪ ϕ. For our analytic work, we neglect the modification of ϕ due to the tide. In general, the Eulerian variation δϕ should be added to U , as we do in our numerical models (see § 5.4 and the Appendix); we find that |δϕ|/|U | ∼ 10−2. In the absence of tidal forces, a given fluid element sits at equilibrium position x with total potential ϕ(x). Gentle inclu- sion of the tidal potential causes the fluid element to move to position x +ξ while preserving the value of the total potential. This is expressed mathematically by ϕ(x) =ϕ(x +ξ) +U(x +ξ, t) =ϕ(x) +ξ ·∇ϕ+U(x, t) +O(ξ2, ξU) . (11) We see that ξ ·∇ϕ = ξrg, where g = GMr/r 2 is the background gravitational acceleration at mass coordinate Mr. To first or- der, the radial displacement of the equilibrium tide is (see also Goldreich & Nicholson 1989) ξr(x, t)≃−U(x, t)/g , (12) which tells us the geometry of the star as a function of time. The tidal potential within the star can be expanded as U(r,θ,φ, t) = − Pℓ(cosψ) , (13) where cosψ = sinθ cos(φp −φ). There is no ℓ = 1 term, since this would give the acceleration of the star’s center of mass, which is already incorporated into the orbital dynamics. The angular expansion of ξr follows immediately from eq. (12): ξr(r,θ,φ, t) Pℓ(cosψ) . (14) In order to express U and ξr in spherical harmonics, we utilize the addition theorem, Pℓ(cosψ) = 2ℓ+ 1 ℓm(π/2,φp) Yℓm(θ,φ) , (15) where “∗” denotes the complex conjugate. Note that Yℓm(π/2,φp) is nonzero only when ℓ−m is even. For the dom- 4 PFAHL, ARRAS, & PAXTON inant ℓ = 2 components of U and ξr, the surface values of U/ϕ and ξr/R are ∼ε, as expected. From eqs. (7), (14), and (15), the components ξor,ℓ/R of the surface radial displacement toward the observer are immedi- ately apparent. As we will see in § 5, the computation of ∆F/F is, in general, rather technical. However, in the special case where the stellar fluid responds adiabatically to a slowly varying tidal potential, ∆Fℓ/F varies in phase with and in pro- portion to ξr,ℓ/r in the linear approximation of the equilibrium tide. Making this assumption, we write ∆Fℓ/F = −λℓξr,ℓ/R at the surface, where the λℓ are real constants that depend on the stellar structure (see § 5.1). We will see in § 4 that λℓ = ℓ+ 2 is a good first guess for radiative stars, and so we might gen- erally expect λℓ to be positive and O(ℓ). We now have the ingredients for the fractional variability (eq. [6]), and we obtain )ℓ−2( fℓ Pℓ(cosψo) , (16) where fℓ = (2−λℓ)bℓ−cℓ, and cosψo = sinθo cos(φp −φo). The ℓ = 2 and 3 Legendre polynomials can be expanded as P2(cosψo) = − (3cos2 I − 1) + 3sin2 I cos2(φp −φo) , (17) P3(cosψo) = sin I − 3(5cos2 I − 1)cos(φp −φo) + 5sin2 I cos3(φp −φo) , (18) where we have substituted θo = π − I. The Eddington limb- darkening formula gives (see Table 1) f2 = − , f3 = − . (19) It is important to note that f2 < 0 when λ2 ≥ 0 (see below). In eq. (16), the orbital dynamics are described by the evolu- tion of d and φp (see § 2). For a circular orbit, we have d = a and φp =Ωt, where Ω = 2π/Porb, and t is the time since perias- tron (modulo Porb). Example lightcurves with e = 0, γ = 3/5, λℓ = 0, and I = π/2 are shown in Fig. 1 for a/R = {2,4,8,16}. When R/a ≪ 1, the ℓ = 2 piece of δF/F is a good approxi- mation, and the temporal flux variation approaches a pure co- sine with angular frequency 2Ω (see eq. [17]). Because f2 < 0, the dominant ℓ = 2 component of the ellipsoidal variability has minimum light when tidal bulge is aligned with the direction to the observer. As R/a increases, so does the importance of ℓ > 2 terms and their extra harmonic content, as seen in eq. (18) and Fig. 1. Additional harmonics in δF/F also result from a finite eccentricity. At the O(e) level, signals with frequenciesΩ and 3Ω, and amplitudes of ∼εe, are present in the ℓ = 2 component of δF/F , which compete with the ℓ = 3 piece when e ∼ R/a. Notice that when e> 0 the flux is variable even when the orbit is viewed face-on (I = 0 or π), by virtue of changes in d−3 = 1+ 3ecosΩt +O(e2). For I = 0, we see that P3(cosψo) vanishes, leaving the largest contribution δF/F ≃ −1.5εe f2 cos(Ωt). 4. AN ASIDE ON VON ZEIPEL’S THEOREM FIG. 1.— Disk-averaged flux variation (eq. [16]) for an edge-on circu- lar orbit under the equilibrium-tide approximation (eqs. [6] and [14]) with ∆F/F = 0 at the surface. The four curves correspond to a/R = 2 (black), 4 (red), 8 (blue) and 16 (green). In order to compare the shapes of the curves, δF/F has been multiplied by (a/R)3(M/Mp). As a/R increases, higher har- monics decrease in strength and the lightcurve approaches a pure cosine with frequency 2/Porb. The tidal bulge closest to the companion points toward the observer at integer values of t/Porb. Our equilibrium calculation in the last section used the sim- ple prescription ∆Fℓ/F = −λℓξr,ℓ/R. There remains the ques- tion of what physics determines ∆F/F . A common practice in empirical studies of close eclipsing binaries—systems that tend to be nearly in tidal equilibrium—is to use some vari- ant of the von Zeipel (1924) theorem, which was originally formulated for purely radiative, strictly hydrostatic stars. In equilibrium, all the thermodynamic variables depend only on the local value of the total potential Φ. Thus, the radiative flux can be written as (e.g., Hansen & Kawaler 1994) ∇Φ , (20) where ρ is the mass density, T is the effective temperature, and κ(ρ,T ) is the opacity. Equation (20) is the essence of von Zeipel’s theorem, which says that the magnitude F of the radiative flux is proportional to the magnitude of the net ac- celeration A = |∇Φ|. When Φ = ϕ + U (see § 3), we obtain A = g + ∂U/∂r +O(ξ2), so that the Lagrangian flux perturba- tion about equilibrium is , (21) where ∆g/g = −2ξr/r, due to the change in radius at approxi- mately constant enclosed mass. Substituting the equilibrium- tide result U = −ξrg into eq. (21), we obtain the compact ex- pression ∆F/F = −∂ξr/∂r. Using eq. (14), we find = −(ℓ+ 2) , (22) from which we identify λℓ = ℓ+ 2. Although the application of von Zeipel’s theorem is instruc- tive, the underlying physical assumptions are inaccurate for ELLIPSOIDAL OSCILLATIONS 5 FIG. 2.— Important oscillations frequencies and time scales as a function of pressure for a 1M⊙ main-sequence star. The four curves show the Brunt- Väisällä frequency N (black), Lamb frequency Sl (red; ℓ = 2 is shown), in- verse thermal time t−1th (blue), and inverse eddy turnover time t ed (green). Large, real values of N occur in the radiative core and very near the photo- sphere, while N2 < 0 in the convective envelope. Gravity waves propagate only where the angular frequency is below both N and Lℓ. The two horizontal lines delimit the range of tidal forcing frequencies of interest here. slowly rotating main-sequence stars of mass 1.0–1.6M⊙ with tidal forcing periods of days. We are now led to investigate the general problem of forced nonadiabatic stellar oscillations. 5. FORCED NONADIABATIC OSCILLATIONS The equilibrium analysis ignores fluid inertia and the exci- tation of the star’s natural oscillation modes. While this as- sumption may be valid near the surface of the star, it does not hold deeper in the interior. Gravity waves (g-modes; re- stored by buoyancy) can propagate in the radiative interiors of Sun-like stars with a range of oscillation periods that in- cludes the tidal forcing periods of interest (.3 days). Tidal forcing of radiative regions may produce substantial devia- tions from hydrostatic balance, as well as large surface am- plitudes of ∆F/F , in particular if resonant oscillations are excited. This is especially relevant for main-sequence stars of mass M & 1.4–1.5M⊙ with mainly radiative envelopes. Less massive stars (M . 1.3–1.4M⊙) have rather deep convective envelopes that can block information about the dynamic inte- rior from being conveyed to the surface. Here we investigate each of these regimes with both analytic estimates and numer- ical models of oscillating stars. Our calculations employ realistic models of 0.9–1.6M⊙ main-sequence stars, constructed with the EZ stellar evolu- tion code (Paxton 2004), a distilled and rewritten version of the program originally created by Peter Eggleton. We adopt Solar metallicity and a convective mixing length of 1.6 times the pressure scale height. All stars are evolved to an age when the core hydrogen abundance has the Solar value of XH = 0.35. Models with 199 radial grid points are interpolated to yield &104 points in which the g-mode radial wavelength is well resolved in the core. Figures 2 and 3 illustrate some of the differences between FIG. 3.— Same as Fig. 2, but for a 1.6M⊙ main-sequence star. Note two geometrically thin, relatively inefficient (tth ∼ ted) convection zones near the surface. The spike in N near the center is at the edge of the convective core, and signals a steep gradient in the mean molecular weight. 1M⊙ and 1.6M⊙ stars, and serve to introduce several impor- tant physical quantities used in the remainder of this section. The Lamb frequency, Sℓ = [ℓ(ℓ+ 1)]1/2 , (23) is the inverse of the horizontal sound-crossing time scale, where cs is the sound speed, and [ℓ(ℓ + 1)]1/2/r ≡ kh is the horizontal wavenumber of the oscillation. For fixed chemical composition, the squared Brunt-Väisällä frequency is (∇ad −∇) , (24) where Hp = −(d ln p/dr)−1 = p/(ρg) is the pressure scale height, and ∇ = d lnT/d ln p is the temperature gradient6 (∇ad is the adiabatic value). Radiative regions have ∇ad −∇ > 0 (N2 > 0), and N represents the frequency of buoyancy oscilla- tions. In convection zones, ∇ad −∇< 0 and N2 < 0, indicating that g-modes are evanescent. When N2 < 0, the time scale ted ∼ |N| −1 , (25) approximates the turnover time of convective motions (for de- tails and modifications for radiative losses, see, e.g., Kippen- hahn & Weigert 1990). A shell of radius r, thickness Hp (size of the largest convective eddies), and radiative luminosity L cools on the thermal time scale tth = 4πr2HpρCpT , (26) where Cp is the specific heat at constant pressure. The 1M⊙ model (Fig. 2) has one deep convection zone with ted ≪ tth over most of the region, indicating that convection 6 Do not confuse the temperature gradient ∇ with the spatial gradient ∇ used in § 3. 6 PFAHL, ARRAS, & PAXTON FIG. 4.— Eddy turnover time at base of the convective envelope versus stellar age for a range of stellar masses. very efficiently transports energy and causes the zone to be essentially isentropic. By consrast, the 1.6M⊙ star (Fig. 3) has two thin surface convection zones with ted ∼ tth, and thus the radiative and convective fluxes are comparable. Gravity waves with frequency ω propagate only in radiative regions where ω < N and ω < Sℓ. For the 1M⊙ star, heat and en- tropy generated by g-modes in the radiative interior may be strongly mitigated owing to the long thermal time at the base of the deep convection zone. On the other hand, g-modes in a 1.6M⊙ star can propagate very near the surface, producing qualitatively different results. We now go on to elucidate the physics of the flux per- turbations. All the analytic and numerical work that fol- lows assumes that the tidal potential has the generic form U ∝ rℓYℓm(θ,φ)exp(−iωt) with forcing frequency ω. 5.1. Heat Transfer in a Convective Envelope Calculation of the perturbed convective flux in oscillating stars is a thorny issue. For the purposes of our study, we argue for an especially simple treatment that draws from previous work on this subject. Specifically, we modify the prescription of Brickhill (1983, 1990; see also Goldreich & Wu 1999a,b), which was originally applied to white-dwarf pulsations, into a form appropriate for the tidal flow problem. In the mixing-length theory of convection, heat is trans- ported by eddies with a spectrum of sizes l . Hp, speeds vl , and turnover times ted(l) = l/vl . The Kolmogorov scalings for turbulent motions give vl ∝ l 1/3, ted ∝ l 2/3, and an energy den- sity per unit mixing length interval ∝l−1/3. We see that in the unperturbed star most of the convective energy flux (∝v3l at scale l) is carried by the largest eddies (l ∼ Hp). Convection efficiently transports energy when the radiative thermal time scale associated with the dominant eddies is much longer than ted. Alternatively, efficient convection implies that the gradi- ent of the specific entropy s is small; i.e., d lns/d ln p ≪ 1. If all the convective energy flux F is carried by eddies with mix- ing length l, the flux and entropy gradient are related by (e.g., FIG. 5.— Thermal time at the base of the convection envelope versus stellar age for a range of stellar masses. Kippenhahn & Weigert 1990) d ln p (l) = (∇−∇ad) ∼ . (27) Efficient convection enforces ∇ − ∇ad ≪ 1, which implies d lns/d ln p ≪ 1, since s & Cp in the convective regions of our background models. Gravity waves with the tidal forcing frequency ω are ex- cited in the radiative region below the convection zone. Con- vective eddies can transport heat during a forcing period only if ted < 2π/ω (e.g., Brickhill 1990; Goldreich & Wu 1999b). Inspection of Fig. 2 shows that in the 1M⊙ model, the largest eddies have ted ≃ 20(p/pbcz) 0.5 days, where pbcz ≃ 1013.5 dyne cm−2 is the pressure at the base of convection zone. Using the Kolmogorov scaling, the “resonant” length lres for which ωted/2π = 1 is ∼ 10−2 1 day )3/2( , (28) which is >1 for all periods 2π/ω > 1 day when p . 1012 dyne cm−2, which still encompasses much of the convec- tion zone. Now imagine the situation where all the convective flux is carried by eddies of size .lres. The entropy gradient for this range of mixing lengths is d ln p (lres)∼ 10 1 day , (29) where we have adopted F/pcs ∼ 10 −8 at the base of the con- vection zone, as indicated by our 1M⊙ stellar model. These arguments suggest that convection is efficient in a 1M⊙ star at the forcing periods of interest even if small “res- onant” eddies carry all the energy flux near the base of the convection zone. At larger radii, but not too near the photo- sphere, convection is both efficient and rapid (ωted/2π < 1) ELLIPSOIDAL OSCILLATIONS 7 over the full spectrum of eddies. Rapid convection on all scales l . Hp enforces isentropy in the convection zone, such that s and its Lagrangian perturbation ∆s are nearly constant, as in the Brickhill (1983, 1990) picture. While convection at the base is rapid only on small scales, it is still highly efficient, which yields s ≃ constant and further indicates that ∆s/Cp is small in magnitude, as we demonstrate in § 5.2. As the stellar mass increases, the convection zone thins and ted at the base decreases (see Figs. 3 and 4). Rapid convection holds over the bulk of the convection zone for masses &1M⊙. However, the assumption that the convection is efficient starts to break down at 1.4–1.5M⊙, since tth ∼ ted at the base (see Figs. 3 and 5). For the full range of stellar masses consid- ered here, we assume that s and ∆s are constant in convection zones. 5.2. Analytic Result for Thick Convection Zones In a fully convective star, the emergent luminosity is de- termined entirely by the surface boundary conditions. Under our assumption that ∆s is constant in the convection zone, the perturbed luminosity is likewise a function only of the bound- ary conditions. Stars of mass .1.3–1.4M⊙ have long thermal times (ωtth ≫ 1) at the top of the interior radiative region (see Fig. 5), so that the flux perturbation ∆F is approximately the “quasi-adiabatic” value, derived by ignoring ∆s ∝ (ωtth) −1 in eq. (A5). We assume efficient convection continues to just below the photosphere. At the photosphere, we adopt the usual Stefan-Boltzmann relation, F = σT 4, and the hydrostatic condition, pκ/A = 2/3, where A is the total acceleration defined in § 4, and 2/3 is the photospheric optical depth. Taking the photosphere to define the stellar surface, we compute the Lagrangian perturbations, , (30) = 0 . (31) Using s and p as our independent thermodynamic variables, we write ∆κ/κ = κad∆p/p +κs∆s/Cp and ∆T/T = ∆s/Cp + ∇ad∆p/p. In our numerical work (see § 5.4), we self- consistently compute the perturbation ∆A to the effective sur- face gravity, in order to follow resonant oscillations, where the equilibrium-tide result fails. However, we are now addressing non-resonant forcing, for which we use the equilibrium-tide approximation at the surface, giving ∆A/g = −∂ξr/∂r (see § 4). We now have κs∆s/Cp +∂ξr/∂r 1 +κad , (32) and upon substitution, 1 +κad −∇adκs 1 +κad 1 +κad . (33) Equation (33) differs from Goldreich & Wu (1999a) in that we retain the gravity perturbation in eq. (31), whereas they consider a constant-gravity atmosphere (and no tidal pertur- bation). For g-modes in white dwarfs, the interesting region is near the surface and the motion is mainly horizontal, so that ∆A = ∆g = 0 is a good approximation. Since the equilibrium tide has large vertical motions, the ∆A term must be retained. The luminosity change across the convection zone is de- rived from the entropy equation (eq. [A6]). If we ignore horizontal flux perturbations (set ℓ = 0 in eq. [A6]) and en- ergy generation, the equation for the luminosity perturbation ∆L/L = 2ξr/r +∆F/F is d(∆L/L) = iωT∆s/L . (34) Integrating over the convection zone with constant ∆s, we obtain ∆Lbcz = iω∆s dMrT/L , (35) where the subscript “ph” refers to the photosphere. We de- fine tcz = Cp,ph dMrT/L to be the mean thermal time of the convection zone, so that the right-hand side of eq. (35) is iωtcz∆s/Cp,ph. Figure 5 shows that the thermal time at the base of the con- vection zone (of order tcz) for M . 1.3M⊙ is orders of magni- tude longer than the forcing periods of 1–10 days. Insofar as |∆L|/L ∼ |ξr|/r at any location in the star (i.e., if resonances are neglected), we see that (|ξr|/r) −1|∆s|/Cp ∼ (ωtcz) −1 ≪ 1 in stars with deep convective envelopes. In this limit, eq. (33) becomes 1 +κad . (36) If we had set ∆A = 0, the amplitude of the photospheric flux perturbation would have been ∼|∆s|/Cp rather than the much larger value ∼|ξr|/R. Photospheric flux perturbations in tidally forced solar- type stars with thick convective envelopes arise mainly from changes in the local effective gravity. This statement is remi- niscent of, but physically distinct from, von Zeipel’s theorem (eqs. [21] and [22]). We have recovered our equilibrium-tide scaling, ∆Fℓ/F = −λℓξr,ℓ/R, where eq. (36) gives λℓ = 4(ℓ+ 2) 1 +κad . (37) For M = 1.0–1.4M⊙, we find λ2 ≃ 1.9–1.1. These estimates neglect resonant excitation of g-modes, a point addressed in § 5.4. 5.3. Analytic Result for Radiative Envelopes As the stellar mass increases beyond 1.4M⊙, the outer convective region thins and sits close to the surface, where ted ∼ tth. Figure 3 shows that the 1.6M⊙ model has two thin, inefficient surface convection zones, as well as a convective core. Radiative energy transport is important throughout the envelopes of these more massive stars. We now consider the idealized case of a completely radiative envelope, and obtain an analytic approximation for ∆L/L at the surface. Near the surface of a radiative star, we have Hp/r ≪ 1, 4πr3ρ/Mr ≪ 1, and ω 2r/g ≪ 1 for 2π/ω = 1–10 days. Under these conditions, the quasi-adiabatic luminosity perturbation becomes (e.g., Unno et al. 1989) ∆Lqad gk2hHp ξr − ξr,eq 8 PFAHL, ARRAS, & PAXTON where ζ = κad − 4∇ad + d lnT , (39) and ξr,eq is the equilibrium-tide radial displacement (eq. [12]). Nonzero values of ∆p/p and (ξr − ξr,eq)/Hp indicate devia- tions from hydrostatic equilibrium. Care must be taken with these terms, because the denominators p and Hp become very small close to the surface. With the help of the Appendix, we define the variables α= y1 − y2 + y3 = − , (40) β = y2 + ξr − ξr,eq , (41) which satisfy the differential equations d lnρ β + (ℓ+ 4) , (42) . (43) When ω2 ≪ gk2hHp, these equations produce the g-mode dispersion relation k2r = k 2/ω2 (in the limit k2r/k h ≪ 1) for radial wavenumber kr. For these propagating waves, the surface amplitudes of α and β are determined at the core radiative-convective boundary, where g-modes are driven (e.g., Goldreich & Nicholson 1989). On the other hand, when ω2 ≫ gk2hHp, the g-modes are evanescent (see Unno et al. 1989) and we neglect the term gk2hβ/ω 2 in eq. (42). This limit yields the approximate solution α≃ −(ℓ+ 4)(HpU/gR2), or ∆p/p ≃ (4 + ℓ)U/gR. In this case, ∆p/p is not small compared to the fractional fluid displacement, and thus the equilibrium-tide approximation loses validity. From our stellar models, we find that the evanescent regime corresponds to forcing periods of .4–8 days for M = 1– 1.6M⊙, most of the range of interest. The high-frequency limit of eq. (38) is ∆Lqad ≃ −ζ(ℓ+ 4) . (44) This relation should be evaluated at the layer where ωtth ≃ 1, above which the luminosity effectively “freezes out.” Fig- ure 6 shows the quasi-adiabic flux perturbation ∆Fqad/F = ∆Lqad/L−2ξr/R, evaluated whereωtth = 1, for a range of forc- ing periods and M = 1.5–1.7M⊙. Note that |∆F/F| can be an order of magnitude larger than |U |/gR, because of the rather large values of |ζ|(ℓ+ 4) for ℓ ≥ 2. Much larger perturbations are possible when g-modes are resonantly excited in a radia- tive star, as we discuss in the next section. We must point out that the quasi-adiabatic approximation is technically inappropriate when ωtth ∼ 1. Equation (44) should be viewed as an estimate of the modulus of the luminosity per- turbation at the surface. If, for instance, |∆s|/Cp & |U |/gR where ωtth ∼ 1, then ∆L/L at the surface will have a substan- tial imaginary part (see eq. [34]). This is what we find in the numerical calculations summarized in the next section. 5.4. Numerical Examples Here we show solutions of the perturbed mass, momentum, and energy equations that describe linear, nonadiabatic oscil- lations of a star subject to a varying tidal force. The equations FIG. 6.— Ratio of surface Lagrangian flux perturbation ∆F/F to equilibrium-tide displacement −U/gR for a range of forcing periods in the limit where surface g-modes are evanescent. The flux is evaluated at the loca- tion where ωtth = 1. Dashed, solid, and dotted curves correspond to M = 1.5, 1.6, and 1.7M⊙ , respectively. listed in the Appendix are the same as in Unno et al. (1989) for radiative regions, but augmented to include the tidal ac- celeration. In convection zones, we apply the prescription ∆s = constant based on our conclusions in § 5.1. Figure 7 summarizes how the interiors of 1M⊙ and 1.6M⊙ stars re- spond to resonant and non-resonant tidal forcing. The tidal potential has been scaled so that ξr/R = 1 corresponds to the equilibrium-tide surface displacement. For our 1M⊙ model, the non-resonant response to a forc- ing period of ≃3 days is shown in Fig. 7a. We see that ξr/R matches the equilibrium-tide result at the surface; the imag- inary piece is completely negligible. We also find that our approximation for ∆F/F at the surface (eq. [36]) works very well. A factor of ∼10 decay in |∆L|/L occurred in order for |∆F |/F ∼ ξr/R at the surface. Variation of ∆s/Cp in the con- vection zone (log[p/(GM2/R4)]> −2.5) is due to changes in Cp. In the radiative interior, the oscillations are caused by most nearly resonant g-modes, whose amplitudes rise rapidly as the core is approached, due to conservation of wave lumi- nosity. We have checked that the quasi-adiabatic approxima- tion of ∆L/L is valid in the radiative region; the ratio of the real and imaginary parts is found to be roughly constant for the ingoing gravity-wave (see also Zahn 1975). In order to model the resonant response of a 1M⊙ star, we tuned the forcing period to ≃1 day (see Figs. 7b and 8). At the surface, both ξr and ∆L have dominant imaginary parts, due to the short radial wavelength of the g-mode compared to the equilibrium-tide fluid displacement. The entropy at the base of the convection zone is very strongly perturbed in compari- son to the non-resonant case, but ∆L is still damped by orders of magnitude as the surface is approached. Figure 8 shows the surface values of the complex modu- lus and phase of ξr/R and ∆F/F versus forcing period. The phase is tan−1(Imaginary/Real) ∈ (−π,π). Solid lines connect points halfway between g-mode resonant periods. We find ELLIPSOIDAL OSCILLATIONS 9 FIG. 7.— Responses of tidally forced 1M⊙ and 1.6M⊙ main-sequence stars. Black, red, blue, and green curves denote, respectively, the logarithms of ξr/r, (δp + ρδϕ)/ρgr, ∆s/Cp, and ∆L/L. Solid (dashed) curves show the real (imaginary) parts. The tidal potential has been scaled so that ξr/R = 1 corresponds to the equilibrium-tide value. The four panels show the following: (a) non-resonant response of a 1M⊙ star tidally forced at a period of 2π/ω ≃ 2.91day; (b) resonant response of a 1M⊙ star with 2π/ω ≃ 1.00day; (c) non-resonant response of a 1.6M⊙ star with 2π/ω ≃ 3.00day; (d) resonant response of a 1.6M⊙ star with 2π/ω ≃ 1.02day that the equilibrium-tide approximation given by eqs. (12) and (36) is excellent for non-resonant forcing. Dashed curves give the maximum and minimum values that occur on resonance. One example of a resonance is shown in the insets. Resonant forcing at periods of <2 days yields surface values of ξr/R and ∆F/F that differ substantially from the equilibrium-tide results. However, the ratio of resonance width to the spac- ing between adjacent resonances is ∼10−4, making resonant forcing very unlikely. It is noteworthy that at forcing periods of >2 days, the equilibrium-tide result holds extremely well even when precisely on a resonance. As explained by Zahn (1975), the resonant response can be considered as the sum of the equilibrium tide and the most nearly resonant wave. As the period increases, the g-mode radial wavelength decreases, resulting in a reduction of the overlap integral for the mode and the tidal force, which in turn gives a decreased amplitude 10 PFAHL, ARRAS, & PAXTON FIG. 8.— Surface radial displacement and Lagrangian flux perturbation versus forcing period for a 1.0M⊙ star. Solid lines connect points halfway be- tween resonant g-mode periods, while dashed curves give the maximum and minimum values found on resonance. The equilibrium-tide approximation is extremely good, except when the forcing period is <2 days and resonant. of the wave component relative to the equilibrium tide. The non-resonant response of the 1.6M⊙ star is shown in Fig. 7c. We see that the equilibrium-tide result pro- vides a good match to ξr/R. Our estimate for the modu- lus of the radiative luminosity perturbation in the evanes- cent limit (eq. [44]) agrees reasonably well with what is in Fig. 7c. We also see that ∆L/L does roughly “freeze-out” when ωtth ≃ 1, just below the base of the convection zone at log p/(GM2/R4) ≃ −9 (see Fig. 3). Our expectations in § 5.3 regarding the imaginary part of ∆L/L are borne out in Fig. 7c A resonantly excited 1.6M⊙ star exhibits huge surface flux perturbations, radial displacements, and phase lags, as seen in Fig. 7d. In Fig. 9, surface values of |ξr|/R and |∆F |/F are plotted as a function of forcing period, where we have taken care to resolve resonances. Resonant amplitudes vary non- monotonically with period, in contrast to the smooth behavior of the 1M⊙ star (Fig. 8). Although we do not show the re- sults here, similar plots for masses between 1M⊙ and 1.6M⊙ show progressively more structure as the mass increases. The cause of this irregularity is not clear, but may have to do with the two thin surface convection zones changing the overlap of successive g-modes with the tidal force. 6. SUMMARY We have investigated in detail the ellipsoidal oscillations of 0.9–1.6M⊙ main-sequence stars induced by substellar com- panions. Classical models of ellipsoidal variability (e.g., Wilson 1994) are built on the assumption of hydrostatic bal- ance in a frame corotating with the binary orbit. This ap- proach is justified in the context of short-period (Porb . 10 days) binaries containing two stars of comparable mass, where tidal dissipation circularizes the orbits and synchro- nizes the stellar spins with the orbital frequency. However, when the companion has a very low mass, we cannot assume that the binary is in complete tidal equilibrium; in fact, this FIG. 9.— Surface radial displacement and Lagrangian flux perturbation versus forcing period for a 1.6M⊙ star. Curves connect evenly spaced points away from resonances, with finer spacing near resonance periods. state may be unattainable (see § 2). In this case, one must, in general, appeal to a dynamical description of the tidal inter- action. A substellar companion with Porb & 1 day raises tides on the star that are a small fraction of the stellar radius (see eq. [1]), permitting a linear analysis of the stellar oscillations. While the root of our study is a dynamical treatment of stellar tidal perturbations, the equilibrium-tide approximation does have an important realm of validity (see below). For this reason, we derived in § 3 a general expression (eq. [16]) for the measurable flux variation of a star that remains in hydrostatic equilibrium under the influence of a small exter- nal tidal force. This formula (1) assumes that the local per- turbation to the energy flux at the stellar surface is propor- tional to and in phase with the equilibrium-tide radial fluid displacement at each angular order ℓ (eq. [12]), (2) neglects stellar rotation, and (3) applies to inclined and eccentric or- bits. As expected, the fractional amplitude of the modulation is ∼ε≡ (Mp/M)(R/a) 3 for small eccentricities and I = 90◦, or ∼10−5(Mp/MJ)(Porb/1day) −2 for a star like the Sun (see § 2). A common practice is to use von Zeipel’s theorem when computing the surface radiative flux from a tidally distorted star (see § 4). The theorem assumes that the star is in hydro- static equilibrium and that the energy transport in the outer layers is purely by radiative diffusion. As already mentioned, the hydrostatic assumption is technically unjustified for sub- stellar perturbers. Moreover, the majority of Kepler targets will be main-sequence stars with masses of <1.4M⊙, which have substantial surface convection zones. Evidently, von Zeipel’s theorem is an inappropriate starting point for the con- ditions of interest. Section 5.1 discusses heat transport in perturbed stars with convective envelopes. Heuristic arguments are used to de- velop a simple treatment of the perturbed convection zone in main-sequence stars of mass<1.6M⊙ with forcing periods of 1–10 days. We suggest that both the specific entropy s and its Lagrangian perturbation ∆s are spatially constant in convec- ELLIPSOIDAL OSCILLATIONS 11 tive regions, a model partly inspired by the ideas of Brickhill (1983, 1990). Using this prescription, we analytically compute in § 5.2 the perturbed flux at the photosphere of deeply convective stars (M . 1.4M⊙), where the thermal time scale at the base of the convection zone is much longer than the forcing period. We find that ∆s/Cp is negligible near the top of the convection zone, and that the photospheric flux perturbation is propor- tional to changes in the effective surface gravity. Thus, we re- cover the equilibrium-tide result, ∆F/F = −λℓξr/R, at the sur- face, where λℓ depends on the adiabatic derivatives of opacity and temperature with respect to pressure (see eq. [37]). Nu- merical solutions of the equations of linear, nonadiabatic stel- lar oscillations (see § 5.4 and Fig. 7a) corroborate our analytic estimates in the non-resonant regime. Resonant excitations of g-modes in the radiative stellar interior cause large departures from the equilibrium-tide approximation when the forcing pe- riod is <2 days (Figs. 7b and 8). However, the likelihood of being on a resonance is small, and at periods of >2 day the equilibrium-tide result holds for M ≃ 1M⊙ even with reso- nant forcing. Stars of mass &1.4M⊙ have thin, relatively inefficient sur- face convection zones. Thus, g-modes can propagate very close to the surface and produce large flux perturbations and fluid displacements. Analytic arguments in § 5.3 indicate that the surface flux perturbations in these stars have non-resonant amplitudes of ∼10ε (eq. [44] and Fig. 6), in rough agree- ment with our numerical calculations (Fig. 7c). As seen in Figs. 7d and 9, a resonantly forced 1.6M⊙ star can exhibit flux perturbation amplitudes of >100ε at forcing periods of ≃1 day. While the amplitudes are not as extreme at longer periods, their dependence on period is rather erratic (Fig. 9), an issue that deserves further study. It will be difficult to de- rive physical interpretations from the ellipsoidal variability of these more massive stars. 7. DETECTION PROSPECTS The dominant sources of periodic variability of a star with a substellar companion are transit occultations (when |cos I| < [R + Rp]/a), Doppler flux modulations, reflection of starlight from the companion, and ellipsoidal oscillations. For each of these signals, Table 2 lists the characteristic amplitude, period with the largest power in the Fourier spectrum, and or- bital phase(s) at which the light is a maximum or minimum. The transit contribution is included for completeness, but its duration is sufficiently short—a fraction ≃(R + Rp)/(πa) of Porb—that it should often be possible to excise it from the data (see Sirko & Paczyński 2003). Of the remaining signals, the Doppler variability is the simplest, being purely sinusoidal with period Porb when the orbit is circular. The dominant ℓ = 2 piece of the equilibrium-tide approximation to the ellipsoidal variability (see eqs. [16] and [17]) is also sinusoidal when e = 0, but with period Porb/2. Reflection is more problem- atic, as its time dependence is generally not sinusoidal and not known a priori. If the companion scatters light as a Lambert sphere (e.g., Seager et al. 2000), the Fourier spectrum of the reflection variability has finite amplitude at all harmonics of the orbital frequency Ω, but the amplitude at 2Ω is roughly 1/5 of the amplitude at Ω. Therefore, the reflection and ellipsoidal vari- ability amplitudes may be similar at a frequency of 2Ω when α = 0.1, Mp ∼ MJ , and Porb ≃ 1 day. Also, the orbital phase at which the reflected light is a maximum is distinct from both the Doppler and ellipsoidal cases, further distinguishing the signals. However, Lambert scattering is probably never appropriate in real planetary atmospheres. Infrared reemis- sion of absorbed optical light, multiple photon scattering, and anisotropic scattering typically conspire to narrow the peak in the reflection lightcurve and lower the albedo, decreasing the prominence of the reflection signal. These issues are sensi- tive to the atmospheric chemistry and the uncertain details in models of irradiated giant planets. For reasonable choices re- garding the atmospheric composition, calculated optical albe- dos of Jovian planets range from<0.01 to ≃0.5 (Seager et al. 2000; Sudarsky et al. 2000). Recent photometric observations of HD 209458, the star hosting the first-detected transiting gi- ant planet (Porb ≃ 3.5 days), constrain the planetary albedo to be <0.25 (Rowe et al. 2006). Detailed lightcurve simulations will be required to say how well the different periodic signals can be extracted from the data. This is beyond the scope of the current study. We now do the simpler exercise of isolating the ellipsoidal modulations and assessing when this effect alone should be detectable. For a star of apparent visual magnitude V and an integration time of T = 6hr, Kepler’s photon shot noise is7 ∼ 10−5100.2(V −12) )−1/2 . (45) Instrumental noise should contribute at a similar level (e.g., Koch et al. 2006). If the data is folded at the orbital period and binned in time intervals T ≪ Porb, the shot noise is sup- pressed by a factor of ∼n orb , where norb is the number of folded cycles. After folding 1 year of continuous photomet- ric data using T = 6 hr, a star with V < 12 orbited by a giant planet with Porb . 3 days may have a fractional shot noise per time bin of .10−6. This is less than the ellipsoidal amplitude, (δF/F )ell, when I is not too small. The actual situation is not so simple when the data spans of weeks or months, because the intrinsic stochastic variability of the star will not have a white-noise power spectrum. Over times of .1 day, the Sun shows variability of (δF/F )int ∼ 10−5, but the amplitude rises steeply between ∼1 and 10 days to ∼10−3. Intrinsic variability tends to be large near the rota- tion period of the star, due mainly to starspots. Low-frequency variability may not too damaging for the study of ellipsoidal oscillations induced by planets with Porb . 3 days, but more study is needed. Kepler’s target list will contain ≃105 main-sequence FGK stars with V = 8–14. The statistics of known exoplanets in- dicate that 1–2% of all such stars host a giant planet (Mp & MJ) with Porb < 10 days (e.g., Marcy et al. 2005). Of these “hot Jupiters,” ≃30% have Porb = 1–3 days. It seems that a maximum of ∼103 Kepler stars will have detectable ellip- soidal modulations. If we neglect intrinsic stellar variabil- ity and consider only shot noise, then many systems with Porb . 3 days and V < 14 will have signal-to-noise S/N > 1 after ∼100 cycles are monitored; this may amount to >100 stars. Obviously, the number drops when we place higher de- mands on S/N and include the intrinsic variability. The results depend critically on the distributions of Mp and Porb. In order to better estimate the number of stars with poten- tially detectable ellipsoidal oscillation, we perform a simple population synthesis calculation. Denote the set of star-planet 7 An integration time of T = 6 hr is chosen for convenience; Kepler’s nom- inal exposure time is 30 min. Here we use the V -band flux as a reference, but, in fact, the Kepler bandpass is 430–890 nm, which spans B, V , and R colors. 12 PFAHL, ARRAS, & PAXTON TABLE 2 PERIODIC FLUX MODULATIONS Variability Dominant Phase at Source Amplitudea,b Harmonic Maximum/Minimumc References Ellipsoidald . . . . . . . 2× 10−5mpmP−21 sin 2 I Porb/2 0.25(0.75)/0.00(0.50) · · · Dopplere . . . . . . . . . . 3× 10−6mpm−2/3P 1 sin I Porb 0.25/0.75 1 Reflectionf . . . . . . . . 6× 10−5(α/0.1)m−2/3P 1 sin I Porb 0.50/0.00 2,3 Transit . . . . . . . . . . . 10−2m−2 Porb · · · /0.00 4 REFERENCES. — (1) Loeb & Gaudi 2003; (2) Seager, Whitney, & Sasselov 2000; (3) Sudarsky, Burrows, & Pinto 2000; (4) Seager & Mallén-Ornelas 2003 a We assume that the orbit is circular in our estimates. b The dimensionless variables used are mp = Mp/(10−3 M⊙), m = M/M⊙ and P1 = Porb/1day. We have as- sumed that the star and companion have respective radii of R/R⊙ = m and 0.1R⊙. c The phase is in the range 0–1, where at phase 0 the planet is closest to the observer. d Only the ℓ = 2 component of eq. (16), with λ2 = 2, is considered here. e We approximate the amplitude as 4vr/c, where vr is the reflex speed of the star along the line of sight, and the factor of 4 is approximately what one obtains for a V -band spectrum similar to the Sun. f Here α is the geometric albedo of the companion. The inclination dependence is an approximation for I ≃ 90◦ and the Lambert phase function. system parameters by P = {M,Mp,Porb, I}, and let f (P)dP be the probability of having a system in the 4-dimensional vol- ume dP. We assume that the planetary orbits are circular and obtain (δF/F )ell from the equilibrium-tide estimate in Ta- ble 2. Given the mass of the star, we compute its absolute V magnitude on the main-sequence using the approximation (see also Henry & McCarthy 1993) MV = 4.8 − 10.3log(M/M⊙) , (46) which is in accord with the usual mass-luminosity relation log(L/L⊙) ≃ 4log(M/M⊙) for M ≃ 1M⊙. With a maximum apparent magnitude of Vmax = 14 for the Kepler targets, the maximum distance of the star is Dmax = 10 1+0.2(14−MV ) pc . (47) With a certain signal-to-noise threshold (S/N)min, there is a maximum distance Dd < Dmax to which the ellipsoidal vari- ability is detectable. For a spatially uniform population, the detectable fraction of systems is (Dd/Dmax) 3. Thus, the net detectable fraction among all systems is dP f (P) , (48) an integral over all relevant P space. When the only noise is intrinsic to the star, N = (δF/F )int and S/N is independent of distance, so that Dd,int = Dmax when S/N > (S/N)min, and Dd,int = 0 otherwise. In the case of pure shot noise, there is a maximum magnitude Vd for which the ellipsoidal oscillations are detectable: Vd = 5log (δF/F )ell χ (S/N)min , (49) where χ ∼ 10−8.4(T6n100) −1/2 is the value of (δF/F )shot for V = 0, T = 6T6 hr, and norb = 100n100. The corresponding dis- tance is given by log[Dd,shot/10pc] = 0.2(Vd − MV ) if Vd < Vmax, and is Dd,shot = Dmax when Vd >Vmax. We take the max- imum detectable distance to be Dd = min{Dd,int,Dd,shot}. At this point the simplest approach is to assume that the parameters {M,Mp,Porb, I} are statistically independent and carry out a Monte Carlo integration to obtain E . To this end, we draw M from the Kroupa et al. (1993) initial mass TABLE 3 NUMBER OF Kepler STARS WITH DETECTABLE ELLIPSOIDAL OSCILLATIONS (S/N)min = 1 (S/N)min = 3 (S/N)min = 5 y x = 1 2 x = 1 2 x = 1 2 1 240 166 76 35 33 13 0 99 62 26 12 11 4 -1 33 19 7 3 2 1 function in the range of 0.5–1.5M⊙. The planetary mass is chosen from the distribution f (Mp) ∝ M p for Mp = 1–10MJ. Marcy et al. (2005) find that x ≃ 1 when considering all de- tected planets; the shape of f (Mp) is not well constrained at Porb < 10 days. We let x = 1 and 2. We adopt f (Porb) ∝ P over 1–10 days. Multiplying the resulting value of E by 1000 provides a crude estimate of the actual number of Kepler tar- gets with detectable ellipsoidal variability. No single value of y is consistent with the data, and so we consider the reason- able range y = −1, 0, and +1. Inclinations are chosen under the assumption that the orbits are randomly oriented, such that f (cos I) = 1/2 for I ∈ (0,π). Our calculations use fixed values of (δF/F )int = 10 −5 and T6 = n100 = 1. Results of our Monte Carlo integrations are shown in Ta- ble 3 as actual numbers of Kepler targets. The largest number of detectable systems is obtained when x = y = 1, parameters that yield the largest proportions short periods and massive planets. We expect that ∼10–100 Kepler stars may exhibit ellipsoidal oscillations with S/N & 5. A handful of systems might have S/N & 10. Higher harmonics from the ℓ = 3 com- ponents of eq. (16) or modest eccentricities might be accessi- ble for at most a few stars. Our integrations also check for cases where the planet is transiting. As (S/N)min increases from 1 to 5, the fraction of systems in Table 3 with |cos I| < (R + Rp)/a runs from ≃30% to ≃50%, with a weak dependence on x and y. Such significant fractions stand to reason, since systems with the shortest periods have the highest ellipsoidal amplitudes and transit probabilities. Transit measurements directly give Porb, sin I & 0.95 (for Porb & 1 day), and (Rp/R) 2. The planet mass Mp can be determined with the addition of spectroscopic ra- ELLIPSOIDAL OSCILLATIONS 13 dial velocity measurements, which should be possible for most of the Kepler targets with detectable ellipsoidal oscil- lations. The ellipsoidal amplitude then depends on the un- measured stellar mass and radius via ε∝ R3/M2 (eq. [1]), as well as the stellar photospheric conditions (eq. [36]). If M and R are obtained from stellar models, ellipsoidal variabil- ity may provide an interesting consistency check on all the system parameters, as well as test the theory of forced stellar oscillations. As a last point, we emphasize that stars of mass &1.4M⊙ may have typical ellipsoidal amplitudes of ∼10ε. However, such stars will also be younger than most Kepler targets and probably have intrinsic variability ≫10−5. We carried out Monte Carlo integrations with M = 1.4–1.6M⊙, (δF/F )ell = 10εsin2 I, and x = y = 1. As we vary (δF/F )int from 10 to 10−4, E decreases from large values of ≃0.4 to a small fraction of ≃0.03 for (S/N)min = 10. Unfortunately, we do not know how many such stars will be included in the Kepler target list. Also, there has not yet been a discovery of a giant planet with Porb < 10 days around a star of mass ≥1.4M⊙, but exoplanet surveys tend to exclude these more massive stars. We thank Tim Brown for general discussions and address- ing Kepler questions, Jørgen Christensen-Dalsgaard for guid- ance on stellar luminosity perturbations, and Mike Muno for advice on signal processing. This work was supported by NSF grant PHY05-51164. APPENDIX OSCILLATION EQUATIONS Here we list the nonadiabatic, linearized fluid equations that we solve numerically. The reader is referred to Unno et al. (1989) for a complete discussion. Scalar and vector quantities are expanded in spherical harmonics Yℓm and poloidal vector harmonics, respectively. The momentum, mass, and energy equations are written in terms of the dimensionless variables y1 = ξr/r, y2 = (δp/ρ + δϕ)/gr, y3 = δϕ/gr, y4 = g−1dδϕ/dr, y5 = ∆s/Cp, and y6 = ∆L/L. Here L is the total (radiative plus convective) luminosity. The radial flux perturbation is ∆F/F = ∆L/L − 2ξr/r. In radiative zones, the nonadiabatic equations are d lnr gk2hr − y5ρs + U , (A1) d lnr ω2 − N2 1 − η+ −ρsy5 − , (A2) d lnr = y3 (1 − η)+ y4 , (A3) d lnr = y1η + y2η ℓ(ℓ+ 1) − η − y4η + y5ρsη , (A4) d lnr + 4(∇−∇ad) + c2 (∇ad −∇) gk2hr c2 + y4 ∇ad + y5 ∇ (4 −κs) − y6 dU/dr , (A5) d lnr = y1ℓ(ℓ+ 1) − y2ℓ(ℓ+ 1) + y3ℓ(ℓ+ 1) 4πr3ρCpT ℓ(ℓ+ 1) , (A6) where cs is the sound speed, η = d lnMr/d lnr, c2 = (r/Hp)∇(κad −4∇ad)+∇ad(d ln∇ad/d lnr+r/Hp), and we have ignored energy generation terms. Note that the tidal acceleration −∇U has been added to the momentum equations. In convection zones, we ignore turbulent viscosity effects and replace the radiative diffusion equation (eq. [A5]) with the prescription ∆s = constant (see § 5.1), or more precisely = 0 . (A7) Equation (A6) still involves the total (convective plus radiative) luminosity. We ignore energy generation and horizontal flux perturbation terms, i.e. we ignore all terms with spherical harmonic index ℓ in eq. (A6) in convection zones. At the center of the star, we require the solutions to be finite, and also set ∆s = 0. At the surface, we set δp = ρgξr and we require δϕ to decrease outward. This boundary condition is only approximate, as g-modes may propagate above the convection zone for wave periods of &4days in our 1M⊙ model. The final surface boundary condition is given by eq. (31). Care must be used in the radiative zone just below the photosphere, since the entropy perturbation is far from the quasi-adiabatic value. If we solve the radiative diffusion equation in this region, we find that the entropy increases by ∼10 orders of magnitude in just a few grid points. However, we regard this behavior as unphysical, because the region at the top of the convection zone is optically thin. 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Hundreds of substellar companions to solar-type stars will be discovered with the Kepler satellite. Kepler's extreme photometric precision gives access to low-amplitude stellar variability contributed by a variety of physical processes. We discuss in detail the periodic flux modulations arising from the tidal force on the star due to a substellar companion. An analytic expression for the variability is derived in the equilibrium-tide approximation. We demonstrate analytically and through numerical solutions of the linear, nonadiabatic stellar oscillation equations that the equilibrium-tide formula works extremely well for stars of mass <1.4 Msun with thick surface convection zones. More massive stars with largely radiative envelopes do not conform to the equilibrium-tide approximation and can exhibit flux variations $\ga$10 times larger than naive estimates. Over the full range of stellar masses considered, we treat the oscillatory response of the convection zone by adapting a prescription that A. J. Brickhill developed for pulsating white dwarfs. Compared to other sources of periodic variability, the ellipsoidal lightcurve has a distinct dependence on time and system parameters. We suggest that ellipsoidal oscillations induced by giant planets may be detectable from as many as ~100 of the 10^5 Kepler target stars. (Abridged)

DRAFT VERSION NOVEMBER 1, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 ELLIPSOIDAL OSCILLATIONS INDUCED BY SUBSTELLAR COMPANIONS: A PROSPECT FOR THE KEPLER MISSION ERIC PFAHL1, PHIL ARRAS1,2 , AND BILL PAXTON1 Draft version November 1, 2018 ABSTRACT Hundreds of substellar companions to solar-type stars will be discovered with the Kepler satellite. Kepler’s extreme photometric precision gives access to low-amplitude stellar variability contributed by a variety of physical processes. We discuss in detail the periodic flux modulations arising from the tidal force on the star due to a substellar companion. An analytic expression for the variability is derived in the equilibrium-tide approximation. We demonstrate analytically and through numerical solutions of the linear, nonadiabatic stellar oscillation equations that the equilibrium-tide formula works extremely well for stars of mass <1.4M⊙ with thick surface convection zones. More massive stars with largely radiative envelopes do not conform to the equilibrium-tide approximation and can exhibit flux variations &10 times larger than naive estimates. Over the full range of stellar masses considered, we treat the oscillatory response of the convection zone by adapting a prescription that A. J. Brickhill developed for pulsating white dwarfs. Compared to other sources of periodic variability, the ellipsoidal lightcurve has a distinct dependence on time and system parameters. We suggest that ellipsoidal oscillations induced by giant planets may be detectable from as many as ∼100 of the 105 Kepler target stars. For the subset of these stars that show transits and have radial-velocity measurements, all system parameters are well constrained, and measurement of ellipsoidal variation provides a consistency check, as well as a test of the theory of forced stellar oscillations in a challenging regime. Subject headings: planetary systems — stars: oscillations — techniques: photometric 1. INTRODUCTION The upcoming Kepler3 satellite will continuously monitor ∼105 main-sequence stars of mass ≃0.5–1.5M⊙ over 4–6 years with fractional photometric precisions of ∼10−5. Such high sensitivity, which is unattainable from the ground, will allow for the robust detection of Earth-size planets that transit their host stars, and the measurement of asteroseismic oscilla- tions as a probe of stellar structure (e.g., Borucki et al. 2004; Basri et al. 2005). These missions will also discover hundreds of “hot Jupiters” with orbital periods of <10 days, revealed by their transits or reflected starlight (e.g., Jenkins & Doyle 2003). Continuous observations of these systems are likely to show a myriad of novel physical effects, including Doppler flux variability of the host stars (Loeb & Gaudi 2003), pho- tometric dips due to moons or rings around the planets (Sartoretti & Schneider 1999; Brown et al. 2001), and the impact of additional perturbing planets on transit timing (Miralda-Escudé 2002; Agol et al. 2005; Holman & Murray 2005). The same ideas apply if the companion is a more mas- sive brown dwarf, but these are rarely found in close orbits around solar-type stars (e.g., Grether & Lineweaver 2006). Here we scrutinize another mechanism for generating peri- odic variability of a star closely orbited by a giant planet or brown dwarf. A star subject to the tidal gravity of a binary companion has a nonspherical shape and surface-brightness distribution. In the simplest approximation, the stellar surface is a prolate ellipsoid with its long axis on the line connecting the two objects. As the tidal bulge tracks the orbital motion, differing amounts of light reach the observer. For a solar-type 1 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106; pfahl@kitp.ucsb.edu, paxton@kitp.ucsb.edu 2 Department of Astronomy, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904-4325; arras@virginia.edu 3 http://kepler.nasa.gov star orbited by a perturbing companion of mass Mp with pe- riod Porb, the expected fractional amplitude of this ellipsoidal variability is ∼10−2(Mp/M⊙)(1day/Porb) 2. This effect has a long history in the study of eclipsing binary stars (see the re- view by Wilson 1994), but was mentioned only recently in the exoplanet context. Udalski et al. (2002), Drake (2003), and Sirko & Paczyński (2003) noted that if ellipsoidal light variations are detected from the ground, where the fractional photometric precision is &10−3, then the perturber must be fairly massive (e.g., &0.1M⊙). They offered this idea as a test to distinguish be- tween planetary transits and eclipses by low-mass stars. The superior sensitivity of Kepler offers the possibility of mea- suring ellipsoidal variability induced by giant planets (Mp ∼ 10−3–10−2 M⊙) with orbital periods of .10 days. Loeb & Gaudi (2003) compare the ellipsoidal variability in- duced by a planetary companion to flux modulations arising from reflected starlight and the Doppler effect. The three am- plitudes are similar when the companion has an orbital period of .3 days and an optical albedo of .0.1. In a sufficiently long observation it should be possible to separately extract each of the signals, since their Fourier decompositions are dis- tinct. Precise physical modeling of the ellipsoidal lightcurve could provide an independent constraint on the mass of the companion, as well as important clues regarding stellar tidal interactions. Ellipsoidal variability is typically modeled under the as- sumption that the distorted star maintains hydrostatic balance and precisely fills a level surface of an appropriate poten- tial (e.g., the Roche potential). The measured flux is then just an integral of the intensity over the visible stellar sur- face, where the intensity includes the effects of limb darken- ing and gravity darkening (e.g., Kopal 1942). This approach is strictly valid only when the orbit is circular and the star rotates at the orbital frequency, so that a stationary configura- http://arxiv.org/abs/0704.1910v1 http://kepler.nasa.gov 2 PFAHL, ARRAS, & PAXTON tion exists in the coorbital frame. These conditions may not be satisfied when the companion has a low mass or long pe- riod, because of the weak tidal interaction. In fact, a state of tidal equilibrium may not be attainable in the case of a plan- etary companion (e.g., Rasio et al. 1996). Equilibrium mod- els of ellipsoidal lightcurves do have a realm of validity for noncircular orbits and asynchronously rototating stars, and have been applied successfully to somewhat eccentric bina- ries (e.g., Soszynski et al. 2004). However, by construction, such models ignore fluid inertia and the possibility exciting normal modes of oscillation, effects that may be of critical importance in a wide range of observationally relevant cir- cumstances. Here we apply the machinery of linear stellar oscillation theory to the weak tidal forcing of stars by sub- stellar companions. Conceptually, our investigation bridges Kepler’s planetary and astroseismology programs. Section 2 describes the geometry of the problem, provides quantitative measures for the strength of the tidal interaction, discusses our simplifying assumptions, and presents the math- ematical framework for calculating ellipsoidal variability. In § 3, we consider the equilibrium-tide approximation and de- rive an analytic expression for the ellipsoidal lightcurve. A brief review of von Zeipel’s theorem and its limitations is given in § 4. Tidally forced, nonadiabatic stellar oscillations are addressed in § 5, where we argue for a simple treatment of perturbed surface convection zones, use this prescription to calculate the ellipsoidal variability of deeply convective stars, estimate analytically the surface flux perturbation in mainly radiative stars, and show select numerical results. Our main conclusions are summarized in § 6. We conclude in § 7 with remarks on the measurement of ellipsoidal oscillations in the presence of other sources of periodic variability. 2. PRELIMINARIES Consider a star of mass M and radius R is orbited by a sub- stellar companion of mass Mp and radius Rp. We work in spherical coordinates (r,θ,φ) with the origin at the star’s cen- ter and the pole direction (θ = 0) parallel to the orbital angular momentum vector. The orbit is then described by (d,π/2,φp), where d and φp are, respectively, the time-dependent or- bital separation and true anomaly; φp = 0 marks the phase of periastron. We assume that the orbit is strictly Keple- rian with fixed semimajor axis a and eccentricity e, such that d = a(1 − e2)/(1 + ecosφp). The direction to the observer from the center of the star is (θo,φo), so that the conventional orbital inclination is I = π − θo. We imagine that the gravity of the companion raises nearly symmetrical tidal bulges on opposite sides of the star that ro- tate at the orbital frequency. A rough measure of both the height of the tides relative to the unperturbed stellar radius and the fractional amplitude of the ellipsoidal variability is given by the ratio of the tidal acceleration to the star’s surface gravity: ∼ 10−5 2.8 hr 1 day , (1) where MJ ≃ 10 −3 M⊙ is the mass of Jupiter, P∗ = 2π(R3/GM)1/2 = 2.8[(R/R⊙) 3(M⊙/M)] 1/2 hr is the dynami- cal time of the star. For main-sequence stars with R/R⊙ ≃ M/M⊙, we see that ε ∝ MpMP orb. The maximum value of ε is attained when the companion fills its Roche lobe at an orbital separation of a ≃ 2Rp(M/Mp) 1/3, which gives εmax ≃ ≃ 10−4 0.1R⊙ , (2) where we have applied a fixed value of Rp = 0.1R⊙, appro- priate for both giant planets and old brown dwarfs. Note that εmax ∼ 1 for massive brown dwarfs (Mp/MJ ∼ 80). Hereafter, we consider only cases with ε≪ 1. For orbital periods as short as ≃1 day, tidal torques on the star from a planetary companion are rather ineffective at alter- ing the stellar rotation rate (e.g., Rasio et al. 1996). Therefore, as already mentioned in § 1, we should not generally expect the star to rotate synchronously with the orbit, and so there is no frame in which the star appears static. This holds when the orbit is circular, and is obviously true when the there is a finite eccentricity. In fact, ≃30% of the known exoplanets4 with Porb < 10 days have eccentricities of >0.1. Small vari- able distortions of the star from its equilibrium state, due to a combination of asynchronous rotation and orbital eccentric- ity, should be viewed as waves excited by the tidal force of the companion. Our task is to study such tidally forced stel- lar oscillations in the linear domain in order to understand the corresponding lightcurves. In order to greatly simplify the mathematical description of the stellar oscillations, we assume that the star is nonrotating in the inertial frame. When the stellar rotation frequency is nonzero, but much smaller than the tidal forcing frequency, the effect of rotation is to introduce fine structure into the os- cillation frequency spectrum, and cause the oscillation eigen- functions to be slightly modified as a result of the Coriolis force (for a discussion, see Unno et al. 1989). Tidal pump- ing of a slowly rotating star by an orbiting companion has a dominant period of Porb/2—a few days in the cases of inter- est. By contrast, single solar-type stars with ages>1 Gyr tend to have rotation periods of >10 days (e.g., Skumanich 1972; Pace & Pasquini 2004); the Sun has an equatorial rotation pe- riod of ≃25 days. Slowly rotating stars with masses of ≃1M⊙ are prime targets for Kepler, since they exhibit low intrinsic variability. Based on this selection effect, and the inability of tidal torques to spin up the star, our assumption of vanishing stellar rotation seems generally justified. The general framework for calculating the measurable flux modulations associated with ellipsoidal stellar oscillations is as follows. We consider small perturbations to a spherical, nonrotating background stellar model, such that fluid ele- ments at equilibrium position x are displaced in a Lagrangian fashion to position x +ξ. Variations in the measured flux from an oscillating star arise from two physically distinct contribu- tions (e.g., Dziembowski 1977): (1) changes in the shape of the star due to radial fluid displacements ξr = ξ · er, where er is the radial unit vector, and (2) hot and cold spots generated by local Lagrangian perturbations ∆F to the heat flux. Our main task in §§ 3 and 5 is to compute ξr and ∆F according to the relevant physics. Given the dependences of ξr and ∆F on (r,θ,φ), it is straightforward to compute the time varying component of the measured flux. The flux5 received from a star at distance D is 4 http://vo.obspm.fr/exoplanetes/encyclo/encycl.html 5 Our calculations concern the bolometric flux, although is relatively http://vo.obspm.fr/exoplanetes/encyclo/encycl.html ELLIPSOIDAL OSCILLATIONS 3 (e.g., Robinson et al. 1982) dS n ·no F h(n ·no) (3) where dS is an area element at the stellar photosphere, F is the net flux of radiation out of the surface element, h is the limb-darkening function, n and no are unit vectors normal to the surface and toward the observer, respectively, and the in- tegration is over the visible stellar disk. Vertical displacement at the surface yields changes in F through changes in surface area and n · no. Following Dziembowski (1977), we expand ξr and ∆F in spherical harmonics, ξr(r,θ,φ, t) = ξr,ℓm(r, t)Yℓm(θ,φ) , (4) ∆F(r,θ,φ, t) = ∆Fℓm(r, t)Yℓm(θ,φ) , (5) and carry out the appropriate linear expansions to obtain the fractional variability (2bℓ − cℓ) ξor,ℓ . (6) Here ξor,ℓ and ∆F are components evaluated at the surface (r = R) and in the direction of the observer: ξor,ℓ ξr,ℓm(R, t) Yℓm(θo,φo) , (7) ∆Fℓm(R, t) Yℓm(θo,φo) . (8) The terms bℓ and cℓ are given by dµµPℓ h , cℓ = dµ(1 −µ2) where µ = n ·no, the Pℓ(µ) are ordinary Legendre polynomials, and h(µ) is normalized such that dµµh = 1. The linear limb-darkening function is h(µ) = (3 −γ) 1 −γ(1 −µ) ; (10) more general nonlinear functions of µ (e.g., Claret 2000) will not be considered here. The classical Eddington limb- darkening function is h = 1 + 3µ/2 (γ = 3/5; e.g., Mihalas 1970). Table 1 shows shows functional forms and particular values of bℓ and cℓ for ℓ = 2 and 3. 3. EQUILIBRIUM TIDE Vertical displacement of the stellar surface is often accu- rately modeled by assuming that the tidally perturbed fluid remains in hydrostatic balance. The cause and magnitude of the surface flux perturbation is a more complicated affair. In this section, we apply a simple parameterization of the flux perturbation and obtain a complete set of formulae for com- puting the ellipsoidal lightcurve. Subsequent sections provide more detailed calculations. In particular, we show in § 5.2 straightforward to modify the analysis for narrow-band measurements. TABLE 1 LIMB DARKENING PARAMETERS bℓ cℓ ℓ General γ γ = 3/5 General γ γ = 3/5 2 (1 +γ)/[20(3 −γ)] 13/40 3(1 + 3γ)[10(3 −γ)] 39/20 3 γ/[4(3 −γ)] 1/16 3γ/(3 −γ) 3/4 that stars with deep convective envelopes (the majority of Ke- pler targets) have surface flux variations that conform to the equilibrium-tide approximation. When the tidal forces on the stellar fluid change sufficiently slowly, the star can stay very nearly in hydrostatic equilib- rium. If the net acceleration required to balance the pressure gradient is derivable from a potential, then equilibrium im- plies that a fluid element remains on an equipotential surface. Since we neglect stellar rotation, there is no centrifugal force, and the total potential is the sum of the gravitational potential ϕ from the spherical background stellar model and the per- turbing tidal potential U ∼ εϕ≪ ϕ. For our analytic work, we neglect the modification of ϕ due to the tide. In general, the Eulerian variation δϕ should be added to U , as we do in our numerical models (see § 5.4 and the Appendix); we find that |δϕ|/|U | ∼ 10−2. In the absence of tidal forces, a given fluid element sits at equilibrium position x with total potential ϕ(x). Gentle inclu- sion of the tidal potential causes the fluid element to move to position x +ξ while preserving the value of the total potential. This is expressed mathematically by ϕ(x) =ϕ(x +ξ) +U(x +ξ, t) =ϕ(x) +ξ ·∇ϕ+U(x, t) +O(ξ2, ξU) . (11) We see that ξ ·∇ϕ = ξrg, where g = GMr/r 2 is the background gravitational acceleration at mass coordinate Mr. To first or- der, the radial displacement of the equilibrium tide is (see also Goldreich & Nicholson 1989) ξr(x, t)≃−U(x, t)/g , (12) which tells us the geometry of the star as a function of time. The tidal potential within the star can be expanded as U(r,θ,φ, t) = − Pℓ(cosψ) , (13) where cosψ = sinθ cos(φp −φ). There is no ℓ = 1 term, since this would give the acceleration of the star’s center of mass, which is already incorporated into the orbital dynamics. The angular expansion of ξr follows immediately from eq. (12): ξr(r,θ,φ, t) Pℓ(cosψ) . (14) In order to express U and ξr in spherical harmonics, we utilize the addition theorem, Pℓ(cosψ) = 2ℓ+ 1 ℓm(π/2,φp) Yℓm(θ,φ) , (15) where “∗” denotes the complex conjugate. Note that Yℓm(π/2,φp) is nonzero only when ℓ−m is even. For the dom- 4 PFAHL, ARRAS, & PAXTON inant ℓ = 2 components of U and ξr, the surface values of U/ϕ and ξr/R are ∼ε, as expected. From eqs. (7), (14), and (15), the components ξor,ℓ/R of the surface radial displacement toward the observer are immedi- ately apparent. As we will see in § 5, the computation of ∆F/F is, in general, rather technical. However, in the special case where the stellar fluid responds adiabatically to a slowly varying tidal potential, ∆Fℓ/F varies in phase with and in pro- portion to ξr,ℓ/r in the linear approximation of the equilibrium tide. Making this assumption, we write ∆Fℓ/F = −λℓξr,ℓ/R at the surface, where the λℓ are real constants that depend on the stellar structure (see § 5.1). We will see in § 4 that λℓ = ℓ+ 2 is a good first guess for radiative stars, and so we might gen- erally expect λℓ to be positive and O(ℓ). We now have the ingredients for the fractional variability (eq. [6]), and we obtain )ℓ−2( fℓ Pℓ(cosψo) , (16) where fℓ = (2−λℓ)bℓ−cℓ, and cosψo = sinθo cos(φp −φo). The ℓ = 2 and 3 Legendre polynomials can be expanded as P2(cosψo) = − (3cos2 I − 1) + 3sin2 I cos2(φp −φo) , (17) P3(cosψo) = sin I − 3(5cos2 I − 1)cos(φp −φo) + 5sin2 I cos3(φp −φo) , (18) where we have substituted θo = π − I. The Eddington limb- darkening formula gives (see Table 1) f2 = − , f3 = − . (19) It is important to note that f2 < 0 when λ2 ≥ 0 (see below). In eq. (16), the orbital dynamics are described by the evolu- tion of d and φp (see § 2). For a circular orbit, we have d = a and φp =Ωt, where Ω = 2π/Porb, and t is the time since perias- tron (modulo Porb). Example lightcurves with e = 0, γ = 3/5, λℓ = 0, and I = π/2 are shown in Fig. 1 for a/R = {2,4,8,16}. When R/a ≪ 1, the ℓ = 2 piece of δF/F is a good approxi- mation, and the temporal flux variation approaches a pure co- sine with angular frequency 2Ω (see eq. [17]). Because f2 < 0, the dominant ℓ = 2 component of the ellipsoidal variability has minimum light when tidal bulge is aligned with the direction to the observer. As R/a increases, so does the importance of ℓ > 2 terms and their extra harmonic content, as seen in eq. (18) and Fig. 1. Additional harmonics in δF/F also result from a finite eccentricity. At the O(e) level, signals with frequenciesΩ and 3Ω, and amplitudes of ∼εe, are present in the ℓ = 2 component of δF/F , which compete with the ℓ = 3 piece when e ∼ R/a. Notice that when e> 0 the flux is variable even when the orbit is viewed face-on (I = 0 or π), by virtue of changes in d−3 = 1+ 3ecosΩt +O(e2). For I = 0, we see that P3(cosψo) vanishes, leaving the largest contribution δF/F ≃ −1.5εe f2 cos(Ωt). 4. AN ASIDE ON VON ZEIPEL’S THEOREM FIG. 1.— Disk-averaged flux variation (eq. [16]) for an edge-on circu- lar orbit under the equilibrium-tide approximation (eqs. [6] and [14]) with ∆F/F = 0 at the surface. The four curves correspond to a/R = 2 (black), 4 (red), 8 (blue) and 16 (green). In order to compare the shapes of the curves, δF/F has been multiplied by (a/R)3(M/Mp). As a/R increases, higher har- monics decrease in strength and the lightcurve approaches a pure cosine with frequency 2/Porb. The tidal bulge closest to the companion points toward the observer at integer values of t/Porb. Our equilibrium calculation in the last section used the sim- ple prescription ∆Fℓ/F = −λℓξr,ℓ/R. There remains the ques- tion of what physics determines ∆F/F . A common practice in empirical studies of close eclipsing binaries—systems that tend to be nearly in tidal equilibrium—is to use some vari- ant of the von Zeipel (1924) theorem, which was originally formulated for purely radiative, strictly hydrostatic stars. In equilibrium, all the thermodynamic variables depend only on the local value of the total potential Φ. Thus, the radiative flux can be written as (e.g., Hansen & Kawaler 1994) ∇Φ , (20) where ρ is the mass density, T is the effective temperature, and κ(ρ,T ) is the opacity. Equation (20) is the essence of von Zeipel’s theorem, which says that the magnitude F of the radiative flux is proportional to the magnitude of the net ac- celeration A = |∇Φ|. When Φ = ϕ + U (see § 3), we obtain A = g + ∂U/∂r +O(ξ2), so that the Lagrangian flux perturba- tion about equilibrium is , (21) where ∆g/g = −2ξr/r, due to the change in radius at approxi- mately constant enclosed mass. Substituting the equilibrium- tide result U = −ξrg into eq. (21), we obtain the compact ex- pression ∆F/F = −∂ξr/∂r. Using eq. (14), we find = −(ℓ+ 2) , (22) from which we identify λℓ = ℓ+ 2. Although the application of von Zeipel’s theorem is instruc- tive, the underlying physical assumptions are inaccurate for ELLIPSOIDAL OSCILLATIONS 5 FIG. 2.— Important oscillations frequencies and time scales as a function of pressure for a 1M⊙ main-sequence star. The four curves show the Brunt- Väisällä frequency N (black), Lamb frequency Sl (red; ℓ = 2 is shown), in- verse thermal time t−1th (blue), and inverse eddy turnover time t ed (green). Large, real values of N occur in the radiative core and very near the photo- sphere, while N2 < 0 in the convective envelope. Gravity waves propagate only where the angular frequency is below both N and Lℓ. The two horizontal lines delimit the range of tidal forcing frequencies of interest here. slowly rotating main-sequence stars of mass 1.0–1.6M⊙ with tidal forcing periods of days. We are now led to investigate the general problem of forced nonadiabatic stellar oscillations. 5. FORCED NONADIABATIC OSCILLATIONS The equilibrium analysis ignores fluid inertia and the exci- tation of the star’s natural oscillation modes. While this as- sumption may be valid near the surface of the star, it does not hold deeper in the interior. Gravity waves (g-modes; re- stored by buoyancy) can propagate in the radiative interiors of Sun-like stars with a range of oscillation periods that in- cludes the tidal forcing periods of interest (.3 days). Tidal forcing of radiative regions may produce substantial devia- tions from hydrostatic balance, as well as large surface am- plitudes of ∆F/F , in particular if resonant oscillations are excited. This is especially relevant for main-sequence stars of mass M & 1.4–1.5M⊙ with mainly radiative envelopes. Less massive stars (M . 1.3–1.4M⊙) have rather deep convective envelopes that can block information about the dynamic inte- rior from being conveyed to the surface. Here we investigate each of these regimes with both analytic estimates and numer- ical models of oscillating stars. Our calculations employ realistic models of 0.9–1.6M⊙ main-sequence stars, constructed with the EZ stellar evolu- tion code (Paxton 2004), a distilled and rewritten version of the program originally created by Peter Eggleton. We adopt Solar metallicity and a convective mixing length of 1.6 times the pressure scale height. All stars are evolved to an age when the core hydrogen abundance has the Solar value of XH = 0.35. Models with 199 radial grid points are interpolated to yield &104 points in which the g-mode radial wavelength is well resolved in the core. Figures 2 and 3 illustrate some of the differences between FIG. 3.— Same as Fig. 2, but for a 1.6M⊙ main-sequence star. Note two geometrically thin, relatively inefficient (tth ∼ ted) convection zones near the surface. The spike in N near the center is at the edge of the convective core, and signals a steep gradient in the mean molecular weight. 1M⊙ and 1.6M⊙ stars, and serve to introduce several impor- tant physical quantities used in the remainder of this section. The Lamb frequency, Sℓ = [ℓ(ℓ+ 1)]1/2 , (23) is the inverse of the horizontal sound-crossing time scale, where cs is the sound speed, and [ℓ(ℓ + 1)]1/2/r ≡ kh is the horizontal wavenumber of the oscillation. For fixed chemical composition, the squared Brunt-Väisällä frequency is (∇ad −∇) , (24) where Hp = −(d ln p/dr)−1 = p/(ρg) is the pressure scale height, and ∇ = d lnT/d ln p is the temperature gradient6 (∇ad is the adiabatic value). Radiative regions have ∇ad −∇ > 0 (N2 > 0), and N represents the frequency of buoyancy oscilla- tions. In convection zones, ∇ad −∇< 0 and N2 < 0, indicating that g-modes are evanescent. When N2 < 0, the time scale ted ∼ |N| −1 , (25) approximates the turnover time of convective motions (for de- tails and modifications for radiative losses, see, e.g., Kippen- hahn & Weigert 1990). A shell of radius r, thickness Hp (size of the largest convective eddies), and radiative luminosity L cools on the thermal time scale tth = 4πr2HpρCpT , (26) where Cp is the specific heat at constant pressure. The 1M⊙ model (Fig. 2) has one deep convection zone with ted ≪ tth over most of the region, indicating that convection 6 Do not confuse the temperature gradient ∇ with the spatial gradient ∇ used in § 3. 6 PFAHL, ARRAS, & PAXTON FIG. 4.— Eddy turnover time at base of the convective envelope versus stellar age for a range of stellar masses. very efficiently transports energy and causes the zone to be essentially isentropic. By consrast, the 1.6M⊙ star (Fig. 3) has two thin surface convection zones with ted ∼ tth, and thus the radiative and convective fluxes are comparable. Gravity waves with frequency ω propagate only in radiative regions where ω < N and ω < Sℓ. For the 1M⊙ star, heat and en- tropy generated by g-modes in the radiative interior may be strongly mitigated owing to the long thermal time at the base of the deep convection zone. On the other hand, g-modes in a 1.6M⊙ star can propagate very near the surface, producing qualitatively different results. We now go on to elucidate the physics of the flux per- turbations. All the analytic and numerical work that fol- lows assumes that the tidal potential has the generic form U ∝ rℓYℓm(θ,φ)exp(−iωt) with forcing frequency ω. 5.1. Heat Transfer in a Convective Envelope Calculation of the perturbed convective flux in oscillating stars is a thorny issue. For the purposes of our study, we argue for an especially simple treatment that draws from previous work on this subject. Specifically, we modify the prescription of Brickhill (1983, 1990; see also Goldreich & Wu 1999a,b), which was originally applied to white-dwarf pulsations, into a form appropriate for the tidal flow problem. In the mixing-length theory of convection, heat is trans- ported by eddies with a spectrum of sizes l . Hp, speeds vl , and turnover times ted(l) = l/vl . The Kolmogorov scalings for turbulent motions give vl ∝ l 1/3, ted ∝ l 2/3, and an energy den- sity per unit mixing length interval ∝l−1/3. We see that in the unperturbed star most of the convective energy flux (∝v3l at scale l) is carried by the largest eddies (l ∼ Hp). Convection efficiently transports energy when the radiative thermal time scale associated with the dominant eddies is much longer than ted. Alternatively, efficient convection implies that the gradi- ent of the specific entropy s is small; i.e., d lns/d ln p ≪ 1. If all the convective energy flux F is carried by eddies with mix- ing length l, the flux and entropy gradient are related by (e.g., FIG. 5.— Thermal time at the base of the convection envelope versus stellar age for a range of stellar masses. Kippenhahn & Weigert 1990) d ln p (l) = (∇−∇ad) ∼ . (27) Efficient convection enforces ∇ − ∇ad ≪ 1, which implies d lns/d ln p ≪ 1, since s & Cp in the convective regions of our background models. Gravity waves with the tidal forcing frequency ω are ex- cited in the radiative region below the convection zone. Con- vective eddies can transport heat during a forcing period only if ted < 2π/ω (e.g., Brickhill 1990; Goldreich & Wu 1999b). Inspection of Fig. 2 shows that in the 1M⊙ model, the largest eddies have ted ≃ 20(p/pbcz) 0.5 days, where pbcz ≃ 1013.5 dyne cm−2 is the pressure at the base of convection zone. Using the Kolmogorov scaling, the “resonant” length lres for which ωted/2π = 1 is ∼ 10−2 1 day )3/2( , (28) which is >1 for all periods 2π/ω > 1 day when p . 1012 dyne cm−2, which still encompasses much of the convec- tion zone. Now imagine the situation where all the convective flux is carried by eddies of size .lres. The entropy gradient for this range of mixing lengths is d ln p (lres)∼ 10 1 day , (29) where we have adopted F/pcs ∼ 10 −8 at the base of the con- vection zone, as indicated by our 1M⊙ stellar model. These arguments suggest that convection is efficient in a 1M⊙ star at the forcing periods of interest even if small “res- onant” eddies carry all the energy flux near the base of the convection zone. At larger radii, but not too near the photo- sphere, convection is both efficient and rapid (ωted/2π < 1) ELLIPSOIDAL OSCILLATIONS 7 over the full spectrum of eddies. Rapid convection on all scales l . Hp enforces isentropy in the convection zone, such that s and its Lagrangian perturbation ∆s are nearly constant, as in the Brickhill (1983, 1990) picture. While convection at the base is rapid only on small scales, it is still highly efficient, which yields s ≃ constant and further indicates that ∆s/Cp is small in magnitude, as we demonstrate in § 5.2. As the stellar mass increases, the convection zone thins and ted at the base decreases (see Figs. 3 and 4). Rapid convection holds over the bulk of the convection zone for masses &1M⊙. However, the assumption that the convection is efficient starts to break down at 1.4–1.5M⊙, since tth ∼ ted at the base (see Figs. 3 and 5). For the full range of stellar masses consid- ered here, we assume that s and ∆s are constant in convection zones. 5.2. Analytic Result for Thick Convection Zones In a fully convective star, the emergent luminosity is de- termined entirely by the surface boundary conditions. Under our assumption that ∆s is constant in the convection zone, the perturbed luminosity is likewise a function only of the bound- ary conditions. Stars of mass .1.3–1.4M⊙ have long thermal times (ωtth ≫ 1) at the top of the interior radiative region (see Fig. 5), so that the flux perturbation ∆F is approximately the “quasi-adiabatic” value, derived by ignoring ∆s ∝ (ωtth) −1 in eq. (A5). We assume efficient convection continues to just below the photosphere. At the photosphere, we adopt the usual Stefan-Boltzmann relation, F = σT 4, and the hydrostatic condition, pκ/A = 2/3, where A is the total acceleration defined in § 4, and 2/3 is the photospheric optical depth. Taking the photosphere to define the stellar surface, we compute the Lagrangian perturbations, , (30) = 0 . (31) Using s and p as our independent thermodynamic variables, we write ∆κ/κ = κad∆p/p +κs∆s/Cp and ∆T/T = ∆s/Cp + ∇ad∆p/p. In our numerical work (see § 5.4), we self- consistently compute the perturbation ∆A to the effective sur- face gravity, in order to follow resonant oscillations, where the equilibrium-tide result fails. However, we are now addressing non-resonant forcing, for which we use the equilibrium-tide approximation at the surface, giving ∆A/g = −∂ξr/∂r (see § 4). We now have κs∆s/Cp +∂ξr/∂r 1 +κad , (32) and upon substitution, 1 +κad −∇adκs 1 +κad 1 +κad . (33) Equation (33) differs from Goldreich & Wu (1999a) in that we retain the gravity perturbation in eq. (31), whereas they consider a constant-gravity atmosphere (and no tidal pertur- bation). For g-modes in white dwarfs, the interesting region is near the surface and the motion is mainly horizontal, so that ∆A = ∆g = 0 is a good approximation. Since the equilibrium tide has large vertical motions, the ∆A term must be retained. The luminosity change across the convection zone is de- rived from the entropy equation (eq. [A6]). If we ignore horizontal flux perturbations (set ℓ = 0 in eq. [A6]) and en- ergy generation, the equation for the luminosity perturbation ∆L/L = 2ξr/r +∆F/F is d(∆L/L) = iωT∆s/L . (34) Integrating over the convection zone with constant ∆s, we obtain ∆Lbcz = iω∆s dMrT/L , (35) where the subscript “ph” refers to the photosphere. We de- fine tcz = Cp,ph dMrT/L to be the mean thermal time of the convection zone, so that the right-hand side of eq. (35) is iωtcz∆s/Cp,ph. Figure 5 shows that the thermal time at the base of the con- vection zone (of order tcz) for M . 1.3M⊙ is orders of magni- tude longer than the forcing periods of 1–10 days. Insofar as |∆L|/L ∼ |ξr|/r at any location in the star (i.e., if resonances are neglected), we see that (|ξr|/r) −1|∆s|/Cp ∼ (ωtcz) −1 ≪ 1 in stars with deep convective envelopes. In this limit, eq. (33) becomes 1 +κad . (36) If we had set ∆A = 0, the amplitude of the photospheric flux perturbation would have been ∼|∆s|/Cp rather than the much larger value ∼|ξr|/R. Photospheric flux perturbations in tidally forced solar- type stars with thick convective envelopes arise mainly from changes in the local effective gravity. This statement is remi- niscent of, but physically distinct from, von Zeipel’s theorem (eqs. [21] and [22]). We have recovered our equilibrium-tide scaling, ∆Fℓ/F = −λℓξr,ℓ/R, where eq. (36) gives λℓ = 4(ℓ+ 2) 1 +κad . (37) For M = 1.0–1.4M⊙, we find λ2 ≃ 1.9–1.1. These estimates neglect resonant excitation of g-modes, a point addressed in § 5.4. 5.3. Analytic Result for Radiative Envelopes As the stellar mass increases beyond 1.4M⊙, the outer convective region thins and sits close to the surface, where ted ∼ tth. Figure 3 shows that the 1.6M⊙ model has two thin, inefficient surface convection zones, as well as a convective core. Radiative energy transport is important throughout the envelopes of these more massive stars. We now consider the idealized case of a completely radiative envelope, and obtain an analytic approximation for ∆L/L at the surface. Near the surface of a radiative star, we have Hp/r ≪ 1, 4πr3ρ/Mr ≪ 1, and ω 2r/g ≪ 1 for 2π/ω = 1–10 days. Under these conditions, the quasi-adiabatic luminosity perturbation becomes (e.g., Unno et al. 1989) ∆Lqad gk2hHp ξr − ξr,eq 8 PFAHL, ARRAS, & PAXTON where ζ = κad − 4∇ad + d lnT , (39) and ξr,eq is the equilibrium-tide radial displacement (eq. [12]). Nonzero values of ∆p/p and (ξr − ξr,eq)/Hp indicate devia- tions from hydrostatic equilibrium. Care must be taken with these terms, because the denominators p and Hp become very small close to the surface. With the help of the Appendix, we define the variables α= y1 − y2 + y3 = − , (40) β = y2 + ξr − ξr,eq , (41) which satisfy the differential equations d lnρ β + (ℓ+ 4) , (42) . (43) When ω2 ≪ gk2hHp, these equations produce the g-mode dispersion relation k2r = k 2/ω2 (in the limit k2r/k h ≪ 1) for radial wavenumber kr. For these propagating waves, the surface amplitudes of α and β are determined at the core radiative-convective boundary, where g-modes are driven (e.g., Goldreich & Nicholson 1989). On the other hand, when ω2 ≫ gk2hHp, the g-modes are evanescent (see Unno et al. 1989) and we neglect the term gk2hβ/ω 2 in eq. (42). This limit yields the approximate solution α≃ −(ℓ+ 4)(HpU/gR2), or ∆p/p ≃ (4 + ℓ)U/gR. In this case, ∆p/p is not small compared to the fractional fluid displacement, and thus the equilibrium-tide approximation loses validity. From our stellar models, we find that the evanescent regime corresponds to forcing periods of .4–8 days for M = 1– 1.6M⊙, most of the range of interest. The high-frequency limit of eq. (38) is ∆Lqad ≃ −ζ(ℓ+ 4) . (44) This relation should be evaluated at the layer where ωtth ≃ 1, above which the luminosity effectively “freezes out.” Fig- ure 6 shows the quasi-adiabic flux perturbation ∆Fqad/F = ∆Lqad/L−2ξr/R, evaluated whereωtth = 1, for a range of forc- ing periods and M = 1.5–1.7M⊙. Note that |∆F/F| can be an order of magnitude larger than |U |/gR, because of the rather large values of |ζ|(ℓ+ 4) for ℓ ≥ 2. Much larger perturbations are possible when g-modes are resonantly excited in a radia- tive star, as we discuss in the next section. We must point out that the quasi-adiabatic approximation is technically inappropriate when ωtth ∼ 1. Equation (44) should be viewed as an estimate of the modulus of the luminosity per- turbation at the surface. If, for instance, |∆s|/Cp & |U |/gR where ωtth ∼ 1, then ∆L/L at the surface will have a substan- tial imaginary part (see eq. [34]). This is what we find in the numerical calculations summarized in the next section. 5.4. Numerical Examples Here we show solutions of the perturbed mass, momentum, and energy equations that describe linear, nonadiabatic oscil- lations of a star subject to a varying tidal force. The equations FIG. 6.— Ratio of surface Lagrangian flux perturbation ∆F/F to equilibrium-tide displacement −U/gR for a range of forcing periods in the limit where surface g-modes are evanescent. The flux is evaluated at the loca- tion where ωtth = 1. Dashed, solid, and dotted curves correspond to M = 1.5, 1.6, and 1.7M⊙ , respectively. listed in the Appendix are the same as in Unno et al. (1989) for radiative regions, but augmented to include the tidal ac- celeration. In convection zones, we apply the prescription ∆s = constant based on our conclusions in § 5.1. Figure 7 summarizes how the interiors of 1M⊙ and 1.6M⊙ stars re- spond to resonant and non-resonant tidal forcing. The tidal potential has been scaled so that ξr/R = 1 corresponds to the equilibrium-tide surface displacement. For our 1M⊙ model, the non-resonant response to a forc- ing period of ≃3 days is shown in Fig. 7a. We see that ξr/R matches the equilibrium-tide result at the surface; the imag- inary piece is completely negligible. We also find that our approximation for ∆F/F at the surface (eq. [36]) works very well. A factor of ∼10 decay in |∆L|/L occurred in order for |∆F |/F ∼ ξr/R at the surface. Variation of ∆s/Cp in the con- vection zone (log[p/(GM2/R4)]> −2.5) is due to changes in Cp. In the radiative interior, the oscillations are caused by most nearly resonant g-modes, whose amplitudes rise rapidly as the core is approached, due to conservation of wave lumi- nosity. We have checked that the quasi-adiabatic approxima- tion of ∆L/L is valid in the radiative region; the ratio of the real and imaginary parts is found to be roughly constant for the ingoing gravity-wave (see also Zahn 1975). In order to model the resonant response of a 1M⊙ star, we tuned the forcing period to ≃1 day (see Figs. 7b and 8). At the surface, both ξr and ∆L have dominant imaginary parts, due to the short radial wavelength of the g-mode compared to the equilibrium-tide fluid displacement. The entropy at the base of the convection zone is very strongly perturbed in compari- son to the non-resonant case, but ∆L is still damped by orders of magnitude as the surface is approached. Figure 8 shows the surface values of the complex modu- lus and phase of ξr/R and ∆F/F versus forcing period. The phase is tan−1(Imaginary/Real) ∈ (−π,π). Solid lines connect points halfway between g-mode resonant periods. We find ELLIPSOIDAL OSCILLATIONS 9 FIG. 7.— Responses of tidally forced 1M⊙ and 1.6M⊙ main-sequence stars. Black, red, blue, and green curves denote, respectively, the logarithms of ξr/r, (δp + ρδϕ)/ρgr, ∆s/Cp, and ∆L/L. Solid (dashed) curves show the real (imaginary) parts. The tidal potential has been scaled so that ξr/R = 1 corresponds to the equilibrium-tide value. The four panels show the following: (a) non-resonant response of a 1M⊙ star tidally forced at a period of 2π/ω ≃ 2.91day; (b) resonant response of a 1M⊙ star with 2π/ω ≃ 1.00day; (c) non-resonant response of a 1.6M⊙ star with 2π/ω ≃ 3.00day; (d) resonant response of a 1.6M⊙ star with 2π/ω ≃ 1.02day that the equilibrium-tide approximation given by eqs. (12) and (36) is excellent for non-resonant forcing. Dashed curves give the maximum and minimum values that occur on resonance. One example of a resonance is shown in the insets. Resonant forcing at periods of <2 days yields surface values of ξr/R and ∆F/F that differ substantially from the equilibrium-tide results. However, the ratio of resonance width to the spac- ing between adjacent resonances is ∼10−4, making resonant forcing very unlikely. It is noteworthy that at forcing periods of >2 days, the equilibrium-tide result holds extremely well even when precisely on a resonance. As explained by Zahn (1975), the resonant response can be considered as the sum of the equilibrium tide and the most nearly resonant wave. As the period increases, the g-mode radial wavelength decreases, resulting in a reduction of the overlap integral for the mode and the tidal force, which in turn gives a decreased amplitude 10 PFAHL, ARRAS, & PAXTON FIG. 8.— Surface radial displacement and Lagrangian flux perturbation versus forcing period for a 1.0M⊙ star. Solid lines connect points halfway be- tween resonant g-mode periods, while dashed curves give the maximum and minimum values found on resonance. The equilibrium-tide approximation is extremely good, except when the forcing period is <2 days and resonant. of the wave component relative to the equilibrium tide. The non-resonant response of the 1.6M⊙ star is shown in Fig. 7c. We see that the equilibrium-tide result pro- vides a good match to ξr/R. Our estimate for the modu- lus of the radiative luminosity perturbation in the evanes- cent limit (eq. [44]) agrees reasonably well with what is in Fig. 7c. We also see that ∆L/L does roughly “freeze-out” when ωtth ≃ 1, just below the base of the convection zone at log p/(GM2/R4) ≃ −9 (see Fig. 3). Our expectations in § 5.3 regarding the imaginary part of ∆L/L are borne out in Fig. 7c A resonantly excited 1.6M⊙ star exhibits huge surface flux perturbations, radial displacements, and phase lags, as seen in Fig. 7d. In Fig. 9, surface values of |ξr|/R and |∆F |/F are plotted as a function of forcing period, where we have taken care to resolve resonances. Resonant amplitudes vary non- monotonically with period, in contrast to the smooth behavior of the 1M⊙ star (Fig. 8). Although we do not show the re- sults here, similar plots for masses between 1M⊙ and 1.6M⊙ show progressively more structure as the mass increases. The cause of this irregularity is not clear, but may have to do with the two thin surface convection zones changing the overlap of successive g-modes with the tidal force. 6. SUMMARY We have investigated in detail the ellipsoidal oscillations of 0.9–1.6M⊙ main-sequence stars induced by substellar com- panions. Classical models of ellipsoidal variability (e.g., Wilson 1994) are built on the assumption of hydrostatic bal- ance in a frame corotating with the binary orbit. This ap- proach is justified in the context of short-period (Porb . 10 days) binaries containing two stars of comparable mass, where tidal dissipation circularizes the orbits and synchro- nizes the stellar spins with the orbital frequency. However, when the companion has a very low mass, we cannot assume that the binary is in complete tidal equilibrium; in fact, this FIG. 9.— Surface radial displacement and Lagrangian flux perturbation versus forcing period for a 1.6M⊙ star. Curves connect evenly spaced points away from resonances, with finer spacing near resonance periods. state may be unattainable (see § 2). In this case, one must, in general, appeal to a dynamical description of the tidal inter- action. A substellar companion with Porb & 1 day raises tides on the star that are a small fraction of the stellar radius (see eq. [1]), permitting a linear analysis of the stellar oscillations. While the root of our study is a dynamical treatment of stellar tidal perturbations, the equilibrium-tide approximation does have an important realm of validity (see below). For this reason, we derived in § 3 a general expression (eq. [16]) for the measurable flux variation of a star that remains in hydrostatic equilibrium under the influence of a small exter- nal tidal force. This formula (1) assumes that the local per- turbation to the energy flux at the stellar surface is propor- tional to and in phase with the equilibrium-tide radial fluid displacement at each angular order ℓ (eq. [12]), (2) neglects stellar rotation, and (3) applies to inclined and eccentric or- bits. As expected, the fractional amplitude of the modulation is ∼ε≡ (Mp/M)(R/a) 3 for small eccentricities and I = 90◦, or ∼10−5(Mp/MJ)(Porb/1day) −2 for a star like the Sun (see § 2). A common practice is to use von Zeipel’s theorem when computing the surface radiative flux from a tidally distorted star (see § 4). The theorem assumes that the star is in hydro- static equilibrium and that the energy transport in the outer layers is purely by radiative diffusion. As already mentioned, the hydrostatic assumption is technically unjustified for sub- stellar perturbers. Moreover, the majority of Kepler targets will be main-sequence stars with masses of <1.4M⊙, which have substantial surface convection zones. Evidently, von Zeipel’s theorem is an inappropriate starting point for the con- ditions of interest. Section 5.1 discusses heat transport in perturbed stars with convective envelopes. Heuristic arguments are used to de- velop a simple treatment of the perturbed convection zone in main-sequence stars of mass<1.6M⊙ with forcing periods of 1–10 days. We suggest that both the specific entropy s and its Lagrangian perturbation ∆s are spatially constant in convec- ELLIPSOIDAL OSCILLATIONS 11 tive regions, a model partly inspired by the ideas of Brickhill (1983, 1990). Using this prescription, we analytically compute in § 5.2 the perturbed flux at the photosphere of deeply convective stars (M . 1.4M⊙), where the thermal time scale at the base of the convection zone is much longer than the forcing period. We find that ∆s/Cp is negligible near the top of the convection zone, and that the photospheric flux perturbation is propor- tional to changes in the effective surface gravity. Thus, we re- cover the equilibrium-tide result, ∆F/F = −λℓξr/R, at the sur- face, where λℓ depends on the adiabatic derivatives of opacity and temperature with respect to pressure (see eq. [37]). Nu- merical solutions of the equations of linear, nonadiabatic stel- lar oscillations (see § 5.4 and Fig. 7a) corroborate our analytic estimates in the non-resonant regime. Resonant excitations of g-modes in the radiative stellar interior cause large departures from the equilibrium-tide approximation when the forcing pe- riod is <2 days (Figs. 7b and 8). However, the likelihood of being on a resonance is small, and at periods of >2 day the equilibrium-tide result holds for M ≃ 1M⊙ even with reso- nant forcing. Stars of mass &1.4M⊙ have thin, relatively inefficient sur- face convection zones. Thus, g-modes can propagate very close to the surface and produce large flux perturbations and fluid displacements. Analytic arguments in § 5.3 indicate that the surface flux perturbations in these stars have non-resonant amplitudes of ∼10ε (eq. [44] and Fig. 6), in rough agree- ment with our numerical calculations (Fig. 7c). As seen in Figs. 7d and 9, a resonantly forced 1.6M⊙ star can exhibit flux perturbation amplitudes of >100ε at forcing periods of ≃1 day. While the amplitudes are not as extreme at longer periods, their dependence on period is rather erratic (Fig. 9), an issue that deserves further study. It will be difficult to de- rive physical interpretations from the ellipsoidal variability of these more massive stars. 7. DETECTION PROSPECTS The dominant sources of periodic variability of a star with a substellar companion are transit occultations (when |cos I| < [R + Rp]/a), Doppler flux modulations, reflection of starlight from the companion, and ellipsoidal oscillations. For each of these signals, Table 2 lists the characteristic amplitude, period with the largest power in the Fourier spectrum, and or- bital phase(s) at which the light is a maximum or minimum. The transit contribution is included for completeness, but its duration is sufficiently short—a fraction ≃(R + Rp)/(πa) of Porb—that it should often be possible to excise it from the data (see Sirko & Paczyński 2003). Of the remaining signals, the Doppler variability is the simplest, being purely sinusoidal with period Porb when the orbit is circular. The dominant ℓ = 2 piece of the equilibrium-tide approximation to the ellipsoidal variability (see eqs. [16] and [17]) is also sinusoidal when e = 0, but with period Porb/2. Reflection is more problem- atic, as its time dependence is generally not sinusoidal and not known a priori. If the companion scatters light as a Lambert sphere (e.g., Seager et al. 2000), the Fourier spectrum of the reflection variability has finite amplitude at all harmonics of the orbital frequency Ω, but the amplitude at 2Ω is roughly 1/5 of the amplitude at Ω. Therefore, the reflection and ellipsoidal vari- ability amplitudes may be similar at a frequency of 2Ω when α = 0.1, Mp ∼ MJ , and Porb ≃ 1 day. Also, the orbital phase at which the reflected light is a maximum is distinct from both the Doppler and ellipsoidal cases, further distinguishing the signals. However, Lambert scattering is probably never appropriate in real planetary atmospheres. Infrared reemis- sion of absorbed optical light, multiple photon scattering, and anisotropic scattering typically conspire to narrow the peak in the reflection lightcurve and lower the albedo, decreasing the prominence of the reflection signal. These issues are sensi- tive to the atmospheric chemistry and the uncertain details in models of irradiated giant planets. For reasonable choices re- garding the atmospheric composition, calculated optical albe- dos of Jovian planets range from<0.01 to ≃0.5 (Seager et al. 2000; Sudarsky et al. 2000). Recent photometric observations of HD 209458, the star hosting the first-detected transiting gi- ant planet (Porb ≃ 3.5 days), constrain the planetary albedo to be <0.25 (Rowe et al. 2006). Detailed lightcurve simulations will be required to say how well the different periodic signals can be extracted from the data. This is beyond the scope of the current study. We now do the simpler exercise of isolating the ellipsoidal modulations and assessing when this effect alone should be detectable. For a star of apparent visual magnitude V and an integration time of T = 6hr, Kepler’s photon shot noise is7 ∼ 10−5100.2(V −12) )−1/2 . (45) Instrumental noise should contribute at a similar level (e.g., Koch et al. 2006). If the data is folded at the orbital period and binned in time intervals T ≪ Porb, the shot noise is sup- pressed by a factor of ∼n orb , where norb is the number of folded cycles. After folding 1 year of continuous photomet- ric data using T = 6 hr, a star with V < 12 orbited by a giant planet with Porb . 3 days may have a fractional shot noise per time bin of .10−6. This is less than the ellipsoidal amplitude, (δF/F )ell, when I is not too small. The actual situation is not so simple when the data spans of weeks or months, because the intrinsic stochastic variability of the star will not have a white-noise power spectrum. Over times of .1 day, the Sun shows variability of (δF/F )int ∼ 10−5, but the amplitude rises steeply between ∼1 and 10 days to ∼10−3. Intrinsic variability tends to be large near the rota- tion period of the star, due mainly to starspots. Low-frequency variability may not too damaging for the study of ellipsoidal oscillations induced by planets with Porb . 3 days, but more study is needed. Kepler’s target list will contain ≃105 main-sequence FGK stars with V = 8–14. The statistics of known exoplanets in- dicate that 1–2% of all such stars host a giant planet (Mp & MJ) with Porb < 10 days (e.g., Marcy et al. 2005). Of these “hot Jupiters,” ≃30% have Porb = 1–3 days. It seems that a maximum of ∼103 Kepler stars will have detectable ellip- soidal modulations. If we neglect intrinsic stellar variabil- ity and consider only shot noise, then many systems with Porb . 3 days and V < 14 will have signal-to-noise S/N > 1 after ∼100 cycles are monitored; this may amount to >100 stars. Obviously, the number drops when we place higher de- mands on S/N and include the intrinsic variability. The results depend critically on the distributions of Mp and Porb. In order to better estimate the number of stars with poten- tially detectable ellipsoidal oscillation, we perform a simple population synthesis calculation. Denote the set of star-planet 7 An integration time of T = 6 hr is chosen for convenience; Kepler’s nom- inal exposure time is 30 min. Here we use the V -band flux as a reference, but, in fact, the Kepler bandpass is 430–890 nm, which spans B, V , and R colors. 12 PFAHL, ARRAS, & PAXTON TABLE 2 PERIODIC FLUX MODULATIONS Variability Dominant Phase at Source Amplitudea,b Harmonic Maximum/Minimumc References Ellipsoidald . . . . . . . 2× 10−5mpmP−21 sin 2 I Porb/2 0.25(0.75)/0.00(0.50) · · · Dopplere . . . . . . . . . . 3× 10−6mpm−2/3P 1 sin I Porb 0.25/0.75 1 Reflectionf . . . . . . . . 6× 10−5(α/0.1)m−2/3P 1 sin I Porb 0.50/0.00 2,3 Transit . . . . . . . . . . . 10−2m−2 Porb · · · /0.00 4 REFERENCES. — (1) Loeb & Gaudi 2003; (2) Seager, Whitney, & Sasselov 2000; (3) Sudarsky, Burrows, & Pinto 2000; (4) Seager & Mallén-Ornelas 2003 a We assume that the orbit is circular in our estimates. b The dimensionless variables used are mp = Mp/(10−3 M⊙), m = M/M⊙ and P1 = Porb/1day. We have as- sumed that the star and companion have respective radii of R/R⊙ = m and 0.1R⊙. c The phase is in the range 0–1, where at phase 0 the planet is closest to the observer. d Only the ℓ = 2 component of eq. (16), with λ2 = 2, is considered here. e We approximate the amplitude as 4vr/c, where vr is the reflex speed of the star along the line of sight, and the factor of 4 is approximately what one obtains for a V -band spectrum similar to the Sun. f Here α is the geometric albedo of the companion. The inclination dependence is an approximation for I ≃ 90◦ and the Lambert phase function. system parameters by P = {M,Mp,Porb, I}, and let f (P)dP be the probability of having a system in the 4-dimensional vol- ume dP. We assume that the planetary orbits are circular and obtain (δF/F )ell from the equilibrium-tide estimate in Ta- ble 2. Given the mass of the star, we compute its absolute V magnitude on the main-sequence using the approximation (see also Henry & McCarthy 1993) MV = 4.8 − 10.3log(M/M⊙) , (46) which is in accord with the usual mass-luminosity relation log(L/L⊙) ≃ 4log(M/M⊙) for M ≃ 1M⊙. With a maximum apparent magnitude of Vmax = 14 for the Kepler targets, the maximum distance of the star is Dmax = 10 1+0.2(14−MV ) pc . (47) With a certain signal-to-noise threshold (S/N)min, there is a maximum distance Dd < Dmax to which the ellipsoidal vari- ability is detectable. For a spatially uniform population, the detectable fraction of systems is (Dd/Dmax) 3. Thus, the net detectable fraction among all systems is dP f (P) , (48) an integral over all relevant P space. When the only noise is intrinsic to the star, N = (δF/F )int and S/N is independent of distance, so that Dd,int = Dmax when S/N > (S/N)min, and Dd,int = 0 otherwise. In the case of pure shot noise, there is a maximum magnitude Vd for which the ellipsoidal oscillations are detectable: Vd = 5log (δF/F )ell χ (S/N)min , (49) where χ ∼ 10−8.4(T6n100) −1/2 is the value of (δF/F )shot for V = 0, T = 6T6 hr, and norb = 100n100. The corresponding dis- tance is given by log[Dd,shot/10pc] = 0.2(Vd − MV ) if Vd < Vmax, and is Dd,shot = Dmax when Vd >Vmax. We take the max- imum detectable distance to be Dd = min{Dd,int,Dd,shot}. At this point the simplest approach is to assume that the parameters {M,Mp,Porb, I} are statistically independent and carry out a Monte Carlo integration to obtain E . To this end, we draw M from the Kroupa et al. (1993) initial mass TABLE 3 NUMBER OF Kepler STARS WITH DETECTABLE ELLIPSOIDAL OSCILLATIONS (S/N)min = 1 (S/N)min = 3 (S/N)min = 5 y x = 1 2 x = 1 2 x = 1 2 1 240 166 76 35 33 13 0 99 62 26 12 11 4 -1 33 19 7 3 2 1 function in the range of 0.5–1.5M⊙. The planetary mass is chosen from the distribution f (Mp) ∝ M p for Mp = 1–10MJ. Marcy et al. (2005) find that x ≃ 1 when considering all de- tected planets; the shape of f (Mp) is not well constrained at Porb < 10 days. We let x = 1 and 2. We adopt f (Porb) ∝ P over 1–10 days. Multiplying the resulting value of E by 1000 provides a crude estimate of the actual number of Kepler tar- gets with detectable ellipsoidal variability. No single value of y is consistent with the data, and so we consider the reason- able range y = −1, 0, and +1. Inclinations are chosen under the assumption that the orbits are randomly oriented, such that f (cos I) = 1/2 for I ∈ (0,π). Our calculations use fixed values of (δF/F )int = 10 −5 and T6 = n100 = 1. Results of our Monte Carlo integrations are shown in Ta- ble 3 as actual numbers of Kepler targets. The largest number of detectable systems is obtained when x = y = 1, parameters that yield the largest proportions short periods and massive planets. We expect that ∼10–100 Kepler stars may exhibit ellipsoidal oscillations with S/N & 5. A handful of systems might have S/N & 10. Higher harmonics from the ℓ = 3 com- ponents of eq. (16) or modest eccentricities might be accessi- ble for at most a few stars. Our integrations also check for cases where the planet is transiting. As (S/N)min increases from 1 to 5, the fraction of systems in Table 3 with |cos I| < (R + Rp)/a runs from ≃30% to ≃50%, with a weak dependence on x and y. Such significant fractions stand to reason, since systems with the shortest periods have the highest ellipsoidal amplitudes and transit probabilities. Transit measurements directly give Porb, sin I & 0.95 (for Porb & 1 day), and (Rp/R) 2. The planet mass Mp can be determined with the addition of spectroscopic ra- ELLIPSOIDAL OSCILLATIONS 13 dial velocity measurements, which should be possible for most of the Kepler targets with detectable ellipsoidal oscil- lations. The ellipsoidal amplitude then depends on the un- measured stellar mass and radius via ε∝ R3/M2 (eq. [1]), as well as the stellar photospheric conditions (eq. [36]). If M and R are obtained from stellar models, ellipsoidal variabil- ity may provide an interesting consistency check on all the system parameters, as well as test the theory of forced stellar oscillations. As a last point, we emphasize that stars of mass &1.4M⊙ may have typical ellipsoidal amplitudes of ∼10ε. However, such stars will also be younger than most Kepler targets and probably have intrinsic variability ≫10−5. We carried out Monte Carlo integrations with M = 1.4–1.6M⊙, (δF/F )ell = 10εsin2 I, and x = y = 1. As we vary (δF/F )int from 10 to 10−4, E decreases from large values of ≃0.4 to a small fraction of ≃0.03 for (S/N)min = 10. Unfortunately, we do not know how many such stars will be included in the Kepler target list. Also, there has not yet been a discovery of a giant planet with Porb < 10 days around a star of mass ≥1.4M⊙, but exoplanet surveys tend to exclude these more massive stars. We thank Tim Brown for general discussions and address- ing Kepler questions, Jørgen Christensen-Dalsgaard for guid- ance on stellar luminosity perturbations, and Mike Muno for advice on signal processing. This work was supported by NSF grant PHY05-51164. APPENDIX OSCILLATION EQUATIONS Here we list the nonadiabatic, linearized fluid equations that we solve numerically. The reader is referred to Unno et al. (1989) for a complete discussion. Scalar and vector quantities are expanded in spherical harmonics Yℓm and poloidal vector harmonics, respectively. The momentum, mass, and energy equations are written in terms of the dimensionless variables y1 = ξr/r, y2 = (δp/ρ + δϕ)/gr, y3 = δϕ/gr, y4 = g−1dδϕ/dr, y5 = ∆s/Cp, and y6 = ∆L/L. Here L is the total (radiative plus convective) luminosity. The radial flux perturbation is ∆F/F = ∆L/L − 2ξr/r. In radiative zones, the nonadiabatic equations are d lnr gk2hr − y5ρs + U , (A1) d lnr ω2 − N2 1 − η+ −ρsy5 − , (A2) d lnr = y3 (1 − η)+ y4 , (A3) d lnr = y1η + y2η ℓ(ℓ+ 1) − η − y4η + y5ρsη , (A4) d lnr + 4(∇−∇ad) + c2 (∇ad −∇) gk2hr c2 + y4 ∇ad + y5 ∇ (4 −κs) − y6 dU/dr , (A5) d lnr = y1ℓ(ℓ+ 1) − y2ℓ(ℓ+ 1) + y3ℓ(ℓ+ 1) 4πr3ρCpT ℓ(ℓ+ 1) , (A6) where cs is the sound speed, η = d lnMr/d lnr, c2 = (r/Hp)∇(κad −4∇ad)+∇ad(d ln∇ad/d lnr+r/Hp), and we have ignored energy generation terms. Note that the tidal acceleration −∇U has been added to the momentum equations. In convection zones, we ignore turbulent viscosity effects and replace the radiative diffusion equation (eq. [A5]) with the prescription ∆s = constant (see § 5.1), or more precisely = 0 . (A7) Equation (A6) still involves the total (convective plus radiative) luminosity. We ignore energy generation and horizontal flux perturbation terms, i.e. we ignore all terms with spherical harmonic index ℓ in eq. (A6) in convection zones. At the center of the star, we require the solutions to be finite, and also set ∆s = 0. At the surface, we set δp = ρgξr and we require δϕ to decrease outward. This boundary condition is only approximate, as g-modes may propagate above the convection zone for wave periods of &4days in our 1M⊙ model. The final surface boundary condition is given by eq. (31). Care must be used in the radiative zone just below the photosphere, since the entropy perturbation is far from the quasi-adiabatic value. If we solve the radiative diffusion equation in this region, we find that the entropy increases by ∼10 orders of magnitude in just a few grid points. However, we regard this behavior as unphysical, because the region at the top of the convection zone is optically thin. 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Motor driven microtubule shape fluctuations - force from within the lattice Hervé Mohrbach1 and Igor M. Kulić2 1Laboratoire de Physique Moléculaire et des Collisions, Université Paul Verlaine - 57012 Metz, France 2School of Engineering and Applied Sciences, Harvard University, Massachusetts 02138, USA (Dated: October 31, 2018) We develop a general theory of microtubule (MT) deformations by molecular motors generating internal force doublets within the MT lattice. We describe two basic internal excitations, the S and V shape, and compare them with experimental observations from literature. We explain the special role of tubulin vacancies and the dramatic deformation amplifying effect observed for katanin acting at positions of defects. Experimentally observed shapes are used to determine the ratio of MT shear and stretch moduli (≈ 6 × 10−5) and to estimate the forces induced in the MT lattice by katanin (10’s of pN). For many motors acting on a single MT we derive expressions for the end-to-end distance reduction and provide criteria for dominance of this new effect over thermal fluctuations. We conclude that molecular motors if acting cooperatively can ”animate” MTs from within the lattice and induce slack even without cross-bridging to other structures, a scenario very much reminiscent of the motor driven axoneme. PACS numbers: 87.15.-v 87.16.Ka 87.16.Nn Microtubules are the stiffest cytoskelletal component and constitute the main routes for motor mediated in- tracellular cargo transport in higher organisms [1]. Un- derstanding their physical properties is at the heart of many biological problems from cellular mechanics to in- formation and material trafficking in the cell. Since the discovery of their high elastic anisotropy [2] it be- came increasingly clear that MTs are mechanically more complex than other semiflexible biofilaments. The high anisotropy has been impressively confirmed by thermal fluctuation analysis of beads attached to MTs of different lengths[3]. The emerging picture of the MT is that of an anisotropic fiber reinforced material [2, 3] with the tubu- lin protofilaments (PF) acting as strong fibers weakly linked with easily shearable lateral bonds. Remarkably this type of design is also found in higher structures like axonemes (constituting the backbone of flagella and cilia) where relatively inextensible MTs are held together with highly stretchable nexin connections [1]. This remark- able structural self-similarity of the two nested structures (MT and axoneme) indicates further analogies in the way they respond to external and internal forces. We explore here important consequences of MT geometry and elas- tic properties and show that motors acting on the MT surface can generate internal lattice strains sufficient to induce observable lateral and longitudinal deformations of the MT backbone. In the following we describe a twist-free MT of length L consisting of N identical PFa with constant dis- tance a and a circular cross section, Fig 1. Each PF, parametrized by the MT backbone arc length s has a position dependent displacement uk (s) from its equilib- rium position, k = 1, ...N . The backbone shape is de- scribed by a curvature vector ~κ (s) = d ~t (s) with ~t (s) the bundle centerline tangent. The elastic properties of the MT are characterized by a PF bending stiffness B = 1 πa4Y and compressional modulus Kc = with Y ≈ 0.1−1.5GPa [2, 3] being the PF Young’s mod- ulus. Additionally there are shear elastic forces restoring the longitudinal displacement between the PFs governed by a very soft elastic shear modulus Ks ≈ 10 −3 − 1MPa [2, 3]. The elastic energy is given by EMT = ∫ L/2 B~κ2 +Kcu k +Ks∆ ds (1) with the first term being the bending energy, the second the PF compression and the third describing the relative shear energy between the neighboring PFs. The shear displacement ∆k is related to the difference of PF dis- placements uk−uk−1 and a curvature induced additional displacement via ∆k (s) = uk (s)− uk−1 (s) + ~κ (s′) ·∆~rkds ′ (2) With ∆~rk = ~rk − ~rk−1 and ~rk = RMT cos 2πk , sin 2πk the vector pointing from the MT center to the k-th PF, cf. Fig 1. Equations 1-2 are 3-D analogues of the previ- ously proposed stretchable railway-track [4] or wormlike- bundle [5] model for the case of a hollow circular bundle. While in general all the N + 3 fields , i.e. the 3 compo- nents of ~κ (s) and the PF displacements {uk (s)}k=1,..,N enter the eqs 1-2 in the limit of small MT deviations from a straight line the problem can be drastically sim- plified. We first expand the tangent ~t ≈ (θx, θy, 1) and θ′x, θ in terms of two angular projections θx and θy of ~t in x and y direction respectively. Exploit- ing the circular geometry of the PF arrangement and the Fourier representation uk (s) = ûq (s) e 2πikq N over k we quickly realize that only the longest wavelength mode û1 (s) couples to overall MT backbone shape given by the curvature ~κ. This leads to total energy decoupling EMT = E MT + E MT + E MT into a shape- independent http://arxiv.org/abs/0704.1911v1 FIG. 1: (Color online) The basic geometry of motors induc- ing internal force doublets along the MT backbone: between two PFs (S-let) and along the same PF (V-let). The red and blue ovals represent two coupled motors or two motor sub- units (legs) of the same motor. Tubulin lattice vacancies at the motor position strongly amplify the MT backbone defor- mation (V gap -let). component E0MT and two shape dependent contributions (in x and y direction) given by: EiMT = ∫ L/2 B̂∆θ′2i + K̂cU 2 + K̂s∆̂ ds (3) With ∆̂i (s) = a (θi (s)− θi (−L/2)) − Ui (s) , a = |∆~rk| the inter-protofilament distance , i = x, y and ~U (s) = (Ux, Uy) = (Reχû1, Imχû1) with χ = 1 − e−2πi/N and renormalized constants B̂ = NB, K̂c = NKc/ (4− 4 cos (2π/N)) and K̂s = NKs. Visually the new variable ~U (s) is a x-y vector at each MT -crosscut and can be interpreted as the (vectorial) mean over rel- ative PF displacements of neighboring PFs. With this enormous simplification at hand we can consider now ba- sic motor induced MT excitations (Fig 1). There are two elementary configurations in which motors can induce in- ternal MT strains: 1) A motor (or a complex of several motors) acting between two (not necessarily neighboring) PFs and 2) A motor (or a complex of several motors) act- ing at two points within the same PF. For reasons that will soon become clear we call the excitation 1 S-type or simply an ”S-let” and excitation 2 we call an V-type ex- citation or ”V-let”. Both excitations are ”internal” in the sense that there is no net torque or force on the system motor+MT similarly to the case of a beating flagellum[1]. Elementary internal MT excitations. In the following we want to understand the properties of the two basic types of excitations from Fig 1 and focus on the S-type first. We assume a single motor (or a complex of two motors) at position s = s0 bridging between two PF with index k1 and k2 and exerting opposing forces F and −F onto them respectively, Fig 1 a) (left) + b). The total energy is Etot = EMT + ES−mot with EMT given by eqs 1-2 and the potential energy of the motor ES−mot = −F ∑k=k2 ∫ L/2 δ (s− s0)∆k (s) ds . As we had for EMT before ES−mot also decouples into inde- pendent modes in the Fourier representation over k and Emot,S = E S−mot+ E S−mot+E S−mot with E S−mot a cur- vature independent term and the two shape dependent contributions E S−mot (in x and y direction) given by: EiS−mot = −F ∫ L/2 δ (s− s0) ∆̂i (s) ds with i = x, y and ~µ being the vector connecting the two attachment points of the motor (or motor complex) with components µi = ~µ · ~ei , Fig 1 b. The equilibrium solution is given by the Euler Lagrange equations: δEitot/δUi = 0 and δEitot/δθi = 0 with E tot = E S,mot + E MT and bound- ary conditions θ′i (±L/2) = U i (±L/2) = 0 (vanishing bending and shearing stresses at the ends). A short cal- culation leads to ∆̂′′i − λ −2∆̂i = (1 + α) δ (s− s0) with the shear decay length λ = λ−2c + λ )−1/2 Here the two important length scales λc = K̂c/K̂s and λB = B̂/a2K̂shave the physical meaning of a pure compression- / pure bending- induced shear screen- ing length respectively, with their squared ratio α = (λc/λB) ≈ 35 (N = 13 PF). The remaining equations lead to conservation laws (aθi + αUi) = 0 which com- bined with the equation for ∆̂i give for the simplest sym- metric case s0 = 0 the tangent angles (up to an arbitrary constant) θi (s) = Φi cosh (|s| /λ− L/2λ) sinh (L/2λ) With Φi = Φ i = ~µ · ~ei . The resulting MT backbone curvature has a jump at s = 0 and attains its maximal modulus there. The resulting MT shape is planar (con- tained in the plane spanned by ~µ and ~t at any position) and S-shape-like with initial and final angle coinciding θi (−L/2) = θi (+L/2) which explains our nomenclature ”S-type excitation” or ”S-let”. Interestingly the length scale λ over which θi (s) declines allows us to indepen- dently estimate the ratio of stretch and shear moduli from the observation of S-let deformations coming from katanin action[7], Y/Ks ≈ 64π −1 (λ/a) ≈ 6 × 105 for λ ≈ 1µm (cf, Fig 2b) and a ≈ 6nm. This value is close to the result obtained by Pampaloni et al. [3] (Y/Ks ≈ 10 Further the maximal deflection angle of θmax ≈ 38 (= 0.66) measured in Fig 2a gives via 4 an estimate for the involved motor forces F ≈ 2B̂θmax/(λ |µ|) = πa4Y/(λ |µ|) = 20−250pN (for |µ| = a ≈ 6nm and the range of values for Y from literature [2, 3]). This indi- FIG. 2: a: The shapes of V and S-type excitations as given by Eq. 4 and their experimental observations for katanin op- erating on MTs (b,c). b: Adapted and edge enhanced image from [7]. c: Adapted from [6], black bars highlight the MT de- formation. The upper and lower images show the MT before and after kink generation respectively. cates that many katanin motors might act cooperatively to generate the observed shape change in Fig 2b. Another interesting possibility is that katanin might be different from dynein/kinesin by generating only small contractile displacement ”powerstrokes” . 1nm with a more effi- cient chemical-mechanical ATP-energy conversion lead- ing to larger contractile forces F & 15kBT/nm = 60 pN . The second fundamental internal MT excitation ap- pears when a motor (or motor complex) acts along a single PF with index k compressing or stretching it. The motor energy in this case can be written as EV,mot = F (uk (s0 + |µ| /2)− uk (s0 − |µ| /2)) ≈ F |µ| δ (s− s0)u k (s) ds where |µ| is the size of the motor step. Like in the previous case it is sufficient to keep the energy contribution of the mode q = 1 as the others decouple from each other and from the curvature term. Along very similar line of derivation as in the S-type excitation case we obtain a solution which is planar and contained in the plane spanned by the vector ~rk and ~t. The resulting tangent angles for an excitation in the middle of the MT (s0 = 0) are given by eq 4 with Φi = Φ i (s) = (~ei ·∆~rk) γiF |µ| 2(1+α)B̂ which is now s dependent and changes sign at s = 0. Here (γx, γy) = sin (π/N) (1− cos (2π/N)) (sin (2π (k + 1/2)/N) ,− cos (2π (k + 1/2) /N)) . The re- sulting shape, that we call a ”V-let”, is a smooth V- shaped planar kink in the MT backbone with continuous curvature which relaxes on the length-scale λ. While su- perficially similar the S-let and V-let solutions are physi- cally very different for two reasons. First, the s/ |s| factor in the V-let solution changes the symmetry with respect to the S-let which leads to dramatic effects on the end- end distance as we see below. Second difference lies in the different scaling of the numerical prefactors which in the case of a V-let do not contain the screening length λ and involve additionally a very large reduction factor 1/ (1 + α) . In practice this suppresses significantly the involved deformations (θ ≈ 10−5 − 10−6), orders of mag- nitude below that of a S-let corresponding to the same force (θ ≈ 10−1). However the situation changes dramat- ically if the motor is operating at a position of a vacancy in the tubulin lattice, cf. Fig 1. In this case the mo- tor is not hindered by the large rigidity of the short PF portion that the motor acts on. Formally there is no re- quirement of continuity for uk (s) of the involved PF k at the position of the gap. For such a combined defect + motor excitation which we call V gap-let, the motor en- ergy is given by EV gap,mot = F (uk (s0 + 0)− uk (s0 − 0)) and its Ux/y dependent component becomes E V gap,mot = Fγi (Ui (s0 + 0)− Ui (s− 0)) . After a short calcula- tion in direct analogy with the previous cases we obtain the same form as in eq 4 but with Φi = Φ V gap i (s) = (~ei ·∆~rk) .While its functional form and symme- try coincide with that of the V-let, the prefactor of a V gap-let solution is more similar in magnitude to that of a S-let. It is intuitive to think of a V gap - let as a V-let with an effectively renormalized motor force F̃ = 2a−1λ (1 + α)F. The distribution of angles of observed V-shapes measured by Davies et al [6] of θmax ≈ 15 30◦(cf.Fig 2c) would suggest for the defect free case (V- let) a very large required force of F = 6×104−6×105pN . However , the same estimate for the V gap-let case gives much more moderate 10 − 100pN (few tens of motors), showing the prime importance of lattice defects in the case of V type excitations. Interestingly Davies et al. [6] also suggested a crucial role of lattice defects based on the pattern and kinetics of MT decomposition by katanin. Statistical mechanics of multiple excitations. It is particularly interesting to understand the collective contribution of a large number of internal MT ex- citations acting at random positions sj and orien- tations −→µ j in addition to the MT thermal fluctu- ations. The motor energy for S- and V-lets be- comes now respectively EiS,mot = − j∆̂i (sj) and EiV,mot = −F j |µ| γ i (sj). It is convenient to in- troduce the vector −→µ S/V,j such that µ S,j ≡ µ j and µiV,j ≡ |µ| γ j (i = x, y). For a fixed (but arbitrary) distribution of motors the partition functions Zi = DUiDθi exp EiMT + E and the correla- tion functions θi,qθi,p (q, p 6= 0) can be obtained. Here 〈...〉 denotes the average over the random motore distri- bution and (...) is the average over the thermal noise. θi,qθi,p decomposes into the sum of a thermal contri- bution 〈θi,qθi,p〉T = 2kBTL −1G(q)δp,q with the propa- gator G(q) = B̂q2 + q 2K̂sa q2+K̂s/K̂c and a motor con- tribution θi,qθi,p = ΨS/V (p)ΨS/V (q)CS/V (q, p) . For S- and V-let case we have respectively ΨS (p) = 2FK̂cp L(K̂cp2+K̂s) G(p) and ΨV (p) = 2FaK̂sp L(K̂cp2+K̂s) G(p), with the motor position and orientation correla- tor CS/V (q, p) = j,l µ S/V,jµ S/V,l cos (qsj) cos (psl) CS/V (q, p) is easily computed for the simplest choice of a uniform motor position and orientation distribu- tion: P (si) = 1/L , −→µ S/V,j = ∣µS/V ∣ (cosφj , sinφj) with an random angle φj with a probability distribution P (φj) = 1/2π. In this case we obtain the length reduc- /L ≈ 1 θ2x,q θ2y,q which naturally decomposes in a sum of a thermal fluctuation term and a motor term . In the relevant limiting case L/λ ≫ 1 we obtain for the thermal part a2K̂sl0p with the large and small scale persistence lengths given by l∞p = B̂ + a2K̂c /kBT and l (α/ (1 + α)) B̂/kBT . Interestingly the term a can be formally understood as an intrinsic self-tension straightening the MT at small scales. Similar formulas appear in different geometries for the railway-track[4] and wormlike-bundle model[5]. The motor dependent length reduction for the S-, V- and V gap-let excitations with line density ρ is given by: ρF 2µ2 a4K̂2sλ ρF 2µ2L a2K̂2sλ V gap = cV gap ρF 2µ2L a4K̂2sλ with cS = α 2 (1 + α) /16 ≈ 6 × 10−2, cV = 0.18α2 (1 + α) ≈ 1.3 × 10−4 and cV gap = 0.73α2 (1 + α) ≈ 0.7. Remarkably the S- and V/V gap- lets show different scaling. In particular V/V gap grows with L (in analogy to the first term in the ther- mal contribution 5) while /L stays length inde- pendent. The physical reason for this difference becomes obvious from Fig 2, as the relative slack /L induced by a single S-let scales with λ/L, while for an V/V gap- let it is essentially length independent. For longer MTs this effect leads to a strong dominance of V gap-lets over S-lets V gap ∼ (L/λ) ≫ 1. Although hav- ing the same L scaling the minute prefactor of defect free V-lets renders their contribution relatively insignif- icant ∼ a2λ−3L ≈ 10−6 even for very long MTs (L = 100µm), underlining the importance of lattice vacancies transforming a V-let into a V gap- let. Another interesting observation is that in all three cases S/V/V gap ∝ ρF 2. Fixing the number of mo- tors Nmot but regrouping them into Nmot/M clusters of size M we have ρ → M−1ρ, F → MF and therefore S/V/V gap ∝ M, i.e. the slack grows linearly with the cluster size. This indicates that cooperativity (po- sitional correlation) of motor action can lead to strong enhancement of the slack length. From Eqs.5-7 we can derive criteria for the dominance of motor slack over the thermal slack. For instance us- ing the values estimated from Fig 2 b,c for katanin for elastic constants from [3] (F = 20pN , λ = 1µm) and µ = 8nm, L = 20µm we obtain /L = ρ/ρc with ρc,S ≈ 0.25nm −1, ρc,V gap ≈ 1.2 × 10 −3nm−1. For large enough motor densities the katanin action easily domi- nates over the thermal slack /L ≈ 6× 10−4. Being evolutionary specialized for MT deformation and degradation katanin is likely to be among the strongest slack generating motors. We suspect however that classi- cal motors like dynein and kinesin might cause less pro- nounced but observable effects as well. While dynein is known to walk between several PFs, kinesin is very strictly following a single one[1]. Our theory suggest that dynein should induce moving S-lets, yet with quickly fluc- tuating signs which would diminish the effect consider- ably. A battery of many kinesins, however, walking over a MT region with many tubulin vacancies, would give rise to spatially stationary V gap-lets blinking between ”on” and ”off” states. The theoretical and experimental ex- ploration of these issues is an interesting future direction. The authors acknowledge fruitful discussions with E. Frey, C. Heussinger, M.Bathe, O. Campas, J.F. Joanny and P.C. Nelson. I.M.K. acknowledges support by the Max-Planck Society. [1] J. Howard, Mechanics of Motor Proteins and the Cy- toskeleton. Sinauer Press (2001); L.A. Amos and W.G. Amos, Molecules of the Cytoskeletion, Guilford (1991). [2] A. Kis et al. Phys. Rev. Lett. 89: 248101 (2002) [3] F. Pampaloni et al. PNAS 103: 10248 (2006) [4] R. Everaers, R. Bundschuh, and K. Kremer, Europhys. Lett 29, 263 , (1995) [5] C. Heussinger, M. Bathe and E. Frey, [cond-mat/0702097] [6] L.J. Davis, D.J. Odde, S.M. Block, and S.P. Gross, Bio- phys. J. 82, 29162927 (2002) [7] J.J. Hartmann et. al, Cell, Vol. 93, 277287; Movie at http://valelab.ucsf.edu/images/mov-rhomtsvkat.mov with kind permission by R. Vale http://arxiv.org/abs/cond-mat/0702097 http://valelab.ucsf.edu/images/mov-rhomtsvkat.mov

We develop a general theory of microtubule (MT) deformations by molecular motors generating internal force doublets within the MT lattice. We describe two basic internal excitations, the S and V shape, and compare them with experimental observations from literature. We explain the special role of tubulin vacancies and the dramatic deformation amplifying effect observed for katanin acting at positions of defects. Experimentally observed shapes are used to determine the ratio of MT shear and stretch moduli ($\approx 6\times10^{-5}$) and to estimate the forces induced in the MT lattice by katanin (10's of pN). For many motors acting on a single MT we derive expressions for the end-to-end distance reduction and provide criteria for dominance of this new effect over thermal fluctuations. We conclude that molecular motors if acting cooperatively can ''animate'' MTs from within the lattice and induce slack even without cross-bridging to other structures, a scenario very much reminiscent of the motor driven axoneme.

Motor driven microtubule shape fluctuations - force from within the lattice Hervé Mohrbach1 and Igor M. Kulić2 1Laboratoire de Physique Moléculaire et des Collisions, Université Paul Verlaine - 57012 Metz, France 2School of Engineering and Applied Sciences, Harvard University, Massachusetts 02138, USA (Dated: October 31, 2018) We develop a general theory of microtubule (MT) deformations by molecular motors generating internal force doublets within the MT lattice. We describe two basic internal excitations, the S and V shape, and compare them with experimental observations from literature. We explain the special role of tubulin vacancies and the dramatic deformation amplifying effect observed for katanin acting at positions of defects. Experimentally observed shapes are used to determine the ratio of MT shear and stretch moduli (≈ 6 × 10−5) and to estimate the forces induced in the MT lattice by katanin (10’s of pN). For many motors acting on a single MT we derive expressions for the end-to-end distance reduction and provide criteria for dominance of this new effect over thermal fluctuations. We conclude that molecular motors if acting cooperatively can ”animate” MTs from within the lattice and induce slack even without cross-bridging to other structures, a scenario very much reminiscent of the motor driven axoneme. PACS numbers: 87.15.-v 87.16.Ka 87.16.Nn Microtubules are the stiffest cytoskelletal component and constitute the main routes for motor mediated in- tracellular cargo transport in higher organisms [1]. Un- derstanding their physical properties is at the heart of many biological problems from cellular mechanics to in- formation and material trafficking in the cell. Since the discovery of their high elastic anisotropy [2] it be- came increasingly clear that MTs are mechanically more complex than other semiflexible biofilaments. The high anisotropy has been impressively confirmed by thermal fluctuation analysis of beads attached to MTs of different lengths[3]. The emerging picture of the MT is that of an anisotropic fiber reinforced material [2, 3] with the tubu- lin protofilaments (PF) acting as strong fibers weakly linked with easily shearable lateral bonds. Remarkably this type of design is also found in higher structures like axonemes (constituting the backbone of flagella and cilia) where relatively inextensible MTs are held together with highly stretchable nexin connections [1]. This remark- able structural self-similarity of the two nested structures (MT and axoneme) indicates further analogies in the way they respond to external and internal forces. We explore here important consequences of MT geometry and elas- tic properties and show that motors acting on the MT surface can generate internal lattice strains sufficient to induce observable lateral and longitudinal deformations of the MT backbone. In the following we describe a twist-free MT of length L consisting of N identical PFa with constant dis- tance a and a circular cross section, Fig 1. Each PF, parametrized by the MT backbone arc length s has a position dependent displacement uk (s) from its equilib- rium position, k = 1, ...N . The backbone shape is de- scribed by a curvature vector ~κ (s) = d ~t (s) with ~t (s) the bundle centerline tangent. The elastic properties of the MT are characterized by a PF bending stiffness B = 1 πa4Y and compressional modulus Kc = with Y ≈ 0.1−1.5GPa [2, 3] being the PF Young’s mod- ulus. Additionally there are shear elastic forces restoring the longitudinal displacement between the PFs governed by a very soft elastic shear modulus Ks ≈ 10 −3 − 1MPa [2, 3]. The elastic energy is given by EMT = ∫ L/2 B~κ2 +Kcu k +Ks∆ ds (1) with the first term being the bending energy, the second the PF compression and the third describing the relative shear energy between the neighboring PFs. The shear displacement ∆k is related to the difference of PF dis- placements uk−uk−1 and a curvature induced additional displacement via ∆k (s) = uk (s)− uk−1 (s) + ~κ (s′) ·∆~rkds ′ (2) With ∆~rk = ~rk − ~rk−1 and ~rk = RMT cos 2πk , sin 2πk the vector pointing from the MT center to the k-th PF, cf. Fig 1. Equations 1-2 are 3-D analogues of the previ- ously proposed stretchable railway-track [4] or wormlike- bundle [5] model for the case of a hollow circular bundle. While in general all the N + 3 fields , i.e. the 3 compo- nents of ~κ (s) and the PF displacements {uk (s)}k=1,..,N enter the eqs 1-2 in the limit of small MT deviations from a straight line the problem can be drastically sim- plified. We first expand the tangent ~t ≈ (θx, θy, 1) and θ′x, θ in terms of two angular projections θx and θy of ~t in x and y direction respectively. Exploit- ing the circular geometry of the PF arrangement and the Fourier representation uk (s) = ûq (s) e 2πikq N over k we quickly realize that only the longest wavelength mode û1 (s) couples to overall MT backbone shape given by the curvature ~κ. This leads to total energy decoupling EMT = E MT + E MT + E MT into a shape- independent http://arxiv.org/abs/0704.1911v1 FIG. 1: (Color online) The basic geometry of motors induc- ing internal force doublets along the MT backbone: between two PFs (S-let) and along the same PF (V-let). The red and blue ovals represent two coupled motors or two motor sub- units (legs) of the same motor. Tubulin lattice vacancies at the motor position strongly amplify the MT backbone defor- mation (V gap -let). component E0MT and two shape dependent contributions (in x and y direction) given by: EiMT = ∫ L/2 B̂∆θ′2i + K̂cU 2 + K̂s∆̂ ds (3) With ∆̂i (s) = a (θi (s)− θi (−L/2)) − Ui (s) , a = |∆~rk| the inter-protofilament distance , i = x, y and ~U (s) = (Ux, Uy) = (Reχû1, Imχû1) with χ = 1 − e−2πi/N and renormalized constants B̂ = NB, K̂c = NKc/ (4− 4 cos (2π/N)) and K̂s = NKs. Visually the new variable ~U (s) is a x-y vector at each MT -crosscut and can be interpreted as the (vectorial) mean over rel- ative PF displacements of neighboring PFs. With this enormous simplification at hand we can consider now ba- sic motor induced MT excitations (Fig 1). There are two elementary configurations in which motors can induce in- ternal MT strains: 1) A motor (or a complex of several motors) acting between two (not necessarily neighboring) PFs and 2) A motor (or a complex of several motors) act- ing at two points within the same PF. For reasons that will soon become clear we call the excitation 1 S-type or simply an ”S-let” and excitation 2 we call an V-type ex- citation or ”V-let”. Both excitations are ”internal” in the sense that there is no net torque or force on the system motor+MT similarly to the case of a beating flagellum[1]. Elementary internal MT excitations. In the following we want to understand the properties of the two basic types of excitations from Fig 1 and focus on the S-type first. We assume a single motor (or a complex of two motors) at position s = s0 bridging between two PF with index k1 and k2 and exerting opposing forces F and −F onto them respectively, Fig 1 a) (left) + b). The total energy is Etot = EMT + ES−mot with EMT given by eqs 1-2 and the potential energy of the motor ES−mot = −F ∑k=k2 ∫ L/2 δ (s− s0)∆k (s) ds . As we had for EMT before ES−mot also decouples into inde- pendent modes in the Fourier representation over k and Emot,S = E S−mot+ E S−mot+E S−mot with E S−mot a cur- vature independent term and the two shape dependent contributions E S−mot (in x and y direction) given by: EiS−mot = −F ∫ L/2 δ (s− s0) ∆̂i (s) ds with i = x, y and ~µ being the vector connecting the two attachment points of the motor (or motor complex) with components µi = ~µ · ~ei , Fig 1 b. The equilibrium solution is given by the Euler Lagrange equations: δEitot/δUi = 0 and δEitot/δθi = 0 with E tot = E S,mot + E MT and bound- ary conditions θ′i (±L/2) = U i (±L/2) = 0 (vanishing bending and shearing stresses at the ends). A short cal- culation leads to ∆̂′′i − λ −2∆̂i = (1 + α) δ (s− s0) with the shear decay length λ = λ−2c + λ )−1/2 Here the two important length scales λc = K̂c/K̂s and λB = B̂/a2K̂shave the physical meaning of a pure compression- / pure bending- induced shear screen- ing length respectively, with their squared ratio α = (λc/λB) ≈ 35 (N = 13 PF). The remaining equations lead to conservation laws (aθi + αUi) = 0 which com- bined with the equation for ∆̂i give for the simplest sym- metric case s0 = 0 the tangent angles (up to an arbitrary constant) θi (s) = Φi cosh (|s| /λ− L/2λ) sinh (L/2λ) With Φi = Φ i = ~µ · ~ei . The resulting MT backbone curvature has a jump at s = 0 and attains its maximal modulus there. The resulting MT shape is planar (con- tained in the plane spanned by ~µ and ~t at any position) and S-shape-like with initial and final angle coinciding θi (−L/2) = θi (+L/2) which explains our nomenclature ”S-type excitation” or ”S-let”. Interestingly the length scale λ over which θi (s) declines allows us to indepen- dently estimate the ratio of stretch and shear moduli from the observation of S-let deformations coming from katanin action[7], Y/Ks ≈ 64π −1 (λ/a) ≈ 6 × 105 for λ ≈ 1µm (cf, Fig 2b) and a ≈ 6nm. This value is close to the result obtained by Pampaloni et al. [3] (Y/Ks ≈ 10 Further the maximal deflection angle of θmax ≈ 38 (= 0.66) measured in Fig 2a gives via 4 an estimate for the involved motor forces F ≈ 2B̂θmax/(λ |µ|) = πa4Y/(λ |µ|) = 20−250pN (for |µ| = a ≈ 6nm and the range of values for Y from literature [2, 3]). This indi- FIG. 2: a: The shapes of V and S-type excitations as given by Eq. 4 and their experimental observations for katanin op- erating on MTs (b,c). b: Adapted and edge enhanced image from [7]. c: Adapted from [6], black bars highlight the MT de- formation. The upper and lower images show the MT before and after kink generation respectively. cates that many katanin motors might act cooperatively to generate the observed shape change in Fig 2b. Another interesting possibility is that katanin might be different from dynein/kinesin by generating only small contractile displacement ”powerstrokes” . 1nm with a more effi- cient chemical-mechanical ATP-energy conversion lead- ing to larger contractile forces F & 15kBT/nm = 60 pN . The second fundamental internal MT excitation ap- pears when a motor (or motor complex) acts along a single PF with index k compressing or stretching it. The motor energy in this case can be written as EV,mot = F (uk (s0 + |µ| /2)− uk (s0 − |µ| /2)) ≈ F |µ| δ (s− s0)u k (s) ds where |µ| is the size of the motor step. Like in the previous case it is sufficient to keep the energy contribution of the mode q = 1 as the others decouple from each other and from the curvature term. Along very similar line of derivation as in the S-type excitation case we obtain a solution which is planar and contained in the plane spanned by the vector ~rk and ~t. The resulting tangent angles for an excitation in the middle of the MT (s0 = 0) are given by eq 4 with Φi = Φ i (s) = (~ei ·∆~rk) γiF |µ| 2(1+α)B̂ which is now s dependent and changes sign at s = 0. Here (γx, γy) = sin (π/N) (1− cos (2π/N)) (sin (2π (k + 1/2)/N) ,− cos (2π (k + 1/2) /N)) . The re- sulting shape, that we call a ”V-let”, is a smooth V- shaped planar kink in the MT backbone with continuous curvature which relaxes on the length-scale λ. While su- perficially similar the S-let and V-let solutions are physi- cally very different for two reasons. First, the s/ |s| factor in the V-let solution changes the symmetry with respect to the S-let which leads to dramatic effects on the end- end distance as we see below. Second difference lies in the different scaling of the numerical prefactors which in the case of a V-let do not contain the screening length λ and involve additionally a very large reduction factor 1/ (1 + α) . In practice this suppresses significantly the involved deformations (θ ≈ 10−5 − 10−6), orders of mag- nitude below that of a S-let corresponding to the same force (θ ≈ 10−1). However the situation changes dramat- ically if the motor is operating at a position of a vacancy in the tubulin lattice, cf. Fig 1. In this case the mo- tor is not hindered by the large rigidity of the short PF portion that the motor acts on. Formally there is no re- quirement of continuity for uk (s) of the involved PF k at the position of the gap. For such a combined defect + motor excitation which we call V gap-let, the motor en- ergy is given by EV gap,mot = F (uk (s0 + 0)− uk (s0 − 0)) and its Ux/y dependent component becomes E V gap,mot = Fγi (Ui (s0 + 0)− Ui (s− 0)) . After a short calcula- tion in direct analogy with the previous cases we obtain the same form as in eq 4 but with Φi = Φ V gap i (s) = (~ei ·∆~rk) .While its functional form and symme- try coincide with that of the V-let, the prefactor of a V gap-let solution is more similar in magnitude to that of a S-let. It is intuitive to think of a V gap - let as a V-let with an effectively renormalized motor force F̃ = 2a−1λ (1 + α)F. The distribution of angles of observed V-shapes measured by Davies et al [6] of θmax ≈ 15 30◦(cf.Fig 2c) would suggest for the defect free case (V- let) a very large required force of F = 6×104−6×105pN . However , the same estimate for the V gap-let case gives much more moderate 10 − 100pN (few tens of motors), showing the prime importance of lattice defects in the case of V type excitations. Interestingly Davies et al. [6] also suggested a crucial role of lattice defects based on the pattern and kinetics of MT decomposition by katanin. Statistical mechanics of multiple excitations. It is particularly interesting to understand the collective contribution of a large number of internal MT ex- citations acting at random positions sj and orien- tations −→µ j in addition to the MT thermal fluctu- ations. The motor energy for S- and V-lets be- comes now respectively EiS,mot = − j∆̂i (sj) and EiV,mot = −F j |µ| γ i (sj). It is convenient to in- troduce the vector −→µ S/V,j such that µ S,j ≡ µ j and µiV,j ≡ |µ| γ j (i = x, y). For a fixed (but arbitrary) distribution of motors the partition functions Zi = DUiDθi exp EiMT + E and the correla- tion functions θi,qθi,p (q, p 6= 0) can be obtained. Here 〈...〉 denotes the average over the random motore distri- bution and (...) is the average over the thermal noise. θi,qθi,p decomposes into the sum of a thermal contri- bution 〈θi,qθi,p〉T = 2kBTL −1G(q)δp,q with the propa- gator G(q) = B̂q2 + q 2K̂sa q2+K̂s/K̂c and a motor con- tribution θi,qθi,p = ΨS/V (p)ΨS/V (q)CS/V (q, p) . For S- and V-let case we have respectively ΨS (p) = 2FK̂cp L(K̂cp2+K̂s) G(p) and ΨV (p) = 2FaK̂sp L(K̂cp2+K̂s) G(p), with the motor position and orientation correla- tor CS/V (q, p) = j,l µ S/V,jµ S/V,l cos (qsj) cos (psl) CS/V (q, p) is easily computed for the simplest choice of a uniform motor position and orientation distribu- tion: P (si) = 1/L , −→µ S/V,j = ∣µS/V ∣ (cosφj , sinφj) with an random angle φj with a probability distribution P (φj) = 1/2π. In this case we obtain the length reduc- /L ≈ 1 θ2x,q θ2y,q which naturally decomposes in a sum of a thermal fluctuation term and a motor term . In the relevant limiting case L/λ ≫ 1 we obtain for the thermal part a2K̂sl0p with the large and small scale persistence lengths given by l∞p = B̂ + a2K̂c /kBT and l (α/ (1 + α)) B̂/kBT . Interestingly the term a can be formally understood as an intrinsic self-tension straightening the MT at small scales. Similar formulas appear in different geometries for the railway-track[4] and wormlike-bundle model[5]. The motor dependent length reduction for the S-, V- and V gap-let excitations with line density ρ is given by: ρF 2µ2 a4K̂2sλ ρF 2µ2L a2K̂2sλ V gap = cV gap ρF 2µ2L a4K̂2sλ with cS = α 2 (1 + α) /16 ≈ 6 × 10−2, cV = 0.18α2 (1 + α) ≈ 1.3 × 10−4 and cV gap = 0.73α2 (1 + α) ≈ 0.7. Remarkably the S- and V/V gap- lets show different scaling. In particular V/V gap grows with L (in analogy to the first term in the ther- mal contribution 5) while /L stays length inde- pendent. The physical reason for this difference becomes obvious from Fig 2, as the relative slack /L induced by a single S-let scales with λ/L, while for an V/V gap- let it is essentially length independent. For longer MTs this effect leads to a strong dominance of V gap-lets over S-lets V gap ∼ (L/λ) ≫ 1. Although hav- ing the same L scaling the minute prefactor of defect free V-lets renders their contribution relatively insignif- icant ∼ a2λ−3L ≈ 10−6 even for very long MTs (L = 100µm), underlining the importance of lattice vacancies transforming a V-let into a V gap- let. Another interesting observation is that in all three cases S/V/V gap ∝ ρF 2. Fixing the number of mo- tors Nmot but regrouping them into Nmot/M clusters of size M we have ρ → M−1ρ, F → MF and therefore S/V/V gap ∝ M, i.e. the slack grows linearly with the cluster size. This indicates that cooperativity (po- sitional correlation) of motor action can lead to strong enhancement of the slack length. From Eqs.5-7 we can derive criteria for the dominance of motor slack over the thermal slack. For instance us- ing the values estimated from Fig 2 b,c for katanin for elastic constants from [3] (F = 20pN , λ = 1µm) and µ = 8nm, L = 20µm we obtain /L = ρ/ρc with ρc,S ≈ 0.25nm −1, ρc,V gap ≈ 1.2 × 10 −3nm−1. For large enough motor densities the katanin action easily domi- nates over the thermal slack /L ≈ 6× 10−4. Being evolutionary specialized for MT deformation and degradation katanin is likely to be among the strongest slack generating motors. We suspect however that classi- cal motors like dynein and kinesin might cause less pro- nounced but observable effects as well. While dynein is known to walk between several PFs, kinesin is very strictly following a single one[1]. Our theory suggest that dynein should induce moving S-lets, yet with quickly fluc- tuating signs which would diminish the effect consider- ably. A battery of many kinesins, however, walking over a MT region with many tubulin vacancies, would give rise to spatially stationary V gap-lets blinking between ”on” and ”off” states. The theoretical and experimental ex- ploration of these issues is an interesting future direction. The authors acknowledge fruitful discussions with E. Frey, C. Heussinger, M.Bathe, O. Campas, J.F. Joanny and P.C. Nelson. I.M.K. acknowledges support by the Max-Planck Society. [1] J. Howard, Mechanics of Motor Proteins and the Cy- toskeleton. Sinauer Press (2001); L.A. Amos and W.G. Amos, Molecules of the Cytoskeletion, Guilford (1991). [2] A. Kis et al. Phys. Rev. Lett. 89: 248101 (2002) [3] F. Pampaloni et al. PNAS 103: 10248 (2006) [4] R. Everaers, R. Bundschuh, and K. Kremer, Europhys. Lett 29, 263 , (1995) [5] C. Heussinger, M. Bathe and E. Frey, [cond-mat/0702097] [6] L.J. Davis, D.J. Odde, S.M. Block, and S.P. Gross, Bio- phys. J. 82, 29162927 (2002) [7] J.J. Hartmann et. al, Cell, Vol. 93, 277287; Movie at http://valelab.ucsf.edu/images/mov-rhomtsvkat.mov with kind permission by R. Vale http://arxiv.org/abs/cond-mat/0702097 http://valelab.ucsf.edu/images/mov-rhomtsvkat.mov

Bone Cancer Rate in the Fossil Record Bone Cancer Rates in Dinosaurs Compared with Modern Vertebrates L.C. Natarajan1, A.L. Melott1, B.M. Rothschild2, and L.D. Martin2 1 Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045 2 Biodiversity Research Center, University of Kansas, Lawrence, KS 66045 ABSTRACT Data on the prevalence of bone cancer in dinosaurs is available from past radiological examination of preserved bones. We statistically test this data for consistency with rates extrapolated from information on bone cancer in modern vertebrates, and find that there is no evidence of a different rate. Thus, this test provides no support for a possible role of ionizing radiation in the K-T extinction event. INTRODUCTION Cancer is one of the few diseases that can be reliably diagnosed in vertebrate remains (Rothschild and Martin 2006). Cancer prevalence may potentially help shed light on biosphere stress. Some scenarios for biodiversity fluctuation and mass extinctions include radiation effects, which might increase the cancer rate (e.g. Medvedev and Melott 2007). Since dinosaurs were involved in a major mass extinction, it is interesting to see whether there is any evidence of an elevated cancer rate. While primary bone cancer can be defined as cancer that begins in the bone, secondary or metastatic bone cancer is cancer that originates elsewhere in the body and spreads to the bone. Bone is the third most common site for metastasis after lung and liver (Alvarez 1948; Abrams, Spiro and Goldstein 1950; Wierman, Clagett and McDonald 1954; Resnick 2002). Whereas a variety of cancers can spread to the bone, the most common forms in humans are breast, prostate, lung, kidney and thyroid (American Cancer Society 2007). In this study, the prevalence of secondary bone cancer in dinosaurs is examined for consistency with rates in modern vertebrates. While 2-3% benign hemangiomas (proliferation of vascular endothelium) were found in the family Hadrosauridae, only one case of metastatic cancer was found (within the same family). The diseased vertebra was that of an Edmontosaurus (Rothschild et al. 2003) specimen from the Maastrichtian stage. It is particularly interesting that this case came from near the end-Cretaceous, when the extinction event took place. We note that the elevated rate of hemangiomas was also Maastrichtian, which is interesting, but we have not studied this question since it cannot be excluded that, for example, it is a consequence of the diet or some other peculiarity of this family. The focus of our study is to do a statistical test for consistency between the rates in dinosaurs versus modern vertebrates. DATA AND METHODS In order to test for consistency, we need to establish an expected rate based on data from modern vertebrates, and then ask whether the event in the dinosaur fossil record is consistent with that rate. While the spine is the most commonly affected area of bone metastasis in the human skeleton, the next most common areas of spread include the ilium, pubis, ischium, proximal femur, femora and skull (American Cancer Society 2007). Thus, if a more comprehensive study is to be performed, examination of these regions is suggested. Furthermore, as suggested by Rothschild and Rothschild (1995), visual examination is complementary to radiologic examination and therefore should be included if the epidemiology of metastatic disease is to be determined reliably. However, the radiologic method alone was the basis of the sample and control rate estimates used in this study. The prevalence of bone cancer in human skeletons was examined. We justify the comparison based on a superficial similarity of large vertebrates with substantial life spans (Erickson et al. 2006). The rate of bone metastasis was estimated with respect to x-ray identified metastatic disease in the Hamann-Todd Collection (Rothschild and Rothschild 1995). The Hamann-Todd Collection is the largest and best preserved compilation of human skeletons for which a background demographic is known (Rothschild and Woods 1991; Rothschild and Martin 1993). From a total of 2906 defleshed skeletons, 33 cases of metastatic disease were identified fluoroscopically, yielding a probability of 1.14 %. Based on necropsy results of captive wild animals (Effron, Griner and Benirschke 1977), the rate of cancer in reptiles is approximately 1/8th that found in humans. Thus, the rate of cancer in dinosaurs can be tested for consistency with a rate of 0.142%. Another estimate can be made based on the rate in birds. The rate of macroscopically observable cancer in birds was less than 24 in 50,000 (Rothschild and Panza 2005), indicating that the rate of cancer in dinosaurs can be tested for consistency with 0.048%. The Poisson distribution was subsequently applied to both rates in order to calculate probability. Our null hypothesis is that the rate of such cancer in dinosaurs is not higher than either of these rates. The data concerning the prevalence of cancer in the fossil record was presented in Rothschild et al. (2003). The epidemiology of tumors in dinosaurs was investigated by fluoroscopically screening dinosaur vertebrae, a technique which allowed the examination of vertebrae in real time, thus negating the need for film. A total of 10,312 vertebrae from 708 individual dinosaurs of varying families were examined, and one such tumor was found. RESULTS The malignant Edmontosaurus tumor reported by Rothschild et al. (2003) is metastatic cancer of unknown (primary) origin. In the table below, we show the probability of finding various numbers of subjects with tumors (left column) in a sample of 708 examined, assuming the rate for reptiles (center column) and birds (right column) It was determined that the rate of metastatic bone disease in the dinosaur fossil record is in fact consistent with the predicted rate of bone metastasis in the dinosaur record (Table 1). The probability of finding less than one cancer is 37% according to the reptilian cancer rate, and 71% according to the avian rate, so the observed event does not represent an elevated rate. We cannot negate the null hypothesis. Thus, there does not appear to be an elevated rate of bone cancer in the dinosaur fossil record. Note that if the null hypothesis had been formulated as “not lower rate”, we would not be able to reject that either, based on the size of the sample and the expected rates. Dinosaurs (708) Reptiles Birds # with cancer Probability Probability 0 0.36605 0.71188 1 0.36787 0.24193 2 0.18486 0.04111 3 0.06193 0.00466 4 0.01556 0.00040 5 0.00313 0.00003 6 0.00052 0.00000 7 0.00008 0.00000 8 0.00001 0.00000 Table 1 Examination of vertebrae from 708 dinosaurs revealed 1 case of metastatic disease. Thus, the frequency of metastatic bone cancer is consistent with the risk estimated from rates in modern vertebrates, as determined using the Poisson distribution. DISCUSSION We have asked whether the bone cancer rate in dinosaurs is consistent with rates for modern vertebrates. Results of past examination of dinosaur fossil bones for evidence of metastatic bone cancer are tested for consistency with available information on rates in modern vertebrates. We were able to make two rate estimates, not more than a factor of three apart, based on available data. We then ask about the probability of finding one bone cancer in 708 dinosaurs based on these assumed rates. We find that with either rate, this is not an unexpected outcome. We conclude that there is no evidence for an abnormal rate of bone cancer in dinosaurs. Exposure to ionizing radiation can elevate the rate of bone cancers (Brenner et al. 2003). A variety of astrophysical sources of such radiation have been hypothesized to have a role in extinction events (e.g. Fields 2004; Medvedev and Melott 2007). It is therefore interesting to ask whether there is evidence for an elevated cancer rate as a possible signal for a role in ionizing radiation. The answer is no. There is however, a residual question of a somewhat elevated rate of benign hemangiomas in the family Hadrosauridae which is outside the scope of this study, but may deserve further consideration in the future. ACKNOWLEDGEMENTS This research was supported by NASA Astrobiology: Exobiology and Evolutionary Biology grant NNG04GM14G, the University of Kansas Honors Program, and by the KU Student Senate. LITERATURE CITED Abrams, H.L., Spiro, R., Goldstein, N. 1950. Metastases in carcinomas: Analysis of 1000 autopsied cases. Cancer 3 p. 74-85. Alvarez, G.H. 1948. Clinica del cancer de pulmon. Revista de la Asociacion Medica Argentina 62 p. 690-694. American Cancer Society. Cancer Reference Information. Detailed Guide: Bone Metastasis.. URL [http://www.cancer.org/docroot/CRI/content/ CRI_2_4_1X_What_Is_bone_metastasis_66.asp]. Date accessed [April 2007]. Brenner, D.J., Doll R., Goodhead, D.T., Hall, E.J., Land, C.E., Little, J.B., Lubin, J.H., Preston, D.L., Preston, R.J., Puskin, J.S., Ron, E., Sachs, R.K., Samet, J.M., Setlow, R.B., Zaider, M. 2003. Cancer risks attributable to low doses of ionizing radiation: Assessing what we really know. Proceedings of the National Academy of Sciences 100 p 13761-13766. 10.1073/pnas.2235592100 Effron, M., Griner, L., Benirschke, K. 1977. Nature and rate of neoplasia in captive wild mammals, birds and reptiles at necropsy. Journal of the National Cancer Institute, 59 p. 185-198. Erickson, G.M., Currie, P.J., Inouye, B.D., Winn, A.A. 2006. Tyrannosaur life tables: An example of nonavian dinosaur population biology. Science, 313 p. 213-217. Fields, B.D. 2004. Live radioisotopes as signatures of nearby supernovae. New Astronomy Reviews 48 p 119-123. Medvedev, M. V., and Melott, A.L. 2007. Do extragalactic cosmic rays induce cycles in fossil diversity? Astrophysical Journal 664 p 879–889. Resnick, D. 2002. Radiology of Bone and Joint Disease. Saunders, Philadelphia. Rothschild, B.M., Martin, L. 1993. Paleopathology: Disease in the fossil record. CRC Press, London. Rothschild, B.M., Martin, L.D. 2006. Skeletal Impact of Disease. Museum of Natural History Press, Albuquerque, New Mexico. Rothschild, B.M., Panza, R.K. 2005. Epidemiologic assessment of trauma Independent skeletal pathology in non-passerine birds from museum collections. Avian Pathology, 34 p. 212-219. Rothschild, B.M., Rothschild, C. 1995. Comparison of radiologic and gross examination for detection of cancer in defleshed skeletons. American Journal of Physical Anthropology, 96 p. 357-363. Rothschild, B.M., Tanke D.H., Helbling M., Martin L.D. 2003. Epidemiologic study of tumors in dinosaurs. Naturwissenschaften 90 p. 495-500. Rothschild, B.M., Woods, R.J. 1991. Spondyloarthropathy: Erosive arthritis in representative defleshed bones. American Journal of Physical Anthropology, 85 p. 125-134. Wierman, W.H., Clagett, O.T., McDonald, J.R. 1954. Articular manifestations in pulmonary diseases. An analysis of their occurrence in 1024 cases in which pulmonary resection was performed. Journal of the American Medical Association, 155 p. 1459-1463.

Data on the prevalence of bone cancer in dinosaurs is available from past radiological examination of preserved bones. We statistically test this data for consistency with rates extrapolated from information on bone cancer in modern vertebrates, and find that there is no evidence of a different rate. Thus, this test provides no support for a possible role of ionizing radiation in the K-T extinction event.

Bone Cancer Rate in the Fossil Record Bone Cancer Rates in Dinosaurs Compared with Modern Vertebrates L.C. Natarajan1, A.L. Melott1, B.M. Rothschild2, and L.D. Martin2 1 Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045 2 Biodiversity Research Center, University of Kansas, Lawrence, KS 66045 ABSTRACT Data on the prevalence of bone cancer in dinosaurs is available from past radiological examination of preserved bones. We statistically test this data for consistency with rates extrapolated from information on bone cancer in modern vertebrates, and find that there is no evidence of a different rate. Thus, this test provides no support for a possible role of ionizing radiation in the K-T extinction event. INTRODUCTION Cancer is one of the few diseases that can be reliably diagnosed in vertebrate remains (Rothschild and Martin 2006). Cancer prevalence may potentially help shed light on biosphere stress. Some scenarios for biodiversity fluctuation and mass extinctions include radiation effects, which might increase the cancer rate (e.g. Medvedev and Melott 2007). Since dinosaurs were involved in a major mass extinction, it is interesting to see whether there is any evidence of an elevated cancer rate. While primary bone cancer can be defined as cancer that begins in the bone, secondary or metastatic bone cancer is cancer that originates elsewhere in the body and spreads to the bone. Bone is the third most common site for metastasis after lung and liver (Alvarez 1948; Abrams, Spiro and Goldstein 1950; Wierman, Clagett and McDonald 1954; Resnick 2002). Whereas a variety of cancers can spread to the bone, the most common forms in humans are breast, prostate, lung, kidney and thyroid (American Cancer Society 2007). In this study, the prevalence of secondary bone cancer in dinosaurs is examined for consistency with rates in modern vertebrates. While 2-3% benign hemangiomas (proliferation of vascular endothelium) were found in the family Hadrosauridae, only one case of metastatic cancer was found (within the same family). The diseased vertebra was that of an Edmontosaurus (Rothschild et al. 2003) specimen from the Maastrichtian stage. It is particularly interesting that this case came from near the end-Cretaceous, when the extinction event took place. We note that the elevated rate of hemangiomas was also Maastrichtian, which is interesting, but we have not studied this question since it cannot be excluded that, for example, it is a consequence of the diet or some other peculiarity of this family. The focus of our study is to do a statistical test for consistency between the rates in dinosaurs versus modern vertebrates. DATA AND METHODS In order to test for consistency, we need to establish an expected rate based on data from modern vertebrates, and then ask whether the event in the dinosaur fossil record is consistent with that rate. While the spine is the most commonly affected area of bone metastasis in the human skeleton, the next most common areas of spread include the ilium, pubis, ischium, proximal femur, femora and skull (American Cancer Society 2007). Thus, if a more comprehensive study is to be performed, examination of these regions is suggested. Furthermore, as suggested by Rothschild and Rothschild (1995), visual examination is complementary to radiologic examination and therefore should be included if the epidemiology of metastatic disease is to be determined reliably. However, the radiologic method alone was the basis of the sample and control rate estimates used in this study. The prevalence of bone cancer in human skeletons was examined. We justify the comparison based on a superficial similarity of large vertebrates with substantial life spans (Erickson et al. 2006). The rate of bone metastasis was estimated with respect to x-ray identified metastatic disease in the Hamann-Todd Collection (Rothschild and Rothschild 1995). The Hamann-Todd Collection is the largest and best preserved compilation of human skeletons for which a background demographic is known (Rothschild and Woods 1991; Rothschild and Martin 1993). From a total of 2906 defleshed skeletons, 33 cases of metastatic disease were identified fluoroscopically, yielding a probability of 1.14 %. Based on necropsy results of captive wild animals (Effron, Griner and Benirschke 1977), the rate of cancer in reptiles is approximately 1/8th that found in humans. Thus, the rate of cancer in dinosaurs can be tested for consistency with a rate of 0.142%. Another estimate can be made based on the rate in birds. The rate of macroscopically observable cancer in birds was less than 24 in 50,000 (Rothschild and Panza 2005), indicating that the rate of cancer in dinosaurs can be tested for consistency with 0.048%. The Poisson distribution was subsequently applied to both rates in order to calculate probability. Our null hypothesis is that the rate of such cancer in dinosaurs is not higher than either of these rates. The data concerning the prevalence of cancer in the fossil record was presented in Rothschild et al. (2003). The epidemiology of tumors in dinosaurs was investigated by fluoroscopically screening dinosaur vertebrae, a technique which allowed the examination of vertebrae in real time, thus negating the need for film. A total of 10,312 vertebrae from 708 individual dinosaurs of varying families were examined, and one such tumor was found. RESULTS The malignant Edmontosaurus tumor reported by Rothschild et al. (2003) is metastatic cancer of unknown (primary) origin. In the table below, we show the probability of finding various numbers of subjects with tumors (left column) in a sample of 708 examined, assuming the rate for reptiles (center column) and birds (right column) It was determined that the rate of metastatic bone disease in the dinosaur fossil record is in fact consistent with the predicted rate of bone metastasis in the dinosaur record (Table 1). The probability of finding less than one cancer is 37% according to the reptilian cancer rate, and 71% according to the avian rate, so the observed event does not represent an elevated rate. We cannot negate the null hypothesis. Thus, there does not appear to be an elevated rate of bone cancer in the dinosaur fossil record. Note that if the null hypothesis had been formulated as “not lower rate”, we would not be able to reject that either, based on the size of the sample and the expected rates. Dinosaurs (708) Reptiles Birds # with cancer Probability Probability 0 0.36605 0.71188 1 0.36787 0.24193 2 0.18486 0.04111 3 0.06193 0.00466 4 0.01556 0.00040 5 0.00313 0.00003 6 0.00052 0.00000 7 0.00008 0.00000 8 0.00001 0.00000 Table 1 Examination of vertebrae from 708 dinosaurs revealed 1 case of metastatic disease. Thus, the frequency of metastatic bone cancer is consistent with the risk estimated from rates in modern vertebrates, as determined using the Poisson distribution. DISCUSSION We have asked whether the bone cancer rate in dinosaurs is consistent with rates for modern vertebrates. Results of past examination of dinosaur fossil bones for evidence of metastatic bone cancer are tested for consistency with available information on rates in modern vertebrates. We were able to make two rate estimates, not more than a factor of three apart, based on available data. We then ask about the probability of finding one bone cancer in 708 dinosaurs based on these assumed rates. We find that with either rate, this is not an unexpected outcome. We conclude that there is no evidence for an abnormal rate of bone cancer in dinosaurs. Exposure to ionizing radiation can elevate the rate of bone cancers (Brenner et al. 2003). A variety of astrophysical sources of such radiation have been hypothesized to have a role in extinction events (e.g. Fields 2004; Medvedev and Melott 2007). It is therefore interesting to ask whether there is evidence for an elevated cancer rate as a possible signal for a role in ionizing radiation. The answer is no. There is however, a residual question of a somewhat elevated rate of benign hemangiomas in the family Hadrosauridae which is outside the scope of this study, but may deserve further consideration in the future. ACKNOWLEDGEMENTS This research was supported by NASA Astrobiology: Exobiology and Evolutionary Biology grant NNG04GM14G, the University of Kansas Honors Program, and by the KU Student Senate. LITERATURE CITED Abrams, H.L., Spiro, R., Goldstein, N. 1950. Metastases in carcinomas: Analysis of 1000 autopsied cases. Cancer 3 p. 74-85. Alvarez, G.H. 1948. Clinica del cancer de pulmon. Revista de la Asociacion Medica Argentina 62 p. 690-694. American Cancer Society. Cancer Reference Information. Detailed Guide: Bone Metastasis.. URL [http://www.cancer.org/docroot/CRI/content/ CRI_2_4_1X_What_Is_bone_metastasis_66.asp]. Date accessed [April 2007]. Brenner, D.J., Doll R., Goodhead, D.T., Hall, E.J., Land, C.E., Little, J.B., Lubin, J.H., Preston, D.L., Preston, R.J., Puskin, J.S., Ron, E., Sachs, R.K., Samet, J.M., Setlow, R.B., Zaider, M. 2003. Cancer risks attributable to low doses of ionizing radiation: Assessing what we really know. Proceedings of the National Academy of Sciences 100 p 13761-13766. 10.1073/pnas.2235592100 Effron, M., Griner, L., Benirschke, K. 1977. Nature and rate of neoplasia in captive wild mammals, birds and reptiles at necropsy. Journal of the National Cancer Institute, 59 p. 185-198. Erickson, G.M., Currie, P.J., Inouye, B.D., Winn, A.A. 2006. Tyrannosaur life tables: An example of nonavian dinosaur population biology. Science, 313 p. 213-217. Fields, B.D. 2004. Live radioisotopes as signatures of nearby supernovae. New Astronomy Reviews 48 p 119-123. Medvedev, M. V., and Melott, A.L. 2007. Do extragalactic cosmic rays induce cycles in fossil diversity? Astrophysical Journal 664 p 879–889. Resnick, D. 2002. Radiology of Bone and Joint Disease. Saunders, Philadelphia. Rothschild, B.M., Martin, L. 1993. Paleopathology: Disease in the fossil record. CRC Press, London. Rothschild, B.M., Martin, L.D. 2006. Skeletal Impact of Disease. Museum of Natural History Press, Albuquerque, New Mexico. Rothschild, B.M., Panza, R.K. 2005. Epidemiologic assessment of trauma Independent skeletal pathology in non-passerine birds from museum collections. Avian Pathology, 34 p. 212-219. Rothschild, B.M., Rothschild, C. 1995. Comparison of radiologic and gross examination for detection of cancer in defleshed skeletons. American Journal of Physical Anthropology, 96 p. 357-363. Rothschild, B.M., Tanke D.H., Helbling M., Martin L.D. 2003. Epidemiologic study of tumors in dinosaurs. Naturwissenschaften 90 p. 495-500. Rothschild, B.M., Woods, R.J. 1991. Spondyloarthropathy: Erosive arthritis in representative defleshed bones. American Journal of Physical Anthropology, 85 p. 125-134. Wierman, W.H., Clagett, O.T., McDonald, J.R. 1954. Articular manifestations in pulmonary diseases. An analysis of their occurrence in 1024 cases in which pulmonary resection was performed. Journal of the American Medical Association, 155 p. 1459-1463.

arXiv:0704.1913v3 [hep-ph] 31 Oct 2007 Effective Lagrangian for the χ+j χ l interaction in the minimal supersymmetric standard model and neutral Higgs decays Tarek Ibrahim Department of Physics, Northeastern University, Boston, MA 02115-5000, USA Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt 1 Abstract We extend previous analyses of the supersymmetric loop correction to the neutral Higgs couplings to include the coupling χ+j χ l . The analysis completes the pre- vious analyses where similar corrections were computed for the τ̄ τH0l , b̄bH l , c̄cH and for t̄tH0l couplings within the minimal supersymmetric standard model. The effective one loop Lagrangian is then applied to the computation of the neutral Higgs decays. The sizes of the supersymmetric loop corrections of the neutral Higgs decay widths into χ+i χ j (i = 1, 2; j = 1, 2) are investigated and the super- symmetric loop correction is found to be in the range of 7 ∼ 15% in significant regions of the parameter space. By including the loop corrections of the other decay channels b̄b, t̄t, τ̄ τ , c̄c, and χ0iχ j (i = 1 − 4; j = 1 − 4), the corrections to branching ratios for H0l → χ+i χ−j can reach as high as 40%. The effects of CP phases on the branching ratio are also investigated. Permanent address. http://arxiv.org/abs/0704.1913v3 1 INTRODUCTION The neutral Higgs couplings to different fields are of great current interest as they enter in a variety of phenomena which are testable in low energy processes [1]. It is known that supersymmetric corrections can affect the neutral Higgs boson decays into bb̄, τ τ̄ and cc̄. The decay properties of the lightest Higgs boson in MSSM would be different from those of the Standard Model Higgs boson when these corrections are taken into consideration. Specifically the ratio of the branching ratios to bb̄ and τ τ̄ of the Higgs boson is an important piece of evidence that might distinguish between the lightest MSSM Higgs boson and the Standard Model one at colliders. In MSSM there are also other modes for neutral Higgs decays that do not exist in Standard Model such as charginos and neutralinos. In this paper we compute the one loop corrected effective Lagrangian for the neutral Higgs and chargino couplings. We then analyze the effects of the loop corrections to the neutral Higgs decays H0l → χ+j χ−k . In the analysis we also include the effect of CP phases arising from the soft SUSY breaking parameters. It is well known that large CP phases can be made compatible [2, 3, 4] with experimental constraints on the electric dipole moments (edms) of the electron [5], of the neutron [6], and of the Hg199 [7]. Further, if the phases are large they could affect the Higgs sector physics. It is well known that one loop contributions to the Higgs masses from the stop, sbottom, the chargino and neutralino sectors can lift the lightest Higgs mass above MZ . The inclusion of the CP violating phases brings mixings between the CP even and the CP odd Higgs [8, 9, 10, 22, 23, 24]. The CP violating phases modifies the physics of dark matter [11], and of other phenomena [12]. (For a review see Ref.[13].) The current analysis of ∆LH0χ+χ− and neutral Higgs decay into charginos is based on the effective Lagrangian method where the couplings of the electroweak eigen states H11 and H 2 with charginos are radiatively corrected using the zero external momentum approximation. The same technique has been used in calcu- lating the effective Lagrangian and decays ofH0l into quarks and leptons [1, 15, 16]. It has been used also in the analysis of the effective Lagrangian of charged Higgs with quarks [1, 17] and their decays into t̄b and νττ [18] and into chargino + neu- tralino [19]. The neutral Higgs decays into charginos have been investigated before in the CP conserving case [20, 21]. In these analyses, the wave function renormal- ization and the counter terms for the mass matrix elements are calculated beside the vertex corrections of the mass eigen states h0, H0 and A0 with charginos. In the effective Lagrangian technique with zero external momentum approximation, the radiative corrections of the processes considered here originate only from the vertex contributions. Thus our analysis of the neutral Higgs decays into charginos is a partial one. However, as mentioned before the above analyses were carried out in the CP conserving scenario. As far as we know, the analysis for the neutral Higgs decays into charginos, with one loop corrections, in the CP violating case where the neutral Higgs sector is modified in couplings, spectrum and mixings, does not exist. We evaluate the radiative corrections to the Higgs boson masses and mixngs by using the effective potential approximation. We include the corrections from the top and bottom quarks and squarks [22], from the chargino, the W and the charged Higgs sector [23] and from the neutralino, Z boson, and the neutral Higgs bosons [24]. It is important to notice that the corrections to the Higgs effective potential from the different sectors mentioned above are all one-loop corrections. The corrections of the interaction ∆LH0χ+χ− to be considered in this work are all one-loop level ones. So the analysis presented here is a consistent one loop study. The outline of the rest of the paper is as follows: In Sec. 2 we compute the effective Lagrangian for the χ+j χ l interaction. In Sec. 3 we give an analysis of the decay widths of the neutral Higgs bosons into charginos using the effective Lagrangian. In Sec. 4 we give a numerical analysis of the size of the loop effects on the partial decay width and on the branching ratios. Conclusions are given in Sec. 5. 2 LOOP CORRECTIONS TO NEUTRAL HIGGS COUPLINGS The tree-level Lagrangian for χ+j χ 0 interaction is L = φjkχ+j PRχ+kH11 + ψjkχ+j PRχ+kH22 +H.c., (1) where H11 andH 2 are the neutral states of the two Higgs isodoublets in the minimal supersymmetric standard model (MSSM), i.e., (H1) = , (H2) = and the couplings φjk and ψjk are given by φjk = −gUk2Vj1, ψjk = −gUk1Vj2 (3) where U and V diagonalize the chargino mass matrix so that U∗Mχ+V −1 = diag(mχ+ , mχ+ ) (4) The loop corrections produce shifts in the couplings of Eq. (1) and the effective Lagrangian with loop corrected couplings is given by Leff = (φjk + δφjk)χ+j PRχ+kH11 +∆φjkχ+j PLχ+kH22 + (ψjk + δψjk)χ j PRχ 2 +∆ψjkχ j PLχ 1 +H.c. (5) In this work we calculate the loop correction to the χ+j χ l using the zero external momentum approximation. 2.1 Loop analysis of δφjk and ∆ψjk Contributions to δφjk and ∆ψjk arise from the thirteen loop diagram of Fig. 1. We note that the contribution from diagrams which have H+W+H0and H0Z0H0 vertices do not contribute in the effective Lagrangian with zero external momentum approximation since these vertices are proportional to the external momentum. We discuss now in detail the contribution of each of these diagrams in Fig. 1. We begin with the loop diagram of Fig. 1i(a) which contributes to δφjk and ∆ψjk. We calculate the corrections of the amplitude from Fig. 1i(a) δM = iδφjkūjPRvk + i∆ψjkūjPLvk (6) The idea is to extract, from the amplitude correction, the expressions for δφjk and ∆ψjk from those parts that are proportional to ūjPRvk and ūjPLvk respectively. For this purpose we need b̃b̃H11 interaction which is given by Lb̃b̃H1 = Hilb̃ 1 +H.c. (7) where Hil is given by Hil = − 2 cos θW sin2 θW )D b1iDb1l − sin2 θWD b2iDb2l) cos β gm2b√ 2mW cos β (D∗b1iDb1l +D b2iDb2l)− gmbAb√ 2mW cos β D∗b2iDb1l (8) The matrix elements Dq are defined as D+q M q̃Dq = diag(m ) (9) We need also the t̄χ+b̃ interaction which is given by Lt̄χ+b̃ = −gχ̄ k [(U − κbU∗k2D∗b2i)PL −κtVk2D∗b1iPR]tb̃ i +H.c (10) where κt,b are given by 2mW sin β 2mW cos β For external momenta s, q and q − s the amplitude correction from loop 1i(a) is given by δM = −g2Hilū(q − s)[CLjlPL + CRjlPR] (2π)4 [( 6s+ 6ℓ) +mt][C∗LkiPR + C PL]v(s) ((s+ ℓ)2 −m2t + iǫ)(ℓ2 −m2b̃l + iǫ)((ℓ+ q) 2 −m2 + iǫ) where CLjl and CRjl are given by CLjl = U − κbU∗j2D∗b2l CRjl = −κtVj2D The part in the numerator [CLjlPL + CRjlPR][( 6s+ 6ℓ) +mt] (C∗LkiPR + C PL) (14) could be written as [CLjlC PL + CRjlC PR]( 6s+ 6ℓ) +mt[CRjlC PR + CLjlC PL] (15) by using the facts that γµPL = PRγ µ, PLPR = 0, P L = PL and P R = PR. The first term in Eq. (15) does not contribute to δφjk or ∆ψjk since it does not have the same Lorentz structure. The second term of Eq. (15) contributes the part of mtCRjlC to δφjk and mtCLjlC to ∆ψjk. Thus the loop corrections δφjk and ∆ψjk read iδφjk = −g2HilmtCRjlC i∆ψjk = −g2HilmtCLjlC where (2π)4 ((s+ ℓ)2 −m2t + iǫ)(ℓ2 −m2b̃l + iǫ)((ℓ+ q) 2 −m2 + iǫ) Now for zero external momentum approximation we set s = q = 0, and the integral would read (2π)4 (ℓ2 −m2t + iǫ)(ℓ2 −m2b̃l + iǫ)(ℓ 2 −m2 + iǫ) A detailed calculation of this integral is given in the appendix. Using the above one finds for δφjk the contribution: jk = κt HilVj2D (Uk1Db1i − κbUk2Db2i)f(m2t , m2b̃l, m ) (19) where f(x, y, z) = (x− y)(x− z)(z − y) × (zxln + xyln + yzln ), (20) f(x, y, y) = (y − x)2 × (xln + x− y) (21) Similarly one finds for the correction ∆ψjk from the same loop the following con- tribution jk = κt k2Db1i(U − κbU∗j2D∗b2l)f(m t , m ) (22) Next for the loop Fig. 1ii(a) we find jk = 0 jk = 0 (23) For the loop of Fig. 1i(b) we find jk = κb FliUk2D (Vj1Dt1l − κtVj2Dt2l)f(m b , m , m2t̃l) jk = κb j2Dt1l(V − κtV ∗k2D∗t2i)f(m b , m , m2t̃l) (24) where Fli is given by Fli = − 2 cos θW sin2 θW )D t1lDt1i + sin2 θWD t2lDt2i) cos β gmtµ√ 2mW sin β D∗t1lDt2i (25) For the loop of Fig. 1ii(b) we find jk = 0 jk = −κb g2m2b U∗j2Dt1i(V − κtV ∗k2D∗t2i)f(m b , m b , m ) (26) For loop of Fig. 1ii(c) we find jk = 2g ik sin βǫ lj cos β f(m2χ0 , m2χ0 , m2H+) jk = 0 (27) where ǫ and ǫ are given by ǫji = −gX4jV ∗i1 − tan θWX1jV ji = −gX∗3jUi1 + X∗2jUi2 + tan θWX 1jUi2 (28) The parameters Q ij are defined as: [X∗3i(X 2j − tan θWX∗1j)] (29) The matrix elements X are defined as XTMχ0X = diag(mχ0 , mχ0 , mχ0 , mχ0 ) (30) For loop of Fig. 1i(c) we find gmW cos β [1 + 2 sin2 β − cos 2β tan2 θW ] ik sin βǫ ij cos β f(m2χ0 , m2H+ , m gmW cos β [1 + 2 sin2 β − cos 2β tan2 θW ] ǫik cos βǫ ij sin β f(m2χ0 , m2H+ , m H+) (31) For loop of Fig. 1i(d) we find jk = g 3 mZ cos β 2 cos θW ((Ym1 − iYm3 sin β)(3Yl1 + iYl3 sin β) −2(Ym2 − iYm3 cos β)(Yl2 + iYl3 cos β)− 4Ym2(Yl1 − iYl3 sin β) tanβ) (Qki(Yl1 + iYl3 sin β) + Ski(Yl2 + iYl3 cos β)) (Qij(Ym1 + iYm3 sin β) + Sij(Ym2 + iYm3 cos β)) , m2H0m , m jk = g 3 mZ cos β 2 cos θW ((Ym1 − iYm3 sin β)(3Yl1 + iYl3 sin β) −2(Ym2 − iYm3 cos β)(Yl2 + iYl3 cos β)− 4Ym2(Yl1 − iYl3 sin β) tanβ) (Q∗ik(Yl1 − iYl3 sin β) + S∗ik(Yl2 − iYl3 cos β)) (Q∗ji(Ym1 − iYm3 sin β) + S∗ji(Ym2 − iYm3 cos β)) , m2H0m , m ) (32) where Qji = − 1√2gφij and Sji = ψij , and the matrix elements Y are defined as YM2HiggsY T = diag(m2 For loop of Fig. 1ii(d) we find jk = −g2 (Qli(Ym1 + iYm3 sin β) + Slj(Ym2 + iYm3 cos β))(Qki(Ym1 + iYm3 sin β) +Ski(Ym2 + iYm3 cos β)) , m2H0m , m jk = 0 (33) For loop of Fig. 1ii(e) we find jk = 0 cos2 θW , m2Z0 , m ) (34) The parameters L′ and R′ are defined by ij = −Vi1V ∗j1 − j2 + δij sin ij = −U∗i1Uj1 − U∗i2Uj2 + δij sin 2 θW (35) For loop of Fig. 1i(e) we find jk = − 2g3mZ cos β cos3 θW L′jiR , m2Z0 , m jk = − 2g3mZ cos β cos3 θW R′jiL , m2Z0 , m ) (36) For loop of Fig. 1ii(f) we find jk = 0 jk = −4 Q”ilR ljLik f(m2χ0 , m2W+, m ) (37) where L, R and Q” are defined as Lij = − X∗4iV j2 +X Rij = X3iUj2 +X2iUj1 gQ” = (X∗3i(gX 2j − g′X∗1j) + (i↔ j)) (38) For loop of Fig. 1i(f) we find jk = − 4g3mW cos β√ L∗ijRik f(m2χ0 , m2W+, m jk = − 4g3mW cos β√ R∗ijLik f(m2χ0 , m2W+, m W+) (39) For loop of Fig. 1ii(g) we find jk = 0 jk = −g2hτκτU∗j2V ∗k1 f(m2τ , m τ , m ) (40) where 2mW cos β The loop corrections for δφjk and ∆ψjk are given by δφjk = ∆ψjk = jk (42) 2.2 Loop analysis of ∆φjk and δψjk We do the same analysis of Fig. 2 as for Fig. 1. We write down here the final results for both corrections from the thirteen loops together. The corrections are written in the same order of the loops in Fig. 2. ∆φjk = κt k2Db1i(U − κbU∗j2D∗b2l)f(m t , m −κtht g2m2t V ∗k2Db1i(U − κbU∗j2D∗b2i)f(m t , m t , m j2Dt1l(V − κtV ∗k2D∗t2i)f(m b , m , m2t̃l) gmW sin β [1 + 2 cos2 β + cos 2β tan2 θW ] ǫik cos βǫ ij sin β f(m2χ0 , m2H+ , m mZ cos β 2 cos θW (tanβ(Yl2 − iYl3 cos β)(3Ym2 + iYm3 cos β) −4Yl1(Ym2 − iYm3 cos β)− 2 tanβ(Ym1 − iYm3 sin β)(Yl1 + iYl3 sin β)) (Q∗ik(Yl1 − iYl3 sin β) + S∗ik(Yl2 − iYl3 cos β)) (Q∗ji(Ym1 − iYm3 sin β) + S∗ji(Ym2 − iYm3 cos β)) , m2H0m , m cos2 θW , m2Z0 , m 2g3mZ sin β cos3 θW R′jiL , m2Z0 , m S”ilR ljLik f(m2χ0 , m2W+, m 4g3mW sin β√ R∗ijLik f(m2χ0 , m2W+, m +0 (43) where G and E are given by Gij = 2 cos θW sin2 θW )D b1iDb1j − sin2 θWD b2iDb2j) sin β gmbµ√ 2mW cos β D∗b1iDb2j Eij = 2 cos θW sin2 θW )D t1iDt1j + sin2 θWD t2iDt2j) sin β 2mW sin β (D∗t1iDt1j +D t2iDt2j)− gmtAt√ 2mW sin β D∗t2iDt2j (44) and S” is given by S”li = − sin β δli −Q”li cos β − R”li) R”li = ∗X∗1lX 1i + m̃2 ∗X∗2lX 2i − µ∗(X∗3lX∗4i +X∗4lX∗3i)) (45) The corrections δψjk are given by δψjk = κt GilVj2D (Uk1Db1i − κbUk2Db2i)f(m2t , m2b̃l , m EliUk2D (Vj1Dt1l − κtVj2Dt2l)f(m b , m , m2t̃l) ik sin βǫ lj cos β f(m2χ0 , m2χ0 , m2H+) gmW sin β [1 + 2 cos2 β + cos 2β tan2 θW ] ik sin βǫ ij cos β f(m2χ0 , m2H+ , m mZ cos β 2 cos θW (tanβ(Yl2 − iYl3 cos β)(3Ym2 + iYm3 cos β) −4Yl1(Ym2 − iYm3 cos β)− 2 tanβ(Ym1 − iYm3 sin β)(Yl1 + iYl3 sin β)) (Qki(Yl1 + iYl3 sin β) + Ski(Yl2 + iYl3 cos β)) (Qij(Ym1 + iYm3 sin β) + Sij(Ym2 + iYm3 cos β)) , m2H0m , m (Qlj(Ym1 + iYm3 sin β) + Slj(Ym2 + iYm3 cos β))(Qki(Ym1 + iYm3 sin β) +Ski(Ym2 + iYm3 cos β)) , m2H0m , m 2g3mZ sin β cos3 θW L′jiR , m2Z0 , m 3mW sin β√ L∗ijRik f(m2χ0 , m2W+, m +0 (46) where S ′ is given by [X∗4j(X 2i − tan θWX∗1i)] (47) 3 Neutral Higgs decays including loop effects We summarize now the result of the analysis. Thus Leff of Eq.(5) may be written as follows Leff = H0l χ+j (αlSjk + γ5αlPjk)χ+k +H.c (48) where αlSjk = ((Yl1+iYl3 sin β)(φjk+δφjk+∆ψjk)+(Yl2+iYl3 cos β)(ψjk+δψjk+∆φjk)) and where αlPjk = ((Yl1+iYl3 sin β)(φjk+δφjk−∆ψjk)+(Yl2+iYl3 cos β)(ψjk+δψjk−∆φjk)) Next we discuss the implications of the above result for the decay of the neutral Higgs. Γljk(H l → χ+j χ−k ) = )2 − 4m2 ((|αlSjk|)2 + (|αlPjk |)2)(M2H0 ((|αlSjk|)2 − (|αlPjk |)2)(2mχ+ )])(51) There are many channels for H0l decays. The important channels for the decay of the neutral Higgs boson are b̄b, t̄t, s̄s, c̄c, τ̄ τ , χ+i χ j and χ j . There is another set of channels that neutral Higgs can also decay into: these are modes of decaying into the other fermions of the SM, squarks, sleptons, other Higgs bosons, W and Z boson pairs, one Higgs and a vector boson, γγ pairs and finally into the gluonic decay i.e, H0l → gg. We neglect the lightest SM fermions for the smallness of their couplings. We choose the region in the parameter space where we can ignore the other channels which either are not allowed kinematically or suppressed by their couplings. Thus in this work, squarks and sleptons are too heavy to be relevant in neutral Higgs decay. The neutral Higgs decays into nonsupersymmetric final states that involve gauge bosons and/or other Higgs bosons are ignored as well. In the region of large tan β, these decays typically contribute less than 1% of the total Higgs decay rate [25]. Thus we can neglect these final states. We calculate the radiative corrected partial decay widths of the important channels mentioned above. In the case of CP violating case under investigation we use for the radiatively corrected Γ of neutral Higgs into quarks and leptons the analysis of [16], for the radiatively corrected partial widths into charginos we use the current analysis, and for the radiatively corrected decay width into neutralino we use [26]. We define Γ(H0l → χ+i χ−j )− Γ0(H0l → χ+i χ−j ) Γ0(H0l → χ+i χ−j ) where the first term in the numerator is the decay width including the full loop corrections and the second term is the decay width evaluated at the tree level. Finally to quantify the size of the loop effects on the branching ratios of the neutral Higgs decay we define the following quantity Br(H0l → χ+i χ−j )− Br0(H0l → χ+i χ−j ) Br0(H0l → χ+i χ−j ) where the first term in the numerator is the branching ratio including the full loop corrections and the second term is the branching ratio evaluated at the tree level. The analysis of this section is utilized in Sec.(4) where we give a numerical analysis of the size of the loop effects and discuss the effect of the loop corrections on decay widths and branching ratios. 4 NUMERICAL ANALYSIS In this section we discuss in a quantitative fashion the size of loop effects on the partial decay width and the branching ratios of the neutral Higgs bosons into charginos. The analysis of Sec. 2 is quite general and valid for the minimal su- persymmetric standard model. For the sake of numerical analysis we will limit the parameter space by working within the framework of the SUGRA model [14]. Specifically we will work within the framework of the the extended mSUGRA model including CP phases. We take as our parameter space at the grand unifica- tion scale to be the following: the universal scalar mass m0, the universal gaugino mass m1/2, the universal trilinear coupling |A0|, the ratio of the Higgs vacuum ex- pectation values tanβ =< H2 > / < H1 > where H2 gives mass to the up quarks and H1 gives mass to the down quarks and the leptons. In addition, we take for CP phases the following: the phase θµ of the Higgs mixing parameter µ, the phase αA0 of the trilinear coupling A0 and the phases ξi(i = 1, 2, 3) of the SU(3)C , SU(2)L and U(1)Y gaugino masses. In this analysis the electroweak symmetry is broken by radiative effects which allows one to determine the magnitude of µ by fixing MZ . In the analysis we use one loop renormalization group (RGEs) equations for the evolution of the soft susy breaking parameters and for the parameter µ, and two loop RGEs for the gauge and Yukawa couplings. In the numerical analysis we com- pute the loop corrections and also analyze their dependence on the phases. The masses of particles involved in the analysis are ordered as follows: for charginos < mχ+ and for the neutral Higgs (mH1 , mH2, mH3) → (mH , mh, mA) in the limit of no CP mixing where mH is the heavy CP even Higgs, mh is the light CP even Higgs, and mA is the CP odd Higgs. We investigate the question of how large loop corrections are relative to the tree values. We first discuss the magnitude of the loop corrections of the partial decay width defined in Eq.(52). As we mentioned earlier the loop corrections to the partial decay width of the chargino channel have been investigated before in the CP conserving case [20, 21]. The correction in these analyses is of the order of ∼ 10% of the tree level value. Our analysis supports this conclusion. In Figs. (3) and (4) we give a plot of ∆Γ l (l = 1, 3) as a function of tan β for the specific set of inputs given in the captions of these figures. We notice that the partial decay width gets a change of 7 ∼ 15% of its tree level value. We also notice that the CP violating phase θµ can affect the magnitude of this change. This effect has not been addressed in the previous analyses as they are working in the CP conserving scenario. To compare between our analysis and the previous ones we have to notice that these analyses are using the general SUSY parameter space where they put by hand all the parameters that control the analysis. In [20], the authors choose the SUSY parameter set SPS1a of the Snowmass Points and Slopes as a reference point. They choose for the trilinear couplings the values of At = −487 GeV, Ab = −766 GeV and Aτ = −250 GeV. The values of the other parameters are: M = 197.6 GeV, M ′ = 98 GeV, µ = 353.1 GeV, tan β = 10, mA0 = 393.6 GeV, MQ̃1,2 = 558.9 GeV, MŨ1,2 = 540.5 GeV, MD̃1,2 = 538.5 GeV, ML̃1,2 = 197.9 GeV, MẼ1,2 = 137.8 GeV, MQ̃3 = 512.2 GeV, MŨ3 = 432.8 GeV, MD̃3 = 536.5 GeV, ML̃3 = 196.4 GeV and MẼ3 = 134.8 GeV. In all the figures of [20], these values are used, if not specified otherwise. In our mSUGRA analysis the magnitude of all these parameters and others are fixed by the five input parameters m0 = 100 GeV, m1/2 = 250 GeV, tanβ = 10, A0 = −100 GeV and a positive sign of µ in the CP conserving scenario [27]. These parameters are different from those of our Figs. (3) and (4). By using these parameters and fixing some of them by hand when needed to match their values in the analysis of [20], we were able to have a fair agreement with their Figs. (2-9). As an example of this check we show in Table.1 a comparison of the two works. For the input of Fig. 2 of [20] with CP violating phases are set to zero we can see that partial decay widths in both works have the same behavior as functions of masses and their magnitudes are fairly close to each other. However it seems that our loop corrected values of the partial widths are different from those of Eberl et al. This could be understood since our loop analysis of the effective lagrangian includes only the vertex corrections beside the corrections in the Higgs potential. case Γtreeeberl Γ our Γ eberl Γ 2.a mA0 = 700 GeV 0.95 GeV 0.94 GeV 0.85 GeV 0.80 GeV 2.a mA0 = 800 GeV 1.18 GeV 1.17 GeV 1.0 GeV 0.91 GeV 2.b mH0 = 800 GeV 0.7 GeV 0.69 GeV 0.63 GeV 0.58 GeV 2.b mH0 = 900 GeV 0.8 GeV 0.8 GeV 0.73 GeV 0.70 GeV Table 1: A comparison between the current analysis and Eberl et al [20] for bench- mark cases. In the work of Ref. [21] only 8 out of 26 diagrams of the present analysis are calculated and they correspond to the vertex corrections from Figs. (1,2ii(a)), (1,2ii(b)), (1,2i(b)) and (1,2i(a)). By considering these diagrams only in the com- parison, our analysis is in fair agreement with their Figs (2-4) and Figs. (6,8) for their inputs. Now we turn to address the question of how much loop corrections can affect the branching ratios into charginos. The branching ratio of a decay mode is defined to be the ratio between the partial decay rate of this mode and the total decay rate. In the parameter space under investigation this total decay rate includes the rates of decays into charginos, heavy quarks, taus and neutralinos. In Figs. (5) and (6) we give a plot of ∆Br l (l = 1, 3) defined by Eq.(53) as a function of tanβ for the specific set of inputs given in the captions of these figures. Fig. (5) is for the neutral Higgs H1 boson and Fig. (6) is for the neutral Higgs H3 boson. In all regions of the parameter space investigated in this work, the decay of the lightest Higgs boson H2 into charginos is forbidden kinematically, since we have in these regions the fact that 2mχ− > mH2 . The analysis of Figs. (5) and (6) shows that the loop correction varies strongly with tanβ with the correction changing sign for the case of H3 decay. Further, the analysis shows that the loop correction can be as large as about −40% of the tree contribution for both H1 and H3 cases. We also notice that the behavior of ∆Br l (l = 1, 3) as a function of tan β changes considerably by changing the phase of µ. So for some values of this phase we find that this parameter increases as tanβ increases and for other values of θµ we see that it decreases as tan β increases. As shown in the previous figures, the parameter tan β is playing a strong role. This parameter is important at the tree level through the diagonalizing mass matrices of the chargino and neutral Higgs and their spectrum. At the loop level it has extra effect explicitly in α jk and implicitly through the radiatively corrected matrix elements Ylm and through the corrections δφjk, ∆φjk, δψjk, ∆ψjk. The values of the branching ratios themselves at tree and one loop levels are shown in Table.2. θµ(rad) Br 0(H1) Br loop(H1) Br 0(H3) Br loop(H3) 0.5 6% 4.7% 18.2% 13.8% 1.0 8.4% 6.9% 21.3% 18.1% 1.5 9.2% 7.9% 23.4% 22.2% Table 2: Values of branching ratios at tree and one-loop levels of neutral Higgs into the channel χ+1 χ 1 at tan β = 24 for the input of Figs. (5) and (6) We notice that their magnitudes are not negligible for the region of the param- eter space investigated. These non negligible branching ratios for the decay of the neutral Higgs into charginos suggest that these decay modes could be measurable at the soon-to-operate LHC. However, one should also consider the production rates for H1 and H3 bosons to assess whether the change in branching ratios could be detectable at colliders. This analysis goes beyond the scope of the current work. We also notice that the phase of the parameter µ affects the tree level branching ratios as well. This comes mainly from the structure of the chargino matrix. The more important channels in the region of the parameter space investigated are the decay into bottom and top quarks. They have the highest values of branching ratios. The radiative corrections of these channels are also more than those of the charginos and neutralinos. These channels were studied before [1, 15, 16] as mentioned above. However a 20% of branching ratio for the case of neutral Higgs as shown in the above table is not very small and could justify carrying out the current analysis. In Figs. (7) and (8) we give a plot of ∆Br l (l = 1, 3) as a function of |A0| for the specific set of inputs given in the caption of these figures. The analysis of these figures shows that the loop corrections are substantial and reaches the value of −38% of the tree contribution for the case of H1 decay and the value of −43% for the case of H3 decay. Next we investigate the effects of CP violating phases on the loop corrections of the neutral Higgs decays into charginos. In Figs. (9) and (10) we give a plot of ∆Br l (l = 1, 3) as a function of θµ for the specific set of inputs given in the caption of these figures. The analysis of the figures shows that the loop correction has a sharp dependence on θµ. Further, the correction is changing sign as θµ varies from 0 to π for two cases of H3 decay. Thus θµ affects not only the magnitude of l but also its sign depending on the value of θµ. In Figs. (11) and (12) we give a plot of ∆Br l (l = 1, 3) as a function of αA0 for the specific set of inputs given in the caption of these figures. Here also we find a very substantial dependence of ∆Br l on αA0. This dependence is very large in the case of H3 decay and it exceeds −40% of the tree contribution. In Figs. (13) and (14) we give a plot of ∆Br l (l = 1, 3) as a function of ξ2 for the specific set of inputs given in the caption of these figures. Here we find a small effect of this phase on the loop corrections. 5 CONCLUSION In this paper we have carried out an analysis of the supersymmetric loop correc- tions to χ+j χ l couplings within MSSM. In supersymmetry after spontaneous breaking of electroweak symmetry one is left with three neutral Higgs bosons which in the absence of CP phases consist of two CP even Higgs bosons and one CP odd Higgs boson. In the absence of loop corrections, the lightest Higgs bo- son mass satisfies the inequality mh < MZ and by including these corrections the lightest Higgs mass can be lifted above MZ . With the inclusion of CP phases the Higgs boson mass eigenstates are no longer CP even and CP odd states when loop corrections to the Higgs boson mass matrix are included. Further, inclusion of loop corrections to the couplings of charginos and neutral Higgs is in general dependent on CP phases. Thus the decays of neutral Higgs into charginos can be sensitive to the loop corrections and to the CP violating phases. The effect of the supersymmetric loop corrections is found to to be in the range of 7 ∼ 15% for the partial decay width. For the branching ratios it is found to be be rather large, as much as 40% in some regions of the parameter space. The effect of CP phases on the modifications of the partial decay width and the branching ratio is found to be substantial in some regions of the MSSM parameter space. Acknowledgments I wish to acknowledge useful discussions with Professor Pran Nath. The support of the Physics Department at Alexandria University is also acknowledged. 6 APPENDIX The integral of import to this work is (2π)4 (k2 −m21 + iǫ)(k2 −m22 + iǫ)(k2 −m23 + iǫ) It could be written in the form (2π)4 where a = k2 −m21 + iǫ b = k2 −m22 + iǫ c = k2 −m23 + iǫ (56) Using Feynman parametrization, 1 could be written as ∫ 1−x [a + (b− a)x+ (c− a)z]3 The denominator in the above integral could be written in the form k2 +M2 + iǫ where M2 = (m21 − m22)x + (m21 − m23)z −m21. Thus the integral J can take the (2π)4 ∫ 1−x [k2 +M2 + iǫ]3 Now integrating over k and using the standard integral, for n ≥ 3 (2π)4 (k2 + Λ + iǫ)n = iπ2 Γ(n− 2) one can find that the integral J has the form (4π)2 ∫ 1−x α + βz where α = (m21 −m22)x−m21 and β = m21 −m23. Integrating over z one can get for the integral J the form of (4π)2 m21 −m23 dx ln(δ1x−m23)− ln(δ2x−m21) (61) where δ1 = m 3 −m22 and δ2 = m21 −m22. 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D 60, 055006(1999). [23] T. Ibrahim, P. Nath, Phys. Rev. D 63, 035009(2001). [24] T. Ibrahim, P. Nath, Phys. Rev. D 66, 015005(2002). [25] J. F. Gunion, H. E. Haber, Nucl. Phys. B 307, 445 (1988); A. Djouadi, hep/ph/9712334; A. Djouadi, J. Kalinowski, and M. Spira, Comp. Phys. Commun. 108, 56 (1998). [26] The neutralino decay of neutral Higgs with CP phases will be discussed elsewhere. [27] B. C. Allanach et al., Eur. Phys. J. C 25, 113 (2002). [28] M. Carena, M. Olechowski, S. Pokorski and C. E. M. Wagner, Nucl. Phys. B426 (1994) 269; L. J. Hall, R. Rattazzi and O. Sarid, Phys. Rev. D50 (1994) 7048; T. Ibrahim and P. Nath, Phys. Rev. D 67, 095003(2003). s2, v2s1, v1 jf2f1 Figure 1: Set of diagrams contributing to radiative corrections δφjk and ∆ψjk. (i): (a) s1 = b̃ i , s2 = b̃ l , f = t; (b) s1 = t̃i, s2 = t̃l, f = b̄; (c) s1 = H +, s2 = H f = χ0i ; (d) s1 = H l , s2 = H m, f = χ i ; (e) v1 = Z 0, v2 = Z 0, f = χ+i ; (f) v1 = W +, v2 = W +, f = χ0i . (ii): (a) f1 = t, f2 = t, s = b̃ i ; (b) f1 = b̄, f2 = b̄, s = t̃i; (c)f1 = χ i , f2 = χ l , s = H +; (d) f1 = χ i , f2 = χ l , s = H m; (e) f1 = χ f2 = χ l , v = Z 0; (f) f1 = χ i , f2 = χ l , v = W ; (g) f1 = τ +, f2 = τ +, s = ν̃τ . s2, v2s1, v1 jf2f1 Figure 2: Set of diagrams contributing to radiative corrections ∆φjk and δψjk. (i): (a) s1 = b̃ i , s2 = b̃ l , f = t; (b) s1 = t̃i, s2 = t̃l, f = b̄; (c) s1 = H +, s2 = H f = χ0i ; (d) s1 = H l , s2 = H m, f = χ i ; (e) v1 = Z 0, v2 = Z 0, f = χ+i ; (f) v1 = W +, v2 = W +, f = χ0i . (ii): (a) f1 = t, f2 = t, s = b̃ i ; (b) f1 = b̄, f2 = b̄, s = t̃i; (c)f1 = χ i , f2 = χ l , s = H +; (d) f1 = χ i , f2 = χ l , s = H m; (e) f1 = χ f2 = χ l , v = Z 0; (f) f1 = χ i , f2 = χ l , v = W ; (g) f1 = τ +, f2 = τ +, s = ν̃τ . Figure 3: tanβ dependence of ∆Γ1 → χ+1 χ−1 . The curves in ascending order correspond to θµ = 0.2, 0.4, 0.6 (rad). The input is m0 = 350 GeV, m1/2 = 180 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), αA0 = 0.8 (rad) and |A0| = 250 Figure 4: tanβ dependence of ∆Γ3 → χ+1 χ−1 . The curves in ascending order correspond to θµ = 0.2, 0.4, 0.6 (rad). The input is m0 = 350 GeV, m1/2 = 180 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), αA0 = 0.8 (rad) and |A0| = 250 ∆ Br1 Figure 5: tan β dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at tanβ = 40 correspond to θµ = 0.5, 0.1, 1.0, 1.5 and 2.0 (rad). The input is m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), αA0 = 0.3 (rad) and |A0| = 250 GeV. ∆ Br3 Figure 6: tan β dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at tanβ = 40 correspond to θµ = 0.5, 0.1, 1.0, 1.5 and 2.0(rad). The input is m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), αA0 = 0.3 (rad) and |A0| = 250 GeV. ∆ Br1 Figure 7: |A0| dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at |A0| = 0 correspond to tanβ = 40, 35, 30, 25 and 20. The input is m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), θµ = 0.7 (rad) and αA0 = 0.1 (rad). ∆ Br3 Figure 8: |A0| dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at |A0| = 0 correspond to tanβ = 40, 35, 30, 25 and 20. The input is m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), θµ = 0.7 (rad) and αA0 = 0.1 (rad). ∆ Br1 Figure 9: θµ dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at θµ = 2.0 (rad) correspond to |A0| = 100, 250, 500, 750 and 900 GeV. The input is tanβ = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad) and αA0 = 0.2 (rad). 0-505 ∆ Br3 Figure 10: θµ dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at θµ = π (rad) correspond to |A0| = 100, 250, 500, 750 and 900 GeV. The input is tan β = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad) and αA0 = 0.2 (rad). ∆ Br1 Figure 11: α0 dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at αA0 = 2.2 (rad) correspond to |A0| = 500, 450, 400, 100 and 200 GeV. The input is tanβ = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad) and θµ = 0.1 (rad). ∆ Br3 Figure 12: α0 dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at αA0 = 2.2 (rad) correspond to |A0| = 500, 450, 400, 100 and 200 GeV. The input is tanβ = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad) and θµ = 0.1 (rad). ∆ Br1 Figure 13: ξ2 dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at ξ2 = 0.75 (rad) correspond to |A0| = 50, 100, 150, 200 and 250 GeV. The input is tan β = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ3 = 0.6 (rad) and θµ = 0.2 (rad) and αA0 = 0.3 (rad). ∆ Br3 Figure 14: ξ2 dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at ξ2 = 0.75 (rad) correspond to |A0| = 50, 100, 150, 200 and 250 GeV. The input is tan β = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ3 = 0.6 (rad) and θµ = 0.2 (rad) and αA0 = 0.3 (rad).

We extend previous analyses of the supersymmetric loop correction to the neutral Higgs couplings to include the coupling $\chi^{+}_j \chi^{-}_kH^{0}_l$. The analysis completes the previous analyses where similar corrections were computed for the $\bar{\tau} \tau H^{0}_l$, $\bar{b} b H^{0}_l$, $\bar{c} c H^{0}_l$ and for $\bar{t} t H^{0}_l$ couplings within the minimal supersymmetric standard model. The effective one loop Lagrangian is then applied to the computation of the neutral Higgs decays. The sizes of the supersymmetric loop corrections of the neutral Higgs decay widths into $\chi^{+}_i \chi^{-}_j$ ($i=1,2$; $j=1,2$) are investigated and the supersymmetric loop correction is found to be in the range of $7\sim15%$ in significant regions of the parameter space. By including the loop corrections of the other decay channels $\bar{b} b$, $\bar{t} t$, $\bar{\tau} \tau$, $\bar{c} c$, and $\chi^0_i \chi^0_j$ ($i=1-4$; $j=1-4$), the corrections to branching ratios for $H^{0}_l\to \chi^{+}_i \chi^{-}_j$ can reach as high as 40%. The effects of CP phases on the branching ratio are also investigated.

arXiv:0704.1913v3 [hep-ph] 31 Oct 2007 Effective Lagrangian for the χ+j χ l interaction in the minimal supersymmetric standard model and neutral Higgs decays Tarek Ibrahim Department of Physics, Northeastern University, Boston, MA 02115-5000, USA Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt 1 Abstract We extend previous analyses of the supersymmetric loop correction to the neutral Higgs couplings to include the coupling χ+j χ l . The analysis completes the pre- vious analyses where similar corrections were computed for the τ̄ τH0l , b̄bH l , c̄cH and for t̄tH0l couplings within the minimal supersymmetric standard model. The effective one loop Lagrangian is then applied to the computation of the neutral Higgs decays. The sizes of the supersymmetric loop corrections of the neutral Higgs decay widths into χ+i χ j (i = 1, 2; j = 1, 2) are investigated and the super- symmetric loop correction is found to be in the range of 7 ∼ 15% in significant regions of the parameter space. By including the loop corrections of the other decay channels b̄b, t̄t, τ̄ τ , c̄c, and χ0iχ j (i = 1 − 4; j = 1 − 4), the corrections to branching ratios for H0l → χ+i χ−j can reach as high as 40%. The effects of CP phases on the branching ratio are also investigated. Permanent address. http://arxiv.org/abs/0704.1913v3 1 INTRODUCTION The neutral Higgs couplings to different fields are of great current interest as they enter in a variety of phenomena which are testable in low energy processes [1]. It is known that supersymmetric corrections can affect the neutral Higgs boson decays into bb̄, τ τ̄ and cc̄. The decay properties of the lightest Higgs boson in MSSM would be different from those of the Standard Model Higgs boson when these corrections are taken into consideration. Specifically the ratio of the branching ratios to bb̄ and τ τ̄ of the Higgs boson is an important piece of evidence that might distinguish between the lightest MSSM Higgs boson and the Standard Model one at colliders. In MSSM there are also other modes for neutral Higgs decays that do not exist in Standard Model such as charginos and neutralinos. In this paper we compute the one loop corrected effective Lagrangian for the neutral Higgs and chargino couplings. We then analyze the effects of the loop corrections to the neutral Higgs decays H0l → χ+j χ−k . In the analysis we also include the effect of CP phases arising from the soft SUSY breaking parameters. It is well known that large CP phases can be made compatible [2, 3, 4] with experimental constraints on the electric dipole moments (edms) of the electron [5], of the neutron [6], and of the Hg199 [7]. Further, if the phases are large they could affect the Higgs sector physics. It is well known that one loop contributions to the Higgs masses from the stop, sbottom, the chargino and neutralino sectors can lift the lightest Higgs mass above MZ . The inclusion of the CP violating phases brings mixings between the CP even and the CP odd Higgs [8, 9, 10, 22, 23, 24]. The CP violating phases modifies the physics of dark matter [11], and of other phenomena [12]. (For a review see Ref.[13].) The current analysis of ∆LH0χ+χ− and neutral Higgs decay into charginos is based on the effective Lagrangian method where the couplings of the electroweak eigen states H11 and H 2 with charginos are radiatively corrected using the zero external momentum approximation. The same technique has been used in calcu- lating the effective Lagrangian and decays ofH0l into quarks and leptons [1, 15, 16]. It has been used also in the analysis of the effective Lagrangian of charged Higgs with quarks [1, 17] and their decays into t̄b and νττ [18] and into chargino + neu- tralino [19]. The neutral Higgs decays into charginos have been investigated before in the CP conserving case [20, 21]. In these analyses, the wave function renormal- ization and the counter terms for the mass matrix elements are calculated beside the vertex corrections of the mass eigen states h0, H0 and A0 with charginos. In the effective Lagrangian technique with zero external momentum approximation, the radiative corrections of the processes considered here originate only from the vertex contributions. Thus our analysis of the neutral Higgs decays into charginos is a partial one. However, as mentioned before the above analyses were carried out in the CP conserving scenario. As far as we know, the analysis for the neutral Higgs decays into charginos, with one loop corrections, in the CP violating case where the neutral Higgs sector is modified in couplings, spectrum and mixings, does not exist. We evaluate the radiative corrections to the Higgs boson masses and mixngs by using the effective potential approximation. We include the corrections from the top and bottom quarks and squarks [22], from the chargino, the W and the charged Higgs sector [23] and from the neutralino, Z boson, and the neutral Higgs bosons [24]. It is important to notice that the corrections to the Higgs effective potential from the different sectors mentioned above are all one-loop corrections. The corrections of the interaction ∆LH0χ+χ− to be considered in this work are all one-loop level ones. So the analysis presented here is a consistent one loop study. The outline of the rest of the paper is as follows: In Sec. 2 we compute the effective Lagrangian for the χ+j χ l interaction. In Sec. 3 we give an analysis of the decay widths of the neutral Higgs bosons into charginos using the effective Lagrangian. In Sec. 4 we give a numerical analysis of the size of the loop effects on the partial decay width and on the branching ratios. Conclusions are given in Sec. 5. 2 LOOP CORRECTIONS TO NEUTRAL HIGGS COUPLINGS The tree-level Lagrangian for χ+j χ 0 interaction is L = φjkχ+j PRχ+kH11 + ψjkχ+j PRχ+kH22 +H.c., (1) where H11 andH 2 are the neutral states of the two Higgs isodoublets in the minimal supersymmetric standard model (MSSM), i.e., (H1) = , (H2) = and the couplings φjk and ψjk are given by φjk = −gUk2Vj1, ψjk = −gUk1Vj2 (3) where U and V diagonalize the chargino mass matrix so that U∗Mχ+V −1 = diag(mχ+ , mχ+ ) (4) The loop corrections produce shifts in the couplings of Eq. (1) and the effective Lagrangian with loop corrected couplings is given by Leff = (φjk + δφjk)χ+j PRχ+kH11 +∆φjkχ+j PLχ+kH22 + (ψjk + δψjk)χ j PRχ 2 +∆ψjkχ j PLχ 1 +H.c. (5) In this work we calculate the loop correction to the χ+j χ l using the zero external momentum approximation. 2.1 Loop analysis of δφjk and ∆ψjk Contributions to δφjk and ∆ψjk arise from the thirteen loop diagram of Fig. 1. We note that the contribution from diagrams which have H+W+H0and H0Z0H0 vertices do not contribute in the effective Lagrangian with zero external momentum approximation since these vertices are proportional to the external momentum. We discuss now in detail the contribution of each of these diagrams in Fig. 1. We begin with the loop diagram of Fig. 1i(a) which contributes to δφjk and ∆ψjk. We calculate the corrections of the amplitude from Fig. 1i(a) δM = iδφjkūjPRvk + i∆ψjkūjPLvk (6) The idea is to extract, from the amplitude correction, the expressions for δφjk and ∆ψjk from those parts that are proportional to ūjPRvk and ūjPLvk respectively. For this purpose we need b̃b̃H11 interaction which is given by Lb̃b̃H1 = Hilb̃ 1 +H.c. (7) where Hil is given by Hil = − 2 cos θW sin2 θW )D b1iDb1l − sin2 θWD b2iDb2l) cos β gm2b√ 2mW cos β (D∗b1iDb1l +D b2iDb2l)− gmbAb√ 2mW cos β D∗b2iDb1l (8) The matrix elements Dq are defined as D+q M q̃Dq = diag(m ) (9) We need also the t̄χ+b̃ interaction which is given by Lt̄χ+b̃ = −gχ̄ k [(U − κbU∗k2D∗b2i)PL −κtVk2D∗b1iPR]tb̃ i +H.c (10) where κt,b are given by 2mW sin β 2mW cos β For external momenta s, q and q − s the amplitude correction from loop 1i(a) is given by δM = −g2Hilū(q − s)[CLjlPL + CRjlPR] (2π)4 [( 6s+ 6ℓ) +mt][C∗LkiPR + C PL]v(s) ((s+ ℓ)2 −m2t + iǫ)(ℓ2 −m2b̃l + iǫ)((ℓ+ q) 2 −m2 + iǫ) where CLjl and CRjl are given by CLjl = U − κbU∗j2D∗b2l CRjl = −κtVj2D The part in the numerator [CLjlPL + CRjlPR][( 6s+ 6ℓ) +mt] (C∗LkiPR + C PL) (14) could be written as [CLjlC PL + CRjlC PR]( 6s+ 6ℓ) +mt[CRjlC PR + CLjlC PL] (15) by using the facts that γµPL = PRγ µ, PLPR = 0, P L = PL and P R = PR. The first term in Eq. (15) does not contribute to δφjk or ∆ψjk since it does not have the same Lorentz structure. The second term of Eq. (15) contributes the part of mtCRjlC to δφjk and mtCLjlC to ∆ψjk. Thus the loop corrections δφjk and ∆ψjk read iδφjk = −g2HilmtCRjlC i∆ψjk = −g2HilmtCLjlC where (2π)4 ((s+ ℓ)2 −m2t + iǫ)(ℓ2 −m2b̃l + iǫ)((ℓ+ q) 2 −m2 + iǫ) Now for zero external momentum approximation we set s = q = 0, and the integral would read (2π)4 (ℓ2 −m2t + iǫ)(ℓ2 −m2b̃l + iǫ)(ℓ 2 −m2 + iǫ) A detailed calculation of this integral is given in the appendix. Using the above one finds for δφjk the contribution: jk = κt HilVj2D (Uk1Db1i − κbUk2Db2i)f(m2t , m2b̃l, m ) (19) where f(x, y, z) = (x− y)(x− z)(z − y) × (zxln + xyln + yzln ), (20) f(x, y, y) = (y − x)2 × (xln + x− y) (21) Similarly one finds for the correction ∆ψjk from the same loop the following con- tribution jk = κt k2Db1i(U − κbU∗j2D∗b2l)f(m t , m ) (22) Next for the loop Fig. 1ii(a) we find jk = 0 jk = 0 (23) For the loop of Fig. 1i(b) we find jk = κb FliUk2D (Vj1Dt1l − κtVj2Dt2l)f(m b , m , m2t̃l) jk = κb j2Dt1l(V − κtV ∗k2D∗t2i)f(m b , m , m2t̃l) (24) where Fli is given by Fli = − 2 cos θW sin2 θW )D t1lDt1i + sin2 θWD t2lDt2i) cos β gmtµ√ 2mW sin β D∗t1lDt2i (25) For the loop of Fig. 1ii(b) we find jk = 0 jk = −κb g2m2b U∗j2Dt1i(V − κtV ∗k2D∗t2i)f(m b , m b , m ) (26) For loop of Fig. 1ii(c) we find jk = 2g ik sin βǫ lj cos β f(m2χ0 , m2χ0 , m2H+) jk = 0 (27) where ǫ and ǫ are given by ǫji = −gX4jV ∗i1 − tan θWX1jV ji = −gX∗3jUi1 + X∗2jUi2 + tan θWX 1jUi2 (28) The parameters Q ij are defined as: [X∗3i(X 2j − tan θWX∗1j)] (29) The matrix elements X are defined as XTMχ0X = diag(mχ0 , mχ0 , mχ0 , mχ0 ) (30) For loop of Fig. 1i(c) we find gmW cos β [1 + 2 sin2 β − cos 2β tan2 θW ] ik sin βǫ ij cos β f(m2χ0 , m2H+ , m gmW cos β [1 + 2 sin2 β − cos 2β tan2 θW ] ǫik cos βǫ ij sin β f(m2χ0 , m2H+ , m H+) (31) For loop of Fig. 1i(d) we find jk = g 3 mZ cos β 2 cos θW ((Ym1 − iYm3 sin β)(3Yl1 + iYl3 sin β) −2(Ym2 − iYm3 cos β)(Yl2 + iYl3 cos β)− 4Ym2(Yl1 − iYl3 sin β) tanβ) (Qki(Yl1 + iYl3 sin β) + Ski(Yl2 + iYl3 cos β)) (Qij(Ym1 + iYm3 sin β) + Sij(Ym2 + iYm3 cos β)) , m2H0m , m jk = g 3 mZ cos β 2 cos θW ((Ym1 − iYm3 sin β)(3Yl1 + iYl3 sin β) −2(Ym2 − iYm3 cos β)(Yl2 + iYl3 cos β)− 4Ym2(Yl1 − iYl3 sin β) tanβ) (Q∗ik(Yl1 − iYl3 sin β) + S∗ik(Yl2 − iYl3 cos β)) (Q∗ji(Ym1 − iYm3 sin β) + S∗ji(Ym2 − iYm3 cos β)) , m2H0m , m ) (32) where Qji = − 1√2gφij and Sji = ψij , and the matrix elements Y are defined as YM2HiggsY T = diag(m2 For loop of Fig. 1ii(d) we find jk = −g2 (Qli(Ym1 + iYm3 sin β) + Slj(Ym2 + iYm3 cos β))(Qki(Ym1 + iYm3 sin β) +Ski(Ym2 + iYm3 cos β)) , m2H0m , m jk = 0 (33) For loop of Fig. 1ii(e) we find jk = 0 cos2 θW , m2Z0 , m ) (34) The parameters L′ and R′ are defined by ij = −Vi1V ∗j1 − j2 + δij sin ij = −U∗i1Uj1 − U∗i2Uj2 + δij sin 2 θW (35) For loop of Fig. 1i(e) we find jk = − 2g3mZ cos β cos3 θW L′jiR , m2Z0 , m jk = − 2g3mZ cos β cos3 θW R′jiL , m2Z0 , m ) (36) For loop of Fig. 1ii(f) we find jk = 0 jk = −4 Q”ilR ljLik f(m2χ0 , m2W+, m ) (37) where L, R and Q” are defined as Lij = − X∗4iV j2 +X Rij = X3iUj2 +X2iUj1 gQ” = (X∗3i(gX 2j − g′X∗1j) + (i↔ j)) (38) For loop of Fig. 1i(f) we find jk = − 4g3mW cos β√ L∗ijRik f(m2χ0 , m2W+, m jk = − 4g3mW cos β√ R∗ijLik f(m2χ0 , m2W+, m W+) (39) For loop of Fig. 1ii(g) we find jk = 0 jk = −g2hτκτU∗j2V ∗k1 f(m2τ , m τ , m ) (40) where 2mW cos β The loop corrections for δφjk and ∆ψjk are given by δφjk = ∆ψjk = jk (42) 2.2 Loop analysis of ∆φjk and δψjk We do the same analysis of Fig. 2 as for Fig. 1. We write down here the final results for both corrections from the thirteen loops together. The corrections are written in the same order of the loops in Fig. 2. ∆φjk = κt k2Db1i(U − κbU∗j2D∗b2l)f(m t , m −κtht g2m2t V ∗k2Db1i(U − κbU∗j2D∗b2i)f(m t , m t , m j2Dt1l(V − κtV ∗k2D∗t2i)f(m b , m , m2t̃l) gmW sin β [1 + 2 cos2 β + cos 2β tan2 θW ] ǫik cos βǫ ij sin β f(m2χ0 , m2H+ , m mZ cos β 2 cos θW (tanβ(Yl2 − iYl3 cos β)(3Ym2 + iYm3 cos β) −4Yl1(Ym2 − iYm3 cos β)− 2 tanβ(Ym1 − iYm3 sin β)(Yl1 + iYl3 sin β)) (Q∗ik(Yl1 − iYl3 sin β) + S∗ik(Yl2 − iYl3 cos β)) (Q∗ji(Ym1 − iYm3 sin β) + S∗ji(Ym2 − iYm3 cos β)) , m2H0m , m cos2 θW , m2Z0 , m 2g3mZ sin β cos3 θW R′jiL , m2Z0 , m S”ilR ljLik f(m2χ0 , m2W+, m 4g3mW sin β√ R∗ijLik f(m2χ0 , m2W+, m +0 (43) where G and E are given by Gij = 2 cos θW sin2 θW )D b1iDb1j − sin2 θWD b2iDb2j) sin β gmbµ√ 2mW cos β D∗b1iDb2j Eij = 2 cos θW sin2 θW )D t1iDt1j + sin2 θWD t2iDt2j) sin β 2mW sin β (D∗t1iDt1j +D t2iDt2j)− gmtAt√ 2mW sin β D∗t2iDt2j (44) and S” is given by S”li = − sin β δli −Q”li cos β − R”li) R”li = ∗X∗1lX 1i + m̃2 ∗X∗2lX 2i − µ∗(X∗3lX∗4i +X∗4lX∗3i)) (45) The corrections δψjk are given by δψjk = κt GilVj2D (Uk1Db1i − κbUk2Db2i)f(m2t , m2b̃l , m EliUk2D (Vj1Dt1l − κtVj2Dt2l)f(m b , m , m2t̃l) ik sin βǫ lj cos β f(m2χ0 , m2χ0 , m2H+) gmW sin β [1 + 2 cos2 β + cos 2β tan2 θW ] ik sin βǫ ij cos β f(m2χ0 , m2H+ , m mZ cos β 2 cos θW (tanβ(Yl2 − iYl3 cos β)(3Ym2 + iYm3 cos β) −4Yl1(Ym2 − iYm3 cos β)− 2 tanβ(Ym1 − iYm3 sin β)(Yl1 + iYl3 sin β)) (Qki(Yl1 + iYl3 sin β) + Ski(Yl2 + iYl3 cos β)) (Qij(Ym1 + iYm3 sin β) + Sij(Ym2 + iYm3 cos β)) , m2H0m , m (Qlj(Ym1 + iYm3 sin β) + Slj(Ym2 + iYm3 cos β))(Qki(Ym1 + iYm3 sin β) +Ski(Ym2 + iYm3 cos β)) , m2H0m , m 2g3mZ sin β cos3 θW L′jiR , m2Z0 , m 3mW sin β√ L∗ijRik f(m2χ0 , m2W+, m +0 (46) where S ′ is given by [X∗4j(X 2i − tan θWX∗1i)] (47) 3 Neutral Higgs decays including loop effects We summarize now the result of the analysis. Thus Leff of Eq.(5) may be written as follows Leff = H0l χ+j (αlSjk + γ5αlPjk)χ+k +H.c (48) where αlSjk = ((Yl1+iYl3 sin β)(φjk+δφjk+∆ψjk)+(Yl2+iYl3 cos β)(ψjk+δψjk+∆φjk)) and where αlPjk = ((Yl1+iYl3 sin β)(φjk+δφjk−∆ψjk)+(Yl2+iYl3 cos β)(ψjk+δψjk−∆φjk)) Next we discuss the implications of the above result for the decay of the neutral Higgs. Γljk(H l → χ+j χ−k ) = )2 − 4m2 ((|αlSjk|)2 + (|αlPjk |)2)(M2H0 ((|αlSjk|)2 − (|αlPjk |)2)(2mχ+ )])(51) There are many channels for H0l decays. The important channels for the decay of the neutral Higgs boson are b̄b, t̄t, s̄s, c̄c, τ̄ τ , χ+i χ j and χ j . There is another set of channels that neutral Higgs can also decay into: these are modes of decaying into the other fermions of the SM, squarks, sleptons, other Higgs bosons, W and Z boson pairs, one Higgs and a vector boson, γγ pairs and finally into the gluonic decay i.e, H0l → gg. We neglect the lightest SM fermions for the smallness of their couplings. We choose the region in the parameter space where we can ignore the other channels which either are not allowed kinematically or suppressed by their couplings. Thus in this work, squarks and sleptons are too heavy to be relevant in neutral Higgs decay. The neutral Higgs decays into nonsupersymmetric final states that involve gauge bosons and/or other Higgs bosons are ignored as well. In the region of large tan β, these decays typically contribute less than 1% of the total Higgs decay rate [25]. Thus we can neglect these final states. We calculate the radiative corrected partial decay widths of the important channels mentioned above. In the case of CP violating case under investigation we use for the radiatively corrected Γ of neutral Higgs into quarks and leptons the analysis of [16], for the radiatively corrected partial widths into charginos we use the current analysis, and for the radiatively corrected decay width into neutralino we use [26]. We define Γ(H0l → χ+i χ−j )− Γ0(H0l → χ+i χ−j ) Γ0(H0l → χ+i χ−j ) where the first term in the numerator is the decay width including the full loop corrections and the second term is the decay width evaluated at the tree level. Finally to quantify the size of the loop effects on the branching ratios of the neutral Higgs decay we define the following quantity Br(H0l → χ+i χ−j )− Br0(H0l → χ+i χ−j ) Br0(H0l → χ+i χ−j ) where the first term in the numerator is the branching ratio including the full loop corrections and the second term is the branching ratio evaluated at the tree level. The analysis of this section is utilized in Sec.(4) where we give a numerical analysis of the size of the loop effects and discuss the effect of the loop corrections on decay widths and branching ratios. 4 NUMERICAL ANALYSIS In this section we discuss in a quantitative fashion the size of loop effects on the partial decay width and the branching ratios of the neutral Higgs bosons into charginos. The analysis of Sec. 2 is quite general and valid for the minimal su- persymmetric standard model. For the sake of numerical analysis we will limit the parameter space by working within the framework of the SUGRA model [14]. Specifically we will work within the framework of the the extended mSUGRA model including CP phases. We take as our parameter space at the grand unifica- tion scale to be the following: the universal scalar mass m0, the universal gaugino mass m1/2, the universal trilinear coupling |A0|, the ratio of the Higgs vacuum ex- pectation values tanβ =< H2 > / < H1 > where H2 gives mass to the up quarks and H1 gives mass to the down quarks and the leptons. In addition, we take for CP phases the following: the phase θµ of the Higgs mixing parameter µ, the phase αA0 of the trilinear coupling A0 and the phases ξi(i = 1, 2, 3) of the SU(3)C , SU(2)L and U(1)Y gaugino masses. In this analysis the electroweak symmetry is broken by radiative effects which allows one to determine the magnitude of µ by fixing MZ . In the analysis we use one loop renormalization group (RGEs) equations for the evolution of the soft susy breaking parameters and for the parameter µ, and two loop RGEs for the gauge and Yukawa couplings. In the numerical analysis we com- pute the loop corrections and also analyze their dependence on the phases. The masses of particles involved in the analysis are ordered as follows: for charginos < mχ+ and for the neutral Higgs (mH1 , mH2, mH3) → (mH , mh, mA) in the limit of no CP mixing where mH is the heavy CP even Higgs, mh is the light CP even Higgs, and mA is the CP odd Higgs. We investigate the question of how large loop corrections are relative to the tree values. We first discuss the magnitude of the loop corrections of the partial decay width defined in Eq.(52). As we mentioned earlier the loop corrections to the partial decay width of the chargino channel have been investigated before in the CP conserving case [20, 21]. The correction in these analyses is of the order of ∼ 10% of the tree level value. Our analysis supports this conclusion. In Figs. (3) and (4) we give a plot of ∆Γ l (l = 1, 3) as a function of tan β for the specific set of inputs given in the captions of these figures. We notice that the partial decay width gets a change of 7 ∼ 15% of its tree level value. We also notice that the CP violating phase θµ can affect the magnitude of this change. This effect has not been addressed in the previous analyses as they are working in the CP conserving scenario. To compare between our analysis and the previous ones we have to notice that these analyses are using the general SUSY parameter space where they put by hand all the parameters that control the analysis. In [20], the authors choose the SUSY parameter set SPS1a of the Snowmass Points and Slopes as a reference point. They choose for the trilinear couplings the values of At = −487 GeV, Ab = −766 GeV and Aτ = −250 GeV. The values of the other parameters are: M = 197.6 GeV, M ′ = 98 GeV, µ = 353.1 GeV, tan β = 10, mA0 = 393.6 GeV, MQ̃1,2 = 558.9 GeV, MŨ1,2 = 540.5 GeV, MD̃1,2 = 538.5 GeV, ML̃1,2 = 197.9 GeV, MẼ1,2 = 137.8 GeV, MQ̃3 = 512.2 GeV, MŨ3 = 432.8 GeV, MD̃3 = 536.5 GeV, ML̃3 = 196.4 GeV and MẼ3 = 134.8 GeV. In all the figures of [20], these values are used, if not specified otherwise. In our mSUGRA analysis the magnitude of all these parameters and others are fixed by the five input parameters m0 = 100 GeV, m1/2 = 250 GeV, tanβ = 10, A0 = −100 GeV and a positive sign of µ in the CP conserving scenario [27]. These parameters are different from those of our Figs. (3) and (4). By using these parameters and fixing some of them by hand when needed to match their values in the analysis of [20], we were able to have a fair agreement with their Figs. (2-9). As an example of this check we show in Table.1 a comparison of the two works. For the input of Fig. 2 of [20] with CP violating phases are set to zero we can see that partial decay widths in both works have the same behavior as functions of masses and their magnitudes are fairly close to each other. However it seems that our loop corrected values of the partial widths are different from those of Eberl et al. This could be understood since our loop analysis of the effective lagrangian includes only the vertex corrections beside the corrections in the Higgs potential. case Γtreeeberl Γ our Γ eberl Γ 2.a mA0 = 700 GeV 0.95 GeV 0.94 GeV 0.85 GeV 0.80 GeV 2.a mA0 = 800 GeV 1.18 GeV 1.17 GeV 1.0 GeV 0.91 GeV 2.b mH0 = 800 GeV 0.7 GeV 0.69 GeV 0.63 GeV 0.58 GeV 2.b mH0 = 900 GeV 0.8 GeV 0.8 GeV 0.73 GeV 0.70 GeV Table 1: A comparison between the current analysis and Eberl et al [20] for bench- mark cases. In the work of Ref. [21] only 8 out of 26 diagrams of the present analysis are calculated and they correspond to the vertex corrections from Figs. (1,2ii(a)), (1,2ii(b)), (1,2i(b)) and (1,2i(a)). By considering these diagrams only in the com- parison, our analysis is in fair agreement with their Figs (2-4) and Figs. (6,8) for their inputs. Now we turn to address the question of how much loop corrections can affect the branching ratios into charginos. The branching ratio of a decay mode is defined to be the ratio between the partial decay rate of this mode and the total decay rate. In the parameter space under investigation this total decay rate includes the rates of decays into charginos, heavy quarks, taus and neutralinos. In Figs. (5) and (6) we give a plot of ∆Br l (l = 1, 3) defined by Eq.(53) as a function of tanβ for the specific set of inputs given in the captions of these figures. Fig. (5) is for the neutral Higgs H1 boson and Fig. (6) is for the neutral Higgs H3 boson. In all regions of the parameter space investigated in this work, the decay of the lightest Higgs boson H2 into charginos is forbidden kinematically, since we have in these regions the fact that 2mχ− > mH2 . The analysis of Figs. (5) and (6) shows that the loop correction varies strongly with tanβ with the correction changing sign for the case of H3 decay. Further, the analysis shows that the loop correction can be as large as about −40% of the tree contribution for both H1 and H3 cases. We also notice that the behavior of ∆Br l (l = 1, 3) as a function of tan β changes considerably by changing the phase of µ. So for some values of this phase we find that this parameter increases as tanβ increases and for other values of θµ we see that it decreases as tan β increases. As shown in the previous figures, the parameter tan β is playing a strong role. This parameter is important at the tree level through the diagonalizing mass matrices of the chargino and neutral Higgs and their spectrum. At the loop level it has extra effect explicitly in α jk and implicitly through the radiatively corrected matrix elements Ylm and through the corrections δφjk, ∆φjk, δψjk, ∆ψjk. The values of the branching ratios themselves at tree and one loop levels are shown in Table.2. θµ(rad) Br 0(H1) Br loop(H1) Br 0(H3) Br loop(H3) 0.5 6% 4.7% 18.2% 13.8% 1.0 8.4% 6.9% 21.3% 18.1% 1.5 9.2% 7.9% 23.4% 22.2% Table 2: Values of branching ratios at tree and one-loop levels of neutral Higgs into the channel χ+1 χ 1 at tan β = 24 for the input of Figs. (5) and (6) We notice that their magnitudes are not negligible for the region of the param- eter space investigated. These non negligible branching ratios for the decay of the neutral Higgs into charginos suggest that these decay modes could be measurable at the soon-to-operate LHC. However, one should also consider the production rates for H1 and H3 bosons to assess whether the change in branching ratios could be detectable at colliders. This analysis goes beyond the scope of the current work. We also notice that the phase of the parameter µ affects the tree level branching ratios as well. This comes mainly from the structure of the chargino matrix. The more important channels in the region of the parameter space investigated are the decay into bottom and top quarks. They have the highest values of branching ratios. The radiative corrections of these channels are also more than those of the charginos and neutralinos. These channels were studied before [1, 15, 16] as mentioned above. However a 20% of branching ratio for the case of neutral Higgs as shown in the above table is not very small and could justify carrying out the current analysis. In Figs. (7) and (8) we give a plot of ∆Br l (l = 1, 3) as a function of |A0| for the specific set of inputs given in the caption of these figures. The analysis of these figures shows that the loop corrections are substantial and reaches the value of −38% of the tree contribution for the case of H1 decay and the value of −43% for the case of H3 decay. Next we investigate the effects of CP violating phases on the loop corrections of the neutral Higgs decays into charginos. In Figs. (9) and (10) we give a plot of ∆Br l (l = 1, 3) as a function of θµ for the specific set of inputs given in the caption of these figures. The analysis of the figures shows that the loop correction has a sharp dependence on θµ. Further, the correction is changing sign as θµ varies from 0 to π for two cases of H3 decay. Thus θµ affects not only the magnitude of l but also its sign depending on the value of θµ. In Figs. (11) and (12) we give a plot of ∆Br l (l = 1, 3) as a function of αA0 for the specific set of inputs given in the caption of these figures. Here also we find a very substantial dependence of ∆Br l on αA0. This dependence is very large in the case of H3 decay and it exceeds −40% of the tree contribution. In Figs. (13) and (14) we give a plot of ∆Br l (l = 1, 3) as a function of ξ2 for the specific set of inputs given in the caption of these figures. Here we find a small effect of this phase on the loop corrections. 5 CONCLUSION In this paper we have carried out an analysis of the supersymmetric loop correc- tions to χ+j χ l couplings within MSSM. In supersymmetry after spontaneous breaking of electroweak symmetry one is left with three neutral Higgs bosons which in the absence of CP phases consist of two CP even Higgs bosons and one CP odd Higgs boson. In the absence of loop corrections, the lightest Higgs bo- son mass satisfies the inequality mh < MZ and by including these corrections the lightest Higgs mass can be lifted above MZ . With the inclusion of CP phases the Higgs boson mass eigenstates are no longer CP even and CP odd states when loop corrections to the Higgs boson mass matrix are included. Further, inclusion of loop corrections to the couplings of charginos and neutral Higgs is in general dependent on CP phases. Thus the decays of neutral Higgs into charginos can be sensitive to the loop corrections and to the CP violating phases. The effect of the supersymmetric loop corrections is found to to be in the range of 7 ∼ 15% for the partial decay width. For the branching ratios it is found to be be rather large, as much as 40% in some regions of the parameter space. The effect of CP phases on the modifications of the partial decay width and the branching ratio is found to be substantial in some regions of the MSSM parameter space. Acknowledgments I wish to acknowledge useful discussions with Professor Pran Nath. The support of the Physics Department at Alexandria University is also acknowledged. 6 APPENDIX The integral of import to this work is (2π)4 (k2 −m21 + iǫ)(k2 −m22 + iǫ)(k2 −m23 + iǫ) It could be written in the form (2π)4 where a = k2 −m21 + iǫ b = k2 −m22 + iǫ c = k2 −m23 + iǫ (56) Using Feynman parametrization, 1 could be written as ∫ 1−x [a + (b− a)x+ (c− a)z]3 The denominator in the above integral could be written in the form k2 +M2 + iǫ where M2 = (m21 − m22)x + (m21 − m23)z −m21. Thus the integral J can take the (2π)4 ∫ 1−x [k2 +M2 + iǫ]3 Now integrating over k and using the standard integral, for n ≥ 3 (2π)4 (k2 + Λ + iǫ)n = iπ2 Γ(n− 2) one can find that the integral J has the form (4π)2 ∫ 1−x α + βz where α = (m21 −m22)x−m21 and β = m21 −m23. Integrating over z one can get for the integral J the form of (4π)2 m21 −m23 dx ln(δ1x−m23)− ln(δ2x−m21) (61) where δ1 = m 3 −m22 and δ2 = m21 −m22. 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D 60, 055006(1999). [23] T. Ibrahim, P. Nath, Phys. Rev. D 63, 035009(2001). [24] T. Ibrahim, P. Nath, Phys. Rev. D 66, 015005(2002). [25] J. F. Gunion, H. E. Haber, Nucl. Phys. B 307, 445 (1988); A. Djouadi, hep/ph/9712334; A. Djouadi, J. Kalinowski, and M. Spira, Comp. Phys. Commun. 108, 56 (1998). [26] The neutralino decay of neutral Higgs with CP phases will be discussed elsewhere. [27] B. C. Allanach et al., Eur. Phys. J. C 25, 113 (2002). [28] M. Carena, M. Olechowski, S. Pokorski and C. E. M. Wagner, Nucl. Phys. B426 (1994) 269; L. J. Hall, R. Rattazzi and O. Sarid, Phys. Rev. D50 (1994) 7048; T. Ibrahim and P. Nath, Phys. Rev. D 67, 095003(2003). s2, v2s1, v1 jf2f1 Figure 1: Set of diagrams contributing to radiative corrections δφjk and ∆ψjk. (i): (a) s1 = b̃ i , s2 = b̃ l , f = t; (b) s1 = t̃i, s2 = t̃l, f = b̄; (c) s1 = H +, s2 = H f = χ0i ; (d) s1 = H l , s2 = H m, f = χ i ; (e) v1 = Z 0, v2 = Z 0, f = χ+i ; (f) v1 = W +, v2 = W +, f = χ0i . (ii): (a) f1 = t, f2 = t, s = b̃ i ; (b) f1 = b̄, f2 = b̄, s = t̃i; (c)f1 = χ i , f2 = χ l , s = H +; (d) f1 = χ i , f2 = χ l , s = H m; (e) f1 = χ f2 = χ l , v = Z 0; (f) f1 = χ i , f2 = χ l , v = W ; (g) f1 = τ +, f2 = τ +, s = ν̃τ . s2, v2s1, v1 jf2f1 Figure 2: Set of diagrams contributing to radiative corrections ∆φjk and δψjk. (i): (a) s1 = b̃ i , s2 = b̃ l , f = t; (b) s1 = t̃i, s2 = t̃l, f = b̄; (c) s1 = H +, s2 = H f = χ0i ; (d) s1 = H l , s2 = H m, f = χ i ; (e) v1 = Z 0, v2 = Z 0, f = χ+i ; (f) v1 = W +, v2 = W +, f = χ0i . (ii): (a) f1 = t, f2 = t, s = b̃ i ; (b) f1 = b̄, f2 = b̄, s = t̃i; (c)f1 = χ i , f2 = χ l , s = H +; (d) f1 = χ i , f2 = χ l , s = H m; (e) f1 = χ f2 = χ l , v = Z 0; (f) f1 = χ i , f2 = χ l , v = W ; (g) f1 = τ +, f2 = τ +, s = ν̃τ . Figure 3: tanβ dependence of ∆Γ1 → χ+1 χ−1 . The curves in ascending order correspond to θµ = 0.2, 0.4, 0.6 (rad). The input is m0 = 350 GeV, m1/2 = 180 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), αA0 = 0.8 (rad) and |A0| = 250 Figure 4: tanβ dependence of ∆Γ3 → χ+1 χ−1 . The curves in ascending order correspond to θµ = 0.2, 0.4, 0.6 (rad). The input is m0 = 350 GeV, m1/2 = 180 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), αA0 = 0.8 (rad) and |A0| = 250 ∆ Br1 Figure 5: tan β dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at tanβ = 40 correspond to θµ = 0.5, 0.1, 1.0, 1.5 and 2.0 (rad). The input is m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), αA0 = 0.3 (rad) and |A0| = 250 GeV. ∆ Br3 Figure 6: tan β dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at tanβ = 40 correspond to θµ = 0.5, 0.1, 1.0, 1.5 and 2.0(rad). The input is m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), αA0 = 0.3 (rad) and |A0| = 250 GeV. ∆ Br1 Figure 7: |A0| dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at |A0| = 0 correspond to tanβ = 40, 35, 30, 25 and 20. The input is m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), θµ = 0.7 (rad) and αA0 = 0.1 (rad). ∆ Br3 Figure 8: |A0| dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at |A0| = 0 correspond to tanβ = 40, 35, 30, 25 and 20. The input is m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), θµ = 0.7 (rad) and αA0 = 0.1 (rad). ∆ Br1 Figure 9: θµ dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at θµ = 2.0 (rad) correspond to |A0| = 100, 250, 500, 750 and 900 GeV. The input is tanβ = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad) and αA0 = 0.2 (rad). 0-505 ∆ Br3 Figure 10: θµ dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at θµ = π (rad) correspond to |A0| = 100, 250, 500, 750 and 900 GeV. The input is tan β = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad) and αA0 = 0.2 (rad). ∆ Br1 Figure 11: α0 dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at αA0 = 2.2 (rad) correspond to |A0| = 500, 450, 400, 100 and 200 GeV. The input is tanβ = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad) and θµ = 0.1 (rad). ∆ Br3 Figure 12: α0 dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at αA0 = 2.2 (rad) correspond to |A0| = 500, 450, 400, 100 and 200 GeV. The input is tanβ = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad) and θµ = 0.1 (rad). ∆ Br1 Figure 13: ξ2 dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at ξ2 = 0.75 (rad) correspond to |A0| = 50, 100, 150, 200 and 250 GeV. The input is tan β = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ3 = 0.6 (rad) and θµ = 0.2 (rad) and αA0 = 0.3 (rad). ∆ Br3 Figure 14: ξ2 dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at ξ2 = 0.75 (rad) correspond to |A0| = 50, 100, 150, 200 and 250 GeV. The input is tan β = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ3 = 0.6 (rad) and θµ = 0.2 (rad) and αA0 = 0.3 (rad).

Microsoft Word - TI_NEA_070302RSingleSpaced.doc - 1 - Thermal inertia of near-Earth asteroids and implications for the magnitude of the Yarkovsky effect Marco Delbo1,2, Aldo dell’Oro1, Alan W. Harris3, Stefano Mottola3, Michael Mueller3 1INAF-Oss. Astron. di Torino, via Osservatorio 20, 10025 Pino Torinese (TO), Italy 2Observatoire de la Côte d'Azur B.P. 4229, 06034 Nice Cedex 4, France 3DLR Institute of Planetary Research, Rutherfordstrasse 2, 12489 Berlin, Germany Submitted to ICARUS: November 26, 2006 Revised: March 2, 2007. Accepted: March 5, 2007. No. of manuscript pages: 31 No. of figures: 6 No. of tables: 1 - 2 - Running head: The thermal inertia of near-Earth asteroids Editorial correspondence to: Marco Delbo’ Laboratoire Cassiopée Observatoire de la Côte d’Azur BP 4229 06304 Nice, cedex 04 - France Tel.: +33 (0)4 9200 1944 Fax.: +33 (0)4 9200 3121 E-mail: delbo@obs-nice.fr - 3 - Abstract Thermal inertia determines the temperature distribution over the surface of an asteroid and therefore governs the magnitude the Yarkovsky effect. The latter causes gradual drifting of the orbits of km-sized asteroids and plays an important role in the delivery of near-Earth asteroids (NEAs) from the Main Belt and in the dynamical spreading of asteroid families. At present, very little is known about the thermal inertia of asteroids in the km size range. Here we show that the average thermal inertia of a sample of NEAs in the km size range is 200±40 J m-2 s-0.5 K-1. Furthermore, we identify a trend of increasing thermal inertia with decreasing asteroid diameter, D. This indicates that the dependence of the drift rate of the orbital semimajor axis on the size of asteroids due to the Yarkovsky effect is a more complex function than the generally adopted D-1 dependence, and that the size distribution of objects injected by Yarkovsky-driven orbital mobility into the NEA source regions is less skewed to smaller sizes than generally assumed. We discuss how this fact may help to explain the small difference in the slope of the size distribution of km-sized NEAs and main belt asteroids. Keywords: Asteroids; Near-Earth Objects; Infrared Observations; Photometry; Dynamics. - 4 - 1 Introduction Observations of asteroids in the wavelength range of their thermal-infrared emission (>5 μm) have been used since the 1970s (Allen, 1970) to determine the sizes and the albedos of these bodies. In recent years, thanks to the advances in detector technology and the availability of 10-m class telescopes on the ground, thermal-infrared observations of asteroids have improved in sensitivity. Increased efforts have consequently been devoted to deriving the sizes and albedos of near-Earth asteroids (NEAs; for reviews see Harris and Lagerros, 2002; Delbo’ and Harris, 2002; Delbo’, 2004; Harris, 2006 and references therein), in order to better assess the impact hazard these bodies pose to our planet and to improve our understanding of their relation to main-belt asteroids and comets (see Stuart and Binzel, 2004; Morbidelli et al., 2002). Furthermore, improvements in spectral coverage and the possibility of easily obtaining spectrophotometric data through narrow-band filters in the range 5 – 20 μm have allowed information on the surface temperatures of asteroids to be obtained. The spectrum of the thermal-infrared radiation received from a body is related to the temperature distribution on that part of its surface visible to the observer. Several factors play a role in determining the temperature distribution on the surface of an asteroid, such as the heliocentric distance, albedo, obliquity of the spin vector, rotation rate, and a number of thermal properties of the surface such as its thermal inertia. Thermal inertia is a measure of the resistance of a material to temperature change. It is defined as cρκΓ = , where κ is the thermal conductivity, ρ the density and c the specific heat capacity. The thermal inertia of an asteroid depends on regolith particle size and depth, degree of compaction, and exposure of solid rocks and boulders within the top few centimeters of the subsurface (see e.g. Mellon et al., 2000). At the limit of zero thermal inertia (the most simple temperature distribution model for asteroids), a body with a smooth surface would display a temperature distribution which depends only on the solar incidence angle i, (on a sphere, i is also the angular distance of a point from the subsolar point): 1/ 4cos , / 2 0, / 2 SST T i i (1) The subsolar temperature, TSS, is determined by equating the total energy absorbed by a surface element to that emitted in the thermal infrared, i.e.: = (2) where A is the bolometric Bond albedo, S is the solar constant, r is the heliocentric distance of the body, ε is the infrared emissivity, σ is the Stefan-Boltzmann constant and η is the so-called “beaming parameter”, which is equal to one in the case that each point of the surface is in instantaneous thermal equilibrium with solar radiation. The surface temperature distribution that one obtain for η=1 on a spherical shape is that of the so- called Standard Thermal Model (STM, Lebofsky and Spencer, 1989) that was widely used to derive diameters and albedos especially of main-belt asteroids (MBAs). In the more realistic case of a body with finite thermal inertia and rotating with a spin vector not pointing toward the sun, the temperature distribution is no longer symmetric with respect to the subsolar point: each surface element behaves like a capacitor or sink for the solar - 5 - energy such that the body’s diurnal temperature profile becomes more smoothed out in longitude (see Spencer et al., 1989; Delbo’ and Harris, 2002; Delbo’, 2004). The hottest temperatures during the day decrease, whereas those on the night-side do not drop to zero as in the idealistic case of zero thermal inertia, implying non-zero thermal-infrared emission from the dark side of the body. However, the effect of thermal inertia is coupled with the rotation rate of the body. An asteroid rotating slowly with a high thermal inertia displays a similar temperature distribution to one rotating more rapidly but with a lower thermal inertia. The degree to which the surface of an asteroid can respond to changes in insolation can be characterized by a single parameter: this is the so-called thermal parameter Θ (e.g. Spencer et al., 1989), which combines rotation period, P, thermal inertia, Γ, and subsolar surface temperature, TSS, and consequently depends on the heliocentric distance of the body. The thermal parameter is given by: SST P Θ = . (3) Note that objects with the same value of Θ, although with different P or Γ display the same diurnal temperature profile, provided they have the same shape and spin axis obliquity (the angle formed by the object spin vector and the direction to the Sun). In the case of non-zero thermal inertia, because the temperature distribution is no longer symmetric with respect to the direction to the Sun, the momentum carried off by the photons emitted in the thermal infrared has a component along the orbital velocity vector of the body, causing a decrease or increase of the asteroid orbital energy depending on whether the rotation sense of the body is prograde or retrograde. This phenomenon, known as the Yarkovsky effect, (see Bottke et al., 2002) causes a secular variation of the semimajor axis of the orbits of asteroids on a time scale of the order of 10-4 AU/Myr for a main-belt asteroid at 2.5 AU from the Sun with a diameter of 1 km. The Yarkovsky effect is responsible for the slow but continuous transport of small asteroids and meteoroids from the zone of their formation into chaotic resonance regions that can deliver them to near-Earth space (Bottke et al., 2002; Morbidelli and Vokrouhlický, 2003). The Yarkovsky effect is also important to explain the spreading of asteroid dynamical families (Bottke et al., 2001; Bottke et al., 2006; Nesvorný and Bottke, 2004). Moreover, the emission of thermal photons also produces a net torque that alters the spin vector of small bodies in two ways: it accelerates or decelerates the spin rate and also changes the direction of the spin axis. This mechanism was named by Rubincam (2000) as the Yarkovsky-O’Keefe-Radzievskii-Paddack effect, or YORP for short. Knowledge of the thermal inertia of asteroids is thus important for a number of reasons: (a) It can be used to infer the presence or absence of loose material on the surface: thermal inertia of fine dust is very low: ~30 J m-2 s-0.5 K-1 (Putzig et al., 2005); lunar regolith, a layer of fragmentary incoherent rocky debris covering the surface of the Moon, also has a low thermal inertia of about 50 J m-2 s-0.5 K-1 (Spencer et al., 1989). Coarse sand has a higher thermal inertia, i.e. about 400 J m-2 s-0.5 K-1 (Mellon et al., 2000; Christiansen et al., 2003), that of bare rock is larger than 2500 J m-2 s-0.5 K-1 (Jakosky, 1986), whereas the thermal inertia of metal rich asteroidal fragments can be larger than 12000 J m-2 s-0.5 K-1 (Farinella et al., 1998, Table 1). (b) Thermal inertia is the key thermophysical parameter that determines the temperature distribution over the surface of an asteroid and therefore governs the magnitude of the Yarkovsky and YORP effects (Capek and Vokrouhlický, 2004). (c) It allows a better determination of systematic errors - 6 - in diameters and albedos derived using simple thermal models, which make assumptions about the surface temperature distribution and/or neglect the thermal-infrared flux from the non-illuminated fraction of the body (see Spencer et al., 1989, Delbo’, 2004, Harris, 2005). However, at present, very little is known about the thermal inertia of asteroids in general, especially in the case of bodies in the km size range. The thermal inertia of an asteroid can be derived by comparing measurements of its thermal-infrared emission to synthetic fluxes generated by means of a thermophysical model (TPM; Spencer, 1990; Lagerros, 1996; Emery et al., 1998; Delbo’, 2004), which is used to calculate the temperature distribution over the body’s surface as a function of a number of parameters, including the thermal inertia Γ. In these models, the asteroid shape is modeled as a mesh of planar facets. The temperature of each facet is determined by numerically solving the one-dimensional heat diffusion equation using assumed values of the thermal inertia, with the boundary condition given by the time-dependent solar energy absorbed at the surface of the facet (see Delbo’, 2004). This latter quantity is calculated from the heliocentric distance of the asteroid, the value assumed for the albedo, and the solar incident angle. Macroscopic surface roughness is usually modeled by adding hemispherical section craters of variable opening angle and variable surface density to each facet. Shadowing and multiple reflections of incident solar and thermally emitted radiation inside craters are taken into account as described by Spencer (1990), Emery et al. (1998), and Delbo’ (2004). Heat conduction is also accounted for within craters (Spencer et al., 1989; Spencer, 1990; Lagerros, 1996, Delbo’, 2004). Surface roughness can be adjusted by changing the opening angle of the craters, the density of the crater distribution, or a combination of both. However, Emery et al. (1998) have shown that if surface roughness is measured in terms of the mean surface slope, θ , according to the parameterization introduced by Hapke (1984), emission spectra are a function of θ only and not of the crater opening angle and crater surface density. We recall here that tan tana d θ θ θ θ= ∫ (4) where θ is the angle of a given facet from the horizontal, and a(θ) is the distribution of surface slopes. The total observable thermal emission is calculated by summing the contributions from each facet visible to the observer. Model parameters (e.g. Γ, A, θ ) are adjusted until the best agreement is obtained with the observational data, i.e. the least- squares residual of the fit χ2 is minimized, thereby constraining the physical properties (albedo, size, macroscopic roughness, and thermal inertia) of the asteroid. To date, TPMs have been used to derive the thermal inertia of seven large MBAs (Müller, T. G. and Lagerros, 1998; Müller, T. G. and Blommaert, 2004; Mueller, M. et al., 2006b), and five NEAs (Harris et al., 2005; Müller, T. G. et al., 2005; Mueller, M. et al., 2006a, Harris et al., 2007); values derived lie between 5 and ~1000 J m-2 s-0.5 K-1, i.e. Γ varies by more than two orders of magnitude. The applicability of TPMs is limited to the few asteroids for which gross shape, rotation period, and spin axis orientation are known. Multi-epoch observations are also required for obtaining a robust estimation of the thermal properties of asteroids via TPM fit. There is, however, an extensive set of thermal-infrared observations of NEAs in the km size range for which no TPM fit is possible (e.g. Veeder et al., 1989; Harris, 1998; Harris et al., 1998; Harris and Davies, 1999; Delbo’ et al., 2003; Delbo’, 2004; Wolters et al., 2005). In order to overcome this limitation, we have developed a statistical - 7 - inversion method, described in Section 2, enabling the determination of the average value of the thermal inertia of NEAs in the km-size range. Our approach is based on the fact that, even though shapes, rotation periods, and spin axis orientations are not known for every NEA, the distribution of these quantities for the whole population can be inferred from published data (La Spina et al., 2004; Hahn, 2006). In Section 3 we compare the result from our statistical inversion method with the values of the thermal inertias of asteroids determined by means of thermophysical models, and we identify a trend of increasing thermal inertia with decreasing asteroid diameter, D. In Section 4 we describe the implications of the trend of increasing thermal inertia with decreasing asteroid diameter, in particular for the size-dependence of the Yarkovsky effect and the size distribution of NEAs and MBAs. 2 Determination of the mean thermal inertia of NEAs The large majority of asteroids for which we have thermal-infrared observations have been observed at a single epoch and/or information about their gross shape and pole orientation is not available, precluding the use of TPMs. In these cases simpler thermal models such as the near-Earth asteroid thermal model (NEATM; Harris, 1998) are used to derive the sizes and the albedos of these objects. The NEATM assumes that the object has a spherical shape, and its surface temperature distribution is described by Eq. (1) and Eq. (2). However, the parameter η is not kept constant, as in the case of the STM, but is adjusted in the fitting procedure to allow the model spectral energy distribution to match the observed data. In order to derive a robust estimate of the η-value the NEATM requires observations at different, ideally well-spaced, wavelengths in the thermal infrared. The parameter η can be seen as a measure of the departure of the asteroid temperature distribution from that of the STM and is a strong function of the surface thermal inertia (Spencer et al., 1989; Harris, 1998; Delbo’, 2004). However, η depends also on parameters such as the macroscopic surface roughness, θ , the rotation period, P, the bolometric Bond albedo, A, the thermal-infrared emissivity, ε, the heliocentric distance, r, the gross shape of the body, , the sub-solar latitude, θSS, the longitude, φSE, and the latitude, θSE, of the sub-Earth point (Delbo’, 2004). In general we can write that η≡η(ε, A, r, Θ(Γ, P),θ , θSS, θSE, φSE, ). (5) These parameters are usually not known for the individual objects, but their distributions can be estimated (or reasonably assumed) for the entire population. Note that a set of θSS, φSE, and θSE , which depend on the ecliptic longitude λ0 and latitude β0 of the pole of the body, also defines the value of the solar phase angle, α. Delbo’ et al. (2003) noted that qualitative information about the average thermal properties of a sample of NEAs could be obtained from the distribution of the η-values of the sample as function of the phase angle, α. In particular, the absence of large η-values (e.g. η > 2) at small or moderate phase angles (e.g. ≤ 45o), and the fact that η tends to ~ 0.8 for α approaching 0o, was interpreted in terms of the NEAs having low thermal inertias in general. In subsequent work (Delbo’, 2004) it was found that for a synthetic population of spherical asteroids with constant values of A, r, Γ, P, and θ , but with pole directions randomly oriented, the distribution of the points in the (α, η) plane is strongly dependent on Γ. By varying Γ until the distribution of the synthetic points in the (α, η) - 8 - plane matched the one derived from the observations, Delbo’ (2004) obtained a best-fit thermal inertia for the NEAs equal to ~500 J m-2 s-0.5 K-1. Harris (2005), using a similar method on a larger database of η-values and neglecting the effects of surface roughness (θ =0o), derived a best-fit thermal inertia of ~300 J m-2 s-0.5 K-1. Here we improve on the above-mentioned work by determining the mean thermal inertia of NEAs using a rigorous statistical inversion method, based on the comparison of the distributions of NEATM η-values from the current NEA database vs. α, with that of a synthetic population of asteroids generated through a TPM, using realistic distributions of the input parameters P, θSS, θSE, φSE, and A derived from the literature (see Table 1 with published η-values from Harris, 1998, Harris et al. (1998); Harris and Davies (1999); Delbo’ et al. (2003); Delbo’ (2004); and Wolters et al. (2005). La Spina et al. (2004) give the distribution of λ0 and β0 for NEAs, and Hahn (2006) that of NEA rotation rates). In the following section we describe our method in detail. 2.1 Model parameter space As a first step we studied the dependence of η on the relevant parameters of Eq. 5. This was done by choosing typical parameter values and showing how small perturbations of the assumed values affect η. For the purpose of this analysis we assume A0=0.073, r0=1.2 AU (as we will show below, these are the average values of A and r for the NEAs in our sample), 0θ =36 o (the value derived for 433 Eros; Domingue et al., 2002), 0=sphere, and Θ0=1.0. Note that Θ0=1.0 corresponds to a thermal inertia of ~200 J m-2 s-0.5 K-1 for surface temperatures typical of NEAs and P = 6 hours, a rotation period representative of asteroids with sizes between ~0.15 and 10 km (Pravec et al., 2002). We will show in section 2.3 that Γ = 200 J m-2 s-0.5 K-1 is the mean thermal inertia of NEAs. The illumination and observation geometry was varied such that θSS was uniformly distributed in the range between 0 and π/2 and θSE, φSE were varied in such a manner that the resulting sub-Earth vectors were uniformly distributed over the celestial sphere. The values of θSE, φSE were further subject to the constraint that the phase angle be ≤ 100o. Figure 1 shows the sensitivity of η to a change in the model parameters. In particular, for each value of θSS, θSE, and φSE, the variation of η due to a 1% change in each parameter is plotted. We have also calculated, for some fixed illumination and observation geometries (e.g. θSE = 0o, φSE = 45o and θSS = 0o), how the variations Δη scale with changes in the model parameters. We found that Δη is proportional to ΔA, Δ 0θ , ΔΘ, and Δr within a large range of variation (>100%) of each parameter from its nominal value. Because for common asteroidal material the thermal-infrared emissivity is thought to be relatively constant, it has been fixed for this study at ε = 0.9. It is appropriate for objects with surfaces that emit a substantial portion of their thermal-infrared radiation shortward of 8 μm (Lim et al., 2005). Mustard and Hays (1997) have also shown that the reflectance spectra of fine-grained particulate materials, thought to be representative of planetary regoliths, have values around 0.1 and in general smaller than 0.2 in the region 8 – 24 μm. Because the reflectance, R, and the emissivity are related by Kirchhoff’s law (R=1-ε), the measurements cited above implies that ε = 0.9 is a reasonable estimate for the thermal- infrared emissivity of NEA surfaces. Moreover, from Eq. 2 one can calculate that Δη≈1.6Δε for ε≈0.9 and η≈1.5 (the average η for the NEAs for which this parameter was derived from observations; see Table 1). This implies that variations of ε in the range 0.8 – 1.0 cause changes of η that are within the typical uncertainty of ~20% in the estimation - 9 - of η from observations. Note that the value of Δη/η = 20%, where Δη is the uncertainty in η, is based on the reproducibility of η for those objects for which more than one measurement is available from independent data sets. Moreover, 20% is also the mean value of Δη/η of the “Delbo’ Thermal Infrared Asteroid Diameters and Albedos” database at the NASA PDS (Delbo’, 2006). In this dataset for those observations where Δη is present, its value was formally calculated from the measurements of the asteroids’ thermal infrared fluxes. The vast majority of the observations in our sample was obtained at a phase angle smaller than 80o, and within this range, Fig. 1 shows that the largest variation of η caused by a 1% change of A (the bolometric Bond albedo) is approximately 0.1%. Because the mean value of A for our sample is A =0.073 and the standard deviation is 0.04 (see Table 1), the variation of η due to the distribution of the albedos is smaller than 5% and thus small compared to the typical uncertainty of Δη/η ~ 20%. For this reason we have utilized a constant value of 0.073 for A in our statistical inversion method. Moreover, the variation of η due to a 1% change in the macroscopic surface roughness is strongly phase angle dependent, but in general smaller than 0.2% for α in the range 0-60o. This implies that even a ±100% change in θ causes a variation of η within the typical 20% uncertainty. Note that a ±100% change in θ corresponds to a large variation of the macroscopic roughness, ranging from that of a completely smooth surface to one oversaturated by hemispherical craters. For those observations carried out at α > 60o, η is more sensitive to variations of θ . For the reasons above we have treated θ as a free parameter in the inversion method and searched for the value that best fits the observational data. The sensitivity of η to changes of the objects’ heliocentric distances is such that a 1% change of r corresponds to a maximum 0.7% change of η. As calculated for the values in Table 1, the heliocentric distances in our sample have a mean value of 1.2 AU and a standard deviation of 0.1 AU (~8%). The corresponding variation of η is approximately 6% and therefore small. We thus took a constant value of 1.2 AU for r in our statistical inversion method. Only in two cases, namely those of the 29-06-1998 observation of (433) Eros and for the 22-03-2002 observation of (6455) 1992 HE is the variation of η due to the deviation of the heliocentric distance from the nominal value of 1.2 AU slightly larger than the error bars. We note here that Eq. (5) implicitly contains the assumption that seasonal effects do not affect asteroid surface temperatures. However, when Θ ≠ 0, asteroid temperatures always depend on the previous thermal history of the surface. Since NEAs have in general large orbital eccentricities, these bodies experience large variations of insolation as a function of their orbital position, which may lead to a seasonal component of the variation of their surface temperatures and thus of the corresponding η-values. To demonstrate that our working hypothesis of Eq. (5) is valid (i.e. seasonal components are negligible), we calculated η for several synthetic asteroids with the same physical characteristics, but with different orbits, in order to explore the effect of different levels of insolation. Orbits were chosen with eccentricities in the range 0 to 0.8 but with a common perihelion distance rp. For different values of the asteroid thermal inertia in the range 200-5000 J m-2 s-0.5 K-1 and rp in the range 0.5 - 1.5 AU, we found variations of only a few percent in the η-values calculated at rp. This leads us to conclude that seasonal variations in the η-values are small and that Eq. (5) is valid. - 10 - In general NEAs have elongated shapes, which may cause their surface temperature distributions to differ from that of a spherical object with the same surface properties and illumination geometry. We studied the sensitivity of η to deviations from the spherical shape by calculating η-values of tri-axial ellipsoids, the semiaxes of which were varied in the ratio ( )1// aa with 1 ≤ a ≤ 6, assuming A0=0.073, r0=1.2 AU, 0θ =36o, Θ0=1.0, and for random orientations of the shape with respect to the Sun and the Earth. We found that the distribution of Δη is a function of a (with values of Δη increasing with increasing a ), where Δη is the deviation of η from that calculated using a sphere under the same illumination and viewing geometry. However, the relative error on η, Δη/η, is always smaller than ±10% for a ≤ 5 and α ≤ 45o. For α > 45o, the mean value of the relative error, 〈Δη/η〉, is smaller than +15% and its standard deviation, σΔη/η, is smaller than 5% for a ≤ 5. Because the maximum lightcurve amplitude of our model ellipsoid is L≈1.25log a mag, Δη/η is smaller than 20% if L ≤ 0.873 mag. This condition is in general satisfied for the NEAs in Table 1, for which the average value of L is around 0.6 mag. We expect that the contributions to Δη due to variations of the model parameters A, r, θ , and the ellipsoid axial ratio a, stack up randomly, since deviations of these parameters from their mean values are fully uncorrelated (e.g. there is no apparent reason that an NEA with an albedo higher than the average is also observed at an heliocentric distance higher than the its average value). We performed some numerical experiments in order to cross check this assumption and found that the value of Δη is in general a good proxy of [(∂η/∂AΔA)2 + (∂η/∂rΔr)2 (∂η/∂θ Δθ )]1/2. Adding the effect of non-spherical shapes increases the value of Δη, but never systematically at phase angles < ~60o. It is clear that ellipsoids are highly idealized shapes and larger contributions to Δη may be expected in the case of real NEAs. Figure 1 shows that the sensitivity of η to changes in the thermal parameter is very similar to the sensitivity to changes in r, with variations of η in general no larger than 0.5% for a 1% change of Θ. However, while the value of r for the asteroids in Table 1 is rather constant around the mean value of 1.2 AU, Θ can range between 0.1 and 20 considering that thermal inertia can be anywhere between 10 J m-2 s-0.5 K-1 (the thermal inertia of large main-belt asteroids) and 2500 J m-2 s-0.5 K-1 (that of bare rock). This implies that the scatter in the η-values that we observe in the NEAs of Table 1 is mainly a function of α and Θ, which depends on the thermal inertia. If we assume the thermal inertia to be roughly constant within the NEA population for a given size, its value can be inferred from the distribution of the measured η-values versus α. This is the idea on which our statistical inversion method is based. 2.2 Model populations Our inversion method requires η to be computed for all members of a synthetic population of NEAs as a function of Γ. The calculation of η was performed by numerically generating thermal-infrared spectra by means of a TPM and fitting them with the NEATM. As discussed in the previous sections, the parameters A, ε, r, θ , and contribute little to the variation of η within the expected parameter ranges. Therefore, they have been kept fixed to their nominal values throughout the modeling process. In order to keep the amount of computing time required for the inversion within reasonable limits, the values of η have been computed only once for all possible combinations of the - 11 - remaining parameters, and the results have been stored in a four-dimensional look up table. The granularity of the look up table was chosen to be small enough to cause changes of η of about 0.1 between two consecutive parameter steps. For each value of Γ, we then generated a large number (30,000) of synthetic objects whose parameters have random values with distributions that have been chosen to provide a reasonable match to the observed population of NEAs. In particular: (i) the distributions of the angles θSS, θSE, and φSE were computed starting from the distributions of the spin-axis orientation (λ0, β0) from La Spina et al. (2004), the phase angle α, the heliocentric ecliptic latitude βH, and the geocentric ecliptic latitude βE of the asteroids at the time of the infrared observations (see Table 1 and Fig. 2); (ii) the distribution of the thermal parameter was calculated starting from the distribution of the NEA rotation periods (Hahn, 2006) and by using a constant value of Γ. In Fig. 3 three such populations are shown that correspond to the Γ values of 15 (green), 200 (red), and 1000 (blue) J m-2 s-0.5 K-1, respectively. We have superimposed the η values for the NEAs in Table 1 on the synthetic data plot. 2.3 Best-fit procedure Figure 3 gives a clear visual impression of the dependence of η on Γ. We therefore used a formal best-fit technique to estimate the value of Γ for which a synthetic population best fits the observed data, under the assumption that Γ is constant for all objects in the observed sample. The method that we used to compare the observed data with the bi-dimensional distributions of the synthetic points in the (α, η) plane is based on the two-dimensional Kolmogorov-Smirnov metric (K-S; Press et al., 1992). The distance D of the K-S metric is used as the goodness of fit estimator (Press et al., 1992). Our best-fit procedure consisted of finding the value of Γ that minimizes the K-S distance D. From here on, we indicate this value with the symbol Γ*. Figure 4, where we have plotted the K-S distance D as a function of Γ, shows that the function D (Γ) has a minimum at Γ=200 J m-2 s-0.5 K-1, which is the value of thermal inertia that we take for Γ*. We expect Γ* to depend upon the assumed value for θ , the distributions of NEA rotation rates, and also on the spin-axis orientations that we have used to produce the distribution of the input parameters θSS, θSE, and φSE. Moreover, the value of Γ* must be affected by the errors in the measurements of the thermal infrared fluxes, i.e. by the errors on the η-values taken from the literature. In order to study the sensitivity of Γ* to changes applied to the nominal values of the input parameters, we first varied θ in the range between 0° (perfectly smooth surface) and 58° (corresponding to the surface completely covered by hemispherical craters). Figure 4 shows the function D (Γ) for three different values of θ . It clearly demonstrates that the value of Γ* only weakly depends on θ and that a high degree of surface roughness produces a better fit to the observed data. We also investigated the sensitivity of Γ* to changes in the input distributions of asteroids’ spin-axis orientations and rotation rates. Figure 4 shows the function D (Γ) obtained by using random spin-axis orientations uniformly distributed over the sphere instead of the nominal distribution. In that case, the best-fit thermal inertia increases to - 12 - 250 J m-2 s-0.5 K-1, and to 230 J m-2 s-0.5 K-1 if the distribution of the rotation rates are assumed to be uniformly distributed between 4 and 10 hours, a case which we believe to be very extreme. The sensitivity of Γ* to the errors affecting the η-values from Table 1 was studied by performing extended Monte Carlo simulations, in which we randomly varied the values of the η-values within their error bars (using normally-distributed random numbers), and for each simulation of noise-corrupted data we calculated the best-fit thermal inertia. The standard deviation of Γ* was found to be 40 J m-2 s-0.5 K-1. Of course, we expect that the distribution of the data points in Fig. 3 derives from a population with a range of thermal inertias, and further investigation is required to understand what the relations are between Γ* and the parameters defining the population, such as the mean value of Γ and the standard deviation of its distribution. In order to answer this question, we applied our inversion method on (α, η) points obtained from synthetic populations of NEAs with known distributions of thermal inertia. We used random values of Γ uniformly and normally distributed, varying both the mean value and the standard deviation of the populations. We found that our fitting procedure, based on the minimization of the K-S distance D, is capable of retrieving a good estimate of the mean value for Γ of the populations in all cases. We conclude that the average value of the thermal inertia for km-sized NEAs is 200±40 J m-2 s-0.5 K-1, which is about four times that of the lunar soil and corresponds to a surface thermal conductivity of 0.0150.0100.027 − W m-1 K-1 assuming that the surface material density and specific heat capacity are in the range 1500-3500 kg m-3 and 500-680 J kg-1 K-1, respectively (Britt et al., 2002; Farinella et al., 1998). The value of Γ* that we have derived by means of the best fit procedure is less than 10% of that expected for a bare-rock surface (Jakosky, 1986). This implies that the surfaces of NEAs have in general significant quantities of thermally-insulating regolith. However, Γ* is also about four times higher than the value that has been determined for the lunar soil and more than ten times higher than the thermal inertia typical of large main-belt asteroids. This effect may be due to the fact that the regolith present on NEA surfaces is less mature and/or less thick than that of the Moon and the largest MBAs. The higher NEA thermal inertia can also be explained in terms of a coarser regolith and the exposure of rocks and boulders on the surface of these bodies, as clearly shown in the high resolution images of (433) Eros and (25143) Itokawa obtained by the NEAR Shoemaker and the Hayabusa missions, respectively. A population of asteroids with constant Γ=200 J m-2 s-0.5 K-1 gives the best fit to the dataset. Figure 3 shows, however, that five points with η > 2 are clearly significantly higher than the majority, indicating that these objects presented unusually low color temperatures to the observer, possibly due to higher-than-average thermal inertia (see Delbo’ et al., 2003 and Delbo’, 2004). To gain insights into the width of the distribution of the thermal inertia of km-sized NEAs, we fitted the observed distribution of the data points with a synthetic population in which Γ was assumed to be uniformly distributed between 0 and ΓMAX. The best fit was obtained for ΓMAX ~ 600 J m-2 s-0.5 K-1. This suggests that the large majority of km-sized NEAs in our sample have thermal inertia below this value. The average value of the thermal inertia was derived for a sample of objects whose diameter distribution is shown in Fig. 5. We use here the radiometric diameters as derived by the NEATM. The mean diameter of the sample is 3 km, but if we remove the asteroid - 13 - 433 Eros, the mean diameter value decreases to 2 km. 433 Eros is much larger than the average size of the sample (see Fig. 5). In fact the median value of the diameter distribution (including 433 Eros) is 1.8 km. We note that the distribution of log D (where D is the diameter measured in km) is well fitted by a Gaussian distribution with a central value of 1.7 km. The standard deviation of the best-fit Gaussian function is 0.31 (in log D). We can thus conclude that the average value of the thermal inertia is representative of NEAs in the size range 0.8 – 3.4 km. 3 Size dependence of asteroid thermal inertia The mean thermal inertia for the sample of NEAs with published η-values is consistent with the values derived by means of TPMs for (433) Eros (Mueller, M. et al., 2006a), (1580) Betulia (Harris et al., 2005), (25143) Itokawa (Mueller, M. et al., 2006a; Müller, T. G. et al., 2005), and (33342) 1998 WT24 (Harris et al., 2007) for which values around 150, 180, 350, 630, and 200 J m-2 s-0.5 K-1 have been obtained respectively. Note that in the case of (25143) Itokawa, Müller, T. G. et al. (2005) have obtained a thermal inertia value of 750 J m-2 s-0.5 K-1 combining thermal-infrared observations gathered at ESO in 2004 with those obtained by Delbo’ (2004) in 2001. On the other hand, from the latter dataset of observations and a series of further observations of (25143) Itokawa obtained at the NASA-IRTF 3 m telescope with MIRSI in 2004, Mueller, M. et al. (2006a) derived a thermal inertia of ~350 J m-2 s-0.5 K-1 or ~800 J m-2 s-0.5 K-1 depending on whether the size of the body was obtained from the TPM or was forced to the radar value of Ostro et al. (2004). In this work we have taken the mean value and the extreme values of 350, 750, and 800 J m-2 s-0.5 K-1 as our best estimate for the thermal inertia of Itokawa and its uncertainty. Müller T. G. et al. (2004) have also attempted at deriving the thermal inertia of the small (~0.28 km) NEA 2002 NY40. They obtained a value of 100 J m-2 s-0.5 K-1 in the case that the size of the object was derived from the TPM, or 1000 J m- 2 s-0.5 K-1 if the body’s size was forced to the value obtained from radar observations. However, it is important to note that that the thermal inertia of 2002 NY40 was derived by assuming an equator-on view and a spherical shape for this object. The value of the thermal inertia derived from the TPM is in general strongly dependent on the pole orientation of the body. For this reason we expect the value of Γ derived for 2002 NY40 be less reliable than the values obtained for the other NEAs, for which the pole orientation derived from lightcurve inversion was adopted. From thermophysical modeling, Müller, T.G. and Lagerros (1998) derived the thermal inertias of a number of the largest MBAs, namely (1) Ceres, (2) Pallas, (3) Juno, (4) Vesta, and (532) Herculina, obtaining the values of 10, 10, 5, 25, and 15 Jm-2 s-0.5 K-1, respectively. Using the same approach, Müller, T.G. and Blommaert (2004) derived a thermal inertia of 15 J m-2 s-0.5 K-1 for (65) Cybele, and Mueller, M. et al. (2006b) obtained Γ~50 J m-2 s-0.5 K-1 for (21) Lutetia. From the published plots of the goodness of the TPM fit to the thermal-infrared data as a function of Γ it is possible to deduce that the relative uncertainties for the thermal inertias of these asteroids are around 50%. From the comparison of the values of Γ mentioned above, it is clear that there is an increase in the thermal inertia from that of large MBAs with diameters of several hundred km to that of much smaller km-sized NEAs, and that the values of Γ obtained for km- sized NEAs are about one order of magnitude or more higher than the values derived for large MBAs, but still an order of magnitude lower than the thermal inertia of bare rock (~2500 J m-2 s-0.5 K-1; Jakosky, 1986). In order to highlight the behavior of the thermal - 14 - inertia of asteroids as a function of their size, we have plotted the mean value of thermal inertia for NEAs and the values of the thermal inertia derived by means of TPMs against object diameters in Fig. 6. Small open circles represent the literature values derived from the application of TPMs. The large open diamond is the result from this work. The axis on the right-hand side gives the asteroid surface thermal conductivity k as a function of size, on the basis of k=Γ2/(ρc), with constant surface density ρ = 2500 kg m-3 and specific heat capacity c = 600 J kg-1 K-1. These values are reasonable assumptions for asteroid surfaces (Britt et al., 2002; Farinella et al., 1998). For the asteroid 2002 NY40 a bar between 100 and 1000 J m-2 s-0.5 K-1 is drawn. The thermal conductivity has also been constrained in the cases of (6489) Golevka (Chesley et al., 2003) and for asteroids in the Karin cluster (Nesvorný and Bottke, 2004). The values of the thermal conductivities derived by these authors have been converted to values of Γ assuming ρ=2500 kg m-3 and c=600 J kg-1 K-1. Fig. 6 shows that the resulting limits, based on the measurements of the Yarkovsky effect on these bodies, are in general agreement with our results. Figure 6 reveals a convincing trend of increasing thermal inertia with decreasing asteroid diameter, D, confirming the intuitive view that large main-belt asteroids, over many hundreds of millions of years, have developed substantial insulating regolith layers, responsible for the low values of their surface thermal inertia. On the other hand, much smaller bodies, with shorter collisional lifetimes, presumably have less regolith, or less mature regolith, and therefore display a larger thermal inertia. Deriving a functional dependence of the thermal inertia as a function of the size of the body has important implications for improving the models of the orbital mobility of asteroids due to the Yarkovsky effect and to better quantify systematic errors in radiometric diameters and albedos of small bodies based on the use of thermal models that neglect the effects of heat conduction, such as the STM. The graph in Fig. 6 suggests that, to the first order, thermal inertia in this size range follows a power law. Expressing Γ as 0d D ξ−Γ = (6) (a linear relation in the log Γ – log D plot), a linear regression gives best-fit values of ξ=0.48±0.04 and d0=300±47, where D is km and Γ in S. I. units (J m-2 s-0.5 K-1), and the 1σ uncertainty is based on the assumption that the errors on the thermal inertia and diameter values are normally distributed. (The values of Γ for 2002 NY40, 6489 Golevka and the Karin cluster asteroids were excluded from the linear regression analysis). However, the slope ξ of Eq. (6) may assume different values in different size ranges, since there are reasons to suspect that the surface properties of large asteroids may be different to those of smaller bodies: for example, Bottke et al. (2005) showed that asteroids with D > 100 km and most bodies with D > 50 km in size are likely to be primordial objects that have not suffered collisional disruption in the past 4 Gy. These large bodies have spent sufficient time in the asteroid belt to build a regolith such that they would display a low thermal inertia independent of size. In this case ξ should be about zero for D larger than about 50 km. In the same study it was shown that asteroids smaller than ~30 km are statistically the remnants of catastrophic collisional disruption of larger parent bodies, and the smaller the object, the fresher the surface. In this latter case one may intuitively expect that a dependence of the thermal inertia on the asteroid diameter would be more likely to occur, implying ξ > 0 for D < 30 km. For these reasons we tried to fit the data piecewise, separating the NEAs from the MBAs: a linear regression of Eq. (6) for the MBAs only of Fig. 6, gives best-fit values of ξ=0.49±0.27 and d0=300±150 (Fig. 6, dotted-line) in good agreement with the trend obtained by fitting - 15 - the whole dataset of thermal inertias. However, we note that the accuracy of this fit is poor and that the value of ξ is strongly influenced by the thermal inertia of 21 Lutetia. On the other hand, a fit of Eq. (6) for near-Earth asteroids only, gives best-fit values of ξ=0.36±0.09 and d0=300±45 (Fig. 6, dashed-line) which corresponds to a shallower dependence of Γ on D for sizes up to 20 km. A further distinction in the thermal properties of MBAs compared to that of NEAs is given by the different mean heliocentric distances of the two classes of body, causing NEAs to have average temperatures ~200 K higher than those of MBAs. The thermal conductivity in the regolith is temperature dependent (Keihm, 1984), and so is thermal inertia. This temperature dependence of Γ may alter the slope ξ of Eq. (6) when both NEAs and main-belt asteroids are included in the fit. Under the assumption that heat is transported in the regolith mainly by radiative conduction between grains, the thermal conductivity is proportional to T3, with T being the temperature of the regolith grains (Kührt and Giese, 1989; Jakosky 1986). In this case 3 2TΓ ∝ and, from Eq. (2), 3 4r−Γ ∝ , where r is the heliocentric distance of the body. On the basis of this dependence of Γ with respect to r, we corrected the values of the thermal inertias of the asteroids of Fig. 6 to the mean heliocentric distance rref of 1.7 AU. Although the correction factors are in general smaller than the errors affecting the values of Γ, the thermal inertia values of NEAs (r < rref) are reduced, whereas those of MBAs (r > rref) are increased, yielding a smaller value of the slope ξ=0.37±0.04 and d0=230±30. Furthermore, the make up of NEA surfaces can be modified by processes such as close encounters with planets causing tidal disruption that do not affect asteroids in the Main Belt. Such processes might have been able to alter or strip off the regolith of some NEAs. Thus, while NEAs may be a good proxy for small main-belt asteroids, more observations are needed to confirm this point. It is clear that with the present small number of asteroids for which we have an estimate of the thermal inertia it is difficult to reveal possible variations of ξ with respect to the mean trend, ξ ~0.4, in different size ranges. Nevertheless, Fig. 6 shows a clear correlation of Γ with asteroid size and that asteroids in the 1 – 30 km size range have values of Γ in general larger than 100 J m-2 s-0.5 K-1. The fact that thermal inertia increases with decreasing size and that the value of Γ for km and multi-km sized asteroids is at least ten times larger than the value derived for the largest main-belt asteroids, has a number of important implications. First of all, radiometric diameters and albedos of asteroids derived by means of thermal models neglecting the effects of thermal inertia, such as the STM, are likely to be affected by increasing systematic errors with decreasing size. Spencer et al. (1989) have studied systematic biases in radiometric diameter determinations as a result of the effects of thermal inertia, rotation rate, pole orientation, and temperature. They concluded that the STM systematically underestimates the diameters of objects with non-negligible thermal inertia, while overestimating their albedos. Because we find that thermal inertia increases with decreasing asteroid diameter, it is likely that the systematic underestimation of asteroid diameters (and overestimation of asteroid albedos) obtained from the STM increases for decreasing asteroid size. Moreover, the absolute value and size dependence of the thermal inertia for asteroids with diameters smaller than about 10-20 km have crucial implications for the magnitude of the Yarkovsky effect, which is an important phenomenon that offers an explanation for the dispersion of asteroid dynamical families and the slow but steady injection of bodies - 16 - into the dynamical resonances that eventually transport them from the main belt to near- Earth space. 4 Implications for the magnitude of the Yarkovsky effect Current models of Yarkovsky-assisted delivery of NEAs from the Main Belt (Morbidelli and Vokrouhlický, 2003) and the spreading of asteroids families (Bottke et al., 2001; Nesvorný and Bottke, 2004), assume that thermal inertia is independent of object size. In this case, the theory of the Yarkovsky effect predicts that the orbital semimajor axis drift rate of an asteroid, /da dt , is proportional to D-1 (Bottke et al., 2002). However, the mean value of the thermal inertia derived for NEAs and the inverse correlation of this thermophysical property with asteroid size, demonstrated in this work, give rise to a different magnitude of the Yarkovsky effect and a modified dependence of /da dt on the object diameter. In order to derive this modification one can directly insert the function 0d D ξ−Γ = in the formulas given by Bottke et al. (2002) and Vokrouhlický (1999) to explicitly calculate /da dt as a function of the relevant parameters. Here we discuss the case of MBAs with D < 10 km, assuming the linearized theory of the diurnal component of the Yarkovsky effect of Vokrouhlický (1999), which yields 1 0.5 +Θ+ Θ . (7) Because Θ is directly proportional to Γ, for Θ 1, 1 1da dt D− −∝ Γ and hence 1da dt Dξ −∝ . We found that this condition holds in general for small asteroids in the Main Belt: in fact for objects with D smaller than ~10 km Fig. 6 shows that the thermal inertia is in general >100 Jm-2s-0.5K-1, and at heliocentric distances >2 AU the surface temperatures of these bodies are in general smaller than 250 K (see e.g. Delbo’, 2004). Pravec et al. (2002) have also shown that asteroids with sizes between ~0.15 and 10 km have a typical rotation rate around 6 hours. Inserting these values in Eq. (3), we find that Θ is in general larger than ~2 for main-belt asteroids in this size range. By taking ξ ~ 0.4, as derived in the previous section for the Γ= d0D-ξ relation, we obtain that the size dependence of /da dt due to the diurnal component of the Yarkovsky effect is proportional to D-0.6 rather than proportional to D-1, as generally assumed, which would hold true for a thermal inertia independent of asteroid size. We caution, however, that there are currently no reliable estimates of thermal inertia available for any object smaller than Itokawa (D ~ 350 m), so the relation derived for 0.6/da dt D−∝ should be assumed to hold for objects with diameters in the range between ~0.35 and ~10 km. Moreover, for asteroids smaller than 350 m and/or higher values of the thermal inertia, the seasonal component of the Yarkovsky effect may become significant and contribute to the average value of da/dt. From this analysis we conclude that, in the Main Belt, the drift rate in semimajor axis due to the diurnal component of the Yarkovsky effect increases with decreasing asteroid size more slowly than is normally assumed in models of the origin of NEAs and the spreading of asteroids families. The shallower dependence of the Yarkovsky effect on the diameter of the bodies caused by the inverse correlation of Γ with D has the important implication that the size distribution of the asteroids injected into the NEA source regions is less skewed to small - 17 - objects than generally assumed. In the following we briefly discuss some of the consequences of this: there is general consensus that the large majority of NEAs originate from the Main Belt via well defined “feeding zones” of dynamical instability (Morbidelli et al., 2002). Asteroidal material can gradually drift towards these NEA source regions as a result of Yarkovsky-driven semimajor axis mobility (Morbidelli et al., 2002; Morbidelli and Vokrouhlický, 2003). The cumulative size distribution of a population of asteroids in a given diameter range (e.g. 0.35 < D < 10 km) can be approximated by a simple exponential function of the form 0( )N D N D α−> = , (8) Therefore, according to this asteroid delivery model, the difference in the exponent α between the bodies injected into the NEA source regions and the remaining population of asteroids in the Main Belt is of the order of ~1 if the semimajor axis mobility is proportional to D-1. The same difference in the value of the exponents holds for the NEA and the MBA populations in a comparable size range, assuming that the large majority of NEAs come from the Main Belt. The results of the latest studies that have analyzed the size distributions of NEAs and km-sized MBAs imply that this difference is closer to 0.5-0.7, in favor of a Yarkovsky effect less effective for smaller asteroids, which is consistent with the results of this work. We recall that Eq. (8) can be converted into a cumulative absolute visual magnitude H distribution of a population of asteroids with the form 0( ) 10 HN H N β′< = , (9) where the exponential slope of the absolute magnitude distribution, β, can be converted into the power-law slope of the diameter distribution via α = 5β (see, e.g., Stuart and Binzel, 2004) . Several authors (Rabinowitz et al., 2000; Bottke et al., 2000; Stuart and Binzel, 2004) agree that β is in the range 0.35 – 0.39 for the NEA population, which implies a value of αNEA between 1.75 and 1.95. The size distribution of km- and sub km- sized MBAs is less constrained than that of NEAs, since the known population is still rather incomplete for H > 14 – 15 (corresponding to values of D between 6 and 3 km for a geometric visible albedo of 0.11), so that beyond this threshold only extrapolations of the known distribution can be made. Morbidelli and Vokrouhlický (2003) used the slopes derived by Ivezic et al. (2001) from the Sloan Digital Sky Survey (SDSS), namely, β=0.61 for 13 < H < 15.5 and β=0.25 for 15.5 < H < 18, to extrapolate the observed H cumulative distribution (as given by the Astorb catalog) to km sized asteroids, and use it in their model of the Yarkovsky-driven origin of near-Earth asteroids. Assuming β=0.25 for 15.5 < H < 18 this would give a value of αMBA=1.25 for the slope of the cumulative size distribution of MBAs in the range 1 < D < 3 km. This value is in good agreement with the even slightly shallower size distribution (αMBA~1.2) of km and sub km sized MBAs (for 0.5 < D < 1 km) found by the SMBAS survey (Sub-km Main-Belt Asteroid Survey) obtained by Yoshida et al., (2003). Taking the values of α from the studies above, we find that αNEA - αMBA = 0.5 – 0.7, in good agreement with a drifting population of asteroids with 0.6/da dt D−∝ . However, Morbidelli and Vokrouhlický (2003) have also shown that the collisional re-orientation of asteroid spin axes (which resets the drift speed due to the Yarkovsky effect), the collisional disruption of the bodies during their slow drift towards the NEA source regions, and the YORP effect, tend to decrease the difference between αNEA and - 18 - αMBA. The addition of these phenomena along with the revised dependence of 0.6/da dt D−∝ due to the Yarkovsky effect may help to explain the even steeper size distribution of small MBAs, and thus a difference between αNEA and αMBA smaller than 0.5 – 0.7 implied by the recent results of the Sub-Kilometer Asteroid Diameter Survey (SKADS; Davis et al., 2006), which found βMBA = 0.38, corresponding to αMBA = 1.9, for 13 < H < 17. The value of the thermal inertia also plays an important role in the YORP effect (Rubincam, 2000; Vokrouhlický and Capek, 2002), which is a torque produced by the thermal radiation emitted by asteroids with irregular shapes causing a slow spin-up/spin- down and a change of the spin axis obliquities of these bodies. In contrast to the Yarkovsky effect, YORP also acts on bodies with zero surface thermal conductivity. However, in the case of a thermal inertia significantly larger than zero (in contrast to the case of zero-conductivity), YORP preferentially drives obliquity toward two asymptotic states perpendicular to the orbital plane, and asymptotically decelerates and accelerates rotation rate in about an equal number of cases (Capek and Vokrouhlický, 2005). Capek and Vokrouhlický (2005) have shown that the acceleration of the rotation rate, /d dtω , is largely independent of the thermal inertia, whereas its value significantly affects the rate of change of the obliquity, /SSd dtθ , in the sense that the higher the thermal inertia the larger the mean value of /SSd dtθ . Capek and Vokrouhlický (2005) found the median value of the distribution of /SSd dtθ for populations of Gaussian spheres increased from 3.33 deg/My for Γ=0 J m-2 s-0.5 K-1 to 5.94 deg/My for Γ=39 J m-2 s-0.5 K-1 and to 8.60 deg/My in the case of Γ=122 J m-2 s-0.5 K-1, which is a suitable value for bodies of about 5 km according to our Fig. 6. Because our Γ (D) relation predicts an even larger value of surface thermal inertia for asteroids of 1 km in diameter, the YORP reorientation of the spin vector of asteroids becomes a more effective mechanism in the case of km-sized asteroids, capable of driving the rotation axis to the asymptotic state perpendicular to the orbital plane in just a few tens of millions of years. 5 Conclusions The thermal inertia of an asteroid can be derived by comparing measurements of its thermal-infrared flux, at wavelengths typically between 5 and 20 μm, to synthetic fluxes generated by means of a thermophysical model (TPM). To date TPMs have been used to derive the thermal inertia of seven large MBAs and five NEAs. Although an extensive set of thermal-infrared observations of NEAs exists, application of TPMs is limited to the few asteroids for which the gross shape, the rotation period, and the spin axis orientation are known. In order to overcome this limitation, we have developed a statistical method enabling the determination of the thermal inertia of a sample of objects for which such information is not available. This method has been applied to a sample of NEAs with diameters generally between 0.8 km and 3.4 km. The resulting value, Γ = 200 ± 40 J m-2 s-0.5 K-1, corresponds to a surface thermal conductivity of about 0.03 W m-1 K-1. This value of thermal inertia and those derived by means of TPMs reveal a significant trend of increasing thermal inertia with decreasing asteroid diameter, D. Assuming that Γ is proportional to D-ξ we derive a best-fit value for the exponent of ξ ~ 0.4. - 19 - The dependence Γ(D) has important implications for the magnitude of the Yarkovsky effect. On the basis of our results, the size dependence of the orbital semimajor axis drift rate /da dt of MBAs for ~0.35 < D < ~10 due to the diurnal component of the Yarkovsky effect is proportional to D-0.6, rather than the generally assumed D-1 dependence for size-independent thermal inertia. The modified dependence, 0.6/da dt D−∝ , implies that the size distribution of the objects injected by Yarkovsky-driven orbital mobility into the NEA source regions is less skewed to smaller sizes than generally assumed. This may help to explain the smaller- than-one difference in the value of the exponents of the cumulative size distribution of NEAs and MBAs. 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YORP-Induced Long-Term Evolution of the Spin State of Small Asteroids and Meteoroids: Rubincam's Approximation. Icarus 159, 449-467. Vokrouhlický, D. 1999. A complete linear model for the Yarkovsky thermal force on spherical asteroid fragments. Astronomy and Astrophysics 344, 362-366. Wolters, S. D., Green, S. F., McBride, N., Davies, J. K. 2005. Optical and thermal infrared observations of six near-Earth asteroids in 2002. Icarus 175, 92-110. Yoshida, F., Nakamura, T., Watanabe, J.-I., Kinoshita, D., Yamamoto, N., Fuse, T. 2003. Size and Spatial Distributionsof Sub-km Main-Belt Asteroids. Publications of the Astronomical Society of Japan 55, 701-715. - 24 - Table 1 D α P spin Asteroid (km) pV A η r (AU) (°) (hrs) Obs. Date axis 433 Eros 23.60 0.200 0.079 1.05 1.135 10 5.27 17-01-1975 Y a 433 Eros 23.60 0.210 0.082 1.07 1.619 31 5.27 29-06-1998 Y a 1580 Betulia 3.82 0.110 0.043 1.09 1.199 53 6.138 22-06-2002 Y b 1862 Apollo 1.40 0.260 0.102 1.15 1.063 35 3.065 26-11-1980 - c 1866 Sisyphus 8.90 0.140 0.055 1.14 1.609 35 2.4 29-06-1998 - d 1980 Tezcatlipoca 6.60 0.150 0.059 1.64 1.129 63 7.252 31-08-1997 Y a 2100 Ra-Shalom 2.79 0.080 0.031 2.32 1.174 39 19.8 21-08-2000 Y e 2100 Ra-Shalom 2.50 0.130 0.051 1.80 1.195 41 19.8 30-08-1997 Y f 3200 Phaethon 5.10 0.110 0.043 1.60 1.132 48 3.604 20-12-1984 - c 3554 Amun 2.10 0.170 0.067 1.20 1.243 16 2.53 15-03-1986 - c 3671 Dionysus 1.50 0.160 0.063 3.10 1.126 58 2.705 02-06-1997 - a 5381 Sekhmet 1.50 0.220 0.086 1.90 1.213 44 3 22-06-2003 - d 5381 Sekhmet 1.40 0.240 0.094 1.50 1.119 35 3 14-05-2003 - d 5587 1990 SB 4.00 0.250 0.098 1.10 1.399 19 5.052 09-04-2001 Y d 5587 1990 SB 3.57 0.320 0.126 0.84 1.210 42 5.052 10-05-2001 Y e 6455 1992 HE 3.43 0.280 0.110 0.80 1.641 22 - 22-03-2002 - g 6455 1992 HE 3.55 0.240 0.094 0.70 1.364 29 - 29-09-2002 - g 9856 1991 EE 1.00 0.300 0.118 1.15 1.093 36 3.045 11-09-1991 - f 14402 1991 DB 0.60 0.140 0.055 1.04 1.025 36 2.266 16-04-2000 - e 19356 1997 GH3 0.91 0.340 0.133 0.98 1.406 5 6.714 11-05-2001 - e 25330 1999 KV4 2.55 0.080 0.031 1.06 1.392 3 4.919 14-05-2003 - d 25330 1999 KV4 2.70 0.080 0.031 1.30 1.495 16 4.919 02-06-2003 - d 25330 1999 KV4 3.21 0.050 0.020 1.50 1.197 54 4.919 10-05-2001 - e 33342 1998 WT24 0.34 0.590 0.232 0.90 0.990 67 3.697 18-12-2001 - d 33342 1998 WT24 0.44 0.350 0.137 1.50 0.987 79 3.697 19-12-2001 - d 33342 1998 WT24 0.50 0.270 0.106 1.85 0.981 93 3.697 21-12-2001 - d 35396 1997 XF11 0.89 0.320 0.126 1.30 1.215 30 3.257 28-11-2002 - d 35396 1997 XF11 0.91 0.310 0.122 1.20 1.018 53 3.257 03-11-2002 - d 35396 1997 XF11 1.18 0.180 0.071 1.80 1.034 63 3.257 05-11-2002 - d 53789 2000 ED104 1.20 0.180 0.071 1.68 1.089 60 - 29-09-2002 - g 85953 1999 FK21 0.59 0.320 0.126 0.91 1.140 35 - 21-02-2002 - e 86039 1999 NC43 2.22 0.140 0.055 2.86 1.116 59 34.49 17-03-2000 - e 99935 2002 AV4 1.50 0.370 0.145 1.60 1.086 70 - 01-06-2003 - d 1999 HF1 4.74 0.110 0.043 1.68 0.957 91 - 22-03-2002 - g 2000 BG19 1.77 0.040 0.016 0.74 1.388 17 - 17-03-2000 - e 2001 LF 2.00 0.050 0.020 1.40 1.172 45 - 02-06-2003 - d 2002 BM26 0.84 0.020 0.008 3.10 1.023 60 - 21-02-2002 - e 2002 HK12 0.80 0.170 0.067 2.84 1.138 33 - 28-09-2002 - g 2002 NX18 2.40 0.030 0.012 1.19 1.158 54 - 29-09-2002 - g 2002 QE15 1.94 0.150 0.059 1.53 1.131 62 - 28-09-2002 - g 2003 YT1 1.50 0.270 0.106 1.92 1.035 74 - 08-05-2004 - d Table 1 Near Earth-asteroids with η-values derived from spectral fitting to multi-wavelength mid- infrared observations. The object effective diameter, D, the geometric visible albedo pV, and the η-values have been derived by using the NEATM. α is the phase angle at the epoch of the observations, which is given in the “Obs. Date” column. P is the rotation period in hours. In the column “Spin axis” a “Y” indicates that the spin axis orientation of the asteroid is known. In the column “Re” we give the original publication reference: a) Harris and Davies (1999); b) Harris et al. (2005); c) Harris (1998); d) Delbo’ (2004); e) Delbo’ et al. (2003); f) Harris et al. (1998); g) Wolters et al. (2005). - 25 - Figure captions Fig. 1 Sensitivity of η to model parameter variations. Δη/η (%) caused by a change of 1% in the bolometric Bond albedo A (upper left), macroscopic surface roughness 0θ (upper right), heliocentric distance r (lower left), and the thermal parameter Θ (lower right). See text, section 2.1 for details. Fig. 2 Distribution of the input parameters used in our statistical inversion method. Upper left: distributions of NEA rotation rates from “Physical parameters of NEOs (Hahn, 2006, http://earn.dlr.de”). Upper right: distribution of NEA phase angles. Lower left: distribution of NEA sub-solar latitudes θSS. Lower right, solid line: distribution of the geometric albedos (pV) and, dashed line: the bolometric Bond albedos (A) for the asteroids of Table 1 having η-values determined from observations. Fig. 3 Dependence of η-value on phase angle, α. Black diamonds: η-values derived from the NEATM for a set of NEAs with adequate multi-filter photometric data to enable η to be derived via spectral fitting (the data set includes multiple values of η for some objects observed at more than one phase angle; for the original data sources see Table 1). The error bars represent a 20% uncertainty, which is based on the reproducibility of η for those objects for which more than one measurement is available from independent data sets. Colored points: distributions of (α, η) calculated by means of our model for different values of thermal inertia: i.e. 15 (green), 200 (red), and 1000 (blue) J m-2 s-0.5 K-1. The distribution of the measured η-values is best described by the red points. Fig. 4 Plot of the function D (Γ), i.e. the distance D of the two-dimensional Kolmogorov-Smirnov best-fit procedure against the thermal inertia Γ. The three curves were generated assuming three different values of the surface roughness: solid line θ =58°; dotted line θ =36°; dashed line θ =0° i.e. a smooth surface. The dashed-dotted line shows the function D (Γ) obtained by using θ =58° and a random distribution of asteroid spin-axis orientations uniformly distributed over the celestial sphere, instead of the nominal one, as input for our model. Fig. 5 Histogram of the distribution of the log of the diameters, D, of the NEAs for which we have η-values determined from observations. The best-fit Gaussian function, 0.37exp(-z2/2), where z=(log D - 0.23)/0.31, with D in km, is also shown. Fig. 6 Thermal inertia as a function of asteroid diameter. Small open circles represent values from the literature derived by means of thermophysical models. The large open diamond is the result from this work (see text for details). The straight (continuous) line which gives the best fit to the trend of increasing thermal inertia, Γ, with decreasing asteroid diameter, D, is given by the expression Γ=300×D-0.48. The axis on the right-hand side gives the asteroid surface thermal conductivity k on the basis of k=Γ2/(ρc), assuming constant surface density, ρ, equal to 2500 kg m-3 and specific heat capacity, c, equal to 600 J kg-1 K-1. These values are reasonable assumptions for asteroid surfaces (Britt et al., 2002; Farinella et al., 1998). The thermal conductivities of (6489) Golevka (Chesley et al., 2003) and for Karin cluster asteroids (Nesvorný and Bottke, 2004) are indicated with arrows. The two values of Γ derived for 2002 NY40 are indicated as the lower and the upper limits of the error bar on the extreme left of the plot. Dotted line: linear regression of Eq. (6) for MBAs only; dashed line: linear regression of Eq. (6) for NEAs only. - 26 - Figures 0 20 40 60 80 100 α (deg) 0 20 40 60 80 100 α (deg) 0 20 40 60 80 100 α (deg) 0 20 40 60 80 100 α (deg) Fig. 1 - 27 - 0 1 2 3 4 5 6 7 8 9 10 11 12 NEA rotation rate, P (hours) 0 10 20 30 40 50 60 70 80 90 100 Solar phase angle, α (degrees) 0 10 20 30 40 50 60 70 80 90 100 Sub-solar latitude, θSS (degrees) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Albedo Fig. 2 - 28 - Fig. 3 - 29 - 0 100 200 300 400 500 600 700 800 900 1000 Thermal Inertia (J m-2 s-0.5 K-1) Fig. 4 - 30 - -2 -1 0 1 2 3 log D (D in km) Fig. 5 - 31 - 1 10 100 1000 Asteroid diameter (km) 1 10 100 1000 Asteroid diameter (km) Fig. 6

Thermal inertia determines the temperature distribution over the surface of an asteroid and therefore governs the magnitude the Yarkovsky effect. The latter causes gradual drifting of the orbits of km-sized asteroids and plays an important role in the delivery of near-Earth asteroids (NEAs) from the main belt and in the dynamical spreading of asteroid families. At present, very little is known about the thermal inertia of asteroids in the km size range. Here we show that the average thermal inertia of a sample of NEAs in the km-size range is 200 $\pm$ 40 J m&#8722;2 s&#8722;0.5 K&#8722;1. Furthermore, we identify a trend of increasing thermal inertia with decreasing asteroid diameter, D. This indicates that the dependence of the drift rate of the orbital semimajor axis on the size of asteroids due to the Yarkovsky effect is a more complex function than the generally adopted D^(&#8722;1) dependence, and that the size distribution of objects injected by Yarkovsky-driven orbital mobility into the NEA source regions is less skewed to smaller sizes than generally assumed. We discuss how this fact may help to explain the small difference in the slope of the size distribution of km-sized NEAs and main-belt asteroids.

Introduction Observations of asteroids in the wavelength range of their thermal-infrared emission (>5 μm) have been used since the 1970s (Allen, 1970) to determine the sizes and the albedos of these bodies. In recent years, thanks to the advances in detector technology and the availability of 10-m class telescopes on the ground, thermal-infrared observations of asteroids have improved in sensitivity. Increased efforts have consequently been devoted to deriving the sizes and albedos of near-Earth asteroids (NEAs; for reviews see Harris and Lagerros, 2002; Delbo’ and Harris, 2002; Delbo’, 2004; Harris, 2006 and references therein), in order to better assess the impact hazard these bodies pose to our planet and to improve our understanding of their relation to main-belt asteroids and comets (see Stuart and Binzel, 2004; Morbidelli et al., 2002). Furthermore, improvements in spectral coverage and the possibility of easily obtaining spectrophotometric data through narrow-band filters in the range 5 – 20 μm have allowed information on the surface temperatures of asteroids to be obtained. The spectrum of the thermal-infrared radiation received from a body is related to the temperature distribution on that part of its surface visible to the observer. Several factors play a role in determining the temperature distribution on the surface of an asteroid, such as the heliocentric distance, albedo, obliquity of the spin vector, rotation rate, and a number of thermal properties of the surface such as its thermal inertia. Thermal inertia is a measure of the resistance of a material to temperature change. It is defined as cρκΓ = , where κ is the thermal conductivity, ρ the density and c the specific heat capacity. The thermal inertia of an asteroid depends on regolith particle size and depth, degree of compaction, and exposure of solid rocks and boulders within the top few centimeters of the subsurface (see e.g. Mellon et al., 2000). At the limit of zero thermal inertia (the most simple temperature distribution model for asteroids), a body with a smooth surface would display a temperature distribution which depends only on the solar incidence angle i, (on a sphere, i is also the angular distance of a point from the subsolar point): 1/ 4cos , / 2 0, / 2 SST T i i (1) The subsolar temperature, TSS, is determined by equating the total energy absorbed by a surface element to that emitted in the thermal infrared, i.e.: = (2) where A is the bolometric Bond albedo, S is the solar constant, r is the heliocentric distance of the body, ε is the infrared emissivity, σ is the Stefan-Boltzmann constant and η is the so-called “beaming parameter”, which is equal to one in the case that each point of the surface is in instantaneous thermal equilibrium with solar radiation. The surface temperature distribution that one obtain for η=1 on a spherical shape is that of the so- called Standard Thermal Model (STM, Lebofsky and Spencer, 1989) that was widely used to derive diameters and albedos especially of main-belt asteroids (MBAs). In the more realistic case of a body with finite thermal inertia and rotating with a spin vector not pointing toward the sun, the temperature distribution is no longer symmetric with respect to the subsolar point: each surface element behaves like a capacitor or sink for the solar - 5 - energy such that the body’s diurnal temperature profile becomes more smoothed out in longitude (see Spencer et al., 1989; Delbo’ and Harris, 2002; Delbo’, 2004). The hottest temperatures during the day decrease, whereas those on the night-side do not drop to zero as in the idealistic case of zero thermal inertia, implying non-zero thermal-infrared emission from the dark side of the body. However, the effect of thermal inertia is coupled with the rotation rate of the body. An asteroid rotating slowly with a high thermal inertia displays a similar temperature distribution to one rotating more rapidly but with a lower thermal inertia. The degree to which the surface of an asteroid can respond to changes in insolation can be characterized by a single parameter: this is the so-called thermal parameter Θ (e.g. Spencer et al., 1989), which combines rotation period, P, thermal inertia, Γ, and subsolar surface temperature, TSS, and consequently depends on the heliocentric distance of the body. The thermal parameter is given by: SST P Θ = . (3) Note that objects with the same value of Θ, although with different P or Γ display the same diurnal temperature profile, provided they have the same shape and spin axis obliquity (the angle formed by the object spin vector and the direction to the Sun). In the case of non-zero thermal inertia, because the temperature distribution is no longer symmetric with respect to the direction to the Sun, the momentum carried off by the photons emitted in the thermal infrared has a component along the orbital velocity vector of the body, causing a decrease or increase of the asteroid orbital energy depending on whether the rotation sense of the body is prograde or retrograde. This phenomenon, known as the Yarkovsky effect, (see Bottke et al., 2002) causes a secular variation of the semimajor axis of the orbits of asteroids on a time scale of the order of 10-4 AU/Myr for a main-belt asteroid at 2.5 AU from the Sun with a diameter of 1 km. The Yarkovsky effect is responsible for the slow but continuous transport of small asteroids and meteoroids from the zone of their formation into chaotic resonance regions that can deliver them to near-Earth space (Bottke et al., 2002; Morbidelli and Vokrouhlický, 2003). The Yarkovsky effect is also important to explain the spreading of asteroid dynamical families (Bottke et al., 2001; Bottke et al., 2006; Nesvorný and Bottke, 2004). Moreover, the emission of thermal photons also produces a net torque that alters the spin vector of small bodies in two ways: it accelerates or decelerates the spin rate and also changes the direction of the spin axis. This mechanism was named by Rubincam (2000) as the Yarkovsky-O’Keefe-Radzievskii-Paddack effect, or YORP for short. Knowledge of the thermal inertia of asteroids is thus important for a number of reasons: (a) It can be used to infer the presence or absence of loose material on the surface: thermal inertia of fine dust is very low: ~30 J m-2 s-0.5 K-1 (Putzig et al., 2005); lunar regolith, a layer of fragmentary incoherent rocky debris covering the surface of the Moon, also has a low thermal inertia of about 50 J m-2 s-0.5 K-1 (Spencer et al., 1989). Coarse sand has a higher thermal inertia, i.e. about 400 J m-2 s-0.5 K-1 (Mellon et al., 2000; Christiansen et al., 2003), that of bare rock is larger than 2500 J m-2 s-0.5 K-1 (Jakosky, 1986), whereas the thermal inertia of metal rich asteroidal fragments can be larger than 12000 J m-2 s-0.5 K-1 (Farinella et al., 1998, Table 1). (b) Thermal inertia is the key thermophysical parameter that determines the temperature distribution over the surface of an asteroid and therefore governs the magnitude of the Yarkovsky and YORP effects (Capek and Vokrouhlický, 2004). (c) It allows a better determination of systematic errors - 6 - in diameters and albedos derived using simple thermal models, which make assumptions about the surface temperature distribution and/or neglect the thermal-infrared flux from the non-illuminated fraction of the body (see Spencer et al., 1989, Delbo’, 2004, Harris, 2005). However, at present, very little is known about the thermal inertia of asteroids in general, especially in the case of bodies in the km size range. The thermal inertia of an asteroid can be derived by comparing measurements of its thermal-infrared emission to synthetic fluxes generated by means of a thermophysical model (TPM; Spencer, 1990; Lagerros, 1996; Emery et al., 1998; Delbo’, 2004), which is used to calculate the temperature distribution over the body’s surface as a function of a number of parameters, including the thermal inertia Γ. In these models, the asteroid shape is modeled as a mesh of planar facets. The temperature of each facet is determined by numerically solving the one-dimensional heat diffusion equation using assumed values of the thermal inertia, with the boundary condition given by the time-dependent solar energy absorbed at the surface of the facet (see Delbo’, 2004). This latter quantity is calculated from the heliocentric distance of the asteroid, the value assumed for the albedo, and the solar incident angle. Macroscopic surface roughness is usually modeled by adding hemispherical section craters of variable opening angle and variable surface density to each facet. Shadowing and multiple reflections of incident solar and thermally emitted radiation inside craters are taken into account as described by Spencer (1990), Emery et al. (1998), and Delbo’ (2004). Heat conduction is also accounted for within craters (Spencer et al., 1989; Spencer, 1990; Lagerros, 1996, Delbo’, 2004). Surface roughness can be adjusted by changing the opening angle of the craters, the density of the crater distribution, or a combination of both. However, Emery et al. (1998) have shown that if surface roughness is measured in terms of the mean surface slope, θ , according to the parameterization introduced by Hapke (1984), emission spectra are a function of θ only and not of the crater opening angle and crater surface density. We recall here that tan tana d θ θ θ θ= ∫ (4) where θ is the angle of a given facet from the horizontal, and a(θ) is the distribution of surface slopes. The total observable thermal emission is calculated by summing the contributions from each facet visible to the observer. Model parameters (e.g. Γ, A, θ ) are adjusted until the best agreement is obtained with the observational data, i.e. the least- squares residual of the fit χ2 is minimized, thereby constraining the physical properties (albedo, size, macroscopic roughness, and thermal inertia) of the asteroid. To date, TPMs have been used to derive the thermal inertia of seven large MBAs (Müller, T. G. and Lagerros, 1998; Müller, T. G. and Blommaert, 2004; Mueller, M. et al., 2006b), and five NEAs (Harris et al., 2005; Müller, T. G. et al., 2005; Mueller, M. et al., 2006a, Harris et al., 2007); values derived lie between 5 and ~1000 J m-2 s-0.5 K-1, i.e. Γ varies by more than two orders of magnitude. The applicability of TPMs is limited to the few asteroids for which gross shape, rotation period, and spin axis orientation are known. Multi-epoch observations are also required for obtaining a robust estimation of the thermal properties of asteroids via TPM fit. There is, however, an extensive set of thermal-infrared observations of NEAs in the km size range for which no TPM fit is possible (e.g. Veeder et al., 1989; Harris, 1998; Harris et al., 1998; Harris and Davies, 1999; Delbo’ et al., 2003; Delbo’, 2004; Wolters et al., 2005). In order to overcome this limitation, we have developed a statistical - 7 - inversion method, described in Section 2, enabling the determination of the average value of the thermal inertia of NEAs in the km-size range. Our approach is based on the fact that, even though shapes, rotation periods, and spin axis orientations are not known for every NEA, the distribution of these quantities for the whole population can be inferred from published data (La Spina et al., 2004; Hahn, 2006). In Section 3 we compare the result from our statistical inversion method with the values of the thermal inertias of asteroids determined by means of thermophysical models, and we identify a trend of increasing thermal inertia with decreasing asteroid diameter, D. In Section 4 we describe the implications of the trend of increasing thermal inertia with decreasing asteroid diameter, in particular for the size-dependence of the Yarkovsky effect and the size distribution of NEAs and MBAs. 2 Determination of the mean thermal inertia of NEAs The large majority of asteroids for which we have thermal-infrared observations have been observed at a single epoch and/or information about their gross shape and pole orientation is not available, precluding the use of TPMs. In these cases simpler thermal models such as the near-Earth asteroid thermal model (NEATM; Harris, 1998) are used to derive the sizes and the albedos of these objects. The NEATM assumes that the object has a spherical shape, and its surface temperature distribution is described by Eq. (1) and Eq. (2). However, the parameter η is not kept constant, as in the case of the STM, but is adjusted in the fitting procedure to allow the model spectral energy distribution to match the observed data. In order to derive a robust estimate of the η-value the NEATM requires observations at different, ideally well-spaced, wavelengths in the thermal infrared. The parameter η can be seen as a measure of the departure of the asteroid temperature distribution from that of the STM and is a strong function of the surface thermal inertia (Spencer et al., 1989; Harris, 1998; Delbo’, 2004). However, η depends also on parameters such as the macroscopic surface roughness, θ , the rotation period, P, the bolometric Bond albedo, A, the thermal-infrared emissivity, ε, the heliocentric distance, r, the gross shape of the body, , the sub-solar latitude, θSS, the longitude, φSE, and the latitude, θSE, of the sub-Earth point (Delbo’, 2004). In general we can write that η≡η(ε, A, r, Θ(Γ, P),θ , θSS, θSE, φSE, ). (5) These parameters are usually not known for the individual objects, but their distributions can be estimated (or reasonably assumed) for the entire population. Note that a set of θSS, φSE, and θSE , which depend on the ecliptic longitude λ0 and latitude β0 of the pole of the body, also defines the value of the solar phase angle, α. Delbo’ et al. (2003) noted that qualitative information about the average thermal properties of a sample of NEAs could be obtained from the distribution of the η-values of the sample as function of the phase angle, α. In particular, the absence of large η-values (e.g. η > 2) at small or moderate phase angles (e.g. ≤ 45o), and the fact that η tends to ~ 0.8 for α approaching 0o, was interpreted in terms of the NEAs having low thermal inertias in general. In subsequent work (Delbo’, 2004) it was found that for a synthetic population of spherical asteroids with constant values of A, r, Γ, P, and θ , but with pole directions randomly oriented, the distribution of the points in the (α, η) plane is strongly dependent on Γ. By varying Γ until the distribution of the synthetic points in the (α, η) - 8 - plane matched the one derived from the observations, Delbo’ (2004) obtained a best-fit thermal inertia for the NEAs equal to ~500 J m-2 s-0.5 K-1. Harris (2005), using a similar method on a larger database of η-values and neglecting the effects of surface roughness (θ =0o), derived a best-fit thermal inertia of ~300 J m-2 s-0.5 K-1. Here we improve on the above-mentioned work by determining the mean thermal inertia of NEAs using a rigorous statistical inversion method, based on the comparison of the distributions of NEATM η-values from the current NEA database vs. α, with that of a synthetic population of asteroids generated through a TPM, using realistic distributions of the input parameters P, θSS, θSE, φSE, and A derived from the literature (see Table 1 with published η-values from Harris, 1998, Harris et al. (1998); Harris and Davies (1999); Delbo’ et al. (2003); Delbo’ (2004); and Wolters et al. (2005). La Spina et al. (2004) give the distribution of λ0 and β0 for NEAs, and Hahn (2006) that of NEA rotation rates). In the following section we describe our method in detail. 2.1 Model parameter space As a first step we studied the dependence of η on the relevant parameters of Eq. 5. This was done by choosing typical parameter values and showing how small perturbations of the assumed values affect η. For the purpose of this analysis we assume A0=0.073, r0=1.2 AU (as we will show below, these are the average values of A and r for the NEAs in our sample), 0θ =36 o (the value derived for 433 Eros; Domingue et al., 2002), 0=sphere, and Θ0=1.0. Note that Θ0=1.0 corresponds to a thermal inertia of ~200 J m-2 s-0.5 K-1 for surface temperatures typical of NEAs and P = 6 hours, a rotation period representative of asteroids with sizes between ~0.15 and 10 km (Pravec et al., 2002). We will show in section 2.3 that Γ = 200 J m-2 s-0.5 K-1 is the mean thermal inertia of NEAs. The illumination and observation geometry was varied such that θSS was uniformly distributed in the range between 0 and π/2 and θSE, φSE were varied in such a manner that the resulting sub-Earth vectors were uniformly distributed over the celestial sphere. The values of θSE, φSE were further subject to the constraint that the phase angle be ≤ 100o. Figure 1 shows the sensitivity of η to a change in the model parameters. In particular, for each value of θSS, θSE, and φSE, the variation of η due to a 1% change in each parameter is plotted. We have also calculated, for some fixed illumination and observation geometries (e.g. θSE = 0o, φSE = 45o and θSS = 0o), how the variations Δη scale with changes in the model parameters. We found that Δη is proportional to ΔA, Δ 0θ , ΔΘ, and Δr within a large range of variation (>100%) of each parameter from its nominal value. Because for common asteroidal material the thermal-infrared emissivity is thought to be relatively constant, it has been fixed for this study at ε = 0.9. It is appropriate for objects with surfaces that emit a substantial portion of their thermal-infrared radiation shortward of 8 μm (Lim et al., 2005). Mustard and Hays (1997) have also shown that the reflectance spectra of fine-grained particulate materials, thought to be representative of planetary regoliths, have values around 0.1 and in general smaller than 0.2 in the region 8 – 24 μm. Because the reflectance, R, and the emissivity are related by Kirchhoff’s law (R=1-ε), the measurements cited above implies that ε = 0.9 is a reasonable estimate for the thermal- infrared emissivity of NEA surfaces. Moreover, from Eq. 2 one can calculate that Δη≈1.6Δε for ε≈0.9 and η≈1.5 (the average η for the NEAs for which this parameter was derived from observations; see Table 1). This implies that variations of ε in the range 0.8 – 1.0 cause changes of η that are within the typical uncertainty of ~20% in the estimation - 9 - of η from observations. Note that the value of Δη/η = 20%, where Δη is the uncertainty in η, is based on the reproducibility of η for those objects for which more than one measurement is available from independent data sets. Moreover, 20% is also the mean value of Δη/η of the “Delbo’ Thermal Infrared Asteroid Diameters and Albedos” database at the NASA PDS (Delbo’, 2006). In this dataset for those observations where Δη is present, its value was formally calculated from the measurements of the asteroids’ thermal infrared fluxes. The vast majority of the observations in our sample was obtained at a phase angle smaller than 80o, and within this range, Fig. 1 shows that the largest variation of η caused by a 1% change of A (the bolometric Bond albedo) is approximately 0.1%. Because the mean value of A for our sample is A =0.073 and the standard deviation is 0.04 (see Table 1), the variation of η due to the distribution of the albedos is smaller than 5% and thus small compared to the typical uncertainty of Δη/η ~ 20%. For this reason we have utilized a constant value of 0.073 for A in our statistical inversion method. Moreover, the variation of η due to a 1% change in the macroscopic surface roughness is strongly phase angle dependent, but in general smaller than 0.2% for α in the range 0-60o. This implies that even a ±100% change in θ causes a variation of η within the typical 20% uncertainty. Note that a ±100% change in θ corresponds to a large variation of the macroscopic roughness, ranging from that of a completely smooth surface to one oversaturated by hemispherical craters. For those observations carried out at α > 60o, η is more sensitive to variations of θ . For the reasons above we have treated θ as a free parameter in the inversion method and searched for the value that best fits the observational data. The sensitivity of η to changes of the objects’ heliocentric distances is such that a 1% change of r corresponds to a maximum 0.7% change of η. As calculated for the values in Table 1, the heliocentric distances in our sample have a mean value of 1.2 AU and a standard deviation of 0.1 AU (~8%). The corresponding variation of η is approximately 6% and therefore small. We thus took a constant value of 1.2 AU for r in our statistical inversion method. Only in two cases, namely those of the 29-06-1998 observation of (433) Eros and for the 22-03-2002 observation of (6455) 1992 HE is the variation of η due to the deviation of the heliocentric distance from the nominal value of 1.2 AU slightly larger than the error bars. We note here that Eq. (5) implicitly contains the assumption that seasonal effects do not affect asteroid surface temperatures. However, when Θ ≠ 0, asteroid temperatures always depend on the previous thermal history of the surface. Since NEAs have in general large orbital eccentricities, these bodies experience large variations of insolation as a function of their orbital position, which may lead to a seasonal component of the variation of their surface temperatures and thus of the corresponding η-values. To demonstrate that our working hypothesis of Eq. (5) is valid (i.e. seasonal components are negligible), we calculated η for several synthetic asteroids with the same physical characteristics, but with different orbits, in order to explore the effect of different levels of insolation. Orbits were chosen with eccentricities in the range 0 to 0.8 but with a common perihelion distance rp. For different values of the asteroid thermal inertia in the range 200-5000 J m-2 s-0.5 K-1 and rp in the range 0.5 - 1.5 AU, we found variations of only a few percent in the η-values calculated at rp. This leads us to conclude that seasonal variations in the η-values are small and that Eq. (5) is valid. - 10 - In general NEAs have elongated shapes, which may cause their surface temperature distributions to differ from that of a spherical object with the same surface properties and illumination geometry. We studied the sensitivity of η to deviations from the spherical shape by calculating η-values of tri-axial ellipsoids, the semiaxes of which were varied in the ratio ( )1// aa with 1 ≤ a ≤ 6, assuming A0=0.073, r0=1.2 AU, 0θ =36o, Θ0=1.0, and for random orientations of the shape with respect to the Sun and the Earth. We found that the distribution of Δη is a function of a (with values of Δη increasing with increasing a ), where Δη is the deviation of η from that calculated using a sphere under the same illumination and viewing geometry. However, the relative error on η, Δη/η, is always smaller than ±10% for a ≤ 5 and α ≤ 45o. For α > 45o, the mean value of the relative error, 〈Δη/η〉, is smaller than +15% and its standard deviation, σΔη/η, is smaller than 5% for a ≤ 5. Because the maximum lightcurve amplitude of our model ellipsoid is L≈1.25log a mag, Δη/η is smaller than 20% if L ≤ 0.873 mag. This condition is in general satisfied for the NEAs in Table 1, for which the average value of L is around 0.6 mag. We expect that the contributions to Δη due to variations of the model parameters A, r, θ , and the ellipsoid axial ratio a, stack up randomly, since deviations of these parameters from their mean values are fully uncorrelated (e.g. there is no apparent reason that an NEA with an albedo higher than the average is also observed at an heliocentric distance higher than the its average value). We performed some numerical experiments in order to cross check this assumption and found that the value of Δη is in general a good proxy of [(∂η/∂AΔA)2 + (∂η/∂rΔr)2 (∂η/∂θ Δθ )]1/2. Adding the effect of non-spherical shapes increases the value of Δη, but never systematically at phase angles < ~60o. It is clear that ellipsoids are highly idealized shapes and larger contributions to Δη may be expected in the case of real NEAs. Figure 1 shows that the sensitivity of η to changes in the thermal parameter is very similar to the sensitivity to changes in r, with variations of η in general no larger than 0.5% for a 1% change of Θ. However, while the value of r for the asteroids in Table 1 is rather constant around the mean value of 1.2 AU, Θ can range between 0.1 and 20 considering that thermal inertia can be anywhere between 10 J m-2 s-0.5 K-1 (the thermal inertia of large main-belt asteroids) and 2500 J m-2 s-0.5 K-1 (that of bare rock). This implies that the scatter in the η-values that we observe in the NEAs of Table 1 is mainly a function of α and Θ, which depends on the thermal inertia. If we assume the thermal inertia to be roughly constant within the NEA population for a given size, its value can be inferred from the distribution of the measured η-values versus α. This is the idea on which our statistical inversion method is based. 2.2 Model populations Our inversion method requires η to be computed for all members of a synthetic population of NEAs as a function of Γ. The calculation of η was performed by numerically generating thermal-infrared spectra by means of a TPM and fitting them with the NEATM. As discussed in the previous sections, the parameters A, ε, r, θ , and contribute little to the variation of η within the expected parameter ranges. Therefore, they have been kept fixed to their nominal values throughout the modeling process. In order to keep the amount of computing time required for the inversion within reasonable limits, the values of η have been computed only once for all possible combinations of the - 11 - remaining parameters, and the results have been stored in a four-dimensional look up table. The granularity of the look up table was chosen to be small enough to cause changes of η of about 0.1 between two consecutive parameter steps. For each value of Γ, we then generated a large number (30,000) of synthetic objects whose parameters have random values with distributions that have been chosen to provide a reasonable match to the observed population of NEAs. In particular: (i) the distributions of the angles θSS, θSE, and φSE were computed starting from the distributions of the spin-axis orientation (λ0, β0) from La Spina et al. (2004), the phase angle α, the heliocentric ecliptic latitude βH, and the geocentric ecliptic latitude βE of the asteroids at the time of the infrared observations (see Table 1 and Fig. 2); (ii) the distribution of the thermal parameter was calculated starting from the distribution of the NEA rotation periods (Hahn, 2006) and by using a constant value of Γ. In Fig. 3 three such populations are shown that correspond to the Γ values of 15 (green), 200 (red), and 1000 (blue) J m-2 s-0.5 K-1, respectively. We have superimposed the η values for the NEAs in Table 1 on the synthetic data plot. 2.3 Best-fit procedure Figure 3 gives a clear visual impression of the dependence of η on Γ. We therefore used a formal best-fit technique to estimate the value of Γ for which a synthetic population best fits the observed data, under the assumption that Γ is constant for all objects in the observed sample. The method that we used to compare the observed data with the bi-dimensional distributions of the synthetic points in the (α, η) plane is based on the two-dimensional Kolmogorov-Smirnov metric (K-S; Press et al., 1992). The distance D of the K-S metric is used as the goodness of fit estimator (Press et al., 1992). Our best-fit procedure consisted of finding the value of Γ that minimizes the K-S distance D. From here on, we indicate this value with the symbol Γ*. Figure 4, where we have plotted the K-S distance D as a function of Γ, shows that the function D (Γ) has a minimum at Γ=200 J m-2 s-0.5 K-1, which is the value of thermal inertia that we take for Γ*. We expect Γ* to depend upon the assumed value for θ , the distributions of NEA rotation rates, and also on the spin-axis orientations that we have used to produce the distribution of the input parameters θSS, θSE, and φSE. Moreover, the value of Γ* must be affected by the errors in the measurements of the thermal infrared fluxes, i.e. by the errors on the η-values taken from the literature. In order to study the sensitivity of Γ* to changes applied to the nominal values of the input parameters, we first varied θ in the range between 0° (perfectly smooth surface) and 58° (corresponding to the surface completely covered by hemispherical craters). Figure 4 shows the function D (Γ) for three different values of θ . It clearly demonstrates that the value of Γ* only weakly depends on θ and that a high degree of surface roughness produces a better fit to the observed data. We also investigated the sensitivity of Γ* to changes in the input distributions of asteroids’ spin-axis orientations and rotation rates. Figure 4 shows the function D (Γ) obtained by using random spin-axis orientations uniformly distributed over the sphere instead of the nominal distribution. In that case, the best-fit thermal inertia increases to - 12 - 250 J m-2 s-0.5 K-1, and to 230 J m-2 s-0.5 K-1 if the distribution of the rotation rates are assumed to be uniformly distributed between 4 and 10 hours, a case which we believe to be very extreme. The sensitivity of Γ* to the errors affecting the η-values from Table 1 was studied by performing extended Monte Carlo simulations, in which we randomly varied the values of the η-values within their error bars (using normally-distributed random numbers), and for each simulation of noise-corrupted data we calculated the best-fit thermal inertia. The standard deviation of Γ* was found to be 40 J m-2 s-0.5 K-1. Of course, we expect that the distribution of the data points in Fig. 3 derives from a population with a range of thermal inertias, and further investigation is required to understand what the relations are between Γ* and the parameters defining the population, such as the mean value of Γ and the standard deviation of its distribution. In order to answer this question, we applied our inversion method on (α, η) points obtained from synthetic populations of NEAs with known distributions of thermal inertia. We used random values of Γ uniformly and normally distributed, varying both the mean value and the standard deviation of the populations. We found that our fitting procedure, based on the minimization of the K-S distance D, is capable of retrieving a good estimate of the mean value for Γ of the populations in all cases. We conclude that the average value of the thermal inertia for km-sized NEAs is 200±40 J m-2 s-0.5 K-1, which is about four times that of the lunar soil and corresponds to a surface thermal conductivity of 0.0150.0100.027 − W m-1 K-1 assuming that the surface material density and specific heat capacity are in the range 1500-3500 kg m-3 and 500-680 J kg-1 K-1, respectively (Britt et al., 2002; Farinella et al., 1998). The value of Γ* that we have derived by means of the best fit procedure is less than 10% of that expected for a bare-rock surface (Jakosky, 1986). This implies that the surfaces of NEAs have in general significant quantities of thermally-insulating regolith. However, Γ* is also about four times higher than the value that has been determined for the lunar soil and more than ten times higher than the thermal inertia typical of large main-belt asteroids. This effect may be due to the fact that the regolith present on NEA surfaces is less mature and/or less thick than that of the Moon and the largest MBAs. The higher NEA thermal inertia can also be explained in terms of a coarser regolith and the exposure of rocks and boulders on the surface of these bodies, as clearly shown in the high resolution images of (433) Eros and (25143) Itokawa obtained by the NEAR Shoemaker and the Hayabusa missions, respectively. A population of asteroids with constant Γ=200 J m-2 s-0.5 K-1 gives the best fit to the dataset. Figure 3 shows, however, that five points with η > 2 are clearly significantly higher than the majority, indicating that these objects presented unusually low color temperatures to the observer, possibly due to higher-than-average thermal inertia (see Delbo’ et al., 2003 and Delbo’, 2004). To gain insights into the width of the distribution of the thermal inertia of km-sized NEAs, we fitted the observed distribution of the data points with a synthetic population in which Γ was assumed to be uniformly distributed between 0 and ΓMAX. The best fit was obtained for ΓMAX ~ 600 J m-2 s-0.5 K-1. This suggests that the large majority of km-sized NEAs in our sample have thermal inertia below this value. The average value of the thermal inertia was derived for a sample of objects whose diameter distribution is shown in Fig. 5. We use here the radiometric diameters as derived by the NEATM. The mean diameter of the sample is 3 km, but if we remove the asteroid - 13 - 433 Eros, the mean diameter value decreases to 2 km. 433 Eros is much larger than the average size of the sample (see Fig. 5). In fact the median value of the diameter distribution (including 433 Eros) is 1.8 km. We note that the distribution of log D (where D is the diameter measured in km) is well fitted by a Gaussian distribution with a central value of 1.7 km. The standard deviation of the best-fit Gaussian function is 0.31 (in log D). We can thus conclude that the average value of the thermal inertia is representative of NEAs in the size range 0.8 – 3.4 km. 3 Size dependence of asteroid thermal inertia The mean thermal inertia for the sample of NEAs with published η-values is consistent with the values derived by means of TPMs for (433) Eros (Mueller, M. et al., 2006a), (1580) Betulia (Harris et al., 2005), (25143) Itokawa (Mueller, M. et al., 2006a; Müller, T. G. et al., 2005), and (33342) 1998 WT24 (Harris et al., 2007) for which values around 150, 180, 350, 630, and 200 J m-2 s-0.5 K-1 have been obtained respectively. Note that in the case of (25143) Itokawa, Müller, T. G. et al. (2005) have obtained a thermal inertia value of 750 J m-2 s-0.5 K-1 combining thermal-infrared observations gathered at ESO in 2004 with those obtained by Delbo’ (2004) in 2001. On the other hand, from the latter dataset of observations and a series of further observations of (25143) Itokawa obtained at the NASA-IRTF 3 m telescope with MIRSI in 2004, Mueller, M. et al. (2006a) derived a thermal inertia of ~350 J m-2 s-0.5 K-1 or ~800 J m-2 s-0.5 K-1 depending on whether the size of the body was obtained from the TPM or was forced to the radar value of Ostro et al. (2004). In this work we have taken the mean value and the extreme values of 350, 750, and 800 J m-2 s-0.5 K-1 as our best estimate for the thermal inertia of Itokawa and its uncertainty. Müller T. G. et al. (2004) have also attempted at deriving the thermal inertia of the small (~0.28 km) NEA 2002 NY40. They obtained a value of 100 J m-2 s-0.5 K-1 in the case that the size of the object was derived from the TPM, or 1000 J m- 2 s-0.5 K-1 if the body’s size was forced to the value obtained from radar observations. However, it is important to note that that the thermal inertia of 2002 NY40 was derived by assuming an equator-on view and a spherical shape for this object. The value of the thermal inertia derived from the TPM is in general strongly dependent on the pole orientation of the body. For this reason we expect the value of Γ derived for 2002 NY40 be less reliable than the values obtained for the other NEAs, for which the pole orientation derived from lightcurve inversion was adopted. From thermophysical modeling, Müller, T.G. and Lagerros (1998) derived the thermal inertias of a number of the largest MBAs, namely (1) Ceres, (2) Pallas, (3) Juno, (4) Vesta, and (532) Herculina, obtaining the values of 10, 10, 5, 25, and 15 Jm-2 s-0.5 K-1, respectively. Using the same approach, Müller, T.G. and Blommaert (2004) derived a thermal inertia of 15 J m-2 s-0.5 K-1 for (65) Cybele, and Mueller, M. et al. (2006b) obtained Γ~50 J m-2 s-0.5 K-1 for (21) Lutetia. From the published plots of the goodness of the TPM fit to the thermal-infrared data as a function of Γ it is possible to deduce that the relative uncertainties for the thermal inertias of these asteroids are around 50%. From the comparison of the values of Γ mentioned above, it is clear that there is an increase in the thermal inertia from that of large MBAs with diameters of several hundred km to that of much smaller km-sized NEAs, and that the values of Γ obtained for km- sized NEAs are about one order of magnitude or more higher than the values derived for large MBAs, but still an order of magnitude lower than the thermal inertia of bare rock (~2500 J m-2 s-0.5 K-1; Jakosky, 1986). In order to highlight the behavior of the thermal - 14 - inertia of asteroids as a function of their size, we have plotted the mean value of thermal inertia for NEAs and the values of the thermal inertia derived by means of TPMs against object diameters in Fig. 6. Small open circles represent the literature values derived from the application of TPMs. The large open diamond is the result from this work. The axis on the right-hand side gives the asteroid surface thermal conductivity k as a function of size, on the basis of k=Γ2/(ρc), with constant surface density ρ = 2500 kg m-3 and specific heat capacity c = 600 J kg-1 K-1. These values are reasonable assumptions for asteroid surfaces (Britt et al., 2002; Farinella et al., 1998). For the asteroid 2002 NY40 a bar between 100 and 1000 J m-2 s-0.5 K-1 is drawn. The thermal conductivity has also been constrained in the cases of (6489) Golevka (Chesley et al., 2003) and for asteroids in the Karin cluster (Nesvorný and Bottke, 2004). The values of the thermal conductivities derived by these authors have been converted to values of Γ assuming ρ=2500 kg m-3 and c=600 J kg-1 K-1. Fig. 6 shows that the resulting limits, based on the measurements of the Yarkovsky effect on these bodies, are in general agreement with our results. Figure 6 reveals a convincing trend of increasing thermal inertia with decreasing asteroid diameter, D, confirming the intuitive view that large main-belt asteroids, over many hundreds of millions of years, have developed substantial insulating regolith layers, responsible for the low values of their surface thermal inertia. On the other hand, much smaller bodies, with shorter collisional lifetimes, presumably have less regolith, or less mature regolith, and therefore display a larger thermal inertia. Deriving a functional dependence of the thermal inertia as a function of the size of the body has important implications for improving the models of the orbital mobility of asteroids due to the Yarkovsky effect and to better quantify systematic errors in radiometric diameters and albedos of small bodies based on the use of thermal models that neglect the effects of heat conduction, such as the STM. The graph in Fig. 6 suggests that, to the first order, thermal inertia in this size range follows a power law. Expressing Γ as 0d D ξ−Γ = (6) (a linear relation in the log Γ – log D plot), a linear regression gives best-fit values of ξ=0.48±0.04 and d0=300±47, where D is km and Γ in S. I. units (J m-2 s-0.5 K-1), and the 1σ uncertainty is based on the assumption that the errors on the thermal inertia and diameter values are normally distributed. (The values of Γ for 2002 NY40, 6489 Golevka and the Karin cluster asteroids were excluded from the linear regression analysis). However, the slope ξ of Eq. (6) may assume different values in different size ranges, since there are reasons to suspect that the surface properties of large asteroids may be different to those of smaller bodies: for example, Bottke et al. (2005) showed that asteroids with D > 100 km and most bodies with D > 50 km in size are likely to be primordial objects that have not suffered collisional disruption in the past 4 Gy. These large bodies have spent sufficient time in the asteroid belt to build a regolith such that they would display a low thermal inertia independent of size. In this case ξ should be about zero for D larger than about 50 km. In the same study it was shown that asteroids smaller than ~30 km are statistically the remnants of catastrophic collisional disruption of larger parent bodies, and the smaller the object, the fresher the surface. In this latter case one may intuitively expect that a dependence of the thermal inertia on the asteroid diameter would be more likely to occur, implying ξ > 0 for D < 30 km. For these reasons we tried to fit the data piecewise, separating the NEAs from the MBAs: a linear regression of Eq. (6) for the MBAs only of Fig. 6, gives best-fit values of ξ=0.49±0.27 and d0=300±150 (Fig. 6, dotted-line) in good agreement with the trend obtained by fitting - 15 - the whole dataset of thermal inertias. However, we note that the accuracy of this fit is poor and that the value of ξ is strongly influenced by the thermal inertia of 21 Lutetia. On the other hand, a fit of Eq. (6) for near-Earth asteroids only, gives best-fit values of ξ=0.36±0.09 and d0=300±45 (Fig. 6, dashed-line) which corresponds to a shallower dependence of Γ on D for sizes up to 20 km. A further distinction in the thermal properties of MBAs compared to that of NEAs is given by the different mean heliocentric distances of the two classes of body, causing NEAs to have average temperatures ~200 K higher than those of MBAs. The thermal conductivity in the regolith is temperature dependent (Keihm, 1984), and so is thermal inertia. This temperature dependence of Γ may alter the slope ξ of Eq. (6) when both NEAs and main-belt asteroids are included in the fit. Under the assumption that heat is transported in the regolith mainly by radiative conduction between grains, the thermal conductivity is proportional to T3, with T being the temperature of the regolith grains (Kührt and Giese, 1989; Jakosky 1986). In this case 3 2TΓ ∝ and, from Eq. (2), 3 4r−Γ ∝ , where r is the heliocentric distance of the body. On the basis of this dependence of Γ with respect to r, we corrected the values of the thermal inertias of the asteroids of Fig. 6 to the mean heliocentric distance rref of 1.7 AU. Although the correction factors are in general smaller than the errors affecting the values of Γ, the thermal inertia values of NEAs (r < rref) are reduced, whereas those of MBAs (r > rref) are increased, yielding a smaller value of the slope ξ=0.37±0.04 and d0=230±30. Furthermore, the make up of NEA surfaces can be modified by processes such as close encounters with planets causing tidal disruption that do not affect asteroids in the Main Belt. Such processes might have been able to alter or strip off the regolith of some NEAs. Thus, while NEAs may be a good proxy for small main-belt asteroids, more observations are needed to confirm this point. It is clear that with the present small number of asteroids for which we have an estimate of the thermal inertia it is difficult to reveal possible variations of ξ with respect to the mean trend, ξ ~0.4, in different size ranges. Nevertheless, Fig. 6 shows a clear correlation of Γ with asteroid size and that asteroids in the 1 – 30 km size range have values of Γ in general larger than 100 J m-2 s-0.5 K-1. The fact that thermal inertia increases with decreasing size and that the value of Γ for km and multi-km sized asteroids is at least ten times larger than the value derived for the largest main-belt asteroids, has a number of important implications. First of all, radiometric diameters and albedos of asteroids derived by means of thermal models neglecting the effects of thermal inertia, such as the STM, are likely to be affected by increasing systematic errors with decreasing size. Spencer et al. (1989) have studied systematic biases in radiometric diameter determinations as a result of the effects of thermal inertia, rotation rate, pole orientation, and temperature. They concluded that the STM systematically underestimates the diameters of objects with non-negligible thermal inertia, while overestimating their albedos. Because we find that thermal inertia increases with decreasing asteroid diameter, it is likely that the systematic underestimation of asteroid diameters (and overestimation of asteroid albedos) obtained from the STM increases for decreasing asteroid size. Moreover, the absolute value and size dependence of the thermal inertia for asteroids with diameters smaller than about 10-20 km have crucial implications for the magnitude of the Yarkovsky effect, which is an important phenomenon that offers an explanation for the dispersion of asteroid dynamical families and the slow but steady injection of bodies - 16 - into the dynamical resonances that eventually transport them from the main belt to near- Earth space. 4 Implications for the magnitude of the Yarkovsky effect Current models of Yarkovsky-assisted delivery of NEAs from the Main Belt (Morbidelli and Vokrouhlický, 2003) and the spreading of asteroids families (Bottke et al., 2001; Nesvorný and Bottke, 2004), assume that thermal inertia is independent of object size. In this case, the theory of the Yarkovsky effect predicts that the orbital semimajor axis drift rate of an asteroid, /da dt , is proportional to D-1 (Bottke et al., 2002). However, the mean value of the thermal inertia derived for NEAs and the inverse correlation of this thermophysical property with asteroid size, demonstrated in this work, give rise to a different magnitude of the Yarkovsky effect and a modified dependence of /da dt on the object diameter. In order to derive this modification one can directly insert the function 0d D ξ−Γ = in the formulas given by Bottke et al. (2002) and Vokrouhlický (1999) to explicitly calculate /da dt as a function of the relevant parameters. Here we discuss the case of MBAs with D < 10 km, assuming the linearized theory of the diurnal component of the Yarkovsky effect of Vokrouhlický (1999), which yields 1 0.5 +Θ+ Θ . (7) Because Θ is directly proportional to Γ, for Θ 1, 1 1da dt D− −∝ Γ and hence 1da dt Dξ −∝ . We found that this condition holds in general for small asteroids in the Main Belt: in fact for objects with D smaller than ~10 km Fig. 6 shows that the thermal inertia is in general >100 Jm-2s-0.5K-1, and at heliocentric distances >2 AU the surface temperatures of these bodies are in general smaller than 250 K (see e.g. Delbo’, 2004). Pravec et al. (2002) have also shown that asteroids with sizes between ~0.15 and 10 km have a typical rotation rate around 6 hours. Inserting these values in Eq. (3), we find that Θ is in general larger than ~2 for main-belt asteroids in this size range. By taking ξ ~ 0.4, as derived in the previous section for the Γ= d0D-ξ relation, we obtain that the size dependence of /da dt due to the diurnal component of the Yarkovsky effect is proportional to D-0.6 rather than proportional to D-1, as generally assumed, which would hold true for a thermal inertia independent of asteroid size. We caution, however, that there are currently no reliable estimates of thermal inertia available for any object smaller than Itokawa (D ~ 350 m), so the relation derived for 0.6/da dt D−∝ should be assumed to hold for objects with diameters in the range between ~0.35 and ~10 km. Moreover, for asteroids smaller than 350 m and/or higher values of the thermal inertia, the seasonal component of the Yarkovsky effect may become significant and contribute to the average value of da/dt. From this analysis we conclude that, in the Main Belt, the drift rate in semimajor axis due to the diurnal component of the Yarkovsky effect increases with decreasing asteroid size more slowly than is normally assumed in models of the origin of NEAs and the spreading of asteroids families. The shallower dependence of the Yarkovsky effect on the diameter of the bodies caused by the inverse correlation of Γ with D has the important implication that the size distribution of the asteroids injected into the NEA source regions is less skewed to small - 17 - objects than generally assumed. In the following we briefly discuss some of the consequences of this: there is general consensus that the large majority of NEAs originate from the Main Belt via well defined “feeding zones” of dynamical instability (Morbidelli et al., 2002). Asteroidal material can gradually drift towards these NEA source regions as a result of Yarkovsky-driven semimajor axis mobility (Morbidelli et al., 2002; Morbidelli and Vokrouhlický, 2003). The cumulative size distribution of a population of asteroids in a given diameter range (e.g. 0.35 < D < 10 km) can be approximated by a simple exponential function of the form 0( )N D N D α−> = , (8) Therefore, according to this asteroid delivery model, the difference in the exponent α between the bodies injected into the NEA source regions and the remaining population of asteroids in the Main Belt is of the order of ~1 if the semimajor axis mobility is proportional to D-1. The same difference in the value of the exponents holds for the NEA and the MBA populations in a comparable size range, assuming that the large majority of NEAs come from the Main Belt. The results of the latest studies that have analyzed the size distributions of NEAs and km-sized MBAs imply that this difference is closer to 0.5-0.7, in favor of a Yarkovsky effect less effective for smaller asteroids, which is consistent with the results of this work. We recall that Eq. (8) can be converted into a cumulative absolute visual magnitude H distribution of a population of asteroids with the form 0( ) 10 HN H N β′< = , (9) where the exponential slope of the absolute magnitude distribution, β, can be converted into the power-law slope of the diameter distribution via α = 5β (see, e.g., Stuart and Binzel, 2004) . Several authors (Rabinowitz et al., 2000; Bottke et al., 2000; Stuart and Binzel, 2004) agree that β is in the range 0.35 – 0.39 for the NEA population, which implies a value of αNEA between 1.75 and 1.95. The size distribution of km- and sub km- sized MBAs is less constrained than that of NEAs, since the known population is still rather incomplete for H > 14 – 15 (corresponding to values of D between 6 and 3 km for a geometric visible albedo of 0.11), so that beyond this threshold only extrapolations of the known distribution can be made. Morbidelli and Vokrouhlický (2003) used the slopes derived by Ivezic et al. (2001) from the Sloan Digital Sky Survey (SDSS), namely, β=0.61 for 13 < H < 15.5 and β=0.25 for 15.5 < H < 18, to extrapolate the observed H cumulative distribution (as given by the Astorb catalog) to km sized asteroids, and use it in their model of the Yarkovsky-driven origin of near-Earth asteroids. Assuming β=0.25 for 15.5 < H < 18 this would give a value of αMBA=1.25 for the slope of the cumulative size distribution of MBAs in the range 1 < D < 3 km. This value is in good agreement with the even slightly shallower size distribution (αMBA~1.2) of km and sub km sized MBAs (for 0.5 < D < 1 km) found by the SMBAS survey (Sub-km Main-Belt Asteroid Survey) obtained by Yoshida et al., (2003). Taking the values of α from the studies above, we find that αNEA - αMBA = 0.5 – 0.7, in good agreement with a drifting population of asteroids with 0.6/da dt D−∝ . However, Morbidelli and Vokrouhlický (2003) have also shown that the collisional re-orientation of asteroid spin axes (which resets the drift speed due to the Yarkovsky effect), the collisional disruption of the bodies during their slow drift towards the NEA source regions, and the YORP effect, tend to decrease the difference between αNEA and - 18 - αMBA. The addition of these phenomena along with the revised dependence of 0.6/da dt D−∝ due to the Yarkovsky effect may help to explain the even steeper size distribution of small MBAs, and thus a difference between αNEA and αMBA smaller than 0.5 – 0.7 implied by the recent results of the Sub-Kilometer Asteroid Diameter Survey (SKADS; Davis et al., 2006), which found βMBA = 0.38, corresponding to αMBA = 1.9, for 13 < H < 17. The value of the thermal inertia also plays an important role in the YORP effect (Rubincam, 2000; Vokrouhlický and Capek, 2002), which is a torque produced by the thermal radiation emitted by asteroids with irregular shapes causing a slow spin-up/spin- down and a change of the spin axis obliquities of these bodies. In contrast to the Yarkovsky effect, YORP also acts on bodies with zero surface thermal conductivity. However, in the case of a thermal inertia significantly larger than zero (in contrast to the case of zero-conductivity), YORP preferentially drives obliquity toward two asymptotic states perpendicular to the orbital plane, and asymptotically decelerates and accelerates rotation rate in about an equal number of cases (Capek and Vokrouhlický, 2005). Capek and Vokrouhlický (2005) have shown that the acceleration of the rotation rate, /d dtω , is largely independent of the thermal inertia, whereas its value significantly affects the rate of change of the obliquity, /SSd dtθ , in the sense that the higher the thermal inertia the larger the mean value of /SSd dtθ . Capek and Vokrouhlický (2005) found the median value of the distribution of /SSd dtθ for populations of Gaussian spheres increased from 3.33 deg/My for Γ=0 J m-2 s-0.5 K-1 to 5.94 deg/My for Γ=39 J m-2 s-0.5 K-1 and to 8.60 deg/My in the case of Γ=122 J m-2 s-0.5 K-1, which is a suitable value for bodies of about 5 km according to our Fig. 6. Because our Γ (D) relation predicts an even larger value of surface thermal inertia for asteroids of 1 km in diameter, the YORP reorientation of the spin vector of asteroids becomes a more effective mechanism in the case of km-sized asteroids, capable of driving the rotation axis to the asymptotic state perpendicular to the orbital plane in just a few tens of millions of years. 5 Conclusions The thermal inertia of an asteroid can be derived by comparing measurements of its thermal-infrared flux, at wavelengths typically between 5 and 20 μm, to synthetic fluxes generated by means of a thermophysical model (TPM). To date TPMs have been used to derive the thermal inertia of seven large MBAs and five NEAs. Although an extensive set of thermal-infrared observations of NEAs exists, application of TPMs is limited to the few asteroids for which the gross shape, the rotation period, and the spin axis orientation are known. In order to overcome this limitation, we have developed a statistical method enabling the determination of the thermal inertia of a sample of objects for which such information is not available. This method has been applied to a sample of NEAs with diameters generally between 0.8 km and 3.4 km. The resulting value, Γ = 200 ± 40 J m-2 s-0.5 K-1, corresponds to a surface thermal conductivity of about 0.03 W m-1 K-1. This value of thermal inertia and those derived by means of TPMs reveal a significant trend of increasing thermal inertia with decreasing asteroid diameter, D. Assuming that Γ is proportional to D-ξ we derive a best-fit value for the exponent of ξ ~ 0.4. - 19 - The dependence Γ(D) has important implications for the magnitude of the Yarkovsky effect. On the basis of our results, the size dependence of the orbital semimajor axis drift rate /da dt of MBAs for ~0.35 < D < ~10 due to the diurnal component of the Yarkovsky effect is proportional to D-0.6, rather than the generally assumed D-1 dependence for size-independent thermal inertia. The modified dependence, 0.6/da dt D−∝ , implies that the size distribution of the objects injected by Yarkovsky-driven orbital mobility into the NEA source regions is less skewed to smaller sizes than generally assumed. This may help to explain the smaller- than-one difference in the value of the exponents of the cumulative size distribution of NEAs and MBAs. We stress that the dataset on which our results are based is small and more multi- wavelength, multi-epoch thermal-infrared observations of asteroids with known spin states are required to refine our conclusions on the size dependence of thermal inertia and its consequences. 6 Acknowledgments We wish to thank Bill Bottke and David Vokrouhlický for earlier suggestions and comments that inspired us to develop the original concept of this work further, and the referees of the present paper, Stephen Wolters and Bill Bottke, for suggestions that led to significant improvements in the presentation. M. D. wishes to acknowledge fruitful discussions with A. Morbidelli and A. Cellino. - 20 - References Allen, D. A. 1970. Infrared Diameter of Vesta. Nature 227, 158. Bottke, W. F., Vokrouhlický, D., Rubincam, D. P., Nesvorný, D. 2006. The Yarkovsky and Yorp Effects: Implications for Asteroid Dynamics. Annual Review of Earth and Planetary Sciences 34, 157-191. Bottke, W. F., Durda, D. 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Date axis 433 Eros 23.60 0.200 0.079 1.05 1.135 10 5.27 17-01-1975 Y a 433 Eros 23.60 0.210 0.082 1.07 1.619 31 5.27 29-06-1998 Y a 1580 Betulia 3.82 0.110 0.043 1.09 1.199 53 6.138 22-06-2002 Y b 1862 Apollo 1.40 0.260 0.102 1.15 1.063 35 3.065 26-11-1980 - c 1866 Sisyphus 8.90 0.140 0.055 1.14 1.609 35 2.4 29-06-1998 - d 1980 Tezcatlipoca 6.60 0.150 0.059 1.64 1.129 63 7.252 31-08-1997 Y a 2100 Ra-Shalom 2.79 0.080 0.031 2.32 1.174 39 19.8 21-08-2000 Y e 2100 Ra-Shalom 2.50 0.130 0.051 1.80 1.195 41 19.8 30-08-1997 Y f 3200 Phaethon 5.10 0.110 0.043 1.60 1.132 48 3.604 20-12-1984 - c 3554 Amun 2.10 0.170 0.067 1.20 1.243 16 2.53 15-03-1986 - c 3671 Dionysus 1.50 0.160 0.063 3.10 1.126 58 2.705 02-06-1997 - a 5381 Sekhmet 1.50 0.220 0.086 1.90 1.213 44 3 22-06-2003 - d 5381 Sekhmet 1.40 0.240 0.094 1.50 1.119 35 3 14-05-2003 - d 5587 1990 SB 4.00 0.250 0.098 1.10 1.399 19 5.052 09-04-2001 Y d 5587 1990 SB 3.57 0.320 0.126 0.84 1.210 42 5.052 10-05-2001 Y e 6455 1992 HE 3.43 0.280 0.110 0.80 1.641 22 - 22-03-2002 - g 6455 1992 HE 3.55 0.240 0.094 0.70 1.364 29 - 29-09-2002 - g 9856 1991 EE 1.00 0.300 0.118 1.15 1.093 36 3.045 11-09-1991 - f 14402 1991 DB 0.60 0.140 0.055 1.04 1.025 36 2.266 16-04-2000 - e 19356 1997 GH3 0.91 0.340 0.133 0.98 1.406 5 6.714 11-05-2001 - e 25330 1999 KV4 2.55 0.080 0.031 1.06 1.392 3 4.919 14-05-2003 - d 25330 1999 KV4 2.70 0.080 0.031 1.30 1.495 16 4.919 02-06-2003 - d 25330 1999 KV4 3.21 0.050 0.020 1.50 1.197 54 4.919 10-05-2001 - e 33342 1998 WT24 0.34 0.590 0.232 0.90 0.990 67 3.697 18-12-2001 - d 33342 1998 WT24 0.44 0.350 0.137 1.50 0.987 79 3.697 19-12-2001 - d 33342 1998 WT24 0.50 0.270 0.106 1.85 0.981 93 3.697 21-12-2001 - d 35396 1997 XF11 0.89 0.320 0.126 1.30 1.215 30 3.257 28-11-2002 - d 35396 1997 XF11 0.91 0.310 0.122 1.20 1.018 53 3.257 03-11-2002 - d 35396 1997 XF11 1.18 0.180 0.071 1.80 1.034 63 3.257 05-11-2002 - d 53789 2000 ED104 1.20 0.180 0.071 1.68 1.089 60 - 29-09-2002 - g 85953 1999 FK21 0.59 0.320 0.126 0.91 1.140 35 - 21-02-2002 - e 86039 1999 NC43 2.22 0.140 0.055 2.86 1.116 59 34.49 17-03-2000 - e 99935 2002 AV4 1.50 0.370 0.145 1.60 1.086 70 - 01-06-2003 - d 1999 HF1 4.74 0.110 0.043 1.68 0.957 91 - 22-03-2002 - g 2000 BG19 1.77 0.040 0.016 0.74 1.388 17 - 17-03-2000 - e 2001 LF 2.00 0.050 0.020 1.40 1.172 45 - 02-06-2003 - d 2002 BM26 0.84 0.020 0.008 3.10 1.023 60 - 21-02-2002 - e 2002 HK12 0.80 0.170 0.067 2.84 1.138 33 - 28-09-2002 - g 2002 NX18 2.40 0.030 0.012 1.19 1.158 54 - 29-09-2002 - g 2002 QE15 1.94 0.150 0.059 1.53 1.131 62 - 28-09-2002 - g 2003 YT1 1.50 0.270 0.106 1.92 1.035 74 - 08-05-2004 - d Table 1 Near Earth-asteroids with η-values derived from spectral fitting to multi-wavelength mid- infrared observations. The object effective diameter, D, the geometric visible albedo pV, and the η-values have been derived by using the NEATM. α is the phase angle at the epoch of the observations, which is given in the “Obs. Date” column. P is the rotation period in hours. In the column “Spin axis” a “Y” indicates that the spin axis orientation of the asteroid is known. In the column “Re” we give the original publication reference: a) Harris and Davies (1999); b) Harris et al. (2005); c) Harris (1998); d) Delbo’ (2004); e) Delbo’ et al. (2003); f) Harris et al. (1998); g) Wolters et al. (2005). - 25 - Figure captions Fig. 1 Sensitivity of η to model parameter variations. Δη/η (%) caused by a change of 1% in the bolometric Bond albedo A (upper left), macroscopic surface roughness 0θ (upper right), heliocentric distance r (lower left), and the thermal parameter Θ (lower right). See text, section 2.1 for details. Fig. 2 Distribution of the input parameters used in our statistical inversion method. Upper left: distributions of NEA rotation rates from “Physical parameters of NEOs (Hahn, 2006, http://earn.dlr.de”). Upper right: distribution of NEA phase angles. Lower left: distribution of NEA sub-solar latitudes θSS. Lower right, solid line: distribution of the geometric albedos (pV) and, dashed line: the bolometric Bond albedos (A) for the asteroids of Table 1 having η-values determined from observations. Fig. 3 Dependence of η-value on phase angle, α. Black diamonds: η-values derived from the NEATM for a set of NEAs with adequate multi-filter photometric data to enable η to be derived via spectral fitting (the data set includes multiple values of η for some objects observed at more than one phase angle; for the original data sources see Table 1). The error bars represent a 20% uncertainty, which is based on the reproducibility of η for those objects for which more than one measurement is available from independent data sets. Colored points: distributions of (α, η) calculated by means of our model for different values of thermal inertia: i.e. 15 (green), 200 (red), and 1000 (blue) J m-2 s-0.5 K-1. The distribution of the measured η-values is best described by the red points. Fig. 4 Plot of the function D (Γ), i.e. the distance D of the two-dimensional Kolmogorov-Smirnov best-fit procedure against the thermal inertia Γ. The three curves were generated assuming three different values of the surface roughness: solid line θ =58°; dotted line θ =36°; dashed line θ =0° i.e. a smooth surface. The dashed-dotted line shows the function D (Γ) obtained by using θ =58° and a random distribution of asteroid spin-axis orientations uniformly distributed over the celestial sphere, instead of the nominal one, as input for our model. Fig. 5 Histogram of the distribution of the log of the diameters, D, of the NEAs for which we have η-values determined from observations. The best-fit Gaussian function, 0.37exp(-z2/2), where z=(log D - 0.23)/0.31, with D in km, is also shown. Fig. 6 Thermal inertia as a function of asteroid diameter. Small open circles represent values from the literature derived by means of thermophysical models. The large open diamond is the result from this work (see text for details). The straight (continuous) line which gives the best fit to the trend of increasing thermal inertia, Γ, with decreasing asteroid diameter, D, is given by the expression Γ=300×D-0.48. The axis on the right-hand side gives the asteroid surface thermal conductivity k on the basis of k=Γ2/(ρc), assuming constant surface density, ρ, equal to 2500 kg m-3 and specific heat capacity, c, equal to 600 J kg-1 K-1. These values are reasonable assumptions for asteroid surfaces (Britt et al., 2002; Farinella et al., 1998). The thermal conductivities of (6489) Golevka (Chesley et al., 2003) and for Karin cluster asteroids (Nesvorný and Bottke, 2004) are indicated with arrows. The two values of Γ derived for 2002 NY40 are indicated as the lower and the upper limits of the error bar on the extreme left of the plot. Dotted line: linear regression of Eq. (6) for MBAs only; dashed line: linear regression of Eq. (6) for NEAs only. - 26 - Figures 0 20 40 60 80 100 α (deg) 0 20 40 60 80 100 α (deg) 0 20 40 60 80 100 α (deg) 0 20 40 60 80 100 α (deg) Fig. 1 - 27 - 0 1 2 3 4 5 6 7 8 9 10 11 12 NEA rotation rate, P (hours) 0 10 20 30 40 50 60 70 80 90 100 Solar phase angle, α (degrees) 0 10 20 30 40 50 60 70 80 90 100 Sub-solar latitude, θSS (degrees) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Albedo Fig. 2 - 28 - Fig. 3 - 29 - 0 100 200 300 400 500 600 700 800 900 1000 Thermal Inertia (J m-2 s-0.5 K-1) Fig. 4 - 30 - -2 -1 0 1 2 3 log D (D in km) Fig. 5 - 31 - 1 10 100 1000 Asteroid diameter (km) 1 10 100 1000 Asteroid diameter (km) Fig. 6

arXiv:0704.1916v2 [math.CA] 7 Aug 2007 SOLUTIONS OF CERTAIN FRACTIONAL KINETIC EQUATIONS AND A FRACTIONAL DIFFUSION EQUATION R.K. SAXENA Department of Mathematics and Statistics, Jai Narain Vyas University Jodhpur-342 004, India A.M. MATHAI Department of Mathematics and Statistics, McGill University Montreal, Canada H3A 2K6 Centre for Mathematical Sciences, Pala Campus, Pala-686 574, Kerala, India H.J. HAUBOLD Office for Outer Space Affairs, United Nations P.O. Box 500, A-1400 Vienna, Austria Centre for Mathematical Sciences, Pala Campus, Pala-686 574, Kerala, India Abstract. In view of the usefulness and importance of kinetic equations in cer- tain physical problems, the authors derive the explicit solution of a fractional kinetic equation of general character, that unifies and extends earlier results. Further, an alternative shorter method based on a result developed by the au- thors is given to derive the solution of a fractional diffusion equation. 1 Introduction Fractional reaction/diffusion equations involve fractional derivatives with re- spect to time and space and are studied to describe anomalous reaction/diffusion of dynamic systems with chaotic motion. Fractional kinetic equation for Hamil- tonian chaos is discussed by Zaslavsky (1994). Solutions and applications of cer- tain kinetic equations are studied by Saichev and Zaslavsky (1997). Solutions of a fractional kinetic equation is investigated by Haubold and Mathai (2000) for a simple production-destruction mechanism. This equation was generalized by Saxena, Mathai, and Haubold (2002). In recent articles, Saxena, Mathai, and Haubold (2002, 2004a, 2004b) discussed the solution of a number of generalized fractional kinetic equations . In the present article, we investigate the solution of a unified fractional kinetic equation, which provides unification and extension of results on fractional kinetic equations given earlier by Haubold and Mathai (2000) and Saxena, Mathai, and Haubold (2002, 2004a). We also present the solution of a fractional integral equation discussed by Miller and Ross (1993). Further, an alternative proof of the solution of a fractional diffusion equation http://arxiv.org/abs/0704.1916v2 given earlier by Kochubei (1990) is investigated, which is based upon a result given by Saxena, Mathai, and Haubold (2006). Most of the results are obtained in terms of generalized Mittag-Leffler functions in elegant and compact forms, which are suitable for numerical computation. The paper is organized as follows. Section 2 contains the solution of a unified fractional kinetic equation while Section 3 considers special cases of the equation. A shorter alternative method for the solution of a diffusion equation discussed earlier by Kochubei (1990) is presented in Section 4. A series representation and asymptotic expansion of the solution are given in Section 5. Incidentally, an H-function representation of a one-sided Lévy stable density is also obtained. 2 Unified fractional kinetic equations In this Section, we present a method based on Laplace transform for deriving the solution of the unified fractional kinetic equations. Theorem 1. If Re(νj) > 0, aj > 0, j ∈ N, and f(t) be a given function, defined on ℜ+, then the equation N(t)−N0f(t) = − aj 0D t N(t), (1) is solvable and its particular solution is given by N(t) = N0 (−1)l r1+...+rn−1=l (r1)! . . . (rn−1)! (aµ+1) f(u)(t− u) νµ+1−1 (l+1) [−a1(t− u) ν1 ]du, (2) where the summation in (2) is taken over all nonnegative integers r1, . . . , rn such that r1 + . . . + rn−1 = l, and provided that the series and integral in (2) are convergent. Here 0D t , j ∈ N are Riemann-Liouville fractional integrals, defined by t f(t) = (t− u)ν−1f(u)du,Re(ν) > 0, (3) with 0D t f(t) = f(t) (Oldham and Spanier, 1974; Miller and Ross, 1993), Eδβ,γ(z) is the generalized Mittag-Leffler function, defined by Prabhakar (1971) in terms of series representation as Eδβ,γ(z) = (δ)τz Γ(βτ + γ)(τ)! (β, γ, δ ∈ C,Re(β) > 0, Re(γ) > 0). (4) Proof. By the application of the convolution theorem of the Laplace transform (Erdélyi et al., 1953, p. 259) to (3), we find that t f(t); s L(f(t)), = s−νf∼(s), (5) where f∼(s) = e−stf(t)dt, s ∈ C,Re(s) > 0. Applying Laplace transform to (1) and using (5), it gives N∼(s) = 1 + a1s−ν1 + . . .+ ans−νn = N0f (−1)l j=1 aj+1s −νj+1 (1 + a1s−ν1)l+1 j=2 ajs 1 + a1s−ν1 If we employ the identity (Abramowitz and Stegun, 1968, p. 823) (x1 + . . .+ xm) r1+...+rn=l (r1)! . . . (rn)! xrµµ , (7) where the summation is taken over all nonnegative integers, r1, . . . , rn, such that r1 + . . .+ rn = l, then for |a1s −ν1 | < 1, (7) transforms into the form N∼(s) = N0f (−1)l r1+...+rn−1=l r1>...rn−1>0 (r1)! . . . (rn−1)! µ=1(aµ+1) (1 + a1s−ν1)l+1 . (8) Taking the inverse Laplace transform of (8) by making use of the formula (Kil- bas, Saigo, and Saxena, 2004, eq. (12)) s−γ(1− as−β)−δ; t = tγ−1Eδβ,γ(at β), (9) where Re(s) > |a|1/Re(γ), Re(γ) > 0, Re(s) > 0, and applying the convolution theorem of the Laplace transform, the result (2) is established. Remark 1. The generalized Mittag-Leffler function defined by (4) is studied by Prabhakar (1971) and Kilbas, Saigo, and Saxena (2004). Recently this function is used in the theory of finite-size scaling of systems with strong anisotropy and long-range interaction by Chamati and Tonshev (2006). 3 Special cases Some special cases of Theorem 1 are of interest to be highlighted. If we set νj = jν, aj = ( jν (j ∈ N), we obtain Theorem 2. If Re(ν) > 0, c > 0 and f(x) ∈ ℜ+, then the equation N(t)−N0f(t) = − (nr )c νrD−νrt N(t), (10) is solvable and its solution has the form N(t) = N0 f(u)Enν,1[−c ν(t− u)ν ]du, (11) where Enν,1(x) is the generalized Mittag-Leffler function defined by (4) and pro- vided that the integral (11) is convergent. When n = 1, we obtain the following result given by Hille and Tamarkin (1930). Corollary 2.1. Let Re(ν) > 0, c > 0 and let f(x) ∈ ℜ+, then for the solution of the integral equation N(t)−N0f(t) = −c t N(t), (12) holds the following formula N(t) = N0 f(u)Eν [−c ν(t− u)ν ]du, (13) where Eν(z) is an entire function of order ρ = and type σ = 1, defined by Eν(z) = Γ(µν + 1) , (ν ∈ C,Re(ν) > 0). (14) Note 1. The above result has also been given by the authors in a different form (Saxena, Mathai, and Haubold, 2004a, 2004b). If we set f(t) = tγ−1Eδν,γ [−(ct) ν ], Theorem 2 yields Corollary 2.2. Let Re(ν) > 0, Re(γ) > 0, c > 0, then for the solution of the kinetic equation N(t)−N0t γ−1Eδν,γ [−(ct) ν ] = − (nr )c t n ∈ N (15) holds the relation N(t) = N0t γ−1Eδ+nν,γ [−(ct) ν ], n ∈ N. (16) For f(t) = tρ−1, Theorem 2 yields the following result Corollary 2.3. If Re(ρ) > 0, Re(ν) > 0, c > 0, then for the solution of the equation N(t)−N0t ρ−1 = − (nr )c t N(t), r ∈ N, (17) holds the relation N(t) = N0t ρ−1Γ(ρ)Enν,ρ[−(ct) ν ], r ∈ N. (18) For n = 1, eq. (18) reduces to a result given by Saxena, Mathai, and Haubold (2002, p. 283, eq. (15)). When aj = a jsνj , for j = 1, . . . , n, we obtain Theorem 3. Let Re(ν) > 0, a > 0, t > 0, n > 1, |an+1s−(n+1)ν | < 1, and f(x) be a given function defined on ℜ+, then the equation N(t)−N0f(t) = − ar 0D t N(t), (19) is solvable and its solution is given by N(t) = N0 f(u)E(n+1)ν,ν [a n+1(t− u)(n+1)ν ]du (t− u)ν−1E(n+1)ν,ν [a n+1(t− u)(n+1)ν ]du , (20) where E(n+1)ν,ν(z) is the generalized Mittag-Leffler function Eα,β(z) defined as Eα,β(z) = Γ(αµ+ β) , (α, β ∈ C,Re(α) > 0, Re(β) > 0) (21) and provided that the integral in (20) is convergent. If we take νj = jν, for j = 1, . . . n, then it is interesting to note that Theorem 1 yields the following result given by (Miller and Ross, 1993) in a different form: Theorem 4. Let Re(ν) > 0, aj > 0, and f(x) be a given function defined on ℜ+, |a1s −ν | < 1, then the fractional kinetic equation N(t)−N0f(t) = − aj 0D t N(t), (22) is solvable and has the solution given by N(t) = N0 (−1)l r1+...+rn−1=l (r1)! . . . (rn−1)! (aµ+1) f(u)(t− u) ν(µ+1)rµ−1 (l+1) ν(µ+1)rµ [−a1(t− u) ν1 ]du, provided that the series and integral in (23) are convergent. 4 Fractional diffusion equation In this Section we present an alternative shorter method for deriving the solution of a diffusion equation discussed earlier by Kochubei (1990). Theorem 5. Consider the Cauchy problem t N(x, t) = −c ν∆N(x, t), (0 < α < 1;x ∈ ℜn; 0 < t ≤ T ), (24) N(x, t = 0) = δ(x), x ∈ ℜ, lim N(x, t) = 0 (25) t is the regularized Caputo (1969) partial fractional derivative with respect to t, defined by t N(x, t) = Γ(1− α) N(x, s)ds (t− x)α N(x, 0) and ∆ is the Laplacian. The fundamental solution of the above Cauchy problem is given by N(x, t) = |x|−nπ− |x|2t−α (1,α) (n/2,1),(1,1) , (26) where H 1,2 (.) is the H-function (Mathai and Saxena, 1978). Proof. Applying the Laplace transform with respect to t, using the result (Caputo, 1969) L {0D t N(x, t)} = s αN∼(x, s)− sα−r−1N (r)(x, 0), m− 1 < α ≤ m, m ∈ N, and Fourier transform with respect to x, gives (k, s)− sα−1 = −cν |k|2N∼ (k, s), where the symbol ”∼“ indicates the Laplace transform with respect to the time variable t and the symbol ”*“ the Fourier transform with respect to the space variable x. Solving for N∼ , we have (k, s) = sα + cν |k|2 . (27) By virtue of the following Fourier transform formula (Samko, Kilbas, andMarichev, 1990, p. 538, eq. (27.1)) |x|(2−n)/2K(n−2)/2(a|x|) (τ) = a2 + τ2 , (τ ∈ ℜn;n ∈ N, a > 0), where the multidimensional Fourier transform with respect to x ∈ ℜn is defined (FxN)(τ, t) = N(x, t)eixτdx (τ ∈ ℜn; t > 0) (29) and Kν(.) is the modified Bessel function of the second kind, yields Ñ(x, s) = c−νsα−1(2π)− |x|cν/2 Kn−2/2 2 |x| . (30) In order to invert the Laplace transform, we employ the following result given by the authors (Saxena, Mathai, and Haubold, 2006) s−ρKν(zs σ); t tρ−1H z2t−2σ (ρ,2σ) ,1)(− ν , (31) where Kν(x) is the modified Bessel function of the second kind, Re(z 0, Re(s) > 0. Thus we obtain the solution in the form N(x, t) = (2π)− 4 |x|1− t−α|x|2 ,1),( . (32) By virtue of a result in Mathai and Saxena (1978), xσHm,np,q (ap,ap) (bq ,bq) = Hm,np,q (ap+σAp,Ap) (bq+σBq ,Bq) , (33) the power of the expression [ t−ν |x|2 /4cν ] can be absorbed inside the H- function and consequently we obtain N(x, t) = |π 2x|−n H t−α|x|2 (1,α) ,1),(1,1) . (34) Remark 1. If we employ the identity (Mathai and Saxena, 1978) Hm,np,q (ap,Ap) (bq ,Bq) Hm,np,q (ap,Ap/λ) (bq,Bq/λ) , λ > 0 (35) the solution given by (32) can be expressed in the form N(x, t) = 2 x|−n H 4cνtα (1,1) ),(1, 1 , (36) where α > 0. We also note that the above form of the solution is due to Schneider and Wyss (1989). There is one importance of our result (32) that it includes the Lévy stable density in terms of the H-function as shown in (45). Similarly, using the identity (35) we arrive at N(c, t) = 2x|−nH 2 |x| ),(1, 1 , (37) where n is not an even integer. This form of the H-function is useful in deter- mining its expansion in powers of x. Due to importance of the solution, we also discuss its series representation and behavior. 5 Series representation of the solution Using the series expansion for the H-function given in Mathai and Saxena (1978), it follows that (1,1) ),(1, 1 )Γ(1 − s Γ(1 − s) xsds (38) Γ(1− n − l)(−1)lxα( Γ(1− an − αl)(l)! − 1− l)(−1)lxα(1+l) Γ(1− α− αl)(l!) where n is not an even integer. Thus for n = 1, we find that N(x, t) = (−1)l Γ(1− α(l + 1)/2)(l!) , (39) where A = x and the duplication formula for the gamma function is used. For n = 2, the H-function of (37) is singular and in this case, the result is explicitly given by Barkai (2001) in the form N(x, t) ∼ πΓ(1− α)tα ]. (40) For n = 3, the series expansion is given by N(x, t) = 4πt3α/2A1/2 (−1)lAl/2 Γ[1− α(1 + l/2)] . (41) From above, it readily follows that for n = 3 and α 6= 1 N(x, t) ∼ , as x → ∞. (42) It will not be out of place to mention that the one sided Lévy stable density can be obtained from Laplace inversion formula (31) by virtue of the identity (x) = e−x, (43) and can be conveniently expressed in terms of the Laplace transform e−utΦρ(t)dt = e −uρ , Re(u) > 0, Re(ρ) > 0. (44) The result is Φρ(t) = (1,1) , (ρ > 0). (45) Note 2. This result is obtained earlier by Schneider and Wyss (1989) by fol- lowing a different procedure. Asymptotic behavior of Φα(t) is also given by Schneider (1986). In conclusion, we mention that some of the results derived in this article may find some applications in problems associated with models of long-memory processes driven by Lévy noise and other related problems, see the article by Anh, Heyde, and Leonenko (2002). Acknowledgment. The authors would like to thank the Department of Science and Technology, Government of India, New Delhi, for the financial assistance for this work under project No. SR/S4/MS:287/05 which enabled this collaboration possible. References Anh, V.V., Heyde, C.C., and Leonenko, N.N.: 2002, Dynamic models driven by Lévy noise, Journal of Applied Probability, 39, 730-747. Abramowitz, M. and Stegun, I.A.: 1968, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc. New York. Barkai, E.: 2001, Fractional Fokker-Planck equation, solution, and application, Physical Review E, 63, 046118. Caputo, M.: 1969, Elasticita e Dissipazione, Zanichelli, Bologna. Chamati, H. and Tonchev, N.S.: 2006, Generalized Mittag-Leffler functions in the theory of finite-size scaling for systems with strong anisotropy and/or long- range interaction, Journal of Physics A. Mathematical and General, 39, 469- Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G.: 1953, Higher Transcendental Functions , Vol. 1, McGraw-Hill, New York, Toronto, and Lon- Haubold, H.J. and Mathai, A.M.: 2000, The fractional kinetic equation and thermonuclear functions, Astrophysics and Space Science, 273, 53-63. Hille, E. and Tamarkin, J.D.: 1930, On the theory of linear integral equations, Annals of Mathematics, 31, 479-528. Kochubei, A.N.: 1990, Diffusion of fractional order, Differential Equations, 26, 485-492. Kilbas, A.A., Saigo, M. and Saxena, R.K.: 2004, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms and Special Functions, 15, 31-49. Mathai, A.M. and Saxena, R.K.: 1978, The H-function with Applications in Statistics and Other Disciplines, John Wiley and Sons, Inc., New York, London, and Sydney. Miller, K.S. and Ross, B.: 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York. Oldham, K.B. and Spanier, J.: 1974, The Fractional Calculus. Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, New York Prabhakar, T.R.: 1971, A singular integral equation with a generalized Mittag- Leffler function in the kernel, Yokohama Journal of Mathematics, 19, 7-15. Saichev, A.I. and Zaslavsky, G.M.: 1997, Fractional kinetic equations: solutions and applications, Chaos, 7, 753-784. Samko, S.G., Kilbas, A.A. and Marichev, O.I.: 1990, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, New York. Saxena, R.K., Mathai, A.M., and Haubold, H.J.: 2002, On fractional kinetic equations, Astrophysics and Space Science, 282, 281-287. Saxena, R.K., Mathai, A.M., and Haubold, H.J.: 2004a, On generalized frac- tional kinetic equations, Physica A, 344, 657-664. Saxena, R.K., Mathai, A.M. and Haubold, H.J.: 2004b, Unified fractional ki- netic equation and a fractional diffusion equation, Astrophysics and Space Sci- ence, 290, 299-310. Saxena, R.K., Mathai, A.M., and Haubold, H.J.: 2006, Solution of generalized fractional reaction-diffusion equations, Astrophysics and Space Science, 305, 305-313. Schneider, W.R.: 1986, in Stochastic Processes in Classical and Quantum Sys- tems, S. Albeverio, G. Casati, and D. Merlini (Eds.), Springer-Verlag, Berlin. Schneider, W.R. and Wyss, W.: 1989, Fractional diffusion and wave equation, Journal of Mathematical Physics, 30, 134-144. Zaslavsky, G.M.: 1994, Fractional kinetic equation for Hamiltonian chaos, Phys- ica D, 76, 110-122.

In view of the usefulness and importance of the kinetic equation in certain physical problems, the authors derive the explicit solution of a fractional kinetic equation of general character, that unifies and extends earlier results. Further, an alternative shorter method based on a result developed by the authors is given to derive the solution of a fractional diffusion equation.

Introduction Fractional reaction/diffusion equations involve fractional derivatives with re- spect to time and space and are studied to describe anomalous reaction/diffusion of dynamic systems with chaotic motion. Fractional kinetic equation for Hamil- tonian chaos is discussed by Zaslavsky (1994). Solutions and applications of cer- tain kinetic equations are studied by Saichev and Zaslavsky (1997). Solutions of a fractional kinetic equation is investigated by Haubold and Mathai (2000) for a simple production-destruction mechanism. This equation was generalized by Saxena, Mathai, and Haubold (2002). In recent articles, Saxena, Mathai, and Haubold (2002, 2004a, 2004b) discussed the solution of a number of generalized fractional kinetic equations . In the present article, we investigate the solution of a unified fractional kinetic equation, which provides unification and extension of results on fractional kinetic equations given earlier by Haubold and Mathai (2000) and Saxena, Mathai, and Haubold (2002, 2004a). We also present the solution of a fractional integral equation discussed by Miller and Ross (1993). Further, an alternative proof of the solution of a fractional diffusion equation http://arxiv.org/abs/0704.1916v2 given earlier by Kochubei (1990) is investigated, which is based upon a result given by Saxena, Mathai, and Haubold (2006). Most of the results are obtained in terms of generalized Mittag-Leffler functions in elegant and compact forms, which are suitable for numerical computation. The paper is organized as follows. Section 2 contains the solution of a unified fractional kinetic equation while Section 3 considers special cases of the equation. A shorter alternative method for the solution of a diffusion equation discussed earlier by Kochubei (1990) is presented in Section 4. A series representation and asymptotic expansion of the solution are given in Section 5. Incidentally, an H-function representation of a one-sided Lévy stable density is also obtained. 2 Unified fractional kinetic equations In this Section, we present a method based on Laplace transform for deriving the solution of the unified fractional kinetic equations. Theorem 1. If Re(νj) > 0, aj > 0, j ∈ N, and f(t) be a given function, defined on ℜ+, then the equation N(t)−N0f(t) = − aj 0D t N(t), (1) is solvable and its particular solution is given by N(t) = N0 (−1)l r1+...+rn−1=l (r1)! . . . (rn−1)! (aµ+1) f(u)(t− u) νµ+1−1 (l+1) [−a1(t− u) ν1 ]du, (2) where the summation in (2) is taken over all nonnegative integers r1, . . . , rn such that r1 + . . . + rn−1 = l, and provided that the series and integral in (2) are convergent. Here 0D t , j ∈ N are Riemann-Liouville fractional integrals, defined by t f(t) = (t− u)ν−1f(u)du,Re(ν) > 0, (3) with 0D t f(t) = f(t) (Oldham and Spanier, 1974; Miller and Ross, 1993), Eδβ,γ(z) is the generalized Mittag-Leffler function, defined by Prabhakar (1971) in terms of series representation as Eδβ,γ(z) = (δ)τz Γ(βτ + γ)(τ)! (β, γ, δ ∈ C,Re(β) > 0, Re(γ) > 0). (4) Proof. By the application of the convolution theorem of the Laplace transform (Erdélyi et al., 1953, p. 259) to (3), we find that t f(t); s L(f(t)), = s−νf∼(s), (5) where f∼(s) = e−stf(t)dt, s ∈ C,Re(s) > 0. Applying Laplace transform to (1) and using (5), it gives N∼(s) = 1 + a1s−ν1 + . . .+ ans−νn = N0f (−1)l j=1 aj+1s −νj+1 (1 + a1s−ν1)l+1 j=2 ajs 1 + a1s−ν1 If we employ the identity (Abramowitz and Stegun, 1968, p. 823) (x1 + . . .+ xm) r1+...+rn=l (r1)! . . . (rn)! xrµµ , (7) where the summation is taken over all nonnegative integers, r1, . . . , rn, such that r1 + . . .+ rn = l, then for |a1s −ν1 | < 1, (7) transforms into the form N∼(s) = N0f (−1)l r1+...+rn−1=l r1>...rn−1>0 (r1)! . . . (rn−1)! µ=1(aµ+1) (1 + a1s−ν1)l+1 . (8) Taking the inverse Laplace transform of (8) by making use of the formula (Kil- bas, Saigo, and Saxena, 2004, eq. (12)) s−γ(1− as−β)−δ; t = tγ−1Eδβ,γ(at β), (9) where Re(s) > |a|1/Re(γ), Re(γ) > 0, Re(s) > 0, and applying the convolution theorem of the Laplace transform, the result (2) is established. Remark 1. The generalized Mittag-Leffler function defined by (4) is studied by Prabhakar (1971) and Kilbas, Saigo, and Saxena (2004). Recently this function is used in the theory of finite-size scaling of systems with strong anisotropy and long-range interaction by Chamati and Tonshev (2006). 3 Special cases Some special cases of Theorem 1 are of interest to be highlighted. If we set νj = jν, aj = ( jν (j ∈ N), we obtain Theorem 2. If Re(ν) > 0, c > 0 and f(x) ∈ ℜ+, then the equation N(t)−N0f(t) = − (nr )c νrD−νrt N(t), (10) is solvable and its solution has the form N(t) = N0 f(u)Enν,1[−c ν(t− u)ν ]du, (11) where Enν,1(x) is the generalized Mittag-Leffler function defined by (4) and pro- vided that the integral (11) is convergent. When n = 1, we obtain the following result given by Hille and Tamarkin (1930). Corollary 2.1. Let Re(ν) > 0, c > 0 and let f(x) ∈ ℜ+, then for the solution of the integral equation N(t)−N0f(t) = −c t N(t), (12) holds the following formula N(t) = N0 f(u)Eν [−c ν(t− u)ν ]du, (13) where Eν(z) is an entire function of order ρ = and type σ = 1, defined by Eν(z) = Γ(µν + 1) , (ν ∈ C,Re(ν) > 0). (14) Note 1. The above result has also been given by the authors in a different form (Saxena, Mathai, and Haubold, 2004a, 2004b). If we set f(t) = tγ−1Eδν,γ [−(ct) ν ], Theorem 2 yields Corollary 2.2. Let Re(ν) > 0, Re(γ) > 0, c > 0, then for the solution of the kinetic equation N(t)−N0t γ−1Eδν,γ [−(ct) ν ] = − (nr )c t n ∈ N (15) holds the relation N(t) = N0t γ−1Eδ+nν,γ [−(ct) ν ], n ∈ N. (16) For f(t) = tρ−1, Theorem 2 yields the following result Corollary 2.3. If Re(ρ) > 0, Re(ν) > 0, c > 0, then for the solution of the equation N(t)−N0t ρ−1 = − (nr )c t N(t), r ∈ N, (17) holds the relation N(t) = N0t ρ−1Γ(ρ)Enν,ρ[−(ct) ν ], r ∈ N. (18) For n = 1, eq. (18) reduces to a result given by Saxena, Mathai, and Haubold (2002, p. 283, eq. (15)). When aj = a jsνj , for j = 1, . . . , n, we obtain Theorem 3. Let Re(ν) > 0, a > 0, t > 0, n > 1, |an+1s−(n+1)ν | < 1, and f(x) be a given function defined on ℜ+, then the equation N(t)−N0f(t) = − ar 0D t N(t), (19) is solvable and its solution is given by N(t) = N0 f(u)E(n+1)ν,ν [a n+1(t− u)(n+1)ν ]du (t− u)ν−1E(n+1)ν,ν [a n+1(t− u)(n+1)ν ]du , (20) where E(n+1)ν,ν(z) is the generalized Mittag-Leffler function Eα,β(z) defined as Eα,β(z) = Γ(αµ+ β) , (α, β ∈ C,Re(α) > 0, Re(β) > 0) (21) and provided that the integral in (20) is convergent. If we take νj = jν, for j = 1, . . . n, then it is interesting to note that Theorem 1 yields the following result given by (Miller and Ross, 1993) in a different form: Theorem 4. Let Re(ν) > 0, aj > 0, and f(x) be a given function defined on ℜ+, |a1s −ν | < 1, then the fractional kinetic equation N(t)−N0f(t) = − aj 0D t N(t), (22) is solvable and has the solution given by N(t) = N0 (−1)l r1+...+rn−1=l (r1)! . . . (rn−1)! (aµ+1) f(u)(t− u) ν(µ+1)rµ−1 (l+1) ν(µ+1)rµ [−a1(t− u) ν1 ]du, provided that the series and integral in (23) are convergent. 4 Fractional diffusion equation In this Section we present an alternative shorter method for deriving the solution of a diffusion equation discussed earlier by Kochubei (1990). Theorem 5. Consider the Cauchy problem t N(x, t) = −c ν∆N(x, t), (0 < α < 1;x ∈ ℜn; 0 < t ≤ T ), (24) N(x, t = 0) = δ(x), x ∈ ℜ, lim N(x, t) = 0 (25) t is the regularized Caputo (1969) partial fractional derivative with respect to t, defined by t N(x, t) = Γ(1− α) N(x, s)ds (t− x)α N(x, 0) and ∆ is the Laplacian. The fundamental solution of the above Cauchy problem is given by N(x, t) = |x|−nπ− |x|2t−α (1,α) (n/2,1),(1,1) , (26) where H 1,2 (.) is the H-function (Mathai and Saxena, 1978). Proof. Applying the Laplace transform with respect to t, using the result (Caputo, 1969) L {0D t N(x, t)} = s αN∼(x, s)− sα−r−1N (r)(x, 0), m− 1 < α ≤ m, m ∈ N, and Fourier transform with respect to x, gives (k, s)− sα−1 = −cν |k|2N∼ (k, s), where the symbol ”∼“ indicates the Laplace transform with respect to the time variable t and the symbol ”*“ the Fourier transform with respect to the space variable x. Solving for N∼ , we have (k, s) = sα + cν |k|2 . (27) By virtue of the following Fourier transform formula (Samko, Kilbas, andMarichev, 1990, p. 538, eq. (27.1)) |x|(2−n)/2K(n−2)/2(a|x|) (τ) = a2 + τ2 , (τ ∈ ℜn;n ∈ N, a > 0), where the multidimensional Fourier transform with respect to x ∈ ℜn is defined (FxN)(τ, t) = N(x, t)eixτdx (τ ∈ ℜn; t > 0) (29) and Kν(.) is the modified Bessel function of the second kind, yields Ñ(x, s) = c−νsα−1(2π)− |x|cν/2 Kn−2/2 2 |x| . (30) In order to invert the Laplace transform, we employ the following result given by the authors (Saxena, Mathai, and Haubold, 2006) s−ρKν(zs σ); t tρ−1H z2t−2σ (ρ,2σ) ,1)(− ν , (31) where Kν(x) is the modified Bessel function of the second kind, Re(z 0, Re(s) > 0. Thus we obtain the solution in the form N(x, t) = (2π)− 4 |x|1− t−α|x|2 ,1),( . (32) By virtue of a result in Mathai and Saxena (1978), xσHm,np,q (ap,ap) (bq ,bq) = Hm,np,q (ap+σAp,Ap) (bq+σBq ,Bq) , (33) the power of the expression [ t−ν |x|2 /4cν ] can be absorbed inside the H- function and consequently we obtain N(x, t) = |π 2x|−n H t−α|x|2 (1,α) ,1),(1,1) . (34) Remark 1. If we employ the identity (Mathai and Saxena, 1978) Hm,np,q (ap,Ap) (bq ,Bq) Hm,np,q (ap,Ap/λ) (bq,Bq/λ) , λ > 0 (35) the solution given by (32) can be expressed in the form N(x, t) = 2 x|−n H 4cνtα (1,1) ),(1, 1 , (36) where α > 0. We also note that the above form of the solution is due to Schneider and Wyss (1989). There is one importance of our result (32) that it includes the Lévy stable density in terms of the H-function as shown in (45). Similarly, using the identity (35) we arrive at N(c, t) = 2x|−nH 2 |x| ),(1, 1 , (37) where n is not an even integer. This form of the H-function is useful in deter- mining its expansion in powers of x. Due to importance of the solution, we also discuss its series representation and behavior. 5 Series representation of the solution Using the series expansion for the H-function given in Mathai and Saxena (1978), it follows that (1,1) ),(1, 1 )Γ(1 − s Γ(1 − s) xsds (38) Γ(1− n − l)(−1)lxα( Γ(1− an − αl)(l)! − 1− l)(−1)lxα(1+l) Γ(1− α− αl)(l!) where n is not an even integer. Thus for n = 1, we find that N(x, t) = (−1)l Γ(1− α(l + 1)/2)(l!) , (39) where A = x and the duplication formula for the gamma function is used. For n = 2, the H-function of (37) is singular and in this case, the result is explicitly given by Barkai (2001) in the form N(x, t) ∼ πΓ(1− α)tα ]. (40) For n = 3, the series expansion is given by N(x, t) = 4πt3α/2A1/2 (−1)lAl/2 Γ[1− α(1 + l/2)] . (41) From above, it readily follows that for n = 3 and α 6= 1 N(x, t) ∼ , as x → ∞. (42) It will not be out of place to mention that the one sided Lévy stable density can be obtained from Laplace inversion formula (31) by virtue of the identity (x) = e−x, (43) and can be conveniently expressed in terms of the Laplace transform e−utΦρ(t)dt = e −uρ , Re(u) > 0, Re(ρ) > 0. (44) The result is Φρ(t) = (1,1) , (ρ > 0). (45) Note 2. This result is obtained earlier by Schneider and Wyss (1989) by fol- lowing a different procedure. Asymptotic behavior of Φα(t) is also given by Schneider (1986). In conclusion, we mention that some of the results derived in this article may find some applications in problems associated with models of long-memory processes driven by Lévy noise and other related problems, see the article by Anh, Heyde, and Leonenko (2002). Acknowledgment. The authors would like to thank the Department of Science and Technology, Government of India, New Delhi, for the financial assistance for this work under project No. SR/S4/MS:287/05 which enabled this collaboration possible. References Anh, V.V., Heyde, C.C., and Leonenko, N.N.: 2002, Dynamic models driven by Lévy noise, Journal of Applied Probability, 39, 730-747. Abramowitz, M. and Stegun, I.A.: 1968, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc. New York. Barkai, E.: 2001, Fractional Fokker-Planck equation, solution, and application, Physical Review E, 63, 046118. Caputo, M.: 1969, Elasticita e Dissipazione, Zanichelli, Bologna. Chamati, H. and Tonchev, N.S.: 2006, Generalized Mittag-Leffler functions in the theory of finite-size scaling for systems with strong anisotropy and/or long- range interaction, Journal of Physics A. Mathematical and General, 39, 469- Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G.: 1953, Higher Transcendental Functions , Vol. 1, McGraw-Hill, New York, Toronto, and Lon- Haubold, H.J. and Mathai, A.M.: 2000, The fractional kinetic equation and thermonuclear functions, Astrophysics and Space Science, 273, 53-63. Hille, E. and Tamarkin, J.D.: 1930, On the theory of linear integral equations, Annals of Mathematics, 31, 479-528. Kochubei, A.N.: 1990, Diffusion of fractional order, Differential Equations, 26, 485-492. Kilbas, A.A., Saigo, M. and Saxena, R.K.: 2004, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms and Special Functions, 15, 31-49. Mathai, A.M. and Saxena, R.K.: 1978, The H-function with Applications in Statistics and Other Disciplines, John Wiley and Sons, Inc., New York, London, and Sydney. Miller, K.S. and Ross, B.: 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York. Oldham, K.B. and Spanier, J.: 1974, The Fractional Calculus. Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, New York Prabhakar, T.R.: 1971, A singular integral equation with a generalized Mittag- Leffler function in the kernel, Yokohama Journal of Mathematics, 19, 7-15. Saichev, A.I. and Zaslavsky, G.M.: 1997, Fractional kinetic equations: solutions and applications, Chaos, 7, 753-784. Samko, S.G., Kilbas, A.A. and Marichev, O.I.: 1990, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, New York. Saxena, R.K., Mathai, A.M., and Haubold, H.J.: 2002, On fractional kinetic equations, Astrophysics and Space Science, 282, 281-287. Saxena, R.K., Mathai, A.M., and Haubold, H.J.: 2004a, On generalized frac- tional kinetic equations, Physica A, 344, 657-664. Saxena, R.K., Mathai, A.M. and Haubold, H.J.: 2004b, Unified fractional ki- netic equation and a fractional diffusion equation, Astrophysics and Space Sci- ence, 290, 299-310. Saxena, R.K., Mathai, A.M., and Haubold, H.J.: 2006, Solution of generalized fractional reaction-diffusion equations, Astrophysics and Space Science, 305, 305-313. Schneider, W.R.: 1986, in Stochastic Processes in Classical and Quantum Sys- tems, S. Albeverio, G. Casati, and D. Merlini (Eds.), Springer-Verlag, Berlin. Schneider, W.R. and Wyss, W.: 1989, Fractional diffusion and wave equation, Journal of Mathematical Physics, 30, 134-144. Zaslavsky, G.M.: 1994, Fractional kinetic equation for Hamiltonian chaos, Phys- ica D, 76, 110-122.

Strong enhancement of transport by interaction on contact links Dan Bohr MIC, Department of Micro- and Nanotechnology, NanoDTU, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark Peter Schmitteckert TKM, Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, D-76128 Karlsruhe, Germany. (Dated: October 26, 2018) Strong repulsive interactions within a one dimensional Fermi system in a two-probe configuration normally lead to a reduced off-resonance conductance. We show that if the repulsive interaction extends to the contact regions, a strong increase of the conductance may occur, even for systems where one would expect to find a reduced conductance. An essential ingredient in our calculations is a momentum-space representation of the leads, which allows a high energy resolution. Further, we demonstrate that these results are independent of the high-energy cutoff and that the relevant scale is set by the Fermi velocity. PACS numbers: 73.63.Kv, 73.23.Hk, 71.10.Pm I. INTRODUCTION Constructing a transport theory for strongly correlated systems is one of the major challenges of condensed mat- ter physics. Even though many interesting ideas have been proposed during recent years, no consensus has yet emerged as to the general validity and applicability of the various schemes. With this state of affairs, it is of high importance to establish reliable benchmarks for simple model systems, which then can be used to validate new approaches. Recently we presented a novel method for calculat- ing linear response conductance1 using the density ma- trix renormalization group (DMRG) method2. A ma- jor challenge in this work consisted in minimizing finite size effects, which was achieved via modified boundary conditions. In this paper we circumvent these technical problems by reformulating the leads in momentum space. This approach enables us to (i) reach a much higher en- ergy resolution (∼ 10−5) and (ii) allows for a greater flexibility in the choice of discretization schemes. In two recent papers Mehta and Andrei3,4 presented nonequilibrium Bethe ansatz results for the interacting resonant-level model (IRLM), where a single spinless level is coupled to a left and a right lead both via a tunneling and a density-density interaction term. How- ever, their work currently excludes the regime of resonant tunneling–i.e., the regime where the conductance is close to unity.5 In this work we study the linear conductance of the IRLM on a lattice to provide a benchmark for the uni- versal properties of the model. In addition, we present results for an extended model, where the central region consists of three sites, with a similar interaction as in the IRLM model. As we will show, this model displays the same qualitative behavior as the IRLM. It should be noted that despite its simplicity, the IRLM captures much of the physics of transport through an arbitrary in- teracting nanostructure provided that only a single level is close to the Fermi energy of the leads, with all other lev- els well separated and outside the voltage window within which the transport takes place. For perfect coupling the IRLM model corresponds to the one-dimensional model studied by Vasseur et al.,6 and Molina et al.,7 obtained by restricting their nanostructure to a single site. Using the embedding method they showed that smoothing the ramp of interaction for perfect contacts can compensate for the decrease of transmission due to interaction on the nanostructure. Here we go far beyond the energy resolu- tion attained in previous work and show that interaction on the contact links can lead to strong renormalization effects, enhancing transport beyond the noninteracting system. II. METHOD AND MODELS We use the DMRG method to evaluate the linear re- sponse conductance of the interacting nanostructure. In previous work1 the leads were modeled in real-space by nearest-neighbor hopping chains. While simple to imple- ment there are several drawbacks of this method, most prominently the need for “damped boundary conditions” and the resulting problem of trapping of fermions on the Wilson chain (the damped region).1 In the present work we introduce a setup where the leads are described in momentum space. Specifically, a short part of the lead close to the nanostructure is rep- resented in real-space, accounting for local (i.e., high en- ergy) physics, while further away from the nanostruc- ture the lead is represented in momentum space; see Fig. 1. Since the low-energy modes of the momentum leads are now directly coupled to the extended structure (the nanostructure plus additional real-space sites), as il- lustrated in Fig. 1, the trapping of fermions on the low energy sites1 is avoided and no scaling sweeps are needed. http://arxiv.org/abs/0704.1917v3 2PSfrag replacements Nanostructure, MS sites Optional real space sites, MAdd sites Mandatory real space sites, 2 sites Left Lead Right LeadME = MS + MAdd tDottDot tDot VVV γVγV tktk t tt FIG. 1: (Color online) Schematics of the nanostructure ex- tended by real space sites and attached to momentum-space leads. This enables much higher energy resolution, and in the current work we resolve resonances of widths O(10−5). By virtue of the momentum representation of the leads the discretization scheme can be chosen arbitrarily to suit the problem at hand. In the present work we use a logarithmic discretization to cover a large energy range, while switching to a linear discretization for the lowest- energy states in order to describe Fermi-surface physics accurately. The linear discretization on the low-energy scale allows for a better representation of the low-energy physics relevant for transport properties–i.e., excitations created by η. The models considered in this work are the IRLM and the natural extension of this model to resonant linear chains, defined by the Hamiltonians HRS = µg ĉ j ĉj − j,j−1∈SE tj ĉ j ĉj−1 + h.c. j,j−1∈SE n̂j − n̂j−1 − , (1) HMS = i∈L,R i ĉi, (2) HT = − tk ĉ ĉ1 + tk ĉ + h.c., (3) where ĉ and ĉℓ are the (spinless) fermionic creation and annihilation operators at site ℓ, n̂ℓ = ĉ . HRS , HMS , and HT denote real-space, momentumspace, and tunnel- ing between real- and momentum-space Hamiltonians, respectively. The symbols S and SE denote the nanos- tructure and the extended nanostructure (the full real- space chain), respectively. The indices 1 and ME denote the first and last site in SE . The general setup and the specific values of the hopping matrix elements tj and the interactions Vj are indicated in Fig. 1, and note specif- ically the interactions on the contact links, γV . The coupling tk of the extended real-space structure to the momentum leads is chosen in such a way that in the case of a cosine band it corresponds to a nearest-neighbor hop- ping chain in real-space with a hopping parameter of t. In the following we measure all energies in units of t = 1. 0.000 0.001 0.002 0.003 IRLM, t’=0.01, γ=1.0 V=0.0 w=2.0 10 V=0.01 w=2.1 10 V=0.03 w=3.2 10 V=1.0 w=2.1 10 V=5.0 w=8.9 10 V=25.0 w=1.0 10 0.000 0.010 0.020 0.030 IRLM, t’=0.03, γ=1.0 V=0.0 w=1.8 10 V=0.03 w=2.1 10 V=0.1 w=2.5 10 V=1.0 w=1.0 10 V=5.0 w=4.5 10 V=25.0 w=9.0 10 FIG. 2: (Color online) Conductance versus gate potential for the interacting resonant level model for a contact hopping of (a) t′ = 0.01 and (b) t′ = 0.03 and contact interaction ranging from zero to 25. To each set of DMRG data a Lorentzian of half width 2w has been added as a guide to the eye. The leads are described with a cosine band between ±2 such that the Fermi velocity is vF = 2. In contrast to intradot interaction the contact interaction enhances the conductance and shows a nonmonotonic behavior versus contact interaction. For a single-site nanostructure and γ = 1 this model reduces to the IRLM. The properties of the leads are defined by the band structure ǫk, which can take any form. In this work we use either the cosine band, ǫk = −2 cos(k), or the linear band, ǫk = 2k. D is a cutoff parameter such that the Fermi velocity vF = 2 is kept constant in all work presented here, and the band ranges between energies −D and D. Throughout this work we use the notion of “contact interaction” for interaction on the link between the nanostructure and the leads. 0 5 10 15 20 25 Local density, t’=0.01 nd, µg=0.0005 nc, µg=0.0005 nd, µg=0.0002 nc, µg=0.0002 FIG. 3: (Color online) Site occupation nd of the resonant level and nc of the real-space sites attached to the level vs the link interaction in the IRLM for t’=0.01 and two different gate voltages. III. RESULTS The aim of this work is to study the effect of contact interaction. It is known from previous work1 that strong repulsive interactions within the nanostructure lead to suppression of the transport off resonance due to the for- mation of a density-wave-like state on the dot. In Fig. 2 we show results for the conductance versus gate potential for different couplings to the leads and different contact interactions for the IRLM (γ = 1). The calculations have been performed with typically 130 sites in total, ME = 10 real-space sites and 120 momentum- space sites. Due to the symmetry of the band, we use a discretization that is symmetric around ǫF = 0, and further use identical discretization of the two leads. To represent the “large” energy span in the band we use 20 logarithmically scaled sites, and thereafter use 10 linearly spaced sites to represent the low-energy scale correctly. In the DMRG calculations presented we used at least 1300 states per block and 10 finite lattice sweeps. To each set of DMRG results in Fig. 2 is added a Lorentzian of half width 2w as a guide to the eye. As the interaction is turned up the width of the reso- nance is increased far beyond the noninteracting result, up to an order of magnitude larger; e.g., for t′ = 0.01 and V = 1 the resonance width is increased by a factor of 10. However, for a larger interaction V > vF = 2, transport is suppressed, and for very large interactions the width even becomes smaller than the noninteracting resonance. A similar nonmonotonic behavior is observed by Borda et al.8 using a perturbative calculation and is opposite to the one originally reported by Mehta and Andrei3 which, however, has been corrected in an erratum.5 Where pre- ceding work3,4,5,8 failed to reach the unitary limit, we demonstrate that indeed the resonant value remains uni- tary. Further by changing the bandwidth D for linear bands we have verified that the relevant energy scale is the Fermi velocity vF of the leads, while the bandwidth D 0.00 0.01 0.02 0.03 0.04 0.05 ERLM, MS=3, tDot=0.5, t’=0.05 V=0.0, γ=0.0 w=2.5 10 V=2.0, γ=0.0 w=1.7 10 V=2.0, γ=0.5 w=8.7 10 V=2.0, γ=1.0 w=1.3 10 FIG. 4: (Color online) Conductance versus gate potential for a resonant three site chain. To each set of DMRG results a Lorentzian of half width 2w has been added as a guide to the eye. The leads are described by a cosine band between ±2 such that vF = 2. The interdot interaction suppresses the transport while the contact interaction is seen to enhance the transport. does not influence the conductance, as long as D ≫ V ; compare Fig. 5. Borda et al.8 conclude in their work that “in the case of repulsive interaction the site next to the occupied d level is empty and thus that electron can easily jump to the conduction band”, while for attractive interaction fermions accumulate close to the impurity. From that reasoning we would expect an asymmetric conductance curve depending on whether the impurity is filled or de- pleted. However, this would violate particle-hole symme- try of the model. In Fig. 3 we plot the site occupation nd of the resonant level and the averaged site occupa- tion nc of the left and right real-space sites attached to the level. The occupations are plotted versus the contact link interaction for the interacting resonant-level model, and for two different gate voltages. The site occupation of the resonant level and the neighboring sites are both enhanced by the repulsive interaction as long as interac- tion is in the range that enhances the conductance. For stronger interaction the site occupancy of the resonant level is indeed reduced; however, this is the regime where the conductance is reduced. We would like to remark that in the noninteracting case and for a weak contact, t′ ≪ 1, the site occupations of the real-space sites in the leads changes only slightly with gate voltage and are all very close to half filling. Thus it seems that the densi- ties of the hybridizing lead levels are not the determining quantity for the interaction-induced changes of transport properties. The strong renormalization of the resonance width and the non-monotonic behavior is, however, not specific to the IRLM. In Fig. 4 we show results for the center peak of a three-site nanostructure. Without a contact interac- -1.0 -0.8 -0.6 -0.4 -0.2 Non particle-hole symmetric IRLM, t’=0.1, V=1.0, gV=1.0 D=1.0 D=2.0 D=4.0 D=10.0 w=4.7 10 FIG. 5: (Color online) Conductance versus gate potential for a single site nanostructure without particle-hole symme- try with a contact interaction of V = 1.0 and a contact hopping of t′ = 0.1 for a linear band with cutoff parame- ter D = 1.0, 2.0, 4.0, and 10.0 and constant Fermi velocity, vF = 2. The conductance is independent of the cutoff. The solid line is a fit with a Lorentzian of half width w= 4.7 10−2. tion we find that the intradot interaction V = 2.0 = 4tDot leads to a suppression of the transport in agreement with previous results.1 As in the single-level case already a small contact interaction increases again the width of the resonance at zero gate potential. The enhancement of the conductance by a contact interaction is stronger than the corresponding suppression by the intradot inter- action. Therefore we conjecture that the enhancement of conductance due to the contact interaction is a universal feature, which should also be present in other systems. These findings may also be relevant for disordered struc- tures, where repulsive interaction was found to enhance transport in the case of strong disorder.9 Finally we have considered a non-particle-hole- symmetric IRLM to address the question of parame- ter renormalization versus bandwidth cutoff. The non- particle-hole-symmetric model is defined by replacing the (n̂j − ) terms in HRS by n̂j . The results are shown in Fig. 5. It is clearly seen from the calculation that vary- ing the cutoff over an order of magnitude does not change the resonance, providing the interaction is not cut off by the band. Neither the position nor the width of the res- onance peak is influenced by the change of the cutoff D, which is in contrast to the the renormalization group flow that follows from the nonequilibrium Bethe ansatz.3 There, all transport quantities depend on the cutoff D and the conductance changes with the cutoff. While it is often difficult to compare a field theoretical model, like the IRLM of Mehta and Andrei, with a lattice model, we can at least conclude that the RG flow found in their work is absent in our model with regularized (tight bind- ing) leads and that the relevant energy scale is the Fermi velocity. IV. SUMMARY A normal paradigm in transport calculations is to make a principal division between transport region, the nanos- tructure or “molecule”, and leads, where all correlation effects are excluded from the leads. In this work we have investigated the influence of an interaction on the contact between a nanostructure and the leads in a simple tight binding model. Using the nonperturbative DMRG method to evaluate the linear conductance we have demonstrated that a contact inter- action significantly influences the transport properties. A repulsive interaction smaller or comparable to the Fermi velocity in the leads enhances the conductance, while a large interaction leads to a suppression of the conduc- tance. Our work shows that even a slight spread of the interaction on the contacts influences the transport strongly. This demonstrates that particular care should be taken in treating the contacts correctly, especially re- garding the interaction. Acknowledgments D.B. acknowledges support from the HPC-EUROPA under Project No. RII3-CT-2003-506079, supported by the European Commission. This work also profited from Project 710 of the Landesstiftung Baden-Württemberg and partial support through project B2.10 of the DFG Center for Functional Nanostructures. Parts of the com- putations were performed on the XC1 and XC2 at the SSC Karlsruhe. The discussion of the the site occupa- tion is attributed to an anonymous referee. 1 D. Bohr, P. Schmitteckert, and P. Wölfle, Europhys. Lett. 73, 246 (2006). 2 S. R. White, Phys. Rev. Lett. 69, 2863 (1992). 3 P. Mehta and N. Andrei, Phys. Rev. Lett. 96, 216802 (2006). 4 P. Mehta and N. Andrei, cond-mat/0702612 (2007). 5 P. Mehta, S. P. Chao, and N. Andrei, cond-mat/0703426 (2007). 6 G. Vasseur, D. Weinmann, and R. A. Jalabert, Eur. Phys. Jour. B 51, 267 (2006). 7 R. A. Molina, D. Weinmann, R. A. Jalabert, G.-L. Ingold, and J.-L. Pichard, Phys. Rev. B 67, 235306 (2003). 8 L. Borda, K. Vladár, and A. Zawadowski, Phys. Rev. B 75, 125107 (2007). 9 R. A. Molina, P. Schmitteckert, D. Weinmann, R. A. Jal- abert, G.-L. Ingold, and J.-L. Pichard, Eur. Phys. Jour. B 39, 107 (2004).

Strong repulsive interactions within a one dimensional Fermi system in a two-probe configuration normally lead to a reduced off-resonance conductance. We show that if the repulsive interaction extends to the contact regions, a strong increase of the conductance may occur, even for systems where one would expect to find a reduced conductance. An essential ingredient in our calculations is a momentum-space representation of the leads, which allows a high energy resolution. Further, we demonstrate that these results are independent of the high-energy cutoff and that the relevant scale is set by the Fermi velocity.

Strong enhancement of transport by interaction on contact links Dan Bohr MIC, Department of Micro- and Nanotechnology, NanoDTU, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark Peter Schmitteckert TKM, Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, D-76128 Karlsruhe, Germany. (Dated: October 26, 2018) Strong repulsive interactions within a one dimensional Fermi system in a two-probe configuration normally lead to a reduced off-resonance conductance. We show that if the repulsive interaction extends to the contact regions, a strong increase of the conductance may occur, even for systems where one would expect to find a reduced conductance. An essential ingredient in our calculations is a momentum-space representation of the leads, which allows a high energy resolution. Further, we demonstrate that these results are independent of the high-energy cutoff and that the relevant scale is set by the Fermi velocity. PACS numbers: 73.63.Kv, 73.23.Hk, 71.10.Pm I. INTRODUCTION Constructing a transport theory for strongly correlated systems is one of the major challenges of condensed mat- ter physics. Even though many interesting ideas have been proposed during recent years, no consensus has yet emerged as to the general validity and applicability of the various schemes. With this state of affairs, it is of high importance to establish reliable benchmarks for simple model systems, which then can be used to validate new approaches. Recently we presented a novel method for calculat- ing linear response conductance1 using the density ma- trix renormalization group (DMRG) method2. A ma- jor challenge in this work consisted in minimizing finite size effects, which was achieved via modified boundary conditions. In this paper we circumvent these technical problems by reformulating the leads in momentum space. This approach enables us to (i) reach a much higher en- ergy resolution (∼ 10−5) and (ii) allows for a greater flexibility in the choice of discretization schemes. In two recent papers Mehta and Andrei3,4 presented nonequilibrium Bethe ansatz results for the interacting resonant-level model (IRLM), where a single spinless level is coupled to a left and a right lead both via a tunneling and a density-density interaction term. How- ever, their work currently excludes the regime of resonant tunneling–i.e., the regime where the conductance is close to unity.5 In this work we study the linear conductance of the IRLM on a lattice to provide a benchmark for the uni- versal properties of the model. In addition, we present results for an extended model, where the central region consists of three sites, with a similar interaction as in the IRLM model. As we will show, this model displays the same qualitative behavior as the IRLM. It should be noted that despite its simplicity, the IRLM captures much of the physics of transport through an arbitrary in- teracting nanostructure provided that only a single level is close to the Fermi energy of the leads, with all other lev- els well separated and outside the voltage window within which the transport takes place. For perfect coupling the IRLM model corresponds to the one-dimensional model studied by Vasseur et al.,6 and Molina et al.,7 obtained by restricting their nanostructure to a single site. Using the embedding method they showed that smoothing the ramp of interaction for perfect contacts can compensate for the decrease of transmission due to interaction on the nanostructure. Here we go far beyond the energy resolu- tion attained in previous work and show that interaction on the contact links can lead to strong renormalization effects, enhancing transport beyond the noninteracting system. II. METHOD AND MODELS We use the DMRG method to evaluate the linear re- sponse conductance of the interacting nanostructure. In previous work1 the leads were modeled in real-space by nearest-neighbor hopping chains. While simple to imple- ment there are several drawbacks of this method, most prominently the need for “damped boundary conditions” and the resulting problem of trapping of fermions on the Wilson chain (the damped region).1 In the present work we introduce a setup where the leads are described in momentum space. Specifically, a short part of the lead close to the nanostructure is rep- resented in real-space, accounting for local (i.e., high en- ergy) physics, while further away from the nanostruc- ture the lead is represented in momentum space; see Fig. 1. Since the low-energy modes of the momentum leads are now directly coupled to the extended structure (the nanostructure plus additional real-space sites), as il- lustrated in Fig. 1, the trapping of fermions on the low energy sites1 is avoided and no scaling sweeps are needed. http://arxiv.org/abs/0704.1917v3 2PSfrag replacements Nanostructure, MS sites Optional real space sites, MAdd sites Mandatory real space sites, 2 sites Left Lead Right LeadME = MS + MAdd tDottDot tDot VVV γVγV tktk t tt FIG. 1: (Color online) Schematics of the nanostructure ex- tended by real space sites and attached to momentum-space leads. This enables much higher energy resolution, and in the current work we resolve resonances of widths O(10−5). By virtue of the momentum representation of the leads the discretization scheme can be chosen arbitrarily to suit the problem at hand. In the present work we use a logarithmic discretization to cover a large energy range, while switching to a linear discretization for the lowest- energy states in order to describe Fermi-surface physics accurately. The linear discretization on the low-energy scale allows for a better representation of the low-energy physics relevant for transport properties–i.e., excitations created by η. The models considered in this work are the IRLM and the natural extension of this model to resonant linear chains, defined by the Hamiltonians HRS = µg ĉ j ĉj − j,j−1∈SE tj ĉ j ĉj−1 + h.c. j,j−1∈SE n̂j − n̂j−1 − , (1) HMS = i∈L,R i ĉi, (2) HT = − tk ĉ ĉ1 + tk ĉ + h.c., (3) where ĉ and ĉℓ are the (spinless) fermionic creation and annihilation operators at site ℓ, n̂ℓ = ĉ . HRS , HMS , and HT denote real-space, momentumspace, and tunnel- ing between real- and momentum-space Hamiltonians, respectively. The symbols S and SE denote the nanos- tructure and the extended nanostructure (the full real- space chain), respectively. The indices 1 and ME denote the first and last site in SE . The general setup and the specific values of the hopping matrix elements tj and the interactions Vj are indicated in Fig. 1, and note specif- ically the interactions on the contact links, γV . The coupling tk of the extended real-space structure to the momentum leads is chosen in such a way that in the case of a cosine band it corresponds to a nearest-neighbor hop- ping chain in real-space with a hopping parameter of t. In the following we measure all energies in units of t = 1. 0.000 0.001 0.002 0.003 IRLM, t’=0.01, γ=1.0 V=0.0 w=2.0 10 V=0.01 w=2.1 10 V=0.03 w=3.2 10 V=1.0 w=2.1 10 V=5.0 w=8.9 10 V=25.0 w=1.0 10 0.000 0.010 0.020 0.030 IRLM, t’=0.03, γ=1.0 V=0.0 w=1.8 10 V=0.03 w=2.1 10 V=0.1 w=2.5 10 V=1.0 w=1.0 10 V=5.0 w=4.5 10 V=25.0 w=9.0 10 FIG. 2: (Color online) Conductance versus gate potential for the interacting resonant level model for a contact hopping of (a) t′ = 0.01 and (b) t′ = 0.03 and contact interaction ranging from zero to 25. To each set of DMRG data a Lorentzian of half width 2w has been added as a guide to the eye. The leads are described with a cosine band between ±2 such that the Fermi velocity is vF = 2. In contrast to intradot interaction the contact interaction enhances the conductance and shows a nonmonotonic behavior versus contact interaction. For a single-site nanostructure and γ = 1 this model reduces to the IRLM. The properties of the leads are defined by the band structure ǫk, which can take any form. In this work we use either the cosine band, ǫk = −2 cos(k), or the linear band, ǫk = 2k. D is a cutoff parameter such that the Fermi velocity vF = 2 is kept constant in all work presented here, and the band ranges between energies −D and D. Throughout this work we use the notion of “contact interaction” for interaction on the link between the nanostructure and the leads. 0 5 10 15 20 25 Local density, t’=0.01 nd, µg=0.0005 nc, µg=0.0005 nd, µg=0.0002 nc, µg=0.0002 FIG. 3: (Color online) Site occupation nd of the resonant level and nc of the real-space sites attached to the level vs the link interaction in the IRLM for t’=0.01 and two different gate voltages. III. RESULTS The aim of this work is to study the effect of contact interaction. It is known from previous work1 that strong repulsive interactions within the nanostructure lead to suppression of the transport off resonance due to the for- mation of a density-wave-like state on the dot. In Fig. 2 we show results for the conductance versus gate potential for different couplings to the leads and different contact interactions for the IRLM (γ = 1). The calculations have been performed with typically 130 sites in total, ME = 10 real-space sites and 120 momentum- space sites. Due to the symmetry of the band, we use a discretization that is symmetric around ǫF = 0, and further use identical discretization of the two leads. To represent the “large” energy span in the band we use 20 logarithmically scaled sites, and thereafter use 10 linearly spaced sites to represent the low-energy scale correctly. In the DMRG calculations presented we used at least 1300 states per block and 10 finite lattice sweeps. To each set of DMRG results in Fig. 2 is added a Lorentzian of half width 2w as a guide to the eye. As the interaction is turned up the width of the reso- nance is increased far beyond the noninteracting result, up to an order of magnitude larger; e.g., for t′ = 0.01 and V = 1 the resonance width is increased by a factor of 10. However, for a larger interaction V > vF = 2, transport is suppressed, and for very large interactions the width even becomes smaller than the noninteracting resonance. A similar nonmonotonic behavior is observed by Borda et al.8 using a perturbative calculation and is opposite to the one originally reported by Mehta and Andrei3 which, however, has been corrected in an erratum.5 Where pre- ceding work3,4,5,8 failed to reach the unitary limit, we demonstrate that indeed the resonant value remains uni- tary. Further by changing the bandwidth D for linear bands we have verified that the relevant energy scale is the Fermi velocity vF of the leads, while the bandwidth D 0.00 0.01 0.02 0.03 0.04 0.05 ERLM, MS=3, tDot=0.5, t’=0.05 V=0.0, γ=0.0 w=2.5 10 V=2.0, γ=0.0 w=1.7 10 V=2.0, γ=0.5 w=8.7 10 V=2.0, γ=1.0 w=1.3 10 FIG. 4: (Color online) Conductance versus gate potential for a resonant three site chain. To each set of DMRG results a Lorentzian of half width 2w has been added as a guide to the eye. The leads are described by a cosine band between ±2 such that vF = 2. The interdot interaction suppresses the transport while the contact interaction is seen to enhance the transport. does not influence the conductance, as long as D ≫ V ; compare Fig. 5. Borda et al.8 conclude in their work that “in the case of repulsive interaction the site next to the occupied d level is empty and thus that electron can easily jump to the conduction band”, while for attractive interaction fermions accumulate close to the impurity. From that reasoning we would expect an asymmetric conductance curve depending on whether the impurity is filled or de- pleted. However, this would violate particle-hole symme- try of the model. In Fig. 3 we plot the site occupation nd of the resonant level and the averaged site occupa- tion nc of the left and right real-space sites attached to the level. The occupations are plotted versus the contact link interaction for the interacting resonant-level model, and for two different gate voltages. The site occupation of the resonant level and the neighboring sites are both enhanced by the repulsive interaction as long as interac- tion is in the range that enhances the conductance. For stronger interaction the site occupancy of the resonant level is indeed reduced; however, this is the regime where the conductance is reduced. We would like to remark that in the noninteracting case and for a weak contact, t′ ≪ 1, the site occupations of the real-space sites in the leads changes only slightly with gate voltage and are all very close to half filling. Thus it seems that the densi- ties of the hybridizing lead levels are not the determining quantity for the interaction-induced changes of transport properties. The strong renormalization of the resonance width and the non-monotonic behavior is, however, not specific to the IRLM. In Fig. 4 we show results for the center peak of a three-site nanostructure. Without a contact interac- -1.0 -0.8 -0.6 -0.4 -0.2 Non particle-hole symmetric IRLM, t’=0.1, V=1.0, gV=1.0 D=1.0 D=2.0 D=4.0 D=10.0 w=4.7 10 FIG. 5: (Color online) Conductance versus gate potential for a single site nanostructure without particle-hole symme- try with a contact interaction of V = 1.0 and a contact hopping of t′ = 0.1 for a linear band with cutoff parame- ter D = 1.0, 2.0, 4.0, and 10.0 and constant Fermi velocity, vF = 2. The conductance is independent of the cutoff. The solid line is a fit with a Lorentzian of half width w= 4.7 10−2. tion we find that the intradot interaction V = 2.0 = 4tDot leads to a suppression of the transport in agreement with previous results.1 As in the single-level case already a small contact interaction increases again the width of the resonance at zero gate potential. The enhancement of the conductance by a contact interaction is stronger than the corresponding suppression by the intradot inter- action. Therefore we conjecture that the enhancement of conductance due to the contact interaction is a universal feature, which should also be present in other systems. These findings may also be relevant for disordered struc- tures, where repulsive interaction was found to enhance transport in the case of strong disorder.9 Finally we have considered a non-particle-hole- symmetric IRLM to address the question of parame- ter renormalization versus bandwidth cutoff. The non- particle-hole-symmetric model is defined by replacing the (n̂j − ) terms in HRS by n̂j . The results are shown in Fig. 5. It is clearly seen from the calculation that vary- ing the cutoff over an order of magnitude does not change the resonance, providing the interaction is not cut off by the band. Neither the position nor the width of the res- onance peak is influenced by the change of the cutoff D, which is in contrast to the the renormalization group flow that follows from the nonequilibrium Bethe ansatz.3 There, all transport quantities depend on the cutoff D and the conductance changes with the cutoff. While it is often difficult to compare a field theoretical model, like the IRLM of Mehta and Andrei, with a lattice model, we can at least conclude that the RG flow found in their work is absent in our model with regularized (tight bind- ing) leads and that the relevant energy scale is the Fermi velocity. IV. SUMMARY A normal paradigm in transport calculations is to make a principal division between transport region, the nanos- tructure or “molecule”, and leads, where all correlation effects are excluded from the leads. In this work we have investigated the influence of an interaction on the contact between a nanostructure and the leads in a simple tight binding model. Using the nonperturbative DMRG method to evaluate the linear conductance we have demonstrated that a contact inter- action significantly influences the transport properties. A repulsive interaction smaller or comparable to the Fermi velocity in the leads enhances the conductance, while a large interaction leads to a suppression of the conduc- tance. Our work shows that even a slight spread of the interaction on the contacts influences the transport strongly. This demonstrates that particular care should be taken in treating the contacts correctly, especially re- garding the interaction. Acknowledgments D.B. acknowledges support from the HPC-EUROPA under Project No. RII3-CT-2003-506079, supported by the European Commission. This work also profited from Project 710 of the Landesstiftung Baden-Württemberg and partial support through project B2.10 of the DFG Center for Functional Nanostructures. Parts of the com- putations were performed on the XC1 and XC2 at the SSC Karlsruhe. The discussion of the the site occupa- tion is attributed to an anonymous referee. 1 D. Bohr, P. Schmitteckert, and P. Wölfle, Europhys. Lett. 73, 246 (2006). 2 S. R. White, Phys. Rev. Lett. 69, 2863 (1992). 3 P. Mehta and N. Andrei, Phys. Rev. Lett. 96, 216802 (2006). 4 P. Mehta and N. Andrei, cond-mat/0702612 (2007). 5 P. Mehta, S. P. Chao, and N. Andrei, cond-mat/0703426 (2007). 6 G. Vasseur, D. Weinmann, and R. A. Jalabert, Eur. Phys. Jour. B 51, 267 (2006). 7 R. A. Molina, D. Weinmann, R. A. Jalabert, G.-L. Ingold, and J.-L. Pichard, Phys. Rev. B 67, 235306 (2003). 8 L. Borda, K. Vladár, and A. Zawadowski, Phys. Rev. B 75, 125107 (2007). 9 R. A. Molina, P. Schmitteckert, D. Weinmann, R. A. Jal- abert, G.-L. Ingold, and J.-L. Pichard, Eur. Phys. Jour. B 39, 107 (2004).

Quantum Filtering of Optical Coherent States C. Wittmann1,∗ D. Elser1, U.L. Andersen1,2, R. Filip1,3, P. Marek4, and G. Leuchs1 Institut für Optik, Information und Photonik, Max-Planck Forschungsgruppe, Universität Erlangen-Nürnberg, Günther-Scharowsky-Straße 1, 91058, Erlangen, Germany Department of Physics, The Technical University of Denmark, 2800 Kongens Lyngby, Denmark Department of Optics, Palacký University, 17. listopadu 50, 772 07 Olomouc, Czech Republic School of Mathematics and Physics, Queens University, Belfast BT7 1NN, United Kingdom (Dated: August 23, 2021) We propose and experimentally demonstrate non-destructive and noiseless removal (filtering) of vacuum states from an arbitrary set of coherent states of continuous variable systems. Errors i.e. vacuum states in the quantum information are diagnosed through a weak measurement, and on that basis, probabilistically filtered out. We consider three different filters based on on/off detection phase stabilized and phase randomized homodyne detection. We find that on/off detection, optimal in the ideal theoretical setting, is superior to the homodyne strategy in a practical setting. PACS numbers: 03.67.-a, 03.67.Hk I. INTRODUCTION Ultra-low noise quantum channels transmitting dis- crete or Continuous-Variable (CV) quantum informa- tion are prerequisite for the successful execution of many quantum information protocols. For example, the secu- rity and the secret key rate of quantum key distribution critically depend on the amount of excess noise added to the quantum state during transmission [1, 2]. All realistic quantum channels are afflicted by such noise: In fiber channels, for example, light scattering by ther- mal phonons causes Gaussian phase noise. On the other hand, noise sources important in atmospheric transmis- sion, such as time jitter and beam pointing noise [3], show a characteristic non-Gaussian behavior. In order to retain security, the errors imposed by the noisy channels must be corrected. Various methods have been developed to combat noise in CV quantum com- munication, examples being entanglement distillation [4] and quantum error correction coding [5], which are rely- ing on highly non-classical resources and complex pro- cessing. An alternative is quantum filtering which is a protocol that probabilistically rejects erroneous quan- tum states through detection. The simplest approach is a classical measure-prepare strategy based on optimal state discrimination using the Neyman-Pearson criterion [6] followed by state recreation. Helstrom found that by using a tailored detection process, it is possible to identify a pure target state in a noisy mixture [7] (see also [8, 9, 10]). Takeoka et al. generalized this strategy and named it unambiguous quantum state filtering since it unambiguously filters out a specific signal from the noise [11]. However, only a single a priori known state is resurrected, which is done destructively and therefore not suitable for quantum communication. In this paper, we propose and experimentally realize ∗Electronic address: cwittmann@optik.uni-erlangen.de a quantum state filter protocol specially tailored to non- Gaussian noise as in atmospheric transmission. The pro- tocol filters a coherent state alphabet non-destructively and noiselessly, i.e. the quantum states are not com- pletely destructed and no excess noise is added by our filter. Our protocol is based on a weak measurement of the corrupted signal followed by a post selection of the remaining part of the signal. We investigate two different weak measurement strategies, namely homodyne detec- tion and on/off detection and compare their efficiencies in filtering out noise. We find that optimum filtering is obtained by the use of an ideal on/off detector. The scheme presented in this paper provides the first imple- mentation of a CV error detection protocol enabled by a photon counting detector. An exemplary application of such a filter is shown in Fig. 1. Suppose a signal is conveyed through two differ- ent quantum channels each possessing different kinds of noises (e.g. a free space channel and a fiber channel). If the first channel is inflicted by the non-Gaussian on/off noise and the following channel by Gaussian noise, the on/off noise might be completely masked by the Gaus- sian noise and cannot easily be filtered out at the receiv- ing station. In order to circumvent a mixing of the two noise sources, the filtration station could be placed be- tween the two channels thus removing the on/off noise before the signal enters the Gaussian noise channel. The filtration protocol can be also used to improve the security of a quantum key distribution scheme based on a coherent state alphabet and heterodyne detection. This is proven at the end of the paper. II. DESCRIPTION OF THE PROTOCOL Let us consider the protocol in detail. Information is encoded into quantum states taken from a coherent state alphabet with a possibly unknown probability distribu- tion. The quantum state is subsequently sent through the quantum channel where it is subject to time jitter http://arxiv.org/abs/0704.1918v3 mailto:cwittmann@optik.uni-erlangen.de on/off noise excess noise X X X P P P classical channel freespace channel fiber channel FIG. 1: Application of the quantum filter device. The filter F is placed between two quantum channels connect- ing sender S and receiver R. We assume, that the channels have non-Gaussian on/off (first part) and Gaussian proper- ties (last part). The on/off behavior of the first channel will be masked by excess noise in the second channel (e.g. |α〉〈α| → |0〉〈0| → ρth). However, a quantum filter in the in- termediate station can sense the channel break and reject the noisy state by sending information over a classical channel to or beam positioning noise. Such non-Gaussian noise oc- curs when the detection time is longer than the signal but shorter than the jitter time or when the aperture of the receiver is larger than the beam but much smaller than the beam pointing noise. This noise can be approx- imated by a mixture of the sent coherent state |α〉 and the vacuum state: ρ(α) = p|α〉〈α|+ (1 − p)|0〉〈0|, (1) where p is an unknown probability for perfect transmis- sion. The task is now to find a protocol that unambigu- ously filters out the vacuum state, while only attenuating the coherent state, e.g. |α〉 → |γα〉, γ < 1. To accomplish a state independent weak measurement adding no excess noise the signal system must be coupled unitarily and phase insensitively to a meter system in which the actual measurement takes place. Due to these requirements, the coupling can be enabled by a beam splitter with the meter system being in the vacuum state before interaction [12]. The signal-meter coupling can therefore simply be described by the transformation ρ(α) ⊗ ρ(0) → ρ( 1−Rα)⊗ ρ( Rα), (2) where R is the reflectivity of the beam splitter. After this interaction, the presence or absence of the vacuum contribution is correlated in the two systems. Thus by detecting the vacuum state in the meter system, filtering of the vacuum in the signal system can be performed by post selecting on the correlated state. The strategy is illustrated in Fig. 1. The next step is thus to find the measurement strategy that optimally and unambiguously detects the vacuum contribution. Let us assume that we use the measurement operators Π⊥ and Π to discriminate the vacuum state and the un- known signal state. We seek a strategy that maximizes the probability 〈 Rα|Π| Rα〉 of measuring | Rα〉 un- FIG. 2: Schematic illustration of a coherent state quantum filter for the non-Gaussian channel: (a) Filter device with ver- ification measurement; (b left) Filter using APD as a detec- tor, (b right) using homodyne detection with a local oscillator (LO). der the condition that the vacuum state is never de- tected incorrectly, that is E = 〈0|Π|0〉 = 0. Such de- cision problem was first encountered by Neyman and Pearson [6] and was further elaborated upon by Hel- strom [7] and Holevo [13]. They found that the maxi- mum probability of detecting the signal correctly (also called the acceptance probability) with no error detec- tions (E = 0) is given by P ( Rα) = 1 − exp(−R|α|2) (Note that E = P (0)). We readily find that measurement operators satisfying these conditions are Π⊥ = |0〉〈0| and Π = 1 − |0〉〈0| for rejecting and accepting the state, re- spectively. Therefore, using these measurement opera- tors the signal states can be unambiguously detected in the meter system and thus perfectly filtered out in the signal system. We stress that since this optimized mea- surement is independent of the signal amplitude and the reflection coefficient, it is the optimal strategy for every coherent state. The physical implementation of these measurement op- erators is known to be an ideal avalanche photodiode (APD) operating in the break down voltage mode. Prac- tical APDs are, however, lossy and possess dark counts which results in a reduced success probability and gives rise to errors, that is, E > 0. Therefore, in addition to the APD, we investigate in the following the filtering performance using a homodyne detector for the decision problem; quadrature values larger than a certain a pri- ori specified threshold value are assumed to stem from the unknown signal state, smaller values from the vac- uum state. Note that a similar strategy was proposed in Ref. [14] and experimentally realized in Ref. [15, 16] to purify non-classical resources. We also note that the incorporation of an APD in a CV system has been im- plemented in previous experiments on state preparation and estimation [17, 18]. In order to quantify the performance of the filtering protocol using different detection methods, we introduce −3 −2 −1 0 1 2 3 X quadrature measurement (shot noise units) vacuum noisy signal filtered signal APD, η=1 HDS, η=1 FIG. 3: Marginal distribution for the perturbed state (p = 0.02) (circles), the vacuum state (crosses) and the filtered state using an APD filter (triangles). The solid and the dotted dashed line correspond to the theoretical performance of a filter with APD (ηAPD = 1) and with homodyne detector (ηHDS = 1) respectively. The mean photon number in the filter is R|α|2 = 1.65 and the error probabilities are identical EAPD = EHDS = 5.3 · 10 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Mean photon number(R|α|2) FIG. 4: Acceptance probability for mean photon number R|α|2 impinging on the filter detector. The triangles, circles and squares show experimental data for APD and homodyne detection with and without stabilized LO, respectively. The solid lines are theoretical predictions for detectors with unit quantum efficiency. EAPD = EHDS = EHDR = 5.3 · 10 two appropriate functions: the sensitivity S and the gain G. The sensitivity quantifies the filtering efficiency near the vacuum state and we define it as d|α|2P ( Rα)|α=0. (3) Since the probablity P must be minimal when α = 0, the sensitivity S is a measure for how quickly the probability increases around α = 0. For the ideal filter we easily find S = R, thus we will be using S/R as the figure of merit. The other parameter that we will use to quantify the per- formance of the filter is the gain G = p′/p where p′ is the probability for the coherent state to occur in the mix- ture after filtering: ρ′ = p′| Tα| + (1 − p′)|0〉〈0|. The success probability for positive filter outputs is PS = Rα)+(1−p)P (0) and the gain can thus be written 1− (1 − p) E . (4) Note that the sensitivity S depends solely on the filter implementation. Thus, it is independent of the chan- nel. In contrast, the gain is a signal-, channel- and filter- dependent parameter, and therefore describes the joint action of channel and filter. III. EXPERIMENTAL SETUP AND RESULTS We proceed with the experimental demonstration of quantum filtering using an APD-based filter. The setup is shown in Fig. 2(a). The source is a diode laser emitting light at 810nm, characterized with a coherence time of 1µs and measured to be shot noise limited in the detected bandwidth. The statistical mixture of the coherent sig- nal states and vacuum states is prepared in a computer controlled electro-optical modulator (EOM). The EOM therefore generates the signal and simulates the noisy channel. To obtain a small excitation of the coherent state, the beam is heavily attenuated after modulation. We set the coherent state probability to p = 2%, thus the probability for vacuum to occur being 1 − p = 98%. The signal duration is defined within 800ns time win- dows, while the rate of the signal preparation is set to 100kHz. We investigate the performance of the APD-based fil- ter by characterizing the state with homodyne detection before and after filtering. First we demonstrate the prin- ciples of the protocol using the APD as a filter and the homodyne detector for characterization. The probability distribution of the mixed input state is shown in Fig. 3 by circles and for comparison the pure vacuum state is shown by crosses. Next, the mixed state is passing the fil- ter beam splitter and the homodyne quadrature data are selected based on the measurement outcomes of the APD detector. The resulting probability distribution is shown by the dotted curve and should be compared with the ideally filtered signal (solid line) and two other curves: the one expected using homodyne detection with equal error probability and one using unit quantum efficiency as filter (dotted dashed line). In the following, we fully characterize the quantum fil- ters. The mixed state is split on a 50/50 beam splitter and subsequently directed to two different detector units: a fiber coupled APD (Perkin-Elmer SPCM CD3017) and a homodyne detector. We therefore simultaneously mea- sure the acceptance probabilities for the two different de- tection schemes; here each detector represented a filter. increasing B increasing R| |α FIG. 5: In all figures the triangles are experimental data points for a filter using APD, the circles and squares show data for filters using homodyne detection with and without stabilized LO respectively; the solid lines are theoretical pre- dictions for filters using unit quantum efficiency detectors, η = 1 (APD: S/R = (1 − pd) 2 for η = 1). (a) Sensitivity as a function of error probability. The dashed line should guide the eye to the error rate where the detectors are compared EAPD = EHDS = EHDR = 5.3 ·10 −3. (b) Gain G as a function of mean photon number R|α|2 impinging on the filter detec- tor. Signal probability fixed to p = 2%. (c) Parametric plot of Gain G and success probability PS for R|α| 2 ∈ [0, 1.65]. The performance decreases from APD to HDS and HDR. The er- ror bars show the statistical errors (three-sigma error bars) and seem to be much smaller than the experimental errors. Let us first discuss the APD-based filter (see Fig. 2(b left)). A gate option is used to precisely determine the detection time. The quantum efficiency is estimated to be ηAPD = 63 ± 3%, while the dark count rate is 180cts/s. Due to these imperfections, the expected ac- ceptance probability is PAPD(β) = 1− (1− pd) exp −ηAPD(1− pd)|β|2 where β = Rα and pd is the dark count probability. The expected error probability is EAPD = PAPD(0) = pd. We measure the acceptance probability by comparing the actual decision (based on the filter measurement out- come) with the a priori known preparation of the state. The results are presented in Fig. 4 as a function of R|α|2. Note that R should be tailored to the actual amplitude α to optimize the performance. The error probability is found to be E = 5.3 · 10−3, which is limited not by the dark count probability 1.4 ·10−4 but by the imperfections in preparing the vacuum state. Next we discuss the filter based on homodyne detec- tion (see Fig. 2(b right)). The detector’s bandwidth is 10MHz. Using a local oscillator (LO) power of about 5mW, the shot noise to electronic noise ratio is 18dB. The detection efficiency, including the mode matching efficiency and the quantum efficiency of the photodiodes is ηHD = 84 ± 3%. We investigate two different kinds of homodyne detectors: one with a phase stabilized lo- cal oscillator (HDS) and one with a phase randomized local oscillator (HDR). The latter scheme should be used when the input alphabet of coherent states is rotationally symmetric in phase space whereas the former scheme is superior if e.g. a binary phase encoding is used where the absolute direction of the displacement is known a priori. The hypothesis whether a signal or a vacuum state was measured is based on the absolute value of the measured quadrature; if it is above a certain threshold value, denoted B, we estimate the state to be |α〉, if not |0〉. Knowing that the signal is encoded into |α〉, the ex- pected acceptance probability for a phase stabilized local oscillator is PHDS(β) = Erfc[ 2(B + a)] + Erfc[ 2(B − a)] where a = ηHDβ, while for the phase randomized local oscillator PHDR(β) = 2(B − a cos θ) dθ. (7) The error is identical for the two approaches and given by EHDR = EHDS = Erfc[ A classical signal is appended to the pulse trains to estimate the phase difference of signal and LO at a given time. Measurement data, acquired with random- ized phase of the LO, is subsequently directly used to evaluate the performance of the scheme based on ran- dom LO. However, due to the appended classical signal the relative phase is known and we selected the data as- sociated with a phase difference of zero corresponding to the phase stabilized case. Using these data and the above hypothesis, we find the acceptance probability for vari- ous excitations R|α|2 and various threshold values B. In Fig. 4 we plot the acceptance probability as a function of R|α|2 with the threshold value set such that the error probability matches the one of the APD. It is clear from the plot that the APD performs better than the homo- dyne detector despite the much higher quantum efficiency of the latter one. The sensitivity of the two filters is obtained by fitting curves to the measured acceptance probabilities corre- sponding to various thresholds (for the homodyne case) and subsequently using equation (3). The results are plotted in Fig. 5(a), where the triangle represents the APD-based filter whereas the circles and squares are as- sociated with the phase randomized and phase locked LO, respectively. For post selection thresholds B close to the coherent state variance the sensitivity of homodyne detectors is maximal. As evident from the plot, for iden- tical error probability the sensitivity of the APD-based filter is much larger than that of the homodyne-based filter. From the measurements we also calculate the gain for different mean photon numbers and different success probabilities as shown in Fig. 5(b) and 5(c), respectively. The former figure clearly shows the superior performance of the APD filter compared with the homodyne filters. From Fig. 5(c) we clearly see that the behavior of the gain as a function of the success probability follows the same curve for the three filters if the error rates are tailored to be the same (by adjusting the error thresholds appropri- ately). This is also what is expected from Eq. (4). IV. FILTERING IN A QUANTUM KEY DISTRIBUTION SCHEME In the final part, we consider the filtering action in a CV quantum key distribution scheme using Gaussian modulated coherent states [1]. To estimate a lower bound for secure transmission we use the recent work of Garcia- Patron and Cerf [19] (see also ref [20]), showing that the lower bound can be directly computed from the covari- ance matrix of the joint state between the sender of infor- mation, Alice, and the receiver, Bob. The lower bound (for reverse reconciliation) is thus given by Klower = I ab − χGbE (8) where χGbE is the maximum information between Bob and the eavesdropper, Eve, corresponding to the Holevo bound and IGab is the Gaussian mutual information be- tween Alice and Bob. These quantities are computed solely from the covariance matrix of the joint state of Alice and Bob. The joint state is found by using the equivalence be- tween the coherent state scheme and an entanglement- based protocol. In an ideal entanglement based proto- col, Alice generates Gaussian two-mode squeezed states and measures one mode using heterodyne detection. The other mode is then being prepared in a coherent state with a Gaussian distributed displacement. A two-mode squeezing variance of V results in a displacement variance of σ = (V + 1/V )/2− 1. The prepared coherent state is sent to Bob through the erasure channel and produces the following density matrix ρAB = p (|V 〉〈V |)AB + (1− p)ρA ⊗ (|0〉〈0|)B , (9) where |V 〉AB represents a two-mode squeezed state and ρA is a thermal state having a variance (V +1/V )/2. The lower bound for secure communication is found numeri- cally by solvingKlower < 0 using the covariance matrix of the joint state ρAB and Eqn. (8). We find security cannot be guaranteed if p < 0.87 (corresponding to Klower < 0). Klower is maximized over the variance V . Let us now consider the security when the filtering pro- tocol is implemented. Assuming first the APD to be ideal, the state after filtering is ρfilter = ΠUBSρABTU Tr(ΠUBSρABTU where UBS is the unitary beam splitter operation, ρABT = ρAB ⊗ |0〉〈0| and Π is the measurement oper- ator of the tap T. The structure of the covariance matrix for this state is given by CV ′AB = (CVAB −P 0CV 0AB), where CVAB is the covariance matrix of the state right after the beam splitter after tracing out the tap mode, ρBS = TrT (UBSρABTU ), and CV 0AB is the covari- ance matrix of the state if the measurement outcome of the filter measurement is associated with Π⊥ = 1 − Π, ρ⊥ = Π⊥UBSρABTU Π⊥/Tr(Π⊥UBSρABTU ). P 0 is the probability for getting a measurement result associ- ated with Π⊥ and PS = 1 − P 0 is the filtering success rate. Using the covariance matrix of the filtered state, CV ′AB , we again compute the lower bound numerically, and find that the probability p for which secure commu- nication can take place is now just required to be larger than zero. This can be also investigated analytically in the limit of weak two-mode squeezing (V ≈ 1), corre- sponding to a small Gaussian alphabet. In this case, the lower bound on the secure key rate is approximately Klower ≈ pPS T (V − 1)2, (11) which is positive for any T > 0 and V 6= 1. We have thus shown that the usages of an ideal filter reestablishes the security of the quantum key distribution system in- dependent on the amount of vacuum noise. We now consider the realistic case where the APD is nonideal. Using an APD with nonzero quantum effi- ciency is not a hindrance for obtaining secure communi- cation, but it will result in a lower success rate. A limiting factor on the security, however, is the presence of dark counts which limits the minimum noise probability pmin for which security can be proven. For a given dark count rate, the transmission T of the filtering beam splitter and the modulation variance σ can be numerically optimized to maximize Klower for which security can be guaran- teed in the protocol with respect to the noisy channel. Using the experimental parameters of the APD used in our experiment (η = 0.63, pd = 0.005) we find that secure communication can be guaranteed if p > pmin = 0.222. By reducing the dark counts to pd = 5 × 10−4 the mini- mum probability is reduced to pmin = 0.028 and for even lower pd = 5× 10−5, the threshold is only pmin = 0.003. V. CONCLUSION In summary, we have investigated a filtering protocol that successfully filters out vacuum states from a set of coherent states of a continuous variable system. A weak measurement scheme consisting of a beam splitter and an optimized measurement was employed to probabilis- tically filter out the unwanted vacuum states. Different measurement strategies based on homodyne and on/off detection were investigated and compared. We therefore provided the first direct comparison between an APD- based and a homodyne-based protocol, and found that the ideal on/off detection is optimal and that the prac- tical (that is non-ideal) on/off detector is superior to homodyne detection despite the much higher quantum efficiency of the latter one. The protocol will be advan- tageous in continuous variable quantum communication wherever beam positioning and time jitter noise are the main obstacles for faithful and secure transmission. Acknowledgments This work was supported by the EU project COV- AQIAL (project no. FP6-511004), the EU-IST network SECOQC, COMPAS project no. 212008, the project 202/07/J040 of the GACR and project MSM6198959213 of Czech Ministry of Education. R.F. acknowledges sup- port from the Alexander von Humboldt foundation. P.M. acknowledges support from the European Social Fund. [1] F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, Nature 421, 238 (2003). [2] M. Heid and N. Lütkenhaus, Physical Review A 73, 052316 (2006). [3] N. Gisin, G. G. Ribordy, W. Tittel, and H. Zbinden, Reviews of Modern Physics 74, 145 (2002). [4] L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Physical Review Letters 84, 4002 (2000). [5] S. L. Braunstein, Nature 394, 47 (1998). [6] J. Neyman and E. Pearson, Philosophical Transactions of the Royal Society of London Series a-Containing Papers of a Mathematical or Physical Character 231, 289 (1933). [7] C. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976). [8] M. G. A. Paris, Physics Letters A 225, 23 (1997). [9] G. M. D’Ariano, M. G. A. Paris, and P. Perinotti, Phys- ical Review A 65, 062106 (2002). [10] M. Takeoka, M. Ban, and M. Sasaki, Physics Letters A 313, 16 (2003). [11] M. Takeoka, M. Ban, and M. Sasaki, Physical Review A 68, 012307 (2003). [12] U. Leonhardt, Measuring the Quantum States of Light (Cambridge University Press, 1997), p. 80. [13] A. Holevo, Probabilistic and Statistical Aspects of Quan- tum Theory (North-Holland, Amsterdam, 1982). [14] S. Suzuki, M. Takeoka, M. Sasaki, U. L. Andersen, and F. Kannari, Physical Review A 73, 042304 (2006). [15] J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, Physical Review Letters 96, 253601 (2006). [16] A. Franzen, B. Hage, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, Physical Review Letters 97, 150505 (2006). [17] J. Wenger, R. Tualle-Brouri, and P. Grangier, Physical Review Letters 92, 153601 (2004). [18] J. Wenger, J. Fiurášek, R. Tualle-Brouri, N. J. Cerf, and P. Grangier, Physical Review A 70, 053812 (2004). [19] R. Garćıa-Patrón and N. J. Cerf, Physical Review Letters 97, 190503 (2006). [20] M. Navascués, F. Grosshans, and A. Acin, Physical Re- view Letters 97, 190502 (2006).

We propose and experimentally demonstrate non-destructive and noiseless removal (filtering) of vacuum states from an arbitrary set of coherent states of continuous variable systems. Errors i.e. vacuum states in the quantum information are diagnosed through a weak measurement, and on that basis, probabilistically filtered out. We consider three different filters based on on/off detection phase stabilized and phase randomized homodyne detection. We find that on/off etection, optimal in the ideal theoretical setting, is superior to the homodyne strategy in a practical setting.

Quantum Filtering of Optical Coherent States C. Wittmann1,∗ D. Elser1, U.L. Andersen1,2, R. Filip1,3, P. Marek4, and G. Leuchs1 Institut für Optik, Information und Photonik, Max-Planck Forschungsgruppe, Universität Erlangen-Nürnberg, Günther-Scharowsky-Straße 1, 91058, Erlangen, Germany Department of Physics, The Technical University of Denmark, 2800 Kongens Lyngby, Denmark Department of Optics, Palacký University, 17. listopadu 50, 772 07 Olomouc, Czech Republic School of Mathematics and Physics, Queens University, Belfast BT7 1NN, United Kingdom (Dated: August 23, 2021) We propose and experimentally demonstrate non-destructive and noiseless removal (filtering) of vacuum states from an arbitrary set of coherent states of continuous variable systems. Errors i.e. vacuum states in the quantum information are diagnosed through a weak measurement, and on that basis, probabilistically filtered out. We consider three different filters based on on/off detection phase stabilized and phase randomized homodyne detection. We find that on/off detection, optimal in the ideal theoretical setting, is superior to the homodyne strategy in a practical setting. PACS numbers: 03.67.-a, 03.67.Hk I. INTRODUCTION Ultra-low noise quantum channels transmitting dis- crete or Continuous-Variable (CV) quantum informa- tion are prerequisite for the successful execution of many quantum information protocols. For example, the secu- rity and the secret key rate of quantum key distribution critically depend on the amount of excess noise added to the quantum state during transmission [1, 2]. All realistic quantum channels are afflicted by such noise: In fiber channels, for example, light scattering by ther- mal phonons causes Gaussian phase noise. On the other hand, noise sources important in atmospheric transmis- sion, such as time jitter and beam pointing noise [3], show a characteristic non-Gaussian behavior. In order to retain security, the errors imposed by the noisy channels must be corrected. Various methods have been developed to combat noise in CV quantum com- munication, examples being entanglement distillation [4] and quantum error correction coding [5], which are rely- ing on highly non-classical resources and complex pro- cessing. An alternative is quantum filtering which is a protocol that probabilistically rejects erroneous quan- tum states through detection. The simplest approach is a classical measure-prepare strategy based on optimal state discrimination using the Neyman-Pearson criterion [6] followed by state recreation. Helstrom found that by using a tailored detection process, it is possible to identify a pure target state in a noisy mixture [7] (see also [8, 9, 10]). Takeoka et al. generalized this strategy and named it unambiguous quantum state filtering since it unambiguously filters out a specific signal from the noise [11]. However, only a single a priori known state is resurrected, which is done destructively and therefore not suitable for quantum communication. In this paper, we propose and experimentally realize ∗Electronic address: cwittmann@optik.uni-erlangen.de a quantum state filter protocol specially tailored to non- Gaussian noise as in atmospheric transmission. The pro- tocol filters a coherent state alphabet non-destructively and noiselessly, i.e. the quantum states are not com- pletely destructed and no excess noise is added by our filter. Our protocol is based on a weak measurement of the corrupted signal followed by a post selection of the remaining part of the signal. We investigate two different weak measurement strategies, namely homodyne detec- tion and on/off detection and compare their efficiencies in filtering out noise. We find that optimum filtering is obtained by the use of an ideal on/off detector. The scheme presented in this paper provides the first imple- mentation of a CV error detection protocol enabled by a photon counting detector. An exemplary application of such a filter is shown in Fig. 1. Suppose a signal is conveyed through two differ- ent quantum channels each possessing different kinds of noises (e.g. a free space channel and a fiber channel). If the first channel is inflicted by the non-Gaussian on/off noise and the following channel by Gaussian noise, the on/off noise might be completely masked by the Gaus- sian noise and cannot easily be filtered out at the receiv- ing station. In order to circumvent a mixing of the two noise sources, the filtration station could be placed be- tween the two channels thus removing the on/off noise before the signal enters the Gaussian noise channel. The filtration protocol can be also used to improve the security of a quantum key distribution scheme based on a coherent state alphabet and heterodyne detection. This is proven at the end of the paper. II. DESCRIPTION OF THE PROTOCOL Let us consider the protocol in detail. Information is encoded into quantum states taken from a coherent state alphabet with a possibly unknown probability distribu- tion. The quantum state is subsequently sent through the quantum channel where it is subject to time jitter http://arxiv.org/abs/0704.1918v3 mailto:cwittmann@optik.uni-erlangen.de on/off noise excess noise X X X P P P classical channel freespace channel fiber channel FIG. 1: Application of the quantum filter device. The filter F is placed between two quantum channels connect- ing sender S and receiver R. We assume, that the channels have non-Gaussian on/off (first part) and Gaussian proper- ties (last part). The on/off behavior of the first channel will be masked by excess noise in the second channel (e.g. |α〉〈α| → |0〉〈0| → ρth). However, a quantum filter in the in- termediate station can sense the channel break and reject the noisy state by sending information over a classical channel to or beam positioning noise. Such non-Gaussian noise oc- curs when the detection time is longer than the signal but shorter than the jitter time or when the aperture of the receiver is larger than the beam but much smaller than the beam pointing noise. This noise can be approx- imated by a mixture of the sent coherent state |α〉 and the vacuum state: ρ(α) = p|α〉〈α|+ (1 − p)|0〉〈0|, (1) where p is an unknown probability for perfect transmis- sion. The task is now to find a protocol that unambigu- ously filters out the vacuum state, while only attenuating the coherent state, e.g. |α〉 → |γα〉, γ < 1. To accomplish a state independent weak measurement adding no excess noise the signal system must be coupled unitarily and phase insensitively to a meter system in which the actual measurement takes place. Due to these requirements, the coupling can be enabled by a beam splitter with the meter system being in the vacuum state before interaction [12]. The signal-meter coupling can therefore simply be described by the transformation ρ(α) ⊗ ρ(0) → ρ( 1−Rα)⊗ ρ( Rα), (2) where R is the reflectivity of the beam splitter. After this interaction, the presence or absence of the vacuum contribution is correlated in the two systems. Thus by detecting the vacuum state in the meter system, filtering of the vacuum in the signal system can be performed by post selecting on the correlated state. The strategy is illustrated in Fig. 1. The next step is thus to find the measurement strategy that optimally and unambiguously detects the vacuum contribution. Let us assume that we use the measurement operators Π⊥ and Π to discriminate the vacuum state and the un- known signal state. We seek a strategy that maximizes the probability 〈 Rα|Π| Rα〉 of measuring | Rα〉 un- FIG. 2: Schematic illustration of a coherent state quantum filter for the non-Gaussian channel: (a) Filter device with ver- ification measurement; (b left) Filter using APD as a detec- tor, (b right) using homodyne detection with a local oscillator (LO). der the condition that the vacuum state is never de- tected incorrectly, that is E = 〈0|Π|0〉 = 0. Such de- cision problem was first encountered by Neyman and Pearson [6] and was further elaborated upon by Hel- strom [7] and Holevo [13]. They found that the maxi- mum probability of detecting the signal correctly (also called the acceptance probability) with no error detec- tions (E = 0) is given by P ( Rα) = 1 − exp(−R|α|2) (Note that E = P (0)). We readily find that measurement operators satisfying these conditions are Π⊥ = |0〉〈0| and Π = 1 − |0〉〈0| for rejecting and accepting the state, re- spectively. Therefore, using these measurement opera- tors the signal states can be unambiguously detected in the meter system and thus perfectly filtered out in the signal system. We stress that since this optimized mea- surement is independent of the signal amplitude and the reflection coefficient, it is the optimal strategy for every coherent state. The physical implementation of these measurement op- erators is known to be an ideal avalanche photodiode (APD) operating in the break down voltage mode. Prac- tical APDs are, however, lossy and possess dark counts which results in a reduced success probability and gives rise to errors, that is, E > 0. Therefore, in addition to the APD, we investigate in the following the filtering performance using a homodyne detector for the decision problem; quadrature values larger than a certain a pri- ori specified threshold value are assumed to stem from the unknown signal state, smaller values from the vac- uum state. Note that a similar strategy was proposed in Ref. [14] and experimentally realized in Ref. [15, 16] to purify non-classical resources. We also note that the incorporation of an APD in a CV system has been im- plemented in previous experiments on state preparation and estimation [17, 18]. In order to quantify the performance of the filtering protocol using different detection methods, we introduce −3 −2 −1 0 1 2 3 X quadrature measurement (shot noise units) vacuum noisy signal filtered signal APD, η=1 HDS, η=1 FIG. 3: Marginal distribution for the perturbed state (p = 0.02) (circles), the vacuum state (crosses) and the filtered state using an APD filter (triangles). The solid and the dotted dashed line correspond to the theoretical performance of a filter with APD (ηAPD = 1) and with homodyne detector (ηHDS = 1) respectively. The mean photon number in the filter is R|α|2 = 1.65 and the error probabilities are identical EAPD = EHDS = 5.3 · 10 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Mean photon number(R|α|2) FIG. 4: Acceptance probability for mean photon number R|α|2 impinging on the filter detector. The triangles, circles and squares show experimental data for APD and homodyne detection with and without stabilized LO, respectively. The solid lines are theoretical predictions for detectors with unit quantum efficiency. EAPD = EHDS = EHDR = 5.3 · 10 two appropriate functions: the sensitivity S and the gain G. The sensitivity quantifies the filtering efficiency near the vacuum state and we define it as d|α|2P ( Rα)|α=0. (3) Since the probablity P must be minimal when α = 0, the sensitivity S is a measure for how quickly the probability increases around α = 0. For the ideal filter we easily find S = R, thus we will be using S/R as the figure of merit. The other parameter that we will use to quantify the per- formance of the filter is the gain G = p′/p where p′ is the probability for the coherent state to occur in the mix- ture after filtering: ρ′ = p′| Tα| + (1 − p′)|0〉〈0|. The success probability for positive filter outputs is PS = Rα)+(1−p)P (0) and the gain can thus be written 1− (1 − p) E . (4) Note that the sensitivity S depends solely on the filter implementation. Thus, it is independent of the chan- nel. In contrast, the gain is a signal-, channel- and filter- dependent parameter, and therefore describes the joint action of channel and filter. III. EXPERIMENTAL SETUP AND RESULTS We proceed with the experimental demonstration of quantum filtering using an APD-based filter. The setup is shown in Fig. 2(a). The source is a diode laser emitting light at 810nm, characterized with a coherence time of 1µs and measured to be shot noise limited in the detected bandwidth. The statistical mixture of the coherent sig- nal states and vacuum states is prepared in a computer controlled electro-optical modulator (EOM). The EOM therefore generates the signal and simulates the noisy channel. To obtain a small excitation of the coherent state, the beam is heavily attenuated after modulation. We set the coherent state probability to p = 2%, thus the probability for vacuum to occur being 1 − p = 98%. The signal duration is defined within 800ns time win- dows, while the rate of the signal preparation is set to 100kHz. We investigate the performance of the APD-based fil- ter by characterizing the state with homodyne detection before and after filtering. First we demonstrate the prin- ciples of the protocol using the APD as a filter and the homodyne detector for characterization. The probability distribution of the mixed input state is shown in Fig. 3 by circles and for comparison the pure vacuum state is shown by crosses. Next, the mixed state is passing the fil- ter beam splitter and the homodyne quadrature data are selected based on the measurement outcomes of the APD detector. The resulting probability distribution is shown by the dotted curve and should be compared with the ideally filtered signal (solid line) and two other curves: the one expected using homodyne detection with equal error probability and one using unit quantum efficiency as filter (dotted dashed line). In the following, we fully characterize the quantum fil- ters. The mixed state is split on a 50/50 beam splitter and subsequently directed to two different detector units: a fiber coupled APD (Perkin-Elmer SPCM CD3017) and a homodyne detector. We therefore simultaneously mea- sure the acceptance probabilities for the two different de- tection schemes; here each detector represented a filter. increasing B increasing R| |α FIG. 5: In all figures the triangles are experimental data points for a filter using APD, the circles and squares show data for filters using homodyne detection with and without stabilized LO respectively; the solid lines are theoretical pre- dictions for filters using unit quantum efficiency detectors, η = 1 (APD: S/R = (1 − pd) 2 for η = 1). (a) Sensitivity as a function of error probability. The dashed line should guide the eye to the error rate where the detectors are compared EAPD = EHDS = EHDR = 5.3 ·10 −3. (b) Gain G as a function of mean photon number R|α|2 impinging on the filter detec- tor. Signal probability fixed to p = 2%. (c) Parametric plot of Gain G and success probability PS for R|α| 2 ∈ [0, 1.65]. The performance decreases from APD to HDS and HDR. The er- ror bars show the statistical errors (three-sigma error bars) and seem to be much smaller than the experimental errors. Let us first discuss the APD-based filter (see Fig. 2(b left)). A gate option is used to precisely determine the detection time. The quantum efficiency is estimated to be ηAPD = 63 ± 3%, while the dark count rate is 180cts/s. Due to these imperfections, the expected ac- ceptance probability is PAPD(β) = 1− (1− pd) exp −ηAPD(1− pd)|β|2 where β = Rα and pd is the dark count probability. The expected error probability is EAPD = PAPD(0) = pd. We measure the acceptance probability by comparing the actual decision (based on the filter measurement out- come) with the a priori known preparation of the state. The results are presented in Fig. 4 as a function of R|α|2. Note that R should be tailored to the actual amplitude α to optimize the performance. The error probability is found to be E = 5.3 · 10−3, which is limited not by the dark count probability 1.4 ·10−4 but by the imperfections in preparing the vacuum state. Next we discuss the filter based on homodyne detec- tion (see Fig. 2(b right)). The detector’s bandwidth is 10MHz. Using a local oscillator (LO) power of about 5mW, the shot noise to electronic noise ratio is 18dB. The detection efficiency, including the mode matching efficiency and the quantum efficiency of the photodiodes is ηHD = 84 ± 3%. We investigate two different kinds of homodyne detectors: one with a phase stabilized lo- cal oscillator (HDS) and one with a phase randomized local oscillator (HDR). The latter scheme should be used when the input alphabet of coherent states is rotationally symmetric in phase space whereas the former scheme is superior if e.g. a binary phase encoding is used where the absolute direction of the displacement is known a priori. The hypothesis whether a signal or a vacuum state was measured is based on the absolute value of the measured quadrature; if it is above a certain threshold value, denoted B, we estimate the state to be |α〉, if not |0〉. Knowing that the signal is encoded into |α〉, the ex- pected acceptance probability for a phase stabilized local oscillator is PHDS(β) = Erfc[ 2(B + a)] + Erfc[ 2(B − a)] where a = ηHDβ, while for the phase randomized local oscillator PHDR(β) = 2(B − a cos θ) dθ. (7) The error is identical for the two approaches and given by EHDR = EHDS = Erfc[ A classical signal is appended to the pulse trains to estimate the phase difference of signal and LO at a given time. Measurement data, acquired with random- ized phase of the LO, is subsequently directly used to evaluate the performance of the scheme based on ran- dom LO. However, due to the appended classical signal the relative phase is known and we selected the data as- sociated with a phase difference of zero corresponding to the phase stabilized case. Using these data and the above hypothesis, we find the acceptance probability for vari- ous excitations R|α|2 and various threshold values B. In Fig. 4 we plot the acceptance probability as a function of R|α|2 with the threshold value set such that the error probability matches the one of the APD. It is clear from the plot that the APD performs better than the homo- dyne detector despite the much higher quantum efficiency of the latter one. The sensitivity of the two filters is obtained by fitting curves to the measured acceptance probabilities corre- sponding to various thresholds (for the homodyne case) and subsequently using equation (3). The results are plotted in Fig. 5(a), where the triangle represents the APD-based filter whereas the circles and squares are as- sociated with the phase randomized and phase locked LO, respectively. For post selection thresholds B close to the coherent state variance the sensitivity of homodyne detectors is maximal. As evident from the plot, for iden- tical error probability the sensitivity of the APD-based filter is much larger than that of the homodyne-based filter. From the measurements we also calculate the gain for different mean photon numbers and different success probabilities as shown in Fig. 5(b) and 5(c), respectively. The former figure clearly shows the superior performance of the APD filter compared with the homodyne filters. From Fig. 5(c) we clearly see that the behavior of the gain as a function of the success probability follows the same curve for the three filters if the error rates are tailored to be the same (by adjusting the error thresholds appropri- ately). This is also what is expected from Eq. (4). IV. FILTERING IN A QUANTUM KEY DISTRIBUTION SCHEME In the final part, we consider the filtering action in a CV quantum key distribution scheme using Gaussian modulated coherent states [1]. To estimate a lower bound for secure transmission we use the recent work of Garcia- Patron and Cerf [19] (see also ref [20]), showing that the lower bound can be directly computed from the covari- ance matrix of the joint state between the sender of infor- mation, Alice, and the receiver, Bob. The lower bound (for reverse reconciliation) is thus given by Klower = I ab − χGbE (8) where χGbE is the maximum information between Bob and the eavesdropper, Eve, corresponding to the Holevo bound and IGab is the Gaussian mutual information be- tween Alice and Bob. These quantities are computed solely from the covariance matrix of the joint state of Alice and Bob. The joint state is found by using the equivalence be- tween the coherent state scheme and an entanglement- based protocol. In an ideal entanglement based proto- col, Alice generates Gaussian two-mode squeezed states and measures one mode using heterodyne detection. The other mode is then being prepared in a coherent state with a Gaussian distributed displacement. A two-mode squeezing variance of V results in a displacement variance of σ = (V + 1/V )/2− 1. The prepared coherent state is sent to Bob through the erasure channel and produces the following density matrix ρAB = p (|V 〉〈V |)AB + (1− p)ρA ⊗ (|0〉〈0|)B , (9) where |V 〉AB represents a two-mode squeezed state and ρA is a thermal state having a variance (V +1/V )/2. The lower bound for secure communication is found numeri- cally by solvingKlower < 0 using the covariance matrix of the joint state ρAB and Eqn. (8). We find security cannot be guaranteed if p < 0.87 (corresponding to Klower < 0). Klower is maximized over the variance V . Let us now consider the security when the filtering pro- tocol is implemented. Assuming first the APD to be ideal, the state after filtering is ρfilter = ΠUBSρABTU Tr(ΠUBSρABTU where UBS is the unitary beam splitter operation, ρABT = ρAB ⊗ |0〉〈0| and Π is the measurement oper- ator of the tap T. The structure of the covariance matrix for this state is given by CV ′AB = (CVAB −P 0CV 0AB), where CVAB is the covariance matrix of the state right after the beam splitter after tracing out the tap mode, ρBS = TrT (UBSρABTU ), and CV 0AB is the covari- ance matrix of the state if the measurement outcome of the filter measurement is associated with Π⊥ = 1 − Π, ρ⊥ = Π⊥UBSρABTU Π⊥/Tr(Π⊥UBSρABTU ). P 0 is the probability for getting a measurement result associ- ated with Π⊥ and PS = 1 − P 0 is the filtering success rate. Using the covariance matrix of the filtered state, CV ′AB , we again compute the lower bound numerically, and find that the probability p for which secure commu- nication can take place is now just required to be larger than zero. This can be also investigated analytically in the limit of weak two-mode squeezing (V ≈ 1), corre- sponding to a small Gaussian alphabet. In this case, the lower bound on the secure key rate is approximately Klower ≈ pPS T (V − 1)2, (11) which is positive for any T > 0 and V 6= 1. We have thus shown that the usages of an ideal filter reestablishes the security of the quantum key distribution system in- dependent on the amount of vacuum noise. We now consider the realistic case where the APD is nonideal. Using an APD with nonzero quantum effi- ciency is not a hindrance for obtaining secure communi- cation, but it will result in a lower success rate. A limiting factor on the security, however, is the presence of dark counts which limits the minimum noise probability pmin for which security can be proven. For a given dark count rate, the transmission T of the filtering beam splitter and the modulation variance σ can be numerically optimized to maximize Klower for which security can be guaran- teed in the protocol with respect to the noisy channel. Using the experimental parameters of the APD used in our experiment (η = 0.63, pd = 0.005) we find that secure communication can be guaranteed if p > pmin = 0.222. By reducing the dark counts to pd = 5 × 10−4 the mini- mum probability is reduced to pmin = 0.028 and for even lower pd = 5× 10−5, the threshold is only pmin = 0.003. V. CONCLUSION In summary, we have investigated a filtering protocol that successfully filters out vacuum states from a set of coherent states of a continuous variable system. A weak measurement scheme consisting of a beam splitter and an optimized measurement was employed to probabilis- tically filter out the unwanted vacuum states. Different measurement strategies based on homodyne and on/off detection were investigated and compared. We therefore provided the first direct comparison between an APD- based and a homodyne-based protocol, and found that the ideal on/off detection is optimal and that the prac- tical (that is non-ideal) on/off detector is superior to homodyne detection despite the much higher quantum efficiency of the latter one. The protocol will be advan- tageous in continuous variable quantum communication wherever beam positioning and time jitter noise are the main obstacles for faithful and secure transmission. Acknowledgments This work was supported by the EU project COV- AQIAL (project no. FP6-511004), the EU-IST network SECOQC, COMPAS project no. 212008, the project 202/07/J040 of the GACR and project MSM6198959213 of Czech Ministry of Education. R.F. acknowledges sup- port from the Alexander von Humboldt foundation. P.M. acknowledges support from the European Social Fund. [1] F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, Nature 421, 238 (2003). [2] M. Heid and N. Lütkenhaus, Physical Review A 73, 052316 (2006). [3] N. Gisin, G. G. Ribordy, W. Tittel, and H. Zbinden, Reviews of Modern Physics 74, 145 (2002). [4] L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Physical Review Letters 84, 4002 (2000). [5] S. L. Braunstein, Nature 394, 47 (1998). [6] J. Neyman and E. Pearson, Philosophical Transactions of the Royal Society of London Series a-Containing Papers of a Mathematical or Physical Character 231, 289 (1933). [7] C. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976). [8] M. G. A. Paris, Physics Letters A 225, 23 (1997). [9] G. M. D’Ariano, M. G. A. Paris, and P. Perinotti, Phys- ical Review A 65, 062106 (2002). [10] M. Takeoka, M. Ban, and M. Sasaki, Physics Letters A 313, 16 (2003). [11] M. Takeoka, M. Ban, and M. Sasaki, Physical Review A 68, 012307 (2003). [12] U. Leonhardt, Measuring the Quantum States of Light (Cambridge University Press, 1997), p. 80. [13] A. Holevo, Probabilistic and Statistical Aspects of Quan- tum Theory (North-Holland, Amsterdam, 1982). [14] S. Suzuki, M. Takeoka, M. Sasaki, U. L. Andersen, and F. Kannari, Physical Review A 73, 042304 (2006). [15] J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, Physical Review Letters 96, 253601 (2006). [16] A. Franzen, B. Hage, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, Physical Review Letters 97, 150505 (2006). [17] J. Wenger, R. Tualle-Brouri, and P. Grangier, Physical Review Letters 92, 153601 (2004). [18] J. Wenger, J. Fiurášek, R. Tualle-Brouri, N. J. Cerf, and P. Grangier, Physical Review A 70, 053812 (2004). [19] R. Garćıa-Patrón and N. J. Cerf, Physical Review Letters 97, 190503 (2006). [20] M. Navascués, F. Grosshans, and A. Acin, Physical Re- view Letters 97, 190502 (2006).

Introduction Extrasolar planets are now routinely discovered orbiting solar-type stars by radial velocimetry, but the discovery of tran- siting planets by photometric surveys is just beginning. Although still marginal, the late success of transit surveys has given an additional impulse to exoplanetology with the possibility to es- timate the radius, density and hence composition of extrasolar planets. Quantitatively, we know to date 206 extrasolar planets with masses below 13 MJup (e.g. Udry et al. 2007; Butler et al. 2006). Among those, a list of 14 currently known transiting planets is presented in table 1. These planets have been discovered by ra- dial velocimetry followed by photometry for 3 of them, and by photometric surveys for the remaining 11. When considering the score of projects devoted to the detec- tion of planets by transit photometry, the present harvest appears meager. The discrepancy between predictions (e.g. Horne 2001) and reality has been attributed to various factors such as: im- perfect duty cycle, a reduced number of stars for which tran- siting planets are detectable (Gould et al. 2006) and the pres- ence of correlated noises that can greatly limit the detectability of small planetary transits (Pont et al. 2006b). Several generic studies have been conducted to understand the yield of different transit surveys. Pepper & Gaudi (2005) studied the optimization of transit searches as a function of the observational setup, the site properties and the planet properties. Gillon et al. (2005) an- alyzed and compared deep field surveys, considering individual stellar ranges and observation windows, but did not include the effects of stellar crowding nor time-correlated noises. Gould et al. (2006) studied the yield of OGLE survey (Udalski et al. 2002), the most successful so far in term of num- ber of transiting planets discovered, with a model populating the line of sight with stars drawn from the Hipparcos Catalogue. They estimated with that model the proportion of stars with sen- sitivity to close-in giant planets to derive from OGLE results the frequency of planets as a function of their period. They find that the yield of the OGLE survey is globally consistent with the detections by radial velocimetry and with planet radii dis- tributed between 1 and 1.25 jovian radii. The aim of the present work is to further test these data sets and the underlying physical model by a forward calculation of transit events with realistic stellar and planetary populations. In particular, we include up- to-date models of the evolution and structure of Pegasids (close- in extrasolar planets) based on models reproducing the observa- tional constraints from known transiting planets (Guillot 2005; Guillot et al. 2006). As a consequence, we should be able to de- termine whether the presently known population of transiting planets represent the “tip of the iceberg”, i.e. that many more small, dense extrasolar giant planets exist and await discovery by the transit method, or whether it is relatively representative of the global population. We first describe the model that is used to simulate transit surveys in general. In Section 3, we describe more particularly the OGLE surveys and the hypothesis chosen for their mod- elling. We then discuss the results of the simulation. A summary of the main conclusions and predictions for future transit surveys are provided in Section 5. 1. Simulating transit surveys 1.1. General remarks The search for planets in transit in front of their star naturally arised with the discovery that a non-negligeable fraction of plan- ets orbit very close to their stars. If orbital planes are randomly oriented, the probability that a planet will transit in front of its star at each orbital revolution is: Ptransit ≃ R⋆/aplanet, (1) where R⋆ is the stellar radius, and aplanet the planet’s orbital semi- major axis. For systems such as 51 Peg b, this probability is close to 10%. Because the probability for a solar-type star to possess such a Pegasid (i.e. a 51 Peg b-like planet, planets semi-major axis up to 0.1 AU) is about 0.5% (e.g. Marcy et al. 2005), 1 in 2000 solar-type star should possess a transiting Pegasid. Using current results from radial velocity surveys and integrating over all periods, we estimate that about 1 in 1100 solar-type stars possesses a transiting giant planet. Of course, depending on the magnitudes and field considered, giant stars may severly out- number the dwarfs, so that in a real field, only one in, say, 3000 stars may harbor a transiting giant planet. Because of the dependence on a, and period distribution, most of the transit events concerning giant planets occur for or- bital periods between 1 and 5 days. The transits typically last for a couple of hours, as this quantity is weakly dependant on the orbital period P: τtransit = 1.82 1 day )1/3 ( )−1/3 (R′⋆ hours, (2) where R′⋆ is the length of the cord traced on the stellar disk by the planet’s trajectory. (more precisely: R′⋆ = R⋆ cos b + Rplanet, where b is the impact parameter of the transit). The depth of the transits themselves is directly given by the ratio of the planetary to the stellar disk surfaces: Rtransit ≃ (Rplanet/R⋆) . (3) This value is of order 1% for a Jupiter-size planet orbiting a Sun-like star. For an F-type star with radius ∼ 1.2 R⊙, the ratio decreases to 0.7%. Furthermore, transiting giant planets discov- ered so far have radii between 0.72 and 1.44 RJup (see table 1). Allowing for stellar radii to vary between 0.8 and 1.3 R⊙ (a typ- ical range, in magnitude limited surveys), this implies that we should expect Rtransit to vary between 0.3% and 3%, for giant planets only. The lower limit is in reality even smaller because for detection purposes we have to account for the fact that plan- ets also orbit stars that are in multiple systems (like HAT-P-1), and hence a dilution factor may apply. Although grazing transits are ignored in this simple analysis, they are included afterwards in our simulations. This altogether implies that in order to detect transiting gi- ant planets, many thousands of dwarf stars have to be monitored over periods of weeks for a photometric precision reaching be- low a fraction of a percent on an equivalent integration time of about one hour. This is typically done by following a relatively dense stellar field over a long time with a stable telescope, and using a camera equiped with a good CCD camera. 1.2. Principle of the simulations On paper, the simulation of the forward problem is simple: one has to generate a complete stellar field, or obtain it from observa- tions, put it on a discrete grid (the CCD), include on probabilistic arguments the planetary companions, calculate lightcurves in- cluding the various sources of noise, and determine which events are detectable. This is the principle of CoRoTlux, a code we first developed to predict the transit yield of CoRoT space telescope http://arxiv.org/abs/0704.1919v1 Table 1. Known transiting planets by 2006⋆ # Name Mplanet Rplanet Period a M⋆ R⋆ Teff Metallicity [MJup] [RJup] [day] [AU] [M⊙] [R⊙] [K] [Fe/H] OGLE planets 6 OGLE-TR-10 0.63±0.14 1.26+0.07 −0.07 3.10129 0.04162 1.18 ±0.04 1.16±0.06 6075±86 0.28±0.10 2 OGLE-TR-56 1.17±0.04 1.32+0.06 −0.06 1.211909 0.0225 1.17±0.04 1.32±0.06 6119±62 0.19±0.07 5 OGLE-TR-111 0.52±0.13 1.067+0.054 −0.054 4.0144479 0.047 0.81±0.02 0.831±0.031 5044±83 0.19±0.07 3 OGLE-TR-113 1.35±0.22 1.09+0.03 −0.03 1.4324757 0.0229 0.78±0.02 0.77±0.02 4804±106 0.15±0.10 4 OGLE-TR-132 1.14±0.12 1.18+0.07 −0.07 1.689868 0.0299 1.26±0.03 1.34±0.08 6210±59 0.37±0.07 Other transit survey planets 7 TrES-1 0.76±0.05 1.081+0.029 −0.029 3.0300737 0.0393 0.89±0.035 0.811±0.020 5250±75 -0.02±0.06 11 TrES-2 1.28±0.07 1.24+0.09 −0.06 2.47063 0.0367 1.08±0.08 1.00±0.05 5960±100 0.15±0.03 10 XO-1 0.90±0.07 1.184+0.028 −0.018 3.941634 0.0488 1.0±0.03 0.928±0.033 5750±13 0.015±0.03 12 HAT-P-1 0.53±0.04 1.36+0.011 −0.09 4.46529 0.0551 1.12±0.09 1.15±0.09 5975±45 0.13±0.02 13 WASP-1 0.867±0.073 1.443+0.039 −0.039 2.519961 0.0382 1.15±0.09 1.453±0.032 6200±200 14 WASP-2 0.88±0.07 1.038+0.05 −0.05 2.152226 0.0307 0.79±0.08 0.813±0.032 5200±200 Transiting planets discovered with Radial velocities 9 HD189733 1.15±0.04 1.154+0.032 −0.032 2.218573 0.0313 0.82±0.03 0.758±0.016 5050±50 -0.03±0.04 8 HD149026 0.330±0.02 0.726+0.064 −0.064 2.87598 0.042 1.3±0.1 1.45±0.1 6147±50 0.36±0.05 1 HD209458 0.657±0.006 1.320+0.025 −0.025 3.52474859 0.047 1.09±0.09 1.148±0.002 6117±26 0±0.02 MJup = 1.8986112 × 10 30 g is the mass of Jupiter. RJup = 71, 492 km is Jupiter’s equatorial radius. OGLE-TR-10: Bouchy et al. (2005); Udalski et al. (2002); Konacki et al. (2005); Santos et al. (2006); Pont et al. (2006a) OGLE-TR-56: Konacki et al. (2003); Udalski et al. (2002); Torres et al. (2003) Bouchy et al. (2005); Santos et al. (2006); Pont et al. (2006a) OGLE-TR-111: Pont et al. (2004); Santos et al. (2006); Udalski et al. (2002); Winn et al. (2007); Bouchy et al. (2005) OGLE-TR-113: Bouchy et al. (2004); Udalski et al. (2002); Konacki et al. (2004); Gillon et al. (2006) OGLE-TR-132: Bouchy et al. (2004); Udalski (2003); Moutou et al. (2004); Magain et al. (2007) TRES-1: Alonso et al. (2004); Laughlin et al. (2005); Winn et al. (2007) TRES-2: O’Donovan et al. (2006) XO-1: McCullough et al. (2006); Holman et al. (2006); Wilson et al. (2006) HAT-P-1: Bakos et al. (2006) WASP-1: Collier Cameron et al. (2006); Shporer et al. (2007); Charbonneau et al. (2006) WASP-2: Collier Cameron et al. (2006); Charbonneau et al. (2006) HD-189733: Bouchy et al. (2005); Bakos et al. (2006) HD-149026: Sato et al. (2005); Charbonneau et al. (2006) HD209458: Brown et al. (2001); Cody & Sasselov (2002); Wittenmyer et al. (2005); Winn et al. (2005); Knutson et al. (2007) # is the label of planets in figures ; they are ranked in the order of their discovery. (Baglin et al. 2002) and quantify the need for follow-up obser- vations, which is here applied to the case of OGLE. The interesting point of such a forward simulation is the pos- sibility to include relatively easily fine details such as the fact that planets are found more frequently around metal-rich stars, or, on the basis of planetary evolution models, the fact that young planets orbiting close to bright stars will be larger than old plan- ets orbiting smaller stars at larger orbital distances. This requires however that a relatively large number of physically relevant parameters (for example, the mass, size, metallicity, age of the stars) be properly defined. We further detail the assumptions that we made for these simulations by describing how we generate the stellar and plane- tary populations, and how we attempt to include realistic sources of noise. 1.3. The stellar population 1.3.1. Main targets and background stars Stellar fields differ enormously in terms of densities and stellar populations. It is therefore most important to properly account for this in order to simulate any given transit survey. It would be very appealing to use direct observations as much as possible to closely match the observed target fields. But as we will see hereafter, many different characteristics of the stars (stellar metallicity, age and subtype ...) are required, and these are difficult to obtain with generic observations. We therefore adopt the following procedure: – The observed stellar densities are obtained from stellar counts by magnitude, on the real stellar fields (see § 2.1) – The characteristics of the stars are obtained following a Monte-Carlo method using the output of the Besançon model of the galaxy (Robin et al. 2003) obtained for the proper lo- cation of the survey. – Where stellar counts are not available, or uncomplete (i.e. for faint stars), we use both stellar counts and characteristics from the Besançon model. Specifically, we keep track of the following parameters ob- tained directtly from the Besançon model: – The mass of each star, used to compute orbital parameters of the transiting object; – The apparent magnitude of the star in the observed spectral range (the I filter in the case of the OGLE survey); – The V magnitude of the star, important to qualify the con- firmability of a transit event with radial velocimetry; – The surface temperature of the star – The luminosity of the star, calculated from its absolute mag- nitude; – The radius of the star, calculated from total luminosity and effective temperature. Stellar radius [RSun] Stellar mass [MSun] Effective temperature [K] Fig. 1. From top to bottom: Distribution functions for the radii, masses and effective temperatures for our fiducial stellar popu- lation corresponding to the simulated OGLE Carina field. The black line represents the ensemble of stars in the field. The filled red region is a subset for dwarf stars with stellar type F4 and later, as these are the only stars for which a transiting planet has a reasonnable chance of being detected by present-day transit surveys. The mass, and effective temperature of the stars are dis- tributed linearly around values given by the Besançon model (at ±20%). Figure 1 shows a simulated distribution of stars for the OGLE Carina field. The ensemble of dwarf stars with types F4 and later are highlighted as these represent targets for which planetary transit events are detectable, and, within observational limits, confirmable by radial velocimetry. The metallicity distribution is obtained from the model of Nordström et al. (2004), which is based on the Geneva- Copenhagen survey of the Solar neighbourhood. These authors find that the distribution of the metallicities [Fe/H] is well ap- proximated by a Gaussian function with a mean of −0.14 and a dispersion of 0.19 dex. We use this gaussian distribution and choose to ignore possible dependencies between stellar parame- ters (e.g. masses, ages...) and the metallicities. (The link between stellar type and metallicity appears to be negligible for F4 and later types anyway (F. Thévenin, pers. communication 2007)). 1.3.2. Binary and triple systems Multiple stellar systems are important especially because of the possibility that stellar eclispes mimic planetary transits (Brown 2003). However, we choose to defer this problem to a later ar- ticle. Multiple systems are taken into account anyway because they can yield a dilution of the planetary transit events that makes them more difficult to detect. Planets may be present in- differently on the primary, secondary or tertiary components of a stellar system. (However, we find that only planets around the primary targets have a non-negligible chance of being discov- ered by current ground based photometric survey.) Specifically, following Duquennoy & Mayor (1991), we consider that 50% of the stars are binaries and 20% of those are ternaries. Multiple systems are considered as individual stars at the same position on the CCD. We choose to estimate their properties more simply than for the other stars, on the basis of DM91: – We randomly add companions to the initial draw of primary stars, without changing their properties. The total mass and luminosity of each multiple system is thus slightly higher than initially. – The mass ratio (primary/secondary) is defined as a gaussian of median value 0.23 and a full width at half maximum of 0.42. Outside a range of 0.05 and 1, we redraw the mass ratio. – The radius is defined as R⋆ = R⊙(M⋆/M⊙) when M⋆ ≤ M⊙ and R⋆ = R⊙(M⋆/M⊙) 1/2 otherwise. – The luminosity is assumed to be proportional to M2 so that: lsecondary = lprimary(Msecondary/Mprimary) – Other stellar parameters are calculated on the basis of these ones and of those of the primary component (same age, same distance, same metallicity). – Triple components are treated with the same method as bi- naries, and are defined relatively to the primary star. 1.4. The planetary companions With more than 200 planets known to orbit stars other than our Sun, we are beginning to have a rather precise view of at least part of this population. We can expect that biases on the detec- tions are small in the case of massive planets (the mass of Saturn and more) and planets that are relatively close to their star (or- bital distances smaller than ∼ 1 AU). These two conditions hap- pen to match quite exactly the requirement for detectability by transit photometry, with one assumption: that only massive giant planets can have large radii. Although not proven, this assump- tion seems quite reasonnable. Hence we choose to focus this study on this well- characterized population of objects. From the current list of 209 detected exoplanets, we select the ones discovered by radial ve- locities with mass higher than 0.3 Jupiter masses and known host star metallicity. Our list of planets includes 153 objects, to which we may add very-close in planets detected by transit photometry, as discussed below. We are using this list as representative of an unbiased sample of giant planets known from radial velocimetry, even though planetary distribution models may have been made from slighlty different samples. 1.4.1. Planet incidence A first important step is the determination of the probability for a star to harbor a planet. As shown by numerous studies (Gonzalez 1998; Santos et al. 2004; Fischer & Valenti 2005), this probabil- ity depends mostly on the metallicity of the parent star. Figure 2 shows one such probability function, as well as the result in terms of planet counts on a simulated stellar field. In this work, we will use the dependency from Santos et al. (2004) shown in Fig. 2. Several points are to be considered how- ever: 1. This probability relation is only valid for solar-like stars, i.e. F, G, K dwarf stars. Although there are strong indications that it may change for other stars (e.g. M dwarfs), we will assume it to hold independently of stellar properties. This [Fe/H] [Fe/H] Fig. 2. Upper panel: Probability for a solar-type star to possess a giant planet companion as a function of the stellar metallicity (from Santos et al. 2004). Lower panel: Relative normalised dis- tributions of stellar metallicities for stars in the field (black line), and for stars with a giant planet companion (red line). assumption is sufficient because F, G and K dwarf stars form by far the majority of stars with detectable planets in photo- metric surveys. 2. This relation has been calculated independently of the prop- erties of the planetary companion, in particular orbital dis- tance. Because in our case we are strongly biased towards short-period planets, the distribution may be different. This point will be considered in § 3.4. 3. The possibility of multiple planetary systems is not consid- ered. This approach is justified because the probability that several planets belonging to the same system are transiting planets is small for giant planets. 1.4.2. Planetary masses and orbits The masses and orbital characteristics of the planet population are inferred almost entirely from the present radial-velocimetry surveys. This technique yields an accurate determination of the orbital period, and less accurately, of the eccentricity of the orbit. It also yields the value of the mass of the planetary companion times the sine of the orbital inclination from the knowledge of the mass of the parent star. With these values, we can then derive from a random inclination of the orbital planes the planets that are transiting and those that are not as well as the characteristics of their orbit. We test several approaches for the derivations of these quan- tities: – An analytical model: In this approach, we consider inde- pendantly the planet period and its mass. The period of the planet Π follows the model of Brown (2003), the proba- bility density P from a piecewise linear fit to the distribu- tion P(logΠ) = {0.509, 0.165, 0.533} for three period inter- vals bounded by logΠ = {0.43, 0.65, 2.3, 3.5}. The distribu- tion in mass is linear in log from 0.3 to 10 Jupiter masses (Zucker & Mazeh 2001). There is no dependency of these two parameters linked to metallicity. – The radial velocity mass-period “carbon-copy” model: A second approach is to make direct use of the list of planets discovered by radial velocimetry. This is possible because in terms of masses and orbital periods the list is almost unbi- ased for the objects that we consider (massive enough to be above detection thresholds, and with periods much shorter than the lifetimes of the surveys). In this case, we select plan- ets randomly in the RV list, and then allow for a small ran- dom deviation of their mass and period (a uniform deviation from −20% to +20%) in order to avoid too much cluster- ing on the same value. This is particularly important in the case of the period because of the importance of stroboscopic effects in planetary transits (e.g. Pont et al. 2005). – The radial velocity mass-period-metallicity “carbon-copy” model: As a modification to the previous approach, we also consider using the metallicity entry in the RV list, because of correlations between metallicity and orbital period that are otherwise not taken into account (see discussion in sec- tion 3). We proceed slightly differently however than for the mass and orbital period because of the limitations caused by the finite number of planets in the RV list. In this case, we choose to split the list into two low- and high-metallicity lists, and then select the mass and periods in the relevant list. Our fiducial cutoff value is [Fe/H]=−0.07. Figure 3 shows a comparison between observations, the car- bon copy model and the analytical model. It is interesting to no- tice at this point that the carbon copy and analytical models are essentially indistinguishable in these diagrams. The differences with the observations arise only because of our choice to smear the masses and orbital periods when generating our planet pop- ulation. Last but not least, we have to consider the existence of plan- ets that orbit extremely close to their star, with periods shorter than 2 days, as discovered by transit surveys (see table 1). Companions with such short orbital periods have been discov- ered by radial velocimetry in two occasions: HD 41004 b, and Gliese 876 d, with respective masses 18.4 and 0.023 Jupiter masses. These objects are outside the mass range considered for this study, and therefore, there is no giant planets with periods shorter than 2 days in the present radial velocimetry list. In or- der to account for these very close-in planets anyway, we add the planets with periods smaller than 2 days discovered by tran- sit photometry to the list, but with a small tunable probability weight. The fiducial value of this parameter is set so that, on av- erage, the planet list contains one and a half such planet (added to the list of 153 RV planets described in § 1.4). Tests on the effect of this parameter are presented in § 3.3.4. Our fiducial model is the mass-period-metallicity carbon copy model, includes addition of very-close in planets and it is that model which is used in all cases except where specified oth- erwise. Other approaches are also tested depending on the model to highlight particular points. 1.4.3. Physical characteristics and the planetary evolution model Because we are focussing on planets more massive than Saturn, we expect most of them to be made of a significant amount of hydrogen and helium. These giant planets thus undergo a pro- gressive contraction and cooling that depends on four quanti- ties: their age, mass, the amount of flux the planet receives from the central star, and the global amount of heavy elements in the planet (e.g. Guillot 2005). Orbital period [days] Planetary mass [MEarth] Planetary radius [RJup] Fig. 3. From top to bottom, distributions of orbital periods, masses and radii, respectively, of the planets observed by radial velocimetry (black lines), simulated as part of the mass-period “carbon copy” model (red lines), and simulated as part of the analytical model (dotted blue lines) (see text). Models attempting to reproduce the radii of known tran- siting giant planets have however had problems in explain- ing the large radii of some of them (Bodenheimer et al. 2001; Guillot & Showman 2002; Baraffe et al. 2005; Laughlin et al. 2005). Several possibilities have been proposed to explain the discrepancy. We can separate them into two categories: – Mecanisms invoking chance configurations of the plane- tary orbits in the case of these anomalously large plan- ets: this includes the tidal circularization of an eccentric orbit (Bodenheimer et al. 2001), and tidal dissipation for a planet locked in a Cassini spin-orbit resonnance with the star (Winn & Holman 2005). – Effects that would apply to all planets, including problems with the equations of state or opacities, and the dissipation by stellar tides of kinetic energy first generated in the atmo- sphere (Showman & Guillot 2002). The first mecanisms appear to have a low probability of oc- curence (Laughlin et al. 2005; Deming et al. 2005; Levrard et al. 2007). The second possibility therefore seems more likely, but requires in some case the presence of relatively large masses of heavy elements to reproduce the observed radii. A model-dependant estimate of the masses of heavy el- ements present in the currently known transiting Pegasids is shown in Fig. 4. This model relies on the hypothesis that 0.5% of the absorbed stellar flux is used to generate kinetic energy that is subsequently dissipated deep into the planetary inte- rior (Guillot & Showman 2002). As proposed by Guillot et al. (2006), there appears to be a correlation between the amount of heavy elements present in the planet and the metallicity of their parent star. This correlation has to be ascertained, but we choose for our fiducial model to adopt a unique relation between metallicity and mass of heavy elements (treated as a central core in our calcula- tions), corresponding to the dotted line in Fig. 4: MZ = 43.75 × 10 [Fe/H] − 23.7 M⊕. (4) Fig. 4. Mass of heavy elements in transiting Pegasids known by 2006 as a function of the metal content of the parent star rela- tive to the Sun. The mass of heavy elements required to fit the measured radii is calculated on the basis of evolution models including an additional heat source slowing the cooling of the planet. This heat source is assumed equal to 0.5% of the incom- ing stellar heat flux (Showman & Guillot 2002). Horizontal er- ror bars correspond to the 1σ errors on the [Fe/H] determina- tion. Vertical error bars are a consequence of the uncertainties on the measured planetary radii and ages. The metallicity of re- cently discovered planets WASP-1 and WASP-2 (right panel) is not precisely known. The dotted line corresponds to a best fit model. [Adapted from Guillot et al. (2006)]. We limit the range of possible values of MZ to [0, 100M⊕]. Similarly, we adopt a simple boundary condition for our evo- lution calculations: T1bar = 1.25Teq0, (5) where T1bar is the temperature at the 1 bar pressure level and Teq0 is the equilibrium temperature for a zero albedo (see Guillot 2005 for a description), calculated as a function of stellar effec- tive temperature and radius and planet semi-major axis. For simplicity, and because it yields only minor changes on the results, we further choose to neglect the time-dependence in the evolution calculations, and to adopt the equilibrium radius, or the 10 Gyr solution, whichever occurs first. Practically, planetary radii are obtained from interpolations in a table based on three parameters: the planetary mass ranging from 100 to 3000M⊕, the core mass from 0 to 100M⊕ and the equilibrium temperature from 100 to 2000 K. Models were not calculated beyond these values of Teq because of convergence problems. However we allowed for a slight extrapolation of the tables to 2600 K to account for rare extremely hot planets. 1 Similarly, because of convergence problems for planets with small total masses and large core masses, we limited the mass of the core to 75 M⊕ for planets with masses smaller than 180 M⊕. More detailed work is required to better simulate this parame- ter space, including planets less massive than considered in this study. 1 An electronic version of the table is available at www.obs- nice.fr/guillot/pegasids/ Figure 5 shows examples of radii obtained for Teq = 1000 and 2000, K, and core masses of 0 and 100Moplus, respectively, compared to available measurements. eq =2000K eq=1000K no core e, n 100 M⊕ core planets brown dwarfs stars Fig. 5. Theoretical and observed mass-radius relations. The black line is applicable to the evolution of solar composition planets, brown dwarfs and stars, when isolated or nearly isolated (as Jupiter, Saturn, Uranus and Neptune, defined by diamonds and their respective symbols), after 5 Ga of evolution. The dotted line shows the effect of a 15M⊕ core on the mass-radius relation. Orange and yellow curves represent the mass-radius relations for heavily irradiated planets with equilibrium temperatures of 1000 and 2000 K, respectively, and assuming that 0.5% of the incoming stellar luminosity is dissipated at the center (see sec- tion 1.4.3). For each irradiation level, two cases are considered: a solar-composition planet with no core (top curve), and one with a 100M⊕ central core (bottom curve). The transiting extrasolar giant planets for which a mass and a radius was measured are shown with points that are color-coded in function of the planet’s equilibrium temperature. The masses and radii of very low mass stars are also indicated as blue points with error bars. 1.5. Modeling transit events and their detectability We now descibe how this population of stars, planets and their interactions during transits are modelled to reproduce real obser- vations. 1.5.1. PSFs and CCDs Each image of a star is spread by the atmosphere and by the telescope to grow to a specific size and shape when reaching the CCD in the focal plane of the telescope, the so-called point spread function (PSF). The CCD being composed of many dis- crete pixels, these PSFs are then effectively discretized, so that the signal to be analyzed for any given star is composed of many different lightcurves corresponding to the many pixels over the size of its PSF. A further complication arises from the fact that the stellar fields generally chosen by transit surveys are dense, so that many PSFs overlap. Recovering individual stellar light curves from real data is a complex problem. Two popular meth- ods are aperture photometry (Stetson 1987) and image subtrac- tion (Alard & Lupton 1998). (An adaptation of the latter was used to extract the OGLE lightcurves). A refined simulation could include possible spatial and tem- poral variations of the PSFs, and a realistic data reduction pipeline. In our case, we choose to simplify the problem by rely- ing on a posteriori analyses of real light curves to provide us with a global noise budget. We however include background stars be- cause of the important effect of signal dilution. In order to do so, we assume that the PSFs are gaussian with a uniform, constant FWHM. (Non-gaussian PSFs are not diffi- cult to include but we tested in the OGLE case that for a fixed equivalent FWHM, they have a negligible effect on the resulting signal-to-noise ratio of simulated transit events). We consider for each target of the survey the global flux from the main star and the background stars in its neighborhood up to magnitude 22 in the spectral band of observation. The neighborhood zone for background stars is defined as a circle of diameter equal to 4 times the PSF’s FWHM around the photocenter of each target star. Each background star whose photocenter is located in that zone is taken into account for the calculation of the global flux. The global flux is calculated as the sum of the pixels located at less than twice the FWHM of the central star. We thus simulate aperture photometry when image subtrac- tion was used for OGLE (Udalski et al. 2002). The choice of the reduction algorithm indeed affects the sensitivity obtained from real observations. In our simulations, i.e. a relatively idealized case, it would have marginal effects since realistic noises are included a posteriori from the analysis of real lightcurves (see hereafter). 1.5.2. Noise budget and event detectability We choose to separate noise sources into two categories: – ‘White noise’ sources, following gaussian and Poisson laws. The main source of white noise is the photon noise of target stars and their background. The level of white noise for a given target star is obtained from the simulation of the flux of that star and its background in the photometric aperture. – ‘Red noise’, or systematic effects on photometry, that un- dergo temporal correlation. The structure of these systemat- ics in the OGLE photometry have been explored in details by Pont et al. (2006b). These noise sources are both instrumen- tal (jitter and breathing of the CCD, frequency spectrum of stellar field moves on the camera, change of the PSF shapes accross the CCD during the night), and environmental (dif- ferential refraction and extinction, changes of airmass and sky brightness, temperature changes). Rather than trying to simulate instrumental and environmental noise sources accu- rately, which is difficult with the relatively poor knowledge we have of the time spectrum of their combined effects, we use the effective global ‘red noise’ measurements of OGLE- III survey real light curves mentionned in Pont et al. (2006b), which consider the combined effect of these noise sources. Pont et al. (2006b) calculated that, in the presence of a mix- ture of white and red noise (i.e. accounting for photometric sys- tematics), the detection threshold for a transit survey is well de- scribed by a limit on the signal-to-noise ratio defined as: S 2r = k=1 nk 2(σw2/nk + σr2) where Ntr is the number of transits sampled, nk the number of data points in the k-th transit. σw and σr are the standard devia- tion of measurement points of white and red noises, respectively, d is the event depth and n the total number of measurement points during the transit. Specifically, we obtain nk by counting for each transit the number of observation points between the middle of ingress and the middle of egress. Equation 6 makes the disctinction between “white” noise sources that decrease with n1/2, where n is the number of succes- sive measurements, and “red” noise sources that are limited by temporal correlation. Pont et al. (2006b) indeed show that taking the red noise into account makes a large difference on the detec- tion threshold – in general as well as in its dependence to the planet parameters – and that models based on the assumption of white noise can be poor approximations of the actual detection threshold. 2. The OGLE survey: input parameters 2.1. Basic parameters and observational procedure The Optical Gravitational Lensing Experiment (OGLE) has done 6 observation campaigns looking for transiting planets to- wards different fields of view from 2001 (Udalski et al. 2002). It took place at the Las Campanas Observatory, Chile, using the 1.3 m Warsaw telescope and the 8k MOSAIC camera, with a to- tal field of view of 0.34◦2. All observations were made through the I filter. We assume for our PSF simulation an average seeing of 1 arcsec. We analyze in this work the first three OGLE-III observation campaigns dedicated to transit search, as their treatment, anal- ysis and follow-up (with current data processing pipelines) has been completed: – OGLE-III-1 (June 12 to July 28, 2001, described in Udalski et al. (2002); Udalski (2002)). More than 800 im- ages of three fields in the direction of the galactic bulge were collected within 32 nights. The exposure time was 120 s, and each field was observed every 12 min. – OGLE-III-2 (February 17 to May 22, described in Udalski (2003)). More than 1100 images of three fields located in the Carina region of the galactic disk were collected in 76 nights. The exposure time was 180 s, and the temporal resolution was about 15 min. – OGLE-III-3 (February 12 to March 26, described in Udalski et al. (2004)). The photometric data were collected during 39 nights spanning the 43 days of the survey. Three fields of the galactic disk were observed with a time resolu- tion of about 15 min. The exposure time was 180 s. In this article, we will refer to these three observation cam- paigns respectively as ‘Bulge’, ‘Carina’, and ‘Centaurus’ fields. The simulations include the real observation windows of each survey, as kindly provided by A. Udalski. For any transit- ing planet in the simulation, the number of effectively observed transits is used in eq. 6. In order to construct a realistic stellar population, we use the stellar counts per magnitude range obtained by Gould et al. (2006) based on OGLE-II data, which have calibrated photome- try. We then randomly select that number of stars per magnitude from the Besançon model. In order to test the validity of our approach, we calculated the fraction of “stars for which tran- sits are detectable” and compared it to the one determined by Gould. This fraction is defined for a given magnitude range as the number of stars around which a planet orbiting edge-on with r = 1.2R jup and a = 7.94R⊙ can be detected, divided by the total number of stars of that magnitude. As shown by table 2, there is an excellent agreement between our results and those of Table 2. Fraction of stars suitable for transit detection Carina Bulge Vmax Gould 2006 This work Gould 2006 This work 15.5 0.11 0.16 0.138 0.141 16 0.14 0.16 0.125 0.128 16.5 0.16 0.15 0.098 0.105 17 0.16 0.15 0.068 0.080 17.5 0.16 0.14 0.041 0.052 Gould et al. (2006). Note however that for the global simulation, the complete star list is used as the above definition for suitable stars is restricted to planets of a given size and orbital distance. We calculated the average flux for target stars, companions and all the background stars near enough to contribute to the target PSF. We then checked that the average photon noise simu- lated for target stars at a given magnitude was close to real values obtained in OGLE light curves at given magnitude presented in figure 4 of Pont et al. (2006b). 2.2. Modelling the detection threshold The candidates in the OGLE survey have been identified with the BLS transit-search algorith of Kovács et al. (2002). A sub- set of the candidates selected with cuts in the α and SDE pa- rameters of the BLS were examined by eye, and only the best were included in the final list. Therefore, the selection thresh- old is mainly defined by subjective appreciation from an ex- perienced specialist. Recently, Pont et al. (2006b) have pointed out that the effective detection threshold of ground-based tran- sit surveys such as OGLE is importantly affected by correlated noise (photometric systematics). The subjective selection of can- didates is in large part necessary because of the presence of this correlated noise, which produce many spurious detections near the threshold. Gould et al. (2006) chose to model the OGLE se- lection threshold with an α > 12 cut (alpha is equivalent to the signal-to-noise ratio of the transit signal assuming uncor- related noise and homogeneous distribution of the data points in phase). Pont et al. (2006b) have included the effect of corre- lated noise in the signal-to- noise calculation and found that the OGLE selection could be better described by a threshold of 8 on the signal-to-noise ratio of the transit signal calculated in- cluding correlated noise (”S r” in their notation, see Sec. 2.5.2), and without the assumption of homogeneous coverage. While the two thresholds have similar effects on the global number of planet detection, they have a very different dependence on some parameters, such as planet period and host star magnitude. Since the objective or our study is to examine the detection statistics in a multi-dimensional parameter space, we use the Pont et al. (2006b) description of the OGLE detection threshold. To calculate S r, one needs an assumption on the level of red noise present in the photometry. Following Pont et al. (2006b), we use a single-parameter description and assume σr = 3.6 mmag in the Bulge fields, σr= 3.1 mmag in the Carina and Centaurus fields, and σr= 2.1 mmag in all fields after applica- tion of decorrelation algorithms. 2.3. Confirmability of transit-like events with follow-up High-resolution spectra allow the confirmation of the planetary events if spectral lines are deep enough. Several scenarios make the follow up of candidates too difficult: early type stars have lines too weak and too broadened by rotation (type F4 and ear- lier). Stars with magnitudes V > 17.5 are too faint for present instruments and telescopes. This is the limit at which observers estimated not being able to provide low-metallicity stars. Those stars having weaker lines, could also be difficult to follow cor- rectly, but as planets are unlikely to be found near this kind of stars in our model, we did not take that parameter into account. To simulate the feasability of follow-up, we only considered in CoRoTlux the stars matching the criteria V < 17.5 and of type F4 and later. 3. Results of the simulations We present hereafter runs for the three OGLE-III campaigns for the fields in the Galactic bulge, in Carina and in Centaurus. In order to obtain a statistically significant population of detected planets, the simulations were run multiple times. We first examine the consistency between the models and ob- servations for relevant physical variables. In doing so, we choose to compare our model population to the global population of transiting planets discovered by OGLE and other surveys. There is a slight inconsistency in assuming that the parameter compari- son is almost independant of the type of survey and observational strategy. In some cases, this is not true, and a clear distinction between the OGLE planets and the other detections has to be made. We then discuss the problem of the detection statistics, whether observations and models are consistent, and whether a constraint on the (low) frequency of very close-in planets can be deduced. 3.1. Deviation of OGLE planets from maximum likelihood of the simulations We use a Maximum-Likelihood (ML) technique in order to test whether model results and observations agree with each other. We do the tests in two-dimension spaces, in order to qualify pos- sible correlation and exclusion zones. The ML technique is our method of choice as it is a powerful tool for fitting a model to a multi-dimentionnal independant-data distribution (Lyons 1986). Instead of determining an approximate analytical law fitting our results, we use the results of a very large Monte-Carlo draw (1000 times the whole OGLE survey, corresponding to ∼ 9000 planets) to get a map of the density of probability in each 2- dimension grid. We bin our data on a 20x20 grid as a compro- mise between resolution of the models and characteristic varia- tions of the parameters.2 The probability of an event in each bin is considered equal to the normalized number of draws in that Figure 6 shows the logarithm of the probability that an event occurs in each of the 20x20 bins of the mass-radius diagram. The likelihood of a draw of several independant events is defined as the sum of the logarithms of the probabilities of these events. In order to compare our results to any n real discoveries, we first estimate the standard deviation of any n-planets-random-draw compared to the maximum likelihood of the model. We ran- domly select n planets among the simulated detections and cal- culate the likelihood of this draw. We repeat this selection 1000 times in order to have the maximum likelihood and its standard deviation σ, then we compare the deviation of the likelihood of the n real detecions calculated the same way in terms of σ. 2 Tests with different grids yield small variations of the results. As an example, the mass-radius deviation from maximum likelihood is re- spectively 0.67, 0.65 and 0.72 σ for 20x20, 30x30 and 40x40 grids. Henceforth, quantitative comparisons between the simulation re- sults and the known planets are systematically given in the figure captions, whilst the text discusses qualitative comparisons and their implications. For the different figures showing the results of our simulation, we compare the distribution of planets over the detection threshold to the 5 OGLE planets. We also compare our results to the 11 planets discovered by all transit surveys, as their detection biases are similar to OGLE, and to the 14 planets which radius is known (11 from transits and 3 from radial ve- locity surveys) to show how our model can reproduce the whole known population. 3.2. Depth of the transit events and magnitude of the targets stars We first attempt to confirm whether the events detected by the model are consistent with those found in the OGLE fields. Figure 7 is a plot showing transit depth as a function of the mag- nitude of the primary star. Model results are considered detected when the signal-to-noise ratio is sufficient for a detection (see § 2.2). We also show events that are considered photometrically detectable but very hard or impossible to confirm by radial ve- locimetry. The figure evidently shows a good correlation between the black crosses and the red circles that indicate real detections by OGLE, with a range of transit depths and V magnitudes that is very similar between the models and the observations. Our models overpredict slightly the number of transit events around faint stars (V ≥ 17), but this may be due to the difficulty of the follow-up work for these targets. Overall, the agreement between models and observations is good. 3.3. Compatibility of transit surveys with radial-velocimetry observations 3.3.1. Compatibility in the mass-period diagram Figure 8 compares the model and observated mass-period rela- tion. As it is independant of the planetary evolution model, it is a direct test of the compatibility between the results of transit sur- veys and those of radial-velocimetry observations that drive our model results. Again, the comparison is very good, assuming a high-enough frequency of very-close in planets (see discussion in § 3.3.4). One can note especially the absence of planets of rel- atively large mass (several times that of Jupiter) at short orbital distances (P < 5 days), and of detectable transit events for peri- ods longer than ∼ 5 days. This is due especially to the fact that only events with a relatively large number of observed transits are detectable, as required by the S r threshold, which, given the day/night interruptions, imposes a constraint of a short orbital period. Note that this feature is not well reproduced by mod- els in which the threshold is computed from white-noise only (Gould et al. 2006; Gillon et al. 2005). 3.3.2. The OGLE yields with a fixed red noise level We have tested the efficiency of the fiducial model at estimat- ing quantitatively the yield of transit surveys. Gillon et al. (2005) have also simulated OGLE yield in their generic study of multi- ple transit surveys, but with restrictive assumptions on transit de- tectability (only complete events matter for detection purposes) and without considering background stars and red noise, also not using OGLE-fields specific stellar population. We also in- cluded in our simulations the recent RV follow-up that has been Fig. 6. Logarithm of the probability that a simulated detection event occurs in each one of the 20x20 bins of the mass/radius diagram. The likelihood of a multiple-events draw is the sum of the logarithms of the probabilities of the events of this draw. Bins without any occuring event in the large Monte-Carlo draw do not have any probability stated. The likelihood of a n-events draw is the sum of the probabilities of its n events. In this mass-radius diagram, OGLE planets are shown as red circles, planets from other surveys are in orange, and planets from radial velocity surveys are in blue. The likelihood of the 5 OGLE discoveries as a result of a Monte-Carlo draw is −8.7, the maximum likelihood is −7 and the standard deviation to maximum likelihood is 2.54. Hence, the result of the OGLE planets mass-radius distribution is at 0.67σ of the maximum likelihood of the model. Table 3. OGLE yields with fixed red noise level Field Mean red RV follow-up Number of planets of view noise level to Vmag detected simulated with 0 1.5 3 VHJ added (P < 2 days) Bulge 3.6 17.5 2 0.4 0.6 0.9 Carina original 3.1 17.5 3 3.4 4.1 4.8 updated 2.1 17.5 +(0 − 1) +1.1 +1.1 +1.1 Centaurus 3.1 17.0 0 1.4 1.8 2.2 Total 6 6.3 7.6 9.0 done on Centaurus and Carina. We use unpublished information from the OGLE/ESO follow-up team, who found one promising planetary candidate among the Carina fields reprocessed with the systematics- removal algorithm from Tamuz et al. (2005) and none in the Centarus fields, with a magnitude limit near V=17 for the radial velocity follow-up. Table 3 compares the average number of planets detected for 1000 Monte-Carlo draws to real detections from the OGLE survey. The total number of simulated discoveries obtained from this quantitative analysis is in good agreement with the real detec- tions. The differences in the number of detections between the Carina and Centaurus surveys are mainly due to the lower duty cycle of the observations towards Centaurus. A red noise level fixed at 3.6 mmag in the direction of the galactic bulge bans most hot Jupiter detections. The agreement between our quan- titative result and the number of real detections is an indicator of the global efficiency of our approach (stellar and planetary distributions, evolution model and noise budget) for estimating transit survey yield. 3.3.3. The OGLE yields with a variable red noise level So far, we have considered the level of red noise to depend only on the field considered. We attempt now to refine this by consid- ering how the stellar density may affect it. Whereas most ground- based transit surveys have a global red noise level from 2 to 3.5 mmag (Superwasp: Smith et al. (2006), Monitor: Irwin et al. (2007), Hatnet: Pont & ISSI team (2007) and OGLE), the causes of these noise levels seem different, with instruments ranging from 10-cm wide field reflectors to deep-sky several-meter tele- scopes. As seen from table 3, the OGLE fields have differ- ent mean red noise levels (σr = 3.6 mmag for the bulge and σr = 3.1 mmag for Centaurus and Carina before SYS-REM), although the instrument and observational strategy were un- Fig. 7. Depth of the planetary transit events versus magnitude of the parent star in the V band. The five confirmed OGLE detections are shown as circles. Model results are shown as black plusses for detectable events and orange crosses for events that are considered undetectable based on the photometric signal (see text). Blue diamonds correspond to events that would be detectable by photometry alone but that cannot be confirmed by radial velocimetry. Note that the model results correspond to 3 times the full OGLE campaign for more statistical significance. The OGLE planets depth-magnitude distribution is at 0.69σ from the maximum likelihood of the model. Fig. 8. Mass versus period of transiting giant planets. (OGLE planets are red circles, other transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets mass-period distribution is at 0.62σ from the maximum likelihood of the model (0.72σ considering the 11 planets discovered by transit surveys and 0.66σ considering the 14 known planets). Fig. 9. Distribution of the crowding index (see text) of target stars in Carina (black) and in the bulge (red). changed. Looking at what distinguishes these fields, it appears that the most significant difference is the stellar density and therefore the amount of crowding: The bulge field is about twice as dense as the Carina and Centaurus fields. Pont & ISSI team (2007) raise the suspicion that the level of red noise depends strongly on the presence and characteristics of contaminating stars, because e.g. of their different colors and differential re- fraction in the atmosphere. It is hence natural to consider a red noise that depends on a crowding index. We define this crowding index as the fraction of the flux coming from background stars versus that from the target in the photometric aperture. Importantly, we do not consider stel- lar companions as contributing to the red noise because they are generally on the same CCD pixel as the target star and should affect the noise budget much less. Figure 9 shows the differences of crowding index for the tar- get stars with planetary transits (detectable or not) in simulations of the Carina and Bulge fields of view. The mean crowding in- dex for target stars of I < 17 is 0.11 in the Carina field and 0.233 in the Bulge field. We can exclude the fact that all red noise is linked with con- tamination as many stars in the Carina fields are unblended by background stars but still show a high noise level. In order to estimate of the influence of the crowding on the red noise level, we use the following simple relation between red noise level and crowding index: σr = α × Fb + β, (7) where Fb is the fraction of total flux from background stars, determined on a star-by-star basis in our simulations, and α and β are parameters to be determined. This is justified by the behaviour of the red noise seen for instance in SuperWASP, showing a linear increase as a function of background flux (Smith et al. 2006). In order to get the same mean red noise values as Pont et al. (2006b), we obtain α = 0.4 mmag and β = 2.65 mmag. This value of β corresponds to the minimum red noise level obtained for non-contaminated stars in the OGLE fields. Table 4 shows the new number of detections when consid- ering this crowding-dependant red noise level. Compared to ta- ble 3, the number of detections is found to be essentially un- changed for the Carina and Centaurus fields, but it increases by a factor ∼ 3 for the bulge field. This result is more satisfactory be- cause in the previous case, only ∼ 5% of the simulations would yield the detection of 2 planets in the bulge, as observed. 3.3.4. Models, observations and the frequency of very close-in planets As discussed in § 1.4.1, three OGLE planets have orbital periods shorter than 2 days and thus belong to a class of objects yet to be detected by radial velocimetry. So far, we have added one such planet (on average) to our carbon copy list of nearly 200 radial velocimetry planets. In Section 3.3, we have shown that with this assumption, radial-velocity and photometric transit surveys are compatible. We now test the range of frequencies of very close- in planets for which this remains true. In order to do so, we compute the deviation from maximum likelihood in the mass-radius diagram like in Section 3.3, as a function of the number of planets which period is less than 2 days added to the RV list. The result is presented in Fig. 10 and shows that a good match is obtained by adding 1 to 3 short- period planets. Larger numbers are also possible from the point of view of the transit surveys, but would conflict with their non-detection by radial-velocimetry. Adding the other transit- ing planets discovered thus far yields smaller probabilities of occurence of these short-period planets, but not by significant amounts. All in all, and assuming that the radial velocity planets sam- ple is unbiased, we constrain the fraction of main-sequence late stars orbited by very hot giant planets with orbital periods less than 2 days to be (1/1265)(1+0.33 −0.33) at a 60 % confidence level or (1/1265)(1+0.83 −0.5 ) at a 90 % confidence level. The distribution of planets in period between 2 and 5 days is directly obtained from the metallicity-linked distribution (Santos et al. 2004) and the RV planets sample. Adding the dis- tribution we found for planets between 1 and 2 days, we obtain a fraction of (1/215) late main-sequence stars orbited by planets in the 1 to 5 days period range, in good agreement with the results obtained in Gould et al. (2006), who obtained (1/220)(1+1.10 −0.45). Similarly, the distribution we obtain by cutting this sample into two parts with the cut-off at 3 days is compatible, showing: – a slightly higher fraction of really short-period planets (1-3 days) of (1/560) instead of (1/710)(1+1.10 −0.54) at a 90 % confi- dence level in Gould et al. (2006). – a similar fraction of short-period planets (3-5 days) of (1/350) instead of (1/320)(1+1.39 −0.59) at a 90 % confidence level in Gould et al. (2006). The results presented hereafter use the variable red noise level approach, and an RV planet list that is complemented with, on average, 1.5 very-close in planets with periods P < 2 days taken from the OGLE detections. 3.4. The metallicity of the stars harboring transiting planets We now compare the metallicity of the parent stars for our ob- served and modelled populations. A first test using the analyti- cal scenario for the radial-velocity population (Fig. 11) yields a clearly different metallicity distribution, with most of the tran- siting planets observed around low-metallicity stars. We veri- fied that this problem occurs independantly of the assumed stel- lar metallicity distribution, for any realistic stellar population. It arises fundamentally because the global metallicity bias as ob- tained by Santos et al. (2004) or Fischer & Valenti (2005) is not Table 4. OGLE yields with variable red noise level Field RV follow-up Number of planets of view to Vmag detected simulated with 0 1.5 3 VHJ added (P < 2days) Bulge 17.5 2 1.2 1.6 2 Carina original 17.5 3 3.6 4.3 4.9 updated 17.5 +(0 − 1) +1.1 +1.1 +1.1 Centaurus 17 0 1.3 1.9 2.3 Total 5-6 7.2 8.9 10.3 Fig. 11. Period of transiting exoplanets versus metallicity of their parent star. The model is based on analytic relations for the mass and period distributions of planetary companions (see § 1.4.2). (OGLE planets are red circles, other transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets period-metallicity distribution is at 2.94σ from the maximum likelihood of the model (2.51σ considering the 11 planets discovered by transit surveys and 2.63σ considering the 14 known planets). strong enough to compensate for the rarity of very metal-rich stars in the Galaxy. As seen in Fig. 12, the problem disappears when one con- siders the carbon-copy model. Thus, we are led to an important conclusion, that the metallicity distribution of pegasids (periods shorter than 10 days) is fundamentally different from the global exoplanet population. More specifically, there are no Pegasids orbiting F, G, K stars with metallicities smaller than [Fe/H]= −0.07. This has strong consequences for planet formation mod- els (see also Guillot et al. 2006). This work shows that this con- clusion is robust, and is needed to explain the results of the pho- tometric surveys. A finer examination of Fig. 12 shows that while our model planets reproduce globally the metallicity of the ensemble of transiting planets, OGLE stars with planets are on average ∼ 0.1 dex more metal-rich. This can tentatively be explained with a metallicity gradient in the galaxy for OGLE TR-10 ([Fe/H] = 0.28±0.10) and OGLE TR-56 ([Fe/H] = 0.19± 0.07), the two planets discovered in the direction of the galactic bulge. The study of galactic cepheids by Andrievsky et al. (2004) shows a metallicity gradient as a func- tion of distance to the galactic center. In the [6.6, 10.6] kpc-range distance from galactic center, this study finds a linear relation be- tween [Fe/H] and galactocentric distance RG: [Fe/H] = −0.044(±0.004)RG + 0.363(±0.032) (8) Following that relation, the two stars with planets discovered in the direction of the galactic bulge both at a distance around 1500 pc would thus be in a 0.04 dex more metal rich region than the solar neighborhood. Concerning the high metallicity of stars with transiting plan- ets discovered by OGLE in the Carina region, we do not have any reason to think that the metallicity distribution would be different from the solar neighborhood. Our only hypothesis is a low-probability draw for metallicity for the 3 OGLE-Carina planets. Fig. 12. Period of transiting exoplanets versus metallicity of their parent star. The figure differs from Fig. 11 in that our fiducial model, i.e. the mass-period-metallicity “carbon-copy” model is used (see § 1.4.2). (OGLE planets are red circles, other transit surveys in orange, planets from radial velocitiy surveys in blue. Simulated planets detected: black plusses, under threshold: or- ange crosses). The OGLE planets period-metallicity distribution is at 0.76σ from the maximum likelihood of the model (0.36σ considering the 11 planets discovered by transit surveys and 0.39σ considering the 14 known planets) . 3.5. Atmospheric potential energy and orbital distances Because evaporation may affect the planet population, it is in- structive to check whether the potential energy of the atmosphere and the orbital period, two crucial quantities for this process (e.g. Lecavelier des Etangs et al. 2004), also possess a relatively con- sistent distribution. We first test the behavior of the analytical model for the distribution of planets (Fig. 13). This results in a prediction of many planets with large radii (small values of the potential energy for atmospheric escape) at small orbital dis- tances, in patent contradiction with the observations. The problem mostly disappears with the carbon-copy model: Fig. 14) shows that in this case, although we do not obtain a linear correlation between the two variables, we get detections in the right area of the diagram. This is explained as stemming from: – The absence of low-mass planets at small orbital distances, with a possible limiting relation between these two quantities (Mazeh et al. 2005); – The difficulty in detecting planets with larger values of po- tential energy per unit mass (smaller radii) at large orbital distances –although we predict that some of these should be detected by future transit surveys. Our results strengthen the case for the existence of a rela- tion between mass and orbital distance for short-period plan- ets, as advocated by Mazeh et al. (2005): Indeed, the analytic model which is characterized by the presence of small mass planets at small distances yields a distribution of detectable plan- ets that is significantly different from the observations (Fig. 13). Our carbon-copy model that includes implicitely this correlation does not (Fig. 14). 3.6. Planetary radii and stellar irradiation Radius and stellar irradiation (or equivalently equilibrium tem- perature) should be positively correlated, as a planet with a higher irradiation dose will tend to cool and contract more slowly than one that endures less stellar insolation. As Fig. 15 shows, the correlation exist, but is weak, and with a signficant scatter. This is well reproduced by the model. However, it can be noted that HD 149026 b lies away from the cloud of points. In general, we find that our fiducial model generates few points in this region. This can be easily accounted for by slightly modifying the metallicity-core mass relation to allow for larger masses. As planets of small masses and large core masses are more difficult to model anyway, we chose not to attempt fine-tuning the model to this level of detail. This should be postponed for further studies, especially with the discovery of more Saturn-mass transiting planets. 3.7. The mass-radius relation We have checked that our fiducial model predicts the detection of transiting planets with properties that are globally consistent with the observations. We can now examine in more detail the mass-radius relation thus obtained, as it is directly tied to as- sumptions on the compositions and evolutionary models of ex- oplanets. The predictions also have implications for transit sur- veys as it is not clear whether they have detected only the “tip of the iceberg”, ie the few largest giant planets while many smaller ones would lie undetected or not. Results with our fiducial model are presented in Fig. 16. We find that planets with low masses (say, less than Jupiter’s Fig. 13. Potential energy per unit mass (Ep = GM/R) versus orbital period of transiting planets. (OGLE planets are red circles, other transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). Observations are compared to models based on the analytical relations for the mass and period distribution of planetary companions (see § 1.4.2). The OGLE planets energy-period distribution is at 2.18σ from the maximum likelihood of the model (1.86σ considering the 11 planets discovered by transit surveys and 2.47σ considering the 14 known planets). Fig. 14. Potential energy per unit mass versus orbital period of transiting planets. The figure is similar to Fig. 13, except for the fact that our fiducial model is used (see § 1.4.2). (OGLE planets are red circles, other transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets energy- period distribution is at 0.55σ from the maximum likelihood of the model. (0.84σ considering the 11 planets discovered by transit surveys and 0.66σ considering the 14 known planets) Fig. 15. Radius as a function of equilibrium temperature of transiting exoplanets. (OGLE planets are red circles, other transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets equilibrium temperature-radius distribution is at 1.22σ from the maximum likelihood of the model (1.05σ considering the 11 planets discovered by transit surveys and 2.25σ considering the 14 known planets). mass) can both have very large or very small radii, depending on whether they contain a significant mass in heavy elements or not. On the other hand, massive planets have radii which are comparatively better defined. This is mostly due to the fact that we assume a maximum mass of heavy elements of 100 M⊕, a hypothesis that will be tested directly by the discovery of a few massive transiting planets. Our model results once again agree well with the detections made by photometry. Importantly, the yellow crosses in Fig. 16 do not lie significantly below the black ones: we predict that future surveys will not discover a population of small-sized giant planets, at least for masses above that of Saturn. The presence of planets with larger masses of heavy ele- ments should remain marginal because otherwise they would have been detected by present-day surveys, Fig. 16 showing that planets below 1 RJup are already detectable, although in fa- vorable cases (small radius of the primary and bright targets). Quantitatively, simulations in the OGLE fields indicate that if planets had radii uniformly distributed between 0.5 and 1.5 RJup, 18.5% of the planets discovered by the survey would have radii below 1 RJup. This fraction is not negligible and is (although marginally) inconsistent with the sample of 0/11 planets with R < RJup discovered by transit surveys thus far. Therefore, although we cannot statistically rule out the pres- ence of a population of small planets, these would require the formation of extremely metal-rich planets. Our prediction is a consequence of evolution models and of our assumption that planets with masses of heavy elements beyond 100 M⊕ should be rare. Figure 17 shows the ensemble of planets obtained for an ex- tremely large number of draws, with our fiducial model. Voids in the ensemble of crosses correspond to the absence of planets with these masses in the radial-velocimetry list. They should not be considered as significant. The contours in the figure indicate the ensemble of masses and radii expected for planets with dif- ferent masses of heavy elements, from 0 to 100 M⊕. Importantly, the location of these contours is linked to our assumption of an energy source in the planetary interior equal to 0.5% times the irradiation received by the planet. Independently of the details of this assumption, this shows that a statistically significant ensem- ble of known transiting planets would allow a determination of the presence or lack of heavy elements in these objects. We have also tested another assumption regarding the plan- etary evolution model: all planets possess 20 M⊕ mass in heavy elements, 70% of them have no extra heat source, whereas 30% have 3 × 1026 erg s−1 dissipated at the center. With this assump- tion, one can qualitatively explain the observed transiting planets (i.e. the “normal” planets and the “anomalously large” ones, re- spectively) with the exception of HD 149026 b, for which one could argue that the planet comes from a different population. In this case, Fig. 18 shows a distribution of radii that is relatively similar to the previous one (Fig. 16), with the exception that no planet has a radius smaller than 0.8 RJup. In this case, the 2 re- gions corresponding to the “standard” model, and to the “heat dissipation” case are clearly different, especially at the low-mass range of the diagram. Present observations cannot distinguish between the two models, showing the need for additional detections of transit- ing giant planets. Particularly important are planets between the mass of Saturn and that of Jupiter, as this is a mass regime where expected compositional differences have the largest impact. Fig. 16. Mass-radius relation for transiting extrasolar giant planets. (OGLE planets are red circles, other transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets mass-radius distribution is at 0.67σ from the maximum likelihood of the model (0.72σ considering the 11 planets discovered by transit surveys and 0.97σ considering the 14 known planets) . 4. Conclusions We have presented a simulation of photometric transiting sur- veys based on basic knowledge of the stellar and planetary pop- ulations in the galactic neighborhood and on a planetary evo- lution model tuned to the information obtained from transiting giant planets with masses above that of Saturn. This simulation was applied to the OGLE survey, and shown to yield a generally excellent agreement with the transiting planets detected by the survey. We have thus shown that radial velocimetry and photometric surveys are compatible within statistical uncertainties, in agree- ment with Gould et al. (2006). We have derived a frequency of very close-in planets with orbital periods shorter than 2 days around solar-type stars, of (1/1265)(1+0.33 −0.33) at a 60 % confidence level or (1/1265)(1+0.83 −0.5 ) at a 90 % confidence level. Using null results by photometric surveys for given ranges of parameters, we are able to strengthen two results already present in the radial velocimetry data: – Stars with low metallicities ([Fe/H]< −0.07) do not, or are very unlikely to harbour close-in giant planets with orbital periods P < 10 days. This is unlike stars above that metallic- ity threshold (see Fig. 12). – There is a lack of small-mass giant planets below the mass of Jupiter and above that of Saturn for orbital periods P < 3 days (see Fig. 8). Further data is required to precisely quantify these empirical re- sults that bear important consequences for our understanding of planet formation and migration. On the basis of our model, and assumptions concerning the composition of giant planets (i.e. masses of heavy elements be- tween 0 and 100 M⊕), we find that the present detections of tran- siting planets have sampled a population that is quite represen- tative of the main population of giant planets, at least for the ones that are above about half the mass of Jupiter. We hence pre- dict that future transit surveys with higher sensitivities will not discover a significant population of yet undetected Jupiter-mass planets with small sizes, i.e. radii smaller than that of Jupiter (see Fig. 16). Many ground-based transit surveys are in progress, and with the space missions CoRoT (Baglin et al. 2002) and Kepler (Borucki et al. 2003), the number of known transiting planets is expected to rise rapidly over the next few years. This will en- able us to better test the models and quantify some of the results presented in this article. We also hope to be able to discriminate between various models of the evolution and compositions of gi- ant planets, a matter of great importance for formation models. We wish to stress however that a continuation of ground- based transit surveys is desirable even in the presence of simi- lar programs from space. CoRoT will survey 60,000 dwarf stars over five 150 days periods and Kepler about 100,000 over 4 years, implying a maximum potential yield of 55 and 90 transit- ing giant planets, respectively, plus many other smaller planets. For what concerns giant planets, quantifying the fraction of very close-in planets with a 10% accuracy at the 3σ level would re- quire the discovery of ∼ 200 transiting planets. Understanding the evolution and compositions of giant planets will require an even larger number of detections. The radius of a giant planet itself depends mainly on four parameters: the planetary mass, equilibrium temperature, age, and its composition (note that the composition can be considered as a simple parameter only in the case of planets mostly made of hydrogen and helium: smaller planets will be more difficult to model!). Additional energy Fig. 17. Mass-radius relation for a very large number of Monte-Carlo trials using the fiducial model. The curves show the ensemble of planets with masses of heavy elements between 0 and 25, 25 and 50, 50 and 75, 75 and 100 M⊕, respectively. Symbols are as in Fig. 8. Fig. 10. Deviations from a maximum likelihood obtained as a function of NVHJ , the number of very hot jupiter of orbital pe- riods shorter than 2 days added to the radial velocities carbon- copy list. Thick line: Deviation from the maximum likelihood obtained in the mass-radius diagram for the OGLE planets. Thin line: Same deviation but when compared to the ensemble of planets. Dashed line: Standard deviation obtained from a com- parison between the number of simulated planets and the number of detected ones for the OGLE survey (see table 4). Dotted line: Standard deviation obtained from the non-detection of these very close-in planets by radial-velocimetry. sources may occur (such as in the presence of tidal heat dissipa- tion), and the initial conditions and formation history may have their say in the matter as well. Furthermore, the observational uncertainties are generally large. For example, the planetary ra- dius is generally only known to∼ 10%, for a global variation that is relatively small (1 to 1.5 RJup). This implies that to constrain a given correlation to, say 10%, and with four independant vari- ables, hundreds of data points are needed, and thousands would be desirable. This motivates us to seek programs capable of detecting thousands of transiting planets in the mid-term future, and ways to reduce the error bars on the different parameters. One direc- tion is to test the Dome C plateau in Antarctica for such an am- bitious program, which is the purpose of A STEP (Fressin et al. 2005). Other directions exist, such as proposals for similar sur- veys from space. In any case, it is most important that a statisti- cally significant population of exoplanets be characterized for a better understanding of planet formation and our origins. Acknowledgments The code used for this work, CoRoTlux, has been devel- opped as part of the CoRoT science program by the au- thors with major contributions by Aurélien Garnier, Maxime Marmier, Martin Vannier, Suzanne Aigrain and help from Claire Moutou, Stéphane Lagarde, Antoine Llebaria, Didier Queloz and François Bouchy. We want to thank Andrzej Udalski and Michael Gillon for their communications on OGLE data, Frédéric Thévenin for his advices on stellar populations sim- ulation. F.F. has been funded by grants from the French Ministère de la Recherche and by the Société des Amis des Sciences. 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A., et al. 2005, ApJ, 632, 1157 Zucker, S. & Mazeh, T. 2001, ApJ, 562, 1038 Simulating transit surveys General remarks Principle of the simulations The stellar population Main targets and background stars Binary and triple systems The planetary companions Planet incidence Planetary masses and orbits Physical characteristics and the planetary evolution model Modeling transit events and their detectability PSFs and CCDs Noise budget and event detectability The OGLE survey: input parameters Basic parameters and observational procedure Modelling the detection threshold Confirmability of transit-like events with follow-up Results of the simulations Deviation of OGLE planets from maximum likelihood of the simulations Depth of the transit events and magnitude of the targets stars Compatibility of transit surveys with radial-velocimetry observations Compatibility in the mass-period diagram The OGLE yields with a fixed red noise level The OGLE yields with a variable red noise level Models, observations and the frequency of very close-in planets The metallicity of the stars harboring transiting planets Atmospheric potential energy and orbital distances Planetary radii and stellar irradiation The mass-radius relation Conclusions

Transiting extrasolar planets are now discovered jointly by photometric surveys and by radial velocimetry. We want to determine whether the different data sets are compatible between themselves and with models of the evolution of extrasolar planets. We simulate directly a population of stars corresponding to the OGLE transit survey and assign them planetary companions based on radial velocimetry discoveries. We use a model of the evolution and structure of giant planets assuming a variable fraction of heavy elements. The output list of detectable planets of the simulations is compared to the real detections. We confirm that the radial velocimetry and photometric survey data sets are compatible within the statistical errors, assuming that planets with periods between 1 and 2 days are approximately 5 times less frequent than planets with periods between 2 and 5 days. We show that evolution models fitting present observational constraints predict a lack of small giant planets with large masses. We also identify distinct populations of planets: those with short periods (P < 10d) are only found in orbit around metal-rich stars with [Fe/H] > -0.07. We further confirm the relative absence of low-mass giant planets at small orbital distances.

Introduction Extrasolar planets are now routinely discovered orbiting solar-type stars by radial velocimetry, but the discovery of tran- siting planets by photometric surveys is just beginning. Although still marginal, the late success of transit surveys has given an additional impulse to exoplanetology with the possibility to es- timate the radius, density and hence composition of extrasolar planets. Quantitatively, we know to date 206 extrasolar planets with masses below 13 MJup (e.g. Udry et al. 2007; Butler et al. 2006). Among those, a list of 14 currently known transiting planets is presented in table 1. These planets have been discovered by ra- dial velocimetry followed by photometry for 3 of them, and by photometric surveys for the remaining 11. When considering the score of projects devoted to the detec- tion of planets by transit photometry, the present harvest appears meager. The discrepancy between predictions (e.g. Horne 2001) and reality has been attributed to various factors such as: im- perfect duty cycle, a reduced number of stars for which tran- siting planets are detectable (Gould et al. 2006) and the pres- ence of correlated noises that can greatly limit the detectability of small planetary transits (Pont et al. 2006b). Several generic studies have been conducted to understand the yield of different transit surveys. Pepper & Gaudi (2005) studied the optimization of transit searches as a function of the observational setup, the site properties and the planet properties. Gillon et al. (2005) an- alyzed and compared deep field surveys, considering individual stellar ranges and observation windows, but did not include the effects of stellar crowding nor time-correlated noises. Gould et al. (2006) studied the yield of OGLE survey (Udalski et al. 2002), the most successful so far in term of num- ber of transiting planets discovered, with a model populating the line of sight with stars drawn from the Hipparcos Catalogue. They estimated with that model the proportion of stars with sen- sitivity to close-in giant planets to derive from OGLE results the frequency of planets as a function of their period. They find that the yield of the OGLE survey is globally consistent with the detections by radial velocimetry and with planet radii dis- tributed between 1 and 1.25 jovian radii. The aim of the present work is to further test these data sets and the underlying physical model by a forward calculation of transit events with realistic stellar and planetary populations. In particular, we include up- to-date models of the evolution and structure of Pegasids (close- in extrasolar planets) based on models reproducing the observa- tional constraints from known transiting planets (Guillot 2005; Guillot et al. 2006). As a consequence, we should be able to de- termine whether the presently known population of transiting planets represent the “tip of the iceberg”, i.e. that many more small, dense extrasolar giant planets exist and await discovery by the transit method, or whether it is relatively representative of the global population. We first describe the model that is used to simulate transit surveys in general. In Section 3, we describe more particularly the OGLE surveys and the hypothesis chosen for their mod- elling. We then discuss the results of the simulation. A summary of the main conclusions and predictions for future transit surveys are provided in Section 5. 1. Simulating transit surveys 1.1. General remarks The search for planets in transit in front of their star naturally arised with the discovery that a non-negligeable fraction of plan- ets orbit very close to their stars. If orbital planes are randomly oriented, the probability that a planet will transit in front of its star at each orbital revolution is: Ptransit ≃ R⋆/aplanet, (1) where R⋆ is the stellar radius, and aplanet the planet’s orbital semi- major axis. For systems such as 51 Peg b, this probability is close to 10%. Because the probability for a solar-type star to possess such a Pegasid (i.e. a 51 Peg b-like planet, planets semi-major axis up to 0.1 AU) is about 0.5% (e.g. Marcy et al. 2005), 1 in 2000 solar-type star should possess a transiting Pegasid. Using current results from radial velocity surveys and integrating over all periods, we estimate that about 1 in 1100 solar-type stars possesses a transiting giant planet. Of course, depending on the magnitudes and field considered, giant stars may severly out- number the dwarfs, so that in a real field, only one in, say, 3000 stars may harbor a transiting giant planet. Because of the dependence on a, and period distribution, most of the transit events concerning giant planets occur for or- bital periods between 1 and 5 days. The transits typically last for a couple of hours, as this quantity is weakly dependant on the orbital period P: τtransit = 1.82 1 day )1/3 ( )−1/3 (R′⋆ hours, (2) where R′⋆ is the length of the cord traced on the stellar disk by the planet’s trajectory. (more precisely: R′⋆ = R⋆ cos b + Rplanet, where b is the impact parameter of the transit). The depth of the transits themselves is directly given by the ratio of the planetary to the stellar disk surfaces: Rtransit ≃ (Rplanet/R⋆) . (3) This value is of order 1% for a Jupiter-size planet orbiting a Sun-like star. For an F-type star with radius ∼ 1.2 R⊙, the ratio decreases to 0.7%. Furthermore, transiting giant planets discov- ered so far have radii between 0.72 and 1.44 RJup (see table 1). Allowing for stellar radii to vary between 0.8 and 1.3 R⊙ (a typ- ical range, in magnitude limited surveys), this implies that we should expect Rtransit to vary between 0.3% and 3%, for giant planets only. The lower limit is in reality even smaller because for detection purposes we have to account for the fact that plan- ets also orbit stars that are in multiple systems (like HAT-P-1), and hence a dilution factor may apply. Although grazing transits are ignored in this simple analysis, they are included afterwards in our simulations. This altogether implies that in order to detect transiting gi- ant planets, many thousands of dwarf stars have to be monitored over periods of weeks for a photometric precision reaching be- low a fraction of a percent on an equivalent integration time of about one hour. This is typically done by following a relatively dense stellar field over a long time with a stable telescope, and using a camera equiped with a good CCD camera. 1.2. Principle of the simulations On paper, the simulation of the forward problem is simple: one has to generate a complete stellar field, or obtain it from observa- tions, put it on a discrete grid (the CCD), include on probabilistic arguments the planetary companions, calculate lightcurves in- cluding the various sources of noise, and determine which events are detectable. This is the principle of CoRoTlux, a code we first developed to predict the transit yield of CoRoT space telescope http://arxiv.org/abs/0704.1919v1 Table 1. Known transiting planets by 2006⋆ # Name Mplanet Rplanet Period a M⋆ R⋆ Teff Metallicity [MJup] [RJup] [day] [AU] [M⊙] [R⊙] [K] [Fe/H] OGLE planets 6 OGLE-TR-10 0.63±0.14 1.26+0.07 −0.07 3.10129 0.04162 1.18 ±0.04 1.16±0.06 6075±86 0.28±0.10 2 OGLE-TR-56 1.17±0.04 1.32+0.06 −0.06 1.211909 0.0225 1.17±0.04 1.32±0.06 6119±62 0.19±0.07 5 OGLE-TR-111 0.52±0.13 1.067+0.054 −0.054 4.0144479 0.047 0.81±0.02 0.831±0.031 5044±83 0.19±0.07 3 OGLE-TR-113 1.35±0.22 1.09+0.03 −0.03 1.4324757 0.0229 0.78±0.02 0.77±0.02 4804±106 0.15±0.10 4 OGLE-TR-132 1.14±0.12 1.18+0.07 −0.07 1.689868 0.0299 1.26±0.03 1.34±0.08 6210±59 0.37±0.07 Other transit survey planets 7 TrES-1 0.76±0.05 1.081+0.029 −0.029 3.0300737 0.0393 0.89±0.035 0.811±0.020 5250±75 -0.02±0.06 11 TrES-2 1.28±0.07 1.24+0.09 −0.06 2.47063 0.0367 1.08±0.08 1.00±0.05 5960±100 0.15±0.03 10 XO-1 0.90±0.07 1.184+0.028 −0.018 3.941634 0.0488 1.0±0.03 0.928±0.033 5750±13 0.015±0.03 12 HAT-P-1 0.53±0.04 1.36+0.011 −0.09 4.46529 0.0551 1.12±0.09 1.15±0.09 5975±45 0.13±0.02 13 WASP-1 0.867±0.073 1.443+0.039 −0.039 2.519961 0.0382 1.15±0.09 1.453±0.032 6200±200 14 WASP-2 0.88±0.07 1.038+0.05 −0.05 2.152226 0.0307 0.79±0.08 0.813±0.032 5200±200 Transiting planets discovered with Radial velocities 9 HD189733 1.15±0.04 1.154+0.032 −0.032 2.218573 0.0313 0.82±0.03 0.758±0.016 5050±50 -0.03±0.04 8 HD149026 0.330±0.02 0.726+0.064 −0.064 2.87598 0.042 1.3±0.1 1.45±0.1 6147±50 0.36±0.05 1 HD209458 0.657±0.006 1.320+0.025 −0.025 3.52474859 0.047 1.09±0.09 1.148±0.002 6117±26 0±0.02 MJup = 1.8986112 × 10 30 g is the mass of Jupiter. RJup = 71, 492 km is Jupiter’s equatorial radius. OGLE-TR-10: Bouchy et al. (2005); Udalski et al. (2002); Konacki et al. (2005); Santos et al. (2006); Pont et al. (2006a) OGLE-TR-56: Konacki et al. (2003); Udalski et al. (2002); Torres et al. (2003) Bouchy et al. (2005); Santos et al. (2006); Pont et al. (2006a) OGLE-TR-111: Pont et al. (2004); Santos et al. (2006); Udalski et al. (2002); Winn et al. (2007); Bouchy et al. (2005) OGLE-TR-113: Bouchy et al. (2004); Udalski et al. (2002); Konacki et al. (2004); Gillon et al. (2006) OGLE-TR-132: Bouchy et al. (2004); Udalski (2003); Moutou et al. (2004); Magain et al. (2007) TRES-1: Alonso et al. (2004); Laughlin et al. (2005); Winn et al. (2007) TRES-2: O’Donovan et al. (2006) XO-1: McCullough et al. (2006); Holman et al. (2006); Wilson et al. (2006) HAT-P-1: Bakos et al. (2006) WASP-1: Collier Cameron et al. (2006); Shporer et al. (2007); Charbonneau et al. (2006) WASP-2: Collier Cameron et al. (2006); Charbonneau et al. (2006) HD-189733: Bouchy et al. (2005); Bakos et al. (2006) HD-149026: Sato et al. (2005); Charbonneau et al. (2006) HD209458: Brown et al. (2001); Cody & Sasselov (2002); Wittenmyer et al. (2005); Winn et al. (2005); Knutson et al. (2007) # is the label of planets in figures ; they are ranked in the order of their discovery. (Baglin et al. 2002) and quantify the need for follow-up obser- vations, which is here applied to the case of OGLE. The interesting point of such a forward simulation is the pos- sibility to include relatively easily fine details such as the fact that planets are found more frequently around metal-rich stars, or, on the basis of planetary evolution models, the fact that young planets orbiting close to bright stars will be larger than old plan- ets orbiting smaller stars at larger orbital distances. This requires however that a relatively large number of physically relevant parameters (for example, the mass, size, metallicity, age of the stars) be properly defined. We further detail the assumptions that we made for these simulations by describing how we generate the stellar and plane- tary populations, and how we attempt to include realistic sources of noise. 1.3. The stellar population 1.3.1. Main targets and background stars Stellar fields differ enormously in terms of densities and stellar populations. It is therefore most important to properly account for this in order to simulate any given transit survey. It would be very appealing to use direct observations as much as possible to closely match the observed target fields. But as we will see hereafter, many different characteristics of the stars (stellar metallicity, age and subtype ...) are required, and these are difficult to obtain with generic observations. We therefore adopt the following procedure: – The observed stellar densities are obtained from stellar counts by magnitude, on the real stellar fields (see § 2.1) – The characteristics of the stars are obtained following a Monte-Carlo method using the output of the Besançon model of the galaxy (Robin et al. 2003) obtained for the proper lo- cation of the survey. – Where stellar counts are not available, or uncomplete (i.e. for faint stars), we use both stellar counts and characteristics from the Besançon model. Specifically, we keep track of the following parameters ob- tained directtly from the Besançon model: – The mass of each star, used to compute orbital parameters of the transiting object; – The apparent magnitude of the star in the observed spectral range (the I filter in the case of the OGLE survey); – The V magnitude of the star, important to qualify the con- firmability of a transit event with radial velocimetry; – The surface temperature of the star – The luminosity of the star, calculated from its absolute mag- nitude; – The radius of the star, calculated from total luminosity and effective temperature. Stellar radius [RSun] Stellar mass [MSun] Effective temperature [K] Fig. 1. From top to bottom: Distribution functions for the radii, masses and effective temperatures for our fiducial stellar popu- lation corresponding to the simulated OGLE Carina field. The black line represents the ensemble of stars in the field. The filled red region is a subset for dwarf stars with stellar type F4 and later, as these are the only stars for which a transiting planet has a reasonnable chance of being detected by present-day transit surveys. The mass, and effective temperature of the stars are dis- tributed linearly around values given by the Besançon model (at ±20%). Figure 1 shows a simulated distribution of stars for the OGLE Carina field. The ensemble of dwarf stars with types F4 and later are highlighted as these represent targets for which planetary transit events are detectable, and, within observational limits, confirmable by radial velocimetry. The metallicity distribution is obtained from the model of Nordström et al. (2004), which is based on the Geneva- Copenhagen survey of the Solar neighbourhood. These authors find that the distribution of the metallicities [Fe/H] is well ap- proximated by a Gaussian function with a mean of −0.14 and a dispersion of 0.19 dex. We use this gaussian distribution and choose to ignore possible dependencies between stellar parame- ters (e.g. masses, ages...) and the metallicities. (The link between stellar type and metallicity appears to be negligible for F4 and later types anyway (F. Thévenin, pers. communication 2007)). 1.3.2. Binary and triple systems Multiple stellar systems are important especially because of the possibility that stellar eclispes mimic planetary transits (Brown 2003). However, we choose to defer this problem to a later ar- ticle. Multiple systems are taken into account anyway because they can yield a dilution of the planetary transit events that makes them more difficult to detect. Planets may be present in- differently on the primary, secondary or tertiary components of a stellar system. (However, we find that only planets around the primary targets have a non-negligible chance of being discov- ered by current ground based photometric survey.) Specifically, following Duquennoy & Mayor (1991), we consider that 50% of the stars are binaries and 20% of those are ternaries. Multiple systems are considered as individual stars at the same position on the CCD. We choose to estimate their properties more simply than for the other stars, on the basis of DM91: – We randomly add companions to the initial draw of primary stars, without changing their properties. The total mass and luminosity of each multiple system is thus slightly higher than initially. – The mass ratio (primary/secondary) is defined as a gaussian of median value 0.23 and a full width at half maximum of 0.42. Outside a range of 0.05 and 1, we redraw the mass ratio. – The radius is defined as R⋆ = R⊙(M⋆/M⊙) when M⋆ ≤ M⊙ and R⋆ = R⊙(M⋆/M⊙) 1/2 otherwise. – The luminosity is assumed to be proportional to M2 so that: lsecondary = lprimary(Msecondary/Mprimary) – Other stellar parameters are calculated on the basis of these ones and of those of the primary component (same age, same distance, same metallicity). – Triple components are treated with the same method as bi- naries, and are defined relatively to the primary star. 1.4. The planetary companions With more than 200 planets known to orbit stars other than our Sun, we are beginning to have a rather precise view of at least part of this population. We can expect that biases on the detec- tions are small in the case of massive planets (the mass of Saturn and more) and planets that are relatively close to their star (or- bital distances smaller than ∼ 1 AU). These two conditions hap- pen to match quite exactly the requirement for detectability by transit photometry, with one assumption: that only massive giant planets can have large radii. Although not proven, this assump- tion seems quite reasonnable. Hence we choose to focus this study on this well- characterized population of objects. From the current list of 209 detected exoplanets, we select the ones discovered by radial ve- locities with mass higher than 0.3 Jupiter masses and known host star metallicity. Our list of planets includes 153 objects, to which we may add very-close in planets detected by transit photometry, as discussed below. We are using this list as representative of an unbiased sample of giant planets known from radial velocimetry, even though planetary distribution models may have been made from slighlty different samples. 1.4.1. Planet incidence A first important step is the determination of the probability for a star to harbor a planet. As shown by numerous studies (Gonzalez 1998; Santos et al. 2004; Fischer & Valenti 2005), this probabil- ity depends mostly on the metallicity of the parent star. Figure 2 shows one such probability function, as well as the result in terms of planet counts on a simulated stellar field. In this work, we will use the dependency from Santos et al. (2004) shown in Fig. 2. Several points are to be considered how- ever: 1. This probability relation is only valid for solar-like stars, i.e. F, G, K dwarf stars. Although there are strong indications that it may change for other stars (e.g. M dwarfs), we will assume it to hold independently of stellar properties. This [Fe/H] [Fe/H] Fig. 2. Upper panel: Probability for a solar-type star to possess a giant planet companion as a function of the stellar metallicity (from Santos et al. 2004). Lower panel: Relative normalised dis- tributions of stellar metallicities for stars in the field (black line), and for stars with a giant planet companion (red line). assumption is sufficient because F, G and K dwarf stars form by far the majority of stars with detectable planets in photo- metric surveys. 2. This relation has been calculated independently of the prop- erties of the planetary companion, in particular orbital dis- tance. Because in our case we are strongly biased towards short-period planets, the distribution may be different. This point will be considered in § 3.4. 3. The possibility of multiple planetary systems is not consid- ered. This approach is justified because the probability that several planets belonging to the same system are transiting planets is small for giant planets. 1.4.2. Planetary masses and orbits The masses and orbital characteristics of the planet population are inferred almost entirely from the present radial-velocimetry surveys. This technique yields an accurate determination of the orbital period, and less accurately, of the eccentricity of the orbit. It also yields the value of the mass of the planetary companion times the sine of the orbital inclination from the knowledge of the mass of the parent star. With these values, we can then derive from a random inclination of the orbital planes the planets that are transiting and those that are not as well as the characteristics of their orbit. We test several approaches for the derivations of these quan- tities: – An analytical model: In this approach, we consider inde- pendantly the planet period and its mass. The period of the planet Π follows the model of Brown (2003), the proba- bility density P from a piecewise linear fit to the distribu- tion P(logΠ) = {0.509, 0.165, 0.533} for three period inter- vals bounded by logΠ = {0.43, 0.65, 2.3, 3.5}. The distribu- tion in mass is linear in log from 0.3 to 10 Jupiter masses (Zucker & Mazeh 2001). There is no dependency of these two parameters linked to metallicity. – The radial velocity mass-period “carbon-copy” model: A second approach is to make direct use of the list of planets discovered by radial velocimetry. This is possible because in terms of masses and orbital periods the list is almost unbi- ased for the objects that we consider (massive enough to be above detection thresholds, and with periods much shorter than the lifetimes of the surveys). In this case, we select plan- ets randomly in the RV list, and then allow for a small ran- dom deviation of their mass and period (a uniform deviation from −20% to +20%) in order to avoid too much cluster- ing on the same value. This is particularly important in the case of the period because of the importance of stroboscopic effects in planetary transits (e.g. Pont et al. 2005). – The radial velocity mass-period-metallicity “carbon-copy” model: As a modification to the previous approach, we also consider using the metallicity entry in the RV list, because of correlations between metallicity and orbital period that are otherwise not taken into account (see discussion in sec- tion 3). We proceed slightly differently however than for the mass and orbital period because of the limitations caused by the finite number of planets in the RV list. In this case, we choose to split the list into two low- and high-metallicity lists, and then select the mass and periods in the relevant list. Our fiducial cutoff value is [Fe/H]=−0.07. Figure 3 shows a comparison between observations, the car- bon copy model and the analytical model. It is interesting to no- tice at this point that the carbon copy and analytical models are essentially indistinguishable in these diagrams. The differences with the observations arise only because of our choice to smear the masses and orbital periods when generating our planet pop- ulation. Last but not least, we have to consider the existence of plan- ets that orbit extremely close to their star, with periods shorter than 2 days, as discovered by transit surveys (see table 1). Companions with such short orbital periods have been discov- ered by radial velocimetry in two occasions: HD 41004 b, and Gliese 876 d, with respective masses 18.4 and 0.023 Jupiter masses. These objects are outside the mass range considered for this study, and therefore, there is no giant planets with periods shorter than 2 days in the present radial velocimetry list. In or- der to account for these very close-in planets anyway, we add the planets with periods smaller than 2 days discovered by tran- sit photometry to the list, but with a small tunable probability weight. The fiducial value of this parameter is set so that, on av- erage, the planet list contains one and a half such planet (added to the list of 153 RV planets described in § 1.4). Tests on the effect of this parameter are presented in § 3.3.4. Our fiducial model is the mass-period-metallicity carbon copy model, includes addition of very-close in planets and it is that model which is used in all cases except where specified oth- erwise. Other approaches are also tested depending on the model to highlight particular points. 1.4.3. Physical characteristics and the planetary evolution model Because we are focussing on planets more massive than Saturn, we expect most of them to be made of a significant amount of hydrogen and helium. These giant planets thus undergo a pro- gressive contraction and cooling that depends on four quanti- ties: their age, mass, the amount of flux the planet receives from the central star, and the global amount of heavy elements in the planet (e.g. Guillot 2005). Orbital period [days] Planetary mass [MEarth] Planetary radius [RJup] Fig. 3. From top to bottom, distributions of orbital periods, masses and radii, respectively, of the planets observed by radial velocimetry (black lines), simulated as part of the mass-period “carbon copy” model (red lines), and simulated as part of the analytical model (dotted blue lines) (see text). Models attempting to reproduce the radii of known tran- siting giant planets have however had problems in explain- ing the large radii of some of them (Bodenheimer et al. 2001; Guillot & Showman 2002; Baraffe et al. 2005; Laughlin et al. 2005). Several possibilities have been proposed to explain the discrepancy. We can separate them into two categories: – Mecanisms invoking chance configurations of the plane- tary orbits in the case of these anomalously large plan- ets: this includes the tidal circularization of an eccentric orbit (Bodenheimer et al. 2001), and tidal dissipation for a planet locked in a Cassini spin-orbit resonnance with the star (Winn & Holman 2005). – Effects that would apply to all planets, including problems with the equations of state or opacities, and the dissipation by stellar tides of kinetic energy first generated in the atmo- sphere (Showman & Guillot 2002). The first mecanisms appear to have a low probability of oc- curence (Laughlin et al. 2005; Deming et al. 2005; Levrard et al. 2007). The second possibility therefore seems more likely, but requires in some case the presence of relatively large masses of heavy elements to reproduce the observed radii. A model-dependant estimate of the masses of heavy el- ements present in the currently known transiting Pegasids is shown in Fig. 4. This model relies on the hypothesis that 0.5% of the absorbed stellar flux is used to generate kinetic energy that is subsequently dissipated deep into the planetary inte- rior (Guillot & Showman 2002). As proposed by Guillot et al. (2006), there appears to be a correlation between the amount of heavy elements present in the planet and the metallicity of their parent star. This correlation has to be ascertained, but we choose for our fiducial model to adopt a unique relation between metallicity and mass of heavy elements (treated as a central core in our calcula- tions), corresponding to the dotted line in Fig. 4: MZ = 43.75 × 10 [Fe/H] − 23.7 M⊕. (4) Fig. 4. Mass of heavy elements in transiting Pegasids known by 2006 as a function of the metal content of the parent star rela- tive to the Sun. The mass of heavy elements required to fit the measured radii is calculated on the basis of evolution models including an additional heat source slowing the cooling of the planet. This heat source is assumed equal to 0.5% of the incom- ing stellar heat flux (Showman & Guillot 2002). Horizontal er- ror bars correspond to the 1σ errors on the [Fe/H] determina- tion. Vertical error bars are a consequence of the uncertainties on the measured planetary radii and ages. The metallicity of re- cently discovered planets WASP-1 and WASP-2 (right panel) is not precisely known. The dotted line corresponds to a best fit model. [Adapted from Guillot et al. (2006)]. We limit the range of possible values of MZ to [0, 100M⊕]. Similarly, we adopt a simple boundary condition for our evo- lution calculations: T1bar = 1.25Teq0, (5) where T1bar is the temperature at the 1 bar pressure level and Teq0 is the equilibrium temperature for a zero albedo (see Guillot 2005 for a description), calculated as a function of stellar effec- tive temperature and radius and planet semi-major axis. For simplicity, and because it yields only minor changes on the results, we further choose to neglect the time-dependence in the evolution calculations, and to adopt the equilibrium radius, or the 10 Gyr solution, whichever occurs first. Practically, planetary radii are obtained from interpolations in a table based on three parameters: the planetary mass ranging from 100 to 3000M⊕, the core mass from 0 to 100M⊕ and the equilibrium temperature from 100 to 2000 K. Models were not calculated beyond these values of Teq because of convergence problems. However we allowed for a slight extrapolation of the tables to 2600 K to account for rare extremely hot planets. 1 Similarly, because of convergence problems for planets with small total masses and large core masses, we limited the mass of the core to 75 M⊕ for planets with masses smaller than 180 M⊕. More detailed work is required to better simulate this parame- ter space, including planets less massive than considered in this study. 1 An electronic version of the table is available at www.obs- nice.fr/guillot/pegasids/ Figure 5 shows examples of radii obtained for Teq = 1000 and 2000, K, and core masses of 0 and 100Moplus, respectively, compared to available measurements. eq =2000K eq=1000K no core e, n 100 M⊕ core planets brown dwarfs stars Fig. 5. Theoretical and observed mass-radius relations. The black line is applicable to the evolution of solar composition planets, brown dwarfs and stars, when isolated or nearly isolated (as Jupiter, Saturn, Uranus and Neptune, defined by diamonds and their respective symbols), after 5 Ga of evolution. The dotted line shows the effect of a 15M⊕ core on the mass-radius relation. Orange and yellow curves represent the mass-radius relations for heavily irradiated planets with equilibrium temperatures of 1000 and 2000 K, respectively, and assuming that 0.5% of the incoming stellar luminosity is dissipated at the center (see sec- tion 1.4.3). For each irradiation level, two cases are considered: a solar-composition planet with no core (top curve), and one with a 100M⊕ central core (bottom curve). The transiting extrasolar giant planets for which a mass and a radius was measured are shown with points that are color-coded in function of the planet’s equilibrium temperature. The masses and radii of very low mass stars are also indicated as blue points with error bars. 1.5. Modeling transit events and their detectability We now descibe how this population of stars, planets and their interactions during transits are modelled to reproduce real obser- vations. 1.5.1. PSFs and CCDs Each image of a star is spread by the atmosphere and by the telescope to grow to a specific size and shape when reaching the CCD in the focal plane of the telescope, the so-called point spread function (PSF). The CCD being composed of many dis- crete pixels, these PSFs are then effectively discretized, so that the signal to be analyzed for any given star is composed of many different lightcurves corresponding to the many pixels over the size of its PSF. A further complication arises from the fact that the stellar fields generally chosen by transit surveys are dense, so that many PSFs overlap. Recovering individual stellar light curves from real data is a complex problem. Two popular meth- ods are aperture photometry (Stetson 1987) and image subtrac- tion (Alard & Lupton 1998). (An adaptation of the latter was used to extract the OGLE lightcurves). A refined simulation could include possible spatial and tem- poral variations of the PSFs, and a realistic data reduction pipeline. In our case, we choose to simplify the problem by rely- ing on a posteriori analyses of real light curves to provide us with a global noise budget. We however include background stars be- cause of the important effect of signal dilution. In order to do so, we assume that the PSFs are gaussian with a uniform, constant FWHM. (Non-gaussian PSFs are not diffi- cult to include but we tested in the OGLE case that for a fixed equivalent FWHM, they have a negligible effect on the resulting signal-to-noise ratio of simulated transit events). We consider for each target of the survey the global flux from the main star and the background stars in its neighborhood up to magnitude 22 in the spectral band of observation. The neighborhood zone for background stars is defined as a circle of diameter equal to 4 times the PSF’s FWHM around the photocenter of each target star. Each background star whose photocenter is located in that zone is taken into account for the calculation of the global flux. The global flux is calculated as the sum of the pixels located at less than twice the FWHM of the central star. We thus simulate aperture photometry when image subtrac- tion was used for OGLE (Udalski et al. 2002). The choice of the reduction algorithm indeed affects the sensitivity obtained from real observations. In our simulations, i.e. a relatively idealized case, it would have marginal effects since realistic noises are included a posteriori from the analysis of real lightcurves (see hereafter). 1.5.2. Noise budget and event detectability We choose to separate noise sources into two categories: – ‘White noise’ sources, following gaussian and Poisson laws. The main source of white noise is the photon noise of target stars and their background. The level of white noise for a given target star is obtained from the simulation of the flux of that star and its background in the photometric aperture. – ‘Red noise’, or systematic effects on photometry, that un- dergo temporal correlation. The structure of these systemat- ics in the OGLE photometry have been explored in details by Pont et al. (2006b). These noise sources are both instrumen- tal (jitter and breathing of the CCD, frequency spectrum of stellar field moves on the camera, change of the PSF shapes accross the CCD during the night), and environmental (dif- ferential refraction and extinction, changes of airmass and sky brightness, temperature changes). Rather than trying to simulate instrumental and environmental noise sources accu- rately, which is difficult with the relatively poor knowledge we have of the time spectrum of their combined effects, we use the effective global ‘red noise’ measurements of OGLE- III survey real light curves mentionned in Pont et al. (2006b), which consider the combined effect of these noise sources. Pont et al. (2006b) calculated that, in the presence of a mix- ture of white and red noise (i.e. accounting for photometric sys- tematics), the detection threshold for a transit survey is well de- scribed by a limit on the signal-to-noise ratio defined as: S 2r = k=1 nk 2(σw2/nk + σr2) where Ntr is the number of transits sampled, nk the number of data points in the k-th transit. σw and σr are the standard devia- tion of measurement points of white and red noises, respectively, d is the event depth and n the total number of measurement points during the transit. Specifically, we obtain nk by counting for each transit the number of observation points between the middle of ingress and the middle of egress. Equation 6 makes the disctinction between “white” noise sources that decrease with n1/2, where n is the number of succes- sive measurements, and “red” noise sources that are limited by temporal correlation. Pont et al. (2006b) indeed show that taking the red noise into account makes a large difference on the detec- tion threshold – in general as well as in its dependence to the planet parameters – and that models based on the assumption of white noise can be poor approximations of the actual detection threshold. 2. The OGLE survey: input parameters 2.1. Basic parameters and observational procedure The Optical Gravitational Lensing Experiment (OGLE) has done 6 observation campaigns looking for transiting planets to- wards different fields of view from 2001 (Udalski et al. 2002). It took place at the Las Campanas Observatory, Chile, using the 1.3 m Warsaw telescope and the 8k MOSAIC camera, with a to- tal field of view of 0.34◦2. All observations were made through the I filter. We assume for our PSF simulation an average seeing of 1 arcsec. We analyze in this work the first three OGLE-III observation campaigns dedicated to transit search, as their treatment, anal- ysis and follow-up (with current data processing pipelines) has been completed: – OGLE-III-1 (June 12 to July 28, 2001, described in Udalski et al. (2002); Udalski (2002)). More than 800 im- ages of three fields in the direction of the galactic bulge were collected within 32 nights. The exposure time was 120 s, and each field was observed every 12 min. – OGLE-III-2 (February 17 to May 22, described in Udalski (2003)). More than 1100 images of three fields located in the Carina region of the galactic disk were collected in 76 nights. The exposure time was 180 s, and the temporal resolution was about 15 min. – OGLE-III-3 (February 12 to March 26, described in Udalski et al. (2004)). The photometric data were collected during 39 nights spanning the 43 days of the survey. Three fields of the galactic disk were observed with a time resolu- tion of about 15 min. The exposure time was 180 s. In this article, we will refer to these three observation cam- paigns respectively as ‘Bulge’, ‘Carina’, and ‘Centaurus’ fields. The simulations include the real observation windows of each survey, as kindly provided by A. Udalski. For any transit- ing planet in the simulation, the number of effectively observed transits is used in eq. 6. In order to construct a realistic stellar population, we use the stellar counts per magnitude range obtained by Gould et al. (2006) based on OGLE-II data, which have calibrated photome- try. We then randomly select that number of stars per magnitude from the Besançon model. In order to test the validity of our approach, we calculated the fraction of “stars for which tran- sits are detectable” and compared it to the one determined by Gould. This fraction is defined for a given magnitude range as the number of stars around which a planet orbiting edge-on with r = 1.2R jup and a = 7.94R⊙ can be detected, divided by the total number of stars of that magnitude. As shown by table 2, there is an excellent agreement between our results and those of Table 2. Fraction of stars suitable for transit detection Carina Bulge Vmax Gould 2006 This work Gould 2006 This work 15.5 0.11 0.16 0.138 0.141 16 0.14 0.16 0.125 0.128 16.5 0.16 0.15 0.098 0.105 17 0.16 0.15 0.068 0.080 17.5 0.16 0.14 0.041 0.052 Gould et al. (2006). Note however that for the global simulation, the complete star list is used as the above definition for suitable stars is restricted to planets of a given size and orbital distance. We calculated the average flux for target stars, companions and all the background stars near enough to contribute to the target PSF. We then checked that the average photon noise simu- lated for target stars at a given magnitude was close to real values obtained in OGLE light curves at given magnitude presented in figure 4 of Pont et al. (2006b). 2.2. Modelling the detection threshold The candidates in the OGLE survey have been identified with the BLS transit-search algorith of Kovács et al. (2002). A sub- set of the candidates selected with cuts in the α and SDE pa- rameters of the BLS were examined by eye, and only the best were included in the final list. Therefore, the selection thresh- old is mainly defined by subjective appreciation from an ex- perienced specialist. Recently, Pont et al. (2006b) have pointed out that the effective detection threshold of ground-based tran- sit surveys such as OGLE is importantly affected by correlated noise (photometric systematics). The subjective selection of can- didates is in large part necessary because of the presence of this correlated noise, which produce many spurious detections near the threshold. Gould et al. (2006) chose to model the OGLE se- lection threshold with an α > 12 cut (alpha is equivalent to the signal-to-noise ratio of the transit signal assuming uncor- related noise and homogeneous distribution of the data points in phase). Pont et al. (2006b) have included the effect of corre- lated noise in the signal-to- noise calculation and found that the OGLE selection could be better described by a threshold of 8 on the signal-to-noise ratio of the transit signal calculated in- cluding correlated noise (”S r” in their notation, see Sec. 2.5.2), and without the assumption of homogeneous coverage. While the two thresholds have similar effects on the global number of planet detection, they have a very different dependence on some parameters, such as planet period and host star magnitude. Since the objective or our study is to examine the detection statistics in a multi-dimensional parameter space, we use the Pont et al. (2006b) description of the OGLE detection threshold. To calculate S r, one needs an assumption on the level of red noise present in the photometry. Following Pont et al. (2006b), we use a single-parameter description and assume σr = 3.6 mmag in the Bulge fields, σr= 3.1 mmag in the Carina and Centaurus fields, and σr= 2.1 mmag in all fields after applica- tion of decorrelation algorithms. 2.3. Confirmability of transit-like events with follow-up High-resolution spectra allow the confirmation of the planetary events if spectral lines are deep enough. Several scenarios make the follow up of candidates too difficult: early type stars have lines too weak and too broadened by rotation (type F4 and ear- lier). Stars with magnitudes V > 17.5 are too faint for present instruments and telescopes. This is the limit at which observers estimated not being able to provide low-metallicity stars. Those stars having weaker lines, could also be difficult to follow cor- rectly, but as planets are unlikely to be found near this kind of stars in our model, we did not take that parameter into account. To simulate the feasability of follow-up, we only considered in CoRoTlux the stars matching the criteria V < 17.5 and of type F4 and later. 3. Results of the simulations We present hereafter runs for the three OGLE-III campaigns for the fields in the Galactic bulge, in Carina and in Centaurus. In order to obtain a statistically significant population of detected planets, the simulations were run multiple times. We first examine the consistency between the models and ob- servations for relevant physical variables. In doing so, we choose to compare our model population to the global population of transiting planets discovered by OGLE and other surveys. There is a slight inconsistency in assuming that the parameter compari- son is almost independant of the type of survey and observational strategy. In some cases, this is not true, and a clear distinction between the OGLE planets and the other detections has to be made. We then discuss the problem of the detection statistics, whether observations and models are consistent, and whether a constraint on the (low) frequency of very close-in planets can be deduced. 3.1. Deviation of OGLE planets from maximum likelihood of the simulations We use a Maximum-Likelihood (ML) technique in order to test whether model results and observations agree with each other. We do the tests in two-dimension spaces, in order to qualify pos- sible correlation and exclusion zones. The ML technique is our method of choice as it is a powerful tool for fitting a model to a multi-dimentionnal independant-data distribution (Lyons 1986). Instead of determining an approximate analytical law fitting our results, we use the results of a very large Monte-Carlo draw (1000 times the whole OGLE survey, corresponding to ∼ 9000 planets) to get a map of the density of probability in each 2- dimension grid. We bin our data on a 20x20 grid as a compro- mise between resolution of the models and characteristic varia- tions of the parameters.2 The probability of an event in each bin is considered equal to the normalized number of draws in that Figure 6 shows the logarithm of the probability that an event occurs in each of the 20x20 bins of the mass-radius diagram. The likelihood of a draw of several independant events is defined as the sum of the logarithms of the probabilities of these events. In order to compare our results to any n real discoveries, we first estimate the standard deviation of any n-planets-random-draw compared to the maximum likelihood of the model. We ran- domly select n planets among the simulated detections and cal- culate the likelihood of this draw. We repeat this selection 1000 times in order to have the maximum likelihood and its standard deviation σ, then we compare the deviation of the likelihood of the n real detecions calculated the same way in terms of σ. 2 Tests with different grids yield small variations of the results. As an example, the mass-radius deviation from maximum likelihood is re- spectively 0.67, 0.65 and 0.72 σ for 20x20, 30x30 and 40x40 grids. Henceforth, quantitative comparisons between the simulation re- sults and the known planets are systematically given in the figure captions, whilst the text discusses qualitative comparisons and their implications. For the different figures showing the results of our simulation, we compare the distribution of planets over the detection threshold to the 5 OGLE planets. We also compare our results to the 11 planets discovered by all transit surveys, as their detection biases are similar to OGLE, and to the 14 planets which radius is known (11 from transits and 3 from radial ve- locity surveys) to show how our model can reproduce the whole known population. 3.2. Depth of the transit events and magnitude of the targets stars We first attempt to confirm whether the events detected by the model are consistent with those found in the OGLE fields. Figure 7 is a plot showing transit depth as a function of the mag- nitude of the primary star. Model results are considered detected when the signal-to-noise ratio is sufficient for a detection (see § 2.2). We also show events that are considered photometrically detectable but very hard or impossible to confirm by radial ve- locimetry. The figure evidently shows a good correlation between the black crosses and the red circles that indicate real detections by OGLE, with a range of transit depths and V magnitudes that is very similar between the models and the observations. Our models overpredict slightly the number of transit events around faint stars (V ≥ 17), but this may be due to the difficulty of the follow-up work for these targets. Overall, the agreement between models and observations is good. 3.3. Compatibility of transit surveys with radial-velocimetry observations 3.3.1. Compatibility in the mass-period diagram Figure 8 compares the model and observated mass-period rela- tion. As it is independant of the planetary evolution model, it is a direct test of the compatibility between the results of transit sur- veys and those of radial-velocimetry observations that drive our model results. Again, the comparison is very good, assuming a high-enough frequency of very-close in planets (see discussion in § 3.3.4). One can note especially the absence of planets of rel- atively large mass (several times that of Jupiter) at short orbital distances (P < 5 days), and of detectable transit events for peri- ods longer than ∼ 5 days. This is due especially to the fact that only events with a relatively large number of observed transits are detectable, as required by the S r threshold, which, given the day/night interruptions, imposes a constraint of a short orbital period. Note that this feature is not well reproduced by mod- els in which the threshold is computed from white-noise only (Gould et al. 2006; Gillon et al. 2005). 3.3.2. The OGLE yields with a fixed red noise level We have tested the efficiency of the fiducial model at estimat- ing quantitatively the yield of transit surveys. Gillon et al. (2005) have also simulated OGLE yield in their generic study of multi- ple transit surveys, but with restrictive assumptions on transit de- tectability (only complete events matter for detection purposes) and without considering background stars and red noise, also not using OGLE-fields specific stellar population. We also in- cluded in our simulations the recent RV follow-up that has been Fig. 6. Logarithm of the probability that a simulated detection event occurs in each one of the 20x20 bins of the mass/radius diagram. The likelihood of a multiple-events draw is the sum of the logarithms of the probabilities of the events of this draw. Bins without any occuring event in the large Monte-Carlo draw do not have any probability stated. The likelihood of a n-events draw is the sum of the probabilities of its n events. In this mass-radius diagram, OGLE planets are shown as red circles, planets from other surveys are in orange, and planets from radial velocity surveys are in blue. The likelihood of the 5 OGLE discoveries as a result of a Monte-Carlo draw is −8.7, the maximum likelihood is −7 and the standard deviation to maximum likelihood is 2.54. Hence, the result of the OGLE planets mass-radius distribution is at 0.67σ of the maximum likelihood of the model. Table 3. OGLE yields with fixed red noise level Field Mean red RV follow-up Number of planets of view noise level to Vmag detected simulated with 0 1.5 3 VHJ added (P < 2 days) Bulge 3.6 17.5 2 0.4 0.6 0.9 Carina original 3.1 17.5 3 3.4 4.1 4.8 updated 2.1 17.5 +(0 − 1) +1.1 +1.1 +1.1 Centaurus 3.1 17.0 0 1.4 1.8 2.2 Total 6 6.3 7.6 9.0 done on Centaurus and Carina. We use unpublished information from the OGLE/ESO follow-up team, who found one promising planetary candidate among the Carina fields reprocessed with the systematics- removal algorithm from Tamuz et al. (2005) and none in the Centarus fields, with a magnitude limit near V=17 for the radial velocity follow-up. Table 3 compares the average number of planets detected for 1000 Monte-Carlo draws to real detections from the OGLE survey. The total number of simulated discoveries obtained from this quantitative analysis is in good agreement with the real detec- tions. The differences in the number of detections between the Carina and Centaurus surveys are mainly due to the lower duty cycle of the observations towards Centaurus. A red noise level fixed at 3.6 mmag in the direction of the galactic bulge bans most hot Jupiter detections. The agreement between our quan- titative result and the number of real detections is an indicator of the global efficiency of our approach (stellar and planetary distributions, evolution model and noise budget) for estimating transit survey yield. 3.3.3. The OGLE yields with a variable red noise level So far, we have considered the level of red noise to depend only on the field considered. We attempt now to refine this by consid- ering how the stellar density may affect it. Whereas most ground- based transit surveys have a global red noise level from 2 to 3.5 mmag (Superwasp: Smith et al. (2006), Monitor: Irwin et al. (2007), Hatnet: Pont & ISSI team (2007) and OGLE), the causes of these noise levels seem different, with instruments ranging from 10-cm wide field reflectors to deep-sky several-meter tele- scopes. As seen from table 3, the OGLE fields have differ- ent mean red noise levels (σr = 3.6 mmag for the bulge and σr = 3.1 mmag for Centaurus and Carina before SYS-REM), although the instrument and observational strategy were un- Fig. 7. Depth of the planetary transit events versus magnitude of the parent star in the V band. The five confirmed OGLE detections are shown as circles. Model results are shown as black plusses for detectable events and orange crosses for events that are considered undetectable based on the photometric signal (see text). Blue diamonds correspond to events that would be detectable by photometry alone but that cannot be confirmed by radial velocimetry. Note that the model results correspond to 3 times the full OGLE campaign for more statistical significance. The OGLE planets depth-magnitude distribution is at 0.69σ from the maximum likelihood of the model. Fig. 8. Mass versus period of transiting giant planets. (OGLE planets are red circles, other transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets mass-period distribution is at 0.62σ from the maximum likelihood of the model (0.72σ considering the 11 planets discovered by transit surveys and 0.66σ considering the 14 known planets). Fig. 9. Distribution of the crowding index (see text) of target stars in Carina (black) and in the bulge (red). changed. Looking at what distinguishes these fields, it appears that the most significant difference is the stellar density and therefore the amount of crowding: The bulge field is about twice as dense as the Carina and Centaurus fields. Pont & ISSI team (2007) raise the suspicion that the level of red noise depends strongly on the presence and characteristics of contaminating stars, because e.g. of their different colors and differential re- fraction in the atmosphere. It is hence natural to consider a red noise that depends on a crowding index. We define this crowding index as the fraction of the flux coming from background stars versus that from the target in the photometric aperture. Importantly, we do not consider stel- lar companions as contributing to the red noise because they are generally on the same CCD pixel as the target star and should affect the noise budget much less. Figure 9 shows the differences of crowding index for the tar- get stars with planetary transits (detectable or not) in simulations of the Carina and Bulge fields of view. The mean crowding in- dex for target stars of I < 17 is 0.11 in the Carina field and 0.233 in the Bulge field. We can exclude the fact that all red noise is linked with con- tamination as many stars in the Carina fields are unblended by background stars but still show a high noise level. In order to estimate of the influence of the crowding on the red noise level, we use the following simple relation between red noise level and crowding index: σr = α × Fb + β, (7) where Fb is the fraction of total flux from background stars, determined on a star-by-star basis in our simulations, and α and β are parameters to be determined. This is justified by the behaviour of the red noise seen for instance in SuperWASP, showing a linear increase as a function of background flux (Smith et al. 2006). In order to get the same mean red noise values as Pont et al. (2006b), we obtain α = 0.4 mmag and β = 2.65 mmag. This value of β corresponds to the minimum red noise level obtained for non-contaminated stars in the OGLE fields. Table 4 shows the new number of detections when consid- ering this crowding-dependant red noise level. Compared to ta- ble 3, the number of detections is found to be essentially un- changed for the Carina and Centaurus fields, but it increases by a factor ∼ 3 for the bulge field. This result is more satisfactory be- cause in the previous case, only ∼ 5% of the simulations would yield the detection of 2 planets in the bulge, as observed. 3.3.4. Models, observations and the frequency of very close-in planets As discussed in § 1.4.1, three OGLE planets have orbital periods shorter than 2 days and thus belong to a class of objects yet to be detected by radial velocimetry. So far, we have added one such planet (on average) to our carbon copy list of nearly 200 radial velocimetry planets. In Section 3.3, we have shown that with this assumption, radial-velocity and photometric transit surveys are compatible. We now test the range of frequencies of very close- in planets for which this remains true. In order to do so, we compute the deviation from maximum likelihood in the mass-radius diagram like in Section 3.3, as a function of the number of planets which period is less than 2 days added to the RV list. The result is presented in Fig. 10 and shows that a good match is obtained by adding 1 to 3 short- period planets. Larger numbers are also possible from the point of view of the transit surveys, but would conflict with their non-detection by radial-velocimetry. Adding the other transit- ing planets discovered thus far yields smaller probabilities of occurence of these short-period planets, but not by significant amounts. All in all, and assuming that the radial velocity planets sam- ple is unbiased, we constrain the fraction of main-sequence late stars orbited by very hot giant planets with orbital periods less than 2 days to be (1/1265)(1+0.33 −0.33) at a 60 % confidence level or (1/1265)(1+0.83 −0.5 ) at a 90 % confidence level. The distribution of planets in period between 2 and 5 days is directly obtained from the metallicity-linked distribution (Santos et al. 2004) and the RV planets sample. Adding the dis- tribution we found for planets between 1 and 2 days, we obtain a fraction of (1/215) late main-sequence stars orbited by planets in the 1 to 5 days period range, in good agreement with the results obtained in Gould et al. (2006), who obtained (1/220)(1+1.10 −0.45). Similarly, the distribution we obtain by cutting this sample into two parts with the cut-off at 3 days is compatible, showing: – a slightly higher fraction of really short-period planets (1-3 days) of (1/560) instead of (1/710)(1+1.10 −0.54) at a 90 % confi- dence level in Gould et al. (2006). – a similar fraction of short-period planets (3-5 days) of (1/350) instead of (1/320)(1+1.39 −0.59) at a 90 % confidence level in Gould et al. (2006). The results presented hereafter use the variable red noise level approach, and an RV planet list that is complemented with, on average, 1.5 very-close in planets with periods P < 2 days taken from the OGLE detections. 3.4. The metallicity of the stars harboring transiting planets We now compare the metallicity of the parent stars for our ob- served and modelled populations. A first test using the analyti- cal scenario for the radial-velocity population (Fig. 11) yields a clearly different metallicity distribution, with most of the tran- siting planets observed around low-metallicity stars. We veri- fied that this problem occurs independantly of the assumed stel- lar metallicity distribution, for any realistic stellar population. It arises fundamentally because the global metallicity bias as ob- tained by Santos et al. (2004) or Fischer & Valenti (2005) is not Table 4. OGLE yields with variable red noise level Field RV follow-up Number of planets of view to Vmag detected simulated with 0 1.5 3 VHJ added (P < 2days) Bulge 17.5 2 1.2 1.6 2 Carina original 17.5 3 3.6 4.3 4.9 updated 17.5 +(0 − 1) +1.1 +1.1 +1.1 Centaurus 17 0 1.3 1.9 2.3 Total 5-6 7.2 8.9 10.3 Fig. 11. Period of transiting exoplanets versus metallicity of their parent star. The model is based on analytic relations for the mass and period distributions of planetary companions (see § 1.4.2). (OGLE planets are red circles, other transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets period-metallicity distribution is at 2.94σ from the maximum likelihood of the model (2.51σ considering the 11 planets discovered by transit surveys and 2.63σ considering the 14 known planets). strong enough to compensate for the rarity of very metal-rich stars in the Galaxy. As seen in Fig. 12, the problem disappears when one con- siders the carbon-copy model. Thus, we are led to an important conclusion, that the metallicity distribution of pegasids (periods shorter than 10 days) is fundamentally different from the global exoplanet population. More specifically, there are no Pegasids orbiting F, G, K stars with metallicities smaller than [Fe/H]= −0.07. This has strong consequences for planet formation mod- els (see also Guillot et al. 2006). This work shows that this con- clusion is robust, and is needed to explain the results of the pho- tometric surveys. A finer examination of Fig. 12 shows that while our model planets reproduce globally the metallicity of the ensemble of transiting planets, OGLE stars with planets are on average ∼ 0.1 dex more metal-rich. This can tentatively be explained with a metallicity gradient in the galaxy for OGLE TR-10 ([Fe/H] = 0.28±0.10) and OGLE TR-56 ([Fe/H] = 0.19± 0.07), the two planets discovered in the direction of the galactic bulge. The study of galactic cepheids by Andrievsky et al. (2004) shows a metallicity gradient as a func- tion of distance to the galactic center. In the [6.6, 10.6] kpc-range distance from galactic center, this study finds a linear relation be- tween [Fe/H] and galactocentric distance RG: [Fe/H] = −0.044(±0.004)RG + 0.363(±0.032) (8) Following that relation, the two stars with planets discovered in the direction of the galactic bulge both at a distance around 1500 pc would thus be in a 0.04 dex more metal rich region than the solar neighborhood. Concerning the high metallicity of stars with transiting plan- ets discovered by OGLE in the Carina region, we do not have any reason to think that the metallicity distribution would be different from the solar neighborhood. Our only hypothesis is a low-probability draw for metallicity for the 3 OGLE-Carina planets. Fig. 12. Period of transiting exoplanets versus metallicity of their parent star. The figure differs from Fig. 11 in that our fiducial model, i.e. the mass-period-metallicity “carbon-copy” model is used (see § 1.4.2). (OGLE planets are red circles, other transit surveys in orange, planets from radial velocitiy surveys in blue. Simulated planets detected: black plusses, under threshold: or- ange crosses). The OGLE planets period-metallicity distribution is at 0.76σ from the maximum likelihood of the model (0.36σ considering the 11 planets discovered by transit surveys and 0.39σ considering the 14 known planets) . 3.5. Atmospheric potential energy and orbital distances Because evaporation may affect the planet population, it is in- structive to check whether the potential energy of the atmosphere and the orbital period, two crucial quantities for this process (e.g. Lecavelier des Etangs et al. 2004), also possess a relatively con- sistent distribution. We first test the behavior of the analytical model for the distribution of planets (Fig. 13). This results in a prediction of many planets with large radii (small values of the potential energy for atmospheric escape) at small orbital dis- tances, in patent contradiction with the observations. The problem mostly disappears with the carbon-copy model: Fig. 14) shows that in this case, although we do not obtain a linear correlation between the two variables, we get detections in the right area of the diagram. This is explained as stemming from: – The absence of low-mass planets at small orbital distances, with a possible limiting relation between these two quantities (Mazeh et al. 2005); – The difficulty in detecting planets with larger values of po- tential energy per unit mass (smaller radii) at large orbital distances –although we predict that some of these should be detected by future transit surveys. Our results strengthen the case for the existence of a rela- tion between mass and orbital distance for short-period plan- ets, as advocated by Mazeh et al. (2005): Indeed, the analytic model which is characterized by the presence of small mass planets at small distances yields a distribution of detectable plan- ets that is significantly different from the observations (Fig. 13). Our carbon-copy model that includes implicitely this correlation does not (Fig. 14). 3.6. Planetary radii and stellar irradiation Radius and stellar irradiation (or equivalently equilibrium tem- perature) should be positively correlated, as a planet with a higher irradiation dose will tend to cool and contract more slowly than one that endures less stellar insolation. As Fig. 15 shows, the correlation exist, but is weak, and with a signficant scatter. This is well reproduced by the model. However, it can be noted that HD 149026 b lies away from the cloud of points. In general, we find that our fiducial model generates few points in this region. This can be easily accounted for by slightly modifying the metallicity-core mass relation to allow for larger masses. As planets of small masses and large core masses are more difficult to model anyway, we chose not to attempt fine-tuning the model to this level of detail. This should be postponed for further studies, especially with the discovery of more Saturn-mass transiting planets. 3.7. The mass-radius relation We have checked that our fiducial model predicts the detection of transiting planets with properties that are globally consistent with the observations. We can now examine in more detail the mass-radius relation thus obtained, as it is directly tied to as- sumptions on the compositions and evolutionary models of ex- oplanets. The predictions also have implications for transit sur- veys as it is not clear whether they have detected only the “tip of the iceberg”, ie the few largest giant planets while many smaller ones would lie undetected or not. Results with our fiducial model are presented in Fig. 16. We find that planets with low masses (say, less than Jupiter’s Fig. 13. Potential energy per unit mass (Ep = GM/R) versus orbital period of transiting planets. (OGLE planets are red circles, other transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). Observations are compared to models based on the analytical relations for the mass and period distribution of planetary companions (see § 1.4.2). The OGLE planets energy-period distribution is at 2.18σ from the maximum likelihood of the model (1.86σ considering the 11 planets discovered by transit surveys and 2.47σ considering the 14 known planets). Fig. 14. Potential energy per unit mass versus orbital period of transiting planets. The figure is similar to Fig. 13, except for the fact that our fiducial model is used (see § 1.4.2). (OGLE planets are red circles, other transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets energy- period distribution is at 0.55σ from the maximum likelihood of the model. (0.84σ considering the 11 planets discovered by transit surveys and 0.66σ considering the 14 known planets) Fig. 15. Radius as a function of equilibrium temperature of transiting exoplanets. (OGLE planets are red circles, other transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets equilibrium temperature-radius distribution is at 1.22σ from the maximum likelihood of the model (1.05σ considering the 11 planets discovered by transit surveys and 2.25σ considering the 14 known planets). mass) can both have very large or very small radii, depending on whether they contain a significant mass in heavy elements or not. On the other hand, massive planets have radii which are comparatively better defined. This is mostly due to the fact that we assume a maximum mass of heavy elements of 100 M⊕, a hypothesis that will be tested directly by the discovery of a few massive transiting planets. Our model results once again agree well with the detections made by photometry. Importantly, the yellow crosses in Fig. 16 do not lie significantly below the black ones: we predict that future surveys will not discover a population of small-sized giant planets, at least for masses above that of Saturn. The presence of planets with larger masses of heavy ele- ments should remain marginal because otherwise they would have been detected by present-day surveys, Fig. 16 showing that planets below 1 RJup are already detectable, although in fa- vorable cases (small radius of the primary and bright targets). Quantitatively, simulations in the OGLE fields indicate that if planets had radii uniformly distributed between 0.5 and 1.5 RJup, 18.5% of the planets discovered by the survey would have radii below 1 RJup. This fraction is not negligible and is (although marginally) inconsistent with the sample of 0/11 planets with R < RJup discovered by transit surveys thus far. Therefore, although we cannot statistically rule out the pres- ence of a population of small planets, these would require the formation of extremely metal-rich planets. Our prediction is a consequence of evolution models and of our assumption that planets with masses of heavy elements beyond 100 M⊕ should be rare. Figure 17 shows the ensemble of planets obtained for an ex- tremely large number of draws, with our fiducial model. Voids in the ensemble of crosses correspond to the absence of planets with these masses in the radial-velocimetry list. They should not be considered as significant. The contours in the figure indicate the ensemble of masses and radii expected for planets with dif- ferent masses of heavy elements, from 0 to 100 M⊕. Importantly, the location of these contours is linked to our assumption of an energy source in the planetary interior equal to 0.5% times the irradiation received by the planet. Independently of the details of this assumption, this shows that a statistically significant ensem- ble of known transiting planets would allow a determination of the presence or lack of heavy elements in these objects. We have also tested another assumption regarding the plan- etary evolution model: all planets possess 20 M⊕ mass in heavy elements, 70% of them have no extra heat source, whereas 30% have 3 × 1026 erg s−1 dissipated at the center. With this assump- tion, one can qualitatively explain the observed transiting planets (i.e. the “normal” planets and the “anomalously large” ones, re- spectively) with the exception of HD 149026 b, for which one could argue that the planet comes from a different population. In this case, Fig. 18 shows a distribution of radii that is relatively similar to the previous one (Fig. 16), with the exception that no planet has a radius smaller than 0.8 RJup. In this case, the 2 re- gions corresponding to the “standard” model, and to the “heat dissipation” case are clearly different, especially at the low-mass range of the diagram. Present observations cannot distinguish between the two models, showing the need for additional detections of transit- ing giant planets. Particularly important are planets between the mass of Saturn and that of Jupiter, as this is a mass regime where expected compositional differences have the largest impact. Fig. 16. Mass-radius relation for transiting extrasolar giant planets. (OGLE planets are red circles, other transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets mass-radius distribution is at 0.67σ from the maximum likelihood of the model (0.72σ considering the 11 planets discovered by transit surveys and 0.97σ considering the 14 known planets) . 4. Conclusions We have presented a simulation of photometric transiting sur- veys based on basic knowledge of the stellar and planetary pop- ulations in the galactic neighborhood and on a planetary evo- lution model tuned to the information obtained from transiting giant planets with masses above that of Saturn. This simulation was applied to the OGLE survey, and shown to yield a generally excellent agreement with the transiting planets detected by the survey. We have thus shown that radial velocimetry and photometric surveys are compatible within statistical uncertainties, in agree- ment with Gould et al. (2006). We have derived a frequency of very close-in planets with orbital periods shorter than 2 days around solar-type stars, of (1/1265)(1+0.33 −0.33) at a 60 % confidence level or (1/1265)(1+0.83 −0.5 ) at a 90 % confidence level. Using null results by photometric surveys for given ranges of parameters, we are able to strengthen two results already present in the radial velocimetry data: – Stars with low metallicities ([Fe/H]< −0.07) do not, or are very unlikely to harbour close-in giant planets with orbital periods P < 10 days. This is unlike stars above that metallic- ity threshold (see Fig. 12). – There is a lack of small-mass giant planets below the mass of Jupiter and above that of Saturn for orbital periods P < 3 days (see Fig. 8). Further data is required to precisely quantify these empirical re- sults that bear important consequences for our understanding of planet formation and migration. On the basis of our model, and assumptions concerning the composition of giant planets (i.e. masses of heavy elements be- tween 0 and 100 M⊕), we find that the present detections of tran- siting planets have sampled a population that is quite represen- tative of the main population of giant planets, at least for the ones that are above about half the mass of Jupiter. We hence pre- dict that future transit surveys with higher sensitivities will not discover a significant population of yet undetected Jupiter-mass planets with small sizes, i.e. radii smaller than that of Jupiter (see Fig. 16). Many ground-based transit surveys are in progress, and with the space missions CoRoT (Baglin et al. 2002) and Kepler (Borucki et al. 2003), the number of known transiting planets is expected to rise rapidly over the next few years. This will en- able us to better test the models and quantify some of the results presented in this article. We also hope to be able to discriminate between various models of the evolution and compositions of gi- ant planets, a matter of great importance for formation models. We wish to stress however that a continuation of ground- based transit surveys is desirable even in the presence of simi- lar programs from space. CoRoT will survey 60,000 dwarf stars over five 150 days periods and Kepler about 100,000 over 4 years, implying a maximum potential yield of 55 and 90 transit- ing giant planets, respectively, plus many other smaller planets. For what concerns giant planets, quantifying the fraction of very close-in planets with a 10% accuracy at the 3σ level would re- quire the discovery of ∼ 200 transiting planets. Understanding the evolution and compositions of giant planets will require an even larger number of detections. The radius of a giant planet itself depends mainly on four parameters: the planetary mass, equilibrium temperature, age, and its composition (note that the composition can be considered as a simple parameter only in the case of planets mostly made of hydrogen and helium: smaller planets will be more difficult to model!). Additional energy Fig. 17. Mass-radius relation for a very large number of Monte-Carlo trials using the fiducial model. The curves show the ensemble of planets with masses of heavy elements between 0 and 25, 25 and 50, 50 and 75, 75 and 100 M⊕, respectively. Symbols are as in Fig. 8. Fig. 10. Deviations from a maximum likelihood obtained as a function of NVHJ , the number of very hot jupiter of orbital pe- riods shorter than 2 days added to the radial velocities carbon- copy list. Thick line: Deviation from the maximum likelihood obtained in the mass-radius diagram for the OGLE planets. Thin line: Same deviation but when compared to the ensemble of planets. Dashed line: Standard deviation obtained from a com- parison between the number of simulated planets and the number of detected ones for the OGLE survey (see table 4). Dotted line: Standard deviation obtained from the non-detection of these very close-in planets by radial-velocimetry. sources may occur (such as in the presence of tidal heat dissipa- tion), and the initial conditions and formation history may have their say in the matter as well. Furthermore, the observational uncertainties are generally large. For example, the planetary ra- dius is generally only known to∼ 10%, for a global variation that is relatively small (1 to 1.5 RJup). This implies that to constrain a given correlation to, say 10%, and with four independant vari- ables, hundreds of data points are needed, and thousands would be desirable. This motivates us to seek programs capable of detecting thousands of transiting planets in the mid-term future, and ways to reduce the error bars on the different parameters. One direc- tion is to test the Dome C plateau in Antarctica for such an am- bitious program, which is the purpose of A STEP (Fressin et al. 2005). Other directions exist, such as proposals for similar sur- veys from space. In any case, it is most important that a statisti- cally significant population of exoplanets be characterized for a better understanding of planet formation and our origins. Acknowledgments The code used for this work, CoRoTlux, has been devel- opped as part of the CoRoT science program by the au- thors with major contributions by Aurélien Garnier, Maxime Marmier, Martin Vannier, Suzanne Aigrain and help from Claire Moutou, Stéphane Lagarde, Antoine Llebaria, Didier Queloz and François Bouchy. We want to thank Andrzej Udalski and Michael Gillon for their communications on OGLE data, Frédéric Thévenin for his advices on stellar populations sim- ulation. F.F. has been funded by grants from the French Ministère de la Recherche and by the Société des Amis des Sciences. 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Global Disk Oscillation Modes in Cataclysmic Variables and Other Newtonian Accretors Manuel Ortega-Rodŕıguez1 Escuela de F́ısica & Centro de Investigaciones Geof́ısicas, Universidad de Costa Rica, San José, Costa Rica; and Gravity Probe B, W. W. Hansen Experimental Physics Lab, Stanford University, Stanford, CA 94305-4085 Robert V. Wagoner2 Dept. of Physics & Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305–4060 ABSTRACT Diskoseismology, the theoretical study of small adiabatic hydrodynamical global perturbations of geometrically thin, optically thick accretion disks around black holes (and other compact objects), is a potentially powerful probe of the gravitational field. For instance, the frequencies of the normal mode oscillations can be used to determine the elusive angular momentum parameter of the black hole. The general formalism developed by diskoseismologists for relativistic sys- tems can be readily applied to the Newtonian case of cataclysmic variables (CVs). Some of these systems (e.g., the dwarf nova SS Cygni) show rapid oscillations in the UV with periods of tens of seconds and high coherence. In this paper, we assess the possibility that these dwarf nova oscillations (DNOs) are diskoseismic modes. Besides its importance in investigating the physical origin of DNOs, the present work could help us to answer the following question. To what extent are the similarities in the oscillation phenomenology of CVs and X-ray binaries (XRBs) indicative of a common physical mechanism? Subject headings: accretion, accretion disks — hydrodynamics — stars: indi- vidual (SS Cygni) — stars: individual (VW Hyi) — stars: novae, cataclysmic variables — white dwarfs 1mortega@cariari.ucr.ac.cr 2wagoner@stanford.edu http://arxiv.org/abs/0704.1920v1 – 2 – 1. Introduction During the outburst phase (timescale ∼ a few days) some non-magnetic (B ≪ 106 G) CVs exhibit fluctuations with periods P & 10 s and Q ≡ 1/|dP/dt| ∼ 103 − 107. The phenomenology of these DNOs is very rich (Warner 2004). Also observed are lpDNOs (long period DNOs): P & 30 s and Q ∼ 103 − 107, as well as QPOs (quasi-periodic oscillations): P & 100 s and Q ∼ 10. At times, all three features can be observed simultaneously. The fact that P ∼ the Keplerian frequency at the radius of the white dwarf has moti- vated several candidate theoretical explanations for DNOs, ranging from ‘hot blobs’ (Popham 1999) to star oscillation modes (Papaloizou & Pringle 1978) to explanations based on the dy- namics of the boundary layer or some other type of transition region located where the disk touches the white dwarf. Examples of the last type include magnetic accretion onto a slipping belt at the equator (Warner & Woudt 2002) and ‘spreading layer’ models (Piro & Bildsten 2004). Hydrodynamic oscillations in and near the boundary layer have been investigated by Carroll et al. (1985) and Collins, Helfer & Van Horn (2000). However, their analyses were local (plane-wave WKB) in radius and height. Indeed, Carroll et al. (1985) mention the need of a global analysis. None of these models is truly successful at the quantitative level, although magnetic models appear to be favored. Lately, other proposals based on the dynamics of the accretion disk have appeared. For example, Kluźniak et al. (2005) propose a nonlinear hydrodynamical disk resonance explanation for the oscillations. Part of the reason this approach is attractive is that it could explain the similarity of the phenomenology in CVs and XRBs. Some DNOs exhibit a 1:2 (e.g., the period halving in SS Cyg) and in one case (VW Hyi) a 1:2:3 harmonic structure. In some sources, a 1:15 relation of the frequencies of a QPO and DNO is observed (Mauche 2002; Warner & Woudt 2005), similar to that seen in XRBs. Some high-frequency QPOs in black hole XRBs are in a 3:2 ratio (Abramowicz & Kluźniak 2001; McClintock & Remillard 2006). All the explanations listed above (except the hot blob and resonance models) are not applicable to black holes, given the absence of a surface (even though some could apply to neutron stars). Conversely, QPO explanations based on general relativistic effects are not applicable to the CV case. The only common explanation is based on hydrodynamical oscillations in the accretion disk (and/or corona). With the intention of adding to this debate, this paper explores the possibility that the DNOs in CVs are caused by global hydrodynamical oscillations (normal modes) in the inner accretion disk. To this end we will use diskoseismology, a formalism previously developed to study small adiabatic perturbations in relativistic accretion disks [see, e.g., Wagoner (1999); Wagoner, Silbergleit, & Ortega-Rodŕıguez (2001)]. The DNOs would arise as the – 3 – fluctuations modulate the outgoing radiation. Although Yamasaki, Kato & Mineshige (1995) mention global hydrodynamical p–modes (trapped acoustic waves) in accretion disks around white dwarfs, these modes lie in a different region from the one studied in this paper; in addition, the physical analysis of these authors is much simpler than ours, as they work with vertically integrated variables and limit themselves to axisymmetric modes. In section 2 we briefly discuss the unperturbed disk model and then outline diskoseis- mology in the general case. The rest of the paper applies diskoseismology to cataclysmic variables. An introduction (section 3) precedes the formal WKB solution in section 4. Fi- nally, section 5 presents the results and discusses their implications. 2. Diskoseismology We take c = 1, and express all distances in units of GM/c2 and all frequencies in units of c3/GM (where M is the mass of the white dwarf) unless otherwise indicated. We work in cylindrical coordinates r, ϕ, z. The stationary (∂/∂t = 0), symmetric about the midplane z = 0, and axially symmetric (∂/∂ϕ = 0) unperturbed disk is taken to be described by the standard thin disk models for the Newtonian (Shakura & Sunyaev 1973) and the relativistic (Novikov & Thorne 1973) cases. In particular, we note that the velocity components vr = vz = 0 (the disk fluid moves in nearly circular orbits), the disk semi-thickness h(r) ∼ cs/Ω ≪ r (where cs and Ω are the speed of sound and the Keplerian frequency, respectively), the disk is optically thick and radiatively efficient, the viscous stress is given by αp (i.e., the pressure multiplied by a constant parameter), and typical values for the disk density and central object mass make the neglect of self-gravity, at both the unperturbed and the perturbed level, a good approximation. Further properties of the unperturbed disk will be discussed in section 5. Akin to helioseismology, diskoseismology is the theoretical study of the normal modes of small adiabatic hydrodynamical global perturbations in geometrically thin, optically thick accretion disks. In this section, we describe diskoseismology in general terms. In the following sections, we will use Newtonian diskoseismology to study oscillations in white dwarf accretion disks. Fortunately, the relativistic and nonrelativistic models are formally very similar. Even though the unperturbed disk is not in equilibrium, this perturbative approach is justified whenever the viscous time scales during which the disk evolves as a whole are much longer than the modes’ periods, and whenever the turbulence has spatial scales which are smaller than the modes’ spatial scales, and it is quasi-steady on the time scales of the modes. (These conditions hold for the modes dealt with in this paper.) – 4 – We consider barotropic disks [p = p(ρ), giving vanishing buoyancy frequency] and as- sume that the results thus obtained are generic, since typically the buoyancy frequency is much less than the characteristic dynamical frequency Ω. In this case hydrostatic equilibrium provides the vertical density and pressure profiles ρ = ρ0(r)(1− y2)g , p = p0(r)(1− y2)g+1 , g ≡ 1/(Γ− 1) > 0 , (1) where Γ > 1 is the adiabatic index. One has Γ = 4/3 within any radiation pressure dominated region of the disk, and Γ = 5/3 within any gas pressure dominated region (such as the white dwarf accretion disks). The disk surfaces are at y = ±1, with y related to the vertical coordinate z by To investigate the eigenmodes of the disk oscillations starting from these assumptions, one applies the formalism that Ipser & Lindblom (1991, 1992) developed for perturbations of purely rotating perfect fluids, starting from the equations of conservation of mass and momentum. One can thus express the Eulerian perturbations of all physical quantities through a single function δV ∝ δp/ρ (the Eulerian perturbation of the pressure divided by the density) which satisfies a second-order partial differential equation. Due to the stationary and axisymmetric background, the angular and time dependences are factored out as δV = V (r, z) exp[i(mϕ+ σt)], where σ is the eigenfrequency, and the governing PDE becomes ω2 − κ2 V = 0 , (2) where β = dt/dτ , grr is the metric coefficient, cs stands for the speed of sound, the corotation frequency ω ≡ σ +mΩ , (3) Ω(r) is the Keplerian frequency, and κ(r) is the radial epicyclic frequency (i.e., the frequency of radial perturbations of free particle circular orbits). An assumption of strong variation of modes in the radial direction (characteristic radial wavelength λr ≪ r) ensures the approximate WKB separability of variables in this PDE, V (r, z) = Vr(r)Vy(r, y). The ‘vertical’ part, Vy, of the functional amplitude V (r, z) varies slowly with r. They obey (Perez et al. 1997, hereafter referred to as RD1) (1− y2) − 2gy + 2gω2 1− y2 Vy = 0 , (4) (ω2 − κ2) (ω2 − κ2) + β2grrc 2 − κ2) Vr = 0 , (5) – 5 – where ω∗ ≡ ω/Ω⊥ ; (6) Ω⊥(r) refers to the vertical epicyclic frequency (i.e., the frequency of vertical perturbations of free particle circular orbits), and the speed of sound cs is evaluated at midplane. In these equations Ψ(r) is the slowly-varying separation function. These equations have been used in the past to reveal the properties of different types of modes in relativistic accretion disks. In this paper we solve the equations in the Newtonian limit. We employ, as in the original papers, WKB methods in a straightforward manner. 3. Diskoseismic Modes for the White Dwarf Case: Physical Parameters and Scope We now apply diskoseismology to the accretion disks of white dwarfs. We are thus working in the Newtonian limit (for a mass M = M⊙ white dwarf, its radius R ∼ 5 × 103, corresponding to 7.4 × 103 km), which amounts to setting grr = β = 1, g = 3/2 in the equations of the previous section. We note in particular that Ω(r) = κ(r) = Ω⊥(r) = 1/r 3/2 , (7) i.e., the rotational, radial epicyclic, and vertical epicyclic frequencies are all equal (up to corrections of relative size ∼ 1/r or smaller). As we will see, this results in a qualitative change in the nature of the solutions with respect to the relativistic case. This is a straightforward problem with few free parameters. In addition to the white dwarf mass M , there is ri [≥ R(M)], the inner radius of the accretion disk. Then, one needs a functional form for the speed of sound. It is fortunate that the properties of the unperturbed disk (viscosity and luminosity) enter into the formalism only through the speed of sound, and that the dependence of the speed of sound on these properties is rather weak. In this exploratory calculation, we ignore the complications related to the nature of the transition from the disk to the star surface or magnetosphere and assume that DNO oscil- lations are confined to the disk. We introduce a disk inner boundary condition parameter, given our ignorance of the physical conditions in the (magnetic) boundary layer. As we will see, the results are rather insensitive to this parameter. The approximate character of our analysis should be stressed. A deeper study of boundary layer dynamics is needed in order to understand better the physical conditions that would allow for the existence of the modes described in this paper. For instance, one could ask how significant is the damping and ‘leakage’ of the modes into the magnetosphere, where the radial epicyclic frequency is given – 6 – by the more general expression κ2 = 2Ωr−1d(r2Ω)/dr, if the modes are still confined to a small vertical extent h ≪ r. As in the case of relativistic diskoseismology (with black holes or weakly magnetic neutron stars), the formalism reveals the existence of different types of modes. In this first diskoseismic investigation of cataclysmic variables, we choose to study the g–modes, since they are trapped near the inner edge of the disk and are therefore the most luminous (and robust). By definition, the g–modes have corotating eigenfrequencies which are smaller than the gravitational radial oscillation (epicyclic) frequencies at the involved radii. In other words, the (smaller) pressure restoring forces act against the gravitational restoring forces. (The ‘g’ in the name comes from a mathematical similarity to the g–modes within slowly rotating stars.) The acoustic (p–) modes extend to the outer region of the disk, thus involving strong effects of viscosity and the uncertain boundary conditions at its outer radius. 4. Formal WKB Solution of the Eigenvalue Problem for G–modes In this section we will proceed to solve formally the eigenvalue problem in order to obtain equation (28), our ‘master’ equation, from which all further results in the paper are derived. Taking into account expressions (7), the separated equations (4) and (5) reduce in the Newtonian limit to (1− y2) d − 3y dVy + 3ω2 1− y2 Vy = 0 , (8) (ω2 − Ω2) (ω2 − Ω2) + c−2s (ω 2 − Ω2) Vr = 0 . (9) (Recall from section 2 that g = 3/2 for white dwarf accretion disks.) Equations (8) and (9) are two coupled eigenvalue equations, the eigenvalues being ω and Ψ (or equivalently, σ and Ψ). We will solve these equations within a straightforward WKB formalism as described by Ortega-Rodŕıguez, Silbergleit, & Wagoner (2002) (hereafter referred to as RD3). With respect to boundary conditions, it turns out that it is sufficient to require that the vertical eigenfunction and its derivative remain finite at the surface of the disk and to introduce an inner boundary parameter for the radial equation, as mentioned. By definition, g–modes exist wherever ω2 < κ2(= Ω2). This means that the mode exists between an inner radius (which is ri, the inner radius of the accretion disk, for the cases studied in this paper) and the radius r+, defined as the radius at which |ω| = Ω. The mode decays exponentially for r > r+. – 7 – 4.1. WKB Solution of the Radial Eigenvalue Problem We will cast equation (9) in a form amenable to a WKB solution, using a procedure which is similar to the one developed in RD3. We introduce the dependent variable W ≡ 1 Ω2 − ω2 , (10) and the independent variable c−2s (r Ψ(r′) dr′ , (11) so that equation (9) can be written in the simpler form + S(τ)W = 0 , S(τ) ≡ c2s(Ω 2 − ω2) . (12) The mode exists in the range ri < r < r+ or, equivalently, 0 < τ < τi ≡ τ(ri). The function S(τ) changes sign at τ = 0 (where r = r+). The boundary conditions are that W decays for τ < 0, while at the other extreme, at τ = τi, we parameterize our ignorance at the inner boundary by means of the quantity θi: (Ω2 − ω2) cos θiW − sin θi = 0 , (13) which is equivalent to cos θi − sin θiVr = 0 . (14) A straightforward WKB solution can be obtained for equation (12) at all points except for τ close to 0: W ∝ S−1/4(τ) cos S1/2(τ ′)dτ ′ + Φ , (15) where Φ is a constant to be determined. For τ near zero, the solution is given by the Airy function of the first kind: W ∝ Ai(−s1/3+ τ) , s+ ≡ dS(0)/dτ > 0 . (16) The asymptotic matching of solutions (15) and (16) in the usual manner yields Φ = π/4+nπ, where n is an integer, and ∫ r+(σ) c−1s (r) (Ω2 − ω2)(Ψ/ω2 − 1)dr = π(n± 1/4) (17) for the cases θi = π/2 and 0, respectively. Thus n is the radial mode number constrained by the fact that the integral in equation (17) is positive. – 8 – 4.2. WKB Solution of the Vertical Eigenvalue Problem Using the new independent variable (1− y′2)−3/2dy′ , (18) we can write equation (8) in the simpler form +Q(ζ)Vy = 0 , Q(ζ) ≡ 3ω2∗ 1− y2 1− y2 , (19) which is amenable to a WKB solution. We now discuss the issue of boundary conditions. In view of the symmetry of the differential equation (8), one has = 0 or Vy = 0 (20) for even and odd modes in the disk, respectively, and it is enough to consider the interval 0 < y < 1. Near the singular boundary point y = 1, equation (8) can be written + . . . + . . . Vy = 0 , (21) where the dots represent non-singular terms. According to the analytical theory of second order ODEs [see, e.g., Olver (1982)], the general solution to equation (8) near y = 1 has the Vy = C1v1(y) + C2(1− y)−1/2v2(y) , (22) where C1 and C2 are arbitrary constants, and v1(y) and v2(y) are converging power series of (1− y). Imposing the boundary condition that the eigenfunction and its derivative be finite at y = 1 implies thus that C2 = 0. As noted in RD1 and implied by equation (21) and the boundary condition, the vertical eigenfunction near the boundary can be written in terms of the Bessel function of the first kind as Vy(y) ∝ (1− y)−1/4J1/2(2|ω∗| (3/2)(1− y) ) , (23) the asymptotic (large argument) expression of which is Vy(y) ∝ (1− y)−1/2 cos 6(1− y)− (π/2) . (24) – 9 – On the other hand, equation (19) can be solved with a WKB method for values of y not close to 1: Vy ∝ Q−1/4(τ) cos Q1/2(ζ ′)dζ ′ − Iπ/2 , (25) where I = 0, 1 for the even and odd eigenfunctions, respectively. Matching this with the asymptotic expression (24) of the boundary solution (23), we arrive at 1 + (Ψ/ω2 − 1)(1− y2) 1− y2 3|ω∗| . (26) The parity (of the vertical eigenfunction) and the vertical mode number j are contained in a single quantity J ≡ j + 3/4 ± 1/4 for the odd and even cases, respectively. Thus J takes positive integer and half-integer values (see Table 1). It should be stressed that, because of the method used to solve the equations, the above results are accurate only when Ψ/ω2 − 1 ≫ 1 for all r in the range ri < r < r+. 4.3. Uncoupling Equations (17) and (26) are two eigenvalue equations (the eigenvalues being Ψ and ω), which need to be uncoupled. We start by inverting the integral I on the left hand side of equation (26) to find Ψ/ω2 − 1 as a function of I: − 1 = f(I) . (27) Once this is done numerically, the function f(I) can be substituted in equation (17): c−1s (r) (Ω2 − ω2)f(πJ/ 3|ω∗|)dr = π(n± 1/4) . (28) The left–hand side of equation (28) now only depends on the eigenfrequency σ and not on Ψ [recall definition (3)]. All further results in this paper are obtained from this equation. 5. Results and Discussion In order to use equation (28) and look for solutions with integer values of n ≥ 0, we need a value for ri and a functional form for the speed of sound cs. We will take ri = 5000, corresponding to 7383(M/M⊙) km. The restriction ri ≥ R, the white dwarf radius, places a – 10 – limit on its mass M . Using a standard mass–radius relation (Nauenberg 1972), this value of ri implies that M ≥ 0.76M⊙. As for the speed of sound, there are different solutions given by Shakura & Sunyaev (1973), depending on the region of the disk in which one is working. For the values of interest in this paper, the relevant expression is the one corresponding to the outermost solution, in which temperatures are low enough that pressure is dominated by gas (not radiation) and opacity is dominated by photon absorption in the form of nonrelativistic thermal bremsstrahlung (‘free-free’ transitions). Their solution for the speed of sound is given by (cs/c) −1 = 300α1/10(M/M⊙) 1/4Ṁ −3/20 3/8 , (29) where c, α, M , and Ṁ16 stand, respectively, for the speed of light, the (dimensionless) viscosity parameter, the mass of the white dwarf, and the mass accretion rate in units of 1016 g/s. The small powers of the three parameters render our results relatively insensitive to them. We remind the reader that the interior structure of the unperturbed accretion disk enters our formulation only through the speed of sound. As mentioned above, we parameterize our ignorance about the radial inner boundary by means of the quantity θi introduced in equation (14). 5.1. Axisymmetric Modes The eigenmodes are described by the azimuthal, radial and vertical mode numbers m, n, j, plus the value of the parity of the vertical eigenfunction. We start by studying the m = 0 modes. For them, the condition Ω2 −ω2 > 0 becomes Ω2 − σ2 > 0, which means that the modes are trapped between ri and r+ = |σ|−2/3. Using the characteristic values ri = 5000, M = M⊙, α = 0.1 (Lasota 2001), and Ṁ16 = 1 (Frank, King & Raine 2002), we calculate σ for different values of n, j and parity for the case θi = π/2. (Note that the values of j and parity are contained in J in the manner described in section 4.2 and Table 1.) Table 2 shows the results. The dimensional frequency f = 3.23 × 104(M⊙/M)|σ| Hz. The n = 0, J = 1 entry thus corresponds to a period of 12 seconds. Table 3 shows ∆r/ri ≡ (r+ − ri)/ri for the same values of the parameters. As in the case of axisymmetric g–modes in relativistic accretion disks (RD1), the value of cs/c is small enough that: a) the small n eigenfrequencies are close to the maximum value of κ(r) [which equals Ω(ri) = 2.83× 10−6 in the present case], approaching it as j increases; b) the modes’ – 11 – radial width decreases as j increases and increases as n increases. Self-consistency requires that the parameter Ψ/ω2 − 1 not be small, and the larger it is, the better the WKB approximation. It turns out that Ψ/ω2 − 1 ∼ 1 when J = 1 and − 1 ≫ 1 when J > 1, so the procedure passes this test. (Incidentally, it is interesting that the minimum value of Ψ/ω2 − 1 over the allowed values of the radius does not depend on n or m, only on J .) There is no J = 1/2 entry in Table 2 because there is no WKB solution. A direct numerical solution of equations (8) and (9) shows that the absence of a solution is not an artifact of the WKB method, although it could result from the assumptions made in obtaining these equations. The possibility remains open that there are modes not detected by our methods. Next we compute the behavior of the eigenfrequency σ as the various parameters change. For the choice J = 1, n = 0, θi = π/2, we obtain ∂ log σ/∂ log ri = −1.5 , (30) ∂ log σ/∂ logα = 0.0041 , (31) ∂ log σ/∂ logM = 0.010 , (32) ∂ log σ/∂ log Ṁ = −0.0061 , (33) which means that the dependence of σ on α, M , and Ṁ can be neglected since it is much weaker than its (expected) dependence on ri. Moreover, a change of θi from π/2 to 0 implies only a change of a few percent in σ. Note, however, that there can be no n = 0 solution with θi = 0 because then the RHS of equation (17) would be negative. Also note that equation (32) assumes a fixed value of ri. Care must be taken then when comparing with observations, since usually the parameters are not independent. For example, for a set of white dwarfs of fixed composition there is a relationship between their radius (R) and mass. So since σ ∼= σ(ri) and ri = ri(M, Ṁ) (for fixed surface magnetic field), one has thus, for fixed Ṁ , ∂ log σ ∂ logM ∂ log ri ∂ logM d log σ d log ri = 0.50 , (34) for the case M ∝ r−3i (Nauenberg 1972) appropriate to low–mass white dwarfs with weak magnetic fields (ri = R). When computing the variation of σ with the luminosity L for any specific cataclysmic variable, if one assumes that ri ∝ L−γ , where γ is a constant, one obtains d logP d logL ∂ log ri ∂ logL d log σ d log ri = −1.5 γ . (35) – 12 – Such a dependence of ri on luminosity could be due to strong magnetic field effects (Shapiro & Teukolsky 1983). The idea is that inside the magnetosphere the accretion flow is controlled by magnetic fields, so one takes ri to be the equatorial radius of the magnetosphere. Its value should be only moderately greater than the white dwarf radius R for those magnetic field strengths thought to be present in many of those white dwarf systems which exhibit DNOs. For exam- ple, the value of d logP/d logL = −0.091 for the dwarf nova SS Cygni reported by Mauche (1996) is reproduced by our formalism provided γ = 0.06. In summary, the model may be consistent with observations. The radial width of the modes may be large enough to make them visible, and the periods are consistent, although this is hardly surprising as they are related to the Keplerian period at ri. This leads to the limit P ≥ 10.94(M⊙/M)1/2[ri(km)/7383 km]3/2 seconds on the periods of axisymmetric modes. However, the physics of the magnetosphere is at present too uncertain to provide a means to test the model accurately. At this point it makes sense to look at the higher m modes. 5.2. Non-Axisymmetric Modes Here we discuss the m > 0 modes. Note that from definition (3) and the fact that only |ω| enters in the equations, we can, without loss of generality, consider only the non-negative m cases provided we allow for negative σ. The condition Ω2 − ω2 > 0 implies that −m− 1 < σ/Ω(r) < −m+ 1 . (36) Therefore, g–modes are trapped between ri and r+ = (m+1) 2/3|σ|−2/3. The inner boundary of the mode (r−) is ri, provided that |σ| ≥ (m− 1)Ω(r). The negative sign of σ reflects the fact that the oscillations’ wavefronts travel in the same direction as the fluid. Table 4 and Figure 1 show some properties of the n = 0, J = 1 modes for the m = 0, 1 and 2 cases. Interestingly, the results are consistent with the observation of a 1:2:3 harmonic structure in the power spectra of the DNO VW Hyi, reported by Warner & Woudt (2005). The observed peaks remain harmonic even during a threefold increase in the fundamental frequency. As in other models, the increase in frequency is due to a decrease in ri induced by an increase in Ṁ (and thus L). If one expresses σ in terms of r+ and works to lowest order in ∆r/ri, from equation (28) one obtains ∆r/ri ∼ (n± 1/4)2/3(m+ 1)−1/3ǫ2/3 . (37) – 13 – Since ǫ ≡ h(ri)/ri ∼ (cs/Ωr)ri, using equation (29) and our chosen set of parameters (which only weakly affect the estimate) gives ǫ ≈ 4.2 × 10−3 r1/8i ≈ 1.2× 10−2. Then equation (37) is seen to agree approximately with the values in Tables 3 and 4. The m > 0 diskoseismic modes are in principle harder to observe than the axisymmetric modes, since in the former the modulation of the outgoing radiation is not as efficient due to cancelations over the entire disk (and their radial extent is somewhat smaller). However, the fact that the white dwarf can eclipse parts of the inner disk renders these modes more visible. We predict that the energy spectrum of the DNOs is characteristic of the very inner disk (and not other regions such as the magnetosphere), and thus should be close to blackbody at the highest disk temperature. As in the case of g–modes in relativistic accretion disks, the eigenfrequency splittings when m changes by unity are much larger than the n and j splittings. The g–modes that are trapped by the general relativistic behavior of the radial epicyclic frequency κ have m splittings that are not as harmonic as those found here (Table 4). In particular, their eigenfrequencies are only proportional to m for m & 2 [an interesting difference between the relativistic and the nonrelativistic eigenfrequencies is that, for m ≫ 1, |σ| → mΩ(ri) for the former (see RD1) while |σ| → (m+ 1)Ω(ri) for the latter]. However, it must be noted that it appears that g–modes with small mode numbers are viscously unstable, at least if the viscosity acts in the usual hydrodynamic way. This result is proven for the relativistic case in Ortega-Rodŕıguez & Wagoner (2000), where the exponential growth rate of the mode is roughly αΩ. For the nonrelativistic g–modes studied in this paper, an application of their method yields the same result. Furthermore, these low–lying modes do not contain a corotation resonance [where ω(r) = 0]. 6. Comments Thus, if the ideas in this paper are correct and DNOs are linear diskoseismic modes then the results could suggest a scenario in which white dwarf DNOs and black hole QPOs are produced through somewhat different physical mechanisms. After all, as Kluźniak et al. (2005) themselves point out, the phenomenology is similar but not identical. While in white dwarf DNOs the simultaneous observation of two or even three of the harmonics is well established, in black holes usually only one mode of oscillation is observed at a given moment. Furthermore, white dwarf DNOs (and neutron star QPOs) have frequencies that vary with time, while QPOs in black holes have fixed frequencies. In neutron star XRBs, the (twin) high–frequency QPOs are analogous to the DNOs, but their frequency ratio is much closer to unity (van der Klis 2006). It appears to be controlled (magnetically) by the neutron – 14 – star, since their frequency difference is close to the neutron star spin frequency or half of it. We note that some DNOs are split at a QPO frequency. We also note that with increasing radius of a neutron star magnetosphere, their g-modes should approach (for ri & 8) those of this paper. We hope to investigate the magnetic boundary layer in more detail in a future paper. How much will the boundary layer contribute to the damping (or growth) of the g–modes? Further work should try to explain why the value of Q is so high. Naively, one would expect Q ∼ 1/α. (It is fortunate that most of our results are fairly insensitive to the value of α.) The nonlinear growth and coupling of the modes must also be investigated. This work was supported by grant 075-2002 (Incentivos) of Ministerio de Ciencia y Tecnoloǵıa, Costa Rica, grant 829-A3-078 of Vicerrectoŕıa de Investigación, Universidad de Costa Rica, and NASA grant NAS 8-39225 to Gravity Probe B. We are also grateful to the Aspen Center for Physics for its support. REFERENCES Abramowicz, M. A. & Kluźniak, W. 2001, A&A, 374, L19 Carroll, B. W., Cabot, W., McDermott, P. N., Savedoff, M. P. & Van Horn, H. M. 1985, ApJ, 296, 529 Collins, T. J. B., Helfer, H. L. & Van Horn, H. M. 2000, ApJ, 534, 944 Frank, J., King, A., & Raine, D. 2002, Accretion Power in Astrophysics (Cambridge: Cam- bridge University Press) Ipser, J. R. & Lindblom, L. 1991, ApJ, 379, 285 Ipser, J. R. & Lindblom, L. 1992, ApJ, 389, 392 Kluźniak, W., Lasota, J.-P., Abramowicz, M. 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L. & Teukolsky, S. A. 1983, Black Holes, White Dwarfs, and Neutron Stars (New York: Wiley) van der Klis, M. 2006, in Compact Stellar X-Ray Sources, ed. W. H. G. Lewin & M. van der Klis (Cambridge: Cambridge Univ. Press), p. 39 Wagoner, R. V. 1999, Phys. Rep., 311, 259 Wagoner, R. V., Silbergleit, A. S., & Ortega-Rodŕıguez, M. 2001, ApJ, 559, L25 Warner, B. 2004, PASP, 116, 115 Warner, B. & Woudt, P. A. 2002, MNRAS, 335, 84 Warner, B. & Woudt, P. A. 2005, ASP Conference Series 330, 227 (San Francisco: ASP) Yamasaki, T., Kato S., & Mineshige, S. 1995, PASJ, 47, 59 This preprint was prepared with the AAS LATEX macros v5.2. – 16 – Table 1. Value of J as a function of j and parity j parity J 0 even 1/2 0 odd 1 1 even 3/2 1 odd 2 – 17 – Table 2. Values of the eigenfrequency σ × 106 for m = 0 g–modes. n 1 3/2 2 0 2.65 2.73 2.75 1 2.39 2.55 2.61 2 2.23 2.43 2.51 Table 3. Values of the modes’ fractional radial width ∆r/ri for m = 0 g–modes n 1 3/2 2 0 0.044 0.024 0.019 1 0.12 0.072 0.055 2 0.17 0.11 0.083 Table 4. Harmonic structure of the J = 1, n = 0 g–modes m −106 σm σm/σ0 ∆r/ri 0 2.65 1 0.044 1 5.39 2.03 0.033 2 8.14 3.07 0.028 – 18 – 1 1.05 1.1 1.15 1.2 m = 0 m = 1 m = 2 Fig. 1.— The horizontal lines show the radial extent and the frequency of the three lowest fundamental (n = 0, J = 1) axial g-modes. Introduction Diskoseismology Diskoseismic Modes for the White Dwarf Case: Physical Parameters and Scope Formal WKB Solution of the Eigenvalue Problem for G–modes WKB Solution of the Radial Eigenvalue Problem WKB Solution of the Vertical Eigenvalue Problem Uncoupling Results and Discussion Axisymmetric Modes Non-Axisymmetric Modes Comments

Diskoseismology, the theoretical study of small adiabatic hydrodynamical global perturbations of geometrically thin, optically thick accretion disks around black holes (and other compact objects), is a potentially powerful probe of the gravitational field. For instance, the frequencies of the normal mode oscillations can be used to determine the elusive angular momentum parameter of the black hole. The general formalism developed by diskoseismologists for relativistic systems can be readily applied to the Newtonian case of cataclysmic variables (CVs). Some of these systems (e.g., the dwarf nova SS Cygni) show rapid oscillations in the UV with periods of tens of seconds and high coherence. In this paper, we assess the possibility that these dwarf nova oscillations (DNOs) are diskoseismic modes. Besides its importance in investigating the physical origin of DNOs, the present work could help us to answer the following question. To what extent are the similarities in the oscillation phenomenology of CVs and X-ray binaries (XRBs) indicative of a common physical mechanism?

Introduction During the outburst phase (timescale ∼ a few days) some non-magnetic (B ≪ 106 G) CVs exhibit fluctuations with periods P & 10 s and Q ≡ 1/|dP/dt| ∼ 103 − 107. The phenomenology of these DNOs is very rich (Warner 2004). Also observed are lpDNOs (long period DNOs): P & 30 s and Q ∼ 103 − 107, as well as QPOs (quasi-periodic oscillations): P & 100 s and Q ∼ 10. At times, all three features can be observed simultaneously. The fact that P ∼ the Keplerian frequency at the radius of the white dwarf has moti- vated several candidate theoretical explanations for DNOs, ranging from ‘hot blobs’ (Popham 1999) to star oscillation modes (Papaloizou & Pringle 1978) to explanations based on the dy- namics of the boundary layer or some other type of transition region located where the disk touches the white dwarf. Examples of the last type include magnetic accretion onto a slipping belt at the equator (Warner & Woudt 2002) and ‘spreading layer’ models (Piro & Bildsten 2004). Hydrodynamic oscillations in and near the boundary layer have been investigated by Carroll et al. (1985) and Collins, Helfer & Van Horn (2000). However, their analyses were local (plane-wave WKB) in radius and height. Indeed, Carroll et al. (1985) mention the need of a global analysis. None of these models is truly successful at the quantitative level, although magnetic models appear to be favored. Lately, other proposals based on the dynamics of the accretion disk have appeared. For example, Kluźniak et al. (2005) propose a nonlinear hydrodynamical disk resonance explanation for the oscillations. Part of the reason this approach is attractive is that it could explain the similarity of the phenomenology in CVs and XRBs. Some DNOs exhibit a 1:2 (e.g., the period halving in SS Cyg) and in one case (VW Hyi) a 1:2:3 harmonic structure. In some sources, a 1:15 relation of the frequencies of a QPO and DNO is observed (Mauche 2002; Warner & Woudt 2005), similar to that seen in XRBs. Some high-frequency QPOs in black hole XRBs are in a 3:2 ratio (Abramowicz & Kluźniak 2001; McClintock & Remillard 2006). All the explanations listed above (except the hot blob and resonance models) are not applicable to black holes, given the absence of a surface (even though some could apply to neutron stars). Conversely, QPO explanations based on general relativistic effects are not applicable to the CV case. The only common explanation is based on hydrodynamical oscillations in the accretion disk (and/or corona). With the intention of adding to this debate, this paper explores the possibility that the DNOs in CVs are caused by global hydrodynamical oscillations (normal modes) in the inner accretion disk. To this end we will use diskoseismology, a formalism previously developed to study small adiabatic perturbations in relativistic accretion disks [see, e.g., Wagoner (1999); Wagoner, Silbergleit, & Ortega-Rodŕıguez (2001)]. The DNOs would arise as the – 3 – fluctuations modulate the outgoing radiation. Although Yamasaki, Kato & Mineshige (1995) mention global hydrodynamical p–modes (trapped acoustic waves) in accretion disks around white dwarfs, these modes lie in a different region from the one studied in this paper; in addition, the physical analysis of these authors is much simpler than ours, as they work with vertically integrated variables and limit themselves to axisymmetric modes. In section 2 we briefly discuss the unperturbed disk model and then outline diskoseis- mology in the general case. The rest of the paper applies diskoseismology to cataclysmic variables. An introduction (section 3) precedes the formal WKB solution in section 4. Fi- nally, section 5 presents the results and discusses their implications. 2. Diskoseismology We take c = 1, and express all distances in units of GM/c2 and all frequencies in units of c3/GM (where M is the mass of the white dwarf) unless otherwise indicated. We work in cylindrical coordinates r, ϕ, z. The stationary (∂/∂t = 0), symmetric about the midplane z = 0, and axially symmetric (∂/∂ϕ = 0) unperturbed disk is taken to be described by the standard thin disk models for the Newtonian (Shakura & Sunyaev 1973) and the relativistic (Novikov & Thorne 1973) cases. In particular, we note that the velocity components vr = vz = 0 (the disk fluid moves in nearly circular orbits), the disk semi-thickness h(r) ∼ cs/Ω ≪ r (where cs and Ω are the speed of sound and the Keplerian frequency, respectively), the disk is optically thick and radiatively efficient, the viscous stress is given by αp (i.e., the pressure multiplied by a constant parameter), and typical values for the disk density and central object mass make the neglect of self-gravity, at both the unperturbed and the perturbed level, a good approximation. Further properties of the unperturbed disk will be discussed in section 5. Akin to helioseismology, diskoseismology is the theoretical study of the normal modes of small adiabatic hydrodynamical global perturbations in geometrically thin, optically thick accretion disks. In this section, we describe diskoseismology in general terms. In the following sections, we will use Newtonian diskoseismology to study oscillations in white dwarf accretion disks. Fortunately, the relativistic and nonrelativistic models are formally very similar. Even though the unperturbed disk is not in equilibrium, this perturbative approach is justified whenever the viscous time scales during which the disk evolves as a whole are much longer than the modes’ periods, and whenever the turbulence has spatial scales which are smaller than the modes’ spatial scales, and it is quasi-steady on the time scales of the modes. (These conditions hold for the modes dealt with in this paper.) – 4 – We consider barotropic disks [p = p(ρ), giving vanishing buoyancy frequency] and as- sume that the results thus obtained are generic, since typically the buoyancy frequency is much less than the characteristic dynamical frequency Ω. In this case hydrostatic equilibrium provides the vertical density and pressure profiles ρ = ρ0(r)(1− y2)g , p = p0(r)(1− y2)g+1 , g ≡ 1/(Γ− 1) > 0 , (1) where Γ > 1 is the adiabatic index. One has Γ = 4/3 within any radiation pressure dominated region of the disk, and Γ = 5/3 within any gas pressure dominated region (such as the white dwarf accretion disks). The disk surfaces are at y = ±1, with y related to the vertical coordinate z by To investigate the eigenmodes of the disk oscillations starting from these assumptions, one applies the formalism that Ipser & Lindblom (1991, 1992) developed for perturbations of purely rotating perfect fluids, starting from the equations of conservation of mass and momentum. One can thus express the Eulerian perturbations of all physical quantities through a single function δV ∝ δp/ρ (the Eulerian perturbation of the pressure divided by the density) which satisfies a second-order partial differential equation. Due to the stationary and axisymmetric background, the angular and time dependences are factored out as δV = V (r, z) exp[i(mϕ+ σt)], where σ is the eigenfrequency, and the governing PDE becomes ω2 − κ2 V = 0 , (2) where β = dt/dτ , grr is the metric coefficient, cs stands for the speed of sound, the corotation frequency ω ≡ σ +mΩ , (3) Ω(r) is the Keplerian frequency, and κ(r) is the radial epicyclic frequency (i.e., the frequency of radial perturbations of free particle circular orbits). An assumption of strong variation of modes in the radial direction (characteristic radial wavelength λr ≪ r) ensures the approximate WKB separability of variables in this PDE, V (r, z) = Vr(r)Vy(r, y). The ‘vertical’ part, Vy, of the functional amplitude V (r, z) varies slowly with r. They obey (Perez et al. 1997, hereafter referred to as RD1) (1− y2) − 2gy + 2gω2 1− y2 Vy = 0 , (4) (ω2 − κ2) (ω2 − κ2) + β2grrc 2 − κ2) Vr = 0 , (5) – 5 – where ω∗ ≡ ω/Ω⊥ ; (6) Ω⊥(r) refers to the vertical epicyclic frequency (i.e., the frequency of vertical perturbations of free particle circular orbits), and the speed of sound cs is evaluated at midplane. In these equations Ψ(r) is the slowly-varying separation function. These equations have been used in the past to reveal the properties of different types of modes in relativistic accretion disks. In this paper we solve the equations in the Newtonian limit. We employ, as in the original papers, WKB methods in a straightforward manner. 3. Diskoseismic Modes for the White Dwarf Case: Physical Parameters and Scope We now apply diskoseismology to the accretion disks of white dwarfs. We are thus working in the Newtonian limit (for a mass M = M⊙ white dwarf, its radius R ∼ 5 × 103, corresponding to 7.4 × 103 km), which amounts to setting grr = β = 1, g = 3/2 in the equations of the previous section. We note in particular that Ω(r) = κ(r) = Ω⊥(r) = 1/r 3/2 , (7) i.e., the rotational, radial epicyclic, and vertical epicyclic frequencies are all equal (up to corrections of relative size ∼ 1/r or smaller). As we will see, this results in a qualitative change in the nature of the solutions with respect to the relativistic case. This is a straightforward problem with few free parameters. In addition to the white dwarf mass M , there is ri [≥ R(M)], the inner radius of the accretion disk. Then, one needs a functional form for the speed of sound. It is fortunate that the properties of the unperturbed disk (viscosity and luminosity) enter into the formalism only through the speed of sound, and that the dependence of the speed of sound on these properties is rather weak. In this exploratory calculation, we ignore the complications related to the nature of the transition from the disk to the star surface or magnetosphere and assume that DNO oscil- lations are confined to the disk. We introduce a disk inner boundary condition parameter, given our ignorance of the physical conditions in the (magnetic) boundary layer. As we will see, the results are rather insensitive to this parameter. The approximate character of our analysis should be stressed. A deeper study of boundary layer dynamics is needed in order to understand better the physical conditions that would allow for the existence of the modes described in this paper. For instance, one could ask how significant is the damping and ‘leakage’ of the modes into the magnetosphere, where the radial epicyclic frequency is given – 6 – by the more general expression κ2 = 2Ωr−1d(r2Ω)/dr, if the modes are still confined to a small vertical extent h ≪ r. As in the case of relativistic diskoseismology (with black holes or weakly magnetic neutron stars), the formalism reveals the existence of different types of modes. In this first diskoseismic investigation of cataclysmic variables, we choose to study the g–modes, since they are trapped near the inner edge of the disk and are therefore the most luminous (and robust). By definition, the g–modes have corotating eigenfrequencies which are smaller than the gravitational radial oscillation (epicyclic) frequencies at the involved radii. In other words, the (smaller) pressure restoring forces act against the gravitational restoring forces. (The ‘g’ in the name comes from a mathematical similarity to the g–modes within slowly rotating stars.) The acoustic (p–) modes extend to the outer region of the disk, thus involving strong effects of viscosity and the uncertain boundary conditions at its outer radius. 4. Formal WKB Solution of the Eigenvalue Problem for G–modes In this section we will proceed to solve formally the eigenvalue problem in order to obtain equation (28), our ‘master’ equation, from which all further results in the paper are derived. Taking into account expressions (7), the separated equations (4) and (5) reduce in the Newtonian limit to (1− y2) d − 3y dVy + 3ω2 1− y2 Vy = 0 , (8) (ω2 − Ω2) (ω2 − Ω2) + c−2s (ω 2 − Ω2) Vr = 0 . (9) (Recall from section 2 that g = 3/2 for white dwarf accretion disks.) Equations (8) and (9) are two coupled eigenvalue equations, the eigenvalues being ω and Ψ (or equivalently, σ and Ψ). We will solve these equations within a straightforward WKB formalism as described by Ortega-Rodŕıguez, Silbergleit, & Wagoner (2002) (hereafter referred to as RD3). With respect to boundary conditions, it turns out that it is sufficient to require that the vertical eigenfunction and its derivative remain finite at the surface of the disk and to introduce an inner boundary parameter for the radial equation, as mentioned. By definition, g–modes exist wherever ω2 < κ2(= Ω2). This means that the mode exists between an inner radius (which is ri, the inner radius of the accretion disk, for the cases studied in this paper) and the radius r+, defined as the radius at which |ω| = Ω. The mode decays exponentially for r > r+. – 7 – 4.1. WKB Solution of the Radial Eigenvalue Problem We will cast equation (9) in a form amenable to a WKB solution, using a procedure which is similar to the one developed in RD3. We introduce the dependent variable W ≡ 1 Ω2 − ω2 , (10) and the independent variable c−2s (r Ψ(r′) dr′ , (11) so that equation (9) can be written in the simpler form + S(τ)W = 0 , S(τ) ≡ c2s(Ω 2 − ω2) . (12) The mode exists in the range ri < r < r+ or, equivalently, 0 < τ < τi ≡ τ(ri). The function S(τ) changes sign at τ = 0 (where r = r+). The boundary conditions are that W decays for τ < 0, while at the other extreme, at τ = τi, we parameterize our ignorance at the inner boundary by means of the quantity θi: (Ω2 − ω2) cos θiW − sin θi = 0 , (13) which is equivalent to cos θi − sin θiVr = 0 . (14) A straightforward WKB solution can be obtained for equation (12) at all points except for τ close to 0: W ∝ S−1/4(τ) cos S1/2(τ ′)dτ ′ + Φ , (15) where Φ is a constant to be determined. For τ near zero, the solution is given by the Airy function of the first kind: W ∝ Ai(−s1/3+ τ) , s+ ≡ dS(0)/dτ > 0 . (16) The asymptotic matching of solutions (15) and (16) in the usual manner yields Φ = π/4+nπ, where n is an integer, and ∫ r+(σ) c−1s (r) (Ω2 − ω2)(Ψ/ω2 − 1)dr = π(n± 1/4) (17) for the cases θi = π/2 and 0, respectively. Thus n is the radial mode number constrained by the fact that the integral in equation (17) is positive. – 8 – 4.2. WKB Solution of the Vertical Eigenvalue Problem Using the new independent variable (1− y′2)−3/2dy′ , (18) we can write equation (8) in the simpler form +Q(ζ)Vy = 0 , Q(ζ) ≡ 3ω2∗ 1− y2 1− y2 , (19) which is amenable to a WKB solution. We now discuss the issue of boundary conditions. In view of the symmetry of the differential equation (8), one has = 0 or Vy = 0 (20) for even and odd modes in the disk, respectively, and it is enough to consider the interval 0 < y < 1. Near the singular boundary point y = 1, equation (8) can be written + . . . + . . . Vy = 0 , (21) where the dots represent non-singular terms. According to the analytical theory of second order ODEs [see, e.g., Olver (1982)], the general solution to equation (8) near y = 1 has the Vy = C1v1(y) + C2(1− y)−1/2v2(y) , (22) where C1 and C2 are arbitrary constants, and v1(y) and v2(y) are converging power series of (1− y). Imposing the boundary condition that the eigenfunction and its derivative be finite at y = 1 implies thus that C2 = 0. As noted in RD1 and implied by equation (21) and the boundary condition, the vertical eigenfunction near the boundary can be written in terms of the Bessel function of the first kind as Vy(y) ∝ (1− y)−1/4J1/2(2|ω∗| (3/2)(1− y) ) , (23) the asymptotic (large argument) expression of which is Vy(y) ∝ (1− y)−1/2 cos 6(1− y)− (π/2) . (24) – 9 – On the other hand, equation (19) can be solved with a WKB method for values of y not close to 1: Vy ∝ Q−1/4(τ) cos Q1/2(ζ ′)dζ ′ − Iπ/2 , (25) where I = 0, 1 for the even and odd eigenfunctions, respectively. Matching this with the asymptotic expression (24) of the boundary solution (23), we arrive at 1 + (Ψ/ω2 − 1)(1− y2) 1− y2 3|ω∗| . (26) The parity (of the vertical eigenfunction) and the vertical mode number j are contained in a single quantity J ≡ j + 3/4 ± 1/4 for the odd and even cases, respectively. Thus J takes positive integer and half-integer values (see Table 1). It should be stressed that, because of the method used to solve the equations, the above results are accurate only when Ψ/ω2 − 1 ≫ 1 for all r in the range ri < r < r+. 4.3. Uncoupling Equations (17) and (26) are two eigenvalue equations (the eigenvalues being Ψ and ω), which need to be uncoupled. We start by inverting the integral I on the left hand side of equation (26) to find Ψ/ω2 − 1 as a function of I: − 1 = f(I) . (27) Once this is done numerically, the function f(I) can be substituted in equation (17): c−1s (r) (Ω2 − ω2)f(πJ/ 3|ω∗|)dr = π(n± 1/4) . (28) The left–hand side of equation (28) now only depends on the eigenfrequency σ and not on Ψ [recall definition (3)]. All further results in this paper are obtained from this equation. 5. Results and Discussion In order to use equation (28) and look for solutions with integer values of n ≥ 0, we need a value for ri and a functional form for the speed of sound cs. We will take ri = 5000, corresponding to 7383(M/M⊙) km. The restriction ri ≥ R, the white dwarf radius, places a – 10 – limit on its mass M . Using a standard mass–radius relation (Nauenberg 1972), this value of ri implies that M ≥ 0.76M⊙. As for the speed of sound, there are different solutions given by Shakura & Sunyaev (1973), depending on the region of the disk in which one is working. For the values of interest in this paper, the relevant expression is the one corresponding to the outermost solution, in which temperatures are low enough that pressure is dominated by gas (not radiation) and opacity is dominated by photon absorption in the form of nonrelativistic thermal bremsstrahlung (‘free-free’ transitions). Their solution for the speed of sound is given by (cs/c) −1 = 300α1/10(M/M⊙) 1/4Ṁ −3/20 3/8 , (29) where c, α, M , and Ṁ16 stand, respectively, for the speed of light, the (dimensionless) viscosity parameter, the mass of the white dwarf, and the mass accretion rate in units of 1016 g/s. The small powers of the three parameters render our results relatively insensitive to them. We remind the reader that the interior structure of the unperturbed accretion disk enters our formulation only through the speed of sound. As mentioned above, we parameterize our ignorance about the radial inner boundary by means of the quantity θi introduced in equation (14). 5.1. Axisymmetric Modes The eigenmodes are described by the azimuthal, radial and vertical mode numbers m, n, j, plus the value of the parity of the vertical eigenfunction. We start by studying the m = 0 modes. For them, the condition Ω2 −ω2 > 0 becomes Ω2 − σ2 > 0, which means that the modes are trapped between ri and r+ = |σ|−2/3. Using the characteristic values ri = 5000, M = M⊙, α = 0.1 (Lasota 2001), and Ṁ16 = 1 (Frank, King & Raine 2002), we calculate σ for different values of n, j and parity for the case θi = π/2. (Note that the values of j and parity are contained in J in the manner described in section 4.2 and Table 1.) Table 2 shows the results. The dimensional frequency f = 3.23 × 104(M⊙/M)|σ| Hz. The n = 0, J = 1 entry thus corresponds to a period of 12 seconds. Table 3 shows ∆r/ri ≡ (r+ − ri)/ri for the same values of the parameters. As in the case of axisymmetric g–modes in relativistic accretion disks (RD1), the value of cs/c is small enough that: a) the small n eigenfrequencies are close to the maximum value of κ(r) [which equals Ω(ri) = 2.83× 10−6 in the present case], approaching it as j increases; b) the modes’ – 11 – radial width decreases as j increases and increases as n increases. Self-consistency requires that the parameter Ψ/ω2 − 1 not be small, and the larger it is, the better the WKB approximation. It turns out that Ψ/ω2 − 1 ∼ 1 when J = 1 and − 1 ≫ 1 when J > 1, so the procedure passes this test. (Incidentally, it is interesting that the minimum value of Ψ/ω2 − 1 over the allowed values of the radius does not depend on n or m, only on J .) There is no J = 1/2 entry in Table 2 because there is no WKB solution. A direct numerical solution of equations (8) and (9) shows that the absence of a solution is not an artifact of the WKB method, although it could result from the assumptions made in obtaining these equations. The possibility remains open that there are modes not detected by our methods. Next we compute the behavior of the eigenfrequency σ as the various parameters change. For the choice J = 1, n = 0, θi = π/2, we obtain ∂ log σ/∂ log ri = −1.5 , (30) ∂ log σ/∂ logα = 0.0041 , (31) ∂ log σ/∂ logM = 0.010 , (32) ∂ log σ/∂ log Ṁ = −0.0061 , (33) which means that the dependence of σ on α, M , and Ṁ can be neglected since it is much weaker than its (expected) dependence on ri. Moreover, a change of θi from π/2 to 0 implies only a change of a few percent in σ. Note, however, that there can be no n = 0 solution with θi = 0 because then the RHS of equation (17) would be negative. Also note that equation (32) assumes a fixed value of ri. Care must be taken then when comparing with observations, since usually the parameters are not independent. For example, for a set of white dwarfs of fixed composition there is a relationship between their radius (R) and mass. So since σ ∼= σ(ri) and ri = ri(M, Ṁ) (for fixed surface magnetic field), one has thus, for fixed Ṁ , ∂ log σ ∂ logM ∂ log ri ∂ logM d log σ d log ri = 0.50 , (34) for the case M ∝ r−3i (Nauenberg 1972) appropriate to low–mass white dwarfs with weak magnetic fields (ri = R). When computing the variation of σ with the luminosity L for any specific cataclysmic variable, if one assumes that ri ∝ L−γ , where γ is a constant, one obtains d logP d logL ∂ log ri ∂ logL d log σ d log ri = −1.5 γ . (35) – 12 – Such a dependence of ri on luminosity could be due to strong magnetic field effects (Shapiro & Teukolsky 1983). The idea is that inside the magnetosphere the accretion flow is controlled by magnetic fields, so one takes ri to be the equatorial radius of the magnetosphere. Its value should be only moderately greater than the white dwarf radius R for those magnetic field strengths thought to be present in many of those white dwarf systems which exhibit DNOs. For exam- ple, the value of d logP/d logL = −0.091 for the dwarf nova SS Cygni reported by Mauche (1996) is reproduced by our formalism provided γ = 0.06. In summary, the model may be consistent with observations. The radial width of the modes may be large enough to make them visible, and the periods are consistent, although this is hardly surprising as they are related to the Keplerian period at ri. This leads to the limit P ≥ 10.94(M⊙/M)1/2[ri(km)/7383 km]3/2 seconds on the periods of axisymmetric modes. However, the physics of the magnetosphere is at present too uncertain to provide a means to test the model accurately. At this point it makes sense to look at the higher m modes. 5.2. Non-Axisymmetric Modes Here we discuss the m > 0 modes. Note that from definition (3) and the fact that only |ω| enters in the equations, we can, without loss of generality, consider only the non-negative m cases provided we allow for negative σ. The condition Ω2 − ω2 > 0 implies that −m− 1 < σ/Ω(r) < −m+ 1 . (36) Therefore, g–modes are trapped between ri and r+ = (m+1) 2/3|σ|−2/3. The inner boundary of the mode (r−) is ri, provided that |σ| ≥ (m− 1)Ω(r). The negative sign of σ reflects the fact that the oscillations’ wavefronts travel in the same direction as the fluid. Table 4 and Figure 1 show some properties of the n = 0, J = 1 modes for the m = 0, 1 and 2 cases. Interestingly, the results are consistent with the observation of a 1:2:3 harmonic structure in the power spectra of the DNO VW Hyi, reported by Warner & Woudt (2005). The observed peaks remain harmonic even during a threefold increase in the fundamental frequency. As in other models, the increase in frequency is due to a decrease in ri induced by an increase in Ṁ (and thus L). If one expresses σ in terms of r+ and works to lowest order in ∆r/ri, from equation (28) one obtains ∆r/ri ∼ (n± 1/4)2/3(m+ 1)−1/3ǫ2/3 . (37) – 13 – Since ǫ ≡ h(ri)/ri ∼ (cs/Ωr)ri, using equation (29) and our chosen set of parameters (which only weakly affect the estimate) gives ǫ ≈ 4.2 × 10−3 r1/8i ≈ 1.2× 10−2. Then equation (37) is seen to agree approximately with the values in Tables 3 and 4. The m > 0 diskoseismic modes are in principle harder to observe than the axisymmetric modes, since in the former the modulation of the outgoing radiation is not as efficient due to cancelations over the entire disk (and their radial extent is somewhat smaller). However, the fact that the white dwarf can eclipse parts of the inner disk renders these modes more visible. We predict that the energy spectrum of the DNOs is characteristic of the very inner disk (and not other regions such as the magnetosphere), and thus should be close to blackbody at the highest disk temperature. As in the case of g–modes in relativistic accretion disks, the eigenfrequency splittings when m changes by unity are much larger than the n and j splittings. The g–modes that are trapped by the general relativistic behavior of the radial epicyclic frequency κ have m splittings that are not as harmonic as those found here (Table 4). In particular, their eigenfrequencies are only proportional to m for m & 2 [an interesting difference between the relativistic and the nonrelativistic eigenfrequencies is that, for m ≫ 1, |σ| → mΩ(ri) for the former (see RD1) while |σ| → (m+ 1)Ω(ri) for the latter]. However, it must be noted that it appears that g–modes with small mode numbers are viscously unstable, at least if the viscosity acts in the usual hydrodynamic way. This result is proven for the relativistic case in Ortega-Rodŕıguez & Wagoner (2000), where the exponential growth rate of the mode is roughly αΩ. For the nonrelativistic g–modes studied in this paper, an application of their method yields the same result. Furthermore, these low–lying modes do not contain a corotation resonance [where ω(r) = 0]. 6. Comments Thus, if the ideas in this paper are correct and DNOs are linear diskoseismic modes then the results could suggest a scenario in which white dwarf DNOs and black hole QPOs are produced through somewhat different physical mechanisms. After all, as Kluźniak et al. (2005) themselves point out, the phenomenology is similar but not identical. While in white dwarf DNOs the simultaneous observation of two or even three of the harmonics is well established, in black holes usually only one mode of oscillation is observed at a given moment. Furthermore, white dwarf DNOs (and neutron star QPOs) have frequencies that vary with time, while QPOs in black holes have fixed frequencies. In neutron star XRBs, the (twin) high–frequency QPOs are analogous to the DNOs, but their frequency ratio is much closer to unity (van der Klis 2006). It appears to be controlled (magnetically) by the neutron – 14 – star, since their frequency difference is close to the neutron star spin frequency or half of it. We note that some DNOs are split at a QPO frequency. We also note that with increasing radius of a neutron star magnetosphere, their g-modes should approach (for ri & 8) those of this paper. We hope to investigate the magnetic boundary layer in more detail in a future paper. How much will the boundary layer contribute to the damping (or growth) of the g–modes? Further work should try to explain why the value of Q is so high. Naively, one would expect Q ∼ 1/α. (It is fortunate that most of our results are fairly insensitive to the value of α.) The nonlinear growth and coupling of the modes must also be investigated. This work was supported by grant 075-2002 (Incentivos) of Ministerio de Ciencia y Tecnoloǵıa, Costa Rica, grant 829-A3-078 of Vicerrectoŕıa de Investigación, Universidad de Costa Rica, and NASA grant NAS 8-39225 to Gravity Probe B. We are also grateful to the Aspen Center for Physics for its support. REFERENCES Abramowicz, M. A. & Kluźniak, W. 2001, A&A, 374, L19 Carroll, B. W., Cabot, W., McDermott, P. N., Savedoff, M. P. & Van Horn, H. M. 1985, ApJ, 296, 529 Collins, T. J. B., Helfer, H. L. & Van Horn, H. M. 2000, ApJ, 534, 944 Frank, J., King, A., & Raine, D. 2002, Accretion Power in Astrophysics (Cambridge: Cam- bridge University Press) Ipser, J. R. & Lindblom, L. 1991, ApJ, 379, 285 Ipser, J. R. & Lindblom, L. 1992, ApJ, 389, 392 Kluźniak, W., Lasota, J.-P., Abramowicz, M. 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L. & Teukolsky, S. A. 1983, Black Holes, White Dwarfs, and Neutron Stars (New York: Wiley) van der Klis, M. 2006, in Compact Stellar X-Ray Sources, ed. W. H. G. Lewin & M. van der Klis (Cambridge: Cambridge Univ. Press), p. 39 Wagoner, R. V. 1999, Phys. Rep., 311, 259 Wagoner, R. V., Silbergleit, A. S., & Ortega-Rodŕıguez, M. 2001, ApJ, 559, L25 Warner, B. 2004, PASP, 116, 115 Warner, B. & Woudt, P. A. 2002, MNRAS, 335, 84 Warner, B. & Woudt, P. A. 2005, ASP Conference Series 330, 227 (San Francisco: ASP) Yamasaki, T., Kato S., & Mineshige, S. 1995, PASJ, 47, 59 This preprint was prepared with the AAS LATEX macros v5.2. – 16 – Table 1. Value of J as a function of j and parity j parity J 0 even 1/2 0 odd 1 1 even 3/2 1 odd 2 – 17 – Table 2. Values of the eigenfrequency σ × 106 for m = 0 g–modes. n 1 3/2 2 0 2.65 2.73 2.75 1 2.39 2.55 2.61 2 2.23 2.43 2.51 Table 3. Values of the modes’ fractional radial width ∆r/ri for m = 0 g–modes n 1 3/2 2 0 0.044 0.024 0.019 1 0.12 0.072 0.055 2 0.17 0.11 0.083 Table 4. Harmonic structure of the J = 1, n = 0 g–modes m −106 σm σm/σ0 ∆r/ri 0 2.65 1 0.044 1 5.39 2.03 0.033 2 8.14 3.07 0.028 – 18 – 1 1.05 1.1 1.15 1.2 m = 0 m = 1 m = 2 Fig. 1.— The horizontal lines show the radial extent and the frequency of the three lowest fundamental (n = 0, J = 1) axial g-modes. Introduction Diskoseismology Diskoseismic Modes for the White Dwarf Case: Physical Parameters and Scope Formal WKB Solution of the Eigenvalue Problem for G–modes WKB Solution of the Radial Eigenvalue Problem WKB Solution of the Vertical Eigenvalue Problem Uncoupling Results and Discussion Axisymmetric Modes Non-Axisymmetric Modes Comments

Quantum molecular dynamics study of the ammonia inversion transition under pressure Quantum molecular dynamics study of the pressure dependence of the ammonia inversion transition I.M. Herbauts and D.J. Dunstan, Physics Department, Queen Mary, University of London, London E1 4NS, England. Abstract: The mechanism of the shift, broadening and quenching of the ammonia inversion frequency with gas pressure has been a problem of lively interest for over seventy years. A simple quantum model of the ammonia molecule perturbed by collisions with ideal gas molecules displays the essential features of the experimental data for NH3 and for ND3. The model does not display the behaviour expected from theories of quantum localisation such as quantum state diffusion and decoherence. On the other hand, models of perturbed classical oscillators do display similar behaviour to our model. The quenching of the ammonia inversion transition cannot therefore be interpreted as spatial localisation of the wavefunction. Introduction:⎯Since the early days of microwave spectroscopy, the inversion transition of the ammonia molecule NH3 and ND3 has been extensively studied experimentally and theoretically. Much of the interest lies in the fact that it is the smallest and simplest of the pyramidal and enantiomorphic molecules whose ground and excited energy eigenstates are the quantum superpositions of two different spatial configurations, and that it is light enough that the transitions between the energy eigenstates are fast enough to be experimentally accessible. The ammonia molecule has two spatial eigenstates L and R with the nitrogen atom on one side or the other of the plane of hydrogen atoms, and its energy ground and first excited states 0 and 1 are the symmetric and antisymmetric quantum superpositions of the spatial eigenstates (ignoring rotational and vibrational states). The ammonia maser is based upon the transition between the energy eigenstates, which may also be described as the Rabi oscillation between the spatial eigenstates. However, the inversion transition is seen only at low gas pressure. As the gas pressure is increased, the transition broadens, shifts to lower frequency and then quenches (the frequency goes to zero). The ammonia molecule appears to undergo spatial localisation as a result of interaction with the environment. This would be of immense theoretical interest. In chemistry and in the classical world generally, enantiomorphic molecules with distinguishable spatial eigenstates L and R are always found in their spatial eigenstates (classical behaviour) rather than their energy ground states (quantum behaviour).3 Whilst ammonia is not enantiomorphic, it does appear to show both behaviours, quantum at low pressure and classical at high pressure, if the quenching is considered to be a direct observation of localisation or collapse of the wave-function into a spatial eigenstate. Within the context of the decoherence programme, it has been treated quantitatively in that way.4 In this paper, we show that interaction with the environment quenches the inversion transition for what might be described as ‘classical’ reasons. The broadening, shift and quenching of the Rabi oscillation are simply consequences of impacts and may be described within the framework of an oscillator subject to white noise from the environment. There is no evidence for localisation onto spatial eigenstates. Background:⎯At low pressures in the gas phase, the transition between the energy eigenstates is observed near 24GHz (0.8 cm–1) in NH3. 1 In ND3 2 the transition is near 1.6 GHz (0.053 cm–1). In NH3, broadening is observed at pressures above a few mm of mercury, with a shift to lower frequency, and quenching is complete at about 1.7 bar. In ND3, pressures about 15 times lower yield the same effects, in proportion to the inversion transition frequency. The first explanation of the shift and broadening of the ammonia inversion transition frequency was given by Anderson5 in terms of perturbation by the electric dipole-dipole interaction between ammonia molecules. Anderson’s discussion was only qualitative, and Margenau investigated the quantum states of two ammonia molecules coupled by their dipole-dipole interaction in more detail.6 He showed that the interaction leads to the splitting of the transition into a higher frequency component with reduced strength and a lower frequency with increased strength. While this accounts for the initial shift to lower frequency, it fails to account for the quenching of the inversion transition at a higher pressure. More recently, the dipole- dipole interaction model has been treated by a quantum mean-field approximation yielding, apparently, a frequency shift, quenching and spatial localisation at pressures for NH3 and ND3 in good agreement with experiment.7 The standard theory of line-broadening by impact is given by Van Vleck and Weisskopf.8 It predicts a line-shape function ( ) ( )202 −− bb f (1) where the width b is given by 1/2πτ for strong impacts occurring at a mean interval of τ, and therefore proportional to the pressure. The theory does not predict any peak shift: ν0 is a constant, the natural frequency of the oscillator. Anderson developed the theory further and obtained a shift of ν0 to lower frequency equal to the width b.9 Fano recast the problem of pressure broadening in the Liouville representation and obtained a shift to lower frequency independent of the broadening.10 Ben-Reuven used the Fano theory to show that the ammonia spectra can be well-fitted with a related expression but with three independent parameters proportional to the pressure. Two of them express the effects of elastic collisions on the width and on the frequency shift, and the third parameter expresses the effect of inelastic collisions.11 We are interested in a dynamical theory of the transition and of quenching and localisation. It is important to know if the dipole-dipole interaction of ammonia molecules is crucial to the quenching, or if it merely influences the collision cross- section while impacts are sufficient to account for the quenching. Accordingly, we have set up a molecular dynamics simulation in which the quantum nature of the ammonia molecule is explicitly taken into account.12 Here we show that the model accounts for the shift, broadening and quenching of the inversions transition purely in terms of perturbation by collision with ambient gas molecules. The Ammonia Quantum Molecular Dynamics Model:⎯We model the problem in one dimension. The ammonia molecule is modelled by a double-well potential, with the two time-dependent spatial wavefunctions ΨL and ΨR. With a weak coupling between the wells the Hamiltonian in the spatial basis is (2) ⎟⎟ Diagonalising, the ground and first excited states of the system are found to be Ψ0 and Ψ1 with a frequency splitting of ω1. The general state of the system is a superposition, with Ψ+Ψ=Ψ (3) Expanding this in the spatial basis set ΨL and ΨR, we have time-varying coefficients, RL tt Ψβ+Ψα=Ψ )()( (4) so that the amplitude of the wave-function beats, or oscillates between the two wells. The squared amplitude 2)(tα = α*α oscillates at the frequency ω1 and with a beat amplitude that depends on the initial values of a and b (from zero for e.g. 2/1== ba to a maximum amplitude of unity for e.g. a = 1, b = 0). This oscillation is the inversion transition or Rabi oscillation of the molecule. We model impacts, or interactions with the environment, by a term which is diagonal in the spatial representation. That is, we suppose that the double well is tilted during an impact. If a gas atom coming in from the left raises the energy of the left- hand well, the Hamiltonian during impact is (5) ⎟⎟ Diagonalising and expanding in the spatial basis set as before, we obtain the normalised eigenvectors u and v of H′ . Equations 3 and 4 become RLPP ttba Ψβ′+Ψα′=Ψ′+Ψ′=Ψ′ )()(10 (6) The Rabi oscillation is now at a much higher frequency and a much smaller amplitude (for ωP >> ω1). In reality, the perturbation rises and falls continuously in an impact, but we approximate with a top-hat function, so that ωP is switched on at a time t0 and switched off again at t1. At these times, we match the coefficients in the spatial basis, using and )()( 00 tt α=α′ )()( 00 tt β=β′ to solve for aP and bP at the onset of the perturbation, and then the new )()( 11 tt α′=α , )()( 11 tt β′=β to solve for the new a and b at the end of the perturbation. These boundary conditions ensure that the amplitude and phase of the wave-function in each well do not change discontinuously at the beginning and end of the perturbation. The resulting time evolution of α*α is illustrated in Fig.1. 0 1 2 Time, t xirta Fig.1. The evolution of the occupancy of the left-hand well is shown with two perturbations occurring at t = 0.7 and t = 1.6. The Rabi angular frequency ω1 is 2π and the perturbation ωP = 60. The initial wavefunction is given by a = b = 1/ 2 ; after the two perturbations the values are a = 0.54 – 0.73i, b = 0.36 + 0.22i. To model NH3 and ND3, we can choose the units of time so that the Rabi frequency is unity (ω1 = 2π). The strength of the perturbation is of the order of kBT, which at room temperature is 208 cm–1. For NH3, therefore, we take ωP = 208 ω1/0.8 = 260 ω1 and for ND3, ωP = 208 ω1/0.0.053 = 3925 ω1. The duration Δt = t0 – t1 of an impact is hard to estimate. However, inspection of Fig.1 shows that to achieve a strong impact (in the sense of Van Vleck and Weisskopf 8), we need something of the order of one cycle of the perturbed Rabi oscillation, i.e. ωPΔt ~2π, while larger values will have no extra effect. We therefore take values of Δt from a random distribution over the range 0 to 2π / ωP. The average frequency of impacts corresponds to the gas pressure. We require an impact cross-section to relate the frequency of impacts to the gas pressure quantitatively. Bleaney and Loubser and other authors obtain impact cross-sections from the pressure-broadening of the transition, assuming strong impacts and using b = 1/2πτ. We shall see below that such estimates are unreliable, and therefore in our simulation we use the measure p impacts per Rabi cycle instead of pressure, and we vary p over a wide range. We calculate the values of α*α at discrete time intervals δt with Δt < δt << 1. At each time interval we have a probability δt/τ of having an impact, so that there are p = 1/τ impacts per cycle. If there is an impact, we use )()( 00 tt α=α′ and to solve for aP and bP at the onset of the perturbation, and then calculated the new , )()( 00 tt β=β′ )()( 11 tt α′=α )()( 11 tt β′=β to solve for the new a and b at the end of the perturbation. Then the calculation of the list of values is resumed. Examples are shown in Fig.2 for medium (a) and high (b) values of p. The numerical Fourier Transforms of the lists are calculated, shown in Fig.2(c) and (d), and fitted with Af(ν) of eq.1, with b, ν0 and amplitude A as fitting parameters. Our interest here is the fitted values of b and ν0 as functions of p. In Figure 3 these are compared with the experimental data for NH3,16 with the constant of proportionality between p and pressure (corresponding to the impact cross-section) as a free parameter. Fig.2. The occupation of the left-hand well, y(t) = α*α, is plotted against time, for (a) p = 3.5 impacts per cycle, below the quenching, and (b) p = 7.5 impacts per cycle, above the quenching. The Fourier transforms Y(ν) are shown in (c) and (d) respectively, together with the fits using eq.1. Fig.3. The solid curves show (a) the peak frequency and (b) the broadening for NH3 as a function of the number of impacts per cycle as described in the text. The data points show (a) the peak frequency and (b) the broadening reported by Bleaney and Loubser,16 scaled as described in the text. The NH3 data is plotted with p = 4.5 impacts per cycle equivalent to a pressure P = 1 bar. For ND3, the data fits equally well but with p = 4.5 equivalent to the pressure P = 1/15 bar, consistent with the fifteen times lower inversion frequency in ND3 but the same impact parameter. In both cases, full quenching is observed at about 6.5 impacts per cycle. The model presented here accounts remarkably well for the shift and quenching of the ammonia inversion transition peak. It accounts less well for the broadening, which occurs initially at the rate b ~ 0.25 p in the simulation and 0.18 p in the experimental data. The experimental broadening is about three-quarters of the model broadening up to the quenching pressure, and above the quenching pressure the experimental broadening continues to increase while the model broadening decreases. To gain a better understanding of this behaviour, we investigate how a simple classical oscillator behaves under similar perturbations. A Classical Perturbed Harmonic Oscillator:⎯A classical oscillator may be perturbed by collision in a large variety of well-defined ways. We evaluate two perturbations here. We calculate the values of a sinusoid of frequency ν = 1 at discrete time intervals δt << 1. At each time interval we have a probability δt/τ of having an impact, so that there are p = 1/τ impacts per cycle. If there is an impact, the sinusoid is modified accordingly, and then the calculation of the list of values is resumed. The numerical Fourier Transform is calculated and we find that af(ν) of eq.1 fits well for a variety of definitions of the impacts, over a very wide range of p, with b, ν0 and a as fitting parameters. Our interest here is the fitted values of b and ν0 as functions of p. Impactsper cycle ycneuqerF 0 1 2 3 4 5 6 Impactsper cycle 0 1 2 3 4 5 6 Impacts per cycle ycneuqerF 0 2 4 6 8 10 12 14 Impacts per cycle 0 2 4 6 8 10 12 14 Fig.4. The peak frequency and the peak width are plotted for the broken sinusoids with unity frequency described in the text. In (a) and (b), results are for the strongest possible impact, with phase and amplitude completely randomised by the impact. In (c) and (d), the boundary condition at impact keeps the sinusoid continuous but changes the phase and amplitude at random within that constraint. The strongest perturbation possible (in the sense of Van Vleck and Weisskopf 8) is a collision that destroys all memory of position and speed (or amplitude and phase). To model this, at impact we pick the new amplitude A of the sinusoid Acos(2πνt + ϕ) at random from the range 0 to 1 and the new phase ϕ at random from the range 0 to 2π. In this model, the peak is shifted to higher frequency, shown in Fig.4(a) until the width of the transition in Fig.4(b) reaches b = 1, at which point quenching occurs, i.e. the frequency collapses to zero. The width continues to increase at still higher values of p, Fig.4(b). This is a stronger impact than the impacts of the ammonia molecule, for the quenching occurs at p = 3.5 impacts per cycle and the initial slope of the broadening is given by b ~ 0.5 p. It is interesting that Van Vleck and Weisskopf give the broadening for strong impacts as 1/2πτ, equivalent to 0.16 p. In an alternative definition of impact which is in closer accordance with the ammonia model of Section 2.1 and Fig.1, we define the impact at t0 by taking the position x(t0) as unchanged by the impact, the new amplitude A as random in the range x(t0) to 1, and the new phase ϕ such that the speed is a random variable in )( 0tx& the range consistent with the new amplitude. In this model, the peak shift in Fig.4(c) and width behave in very much the same way as the ammonia results of Fig.3, with the initial broadening b ~0.2 p. However, the impact is weaker than in the ammonia model, for quenching occurs at p = 10.5 impacts per cycle. Above quenching b decreases again. In these models we may weaken the strength of the impact by letting the new amplitude and phase be given by some amount ε of the amplitude and phase calculated as above plus (1 – ε) of the old amplitude and phase. Not surprisingly, the shift and broadening are identical if plotted as functions of the normalised impact rate εp. Results and Discussion:⎯Fig.4 shows that the details of the peak shift and the broadening are very sensitive to the exact nature of the boundary conditions between the periods of unperturbed free oscillation. A more complete description of the impact (including, for example, inelastic collisions as in Ben-Reuven11) might well account for the discrepancies between data and model in Fig.3. However, we do not know of any way to predict the initial slope of b(p), nor its functional form below and above the quenching, from a specification of the boundary conditions. Neither the mathematics of the noisy oscillator (see, e.g. the book by Gittenberg13) nor of the classical kicked rotor (see, e.g. Ref.) appear to answer this question. The key result is that the ammonia model (Fig.3) and even the broken sinusoid of Fig.4(c) both show the Rabi oscillation frequency shifting to lower frequency, broadening, and quenching – going to zero frequency – as the impact rate is increased, in agreement with experiment. Note that the density matrix shows no evidence of localisation. In particular, the off-diagonal elements do not go towards zero as predicted by the decoherence programme. Nor is there any evidence or quantum state diffusion towards the configurational eigenstates. The ammonia shift and quenching are fully accounted for in terms of a perturbed oscillator, and should not therefore be cited as an experimental observation of quantum localisation. Acknowledgements:⎯We are grateful to Prof. I.C. Percival and Dr T. Prellberg for useful discussions. References 1. C.H. Townes, Phys. Rev. 70, 665 (1946). 2. G. Birnbaum and A.A. Maryott, Phys. Rev. 92, 270 (1953). 3. R.G. Woolley, Adv. Phys. 25, 27 (1976). 4. E. Joos, in Decoherence and the Appearance of a Classical World in Quantum Theory, ed. D. Guilini (Springer, Berlin, 1996). 5. P.W. Anderson, Phys. Rev. 75, 1450 (1949). 6. H. Margenau, Phys. Rev. 76, 1423 (1949). 7. G. Jona-Lasinio, C. Presilla and C. Toninelli, Phys. Rev. Lett. 88, 123001 (2002). 8. J.H. Van Vleck and V.F. Weisskopf, Rev. Mod. Phys. 17, 227 (1945). 9. P.W. Anderson, Phys. Rev. 76, 647 (1949). 10. U. Fano, Phys. Rev. 131, 259 (1963). 11. A. Ben-Reuven, Phys. Rev. Lett. 14, 349 (1965). 12. D.J. Dunstan, archive. 13. M. Gittenberg, The Noisy Oscillator: The First Hundred Years, From Einstein Until Now (World Scientific, Singapore, 2005). 14. B. Bleaney and R.P. Penrose, Proc. Roy. Soc. A189, 358 (1947). 15. B. Bleaney and R.P. Penrose, Proc. Phys. Soc. 60, 540 (1948). 16. B. Bleaney and J.H.N. Loubser, Proc. Phys. Soc. A63, 483 (1950). Acknowledgements:(We are grateful to Prof. I.C. Percival and Dr T. Prellberg for useful discussions.

The mechanism of the shift, broadening and quenching of the ammonia inversion frequency with gas pressure has been a problem of lively interest for over seventy years. A simple quantum model of the ammonia molecule perturbed by collisions with ideal gas molecules displays the essential features of the experimental data for NH3 and for ND3. The model does not display the behaviour expected from theories of quantum localisation such as quantum state diffusion and decoherence. On the other hand, models of perturbed classical oscillators do display similar behaviour to our model. The quenching of the ammonia inversion transition cannot therefore be interpreted as spatial localisation of the wavefunction.

Introduction:⎯Since the early days of microwave spectroscopy, the inversion transition of the ammonia molecule NH3 and ND3 has been extensively studied experimentally and theoretically. Much of the interest lies in the fact that it is the smallest and simplest of the pyramidal and enantiomorphic molecules whose ground and excited energy eigenstates are the quantum superpositions of two different spatial configurations, and that it is light enough that the transitions between the energy eigenstates are fast enough to be experimentally accessible. The ammonia molecule has two spatial eigenstates L and R with the nitrogen atom on one side or the other of the plane of hydrogen atoms, and its energy ground and first excited states 0 and 1 are the symmetric and antisymmetric quantum superpositions of the spatial eigenstates (ignoring rotational and vibrational states). The ammonia maser is based upon the transition between the energy eigenstates, which may also be described as the Rabi oscillation between the spatial eigenstates. However, the inversion transition is seen only at low gas pressure. As the gas pressure is increased, the transition broadens, shifts to lower frequency and then quenches (the frequency goes to zero). The ammonia molecule appears to undergo spatial localisation as a result of interaction with the environment. This would be of immense theoretical interest. In chemistry and in the classical world generally, enantiomorphic molecules with distinguishable spatial eigenstates L and R are always found in their spatial eigenstates (classical behaviour) rather than their energy ground states (quantum behaviour).3 Whilst ammonia is not enantiomorphic, it does appear to show both behaviours, quantum at low pressure and classical at high pressure, if the quenching is considered to be a direct observation of localisation or collapse of the wave-function into a spatial eigenstate. Within the context of the decoherence programme, it has been treated quantitatively in that way.4 In this paper, we show that interaction with the environment quenches the inversion transition for what might be described as ‘classical’ reasons. The broadening, shift and quenching of the Rabi oscillation are simply consequences of impacts and may be described within the framework of an oscillator subject to white noise from the environment. There is no evidence for localisation onto spatial eigenstates. Background:⎯At low pressures in the gas phase, the transition between the energy eigenstates is observed near 24GHz (0.8 cm–1) in NH3. 1 In ND3 2 the transition is near 1.6 GHz (0.053 cm–1). In NH3, broadening is observed at pressures above a few mm of mercury, with a shift to lower frequency, and quenching is complete at about 1.7 bar. In ND3, pressures about 15 times lower yield the same effects, in proportion to the inversion transition frequency. The first explanation of the shift and broadening of the ammonia inversion transition frequency was given by Anderson5 in terms of perturbation by the electric dipole-dipole interaction between ammonia molecules. Anderson’s discussion was only qualitative, and Margenau investigated the quantum states of two ammonia molecules coupled by their dipole-dipole interaction in more detail.6 He showed that the interaction leads to the splitting of the transition into a higher frequency component with reduced strength and a lower frequency with increased strength. While this accounts for the initial shift to lower frequency, it fails to account for the quenching of the inversion transition at a higher pressure. More recently, the dipole- dipole interaction model has been treated by a quantum mean-field approximation yielding, apparently, a frequency shift, quenching and spatial localisation at pressures for NH3 and ND3 in good agreement with experiment.7 The standard theory of line-broadening by impact is given by Van Vleck and Weisskopf.8 It predicts a line-shape function ( ) ( )202 −− bb f (1) where the width b is given by 1/2πτ for strong impacts occurring at a mean interval of τ, and therefore proportional to the pressure. The theory does not predict any peak shift: ν0 is a constant, the natural frequency of the oscillator. Anderson developed the theory further and obtained a shift of ν0 to lower frequency equal to the width b.9 Fano recast the problem of pressure broadening in the Liouville representation and obtained a shift to lower frequency independent of the broadening.10 Ben-Reuven used the Fano theory to show that the ammonia spectra can be well-fitted with a related expression but with three independent parameters proportional to the pressure. Two of them express the effects of elastic collisions on the width and on the frequency shift, and the third parameter expresses the effect of inelastic collisions.11 We are interested in a dynamical theory of the transition and of quenching and localisation. It is important to know if the dipole-dipole interaction of ammonia molecules is crucial to the quenching, or if it merely influences the collision cross- section while impacts are sufficient to account for the quenching. Accordingly, we have set up a molecular dynamics simulation in which the quantum nature of the ammonia molecule is explicitly taken into account.12 Here we show that the model accounts for the shift, broadening and quenching of the inversions transition purely in terms of perturbation by collision with ambient gas molecules. The Ammonia Quantum Molecular Dynamics Model:⎯We model the problem in one dimension. The ammonia molecule is modelled by a double-well potential, with the two time-dependent spatial wavefunctions ΨL and ΨR. With a weak coupling between the wells the Hamiltonian in the spatial basis is (2) ⎟⎟ Diagonalising, the ground and first excited states of the system are found to be Ψ0 and Ψ1 with a frequency splitting of ω1. The general state of the system is a superposition, with Ψ+Ψ=Ψ (3) Expanding this in the spatial basis set ΨL and ΨR, we have time-varying coefficients, RL tt Ψβ+Ψα=Ψ )()( (4) so that the amplitude of the wave-function beats, or oscillates between the two wells. The squared amplitude 2)(tα = α*α oscillates at the frequency ω1 and with a beat amplitude that depends on the initial values of a and b (from zero for e.g. 2/1== ba to a maximum amplitude of unity for e.g. a = 1, b = 0). This oscillation is the inversion transition or Rabi oscillation of the molecule. We model impacts, or interactions with the environment, by a term which is diagonal in the spatial representation. That is, we suppose that the double well is tilted during an impact. If a gas atom coming in from the left raises the energy of the left- hand well, the Hamiltonian during impact is (5) ⎟⎟ Diagonalising and expanding in the spatial basis set as before, we obtain the normalised eigenvectors u and v of H′ . Equations 3 and 4 become RLPP ttba Ψβ′+Ψα′=Ψ′+Ψ′=Ψ′ )()(10 (6) The Rabi oscillation is now at a much higher frequency and a much smaller amplitude (for ωP >> ω1). In reality, the perturbation rises and falls continuously in an impact, but we approximate with a top-hat function, so that ωP is switched on at a time t0 and switched off again at t1. At these times, we match the coefficients in the spatial basis, using and )()( 00 tt α=α′ )()( 00 tt β=β′ to solve for aP and bP at the onset of the perturbation, and then the new )()( 11 tt α′=α , )()( 11 tt β′=β to solve for the new a and b at the end of the perturbation. These boundary conditions ensure that the amplitude and phase of the wave-function in each well do not change discontinuously at the beginning and end of the perturbation. The resulting time evolution of α*α is illustrated in Fig.1. 0 1 2 Time, t xirta Fig.1. The evolution of the occupancy of the left-hand well is shown with two perturbations occurring at t = 0.7 and t = 1.6. The Rabi angular frequency ω1 is 2π and the perturbation ωP = 60. The initial wavefunction is given by a = b = 1/ 2 ; after the two perturbations the values are a = 0.54 – 0.73i, b = 0.36 + 0.22i. To model NH3 and ND3, we can choose the units of time so that the Rabi frequency is unity (ω1 = 2π). The strength of the perturbation is of the order of kBT, which at room temperature is 208 cm–1. For NH3, therefore, we take ωP = 208 ω1/0.8 = 260 ω1 and for ND3, ωP = 208 ω1/0.0.053 = 3925 ω1. The duration Δt = t0 – t1 of an impact is hard to estimate. However, inspection of Fig.1 shows that to achieve a strong impact (in the sense of Van Vleck and Weisskopf 8), we need something of the order of one cycle of the perturbed Rabi oscillation, i.e. ωPΔt ~2π, while larger values will have no extra effect. We therefore take values of Δt from a random distribution over the range 0 to 2π / ωP. The average frequency of impacts corresponds to the gas pressure. We require an impact cross-section to relate the frequency of impacts to the gas pressure quantitatively. Bleaney and Loubser and other authors obtain impact cross-sections from the pressure-broadening of the transition, assuming strong impacts and using b = 1/2πτ. We shall see below that such estimates are unreliable, and therefore in our simulation we use the measure p impacts per Rabi cycle instead of pressure, and we vary p over a wide range. We calculate the values of α*α at discrete time intervals δt with Δt < δt << 1. At each time interval we have a probability δt/τ of having an impact, so that there are p = 1/τ impacts per cycle. If there is an impact, we use )()( 00 tt α=α′ and to solve for aP and bP at the onset of the perturbation, and then calculated the new , )()( 00 tt β=β′ )()( 11 tt α′=α )()( 11 tt β′=β to solve for the new a and b at the end of the perturbation. Then the calculation of the list of values is resumed. Examples are shown in Fig.2 for medium (a) and high (b) values of p. The numerical Fourier Transforms of the lists are calculated, shown in Fig.2(c) and (d), and fitted with Af(ν) of eq.1, with b, ν0 and amplitude A as fitting parameters. Our interest here is the fitted values of b and ν0 as functions of p. In Figure 3 these are compared with the experimental data for NH3,16 with the constant of proportionality between p and pressure (corresponding to the impact cross-section) as a free parameter. Fig.2. The occupation of the left-hand well, y(t) = α*α, is plotted against time, for (a) p = 3.5 impacts per cycle, below the quenching, and (b) p = 7.5 impacts per cycle, above the quenching. The Fourier transforms Y(ν) are shown in (c) and (d) respectively, together with the fits using eq.1. Fig.3. The solid curves show (a) the peak frequency and (b) the broadening for NH3 as a function of the number of impacts per cycle as described in the text. The data points show (a) the peak frequency and (b) the broadening reported by Bleaney and Loubser,16 scaled as described in the text. The NH3 data is plotted with p = 4.5 impacts per cycle equivalent to a pressure P = 1 bar. For ND3, the data fits equally well but with p = 4.5 equivalent to the pressure P = 1/15 bar, consistent with the fifteen times lower inversion frequency in ND3 but the same impact parameter. In both cases, full quenching is observed at about 6.5 impacts per cycle. The model presented here accounts remarkably well for the shift and quenching of the ammonia inversion transition peak. It accounts less well for the broadening, which occurs initially at the rate b ~ 0.25 p in the simulation and 0.18 p in the experimental data. The experimental broadening is about three-quarters of the model broadening up to the quenching pressure, and above the quenching pressure the experimental broadening continues to increase while the model broadening decreases. To gain a better understanding of this behaviour, we investigate how a simple classical oscillator behaves under similar perturbations. A Classical Perturbed Harmonic Oscillator:⎯A classical oscillator may be perturbed by collision in a large variety of well-defined ways. We evaluate two perturbations here. We calculate the values of a sinusoid of frequency ν = 1 at discrete time intervals δt << 1. At each time interval we have a probability δt/τ of having an impact, so that there are p = 1/τ impacts per cycle. If there is an impact, the sinusoid is modified accordingly, and then the calculation of the list of values is resumed. The numerical Fourier Transform is calculated and we find that af(ν) of eq.1 fits well for a variety of definitions of the impacts, over a very wide range of p, with b, ν0 and a as fitting parameters. Our interest here is the fitted values of b and ν0 as functions of p. Impactsper cycle ycneuqerF 0 1 2 3 4 5 6 Impactsper cycle 0 1 2 3 4 5 6 Impacts per cycle ycneuqerF 0 2 4 6 8 10 12 14 Impacts per cycle 0 2 4 6 8 10 12 14 Fig.4. The peak frequency and the peak width are plotted for the broken sinusoids with unity frequency described in the text. In (a) and (b), results are for the strongest possible impact, with phase and amplitude completely randomised by the impact. In (c) and (d), the boundary condition at impact keeps the sinusoid continuous but changes the phase and amplitude at random within that constraint. The strongest perturbation possible (in the sense of Van Vleck and Weisskopf 8) is a collision that destroys all memory of position and speed (or amplitude and phase). To model this, at impact we pick the new amplitude A of the sinusoid Acos(2πνt + ϕ) at random from the range 0 to 1 and the new phase ϕ at random from the range 0 to 2π. In this model, the peak is shifted to higher frequency, shown in Fig.4(a) until the width of the transition in Fig.4(b) reaches b = 1, at which point quenching occurs, i.e. the frequency collapses to zero. The width continues to increase at still higher values of p, Fig.4(b). This is a stronger impact than the impacts of the ammonia molecule, for the quenching occurs at p = 3.5 impacts per cycle and the initial slope of the broadening is given by b ~ 0.5 p. It is interesting that Van Vleck and Weisskopf give the broadening for strong impacts as 1/2πτ, equivalent to 0.16 p. In an alternative definition of impact which is in closer accordance with the ammonia model of Section 2.1 and Fig.1, we define the impact at t0 by taking the position x(t0) as unchanged by the impact, the new amplitude A as random in the range x(t0) to 1, and the new phase ϕ such that the speed is a random variable in )( 0tx& the range consistent with the new amplitude. In this model, the peak shift in Fig.4(c) and width behave in very much the same way as the ammonia results of Fig.3, with the initial broadening b ~0.2 p. However, the impact is weaker than in the ammonia model, for quenching occurs at p = 10.5 impacts per cycle. Above quenching b decreases again. In these models we may weaken the strength of the impact by letting the new amplitude and phase be given by some amount ε of the amplitude and phase calculated as above plus (1 – ε) of the old amplitude and phase. Not surprisingly, the shift and broadening are identical if plotted as functions of the normalised impact rate εp. Results and Discussion:⎯Fig.4 shows that the details of the peak shift and the broadening are very sensitive to the exact nature of the boundary conditions between the periods of unperturbed free oscillation. A more complete description of the impact (including, for example, inelastic collisions as in Ben-Reuven11) might well account for the discrepancies between data and model in Fig.3. However, we do not know of any way to predict the initial slope of b(p), nor its functional form below and above the quenching, from a specification of the boundary conditions. Neither the mathematics of the noisy oscillator (see, e.g. the book by Gittenberg13) nor of the classical kicked rotor (see, e.g. Ref.) appear to answer this question. The key result is that the ammonia model (Fig.3) and even the broken sinusoid of Fig.4(c) both show the Rabi oscillation frequency shifting to lower frequency, broadening, and quenching – going to zero frequency – as the impact rate is increased, in agreement with experiment. Note that the density matrix shows no evidence of localisation. In particular, the off-diagonal elements do not go towards zero as predicted by the decoherence programme. Nor is there any evidence or quantum state diffusion towards the configurational eigenstates. The ammonia shift and quenching are fully accounted for in terms of a perturbed oscillator, and should not therefore be cited as an experimental observation of quantum localisation. Acknowledgements:⎯We are grateful to Prof. I.C. Percival and Dr T. Prellberg for useful discussions. References 1. C.H. Townes, Phys. Rev. 70, 665 (1946). 2. G. Birnbaum and A.A. Maryott, Phys. Rev. 92, 270 (1953). 3. R.G. Woolley, Adv. Phys. 25, 27 (1976). 4. E. Joos, in Decoherence and the Appearance of a Classical World in Quantum Theory, ed. D. Guilini (Springer, Berlin, 1996). 5. P.W. Anderson, Phys. Rev. 75, 1450 (1949). 6. H. Margenau, Phys. Rev. 76, 1423 (1949). 7. G. Jona-Lasinio, C. Presilla and C. Toninelli, Phys. Rev. Lett. 88, 123001 (2002). 8. J.H. Van Vleck and V.F. Weisskopf, Rev. Mod. Phys. 17, 227 (1945). 9. P.W. Anderson, Phys. Rev. 76, 647 (1949). 10. U. Fano, Phys. Rev. 131, 259 (1963). 11. A. Ben-Reuven, Phys. Rev. Lett. 14, 349 (1965). 12. D.J. Dunstan, archive. 13. M. Gittenberg, The Noisy Oscillator: The First Hundred Years, From Einstein Until Now (World Scientific, Singapore, 2005). 14. B. Bleaney and R.P. Penrose, Proc. Roy. Soc. A189, 358 (1947). 15. B. Bleaney and R.P. Penrose, Proc. Phys. Soc. 60, 540 (1948). 16. B. Bleaney and J.H.N. Loubser, Proc. Phys. Soc. A63, 483 (1950). Acknowledgements:(We are grateful to Prof. I.C. Percival and Dr T. Prellberg for useful discussions.

arXiv:0704.1922v4 [math.GT] 4 Aug 2008 RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES MAHAN MJ Abstract. We introduce and study the notion of relative rigidity for pairs (X,J ) where 1) X is a hyperbolic metric space and J a collection of quasiconvex sets 2) X is a relatively hyperbolic group and J the collection of parabolics 3) X is a higher rank symmetric space and J an equivariant collection of maximal flats Relative rigidity can roughly be described as upgrading a uniformly proper map between two such J ’s to a quasi-isometry between the cor- responding X’s. A related notion is that of a C-complex which is the adaptation of a Tits complex to this context. We prove the relative rigidity of the collection of pairs (X,J ) as above. This generalises a re- sult of Schwarz for symmetric patterns of geodesics in hyperbolic space. We show that a uniformly proper map induces an isomorphism of the corresponding C-complexes. We also give a couple of characterizations of quasiconvexity of subgroups of hyperbolic groups on the way. AMS subject classification = 20F67(Primary), 22E40, 57M50(Secondary) Contents 1. Introduction 2 1.1. Relative Rigidity and Statement of Results 2 1.2. Relative Hyperbolicity and Electric Geometry 6 1.3. Height of Subgroups and C-Complexes 8 2. Characterizations of Quasiconvexity 9 2.1. Limit Sets and Quasiconvexity 10 2.2. Quasiconvexity and Relative Hyperbolicity 11 3. Relative Rigidity 13 3.1. Pairing of Limit Sets by Quasi-isometries 13 3.2. C-Complexes 16 3.3. Cross Ratios, Annular Systems and a Dynamical Formulation 17 3.4. Axiomatisation, Relative Hyperbolicity 18 3.5. Symmetric Spaces of Higher Rank 20 References 21 http://arxiv.org/abs/0704.1922v4 2 MAHAN MJ 1. Introduction 1.1. Relative Rigidity and Statement of Results. In this paper, we study a rigidity phenomenon within the framework of coarse geometry. We call it relative rigidity. Much of the work on quasi-isometric rigidity (e.g. Farb-Schwartz [FS96] Kleiner-Leeb [KL97b] Eskin-Farb [EF 3] and Mosher- Sageev-Whyte [MSW03] [MSW04] ) contains a crucial step showing that a self quasi-isometry of a space X coarsely preserves a family J of distin- guished subsets of X. The family J again has a coarse intersection pattern that may be combinatorially coded and these proofs of quasi-isometric rigid- ity often show that the intersection pattern is preserved by a quasi-isometry. In this note, we investigate a sort of a converse to this: When does a uniformly proper map between two families J1 and J2 come from a quasi-isometry φ between X1 and X2? Does such a map preserve intersection patterns? We show that the answer is affirmative when (1) Xi’s are (Cayley graphs of) hyperbolic groups and Ji’s correspond to cosets of a quasiconvex subgroup (2) Xi’s are (Cayley graphs of) relatively hyperbolic groups and Ji’s correspond to parabolic subgroups (3) Xi’s are symmetric spaces of non-positive curvature and Ji’s corre- spond to lifts of a maximal torus in a compact locally symmetric space modeled on Xi. If in addition one can show that a quasi-isometry preserving intersection patterns is close to an isometry, we would be able to conclude that a uni- formly proper map between the Ji’s is induced by an isometry. This latter phenomenon has been investigated by Mosher, Sageev and Whyte [MSW04] and has been termed pattern rigidity. Thus, in a sense, the notion of relative rigidity complements that of pattern rigidity. Some further examples where a family of distinguished subsets of a space and the resulting (combinatorial) configuration yields information about the ambient space are: 1) Collection of flats in a symmetric space of higher rank (Mostow [Mos73]) 2) Collection of maximal abelian subgroups of the mapping class group (Behrstock-Drutu-Mosher [BDM05] ) 3) Collection of hyperbolic spaces in the Cayley complex of the Baumslag- Solitar groups (Farb-Mosher [FM98] , [FM 3] ; see also [FM00] ) 4) Quasi-isometric rigidity of sufficiently complicated patterns of flats in the universal cover of a Haken 3 manifold (Kapovich-Leeb [KL97a] ) 5) We were most influenced by a beautiful result of Schwarz [Sch97] which shows that a uniformly proper map from a symmetric pattern of geodesics in RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 3 Hn to another symmetric pattern of geodesics in Hn (for n > 2) is induced by an isometry. Again as in Mostow, there are two parts to this. A first step is to construct a quasi-isometry of Hn inducing the given pairing. Schwarz terms this ambient extension. The second is to construct an isometry. Let us look at a general form of the situation that Schwarz considers. (X1, d1), (X2, d2) are metric spaces. Let J1,J2 be collections of closed sub- sets of X1,X2 respectively. Then di induces a pseudo-metric (which, by abuse of notation, we continue to refer to as di) on Ji for i = 1, 2. This is just the ordinary (not Hausdorff) distance between closed subsets of a metric space. In [Sch97], X1 = X2 = H n, and Ji are lifts (to the universal cover) of finite collections of closed geodesics in two hyperbolic manifolds. Also, the hypothesis in Schwarz’s paper [Sch97] is the existence of a uni- formly proper map φ between symmetric patterns of geodesics J1 and J2. A uniformly proper map may be thought of as an isomorphism in the so-called coarse category in the sense of John Roe [Roe95]. Thus, we can re-interpret the first step of Schwarz’s result as saying that an isomorphism φ in the coarse category between Ji implies the existence of a quasi-isometry from Hn to itself inducing φ. In the language of [Sch97], uniformly proper pairings come from ambient extensions. In Mostow’s proof of rigidity for higher rank symmetric spaces, he obtains in a crucial step, an isomorphism of Tits complexes [Mos73]. We would like to associate to a pair (X,J ) some such complex just as a Tits complex is associated to a higher rank locally symmetric space and its collection of maximal parablic subgroups. We propose the notion of a C-complex in this paper as the appropriate generalization of a Tits complex to coarse geometry. Then what we would hope for (as a conclusion) is an isomorphism of these C-complexes. This transition from the existence of a uniformly proper map between Ji’s to the existence of a a quasi-isometry between Xi’s inducing an isomorphism of C-complexes is what we term relative rigidity. Schwarz proves the relative rigidity of pairs (X,J ) where X is hyperbolic space and J a symmetric collection of geodesics. Much of what he does extends to the case whereX is a higher rank symmetric space and J a symmetric collection of maximal periodic flats or a symmmetric collection of maximal parabolic subgroups in a non-uniform lattice. The main point of this paper is illustrated first in the context of rela- tive rigidity of the category of pairs (Γ,J ), where Γ is (the Cayley graph of) a hyperbolic group, and J the set of cosets of a quasiconvex subgroup. Throughout this paper we shall assume that the quasiconvex sub- groups are of infinite index in the big groups. Note that the upgrading of a uniformly proper map between J ’s to a quasi-isometry between the Γ’s is the most we can hope for in light of the fact that the Cayley graph of a finitely generated group is only determined up to quasi-isometry. (See Paulin [Pau96] for a proof of this fact.) 4 MAHAN MJ We start with a pair of hyperbolic groups G1, G2 with Cayley graphs Γ1,Γ2, and quasiconvex subgroups H1,H2. Let Λ1, Λ2 be the limit sets of H1,H2 in ∂G1, ∂G2 respectively. For convenience we consider the collection Ji of translates of Ji the join of Λi in Γi rather than cosets of Hi. Recall that the join of Λi is the union of bi-infinite geodesics in Γi with end-points in Λi. This is a uniformly quasiconvex set and lies at a bounded Hausdorff distance from the Cayley graph of the subgroup Hi (Since H has finite index in its commensurator, only finitely many cosets of H are at a finite Hausdorff distance from it. Since Ji is at a bounded Hausdorff distance from Hi the same is true for elements of Ji.) The main theorems of this paper are as follows. Theorem 3.5:Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi-isometry q from Γ1 to Γ2 which pairs the sets J1 and J2 as φ does. The construction of the quasi-isometry q proceeds by constructing a ”coarse barycenter” of some infinite diameter sets (reminiscent of the celebrated measure-theoretic barycenter method discovered by Douady and Earle, and extended greatly by Besson, Courtois, Gallot [BCG98] ). We prove an analogous theorem for pairs (X,J ) when X is (strongly) hyperbolic relative to the collection J . Theorem 3.11:Let Xi be (strongly) hyperbolic relative to collections Ji (i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi-isometry q from X1 to X2 which pairs the sets J1 and J2 as φ does. As a Corollary of Theorem 3.11 and work of Hruska and Kleiner [HK04], we deduce relative rigidity for pairs (X,J ) where X is a CAT(0) space with isolated flats and J is the collection of maximal flats. The third main theorem of this paper is an analog for higher rank sym- metric spaces. Theorem 3.13:Let Xi be symmetric spaces of non-positive curvature, and Ji be equivariant collections of lifts of a maximal torus in a compact locally symmetric space modeled on Xi (i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi-isometry q from X1 to X2 which pairs the sets J1 and J2 as φ does. In fact, combining Theorem 3.13 with the quasi-isometric rigidity theorem of Kleiner-Leeb [KL97b] and Eskin-Farb [EF 3], we may upgrade the quasi- isometry of Theorem 3.13 to an isometry. Let Gi,Hi (i = 1, 2) be hyperbolic groups and quasiconvex subgroups re- spectively. In Section 1.3, we shall construct simplicial complexes (termed C- complexes) from the incidence relations determined by the cosets of Hi. Let C(Gi,Hi) be the C-complexes associated with the pairs (Gi,Hi). Roughly RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 5 speaking, the vertices of C(Gi,Hi) are the translates g iΛi of Λi by distinct coset representatives g i and the (n− 1)-cells are n-tuples {g 1Λ, · · · , g 1Λ} of distinct translates such that ∩n1g 1Λ 6= ∅. Theorem 3.7: Let φ : J1 → J2 be a uniformly proper map. Then φ induces an isomorphism of C(G1,H1) with C(G2,H2). On the way towards proving Theorems 3.5 and 3.7, we prove two Proposi- tions characterizing quasiconvexity. These might be of independent interest. The first is in terms of the Hausdorff topology on the collection C0c (∂G), which is the collection of closed subsets of ∂G having more than one point. Proposition 2.3: Let H be a subgroup of a hyperbolic group G with limit set Λ. Let L be the collection of translates of Λ by elements of distinct cosets of H (one for each coset). Then H is quasiconvex if and only if L is a discrete subset of C0c (∂G). The second characterization is in terms of strong relative hyperbolicity. Definition 1.1. A subgroup H of a group G is said to be malnormal if for all g ∈ G \H, gHg−1 ∩H is trivial. A subgroup H of a group G is said to be almost malnormal if for all g ∈ G \H, gHg−1 ∩H is finite. It was pointed out to us by the referee that the following result follows from work of Farb [Far98], Bowditch ([Bow97] Theorem 7.11) and Drutu- Sapir ([DS05] Lemma 4.15). We shall include a proof for completeness. Proposition 2.9:[Far98] [Bow97] [DS05] Let G be a hyperbolic group and H a subgroup. Then G is strongly relatively hyperbolic with respect to H if and only if H is a malnormal quasiconvex subgroup. The prototypical example is that of (fundamental groups of) a closed hyperbolic manifold with a totally geodesic embedded submanifold. Finally, we give an intrinsic or dynamic reformulation of Theorems 3.5 and 3.7 following Bowditch [Bow98], which makes use of the existence of a cross-ratio on the boundary of a hyperbolic group. The cross-ratio in turn induces a pseudometric on the collection L of translates of Λ. Theorem 3.10: Let G1, G2 be uniform convergence (hence hyperbolic) groups acting on compacta M1,M2 respectively. Also, let Åi (for i = 1, 2) be Gi-invariant annulus systems and let (..|..)i denote the corresponding an- nular cross-ratios. Let H1,H2 be subgroups of G1, G2 with limit sets Λ1,Λ2. Suppose that the set Li of translates of Λi (for i = 1, 2) by essentially distinct elements of Hi in Gi forms a discrete subset of C c (Mi). Also assume that there exists a bijective function φ : L1 → L2 and that this pairing is uniformly proper with respect to the cross-ratios (..|..)1 and (..|..)2. 1) Hi is quasiconvex in Gi 2) There is a homeomorphism q : M1 → M2 which pairs L1 with L2 as φ 6 MAHAN MJ does. Further, q is uniformly proper with respect to the cross-ratios (..|..)1 and (..|..)2 on M1, M2 respectively. 3) q (and hence also φ) induces an isomorphism of C-complexes C(G1,H1) with C(G2,H2). Acknowledgements: My interest in relative hyperbolicity and quasi-isometric rigidity is largely due to Benson Farb. It is a pleasure to acknowledge his help, support and camaraderie, both mathematical and personal. I would also like to thank the referee for suggesting several corrections and for pro- viding additional references. 1.2. Relative Hyperbolicity and Electric Geometry. We start off by fixing notions and notation. Let G (resp. X) be a hyperbolic group (resp. a hyperbolic metric space) with Cayley graph (resp. a net) Γ equipped with a word-metric (resp. a simplicial metric) d. Here a net N is a collection of distinct points xi ∈ X such that there exist 0 < C1 < C2 such that 1) d(xi, xj) ≥ C1 for all i 6= j 2) For all x ∈ X, there exists xi ∈ N such that d(xi, x) ≤ C2 For the net N we construct a graph GN with edges corresponding to pairs xi 6= xj such that d(xi, xj) ≤ 4C2. The simplicial metric on N is obtained by declaring that each edge of GN has length one. Let the Gromov boundary of Γ be denoted by ∂G. (cf.[GdlH90]). We shall have need for the fact that for hyperbolic metric spaces (in the sense of Gromov [Gro85]) the notions of quasiconvexity and qi embeddings coincide [Gro85]. We shall now recall certain notions of relative hyperbolicity due to Gro- mov [Gro85] and Farb [Far98]. Let X be a path metric space. A collection of closed subsets H = {Hα} of X will be said to be uniformly separated if there exists ǫ > 0 such that d(H1,H2) ≥ ǫ for all distinct H1,H2 ∈ H. The electric space (or coned-off space) X̂ corresponding to the pair (X,H) is a metric space which consists of X and a collection of vertices vα (one for each Hα ∈ H) such that each point of Hα is joined to (coned off at) vα by an edge of length Definition 1.2. [Far98] [Bow97] Let X be a geodesic metric space and H be a collection of uniformly separated subsets. Then X is said to be weakly hyperbolic relative to the collection H, if the electric space X̂ is hyperbolic. Lemma 1.3. (See Bowditch [Bow97], generalizing Lemma 4.5 and Propo- sition 4.6 of Farb [Far98]) Given δ, C,D there exists ∆ such that if X is a δ-hyperbolic metric space with a collection H of C-quasiconvex D-separated sets. then, the electric space X̂ is ∆-hyperbolic, i.e. X is weakly hyperbolic relative to the collection H. RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 7 Definitions: Given a collection H of C-quasiconvex, D-separated sets and a number ǫ we shall say that a geodesic (resp. quasigeodesic) γ is a geodesic (resp. quasigeodesic) without backtracking with respect to ǫ neighborhoods if γ does not return to Nǫ(H) after leaving it, for any H ∈ H. A geodesic (resp. quasigeodesic) γ is a geodesic (resp. quasigeodesic) without backtracking if it is a geodesic (resp. quasigeodesic) without backtracking with respect to ǫ neighborhoods for some ǫ ≥ 0. Electric P -quasigeodesics without backtracking are said to have similar intersection patterns if for β, γ electric P -quasigeodesics without backtrack- ing both joining x, y, the following are satisfied. (1) Similar Intersection Patterns 1: if precisely one of {β, γ} meets an ǫ-neighborhood Nǫ(H1) of an electrocuted quasiconvex set H1 ∈ H, then the length (measured in the intrinsic path-metric on Nǫ(H1) ) from the entry point to the exit point is at most D. (2) Similar Intersection Patterns 2: if both {β, γ} meet some Nǫ(H1) then the length (measured in the intrinsic path-metric on Nǫ(H1) ) from the entry point of β to that of γ is at most D; similarly for exit points. Definition 1.4. [Far98] [Bow97] Let X be a geodesic metric space and H be a collection of mutually disjoint uniformly separated subsets such that X is weakly hyperbolic relative to the collection H. If any pair of P - electric quasigeodesics without backtracking starting and ending at the same point have similar intersection patterns with horosphere-like sets (elements of H) then quasigeodesics are said to satisfy Bounded Penetration and X is said to be strongly hyperbolic relative to the collection H. Definition 1.5. [Mj05] A collection H of uniformly C-quasiconvex sets in a δ-hyperbolic metric space X is said to be mutually D-cobounded if for all Hi,Hj ∈ H, πi(Hj) has diameter less than D, where πi denotes a nearest point projection of X onto Hi. A collection is mutually cobounded if it is mutually D-cobounded for some D. Mutual coboundedness was proven by Farb for horoballs in finite volume Hadamard manifolds of pinched negative curvature in Lemma 4.7 of [Far98]. The following generalization is due to Bowditch [Bow97]. Lemma 1.6. (See Bowditch [Bow97] Lemma 7.13 for a proof) Suppose X is a δ-hyperbolic metric space with a collection H of C-quasiconvex K-separated D-mutually cobounded subsets. Then X is strongly hyperbolic relative to the collection H. Gromov gave a different definition of strong relative hyperbolicity. We give a condition below that is equivalent to a special case of Gromov’s definition. Let X be a geodesic metric space with a collection H of uniformly 8 MAHAN MJ separated subsets {Hi}. The hyperbolic cone cHi is the product of Hi and the non-negative reals Hi ×R+, equipped with the metric of the type 2−tds2 + dt2. More precisely, Hi ×{n} is given the path metric of Hi scaled by 2−n. The R+ direction is given the standard Euclidean metric. Let X denote X with hyperbolic cones cHi glued to it along Hi’s. X h will be referred to as the hyperbolically coned off X. This is to be contrasted with the coned off space X̂ in Farb’s definition. Definition 1.7. X is said to be strongly hyperbolic relative to the collec- tion H in the sense of Gromov if the hyperbolically coned off space Xh is a hyperbolic metric space. The equivalence of the two notions of strong relative hyperbolicity was proven by Bowditch in [Bow97]. Theorem 1.8. ( Bowditch [Bow97] ) X is strongly hyperbolic relative to a collection H of uniformly separated subsets {Hi} in the sense of Gromov if and only if X is strongly hyperbolic relative to the collection H in the sense of Farb. 1.3. Height of Subgroups and C-Complexes. The notion of height of a subgroup was introduced by Gitik, Mitra, Rips and Sageev in [GMRS97] and further developed by the author in [Mit04]. Definition 1.9. Let H be a subgroup of a group G. We say that the elements {gi|1 ≤ i ≤ n} of G are essentially distinct if Hgi 6= Hgj for i 6= j. Con- jugates of H by essentially distinct elements are called essentially distinct conjugates. Note that we are abusing notation slightly here, as a conjugate of H by an element belonging to the normalizer of H but not belonging to H is still essentially distinct from H. Thus in this context a conjugate of H records (implicitly) the conjugating element. Definition 1.10. We say that the height of an infinite subgroup H in G is n if there exists a collection of n essentially distinct conjugates of H such that the intersection of all the elements of the collection is infinite and n is maximal possible. We define the height of a finite subgroup to be 0. We say that the width of an infinite subgroup H in G is n if there exists a collection of n essentially distinct conjugates of H such that the pairwise intersection of the elements of the collection is infinite and n is maximal possible. The main theorem of [GMRS97] states: Theorem 1.11. If H is a quasiconvex subgroup of a hyperbolic group G,then H has finite height and finite width. In this context, a theorem we shall be needing several times is the following result from [GMRS97] that is proved using a result of Short [Sho91]. RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 9 Theorem 1.12. (Lemma 2.6 of [GMRS97]) Let G be a hyperbolic group and Hi (for i = 1 · · · k ) be quasiconvex subgroups with limit sets Λi, i = 1 · · · k. Then ∩Hi is a quasiconvex subgroup with limit set ∩Λi. We now proceed to define a simplicial complex C(G,H) for a group G and H a subgroup. For G hyperbolic and H quasiconvex, we give below three equivalent descriptions of a complex C(G,H). In this case, let ∂G denote the boundary of G, Λ the limit set of H, and J the join of Λ. 1) Vertices ( 0-cells ) are conjugates of H by essentially distinct elements, and (n − 1)-cells are n-tuples {g1H, · · · , gnH} of distinct cosets such that ∩n1giHg i is infinite (in fact by Theorem 1.12 an infinite quasiconvex sub- group of G). 2) Vertices ( 0-cells ) are translates of Λ by essentially distinct elements, and (n−1)-cells are n-tuples {g1Λ, · · · , gnΛ} of distinct translates such that ∩n1giΛ 6= ∅. 3) Vertices ( 0-cells ) are translates of J by essentially distinct elements, and (n− 1)-cells are n-tuples {g1J, · · · , gnJ} of distinct translates such that ∩n1giJ is infinite. We shall refer to the complex C(G,H) as the C-complex for the pair G,H. (C stands for “coarse” or “Čech” or “cover”, since C(G,H) is like a coarse nerve of a cover, reminiscent of constructions in Cech cochains.) Note that if h(H) denote the height of H, then (h(H)+1) is the dimension of the C-complex C(G,H). Also, if w(H) denote the width of H, then w(H) = w is equal to the size of the largest complete graph Kw that is embeddable in C(G,H). If C(G,H) is connected then its one-skeleton is closely related to the coned off space Γ̂ with an appropriately chosen set of generators. This definition is inspired by that of the Tits complex for a non-uniform lattice in a higher rank symmetric space. Related constructs in the context of codimension 1 subgroups also occur in work of Sageev [Sag95] where he constructs cubings. 2. Characterizations of Quasiconvexity Let G be a hyperbolic group. Let Cc(∂G) denote the collection of closed subsets of the boundary ∂G equipped with the Hausdorff topology. Let C0c (∂G) ⊂ Cc(∂G) denote the subset obtained from Cc(∂G) by removing the singleton sets {{x} : x ∈ ∂G}. Next fix a subgroup H ⊂ G with limit set Λ ⊂ ∂G. Consider the G-invariant collection L ={ gΛ } ⊂ C0c (∂G) with g ranging over distinct cosets (one for each coset) of H in G. Note that L is (strictly speaking) a multi-set as distinct elements of L may denote the same element of C0c (∂G) in case two distinct translates of Λ coincide. One extreme case is when Λ = ∂G, though H is of infinite index in G (e.g. if H is normal of infinite index in in G.) Then L consists of infinitely many copies of Λ. 10 MAHAN MJ Definition 2.1. The join J(Λ) of Λ is defined as the union of all bi-infinite geodesics whose end-points lie in Λ It is easy to see that J(Λ) is 2δ-quasiconvex if G is δ-hyperbolic. In fact this is true for any subset Λ of the boundary of a δ-hyperbolic metric space X (no equivariance is necessary). For Λ the limit set of H, J(Λ) is H- invariant. The visual diameter dia∂G(Λ) of a subset Λ of ∂G is the same as the diameter in the metric on ∂G obtained from the Gromov inner product. (See [GdlH90] Chapter 7 for details about the visual metric on ∂G.) 2.1. Limit Sets and Quasiconvexity. The next Lemma follows directly from the fact that sets with visual diameter bounded below contain points with Gromov inner product bounded above and conversely[GdlH90]. Lemma 2.2. For all ǫ > 0 there exists N such that if the diameter dia∂G(Λ) ≥ ǫ for a closed subset Λ of ∂G, then there exists p ∈ J(Λ) such that d(p, 1) ≤ N . Conversely, for all N > 0 there exists ǫ > 0 such that if there exists p ∈ J(Λ) with d(p, 1) ≤ N , then dia∂G(Λ) ≥ ǫ. The next Proposition gives our first characterisation of quasiconvex sub- groups of a hyperbolic group. Proposition 2.3. (Characterization of Quasiconvexity I) Let H be a subgroup of a hyperbolic group G with limit set Λ. Let L be the collection of translates of Λ (counted with multiplicity) by elements of distinct cosets of H (one for each coset). Then H is quasiconvex if and only if L is a discrete subset of C0c (∂G). Proof: Suppose H is quasiconvex. We want to show that L is a discrete subset of C0c (∂G). Thus it suffices to show that any limit of elements of L is a singleton set. This in turn follows from the following. Claim: For all ǫ > 0, Lǫ = {Li ∈ L : dia∂G(Li) ≥ ǫ} is finite. Proof of Claim: Let N = N(ǫ) be as in Lemma 2.2. Since dia∂G(Li) ≥ ǫ, therefore by Lemma 2.2, there exists pi ∈ J(Li) such that dG(pi, 1) ≤ N . Also, there exists K > 0 depending on δ (recall that J(Li) is 2δ-qc) and the quasiconvexity constant of H such that if Li = giΛ, then there exists hi ∈ H with dG(pi, gihi) ≤ K. Hence, dG(1, gihi) ≤ K + N . Since G is finitely generated, the number of such elements gihi is finite. Since gi are picked from distinct cosets of H, we conclude that the set Lǫ is finite. ✷ Conversely, suppose that H is not quasiconvex. Assume, without loss of generality, that a finite generating set of H is contained in a finite generating set of G and that ΓH ,ΓG are Cayley graphs with respect to these generating sets. Then there exist pi ∈ J(Λ) such that dG(pi,ΓH) ≥ i. Translating by an appropriate element of H, we may assume that dG(pi,ΓH) = dG(pi, 1) ≥ i. Further, we may assume (by passing to a subsequence if necessary) that the sequence dG(pi, 1) is monotonically increasing. Then p i J(Λ) has limit set p−1i Λ. Further, as pi ∈ J(Λ), therefore, 1 ∈ p i J(Λ). Since J(Λ) RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 11 is 2δ-qc, so is p−1i J(Λ) for all i. Hence, there exists ǫ > 0 by Lemma 2.2 such that dia∂Gp i J(Λ) ≥ ǫ. Since dG(pi, 1) is monotonically strictly increasing, we conclude that pi’s lie in distinbct cosets of H. Further, since Cc(∂G) is compact, we conclude that the collection p i J(Λ) has a convergent subsequence, converging to a subset of diameter greater than or equal to ǫ. Therefore, the collection L is not a discrete subset (strictly speaking a multiset) of C0c (∂G). ✷ We next prove a result about projections of J(Li) on J(Lj). We start off with an elementary fact about hyperbolic metric spaces. See [Mit98] for a proof. Lemma 2.4. [Mit98] Given δ > 0, there exist D,C1, k, ǫ such that if a, b, c, d are points of a δ-hyperbolic metric space (Z, d), with d(a, [b, c]) = d(a, b), d(d, [b, c]) = d(c, d) and d(b, c) ≥ D then [a, b] ∪ [b, c] ∪ [c, d] lies in a C1- neighborhood of any geodesic joining a, d and is a (k, ǫ)-quasigeodesic. Assume that H is quasiconvex and that Lk is the limit set gkΛ of gkH. Let Pj denote the nearest point projection of ΓG onto J(Lj). Also, let Hk = gkΓH be the left translate of ΓH by gk. Proposition 2.5. There exists K > 0 such that Pj(ΓHi) lies in a K- neighborhood of J(Li ∩ Lj) if (Li ∩ Lj) 6= ∅. Else, Pj(ΓHi) has diameter less than K. Proof: Since J(Li) is 2δ-qc and H is quasiconvex, it suffices to show that Pj(J(Li)) lies in a K-neighborhood of J(Li ∩Lj) if the latter is non-empty. By G-equivariance, we may assume that Lj = Λ and gi = 1. We represent Pj by P in this case. First note that by Theorem 1.12, Hi∩Hj is quasiconvex and the limit set of Hi ∩Hj is Li ∩ Lj. Also, J(Li ∩ Lj) ⊂ J(Li). Let a, b ∈ J(Li). Let P (a) = c, P (b) = d. Let D,C1, k, ǫ be as in Lemma 2.4. If dG(c, d) ≥ D, then [a, c]∪ [c, d]∪ [d, b] is a (k, ǫ)-quasigeodesic lying in a C1 neighborhood of [a, b]. Since J(Li), J(Lj) are both 2δ-qc, [a, b] lies in a 2δ -neighborhood of J(Li), and [c, d] lies in a 2δ -neighborhood of J(Lj). In particular c, d lie in a (C1 + 2δ)-neighborhood of J(Lj). Translating by an element of H, we may assume that c = 1. (Note that the argument in this paragraph works independent of whether J(Li) ∩ J(Lj) is empty or not. We proceed now by contradiction. Suppose there exists a sequence of Li’s and bi ∈ J(Li) such that P (bi) = di lies at a distance greater than i from J(Li ∩Lj) (resp. c = 1) according as J(Li)∩ J(Lj) is non-empty or empty. This shows that the sequence Li has a limit point on Λ disjoint from Li ∩Λ for all i and further that J(Li) passes through a bounded neighborhood of 1. Hence the sequence Li is not discrete in C c (∂G). This contradicts Proposition 2.3 and proves our claim. ✷ 2.2. Quasiconvexity and Relative Hyperbolicity. As an immediate corollary of Proposition 2.5 in conjunction with Theorem 1.12 of Short [Sho91], we immediately conclude 12 MAHAN MJ Corollary 2.6. Let H be a malnormal quasiconvex subgroup of a hyperbolic group G with Cayley graph Γ and limit set L. Then the set of joins J of distinct translates of L is a uniformly cobounded collection of uniformly quasiconvex sets in Γ. Combining Lemma 1.6 with Corollary 2.6 above, we have the following Proposition due to Bowditch [Bow97]. Proposition 2.7. (Characterization of Quasiconvexity II) [Bow97] Let H be a malnormal quasiconvex subgroup of a hyperbolic group G. Then G is strongly relatively hyperbolic with respect to H. In fact the converse to Proposition 2.7 is also true. We came to learn from the referee that this follows by combining work of Farb [Far98], Bowditch [Bow97] and Drutu-Sapir [DS05]. We provide a proof below for completeness (and because it is easily done). Malnormality of strongly relatively hyperbolic subgroups is due to Farb [Far98]. In fact this does not require G to be hyperbolic. Lemma 2.8. (Farb [Far98]) Let G be strongly relatively hyperbolic with respect to H. Then H is malnormal in G. It remains to show that H is quasiconvex if a hyperbolic group G be strongly relatively hyperbolic with respect toH. We use Gromov’s definition of strong relative hyperbolicity. Attach hyperbolic cones cH to distinct translates of ΓH in ΓG to obtain the hyperbolically coned off Cayley graph ΓhG. Then Γ G is hyperbolic by Gromov’s definition. If H is not quasi-isometrically embedded in G then for all i ∈ N, there exist pi1, pi2 ∈ ΓH such that dH(pi1, pi2) ≥ idG(pi1, pi2) . Also from the metric dcH on cH, we find that dcH(pi1, pi2) is of the order of log2dH(pi1, pi2). Hence, we can further assume that dH(pi1, pi2) ≥ idcH(pi1, pi2) . Join pi1, pi2 by shortest paths αi, βi in cH, ΓG respectively. Then αi∪βi = σi is a closed loop in Γ G with total length l(σi) = (dcH(pi1, pi2)+dG(pi1, pi2)). Therefore il(σi) ≤ 2dH(pi1, pi2). Since any (combinatorial) disk Di in Γ G spanning σi must contain a path γi in ΓH joining p1i, p2i, therefore the area A(Di) of Di must be at least that of N1(γi), the 1-neighborhood of γi in Di. Therefore there exists C > 0 such that for all i, A(Di) ≥ A(N1(γi)) ≥ dH(pi1, pi2) il(σi) Since i is arbitrary, this shows that ΓhG cannot satisfy a linear isoperi- metric inequality. Hence ΓhG cannot be a hyperbolic metric space. This is a contradiction. Hence H must be quasi-isometrically embedded in G. Hence RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 13 (see for instance [Gro85] ), H is quasiconvex in G. This completes our proof of the following characterisation of strongly relatively hyperbolic subgroups of hyperbolic groups. Proposition 2.9. Let G be a hyperbolic group and H a subgroup. Then G is strongly relatively hyperbolic with respect to H if and only if H is a malnormal quasiconvex subgroup. 3. Relative Rigidity 3.1. Pairing of Limit Sets by Quasi-isometries. We now consider two hyperbolic groupsG1, G2 with quasiconvex subgroupsH1,H2, Cayley graphs Γ1,Γ2. Let Lj for j = 1, 2 denote the collection of translates of limit sets (counted with multiplicity as before) of H1,H2 in ∂G1, ∂G2 respectively. Individual members of the collection Lj will be denoted as L i . Let Jj denote the collection {J i = J(L i ) : L i ∈ Lj}. Following Schwarz [Sch97], we define: Definition 3.1. A bijective map φ from J1 → J2 is said to be uniformly proper if there exists a function f : N → N such that 1) dG1(J(L i ), J(L j )) ≤ n ⇒ dG2(φ(J(L i )), φ(J(L j ))) ≤ f(n) 2) dG2(φ(J(L i )), φ(J(L j ))) ≤ n ⇒ dG1(J(L i ), J(L j )) ≤ f(n). When Ji consists of all singleton subsets of Γ1,Γ2, we shall refer to φ as a uniformly proper map from Γ1 to Γ2. Note: We observe that if Ji is just the collection of singleton sets in Γi, then a uniformly proper map between J ’s is the same as a quasi-isometry between Γi’s. This can be seen by putting n = 1 in conditions 1 and 2 above and then using the fact that graphs have edge length one. Hence what is important here is that J ’s are infinite diameter sets. Definition 3.2. A map q from Γ1 to Γ2 is said to pair the sets J1 and J2 as φ does if there exists a function h : N → N such that dG(p, J j )) ≤ n ⇒ dG(q(p), φ(J(L j ))) ≤ h(n). The following Lemma generalises Lemma 3.1 of Schwarz [Sch97], where the result is proven in the special case of a symmetric pattern of geodesics in Hn. The referee pointed out to us that the Lemma follows from Lemma 7 of [NR03] by Niblo and Reeves. Lemma 3.3. For M,m > 0, there exists R > 0, such that the following holds. Let L1, · · · , LM be distinct translates of the limit set of a quasiconvex sub- group H of a hyperbolic group G, such that dG(Ji, Jj) ≤ m for all i, j = 1 · · · ,M and Ji = J(Li). Then there exists a ball of radius R meeting Ji for all i = 1 · · · ,M . Proof: If ∩M1 Li 6= ∅, choose any point p ∈ J(∩ 1 Li). Then B1(p) intersects all Ji and we are through. 14 MAHAN MJ Suppose therefore that ∩m1 Li = ∅. We proceed by induction on M . There exists RM−1 such that a ball of radius RM−1 meets Ji for i = 1 · · ·M − 1. We now proceed by contradiction. If no such R exists for M , we have collections {Lk1 , · · · , L M}, k ∈ N such that a ball of radius RM−1 meets Jki , i = 1 · · ·M−1 but no ball of radius k meets J i , i = 1 · · ·M . In particular, (since J(∩M−11 L i ) ⊂ ∩ i )), if ∩ i 6= ∅, then Nk(J(∩ i )) ∩ JMi = ∅. For all i, j, k, choose points pkij ∈ J i such that dG(p ij, p ji) ≤ m. Assume by G-invariance of J that the ball of radius RM−1 centered at 1 ∈ ΓG meets J i , i = 1 · · ·M − 1. Therefore J M lies outside a k-ball about Since the collection of Ji’s through 1 is finite, therefore assume after pass- ing to a subsequence if necessary, that 1) {Jki }k is a constant sequence for i = 1 · · ·M − 1. Hence, {L i }k is a con- stant sequence Li (say) for i = 1 · · ·M − 1. 2) pkiM → piM ∈ ∂G for i = 1 · · ·M − 1. Hence p Mi → piM ∈ ∂G. Further, by (1) above, piM ∈ Li. 3) LkM converges to a closed set Z ⊂ ∂G. By Proposition 2.3, Z must be a singleton set {z}. 4) JkM lies outside Bk(1) ∪ Nk(J(∩ i )). If ∩ 1 Li 6= ∅, then assume further by G-invariance, that 1 ∈ J(∩M−11 L i ). Also, using Theorem 1.12 due to Short [Sho91], and translating by an appropriate element of ∩M−11 H we may assume that 1 ∈ J(∩M−11 L i ) is closest to J Now, pkMi ∈ J M and hence by (3) above, p Mi → z ∈ ∂G. Combining this with (2) above, we get z = piM for all i = 1 · · ·M − 1. Therefore, z ∈ ∩M−11 Li 6= ∅. But dG(1, J M ) = dG(J(∩ i ), J M ) ≥ k. Let zk ∈ J M such that dG(1, J M ) = dG(1, zk) = dG(J(∩ i ), J M ) ≥ k. Then the Gromov inner product (zk, p iM )1 is uniformly bounded above. Therefore (zk, piM )1 is uniformly bounded above. Hence finally (z, piM )1 is bounded above. In particular z 6= piM . This is the contradiction that proves the Lemma. ✷ Definition of q Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. We shall now show tha there exists a quasi-isometry q from Γ1 to Γ2 which pairs the sets J1 and J2 as φ does. We will define a map q : Γ1 → Γ2 which pairs J1 with J2 as φ does and prove that q is a quasi-isometry as promised. Choose K > 0 such that the K neighborhood BK(g) of g ∈ Γ1 has greater than w2 ( the width of H2 in G2 ) J i ’s passing through it. RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 15 Let J (for j = 1, 2 ) denote the collection of J i ’s passing through NK(g) for g ∈ Γj, j = 1, 2. By the proof of Proposition 2.3, there exists M = M(K) (independent of g ∈ Γ1) such that at most M J i ’s in J K,g pass through NK(g). Since φ is a bijective pairing, φ(J K,g) has at least (w2 +1) and at most M(K) elements in it. By definition of w2, and by Theorem 1.12 at least two of the limit sets of the φ(J1i )’s are disjoint. Let L 1 and L 2 denote these limit sets. Hence, by Corollary 2.6, for any K1 ≥ f(K), there exists D such that the collection of points {p ∈ Γ2 : d2(p, J 2 ) ≤ K1, d2(p, J 2 ) ≤ K1} has diameter less than D. Also, by uniform properness of φ, d2(φ(J m), φ(J n)) ≤ f(2K) for J1m, J n passing through NK(g) (independent of g). Summarising, 1) L21 and L 2 are disjoint. 2) But, by Lemma 3.3, using m = f(2K) and M = M(K), there exists R = R(K) and a ball of radius R meeting each φ(J1i ). 3) For anyK1, there existsD, such that {p ∈ Γ2 : d2(p, J 2 ) ≤ K1, d2(p, J 2 ) ≤ K1} has diameter less than D.In particular, we may choose K1 = R. Define q(g) to be the center of the ball of radius R obtained in (2) above. By (3), q(g) is thus defined upto a uniformly bounded amount of discrepancy for all g ∈ Γ1. Lemma 3.4. q is uniformly proper with respect to the metrics d1, d2. Proof: The proof is an almost exact replica of Lemma 3.2 of Schwarz [Sch97] and we content ourselves with reproducing the heuristics of his argument here. If x, y are close in Γ1, then the pairwise minimal distances between ele- ments of JxK1 and J is uniformly bounded above. Hence, by Lemma 3.3, there exists a uniform upper bound to the radius of a minimal radius ball intersecting all elements of φ(JxK1) as well as φ(J ). Also, since the center w of such a ball is defined upto a bounded amount of discrepancy, it must be at a bounded distance from both q(x) as well as q(y). Hence d2(q(x), q(y)) is uniformly bounded, i.e. close. Conversely, suppose that q(x), q(y) are close. First, by Lemma 3.3, there exists a uniform upper bound R on radius of minimal radius balls B1, B2 centered at q(x), q(y), intersecting all elements of φ(J xK1), φ(J K1) respec- tively. Then the (R + d2(q(x), q(y))) ball about q(x) (or q(y)) meets every element of φ(J xK1) as well as φ(J K1). Since φ is uniformly proper, this means that there is a uniform upper bound on the minimal radius of a ball meeting 16 MAHAN MJ every element of (J xK1) as well as (J K1). As before, d1(x, y) is uniformly bounded, i.e. x, y are close. ✷ Similarly, we can construct q−1 using the bijective pairing φ−1 such that q−1 is uniformly proper. Also, from Lemma 3.3 q, q−1 composed with each other in either direction is close to the identity. Since φ pairs L1, L2 bijectively and is uniformly proper from J1 to J2, therefore by invariance of J2 under G2, every point of Γ2 lies close to the image of q. Therefore q is uniformly proper, by Lemma 3.4 above, from Γ1 onto a net in Γ2. Hence q is a quasi-isometry. This concludes the proof of the main theorem of this subsection. Theorem 3.5. Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi-isometry q from Γ1 to Γ2 which pairs the sets J1 and J2 as φ does. We have thus shown one aspect of relative rigidity, viz. upgrading a uniformly proper map between Ji’s to a quasi-isometry between Γi’s. In the next subsection, we shall deduce the second aspect, viz. isomorphism of C-complexes. 3.2. C-Complexes. By Theorem 3.5 we obtain a quasi-isometry q from Γ1 to Γ2 which pairs J1 and J2 as φ does. Since q is a quasi-isometry, it extends to a quasiconformal homeomorphism from ∂G1 to ∂G2. Also, for all α > 0, there exists β > 0 such that d1(x, J i ) ≤ α ⇒ d2(q(x), φ(J i )) ≤ β and conversely, d2(y, J i ) ≤ α ⇒ d1(x, φ −1(J2i )) ≤ β In particular, ∂q maps the limit set L1i homeomorphically to the limit set of φ(J1i )). Hence, ∂q preserves intersection patterns of limit sets. Since φ pairs J1 with J2 as q does, summarising we get: Lemma 3.6. The following are equivalent. 1) ∩ki=1L i = ∅ 2) ∩ki=1∂q(L i ) = ∅ 3) ∩ki=1φ(L i ) = ∅ Hence by the definition of the C-complexes C(G1,H1) and C(G2,H2), we find that ∂q induces an isomorphism of C(G1,H1) with C(G2,H2). We conclude: Theorem 3.7. Let φ : J1 → J2 be a uniformly proper map. Then φ induces an isomorphism of C(G1,H1) with C(G2,H2). Note: In Theorem 3.5 and Theorem 3.7 we start with the assumption that there exists a uniformly proper pairing of the collections J1 and J2. This can be translated to a pairing of collections of limit sets L1 and L2. Theorem 3.5 RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 17 then says that the pairing of the Ji’s (or Li’s) is induced by a quasi-isometry from Γ1 to Γ2. Thus, the existence of a uniformly proper pairing implies the existence of a quasi-isometry between the Γi’s, i.e. an ambient extension (or, equivalently, a quasiconformal homeomorphism between ∂Gi’s). Also Theorem 3.7 shows that a uniformly proper pairing induces an iso- morphism of the C-complexes C(Gi,Hi). This is reminiscent of the initial step in the proof of rigidity theorems for higher rank symmetric spaces, where Tits complexes replace C-complexes. 3.3. Cross Ratios, Annular Systems and a Dynamical Formulation. In this subsection, we give a more intrinsic formulation of Theorems 3.5 and 3.7. The hypothesis of these theorems is given in terms of distances between elements of Ji. A more intrinsic way of formulating this hypothesis would be in terms of the action of Gi on ∂Gi, i = 1, 2. In this case, the distance between J il , J m can be approximated by the hyperbolic cross-ratio of their limit sets. This was described in detail by Bowditch [Bow98]. We give the relevant definitions and Theorems below and then dynamically reformulate Theorems 3.5 and 3.7. Let M be a compactum. Definition 3.8. An annulus A is an ordered pair (A−, A+) of disjoint closed subsets of M such that M \ (A− ∪ A+) 6= ∅. An annulus system is a collection of such annulii. If A = (A−, A+), then −A = (A+, A−). An annulus system is symmetric if A ∈ A ⇒ −A ∈ A. Given a closed set K ⊂ M and an annulus A, we say that K < A if K ⊂ intA−. Also, A < K if K < −A. If A,B are annulii, we say that A < B if M = intA− ∪ intB+. Fix an annulus system A. Given closed sets K,L ⊂ M , we say that the annular cross-ratio (K|L)A ∈ N∪∞ for the maximal number n ∈ N such that we can find annulii A1, · · ·An ∈ A such that K < A1 < · · · < An < L . We set (K|L)A = ∞ if there is no such bound. Thus (K|L)A is the length of the maximal chain of nested annulii sep- atrating K,L. For two point sets {x, y} = K and {z, w} = L, we write (K|L)A as (xy|zw)A. One of the crucial results of [Bow98] is: Theorem 3.9. (Bowditch [Bow98]) Suppose a group G acts as a uniform convergence group on a perfect metrizable compactum M . Then there exists a symmetric G-invariant annulus system A such that if x, y, z, w are distinct elements in M , then the theree quantities (xy|zw)A, (xz|yw)A, (xw|zy)A are all finite and at least two of them are zero. Also, if x 6= y, then (x|y)A > 0. Further, G is hyperbolic, and dG(J(K), J(L)) differs from (K,L)A upto bounded additive and multiplicative factors. 18 MAHAN MJ Combining Theorems 3.5 , 3.7 with Proposition 2.3 and Theorem 3.9, we get the dynamical formulation we promised. Let C0c (M) denote the collection of closed subsets of M containing more than one point. (Replacing dGi by cross-ratios (..|..)i in Definition 3.1 we get the corresponding notion of a map being uniformly proper with respect to the cross-ratios (..|..)1 and (..|..)2 in the theorem below. Similarly for the homeomorphism q.) Theorem 3.10. Let G1, G2 be uniform convergence (hence hyperbolic) groups acting on compacta M1,M2 respectively. Also, let Ai (for i = 1, 2 ) be Gi- invariant annulus systems and let (..|..)i denote the corresponding annular cross-ratios. Let H1,H2 be subgroups of G1, G2 with limit sets Λ1,Λ2. Suppose that the set Li of translates of Λi (for i = 1, 2) by essentially distinct elements of Hi in Gi forms a discrete subset of C c (Mi). Also assume that there exists a bijective function φ : L1 → L2 and that this pairing is uniformly proper with respect to the cross-ratios (..|..)1 and (..|..)2. (1) Hi is quasiconvex in Gi (2) There is a homeomorphism q : M1 → M2 which pairs L1 with L2 as φ does. Further, q is uniformly proper with respect to the cross-ratios (..|..)1 and (..|..)2 on M1, M2 respectively. (3) q (and hence also φ) induces an isomorphism of C-complexes C(G1,H1) with C(G2,H2). Thus from a uniformly proper map with respect to the pseudometrics on Li’s induced by cross-ratios we infer a quasi-isometry that is an ambient extension as also a (simplicial) isomorphism of C-complexes. 3.4. Axiomatisation, Relative Hyperbolicity. For classes of pairs (X,J ), what did we really require to ensure relative rigidity? Assume (X, d) is a metric space and let the induced pseudometric on J be also denoted by d. 1) For all k > 0 there exists M ∈ N such that for all x ∈ X, Nk(x) meets at most M of the J ’s in J . (This is a coarsening of the notion of height.) 2) For all K ∈ N, there exists k = k(K) > 0 such that for all x ∈ X, Nk(x) meets at least K of the J ’s in J . (This is the converse condition to (1).) 3) For all k > 0, n ∈ N there exists K > 0 such that for any collection J1, · · · , Jn ∈ J with d(Ji, Jj) ≤ k, there exists a ball of radius at most K meeting all the Ji’s. 4) There exists N ∈ N such that for all k > 0 there exists K = K(k) > 0 such that the following holds. For all n ≥ N and J1, · · · , Jn ∈ J , the set of points {x ∈ X : Nk(x) ∩ Ji 6= ∅, i = 1 · · · n} is either empty or has diameter bounded by K. Given (1)-(4), the construction of q : X1 → X2 from a uniformly proper pairing φ : J1 → J2 goes through as in Theorem 3.5. In short, pick N from (4). From (2), pick k = k(N). Now for all x ∈ X1, consider the collection RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 19 of J ’s in J1 that meet Nk(x). By (1) there is an upper bound M = M(k) on the number of such J ’s. Map these over to J2. Any two of these are at a distance of at most m apart where m depends on φ and k. From (3) choose K = K(M,m) such that a ball of radius K meets all these. Set q(x) to be the center of such a ball. By (4), q(x) is defined upto a uniformly bounded degree of discrepancy. The rest of the proof goes through as before. Hence (1)-(4) define sufficient conditions for relative rigidity for a class of pairs (X,J ). With these conditions, it is easy to extend Theorem 3.5 to pairs (X,J ) where X is (strongly) hyperbolic relative to the collection J . Conditions (1) and (2) are trivial. Condition (3) follows from “bounded penetration” (see Farb [Far98]). For any subcollection J1 of J with d(Ji, Jj) ≤ C0 (for all Ji, Jj ∈ J1), fix any two J1, J2 ∈ J and a geodesic γ12 of length ≤ C0 joining them. Construct an electric triangle for triples J1, J2, J3 ∈ J1 of horosphere-like sets for arbi- trary J3 ∈ J1, such that the hyperbolic geodesics γ13, γ32 joining J1, J3 and J3, J2 respectively have lengths bounded by C0. Then γ12 and γ13 meet J1 at a uniformly bounded distance from each other by bounded penetration. To see this, first note that J1, J2 can be joined by two paths, one consisting of one side of the triangle and the other the union of the two remaining sides of the triangle and both paths have electric length bounded by 2C0; in par- ticular both paths are uniform quasigeodesics (with quasigeodesic constant depending only on C0). They may be converted to quasigeodesics without backtracking by not increasing lengths. Thus γ13 ∪ γ32 decomposes as the union of a quasigeodesic without backtracking γ′12 joining J1, J2 and (possi- bly) a uniformly bounded (≤ C0) number of loops of length not longer than 2C0. The entire quasigeodesic without backtracking γ 12 lies near γ12 for all J3 ∈ J1. The same holds for the loops of bounded length (since they in turn may be regarded as uniform quasigeodesics without backtracking starting and ending at the same point.) In particular J3 lies at a uniformly bounded distance D0 from γ12. Since γ12 has length bounded by C0, and J3 may be chosen arbitrarily satisfying the hypothesis of (3) above, it follows that for any x ∈ γ12, d(x, J3) ≤ (C0 + D0) for all J3 ∈ J1. Condition (3) follows. (Results closely related to the proof of Condition (3) here occur as Lemma 3.11 of [Mj05] and Prop. 8.6 of [HW06].) Condition (4) follows from the fact that for a pair of distinct Ji, Jj , Nk(Ji) ∩Nk(Jj) is either empty or has diameter bounded by some C(k). We have thus shown: Theorem 3.11. Let Xi be (strongly) hyperbolic relative to collections Ji (i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi-isometry q from X1 to X2 which pairs the sets J1 and J2 as φ does. 20 MAHAN MJ By work of Hruska and Kleiner [HK04], CAT(0) spaces with isolated flats are (strongly) hyperbolic relative to maximal flats. Hence we have from Theorem 3.11 above: Corollary 3.12. Let Xi be CAT(0) spaces with isolated flats and let Ji denote the collections of maximal flats (i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi- isometry q from X1 to X2 which pairs the sets J1 and J2 as φ does. 3.5. Symmetric Spaces of Higher Rank. We now consider CAT(0) spaces which are at the other end of the spectrum. Let M be a compact lo- cally symmetric space and T a totally geodesic torus with rank = rank(M). Take X = M̃ and J to be the lifts of T to M̃ . As these are all equivariant examples (i.e. J is invariant under a cocompact group action), it is enough to check (1)-(4) at a point. (1) and (2) are clear. To prove condition (4), we consider ∩iNk(Fi) and it is easy to bound from below the N appearing in Condition (4) (Section 3.4) in terms of the size of the Weyl group and rank. In that case, ∩iNk(Fi) has bounded diameter or is empty. Finally, to prove (3), we proceed as in Lemma 3.3. As in Lemma 3.3 we assume by induction that any k flats {F1, · · · , Fk} that ”coarsely pair- wise intersect at scale D” (i.e. ND(Fi) ∩ ND(Fj) 6= ∅ ) intersect coarsely (i.e. ∩i=1···kND′(Fi) 6= ∅ for some D ′ = D′(D, k)). To get to the induc- tive step, we suppose that for i = k + 1, we have collections of worse and worse counterexamples. Consider a maximal collection F = {F1, · · · , Fk} of maximal flats whose ”coarse intersection at scale D” ∩iND(Fi) = F is non-null. Translate the collection by a group element so that a fixed point 0 (thought of as the origin) lies on the intersection F . Now take a sequence of maximal flats F j whose D-neighborhoods ND(F j) intersect each ND(Fi), but dj = d(F j , F ) = d(0, F ) ≥ j. We scale the metric on (X, d) by a factor of dj to obtain a sequence of metric spaces (X, ) converging (via a non- principal ultrafilter) to a Euclidean building X∞ (this fact is due to Kleiner and Leeb [KL97b], but we shall only mildly need the exact nature of X∞). Fi’s converge to flats F i ⊂ X ∞ and F j ’s converge to a flat G∞ ⊂ X∞. Then the collection G = F∞i , G ∞ satisfy the following: (P1) Each element of G is a flat in X∞ (P2) By induction, the intersection of any i elements of G is non-empty and convex for i ≤ k (P3) The intersection of all the (k + 1) elements of G is empty. Consider the subcomplex K = G∞ i of X ∞. Then K is a union of r-flats, where r = rank(X). In particular, the homology groups Hn(K) = 0 for n > r. On the other hand, if we consider the nerve of the covering of K by the sets G∞, F∞i , then using the properties (P1), (P2), (P3) to compute Cech homology groups, we conclude that K has the same homology groups as the boundary of a k-simplex. In particular, Hk(K) = Z. For k > r this RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 21 is a contradiction, finally proving Condition (3). Thus we conclude: Theorem 3.13. Let Xi be symmetric spaces of non-positive curvature, and Ji be equivariant collections of lifts of a maximal torus in a compact locally symmetric space modeled on Xi (i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi-isometry q from X1 to X2 which pairs the sets J1 and J2 as φ does. Combining Theorem 3.13 with the quasi-isometric rigidity theorem of Kleiner-Leeb [KL97b] and Eskin-Farb [EF 3] we can upgrade the quasi- isometry q to an isometry i. Corollary 3.14. Let Xi be symmetric spaces of non-positive curvature, and Ji be equivariant collections of lifts of a maximal torus in a compact locally symmetric space modeled on Xi (i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists an isometry i from X1 to X2 which pairs the sets J1 and J2 as φ does. Remark 3.15. The technique of using asymptotic cones and the nerve of the covering by flats can be generalised easily to equivariant flats of arbitrary (not necessarily maximal) rank. We conclude this paper with two (related) questions: Question 1: In analogy with a Theorem of Ivanov, Korkmaz, Luo (see for instance [Luo00] ), regarding the automorphism group of the curve complex, we ask: If the C-Complex C(G,H) of a pair (G,H) (for G a hyperbolic group and H a quasiconvex subgroup) is connected, is the automorphism group of C(G,H) commensurable with G? Question 2: Consider the pair (G,H), with G a hyperbolic group and H a quasiconvex subgroup. Let (J , d) be the collection of joins as in Lemma 3.3 with the induced pseudometric. For a uniformly proper map φ from (J , d) to itself, is there an isometry pairing the elements of J as φ? We have proved in Theorem 3.5 that a quasi-isometry q exists pairing the J as φ does. The question is whether q may be upgraded to an isometry, or better, to an element of G? This question is related to the notion of pattern rigidity introduced by Mosher, Sageev and Whyte in [MSW04]. References [BCG98] G. Besson, G. Courtois, and S. Gallot. A real Schwarz lemma and some appli- cations. Rend. Mat. Appl., VII. Ser. 18, No.2, pages 381–410, 1998. [BDM05] J. Behrstock, C. Drutu, and L. Mosher. Thick metric spaces, relative hyper- bolicity, and quasi-isometric rigidity. preprint, arXiv:math.GT/0512592, 2005. [Bow97] B. H. Bowditch. Relatively hyperbolic groups. preprint, Southampton, 1997. [Bow98] B. H. Bowditch. A topological characterization of hyperbolic groups. J. A. M. S. 11, pages 643–667, 1998. [DS05] C. Drutu and M. Sapir. Tree-graded spaces and asymptotic cones of groups. Topology 44, pages 959–1058, 2005. 22 MAHAN MJ [EF 3] A. Eskin and B. Farb. Quasi-flats and rigidity in higher rank symmetric spaces. . J. Amer. Math. Soc., 10, pages pp. 653–692., (1997),no. 3. [Far98] B. Farb. Relatively hyperbolic groups. Geom. Funct. Anal. 8, pages 810–840, 1998. [FM98] B. Farb and L. Mosher. A rigidity theorem for the solvable Baumslag-Solitar groups. Inventiones Math. , Vol. 131, No.2, pages 419–451, 1998. [FM00] B. Farb and L. Mosher. On the asymptotic geometry of abelian-by-cyclic groups. Acta Math. , Vol. 184, No.2, pages 145–202, (2000). [FM 3] B. Farb and L. Mosher. Quasi-isometric rigidity for the solvable Baumslag- Solitar groups, II. Inventiones Math. , Vol. 137, pages 273–296, (1999), No. [FS96] B. Farb and R.E. Schwartz. The large-scale geometry of Hilbert modular groups. J. Differential Geom. 44(3), page 435478, 1996. [GdlH90] E. Ghys and P. de la Harpe(eds.). 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We introduce and study the notion of relative rigidity for pairs $(X,\JJ)$ where 1) $X$ is a hyperbolic metric space and $\JJ$ a collection of quasiconvex sets 2) $X$ is a relatively hyperbolic group and $\JJ$ the collection of parabolics 3) $X$ is a higher rank symmetric space and $\JJ$ an equivariant collection of maximal flats Relative rigidity can roughly be described as upgrading a uniformly proper map between two such $\JJ$'s to a quasi-isometry between the corresponding $X$'s. A related notion is that of a $C$-complex which is the adaptation of a Tits complex to this context. We prove the relative rigidity of the collection of pairs $(X, \JJ)$ as above. This generalises a result of Schwarz for symmetric patterns of geodesics in hyperbolic space. We show that a uniformly proper map induces an isomorphism of the corresponding $C$-complexes. We also give a couple of characterizations of quasiconvexity. of subgroups of hyperbolic groups on the way.

Introduction 2 1.1. Relative Rigidity and Statement of Results 2 1.2. Relative Hyperbolicity and Electric Geometry 6 1.3. Height of Subgroups and C-Complexes 8 2. Characterizations of Quasiconvexity 9 2.1. Limit Sets and Quasiconvexity 10 2.2. Quasiconvexity and Relative Hyperbolicity 11 3. Relative Rigidity 13 3.1. Pairing of Limit Sets by Quasi-isometries 13 3.2. C-Complexes 16 3.3. Cross Ratios, Annular Systems and a Dynamical Formulation 17 3.4. Axiomatisation, Relative Hyperbolicity 18 3.5. Symmetric Spaces of Higher Rank 20 References 21 http://arxiv.org/abs/0704.1922v4 2 MAHAN MJ 1. Introduction 1.1. Relative Rigidity and Statement of Results. In this paper, we study a rigidity phenomenon within the framework of coarse geometry. We call it relative rigidity. Much of the work on quasi-isometric rigidity (e.g. Farb-Schwartz [FS96] Kleiner-Leeb [KL97b] Eskin-Farb [EF 3] and Mosher- Sageev-Whyte [MSW03] [MSW04] ) contains a crucial step showing that a self quasi-isometry of a space X coarsely preserves a family J of distin- guished subsets of X. The family J again has a coarse intersection pattern that may be combinatorially coded and these proofs of quasi-isometric rigid- ity often show that the intersection pattern is preserved by a quasi-isometry. In this note, we investigate a sort of a converse to this: When does a uniformly proper map between two families J1 and J2 come from a quasi-isometry φ between X1 and X2? Does such a map preserve intersection patterns? We show that the answer is affirmative when (1) Xi’s are (Cayley graphs of) hyperbolic groups and Ji’s correspond to cosets of a quasiconvex subgroup (2) Xi’s are (Cayley graphs of) relatively hyperbolic groups and Ji’s correspond to parabolic subgroups (3) Xi’s are symmetric spaces of non-positive curvature and Ji’s corre- spond to lifts of a maximal torus in a compact locally symmetric space modeled on Xi. If in addition one can show that a quasi-isometry preserving intersection patterns is close to an isometry, we would be able to conclude that a uni- formly proper map between the Ji’s is induced by an isometry. This latter phenomenon has been investigated by Mosher, Sageev and Whyte [MSW04] and has been termed pattern rigidity. Thus, in a sense, the notion of relative rigidity complements that of pattern rigidity. Some further examples where a family of distinguished subsets of a space and the resulting (combinatorial) configuration yields information about the ambient space are: 1) Collection of flats in a symmetric space of higher rank (Mostow [Mos73]) 2) Collection of maximal abelian subgroups of the mapping class group (Behrstock-Drutu-Mosher [BDM05] ) 3) Collection of hyperbolic spaces in the Cayley complex of the Baumslag- Solitar groups (Farb-Mosher [FM98] , [FM 3] ; see also [FM00] ) 4) Quasi-isometric rigidity of sufficiently complicated patterns of flats in the universal cover of a Haken 3 manifold (Kapovich-Leeb [KL97a] ) 5) We were most influenced by a beautiful result of Schwarz [Sch97] which shows that a uniformly proper map from a symmetric pattern of geodesics in RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 3 Hn to another symmetric pattern of geodesics in Hn (for n > 2) is induced by an isometry. Again as in Mostow, there are two parts to this. A first step is to construct a quasi-isometry of Hn inducing the given pairing. Schwarz terms this ambient extension. The second is to construct an isometry. Let us look at a general form of the situation that Schwarz considers. (X1, d1), (X2, d2) are metric spaces. Let J1,J2 be collections of closed sub- sets of X1,X2 respectively. Then di induces a pseudo-metric (which, by abuse of notation, we continue to refer to as di) on Ji for i = 1, 2. This is just the ordinary (not Hausdorff) distance between closed subsets of a metric space. In [Sch97], X1 = X2 = H n, and Ji are lifts (to the universal cover) of finite collections of closed geodesics in two hyperbolic manifolds. Also, the hypothesis in Schwarz’s paper [Sch97] is the existence of a uni- formly proper map φ between symmetric patterns of geodesics J1 and J2. A uniformly proper map may be thought of as an isomorphism in the so-called coarse category in the sense of John Roe [Roe95]. Thus, we can re-interpret the first step of Schwarz’s result as saying that an isomorphism φ in the coarse category between Ji implies the existence of a quasi-isometry from Hn to itself inducing φ. In the language of [Sch97], uniformly proper pairings come from ambient extensions. In Mostow’s proof of rigidity for higher rank symmetric spaces, he obtains in a crucial step, an isomorphism of Tits complexes [Mos73]. We would like to associate to a pair (X,J ) some such complex just as a Tits complex is associated to a higher rank locally symmetric space and its collection of maximal parablic subgroups. We propose the notion of a C-complex in this paper as the appropriate generalization of a Tits complex to coarse geometry. Then what we would hope for (as a conclusion) is an isomorphism of these C-complexes. This transition from the existence of a uniformly proper map between Ji’s to the existence of a a quasi-isometry between Xi’s inducing an isomorphism of C-complexes is what we term relative rigidity. Schwarz proves the relative rigidity of pairs (X,J ) where X is hyperbolic space and J a symmetric collection of geodesics. Much of what he does extends to the case whereX is a higher rank symmetric space and J a symmetric collection of maximal periodic flats or a symmmetric collection of maximal parabolic subgroups in a non-uniform lattice. The main point of this paper is illustrated first in the context of rela- tive rigidity of the category of pairs (Γ,J ), where Γ is (the Cayley graph of) a hyperbolic group, and J the set of cosets of a quasiconvex subgroup. Throughout this paper we shall assume that the quasiconvex sub- groups are of infinite index in the big groups. Note that the upgrading of a uniformly proper map between J ’s to a quasi-isometry between the Γ’s is the most we can hope for in light of the fact that the Cayley graph of a finitely generated group is only determined up to quasi-isometry. (See Paulin [Pau96] for a proof of this fact.) 4 MAHAN MJ We start with a pair of hyperbolic groups G1, G2 with Cayley graphs Γ1,Γ2, and quasiconvex subgroups H1,H2. Let Λ1, Λ2 be the limit sets of H1,H2 in ∂G1, ∂G2 respectively. For convenience we consider the collection Ji of translates of Ji the join of Λi in Γi rather than cosets of Hi. Recall that the join of Λi is the union of bi-infinite geodesics in Γi with end-points in Λi. This is a uniformly quasiconvex set and lies at a bounded Hausdorff distance from the Cayley graph of the subgroup Hi (Since H has finite index in its commensurator, only finitely many cosets of H are at a finite Hausdorff distance from it. Since Ji is at a bounded Hausdorff distance from Hi the same is true for elements of Ji.) The main theorems of this paper are as follows. Theorem 3.5:Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi-isometry q from Γ1 to Γ2 which pairs the sets J1 and J2 as φ does. The construction of the quasi-isometry q proceeds by constructing a ”coarse barycenter” of some infinite diameter sets (reminiscent of the celebrated measure-theoretic barycenter method discovered by Douady and Earle, and extended greatly by Besson, Courtois, Gallot [BCG98] ). We prove an analogous theorem for pairs (X,J ) when X is (strongly) hyperbolic relative to the collection J . Theorem 3.11:Let Xi be (strongly) hyperbolic relative to collections Ji (i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi-isometry q from X1 to X2 which pairs the sets J1 and J2 as φ does. As a Corollary of Theorem 3.11 and work of Hruska and Kleiner [HK04], we deduce relative rigidity for pairs (X,J ) where X is a CAT(0) space with isolated flats and J is the collection of maximal flats. The third main theorem of this paper is an analog for higher rank sym- metric spaces. Theorem 3.13:Let Xi be symmetric spaces of non-positive curvature, and Ji be equivariant collections of lifts of a maximal torus in a compact locally symmetric space modeled on Xi (i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi-isometry q from X1 to X2 which pairs the sets J1 and J2 as φ does. In fact, combining Theorem 3.13 with the quasi-isometric rigidity theorem of Kleiner-Leeb [KL97b] and Eskin-Farb [EF 3], we may upgrade the quasi- isometry of Theorem 3.13 to an isometry. Let Gi,Hi (i = 1, 2) be hyperbolic groups and quasiconvex subgroups re- spectively. In Section 1.3, we shall construct simplicial complexes (termed C- complexes) from the incidence relations determined by the cosets of Hi. Let C(Gi,Hi) be the C-complexes associated with the pairs (Gi,Hi). Roughly RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 5 speaking, the vertices of C(Gi,Hi) are the translates g iΛi of Λi by distinct coset representatives g i and the (n− 1)-cells are n-tuples {g 1Λ, · · · , g 1Λ} of distinct translates such that ∩n1g 1Λ 6= ∅. Theorem 3.7: Let φ : J1 → J2 be a uniformly proper map. Then φ induces an isomorphism of C(G1,H1) with C(G2,H2). On the way towards proving Theorems 3.5 and 3.7, we prove two Proposi- tions characterizing quasiconvexity. These might be of independent interest. The first is in terms of the Hausdorff topology on the collection C0c (∂G), which is the collection of closed subsets of ∂G having more than one point. Proposition 2.3: Let H be a subgroup of a hyperbolic group G with limit set Λ. Let L be the collection of translates of Λ by elements of distinct cosets of H (one for each coset). Then H is quasiconvex if and only if L is a discrete subset of C0c (∂G). The second characterization is in terms of strong relative hyperbolicity. Definition 1.1. A subgroup H of a group G is said to be malnormal if for all g ∈ G \H, gHg−1 ∩H is trivial. A subgroup H of a group G is said to be almost malnormal if for all g ∈ G \H, gHg−1 ∩H is finite. It was pointed out to us by the referee that the following result follows from work of Farb [Far98], Bowditch ([Bow97] Theorem 7.11) and Drutu- Sapir ([DS05] Lemma 4.15). We shall include a proof for completeness. Proposition 2.9:[Far98] [Bow97] [DS05] Let G be a hyperbolic group and H a subgroup. Then G is strongly relatively hyperbolic with respect to H if and only if H is a malnormal quasiconvex subgroup. The prototypical example is that of (fundamental groups of) a closed hyperbolic manifold with a totally geodesic embedded submanifold. Finally, we give an intrinsic or dynamic reformulation of Theorems 3.5 and 3.7 following Bowditch [Bow98], which makes use of the existence of a cross-ratio on the boundary of a hyperbolic group. The cross-ratio in turn induces a pseudometric on the collection L of translates of Λ. Theorem 3.10: Let G1, G2 be uniform convergence (hence hyperbolic) groups acting on compacta M1,M2 respectively. Also, let Åi (for i = 1, 2) be Gi-invariant annulus systems and let (..|..)i denote the corresponding an- nular cross-ratios. Let H1,H2 be subgroups of G1, G2 with limit sets Λ1,Λ2. Suppose that the set Li of translates of Λi (for i = 1, 2) by essentially distinct elements of Hi in Gi forms a discrete subset of C c (Mi). Also assume that there exists a bijective function φ : L1 → L2 and that this pairing is uniformly proper with respect to the cross-ratios (..|..)1 and (..|..)2. 1) Hi is quasiconvex in Gi 2) There is a homeomorphism q : M1 → M2 which pairs L1 with L2 as φ 6 MAHAN MJ does. Further, q is uniformly proper with respect to the cross-ratios (..|..)1 and (..|..)2 on M1, M2 respectively. 3) q (and hence also φ) induces an isomorphism of C-complexes C(G1,H1) with C(G2,H2). Acknowledgements: My interest in relative hyperbolicity and quasi-isometric rigidity is largely due to Benson Farb. It is a pleasure to acknowledge his help, support and camaraderie, both mathematical and personal. I would also like to thank the referee for suggesting several corrections and for pro- viding additional references. 1.2. Relative Hyperbolicity and Electric Geometry. We start off by fixing notions and notation. Let G (resp. X) be a hyperbolic group (resp. a hyperbolic metric space) with Cayley graph (resp. a net) Γ equipped with a word-metric (resp. a simplicial metric) d. Here a net N is a collection of distinct points xi ∈ X such that there exist 0 < C1 < C2 such that 1) d(xi, xj) ≥ C1 for all i 6= j 2) For all x ∈ X, there exists xi ∈ N such that d(xi, x) ≤ C2 For the net N we construct a graph GN with edges corresponding to pairs xi 6= xj such that d(xi, xj) ≤ 4C2. The simplicial metric on N is obtained by declaring that each edge of GN has length one. Let the Gromov boundary of Γ be denoted by ∂G. (cf.[GdlH90]). We shall have need for the fact that for hyperbolic metric spaces (in the sense of Gromov [Gro85]) the notions of quasiconvexity and qi embeddings coincide [Gro85]. We shall now recall certain notions of relative hyperbolicity due to Gro- mov [Gro85] and Farb [Far98]. Let X be a path metric space. A collection of closed subsets H = {Hα} of X will be said to be uniformly separated if there exists ǫ > 0 such that d(H1,H2) ≥ ǫ for all distinct H1,H2 ∈ H. The electric space (or coned-off space) X̂ corresponding to the pair (X,H) is a metric space which consists of X and a collection of vertices vα (one for each Hα ∈ H) such that each point of Hα is joined to (coned off at) vα by an edge of length Definition 1.2. [Far98] [Bow97] Let X be a geodesic metric space and H be a collection of uniformly separated subsets. Then X is said to be weakly hyperbolic relative to the collection H, if the electric space X̂ is hyperbolic. Lemma 1.3. (See Bowditch [Bow97], generalizing Lemma 4.5 and Propo- sition 4.6 of Farb [Far98]) Given δ, C,D there exists ∆ such that if X is a δ-hyperbolic metric space with a collection H of C-quasiconvex D-separated sets. then, the electric space X̂ is ∆-hyperbolic, i.e. X is weakly hyperbolic relative to the collection H. RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 7 Definitions: Given a collection H of C-quasiconvex, D-separated sets and a number ǫ we shall say that a geodesic (resp. quasigeodesic) γ is a geodesic (resp. quasigeodesic) without backtracking with respect to ǫ neighborhoods if γ does not return to Nǫ(H) after leaving it, for any H ∈ H. A geodesic (resp. quasigeodesic) γ is a geodesic (resp. quasigeodesic) without backtracking if it is a geodesic (resp. quasigeodesic) without backtracking with respect to ǫ neighborhoods for some ǫ ≥ 0. Electric P -quasigeodesics without backtracking are said to have similar intersection patterns if for β, γ electric P -quasigeodesics without backtrack- ing both joining x, y, the following are satisfied. (1) Similar Intersection Patterns 1: if precisely one of {β, γ} meets an ǫ-neighborhood Nǫ(H1) of an electrocuted quasiconvex set H1 ∈ H, then the length (measured in the intrinsic path-metric on Nǫ(H1) ) from the entry point to the exit point is at most D. (2) Similar Intersection Patterns 2: if both {β, γ} meet some Nǫ(H1) then the length (measured in the intrinsic path-metric on Nǫ(H1) ) from the entry point of β to that of γ is at most D; similarly for exit points. Definition 1.4. [Far98] [Bow97] Let X be a geodesic metric space and H be a collection of mutually disjoint uniformly separated subsets such that X is weakly hyperbolic relative to the collection H. If any pair of P - electric quasigeodesics without backtracking starting and ending at the same point have similar intersection patterns with horosphere-like sets (elements of H) then quasigeodesics are said to satisfy Bounded Penetration and X is said to be strongly hyperbolic relative to the collection H. Definition 1.5. [Mj05] A collection H of uniformly C-quasiconvex sets in a δ-hyperbolic metric space X is said to be mutually D-cobounded if for all Hi,Hj ∈ H, πi(Hj) has diameter less than D, where πi denotes a nearest point projection of X onto Hi. A collection is mutually cobounded if it is mutually D-cobounded for some D. Mutual coboundedness was proven by Farb for horoballs in finite volume Hadamard manifolds of pinched negative curvature in Lemma 4.7 of [Far98]. The following generalization is due to Bowditch [Bow97]. Lemma 1.6. (See Bowditch [Bow97] Lemma 7.13 for a proof) Suppose X is a δ-hyperbolic metric space with a collection H of C-quasiconvex K-separated D-mutually cobounded subsets. Then X is strongly hyperbolic relative to the collection H. Gromov gave a different definition of strong relative hyperbolicity. We give a condition below that is equivalent to a special case of Gromov’s definition. Let X be a geodesic metric space with a collection H of uniformly 8 MAHAN MJ separated subsets {Hi}. The hyperbolic cone cHi is the product of Hi and the non-negative reals Hi ×R+, equipped with the metric of the type 2−tds2 + dt2. More precisely, Hi ×{n} is given the path metric of Hi scaled by 2−n. The R+ direction is given the standard Euclidean metric. Let X denote X with hyperbolic cones cHi glued to it along Hi’s. X h will be referred to as the hyperbolically coned off X. This is to be contrasted with the coned off space X̂ in Farb’s definition. Definition 1.7. X is said to be strongly hyperbolic relative to the collec- tion H in the sense of Gromov if the hyperbolically coned off space Xh is a hyperbolic metric space. The equivalence of the two notions of strong relative hyperbolicity was proven by Bowditch in [Bow97]. Theorem 1.8. ( Bowditch [Bow97] ) X is strongly hyperbolic relative to a collection H of uniformly separated subsets {Hi} in the sense of Gromov if and only if X is strongly hyperbolic relative to the collection H in the sense of Farb. 1.3. Height of Subgroups and C-Complexes. The notion of height of a subgroup was introduced by Gitik, Mitra, Rips and Sageev in [GMRS97] and further developed by the author in [Mit04]. Definition 1.9. Let H be a subgroup of a group G. We say that the elements {gi|1 ≤ i ≤ n} of G are essentially distinct if Hgi 6= Hgj for i 6= j. Con- jugates of H by essentially distinct elements are called essentially distinct conjugates. Note that we are abusing notation slightly here, as a conjugate of H by an element belonging to the normalizer of H but not belonging to H is still essentially distinct from H. Thus in this context a conjugate of H records (implicitly) the conjugating element. Definition 1.10. We say that the height of an infinite subgroup H in G is n if there exists a collection of n essentially distinct conjugates of H such that the intersection of all the elements of the collection is infinite and n is maximal possible. We define the height of a finite subgroup to be 0. We say that the width of an infinite subgroup H in G is n if there exists a collection of n essentially distinct conjugates of H such that the pairwise intersection of the elements of the collection is infinite and n is maximal possible. The main theorem of [GMRS97] states: Theorem 1.11. If H is a quasiconvex subgroup of a hyperbolic group G,then H has finite height and finite width. In this context, a theorem we shall be needing several times is the following result from [GMRS97] that is proved using a result of Short [Sho91]. RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 9 Theorem 1.12. (Lemma 2.6 of [GMRS97]) Let G be a hyperbolic group and Hi (for i = 1 · · · k ) be quasiconvex subgroups with limit sets Λi, i = 1 · · · k. Then ∩Hi is a quasiconvex subgroup with limit set ∩Λi. We now proceed to define a simplicial complex C(G,H) for a group G and H a subgroup. For G hyperbolic and H quasiconvex, we give below three equivalent descriptions of a complex C(G,H). In this case, let ∂G denote the boundary of G, Λ the limit set of H, and J the join of Λ. 1) Vertices ( 0-cells ) are conjugates of H by essentially distinct elements, and (n − 1)-cells are n-tuples {g1H, · · · , gnH} of distinct cosets such that ∩n1giHg i is infinite (in fact by Theorem 1.12 an infinite quasiconvex sub- group of G). 2) Vertices ( 0-cells ) are translates of Λ by essentially distinct elements, and (n−1)-cells are n-tuples {g1Λ, · · · , gnΛ} of distinct translates such that ∩n1giΛ 6= ∅. 3) Vertices ( 0-cells ) are translates of J by essentially distinct elements, and (n− 1)-cells are n-tuples {g1J, · · · , gnJ} of distinct translates such that ∩n1giJ is infinite. We shall refer to the complex C(G,H) as the C-complex for the pair G,H. (C stands for “coarse” or “Čech” or “cover”, since C(G,H) is like a coarse nerve of a cover, reminiscent of constructions in Cech cochains.) Note that if h(H) denote the height of H, then (h(H)+1) is the dimension of the C-complex C(G,H). Also, if w(H) denote the width of H, then w(H) = w is equal to the size of the largest complete graph Kw that is embeddable in C(G,H). If C(G,H) is connected then its one-skeleton is closely related to the coned off space Γ̂ with an appropriately chosen set of generators. This definition is inspired by that of the Tits complex for a non-uniform lattice in a higher rank symmetric space. Related constructs in the context of codimension 1 subgroups also occur in work of Sageev [Sag95] where he constructs cubings. 2. Characterizations of Quasiconvexity Let G be a hyperbolic group. Let Cc(∂G) denote the collection of closed subsets of the boundary ∂G equipped with the Hausdorff topology. Let C0c (∂G) ⊂ Cc(∂G) denote the subset obtained from Cc(∂G) by removing the singleton sets {{x} : x ∈ ∂G}. Next fix a subgroup H ⊂ G with limit set Λ ⊂ ∂G. Consider the G-invariant collection L ={ gΛ } ⊂ C0c (∂G) with g ranging over distinct cosets (one for each coset) of H in G. Note that L is (strictly speaking) a multi-set as distinct elements of L may denote the same element of C0c (∂G) in case two distinct translates of Λ coincide. One extreme case is when Λ = ∂G, though H is of infinite index in G (e.g. if H is normal of infinite index in in G.) Then L consists of infinitely many copies of Λ. 10 MAHAN MJ Definition 2.1. The join J(Λ) of Λ is defined as the union of all bi-infinite geodesics whose end-points lie in Λ It is easy to see that J(Λ) is 2δ-quasiconvex if G is δ-hyperbolic. In fact this is true for any subset Λ of the boundary of a δ-hyperbolic metric space X (no equivariance is necessary). For Λ the limit set of H, J(Λ) is H- invariant. The visual diameter dia∂G(Λ) of a subset Λ of ∂G is the same as the diameter in the metric on ∂G obtained from the Gromov inner product. (See [GdlH90] Chapter 7 for details about the visual metric on ∂G.) 2.1. Limit Sets and Quasiconvexity. The next Lemma follows directly from the fact that sets with visual diameter bounded below contain points with Gromov inner product bounded above and conversely[GdlH90]. Lemma 2.2. For all ǫ > 0 there exists N such that if the diameter dia∂G(Λ) ≥ ǫ for a closed subset Λ of ∂G, then there exists p ∈ J(Λ) such that d(p, 1) ≤ N . Conversely, for all N > 0 there exists ǫ > 0 such that if there exists p ∈ J(Λ) with d(p, 1) ≤ N , then dia∂G(Λ) ≥ ǫ. The next Proposition gives our first characterisation of quasiconvex sub- groups of a hyperbolic group. Proposition 2.3. (Characterization of Quasiconvexity I) Let H be a subgroup of a hyperbolic group G with limit set Λ. Let L be the collection of translates of Λ (counted with multiplicity) by elements of distinct cosets of H (one for each coset). Then H is quasiconvex if and only if L is a discrete subset of C0c (∂G). Proof: Suppose H is quasiconvex. We want to show that L is a discrete subset of C0c (∂G). Thus it suffices to show that any limit of elements of L is a singleton set. This in turn follows from the following. Claim: For all ǫ > 0, Lǫ = {Li ∈ L : dia∂G(Li) ≥ ǫ} is finite. Proof of Claim: Let N = N(ǫ) be as in Lemma 2.2. Since dia∂G(Li) ≥ ǫ, therefore by Lemma 2.2, there exists pi ∈ J(Li) such that dG(pi, 1) ≤ N . Also, there exists K > 0 depending on δ (recall that J(Li) is 2δ-qc) and the quasiconvexity constant of H such that if Li = giΛ, then there exists hi ∈ H with dG(pi, gihi) ≤ K. Hence, dG(1, gihi) ≤ K + N . Since G is finitely generated, the number of such elements gihi is finite. Since gi are picked from distinct cosets of H, we conclude that the set Lǫ is finite. ✷ Conversely, suppose that H is not quasiconvex. Assume, without loss of generality, that a finite generating set of H is contained in a finite generating set of G and that ΓH ,ΓG are Cayley graphs with respect to these generating sets. Then there exist pi ∈ J(Λ) such that dG(pi,ΓH) ≥ i. Translating by an appropriate element of H, we may assume that dG(pi,ΓH) = dG(pi, 1) ≥ i. Further, we may assume (by passing to a subsequence if necessary) that the sequence dG(pi, 1) is monotonically increasing. Then p i J(Λ) has limit set p−1i Λ. Further, as pi ∈ J(Λ), therefore, 1 ∈ p i J(Λ). Since J(Λ) RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 11 is 2δ-qc, so is p−1i J(Λ) for all i. Hence, there exists ǫ > 0 by Lemma 2.2 such that dia∂Gp i J(Λ) ≥ ǫ. Since dG(pi, 1) is monotonically strictly increasing, we conclude that pi’s lie in distinbct cosets of H. Further, since Cc(∂G) is compact, we conclude that the collection p i J(Λ) has a convergent subsequence, converging to a subset of diameter greater than or equal to ǫ. Therefore, the collection L is not a discrete subset (strictly speaking a multiset) of C0c (∂G). ✷ We next prove a result about projections of J(Li) on J(Lj). We start off with an elementary fact about hyperbolic metric spaces. See [Mit98] for a proof. Lemma 2.4. [Mit98] Given δ > 0, there exist D,C1, k, ǫ such that if a, b, c, d are points of a δ-hyperbolic metric space (Z, d), with d(a, [b, c]) = d(a, b), d(d, [b, c]) = d(c, d) and d(b, c) ≥ D then [a, b] ∪ [b, c] ∪ [c, d] lies in a C1- neighborhood of any geodesic joining a, d and is a (k, ǫ)-quasigeodesic. Assume that H is quasiconvex and that Lk is the limit set gkΛ of gkH. Let Pj denote the nearest point projection of ΓG onto J(Lj). Also, let Hk = gkΓH be the left translate of ΓH by gk. Proposition 2.5. There exists K > 0 such that Pj(ΓHi) lies in a K- neighborhood of J(Li ∩ Lj) if (Li ∩ Lj) 6= ∅. Else, Pj(ΓHi) has diameter less than K. Proof: Since J(Li) is 2δ-qc and H is quasiconvex, it suffices to show that Pj(J(Li)) lies in a K-neighborhood of J(Li ∩Lj) if the latter is non-empty. By G-equivariance, we may assume that Lj = Λ and gi = 1. We represent Pj by P in this case. First note that by Theorem 1.12, Hi∩Hj is quasiconvex and the limit set of Hi ∩Hj is Li ∩ Lj. Also, J(Li ∩ Lj) ⊂ J(Li). Let a, b ∈ J(Li). Let P (a) = c, P (b) = d. Let D,C1, k, ǫ be as in Lemma 2.4. If dG(c, d) ≥ D, then [a, c]∪ [c, d]∪ [d, b] is a (k, ǫ)-quasigeodesic lying in a C1 neighborhood of [a, b]. Since J(Li), J(Lj) are both 2δ-qc, [a, b] lies in a 2δ -neighborhood of J(Li), and [c, d] lies in a 2δ -neighborhood of J(Lj). In particular c, d lie in a (C1 + 2δ)-neighborhood of J(Lj). Translating by an element of H, we may assume that c = 1. (Note that the argument in this paragraph works independent of whether J(Li) ∩ J(Lj) is empty or not. We proceed now by contradiction. Suppose there exists a sequence of Li’s and bi ∈ J(Li) such that P (bi) = di lies at a distance greater than i from J(Li ∩Lj) (resp. c = 1) according as J(Li)∩ J(Lj) is non-empty or empty. This shows that the sequence Li has a limit point on Λ disjoint from Li ∩Λ for all i and further that J(Li) passes through a bounded neighborhood of 1. Hence the sequence Li is not discrete in C c (∂G). This contradicts Proposition 2.3 and proves our claim. ✷ 2.2. Quasiconvexity and Relative Hyperbolicity. As an immediate corollary of Proposition 2.5 in conjunction with Theorem 1.12 of Short [Sho91], we immediately conclude 12 MAHAN MJ Corollary 2.6. Let H be a malnormal quasiconvex subgroup of a hyperbolic group G with Cayley graph Γ and limit set L. Then the set of joins J of distinct translates of L is a uniformly cobounded collection of uniformly quasiconvex sets in Γ. Combining Lemma 1.6 with Corollary 2.6 above, we have the following Proposition due to Bowditch [Bow97]. Proposition 2.7. (Characterization of Quasiconvexity II) [Bow97] Let H be a malnormal quasiconvex subgroup of a hyperbolic group G. Then G is strongly relatively hyperbolic with respect to H. In fact the converse to Proposition 2.7 is also true. We came to learn from the referee that this follows by combining work of Farb [Far98], Bowditch [Bow97] and Drutu-Sapir [DS05]. We provide a proof below for completeness (and because it is easily done). Malnormality of strongly relatively hyperbolic subgroups is due to Farb [Far98]. In fact this does not require G to be hyperbolic. Lemma 2.8. (Farb [Far98]) Let G be strongly relatively hyperbolic with respect to H. Then H is malnormal in G. It remains to show that H is quasiconvex if a hyperbolic group G be strongly relatively hyperbolic with respect toH. We use Gromov’s definition of strong relative hyperbolicity. Attach hyperbolic cones cH to distinct translates of ΓH in ΓG to obtain the hyperbolically coned off Cayley graph ΓhG. Then Γ G is hyperbolic by Gromov’s definition. If H is not quasi-isometrically embedded in G then for all i ∈ N, there exist pi1, pi2 ∈ ΓH such that dH(pi1, pi2) ≥ idG(pi1, pi2) . Also from the metric dcH on cH, we find that dcH(pi1, pi2) is of the order of log2dH(pi1, pi2). Hence, we can further assume that dH(pi1, pi2) ≥ idcH(pi1, pi2) . Join pi1, pi2 by shortest paths αi, βi in cH, ΓG respectively. Then αi∪βi = σi is a closed loop in Γ G with total length l(σi) = (dcH(pi1, pi2)+dG(pi1, pi2)). Therefore il(σi) ≤ 2dH(pi1, pi2). Since any (combinatorial) disk Di in Γ G spanning σi must contain a path γi in ΓH joining p1i, p2i, therefore the area A(Di) of Di must be at least that of N1(γi), the 1-neighborhood of γi in Di. Therefore there exists C > 0 such that for all i, A(Di) ≥ A(N1(γi)) ≥ dH(pi1, pi2) il(σi) Since i is arbitrary, this shows that ΓhG cannot satisfy a linear isoperi- metric inequality. Hence ΓhG cannot be a hyperbolic metric space. This is a contradiction. Hence H must be quasi-isometrically embedded in G. Hence RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 13 (see for instance [Gro85] ), H is quasiconvex in G. This completes our proof of the following characterisation of strongly relatively hyperbolic subgroups of hyperbolic groups. Proposition 2.9. Let G be a hyperbolic group and H a subgroup. Then G is strongly relatively hyperbolic with respect to H if and only if H is a malnormal quasiconvex subgroup. 3. Relative Rigidity 3.1. Pairing of Limit Sets by Quasi-isometries. We now consider two hyperbolic groupsG1, G2 with quasiconvex subgroupsH1,H2, Cayley graphs Γ1,Γ2. Let Lj for j = 1, 2 denote the collection of translates of limit sets (counted with multiplicity as before) of H1,H2 in ∂G1, ∂G2 respectively. Individual members of the collection Lj will be denoted as L i . Let Jj denote the collection {J i = J(L i ) : L i ∈ Lj}. Following Schwarz [Sch97], we define: Definition 3.1. A bijective map φ from J1 → J2 is said to be uniformly proper if there exists a function f : N → N such that 1) dG1(J(L i ), J(L j )) ≤ n ⇒ dG2(φ(J(L i )), φ(J(L j ))) ≤ f(n) 2) dG2(φ(J(L i )), φ(J(L j ))) ≤ n ⇒ dG1(J(L i ), J(L j )) ≤ f(n). When Ji consists of all singleton subsets of Γ1,Γ2, we shall refer to φ as a uniformly proper map from Γ1 to Γ2. Note: We observe that if Ji is just the collection of singleton sets in Γi, then a uniformly proper map between J ’s is the same as a quasi-isometry between Γi’s. This can be seen by putting n = 1 in conditions 1 and 2 above and then using the fact that graphs have edge length one. Hence what is important here is that J ’s are infinite diameter sets. Definition 3.2. A map q from Γ1 to Γ2 is said to pair the sets J1 and J2 as φ does if there exists a function h : N → N such that dG(p, J j )) ≤ n ⇒ dG(q(p), φ(J(L j ))) ≤ h(n). The following Lemma generalises Lemma 3.1 of Schwarz [Sch97], where the result is proven in the special case of a symmetric pattern of geodesics in Hn. The referee pointed out to us that the Lemma follows from Lemma 7 of [NR03] by Niblo and Reeves. Lemma 3.3. For M,m > 0, there exists R > 0, such that the following holds. Let L1, · · · , LM be distinct translates of the limit set of a quasiconvex sub- group H of a hyperbolic group G, such that dG(Ji, Jj) ≤ m for all i, j = 1 · · · ,M and Ji = J(Li). Then there exists a ball of radius R meeting Ji for all i = 1 · · · ,M . Proof: If ∩M1 Li 6= ∅, choose any point p ∈ J(∩ 1 Li). Then B1(p) intersects all Ji and we are through. 14 MAHAN MJ Suppose therefore that ∩m1 Li = ∅. We proceed by induction on M . There exists RM−1 such that a ball of radius RM−1 meets Ji for i = 1 · · ·M − 1. We now proceed by contradiction. If no such R exists for M , we have collections {Lk1 , · · · , L M}, k ∈ N such that a ball of radius RM−1 meets Jki , i = 1 · · ·M−1 but no ball of radius k meets J i , i = 1 · · ·M . In particular, (since J(∩M−11 L i ) ⊂ ∩ i )), if ∩ i 6= ∅, then Nk(J(∩ i )) ∩ JMi = ∅. For all i, j, k, choose points pkij ∈ J i such that dG(p ij, p ji) ≤ m. Assume by G-invariance of J that the ball of radius RM−1 centered at 1 ∈ ΓG meets J i , i = 1 · · ·M − 1. Therefore J M lies outside a k-ball about Since the collection of Ji’s through 1 is finite, therefore assume after pass- ing to a subsequence if necessary, that 1) {Jki }k is a constant sequence for i = 1 · · ·M − 1. Hence, {L i }k is a con- stant sequence Li (say) for i = 1 · · ·M − 1. 2) pkiM → piM ∈ ∂G for i = 1 · · ·M − 1. Hence p Mi → piM ∈ ∂G. Further, by (1) above, piM ∈ Li. 3) LkM converges to a closed set Z ⊂ ∂G. By Proposition 2.3, Z must be a singleton set {z}. 4) JkM lies outside Bk(1) ∪ Nk(J(∩ i )). If ∩ 1 Li 6= ∅, then assume further by G-invariance, that 1 ∈ J(∩M−11 L i ). Also, using Theorem 1.12 due to Short [Sho91], and translating by an appropriate element of ∩M−11 H we may assume that 1 ∈ J(∩M−11 L i ) is closest to J Now, pkMi ∈ J M and hence by (3) above, p Mi → z ∈ ∂G. Combining this with (2) above, we get z = piM for all i = 1 · · ·M − 1. Therefore, z ∈ ∩M−11 Li 6= ∅. But dG(1, J M ) = dG(J(∩ i ), J M ) ≥ k. Let zk ∈ J M such that dG(1, J M ) = dG(1, zk) = dG(J(∩ i ), J M ) ≥ k. Then the Gromov inner product (zk, p iM )1 is uniformly bounded above. Therefore (zk, piM )1 is uniformly bounded above. Hence finally (z, piM )1 is bounded above. In particular z 6= piM . This is the contradiction that proves the Lemma. ✷ Definition of q Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. We shall now show tha there exists a quasi-isometry q from Γ1 to Γ2 which pairs the sets J1 and J2 as φ does. We will define a map q : Γ1 → Γ2 which pairs J1 with J2 as φ does and prove that q is a quasi-isometry as promised. Choose K > 0 such that the K neighborhood BK(g) of g ∈ Γ1 has greater than w2 ( the width of H2 in G2 ) J i ’s passing through it. RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 15 Let J (for j = 1, 2 ) denote the collection of J i ’s passing through NK(g) for g ∈ Γj, j = 1, 2. By the proof of Proposition 2.3, there exists M = M(K) (independent of g ∈ Γ1) such that at most M J i ’s in J K,g pass through NK(g). Since φ is a bijective pairing, φ(J K,g) has at least (w2 +1) and at most M(K) elements in it. By definition of w2, and by Theorem 1.12 at least two of the limit sets of the φ(J1i )’s are disjoint. Let L 1 and L 2 denote these limit sets. Hence, by Corollary 2.6, for any K1 ≥ f(K), there exists D such that the collection of points {p ∈ Γ2 : d2(p, J 2 ) ≤ K1, d2(p, J 2 ) ≤ K1} has diameter less than D. Also, by uniform properness of φ, d2(φ(J m), φ(J n)) ≤ f(2K) for J1m, J n passing through NK(g) (independent of g). Summarising, 1) L21 and L 2 are disjoint. 2) But, by Lemma 3.3, using m = f(2K) and M = M(K), there exists R = R(K) and a ball of radius R meeting each φ(J1i ). 3) For anyK1, there existsD, such that {p ∈ Γ2 : d2(p, J 2 ) ≤ K1, d2(p, J 2 ) ≤ K1} has diameter less than D.In particular, we may choose K1 = R. Define q(g) to be the center of the ball of radius R obtained in (2) above. By (3), q(g) is thus defined upto a uniformly bounded amount of discrepancy for all g ∈ Γ1. Lemma 3.4. q is uniformly proper with respect to the metrics d1, d2. Proof: The proof is an almost exact replica of Lemma 3.2 of Schwarz [Sch97] and we content ourselves with reproducing the heuristics of his argument here. If x, y are close in Γ1, then the pairwise minimal distances between ele- ments of JxK1 and J is uniformly bounded above. Hence, by Lemma 3.3, there exists a uniform upper bound to the radius of a minimal radius ball intersecting all elements of φ(JxK1) as well as φ(J ). Also, since the center w of such a ball is defined upto a bounded amount of discrepancy, it must be at a bounded distance from both q(x) as well as q(y). Hence d2(q(x), q(y)) is uniformly bounded, i.e. close. Conversely, suppose that q(x), q(y) are close. First, by Lemma 3.3, there exists a uniform upper bound R on radius of minimal radius balls B1, B2 centered at q(x), q(y), intersecting all elements of φ(J xK1), φ(J K1) respec- tively. Then the (R + d2(q(x), q(y))) ball about q(x) (or q(y)) meets every element of φ(J xK1) as well as φ(J K1). Since φ is uniformly proper, this means that there is a uniform upper bound on the minimal radius of a ball meeting 16 MAHAN MJ every element of (J xK1) as well as (J K1). As before, d1(x, y) is uniformly bounded, i.e. x, y are close. ✷ Similarly, we can construct q−1 using the bijective pairing φ−1 such that q−1 is uniformly proper. Also, from Lemma 3.3 q, q−1 composed with each other in either direction is close to the identity. Since φ pairs L1, L2 bijectively and is uniformly proper from J1 to J2, therefore by invariance of J2 under G2, every point of Γ2 lies close to the image of q. Therefore q is uniformly proper, by Lemma 3.4 above, from Γ1 onto a net in Γ2. Hence q is a quasi-isometry. This concludes the proof of the main theorem of this subsection. Theorem 3.5. Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi-isometry q from Γ1 to Γ2 which pairs the sets J1 and J2 as φ does. We have thus shown one aspect of relative rigidity, viz. upgrading a uniformly proper map between Ji’s to a quasi-isometry between Γi’s. In the next subsection, we shall deduce the second aspect, viz. isomorphism of C-complexes. 3.2. C-Complexes. By Theorem 3.5 we obtain a quasi-isometry q from Γ1 to Γ2 which pairs J1 and J2 as φ does. Since q is a quasi-isometry, it extends to a quasiconformal homeomorphism from ∂G1 to ∂G2. Also, for all α > 0, there exists β > 0 such that d1(x, J i ) ≤ α ⇒ d2(q(x), φ(J i )) ≤ β and conversely, d2(y, J i ) ≤ α ⇒ d1(x, φ −1(J2i )) ≤ β In particular, ∂q maps the limit set L1i homeomorphically to the limit set of φ(J1i )). Hence, ∂q preserves intersection patterns of limit sets. Since φ pairs J1 with J2 as q does, summarising we get: Lemma 3.6. The following are equivalent. 1) ∩ki=1L i = ∅ 2) ∩ki=1∂q(L i ) = ∅ 3) ∩ki=1φ(L i ) = ∅ Hence by the definition of the C-complexes C(G1,H1) and C(G2,H2), we find that ∂q induces an isomorphism of C(G1,H1) with C(G2,H2). We conclude: Theorem 3.7. Let φ : J1 → J2 be a uniformly proper map. Then φ induces an isomorphism of C(G1,H1) with C(G2,H2). Note: In Theorem 3.5 and Theorem 3.7 we start with the assumption that there exists a uniformly proper pairing of the collections J1 and J2. This can be translated to a pairing of collections of limit sets L1 and L2. Theorem 3.5 RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 17 then says that the pairing of the Ji’s (or Li’s) is induced by a quasi-isometry from Γ1 to Γ2. Thus, the existence of a uniformly proper pairing implies the existence of a quasi-isometry between the Γi’s, i.e. an ambient extension (or, equivalently, a quasiconformal homeomorphism between ∂Gi’s). Also Theorem 3.7 shows that a uniformly proper pairing induces an iso- morphism of the C-complexes C(Gi,Hi). This is reminiscent of the initial step in the proof of rigidity theorems for higher rank symmetric spaces, where Tits complexes replace C-complexes. 3.3. Cross Ratios, Annular Systems and a Dynamical Formulation. In this subsection, we give a more intrinsic formulation of Theorems 3.5 and 3.7. The hypothesis of these theorems is given in terms of distances between elements of Ji. A more intrinsic way of formulating this hypothesis would be in terms of the action of Gi on ∂Gi, i = 1, 2. In this case, the distance between J il , J m can be approximated by the hyperbolic cross-ratio of their limit sets. This was described in detail by Bowditch [Bow98]. We give the relevant definitions and Theorems below and then dynamically reformulate Theorems 3.5 and 3.7. Let M be a compactum. Definition 3.8. An annulus A is an ordered pair (A−, A+) of disjoint closed subsets of M such that M \ (A− ∪ A+) 6= ∅. An annulus system is a collection of such annulii. If A = (A−, A+), then −A = (A+, A−). An annulus system is symmetric if A ∈ A ⇒ −A ∈ A. Given a closed set K ⊂ M and an annulus A, we say that K < A if K ⊂ intA−. Also, A < K if K < −A. If A,B are annulii, we say that A < B if M = intA− ∪ intB+. Fix an annulus system A. Given closed sets K,L ⊂ M , we say that the annular cross-ratio (K|L)A ∈ N∪∞ for the maximal number n ∈ N such that we can find annulii A1, · · ·An ∈ A such that K < A1 < · · · < An < L . We set (K|L)A = ∞ if there is no such bound. Thus (K|L)A is the length of the maximal chain of nested annulii sep- atrating K,L. For two point sets {x, y} = K and {z, w} = L, we write (K|L)A as (xy|zw)A. One of the crucial results of [Bow98] is: Theorem 3.9. (Bowditch [Bow98]) Suppose a group G acts as a uniform convergence group on a perfect metrizable compactum M . Then there exists a symmetric G-invariant annulus system A such that if x, y, z, w are distinct elements in M , then the theree quantities (xy|zw)A, (xz|yw)A, (xw|zy)A are all finite and at least two of them are zero. Also, if x 6= y, then (x|y)A > 0. Further, G is hyperbolic, and dG(J(K), J(L)) differs from (K,L)A upto bounded additive and multiplicative factors. 18 MAHAN MJ Combining Theorems 3.5 , 3.7 with Proposition 2.3 and Theorem 3.9, we get the dynamical formulation we promised. Let C0c (M) denote the collection of closed subsets of M containing more than one point. (Replacing dGi by cross-ratios (..|..)i in Definition 3.1 we get the corresponding notion of a map being uniformly proper with respect to the cross-ratios (..|..)1 and (..|..)2 in the theorem below. Similarly for the homeomorphism q.) Theorem 3.10. Let G1, G2 be uniform convergence (hence hyperbolic) groups acting on compacta M1,M2 respectively. Also, let Ai (for i = 1, 2 ) be Gi- invariant annulus systems and let (..|..)i denote the corresponding annular cross-ratios. Let H1,H2 be subgroups of G1, G2 with limit sets Λ1,Λ2. Suppose that the set Li of translates of Λi (for i = 1, 2) by essentially distinct elements of Hi in Gi forms a discrete subset of C c (Mi). Also assume that there exists a bijective function φ : L1 → L2 and that this pairing is uniformly proper with respect to the cross-ratios (..|..)1 and (..|..)2. (1) Hi is quasiconvex in Gi (2) There is a homeomorphism q : M1 → M2 which pairs L1 with L2 as φ does. Further, q is uniformly proper with respect to the cross-ratios (..|..)1 and (..|..)2 on M1, M2 respectively. (3) q (and hence also φ) induces an isomorphism of C-complexes C(G1,H1) with C(G2,H2). Thus from a uniformly proper map with respect to the pseudometrics on Li’s induced by cross-ratios we infer a quasi-isometry that is an ambient extension as also a (simplicial) isomorphism of C-complexes. 3.4. Axiomatisation, Relative Hyperbolicity. For classes of pairs (X,J ), what did we really require to ensure relative rigidity? Assume (X, d) is a metric space and let the induced pseudometric on J be also denoted by d. 1) For all k > 0 there exists M ∈ N such that for all x ∈ X, Nk(x) meets at most M of the J ’s in J . (This is a coarsening of the notion of height.) 2) For all K ∈ N, there exists k = k(K) > 0 such that for all x ∈ X, Nk(x) meets at least K of the J ’s in J . (This is the converse condition to (1).) 3) For all k > 0, n ∈ N there exists K > 0 such that for any collection J1, · · · , Jn ∈ J with d(Ji, Jj) ≤ k, there exists a ball of radius at most K meeting all the Ji’s. 4) There exists N ∈ N such that for all k > 0 there exists K = K(k) > 0 such that the following holds. For all n ≥ N and J1, · · · , Jn ∈ J , the set of points {x ∈ X : Nk(x) ∩ Ji 6= ∅, i = 1 · · · n} is either empty or has diameter bounded by K. Given (1)-(4), the construction of q : X1 → X2 from a uniformly proper pairing φ : J1 → J2 goes through as in Theorem 3.5. In short, pick N from (4). From (2), pick k = k(N). Now for all x ∈ X1, consider the collection RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 19 of J ’s in J1 that meet Nk(x). By (1) there is an upper bound M = M(k) on the number of such J ’s. Map these over to J2. Any two of these are at a distance of at most m apart where m depends on φ and k. From (3) choose K = K(M,m) such that a ball of radius K meets all these. Set q(x) to be the center of such a ball. By (4), q(x) is defined upto a uniformly bounded degree of discrepancy. The rest of the proof goes through as before. Hence (1)-(4) define sufficient conditions for relative rigidity for a class of pairs (X,J ). With these conditions, it is easy to extend Theorem 3.5 to pairs (X,J ) where X is (strongly) hyperbolic relative to the collection J . Conditions (1) and (2) are trivial. Condition (3) follows from “bounded penetration” (see Farb [Far98]). For any subcollection J1 of J with d(Ji, Jj) ≤ C0 (for all Ji, Jj ∈ J1), fix any two J1, J2 ∈ J and a geodesic γ12 of length ≤ C0 joining them. Construct an electric triangle for triples J1, J2, J3 ∈ J1 of horosphere-like sets for arbi- trary J3 ∈ J1, such that the hyperbolic geodesics γ13, γ32 joining J1, J3 and J3, J2 respectively have lengths bounded by C0. Then γ12 and γ13 meet J1 at a uniformly bounded distance from each other by bounded penetration. To see this, first note that J1, J2 can be joined by two paths, one consisting of one side of the triangle and the other the union of the two remaining sides of the triangle and both paths have electric length bounded by 2C0; in par- ticular both paths are uniform quasigeodesics (with quasigeodesic constant depending only on C0). They may be converted to quasigeodesics without backtracking by not increasing lengths. Thus γ13 ∪ γ32 decomposes as the union of a quasigeodesic without backtracking γ′12 joining J1, J2 and (possi- bly) a uniformly bounded (≤ C0) number of loops of length not longer than 2C0. The entire quasigeodesic without backtracking γ 12 lies near γ12 for all J3 ∈ J1. The same holds for the loops of bounded length (since they in turn may be regarded as uniform quasigeodesics without backtracking starting and ending at the same point.) In particular J3 lies at a uniformly bounded distance D0 from γ12. Since γ12 has length bounded by C0, and J3 may be chosen arbitrarily satisfying the hypothesis of (3) above, it follows that for any x ∈ γ12, d(x, J3) ≤ (C0 + D0) for all J3 ∈ J1. Condition (3) follows. (Results closely related to the proof of Condition (3) here occur as Lemma 3.11 of [Mj05] and Prop. 8.6 of [HW06].) Condition (4) follows from the fact that for a pair of distinct Ji, Jj , Nk(Ji) ∩Nk(Jj) is either empty or has diameter bounded by some C(k). We have thus shown: Theorem 3.11. Let Xi be (strongly) hyperbolic relative to collections Ji (i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi-isometry q from X1 to X2 which pairs the sets J1 and J2 as φ does. 20 MAHAN MJ By work of Hruska and Kleiner [HK04], CAT(0) spaces with isolated flats are (strongly) hyperbolic relative to maximal flats. Hence we have from Theorem 3.11 above: Corollary 3.12. Let Xi be CAT(0) spaces with isolated flats and let Ji denote the collections of maximal flats (i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi- isometry q from X1 to X2 which pairs the sets J1 and J2 as φ does. 3.5. Symmetric Spaces of Higher Rank. We now consider CAT(0) spaces which are at the other end of the spectrum. Let M be a compact lo- cally symmetric space and T a totally geodesic torus with rank = rank(M). Take X = M̃ and J to be the lifts of T to M̃ . As these are all equivariant examples (i.e. J is invariant under a cocompact group action), it is enough to check (1)-(4) at a point. (1) and (2) are clear. To prove condition (4), we consider ∩iNk(Fi) and it is easy to bound from below the N appearing in Condition (4) (Section 3.4) in terms of the size of the Weyl group and rank. In that case, ∩iNk(Fi) has bounded diameter or is empty. Finally, to prove (3), we proceed as in Lemma 3.3. As in Lemma 3.3 we assume by induction that any k flats {F1, · · · , Fk} that ”coarsely pair- wise intersect at scale D” (i.e. ND(Fi) ∩ ND(Fj) 6= ∅ ) intersect coarsely (i.e. ∩i=1···kND′(Fi) 6= ∅ for some D ′ = D′(D, k)). To get to the induc- tive step, we suppose that for i = k + 1, we have collections of worse and worse counterexamples. Consider a maximal collection F = {F1, · · · , Fk} of maximal flats whose ”coarse intersection at scale D” ∩iND(Fi) = F is non-null. Translate the collection by a group element so that a fixed point 0 (thought of as the origin) lies on the intersection F . Now take a sequence of maximal flats F j whose D-neighborhoods ND(F j) intersect each ND(Fi), but dj = d(F j , F ) = d(0, F ) ≥ j. We scale the metric on (X, d) by a factor of dj to obtain a sequence of metric spaces (X, ) converging (via a non- principal ultrafilter) to a Euclidean building X∞ (this fact is due to Kleiner and Leeb [KL97b], but we shall only mildly need the exact nature of X∞). Fi’s converge to flats F i ⊂ X ∞ and F j ’s converge to a flat G∞ ⊂ X∞. Then the collection G = F∞i , G ∞ satisfy the following: (P1) Each element of G is a flat in X∞ (P2) By induction, the intersection of any i elements of G is non-empty and convex for i ≤ k (P3) The intersection of all the (k + 1) elements of G is empty. Consider the subcomplex K = G∞ i of X ∞. Then K is a union of r-flats, where r = rank(X). In particular, the homology groups Hn(K) = 0 for n > r. On the other hand, if we consider the nerve of the covering of K by the sets G∞, F∞i , then using the properties (P1), (P2), (P3) to compute Cech homology groups, we conclude that K has the same homology groups as the boundary of a k-simplex. In particular, Hk(K) = Z. For k > r this RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 21 is a contradiction, finally proving Condition (3). Thus we conclude: Theorem 3.13. Let Xi be symmetric spaces of non-positive curvature, and Ji be equivariant collections of lifts of a maximal torus in a compact locally symmetric space modeled on Xi (i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists a quasi-isometry q from X1 to X2 which pairs the sets J1 and J2 as φ does. Combining Theorem 3.13 with the quasi-isometric rigidity theorem of Kleiner-Leeb [KL97b] and Eskin-Farb [EF 3] we can upgrade the quasi- isometry q to an isometry i. Corollary 3.14. Let Xi be symmetric spaces of non-positive curvature, and Ji be equivariant collections of lifts of a maximal torus in a compact locally symmetric space modeled on Xi (i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from J1 → J2. There exists an isometry i from X1 to X2 which pairs the sets J1 and J2 as φ does. Remark 3.15. The technique of using asymptotic cones and the nerve of the covering by flats can be generalised easily to equivariant flats of arbitrary (not necessarily maximal) rank. We conclude this paper with two (related) questions: Question 1: In analogy with a Theorem of Ivanov, Korkmaz, Luo (see for instance [Luo00] ), regarding the automorphism group of the curve complex, we ask: If the C-Complex C(G,H) of a pair (G,H) (for G a hyperbolic group and H a quasiconvex subgroup) is connected, is the automorphism group of C(G,H) commensurable with G? Question 2: Consider the pair (G,H), with G a hyperbolic group and H a quasiconvex subgroup. 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8 Theory of enhanced second-harmonic generation by the quadrupole-dipole hybrid exciton Oleksiy Roslyak Physics Department, The City College, CUNY Convent Ave. at 138 St, New York, N.Y. 10031, USA E-mail: avroslyak@gmail.com Joseph L. Birman Physics Department, The City College, CUNY Convent Ave. at 138 St, New York, N.Y. 10031, USA Abstract. We report calculated substantial enhancement of the second harmonic generation (SHG) in cuprous oxide crystals resonantly hybridized with an appropriate organic material (DCM2:CA:PS ’solid-state solvent’). The quadrupole origin of the inorganic part of the quadrupole-dipole hybrid provides inversion symmetry breaking and the organic part contributes to the oscillator strength of the hybrid. We show that the enhancement of the SHG, compared to bulk cuprous oxide crystal, is proportional to the ratio of the DCM2 dipole moment and the effective dipole moment of the quadrupole transitions in the cuprous oxide. It is also inversely proportional to the line-width of the hybrid and bulk excitons. The enhancement may be regulated by adjusting the organic blend (mutual concentration of the DCM2 and CA part of the solvent) and pumping conditions(varying the angle of incidence in case of optical pumping or populating the minimum of the lower branch of the hybrid in case of electrical pumping). PACS numbers: 73.21.La, 73.22.Dj, 78.67.Hc http://arxiv.org/abs/0704.1923v3 Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton2 1. Introduction Considerable attention has been paid to the relatively strong optical second-harmonic generation (SHG) in thin films (D4h symmetry) and bulk (Oh symmetry) of cuprous oxide crystals. Which was first addressed in the pioneering work of Shen [Shen et al. (1996)]. This effect is attributed to the electric-quadrupole h̄ω1S = 2.05 eV exciton effect. The quadrupole exciton has very small oscillator strength but it possess rather narrow line-width h̄γ1S. So the effect is well pronounced when the exciting laser energy is close to one h̄ω1S − h̄ω ≪ h̄γ1S or two photon resonance h̄ω1S − 2h̄ω ≪ h̄γ1S. In the dipole approximation this effect disappears [Atanasov et al. (1994)]. We propose to amplify the SHG characteristic of the 1S quadrupole Wannier exciton (WE) in cuprous oxide by making a hybrid with an organic Frenkel exciton (FE) (See next section for more details). The idea of resonant enhancement of some non- linear properties generic to semiconductor dipole-allowed Wannier-Mott (WE) excitons was presented in pioneering work of [Agranovich et al. (1998)] for the layered organic- inorganic heterostructures. It was also developed for quantum wires and dots embedded into organic shell [Engelmann et al. (1998)],[Gao et al. (2004)] or attached to dendrimer structure [Huong and Birman (2000)],[Huong and Birman (2003)]. In our previous work [Roslyak and Birman (2007a)] we demonstrated considerable enhancement of another non-linear effect in cuprous oxide, photo-thermal bi-stability [Dasbach (2004)]. We demonstrated a considerable enhancement in the hysteresis-like region size (from µeV for bulk cuprous oxide to meV for the hybrid). The enhancement was attributed to the large oscillator strength of the hybrid exciton inherited from the organic part and still rather narrow line-width of the same order as the coupling. Analogous enhancement can be expected for the SHG, which is the subject of this paper. In Section 2 we propose a pump-prob experiment to reveal the SHG enhancement due to the resonant dynamical hybridization and briefly discuss relevant quadrupole hybrid exciton properties. In the next Section 3 we address the question how this resonant‡ enhancement depends on such parameters of the system as oscillator strengths and damping of the FE and WE constituting the hybrid. Using a classical model of nonlinear coupled oscillators, we demonstrate that while the big ratio of the hybrid oscillator strength suggests many orders of the enhancement magnitude it is actually somehow reduced by the rather small coupling parameter and density of the FE. Because the FE is dynamically brought into resonance with the WE there is an important hybridization time τh parameter. Hence, in the Section 4, we develop more sophisticated quantum mechanical model to address the dynamics of the hybrid SHG. Namely we show that the signal enhancement drastically depends on the either the one probes the system before or after the hybridization occurred. ‡ The resonance occurs between the FE and WE Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton3 2. Proposed experimental set-up for the SHG In this work we adopt the concept of a layered organic-inorganic heterostructure. The inorganic component of the hybrid is a thin layer of Cu2O (quantum well, latter in the text referred to as QW) grown upon a film of the organic composite (See Fig.1). Due to the small radius of both the WE and FE exciton part of the hybrid one can neglect the effect of confinement. In this case one can not tune the two types of excitons in resonance by adjusting the confinement (Lw > a B ≈ to the cuprous oxide unit cell a = 4.6 Å). The QW confinement just assures the WE propagate along the interface and is the subjected to the electric field gradient of the FE propagating along the adjacent chain of the DCM2 molecules. x, kx, Eg 143 meV 2.17 eV ≫ 2.17 eV Cu2O PS :DCM2: CA Lw ≈ 4.6 Å probe Figure 1. Schematic representation and the energy offset of a possible experimental set-up to observe the enhanced SHG by the quadrupole-dipole exciton. Here the inorganic Cu2O quantum well provides the 1S quadrupole WE. The DCM2 part of the organic ’solid state solute’ provides dipole allowed FE (set of small arrows); the PS host prevents wave function overlapping between organic and inorganic excitons; CA under proper concentration allows tuning of the excitons into the resonance. To provide resonance between WE in cuprous oxide and FE in the organic, we pro- posed utilization of ’solid state solvation’ (SSS) of the DCM2 § molecules in transpar- ent polystyrene (PS) host doped with camorphic anhydride (CA)[Bulovic et al. (1999)]. The SSS is a type of solvatochromism manifesting itself as some change in the spec- tral position of the absorption/luminescence band due to change in the polarity of the medium. The Förster dipole-dipole non-resonant interaction between DCM2 and CA modifies the energy structure of the involved molecules. § [2-methyl-6-2-(2,3,6,7-tetrahydro-1H, 5H - benzo[i,j] - quinolizin - 9 - yl) - ethenyl] - 4H - pyran - 4 - ylidene] propane dinitritle. Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton4 During the ’slow’ phase (τs ≈ 3.3ns) the energy of the FE ‖ experiences a red shift linear with the CA concentration due to non-resonant dipole-dipole interaction with the CA molecules. Not that our model capitalizes on the fact that DCM2 molecules form a 2D layer rather than been diluted in the PS:CA solvent which is the case of currently manufactured optical light emitting devices (OLED). This allows us to neglect rather complicated problem of the inhomogeneous broadening of the FE energy by utilizing a mean field approximation ¶. For the mean field approximation the red spectral shift of the FE energy in resonance with the quadrupole WE can be accomplished with ρCA ≈ 22% CA concentration. To avoid complicated problems of time dependent hybridization and stay within the analytical model framework, we assume that the FE and WE are in exact resonance once the DCM2 energy is in close proximity to the WE energy i.e. h̄ωDCM2 − h̄ω1S ≤ Γk. We introduced the quadrupole-dipole coupling parameter Γk ≤ 4 µeV [Roslyak and Birman (2007b)] (See also Appendix (A.1)). This resonant coupling gives rise to the upper and lower branches of the quadrupole-dipole hybrid (QDH) dispersion +: h̄ωu,l = h̄ω1S ± Γk. To populate both of the branches one needs a second pumping photon tuned into resonance with the 1S transition. The radiation field interacts through both dipole and quadrupole part of the hybrid. The dipole interaction can be utilized to produce linear response signal due to the pumping [Roslyak and Birman (2007a)]. By using the non-linear response to the prob signal, the second harmonic can be generated through the quadrupole part of the hybrid. Different SHG regimes can be achieved by changing the timing between pumping and prob signals (See section 4 for more details). According to the selection rules for the quadrupole-dipole hybrid the pumping signal, running along the organic-inorganic interface of the heterostructure, induces the linear polarization in the z direction [Roslyak and Birman (2007b)]. The prob signal induces the second order non-linear response in the cuprous oxide. Which is perpendicular to the interface, and defined by the second order polarization along the x direction (See Fig.1). The net polarization is given by a second rank tensor through the following expression: P (1)z + P l=ĵ×x̂ i,zEi + iχ l,i,j,xkxEiEj (1) Here Ei, Ej are the electric field of the pumping and prob lasers correspondingly. The x component of the prob signal wave vector is taken to be close to zero to avoid possible interference in momentum conservation. For the sake of simplicity we are going to omit, x and l indexes of the tensor keeping in mind that the wave vector of the pump signal ‖ in our case we define the FE as DCM2 excitation ¶ Indeed, in our simplified model the DCM2 molecules are not randomly situated but rather form a uniform (homogeneous) thin layer near the interface. Also the experimentally observed FE energy relaxation shows no significant energy fluctuations. This experiments are performed at MIT by the Dr. Bulovic. Although this results are not officially published yet, but reported in the MIT proceedings. + See the eigenvalues of the linearized system (4) or the Hamiltonian (8) Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton5 has only x component and the SH signal is perpendicular to it and the prob signal polarization: iχ l,i,j,xkx = χ i,j . In this paper we develop both classical and quantum mechanical models, which can be used to find a specific form of the hybrid second order nonlinear susceptibility. In section 3 we demonstrate that the second order non linearity (generic to the cuprous oxide and introduced through a small parameter λ) is enhanced due to the resonant quadrupole-dipole hybridization with the organic (See (5)). In section 4 we develop quantum theory of the enhanced SHG. It allows investigation of different regimes of the process defined by the time ordering between the prob pulse and the time when the FE and WE energies are close enough to form the hybrid. We generalize the concept of the double-sided Feynman diagrams [Mukamel (1995)] to include non radiative processes of the energy exchange between DCM2 and CA as well as resonant QDH between DCM2 and cuprous oxide. 3. Anharmonic coupled oscillators model As a first step, we will use the simplest classical model neglecting the non-local effects of the linear χ i,z and non-linear susceptibility χ l,i,j,x to describe the hybrid SHG. Namely, we adopt an extension of the anharmonic oscillator model [Bloembergen (1965), Mukamel (1995)] generalized for the case of resonant coupling between two distinct sets of oscillators. This simplified picture only covers the case when the pumping field is polarized along ẑ ‖ [001] axis (Ei = Ez) and we prob the hybrid system (ω1S = ωF ) with a signal perpendicular to the interface and polarized along ŷ ‖ [010] direction (Ej = Ey). We consider the WE in cuprous oxide as an assembly of the oscillators with the oscillator strength per unit cell given by fxz,k ∝ kx (See for example reference [Moskalenko and Liberman (2002)]). The second set of the oscillators with the oscillator strength given by fF corresponds to the FE in the organic. Treating the wave vector k as just another parameter∗, the polarization PW , P F due to WE and FE can be written in terms of the effective electron-hole displacements X, Y as: aWB S fxz,keX (2) P F = ρDCM2N fFeY (3) Here S is the area of the interface and aWB , a B are the WE and FE Bohr radius, e is the electron charge. The surface density of the WE and FE excitons are Nfxz,k/ (aBS) and ρDCM2Nf F/ (aBS) correspondingly and N is the total number of the oscillators. Here we also took into account the low density (ρDCM2 = 0.05%) of the DCM2 molecules in the organic to avoid the aggregation effect [Madigan and Bulovic (2003)]. ∗ In the text we are going to omit index k unless we put an emphasis on it Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton6 In the time frame of the hybridization τs−τh < t < τs, the WE and FE energies are at perfect resonance. Hence, the system of equations governing the oscillators dynamics can be written in the form: Ẍ + ω21SX + γẊ − 2ω1SΓk Y − ω21SλX2 = 0 Ÿ + ω21SY + γẎ − 2ω1SΓk The nonlinear factor ω21Sλ appears due to the prob signal. It is defined such that λ has dimensions of reciprocal length and is considered to be small in a sense that it is much less than the reciprocal of the maximum displacement of FE (Y ) and WE (X) oscillator. The exact value of λ can be obtained either from an experiment or from the microscopic quantum theory (See next section for more details). The terms proportional to γ describe the QDH damping. The terms proportional to 2ω1SΓk/h̄ describe the quadrupole-dipole coupling. Hence, the eigenvalues of the linearized system of equations (4) give both branches of the QDH. The system is driven dominantly by the light-dipole interaction in the organic and the quadrupole-light interaction is neglected (m is the electron mass). Using standard perturbation theory with respect to the small parameter λ in zero order (neglecting the quadratic term) and combining equations (1,2,4) one has the linear response of the hybrid and bulk cuprous oxide given by the following expressions: Hy (ω) = ρDCM2N fFe2/m (ω21S − ω2 + iωγ) (ω21S − ω2 + iωγ) 2 − (2ω1SΓk/h̄)2 Cu2O (ω) = aWB S fxz,ke ω21S − ω2 + iγ Including the nonlinear term as a source for the SHG to first order in the perturbation parameter, there is a displacement at 2ω. The SHG response is given by a solution of the following system: Ẍ + ω21SX + γẊ − 2ω1SΓk Y − ω21SλX2λ=0 = 0 Ÿ + ω21SY + γẎ − 2ω1SΓk X = 0 Using the definitions (1) and (2) one gets the following non-linear second order response function for the hybrid and bulk cuprous oxide correspondingly: Hy (2ω;ω, ω) = ρDCM2N fFe3/m2ω21Sλ (2ω1SΓk/h̄) ω21S − (2ω) + i2ωγ − (2ω1SΓk/h̄)2 (ω21S − ω2 + iωγ) (ω21S − ω2 + iωγ) 2 − (2ω1SΓk/h̄)2 Cu2O (2ω;ω, ω) = aWB S fxz,ke 3/m2ω21Sλ ω21S − (2ω) + i2γ (ω21S − ω2 + iγ) Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton7 Straightforward comparison of the expressions above evinces the resonant rise of the second order nonlinearity owing to hybridization. There are several competing factors involved. The enhancement by means of big oscillator strength ratio fF/fxz,k is reduced by rather small coupling parameter ΓK and small DCM2 density ρDCM2 (see more numerical details in Section 5). 4. Quantum theory of SHG due to the QDH Although the system of non-linear susceptibilities (5) in principle solves the problem of SHG due to the hybrid it does not clarify the origin of the nonlinearity λ. Also, such an important parameter as the hybridization time τh is left out of the classical description. Hence, in this section we propose a unified quantum theory of the hybrid SHG. The linear response of the hybrid is due to dipole transitions from the ground |g〉 state♯ to the FE |F 〉 in the organic and due to quadrupole transitions to the WE |1S〉 in the cuprous oxide. The non-linearities are the result of some intermediate inter-band transitions in the cuprous oxide [Mukamel (1995)]. In cuprous oxide the nearest state in energy to the quadrupole ortho-exciton h̄ω1S is the h̄ω2P dipole allowed excitonic band |2P 〉, Eg > h̄ω2P > h̄ωF > h̄ω1S. Hence it plays the main role in formation of the non-linear response and can be excited by the properly tuned prob signal. We neglect all the rest of inter-band and intra-band†† transitions. Therefore, the states above form a complete basis for the SHG problem: |g〉 , |1S〉 , |F 〉 , |2P 〉 (6) Inversion symmetry of the DCM2 is also broken by the CA induced local field and the interface effect. Therefore, unlike in classical model, the contribution from the organic to the SHG has to be consider as well. But due to the smallness of the symmetry breaking local field it contributes a little to the SHG enhancement. Using the basis above let us introduce creation operators for the FE and the 1S and the 2P WE exciton b† = |F 〉 〈g|, B†1S = |1S〉 〈g|, B 2P = |2P 〉 〈g| respectively. The commutation algebra of the operators is presented in the Appendix (B.1). The net polarization of the sample is defined as [Mukamel (1995)]: P = µi1S,k 1S +B1S + µi2P 2P +B2P b† + b 1S,2P 1SB2P +B1SB Here µi1S,k = î · ẑ kxQx,z = 3 · 10−5(kx/k0,x) D is an effective dipole moment [Moskalenko and Liberman (2002), Roslyak and Birman (2007b)] due to the quadrupole transitions associated with the oscillator strength; k0 is the resonant wave vector for bulk ♯ when no excitations are present in the system ††due to small radius of the quadrupole WE Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton8 cuprous oxide( Appendix (A.2)). The dipole moment of the transitions from |1S〉 to 〈2P | is defined by [Artoni et al. (2002), Elliott (1961)]: 1S,2P Ne2h̄2f2P SaWB 2m ĵ × x̂ = 6 · 10−3 D2 Finally, the DCM2 dipole moment of the transition from |g〉 to 〈F | per unit area of the interface is given by [Madigan and Bulovic (2004)]: ρDCM2Ne 2h̄2fF SaFB2m ∗h̄ω1S = 0.2 D2 Using equation (7) and the rotating wave approximation for the resonant wave vector k, the hybrid Hamiltonian can be written as: H = h̄ωF b †b+ h̄ω1SB 1SB1S + E2PB 2PB2P + Γk 1Sb+B1Sb + (8) i + bEi + µi1S,k i +B1SEi + µi2P i +B2PEi 1S,2P 1SB2PE j +B1SB The linear response from both branches of the hybrid may be observed by pumping the hybrid with two signals Ei||ẑ ∝ eiωt. The first photon h̄ω = EDCM2 excites DCM2 molecules. During the time period τs − τh the system relaxes to the FE exciton energy close to h̄ω1S thus providing resonance between WE and FE. Then the second pumping photon h̄ω = h̄ω1S enters and excites quadrupole WE so that both QDH branches are populated. The QDH exciton lives for τh nano-seconds and then both branches of the hybrid relaxes to the ground state emitting photons of the energy h̄ω1S ± Γk. Generalizing conventional double-sided Feynman diagrams [Mukamel (1995)] to include the non-radiative processes, the linear response from the QDH can be represented by the following diagram: |g〉 〈g| 〈1S ⊕ F 〉 h̄ω1S−Γk dd d$ d$ h̄ω1S+Γk 8x8x8x 8x8x8x h̄ω1S 〈DCM2| // h̄ω1S EDCM2 τs−τh i (ω, k) = µ 1S,0 +B1S,0 + µiF 0 + b0 (µiF ) (h̄ω − h̄ω1S + ih̄γ) + µiFµi1SΓk (h̄ω − h̄ω1S + ih̄γ)2 − Γ2k + c.c. (10) On the diagram the wavy lines represent the incoming and outgoing photons; the straight lines stand for the non-radiative transitions. The diagram shows energy exchange between photon-exciton and exciton-exciton as well as the time separation between two pumping signals. Time increases from bottom to the top of the diagram as for the Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton9 conventional Feynman diagram. The hybrid life time is denoted as τh = 1/γ and the hybridization between FE and WE is denoted as ⊕. In the derivation of the linear response χ i (ω, k) we used equation (7) along with solutions of the Heisenberg equations of motion presented in the Appendix II (B.2). Formally the linear response can be written in terms of the hybrid Green’s functions as: i (ω, k) = a,b={g,1S,F} µiabµ baIab (ω) I1S,g = IF,g = h̄ω − h̄ω1S + ih̄γ (h̄ω − h̄ω1S + ih̄γ)2 − Γ2k I1S,F = (h̄ω − h̄ω1S + ih̄γ)2 − Γ2k Iab = I Here the dipole matrix elements in the corresponding basis (6) are given by: 0 µ1S µF 0 µ1S 0 µ1SµF 0 µFµ1S 0 0 0 0 0 0 Note that we neglected the non-resonant term associated with ground state dipole moment of the organic µg. The SHG is due to second order response Ej⊥Ei||z and given by the last term in the equation (7) and the solutions of the equations of motion (B.2,B.3). The first type of the SHG is formed when the branches of the hybrid interacts with the |2P 〉 level excited by the prob signal. Using all the diagram conventions we adopted above, the diagram for this non linear process is given below: |g〉 〈g| |2P 〉 oo µ1S,2P // 〈1S ⊕ F 〉 h̄ω1S−Γkd$ h̄ω1S+Γk8x8x 888x8x8x h̄ω2P h̄ω1S 〈DCM2| // h̄ω1S τ2P>τs−τh 〈g| 〈g| EDCM2 τs−τh ij (2ω;ω, ω) = µ1S,2P 1S,0B2P,1 + c.c. = (11) µi2Pµ 1S,2P (µ 1S (h̄ω − h̄ω1S + ih̄γ) + µiFΓk) (2h̄ω − h̄ω2P ) (h̄ω − h̄ω1S + ih̄γ)2 − Γ2k ) + c.c. (12) Here the prob signal comes after the hybrid is formed: τ2P > τs − τh. Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton10 Another second order non linear response can be formed if the prob signal is coming before the hybridization τ2P < τs − τh. It can be represented by the following diagram: |g〉 |g〉 〈g| |g〉 h̄ω1S///o/o/o 〈1S ⊕ F 〉 1S±Γk 1S∓Γk h̄ω2P µ1S,2P |2P 〉 〈DCM2| // |CA〉 τs−τh τ2P<τs−τh h̄ω2P EDCM2 ij (2ω;ω, ω) = µ1S,2P 1S,1B2P,0 + c.c. µi2Pµ 1S,2P (µ 1S (2h̄ω − h̄ω1S + ih̄γ) + µiFΓk) (h̄ω − h̄ω2P ) (2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k The Green’s function representation of the SHG due to the second order response is given by the following expression: ij (2ω;ω, ω) = µ 1S,2P a={g,1S,F,2P} µia,1Sµ 2P,a × × [Ia,1S (ω) Ia,2P (2ω) + I1S,a (2ω) I2P,a (ω)] I2P,g = h̄ω − h̄ω2P The dipole matrix elements on the basis (6) are given by: 0 µ1S µF µ2P µ1S 0 µ1SµF µ1S,2P µFµ1S 0 0 µ2P µ1S,2P 0 0 According to the last term in the equation (7), the signal at 2h̄ω = h̄ω1S ± Γk may generate the signal at h̄ω = h̄ω1S ± Γk: ij (ω; 2ω,−ω) = 1S,2P (2h̄ω − h̄ω1S + ih̄γ) (h̄ω − h̄ω2P )2 (2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k ) + c.c. This type of signal has been experimentally detected [Shen et al. (1996)] in bulk cuprous oxide (Γk = 0) when the pumping signal was tuned to the wave length between 12285 Å and 12195 Å. A strong SH signal was detected at 6096 Å which has to be attributed not only to the narrow line-width of the quadrupole exciton but to the fact that µ1S,2P ≫ µ1S as well. From the last expression it follows that in this case no increment in the outgoing signal can be expected due to the hybridization effect. Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton11 The third order nonlinearity is responsible as well for some small contribution to the SHG due to the non-zero ground state dipole moment of the DCM2 molecules [Kishida et al. (1994)]. In the local electric field created by the polar CA molecules on the interface Eloc (0) ,the SH signal is due to the third order susceptibility ij (2ω;ω, ω, 0). The exact expression in terms of the corresponding Green’s functions is too lengthy to be listed here [Mukamel (1995)], therefore we provide numerical calculations of the total SHG including the above correction in the next section. 5. Results and discussion In order to make a numerical comparison of the hybrid and bulk SHG the life- time of the hybrid plays a major role. Considering the bi-stability effect in the hybrid [Roslyak and Birman (2007a)] we assumed that the cuprous oxide has purity of 99.99% with the reported line-width of h̄γ1S = 0.1 meV (pico-second lifetime) [Shen et al. (1996)]. Therefore the hybrid life-time is dominated by its inorganic part h̄γ ≈ h̄γ1S. To compensate for such big line-width we also assumed that the DCM2 is presented as a thin film embedded into PS host close to the interface with the cuprous oxide. For the non-linear absorption experiment this assumption can be justified as it makes the absorption length of the hybrid equal to the narrow region around the interface, of the size of the hybrid itself. But there is a drawback in that model due to possible aggregation of the DCM2. Hence in this article we adopted the picture of disordered organic and higher purity of the inorganic crystal. This will bring the line-width and the coupling parameter to the same order. For pure cuprous oxide crystal the life-time of the quadrupole 1S exciton is reported to be τ1S = 1.7 . . . 3.0 ns (h̄γ = 1 . . . 0.5 µeV ) [Dasbach (2004), Frohlich et al. (2005), Elliott (1961)]. Such crystals and thin films are widely used in searching for BEC of excitons. In this case the life-time of the 1S quadrupole exciton is mainly determined by the ortho-para exciton conversion. The life-time of the organic part of the hybrid is determined by the time the excited DCM2 molecule reaches an equilibrium with the bath of polar CA molecules. The life-time for the given concentration of the CA is reported to be 3.3 ns [Madigan and Bulovic (2003), Madigan and Bulovic (2004)]. Because these processes are of the same order, the effective life-time of the hybrid is a non-trivial combination of the effects described above and will be reported elsewhere. Here we assume the simplest case of non-coherent life-time of the hybrid h̄γ = 0.29 µeV [Roslyak and Birman (2007b)]. The intensity of the second-harmonic is proportional to ∣χ(2)kx (See for example [Haueisen and Mahr (1973)]). Therefore an important measurable quantity is a relative value of nonlinear susceptibility ∣χ(2)kx ∣ presented in Fig.2. The SHG signal is split according to the response from the lower and upper branch of the hybrid. Asymmetry between this two branches is a result of quantum effects and not present in the classical anharmonic oscillator picture. We also included the Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton12 −3 −2 −1 0 1 2 3 (2ω − ω1S) /2γ −1 −0.5 0 0.5 1 (2ω − ω1S) /2Γk Figure 2. (Color on-line) Relative value of the nonlinear susceptibility in case of bulk cuprous oxide (dotted curves) and the quadrupole-dipole hybrid (solid curves). The density of the disordered DCM2 is taken ρDCM2 = 0.005% while the CA density is ρCA = 22%. The Fig.2a represents moderate coupling Γk = h̄γ1S = 0.29 µeV and Fig.2b corresponds to strong coupling regime Γk = 3.5 µeV . In the last case the enhancement is evident and indicated by the different scales for the bare cuprous oxide (left) and hybrid (right) SHG corrections due to interface effect in the organic in our numerical simulation. For the sake of simplicity let us consider two distinct cases. First, the pump laser is perpendicular to the interface. The states up to ka = k0a are populated thermally. No hybridization occurs and it is equivalent to the bulk case SHG (See Fig.2 dotted curve). The maximum power generated by the second-harmonic is proportional to the square of the following expression: ij,max (2h̄ω = h̄ω1S) µ2Pµ1S,2P h̄ω − h̄ω2P µ1S,kkx h̄γ1S The small relative value of the SHG is due to the narrowness of the cuprous oxide quantum well. Second, the pump laser incidence angle is reduced to acquire the wave vector Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton13 k0a ≪ ka. The maximum power generated by the second-harmonic is proportional to the square of the following expression: ij,max (2h̄ω = h̄ω1S ± Γk) µ2Pµ1S,2P h̄ω − h̄ω2P αkh̄γ Here the incidence angle dependent coefficient αk = 5 (See Fig.2a) for ka = 0.13 (Γk = h̄γ = 0.29 µeV ) and αk = 2 for maximum value of the coupling Γk = 3.5 µeV at ka = 1.57 (See Fig.2b). Finally, comparing the last expression (14) to the bulk cuprous oxide (13) the second order response of the hybrid is amplified by the factor: h̄γ1S Therefore the amplification can be adjusted by manipulating the organic composition (DCM2 and CA densities) or changing the pump laser incidence angle. Finally, we would like to note that there is another merit in using the hybrid structure for the SHG. Namely the fact that the optical pumping can be replaced by an electrical pumping. For this the hybrid sample has to be placed between Alq3 and a-NPD [Madigan and Bulovic (2004)] semiconductor plates. The bond structure and offset of these materials provide electrons and holes to form the hybrid exciton on the interface. Although, in this case one can expect the SHG only from the lower branch of the hybrid as the excitons are accumulated in the minimum of the hybrid dispersion [Roslyak and Birman (2007b)]. 6. Conclusion In this paper we addressed possibility to enhance SHG signal χ(2) generic to a cuprous oxide bulk crystal as the lowest excitation in this material has quadrupole origin. To demonstrate the concept we proposed to consider a pump-prob experiment performed on cuprous oxide sandwiched between the organic composite. An intense pump signal excites one part of the organic known as DCM2 (FE). Non-resonant (Förster) energy transfer in the organic layer (”solid state solvation” effect) provides dynamical red shift of the FE. When the FE energy is close enough to the quadrupole allowed 1S exciton in the adjacent cuprous oxide the quadrupole-dipole hybridization occurs due to the FE induced gradient of the electric field penetrating into the inorganic layer. The prob signal is designed to reveal the SHG signal. The resonant enhancement of the χ(2) occurs for the hybrid exciton shares properties of both quadrupole WE (long radiative lifetime) and organic FE (big oscillator strength). I’t quadrupole part allows χ(2) to be non-vanishing. While it’s FE part provides the enhancement of the SHG signal compared to the bare cuprous oxide crystal due to more efficient absorption of the pump signal by means of large oscillator strength of the hybrid. However, as we demonstrated in the classical coupled oscillator model framework,the enhancement is determined not only by the big ratio of the corresponding Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton14 organic/inorganic oscillator strengths but somehow quenched by the small coupling parameter and low DCM2 density. By varying those parameters one the hybrid SHG signal may be enhanced by orders of magnitude compared with the generic (cuprous oxide) one. To reveal the enhancement dependence on such an important parameter of the hybrid as the hybridization time τh we proposed more sophisticated quantum theory. It suggests that there is substantial difference in the hybrid SHG signal provided one probes the system before or after the hybridization occurred. In the first case the SH is generated at ω2P/2 frequency. Hence it is vastly suppressed by the short life time of the dipole-allowed 2P WE in cuprous oxide. Nevertheless, if one probes the system after the hybridization had happened, the SH is generated at ω1S/2 and is heightened by the big oscillator strength of the hybrid and it’s small damping coefficient. 7. Acknowledgments We would like to thank Ms. Upali Aparajita for helpful discussion and comments. The project was supported in part by PCS-CUNY. References [Shen et al. (1996)] M. Shen, S. Koyama, M. Saito, T. Goto, and N. Kuroda, Phys. Rev. B 53, 13477 (1996). [Atanasov et al. (1994)] R. Atanasov, F. Bassani, and V. Agranovich, Phys. Rev. B 50, 7809 (1994). [Agranovich et al. (1998)] V. Agranovich, D. Basko, G. L. Rocca, and F. Bassani, J. Phys.: Condens. Matter 13, 9369 (1998). [Engelmann et al. (1998)] A. Engelmann, V. I. Yudson, and P. Reineker, Phys. Rev. B 57, 1784 (1998). [Gao et al. (2004)] Y. Gao, J. Birman, N. Huong, and M. Potasek, Journal of applied physics 96, 1 (2004). [Huong and Birman (2000)] N. Q. Huong and J. L. Birman, Phys. Rev. B 61, 13131 (2000). [Huong and Birman (2003)] N. Q. Huong and J. L. Birman, Phys. Rev. B 67, 075313 (2003). [Roslyak and Birman (2007a)] O. Roslyak and J. Birman, Solid State Communications 143, 487 (2007a). [Dasbach (2004)] G. Dasbach, Phys. Rev. B 70, 121202 (2004). [Bulovic et al. (1999)] V. Bulovic, R. Deshpande, and S. Forrest, Chem. Phys. Lett. 308, 317 (1999). [Roslyak and Birman (2007b)] O. Roslyak and J. L. Birman, Physical Review B (Condensed Matter and Materials Physics) 75, 245309 (pages 11) (2007b). [Mukamel (1995)] S. Mukamel, Principles of nonlinear optical spectroscopy (Oxford Press, New York, 1995). [Bloembergen (1965)] N. Bloembergen, Nonlinear optics (Benjamin, New York, 1965). [Moskalenko and Liberman (2002)] S. Moskalenko and M. Liberman, Phys. Rev. B 65, 064303 (2002). [Madigan and Bulovic (2003)] C. Madigan and V. Bulovic, Physical review letters 91, 247403 (2003). [Artoni et al. (2002)] M. Artoni, I. Carusotto, G. La Rocca, and F. Bassani, J. Opt. B.: Quantum Semiclass. Opt. 4, S345 (2002). [Elliott (1961)] R. J. Elliott, Phys. Rev. 124, 340 (1961). [Madigan and Bulovic (2004)] C. Madigan and V. Bulovic (Bringing Materials Research Together, 2004). Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton15 [Kishida et al. (1994)] H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto, S. Wada, T. Lay, and H. Tashiro, Phys. Rev. B 50, 7786 (1994). [Frohlich et al. (2005)] D. Frohlich, G. Dasbach, G. B. Hogersthal, M. Bayer, R. Kliebera, D. Sutera, and H. Stolzb, Solid State Communications 134, 139 (2005). [Haueisen and Mahr (1973)] D. Haueisen and H. Mahr, Phys. Rev. B 8, 734 (1973). Appendix A. Explicit expression for the quadrupole-dipole coupling is given below: (ε+ ε̃)Lw ke−kz (A.1) Here aFB, a B are the Bohr radius of the FE and WE exciton; ε̃ and ε are the corresponding dielectric constants, z′ is the distance to the DCM2 layer, Lw is the quantum well width. The quadrupole transition matrix element Qxz may be estimated from the corresponding oscillator strength per unit cell through the following identity [Moskalenko and Liberman (2002)] and depends on polarization of the pumping laser field: fxz,k0 = 4πmEg 3e2h̄2 (z · k0,x ·Qx,z)2 (A.2) fxz,k0||[1,1,0] = 3.9× 10−9 fxz,k0||[1,1,1] = 3.9× 10−9 Here the energy gap of cuprous oxide is denoted as Eg = 2.173 eV ; k0 = 2.62×105 cm−1 is the resonant wave vector; a is the unit cell size; the unit vector in the pumping field polarization is z. Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton16 Appendix B. The non-zero commutator relations for the organic and inorganic parts of the hybrid yield [Mukamel (1995)]: 1S, B1S = −1 +B†2PB2P + b†b; 1S, B2P 1SB2P ; 2P , B1S 2PB1S (B.1) 2P , B2P = −1 +B†1SB1S + b†b; b†, B1S = b†B1S; b†, B2P = b†B2P b†, b = −1 +B†2PB2P +B 1SB1S; 1S, b 2P , b In the TDHF approximate factorization for the averages, the corresponding Heisenberg equations up to the second order in the creation and annihilation operators are: = h̄ω1SB 1S + Γkb † − µFEiB†1Sb+ µ1S,kEi 1− B†2PB2P − b†b µ2PEiB 1SB2P + µ1S,2PEjB = −h̄ω1SB1S − Γkb+ µFE⋆i b†B1S − µ1S,kE⋆i 1− B†2PB2P − b†b 2PB1S − µ1S,2PE⋆jB2P = h̄ω2PB 2P − µFEiB 2P b− µ1S,kEiB 2PB1S + µ2PEi 1− B†1SB1S − b†b +µ1S,2PEjB = −h̄ω2PB2P + µFE⋆i b†B2P + µ1S,kE⋆i B 1SB2P − µ2PE⋆i 1−B†1SB1S − b†b −µ1S,2PE⋆jB1S = EF b† + ΓkB 1S + µ 1−B†1SB1S −B 2PB2P − µ1S,kEib†B1S − µ2PEib†B2P = −EF b− ΓkB1S − µFE⋆i 1−B†1SB1S −B 2PB2P + µ1S,kE 1Sb+ µ2PE Here we omitted the average brackets for the shorter notation. In the exact resonance between FE and WE excitons h̄ω1S = h̄ωF the linear approximation is straightforward. The creation operators are proportional to ∝ eiωt and the system above is reduced to: 1S,0 = (h̄ω1S − ih̄γ)B 1S,0 + Γkb 0 + µ1S,kEi − h̄ωB1S,0 = − (h̄ω1S + ih̄γ)B1S,0 − Γkb0 − µ1S,kE⋆i 2P,0 = h̄ω2PB 2P,0 + µ2PEi − h̄ωB2P,0 = −h̄ω2PB2P,0 − µ2PE⋆i 0 = (h̄ω1S − ih̄γ) b 0 + ΓkB 1S,0 + µ FEi − h̄ωb0 = − (h̄ω1S + ih̄γ) b0 − ΓkB1S,0 − µFE⋆i Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton17 The system above has a solution: 2P,0 = µ2PEi h̄ω − h̄ω2P (B.2) 1S,0 = µ1SEi (h̄ω − h̄ω1S + ih̄γ) + µFΓkEi (h̄ω − h̄ω1S + ih̄γ)2 − Γ2k µFEi (h̄ω − h̄ω1S + ih̄γ) + µ1SΓkEi (h̄ω − h̄ω1S + ih̄γ)2 − Γ2k The SHG is due to response to induced polarization and is proportional to ∝ ei2ωt: 2h̄ωB 1S,1 = (h̄ω1S − ih̄γ)B 1S,1 + Γkb 1 + µ1S,2PEjB 2h̄ωB1S,1 = (h̄ω1S + ih̄γ)B1S,1 + Γkb1 + µ1S,2PE 2h̄ωB 2P,1 = h̄ω2PB 2P,1 + µ1S,2PEjB 2h̄ωB2P,1 = h̄ω2PB2P,1 + µ1S,2PE 2h̄ωb 1 = (h̄ω1S − ih̄γ) b 1 + ΓkB 2h̄ωb1 = (h̄ω1S + ih̄γ) b1 + ΓkB1S,1 The system has a solution: 2P,1 = µ1S,2PEjB 2h̄ω − h̄ω2P (B.3) 1S,1 = µ1S,2PEj (2h̄ω − h̄ω1S + ih̄γ)B†2P,0 (2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k µ1S,2PEjΓkB (2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k This solutions are implemented in the main text to calculate the linear and nonlinear responses of the hybrid. Introduction Proposed experimental set-up for the SHG Anharmonic coupled oscillators model Quantum theory of SHG due to the QDH Results and discussion Conclusion Acknowledgments

We report calculated substantial enhancement of the second harmonic generation (SHG) in cuprous oxide crystals resonantly hybridized with an appropriate organic material (DCM2:CA:PS 'solid-state solvent'). The quadrupole origin of the inorganic part of the quadrupole-dipole hybrid provides inversion symmetry breaking and the organic part contributes to the oscillator strength of the hybrid. We show that the enhancement of the SHG, compared to bulk cuprous oxide crystal, is proportional to the ratio of the DCM2 dipole moment and the effective dipole moment of the quadrupole transitions in the cuprous oxide. It is also inversely proportional to the line-width of the hybrid and bulk excitons. The enhancement may be regulated by adjusting the organic blend (mutual concentration of the DCM2 and CA part of the solvent) and pumping conditions(varying the angle of incidence in case of optical pumping or populating the minimum of the lower branch of the hybrid in case of electrical pumping).

Introduction Considerable attention has been paid to the relatively strong optical second-harmonic generation (SHG) in thin films (D4h symmetry) and bulk (Oh symmetry) of cuprous oxide crystals. Which was first addressed in the pioneering work of Shen [Shen et al. (1996)]. This effect is attributed to the electric-quadrupole h̄ω1S = 2.05 eV exciton effect. The quadrupole exciton has very small oscillator strength but it possess rather narrow line-width h̄γ1S. So the effect is well pronounced when the exciting laser energy is close to one h̄ω1S − h̄ω ≪ h̄γ1S or two photon resonance h̄ω1S − 2h̄ω ≪ h̄γ1S. In the dipole approximation this effect disappears [Atanasov et al. (1994)]. We propose to amplify the SHG characteristic of the 1S quadrupole Wannier exciton (WE) in cuprous oxide by making a hybrid with an organic Frenkel exciton (FE) (See next section for more details). The idea of resonant enhancement of some non- linear properties generic to semiconductor dipole-allowed Wannier-Mott (WE) excitons was presented in pioneering work of [Agranovich et al. (1998)] for the layered organic- inorganic heterostructures. It was also developed for quantum wires and dots embedded into organic shell [Engelmann et al. (1998)],[Gao et al. (2004)] or attached to dendrimer structure [Huong and Birman (2000)],[Huong and Birman (2003)]. In our previous work [Roslyak and Birman (2007a)] we demonstrated considerable enhancement of another non-linear effect in cuprous oxide, photo-thermal bi-stability [Dasbach (2004)]. We demonstrated a considerable enhancement in the hysteresis-like region size (from µeV for bulk cuprous oxide to meV for the hybrid). The enhancement was attributed to the large oscillator strength of the hybrid exciton inherited from the organic part and still rather narrow line-width of the same order as the coupling. Analogous enhancement can be expected for the SHG, which is the subject of this paper. In Section 2 we propose a pump-prob experiment to reveal the SHG enhancement due to the resonant dynamical hybridization and briefly discuss relevant quadrupole hybrid exciton properties. In the next Section 3 we address the question how this resonant‡ enhancement depends on such parameters of the system as oscillator strengths and damping of the FE and WE constituting the hybrid. Using a classical model of nonlinear coupled oscillators, we demonstrate that while the big ratio of the hybrid oscillator strength suggests many orders of the enhancement magnitude it is actually somehow reduced by the rather small coupling parameter and density of the FE. Because the FE is dynamically brought into resonance with the WE there is an important hybridization time τh parameter. Hence, in the Section 4, we develop more sophisticated quantum mechanical model to address the dynamics of the hybrid SHG. Namely we show that the signal enhancement drastically depends on the either the one probes the system before or after the hybridization occurred. ‡ The resonance occurs between the FE and WE Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton3 2. Proposed experimental set-up for the SHG In this work we adopt the concept of a layered organic-inorganic heterostructure. The inorganic component of the hybrid is a thin layer of Cu2O (quantum well, latter in the text referred to as QW) grown upon a film of the organic composite (See Fig.1). Due to the small radius of both the WE and FE exciton part of the hybrid one can neglect the effect of confinement. In this case one can not tune the two types of excitons in resonance by adjusting the confinement (Lw > a B ≈ to the cuprous oxide unit cell a = 4.6 Å). The QW confinement just assures the WE propagate along the interface and is the subjected to the electric field gradient of the FE propagating along the adjacent chain of the DCM2 molecules. x, kx, Eg 143 meV 2.17 eV ≫ 2.17 eV Cu2O PS :DCM2: CA Lw ≈ 4.6 Å probe Figure 1. Schematic representation and the energy offset of a possible experimental set-up to observe the enhanced SHG by the quadrupole-dipole exciton. Here the inorganic Cu2O quantum well provides the 1S quadrupole WE. The DCM2 part of the organic ’solid state solute’ provides dipole allowed FE (set of small arrows); the PS host prevents wave function overlapping between organic and inorganic excitons; CA under proper concentration allows tuning of the excitons into the resonance. To provide resonance between WE in cuprous oxide and FE in the organic, we pro- posed utilization of ’solid state solvation’ (SSS) of the DCM2 § molecules in transpar- ent polystyrene (PS) host doped with camorphic anhydride (CA)[Bulovic et al. (1999)]. The SSS is a type of solvatochromism manifesting itself as some change in the spec- tral position of the absorption/luminescence band due to change in the polarity of the medium. The Förster dipole-dipole non-resonant interaction between DCM2 and CA modifies the energy structure of the involved molecules. § [2-methyl-6-2-(2,3,6,7-tetrahydro-1H, 5H - benzo[i,j] - quinolizin - 9 - yl) - ethenyl] - 4H - pyran - 4 - ylidene] propane dinitritle. Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton4 During the ’slow’ phase (τs ≈ 3.3ns) the energy of the FE ‖ experiences a red shift linear with the CA concentration due to non-resonant dipole-dipole interaction with the CA molecules. Not that our model capitalizes on the fact that DCM2 molecules form a 2D layer rather than been diluted in the PS:CA solvent which is the case of currently manufactured optical light emitting devices (OLED). This allows us to neglect rather complicated problem of the inhomogeneous broadening of the FE energy by utilizing a mean field approximation ¶. For the mean field approximation the red spectral shift of the FE energy in resonance with the quadrupole WE can be accomplished with ρCA ≈ 22% CA concentration. To avoid complicated problems of time dependent hybridization and stay within the analytical model framework, we assume that the FE and WE are in exact resonance once the DCM2 energy is in close proximity to the WE energy i.e. h̄ωDCM2 − h̄ω1S ≤ Γk. We introduced the quadrupole-dipole coupling parameter Γk ≤ 4 µeV [Roslyak and Birman (2007b)] (See also Appendix (A.1)). This resonant coupling gives rise to the upper and lower branches of the quadrupole-dipole hybrid (QDH) dispersion +: h̄ωu,l = h̄ω1S ± Γk. To populate both of the branches one needs a second pumping photon tuned into resonance with the 1S transition. The radiation field interacts through both dipole and quadrupole part of the hybrid. The dipole interaction can be utilized to produce linear response signal due to the pumping [Roslyak and Birman (2007a)]. By using the non-linear response to the prob signal, the second harmonic can be generated through the quadrupole part of the hybrid. Different SHG regimes can be achieved by changing the timing between pumping and prob signals (See section 4 for more details). According to the selection rules for the quadrupole-dipole hybrid the pumping signal, running along the organic-inorganic interface of the heterostructure, induces the linear polarization in the z direction [Roslyak and Birman (2007b)]. The prob signal induces the second order non-linear response in the cuprous oxide. Which is perpendicular to the interface, and defined by the second order polarization along the x direction (See Fig.1). The net polarization is given by a second rank tensor through the following expression: P (1)z + P l=ĵ×x̂ i,zEi + iχ l,i,j,xkxEiEj (1) Here Ei, Ej are the electric field of the pumping and prob lasers correspondingly. The x component of the prob signal wave vector is taken to be close to zero to avoid possible interference in momentum conservation. For the sake of simplicity we are going to omit, x and l indexes of the tensor keeping in mind that the wave vector of the pump signal ‖ in our case we define the FE as DCM2 excitation ¶ Indeed, in our simplified model the DCM2 molecules are not randomly situated but rather form a uniform (homogeneous) thin layer near the interface. Also the experimentally observed FE energy relaxation shows no significant energy fluctuations. This experiments are performed at MIT by the Dr. Bulovic. Although this results are not officially published yet, but reported in the MIT proceedings. + See the eigenvalues of the linearized system (4) or the Hamiltonian (8) Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton5 has only x component and the SH signal is perpendicular to it and the prob signal polarization: iχ l,i,j,xkx = χ i,j . In this paper we develop both classical and quantum mechanical models, which can be used to find a specific form of the hybrid second order nonlinear susceptibility. In section 3 we demonstrate that the second order non linearity (generic to the cuprous oxide and introduced through a small parameter λ) is enhanced due to the resonant quadrupole-dipole hybridization with the organic (See (5)). In section 4 we develop quantum theory of the enhanced SHG. It allows investigation of different regimes of the process defined by the time ordering between the prob pulse and the time when the FE and WE energies are close enough to form the hybrid. We generalize the concept of the double-sided Feynman diagrams [Mukamel (1995)] to include non radiative processes of the energy exchange between DCM2 and CA as well as resonant QDH between DCM2 and cuprous oxide. 3. Anharmonic coupled oscillators model As a first step, we will use the simplest classical model neglecting the non-local effects of the linear χ i,z and non-linear susceptibility χ l,i,j,x to describe the hybrid SHG. Namely, we adopt an extension of the anharmonic oscillator model [Bloembergen (1965), Mukamel (1995)] generalized for the case of resonant coupling between two distinct sets of oscillators. This simplified picture only covers the case when the pumping field is polarized along ẑ ‖ [001] axis (Ei = Ez) and we prob the hybrid system (ω1S = ωF ) with a signal perpendicular to the interface and polarized along ŷ ‖ [010] direction (Ej = Ey). We consider the WE in cuprous oxide as an assembly of the oscillators with the oscillator strength per unit cell given by fxz,k ∝ kx (See for example reference [Moskalenko and Liberman (2002)]). The second set of the oscillators with the oscillator strength given by fF corresponds to the FE in the organic. Treating the wave vector k as just another parameter∗, the polarization PW , P F due to WE and FE can be written in terms of the effective electron-hole displacements X, Y as: aWB S fxz,keX (2) P F = ρDCM2N fFeY (3) Here S is the area of the interface and aWB , a B are the WE and FE Bohr radius, e is the electron charge. The surface density of the WE and FE excitons are Nfxz,k/ (aBS) and ρDCM2Nf F/ (aBS) correspondingly and N is the total number of the oscillators. Here we also took into account the low density (ρDCM2 = 0.05%) of the DCM2 molecules in the organic to avoid the aggregation effect [Madigan and Bulovic (2003)]. ∗ In the text we are going to omit index k unless we put an emphasis on it Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton6 In the time frame of the hybridization τs−τh < t < τs, the WE and FE energies are at perfect resonance. Hence, the system of equations governing the oscillators dynamics can be written in the form: Ẍ + ω21SX + γẊ − 2ω1SΓk Y − ω21SλX2 = 0 Ÿ + ω21SY + γẎ − 2ω1SΓk The nonlinear factor ω21Sλ appears due to the prob signal. It is defined such that λ has dimensions of reciprocal length and is considered to be small in a sense that it is much less than the reciprocal of the maximum displacement of FE (Y ) and WE (X) oscillator. The exact value of λ can be obtained either from an experiment or from the microscopic quantum theory (See next section for more details). The terms proportional to γ describe the QDH damping. The terms proportional to 2ω1SΓk/h̄ describe the quadrupole-dipole coupling. Hence, the eigenvalues of the linearized system of equations (4) give both branches of the QDH. The system is driven dominantly by the light-dipole interaction in the organic and the quadrupole-light interaction is neglected (m is the electron mass). Using standard perturbation theory with respect to the small parameter λ in zero order (neglecting the quadratic term) and combining equations (1,2,4) one has the linear response of the hybrid and bulk cuprous oxide given by the following expressions: Hy (ω) = ρDCM2N fFe2/m (ω21S − ω2 + iωγ) (ω21S − ω2 + iωγ) 2 − (2ω1SΓk/h̄)2 Cu2O (ω) = aWB S fxz,ke ω21S − ω2 + iγ Including the nonlinear term as a source for the SHG to first order in the perturbation parameter, there is a displacement at 2ω. The SHG response is given by a solution of the following system: Ẍ + ω21SX + γẊ − 2ω1SΓk Y − ω21SλX2λ=0 = 0 Ÿ + ω21SY + γẎ − 2ω1SΓk X = 0 Using the definitions (1) and (2) one gets the following non-linear second order response function for the hybrid and bulk cuprous oxide correspondingly: Hy (2ω;ω, ω) = ρDCM2N fFe3/m2ω21Sλ (2ω1SΓk/h̄) ω21S − (2ω) + i2ωγ − (2ω1SΓk/h̄)2 (ω21S − ω2 + iωγ) (ω21S − ω2 + iωγ) 2 − (2ω1SΓk/h̄)2 Cu2O (2ω;ω, ω) = aWB S fxz,ke 3/m2ω21Sλ ω21S − (2ω) + i2γ (ω21S − ω2 + iγ) Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton7 Straightforward comparison of the expressions above evinces the resonant rise of the second order nonlinearity owing to hybridization. There are several competing factors involved. The enhancement by means of big oscillator strength ratio fF/fxz,k is reduced by rather small coupling parameter ΓK and small DCM2 density ρDCM2 (see more numerical details in Section 5). 4. Quantum theory of SHG due to the QDH Although the system of non-linear susceptibilities (5) in principle solves the problem of SHG due to the hybrid it does not clarify the origin of the nonlinearity λ. Also, such an important parameter as the hybridization time τh is left out of the classical description. Hence, in this section we propose a unified quantum theory of the hybrid SHG. The linear response of the hybrid is due to dipole transitions from the ground |g〉 state♯ to the FE |F 〉 in the organic and due to quadrupole transitions to the WE |1S〉 in the cuprous oxide. The non-linearities are the result of some intermediate inter-band transitions in the cuprous oxide [Mukamel (1995)]. In cuprous oxide the nearest state in energy to the quadrupole ortho-exciton h̄ω1S is the h̄ω2P dipole allowed excitonic band |2P 〉, Eg > h̄ω2P > h̄ωF > h̄ω1S. Hence it plays the main role in formation of the non-linear response and can be excited by the properly tuned prob signal. We neglect all the rest of inter-band and intra-band†† transitions. Therefore, the states above form a complete basis for the SHG problem: |g〉 , |1S〉 , |F 〉 , |2P 〉 (6) Inversion symmetry of the DCM2 is also broken by the CA induced local field and the interface effect. Therefore, unlike in classical model, the contribution from the organic to the SHG has to be consider as well. But due to the smallness of the symmetry breaking local field it contributes a little to the SHG enhancement. Using the basis above let us introduce creation operators for the FE and the 1S and the 2P WE exciton b† = |F 〉 〈g|, B†1S = |1S〉 〈g|, B 2P = |2P 〉 〈g| respectively. The commutation algebra of the operators is presented in the Appendix (B.1). The net polarization of the sample is defined as [Mukamel (1995)]: P = µi1S,k 1S +B1S + µi2P 2P +B2P b† + b 1S,2P 1SB2P +B1SB Here µi1S,k = î · ẑ kxQx,z = 3 · 10−5(kx/k0,x) D is an effective dipole moment [Moskalenko and Liberman (2002), Roslyak and Birman (2007b)] due to the quadrupole transitions associated with the oscillator strength; k0 is the resonant wave vector for bulk ♯ when no excitations are present in the system ††due to small radius of the quadrupole WE Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton8 cuprous oxide( Appendix (A.2)). The dipole moment of the transitions from |1S〉 to 〈2P | is defined by [Artoni et al. (2002), Elliott (1961)]: 1S,2P Ne2h̄2f2P SaWB 2m ĵ × x̂ = 6 · 10−3 D2 Finally, the DCM2 dipole moment of the transition from |g〉 to 〈F | per unit area of the interface is given by [Madigan and Bulovic (2004)]: ρDCM2Ne 2h̄2fF SaFB2m ∗h̄ω1S = 0.2 D2 Using equation (7) and the rotating wave approximation for the resonant wave vector k, the hybrid Hamiltonian can be written as: H = h̄ωF b †b+ h̄ω1SB 1SB1S + E2PB 2PB2P + Γk 1Sb+B1Sb + (8) i + bEi + µi1S,k i +B1SEi + µi2P i +B2PEi 1S,2P 1SB2PE j +B1SB The linear response from both branches of the hybrid may be observed by pumping the hybrid with two signals Ei||ẑ ∝ eiωt. The first photon h̄ω = EDCM2 excites DCM2 molecules. During the time period τs − τh the system relaxes to the FE exciton energy close to h̄ω1S thus providing resonance between WE and FE. Then the second pumping photon h̄ω = h̄ω1S enters and excites quadrupole WE so that both QDH branches are populated. The QDH exciton lives for τh nano-seconds and then both branches of the hybrid relaxes to the ground state emitting photons of the energy h̄ω1S ± Γk. Generalizing conventional double-sided Feynman diagrams [Mukamel (1995)] to include the non-radiative processes, the linear response from the QDH can be represented by the following diagram: |g〉 〈g| 〈1S ⊕ F 〉 h̄ω1S−Γk dd d$ d$ h̄ω1S+Γk 8x8x8x 8x8x8x h̄ω1S 〈DCM2| // h̄ω1S EDCM2 τs−τh i (ω, k) = µ 1S,0 +B1S,0 + µiF 0 + b0 (µiF ) (h̄ω − h̄ω1S + ih̄γ) + µiFµi1SΓk (h̄ω − h̄ω1S + ih̄γ)2 − Γ2k + c.c. (10) On the diagram the wavy lines represent the incoming and outgoing photons; the straight lines stand for the non-radiative transitions. The diagram shows energy exchange between photon-exciton and exciton-exciton as well as the time separation between two pumping signals. Time increases from bottom to the top of the diagram as for the Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton9 conventional Feynman diagram. The hybrid life time is denoted as τh = 1/γ and the hybridization between FE and WE is denoted as ⊕. In the derivation of the linear response χ i (ω, k) we used equation (7) along with solutions of the Heisenberg equations of motion presented in the Appendix II (B.2). Formally the linear response can be written in terms of the hybrid Green’s functions as: i (ω, k) = a,b={g,1S,F} µiabµ baIab (ω) I1S,g = IF,g = h̄ω − h̄ω1S + ih̄γ (h̄ω − h̄ω1S + ih̄γ)2 − Γ2k I1S,F = (h̄ω − h̄ω1S + ih̄γ)2 − Γ2k Iab = I Here the dipole matrix elements in the corresponding basis (6) are given by: 0 µ1S µF 0 µ1S 0 µ1SµF 0 µFµ1S 0 0 0 0 0 0 Note that we neglected the non-resonant term associated with ground state dipole moment of the organic µg. The SHG is due to second order response Ej⊥Ei||z and given by the last term in the equation (7) and the solutions of the equations of motion (B.2,B.3). The first type of the SHG is formed when the branches of the hybrid interacts with the |2P 〉 level excited by the prob signal. Using all the diagram conventions we adopted above, the diagram for this non linear process is given below: |g〉 〈g| |2P 〉 oo µ1S,2P // 〈1S ⊕ F 〉 h̄ω1S−Γkd$ h̄ω1S+Γk8x8x 888x8x8x h̄ω2P h̄ω1S 〈DCM2| // h̄ω1S τ2P>τs−τh 〈g| 〈g| EDCM2 τs−τh ij (2ω;ω, ω) = µ1S,2P 1S,0B2P,1 + c.c. = (11) µi2Pµ 1S,2P (µ 1S (h̄ω − h̄ω1S + ih̄γ) + µiFΓk) (2h̄ω − h̄ω2P ) (h̄ω − h̄ω1S + ih̄γ)2 − Γ2k ) + c.c. (12) Here the prob signal comes after the hybrid is formed: τ2P > τs − τh. Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton10 Another second order non linear response can be formed if the prob signal is coming before the hybridization τ2P < τs − τh. It can be represented by the following diagram: |g〉 |g〉 〈g| |g〉 h̄ω1S///o/o/o 〈1S ⊕ F 〉 1S±Γk 1S∓Γk h̄ω2P µ1S,2P |2P 〉 〈DCM2| // |CA〉 τs−τh τ2P<τs−τh h̄ω2P EDCM2 ij (2ω;ω, ω) = µ1S,2P 1S,1B2P,0 + c.c. µi2Pµ 1S,2P (µ 1S (2h̄ω − h̄ω1S + ih̄γ) + µiFΓk) (h̄ω − h̄ω2P ) (2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k The Green’s function representation of the SHG due to the second order response is given by the following expression: ij (2ω;ω, ω) = µ 1S,2P a={g,1S,F,2P} µia,1Sµ 2P,a × × [Ia,1S (ω) Ia,2P (2ω) + I1S,a (2ω) I2P,a (ω)] I2P,g = h̄ω − h̄ω2P The dipole matrix elements on the basis (6) are given by: 0 µ1S µF µ2P µ1S 0 µ1SµF µ1S,2P µFµ1S 0 0 µ2P µ1S,2P 0 0 According to the last term in the equation (7), the signal at 2h̄ω = h̄ω1S ± Γk may generate the signal at h̄ω = h̄ω1S ± Γk: ij (ω; 2ω,−ω) = 1S,2P (2h̄ω − h̄ω1S + ih̄γ) (h̄ω − h̄ω2P )2 (2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k ) + c.c. This type of signal has been experimentally detected [Shen et al. (1996)] in bulk cuprous oxide (Γk = 0) when the pumping signal was tuned to the wave length between 12285 Å and 12195 Å. A strong SH signal was detected at 6096 Å which has to be attributed not only to the narrow line-width of the quadrupole exciton but to the fact that µ1S,2P ≫ µ1S as well. From the last expression it follows that in this case no increment in the outgoing signal can be expected due to the hybridization effect. Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton11 The third order nonlinearity is responsible as well for some small contribution to the SHG due to the non-zero ground state dipole moment of the DCM2 molecules [Kishida et al. (1994)]. In the local electric field created by the polar CA molecules on the interface Eloc (0) ,the SH signal is due to the third order susceptibility ij (2ω;ω, ω, 0). The exact expression in terms of the corresponding Green’s functions is too lengthy to be listed here [Mukamel (1995)], therefore we provide numerical calculations of the total SHG including the above correction in the next section. 5. Results and discussion In order to make a numerical comparison of the hybrid and bulk SHG the life- time of the hybrid plays a major role. Considering the bi-stability effect in the hybrid [Roslyak and Birman (2007a)] we assumed that the cuprous oxide has purity of 99.99% with the reported line-width of h̄γ1S = 0.1 meV (pico-second lifetime) [Shen et al. (1996)]. Therefore the hybrid life-time is dominated by its inorganic part h̄γ ≈ h̄γ1S. To compensate for such big line-width we also assumed that the DCM2 is presented as a thin film embedded into PS host close to the interface with the cuprous oxide. For the non-linear absorption experiment this assumption can be justified as it makes the absorption length of the hybrid equal to the narrow region around the interface, of the size of the hybrid itself. But there is a drawback in that model due to possible aggregation of the DCM2. Hence in this article we adopted the picture of disordered organic and higher purity of the inorganic crystal. This will bring the line-width and the coupling parameter to the same order. For pure cuprous oxide crystal the life-time of the quadrupole 1S exciton is reported to be τ1S = 1.7 . . . 3.0 ns (h̄γ = 1 . . . 0.5 µeV ) [Dasbach (2004), Frohlich et al. (2005), Elliott (1961)]. Such crystals and thin films are widely used in searching for BEC of excitons. In this case the life-time of the 1S quadrupole exciton is mainly determined by the ortho-para exciton conversion. The life-time of the organic part of the hybrid is determined by the time the excited DCM2 molecule reaches an equilibrium with the bath of polar CA molecules. The life-time for the given concentration of the CA is reported to be 3.3 ns [Madigan and Bulovic (2003), Madigan and Bulovic (2004)]. Because these processes are of the same order, the effective life-time of the hybrid is a non-trivial combination of the effects described above and will be reported elsewhere. Here we assume the simplest case of non-coherent life-time of the hybrid h̄γ = 0.29 µeV [Roslyak and Birman (2007b)]. The intensity of the second-harmonic is proportional to ∣χ(2)kx (See for example [Haueisen and Mahr (1973)]). Therefore an important measurable quantity is a relative value of nonlinear susceptibility ∣χ(2)kx ∣ presented in Fig.2. The SHG signal is split according to the response from the lower and upper branch of the hybrid. Asymmetry between this two branches is a result of quantum effects and not present in the classical anharmonic oscillator picture. We also included the Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton12 −3 −2 −1 0 1 2 3 (2ω − ω1S) /2γ −1 −0.5 0 0.5 1 (2ω − ω1S) /2Γk Figure 2. (Color on-line) Relative value of the nonlinear susceptibility in case of bulk cuprous oxide (dotted curves) and the quadrupole-dipole hybrid (solid curves). The density of the disordered DCM2 is taken ρDCM2 = 0.005% while the CA density is ρCA = 22%. The Fig.2a represents moderate coupling Γk = h̄γ1S = 0.29 µeV and Fig.2b corresponds to strong coupling regime Γk = 3.5 µeV . In the last case the enhancement is evident and indicated by the different scales for the bare cuprous oxide (left) and hybrid (right) SHG corrections due to interface effect in the organic in our numerical simulation. For the sake of simplicity let us consider two distinct cases. First, the pump laser is perpendicular to the interface. The states up to ka = k0a are populated thermally. No hybridization occurs and it is equivalent to the bulk case SHG (See Fig.2 dotted curve). The maximum power generated by the second-harmonic is proportional to the square of the following expression: ij,max (2h̄ω = h̄ω1S) µ2Pµ1S,2P h̄ω − h̄ω2P µ1S,kkx h̄γ1S The small relative value of the SHG is due to the narrowness of the cuprous oxide quantum well. Second, the pump laser incidence angle is reduced to acquire the wave vector Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton13 k0a ≪ ka. The maximum power generated by the second-harmonic is proportional to the square of the following expression: ij,max (2h̄ω = h̄ω1S ± Γk) µ2Pµ1S,2P h̄ω − h̄ω2P αkh̄γ Here the incidence angle dependent coefficient αk = 5 (See Fig.2a) for ka = 0.13 (Γk = h̄γ = 0.29 µeV ) and αk = 2 for maximum value of the coupling Γk = 3.5 µeV at ka = 1.57 (See Fig.2b). Finally, comparing the last expression (14) to the bulk cuprous oxide (13) the second order response of the hybrid is amplified by the factor: h̄γ1S Therefore the amplification can be adjusted by manipulating the organic composition (DCM2 and CA densities) or changing the pump laser incidence angle. Finally, we would like to note that there is another merit in using the hybrid structure for the SHG. Namely the fact that the optical pumping can be replaced by an electrical pumping. For this the hybrid sample has to be placed between Alq3 and a-NPD [Madigan and Bulovic (2004)] semiconductor plates. The bond structure and offset of these materials provide electrons and holes to form the hybrid exciton on the interface. Although, in this case one can expect the SHG only from the lower branch of the hybrid as the excitons are accumulated in the minimum of the hybrid dispersion [Roslyak and Birman (2007b)]. 6. Conclusion In this paper we addressed possibility to enhance SHG signal χ(2) generic to a cuprous oxide bulk crystal as the lowest excitation in this material has quadrupole origin. To demonstrate the concept we proposed to consider a pump-prob experiment performed on cuprous oxide sandwiched between the organic composite. An intense pump signal excites one part of the organic known as DCM2 (FE). Non-resonant (Förster) energy transfer in the organic layer (”solid state solvation” effect) provides dynamical red shift of the FE. When the FE energy is close enough to the quadrupole allowed 1S exciton in the adjacent cuprous oxide the quadrupole-dipole hybridization occurs due to the FE induced gradient of the electric field penetrating into the inorganic layer. The prob signal is designed to reveal the SHG signal. The resonant enhancement of the χ(2) occurs for the hybrid exciton shares properties of both quadrupole WE (long radiative lifetime) and organic FE (big oscillator strength). I’t quadrupole part allows χ(2) to be non-vanishing. While it’s FE part provides the enhancement of the SHG signal compared to the bare cuprous oxide crystal due to more efficient absorption of the pump signal by means of large oscillator strength of the hybrid. However, as we demonstrated in the classical coupled oscillator model framework,the enhancement is determined not only by the big ratio of the corresponding Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton14 organic/inorganic oscillator strengths but somehow quenched by the small coupling parameter and low DCM2 density. By varying those parameters one the hybrid SHG signal may be enhanced by orders of magnitude compared with the generic (cuprous oxide) one. To reveal the enhancement dependence on such an important parameter of the hybrid as the hybridization time τh we proposed more sophisticated quantum theory. It suggests that there is substantial difference in the hybrid SHG signal provided one probes the system before or after the hybridization occurred. In the first case the SH is generated at ω2P/2 frequency. Hence it is vastly suppressed by the short life time of the dipole-allowed 2P WE in cuprous oxide. Nevertheless, if one probes the system after the hybridization had happened, the SH is generated at ω1S/2 and is heightened by the big oscillator strength of the hybrid and it’s small damping coefficient. 7. Acknowledgments We would like to thank Ms. Upali Aparajita for helpful discussion and comments. The project was supported in part by PCS-CUNY. References [Shen et al. (1996)] M. Shen, S. Koyama, M. Saito, T. Goto, and N. Kuroda, Phys. Rev. B 53, 13477 (1996). [Atanasov et al. (1994)] R. Atanasov, F. Bassani, and V. Agranovich, Phys. Rev. B 50, 7809 (1994). [Agranovich et al. (1998)] V. Agranovich, D. Basko, G. L. Rocca, and F. Bassani, J. Phys.: Condens. Matter 13, 9369 (1998). [Engelmann et al. (1998)] A. Engelmann, V. I. Yudson, and P. Reineker, Phys. Rev. B 57, 1784 (1998). [Gao et al. (2004)] Y. Gao, J. Birman, N. Huong, and M. Potasek, Journal of applied physics 96, 1 (2004). [Huong and Birman (2000)] N. Q. Huong and J. L. Birman, Phys. Rev. B 61, 13131 (2000). [Huong and Birman (2003)] N. Q. Huong and J. L. Birman, Phys. Rev. B 67, 075313 (2003). [Roslyak and Birman (2007a)] O. Roslyak and J. Birman, Solid State Communications 143, 487 (2007a). [Dasbach (2004)] G. Dasbach, Phys. Rev. B 70, 121202 (2004). [Bulovic et al. (1999)] V. Bulovic, R. Deshpande, and S. Forrest, Chem. Phys. Lett. 308, 317 (1999). [Roslyak and Birman (2007b)] O. Roslyak and J. L. Birman, Physical Review B (Condensed Matter and Materials Physics) 75, 245309 (pages 11) (2007b). [Mukamel (1995)] S. Mukamel, Principles of nonlinear optical spectroscopy (Oxford Press, New York, 1995). [Bloembergen (1965)] N. Bloembergen, Nonlinear optics (Benjamin, New York, 1965). [Moskalenko and Liberman (2002)] S. Moskalenko and M. Liberman, Phys. Rev. B 65, 064303 (2002). [Madigan and Bulovic (2003)] C. Madigan and V. Bulovic, Physical review letters 91, 247403 (2003). [Artoni et al. (2002)] M. Artoni, I. Carusotto, G. La Rocca, and F. Bassani, J. Opt. B.: Quantum Semiclass. Opt. 4, S345 (2002). [Elliott (1961)] R. J. Elliott, Phys. Rev. 124, 340 (1961). [Madigan and Bulovic (2004)] C. Madigan and V. Bulovic (Bringing Materials Research Together, 2004). Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton15 [Kishida et al. (1994)] H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto, S. Wada, T. Lay, and H. Tashiro, Phys. Rev. B 50, 7786 (1994). [Frohlich et al. (2005)] D. Frohlich, G. Dasbach, G. B. Hogersthal, M. Bayer, R. Kliebera, D. Sutera, and H. Stolzb, Solid State Communications 134, 139 (2005). [Haueisen and Mahr (1973)] D. Haueisen and H. Mahr, Phys. Rev. B 8, 734 (1973). Appendix A. Explicit expression for the quadrupole-dipole coupling is given below: (ε+ ε̃)Lw ke−kz (A.1) Here aFB, a B are the Bohr radius of the FE and WE exciton; ε̃ and ε are the corresponding dielectric constants, z′ is the distance to the DCM2 layer, Lw is the quantum well width. The quadrupole transition matrix element Qxz may be estimated from the corresponding oscillator strength per unit cell through the following identity [Moskalenko and Liberman (2002)] and depends on polarization of the pumping laser field: fxz,k0 = 4πmEg 3e2h̄2 (z · k0,x ·Qx,z)2 (A.2) fxz,k0||[1,1,0] = 3.9× 10−9 fxz,k0||[1,1,1] = 3.9× 10−9 Here the energy gap of cuprous oxide is denoted as Eg = 2.173 eV ; k0 = 2.62×105 cm−1 is the resonant wave vector; a is the unit cell size; the unit vector in the pumping field polarization is z. Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton16 Appendix B. The non-zero commutator relations for the organic and inorganic parts of the hybrid yield [Mukamel (1995)]: 1S, B1S = −1 +B†2PB2P + b†b; 1S, B2P 1SB2P ; 2P , B1S 2PB1S (B.1) 2P , B2P = −1 +B†1SB1S + b†b; b†, B1S = b†B1S; b†, B2P = b†B2P b†, b = −1 +B†2PB2P +B 1SB1S; 1S, b 2P , b In the TDHF approximate factorization for the averages, the corresponding Heisenberg equations up to the second order in the creation and annihilation operators are: = h̄ω1SB 1S + Γkb † − µFEiB†1Sb+ µ1S,kEi 1− B†2PB2P − b†b µ2PEiB 1SB2P + µ1S,2PEjB = −h̄ω1SB1S − Γkb+ µFE⋆i b†B1S − µ1S,kE⋆i 1− B†2PB2P − b†b 2PB1S − µ1S,2PE⋆jB2P = h̄ω2PB 2P − µFEiB 2P b− µ1S,kEiB 2PB1S + µ2PEi 1− B†1SB1S − b†b +µ1S,2PEjB = −h̄ω2PB2P + µFE⋆i b†B2P + µ1S,kE⋆i B 1SB2P − µ2PE⋆i 1−B†1SB1S − b†b −µ1S,2PE⋆jB1S = EF b† + ΓkB 1S + µ 1−B†1SB1S −B 2PB2P − µ1S,kEib†B1S − µ2PEib†B2P = −EF b− ΓkB1S − µFE⋆i 1−B†1SB1S −B 2PB2P + µ1S,kE 1Sb+ µ2PE Here we omitted the average brackets for the shorter notation. In the exact resonance between FE and WE excitons h̄ω1S = h̄ωF the linear approximation is straightforward. The creation operators are proportional to ∝ eiωt and the system above is reduced to: 1S,0 = (h̄ω1S − ih̄γ)B 1S,0 + Γkb 0 + µ1S,kEi − h̄ωB1S,0 = − (h̄ω1S + ih̄γ)B1S,0 − Γkb0 − µ1S,kE⋆i 2P,0 = h̄ω2PB 2P,0 + µ2PEi − h̄ωB2P,0 = −h̄ω2PB2P,0 − µ2PE⋆i 0 = (h̄ω1S − ih̄γ) b 0 + ΓkB 1S,0 + µ FEi − h̄ωb0 = − (h̄ω1S + ih̄γ) b0 − ΓkB1S,0 − µFE⋆i Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton17 The system above has a solution: 2P,0 = µ2PEi h̄ω − h̄ω2P (B.2) 1S,0 = µ1SEi (h̄ω − h̄ω1S + ih̄γ) + µFΓkEi (h̄ω − h̄ω1S + ih̄γ)2 − Γ2k µFEi (h̄ω − h̄ω1S + ih̄γ) + µ1SΓkEi (h̄ω − h̄ω1S + ih̄γ)2 − Γ2k The SHG is due to response to induced polarization and is proportional to ∝ ei2ωt: 2h̄ωB 1S,1 = (h̄ω1S − ih̄γ)B 1S,1 + Γkb 1 + µ1S,2PEjB 2h̄ωB1S,1 = (h̄ω1S + ih̄γ)B1S,1 + Γkb1 + µ1S,2PE 2h̄ωB 2P,1 = h̄ω2PB 2P,1 + µ1S,2PEjB 2h̄ωB2P,1 = h̄ω2PB2P,1 + µ1S,2PE 2h̄ωb 1 = (h̄ω1S − ih̄γ) b 1 + ΓkB 2h̄ωb1 = (h̄ω1S + ih̄γ) b1 + ΓkB1S,1 The system has a solution: 2P,1 = µ1S,2PEjB 2h̄ω − h̄ω2P (B.3) 1S,1 = µ1S,2PEj (2h̄ω − h̄ω1S + ih̄γ)B†2P,0 (2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k µ1S,2PEjΓkB (2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k This solutions are implemented in the main text to calculate the linear and nonlinear responses of the hybrid. Introduction Proposed experimental set-up for the SHG Anharmonic coupled oscillators model Quantum theory of SHG due to the QDH Results and discussion Conclusion Acknowledgments

ERROR ESTIMATION AND ATOMISTIC-CONTINUUM ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL MARCEL ARNDT AND MITCHELL LUSKIN Abstract. We propose and analyze a goal-oriented a posteriori error estimator for the atomistic- continuum modeling error in the quasicontinuum method. Based on this error estimator, we develop an algorithm which adaptively determines the atomistic and continuum regions to compute a quan- tity of interest to within a given tolerance. We apply the algorithm to the computation of the structure of a crystallographic defect described by a Frenkel-Kontorova model and present the re- sults of numerical experiments. The numerical results show that our method gives an efficient estimate of the error and a nearly optimal atomistic-continuum modeling strategy. 1. Introduction The quasicontinuum (QC) method [22, 23, 24] has been successfully used to efficiently cou- ple atomistic and continuum models for crystalline solids and offers the possibility of computing mesoscale or macroscale properties by a nearly minimal number of degrees of freedom. Accurate modeling requires that an atomistic model be used in regions with highly non-uniform deformations such as around dislocations, whereas a continuum model can be used in regions with nearly uniform deformations to reduce the number of degrees of freedom. It is usually not known a priori which regions of some specimen undergo uniform deformations and which do not, so a posteriori error estimation is important for the design of efficient numerical approximations by the quasicontinuum method. Since the purpose of a computation is often to obtain the value of a (usually local) quantity of interest to a desired error tolerance rather than to obtain a solution to a desired error tolerance for a global norm, there has been great interest in the development of goal-oriented error estimators for many problems. They are based on duality techniques and have been developed and used to adaptively refine finite element approximations of continuum problems [1, 3] and to study and control modeling error [19]. In this paper, we extend this approach to develop an a posteriori error estimator for the qua- sicontinuum method which quantifies the atomistic-continuum modeling error for a goal function and allows for an adaptive decision about which regions can be accurately modeled as a continuum and which regions need to be modeled atomistically. Methods to determine the optimal mesh size within the continuum region will be studied in a forthcoming paper. Crystallographic defects [5] provide a challenge to validate atomistic-continuum error estimators and adaptivity. No such error estimators and adaptive methods currently exist for fully three- dimensional crystals. As a step in this direction, we develop a rigorous theory for a simple one- dimensional atomistic model for a defect that is a modification of the Frenkel-Kontorova model [15]. We add next-nearest-neighbor harmonic interactions between the atoms to the nearest-neighbor harmonic interactions between the atoms in the classical Frenkel-Kontorova model. A priori analyses for various quasicontinuum approximations have been given in [4, 8, 9, 10, 12, 13, 14, 21]. An a posteriori analysis for a slightly different one-dimensional quasicontinuum http://arxiv.org/abs/0704.1924v1 ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 2 approximation is given in [20]. The development and application of a goal-oriented error estimator for mesh coarsening in a two-dimensional quasicontinuum method is reported in [17, 18]. Let us mention that the continuum model used in the QC method, which coincides with the model obtained by the classical thermodynamic limit, is by far not the only reasonable continuum model to use. A method to derive continuum models which approximate atomistic models up to an arbitrarily high order has been proposed in [2]. The paper is organized as follows. In Section 2, we give a general formulation of the one- dimensional quasicontinuum approximation [23] that includes not only two-body and three-body potentials, but also many body potentials such as the embedded atom potential [6, 7]. In Section 3, we describe our extension of the Frenkel-Kontorova model and its quasicontinuum approximation. In Section 4, we introduce the primal and dual problems for our model and formulate our approach to goal-oriented error estimation. Next, in Section 5 we extend the approach in [16] to develop an error estimator for atomistic- continuum modeling. This first error estimator does not allow a decomposition among the atoms that can be used for atomistic-continuum adaptivity, so we propose and analyze a less accurate second error estimator that does allow such a decomposition. Finally, in Section 6 we propose an adaptive atomistic-continuum modeling algorithm and show that it gives an efficient estimate of the modeling error and a nearly optimal atomistic-continuum modeling strategy for the computation of defect structure. 2. Quasicontinuum Approximation The departure point for the QC approximation is the potential energy of the atomistic system. The potential energy that is utilized fully models the properties of the system. The local minima of the potential energy model the metastable states of the system, and the potential energy can be used in Newton’s equations of motion to model the dynamical behavior. The QC method approximates the potential energy of the atomistic system in two steps. First, we develop a continuum potential energy that will be used in the adaptively determined continuum region, and we then show how to reduce the degrees of freedom in the continuum region. 2.1. The Atomistic System. We assume that the atomistic system has 2M atoms with deforma- tion given by ya = (ya−M+1, . . . y M) ∈ R 2M . Without loss of generality, we assume that the atoms are ordered so that their positions satisfy yai < y i+1. Furthermore, we assume that the atomistic total potential energy, Ea(ya), can be written as a sum over potential energies associated with each atom, Eai (y a), so that Ea(ya) = i=−M+1 Eai (y a). (2.1) This decomposition can be found for most empirical potentials, including embedded atom potential energies [6, 7]. For example, if the atomistic total potential energy Ea(ya) is given by Ea(ya) = ψ(yaj − y i ), (2.2) where ψ(r) is an empirical two-body potential energy, then we can obtain the decomposition (2.1) by taking Eai (y j 6=i ψ(yaj − y i ). (2.3) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 3 We note that Eai (y a) can also contain contributions from external forces, such as for the Frenkel- Kontorova model described in Section 3, and can thus depend on i. 2.2. The Atomistic-Continuum Energy. For any deformation ya ∈ R2M , we let Li,i+1ya ∈ RZ denote the linear extrapolation of the atomistic positions yai and y i+1 given by (Li,i+1ya)k = (k − i)y i+1 + (i+ 1− k)y i for k = −∞, . . . ,∞. (2.4) The continuum potential energy Eci (y a) of atom i is obtained from the average of the atomistic potential energy Eai evaluated at the extrapolations L i−1,iya and Li,i+1ya by Eci (y a) := 1 Eai (L i−1,i a) + 1 Eai (L i,i+1 a), (2.5) where we note that the domain of Eai has been expanded to the infinite periodic atomistic systems in the range of Li−1,i and Li,i+1. We assume that Eai is finite for infinite periodic atomistic systems, which is true for (2.3) when the two-body potential ψ(r) decays fast enough so that k=1 ψ(kr) is finite for r 6= 0. At the endpoints of the chain, the extrapolation can be done only to one side, so we neglect the undefined part and define Ec−M+1(y a) := 1 Ea−M+1(L −M+1,−M+2 a) and EcM (y a) := 1 EaM (L M−1,M a). (2.6) We then decide for each atom i whether to model its energy atomistically by Eai (y a) or as a continuum by Eci (y a). We thus obtain for the whole chain the atomistic-continuum energy Eac(ya) := i=−M+1 δai E i=−M+1 δci E i=−M+1 δai E a) + 1 i=−M+2 i−1,i a) + 1 i=−M+1 δci E i,i+1 (2.7) where δai = 1 if atom i is modeled atomistically, 0 if atom i is modeled as continuum, and δci = 1− δ i . (2.8) This approximation allows for a slightly faster evaluation of the energy and its derivatives, especially if Eai is long-ranged. However, it reveals its full strength only after the quasicontinuum coarsening to be described next. We note that sometimes atomistic degrees of freedom and energies are referred to as nonlocal and continuum degrees of freedom and energies are referred to as local [23]. 2.3. Repatoms: Reduction of Degrees of Freedom. The quasicontinuum method allows a reduction of the number of degrees of freedom in the continuum region. To this end, we choose so-called representative atoms, or more briefly called repatoms. The repatoms are a subset of the original atoms. The quasicontinuum approximation of the energy is defined completely in terms of the repatoms. We choose the repatoms by defining indices ℓj for j = −N + 1, . . . , N where −M + 1 = ℓ−N+1 < · · · < ℓj < ℓj+1 < · · · < ℓN =M. The atoms at yai for i = ℓ−N+1, ℓ−N+2, . . . , ℓN are repatoms, and all of the remaining atoms are non-repatoms. We have that νj = ℓj+1 − ℓj (2.9) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 4 gives the number of atomistic intervals between the repatoms ℓj and ℓj+1.We require that the chain not be coarsened in the atomistic regions, which precisely means that δc = . . . = δc whenever νj > 1. Finally, the interactions of the atomistic energy only partially reach into the continuum part if the atomistic potential has a finite cutoff radius. To allow for an exact calculation of this energy without atomistic interpolation, we require that these regions are not coarsened as well. As we will see in the next subsection, the atomistic next-nearest-neighbor interactions from the Frenkel- Kontorova model studied in this paper reach two atoms into the continuum part. Hence, we require that νj−2 = νj−1 = νj = νj+1 = 1 whenever δ = 1. Other potential energies in general require similar conditions that depend on their cut-off radius. We denote the position of the j-th repatom by y j = y and the vector of all repatoms by yqc ∈ R2N . 2.4. The Quasicontinuum Energy. Now we define the quasicontinuum energy. To this end, the missing non-repatoms are implicitly reconstructed. We will see later that this helps to set up the QC model, but needs not be done for the actual computation. The reconstruction is done by a linear interpolation between the nearest repatom to the right and to the left. That is, the vector of all atomistic positions is computed from the vector yqc of repatom positions by I : R2N → R2M , (Iyqc)ℓj+m := νj −m j+1, m = 0, . . . , νj . (2.10) We note that j = (Iy qc)ℓj . (2.11) The underlying idea is that in regions where the lattice spacing of the atoms is nearly constant, this interpolation is very close to the actual atomistic positions and therefore leads to a good approximation of the total energy. Only a few repatoms are needed in these regions. This exactly corresponds to mesh coarsening in classical finite element approximations of continuum models. On the other hand, in regions where the lattice spacing is non-uniform, such as around a dislocation, all atoms must be chosen to be repatoms to obtain sufficient accuracy. This guarantees that the full resolution of the atomistic model in the critical regions is retained and corresponds to a high refinement in classical finite element continuum models. We define the QC approximation of the total energy to be Eqc(yqc) := Eac(Iyqc). (2.12) Now (2.12) has to be reformulated such that it can be computed efficiently, without the overhead of the interpolation. Most atomistic potentials are invariant to translations, a property that al- lows us to simplify (2.12) considerably. For any translationally invariant energy Eai , we have that Eai (L i,i+1ya) = φi(y i+1 − y i ) and E i−1,iya) = φi(y i − y i−1) for some function φi. If these functions φi coincide, that is, φi = φj for all i and j, we can write Eai (L i−1,i a) = φ yai − y and Eai (L i,i+1 a) = φ yai+1 − y (2.13) for some function φ : R→ R. Here φ plays the role of a continuum energy density and is given for the two-body potential (2.2) by φ(r) = ψ(kr). ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 5 Equations (2.7), (2.12), and (2.13) lead to Eqc(yqc) = i=−M+1 δai E i (Iy qc) + 1 i=−M+2 i−1,iIyqc) + 1 i=−M+1 δci E i,i+1Iyqc) i=−M+1 δai E i (Iy qc) + 1 i=−M+2 δciφ((Iy qc)i − (Iy qc)i−1) (2.14) i=−M+1 δciφ((Iy qc)i+1 − (Iy qc)i). Because Iyqc is the linear interpolation between two repatoms y j and y j+1, we have (Iyqc)i+1 − (Iy qc)i = j+1 − y , i = ℓj, . . . , ℓj+1 − 1. (2.15) Hence, Eqc(yqc) = i=−M+1 δai E i (Iy qc) + j=−N+1 j+1 − y (2.16) with weight factors δcℓj + δ 0 if both y j and y j+1 are atomistic, if exactly one of y j and y j+1 is continuum, νj if both y and y j+1 are continuum. (2.17) The first sum corresponds to the atomistic region which will be a small region and is thus com- putationally inexpensive. The second sum only involves at most 2N terms which is a considerable reduction when N ≪M. Note that the second term in formula (2.16) coincides with an integral over the energy density φ as it occurs in finite element discretizations of classical continuum mechanical models. Hence the apparently unmotivated definitions (2.7) and (2.5) of the continuum energy here result in what is commonly understood as a continuum energy. The linear interpolation operator I resembles the Cauchy-Born hypothesis. 3. Frenkel-Kontorova Model Dislocations are lines in crystals which represent a defect in the lattice structure [15], see Fig- ure 1. Typically, there is a core of small radius surrounding the dislocation line where the lattice structure is highly deformed, but the lattice structure is nearly uniform outside the core. A sim- ple one-dimensional model for a defect such as a dislocation is given by the Frenkel-Kontorova model [15]. Here, the elastic energy is modeled by harmonic interactions between the atoms in the one-dimensional chain and the misfit energy of the slip plane is modeled by a periodic potential. A more accurate model of the same form is given by the Peierls-Nabarro model [11]. 3.1. Atomistic Frenkel-Kontorova Model. We study a single defect in the middle of the chain of 2M atoms. To achieve a symmetric description in terms of bonds, we number the atoms from −M + 1 to M . The defect is situated between the atoms numbered 0 and 1 (Figure 2). Recall that the atomistic positions are denoted by ya = (ya−M+1, . . . , y M ) ∈ R 2M . The total potential energy for this atomistic system is then a function Ea : R2M → R of the atomistic ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 6 Figure 1: Cross-section through a dislocation in a three-dimensional cubic lattice. The displayed plane repeats periodically in the three-dimensional crystal. Vertical bonds are shown by lines to emphasize the topological defect. (−K−1)a (K+1)a = atomistic = continuum Figure 2: Numbering of the atoms. The dislocation is situated in the middle of the chain between atoms ya and ya positions. For the Frenkel-Kontorova model, the energy, Ea = Ea,e + Ea,m, consists of two parts, namely the part which models the elastic energy of the defect, Ea,e, and the part which models the misfit energy on the slip plane, Ea,m. The elastic energy is modeled by Hookean (harmonic) springs between nearest-neighbors (NN) and next-nearest neighbors (NNN), and the total elastic energy is given by Ea,e(ya) = i=−M+1 i+1 − y i − a0) i=−M+2 i+1 − y i−1 − 2a0) 2, (3.1) where the moduli k1 > 0 and k2 > 0 describe the strength of the elastic interactions, and where a0 ∈ R denotes the equilibrium distance. We note that the asymptotic expansion to second order of any nonlinear NN/NNN potential energy E(ya) = i=−M+1 ψ(yai+1 − y i ) + i=−M+2 ψ(yai+1 − y i−1) (3.2) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 7 substrate modeled layer misfit energy Figure 3: Frenkel-Kontorova model. The wells depict the misfit energy (3.4). about aa = [(−M + 1)a0, (−M + 2)a0, · · · , (M − 1)a0,Ma0] ∈ R2M has the form E(ya) ≈ E(aa) + ψ′(a0) + 2ψ ′(2a0) i=−M+1 (yai+1 − y i − a0) − ψ′(2a0)(y M − y M−1 − a0)− ψ ′(2a0)(y −M+2 − y −M+1 − a0) ψ′′(a0) i=−M+1 (yai+1 − y i − a0) 2 + 1 ψ′′(2a0) i=−M+2 (yai+1 − y i−1 − 2a0) (3.3) We thus see that the elastic energy (3.1) with k1 = ψ ′′(a0) and k2 = ψ ′′(2a0) approximates the energy (3.2) to second order if ψ′(a0) + 2ψ ′(2a0) = 0 and if we ignore the boundary terms in the second line of (3.3). The misfit energy of the slip plane is modeled by a periodic potential (Figure 3). We model this misfit energy by Ea,m(ya) = i=−M+1 yai − a0 , (3.4) where ⌊x⌋ denotes the largest integer smaller than or equal to x, and where the constant k0 > 0 determines the strength of the misfit energy. Altogether, the total potential energy of the atomistic system is given by Ea(ya) = Ea,e(ya) + Ea,m(ya) i=−M+1 (yai+1 − y i − a0) 2 + 1 i=−M+2 (yai+1 − y i−1 − 2a0) i=−M+1 yai − a0 (3.5) We restrict ourselves to configurations in which the M leftmost atoms yai for −M + 1 ≤ i ≤ 0 are situated in the interval , whereas the M rightmost atoms yai for 1 ≤ i ≤ M are situated in the interval . The defect is situated between atoms ya0 and ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 8 ya1 . In this case, the total energy simplifies to Ea(ya) = 1 i=−M+1 (yai+1 − y i − a0) 2 + 1 i=−M+2 (yai+1 − y i−1 − 2a0) i=−M+1 (yai − (i− 1)a0) (yai − ia0) (3.6) 3.2. Quasicontinuum Approximation of the Frenkel-Kontorova Model. We now apply the quasicontinuum method to the dislocation model described in Section 3.1. The total energy (3.6) is split up into atom-wise contributions, separately for the elastic interactions and the misfit interactions: a) = 1 i − y i−1 − a0) 2 + 1 i+1 − y i − a0) i − y i−2 − 2a0) 2 + 1 i+2 − y i − 2a0) k0 (y i − (i− 1)a0) , i = −M + 1, . . . , 0, k0 (y i − ia0) , i = 1, . . . ,M. (3.7) To simplify notation, we use the convention that the undefined terms at the endpoints of the chain are neglected. We thus have that Ea(ya) = Ea,m(ya) + Ea,e(ya) = i=−M+1 a) + E a)] . (3.8) Since the largest displacement of the atoms is to be expected near the defect, we deem the atoms −K +1, . . . ,K atomistic and the remaining atoms −M +1, . . . ,−K and K +1, . . . ,M continuum. Here K < M is some constant whose optimal value will be determined by the algorithm given in Section 5. The optimal choice of the repatoms for coarsening is investigated in the second paper of this series, so we work with a general formulation which holds for any values of ℓj for now. However, there are two restrictions on the coarsening. Since the atomistic region must not be coarsened and since we need full refinement in the vicinity of two atoms around the atomistic region due to the NNN interactions, we have that ℓj = j, j = −K − 1, . . . ,K + 2. (3.9) Second, we require that ℓ−N+1 = −M + 1, ℓ−N+2 = −M + 2, ℓN−1 =M − 1, and ℓN =M (3.10) to incorporate the boundary conditions later. The elastic part E i is translationally invariant, so we perform its QC approximation as described in the previous section. This leads to the continuum energy density φe(r) = 1 k1(r − a0) 2 + 1 k2(2r − 2a0) k12(r − a0) 2 (3.11) where k12 := k1 + 4k2. Regarding the misfit part E i , the above technique cannot be applied since the potential is not translationally invariant. However, there is a different summation technique to achieve a computationally efficient formulation which avoids the costly interpolation operator. ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 9 To shorten the notation, we let indicate the sum in which the first term and the last term are only counted half: xi := i=m+1 xn (3.12) where m < n and xi ∈ R. It is easy to verify that 2m3 +m i(m− i) = m3 −m (3.13) for m > 0. For all pairs (j, j + 1) of continuum repatoms, we now reformulate all terms from (2.7) which involve the interaction between ℓj and ℓj+1. For j > 0, we get by definition (2.4) of the operator L, by definition (3.7) of E i , and by (3.13) that i=ℓj+1 i−1,iIyqc) + 1 ℓj+1−1 i,i+1Iyqc) i=ℓj+1 i (Iy qc) + 1 ℓj+1−1 i (Iy ℓj+1 − i i− ℓj j+1 − ia0 ℓj+1 − i j − ℓja0) + i− ℓj j+1 − ℓj+1a0) j − ℓja0) (ℓj+1 − i) 2 + 1 j+1 − ℓj+1a0) (i− ℓj) − ℓja0)(y j+1 − ℓj+1a0) (ℓj+1 − i)(i − ℓj) j − ℓja0) 2ν3j + νj j+1 − ℓj+1a0) 2ν3j + νj j − ℓja0)(y j+1 − ℓj+1a0) ν3j − νj =: φmj+(y j , y j+1). ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 10 For j < 0, we similarly obtain i=ℓj+1 (Li−1,iIyqc) + 1 ℓj+1−1 (Li,i+1Iyqc) j − (ℓj − 1)a0) 2ν3j + νj j+1 − (ℓj+1 − 1)a0) 2ν3j + νj j − (ℓj − 1)a0)(y j+1 − (ℓj+1 − 1)a0) ν3j − νj =: φmj−(y j , y j+1). Since E −K (L −K,−K+1Iyqc) = E −K (Iy qc) and E K+1(L K,K+1Iyqc) = E K+1(Iy qc), the QC approx- imation of the chain can be given by Eqc(yqc) = j=−N+1 j+1 − y + φmj−(y j , y −K (Iy qc) + j=−K+1 Eai (Iy qc) + 1 K+1(Iy j+1 − y + φmj+(y j , y (3.14) Note that the interpolation Iyqc does not have to be computed here since the relevant terms only depend on uncoarsened parts of the chain. Additionally, we will consider the atomistic-continuum approximation Eac : R2M → R (3.15) of the atomistic energy without coarsening. It is given exactly like the QC approximation (3.14) with the only difference being that νj = 1 and ℓj = j everywhere. 4. Primal and Dual Problems 4.1. Problem Setup. We are now ready to set up the problems we will solve. We are interested in finding the minimum of the energy (3.14) subject to given boundary conditions. We give the boundary conditions by constraining the deformation of two atoms at each end of the chain. This guarantees that the potential with next-nearest-neighbor interactions can be directly applied to all non-boundary atoms without having to neglect interactions. We define the spaces V a := R2M , V a0 := R 2M−4, V qc := R2N , V 0 := R 2N−4. (4.1) The spaces V a and V a0 will also be used for the uncoarsened atomistic-continuum potential E so there is no need to define spaces V ac and V ac0 . We let y bc ∈ V a denote any vector which has the desired boundary values ybc−M+1, y −M+2, y M−1, and y M , and we let y bcq ∈ V qc by any vector satisfying (recall (3.10)) −N+1 = y −M+1, y N−1 = y −N+2 = y −M+2, y = ybcM . (4.2) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 11 For any vector y ∈ V a0 , we denote the extension by zero boundary conditions to be Jy ∈ V a, so Jy := 0 0 yT 0 0 ∈ R2M , (4.3) and similarly we denote the extension by zero boundary conditions of y ∈ V 0 to be J qcy ∈ V qc. The spaces of admissible solutions are then given by JV a0 + y bc ⊂ V a and JqcV 0 + y bcq ⊂ V qc, respectively. We note that JT : V a → V a0 is the restriction operator defined by (JTy)j = yj for j = −M + 3, . . . ,M − 2. The minima ȳa, ȳac, and ȳqc of the energy functionals Ea, Eac, and Eqc given by (3.6), (3.15), and (3.14) subject to the above “clamped” boundary conditions are characterized as a := argmin y∈JV a Ea(y) ∈ V a, (4.4) ac := argmin y∈JV a Eac(y) ∈ V a, (4.5) qc := argmin y∈JqcV Eqc(y) ∈ V qc. (4.6) We note that the minima are uniquely determined because Ea, Eac, and Eqc are strictly convex. 4.2. Matrix Formulation. For the subsequent discussion, it will be convenient to reformulate the total energies in matrix notation: Ea(y) = 1 (y − aa)TDaTEaDa(y − aa) + 1 (y − ba)TKa(y − ba), (4.7a) Eac(y) = 1 (y − aa)TDaTEacDa(y − aa) + 1 (y − ba)TKa(y − ba), (4.7b) Eqc(y) = 1 (y − aqc)TDqcTEqcDqc(y − aqc) + 1 (y − bqc)TKqc(y − bqc). (4.7c) The matrices Da ∈ R(2M−1)×2M and Dqc ∈ R(2N−1)×2N compute the distance between two adja- cent atomistic positions; the matrices Ea ∈ R(2M−1)×(2M−1), Eac ∈ R(2M−1)×(2M−1), and Eqc ∈ (2N−1)×(2N−1) contain the spring constants k1, k2, and k12; and the matricesK a ∈ R(2M−1)×(2M−1) and Kqc ∈ R(2N−1)×(2N−1) contain the misfit constant k0. The vectors a a,ba ∈ R2M and aqc,bqc ∈ 2N are constants describing the minimum energy deformations for the elastic energy and mis- fit energy. The precise and lengthy definitions for all of these matrices and vectors are given in Appendix A. If we decompose ȳa = Jya + ybc for ya ∈ V a0 , then the minimization problem is given as a = argmin y∈V a Ea(Jy + ybc) = argmin y∈V a Jy + ybc − aa DaTEaDa Jy + ybc − aa Jy + ybc − ba Jy + ybc − ba . (4.8) We also decompose ȳac = Jyac + ybc and ȳqc = Jqcyqc + ybcq for yac ∈ V a0 and y qc ∈ V 0 , and we then formulate similar minimization problems for yac and yqc. Therefore, ya, yac, and yqc are determined by the linear systems Maya = fa, (4.9a) Macyac = fac, (4.9b) M qcyqc = f qc, (4.9c) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 12 where Ma := JT (DaTEaDa +Ka)J, Mac := JT (DaTEacDa +Ka)J, M qc := JqcT (DqcTEqcDqc +Kqc)Jqc, a := −JTDaTEaDa(ybc − aa)− JTKa(ybc − ba), ac := −JTDaTEacDa(ybc − aa)− JTKa(ybc − ba), qc := −JqcTDqcTEqcDqc(ybcq − aqc)− JqcTKqc(ybcq − bqc). (4.10) We note that the matrices Ma, Mac, and M qc are positive definite, so the total energies admit a single global minimum and no other local minimum. 4.3. Goal-Oriented Error Estimation. To compare the approximate QC model to the original atomistic model, we have to analyze how much the solution ya of the atomistic model deviates from the solution yqc of the QC model. This deviation, which can be viewed as an approximation error, can be measured in different ways, for example as ‖ya−JT IJqcyqc‖ for some norm ‖·‖. Here we follow a different approach, namely we measure the error of a quantity of interest denoted by Q(y) for some function Q : R2M−4 → R. Hence, we intend to estimate Q(ya)−Q(JT IJqcyqc). (4.11) We will assume for simplicity that Q is linear and thus has a representation Q(y) = qTy for some vector q ∈ V a0 . For our application, a natural quantity of interest is the size of the dislocation, that is, the distance between the two atoms y0 and y1 to the left and right of the dislocation. This gives us Q(y) = qTy = y1 − y0 with q = [0, . . . , 0,−1, 1, 0, . . . , 0] T . (4.12) Two different sources of error arise during the QC approximation, namely the localization of the potential energy, that is, the passage from the atomistic to the continuum formulation on the one hand, and the coarsening in the continuum region by the restriction to the repatoms on the other hand. We denote these two errors by e := ya − yac and eacqc := yac − JT IJqcyqc. (4.13) It makes sense to study these sources independently. Employing the linearity of Q, we have that |Q(ya)−Q(JT IJqcyqc)| = |Q(e) +Q(eacqc)| ≤ |Q(e)|+ |Q(eacqc)|. (4.14) The error term |Q(e)| will be studied in Section 5, and the error term |Q(eacqc)| will be studied in the second part of this paper series. 4.4. Dual Problems. To facilitate the goal-oriented error analysis, we introduce the dual problems Maga = q, (4.15a) Macgac = q, (4.15b) M qcgqc = JqcT ITJq, (4.15c) for ga,gac ∈ R2M−4, and gqc ∈ R2N−4. We note that the dual problems differ from the primal problems only by the right hand side since the matrices Ma, Mac, and M qc are symmetric. The solutions ga, gac and gqc can be viewed as influence functions: They describe how the error at a specific point in the domain influences the error measured in terms of the goal function. ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 13 Analogously to the primal errors (4.13), we define the dual errors ê := ga − gac and êacqc := gac − JT IJqcgqc. (4.16) In addition, we will need the primal and dual residuals Ra(y) :=Ma (ya − y) = fa −May, Rac(y) :=Mac (yac − y) = fac −Macy, R̂a(g) :=Ma (ga − g) = q−Mag, R̂ac(g) :=Mac (gac − g) = q−Macg. (4.17) 5. Error Estimation for Atomistic vs. Continuum Modeling In this section, we estimate the error |Q(e)| arising from the approximation of an atomistic model by a continuum model. We consider yac and gac to be computable, although in practice we can only compute the coarsened approximations yqc and gqc. To this end, we adapt a technique introduced in [16] and [19] to estimate the modeling error for an elasticity model with rapidly oscillating coefficients and its homogenized version. We generalize this technique such that it allows for different right hand sides fa and fac of the primal problem (4.9) instead of a common right hand side as it is used in the above-mentioned works. We have Q(ya)−Q(yac) = qTe = gaTMae = (gacT + ê)Mae = gacTRa(yac) + êTMae. (5.1) The term gacTRa(yac) can be computed, whereas êTMae cannot because both e and ê are nu- merically unknown. Instead, we estimate êTMae from above and from below by quantities that actually can be computed. We will give two different error estimators η1 and η2. Before, we need to derive some auxiliary estimates to facilitate their development and analysis. 5.1. Auxiliary Estimates. We reformulate the difference ya − yac of the respective solutions in terms of a difference of the energy matrices. To this end, we define the perturbation matrix P := I − (Ea)−1Eac (5.2) where I denotes the identity matrix. Note that EaP = E a − Eac. Lemma 5.1. For any α, β ∈ R, we have that Ma(αe+ βê) = −JTDaTEaPDa α(Jyac + ybc − aa) + βJgac . (5.3) Proof. We conclude from (4.9) that Mae =Maya −Macyac + (Mac −Ma)yac = fa − fac + (Mac −Ma)yac, (5.4) and similarly since Maga =Macgac = q that Maê =Ma(ga − gac) = (Mac −Ma)gac. (5.5) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 14 Thus, it follows from (4.10) and (5.2) that Ma(αe+ βê) = α [(Mac −Ma)yac + fa − fac] + β(Mac −Ma)gac = JTDaT (Eac − Ea)Da α(Jyac + ybc − aa) + βJgac = −JTDaTEaPDa α(Jyac + ybc − aa) + βJgac (5.6) We note that the Ka-related terms cancel here, because they coincide for the atomistic model and the continuum model. � Lemma 5.2. We have that ‖αe+ βê‖Ma ≤ α(Jyac + ybc − aa) + βJgac . (5.7) We note that the right hand side is numerically computable. Proof. To shorten the notation, we abbreviate z = α(Jyac +ybc− aa)+ βJgac. By Lemma 5.1, we ‖αe + βê‖Ma = sup v∈V a vTMa(αe+ βê) ‖v‖Ma = sup v∈V a −vTJTDaTEaPDaz ‖v‖Ma ≤ sup v∈V a ‖DaJv‖Ea‖PD az‖Ea ‖DaJv‖Ea = ‖PDaz‖Ea . (5.8) Here we have used that ‖DaJv‖Ea ≤ ‖v‖Ma because the matrixK a in (4.10) is positive definite. � 5.2. First Error Estimator. We are now ready to derive the first error estimator, η1. By the parallelogram identity, we have for all σ 6= 0 that TMae = (σ−1êT )Ma(σe) ‖σe+ σ−1ê‖2Ma − ‖σe− σ−1ê‖2Ma . (5.9) In the following, we will determine computable constants η+low, η low, η upp and η upp such that η+low ≤ ‖σe+ σ ê‖Ma ≤ η η−low ≤ ‖σe− σ ê‖Ma ≤ η (5.10) From Lemma 5.2, we immediately get the upper estimates η+upp and η η+upp := σ(Jyac + ybc − aa) + σ−1Jgac η−upp := σ(Jyac + ybc − aa)− σ−1Jgac (5.11) We note that η+low, η low, η upp and η upp will depend on σ, but the estimates (5.10) will hold for any σ 6= 0. We will now choose σ in such a way that the estimates are as sharp as possible, that is, such that η+upp and η upp are smallest. Lemma 5.3. Both η+upp and η upp given by (5.11) attain their minima for σ̄ := ‖PDaJgac‖Ea ‖PDa(Jyac + ybc − aa)‖Ea . (5.12) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 15 Proof. We have that (η±upp) 2 = σ2 ∥PDa(Jyac + ybc − aa) ± 2gacTJTDaTP TEaPDa(Jyac + ybc − aa) + σ−2 ∥PDaJgac . (5.13) Setting the first derivative of the mapping σ 7→ (η±upp) 2 to zero, we obtain the condition ∥PDa(Jyac + ybc − aa) − 2σ̄−3 ∥PDaJgac = 0 (5.14) for critical points of (η±upp) 2. This equation has the unique positive solution (5.12). Because lim|σ|→∞ η upp = limσ→0 η upp =∞, this point corresponds to a minimum. Hence the quantities η attain their minima at σ = σ̄. � Regarding the lower bounds η+low and η low, we have ‖σ̄e± σ̄−1ê‖Ma = sup v∈V a vTMa(σ̄e± σ̄−1ê) ‖v‖Ma = sup v∈V a vT (σ̄Ra(yac)± σ̄−1R̂a(gac)) ‖v‖Ma vT0 (σ̄R a(yac)± σ̄−1R̂a(gac)) ‖v0‖Ma (5.15) for any vector v0 ∈ V 0 \ {0}. Numerically, we have the two vectors y ac and gac at our disposal, hence it makes sense to take a linear combination v0 = y ac + θ±gac. Here we follow the strategy of [19] and choose θ± as the critical points of η±low. Lemma 5.4. Let ± = σ̄Ra(yac)± σ̄−1R̂a(gac). (5.16) Then the lower bounds (yac + θ±gac)T r± ‖yac + θ±gac‖Ma (5.17) have a unique critical point for θ̄± := r±Tyac gacTMayac − r±Tgac ‖yac‖2Ma r±Tgac gacTMayac − r±Tyac ‖gac‖2 . (5.18) Proof. We have r±Tgac‖yac + θ±gac‖Ma − r ±T (yac + θ±gac) (yac+θ±gac)TMagac ‖yac+θ±gac‖Ma ‖yac + θ±gac‖2 . (5.19) Setting this expression to zero and solving for θ± leads to the above condition. � However, let us note that this critical point is not necessarily a maximum of η±low, which would be optimal for bound (5.10). Depending on the actual vectors yac and gac, it can be shown that this critical point could be a minimum. Now we have all necessary ingredients to construct the error estimator η1. From (5.1) and (5.9), we get the computable estimate gacTRa(yac) + 1 )2 − 1 (η−upp) 2 ≤ Q(ya)−Q(yac) ≤ gacTRa(yac) + 1 (η+upp) 2 − 1 (5.20) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 16 At first sight, this looks like we could get an estimate for |Q(ya)−Q(yac)| from both above and below. However, this is only true if both the left hand side and the right hand side have the same sign, which in general does not hold. But we get the following estimate. Theorem 5.1. We have that |Q(ya)−Q(yac)| ≤ η1, (5.21) where the computable error estimator is defined as η1 := max ∣gacTRa(yac) + 1 )2 − 1 (η−upp) ∣gacTRa(yac) + 1 (η+upp) 2 − 1 . (5.22) We note that the computation of the η±upp terms involves the solution of a linear system with matrix Ea as its inverse appears in the operator P . The matrix Ea is not diagonal, but has condition number O(1). So this is negligible compared to what would be necessary to solve the original atomistic problem which includes the operator DaTEaDa with condition number O(M2). 5.3. Second Error Estimator. There is no reasonable way to decompose the error estimator η1 into a sum of element-wise or atom-wise contributions due to the η± terms. Therefore, we derive another error estimator η2 which allows for such a decomposition, at the price of a less accurate estimate than η1. Theorem 5.2. We have that |Q(ya)−Q(yac)| ≤ η2 ≤ i=−M+3 ηat2,i + i=−M+1 ηel2,i (5.23) where the computable global error estimator η2 and the computable local error estimators, η 2,i and ηel2,i, associated with atoms and elements, respectively, are defined as η2 := ∣gacTRa(yac) ∣+ ‖PDa(Jyac + ybc − aa)‖Ea‖PD aJgac‖Ea, ηat2,i := |g a(yac)i| , i = −M + 3, . . . ,M − 2, ηel2,i := PDa(Jyac + ybc − aa) (Ea − Eac)Da(Jyac + ybc − aa) ∣(PDaJgac)i (Ea − Eac)DaJgac ∣ , i = −M + 1, . . . ,M − 1. (5.24) Proof. From (5.1) and Lemma 5.2, we conclude that |Q(ya)−Q(yac)| ≤ acTRa(yac) acTRa(yac) ∣+ ‖ê‖Ma‖e‖Ma acTRa(yac) ∣+ ‖PDa(Jyac + ybc − aa)‖Ea‖PD aJgac‖Ea = η2, (5.25) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 17 which gives us the global estimate. For the decomposition into local contributions, we further estimate acTRa(yac) ∣+ ‖PDa(Jyac + ybc − aa)‖Ea‖PD aJgac‖Ea acTRa(yac) ‖PDa(Jyac + ybc − aa)‖2Ea + ‖PDaJgac‖2Ea i=−M+3 |gaci R a(yac)i|+ i=−M+1 ∣(PDaJgac)i (Ea − Eac)DaJgac i=−M+1 PDa(Jyac + ybc − aa) (Ea − Eac)Da(Jyac + ybc − aa) i=−M+3 ηat2,i + i=−M+1 ηel2,i, (5.26) which completes the proof. � Let us remark that instead of the first inequality in (5.26), one can get an apparently better estimate ‖PDa(Jyac + ybc − aa)‖Ea‖PD aJgac‖Ea γ‖PDa(Jyac + ybc − aa)‖2Ea + γ−1‖PDaJgac‖2Ea (5.27) by introducing the additional weight factor ‖PDaJgac‖Ea ‖PDa(Jyac + ybc − aa)‖Ea , (5.28) and then decomposing the resulting terms similar to the above. However, our numerical results showed that this modification does not significantly improve the decomposed error estimator for the application considered here. 6. Numerics In the preceding sections, we constructed the error estimators η1 and η2. We will now give an algorithm for adaptive atomistic-continuum modeling based on these error estimators. Then we will present and discuss some numerical results. 6.1. Algorithm. The error estimator η1 should give a better estimate of the error than η2, because η2 involves the inequality ∣êTMae ∣ ≤ ‖ê‖Ma‖e‖Ma in (5.25) in contrast to the parallelogram identity for η1. However, η2 can be decomposed into atom-wise and element-wise contributions η 2,i and ηel2,i, whereas the η terms in η1 do not admit a reasonable decomposition that can be used for atomistic-continuum adaptivity. We make use of this by employing the sharper estimate η1 to determine whether a given global error tolerance τgl for the error in an adaptive algorithm has already been achieved or not. If not, we use the decomposed estimates ηat2,i and η 2,i to determine where the more precise atomistic modeling is needed. This leads us to the following algorithm: (1) Choose τgl. Model all atoms as a continuum. Set τat ← τgl. (2) Solve primal problem (4.9b) for yac and dual problem (4.15b) for gac. (3) Compute error estimator η1 from (5.22). (4) If η1 ≤ τgl, then stop. ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 18 M iteration K τat η1 100 1 0 1.000000e-10 3.899207e-02 2 28 1.000000e-11 5.915080e-10 3 32 1.000000e-12 4.878532e-11 1000 1 0 1.000000e-10 3.899208e-02 2 28 1.000000e-11 5.915100e-10 3 32 1.000000e-12 4.878548e-11 10000 1 0 1.000000e-10 3.899208e-02 2 28 1.000000e-11 5.915100e-10 3 32 1.000000e-12 4.878548e-11 100000 1 0 1.000000e-10 3.899208e-02 2 28 1.000000e-11 5.915099e-10 3 32 1.000000e-12 4.878540e-11 1000000 1 0 1.000000e-10 3.899208e-02 2 28 1.000000e-11 5.914422e-10 3 32 1.000000e-12 4.871775e-11 Table 1: Convergence of the algorithm for τgl = 10 −10 and different values of M . (5) Compute local error estimators ηat2,i and η 2,i from (5.24). (6) Set τat ← (7) Make all atoms i atomistic for which ηtot2,i := η 2,i + ηel2,i−1 + η ≥ τat. (6.1) (8) Go to (2). Here τdiv > 1 is a constant factor which describes how fast the atom-wise tolerance τat should decrease during adaption. Our experience has been that τdiv = 10 is a reasonable choice. The crucial adaption step is (7). The adaption criterion (6.1) deems all atoms to be modeled atomistically if the associated error from the decomposition of η2 exceeds the atomistic error toler- ance τat. Here the element-wise errors η 2,i are distributed equally to the two adjacent atoms i and i+ 1. For the dislocation at the center of the chain and the chosen goal function, we expect that the atomistic repatoms always form a symmetric interval around the center. We have used the above adaptive atomistic-continuum algorithm to approximate our Frenkel-Kontorova model and have always found that the atomistic region is the set of atoms −K + 1, . . . ,K for some K depending on M and τgl. Thus, the modeling approach given in Section 3 of restricting to an atomistic region consisting of atoms −K + 1, . . . ,K for some K rather than considering a more general atomistic region is justified a posteriori. 6.2. Numerical Results. The algorithm has been implemented as described above. The boundary conditions were chosen as ybc−M+1 = −M, y −M+2 = −M + 1, y M−1 =M − 1, y M =M. (6.2) The elastic constants are k0 = 1 and k1 = k2 = 2. ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 19 −500 −400 −300 −200 −100 0 100 200 300 400 500 −500 −400 −300 −200 −100 0 100 200 300 400 500 Figure 4: The error estimators ηtot 2,i (left) and η 2,i, η 2,i (right) for M = 500, K = 20. Table 1 shows how the algorithm given above performs. After 3 iterations, the desired accuracy τgl = 10 −10 is achieved. Moreover, we can see from the table that the number of iterations are independent of M , that means the algorithm behaves robustly with respect to the problem size M . Figure 4 (left) shows the decomposition of the error estimator η2 for a typical setting M = 500, K = 20. One can clearly see that the error in the atomistic region is small, whereas the error is large in the continuum regions that border the atomistic region. It then decreases exponentially towards the endpoints. The error in both the atomistic region around the center and the continuum regions far away from the center are in the range of the (relative) machine precision εmach, which accounts for the fluctuations in these regions. The error can be considered to be numerically zero in these regions. In the continuum regions, we observe an error of magnitude O(ε2 ), whereas in the continuum region we have O(εmach), which leads to the different magnitudes of the fluctuations. Figure 4 (right) shows the element-wise contributions ηel2,i and the atom-wise contributions η of the decomposed error estimator ηtot2,i = η 2,i + ηel2,i−1 + η . The atomistic part ηat2,i, which corresponds to the gacTRa(yac) term, is dominant in the sense that it is about ten times larger than ηel2,i, which comes from the estimate for the perturbation term ê TMae. The fluctuations due to the limited machine precion in the atomistic region come from ηat2,i, whereas those in the continuum region away from the defect stem from ηel2,i. Let us note that in other applications of duality-based error estimation, the first term might not always be the dominant term. For example, in mesh refinement for classical linear finite elements, the first term even vanishes due to Galerkin orthogonality. Table 2 and Figure 5, which display the same data, show the efficiency of the error estimators, η1/|Q(y a − yac)| and η2/|Q(y a − yac)|, for M = 1000. For comparison, the actual error is given as well. For the relatively small 1D problem, the actual error can be easily computed, whereas in real world applications it is of course not available. One can clearly see that η1 gives a better estimate than η2, which numerically confirms our conjecture that η1 is a better estimator than η2. We see that η1 overestimates the actual error by a factor of 1.4, while η2 is in a still acceptable range of ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 20 K |Q(ya − yac)| η1 η1/|Q(y a − yac)| η2 η2/|Q(y a − yac)| 0 3.627633e-02 3.899208e-02 1.074863 3.999783e-02 1.102588 2 3.375762e-02 3.872272e-02 1.147081 5.101700e-02 1.511274 4 3.468605e-03 4.343595e-03 1.252260 5.422007e-03 1.563166 6 5.418585e-04 7.156249e-04 1.320686 9.187940e-04 1.695635 8 1.227067e-04 1.675383e-04 1.365356 2.193196e-04 1.787348 10 3.287188e-05 4.540984e-05 1.381419 5.984186e-05 1.820457 15 1.416914e-06 1.966114e-06 1.387603 2.597488e-06 1.833201 20 6.267636e-08 8.695824e-08 1.387417 1.148736e-07 1.832805 25 2.770161e-09 3.843388e-09 1.387424 5.077204e-09 1.832819 30 1.224369e-10 1.698739e-10 1.387440 2.244073e-10 1.832840 35 5.410783e-12 7.508365e-12 1.387667 9.918687e-12 1.833133 40 2.379208e-13 3.318024e-13 1.394592 4.383361e-13 1.842362 45 8.992806e-15 1.430733e-14 1.590975 1.901601e-14 2.114580 50 7.771561e-16 4.120094e-16 0.530150 6.201285e-16 0.797946 Table 2: Efficiency of the error estimators, η1/|Q(y a − yac)| and η2/|Q(y a − yac)|, for M = 1000. For K = 45 and K = 50 the results become inaccurate due to limited machine precision. 0 5 10 15 20 25 30 35 40 45 50 |Q(ya−yac)| Figure 5: Efficiency of the error estimators for M = 1000. ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 21 τgl optimal K K by η1 K by η2 1e-02 3 3 3 1e-03 5 5 5 1e-04 9 9 10 1e-05 12 13 13 1e-06 16 17 17 1e-07 20 20 21 1e-08 23 24 24 1e-09 27 28 28 1e-10 31 31 32 1e-11 35 35 35 1e-12 38 39 39 1e-13 42 42 43 1e-14 45 46 47 Table 3: Efficiency of the error estimators for M = 1000. 2 times the actual error. Moreover, we can see from Table 2 and Figure 5 that the error decreases exponentially with K. Finally, we compare the optimal (smallest) value of K which is needed to achieve a given accuracy τgl with the values for K determined by the error estimators η1 and η2, again taking into account the precise error which is available for the model problem. We see from Table 3 that even η2 only overestimates K by at most 2 atoms. Thus, we get an efficient estimate of the required atomistic region for both error estimators. Appendix A. Matrix Definitions We describe the matrices from Section 4.2. The matrix . . . . . . ∈ R(2M−1)×2M (A.1) transforms atomistic positions to distances between adjacent atoms. Similarly, Dqc = −ν−1−N+1 ν −ν−1−N+2 ν . . . . . . N−1 ν ∈ R(2N−1)×2N (A.2) transforms repatom positions from a coarsened chain to normalized distances between adjacent repatoms. The matrices (Ea)ij = k1 + k2 i = j ∈ {−M + 1,M − 1} k1 + 2k2 i = j ∈ {−M + 2, . . . ,M − 2} k2 |j − i| = 1 0 otherwise, (A.3) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 22 (Eac)ij = k12(δ i + δ i+1) + i + δ i+1) + i−1 + δ i + δ i+1 + δ i+2) i = j i + δ i+2) j = i+ 1 i−1 + δ i+1) j = i− 1 0 otherwise, (A.4) (Eqc)ij = ωik12 + i + δ i+1) + i−1 + δ i + δ i+1 + δ i+2) i = j i + δ i+2) j = i+ 1 i−1 + δ i+1) j = i− 1 0 otherwise, (A.5) for i, j = −M + 1, . . . ,M − 1 and i, j = −N + 1, . . . , N − 1, respectively, describe the spring interactions in terms of the distances between atoms or repatoms. Accordingly, the matrices . . . ∈ R2M×2M (A.6) (Kqc)ij = (2νi−1 + ν i−1) + (2νi + ν i = j ∈ {−N + 2, . . . , N − 1} k0(2ν−N+1 + ν −N+1) i = j = −N + 1 k0(2νN−1 + ν N−1) i = j = N k0(νi − ν i ) j = i+ 1 k0(νj − ν j ) j = i− 1 0 otherwise, (A.7) for i, j = −N + 1, . . . , N describe the misfit interactions for the original atomistic system and the QC approximation. 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Vemaganti, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials: Part I: Error estimates and adaptive algorithms, J. Comput. Phys., 164 (2000), pp. 22–47. [20] C. Ortner and E. Süli, A-posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension, Research Report NA-06/13, Oxford University Computing Laboratory, 2006. [21] , A-priori analysis of the quasicontinuum method in one dimension, Research Report NA-06/12, Oxford University Computing Laboratory, 2006. [22] E. B. Tadmor, R. Miller, R. Phillips, and M. Ortiz, Nanoindentation and incipient plasticity, J. Mater. Res., 14 (1999), pp. 2233–2250. [23] E. B. Tadmor, M. Ortiz, and R. Phillips, Quasicontinuum analysis of defects in solids, Philos. Mag. A, 73 (1996), pp. 1529–1563. [24] E. B. Tadmor, R. Phillips, and M. Ortiz, Mixed atomistic and continuum models of deformation in solids, Langmuir, 12 (1996), pp. 4529–4534. 1. Introduction 2. Quasicontinuum Approximation 2.1. The Atomistic System 2.2. The Atomistic-Continuum Energy 2.3. Repatoms: Reduction of Degrees of Freedom 2.4. The Quasicontinuum Energy 3. Frenkel-Kontorova Model 3.1. Atomistic Frenkel-Kontorova Model 3.2. Quasicontinuum Approximation of the Frenkel-Kontorova Model 4. Primal and Dual Problems 4.1. Problem Setup 4.2. Matrix Formulation 4.3. Goal-Oriented Error Estimation 4.4. Dual Problems 5. Error Estimation for Atomistic vs. Continuum Modeling 5.1. Auxiliary Estimates 5.2. First Error Estimator 5.3. Second Error Estimator 6. Numerics 6.1. Algorithm 6.2. Numerical Results Appendix A. Matrix Definitions References

We propose and analyze a goal-oriented a posteriori error estimator for the atomistic-continuum modeling error in the quasicontinuum method. Based on this error estimator, we develop an algorithm which adaptively determines the atomistic and continuum regions to compute a quantity of interest to within a given tolerance. We apply the algorithm to the computation of the structure of a crystallographic defect described by a Frenkel-Kontorova model and present the results of numerical experiments. The numerical results show that our method gives an efficient estimate of the error and a nearly optimal atomistic-continuum modeling strategy.

Introduction The quasicontinuum (QC) method [22, 23, 24] has been successfully used to efficiently cou- ple atomistic and continuum models for crystalline solids and offers the possibility of computing mesoscale or macroscale properties by a nearly minimal number of degrees of freedom. Accurate modeling requires that an atomistic model be used in regions with highly non-uniform deformations such as around dislocations, whereas a continuum model can be used in regions with nearly uniform deformations to reduce the number of degrees of freedom. It is usually not known a priori which regions of some specimen undergo uniform deformations and which do not, so a posteriori error estimation is important for the design of efficient numerical approximations by the quasicontinuum method. Since the purpose of a computation is often to obtain the value of a (usually local) quantity of interest to a desired error tolerance rather than to obtain a solution to a desired error tolerance for a global norm, there has been great interest in the development of goal-oriented error estimators for many problems. They are based on duality techniques and have been developed and used to adaptively refine finite element approximations of continuum problems [1, 3] and to study and control modeling error [19]. In this paper, we extend this approach to develop an a posteriori error estimator for the qua- sicontinuum method which quantifies the atomistic-continuum modeling error for a goal function and allows for an adaptive decision about which regions can be accurately modeled as a continuum and which regions need to be modeled atomistically. Methods to determine the optimal mesh size within the continuum region will be studied in a forthcoming paper. Crystallographic defects [5] provide a challenge to validate atomistic-continuum error estimators and adaptivity. No such error estimators and adaptive methods currently exist for fully three- dimensional crystals. As a step in this direction, we develop a rigorous theory for a simple one- dimensional atomistic model for a defect that is a modification of the Frenkel-Kontorova model [15]. We add next-nearest-neighbor harmonic interactions between the atoms to the nearest-neighbor harmonic interactions between the atoms in the classical Frenkel-Kontorova model. A priori analyses for various quasicontinuum approximations have been given in [4, 8, 9, 10, 12, 13, 14, 21]. An a posteriori analysis for a slightly different one-dimensional quasicontinuum http://arxiv.org/abs/0704.1924v1 ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 2 approximation is given in [20]. The development and application of a goal-oriented error estimator for mesh coarsening in a two-dimensional quasicontinuum method is reported in [17, 18]. Let us mention that the continuum model used in the QC method, which coincides with the model obtained by the classical thermodynamic limit, is by far not the only reasonable continuum model to use. A method to derive continuum models which approximate atomistic models up to an arbitrarily high order has been proposed in [2]. The paper is organized as follows. In Section 2, we give a general formulation of the one- dimensional quasicontinuum approximation [23] that includes not only two-body and three-body potentials, but also many body potentials such as the embedded atom potential [6, 7]. In Section 3, we describe our extension of the Frenkel-Kontorova model and its quasicontinuum approximation. In Section 4, we introduce the primal and dual problems for our model and formulate our approach to goal-oriented error estimation. Next, in Section 5 we extend the approach in [16] to develop an error estimator for atomistic- continuum modeling. This first error estimator does not allow a decomposition among the atoms that can be used for atomistic-continuum adaptivity, so we propose and analyze a less accurate second error estimator that does allow such a decomposition. Finally, in Section 6 we propose an adaptive atomistic-continuum modeling algorithm and show that it gives an efficient estimate of the modeling error and a nearly optimal atomistic-continuum modeling strategy for the computation of defect structure. 2. Quasicontinuum Approximation The departure point for the QC approximation is the potential energy of the atomistic system. The potential energy that is utilized fully models the properties of the system. The local minima of the potential energy model the metastable states of the system, and the potential energy can be used in Newton’s equations of motion to model the dynamical behavior. The QC method approximates the potential energy of the atomistic system in two steps. First, we develop a continuum potential energy that will be used in the adaptively determined continuum region, and we then show how to reduce the degrees of freedom in the continuum region. 2.1. The Atomistic System. We assume that the atomistic system has 2M atoms with deforma- tion given by ya = (ya−M+1, . . . y M) ∈ R 2M . Without loss of generality, we assume that the atoms are ordered so that their positions satisfy yai < y i+1. Furthermore, we assume that the atomistic total potential energy, Ea(ya), can be written as a sum over potential energies associated with each atom, Eai (y a), so that Ea(ya) = i=−M+1 Eai (y a). (2.1) This decomposition can be found for most empirical potentials, including embedded atom potential energies [6, 7]. For example, if the atomistic total potential energy Ea(ya) is given by Ea(ya) = ψ(yaj − y i ), (2.2) where ψ(r) is an empirical two-body potential energy, then we can obtain the decomposition (2.1) by taking Eai (y j 6=i ψ(yaj − y i ). (2.3) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 3 We note that Eai (y a) can also contain contributions from external forces, such as for the Frenkel- Kontorova model described in Section 3, and can thus depend on i. 2.2. The Atomistic-Continuum Energy. For any deformation ya ∈ R2M , we let Li,i+1ya ∈ RZ denote the linear extrapolation of the atomistic positions yai and y i+1 given by (Li,i+1ya)k = (k − i)y i+1 + (i+ 1− k)y i for k = −∞, . . . ,∞. (2.4) The continuum potential energy Eci (y a) of atom i is obtained from the average of the atomistic potential energy Eai evaluated at the extrapolations L i−1,iya and Li,i+1ya by Eci (y a) := 1 Eai (L i−1,i a) + 1 Eai (L i,i+1 a), (2.5) where we note that the domain of Eai has been expanded to the infinite periodic atomistic systems in the range of Li−1,i and Li,i+1. We assume that Eai is finite for infinite periodic atomistic systems, which is true for (2.3) when the two-body potential ψ(r) decays fast enough so that k=1 ψ(kr) is finite for r 6= 0. At the endpoints of the chain, the extrapolation can be done only to one side, so we neglect the undefined part and define Ec−M+1(y a) := 1 Ea−M+1(L −M+1,−M+2 a) and EcM (y a) := 1 EaM (L M−1,M a). (2.6) We then decide for each atom i whether to model its energy atomistically by Eai (y a) or as a continuum by Eci (y a). We thus obtain for the whole chain the atomistic-continuum energy Eac(ya) := i=−M+1 δai E i=−M+1 δci E i=−M+1 δai E a) + 1 i=−M+2 i−1,i a) + 1 i=−M+1 δci E i,i+1 (2.7) where δai = 1 if atom i is modeled atomistically, 0 if atom i is modeled as continuum, and δci = 1− δ i . (2.8) This approximation allows for a slightly faster evaluation of the energy and its derivatives, especially if Eai is long-ranged. However, it reveals its full strength only after the quasicontinuum coarsening to be described next. We note that sometimes atomistic degrees of freedom and energies are referred to as nonlocal and continuum degrees of freedom and energies are referred to as local [23]. 2.3. Repatoms: Reduction of Degrees of Freedom. The quasicontinuum method allows a reduction of the number of degrees of freedom in the continuum region. To this end, we choose so-called representative atoms, or more briefly called repatoms. The repatoms are a subset of the original atoms. The quasicontinuum approximation of the energy is defined completely in terms of the repatoms. We choose the repatoms by defining indices ℓj for j = −N + 1, . . . , N where −M + 1 = ℓ−N+1 < · · · < ℓj < ℓj+1 < · · · < ℓN =M. The atoms at yai for i = ℓ−N+1, ℓ−N+2, . . . , ℓN are repatoms, and all of the remaining atoms are non-repatoms. We have that νj = ℓj+1 − ℓj (2.9) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 4 gives the number of atomistic intervals between the repatoms ℓj and ℓj+1.We require that the chain not be coarsened in the atomistic regions, which precisely means that δc = . . . = δc whenever νj > 1. Finally, the interactions of the atomistic energy only partially reach into the continuum part if the atomistic potential has a finite cutoff radius. To allow for an exact calculation of this energy without atomistic interpolation, we require that these regions are not coarsened as well. As we will see in the next subsection, the atomistic next-nearest-neighbor interactions from the Frenkel- Kontorova model studied in this paper reach two atoms into the continuum part. Hence, we require that νj−2 = νj−1 = νj = νj+1 = 1 whenever δ = 1. Other potential energies in general require similar conditions that depend on their cut-off radius. We denote the position of the j-th repatom by y j = y and the vector of all repatoms by yqc ∈ R2N . 2.4. The Quasicontinuum Energy. Now we define the quasicontinuum energy. To this end, the missing non-repatoms are implicitly reconstructed. We will see later that this helps to set up the QC model, but needs not be done for the actual computation. The reconstruction is done by a linear interpolation between the nearest repatom to the right and to the left. That is, the vector of all atomistic positions is computed from the vector yqc of repatom positions by I : R2N → R2M , (Iyqc)ℓj+m := νj −m j+1, m = 0, . . . , νj . (2.10) We note that j = (Iy qc)ℓj . (2.11) The underlying idea is that in regions where the lattice spacing of the atoms is nearly constant, this interpolation is very close to the actual atomistic positions and therefore leads to a good approximation of the total energy. Only a few repatoms are needed in these regions. This exactly corresponds to mesh coarsening in classical finite element approximations of continuum models. On the other hand, in regions where the lattice spacing is non-uniform, such as around a dislocation, all atoms must be chosen to be repatoms to obtain sufficient accuracy. This guarantees that the full resolution of the atomistic model in the critical regions is retained and corresponds to a high refinement in classical finite element continuum models. We define the QC approximation of the total energy to be Eqc(yqc) := Eac(Iyqc). (2.12) Now (2.12) has to be reformulated such that it can be computed efficiently, without the overhead of the interpolation. Most atomistic potentials are invariant to translations, a property that al- lows us to simplify (2.12) considerably. For any translationally invariant energy Eai , we have that Eai (L i,i+1ya) = φi(y i+1 − y i ) and E i−1,iya) = φi(y i − y i−1) for some function φi. If these functions φi coincide, that is, φi = φj for all i and j, we can write Eai (L i−1,i a) = φ yai − y and Eai (L i,i+1 a) = φ yai+1 − y (2.13) for some function φ : R→ R. Here φ plays the role of a continuum energy density and is given for the two-body potential (2.2) by φ(r) = ψ(kr). ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 5 Equations (2.7), (2.12), and (2.13) lead to Eqc(yqc) = i=−M+1 δai E i (Iy qc) + 1 i=−M+2 i−1,iIyqc) + 1 i=−M+1 δci E i,i+1Iyqc) i=−M+1 δai E i (Iy qc) + 1 i=−M+2 δciφ((Iy qc)i − (Iy qc)i−1) (2.14) i=−M+1 δciφ((Iy qc)i+1 − (Iy qc)i). Because Iyqc is the linear interpolation between two repatoms y j and y j+1, we have (Iyqc)i+1 − (Iy qc)i = j+1 − y , i = ℓj, . . . , ℓj+1 − 1. (2.15) Hence, Eqc(yqc) = i=−M+1 δai E i (Iy qc) + j=−N+1 j+1 − y (2.16) with weight factors δcℓj + δ 0 if both y j and y j+1 are atomistic, if exactly one of y j and y j+1 is continuum, νj if both y and y j+1 are continuum. (2.17) The first sum corresponds to the atomistic region which will be a small region and is thus com- putationally inexpensive. The second sum only involves at most 2N terms which is a considerable reduction when N ≪M. Note that the second term in formula (2.16) coincides with an integral over the energy density φ as it occurs in finite element discretizations of classical continuum mechanical models. Hence the apparently unmotivated definitions (2.7) and (2.5) of the continuum energy here result in what is commonly understood as a continuum energy. The linear interpolation operator I resembles the Cauchy-Born hypothesis. 3. Frenkel-Kontorova Model Dislocations are lines in crystals which represent a defect in the lattice structure [15], see Fig- ure 1. Typically, there is a core of small radius surrounding the dislocation line where the lattice structure is highly deformed, but the lattice structure is nearly uniform outside the core. A sim- ple one-dimensional model for a defect such as a dislocation is given by the Frenkel-Kontorova model [15]. Here, the elastic energy is modeled by harmonic interactions between the atoms in the one-dimensional chain and the misfit energy of the slip plane is modeled by a periodic potential. A more accurate model of the same form is given by the Peierls-Nabarro model [11]. 3.1. Atomistic Frenkel-Kontorova Model. We study a single defect in the middle of the chain of 2M atoms. To achieve a symmetric description in terms of bonds, we number the atoms from −M + 1 to M . The defect is situated between the atoms numbered 0 and 1 (Figure 2). Recall that the atomistic positions are denoted by ya = (ya−M+1, . . . , y M ) ∈ R 2M . The total potential energy for this atomistic system is then a function Ea : R2M → R of the atomistic ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 6 Figure 1: Cross-section through a dislocation in a three-dimensional cubic lattice. The displayed plane repeats periodically in the three-dimensional crystal. Vertical bonds are shown by lines to emphasize the topological defect. (−K−1)a (K+1)a = atomistic = continuum Figure 2: Numbering of the atoms. The dislocation is situated in the middle of the chain between atoms ya and ya positions. For the Frenkel-Kontorova model, the energy, Ea = Ea,e + Ea,m, consists of two parts, namely the part which models the elastic energy of the defect, Ea,e, and the part which models the misfit energy on the slip plane, Ea,m. The elastic energy is modeled by Hookean (harmonic) springs between nearest-neighbors (NN) and next-nearest neighbors (NNN), and the total elastic energy is given by Ea,e(ya) = i=−M+1 i+1 − y i − a0) i=−M+2 i+1 − y i−1 − 2a0) 2, (3.1) where the moduli k1 > 0 and k2 > 0 describe the strength of the elastic interactions, and where a0 ∈ R denotes the equilibrium distance. We note that the asymptotic expansion to second order of any nonlinear NN/NNN potential energy E(ya) = i=−M+1 ψ(yai+1 − y i ) + i=−M+2 ψ(yai+1 − y i−1) (3.2) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 7 substrate modeled layer misfit energy Figure 3: Frenkel-Kontorova model. The wells depict the misfit energy (3.4). about aa = [(−M + 1)a0, (−M + 2)a0, · · · , (M − 1)a0,Ma0] ∈ R2M has the form E(ya) ≈ E(aa) + ψ′(a0) + 2ψ ′(2a0) i=−M+1 (yai+1 − y i − a0) − ψ′(2a0)(y M − y M−1 − a0)− ψ ′(2a0)(y −M+2 − y −M+1 − a0) ψ′′(a0) i=−M+1 (yai+1 − y i − a0) 2 + 1 ψ′′(2a0) i=−M+2 (yai+1 − y i−1 − 2a0) (3.3) We thus see that the elastic energy (3.1) with k1 = ψ ′′(a0) and k2 = ψ ′′(2a0) approximates the energy (3.2) to second order if ψ′(a0) + 2ψ ′(2a0) = 0 and if we ignore the boundary terms in the second line of (3.3). The misfit energy of the slip plane is modeled by a periodic potential (Figure 3). We model this misfit energy by Ea,m(ya) = i=−M+1 yai − a0 , (3.4) where ⌊x⌋ denotes the largest integer smaller than or equal to x, and where the constant k0 > 0 determines the strength of the misfit energy. Altogether, the total potential energy of the atomistic system is given by Ea(ya) = Ea,e(ya) + Ea,m(ya) i=−M+1 (yai+1 − y i − a0) 2 + 1 i=−M+2 (yai+1 − y i−1 − 2a0) i=−M+1 yai − a0 (3.5) We restrict ourselves to configurations in which the M leftmost atoms yai for −M + 1 ≤ i ≤ 0 are situated in the interval , whereas the M rightmost atoms yai for 1 ≤ i ≤ M are situated in the interval . The defect is situated between atoms ya0 and ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 8 ya1 . In this case, the total energy simplifies to Ea(ya) = 1 i=−M+1 (yai+1 − y i − a0) 2 + 1 i=−M+2 (yai+1 − y i−1 − 2a0) i=−M+1 (yai − (i− 1)a0) (yai − ia0) (3.6) 3.2. Quasicontinuum Approximation of the Frenkel-Kontorova Model. We now apply the quasicontinuum method to the dislocation model described in Section 3.1. The total energy (3.6) is split up into atom-wise contributions, separately for the elastic interactions and the misfit interactions: a) = 1 i − y i−1 − a0) 2 + 1 i+1 − y i − a0) i − y i−2 − 2a0) 2 + 1 i+2 − y i − 2a0) k0 (y i − (i− 1)a0) , i = −M + 1, . . . , 0, k0 (y i − ia0) , i = 1, . . . ,M. (3.7) To simplify notation, we use the convention that the undefined terms at the endpoints of the chain are neglected. We thus have that Ea(ya) = Ea,m(ya) + Ea,e(ya) = i=−M+1 a) + E a)] . (3.8) Since the largest displacement of the atoms is to be expected near the defect, we deem the atoms −K +1, . . . ,K atomistic and the remaining atoms −M +1, . . . ,−K and K +1, . . . ,M continuum. Here K < M is some constant whose optimal value will be determined by the algorithm given in Section 5. The optimal choice of the repatoms for coarsening is investigated in the second paper of this series, so we work with a general formulation which holds for any values of ℓj for now. However, there are two restrictions on the coarsening. Since the atomistic region must not be coarsened and since we need full refinement in the vicinity of two atoms around the atomistic region due to the NNN interactions, we have that ℓj = j, j = −K − 1, . . . ,K + 2. (3.9) Second, we require that ℓ−N+1 = −M + 1, ℓ−N+2 = −M + 2, ℓN−1 =M − 1, and ℓN =M (3.10) to incorporate the boundary conditions later. The elastic part E i is translationally invariant, so we perform its QC approximation as described in the previous section. This leads to the continuum energy density φe(r) = 1 k1(r − a0) 2 + 1 k2(2r − 2a0) k12(r − a0) 2 (3.11) where k12 := k1 + 4k2. Regarding the misfit part E i , the above technique cannot be applied since the potential is not translationally invariant. However, there is a different summation technique to achieve a computationally efficient formulation which avoids the costly interpolation operator. ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 9 To shorten the notation, we let indicate the sum in which the first term and the last term are only counted half: xi := i=m+1 xn (3.12) where m < n and xi ∈ R. It is easy to verify that 2m3 +m i(m− i) = m3 −m (3.13) for m > 0. For all pairs (j, j + 1) of continuum repatoms, we now reformulate all terms from (2.7) which involve the interaction between ℓj and ℓj+1. For j > 0, we get by definition (2.4) of the operator L, by definition (3.7) of E i , and by (3.13) that i=ℓj+1 i−1,iIyqc) + 1 ℓj+1−1 i,i+1Iyqc) i=ℓj+1 i (Iy qc) + 1 ℓj+1−1 i (Iy ℓj+1 − i i− ℓj j+1 − ia0 ℓj+1 − i j − ℓja0) + i− ℓj j+1 − ℓj+1a0) j − ℓja0) (ℓj+1 − i) 2 + 1 j+1 − ℓj+1a0) (i− ℓj) − ℓja0)(y j+1 − ℓj+1a0) (ℓj+1 − i)(i − ℓj) j − ℓja0) 2ν3j + νj j+1 − ℓj+1a0) 2ν3j + νj j − ℓja0)(y j+1 − ℓj+1a0) ν3j − νj =: φmj+(y j , y j+1). ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 10 For j < 0, we similarly obtain i=ℓj+1 (Li−1,iIyqc) + 1 ℓj+1−1 (Li,i+1Iyqc) j − (ℓj − 1)a0) 2ν3j + νj j+1 − (ℓj+1 − 1)a0) 2ν3j + νj j − (ℓj − 1)a0)(y j+1 − (ℓj+1 − 1)a0) ν3j − νj =: φmj−(y j , y j+1). Since E −K (L −K,−K+1Iyqc) = E −K (Iy qc) and E K+1(L K,K+1Iyqc) = E K+1(Iy qc), the QC approx- imation of the chain can be given by Eqc(yqc) = j=−N+1 j+1 − y + φmj−(y j , y −K (Iy qc) + j=−K+1 Eai (Iy qc) + 1 K+1(Iy j+1 − y + φmj+(y j , y (3.14) Note that the interpolation Iyqc does not have to be computed here since the relevant terms only depend on uncoarsened parts of the chain. Additionally, we will consider the atomistic-continuum approximation Eac : R2M → R (3.15) of the atomistic energy without coarsening. It is given exactly like the QC approximation (3.14) with the only difference being that νj = 1 and ℓj = j everywhere. 4. Primal and Dual Problems 4.1. Problem Setup. We are now ready to set up the problems we will solve. We are interested in finding the minimum of the energy (3.14) subject to given boundary conditions. We give the boundary conditions by constraining the deformation of two atoms at each end of the chain. This guarantees that the potential with next-nearest-neighbor interactions can be directly applied to all non-boundary atoms without having to neglect interactions. We define the spaces V a := R2M , V a0 := R 2M−4, V qc := R2N , V 0 := R 2N−4. (4.1) The spaces V a and V a0 will also be used for the uncoarsened atomistic-continuum potential E so there is no need to define spaces V ac and V ac0 . We let y bc ∈ V a denote any vector which has the desired boundary values ybc−M+1, y −M+2, y M−1, and y M , and we let y bcq ∈ V qc by any vector satisfying (recall (3.10)) −N+1 = y −M+1, y N−1 = y −N+2 = y −M+2, y = ybcM . (4.2) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 11 For any vector y ∈ V a0 , we denote the extension by zero boundary conditions to be Jy ∈ V a, so Jy := 0 0 yT 0 0 ∈ R2M , (4.3) and similarly we denote the extension by zero boundary conditions of y ∈ V 0 to be J qcy ∈ V qc. The spaces of admissible solutions are then given by JV a0 + y bc ⊂ V a and JqcV 0 + y bcq ⊂ V qc, respectively. We note that JT : V a → V a0 is the restriction operator defined by (JTy)j = yj for j = −M + 3, . . . ,M − 2. The minima ȳa, ȳac, and ȳqc of the energy functionals Ea, Eac, and Eqc given by (3.6), (3.15), and (3.14) subject to the above “clamped” boundary conditions are characterized as a := argmin y∈JV a Ea(y) ∈ V a, (4.4) ac := argmin y∈JV a Eac(y) ∈ V a, (4.5) qc := argmin y∈JqcV Eqc(y) ∈ V qc. (4.6) We note that the minima are uniquely determined because Ea, Eac, and Eqc are strictly convex. 4.2. Matrix Formulation. For the subsequent discussion, it will be convenient to reformulate the total energies in matrix notation: Ea(y) = 1 (y − aa)TDaTEaDa(y − aa) + 1 (y − ba)TKa(y − ba), (4.7a) Eac(y) = 1 (y − aa)TDaTEacDa(y − aa) + 1 (y − ba)TKa(y − ba), (4.7b) Eqc(y) = 1 (y − aqc)TDqcTEqcDqc(y − aqc) + 1 (y − bqc)TKqc(y − bqc). (4.7c) The matrices Da ∈ R(2M−1)×2M and Dqc ∈ R(2N−1)×2N compute the distance between two adja- cent atomistic positions; the matrices Ea ∈ R(2M−1)×(2M−1), Eac ∈ R(2M−1)×(2M−1), and Eqc ∈ (2N−1)×(2N−1) contain the spring constants k1, k2, and k12; and the matricesK a ∈ R(2M−1)×(2M−1) and Kqc ∈ R(2N−1)×(2N−1) contain the misfit constant k0. The vectors a a,ba ∈ R2M and aqc,bqc ∈ 2N are constants describing the minimum energy deformations for the elastic energy and mis- fit energy. The precise and lengthy definitions for all of these matrices and vectors are given in Appendix A. If we decompose ȳa = Jya + ybc for ya ∈ V a0 , then the minimization problem is given as a = argmin y∈V a Ea(Jy + ybc) = argmin y∈V a Jy + ybc − aa DaTEaDa Jy + ybc − aa Jy + ybc − ba Jy + ybc − ba . (4.8) We also decompose ȳac = Jyac + ybc and ȳqc = Jqcyqc + ybcq for yac ∈ V a0 and y qc ∈ V 0 , and we then formulate similar minimization problems for yac and yqc. Therefore, ya, yac, and yqc are determined by the linear systems Maya = fa, (4.9a) Macyac = fac, (4.9b) M qcyqc = f qc, (4.9c) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 12 where Ma := JT (DaTEaDa +Ka)J, Mac := JT (DaTEacDa +Ka)J, M qc := JqcT (DqcTEqcDqc +Kqc)Jqc, a := −JTDaTEaDa(ybc − aa)− JTKa(ybc − ba), ac := −JTDaTEacDa(ybc − aa)− JTKa(ybc − ba), qc := −JqcTDqcTEqcDqc(ybcq − aqc)− JqcTKqc(ybcq − bqc). (4.10) We note that the matrices Ma, Mac, and M qc are positive definite, so the total energies admit a single global minimum and no other local minimum. 4.3. Goal-Oriented Error Estimation. To compare the approximate QC model to the original atomistic model, we have to analyze how much the solution ya of the atomistic model deviates from the solution yqc of the QC model. This deviation, which can be viewed as an approximation error, can be measured in different ways, for example as ‖ya−JT IJqcyqc‖ for some norm ‖·‖. Here we follow a different approach, namely we measure the error of a quantity of interest denoted by Q(y) for some function Q : R2M−4 → R. Hence, we intend to estimate Q(ya)−Q(JT IJqcyqc). (4.11) We will assume for simplicity that Q is linear and thus has a representation Q(y) = qTy for some vector q ∈ V a0 . For our application, a natural quantity of interest is the size of the dislocation, that is, the distance between the two atoms y0 and y1 to the left and right of the dislocation. This gives us Q(y) = qTy = y1 − y0 with q = [0, . . . , 0,−1, 1, 0, . . . , 0] T . (4.12) Two different sources of error arise during the QC approximation, namely the localization of the potential energy, that is, the passage from the atomistic to the continuum formulation on the one hand, and the coarsening in the continuum region by the restriction to the repatoms on the other hand. We denote these two errors by e := ya − yac and eacqc := yac − JT IJqcyqc. (4.13) It makes sense to study these sources independently. Employing the linearity of Q, we have that |Q(ya)−Q(JT IJqcyqc)| = |Q(e) +Q(eacqc)| ≤ |Q(e)|+ |Q(eacqc)|. (4.14) The error term |Q(e)| will be studied in Section 5, and the error term |Q(eacqc)| will be studied in the second part of this paper series. 4.4. Dual Problems. To facilitate the goal-oriented error analysis, we introduce the dual problems Maga = q, (4.15a) Macgac = q, (4.15b) M qcgqc = JqcT ITJq, (4.15c) for ga,gac ∈ R2M−4, and gqc ∈ R2N−4. We note that the dual problems differ from the primal problems only by the right hand side since the matrices Ma, Mac, and M qc are symmetric. The solutions ga, gac and gqc can be viewed as influence functions: They describe how the error at a specific point in the domain influences the error measured in terms of the goal function. ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 13 Analogously to the primal errors (4.13), we define the dual errors ê := ga − gac and êacqc := gac − JT IJqcgqc. (4.16) In addition, we will need the primal and dual residuals Ra(y) :=Ma (ya − y) = fa −May, Rac(y) :=Mac (yac − y) = fac −Macy, R̂a(g) :=Ma (ga − g) = q−Mag, R̂ac(g) :=Mac (gac − g) = q−Macg. (4.17) 5. Error Estimation for Atomistic vs. Continuum Modeling In this section, we estimate the error |Q(e)| arising from the approximation of an atomistic model by a continuum model. We consider yac and gac to be computable, although in practice we can only compute the coarsened approximations yqc and gqc. To this end, we adapt a technique introduced in [16] and [19] to estimate the modeling error for an elasticity model with rapidly oscillating coefficients and its homogenized version. We generalize this technique such that it allows for different right hand sides fa and fac of the primal problem (4.9) instead of a common right hand side as it is used in the above-mentioned works. We have Q(ya)−Q(yac) = qTe = gaTMae = (gacT + ê)Mae = gacTRa(yac) + êTMae. (5.1) The term gacTRa(yac) can be computed, whereas êTMae cannot because both e and ê are nu- merically unknown. Instead, we estimate êTMae from above and from below by quantities that actually can be computed. We will give two different error estimators η1 and η2. Before, we need to derive some auxiliary estimates to facilitate their development and analysis. 5.1. Auxiliary Estimates. We reformulate the difference ya − yac of the respective solutions in terms of a difference of the energy matrices. To this end, we define the perturbation matrix P := I − (Ea)−1Eac (5.2) where I denotes the identity matrix. Note that EaP = E a − Eac. Lemma 5.1. For any α, β ∈ R, we have that Ma(αe+ βê) = −JTDaTEaPDa α(Jyac + ybc − aa) + βJgac . (5.3) Proof. We conclude from (4.9) that Mae =Maya −Macyac + (Mac −Ma)yac = fa − fac + (Mac −Ma)yac, (5.4) and similarly since Maga =Macgac = q that Maê =Ma(ga − gac) = (Mac −Ma)gac. (5.5) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 14 Thus, it follows from (4.10) and (5.2) that Ma(αe+ βê) = α [(Mac −Ma)yac + fa − fac] + β(Mac −Ma)gac = JTDaT (Eac − Ea)Da α(Jyac + ybc − aa) + βJgac = −JTDaTEaPDa α(Jyac + ybc − aa) + βJgac (5.6) We note that the Ka-related terms cancel here, because they coincide for the atomistic model and the continuum model. � Lemma 5.2. We have that ‖αe+ βê‖Ma ≤ α(Jyac + ybc − aa) + βJgac . (5.7) We note that the right hand side is numerically computable. Proof. To shorten the notation, we abbreviate z = α(Jyac +ybc− aa)+ βJgac. By Lemma 5.1, we ‖αe + βê‖Ma = sup v∈V a vTMa(αe+ βê) ‖v‖Ma = sup v∈V a −vTJTDaTEaPDaz ‖v‖Ma ≤ sup v∈V a ‖DaJv‖Ea‖PD az‖Ea ‖DaJv‖Ea = ‖PDaz‖Ea . (5.8) Here we have used that ‖DaJv‖Ea ≤ ‖v‖Ma because the matrixK a in (4.10) is positive definite. � 5.2. First Error Estimator. We are now ready to derive the first error estimator, η1. By the parallelogram identity, we have for all σ 6= 0 that TMae = (σ−1êT )Ma(σe) ‖σe+ σ−1ê‖2Ma − ‖σe− σ−1ê‖2Ma . (5.9) In the following, we will determine computable constants η+low, η low, η upp and η upp such that η+low ≤ ‖σe+ σ ê‖Ma ≤ η η−low ≤ ‖σe− σ ê‖Ma ≤ η (5.10) From Lemma 5.2, we immediately get the upper estimates η+upp and η η+upp := σ(Jyac + ybc − aa) + σ−1Jgac η−upp := σ(Jyac + ybc − aa)− σ−1Jgac (5.11) We note that η+low, η low, η upp and η upp will depend on σ, but the estimates (5.10) will hold for any σ 6= 0. We will now choose σ in such a way that the estimates are as sharp as possible, that is, such that η+upp and η upp are smallest. Lemma 5.3. Both η+upp and η upp given by (5.11) attain their minima for σ̄ := ‖PDaJgac‖Ea ‖PDa(Jyac + ybc − aa)‖Ea . (5.12) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 15 Proof. We have that (η±upp) 2 = σ2 ∥PDa(Jyac + ybc − aa) ± 2gacTJTDaTP TEaPDa(Jyac + ybc − aa) + σ−2 ∥PDaJgac . (5.13) Setting the first derivative of the mapping σ 7→ (η±upp) 2 to zero, we obtain the condition ∥PDa(Jyac + ybc − aa) − 2σ̄−3 ∥PDaJgac = 0 (5.14) for critical points of (η±upp) 2. This equation has the unique positive solution (5.12). Because lim|σ|→∞ η upp = limσ→0 η upp =∞, this point corresponds to a minimum. Hence the quantities η attain their minima at σ = σ̄. � Regarding the lower bounds η+low and η low, we have ‖σ̄e± σ̄−1ê‖Ma = sup v∈V a vTMa(σ̄e± σ̄−1ê) ‖v‖Ma = sup v∈V a vT (σ̄Ra(yac)± σ̄−1R̂a(gac)) ‖v‖Ma vT0 (σ̄R a(yac)± σ̄−1R̂a(gac)) ‖v0‖Ma (5.15) for any vector v0 ∈ V 0 \ {0}. Numerically, we have the two vectors y ac and gac at our disposal, hence it makes sense to take a linear combination v0 = y ac + θ±gac. Here we follow the strategy of [19] and choose θ± as the critical points of η±low. Lemma 5.4. Let ± = σ̄Ra(yac)± σ̄−1R̂a(gac). (5.16) Then the lower bounds (yac + θ±gac)T r± ‖yac + θ±gac‖Ma (5.17) have a unique critical point for θ̄± := r±Tyac gacTMayac − r±Tgac ‖yac‖2Ma r±Tgac gacTMayac − r±Tyac ‖gac‖2 . (5.18) Proof. We have r±Tgac‖yac + θ±gac‖Ma − r ±T (yac + θ±gac) (yac+θ±gac)TMagac ‖yac+θ±gac‖Ma ‖yac + θ±gac‖2 . (5.19) Setting this expression to zero and solving for θ± leads to the above condition. � However, let us note that this critical point is not necessarily a maximum of η±low, which would be optimal for bound (5.10). Depending on the actual vectors yac and gac, it can be shown that this critical point could be a minimum. Now we have all necessary ingredients to construct the error estimator η1. From (5.1) and (5.9), we get the computable estimate gacTRa(yac) + 1 )2 − 1 (η−upp) 2 ≤ Q(ya)−Q(yac) ≤ gacTRa(yac) + 1 (η+upp) 2 − 1 (5.20) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 16 At first sight, this looks like we could get an estimate for |Q(ya)−Q(yac)| from both above and below. However, this is only true if both the left hand side and the right hand side have the same sign, which in general does not hold. But we get the following estimate. Theorem 5.1. We have that |Q(ya)−Q(yac)| ≤ η1, (5.21) where the computable error estimator is defined as η1 := max ∣gacTRa(yac) + 1 )2 − 1 (η−upp) ∣gacTRa(yac) + 1 (η+upp) 2 − 1 . (5.22) We note that the computation of the η±upp terms involves the solution of a linear system with matrix Ea as its inverse appears in the operator P . The matrix Ea is not diagonal, but has condition number O(1). So this is negligible compared to what would be necessary to solve the original atomistic problem which includes the operator DaTEaDa with condition number O(M2). 5.3. Second Error Estimator. There is no reasonable way to decompose the error estimator η1 into a sum of element-wise or atom-wise contributions due to the η± terms. Therefore, we derive another error estimator η2 which allows for such a decomposition, at the price of a less accurate estimate than η1. Theorem 5.2. We have that |Q(ya)−Q(yac)| ≤ η2 ≤ i=−M+3 ηat2,i + i=−M+1 ηel2,i (5.23) where the computable global error estimator η2 and the computable local error estimators, η 2,i and ηel2,i, associated with atoms and elements, respectively, are defined as η2 := ∣gacTRa(yac) ∣+ ‖PDa(Jyac + ybc − aa)‖Ea‖PD aJgac‖Ea, ηat2,i := |g a(yac)i| , i = −M + 3, . . . ,M − 2, ηel2,i := PDa(Jyac + ybc − aa) (Ea − Eac)Da(Jyac + ybc − aa) ∣(PDaJgac)i (Ea − Eac)DaJgac ∣ , i = −M + 1, . . . ,M − 1. (5.24) Proof. From (5.1) and Lemma 5.2, we conclude that |Q(ya)−Q(yac)| ≤ acTRa(yac) acTRa(yac) ∣+ ‖ê‖Ma‖e‖Ma acTRa(yac) ∣+ ‖PDa(Jyac + ybc − aa)‖Ea‖PD aJgac‖Ea = η2, (5.25) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 17 which gives us the global estimate. For the decomposition into local contributions, we further estimate acTRa(yac) ∣+ ‖PDa(Jyac + ybc − aa)‖Ea‖PD aJgac‖Ea acTRa(yac) ‖PDa(Jyac + ybc − aa)‖2Ea + ‖PDaJgac‖2Ea i=−M+3 |gaci R a(yac)i|+ i=−M+1 ∣(PDaJgac)i (Ea − Eac)DaJgac i=−M+1 PDa(Jyac + ybc − aa) (Ea − Eac)Da(Jyac + ybc − aa) i=−M+3 ηat2,i + i=−M+1 ηel2,i, (5.26) which completes the proof. � Let us remark that instead of the first inequality in (5.26), one can get an apparently better estimate ‖PDa(Jyac + ybc − aa)‖Ea‖PD aJgac‖Ea γ‖PDa(Jyac + ybc − aa)‖2Ea + γ−1‖PDaJgac‖2Ea (5.27) by introducing the additional weight factor ‖PDaJgac‖Ea ‖PDa(Jyac + ybc − aa)‖Ea , (5.28) and then decomposing the resulting terms similar to the above. However, our numerical results showed that this modification does not significantly improve the decomposed error estimator for the application considered here. 6. Numerics In the preceding sections, we constructed the error estimators η1 and η2. We will now give an algorithm for adaptive atomistic-continuum modeling based on these error estimators. Then we will present and discuss some numerical results. 6.1. Algorithm. The error estimator η1 should give a better estimate of the error than η2, because η2 involves the inequality ∣êTMae ∣ ≤ ‖ê‖Ma‖e‖Ma in (5.25) in contrast to the parallelogram identity for η1. However, η2 can be decomposed into atom-wise and element-wise contributions η 2,i and ηel2,i, whereas the η terms in η1 do not admit a reasonable decomposition that can be used for atomistic-continuum adaptivity. We make use of this by employing the sharper estimate η1 to determine whether a given global error tolerance τgl for the error in an adaptive algorithm has already been achieved or not. If not, we use the decomposed estimates ηat2,i and η 2,i to determine where the more precise atomistic modeling is needed. This leads us to the following algorithm: (1) Choose τgl. Model all atoms as a continuum. Set τat ← τgl. (2) Solve primal problem (4.9b) for yac and dual problem (4.15b) for gac. (3) Compute error estimator η1 from (5.22). (4) If η1 ≤ τgl, then stop. ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 18 M iteration K τat η1 100 1 0 1.000000e-10 3.899207e-02 2 28 1.000000e-11 5.915080e-10 3 32 1.000000e-12 4.878532e-11 1000 1 0 1.000000e-10 3.899208e-02 2 28 1.000000e-11 5.915100e-10 3 32 1.000000e-12 4.878548e-11 10000 1 0 1.000000e-10 3.899208e-02 2 28 1.000000e-11 5.915100e-10 3 32 1.000000e-12 4.878548e-11 100000 1 0 1.000000e-10 3.899208e-02 2 28 1.000000e-11 5.915099e-10 3 32 1.000000e-12 4.878540e-11 1000000 1 0 1.000000e-10 3.899208e-02 2 28 1.000000e-11 5.914422e-10 3 32 1.000000e-12 4.871775e-11 Table 1: Convergence of the algorithm for τgl = 10 −10 and different values of M . (5) Compute local error estimators ηat2,i and η 2,i from (5.24). (6) Set τat ← (7) Make all atoms i atomistic for which ηtot2,i := η 2,i + ηel2,i−1 + η ≥ τat. (6.1) (8) Go to (2). Here τdiv > 1 is a constant factor which describes how fast the atom-wise tolerance τat should decrease during adaption. Our experience has been that τdiv = 10 is a reasonable choice. The crucial adaption step is (7). The adaption criterion (6.1) deems all atoms to be modeled atomistically if the associated error from the decomposition of η2 exceeds the atomistic error toler- ance τat. Here the element-wise errors η 2,i are distributed equally to the two adjacent atoms i and i+ 1. For the dislocation at the center of the chain and the chosen goal function, we expect that the atomistic repatoms always form a symmetric interval around the center. We have used the above adaptive atomistic-continuum algorithm to approximate our Frenkel-Kontorova model and have always found that the atomistic region is the set of atoms −K + 1, . . . ,K for some K depending on M and τgl. Thus, the modeling approach given in Section 3 of restricting to an atomistic region consisting of atoms −K + 1, . . . ,K for some K rather than considering a more general atomistic region is justified a posteriori. 6.2. Numerical Results. The algorithm has been implemented as described above. The boundary conditions were chosen as ybc−M+1 = −M, y −M+2 = −M + 1, y M−1 =M − 1, y M =M. (6.2) The elastic constants are k0 = 1 and k1 = k2 = 2. ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 19 −500 −400 −300 −200 −100 0 100 200 300 400 500 −500 −400 −300 −200 −100 0 100 200 300 400 500 Figure 4: The error estimators ηtot 2,i (left) and η 2,i, η 2,i (right) for M = 500, K = 20. Table 1 shows how the algorithm given above performs. After 3 iterations, the desired accuracy τgl = 10 −10 is achieved. Moreover, we can see from the table that the number of iterations are independent of M , that means the algorithm behaves robustly with respect to the problem size M . Figure 4 (left) shows the decomposition of the error estimator η2 for a typical setting M = 500, K = 20. One can clearly see that the error in the atomistic region is small, whereas the error is large in the continuum regions that border the atomistic region. It then decreases exponentially towards the endpoints. The error in both the atomistic region around the center and the continuum regions far away from the center are in the range of the (relative) machine precision εmach, which accounts for the fluctuations in these regions. The error can be considered to be numerically zero in these regions. In the continuum regions, we observe an error of magnitude O(ε2 ), whereas in the continuum region we have O(εmach), which leads to the different magnitudes of the fluctuations. Figure 4 (right) shows the element-wise contributions ηel2,i and the atom-wise contributions η of the decomposed error estimator ηtot2,i = η 2,i + ηel2,i−1 + η . The atomistic part ηat2,i, which corresponds to the gacTRa(yac) term, is dominant in the sense that it is about ten times larger than ηel2,i, which comes from the estimate for the perturbation term ê TMae. The fluctuations due to the limited machine precion in the atomistic region come from ηat2,i, whereas those in the continuum region away from the defect stem from ηel2,i. Let us note that in other applications of duality-based error estimation, the first term might not always be the dominant term. For example, in mesh refinement for classical linear finite elements, the first term even vanishes due to Galerkin orthogonality. Table 2 and Figure 5, which display the same data, show the efficiency of the error estimators, η1/|Q(y a − yac)| and η2/|Q(y a − yac)|, for M = 1000. For comparison, the actual error is given as well. For the relatively small 1D problem, the actual error can be easily computed, whereas in real world applications it is of course not available. One can clearly see that η1 gives a better estimate than η2, which numerically confirms our conjecture that η1 is a better estimator than η2. We see that η1 overestimates the actual error by a factor of 1.4, while η2 is in a still acceptable range of ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 20 K |Q(ya − yac)| η1 η1/|Q(y a − yac)| η2 η2/|Q(y a − yac)| 0 3.627633e-02 3.899208e-02 1.074863 3.999783e-02 1.102588 2 3.375762e-02 3.872272e-02 1.147081 5.101700e-02 1.511274 4 3.468605e-03 4.343595e-03 1.252260 5.422007e-03 1.563166 6 5.418585e-04 7.156249e-04 1.320686 9.187940e-04 1.695635 8 1.227067e-04 1.675383e-04 1.365356 2.193196e-04 1.787348 10 3.287188e-05 4.540984e-05 1.381419 5.984186e-05 1.820457 15 1.416914e-06 1.966114e-06 1.387603 2.597488e-06 1.833201 20 6.267636e-08 8.695824e-08 1.387417 1.148736e-07 1.832805 25 2.770161e-09 3.843388e-09 1.387424 5.077204e-09 1.832819 30 1.224369e-10 1.698739e-10 1.387440 2.244073e-10 1.832840 35 5.410783e-12 7.508365e-12 1.387667 9.918687e-12 1.833133 40 2.379208e-13 3.318024e-13 1.394592 4.383361e-13 1.842362 45 8.992806e-15 1.430733e-14 1.590975 1.901601e-14 2.114580 50 7.771561e-16 4.120094e-16 0.530150 6.201285e-16 0.797946 Table 2: Efficiency of the error estimators, η1/|Q(y a − yac)| and η2/|Q(y a − yac)|, for M = 1000. For K = 45 and K = 50 the results become inaccurate due to limited machine precision. 0 5 10 15 20 25 30 35 40 45 50 |Q(ya−yac)| Figure 5: Efficiency of the error estimators for M = 1000. ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 21 τgl optimal K K by η1 K by η2 1e-02 3 3 3 1e-03 5 5 5 1e-04 9 9 10 1e-05 12 13 13 1e-06 16 17 17 1e-07 20 20 21 1e-08 23 24 24 1e-09 27 28 28 1e-10 31 31 32 1e-11 35 35 35 1e-12 38 39 39 1e-13 42 42 43 1e-14 45 46 47 Table 3: Efficiency of the error estimators for M = 1000. 2 times the actual error. Moreover, we can see from Table 2 and Figure 5 that the error decreases exponentially with K. Finally, we compare the optimal (smallest) value of K which is needed to achieve a given accuracy τgl with the values for K determined by the error estimators η1 and η2, again taking into account the precise error which is available for the model problem. We see from Table 3 that even η2 only overestimates K by at most 2 atoms. Thus, we get an efficient estimate of the required atomistic region for both error estimators. Appendix A. Matrix Definitions We describe the matrices from Section 4.2. The matrix . . . . . . ∈ R(2M−1)×2M (A.1) transforms atomistic positions to distances between adjacent atoms. Similarly, Dqc = −ν−1−N+1 ν −ν−1−N+2 ν . . . . . . N−1 ν ∈ R(2N−1)×2N (A.2) transforms repatom positions from a coarsened chain to normalized distances between adjacent repatoms. The matrices (Ea)ij = k1 + k2 i = j ∈ {−M + 1,M − 1} k1 + 2k2 i = j ∈ {−M + 2, . . . ,M − 2} k2 |j − i| = 1 0 otherwise, (A.3) ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 22 (Eac)ij = k12(δ i + δ i+1) + i + δ i+1) + i−1 + δ i + δ i+1 + δ i+2) i = j i + δ i+2) j = i+ 1 i−1 + δ i+1) j = i− 1 0 otherwise, (A.4) (Eqc)ij = ωik12 + i + δ i+1) + i−1 + δ i + δ i+1 + δ i+2) i = j i + δ i+2) j = i+ 1 i−1 + δ i+1) j = i− 1 0 otherwise, (A.5) for i, j = −M + 1, . . . ,M − 1 and i, j = −N + 1, . . . , N − 1, respectively, describe the spring interactions in terms of the distances between atoms or repatoms. Accordingly, the matrices . . . ∈ R2M×2M (A.6) (Kqc)ij = (2νi−1 + ν i−1) + (2νi + ν i = j ∈ {−N + 2, . . . , N − 1} k0(2ν−N+1 + ν −N+1) i = j = −N + 1 k0(2νN−1 + ν N−1) i = j = N k0(νi − ν i ) j = i+ 1 k0(νj − ν j ) j = i− 1 0 otherwise, (A.7) for i, j = −N + 1, . . . , N describe the misfit interactions for the original atomistic system and the QC approximation. Finally, the constant vectors (−M + 1)a0 (−M + 2)a0 · · · (M − 1)a0 Ma0 ∈ R2M , (A.8a) ℓ−N+1a0 ℓ−N+2a0 · · · ℓN−1a0 ℓNa0 ∈ R2N , (A.8b) −Ma0 (−M + 1)a0 · · · −a0 a0 · · · (M − 1)a0 Ma0 ∈ R2M , (A.8c) (ℓ−N+1 − 1)a0 (ℓ−N+2 − 1)a0 · · · (ℓ0 − 1)a0 ℓ1a0 · · · ℓN−1a0 ℓNa0 ∈ R2M , (A.8d) fix the equilibrium positions for the spring interactions and the misfit energy, respectively. References [1] M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley, New York, 2000. [2] M. Arndt and M. Griebel, Derivation of higher order gradient continuum models from atomistic models for crystalline solids, Multiscale Model. Simul., 4 (2005), pp. 531–562. [3] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel, 2003. [4] X. Blanc, C. Le Bris, and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics, Math. Model. Numer. Anal., 39 (2005), pp. 797–826. [5] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, 2000. ADAPTIVITY FOR THE QUASICONTINUUM APPROXIMATION OF A FRENKEL-KONTOROVA MODEL 23 [6] M. S. Daw and M. I. Baskes, Semiempirical, quantum mechanical calculation of hydrogen embrittlement in metals, Phys. Rev. Lett., 50 (1983), pp. 1285–1288. [7] , Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals, Phys. Rev. B, 29 (1984), pp. 6443–6453. [8] M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum approximation, 2006, arXiv:math.NA/0611543. [9] W. E, J. Lu, and J. Z. Yang, Uniform accuracy of the quasicontinuum method, Phys. Rev. B, 74 (2006), p. 214115. [10] W. E and P. Ming, Analysis of multiscale methods, J. Comput. Math., 22 (2004), pp. 210–219. [11] E. Kaxiras, Atomic and Electronic Structure of Solids, Cambridge University Press, 2003. [12] J. Knap and M. Ortiz, An analysis of the quasicontinuum method, J. Mech. Phys. Solids, 49 (2001), pp. 1899– 1923. [13] P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model, Math. Comput., 72 (2003), pp. 657–675. [14] , Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defects, SIAM J. Numer. Anal., 45 (2007), pp. 313–332. [15] M. Marder, Condensed Matter Physics, John Wiley & Sons, 2000. [16] J. T. Oden and S. Prudhomme, Estimation of modeling error in computational mechanics, J. Comput. Phys., 182 (2002), pp. 496–515. [17] J. T. Oden, S. Prudhomme, and P. Bauman, Error control for molecular statics problems, Int. J. Multiscale Comput. Eng., 4 (2006), pp. 647–662. [18] J. T. Oden, S. Prudhomme, A. Romkes, and P. Bauman, Multiscale modeling of physical phenomena: Adaptive control of models, SIAM J. Sci. Comput., 28 (2006), pp. 2359–2389. [19] J. T. Oden and K. S. Vemaganti, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials: Part I: Error estimates and adaptive algorithms, J. Comput. Phys., 164 (2000), pp. 22–47. [20] C. Ortner and E. Süli, A-posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension, Research Report NA-06/13, Oxford University Computing Laboratory, 2006. [21] , A-priori analysis of the quasicontinuum method in one dimension, Research Report NA-06/12, Oxford University Computing Laboratory, 2006. [22] E. B. Tadmor, R. Miller, R. Phillips, and M. Ortiz, Nanoindentation and incipient plasticity, J. Mater. Res., 14 (1999), pp. 2233–2250. [23] E. B. Tadmor, M. Ortiz, and R. Phillips, Quasicontinuum analysis of defects in solids, Philos. Mag. A, 73 (1996), pp. 1529–1563. [24] E. B. Tadmor, R. Phillips, and M. Ortiz, Mixed atomistic and continuum models of deformation in solids, Langmuir, 12 (1996), pp. 4529–4534. 1. Introduction 2. Quasicontinuum Approximation 2.1. The Atomistic System 2.2. The Atomistic-Continuum Energy 2.3. Repatoms: Reduction of Degrees of Freedom 2.4. The Quasicontinuum Energy 3. Frenkel-Kontorova Model 3.1. Atomistic Frenkel-Kontorova Model 3.2. Quasicontinuum Approximation of the Frenkel-Kontorova Model 4. Primal and Dual Problems 4.1. Problem Setup 4.2. Matrix Formulation 4.3. Goal-Oriented Error Estimation 4.4. Dual Problems 5. Error Estimation for Atomistic vs. Continuum Modeling 5.1. Auxiliary Estimates 5.2. First Error Estimator 5.3. Second Error Estimator 6. Numerics 6.1. Algorithm 6.2. Numerical Results Appendix A. Matrix Definitions References

BLIND IDENTIFICATION OF DISTRIBUTED ANTENNA SYSTEMS WITH MULTIPLE CARRIER FREQUENCY OFFSETS Yuanning Yu, Athina P. Petropulu and H. Vincent Poor+ Electrical & Computer Engineering Department, Drexel University +School of Engineering and Applied Science, Princeton University ABSTRACT In spatially distributed multiuser antenna systems, the received signal contains multiple carrier-frequency offsets (CFOs) arising from mismatch between the oscillators of transmitters and receivers. This results in a time-varying rotation of the data constellation, which needs to be compensated at the receiver before symbol re- covery. In this paper, a new approach for blind CFO estimation and symbol recovery is proposed. The received base-band signal is over-sampled, and its polyphase components are used to formu- late a virtual Multiple-Input Multiple-Output (MIMO) problem. By applying blind MIMO system estimation techniques, the system re- sponse can be estimated and decoupled versions of the user symbols can be recovered, each one of which contains a distinct CFO. By applying a decision feedback Phase Lock Loop (PLL), the CFO can be mitigated and the transmitted symbols can be recovered. The esti- mated MIMO system response provides information about the CFOs that can be used to initialize the PLL, speed up its convergence, and avoid ambiguities usually linked with PLL. keywords-Multi-user Systems, Distributed Antenna Systems, Carrier Frequency Offset, Blind MIMO System Identification 1. BACKGROUND In both wireless and wireline communication systems, received sig- nals are often corrupted by carrier-frequency offsets (CFOs), due to Doppler shift and/or local oscillator drift. The CFO causes a time-varying rotation of the data symbols, and thus before symbol recovery, it must be estimated and accurately compensated for by the receiver. The CFO can be estimated via the use of pilots sym- bols; however, even a small error in this estimation tends to cause large data recovery errors. This necessitates transmission of pilot symbols rather often. In single user systems, or in multiuser sys- tems where the transmitters are physically connected to the same oscillator, there is only one CFO that needs to be estimated. This is typically done via a decision feedback Phase Lock Loop (PLL) at the receiver. The PLL is a closed-loop feedback control system that can adaptively track both frequency and phase offsets between the equalized signals and the reference constellation. However, de- pending on the constellation used during transmission, the PLL can have an M -fold symmetric ambiguity, and thus it has limited CFO acquisition range; e.g., |fk| < 1/8 for 4QAM signals. Moreover, the PLL require a long convergence time. To solve these problems, several methods have been proposed [3], [5], [6], [9] [11] that allow for blind estimation of the CFO and symbols using the second-order cyclo-stationary statistics of the over-sampled received signal. Blind CFO estimation has also been studied in the context of orthogonal This work was supported by the U. S. National Science Foundation under Grants ANI-03-38807, CNS-06-25637 and CNS-04-35052. frequency-division multiplexing (OFDM) systems, where the CFO destroys the orthogonality between the carriers (see [4] and the ref- erences therein). In a spatially distributed multiuser antenna system where data are transmitted simultaneously from multiple antennas, the received signal contains multiple CFOs, one for each transmit antenna. A PLL does not work in this case as there is no single frequency to lock onto. The literature on estimation of multiple CFOs is rather sparse. In [8], multiple CFOs were estimated by using pilot sym- bols that were uncorrelated among the different users. To account for multiple offsets, [10] proposed that multiple nodes transmit the same copy of the data with an artificial delay at each node. The resulting system was modeled as a convolutive single-input/single- output (SISO) system with time-varying system response caused by the multiple CFOs. A minimum mean-square error (MMSE) deci- sion feedback equalizer was used to track and equalize the channel and to recover the input data. Training symbols were required in order to obtain a channel estimate, which was used to initialize the equalizer. In this paper, a new approach to blind CFO estimation and sym- bol recovery is proposed. The received base-band signal is over- sampled, and its polyphase components are used to formulate a vir- tual MIMO problem. By applying blind MIMO system estimation techniques, the system response can be estimated, and decoupled versions of the user symbols can be recovered, each one of which contains a distinct CFO. By applying a PLL, the CFO can be miti- gated and the transmitted symbols can be recovered. The estimated MIMO system response provides information about the CFOs that can be used to initialize the PLL, speed up its convergence, and avoid ambiguities usually linked with PLLs. 2. SYSTEM MODEL We consider a distributed antenna system, where K users trans- mit simultaneously to a base station. Narrow-band transmission is assumed here, where the channel between any user and the base station is frequency non-selective. In addition, quasi-static fading is assumed, i.e., the channel gains remain fixed during the packet length. The continuous-time base-band received signal y(t) can be expressed as y(t) = akxk(t− τk)e j2πFkt + w(t) , (1) where ak represents the effect of channel fading between the k−th user and the base station and also phase offset; τk is the delay asso- ciated with the path between the k−th user and the base station; Fk is the frequency offset of the k−th user and w(t) represents noise; http://arxiv.org/abs/0704.1925v1 xk(t) denotes the transmitted signal of user k: xk(t) = sk(i)p(t− iTs) , (2) where sk(i) is the i−th symbol of user k; Ts is the symbol period; and p(t) is a pulse function with support [0, Ts]. Our objective is to obtain an estimate of s(i) = [s1(i), ..., sK(i)] T in the form ŝ(i) = Λ̂P s(i) , (3) where P is a column permutation matrix and Λ̂ a constant diago- nal matrix. These are considered to be trivial ambiguities, and are typical in any blind problem. 3. FORMATION OF THE MIMO PROBLEM The received signal y(t) is sampled at rate 1/T = P/Ts, where the over-sampling factor P ≥ K is an integer. In order to guar- antee that all the users’ pulses overlap at the sampling times, the over-sampling period should satisfy: Ts/P ≥ τk, k = 1, ...K. Or, in other words, the over-sampling factor P is upper bounded by Ts/min{τ1, ..., τK}. Let t = iTs +mT, m = 1, . . . , , P, denote the sampling times. The over-sampled received signal can be expressed as ym(i) = y(iTs +mT ) j2πfk(i+ xk((i+ )Ts − τk) + w((i+ j2πfk(i+ sk(i)p( Ts − τk) + w(iTs + am,k(sk(i)e j2πfki) + w(i+ ), m = 1, . . . , P , (4) where fk = FkTs is the normalized frequency offset between the k−th user and the base station, and the typical element of the virtual MIMO channel matrix A is given by am,k = ake Ts − τk . (5) Define the following: y(i) = [y1(i), ..., yP (i)] A = {am,k}, a tall matrix of dimension P × K; s̃(i) = [s1(i)e j2πf1i, ..., sK(i)e j2πfK i]T ; and w(i) [w(i + 1 ), ..., w(i + P )]T . Then, (4) can be written in ma- trix form as y(i) = As̃(i) +w(i) . (6) We could use the training based method of [8] to solve the MIMO system (4). That method assumes that the pilot symbols of different users are uncorrelated. The CFOs are obtained by search- ing for the location of a peak in the cross-correlation between the Discrete-Time Fourier Transform (DTFT) of a pilot sequence and that of the received signal. In the following we show how to estimate CFOs and recover the transmitted signals in a bind fashion, i.e., without the need for pilot symbols. The advantage of a blind approach is bandwidth efficiency since no bandwidth is wasted transmitting pilot symbols. 4. BLIND CHANNEL ESTIMATION AND COMPENSATION OF THE CFOS Let us make the following assumptions. • A1) For each m = 1, . . . , P , wm(.) is a zero-mean Gaus- sian stationary random processes with variance σ2w, and is independent of the channel inputs. • A2) For each k, the sequence sk(i) is a zero mean with independent and identically distributed (i.i.d.) elements having nonzero kurtosis; i.e., γ4sk = Cum[sk(i), s k(i), sk(i), s k(i)] 6= 0. The sequences sk’s are also mutually independent. • A3) The over-sampling factor P is no less than K. Under assumption (A2), it is easy to verify that the rotated input sig- nals s̃k(.) are also zero mean and i.i.d with nonzero kurtosis. Also, the s̃k(i)’s are mutually independent for different k’s. Assumption (A3) guarantees that the virtual MIMO channel matrix A in (6) has full rank with probability one. If the delays of users are randomly distributed in the interval [0, Ts/P ), then each row of the channel matrix can be viewed as having been drawn randomly from a contin- uous distribution so that the channel matrix has full rank with prob- ability one. One can apply any blind source separation algorithm (e.g., [1],[2] or [7] ) to obtain = APΛ . (7) Subsequently, using a least-squares equalizer we can obtain an estimate of the de-coupled signals s̃(i), within permutation and scalar ambiguities as ˆ̃s(i) = (Â y(i) = e jArg{−Λ} s̃(i) . (8) Without loss of generality we can assume that the transmitted signal has unit power. Then, on denoting by θk the k−th diagonal element of Arg{Λ}, the j−th separated input signal can be expressed as ˆ̃sk(i) = sk(i)e j(−θk+2πfki) . (9) In order to recover the transmitted signals, we still need to mit- igate the effect of CFO in each decoupled signal. This can be done via a PLL. By using the decoupled signals as inputs, and the constel- lation used in transmission as a reference, the PLL can effectively mitigate the CFO by minimizing the feedback error, which is calcu- lated based on the distance of the recovered signal and the closest valid constellation point. However, depending on the constellation used in transmission, there is a four-fold symmetric ambiguity for MQAM signals, or M -fold symmetric ambiguity for MPSK signals. For example, for 4QAM signals, and an initial CFO value of fk = 0, the effective tracking range for fk is |fk| < 1/8. Moreover, de- pending on the value of the CFO, the PLL generally needs a long convergence time, during which the input signals are not correctly recovered. Next we will show that by exploring the structure of the virtual channel matrix, we can obtain an estimate of the CFOs, which can then be used to initialize the PLL. By doing this, we can prevent the symmetric ambiguity problem and enlarge the effective tracking range of the PLL from |fk| < 1/8 to |fk| < 1/2. Also, the conver- gence time of the PLL can be greatly reduced. By taking the phase of the estimated channel matrix Â, we ob- Ψ = Arg + φ1 . . . . . . 2πf1P + φ1 . . . 2πfKP where φk = Arg{ak} + θk, which accounts for both the phase of ak and the estimated phase ambiguity in (9). By applying linear fitting on the j−th column of Ψ we obtain the least squares estimate of fj as f̂j = p=1 pΨp,j)− ( p=1 p)( p=1 Ψp,j) p2)− ( . (11) We can write f̂j = fj + ǫj where ǫj represents estimation error. On noting that the de-coupled signals ˆ̃sj(i) in (9) are shuffled in the same manner as the estimated CFOs in (11), we can use the esti- mated CFOs to compensate for the effects of CFO in the decoupled signals (9) and thereby obtain estimates of the input signals as ŝ(i) = e jArg{−Λ} s(i) . (12) Due to errors in the channel estimates, we can only compensate for most, but not all, of the CFO effects in (9) and so we can write ŝk(i) = sk(i)e j(−θk−2πǫki) . (13) By subsequently applying a PLL to ŝj(i), we can further mitigate the effects of the residual CFO ǫk. For 4QAM signals, as long as |ǫk| < 1/8, the residual CFO can be effectively removed by the PLL. The initial CFO estimator (11) can prevent the symmetric ambiguity of the PLL, and can also greatly reduce the convergence time of the PLL. From (10), we can see that the CFO estimator will achieve full acquisition range for the normalized CFO. 5. SIMULATION RESULTS In this section, we verify the validity of the proposed method via sim- ulations, under the following assumptions. The channel coefficients ak, k = 1, . . . ,K are zero-mean Gaussian random variables. The waveform p(·) is a Hamming window. The delays, τk, k = 1, . . . ,K are uniformly distributed in the range [0, Ts/P ). The input signals are 4QAM signals. The blind source separation algorithm used here is the JADE method, which was downloaded from http://www.tsi.enst.fr/c̃ardoso/guidesepsou.html. First we show results for a two-user systems with f1 = −0.1552, f2 = 0.4335, a1 = 0.3173 − 0.6483i, a2 = 0.1625 + 0.5867i, with SNR = 20dB, and N = 1024. In Fig. 1 we show the polyphase outputs y1, y2. Due to the mixing and the CFOs no obvious constellation is visible. In Fig. 2, we show the de-coupled signals ˆ̃sk, k = 1, 2 right after JADE. Although still rotated by the CFOs, two signals s̃k, k = 1, 2 are clearly separated. In Fig. 3, we show the recovered input signals ŝk, k = 1, 2, where we can see that after compensating for the effect of CFOs the constellations are recovered. Next we show estimation results averaged over 300 indepen- dent channel runs, and 20 Monte-Carlo runs for each channel. For each channel case, the coefficients ak, k = 1, 2 were generated randomly, and the continuous CFOs where chosen randomly in the range [− 1 ). The delays, τk, k = 1, ..., K where chosen uni- formly in the range of [0, Ts/P ). The transmitted signal was 4QAM. The performance of both the pilot-based method and the pro- posed method at different data lengths and with SNR set to 30dB is shown in Fig. 4. For the pilot-based method, each user trans- mits a pilot sequence of length 32, and the pilots are random se- quences uncorrelated between different users. Fig. 4 shows the mean-square error (MSE) for the CFO estimate (11) for different values of the over-sampling factor P . The MSE is calculated usings [(f̂k − fk)] 2 = 1 [(F̂k − Fk)Ts] 2. We can see that by increasing P we can improve the estimation accuracy. Fig. 5 shows the Bit Error Rate (BER) for different values of P . For both blind and training methods, the BER is calculated based on the re- covered signals after the PLL. As expected, the BER performance also improves by increasing P . The proposed method appears to work well even for short data length. Next we show the performance of both methods at various noise levels. We use packet length N = 1, 024. In Fig. 6 we show the MSE of the blind CFO estimator (11) as well as that of the training based method. We can see that by increasing P we improve estima- tion accuracy. In Fig. 7, we show the BER performance after the PLL for both blind and training based methods. We see that the pro- posed blind method has almost the same performance as the training based method for SNR values lower than 20dB, while the training based method can achieve better BER performance for higher values of SNR. 6. CONCLUSION In this paper we have proposed a novel blind approach for identifica- tion of a distributed multiuser antenna system with multiple CFOs. By over-sampling of the received base-band signal, the MISO prob- lem is converted into a MIMO one. Blind MIMO system estimation then yields the system response, and MIMO input recovery yields the decoupled transmitted signals, each one containing a CFO. By exploring the structure of the MIMO system response we obtain a coarse estimate of the CFOs, which is then combined with a deci- sion feedback PLL to compensate for the CFOs in the decoupled transmitted signals. The proposed blind method has full acquisition range for normalized CFOs. 7. REFERENCES [1] T. Acar, Y. Yu and A. P. Petropulu, “Blind MIMO system estimation based on PARAFAC decomposition of higher order output tensors” IEEE Trans. Signal Process., Vol. 54, No. 11, pp. 4156 - 4168, Nov. [2] J. F. Cardoso and A. Souloumiac, “Blind beamforming for non- Gaussian signals,” IEE Proceedings: Radar and Signal Process., Vol. 140, No. 6, pp. 362-370 Dec. 1993. [3] P. Ciblat, P. Loubaton, E. Serpedin and G. B. Giannakis, “Perfor- mance analysis of blind carrier frequency offset estimators for non- circular transmissions through frequency-selective channels,” IEEE Trans. Signal Process., Vol. 50, No. 1, pp. 130 - 140, Jan. 2002. [4] T. Roman, S. Visuri and V. Koivunen, “Blind frequency synchroniza- tion in OFDM via diagonality criterion,” IEEE Trans. Signal Pro- cess., Vol. 54, No. 8, pp. 3125 - 3135, Aug. 2006. [5] M. Ghogho, A. Swami and T. Durrani, “On blind carrier recovery in time-selective fading channels,” in Proc. 33rd Asilomar Conf. Sig- nals, Systems and Computers, Vol. 1, pp. 243-247, Pacific Grove, CA, Oct. 1999, http://www.tsi.enst.fr/~cardoso/guidesepsou.html [6] F. Gini and G. B. Giannakis, “Frequency offset and symbol timing recovery in flat-fading channels: A cyclostationary approach,” IEEE Trans. Commun., Vol. 46, No. 3, pp. 400 - 411, Mar. 1998. [7] L. de Lathauwer, B. de Moor and J. 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Signal Process., Vol. 50, No. 9, pp. 2295- 2305, Sept. 2002. −5 0 5 received signals y −5 0 5 received signals y Fig. 1. Received mixing signal y −4 −2 0 2 4 decoupled signals 1 −5 0 5 decoupled signals 2 Fig. 2. De-coupled inputs ˆ̃s −4 −3 −2 −1 0 1 2 3 4 recovered input signal 1 −4 −3 −2 −1 0 1 2 3 4 recovered input signal 2 Fig. 3. Recovered input signals ŝ with P = 2 0 500 1000 1500 2000 2500 3000 3500 4000 4500 packet length N MSE of CFOs at SNR=30dB P=2, bind P=2, pilots P=4, blind P=4, pilots Fig. 4. MSE of CFOs vs N for K=2, with SNR=30dB, 4QAM 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Packet length N BER Comparison for SNR 30dB P=2, blind P=2, pilots P=4, blind P=4, pilots Fig. 5. BER vs N for K=2, with SNR=30dB, 4QAM 0 5 10 15 20 25 30 SNR dB MSE of CFOs at N=1024 P=2, blind P=2, pilots P=4, blind P=4, pilots Fig. 6. MSE of CFOs vs SNR for K=2, 4QAM 0 5 10 15 20 25 30 SNR dB BER Comparison at N=1024 P=2, blind P=2, pilots P=4, blind P=4, pilots Fig. 7. BER vs SNR for K=2, 4QAM, T=1024 Background System Model Formation of the MIMO Problem Blind channel estimation and compensation of the CFOs Simulation Results Conclusion References

In spatially distributed multiuser antenna systems, the received signal contains multiple carrier-frequency offsets (CFOs) arising from mismatch between the oscillators of transmitters and receivers. This results in a time-varying rotation of the data constellation, which needs to be compensated at the receiver before symbol recovery. In this paper, a new approach for blind CFO estimation and symbol recovery is proposed. The received base-band signal is over-sampled, and its polyphase components are used to formulate a virtual Multiple-Input Multiple-Output (MIMO) problem. By applying blind MIMO system estimation techniques, the system response can be estimated and decoupled versions of the user symbols can be recovered, each one of which contains a distinct CFO. By applying a decision feedback Phase Lock Loop (PLL), the CFO can be mitigated and the transmitted symbols can be recovered. The estimated MIMO system response provides information about the CFOs that can be used to initialize the PLL, speed up its convergence, and avoid ambiguities usually linked with PLL.

BLIND IDENTIFICATION OF DISTRIBUTED ANTENNA SYSTEMS WITH MULTIPLE CARRIER FREQUENCY OFFSETS Yuanning Yu, Athina P. Petropulu and H. Vincent Poor+ Electrical & Computer Engineering Department, Drexel University +School of Engineering and Applied Science, Princeton University ABSTRACT In spatially distributed multiuser antenna systems, the received signal contains multiple carrier-frequency offsets (CFOs) arising from mismatch between the oscillators of transmitters and receivers. This results in a time-varying rotation of the data constellation, which needs to be compensated at the receiver before symbol re- covery. In this paper, a new approach for blind CFO estimation and symbol recovery is proposed. The received base-band signal is over-sampled, and its polyphase components are used to formu- late a virtual Multiple-Input Multiple-Output (MIMO) problem. By applying blind MIMO system estimation techniques, the system re- sponse can be estimated and decoupled versions of the user symbols can be recovered, each one of which contains a distinct CFO. By applying a decision feedback Phase Lock Loop (PLL), the CFO can be mitigated and the transmitted symbols can be recovered. The esti- mated MIMO system response provides information about the CFOs that can be used to initialize the PLL, speed up its convergence, and avoid ambiguities usually linked with PLL. keywords-Multi-user Systems, Distributed Antenna Systems, Carrier Frequency Offset, Blind MIMO System Identification 1. BACKGROUND In both wireless and wireline communication systems, received sig- nals are often corrupted by carrier-frequency offsets (CFOs), due to Doppler shift and/or local oscillator drift. The CFO causes a time-varying rotation of the data symbols, and thus before symbol recovery, it must be estimated and accurately compensated for by the receiver. The CFO can be estimated via the use of pilots sym- bols; however, even a small error in this estimation tends to cause large data recovery errors. This necessitates transmission of pilot symbols rather often. In single user systems, or in multiuser sys- tems where the transmitters are physically connected to the same oscillator, there is only one CFO that needs to be estimated. This is typically done via a decision feedback Phase Lock Loop (PLL) at the receiver. The PLL is a closed-loop feedback control system that can adaptively track both frequency and phase offsets between the equalized signals and the reference constellation. However, de- pending on the constellation used during transmission, the PLL can have an M -fold symmetric ambiguity, and thus it has limited CFO acquisition range; e.g., |fk| < 1/8 for 4QAM signals. Moreover, the PLL require a long convergence time. To solve these problems, several methods have been proposed [3], [5], [6], [9] [11] that allow for blind estimation of the CFO and symbols using the second-order cyclo-stationary statistics of the over-sampled received signal. Blind CFO estimation has also been studied in the context of orthogonal This work was supported by the U. S. National Science Foundation under Grants ANI-03-38807, CNS-06-25637 and CNS-04-35052. frequency-division multiplexing (OFDM) systems, where the CFO destroys the orthogonality between the carriers (see [4] and the ref- erences therein). In a spatially distributed multiuser antenna system where data are transmitted simultaneously from multiple antennas, the received signal contains multiple CFOs, one for each transmit antenna. A PLL does not work in this case as there is no single frequency to lock onto. The literature on estimation of multiple CFOs is rather sparse. In [8], multiple CFOs were estimated by using pilot sym- bols that were uncorrelated among the different users. To account for multiple offsets, [10] proposed that multiple nodes transmit the same copy of the data with an artificial delay at each node. The resulting system was modeled as a convolutive single-input/single- output (SISO) system with time-varying system response caused by the multiple CFOs. A minimum mean-square error (MMSE) deci- sion feedback equalizer was used to track and equalize the channel and to recover the input data. Training symbols were required in order to obtain a channel estimate, which was used to initialize the equalizer. In this paper, a new approach to blind CFO estimation and sym- bol recovery is proposed. The received base-band signal is over- sampled, and its polyphase components are used to formulate a vir- tual MIMO problem. By applying blind MIMO system estimation techniques, the system response can be estimated, and decoupled versions of the user symbols can be recovered, each one of which contains a distinct CFO. By applying a PLL, the CFO can be miti- gated and the transmitted symbols can be recovered. The estimated MIMO system response provides information about the CFOs that can be used to initialize the PLL, speed up its convergence, and avoid ambiguities usually linked with PLLs. 2. SYSTEM MODEL We consider a distributed antenna system, where K users trans- mit simultaneously to a base station. Narrow-band transmission is assumed here, where the channel between any user and the base station is frequency non-selective. In addition, quasi-static fading is assumed, i.e., the channel gains remain fixed during the packet length. The continuous-time base-band received signal y(t) can be expressed as y(t) = akxk(t− τk)e j2πFkt + w(t) , (1) where ak represents the effect of channel fading between the k−th user and the base station and also phase offset; τk is the delay asso- ciated with the path between the k−th user and the base station; Fk is the frequency offset of the k−th user and w(t) represents noise; http://arxiv.org/abs/0704.1925v1 xk(t) denotes the transmitted signal of user k: xk(t) = sk(i)p(t− iTs) , (2) where sk(i) is the i−th symbol of user k; Ts is the symbol period; and p(t) is a pulse function with support [0, Ts]. Our objective is to obtain an estimate of s(i) = [s1(i), ..., sK(i)] T in the form ŝ(i) = Λ̂P s(i) , (3) where P is a column permutation matrix and Λ̂ a constant diago- nal matrix. These are considered to be trivial ambiguities, and are typical in any blind problem. 3. FORMATION OF THE MIMO PROBLEM The received signal y(t) is sampled at rate 1/T = P/Ts, where the over-sampling factor P ≥ K is an integer. In order to guar- antee that all the users’ pulses overlap at the sampling times, the over-sampling period should satisfy: Ts/P ≥ τk, k = 1, ...K. Or, in other words, the over-sampling factor P is upper bounded by Ts/min{τ1, ..., τK}. Let t = iTs +mT, m = 1, . . . , , P, denote the sampling times. The over-sampled received signal can be expressed as ym(i) = y(iTs +mT ) j2πfk(i+ xk((i+ )Ts − τk) + w((i+ j2πfk(i+ sk(i)p( Ts − τk) + w(iTs + am,k(sk(i)e j2πfki) + w(i+ ), m = 1, . . . , P , (4) where fk = FkTs is the normalized frequency offset between the k−th user and the base station, and the typical element of the virtual MIMO channel matrix A is given by am,k = ake Ts − τk . (5) Define the following: y(i) = [y1(i), ..., yP (i)] A = {am,k}, a tall matrix of dimension P × K; s̃(i) = [s1(i)e j2πf1i, ..., sK(i)e j2πfK i]T ; and w(i) [w(i + 1 ), ..., w(i + P )]T . Then, (4) can be written in ma- trix form as y(i) = As̃(i) +w(i) . (6) We could use the training based method of [8] to solve the MIMO system (4). That method assumes that the pilot symbols of different users are uncorrelated. The CFOs are obtained by search- ing for the location of a peak in the cross-correlation between the Discrete-Time Fourier Transform (DTFT) of a pilot sequence and that of the received signal. In the following we show how to estimate CFOs and recover the transmitted signals in a bind fashion, i.e., without the need for pilot symbols. The advantage of a blind approach is bandwidth efficiency since no bandwidth is wasted transmitting pilot symbols. 4. BLIND CHANNEL ESTIMATION AND COMPENSATION OF THE CFOS Let us make the following assumptions. • A1) For each m = 1, . . . , P , wm(.) is a zero-mean Gaus- sian stationary random processes with variance σ2w, and is independent of the channel inputs. • A2) For each k, the sequence sk(i) is a zero mean with independent and identically distributed (i.i.d.) elements having nonzero kurtosis; i.e., γ4sk = Cum[sk(i), s k(i), sk(i), s k(i)] 6= 0. The sequences sk’s are also mutually independent. • A3) The over-sampling factor P is no less than K. Under assumption (A2), it is easy to verify that the rotated input sig- nals s̃k(.) are also zero mean and i.i.d with nonzero kurtosis. Also, the s̃k(i)’s are mutually independent for different k’s. Assumption (A3) guarantees that the virtual MIMO channel matrix A in (6) has full rank with probability one. If the delays of users are randomly distributed in the interval [0, Ts/P ), then each row of the channel matrix can be viewed as having been drawn randomly from a contin- uous distribution so that the channel matrix has full rank with prob- ability one. One can apply any blind source separation algorithm (e.g., [1],[2] or [7] ) to obtain = APΛ . (7) Subsequently, using a least-squares equalizer we can obtain an estimate of the de-coupled signals s̃(i), within permutation and scalar ambiguities as ˆ̃s(i) = (Â y(i) = e jArg{−Λ} s̃(i) . (8) Without loss of generality we can assume that the transmitted signal has unit power. Then, on denoting by θk the k−th diagonal element of Arg{Λ}, the j−th separated input signal can be expressed as ˆ̃sk(i) = sk(i)e j(−θk+2πfki) . (9) In order to recover the transmitted signals, we still need to mit- igate the effect of CFO in each decoupled signal. This can be done via a PLL. By using the decoupled signals as inputs, and the constel- lation used in transmission as a reference, the PLL can effectively mitigate the CFO by minimizing the feedback error, which is calcu- lated based on the distance of the recovered signal and the closest valid constellation point. However, depending on the constellation used in transmission, there is a four-fold symmetric ambiguity for MQAM signals, or M -fold symmetric ambiguity for MPSK signals. For example, for 4QAM signals, and an initial CFO value of fk = 0, the effective tracking range for fk is |fk| < 1/8. Moreover, de- pending on the value of the CFO, the PLL generally needs a long convergence time, during which the input signals are not correctly recovered. Next we will show that by exploring the structure of the virtual channel matrix, we can obtain an estimate of the CFOs, which can then be used to initialize the PLL. By doing this, we can prevent the symmetric ambiguity problem and enlarge the effective tracking range of the PLL from |fk| < 1/8 to |fk| < 1/2. Also, the conver- gence time of the PLL can be greatly reduced. By taking the phase of the estimated channel matrix Â, we ob- Ψ = Arg + φ1 . . . . . . 2πf1P + φ1 . . . 2πfKP where φk = Arg{ak} + θk, which accounts for both the phase of ak and the estimated phase ambiguity in (9). By applying linear fitting on the j−th column of Ψ we obtain the least squares estimate of fj as f̂j = p=1 pΨp,j)− ( p=1 p)( p=1 Ψp,j) p2)− ( . (11) We can write f̂j = fj + ǫj where ǫj represents estimation error. On noting that the de-coupled signals ˆ̃sj(i) in (9) are shuffled in the same manner as the estimated CFOs in (11), we can use the esti- mated CFOs to compensate for the effects of CFO in the decoupled signals (9) and thereby obtain estimates of the input signals as ŝ(i) = e jArg{−Λ} s(i) . (12) Due to errors in the channel estimates, we can only compensate for most, but not all, of the CFO effects in (9) and so we can write ŝk(i) = sk(i)e j(−θk−2πǫki) . (13) By subsequently applying a PLL to ŝj(i), we can further mitigate the effects of the residual CFO ǫk. For 4QAM signals, as long as |ǫk| < 1/8, the residual CFO can be effectively removed by the PLL. The initial CFO estimator (11) can prevent the symmetric ambiguity of the PLL, and can also greatly reduce the convergence time of the PLL. From (10), we can see that the CFO estimator will achieve full acquisition range for the normalized CFO. 5. SIMULATION RESULTS In this section, we verify the validity of the proposed method via sim- ulations, under the following assumptions. The channel coefficients ak, k = 1, . . . ,K are zero-mean Gaussian random variables. The waveform p(·) is a Hamming window. The delays, τk, k = 1, . . . ,K are uniformly distributed in the range [0, Ts/P ). The input signals are 4QAM signals. The blind source separation algorithm used here is the JADE method, which was downloaded from http://www.tsi.enst.fr/c̃ardoso/guidesepsou.html. First we show results for a two-user systems with f1 = −0.1552, f2 = 0.4335, a1 = 0.3173 − 0.6483i, a2 = 0.1625 + 0.5867i, with SNR = 20dB, and N = 1024. In Fig. 1 we show the polyphase outputs y1, y2. Due to the mixing and the CFOs no obvious constellation is visible. In Fig. 2, we show the de-coupled signals ˆ̃sk, k = 1, 2 right after JADE. Although still rotated by the CFOs, two signals s̃k, k = 1, 2 are clearly separated. In Fig. 3, we show the recovered input signals ŝk, k = 1, 2, where we can see that after compensating for the effect of CFOs the constellations are recovered. Next we show estimation results averaged over 300 indepen- dent channel runs, and 20 Monte-Carlo runs for each channel. For each channel case, the coefficients ak, k = 1, 2 were generated randomly, and the continuous CFOs where chosen randomly in the range [− 1 ). The delays, τk, k = 1, ..., K where chosen uni- formly in the range of [0, Ts/P ). The transmitted signal was 4QAM. The performance of both the pilot-based method and the pro- posed method at different data lengths and with SNR set to 30dB is shown in Fig. 4. For the pilot-based method, each user trans- mits a pilot sequence of length 32, and the pilots are random se- quences uncorrelated between different users. Fig. 4 shows the mean-square error (MSE) for the CFO estimate (11) for different values of the over-sampling factor P . The MSE is calculated usings [(f̂k − fk)] 2 = 1 [(F̂k − Fk)Ts] 2. We can see that by increasing P we can improve the estimation accuracy. Fig. 5 shows the Bit Error Rate (BER) for different values of P . For both blind and training methods, the BER is calculated based on the re- covered signals after the PLL. As expected, the BER performance also improves by increasing P . The proposed method appears to work well even for short data length. Next we show the performance of both methods at various noise levels. We use packet length N = 1, 024. In Fig. 6 we show the MSE of the blind CFO estimator (11) as well as that of the training based method. We can see that by increasing P we improve estima- tion accuracy. In Fig. 7, we show the BER performance after the PLL for both blind and training based methods. We see that the pro- posed blind method has almost the same performance as the training based method for SNR values lower than 20dB, while the training based method can achieve better BER performance for higher values of SNR. 6. CONCLUSION In this paper we have proposed a novel blind approach for identifica- tion of a distributed multiuser antenna system with multiple CFOs. By over-sampling of the received base-band signal, the MISO prob- lem is converted into a MIMO one. Blind MIMO system estimation then yields the system response, and MIMO input recovery yields the decoupled transmitted signals, each one containing a CFO. By exploring the structure of the MIMO system response we obtain a coarse estimate of the CFOs, which is then combined with a deci- sion feedback PLL to compensate for the CFOs in the decoupled transmitted signals. The proposed blind method has full acquisition range for normalized CFOs. 7. REFERENCES [1] T. Acar, Y. Yu and A. P. Petropulu, “Blind MIMO system estimation based on PARAFAC decomposition of higher order output tensors” IEEE Trans. Signal Process., Vol. 54, No. 11, pp. 4156 - 4168, Nov. [2] J. F. Cardoso and A. Souloumiac, “Blind beamforming for non- Gaussian signals,” IEE Proceedings: Radar and Signal Process., Vol. 140, No. 6, pp. 362-370 Dec. 1993. [3] P. Ciblat, P. Loubaton, E. Serpedin and G. B. Giannakis, “Perfor- mance analysis of blind carrier frequency offset estimators for non- circular transmissions through frequency-selective channels,” IEEE Trans. Signal Process., Vol. 50, No. 1, pp. 130 - 140, Jan. 2002. [4] T. Roman, S. Visuri and V. Koivunen, “Blind frequency synchroniza- tion in OFDM via diagonality criterion,” IEEE Trans. Signal Pro- cess., Vol. 54, No. 8, pp. 3125 - 3135, Aug. 2006. [5] M. Ghogho, A. Swami and T. Durrani, “On blind carrier recovery in time-selective fading channels,” in Proc. 33rd Asilomar Conf. Sig- nals, Systems and Computers, Vol. 1, pp. 243-247, Pacific Grove, CA, Oct. 1999, http://www.tsi.enst.fr/~cardoso/guidesepsou.html [6] F. Gini and G. B. Giannakis, “Frequency offset and symbol timing recovery in flat-fading channels: A cyclostationary approach,” IEEE Trans. Commun., Vol. 46, No. 3, pp. 400 - 411, Mar. 1998. [7] L. de Lathauwer, B. de Moor and J. Vandewalle, “Independent com- ponent analysis and (simultaneous) third-order tensor diagonaliza- tion,” IEEE Trans. Signal Process., Vol. 49, No. 10, pp. 2262-2271, Oct. 2001. [8] F. Prihoda, E. Garbarine and A. P. Petropulu, “Resolving wireless collisions in random access networks,” Proc. 2006 Asilomar Conf. Signals, Systems and Computers, Pacific Grove, CA, Oct. 2006. [9] K. E. Scott and E. B. Olasz, “Simultaneous clock phase and frequency offset estimation,” IEEE Trans. Commun., Vol. 43, No. 7, pp. 2263 - 2270, July 1995. [10] D. Veronesi and D. L. Goeckel, “Multiple frequency offset compen- sation in cooperative wireless systems,” Proc. IEEE Global Telecom- mun. Conf., San Francisco, CA, Nov. 2006. [11] Y. Wang, P. Ciblat, E. Serpedin and P. Loubaton, “Performance anal- ysis of a class of nondata-aided frequency offset and symbol timing estimators for flat-fading channels,” IEEE Trans. Signal Process., Vol. 50, No. 9, pp. 2295- 2305, Sept. 2002. −5 0 5 received signals y −5 0 5 received signals y Fig. 1. Received mixing signal y −4 −2 0 2 4 decoupled signals 1 −5 0 5 decoupled signals 2 Fig. 2. De-coupled inputs ˆ̃s −4 −3 −2 −1 0 1 2 3 4 recovered input signal 1 −4 −3 −2 −1 0 1 2 3 4 recovered input signal 2 Fig. 3. Recovered input signals ŝ with P = 2 0 500 1000 1500 2000 2500 3000 3500 4000 4500 packet length N MSE of CFOs at SNR=30dB P=2, bind P=2, pilots P=4, blind P=4, pilots Fig. 4. MSE of CFOs vs N for K=2, with SNR=30dB, 4QAM 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Packet length N BER Comparison for SNR 30dB P=2, blind P=2, pilots P=4, blind P=4, pilots Fig. 5. BER vs N for K=2, with SNR=30dB, 4QAM 0 5 10 15 20 25 30 SNR dB MSE of CFOs at N=1024 P=2, blind P=2, pilots P=4, blind P=4, pilots Fig. 6. MSE of CFOs vs SNR for K=2, 4QAM 0 5 10 15 20 25 30 SNR dB BER Comparison at N=1024 P=2, blind P=2, pilots P=4, blind P=4, pilots Fig. 7. BER vs SNR for K=2, 4QAM, T=1024 Background System Model Formation of the MIMO Problem Blind channel estimation and compensation of the CFOs Simulation Results Conclusion References

THE DISTRIBUTION OF THE FIRST EIGENVALUE SPACING AT THE HARD EDGE OF THE LAGUERRE UNITARY ENSEMBLE PETER J. FORRESTER AND NICHOLAS S. WITTE Abstract. The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlevé V system, and the solution of its associated linear isomonodromic system. In particular it is characterised by the polynomial solutions to the isomonodromic equations which are also orthogonal with respect to a defor- mation of the Laguerre weight. In the scaling to the hard edge regime we find an analogous situation where a certain Painlevé III′ system and its asso- ciated linear isomonodromic system characterise the scaled distribution. We undertake extensive analytical studies of this system and use this knowledge to accurately compute the distribution and its moments for various values of the parameter a. In particular choosing a = ±1/2 allows the first eigenvalue spacing distribution for random real orthogonal matrices to be computed. 1. Introduction The Laguerre unitary ensemble (LUEn,a) of random matrices is specified by the eigenvalue probability density function (p.d.f.) (1.1) p(λ1, . . . , λn) n!cn,n+a e−λjλaj 1≤j<k≤n (λk − λj)2, λ1, . . . , λn ∈ [0,∞), where (1.2) cm,n := Γ(n− j + 1)Γ(m− j + 2). The naming relates to the fact that (1.1) is the eigenvalue p.d.f. of complex Hermit- ian matrices X with measure invariant under unitary conjugation X 7→ UXU−1, proportional to the generalised Laguerre form (1.3) (detX) e−Tr(X). 2000 Mathematics Subject Classification. 15A52, 33C45, 33E17, 42C05, 60K35, 62E15. Key words and phrases. random matrices, eigenvalue distribution, Wishart matrices, Painlevé equations, isomonodromic deformations. http://arxiv.org/abs/0704.1926v2 2 PETER J. FORRESTER AND NICHOLAS S. WITTE In multivariate statistics (1.1) is realised as the eigenvalue p.d.f. for the complex case of the so-called Wishart matrices X = Y †Y . Here Y is an N × n (N ≥ n) rectangular matrix of i.i.d. entries with distribution N[0, 1]+iN[0, 1]. In this setting a = N − n, and so a is naturally a non-negative integer. The spectrum of complex Wishart matrices has found recent application in studies of wireless communication systems [34], where the matrix Y consists of the complex amplitudes of various channels of transmitted waves as received by the antennas. The matrix structure Y †Y is relevant to the study of the eigenvalues of the (n+N)× (n+N) Hermitian matrix (1.4) X̃ := 0N×N Y Y † 0n×n Thus one has that X̃ has in general N − n zero eigenvalues, with the remaining eigenvalues given by ± the positive square roots of the eigenvalues of Y †Y (see e.g. [9]). This matrix structure has application to the study of the Dirac equation in the context of quantum chromodynamics [36]. There most interest is in the scaling behaviour of the smallest eigenvalues. In the study of matrix Lie algebras one encounters antisymmetric matrices (XT = −X) with pure imaginary complex elements. Specifically, such matrices are the Hermitian part of the matrix Lie algebra (1.5) i× (so(n,C)) := {i times n× n skew symmetric complex matrices}. If the independent imaginary complex elements are i.i.d with distribution iN[0, 1], then the p.d.f. of the positive eigenvalues is proportional to n = 2m even: exp(−λ2j) 1≤j<k≤m (λ2k − λ2j)2,(1.6) n = 2m+ 1 odd: λ2j exp(−λ2j ) 1≤j<k≤m (λ2k − λ2j )2.(1.7) This ensemble will be denoted by AS(n). Under the change of variables λ2j 7→ λj these reduce to the LUEn,a with parameters a = −1/2, 1/2 respectively. Antisymmetric Hermitian matrices X can be used to parameterise real orthog- onal matrices R with determinant +1 (and thus, by definition, members of the classical group O+(n)). Thus we can write R according to a Cayley transformation (1.8) R = In + iX In − iX Note from this that the property that the eigenvalues of X come in ± pairs is consistent with the property that the eigenvalues of R come in complex conjugate pairs e±iθ. This can be used (see e.g. [9]) to show that with the matrix R chosen DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 3 with uniform (Haar) measure, the eigenvalue p.d.f for the eigenvalues with angles 0 ≤ θ ≤ π is proportional to n = 2m even: 1≤j<k≤m (cos θk − cos θj)2,(1.9) n = 2m+ 1 odd: (1− cos θj) 1≤j<k≤m (cos θk − cos θj)2.(1.10) Note that for θl → 0 these have the same leading behaviour as (1.6), (1.7) with λl → 0. This is consistent with the fact that the m → ∞ scaled joint distribution function for the k smallest eigenvalues, p(k) say, is the same for both ensembles, (1.11) AS(n) , . . . , ) = lim O+(n) , . . . , where n = 2m, 2m + 1. This scaling is such that the average spacing between eigenvalues approaches unity as k → ∞. The distribution (1.11) is of primary importance in the study of the spectral interpretation of L-functions [20],[30]. From the remark below (1.7) we know that (1.12) · · · AS(n) x1, . . . , xk) = p LUEm,a (x1, . . . , xk), where for n = 2m, a = −1/2, while for n = 2m+ 1, a = 1/2. Consequently (1.13) lim O+(n) , . . . , = lim X1 . . . Xk p LUEm,a π2X21 , . . . , π2X2k Thus knowledge of the distribution p LUEm,a for a = ±1/2 suffices to compute the scaled limit of p O+(n) In the case k = 1 a number of different characterisations of p LUEm,a , which is the distribution of the smallest eigenvalue in LUEm,a, are known. First, with n 7→ n+1 in (1.1) for convenience, the p.d.f. of the smallest eigenvalue is given by fixing one of the coordinates at x1, and integrating the remaining over [x1,∞). Thus (1.14) p LUEn+1,a (x1) = n!cn+1,n+1+a e−x1xa1 dλ1 . . . e−λjλaj (λj − x1)2 1≤j<k≤n (λk − λj)2. One has that (1.15) p LUEn+1,a (x1) = − ELUEn+1,a(x1), 4 PETER J. FORRESTER AND NICHOLAS S. WITTE where (1.16) ELUEn+1,a(t) := dλ1 · · · dλn+1p(λ1, . . . , λn+1), is the probability (gap probability) that no eigenvalues are in the interval (0, t). For integer values of the parameter a in (1.1), ELUEn,a(t) was studied by orthog- onal polynomial techniques in [11], where it was evaluated as an a×a determinant, and by the method of Jack polynomials in [10], giving an a-dimensional integral form. In [32] (see also [13]), for general Re(a) > −1, it was expressed in terms of a fifth Painlevé transcendent. Explicitly, it was found that (1.17) ELUEn,a(t) = exp UV(s) where UV(s) satisfies the Jumbo-Miwa-Okamoto σ-form of the Painlevé V equation (1.18) (tσ′′)2 − σ − tσ′ + 2(σ′)2 + (ν0 + ν1 + ν2 + ν3)σ′ + 4(ν0 + σ ′)(ν1 + σ ′)(ν2 + σ ′)(ν3 + σ ′) = 0, with parameters (1.19) ν0 = ν1 = 0, ν2 = n+ a, ν3 = n. Alternatively the conventional Painlevé V parameters are (1.20) α = a2, β = 0, γ = −2n− a− 1, δ = −1 and in terms of the Okamoto parameters they are (1.21) v2 − v1 = 0, v3 − v1 = n+ a, v4 − v1 = n, v3 − v4 = a. Because the eigenvalue density is strictly zero for λ < 0, the neighbourhood of the smallest eigenvalue is referred to as the hard edge, and is denoted by HEa. As is consistent with (1.13), a well defined limit of (1.17) is obtained by the scaling t 7→ t/4n and n→ ∞. Thus [33] (1.22) lim ELUEn,a(t/4n) := EHEa(t) = exp UIII′(s) where UIII′(t) satisfies the Jimbo-Miwa-Okamoto σ-form of the Painlevé III ′ equation (1.23) (tσ′′)2 − v1v2(σ′)2 + σ′(4σ′ − 1)(σ − tσ′)− (v1 − v2)2 = 0, with parameters (1.24) v1 = v2 = a, and subject to the boundary condition (1.25) UIII′(t) ∼ 22a+2Γ(a+ 1)Γ(a+ 2) ta+1. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 5 Our interest in this paper is the distribution p LUEm,a and its hard edge scaled limit. Analogous to (1.14), we see from (1.1) that (1.26) p LUEn+2,a (x1, x2) = n!cn+2,n+2+a e−x1−x2(x1 − x2)2(x1x2)a dλ1 . . . e−λjλaj (λj − x1)2(λj − x2)2 1≤j<k≤n (λk − λj)2, where x1 denotes the smallest eigenvalue and x2 the second smallest eigenvalue. In [12], for a integer, this was expressed as an (a + 2) × (a + 2) determinant. In the hard edge scaled limit this gave (1.27) lim LUEn+2,a ) =: pHEa (x1, x2) = 2−4 e−x2/4 × det Ij+2−k( j=1,...,a k=1,...a+2 x2 − x1 )(k−j)/2 Ij+2−k( x2 − x1) j=1,2 k=1,...a+2 We seek a Painlevé type characterisation of (1.26) and its scaled limit, valid for general Re(a) > −1. One use of knowledge of p LUEm,a is the computation of the distribution of the spacing between the smallest and the second smallest eigenvalues. Denoting this distribution by An,a for LUEn+2,a, we have (1.28) An,a(y) := dx1 p LUEn+2,a (x1, x1 + y), y ∈ R+. Important to our subsequent working is a rewrite of (1.26) and (1.28) in terms of an integral of the form (1.29) Dn(x1, x2)[w(λ)] dλ1 . . . w(λl) (λl − x1)(λl − x2) 1≤j<k≤n (λk − λj)2, where I denotes the support of the weight w(λ). We have (1.30) p(2)(x, x+ y) = cn+2,n+2+a e−(n+1)(x+y)−xy2[x(x + y)]aDn(−y,−y)[λ2(λ+ x+ y)ae−λχ>0], (1.31) An,a(y) = cn+2,n+2+a dt ta(t− y)ae−(n+2)tDn(−y,−y)[λ2(λ+ t)ae−λχ>0]. 6 PETER J. FORRESTER AND NICHOLAS S. WITTE These equations exhibit the occurrence of a deformation of the Laguerre weight (1.32) w(x; t) := x2(x+ t)ae−x, x ∈ R+. This deformed weight actually interpolates between two Laguerre weights - when t→ 0 then we have the general parameter a+ 2 case, whilst if t→ ∞ in the sector −π < arg(t) ≤ π we have the special parameter situation with an exponent of 2. In fact virtually all of our analysis can be carried over to the more general situation where the exponent 2 is an arbitrary complex parameter suitably restricted. We begin in Section 2 by revising appropriate results from orthogonal poly- nomial system theory and apply this to the particular deformed Laguerre weight (1.32). This allows us, in Section 3, to characterise the distributions (1.30) and (1.31) by a solution of the fifth Painlevé equation and its associated linear isomon- odromic system (see Proposition 3.2). Section 4 is devoted to the determinant evaluations of those distributions for positive integer values of the parameter a. We proceed in Section 5 to the study of the hard edge limits (x1, x2) := lim LUEn,a ),(1.33) Aa(z) := lim An,a( ).(1.34) It is found that these scaled distributions can be characterised by the solution of a certain Painlevé III′ equation and its associated linear isomonodromic system (see Propositions 5.2, 5.5 and Remark 5.3). In Section 6 this characterisation is used to obtain the high precision numerical values of statistical characteristics of Aa(z) for various integer values of a and for the values a = ±1/2, the latter being relevant to (1.6), (1.7) with the change of variable λ2j 7→ λj . Let p (x1, x2) denote the scaled distribution of the eigenvalues eiθ1 , eiθ2(θ1, θ2 > 0) closest to the origin in O+(2n + 1) and O+(2n) respectively. With the scaling chosen so that the bulk density is unity, it follows from (1.13) and (1.33) that (1.35) p± (x1, x2) = 4π 2x1x2p HE±1/2 (π2x21, π 2x22). Consequently A±(y) := dx p± (x, x+ y) = 4π2 dxx(x + y)p HE±1/2 (π2x2, π2(x+ y)2).(1.36) We use our results for p HE±1/2 to provide the high precision numerical values of statistical characteristics of A±(y). DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 7 2. Orthogonal Polynomial System 2.1. Semi-classical Orthogonal Polynomials. Consider the general orthogonal polynomial system {pn(x)}∞n=0 defined by the orthogonality relations (2.1) dx w(x)pn(x)x 0 0 ≤ m < n hn m = n with I denoting the support of the weight w(x). We give special notation for the coefficients of xn and xn−1 in pn(x), (2.2) pn(x) = γnx n + γn,1x n−1 + . . . . The corresponding monic polynomials are then (2.3) πn(x) = pn(x). It follows from (2.1) that dx w(x)(pn(x)) 2 = γnhn, and thus for pn(x) to be normalised as well as orthogonal we set γnhn = 1. A consequence of the orthogonality relation is the three term recurrence relation (2.4) an+1pn+1(x) = (x− bn)pn(x) − anpn−1(x), n ≥ 1, and we consider the set of orthogonal polynomials with initial values p−1 = 0 and p0 = γ0. The three term recurrence coefficients are related to the polynomial coefficients by [31], [14] (2.5) an = , bn = γn+1,1 , n ≥ 1, along with (2.6) b0 = − , a0 = 0, γ0,1 = 0. A well known consequence of (2.4) is the Christoffel-Darboux summation (2.7) pj(x)pj(y) = an [pn(x)pn−1(y)− pn−1(x)pn(y)] Central objects in our probabilistic model are the Hankel determinants (2.8) ∆n := det[µj+k−2]j,k=1,...,n, n ≥ 1, ∆0 := 1, (2.9) Σn := det µ0 · · · µn−2 µn µn−1 · · · µ2n−3 µ2n−1 , n ≥ 1, Σ0 := 0, 8 PETER J. FORRESTER AND NICHOLAS S. WITTE defined in terms of the moments {µn}n=0,1,...,∞ of the weight, (2.10) µn := dx w(x)xn. We have integral representations for ∆n (2.11) ∆n = dx1 . . . w(xl) 1≤j<k≤n (xk − xj)2, n ≥ 1, and Σn (2.12) Σn = dx1 . . . w(xl) 1≤j<k≤n (xk − xj)2, n ≥ 1. The three-term recurrence coefficients are related to these determinants by standard result in orthogonal polynomial theory [31], [14] a2n = ∆n+1∆n−1 , n ≥ 1,(2.13) , n ≥ 0,(2.14) γ2n = , n ≥ 0,(2.15) with initial values (2.16) a21 = µ0µ2 − µ21 , b0 = , µ0γ 0 = 1. The orthogonal polynomials themselves also have a determinantal representation (2.17) ∆n∆n+1pn(x) = det µ0 · · · µn µn−1 · · · µ2n−1 1 · · · xn , n ≥ 1, and the integral representation (2.18) ∆n∆n+1pn(x) = dx1 . . . w(xl)(x−xl) 1≤j<k≤n (xk−xj)2. Another set of polynomial solutions to the three term recurrence relation are the associated polynomials {p(1)n (x)}∞n=0, defined by (2.19) p n−1(x) := ds w(s) pn(s)− pn(x) , n ≥ 0. In particular these polynomials satisfy (2.20) an+1p n (x) = (x − bn)p n−1(x)− anp n−2(x), with the initial conditions p −1(x) = 0, p 0 (x) = µ0γ1. Note the shift by one decrement in comparison to the three-term recurrence (2.4) for the polynomials DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 9 {pn(x)}∞n=0. We also need the definition of the moment generating function or Stieltjes function f(x) : = , x /∈ I,(2.21) , x /∈ I, x→ ∞.(2.22) We define non-polynomial associated functions {ǫn(x)}∞n=0 by (2.23) ǫn(x) := f(x)pn(x) − p(1)n−1(x), which also satisfy the three term recurrence relation (2.4), namely (2.24) an+1ǫn+1(x) = (x− bn)ǫn(x) − anǫn−1(x), subject to the initial values ǫ−1(x) = 0, ǫ0(x) = γ0f(x). The associated functions have an integral representation analogous to (2.18) (2.25) ∆n∆n+1ǫn(x) (n+ 1)! dx1 . . . dxn+1 w(xl) x− xl 1≤j<k≤n+1 (xk − xj)2, x /∈ I. The polynomials and their associated functions satisfy the Casoratian relation (2.26) pn(x)ǫn−1(x)− pn−1(x)ǫn(x) = , n ≥ 1. Extending (2.2) and (2.5) we have (2.27) pn(x) = γn 0≤i<j<n bibj − xn−2 +O(xn−3) valid for n ≥ 1, while for the associated functions (2.28) ǫn(x) = γ x−n−1 + x−n−2 0≤i≤j≤n bibj + x−n−3 +O(x−n−4) valid for n ≥ 0. Proposition 2.1 ([6],[4],[24]). Let (2.29) w(x) = 2V (x) W (x) 10 PETER J. FORRESTER AND NICHOLAS S. WITTE for V,W irreducible. The orthogonal polynomials and associated functions satisfy a system of coupled first order linear differential equations with respect to x ( ′ ≡ d/dx) Wp′n = (Ωn − V )pn − anΘnpn−1, n ≥ 1,(2.30) Wp′n−1 = anΘn−1pn − (Ωn + V )pn−1, n ≥ 0,(2.31) for certain coefficient functions V (x),W (x),Θn(x),Ωn(x). The associated func- tions ǫn, ǫn−1 satisfy precisely the same set of equations. If we define the 2× 2 matrix variable (2.32) Yn(x; t) = pn(x) ǫn(x) pn−1(x) ǫn−1(x) then the above coupled system can be written as (2.33) Yn(x) = W (x) Ωn(x) − V (x) −anΘn(x) anΘn−1(x) −Ωn(x)− V (x) Yn(x) It follows that the coefficient functions are specified by Θn =W [ǫnp n − ǫ′npn] + 2V ǫnpn, n ≥ 0, Θ−1 = 0, (2.34) Ωn = anW [ǫn−1p n − ǫ′npn−1] + anV [ǫnpn−1 + ǫn−1pn] , n ≥ 1, Ω0 = 0. (2.35) Proposition 2.2 ([24]). The coefficient functions arising in Proposition 2.1 satisfy the recurrence relations (Ωn+1 − Ωn)(x − bn) =W + a2n+1Θn+1 − a2nΘn−1, n ≥ 0,(2.36) Ωn+1 +Ωn = (x − bn)Θn, n ≥ 0(2.37) We will find it necessary to study the zeros of the orthogonal polynomial pn(x) which we denote {x1,n < . . . < xj,n < . . . < xn,n}. They have an electrostatic interpretation as the equilibrium positions of the mobile unit charges, and there is a set of equations governing these equilibrium positions known as the Bethe Ansatz equations. Proposition 2.3 ([16]). The zeros {xj,n}nj=1 of the polynomial pn(x) satisfy the coupled functional equations (2.38) 2 k 6=j xj,n − xk,n Θ′n(xj,n) Θn(xj,n) ′(xj,n) + 2V (xj,n) W (xj,n) for all 1 ≤ j ≤ n. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 11 One can also represent many useful quantities in terms of sums over the ze- ros and we illustrate this with an example. Firstly the consecutive ratios of the orthogonal polynomials have a partial fraction decomposition (2.39) pn−1(x) pn(x) = − 1 W (xj,n) Θn(xj,n) x− xj,n along with (2.40) a2n = − W (xj,n) Θn(xj,n) Of particular relevance to our application are the semi-classical class of orthog- onal polynomial systems [25] defined by the property that V (x) andW (x) in (2.29) are polynomials in x. The zeros of W (x) define finite singularities of the system of ordinary differential equations (2.30), (2.31) and will feature prominently in this study. Let xr be such a point with r ≥ 1. Then at xr the relations (2.36) and (2.37) can be combined and integrated to yield (2.41) Ω2n(xr)− V 2(xr) = a2nΘn(xr)Θn−1(xr), n ≥ 1. In fact, at a given finite singular point xr, we can deduce the following bi-linear identities, that factorise the one above. Corollary 2.1. The coefficient functions evaluated at a finite singular point xr are related to evaluations of the orthogonal polynomials and associated functions by the relations Ωn(xr) + V (xr) = 2anV (xr)pn(xr)ǫn−1(xr), n ≥ 1,(2.42) Ωn(xr)− V (xr) = 2anV (xr)pn−1(xr)ǫn(xr), n ≥ 1,(2.43) Θn(xr) = 2V (xr)pn(xr)ǫn(xr). n ≥ 0,(2.44) From the theory of Uvarov the following general result for (1.29) is known. Proposition 2.4 ([35]). The quantity Dn(x, x)[w(λ)], defined by the equal argu- ment form of (1.29), is evaluated in terms of the polynomials pn(x) orthogonal with respect to w(x) and coefficients γn,∆n of this system as (2.45) Dn(x, x)[w(λ)] = γnγn+1 [pn(x)p n+1(x)− pn+1(x)p′n(x)]. Proof. This is a specialisation of Uvarov’s general result to the case k = 0 and l = 2 where the integral Dn(x1, x2)[w(λ)] is a Hankel determinant with respect to the weight w0,2(x), defined by (2.46) w0,2(x)dx = dρ0,2(x), ρ0,2(x) = (s− x1)(s− x2)w(s)ds. 12 PETER J. FORRESTER AND NICHOLAS S. WITTE The non-confluent form of the corresponding identity states that (2.47) Dn(x1, x2)[w(λ)] = (∆n+2∆n) 1/2 det[pn+k−1(xj)]j,k=1,2 x2 − x1 and the result follows under the confluence x2 → x1. � We see from (2.45) that our main task is to obtain appropriate characterisations of the orthogonal polynomials and their derivatives associated with the weight (1.32). 2.2. Deformed Laguerre Orthogonal Polynomials. As we noted in the intro- duction we see the appearance of a deformed Laguerre weight (1.32) which is a member of the semi-classical class with the polynomials V,W in (2.29) specified by (2.48) 2V (x; t) = −x2 + (a+ 2− t)x+ 2t, W (x; t) = x(x+ t), and has finite singularities at x = 0,−t. The moments have the simple evaluation (2.49) µn(t) = t a+n+3Γ(n+ 3)U(n+ 3, a+ n+ 4; t), n ≥ 0, |arg(t)| < π, where U(α, γ; z) is the confluent hypergeometric that is not analytic at z = 0. The moments can be written as a sum of two parts one of which is analytic and the other non-analytic about t = 0 (2.50) µn(t) = Γ(a+ n+ 3)1F1(−a;−a− n− 2; t) + (−1)n+3Γ(a+ 1)Γ(n+ 3) Γ(a+ n+ 4) ta+n+31F1(n+ 3; a+ n+ 4; t), where we have to exclude the cases a ∈ Z≥0. Within the semi-classical class the coefficient functions Θn(x),Ωn(x) are poly- nomials with degree fixed independently of the index n. In particular we can relate these polynomials to the coefficients of the orthogonal polynomials themselves. Proposition 2.5. The coefficient functions are Θn(x) = 2n+ a+ 3− t− bn − x, n ≥ 0,(2.51) Ωn(x) = − (2n+ a+ 2− t)x + (n+ 1)t− a2n − , n ≥ 1.(2.52) Proof. From the theory of [24] we note that the degrees of the coefficient functions are degΘn ≤ max{degW − 2, degV − 1} = 1 and degΩn ≤ max{degW − 1, degV − 1, degΘn−1, degU+1} = 2. To obtain explicit forms for these we use the definitions (2.34) and (2.35) and the large x→ ∞ expansions of the polynomials and associated functions given in (2.27) and (2.28). The first equalities in (2.51) and (2.52) then follow. � DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 13 We will also find it convenient to make the following definitions motivated by the above result, θn := 2n+ a+ 3− t− bn,(2.53) κn := (n+ 1)t− a2n − .(2.54) Proposition 2.6. The spectral derivatives of the polynomials pn(x) are ( ∂/∂x) x(x+ t)p′n = (nx+ κn − t)pn − an(θn − x)pn−1(2.55) x(x+ t)p′n−1 = an(θn−1 − x)pn − (−x2 + (n+ a+ 2− t)x + t+ κn)pn−1(2.56) Proof. This follows from the general form of the spectral derivatives (2.30) and (2.30), along with the explicit particular forms (2.51) and (2.52). � Proposition 2.7. The deformation derivatives of the orthogonal polynomials are ( ˙ ≡ ∂/∂t) t(x+ t)ṗn = (n+ 1)t− κn − (x+ t)(θn + t) pn + an(θn + t)pn−1 (2.57) t(x+ t)ṗn−1 = −an(θn−1 + t)pn + (x+ t)(θn−1 + t) + κn − (n+ a+ 1)t (2.58) Proof. We will opt to establish this relation directly from the orthonormality con- ditions on the polynomials dx w(x)pn(x)pn−i(x) = δi,0, 0 ≤ i ≤ n. Differentiating this with respect to t leaves us with the relation (2.59) 0 = a dx w(x) pnpn−i dx w(x)ṗnpn−i + dx w(x)pnṗn−i, where use of the logarithmic derivative of w(x) has been made. Now we employ ṗn−i = γ̇n−i pn−i +Πn−i−1, to write the last term of (2.59) as dx w(x)pnṗn−i = γ̇n−i δi,0 = dx w(x)pnpn−i. 14 PETER J. FORRESTER AND NICHOLAS S. WITTE Considering the first term of (2.59) we note dx w(x) pn(x)pn−i(x) dx w(x)pn(x) pn−i(x)− pn−i(−t) + pn−i(−t) dx w(x) pn(x) x + t = pn−i(−t) dx w(x) pn(x) x + t =− pn−i(−t)ǫn(−t). But we can recast pn−i(−t) as pn−i(−t) = pn−j(−t)δi,j , pn−j(−t) dx w(x)pn−j(x)pn−i(x), dx w(x)pn−i(x) pn−j(−t)pn−j(x), dx w(x)pn−i(x) pj(−t)pj(x). Combining these we deduce that dx w(x)pn−i(x) ṗn(x) + pn(x) − aǫn(−t) pj(−t)pj(x) for i = 0, . . . , n. The factor in curly brackets in the integrand must be a polynomial in x with degree less than or equal to n, and yet is orthogonal to all polynomials pj for j = 0, . . . , n, and thus must be identically zero. This gives us our first form for the deformation derivative of pn, (2.60) ṗn(x) = − pn(x) + aǫn(−t) pj(−t)pj(x). Equating coefficients of pn(x) in this relation we find (2.61) 2 = apn(−t)ǫn(−t). Furthermore using the Christoffel-Darboux formula (2.7), and the above equation, we can express this derivative solely as a linear combination of pn and pn−1 (2.62) ṗn(x) = aǫn(−t) pn(−t) + an pn−1(−t) pn(x) − aanǫn(−t) pn(−t) pn−1(x). DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 15 Using the bilinear product relations (2.43) and (2.44) we arrive at the result (2.57). The second of the two relations can be found by shifting n 7→ n− 1 in (2.62) and using the three term recurrence relation. � In the matrix formulation the spectral derivative take the particular form (2.63) ∂xYn(x; t) = Yn(x; t). The residue matrices are explicitly given by κn − t −anθn anθn−1 −κn − t , χ0 = 0,−2(2.64) (n+ 1)t− κn an(θn + t) −an(θn−1 + t) κn − (n+ a+ 1)t , χt = 0,−a(2.65) (2.66) Our linear system has two regular singularities at x = 0,−t and an irregular sin- gularity at x = ∞ with Poincaré index 1. In this formulation the deformation derivative is (2.67) ∂tYn(x; t) = B + At Yn(x; t) (2.68) B = 1 −θn − t 0 0 θn−1 + t As we will see in Section 3.2 (2.63) and (2.67) form the monodromy preserving system corresponding to the fifth Painlevé equation with a form equivalent to that discussed by Jimbo [18], in contrast to other forms studied in [19], [7], [8] or [17]. Corollary 2.2. The polynomial coefficients satisfy the following coupled, first order mixed deformation derivative and difference equations = 2 + bn−1 − bn, n ≥ 1,(2.69) tḃn = a n − a2n+1 + bn, n ≥ 0,(2.70) with the initial t = 0 values for bn and a n given by (2.87) and (2.88) respectively. This system of differential equations is equivalent to the Schlesinger equations. Proof. There are several methods of proof available here. The first is using the general result of [24], expressing the deformation derivatives of the polynomial coefficients in terms of a sum of the coefficient functions over the movable finite 16 PETER J. FORRESTER AND NICHOLAS S. WITTE singular points Θn(xr)−Θn−1(xr) W ′(xr) ẋr , n ≥ 1,(2.71) ḃn = Ωn+1(xr)− Ωn(xr) W ′(xr) ẋr n ≥ 1.(2.72) The only finite singular point contributing here is x = −t. Alternatively one can find these derivatives from the polynomial derivatives by examining selected coefficients. For example by considering the coefficients of xn−1 in (2.60) we have (2.73) γ̇n,1 = aǫn(−t)pn(−t)γn,1 + aǫn(−t)pn−1(−t)γn−1, and therefore (2.74) ˙(γn,1 = aǫn(−t)pn−1(−t). Consequently, using (2.5), we find that a [ǫn−1(−t)pn−1(−t)− ǫn(−t)pn(−t)] ,(2.75) ḃn = a [anǫn(−t)pn−1(−t)− an+1ǫn+1(−t)pn(−t)] ,(2.76) which are identical to (2.69) and (2.70) respectively. � Corollary 2.3. The recurrences for the polynomial coefficients are (2.77) a2n+2 − a2n = 2t+ bn+1 [2n+ a+ 6− t− bn+1]− bn [2n+ a+ 2− t− bn] , n ≥ 1. (2.78) a2n+1 [2n+ a+ 5− t− bn − bn+1]− a2n [2n+ a+ 1− t− bn−1 − bn] = −bn(bn + t), n ≥ 1. The initial data b0(t) and a 1(t) are given by (2.16) with the evaluation of the mo- ments (2.49). Proof. The first relation (2.77) follows by substituting the explicit forms for the co- efficient functions, (2.51) and (2.52), into the relation (2.37) and requiring equality of the polynomials in x. Equality is trivial for x2 and x1, whilst the non-trivial equality for x0 gives (2.77). The second relation (2.78) follows from the same pro- cedure applied to the recurrence relation (2.36) and again the only nontrivial result occurs for the x0 part. � DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 17 This result can also be recovered from a specialisation of work in [5]. In their sys- tem of monic orthogonal polynomials the three term recurrence coefficients βn, γn (not to be confused with our use of these symbols subsequently) are related to ours (2.79) βn = bn, γn = a Equation (39) of this work implies (2.80) γn+1 + γn = (2n+ 3)t+ (2n+ a+ 4− t)βn + 2 βk − β2n, and differencing this once leads directly to (2.77). In addition their equation (40) implies (2.81) γn+1βn+1 = (2n+ a+ 5− t− βn)γn+1 + 2 βk(βk + t). Again differencing this once one finds precisely (2.78). A check of the above results can be made when t → 0 whilst all parameters are kept fixed since this, as noted before, corresponds to the Laguerre weight with parameter a+ 2. Therefore we have a2n(0) = n(n+ a+ 2),(2.82) bn(0) = 2n+ a+ 3,(2.83) ∆n(0) = j=1 j!Γ(j + a+ 2) .(2.84) In our normalisation we have (2.85) pn(x; 0) = (−1)n Γ(n+ a+ 3) L(a+2)n (x), where L n (x) are the standard associated Laguerre polynomials of degree n and index α. We see that the spectral derivative equations (2.55) and (2.56) reduce to the standard expressions for the derivative of the Laguerre polynomials, the coefficients in the right-hand sides of the deformation derivatives (2.57) and (2.58) vanish, the recurrence relations (2.77) and (2.78) are identically satisfied, and the right-hand sides of (2.69) and (2.70) are zero. In fact we will need to develop expansions about t = 0 in order to charac- terise our quantities as particular solutions of difference and differential equations in Section 3. To this end we have the following result. 18 PETER J. FORRESTER AND NICHOLAS S. WITTE Proposition 2.8. For fixed n, a /∈ Z≥0 the Hankel determinant (2.8) with the weight (1.32) has the expansion about t = 0 (2.86) ∆n(t) = ∆n(0) (a− 1)(n+ 1) (a+ 1)(n− 1) nt2 +O(t3) − 2Γ(a+ 1) Γ(a+ 3)Γ2(a+ 4) Γ(a+ n+ 3) ta+3 (1 + O(t)) + O(t2a+6) with |arg(t)| < π. Consequently, under the same conditions, the three-term recur- rence coefficients have the expansions about t = 0 (2.87) bn(t) = 2n+ a+ 3− 2a(2n+ a+ 3) (a+ 3)(a+ 2)2(a+ 1) t2 +O(t3) (a+ 2)(a+ 1)Γ2(a+ 3) Γ(a+ n+ 3) Γ(n+ 1) ta+3 (1 + O(t)) + O(t2a+6), (2.88) a2n(t) = n(n+ a+ 2) 1− 2a (a+ 3)(a+ 2)2(a+ 1) t2 +O(t3) (a+ 1)Γ(a+ 3)Γ(a+ 4) Γ(a+ n+ 2) Γ(n+ 1) ta+3 (1 + O(t)) + O(t2a+6) Proof. We adopt the method of expanding the Hankel determinant by expanding the moments to leading order (2.89) µn(t) = Γ(a+ n+ 3) + aΓ(a+ n+ 2)t+ a(a− 1)Γ(a+ n+ 1)t2 +O(t3) + (−1)n+1Γ(a+ 1)Γ(n+ 3) Γ(a+ n+ 4) tn+a+3 +O(tn+a+4), as t→ 0 using (2.50). The determinant can be expanded to leading orders in t, ta+3 and the resulting determinants evaluated using the identity [26] (2.90) det(Γ(zk + j))j,k=0,...,n−1 = Γ(zj) 0≤j<k≤n−1 (zk − zj), where {z0, . . . , zn−1} is an arbitrary sequence not necessarily in arithmetic progres- sion. � We conclude this section by noting some identities relating the polynomial coeffi- cients and the zeros of the polynomials. Firstly we give the Bethe Ansatz equations for the zeros of the deformed Laguerre orthogonal polynomials which can be directly deduced from Proposition 2.3. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 19 Corollary 2.4. The zeros xj,n of the deformed Laguerre orthogonal polynomials pn(x) satisfy the functional equations (2.91) xj,n + t xj,n − θn k 6=j xj,n − xk,n = 1, 1 ≤ j ≤ n. According to the electrostatic interpretation, the terms of (2.91) can be inter- preted in the following way - the first is the interaction of the mobile unit charge at xj,n with the fixed charge of size 3 at the singularity x = 0, the second with the fixed charge of size a + 1 at the singularity x = −t, the third with a fixed charge of size −1 at the apparent singularity x = θn, the fourth the mutual repulsion with the other mobile charges and the term on the right-hand side is the linear confining potential. From the partial fraction decomposition (2.39) specialised to the arguments x = 0,−t we have the summation identities. Proposition 2.9. The following summations over the zeros have the explicit eval- uations κn − (n+ 1)t θn + t θn − xj,n ,(2.92) κn − t t+ xj,n θn − xj,n ,(2.93) θn + t κn − t θn − xj,n ,(2.94) a2n = xj,n(xj,n + t) xj,n − θn .(2.95) In addition we can characterise the motion of the zeros with respect to the deformation variable. Proposition 2.10. The zeros xj,n(t) satisfy the differential equation with respect (2.96) tẋj,n = θn + t θn − xj,n xj,n. Proof. This follows by equating (2.97) ṗn(x) pn(x) ẋj,n x− xj,n and (2.57), and then employing (2.39) along with (2.48), (2.51). � 3. Difference and Differential Equations 3.1. Difference Equations. In the first subsection we derive an alternative dif- ference system in terms of the new variables θn(t), κn(t) as specified by (2.53), (2.54). 20 PETER J. FORRESTER AND NICHOLAS S. WITTE Proposition 3.1. The auxiliary functions θn(t), κn(t) satisfy a system of coupled first order recurrence relations κn+1 + κn = θn(θn + t− 2n− a− 3), n ≥ 0,(3.1) θn + t θn−1 + t (κn − t)(κn + t) [κn − (n+ a+ 1)t][κn − (n+ 1)t] , n ≥ 1.(3.2) The initial values θ0 and κ0 are given by (3.3) θ0(t) = −2t dx e−xx(t+ x)a dx e−xx2(t+ x)a , κ0 = t. Proof. The first of the recurrence relations (2.77) can be exactly summed and the result is (3.4) a2n+1 + a n = (2n+ 3)t+ (2n+ a+ 4− t)bn + 2 bi − b2n. Recalling the second relation of (2.5) the summation appearing here can done by recasting the equation in terms of the new variables and yields (3.5) κn+1 + κn = −θnbn, which is (3.1). The second member of the coupled set is most easily found from the general relation (2.41) evaluated at the finite singular points x = 0,−t and employing the new variables. These two key identities are (κn + t)(κn − t) = a2nθnθn−1,(3.6) [κn − (n+ a+ 1)t][κn − (n+ 1)t] = a2n(θn + t)(θn−1 + t).(3.7) The ratio of these two identities yields the relation (3.2). � There are other recurrence relations which will be used subsequently, and the first is (3.8) a2n(θn + θn−1 + t) = −(2n+ a+ 2)κn + [n2 + (n+ 1)(a+ 2)]t. This follows from the subtraction of (3.6) from (3.7). The second relation (3.9) a2n+1 − a2n − bn − t = 2κn + bnθn, is derived by writing the definition of κn+1 − κn in terms of the old variables and then employing (3.1). The last relation (3.10) a2n+1θn+1 − a2nθn−1 = bn(2κn + bnθn), is a consequence of (2.78) along with the use of (3.9). DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 21 As a consequence of relations (3.6) from (3.7) we have θn + t κn − t κn − t κn − (n+ 1)t θn + t κn + t − a2n θn−1 + t κn − (n+ a+ 1)t κn − t κn − (n+ a+ 1)t n+ a+ 2 + κn + t ,(3.11) (3.12) n+ κn − t n+ a+ 2 + κn + t n7→n+1 = θn + t. 3.2. Reduction to Painlevé V. Here we will identify the fifth Painlevé system as the solution to our system of equations characterising the deformed Laguerre or- thogonal polynomial system. This is most simply seen in terms of the new variables θn, κn rather than the basic orthogonal polynomial variables an, bn. Proposition 3.2. The auxiliary quantities θn(t), κn(t) satisfy the coupled first or- der ordinary differential equations (3.13) tθ̇n = 2κn + θn(2n+ a+ 3− t− θn), (3.14) tκ̇n = θn + t κ2n + 2n+ a+ 3− (2n+ a+ 2) t θn + t − [n2 + (n+ 1)(a+ 2)]t− t + (n+ 1)(n+ a+ 1) θn + t Equations (3.13) and (3.14) can be solved in terms of the fifth Painlevé system (3.15) θn = t , κn = t(1 + qp), where q, p are the Hamiltonian variables of the Okamoto PV [27] system with the parameters (3.16) α = , β = −2, γ = −(2n+ a+ 3), δ = − (3.17) v2 − v1 = −2, v3 − v1 = n+ a, v4 − v1 = n, v3 − v4 = a. The solutions satisfy the boundary value data at t = 0 (3.18) θn(t) = 2a(2n+ a+ 3) (a+ 3)(a+ 2)2(a+ 1) t2 +O(t3) (a+ 2)(a+ 1)Γ2(a+ 3) Γ(a+ n+ 3) Γ(n+ 1) ta+3 (1 + O(t)) + O(t2a+6), 22 PETER J. FORRESTER AND NICHOLAS S. WITTE (3.19) κn(t) = 2n+ a+ 2 4n(n+ a+ 2)a (a+ 3)(a+ 2)2(a+ 1) t2 +O(t3) (a+ 2)(a+ 1)Γ2(a+ 3) Γ(a+ n+ 3) ta+3 (1 + O(t)) + O(t2a+6), provided a /∈ Z≥0 and |arg(t)| < π. Proof. If we employ (3.9) in (2.70) and then substitute for bn in terms of θn then the result is (3.13). Let us define the shorthand notation (3.20) Γn := Furthermore when the deformation derivative (2.70) is summed on the free index the result is (3.21) tΓ̇n = a n + Γn. Now if compute the deformation derivative of κn and use (2.69) along with (3.21) we arrive at (3.22) tκ̇n = κn − a2n(θn − θn−1). Now the idea is to eliminate a2n and θn−1, which appear in (3.22), through use of the recurrence relations. Equation (3.2) is a linear equation for θn−1 in terms of the unshifted variables and the solution can be substituted into (3.8) yielding a linear relation for a2n in terms of unshifted variables. Then both solutions can be substituted into (3.22) and the result is (3.14). One can easily verify that the transformation to the Hamiltonian variables q, p (3.15) with the parameters (3.16) yields the Hamilton equations of motion for the Hamiltonian in [27]. � Remark 3.1. A few remarks can now be made regarding the identification of the recurrences (3.1) and (3.2). This is different in appearance from the discrete in- tegrable equations that arose in the study of the Laguerre unitary ensemble [13] which were explicitly identified with the system in the Sakai scheme, with the ra- tional surface D 5 → E 6 , and has a continuous limit of Painlevé IV . In fact we find that the variables of the latter system can be expressed in terms of our own (3.23) xn = κn + θn(n+ a+ 1− t− θn) θn + t , yn = − and it is clear that one cannot transform (3.1) and (3.2) into this system using such a transformation. Also the two systems arise as different Schlesinger-type transformations - in our case as a sequence where α0 7→ α0 + 1, α2 7→ α2 − 1 whereas in the other case as α0 7→ α0 + 1, α3 7→ α3 − 1. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 23 Remark 3.2. In [13] two fundamental quantities were studied - the τ -function τ [n](t) and its logarithmic derivative the σ-function Vn(t; a, µ). Their relation to the ob- jects of the present work are (3.24) τ [n] = c(n, a)n!e−ntt−n 2−n(a+4)∆n(t), where c(n, a) is an unspecified constant and (3.25) Vn(t; a, 2) = −nt− 4n+ t log∆n(t). We also note that (3.26) Vn(t; a, 2) = Γn + n(n+ a− 2− t) and consequently Γn = −n(n+ a+ 2) + t log∆n(t)(3.27) bn = 2n+ a+ 3 + t ∆n(t) ∆n+1(t) (3.28) Remark 3.3. The new variable Γn(t) possess an expansion as t → 0 which can be directly found from (2.86) and (2.87) (3.29) n(n+ a+ 2) + Γn(t) = nt− 2n(n+ a+ 2)a (a+ 3)(a+ 2)2(a+ 1) t2 +O(t3) (a+ 2)(a+ 1)Γ(a+ 3)Γ(a+ 4) Γ(a+ n+ 3) ta+3 (1 + O(t)) + O(t2a+6), again provided a /∈ Z≥0. Remark 3.4. The use of Proposition 3.2 is in the computation of the orthogonal polynomials in (2.4) corresponding to the weight (1.32). For this we note from (2.53) that (3.30) bn = 2n+ a+ 3− t− θn, while (2.54) together with (2.5), (2.6) show a2n = (n+ 1)t− − κn(3.31) = (n+ 1)t+ bj − κn,(3.32) (the quantity an is positive, so the positive square root of this equation is to be taken). Further, it follows from (2.13) and (2.5) that (3.33) 1 · · · γ2n−1 = ∆n, where each γj is the coefficient of x j in pj(x) as specified by (2.2). All terms in the equation (2.45) for Dn(x, x) are then known, and the task is then to compute the integral as required by (1.31). 24 PETER J. FORRESTER AND NICHOLAS S. WITTE An alternative system of coupled first order ordinary differential equations which will be used for scaling to the hard edge is given in the following proposition. Proposition 3.3. The variables θn(t),Γn(t) satisfy the coupled first order ordinary differential equations (3.34) tθ̇n = θn − θ4n − 2(2n+ a+ 2− t)θ3n + [4Γn + (2n+ a+ 2− t)2 − 4(n+ 1)t]θ2n + 4t[Γn + n(n+ a+ 2− t) + a+ 2− t]θn + 4t2 (3.35) tΓ̇n = (n+ 1)t+ θn(2n+ a+ 2− t− θn) + θ4n − 2(2n+ a+ 2− t)θ3n + [4Γn + (2n+ a+ 2− t)2 − 4(n+ 1)t]θ2n + 4t[Γn + n(n+ a+ 2− t) + a+ 2− t]θn + 4t2 Proof. We proceed in a series of steps. Firstly we use (3.6) to solve for θn−1 in terms of θn, κn and Γn. In the second step we substitute this solution for θn−1 into (3.7) and solve the following quadratic equation for κn in terms of θn and Γn, (3.36) κ2n+θn(2n+a+2−t−θn)κn+[(n+1)t−Γn]θn(θn+t)−[n2+(n+1)(a+2)]tθn−t2 = 0. The choice of the sign of the square-root branch follows from the expansions (3.18) and (3.29) on one hand, and on the other hand noting that as t→ 0 (3.37) θ4n − 2(2n+ a+ 2− t)θ3n + [4Γn + (2n+ a+ 2− t)2 − 4(n+ 1)t]θ2n + 4t[Γn + n(n+ a+ 2− t) + a+ 2− t]θn + 4t2 2a(2n+ a+ 3) (a+ 3)(a+ 2)2(a+ 1) t2 +O(t3). In the final step we use these solutions for θn−1 and κn in (3.22) and (3.21). � In preparation for the hard edge scaling limit we need to make evaluations of the polynomials at the finite singular points. Firstly considering x = 0 we note (3.38) πn(0) πn−1(0) pn(0) pn−1(0) κn + t = a2n κn − t as follows immediately from (2.55). Furthermore we also have (3.39) t ˙(log πn(0)) = t ˙(log pn(0)) + (θn + t) = n+ κn − t DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 25 which follows from (2.57) and the above result. The corresponding result for the polynomial ratio at x = −t is (3.40) pn(−t) pn−1(−t) κn − (n+ a+ 1)t θn−1 + t θn + t κn − (n+ 1)t After [1] we define the orthogonal polynomial ratios (3.41) Qn(x; t) := pn(x; t) pn(0; t) because we are interested in the scaling properties of the orthogonal polynomial system at the edge of their interval of orthogonality, x = 0. Then (2.4) and Propo- sitions 2.6 and 2.7 can be translated into the following three corollaries. Corollary 3.1. The three-term recurrence for {Qn}n=0,1,... system is (3.42) bn(Qn+1 +Qn−1 − 2Qn) + (bn + 2 κn − t )(Qn+1 −Qn−1) + 2xQn = 0. Corollary 3.2. The spectral derivatives of Qn, Qn−1 are x(x+ t)Q′n = nxQn + (κn − t) Qn −Qn−1 + (3.43) x(x+ t)Q′n−1 = x[x − (n+ a+ 2− t)]Qn−1 + (κn + t) Qn −Qn−1 − (3.44) Corollary 3.3. The deformation derivatives of Qn, Qn−1 are t(x+ t)Q̇n = −x(n+ κn − t )Qn + (κn − t) θn + t [Qn−1 −Qn] ,(3.45) t(x+ t)Q̇n−1 = x[n+ a+ 2 + κn + t ]Qn−1 + (κn + t) θn−1 + t [Qn−1 −Qn] . (3.46) As we noted earlier the polynomial ratio Qn(x; t) has a product representation (3.47) Qn(x; t) = where again xj,n is the j-th zero of the polynomial. We can use this fact to compute sums of the inverse powers of the zeros from the above differential equations. Proposition 3.4. The increment of the sum of the reciprocals of the zeros going from n− 1 to n is given by (3.48) 3t xj,n−1 κn − t + n+ a+ 2 + κn + t 26 PETER J. FORRESTER AND NICHOLAS S. WITTE Proof. The required increment of the sum of the reciprocals of the zeros is the order x term in the expansion of ψn(x; t) := Qn(x; t)/Qn−1(x; t) about x = 0, and this can evaluated from the same expansion of the differential equation for ψn with respect to x. This latter differential equation is easily found from (3.43) and (3.44) and is (3.49) x(x+t)ψ′n = x− θn (κn−t)+ 2κn+x[−x+2n+a+2−t] x− θn−1 (κn+t)ψ At the other finite singular point, x = −t, we have as a consequence of these corollaries (3.50) Qn(−t; t) Qn−1(−t; t) κn − t κn − (n+ 1)t θn + t κn − (n+ a+ 1)t κn + t θn−1 + t (3.51) Qn(−t; t) = t− κn − nθn θn(θn + t) Qn(−t; t). Note that the derivative with respect to t in the latter equation is a total derivative. To complete our preparations for the hard edge scaling we need to identify two polynomial variables that will scale to independent variables in the scaling limit. The first is the orthogonal polynomial ratio Qn, and for the second a number of choices could be made but a simple choice is (3.52) Rn := Qn −Qn−1. Corollary 3.4. The spectral derivatives of Qn, Rn are x(x + t)Q′n = x κn − t Qn + (κn − t) θn − x Rn,(3.53) x(x+ t)R′n = x κn − t + n+ a+ 2 + κn + t − x− t Qn(3.54) κn − t + x(x + t)− (a+ 2)x− 2t Proof. This follows from Corollary 3.2. � Corollary 3.5. The deformation derivatives of Qn, Rn are t(x + t)Q̇n = −x κn − t Qn − (κn − t) θn + t (3.55) t(x+ t)Ṙn = −x κn − t + n+ a+ 2 + κn + t (3.56) n+ a+ 2 + κn + t + (κn + t) θn−1 + t − (κn − t) θn + t DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 27 Proof. This follows from Corollary 3.3. � 3.3. Inequalities and Bounds. A key step in proving our hard edge scaling limits will be bounds on the variables θn, κn and some auxiliary quantities. The first step is the following result. Lemma 3.1. The variables θn(t), κn(t) satisfy the inequalities θn + t κn − (n+ 1)t κn − t < 0,(3.57) κn + t θn−1 + t κn − (n+ a+ 1)t < 0,(3.58) for all positive, real and bounded t and n ≥ 1. Proof. That the ratios given in (3.57) and (3.58) are negative is a consequence of the fact that the polynomial pn(x) evaluated on the negative real axis, i.e. exterior to the interval of orthogonality, has a fixed sign. Specifically (−1)npn(−y) > 0 for real, positive y. Using the ratio relations (3.38) and (3.40) we have the upper bounds. From the Christoffel-Darboux formula (2.7) at equal arguments we note (3.59) pn−1(x)p n(x)− p′n−1(x)pn(x) > 0, and from the above pn−1(x)pn(x) < 0 for x ∈ −R+ we conclude (3.60) p′n(x) pn(x) p′n−1(x) pn−1(x) under the conditions on x. Integrating this inequality from 0 to −y ∈ −R+ we arrive at (3.61) pn(−y) pn−1(−y) pn(0) pn−1(0) Then identifying these ratios with (3.38) and (3.40) in the case y = t leads to the relative inequalities. � The above set of inequalities must all apply simultaneously and we see in fact that it implies restrictions on the variables θn, κn. Lemma 3.2. For bounded t ∈ R+ and n ≥ 0 the variables θn, κn satisfy inequalities which place them in one of three cases, as illustrated in Figure 1 - Case I: 0 < θn,(3.62) θn(θn + t− 2n− a− 3) + t < κn < t− nθn,(3.63) Case II: −t ≤ θn ≤ 0,(3.64) t− nθn ≤ κn ≤ t+min{nt, θn(θn + t− 2n− a− 3)} ≤ (n+ 1)t,(3.65) 28 PETER J. FORRESTER AND NICHOLAS S. WITTE (n+1)t t−nθn (θn+t)(θn−2n−a−3)+(n+1)t θn(θn+t−n)+t θn(θn+t−2n−a−3)+t ......... .......... ......... .......... .......... ......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... ......... ....... ....... ....... ....... ....... ....... ...... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ...... ....... ........ ....... ........ ........ ..... ........ ........ .......... ....... ............. ............... .............. .............. ............... ................ ................. .................. ..................... ........................ ........................... .............................. ................................ ................................... ...................................... ......................................... ....... ....... ....... ....... ....... ....... ....... ....... ....... ...... ....... ....... ...... ....... ....... ....... ....... ....... ....... ...... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ..... ....... ....... ....... ....... ....... ....... ....... ...... ....... ....... ....... ....... ........ ...... ........ ........ .......... ........ ............. ................ ............... ............... ................ ................. .................. ................... ..................... ........................ ........................... .............................. ................................. .................................... ....................................... .......................................... ............................................. ................................................ ................................................... ...... ....... ...... ....... ....... ...... ....... ...... ....... ....... ....... ....... ...... ....... ....... ....... ....... ..... ...... ....... ....... ....... ....... ...... ....... ....... ...... ....... ....... ...... ....... ....... ...... ....... ....... ...... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ...... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ...... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ........ ........ ....... ....... ........ ........ .......... ........ ............ ..... ................ ............... ............... ................ ................. .................. ................... ...................... .......................... ............................. ................................ ................................... ...................................... ......................................... ............................................ ................................................ ................................................... ...................................................... ......................................................... ............................................................ ............................................................... Figure 1. A pictorial form of the inequalities taking the example of a = 1/2, n = 2 and t = 5/3, which illustrates the generic situation for n+ a+ 3 > t. Case III: θn < −t,(3.66) (n+ 1)t < κn < t− nθn.(3.67) Proof. From the three inequalities implied by (3.57) we see that θn ≷ 0 according as κn ≶ t, θn + t ≷ 0 according as κn ≶ (n+ 1)t, and θn(θn + t) ≷ 0 according as κn ≶ t − nθn. In (3.58) we make the replacement n 7→ n + 1 and employ (3.1) to eliminate κn+1. This inequality now reads (3.68) − t− κn − (n+ 1)t θn + t < −κn − t < bn = 2n+ a+ 3− t− θn. Consequently these three inequalities imply θn ≷ 0 according as κn − t ≷ θn(θn + t− 2n− a− 3), θn + t ≷ 0 according as κn − (n+ 1)t ≷ (θn + t)(θn − 2n− a− 3), and θn(θn + t) ≷ 0 according as κn ≶ θn(θn + t− n) + t. Combining these sets of inequalities leads to the three cases above. � We see that Case II applies in our situation. Lemma 3.3. For all n and t ∈ R+ we have −t ≤ θn ≤ 0 and t − nθn ≤ κn ≤ (n+ 1)t. Proof. From the residue formula (2.44) we recall that Θn(0) = θn = 2tpn(0)ǫn(0) and Θn(−t) = θn + t = −atpn(−t)ǫn(−t). As we noted in the proof of Lemma DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 29 (3.1) it is immediate from the integral representations of the polynomials and their associated functions, (2.18) and (2.25), that (−1)npn(−t) ≥ 0 and (−1)n+1ǫn(−t) ≥ 0 for all real t ≥ 0. This places θn in the range applying to Case II. � We can also draw some conclusions concerning the zeros of the orthogonal poly- nomials which will be important subsequently. Corollary 3.6. Each zero xj,n(t) is a monotonically decreasing function of t and interpolates between the Laguerre zero with t = 0 and exponent a + 2 and the Laguerre zero with t = ∞ and exponent 2, (3.69) xj,n(0) > xj,n(t) > xj,n(∞) > 0, for all 1 ≤ j ≤ n and bounded t > 0. Corollary 3.7. The following bounds on the reciprocal sums over the zeros hold xj,n(t) ,(3.70) xj,n(t) + t .(3.71) Proof. From the two-sided bound on θn we can deduce (3.72) xj,n(t) + t xj,n(t)− θn xj,n(t) Employing this in the Bethe Ansatz (2.91) summed over j we arrive at the above bounds. � 4. Special Case a ∈ Z≥0 Our evaluation of the distribution function in terms of the fifth Painlevé sys- tem is with all three free parameters variable in some sense - one is fixed in this application at a positive integer, one is the index n ∈ Z and the remaining one is a ∈ C. Up to this point we have studied in some depth the recurrence relations with respect to n while a has been left arbitrary other than being restricted because of the existence considerations. From the point of view of the Painlevé theory it is quite natural that the transcendental objects become classical when a ∈ Z for either positive or negative subsets of the integers. In particular it is expected that the τ functions in the theory will have Hankel determinantal forms of classical function entries with a rank dependent on a. It is these cases which have been studied in the past [12],[11] using methods which transform the integral into the determinantal representations and then employ confluent Vandermonde identities. 30 PETER J. FORRESTER AND NICHOLAS S. WITTE Proposition 4.1. When a ∈ Z>0 we have the evaluation for the Hankel determi- (4.1) ∆n(t) = cn+1,n+1+a (n+ a)! det[L (j+k+1−a) n+a+1−j−k(−t)]j,k=1,...,a, and for a = 0 (4.2) ∆n(t) = cn+1,n+1 Proof. In [12] Eq. (3.18) (after correcting) states (4.3) ∆n(t) = cn+1,n+1+a (n+ a)! (−1)a(a−1)/2 det[Dj+k−2x L −(a−3) n+a−1 (x)|x=−t]j,k=1,...,a, where Dx := d/dx. Using the Laguerre polynomial identity (4.4) Dmx L n (x) = (−1)mL (α+m) n−m (x), m ∈ Z≥0, with the proviso L n (x) = 0 for n < 0, we arrive at (4.1). � As a consequence of the relations (3.28) and (3.27) the variables θn(t),Γn(t) will have a× a determinant forms, and in particular for a = 0 (4.5) θn(t) = −t, κn(t) = (n+ 1)t, Γn(t) = −n(n+ 2). The orthogonal polynomials also have determinantal forms of the following type. Proposition 4.2. When a ∈ Z>0 the orthogonal polynomials are given by (4.6) ∆n∆n+1pn(x; t) = (−1)n+a+⌊ ⌋a! . . . (n+ a)!1! . . . (n+ 1)! × (x+ t)−a det (k+1−a) n+a+1−k(x) k=1,...a+1 (j+k−a) n+a+2−j−k(−t) j=1,...,a k=1,...a+1 and for a = 0 (4.7) pn(x; t) = (−1)n (n+ 2)(n+ 1) L(2)n (x). If a > n+1 we note that L (a+1) n+1−a(−t) = 0. Consequently, under the same condition, the polynomial ratio is given by (4.8) Qn(x; t) = (k+1−a) n+a+1−k(x) k=1,...a+1 (j+k−a) n+a+2−j−k(−t) j=1,...,a k=1,...a+1 n+a+1−k k=1,...a+1 (j+k−a) n+a+2−j−k(−t) j=1,...,a k=1,...a+1 DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 31 Proof. Starting with the integral representation (2.18) we follow the procedures used in [12]. Taking one factor of the squared product of differences we write it like (4.9) 1≤j<k≤n (xk − xj) = l=0 cl det[L k−1(xj)]j,k=1,...,n, using the Vandermonde identity and where cn is the leading coefficient of L n (x) and α is a parameter to be fixed later. Of the remaining factors in the integrand we write (4.10) (xj + t) a(x− xj) 1≤j<k≤n (xk − xj) = (−1)a(n+1) l=0 l! l=0 cl (x+ t)−a × det k−1(xj) j=1,...,n k=1,...,N k−1(x) k=1,...,N Dj−1y L k−1(y)|y=−t j=1,...,a k=1,...,N where the confluent Vandermonde identity has been used and N = n+ a+ 1. Reassembling the integral with these two factors, then expanding the determi- nant in (4.9) we multiply each of n factors into the determinant of (4.10). Making use of the antisymmetry of the row ordering in the first n rows of the determinant we can perform the n integrals as long as we choose α = 2. Then (4.11) ∆n∆n+1pn(x; t) = (−1)a(n+1) l=0 l! l=0 cl l=0 cl (j + 2)! (x+ t)−a × det n+k−1(x) k=1,...,a+1 Dj−1y L n+k−1(y)|y=−t j=1,...,a k=1,...,a+1 Using the identities (4.12) L(α−1)n (x) = L n (x)− L n−1(x), along with elementary column operations, and then identity (4.4) we are lead to (4.6). The evaluation for the polynomial ratio (4.8) is a simple consequence of the first evaluation along with (4.13) L (−a+1+k) n+a+1−k (0) = n+ a+ 1− k 32 PETER J. FORRESTER AND NICHOLAS S. WITTE Proposition 4.3. When a ∈ Z≥0 the deformed Hankel determinant is given by (4.14) Dn(x, x) = cn+2,n+2+a(−1) (a+1)(a+2)(x+ t)−2a × det (j+k−1−a) n+a+3−j−k(−t) j=1,...,a k=1,...a+2 (j+k−1−a) n+a+3−j−k(x) j=1,2 k=1,...a+2 Proof. The quantity Dn(x, x) was essentially computed in [12] in Eq. (3.20), which can be recast as (4.15) Dn(x, x) = cn+2,n+2+a(−1) (a+1)(a+2)(x+ t)−2a × det Dj+k−2u L −(a−1) n+a+1 (u)|u=−t j=1,...,a k=1,...a+2 Dj+k−2u L −(a−1) n+a+1 (u)|u=x j=1,2 k=1,...a+2 Then (4.14) follows by application of the identity (4.4). � As a consequence the distribution of the first eigenvalue spacing is (4.16) An,a(y) = (−1) (a+1)(a+2)y2ey dt ta(t− y)−ae−(n+2)t × det (j+k−a−1) n+a+3−j−k(−t) j=1,...,a k=1,...a+2 (j+k−a−1) n+a+3−j−k(−y) j=1,2 k=1,...a+2 for a = 1, 2, 3, . . . and (4.17) An,0(y) = y2e−(n+1)y n+1(−y)L n−1(−y)− L(2)n (−y) for a = 0. 5. Hard Edge Scaling 5.1. General Case. We define new scaled spectral variables s, z by (5.1) t = , x = − in the triangular domain s > z > 0 and study the scaling of the finite distribu- tion (1.31) as the polynomial degree n → ∞. What is required here is not just the asymptotic scaling of the orthogonal polynomial coefficients but also of the polynomials themselves in the neighbourhood of an endpoint of the interval of or- thogonality. For the deformed Laguerre polynomials this would be a generalisation of the asymptotics of Hilb’s type for the Laguerre polynomials as found in Szegö (5.2) ex/2xµ/2L(µ)n (−x) = Γ(µ+ n+ 1) M)µ/2 Mx) + O(n DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 33 with M = 4n + 2µ + 2 as n → ∞ and Iµ(z) the standard modified Bessel func- tion. Despite a resurgence in activity around these questions, especially the use of Riemann-Hilbert techniques on these problems, there are no results available for our particular problem. A general review of the asymptotics of orthogonal polyno- mials can be found in [23], and an introduction to the Riemann-Hilbert approach to the asymptotics is the chapter in [22]. However to establish the nature of and the existence of limits for our variables we do not require such techniques. Lemma 5.1. Under the scaling of the (5.1) 4nθn(t) and κn(t) are bounded for all real positive t and n ≥ 1. Proof. From the result of Lemma 3.3 we see that (5.3) − s ≤ 4nθn(s/4n) ≤ 0, s/4n− 4nθn(s/4n) ≤ κn(s/4n) ≤ (n+ 1)s/4n, and the assertion follows. � Corollary 5.1. Under the above conditions (5.4) n+ κn(t)− t θn(t) t=s/4n = O(1), as n→ ∞. Proof. The above lemma states that (5.5) θn(s/4n) = O(1), as n→ ∞, and we find as a consequence that also (5.6) ε := n(n+ a+ 2)− s/4 + Γn(s/4n) = O(1), as n→ ∞. The discriminantD appearing in the workings of Proposition 3.3 can then be written (5.7) D2 := θ4n−2(2n+a+2−t)θ3n+[4(ε−t)+(a+2−t)2−4nt]θ2n+4t[ε+a+2−t]θn+4t2, and therefore (5.8) θn(t) t=s/4n = O(1), as n→ ∞. From the formula for κn in Proposition 3.3 we note that (5.9) n+ κn − t = −a+ 2 (θn + t)− and the result then follows. � Proposition 5.1. For bounded s ∈ R+ under the scaling (5.1) the variables θn(t) and Γn(t) converge to limits in the following manner 4nθn(t)|t=s/4n = µ(s),(5.10) n(n+ a+ 2) + Γn(t)|t=s/4n = ν(s).(5.11) 34 PETER J. FORRESTER AND NICHOLAS S. WITTE The variable κn(t) converges like (5.12) lim κn(t)|t=s/4n = − µ(s). Proof. Firstly we note that η := a2n(t)− n(n+ a+ 2) t=s/4n = O(1), as n→ ∞, as follows from (5.6). Starting with (3.8) we see that θn − θn−1 = 2θn + t+ (2n+ a+ 2)κn − [n2 + (n+ 1)(a+ 2)]t (2n+ a+ 2 + η)t+ 2(n+ a+ 2)θn κn − t ,(5.13) and thus nθn−(n−1)θn−1 = θn−1+ (2n+ a+ 2 + η)t+ 2(n+ a+ 2)θn κn − t Thus we have shown nθn(t)− (n− 1)θn−1(t)|t=s/4n = O(n −1), as n→ ∞. Now we can write the quantity of interest nθn(t)|t=s/4n − (n− 1)θn−1(t)|t=s/4(n−1) = [nθn(t)− (n− 1)θn−1(t)]|t=s/4n + (n− 1)θn−1(t)|t=s/4n − (n− 1)θn−1(t)|t=s/4(n−1) , so we require bounds on the difference of the last two terms on the right-hand side. Let t = s/4n and t> = s/4(n− 1). Because θn(t) is continuously differentiable and its derivative is given by (3.13) (n− 1) |θn−1(t)− θn−1(t>)| ≤ (n− 1)(t> − t) max u∈(t,t>) |θ̇n−1(u)|, u∈(t,t>) |θ̇n−1(u)|, u∈(t,t>) u−1|2κn−1(u) + θn−1(u)(2n+ a+ 1− u− θn−1(u))|, u∈(t,t>) u−1|θn−1(u)| n− 1 + κn−1(u)− u θn−1(u) + a+ 3− u− θn−1(u) + θn−1(u) ≤ t> max u∈(t,t>) n− 1 + κn−1(u)− u θn−1(u) + a+ 3 + u+ |θn−1(u)|+ |θn−1(u)| = O(n−1), as n→ ∞. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 35 Thus the limit shown in (5.10) exists. Turning our attention to (5.12) we can use (3.1) to compute that κn+1 − κn = −2κn − bnθn, = −2κn − (2n+ a+ 3− t− θn)θn, = −2θn κn − t − 2t− (a+ 3)θn + θn(θn + t), and therefore [κn+1(t)− κn(t)]|t=s/4n = O(n −1), as n→ ∞. Now in this case the quantity we require is κn+1(t)|t=s/4(n+1) − κn(t)|t=s/4n = κn+1(t)|t=s/4(n+1) − κn+1(t)|t=s/4n + [κn+1(t)− κn(t)]|t=s/4n , and therefore we need to bound the difference of first two terms on the right- hand side. Let us denote t< = s/4(n + 1). Again because κn(t) is continuously differentiable with derivative (3.22) we have |κn+1(t<)− κn+1(t)| ≤ (t− t<) max u∈(t<,t) |κ̇n+1(u)|, 4n(n+ 1) u∈(t<,t) |κ̇n+1(u)|, 4n(n+ 1) u∈(t<,t) u−1|κn+1(u)− a2n+1(u)(θn+1(u)− θn(u))|, u∈(t<,t) |κn+1(u)|+ (2n+ a+ 4 + |η(u)|)u + 2(n+ a+ 3)|θn+1(u)| n+ 1 + κn+1(u)− u θn+1(u) = O(n−1), as n→ ∞. In the last two steps we have used (5.13) and the subsequent estimates. Thus the limit in (5.12) follows. The fact that the limit of κn(t) under the hard edge scaling is related to the limit given in (5.10) follows from the relation (3.2). The limit (5.11) is a consequence of the limits in the primary variables. � In addition the following combinations of variables possess scaling limits which will subsequently be useful. 36 PETER J. FORRESTER AND NICHOLAS S. WITTE Corollary 5.2. For bounded s ∈ R+ the following limits as n→ ∞ exist 2κn(t) + θn(t)bn(t) θn(t) t=s/4n ,(5.14) κn(t)− t θn(t) t=s/4n C(s),(5.15) n+ a+ 2 + κn(t) + t θn−1(t) t=s/4n −C(s),(5.16) κn(t)− t θn(t) + n+ a+ 2 + κn(t) + t θn−1(t) t=s/4n ξ(s).(5.17) Proof. The limit in (5.15) is a consequence of (5.4) in a previous corollary. The scaling limit of (5.16) follows from that of (5.15) and the identity (3.11). The limit (5.17) can be derived from (3.12) and as a result one can deduce that κn − t + n+ a+ 2 + κn + t is of order O(n−1). � We note other relations amongst the scaling limit functions, namely (5.18) 2µC(s) = −[(a+ 2)µ+ 2s] [(a+ 2)µ+ 2s]2 + 4µ(µ+ s)ν − µ(µ+ s)2 (5.19) ξ(s) = − sC(C + a) (5.20) 2C + a+ 3 = s µ̇− 2 Proposition 5.2. The scaled variables µ(s), ν(s) are characterised by solutions to the PIII′ system with parameters v1 = a+ 2, v2 = a− 2. In particular (5.21) ν(s) = −σIII′(s) + s− a− 2, where σIII′(s) satisfies the Jimbo-Miwa-Okamoto σ-form for PIII′ with above param- eters. The boundary conditions to uniquely specify the solution ν(s) are (5.22) ν(s) = 8(a+ 3)(a+ 2)2(a+ 1) s2 +O(s3) 4a+3(a+ 2)(a+ 1)Γ(a+ 3)Γ(a+ 4) sa+3 (1 + O(s)) + O(s2a+6), assuming a /∈ Z≥0 and |arg(s)| < π. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 37 Proof. Introducing the scaling ansatzes (5.10) and (5.11) formally into the differen- tial equation (3.34) under the scaling (5.1) we find that the highest order nontrivial relation (5.23) µ(s) + s = 4sν̇(s), at order n−1. Proceeding in the same manner with the differential equation (3.35) we find the highest order relation is (5.24) sµ̇(s) = µ+ [(a+ 2)µ+ 2s]2 + 4µ(µ+ s)ν − µ(µ+ s)2 which occurs at order n−1 as well. Eliminating µ(s) using (5.23) we find that (5.24) yields (5.25) s2(ν̈)2 − (a+ 2)2(ν̇)2 + ν̇(4ν̇ − 1)(sν̇ − ν) + a(a+ 2)ν̇ − a2 = 0, which is almost the Jimbo-Miwa-Okamoto σ-form for PIII′ [28]. The boundary conditions follow from the application of the scaling limit (5.11) to the expansion about t = 0, Equation (3.29). � In addition we find the scaling behaviour of the Hankel determinants and poly- nomial evaluations to be given by the following propositions. Proposition 5.3. As n→ ∞ under the hard edge scaling the Hankel determinants scale as (5.26) ∆n(t)|t=s/4n ∼ 1! . . . (n− 1)!Γ(a+3) . . .Γ(n+ a+2) exp and the monic polynomials evaluated at x = 0 scale as (5.27) πn(0; t)|t=s/4n ∼ (−1)n(a+ 3)n exp Proof. The relation (5.26) arises from integrating (3.27) with respect to t and then employing the scaling form for the integrand as given by (5.11). The second relation is derived by integrating (3.39) and using (5.15) for the scaling of the resulting integrand. The factorial and Pochhammer prefactors arise from the normalisations at t = 0. � We find that the polynomial ratios have well defined scaling behaviour rather than the polynomial themselves. Proposition 5.4. The polynomial ratios Qn(x; t), Rn(x; t) scale as Qn(x; t)|x=−z/4n,t=s/4n = q(z; s),(5.28) nRn(x; t)|x=−z/4n,t=s/4n = p(z; s),(5.29) where q(z; s), p(z; s) are entire functions of z. 38 PETER J. FORRESTER AND NICHOLAS S. WITTE Proof. We start with the product form of the scaled polynomial (5.30) Qn(− 4nxj,n(s/4n) and seek bounds for the logarithm of the ratio of the n-th scaled polynomial to the (n− 1)st. When 0 < s, z < ∞ we can write the general bound as the sum of four contributions (5.31) Qn(−z/4n; s/4n) Qn−1(−z/4(n− 1); s/4(n− 1)) 4nxn,n(t) 4nxj,n(t) − log 4(n− 1)xj,n(t) 4(n− 1)xj,n(t) − log 4(n− 1)xj,n(t>) 4(n− 1)xj,n(t>) − log 4(n− 1)xj,n−1(t>) where t = s/4n and t> = s/4(n− 1). Using the inequality < A−B for A > B > 0 we can find a simpler bound (5.32) LHS of (5.31) ≤ z 4nxn,n(t) 4nxj,n(t) 4(n− 1)xj,n(t) 4(n− 1)xj,n(t) 4(n− 1)xj,n(t>) 4(n− 1)xj,n(t>) 4(n− 1)xj,n−1(t>) Considering the first sum of (5.32) we see that this is 4nxj,n(t) 4(n− 1)xj,n(t) 4n(n− 1) xj,n(t) 4n(n− 1) xj,n(t) 4n(n− 1) = zO(n−1), as n→ ∞, DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 39 for all bounded s. The second term of (5.32) is 4(n− 1)xj,n(t) 4(n− 1)xj,n(t>) 4(n− 1) xj,n(t) xj,n(t>) The summand appearing here can be bounded in the following way xj,n(t) xj,n(t>) |xj,n(t>)− xj,n(t)| xj,n(t>)xj,n(t) xj,n(t>)xj,n(t) (t> − t) max u∈(t,t>) |ẋj,n(u)|.(5.33) Now from (2.96) we note that |ẋj,n(u)| = u−1 θn(u) + u −θn(u) + xj,n(u) xj,n(u), and so furnishes a bound on (5.33) xj,n(t) xj,n(t>) xj,n(t>)xj,n(t) 4n(n− 1) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) xj,n(u), xj,n(t>)xj,n(t) xj,n(t) max u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) xj,n(t>) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) This means that xj,n(t) xj,n(t>) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) xj,n(t>) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) xj,n(t>) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) xj,n(t>) 2(n− 1) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) 2(n− 1) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) 2(n− 1) maxu∈(t,t>) κn(u)− u θn(u) where we have used (2.94) in the last step. The total contribution of the second term is therefore bounded by 8(n− 1)2 O(1) = zO(n−1). 40 PETER J. FORRESTER AND NICHOLAS S. WITTE The third sum in (5.32) is 4(n− 1) xj,n(t>) xj,n−1(t>) 4(n− 1) xj,n(t>) xj,n−1(t>) 4(n− 1) xj,n(t>) xj,n−1(t>) From the identity (3.48) and the scaling of the variables involved as given in (5.17) we conclude that the contribution of this sum is bounded by 4(n− 1) O(1) = zO(n−1). Finally we note that the isolated term in (5.32) is of order zO(n−2) as a leading estimate of the largest zero xn,n is of order O(n). This establishes that Qn under the hard edge scaling of the independent variables converges to a limit as n → ∞, for all real, positive and bounded z, s. � The spectral and deformation derivatives of the Qn, Rn system scale to the corresponding derivatives of the q, p system as in the following result. Proposition 5.5. Specify scaled quantities as in Proposition 5.1, Corollary 5.2 and Proposition 5.2. The spectral derivatives of the q, p system are (s− z)z∂zq = −zCq − (µ+ z)p,(5.34) (s− z)z∂zp = −z (z − s) q + [−2s+ z(C + a+ 2)]p,(5.35) and their deformation derivatives are (s− z)s∂sq = zCq + (µ+ s)p,(5.36) (s− z)s∂sp = zξq − [s(2C + a)− zC]p,(5.37) The boundary conditions satisfied by the solutions q(z; s) and p(z; s) of the above system on the domain s ≥ z ≥ 0 along z = 0 are q(0; s) = 1,(5.38) p(0; s) = 0,(5.39) for all s > 0. Proof. The first spectral derivative (5.34) follows from the scaling of (3.53) and employing (5.10), (5.12) and (5.15). The second member (5.35) is derived from the scaling of (3.54) and using (5.15) and (5.17). The first deformation derivative (5.36) follows from the scaling of (3.55) and utilising (5.10), (5.12) and (5.15). The DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 41 second deformation derivative (5.37) arises from the scaling of (3.56), employing (5.16) and (5.17) and noting that (κn + t) θn−1 + t − (κn − t) θn + t t=s/4n −s(2C + a) The boundary conditions (5.38) and (5.39) follow from the definitions (3.41) and (3.52) respectively and the scalings in Proposition 5.4. � Remark 5.1. Both the spectral derivative (3.43) and the deformation derivative (3.45) scale to (5.40) (µ+ s)z∂zq + (µ+ z)s∂sq + zC(s)q = 0, and this mixed derivative equation can be easily found from Proposition 5.5 by eliminating the variable p between (5.34) and (5.36), i.e. (µ+ s) times (5.34) plus (µ+ z) times (5.36). The three-term recurrence relation (3.42) scales to (5.41) z2∂2zq + 2sz∂z∂sq + s 2∂2sq + s µ̇− 2 [z∂zq + s∂sq]− zq = 0. This can also be recovered from the equations of Proposition 5.5. If we eliminate q between (5.34) and (5.36) by adding them we find that (5.42) p = z∂zq + s∂sq. Employing this we find that (5.41) is equivalent to (5.43) z∂zp+ s∂sp+ (−1 + s µ̇− 2 )p− 1 zq = 0, which can be found by adding (5.35) and (5.37) and noting the relation (5.20). Remark 5.2. In addition to the boundary conditions at z = 0 given by (5.38) and (5.39) there are also relations along s = z (5.44) q(s; s) = − C q(s; s), and further (5.45) p(s; s) q(s; s) C + a = − sC However these are a consequence of the spectral and deformation derivatives and so do not constitute independent boundary conditions. Remark 5.3. The compatibility of the two sets of derivatives, (5.34) and (5.35) on the one hand, and (5.36) and (5.37) on the other hand, affords a check on our results. We find that compatibility of (5.34) and (5.36) leads us to conclude that (5.46) ξ = sĊ + (µ+ s), 42 PETER J. FORRESTER AND NICHOLAS S. WITTE and we also recover (5.20). Similar considerations applied to (5.35) and (5.37) imply (5.47) sξ̇ = −(2C + a+ 2)ξ + 1 s(2C + a), and we get (5.46) again. Using (5.19) to eliminate µ we arrive at a coupled pair of first order ordinary differential equations sĊ = ξ + C(C + a),(5.48) sξ̇ = −(2C + a+ 2)ξ + 1 s(2C + a).(5.49) Using the latter equation to eliminate C we obtain a second order ordinary differ- ential equation for ξ, which by means of the transformation (5.50) ξ(s) = sy(s) y(s)− 1 is transformed into the standard equation for the fifth Painlevé transcendent. This is a degenerate case of PV which reduces to the third Painlevé transcendent be- cause the parameters are αV = 9/2, βV = −a2/2, γV = 1/2, δV = 0. Making an independent variable transformation s 7→ 2s so that γV = 1 we determine the PIII parameters to be αIII = 2(2 − a), βIII = 2(a + 3), γIII = 1, δIII = −1 which is consistent with those in Proposition 5.2. The matrix form of the spectral derivatives (5.34,5.35) and deformation deriva- tives (5.36,5.37) yield the Lax pairs ∂zΨ = z − s Ψ,(5.51) ∂sΨ = B − As z − s Ψ,(5.52) in the matrix variable (5.53) Ψ(z; s) = q(z; s) p(z; s) The system has two regular singularities at z = 0, s and an irregular one at z = ∞ with a Poincaré index of 1. This system is essentially equivalent to the isomon- odromic system of the fifth Painlevé equation but is the degenerate case. The DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 43 residue matrices are 0 −µ(s)s ,(5.54) µ(s) + s ξ(s) −C(s)− a ,(5.55) ,(5.56) −C(s) 0 −ξ(s) −C(s) .(5.57) Local convergent expansions about the regular singularities take the form for z = 0 q(z; s) = r0m(s)z χ0+m,(5.58) p(z; s) = u0m(s)z χ0+m,(5.59) for |z| < s and the initial relations amongst the coefficients are found to be (5.60) sχ0r 0 = −µu00, sχ0u00 = −2su00. This implies that χ0 = −2 or χ0 = 0 and u00 = 0. The latter case applies here as both q, p are analytic at z = 0, and in addition we also require p = 0 on z = 0. In addition r00 = 1. The recurrence relations for general m are s(m+ 2)u0m = (C + a+m+ 1)u m−1 + r0m−1 − r0m−2,(5.61) smr0m = −µu0m + (m− 1− C)r0m−1 − u0m−1.(5.62) For z = s we have the convergent expansion q(z; s) = rsm(s)(s− z)χs+m,(5.63) p(z; s) = usm(s)(s− z)χs+m,(5.64) for |z − s| < s and where in this case the initial relations are (5.65) s(−χs + C)rs0 = −(µ+ s)us0, (C + a+ χs)us0 = ξrs0. Combining these we get a relation which is identical to the second equality in (5.45) only if χs = 0,−a. The former case is the one we must choose as q, p are well-defined 44 PETER J. FORRESTER AND NICHOLAS S. WITTE and finite on z = s. For general m the recurrence relations are s(C −m)rsm + (µ+ s)usm = (C −m+ 1)rsm−1 + usm−1, (5.66) −ξrsm + s(C + a+m)usm = − rsm−1 + (C + a+m+ 1)u m−1 + rsm−2. (5.67) This system has a unique solution only if s2m(a + m) 6= 0 for s > 0 and m ≥ 1 which in turn means that a 6= −N. We also note that the two sets of coefficients are related by (5.68) rsm(s) = (−1)m snr0m+n(s), and an identical relation for usm(s). Proposition 5.6. The determinant Dn(x, x) scales as (5.69) Dn(x, x)|x=−z/4n,t=s/4n ∼ −4∆nπn(0)πn+1(0) [q∂zp− p∂zq] . Proof. We can employ the polynomials Qn, Rn in the evaluation (2.45) and find Dn(x, x) = ∆nπn(0)πn+1(0)[Qn+1R n+1 − Rn+1Q′n+1]. Applying the scaling of Proposition 5.4 to this expression we arrive at (5.69). � Proposition 5.7. The distribution An,a(y) scales to (5.70) An,a(y)|y=z/4n ∼ Aa(z). Proof. This is the only possible scaling consistent with the scaling of the indepen- dent variable to the hard edge, y = z/4n. � Proposition 5.8. The distribution of the first eigenvalue spacing at the hard edge is given by (5.71) Aa(z) = 42a+3Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3) ds sa(s− z)a exp [ν(v) + 2C(v)] [q∂zp− p∂zq] . Proof. We apply the scaling (5.70) to Eq. (1.31) and utilise the previous relation for the scaling of the integrand (5.69). For the first three factors on the right-hand side of (5.69) we can use the scaling results of (5.26) and (5.27), yielding the above integral representation. � We note that the factor of the integrand of (5.71) can be written as (5.72) q∂zp− p∂zq = q2 − 2 p2 − 1 (ξq − Cp)ξq − (C + a)p DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 45 The corresponding distribution A±(z) defined in (1.35) and (1.36) for the special cases a = ±1/2 is given by (5.73) A±(z) = 42a+2Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3) ds s2a+1(s− z)2a+1(2s− z)2 exp [ν(v) + 2C(v)] × [q∂zp− p∂zq]| s7→s2 z 7→z(2s−z) 5.2. Special Case a ∈ Z≥0. Proposition 5.9. In the special case a ∈ Z≥0 we have (5.74) ∆n(t)|t=s/4n ∼ cn+1,n+1+a (n+ a)! (−1)⌊a/2⌋(2n)as−a det Ij+2−k( j,k=1,...,a (5.75) ν(s) = s log s−a det Ij+2−k( j,k=1,...,a whilst for a = 0 we find ν(s) = 0. Proof. The first relation (5.74) follows from an application of the Hilb type asymp- totic formula (5.2) to (4.1) and the second follows by using this result in (3.27). � Another consequence of the Hilb formula is the scaling of the orthogonal poly- nomial ratio as given by (4.8). Proposition 5.10. The scaled orthogonal polynomial ratio q(z; s) is (5.76) q(z; s) = z−3/2 (z/s)k/2I3−k( k=1,...a+1 [Ij+2−k( s)] j=1,...,a k=1,...a+1 s, 1/2s, 1/s3/2, 0, . . . , 0 [Ij+2−k( s)] j=1,...,a k=1,...a+1 , a ≥ 1 and for a = 0 is (5.77) q(z; s) = Finally the eigenvalue spacing distribution (4.16) takes the following form in the scaling limit. 46 PETER J. FORRESTER AND NICHOLAS S. WITTE Proposition 5.11. The distribution of the first eigenvalue spacing at the hard edge for a ∈ Z>0 is (5.78) Aa(z) = 2 −4z1/2 e−s/4 × det [Ij+2−k( s)] j=1,...,a k=1,...a+2 (s/z)(2−k)/2Ij+2−k( j=1,2 k=1,...a+2 for a ≥ 1 and for a = 0 is (5.79) A0(z) = e−z/4 I22 ( z)− I1( z)I3( Proof. We apply the Hilb asymptotic formula (5.2) to (4.16). As 1 (a+1)(a+2)+ 1 + ⌊a/2⌋ is always even for a ∈ Z we have (5.78). � 6. Analytical Studies at the Hard Edge In this section of our study we intend to develop the analytical and non-formal theory of the solutions to the defining ordinary and partial differential equations described in the previous sections. This is because we wish to compute precision numerical data characterising the distribution function of the first eigenvalue spac- ing at the hard edge Aa(z) for arbitrary parameter a. For this purpose it is not sufficient to employ a single local expansion of the σ-function, about s = 0 say, because it has a finite convergence domain and one cannot use this to evaluate the s-integrals on the interval [0,∞). For this reason we construct a patchwork of over- lapping local expansions including Taylor series expansions about regular points s0 for positive and real values. A similar approach was undertaken by Prähofer and Spohn in their study [29] of the exact scaling functions for one-dimensional stationary KPZ growth. 6.1. The σ-function expansion about s = 0. The terms given in the expansion of the Painlevé III′ σ-function (5.22) are the minimum required to specify the full non-analytic Puiseux-type expansion of the particular solution for ν(s) about s = 0 in the sector −π ≤ arg(s) < π. This is the primary data specifying our particular solution and we need to use this in various ways in order to compute the distribution Aa(z). Proposition 6.1 ([18],[21]). The Painlevé III′ σ-function ν(s) has a Puiseux-type expansion about the fixed regular singular point s = 0 of the form (6.1) ν(s) = ck,js j+ka, a ∈ C, 0 ≤ Re(a) < 1, with |arg(s)| < π and is convergent in a finite domain s ∈ {z ∈ C : |z| < R, |za| < DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 47 The coefficients ck,j are determined by recurrences which follow from the sub- stitution of expansion (6.1) into the relation (5.25). We will assume that a is not rational for simplicity. Considering terms in the resulting equation with sp and p ∈ Z≥0 then the p = 0 case implies that if c0,0 = 0 then this choice fixes (6.2) c0,1 = 4(a+ 2) The p = 1 is automatically zero but for the p = 2 case one has the two options (6.3) c0,2 = − 8(a+ 3)(a+ 2)2(a+ 1) , or 0. The former case applies here by comparison with (5.22). For p ≥ 3 the recurrence (6.4) − a(p+ a+ 1)(p− a− 3) (a+ 3)(a+ 2)2(a+ 1) c0,p − j(p− j)c0,jc0,p+1−j + 8c0,2 (j − 1)(p− j)c0,j−1c0,p+1−j j(p+ 2− j)[(j − 1)(p+ 1− j)− (a+ 2)2]c0,jc0,p+2−j p+2−j jm(p+ 1− j −m)c0,jc0,mc0,p+2−j−m = 0, which allows for c0,p to be recursively found, valid for a /∈ Z. Terms with s−2+qa for q = 2k ≥ 2 imply that ck,0 = 0 given that c0,0 = 0 and a 6= 3/(k − 1),−1/(k + 1). Consequently all terms with s−1+qa vanish also. The generic recurrence relation following from examining the sp+qa term is (6.5) a(a+ 2)(p+ 1 + qa)cq,p+1 = (j + ka)[p− j + (q − k)a]ck,jcq−k,p+1−j (j+ka)[p+2−j+(q−k)a][(j−1+ka)[p+1−j+(q−k)a]−(a+2)2]ck,jcq−k,p+2−j p+2−j (j+ka)(m+la)[p+1−j−m+(q−k−l)a]ck,jcl,mcq−k−l,p+2−j−m, for q, p ≥ 0. The convention is taken that sums with upper limits less than their lower limits are zero, or equivalently coefficients with negative indices are zero. Contrary to appearances the highest coefficient in (6.5) will turn out to be cq,p as cq,p+2 occurs with c0,0 as a factor and cq,p+1 has a factor of (6.6) a(a+ 2) + c0,0 − 2(a+ 2)2c0,1 − 8c0,0c0,1, 48 PETER J. FORRESTER AND NICHOLAS S. WITTE which is zero as a consequence of c0,0 = 0 and (6.2). We find that the coefficient cq,p occurs as a linear term and has a factor of (6.7) (p−1+qa)c0,1(4c0,1−1)+4(p+qa)(p−1+qa−(a+2)2)c0,2−16(p+qa)c0,0c0,2. With the above evaluations (6.2) and (6.3) this is (6.8) − a(p+ 1 + (q + 1)a)(p− 3 + (q − 1)a) 2(a+ 3)(a+ 2)2(a+ 1) This vanishes when q = 1, p = 3 and we find that c1,3 is undetermined and therefore is a free parameter. With these choices a triangular subset of the coefficients vanish (6.9) ck,j = 0, j = 0, . . . , 3k − 1 for k = 1, . . . ,∞, so that the initial non-zero term in the j−sum is ck,3k. All other terms are fixed by c1,3 and a. For the variable µ(s) we can deduce the following Puiseux-type expansion from the above work (6.10) µ(s) ∼ (a+ 3)(a+ 2)2(a+ 1) 4a+2(a+ 2)(a+ 1)Γ2(a+ 3) sa+3. In regard to the quantity 2C(s) we require as much detail about this as for ν(s). This variable is a σ-function for an identical problem, where the fixed exponent 2 in the weight (1.32) is replaced by 3. If we define the expansion coefficients for this object (6.11) 2C(s) = ak,js j+ka, a ∈ C, 0 ≤ Re(a) < 1, then the defining recurrences for these using (5.20) are (6.12) a0,0 = 0, a0,1 = 4(a+ 2)(a+ 1)c0,2, and for j ≥ 2 (6.13) a0,j = 2(a+ 2) − (j + 1)(j − 2− a)c0,j+1 + la0,j+1−lc0,l For the case k ≥ 1 we have ak,j = 0 for j = 0, .., 3k− 1 and the remaining non-zero terms are given by (6.14) a1,j = 2(a+ 2) − (j + 1 + a)(j − 2)c1,j+1 (l + a)a0,j+1−lc1,l + la1,j+1−lc0,l DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 49 for k = 1 and the general case k ≥ 2 by (6.15) ak,j = 2(a+ 2) − (j + 1 + ka)(j − 2 + (k − 1)a)ck,j+1 (l + ka)a0,j+1−lck,l + lak,j+1−lc0,l (l +ma)ak−m,j+1−lcm,l The first few terms are (6.16) 2C(s) ∼ 2(a+ 3)(a+ 2) s− a(a 2 − 5a− 18) 8(a+ 4)(a+ 3)2(a+ 2)2(a+ 1) 4a+2(a+ 2)(a+ 1)Γ(a+ 3)Γ(a+ 4) sa+3. (6.17) ξ(s) ∼ 4(a+ 3) s− 3a 8(a+ 4)(a+ 3)2(a+ 2) 4a+7(a+ 3)(a+ 2)(a+ 1)Γ2(a+ 4) sa+4. 6.2. The q, p expansion about s = 0. In this subsection we seek local expansions about s = 0 for the coefficients functions r0m(s), u m(s), r m(s), u m(s) appearing in (5.58), (5.59), (5.63) and (5.64). In parallel with the transcendent quantities these will have Puisuex-type expansions of the form r0m(s) = k,j≥0 r0m,k,js j+ka,(6.18) rsm(s) = k,j≥0 rsm,k,js j+ka,(6.19) with analogous expansions for the remaining two coefficients. This is immediately clear because the recurrence relations for these imply that they are polynomial functions of the variables µ(s), C(s), ξ(s). These recurrence relations imply the following ones for the z = 0 coefficients r0m,k,j , u m,k,j (6.20) (a+m+ 2)u0m,k,0 = r0m−1,k,0, mr m,k,0 = u m,k,0, for m ≥ 1, k ≥ 0. For the general case j ≥ 1, k ≥ 0 we have (6.21) (m+ 2)u0m,k,j−1 = (a+m+ 1)u m−1,k,j + ak−q,j−pu m−1,q,p r0m−1,k,j−1− [j−p+(k−q)a] ck−q,j−p + ak−q,j−p r0m−1,q,p− r0m−2,k,j , 50 PETER J. FORRESTER AND NICHOLAS S. WITTE (6.22) mr0m,k,j−1 = (m− 1)r0m−1,k,j − ak−q,j−pr m−1,q,p + u0m,k,j−1 − 4 [j − p+ (k − q)a]ck−q,j−pu0m,q,p − u0m−1,k,j, These equations can be solved for successive values of m starting with the m = 0 values for all k, j (6.23) u00,k,j = 0, r 0,k,j = 0, except for k = j = 0 where r 0,0,0 = 1, which follow from r00(s) = 1, u 0(s) = 0. The next few coefficients can be read off (6.24) r01(s) ∼ 4(a+ 3) 16(a+ 4)(a+ 3)2(a+ 2) 128.4a(a+ 3)(a+ 2)(a+ 1)Γ(a+ 5)Γ(a+ 4) sa+3, (6.25) u01(s) ∼ 4(a+ 3) 8(a+ 4)(a+ 3)2(a+ 2) 128.4a(a+ 3)(a+ 2)(a+ 1)Γ2(a+ 4) sa+3, (6.26) r02(s) ∼ 32(a+ 4)(a+ 3) 64(a+ 5)(a+ 4)(a+ 3)2(a+ 2) 512.4a(a+ 3)(a+ 2)(a+ 1)Γ(a+ 6)Γ(a+ 4) sa+3, (6.27) u02(s) ∼ 16(a+ 4)(a+ 3) 64(a+ 5)(a+ 4)(a+ 3)2(a+ 2) 512.4a(a+ 3)(a+ 2)(a+ 1)Γ(a+ 5)Γ(a+ 4) sa+3. The analogous results for rsm,k,j , u m,k,j are (6.28) (a+m+ 2)usm,k,0 = − rsm−1,k,0, mr m,k,0 = u m,k,0, DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 51 for m ≥ 1, k ≥ 0. Again for the general case j ≥ 1, k ≥ 0 we have (6.29) (a+m)usm,k,j−1 + ak−q,j−1−pu m,q,p = (a+m+ 1)usm−1,k,j + ak−q,j−pu m−1,q,p [j − 1− p+ (k − q)a] ck−q,j−1−p + ak−q,j−1−p rsm,q,p rsm−1,k,j−1 − [j − p+ (k − q)a] ck−q,j−p + ak−q,j−p rsm−1,q,p r0m−2,k,j , (6.30) mrsm,k,j−1 − ak−q,j−1−pr m,q,p = (m− 1)rsm−1,k,j − ak−q,j−pr m−1,q,p − usm−1,k,j + 4 [j − p+ (k − q)a]ck−q,j−pusm,q,p, Again these can be solved for successive values of m starting with the initial values given by the relations (6.31) rs0,k,j = r0n,k,j−n, along with an identical formula for us0,k,j. 6.3. The σ-function expansion about s = ∞. The nature of the expansion of ν(s) about s = ∞ is rather different because this fixed singular point is irregular in the case of the third Painlevé transcendent. Proposition 6.2. The formal asymptotic expansion of ν(s) about s = ∞ has the (6.32) ν(s) ∼ −j/2, where d−1 = ± 12a. Proof. We start with the general ansatz of (6.33) ν(s) = dskα +O(s(k−1)α), 52 PETER J. FORRESTER AND NICHOLAS S. WITTE where k ∈ N and α ∈ C with 0 < Re(α) < 1. Using (5.25) we find that the only terms which can balance the O(1) term are the O(s2kα−1) and O(s3kα−2) terms. If kα < 1 then the former choice applies and we find kα = 1/2. If we assume kα > 1 then the latter case must be chosen but this leads to kα = 2/3 in contradiction to the hypothesis. With the correct choice of kα we find the above relation for leading coefficient, d−1. Considering the sub-leading terms we note that it is only possible for the terms of O(s−α) to balance those of O(s−1/2) so that in fact α = 1/2 and k = 1. This is entirely consistent with the expansion for the Painlevé III transcendent qIII(t) about t = ∞ as found in [15]. � The choice of the sign of the leading coefficient is positive in our application. Consequently we find the first few terms of the asymptotic expansions of the σ- function (6.34) ν(s) = as1/2 − 1 a(a+ 4) + as−1/2 + a2s−1 a(16a2 − 7)s−3/2 +O(s−2), and the auxiliary variables (6.35) µ(s) + s = as1/2 − 15 as−1/2 − 15 a2s−1 a(16a2 − 7)s−3/2 +O(s−2), (6.36) 2C(s) + a = as−1/2 + a2s−1 + a(8a2 − 21)s−3/2 +O(s−2), (6.37) ξ(s) = as1/2 − 35 as−1/2 − 35 a2s−1 +O(s−3/2). The large s-regime also implies a simplification of spectral derivative (5.51) which becomes (6.38) ∂zΨ = 2(z − s) 2(z − s) − I+O(s−1/2) Using the substitution Ψ 7→ exp(−a ln(s−z)I/2)Ψ this decouples and can be solved in terms of the modified Bessel functions. An application of the boundary condition (5.38) implies that q(z; s) ∼ z),(6.39) p(z; s) ∼ .(6.40) DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 53 6.4. The σ-function expansion about a regular point. In this subsection we seek Taylor series expansions for the sigma function and derived variables about regular points s0, taken to be positive and real without any loss of generality. Let us write (6.41) ν(s) = dj(s− s0)j , and using (5.25) we find the following recurrence relations for the coefficients dj (6.42) 4s20d a(a+2)d1+ d0d1 − [s0+(a+2)2]d21 +4(s0d1 − d0)d21 = 0, and for n ≥ 1, d2 6= 0 (6.43) 4s20(n+ 2)(n+ 1)d2dn+2 = − a(a+ 2)(n+ 1)dn+1 + [s0 + (a+ 2) (j + 1)(n− j + 1)dj+1dn−j+1 (j + 1)(n− j − 1)dj+1dn−j (j + 1)j(n− j + 1)(n− j)dj+1dn−j+1 − 2s0 (j + 2)(j + 1)(n− j + 1)(n− j)dj+2dn−j+1 − s20 (j + 2)(j + 1)(n− j + 2)(n− j + 1)dj+2dn−j+2 (i+ 1)(j + 1)(n− i− j − 1)di+1dj+1dn−i−j − 4s0 (i+ 1)(j + 1)(n− i− j + 1)di+1dj+1dn−i−j+1. These recurrences are solved subject to the initial values of ck,js 0 ,(6.44) (j + ka)ck,js j−1+ka 0 ,(6.45) 54 PETER J. FORRESTER AND NICHOLAS S. WITTE which in turn can be found from the solutions to the recurrences (6.4) and (6.5). In addition we define µ(s) = fj(s− s0)j ,(6.46) C(s) = gj(s− s0)j ,(6.47) ξ(s) = hj(s− s0)j .(6.48) The coefficients appearing here are computed using the recurrences (6.49) f0 = s0(4d1 − 1), f1 = 4d1 − 1 + 8s0d2, fj = 4[jdj + s0(j + 1)dj+1], j ≥ 2, and subject to f0 6= 0 2f0g0 = −2s0 − (a+ 3)f0 + s0f1,(6.50) 2f0g1 = −2− (a+ 2)f1 + 2s0f2 − 2f1g0,(6.51) 2f0gj = (j − a− 3)fj + (j + 1)s0fj+1 − 2 fj−kgk, j ≥ 2,(6.52) and provided s0 + f0 6= 0 (s0 + f0)h0 = −s0g0(a+ g0),(6.53) (s0 + f0)h1 = −as0g1 − 2s0g0g1 − (a+ g0)g0 − (1 + f1)h0,(6.54) (s0 + f0)hj = −as0gj − agj−1 − s0 gj−kgk − gj−kgk−1(6.55) − (1 + f1)hj−1 − fj−khk, j ≥ 2. 6.5. The q, p expansion about a regular point. We will seek a Taylor series ap- proximation for the scaled polynomial and associated function q(z; s), p(z; s) about the regular point (z0, s0) with 0 < z0 < s0. Let us write q(z; s) = j,k≥0 rj,k(z − z0)j(s− s0)k,(6.56) p(z; s) = j,k≥0 uj,k(z − z0)j(s− s0)k,(6.57) Using the first spectral derivative (5.34) we obtain the recurrence relation (6.58) z0(s0−z0)(j+1)rj+1,k+(s0−2z0)jrj,k−(j−1)rj−1,k+z0(j+1)rj+1,k−1+jrj,k−1 = −z0uj,k − uj−1,k − [fk−luj,l + gk−lrj−1,l + z0gk−lrj,l] , DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 55 for j, k ≥ 1. When k = 0 and j ≥ 1 we have the specialisation (6.59) z0(s0 − z0)(j + 1)rj+1,0 + [(s0 − 2z0)j + z0g0]rj,0 + [g0 − j + 1]rj−1,0 = −(f0 + z0)uj,0 − uj−1,0. The second spectral derivative (5.35) yields the recurrence relation (6.60) z0(s0 − z0)(j + 1)uj+1,k + [(s0 − 2z0)j + 2s0 − (a+ 2)z0]uj,k − (a+ j + 1)uj−1,k + z0(j + 1)uj+1,k−1 + (j + 2)uj,k−1 z0(s0 − z0)rj,k + (s0 − 2z0)rj−1,k − rj−2,k + z0rj,k−1 + rj−1,k−1 hk−l(rj−1,l + z0rj,l) + gk−l(uj−1,l + z0uj,l), for j, k ≥ 1. When k = 0 and j ≥ 1 we have the specialisation (6.61) z0(s0−z0)(j+1)uj+1,0+[(s0−2z0)j+2s0−(a+2)z0−z0g0]uj,0−[a+1+g0+j]uj−1,0 z0(s0 − z0)− h0z0]rj,0 + [ (s0 − 2z0)− h0]rj−1,0 − rj−2,0. Using the first deformation derivative (5.36) we obtain the recurrence relation (6.62) s0(s0−z0)(k+1)rj,k+1−s0(k+1)rj−1,k+1+(2s0−z0)krj,k−krj−1,k+(k−1)rj,k−1 = s0uj,k + uj,k−1 + [fk−luj,l + gk−lrj−1,l + z0gk−lrj,l] , for j, k ≥ 1. When j = 0 and k ≥ 1 we have the specialisation (6.63) s0(s0 − z0)(k + 1)r0,k+1 + (2s0 − z0)kr0,k + (k − 1)r0,k−1 = s0u0,k + u0,k−1 + [fk−lu0,l + z0gk−lr0,l]. The second deformation derivative (5.37) in turn gives us the recurrence relation (6.64) s0(s0 − z0)(k + 1)uj,k+1 − s0(k + 1)uj−1,k+1 + [(2s0 − z0)k + as0]uj,k − kuj−1,k + (a+ k − 1)uj,k−1 hk−l(rj−1,l+z0rj,l)−(2s0−z0) gk−luj,l+ gk−luj−1,l−2 gk−1−luj,l, 56 PETER J. FORRESTER AND NICHOLAS S. WITTE for j, k ≥ 1. When j = 0 and k ≥ 1 we have the specialisation (6.65) s0(s0 − z0)(k + 1)u0,k+1 + [(2s0 − z0)k + as0]u0,k + [a+ k − 1]u0,k−1 hk−lr0,l − (2s0 − z0) gk−lu0,l − 2 gk−1−lu0,l. These recurrences can be solved in the following way. First (6.59) and (6.61) are solved for rj,0, uj,0 for j ≥ 1 in terms of r0,0, u0,0. Then these solutions can be substituted into the boundary conditions rj,0(−z0)j = 1,(6.66) uj,0(−z0)j = 0,(6.67) and this allows for r0,0, u0,0 to be found. Next the sequence r0,k, u0,k can be found for k ≥ 1 using (6.63) and (6.65). Finally the two general systems (6.58),(6.60) and (6.62),(6.64) can be employed to compute rj,k, uj,k for j, k ≥ 1. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 57 6.6. Numerical studies at the Hard Edge. Using the integral formula for the distribution Aa(z) as given by (5.71) it is possible to compute values of this and the examples of a = 0, 1, 2 are plotted in Figure 2. However we wish to characterise it in a precise quantitative way and evaluate the moments of this distribution (6.68) mk := dz zkAa(z), k ∈ Z≥0. These are easily seen to be (6.69) mk = 42a+3Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3) ds sk+2a+3e−s/4eF (s) du uk+2(1− u)aG(us; s), where (6.70) F (s) := [ν(v) + 2C(v)], (6.71) G(z; s) := q∂zp− p∂zq. We note that by employing the large s asymptotic form of q(z; s), p(z; s) as given in (6.39,6.40) we can deduce the asymptotic form of the spacing distribution is given (6.72) Aa(z) ∼ Ce−z/4+(a+2) zz−1/2−a where C is a constant which cannot be found from our methods. For small z we find that (6.73) Aa(z) ∼ 42a+3Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3) ds s2a − ξ(s) [ν(v) + 2C(v)] where we have used (5.58,5.59) and the fact that u01(s) = 1/12−ξ(s)/3s. Therefore we can conclude that the moments exist for Re(k) > −3, Re(a) > −1 and Re(k + 2a) > −4. An instance where exact evaluation of the moments can be made is the case a = 0 and the first four of these are m1 = 4e 2 [I0(2)− I1(2)] , m2 = 32e2I0(2),(6.74) m3 = 384e 2 [2I0(2) + I1(2)] , m4 = 2048e 2 [13I0(2) + 9I1(2)] .(6.75) We investigated the distributions Aa(z) and A ±(z) for the two special cases of a = ±1/2 in some detail because of the motivations provided by (1.6) and (1.7). 58 PETER J. FORRESTER AND NICHOLAS S. WITTE The analogue of (6.69) for A±(z) is (6.76) m±k = 24a+5Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3) ds sk/2+2a+3e−s/4eF (s) du uk+2(1 − u)2a+1(2− u)2G(u(2− u)s; s), for a = ± 1 . The statistical data for Aa(z) for the cases a = −1/2, 0, 1/2, 1, 2 are given in Table 1 and the data for A±(z) is given in Table 2. Our strategy is that by employing local Puiseux-type and Taylor expansions for the two factors in the integrand, namely eF (s) and G(z; s), within a given finite member of the patchwork of local expansion domains the above integrals restricted to this domain can be exactly evaluated. This is essential as numerical quadrature algorithms implemented in either computer algebra software or compiled language packages (e.g. QUADPACK) have minimal attained error tolerances which cannot be reduced below a fixed bound. For the compiled language option with a floating point representation of 64 bits the best one could expect is a relative error of around 10−15 but often it is far worse and around 10−8 − 10−9. To illustrate this we have computed the statistical data for the a = 1, 2 cases using QUADPACK routines and the results are displayed in Table 1. In the case of the Puiseux-type expansions the integrals are (6.77) ds sk+3+l+n+(m+2)ae−s/4, (6.78) ∫ 1/2 du uk+2+n(1− u)a or du uk+2(1 − u)a+n, which for k, l,m, n ∈ Z≥0 can be evaluated in terms of radicals, the Gamma func- tion at integer arguments, a rational function of a and the Whittaker function M(α, β;S/4) or its specialisations depending on a. For example in the case a = 1/2 the s-integral reduces to the error and exponential functions. For the Taylor ex- pansion case we have the double integral (6.79) ds sk+3+2a(s− s0)me−s/4 du uk+2(1− u)a(us− z0)l, which for k, l,m ∈ Z≥0 and s0 ∈ (s1, s2) is evaluated in terms of a rational function of a, a polynomial function of s0, z0 and the Whittaker functions M(α, β; s1,2/4). Therefore the only sources of error are from the truncation of the expansions and the finite number of intervals, both of which can be adjusted to reduce the contributing errors. The computations were performed, in most cases, using the computer algebra system Maple with a sufficiently large number of decimal digits and found that 250 digits was more than adequate. In addition we found that we had to tailor DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 59 the numerical parameters for each case of a = ±1/2 differently as the errors varied quite strongly with a (this was especially pronounced as a approached −1). We discuss the case of a = 1/2 first. In regard to the truncations about the singular point s = 0, we found that 1890 terms in the expansion of the transcendent vari- able (6.1) with k ≤ 30, j + k ≤ 120 yielded an error for the second derivative of ν(s) at s = 2 which was estimated to be 4.6 × 10−98. At most 100 terms were re- tained in each of the expansions of the transcendent variables about regular points (6.41),(6.46),(6.47) and (6.48) because much fewer, of the order of j, k ≤ 20, were required in the corresponding expansions of the linear variables. The number of intervals in the s-direction was taken to be 19 with the sequence of s0 values being {0, 2, 5/2, 3, 4, 6, 9, 13, 19, 25, 30, 38, 54, 72, 90, 115, 150, 200, 300, 500}. The bound- aries of the s-interval with node s0 were taken to be located at the midpoints of s0 and its neighbouring nodes. This sequence of nodes was chosen to be close to an optimal situation yielding the largest separation of each node from its preceding node, yet close enough so to ensure that the error in ν′′(s) at the node was less than 9.8 × 10−37. For each s-interval with node s0 two expansion points in the z-direction were chosen because a single expansion point could never ensure that all of the integration region would fall within the domain of convergence about that point. The two points that together yielded the largest convergence domain were found to be located at z0 = 0 and z0 = s0. Subdividing the z-interval into three sub-intervals was found to contribute a variation of less than 3.2 × 10−19 to the normalisation. Another criteria that the sequence of s0 nodes had to sat- isfy was that each (s, z) integration region fell completely within the union of the convergence domains about (s0, 0) and (s0, s0). For the expansions of the linear variables about the lines z = 0, s and about the singular point s = 0, as defined in (6.18,6.19) along with (5.58,5.59,5.63,5.64), we chose the cut-off in the sum to be 20. The expansions of linear variables about the lines z = 0, s and about a regular point s = s0 > 0 as defined in (6.56,6.57) were cutoff at 25. An overall es- timate of the accuracy is provided by the normalisation, which was unity to within 1.6×10−18. The second case with a = −1/2 was more demanding computationally. We needed 5150 terms in the expansion of the transcendent variable (6.1) with k ≤ 50, j + k ≤ 200 and computed these with compiled code using the multiple- precision library MPFUN [3],[2]. The error for ν′′(s) at s = 2 was estimated to be around 7.9×10−59. Again only 100 terms were retained in each of the expansions of the transcendent variables about regular points. A larger number of intervals in the s-direction were employed, namely 24, and the sequence of s0 values was taken to be {0, 2, 5/2, 3, 7/2, 4, 5, 6, 7, 9, 11, 14, 18, 22, 28, 36, 46, 58, 72, 90, 114, 144, 180, 220, 300}. This time the sequence of nodes was chosen to ensure that the error in ν′′(s) at each node was less than 3.6×10−60. And again for each s-interval with node s0 two expansion points in the z-direction were chosen at z0 = 0, s0. In the expansions of 60 PETER J. FORRESTER AND NICHOLAS S. WITTE the linear variables about the lines z = 0, s and about the singular point s = 0 the cut-off in the sum was chosen to be 20 as before. The expansions of these variables about the lines z = 0, s and about a regular point s = s0 > 0 was terminated at the cutoff of 25 also. The estimate of the accuracy provided by the normalisation was 4.6× 10−20. Because the raw moments grow rapidly with order we have computed some standard statistical quantities instead using the definitions of the variance σ2, the skewness γ1 and the kurtosis excess γ2 (6.80) σ2 = µ2, γ1 = , γ2 = in terms of the central moments µ2 = m2 −m21,(6.81) µ3 = m3 − 3m1m2 + 2m31,(6.82) µ4 = m4 − 4m1m3 + 6m21m2 − 3m41.(6.83) DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 61 Figure 2. The distribution of the first eigenvalue spacing at the hard edge of random hermitian matrices Aa(z) for integral values of a = 0, 1, 2. Acknowledgements This work was supported by the Australian Research Council. 62 PETER J. FORRESTER AND NICHOLAS S. WITTE Table 1. Low order statistics of the distribution Aa(z) for various values of the parameter a. References [1] A. I. Aptekarev. Asymptotics of orthogonal polynomials in a neighborhood of endpoints of the interval of orthogonality. Sb. Math., 76(1):35–50, 1993. [2] D.H. Bailey. A Portable High Performance Multiprecision Package. 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Colloquium Publications 23. American Mathematical Society, Providence, Rhode Island, third edition, 1967. [32] C. A. Tracy and H. Widom. Fredholm determinants, differential equations and matrix models. Comm. Math. Phys., 163(1):33–72, 1994. [33] C. A. Tracy and H. Widom. Level spacing distributions and the Bessel kernel. Comm. Math. Phys., 161(2):289–309, 1994. [34] A. M. Tulino and S. Verdú. Random matrix theory and wireless communications. Found. Trends Commun. Inf. Theory, 1(1):1–182, 2004. [35] V. B. Uvarov. Relation between polynomials orthogonal with different weights. Dokl. Akad. Nauk SSSR, 126:33–36, 1959. [36] J. J. M. Verbaarschot. The spectrum of the Dirac operator near zero virtuality for Nc = 2 and chiral random matrix theory. Nuclear Phys. B, 426(3):559–574, 1994. Department of Mathematics and Statistics, University of Melbourne,Victoria 3010, Australia E-mail address: p.forrester@ms.unimelb.edu.au E-mail address: n.witte@ms.unimelb.edu.au 1. Introduction 2. Orthogonal Polynomial System 2.1. Semi-classical Orthogonal Polynomials 2.2. Deformed Laguerre Orthogonal Polynomials 3. Difference and Differential Equations 3.1. Difference Equations 3.2. Reduction to Painlevé V 3.3. Inequalities and Bounds 4. Special Case a Z0 5. Hard Edge Scaling 5.1. General Case 5.2. Special Case a Z0 6. Analytical Studies at the Hard Edge 6.1. The -function expansion about s=0 6.2. The q, p expansion about s=0 6.3. The -function expansion about s= 6.4. The -function expansion about a regular point 6.5. The q, p expansion about a regular point 6.6. Numerical studies at the Hard Edge Acknowledgements References

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlev\'e V system, and the solution of its associated linear isomonodromic system. In particular it is characterised by the polynomial solutions to the isomonodromic equations which are also orthogonal with respect to a deformation of the Laguerre weight. In the scaling to the hard edge regime we find an analogous situation where a certain Painlev\'e \IIId system and its associated linear isomonodromic system characterise the scaled distribution. We undertake extensive analytical studies of this system and use this knowledge to accurately compute the distribution and its moments for various values of the parameter $ a $. In particular choosing $ a=\pm 1/2 $ allows the first eigenvalue spacing distribution for random real orthogonal matrices to be computed.

Introduction The Laguerre unitary ensemble (LUEn,a) of random matrices is specified by the eigenvalue probability density function (p.d.f.) (1.1) p(λ1, . . . , λn) n!cn,n+a e−λjλaj 1≤j<k≤n (λk − λj)2, λ1, . . . , λn ∈ [0,∞), where (1.2) cm,n := Γ(n− j + 1)Γ(m− j + 2). The naming relates to the fact that (1.1) is the eigenvalue p.d.f. of complex Hermit- ian matrices X with measure invariant under unitary conjugation X 7→ UXU−1, proportional to the generalised Laguerre form (1.3) (detX) e−Tr(X). 2000 Mathematics Subject Classification. 15A52, 33C45, 33E17, 42C05, 60K35, 62E15. Key words and phrases. random matrices, eigenvalue distribution, Wishart matrices, Painlevé equations, isomonodromic deformations. http://arxiv.org/abs/0704.1926v2 2 PETER J. FORRESTER AND NICHOLAS S. WITTE In multivariate statistics (1.1) is realised as the eigenvalue p.d.f. for the complex case of the so-called Wishart matrices X = Y †Y . Here Y is an N × n (N ≥ n) rectangular matrix of i.i.d. entries with distribution N[0, 1]+iN[0, 1]. In this setting a = N − n, and so a is naturally a non-negative integer. The spectrum of complex Wishart matrices has found recent application in studies of wireless communication systems [34], where the matrix Y consists of the complex amplitudes of various channels of transmitted waves as received by the antennas. The matrix structure Y †Y is relevant to the study of the eigenvalues of the (n+N)× (n+N) Hermitian matrix (1.4) X̃ := 0N×N Y Y † 0n×n Thus one has that X̃ has in general N − n zero eigenvalues, with the remaining eigenvalues given by ± the positive square roots of the eigenvalues of Y †Y (see e.g. [9]). This matrix structure has application to the study of the Dirac equation in the context of quantum chromodynamics [36]. There most interest is in the scaling behaviour of the smallest eigenvalues. In the study of matrix Lie algebras one encounters antisymmetric matrices (XT = −X) with pure imaginary complex elements. Specifically, such matrices are the Hermitian part of the matrix Lie algebra (1.5) i× (so(n,C)) := {i times n× n skew symmetric complex matrices}. If the independent imaginary complex elements are i.i.d with distribution iN[0, 1], then the p.d.f. of the positive eigenvalues is proportional to n = 2m even: exp(−λ2j) 1≤j<k≤m (λ2k − λ2j)2,(1.6) n = 2m+ 1 odd: λ2j exp(−λ2j ) 1≤j<k≤m (λ2k − λ2j )2.(1.7) This ensemble will be denoted by AS(n). Under the change of variables λ2j 7→ λj these reduce to the LUEn,a with parameters a = −1/2, 1/2 respectively. Antisymmetric Hermitian matrices X can be used to parameterise real orthog- onal matrices R with determinant +1 (and thus, by definition, members of the classical group O+(n)). Thus we can write R according to a Cayley transformation (1.8) R = In + iX In − iX Note from this that the property that the eigenvalues of X come in ± pairs is consistent with the property that the eigenvalues of R come in complex conjugate pairs e±iθ. This can be used (see e.g. [9]) to show that with the matrix R chosen DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 3 with uniform (Haar) measure, the eigenvalue p.d.f for the eigenvalues with angles 0 ≤ θ ≤ π is proportional to n = 2m even: 1≤j<k≤m (cos θk − cos θj)2,(1.9) n = 2m+ 1 odd: (1− cos θj) 1≤j<k≤m (cos θk − cos θj)2.(1.10) Note that for θl → 0 these have the same leading behaviour as (1.6), (1.7) with λl → 0. This is consistent with the fact that the m → ∞ scaled joint distribution function for the k smallest eigenvalues, p(k) say, is the same for both ensembles, (1.11) AS(n) , . . . , ) = lim O+(n) , . . . , where n = 2m, 2m + 1. This scaling is such that the average spacing between eigenvalues approaches unity as k → ∞. The distribution (1.11) is of primary importance in the study of the spectral interpretation of L-functions [20],[30]. From the remark below (1.7) we know that (1.12) · · · AS(n) x1, . . . , xk) = p LUEm,a (x1, . . . , xk), where for n = 2m, a = −1/2, while for n = 2m+ 1, a = 1/2. Consequently (1.13) lim O+(n) , . . . , = lim X1 . . . Xk p LUEm,a π2X21 , . . . , π2X2k Thus knowledge of the distribution p LUEm,a for a = ±1/2 suffices to compute the scaled limit of p O+(n) In the case k = 1 a number of different characterisations of p LUEm,a , which is the distribution of the smallest eigenvalue in LUEm,a, are known. First, with n 7→ n+1 in (1.1) for convenience, the p.d.f. of the smallest eigenvalue is given by fixing one of the coordinates at x1, and integrating the remaining over [x1,∞). Thus (1.14) p LUEn+1,a (x1) = n!cn+1,n+1+a e−x1xa1 dλ1 . . . e−λjλaj (λj − x1)2 1≤j<k≤n (λk − λj)2. One has that (1.15) p LUEn+1,a (x1) = − ELUEn+1,a(x1), 4 PETER J. FORRESTER AND NICHOLAS S. WITTE where (1.16) ELUEn+1,a(t) := dλ1 · · · dλn+1p(λ1, . . . , λn+1), is the probability (gap probability) that no eigenvalues are in the interval (0, t). For integer values of the parameter a in (1.1), ELUEn,a(t) was studied by orthog- onal polynomial techniques in [11], where it was evaluated as an a×a determinant, and by the method of Jack polynomials in [10], giving an a-dimensional integral form. In [32] (see also [13]), for general Re(a) > −1, it was expressed in terms of a fifth Painlevé transcendent. Explicitly, it was found that (1.17) ELUEn,a(t) = exp UV(s) where UV(s) satisfies the Jumbo-Miwa-Okamoto σ-form of the Painlevé V equation (1.18) (tσ′′)2 − σ − tσ′ + 2(σ′)2 + (ν0 + ν1 + ν2 + ν3)σ′ + 4(ν0 + σ ′)(ν1 + σ ′)(ν2 + σ ′)(ν3 + σ ′) = 0, with parameters (1.19) ν0 = ν1 = 0, ν2 = n+ a, ν3 = n. Alternatively the conventional Painlevé V parameters are (1.20) α = a2, β = 0, γ = −2n− a− 1, δ = −1 and in terms of the Okamoto parameters they are (1.21) v2 − v1 = 0, v3 − v1 = n+ a, v4 − v1 = n, v3 − v4 = a. Because the eigenvalue density is strictly zero for λ < 0, the neighbourhood of the smallest eigenvalue is referred to as the hard edge, and is denoted by HEa. As is consistent with (1.13), a well defined limit of (1.17) is obtained by the scaling t 7→ t/4n and n→ ∞. Thus [33] (1.22) lim ELUEn,a(t/4n) := EHEa(t) = exp UIII′(s) where UIII′(t) satisfies the Jimbo-Miwa-Okamoto σ-form of the Painlevé III ′ equation (1.23) (tσ′′)2 − v1v2(σ′)2 + σ′(4σ′ − 1)(σ − tσ′)− (v1 − v2)2 = 0, with parameters (1.24) v1 = v2 = a, and subject to the boundary condition (1.25) UIII′(t) ∼ 22a+2Γ(a+ 1)Γ(a+ 2) ta+1. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 5 Our interest in this paper is the distribution p LUEm,a and its hard edge scaled limit. Analogous to (1.14), we see from (1.1) that (1.26) p LUEn+2,a (x1, x2) = n!cn+2,n+2+a e−x1−x2(x1 − x2)2(x1x2)a dλ1 . . . e−λjλaj (λj − x1)2(λj − x2)2 1≤j<k≤n (λk − λj)2, where x1 denotes the smallest eigenvalue and x2 the second smallest eigenvalue. In [12], for a integer, this was expressed as an (a + 2) × (a + 2) determinant. In the hard edge scaled limit this gave (1.27) lim LUEn+2,a ) =: pHEa (x1, x2) = 2−4 e−x2/4 × det Ij+2−k( j=1,...,a k=1,...a+2 x2 − x1 )(k−j)/2 Ij+2−k( x2 − x1) j=1,2 k=1,...a+2 We seek a Painlevé type characterisation of (1.26) and its scaled limit, valid for general Re(a) > −1. One use of knowledge of p LUEm,a is the computation of the distribution of the spacing between the smallest and the second smallest eigenvalues. Denoting this distribution by An,a for LUEn+2,a, we have (1.28) An,a(y) := dx1 p LUEn+2,a (x1, x1 + y), y ∈ R+. Important to our subsequent working is a rewrite of (1.26) and (1.28) in terms of an integral of the form (1.29) Dn(x1, x2)[w(λ)] dλ1 . . . w(λl) (λl − x1)(λl − x2) 1≤j<k≤n (λk − λj)2, where I denotes the support of the weight w(λ). We have (1.30) p(2)(x, x+ y) = cn+2,n+2+a e−(n+1)(x+y)−xy2[x(x + y)]aDn(−y,−y)[λ2(λ+ x+ y)ae−λχ>0], (1.31) An,a(y) = cn+2,n+2+a dt ta(t− y)ae−(n+2)tDn(−y,−y)[λ2(λ+ t)ae−λχ>0]. 6 PETER J. FORRESTER AND NICHOLAS S. WITTE These equations exhibit the occurrence of a deformation of the Laguerre weight (1.32) w(x; t) := x2(x+ t)ae−x, x ∈ R+. This deformed weight actually interpolates between two Laguerre weights - when t→ 0 then we have the general parameter a+ 2 case, whilst if t→ ∞ in the sector −π < arg(t) ≤ π we have the special parameter situation with an exponent of 2. In fact virtually all of our analysis can be carried over to the more general situation where the exponent 2 is an arbitrary complex parameter suitably restricted. We begin in Section 2 by revising appropriate results from orthogonal poly- nomial system theory and apply this to the particular deformed Laguerre weight (1.32). This allows us, in Section 3, to characterise the distributions (1.30) and (1.31) by a solution of the fifth Painlevé equation and its associated linear isomon- odromic system (see Proposition 3.2). Section 4 is devoted to the determinant evaluations of those distributions for positive integer values of the parameter a. We proceed in Section 5 to the study of the hard edge limits (x1, x2) := lim LUEn,a ),(1.33) Aa(z) := lim An,a( ).(1.34) It is found that these scaled distributions can be characterised by the solution of a certain Painlevé III′ equation and its associated linear isomonodromic system (see Propositions 5.2, 5.5 and Remark 5.3). In Section 6 this characterisation is used to obtain the high precision numerical values of statistical characteristics of Aa(z) for various integer values of a and for the values a = ±1/2, the latter being relevant to (1.6), (1.7) with the change of variable λ2j 7→ λj . Let p (x1, x2) denote the scaled distribution of the eigenvalues eiθ1 , eiθ2(θ1, θ2 > 0) closest to the origin in O+(2n + 1) and O+(2n) respectively. With the scaling chosen so that the bulk density is unity, it follows from (1.13) and (1.33) that (1.35) p± (x1, x2) = 4π 2x1x2p HE±1/2 (π2x21, π 2x22). Consequently A±(y) := dx p± (x, x+ y) = 4π2 dxx(x + y)p HE±1/2 (π2x2, π2(x+ y)2).(1.36) We use our results for p HE±1/2 to provide the high precision numerical values of statistical characteristics of A±(y). DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 7 2. Orthogonal Polynomial System 2.1. Semi-classical Orthogonal Polynomials. Consider the general orthogonal polynomial system {pn(x)}∞n=0 defined by the orthogonality relations (2.1) dx w(x)pn(x)x 0 0 ≤ m < n hn m = n with I denoting the support of the weight w(x). We give special notation for the coefficients of xn and xn−1 in pn(x), (2.2) pn(x) = γnx n + γn,1x n−1 + . . . . The corresponding monic polynomials are then (2.3) πn(x) = pn(x). It follows from (2.1) that dx w(x)(pn(x)) 2 = γnhn, and thus for pn(x) to be normalised as well as orthogonal we set γnhn = 1. A consequence of the orthogonality relation is the three term recurrence relation (2.4) an+1pn+1(x) = (x− bn)pn(x) − anpn−1(x), n ≥ 1, and we consider the set of orthogonal polynomials with initial values p−1 = 0 and p0 = γ0. The three term recurrence coefficients are related to the polynomial coefficients by [31], [14] (2.5) an = , bn = γn+1,1 , n ≥ 1, along with (2.6) b0 = − , a0 = 0, γ0,1 = 0. A well known consequence of (2.4) is the Christoffel-Darboux summation (2.7) pj(x)pj(y) = an [pn(x)pn−1(y)− pn−1(x)pn(y)] Central objects in our probabilistic model are the Hankel determinants (2.8) ∆n := det[µj+k−2]j,k=1,...,n, n ≥ 1, ∆0 := 1, (2.9) Σn := det µ0 · · · µn−2 µn µn−1 · · · µ2n−3 µ2n−1 , n ≥ 1, Σ0 := 0, 8 PETER J. FORRESTER AND NICHOLAS S. WITTE defined in terms of the moments {µn}n=0,1,...,∞ of the weight, (2.10) µn := dx w(x)xn. We have integral representations for ∆n (2.11) ∆n = dx1 . . . w(xl) 1≤j<k≤n (xk − xj)2, n ≥ 1, and Σn (2.12) Σn = dx1 . . . w(xl) 1≤j<k≤n (xk − xj)2, n ≥ 1. The three-term recurrence coefficients are related to these determinants by standard result in orthogonal polynomial theory [31], [14] a2n = ∆n+1∆n−1 , n ≥ 1,(2.13) , n ≥ 0,(2.14) γ2n = , n ≥ 0,(2.15) with initial values (2.16) a21 = µ0µ2 − µ21 , b0 = , µ0γ 0 = 1. The orthogonal polynomials themselves also have a determinantal representation (2.17) ∆n∆n+1pn(x) = det µ0 · · · µn µn−1 · · · µ2n−1 1 · · · xn , n ≥ 1, and the integral representation (2.18) ∆n∆n+1pn(x) = dx1 . . . w(xl)(x−xl) 1≤j<k≤n (xk−xj)2. Another set of polynomial solutions to the three term recurrence relation are the associated polynomials {p(1)n (x)}∞n=0, defined by (2.19) p n−1(x) := ds w(s) pn(s)− pn(x) , n ≥ 0. In particular these polynomials satisfy (2.20) an+1p n (x) = (x − bn)p n−1(x)− anp n−2(x), with the initial conditions p −1(x) = 0, p 0 (x) = µ0γ1. Note the shift by one decrement in comparison to the three-term recurrence (2.4) for the polynomials DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 9 {pn(x)}∞n=0. We also need the definition of the moment generating function or Stieltjes function f(x) : = , x /∈ I,(2.21) , x /∈ I, x→ ∞.(2.22) We define non-polynomial associated functions {ǫn(x)}∞n=0 by (2.23) ǫn(x) := f(x)pn(x) − p(1)n−1(x), which also satisfy the three term recurrence relation (2.4), namely (2.24) an+1ǫn+1(x) = (x− bn)ǫn(x) − anǫn−1(x), subject to the initial values ǫ−1(x) = 0, ǫ0(x) = γ0f(x). The associated functions have an integral representation analogous to (2.18) (2.25) ∆n∆n+1ǫn(x) (n+ 1)! dx1 . . . dxn+1 w(xl) x− xl 1≤j<k≤n+1 (xk − xj)2, x /∈ I. The polynomials and their associated functions satisfy the Casoratian relation (2.26) pn(x)ǫn−1(x)− pn−1(x)ǫn(x) = , n ≥ 1. Extending (2.2) and (2.5) we have (2.27) pn(x) = γn 0≤i<j<n bibj − xn−2 +O(xn−3) valid for n ≥ 1, while for the associated functions (2.28) ǫn(x) = γ x−n−1 + x−n−2 0≤i≤j≤n bibj + x−n−3 +O(x−n−4) valid for n ≥ 0. Proposition 2.1 ([6],[4],[24]). Let (2.29) w(x) = 2V (x) W (x) 10 PETER J. FORRESTER AND NICHOLAS S. WITTE for V,W irreducible. The orthogonal polynomials and associated functions satisfy a system of coupled first order linear differential equations with respect to x ( ′ ≡ d/dx) Wp′n = (Ωn − V )pn − anΘnpn−1, n ≥ 1,(2.30) Wp′n−1 = anΘn−1pn − (Ωn + V )pn−1, n ≥ 0,(2.31) for certain coefficient functions V (x),W (x),Θn(x),Ωn(x). The associated func- tions ǫn, ǫn−1 satisfy precisely the same set of equations. If we define the 2× 2 matrix variable (2.32) Yn(x; t) = pn(x) ǫn(x) pn−1(x) ǫn−1(x) then the above coupled system can be written as (2.33) Yn(x) = W (x) Ωn(x) − V (x) −anΘn(x) anΘn−1(x) −Ωn(x)− V (x) Yn(x) It follows that the coefficient functions are specified by Θn =W [ǫnp n − ǫ′npn] + 2V ǫnpn, n ≥ 0, Θ−1 = 0, (2.34) Ωn = anW [ǫn−1p n − ǫ′npn−1] + anV [ǫnpn−1 + ǫn−1pn] , n ≥ 1, Ω0 = 0. (2.35) Proposition 2.2 ([24]). The coefficient functions arising in Proposition 2.1 satisfy the recurrence relations (Ωn+1 − Ωn)(x − bn) =W + a2n+1Θn+1 − a2nΘn−1, n ≥ 0,(2.36) Ωn+1 +Ωn = (x − bn)Θn, n ≥ 0(2.37) We will find it necessary to study the zeros of the orthogonal polynomial pn(x) which we denote {x1,n < . . . < xj,n < . . . < xn,n}. They have an electrostatic interpretation as the equilibrium positions of the mobile unit charges, and there is a set of equations governing these equilibrium positions known as the Bethe Ansatz equations. Proposition 2.3 ([16]). The zeros {xj,n}nj=1 of the polynomial pn(x) satisfy the coupled functional equations (2.38) 2 k 6=j xj,n − xk,n Θ′n(xj,n) Θn(xj,n) ′(xj,n) + 2V (xj,n) W (xj,n) for all 1 ≤ j ≤ n. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 11 One can also represent many useful quantities in terms of sums over the ze- ros and we illustrate this with an example. Firstly the consecutive ratios of the orthogonal polynomials have a partial fraction decomposition (2.39) pn−1(x) pn(x) = − 1 W (xj,n) Θn(xj,n) x− xj,n along with (2.40) a2n = − W (xj,n) Θn(xj,n) Of particular relevance to our application are the semi-classical class of orthog- onal polynomial systems [25] defined by the property that V (x) andW (x) in (2.29) are polynomials in x. The zeros of W (x) define finite singularities of the system of ordinary differential equations (2.30), (2.31) and will feature prominently in this study. Let xr be such a point with r ≥ 1. Then at xr the relations (2.36) and (2.37) can be combined and integrated to yield (2.41) Ω2n(xr)− V 2(xr) = a2nΘn(xr)Θn−1(xr), n ≥ 1. In fact, at a given finite singular point xr, we can deduce the following bi-linear identities, that factorise the one above. Corollary 2.1. The coefficient functions evaluated at a finite singular point xr are related to evaluations of the orthogonal polynomials and associated functions by the relations Ωn(xr) + V (xr) = 2anV (xr)pn(xr)ǫn−1(xr), n ≥ 1,(2.42) Ωn(xr)− V (xr) = 2anV (xr)pn−1(xr)ǫn(xr), n ≥ 1,(2.43) Θn(xr) = 2V (xr)pn(xr)ǫn(xr). n ≥ 0,(2.44) From the theory of Uvarov the following general result for (1.29) is known. Proposition 2.4 ([35]). The quantity Dn(x, x)[w(λ)], defined by the equal argu- ment form of (1.29), is evaluated in terms of the polynomials pn(x) orthogonal with respect to w(x) and coefficients γn,∆n of this system as (2.45) Dn(x, x)[w(λ)] = γnγn+1 [pn(x)p n+1(x)− pn+1(x)p′n(x)]. Proof. This is a specialisation of Uvarov’s general result to the case k = 0 and l = 2 where the integral Dn(x1, x2)[w(λ)] is a Hankel determinant with respect to the weight w0,2(x), defined by (2.46) w0,2(x)dx = dρ0,2(x), ρ0,2(x) = (s− x1)(s− x2)w(s)ds. 12 PETER J. FORRESTER AND NICHOLAS S. WITTE The non-confluent form of the corresponding identity states that (2.47) Dn(x1, x2)[w(λ)] = (∆n+2∆n) 1/2 det[pn+k−1(xj)]j,k=1,2 x2 − x1 and the result follows under the confluence x2 → x1. � We see from (2.45) that our main task is to obtain appropriate characterisations of the orthogonal polynomials and their derivatives associated with the weight (1.32). 2.2. Deformed Laguerre Orthogonal Polynomials. As we noted in the intro- duction we see the appearance of a deformed Laguerre weight (1.32) which is a member of the semi-classical class with the polynomials V,W in (2.29) specified by (2.48) 2V (x; t) = −x2 + (a+ 2− t)x+ 2t, W (x; t) = x(x+ t), and has finite singularities at x = 0,−t. The moments have the simple evaluation (2.49) µn(t) = t a+n+3Γ(n+ 3)U(n+ 3, a+ n+ 4; t), n ≥ 0, |arg(t)| < π, where U(α, γ; z) is the confluent hypergeometric that is not analytic at z = 0. The moments can be written as a sum of two parts one of which is analytic and the other non-analytic about t = 0 (2.50) µn(t) = Γ(a+ n+ 3)1F1(−a;−a− n− 2; t) + (−1)n+3Γ(a+ 1)Γ(n+ 3) Γ(a+ n+ 4) ta+n+31F1(n+ 3; a+ n+ 4; t), where we have to exclude the cases a ∈ Z≥0. Within the semi-classical class the coefficient functions Θn(x),Ωn(x) are poly- nomials with degree fixed independently of the index n. In particular we can relate these polynomials to the coefficients of the orthogonal polynomials themselves. Proposition 2.5. The coefficient functions are Θn(x) = 2n+ a+ 3− t− bn − x, n ≥ 0,(2.51) Ωn(x) = − (2n+ a+ 2− t)x + (n+ 1)t− a2n − , n ≥ 1.(2.52) Proof. From the theory of [24] we note that the degrees of the coefficient functions are degΘn ≤ max{degW − 2, degV − 1} = 1 and degΩn ≤ max{degW − 1, degV − 1, degΘn−1, degU+1} = 2. To obtain explicit forms for these we use the definitions (2.34) and (2.35) and the large x→ ∞ expansions of the polynomials and associated functions given in (2.27) and (2.28). The first equalities in (2.51) and (2.52) then follow. � DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 13 We will also find it convenient to make the following definitions motivated by the above result, θn := 2n+ a+ 3− t− bn,(2.53) κn := (n+ 1)t− a2n − .(2.54) Proposition 2.6. The spectral derivatives of the polynomials pn(x) are ( ∂/∂x) x(x+ t)p′n = (nx+ κn − t)pn − an(θn − x)pn−1(2.55) x(x+ t)p′n−1 = an(θn−1 − x)pn − (−x2 + (n+ a+ 2− t)x + t+ κn)pn−1(2.56) Proof. This follows from the general form of the spectral derivatives (2.30) and (2.30), along with the explicit particular forms (2.51) and (2.52). � Proposition 2.7. The deformation derivatives of the orthogonal polynomials are ( ˙ ≡ ∂/∂t) t(x+ t)ṗn = (n+ 1)t− κn − (x+ t)(θn + t) pn + an(θn + t)pn−1 (2.57) t(x+ t)ṗn−1 = −an(θn−1 + t)pn + (x+ t)(θn−1 + t) + κn − (n+ a+ 1)t (2.58) Proof. We will opt to establish this relation directly from the orthonormality con- ditions on the polynomials dx w(x)pn(x)pn−i(x) = δi,0, 0 ≤ i ≤ n. Differentiating this with respect to t leaves us with the relation (2.59) 0 = a dx w(x) pnpn−i dx w(x)ṗnpn−i + dx w(x)pnṗn−i, where use of the logarithmic derivative of w(x) has been made. Now we employ ṗn−i = γ̇n−i pn−i +Πn−i−1, to write the last term of (2.59) as dx w(x)pnṗn−i = γ̇n−i δi,0 = dx w(x)pnpn−i. 14 PETER J. FORRESTER AND NICHOLAS S. WITTE Considering the first term of (2.59) we note dx w(x) pn(x)pn−i(x) dx w(x)pn(x) pn−i(x)− pn−i(−t) + pn−i(−t) dx w(x) pn(x) x + t = pn−i(−t) dx w(x) pn(x) x + t =− pn−i(−t)ǫn(−t). But we can recast pn−i(−t) as pn−i(−t) = pn−j(−t)δi,j , pn−j(−t) dx w(x)pn−j(x)pn−i(x), dx w(x)pn−i(x) pn−j(−t)pn−j(x), dx w(x)pn−i(x) pj(−t)pj(x). Combining these we deduce that dx w(x)pn−i(x) ṗn(x) + pn(x) − aǫn(−t) pj(−t)pj(x) for i = 0, . . . , n. The factor in curly brackets in the integrand must be a polynomial in x with degree less than or equal to n, and yet is orthogonal to all polynomials pj for j = 0, . . . , n, and thus must be identically zero. This gives us our first form for the deformation derivative of pn, (2.60) ṗn(x) = − pn(x) + aǫn(−t) pj(−t)pj(x). Equating coefficients of pn(x) in this relation we find (2.61) 2 = apn(−t)ǫn(−t). Furthermore using the Christoffel-Darboux formula (2.7), and the above equation, we can express this derivative solely as a linear combination of pn and pn−1 (2.62) ṗn(x) = aǫn(−t) pn(−t) + an pn−1(−t) pn(x) − aanǫn(−t) pn(−t) pn−1(x). DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 15 Using the bilinear product relations (2.43) and (2.44) we arrive at the result (2.57). The second of the two relations can be found by shifting n 7→ n− 1 in (2.62) and using the three term recurrence relation. � In the matrix formulation the spectral derivative take the particular form (2.63) ∂xYn(x; t) = Yn(x; t). The residue matrices are explicitly given by κn − t −anθn anθn−1 −κn − t , χ0 = 0,−2(2.64) (n+ 1)t− κn an(θn + t) −an(θn−1 + t) κn − (n+ a+ 1)t , χt = 0,−a(2.65) (2.66) Our linear system has two regular singularities at x = 0,−t and an irregular sin- gularity at x = ∞ with Poincaré index 1. In this formulation the deformation derivative is (2.67) ∂tYn(x; t) = B + At Yn(x; t) (2.68) B = 1 −θn − t 0 0 θn−1 + t As we will see in Section 3.2 (2.63) and (2.67) form the monodromy preserving system corresponding to the fifth Painlevé equation with a form equivalent to that discussed by Jimbo [18], in contrast to other forms studied in [19], [7], [8] or [17]. Corollary 2.2. The polynomial coefficients satisfy the following coupled, first order mixed deformation derivative and difference equations = 2 + bn−1 − bn, n ≥ 1,(2.69) tḃn = a n − a2n+1 + bn, n ≥ 0,(2.70) with the initial t = 0 values for bn and a n given by (2.87) and (2.88) respectively. This system of differential equations is equivalent to the Schlesinger equations. Proof. There are several methods of proof available here. The first is using the general result of [24], expressing the deformation derivatives of the polynomial coefficients in terms of a sum of the coefficient functions over the movable finite 16 PETER J. FORRESTER AND NICHOLAS S. WITTE singular points Θn(xr)−Θn−1(xr) W ′(xr) ẋr , n ≥ 1,(2.71) ḃn = Ωn+1(xr)− Ωn(xr) W ′(xr) ẋr n ≥ 1.(2.72) The only finite singular point contributing here is x = −t. Alternatively one can find these derivatives from the polynomial derivatives by examining selected coefficients. For example by considering the coefficients of xn−1 in (2.60) we have (2.73) γ̇n,1 = aǫn(−t)pn(−t)γn,1 + aǫn(−t)pn−1(−t)γn−1, and therefore (2.74) ˙(γn,1 = aǫn(−t)pn−1(−t). Consequently, using (2.5), we find that a [ǫn−1(−t)pn−1(−t)− ǫn(−t)pn(−t)] ,(2.75) ḃn = a [anǫn(−t)pn−1(−t)− an+1ǫn+1(−t)pn(−t)] ,(2.76) which are identical to (2.69) and (2.70) respectively. � Corollary 2.3. The recurrences for the polynomial coefficients are (2.77) a2n+2 − a2n = 2t+ bn+1 [2n+ a+ 6− t− bn+1]− bn [2n+ a+ 2− t− bn] , n ≥ 1. (2.78) a2n+1 [2n+ a+ 5− t− bn − bn+1]− a2n [2n+ a+ 1− t− bn−1 − bn] = −bn(bn + t), n ≥ 1. The initial data b0(t) and a 1(t) are given by (2.16) with the evaluation of the mo- ments (2.49). Proof. The first relation (2.77) follows by substituting the explicit forms for the co- efficient functions, (2.51) and (2.52), into the relation (2.37) and requiring equality of the polynomials in x. Equality is trivial for x2 and x1, whilst the non-trivial equality for x0 gives (2.77). The second relation (2.78) follows from the same pro- cedure applied to the recurrence relation (2.36) and again the only nontrivial result occurs for the x0 part. � DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 17 This result can also be recovered from a specialisation of work in [5]. In their sys- tem of monic orthogonal polynomials the three term recurrence coefficients βn, γn (not to be confused with our use of these symbols subsequently) are related to ours (2.79) βn = bn, γn = a Equation (39) of this work implies (2.80) γn+1 + γn = (2n+ 3)t+ (2n+ a+ 4− t)βn + 2 βk − β2n, and differencing this once leads directly to (2.77). In addition their equation (40) implies (2.81) γn+1βn+1 = (2n+ a+ 5− t− βn)γn+1 + 2 βk(βk + t). Again differencing this once one finds precisely (2.78). A check of the above results can be made when t → 0 whilst all parameters are kept fixed since this, as noted before, corresponds to the Laguerre weight with parameter a+ 2. Therefore we have a2n(0) = n(n+ a+ 2),(2.82) bn(0) = 2n+ a+ 3,(2.83) ∆n(0) = j=1 j!Γ(j + a+ 2) .(2.84) In our normalisation we have (2.85) pn(x; 0) = (−1)n Γ(n+ a+ 3) L(a+2)n (x), where L n (x) are the standard associated Laguerre polynomials of degree n and index α. We see that the spectral derivative equations (2.55) and (2.56) reduce to the standard expressions for the derivative of the Laguerre polynomials, the coefficients in the right-hand sides of the deformation derivatives (2.57) and (2.58) vanish, the recurrence relations (2.77) and (2.78) are identically satisfied, and the right-hand sides of (2.69) and (2.70) are zero. In fact we will need to develop expansions about t = 0 in order to charac- terise our quantities as particular solutions of difference and differential equations in Section 3. To this end we have the following result. 18 PETER J. FORRESTER AND NICHOLAS S. WITTE Proposition 2.8. For fixed n, a /∈ Z≥0 the Hankel determinant (2.8) with the weight (1.32) has the expansion about t = 0 (2.86) ∆n(t) = ∆n(0) (a− 1)(n+ 1) (a+ 1)(n− 1) nt2 +O(t3) − 2Γ(a+ 1) Γ(a+ 3)Γ2(a+ 4) Γ(a+ n+ 3) ta+3 (1 + O(t)) + O(t2a+6) with |arg(t)| < π. Consequently, under the same conditions, the three-term recur- rence coefficients have the expansions about t = 0 (2.87) bn(t) = 2n+ a+ 3− 2a(2n+ a+ 3) (a+ 3)(a+ 2)2(a+ 1) t2 +O(t3) (a+ 2)(a+ 1)Γ2(a+ 3) Γ(a+ n+ 3) Γ(n+ 1) ta+3 (1 + O(t)) + O(t2a+6), (2.88) a2n(t) = n(n+ a+ 2) 1− 2a (a+ 3)(a+ 2)2(a+ 1) t2 +O(t3) (a+ 1)Γ(a+ 3)Γ(a+ 4) Γ(a+ n+ 2) Γ(n+ 1) ta+3 (1 + O(t)) + O(t2a+6) Proof. We adopt the method of expanding the Hankel determinant by expanding the moments to leading order (2.89) µn(t) = Γ(a+ n+ 3) + aΓ(a+ n+ 2)t+ a(a− 1)Γ(a+ n+ 1)t2 +O(t3) + (−1)n+1Γ(a+ 1)Γ(n+ 3) Γ(a+ n+ 4) tn+a+3 +O(tn+a+4), as t→ 0 using (2.50). The determinant can be expanded to leading orders in t, ta+3 and the resulting determinants evaluated using the identity [26] (2.90) det(Γ(zk + j))j,k=0,...,n−1 = Γ(zj) 0≤j<k≤n−1 (zk − zj), where {z0, . . . , zn−1} is an arbitrary sequence not necessarily in arithmetic progres- sion. � We conclude this section by noting some identities relating the polynomial coeffi- cients and the zeros of the polynomials. Firstly we give the Bethe Ansatz equations for the zeros of the deformed Laguerre orthogonal polynomials which can be directly deduced from Proposition 2.3. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 19 Corollary 2.4. The zeros xj,n of the deformed Laguerre orthogonal polynomials pn(x) satisfy the functional equations (2.91) xj,n + t xj,n − θn k 6=j xj,n − xk,n = 1, 1 ≤ j ≤ n. According to the electrostatic interpretation, the terms of (2.91) can be inter- preted in the following way - the first is the interaction of the mobile unit charge at xj,n with the fixed charge of size 3 at the singularity x = 0, the second with the fixed charge of size a + 1 at the singularity x = −t, the third with a fixed charge of size −1 at the apparent singularity x = θn, the fourth the mutual repulsion with the other mobile charges and the term on the right-hand side is the linear confining potential. From the partial fraction decomposition (2.39) specialised to the arguments x = 0,−t we have the summation identities. Proposition 2.9. The following summations over the zeros have the explicit eval- uations κn − (n+ 1)t θn + t θn − xj,n ,(2.92) κn − t t+ xj,n θn − xj,n ,(2.93) θn + t κn − t θn − xj,n ,(2.94) a2n = xj,n(xj,n + t) xj,n − θn .(2.95) In addition we can characterise the motion of the zeros with respect to the deformation variable. Proposition 2.10. The zeros xj,n(t) satisfy the differential equation with respect (2.96) tẋj,n = θn + t θn − xj,n xj,n. Proof. This follows by equating (2.97) ṗn(x) pn(x) ẋj,n x− xj,n and (2.57), and then employing (2.39) along with (2.48), (2.51). � 3. Difference and Differential Equations 3.1. Difference Equations. In the first subsection we derive an alternative dif- ference system in terms of the new variables θn(t), κn(t) as specified by (2.53), (2.54). 20 PETER J. FORRESTER AND NICHOLAS S. WITTE Proposition 3.1. The auxiliary functions θn(t), κn(t) satisfy a system of coupled first order recurrence relations κn+1 + κn = θn(θn + t− 2n− a− 3), n ≥ 0,(3.1) θn + t θn−1 + t (κn − t)(κn + t) [κn − (n+ a+ 1)t][κn − (n+ 1)t] , n ≥ 1.(3.2) The initial values θ0 and κ0 are given by (3.3) θ0(t) = −2t dx e−xx(t+ x)a dx e−xx2(t+ x)a , κ0 = t. Proof. The first of the recurrence relations (2.77) can be exactly summed and the result is (3.4) a2n+1 + a n = (2n+ 3)t+ (2n+ a+ 4− t)bn + 2 bi − b2n. Recalling the second relation of (2.5) the summation appearing here can done by recasting the equation in terms of the new variables and yields (3.5) κn+1 + κn = −θnbn, which is (3.1). The second member of the coupled set is most easily found from the general relation (2.41) evaluated at the finite singular points x = 0,−t and employing the new variables. These two key identities are (κn + t)(κn − t) = a2nθnθn−1,(3.6) [κn − (n+ a+ 1)t][κn − (n+ 1)t] = a2n(θn + t)(θn−1 + t).(3.7) The ratio of these two identities yields the relation (3.2). � There are other recurrence relations which will be used subsequently, and the first is (3.8) a2n(θn + θn−1 + t) = −(2n+ a+ 2)κn + [n2 + (n+ 1)(a+ 2)]t. This follows from the subtraction of (3.6) from (3.7). The second relation (3.9) a2n+1 − a2n − bn − t = 2κn + bnθn, is derived by writing the definition of κn+1 − κn in terms of the old variables and then employing (3.1). The last relation (3.10) a2n+1θn+1 − a2nθn−1 = bn(2κn + bnθn), is a consequence of (2.78) along with the use of (3.9). DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 21 As a consequence of relations (3.6) from (3.7) we have θn + t κn − t κn − t κn − (n+ 1)t θn + t κn + t − a2n θn−1 + t κn − (n+ a+ 1)t κn − t κn − (n+ a+ 1)t n+ a+ 2 + κn + t ,(3.11) (3.12) n+ κn − t n+ a+ 2 + κn + t n7→n+1 = θn + t. 3.2. Reduction to Painlevé V. Here we will identify the fifth Painlevé system as the solution to our system of equations characterising the deformed Laguerre or- thogonal polynomial system. This is most simply seen in terms of the new variables θn, κn rather than the basic orthogonal polynomial variables an, bn. Proposition 3.2. The auxiliary quantities θn(t), κn(t) satisfy the coupled first or- der ordinary differential equations (3.13) tθ̇n = 2κn + θn(2n+ a+ 3− t− θn), (3.14) tκ̇n = θn + t κ2n + 2n+ a+ 3− (2n+ a+ 2) t θn + t − [n2 + (n+ 1)(a+ 2)]t− t + (n+ 1)(n+ a+ 1) θn + t Equations (3.13) and (3.14) can be solved in terms of the fifth Painlevé system (3.15) θn = t , κn = t(1 + qp), where q, p are the Hamiltonian variables of the Okamoto PV [27] system with the parameters (3.16) α = , β = −2, γ = −(2n+ a+ 3), δ = − (3.17) v2 − v1 = −2, v3 − v1 = n+ a, v4 − v1 = n, v3 − v4 = a. The solutions satisfy the boundary value data at t = 0 (3.18) θn(t) = 2a(2n+ a+ 3) (a+ 3)(a+ 2)2(a+ 1) t2 +O(t3) (a+ 2)(a+ 1)Γ2(a+ 3) Γ(a+ n+ 3) Γ(n+ 1) ta+3 (1 + O(t)) + O(t2a+6), 22 PETER J. FORRESTER AND NICHOLAS S. WITTE (3.19) κn(t) = 2n+ a+ 2 4n(n+ a+ 2)a (a+ 3)(a+ 2)2(a+ 1) t2 +O(t3) (a+ 2)(a+ 1)Γ2(a+ 3) Γ(a+ n+ 3) ta+3 (1 + O(t)) + O(t2a+6), provided a /∈ Z≥0 and |arg(t)| < π. Proof. If we employ (3.9) in (2.70) and then substitute for bn in terms of θn then the result is (3.13). Let us define the shorthand notation (3.20) Γn := Furthermore when the deformation derivative (2.70) is summed on the free index the result is (3.21) tΓ̇n = a n + Γn. Now if compute the deformation derivative of κn and use (2.69) along with (3.21) we arrive at (3.22) tκ̇n = κn − a2n(θn − θn−1). Now the idea is to eliminate a2n and θn−1, which appear in (3.22), through use of the recurrence relations. Equation (3.2) is a linear equation for θn−1 in terms of the unshifted variables and the solution can be substituted into (3.8) yielding a linear relation for a2n in terms of unshifted variables. Then both solutions can be substituted into (3.22) and the result is (3.14). One can easily verify that the transformation to the Hamiltonian variables q, p (3.15) with the parameters (3.16) yields the Hamilton equations of motion for the Hamiltonian in [27]. � Remark 3.1. A few remarks can now be made regarding the identification of the recurrences (3.1) and (3.2). This is different in appearance from the discrete in- tegrable equations that arose in the study of the Laguerre unitary ensemble [13] which were explicitly identified with the system in the Sakai scheme, with the ra- tional surface D 5 → E 6 , and has a continuous limit of Painlevé IV . In fact we find that the variables of the latter system can be expressed in terms of our own (3.23) xn = κn + θn(n+ a+ 1− t− θn) θn + t , yn = − and it is clear that one cannot transform (3.1) and (3.2) into this system using such a transformation. Also the two systems arise as different Schlesinger-type transformations - in our case as a sequence where α0 7→ α0 + 1, α2 7→ α2 − 1 whereas in the other case as α0 7→ α0 + 1, α3 7→ α3 − 1. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 23 Remark 3.2. In [13] two fundamental quantities were studied - the τ -function τ [n](t) and its logarithmic derivative the σ-function Vn(t; a, µ). Their relation to the ob- jects of the present work are (3.24) τ [n] = c(n, a)n!e−ntt−n 2−n(a+4)∆n(t), where c(n, a) is an unspecified constant and (3.25) Vn(t; a, 2) = −nt− 4n+ t log∆n(t). We also note that (3.26) Vn(t; a, 2) = Γn + n(n+ a− 2− t) and consequently Γn = −n(n+ a+ 2) + t log∆n(t)(3.27) bn = 2n+ a+ 3 + t ∆n(t) ∆n+1(t) (3.28) Remark 3.3. The new variable Γn(t) possess an expansion as t → 0 which can be directly found from (2.86) and (2.87) (3.29) n(n+ a+ 2) + Γn(t) = nt− 2n(n+ a+ 2)a (a+ 3)(a+ 2)2(a+ 1) t2 +O(t3) (a+ 2)(a+ 1)Γ(a+ 3)Γ(a+ 4) Γ(a+ n+ 3) ta+3 (1 + O(t)) + O(t2a+6), again provided a /∈ Z≥0. Remark 3.4. The use of Proposition 3.2 is in the computation of the orthogonal polynomials in (2.4) corresponding to the weight (1.32). For this we note from (2.53) that (3.30) bn = 2n+ a+ 3− t− θn, while (2.54) together with (2.5), (2.6) show a2n = (n+ 1)t− − κn(3.31) = (n+ 1)t+ bj − κn,(3.32) (the quantity an is positive, so the positive square root of this equation is to be taken). Further, it follows from (2.13) and (2.5) that (3.33) 1 · · · γ2n−1 = ∆n, where each γj is the coefficient of x j in pj(x) as specified by (2.2). All terms in the equation (2.45) for Dn(x, x) are then known, and the task is then to compute the integral as required by (1.31). 24 PETER J. FORRESTER AND NICHOLAS S. WITTE An alternative system of coupled first order ordinary differential equations which will be used for scaling to the hard edge is given in the following proposition. Proposition 3.3. The variables θn(t),Γn(t) satisfy the coupled first order ordinary differential equations (3.34) tθ̇n = θn − θ4n − 2(2n+ a+ 2− t)θ3n + [4Γn + (2n+ a+ 2− t)2 − 4(n+ 1)t]θ2n + 4t[Γn + n(n+ a+ 2− t) + a+ 2− t]θn + 4t2 (3.35) tΓ̇n = (n+ 1)t+ θn(2n+ a+ 2− t− θn) + θ4n − 2(2n+ a+ 2− t)θ3n + [4Γn + (2n+ a+ 2− t)2 − 4(n+ 1)t]θ2n + 4t[Γn + n(n+ a+ 2− t) + a+ 2− t]θn + 4t2 Proof. We proceed in a series of steps. Firstly we use (3.6) to solve for θn−1 in terms of θn, κn and Γn. In the second step we substitute this solution for θn−1 into (3.7) and solve the following quadratic equation for κn in terms of θn and Γn, (3.36) κ2n+θn(2n+a+2−t−θn)κn+[(n+1)t−Γn]θn(θn+t)−[n2+(n+1)(a+2)]tθn−t2 = 0. The choice of the sign of the square-root branch follows from the expansions (3.18) and (3.29) on one hand, and on the other hand noting that as t→ 0 (3.37) θ4n − 2(2n+ a+ 2− t)θ3n + [4Γn + (2n+ a+ 2− t)2 − 4(n+ 1)t]θ2n + 4t[Γn + n(n+ a+ 2− t) + a+ 2− t]θn + 4t2 2a(2n+ a+ 3) (a+ 3)(a+ 2)2(a+ 1) t2 +O(t3). In the final step we use these solutions for θn−1 and κn in (3.22) and (3.21). � In preparation for the hard edge scaling limit we need to make evaluations of the polynomials at the finite singular points. Firstly considering x = 0 we note (3.38) πn(0) πn−1(0) pn(0) pn−1(0) κn + t = a2n κn − t as follows immediately from (2.55). Furthermore we also have (3.39) t ˙(log πn(0)) = t ˙(log pn(0)) + (θn + t) = n+ κn − t DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 25 which follows from (2.57) and the above result. The corresponding result for the polynomial ratio at x = −t is (3.40) pn(−t) pn−1(−t) κn − (n+ a+ 1)t θn−1 + t θn + t κn − (n+ 1)t After [1] we define the orthogonal polynomial ratios (3.41) Qn(x; t) := pn(x; t) pn(0; t) because we are interested in the scaling properties of the orthogonal polynomial system at the edge of their interval of orthogonality, x = 0. Then (2.4) and Propo- sitions 2.6 and 2.7 can be translated into the following three corollaries. Corollary 3.1. The three-term recurrence for {Qn}n=0,1,... system is (3.42) bn(Qn+1 +Qn−1 − 2Qn) + (bn + 2 κn − t )(Qn+1 −Qn−1) + 2xQn = 0. Corollary 3.2. The spectral derivatives of Qn, Qn−1 are x(x+ t)Q′n = nxQn + (κn − t) Qn −Qn−1 + (3.43) x(x+ t)Q′n−1 = x[x − (n+ a+ 2− t)]Qn−1 + (κn + t) Qn −Qn−1 − (3.44) Corollary 3.3. The deformation derivatives of Qn, Qn−1 are t(x+ t)Q̇n = −x(n+ κn − t )Qn + (κn − t) θn + t [Qn−1 −Qn] ,(3.45) t(x+ t)Q̇n−1 = x[n+ a+ 2 + κn + t ]Qn−1 + (κn + t) θn−1 + t [Qn−1 −Qn] . (3.46) As we noted earlier the polynomial ratio Qn(x; t) has a product representation (3.47) Qn(x; t) = where again xj,n is the j-th zero of the polynomial. We can use this fact to compute sums of the inverse powers of the zeros from the above differential equations. Proposition 3.4. The increment of the sum of the reciprocals of the zeros going from n− 1 to n is given by (3.48) 3t xj,n−1 κn − t + n+ a+ 2 + κn + t 26 PETER J. FORRESTER AND NICHOLAS S. WITTE Proof. The required increment of the sum of the reciprocals of the zeros is the order x term in the expansion of ψn(x; t) := Qn(x; t)/Qn−1(x; t) about x = 0, and this can evaluated from the same expansion of the differential equation for ψn with respect to x. This latter differential equation is easily found from (3.43) and (3.44) and is (3.49) x(x+t)ψ′n = x− θn (κn−t)+ 2κn+x[−x+2n+a+2−t] x− θn−1 (κn+t)ψ At the other finite singular point, x = −t, we have as a consequence of these corollaries (3.50) Qn(−t; t) Qn−1(−t; t) κn − t κn − (n+ 1)t θn + t κn − (n+ a+ 1)t κn + t θn−1 + t (3.51) Qn(−t; t) = t− κn − nθn θn(θn + t) Qn(−t; t). Note that the derivative with respect to t in the latter equation is a total derivative. To complete our preparations for the hard edge scaling we need to identify two polynomial variables that will scale to independent variables in the scaling limit. The first is the orthogonal polynomial ratio Qn, and for the second a number of choices could be made but a simple choice is (3.52) Rn := Qn −Qn−1. Corollary 3.4. The spectral derivatives of Qn, Rn are x(x + t)Q′n = x κn − t Qn + (κn − t) θn − x Rn,(3.53) x(x+ t)R′n = x κn − t + n+ a+ 2 + κn + t − x− t Qn(3.54) κn − t + x(x + t)− (a+ 2)x− 2t Proof. This follows from Corollary 3.2. � Corollary 3.5. The deformation derivatives of Qn, Rn are t(x + t)Q̇n = −x κn − t Qn − (κn − t) θn + t (3.55) t(x+ t)Ṙn = −x κn − t + n+ a+ 2 + κn + t (3.56) n+ a+ 2 + κn + t + (κn + t) θn−1 + t − (κn − t) θn + t DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 27 Proof. This follows from Corollary 3.3. � 3.3. Inequalities and Bounds. A key step in proving our hard edge scaling limits will be bounds on the variables θn, κn and some auxiliary quantities. The first step is the following result. Lemma 3.1. The variables θn(t), κn(t) satisfy the inequalities θn + t κn − (n+ 1)t κn − t < 0,(3.57) κn + t θn−1 + t κn − (n+ a+ 1)t < 0,(3.58) for all positive, real and bounded t and n ≥ 1. Proof. That the ratios given in (3.57) and (3.58) are negative is a consequence of the fact that the polynomial pn(x) evaluated on the negative real axis, i.e. exterior to the interval of orthogonality, has a fixed sign. Specifically (−1)npn(−y) > 0 for real, positive y. Using the ratio relations (3.38) and (3.40) we have the upper bounds. From the Christoffel-Darboux formula (2.7) at equal arguments we note (3.59) pn−1(x)p n(x)− p′n−1(x)pn(x) > 0, and from the above pn−1(x)pn(x) < 0 for x ∈ −R+ we conclude (3.60) p′n(x) pn(x) p′n−1(x) pn−1(x) under the conditions on x. Integrating this inequality from 0 to −y ∈ −R+ we arrive at (3.61) pn(−y) pn−1(−y) pn(0) pn−1(0) Then identifying these ratios with (3.38) and (3.40) in the case y = t leads to the relative inequalities. � The above set of inequalities must all apply simultaneously and we see in fact that it implies restrictions on the variables θn, κn. Lemma 3.2. For bounded t ∈ R+ and n ≥ 0 the variables θn, κn satisfy inequalities which place them in one of three cases, as illustrated in Figure 1 - Case I: 0 < θn,(3.62) θn(θn + t− 2n− a− 3) + t < κn < t− nθn,(3.63) Case II: −t ≤ θn ≤ 0,(3.64) t− nθn ≤ κn ≤ t+min{nt, θn(θn + t− 2n− a− 3)} ≤ (n+ 1)t,(3.65) 28 PETER J. FORRESTER AND NICHOLAS S. WITTE (n+1)t t−nθn (θn+t)(θn−2n−a−3)+(n+1)t θn(θn+t−n)+t θn(θn+t−2n−a−3)+t ......... .......... ......... .......... .......... ......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... .......... ......... .......... ......... ....... ....... ....... ....... ....... ....... ...... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ...... ....... ........ ....... ........ ........ ..... ........ ........ .......... ....... ............. ............... .............. .............. ............... ................ ................. .................. ..................... ........................ ........................... .............................. ................................ ................................... ...................................... ......................................... ....... ....... ....... ....... ....... ....... ....... ....... ....... ...... ....... ....... ...... ....... ....... ....... ....... ....... ....... ...... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ..... ....... ....... ....... ....... ....... ....... ....... ...... ....... ....... ....... ....... ........ ...... ........ ........ .......... ........ ............. ................ ............... ............... ................ ................. .................. ................... ..................... ........................ ........................... .............................. ................................. .................................... ....................................... .......................................... ............................................. ................................................ ................................................... ...... ....... ...... ....... ....... ...... ....... ...... ....... ....... ....... ....... ...... ....... ....... ....... ....... ..... ...... ....... ....... ....... ....... ...... ....... ....... ...... ....... ....... ...... ....... ....... ...... ....... ....... ...... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ...... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ...... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ........ ........ ....... ....... ........ ........ .......... ........ ............ ..... ................ ............... ............... ................ ................. .................. ................... ...................... .......................... ............................. ................................ ................................... ...................................... ......................................... ............................................ ................................................ ................................................... ...................................................... ......................................................... ............................................................ ............................................................... Figure 1. A pictorial form of the inequalities taking the example of a = 1/2, n = 2 and t = 5/3, which illustrates the generic situation for n+ a+ 3 > t. Case III: θn < −t,(3.66) (n+ 1)t < κn < t− nθn.(3.67) Proof. From the three inequalities implied by (3.57) we see that θn ≷ 0 according as κn ≶ t, θn + t ≷ 0 according as κn ≶ (n+ 1)t, and θn(θn + t) ≷ 0 according as κn ≶ t − nθn. In (3.58) we make the replacement n 7→ n + 1 and employ (3.1) to eliminate κn+1. This inequality now reads (3.68) − t− κn − (n+ 1)t θn + t < −κn − t < bn = 2n+ a+ 3− t− θn. Consequently these three inequalities imply θn ≷ 0 according as κn − t ≷ θn(θn + t− 2n− a− 3), θn + t ≷ 0 according as κn − (n+ 1)t ≷ (θn + t)(θn − 2n− a− 3), and θn(θn + t) ≷ 0 according as κn ≶ θn(θn + t− n) + t. Combining these sets of inequalities leads to the three cases above. � We see that Case II applies in our situation. Lemma 3.3. For all n and t ∈ R+ we have −t ≤ θn ≤ 0 and t − nθn ≤ κn ≤ (n+ 1)t. Proof. From the residue formula (2.44) we recall that Θn(0) = θn = 2tpn(0)ǫn(0) and Θn(−t) = θn + t = −atpn(−t)ǫn(−t). As we noted in the proof of Lemma DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 29 (3.1) it is immediate from the integral representations of the polynomials and their associated functions, (2.18) and (2.25), that (−1)npn(−t) ≥ 0 and (−1)n+1ǫn(−t) ≥ 0 for all real t ≥ 0. This places θn in the range applying to Case II. � We can also draw some conclusions concerning the zeros of the orthogonal poly- nomials which will be important subsequently. Corollary 3.6. Each zero xj,n(t) is a monotonically decreasing function of t and interpolates between the Laguerre zero with t = 0 and exponent a + 2 and the Laguerre zero with t = ∞ and exponent 2, (3.69) xj,n(0) > xj,n(t) > xj,n(∞) > 0, for all 1 ≤ j ≤ n and bounded t > 0. Corollary 3.7. The following bounds on the reciprocal sums over the zeros hold xj,n(t) ,(3.70) xj,n(t) + t .(3.71) Proof. From the two-sided bound on θn we can deduce (3.72) xj,n(t) + t xj,n(t)− θn xj,n(t) Employing this in the Bethe Ansatz (2.91) summed over j we arrive at the above bounds. � 4. Special Case a ∈ Z≥0 Our evaluation of the distribution function in terms of the fifth Painlevé sys- tem is with all three free parameters variable in some sense - one is fixed in this application at a positive integer, one is the index n ∈ Z and the remaining one is a ∈ C. Up to this point we have studied in some depth the recurrence relations with respect to n while a has been left arbitrary other than being restricted because of the existence considerations. From the point of view of the Painlevé theory it is quite natural that the transcendental objects become classical when a ∈ Z for either positive or negative subsets of the integers. In particular it is expected that the τ functions in the theory will have Hankel determinantal forms of classical function entries with a rank dependent on a. It is these cases which have been studied in the past [12],[11] using methods which transform the integral into the determinantal representations and then employ confluent Vandermonde identities. 30 PETER J. FORRESTER AND NICHOLAS S. WITTE Proposition 4.1. When a ∈ Z>0 we have the evaluation for the Hankel determi- (4.1) ∆n(t) = cn+1,n+1+a (n+ a)! det[L (j+k+1−a) n+a+1−j−k(−t)]j,k=1,...,a, and for a = 0 (4.2) ∆n(t) = cn+1,n+1 Proof. In [12] Eq. (3.18) (after correcting) states (4.3) ∆n(t) = cn+1,n+1+a (n+ a)! (−1)a(a−1)/2 det[Dj+k−2x L −(a−3) n+a−1 (x)|x=−t]j,k=1,...,a, where Dx := d/dx. Using the Laguerre polynomial identity (4.4) Dmx L n (x) = (−1)mL (α+m) n−m (x), m ∈ Z≥0, with the proviso L n (x) = 0 for n < 0, we arrive at (4.1). � As a consequence of the relations (3.28) and (3.27) the variables θn(t),Γn(t) will have a× a determinant forms, and in particular for a = 0 (4.5) θn(t) = −t, κn(t) = (n+ 1)t, Γn(t) = −n(n+ 2). The orthogonal polynomials also have determinantal forms of the following type. Proposition 4.2. When a ∈ Z>0 the orthogonal polynomials are given by (4.6) ∆n∆n+1pn(x; t) = (−1)n+a+⌊ ⌋a! . . . (n+ a)!1! . . . (n+ 1)! × (x+ t)−a det (k+1−a) n+a+1−k(x) k=1,...a+1 (j+k−a) n+a+2−j−k(−t) j=1,...,a k=1,...a+1 and for a = 0 (4.7) pn(x; t) = (−1)n (n+ 2)(n+ 1) L(2)n (x). If a > n+1 we note that L (a+1) n+1−a(−t) = 0. Consequently, under the same condition, the polynomial ratio is given by (4.8) Qn(x; t) = (k+1−a) n+a+1−k(x) k=1,...a+1 (j+k−a) n+a+2−j−k(−t) j=1,...,a k=1,...a+1 n+a+1−k k=1,...a+1 (j+k−a) n+a+2−j−k(−t) j=1,...,a k=1,...a+1 DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 31 Proof. Starting with the integral representation (2.18) we follow the procedures used in [12]. Taking one factor of the squared product of differences we write it like (4.9) 1≤j<k≤n (xk − xj) = l=0 cl det[L k−1(xj)]j,k=1,...,n, using the Vandermonde identity and where cn is the leading coefficient of L n (x) and α is a parameter to be fixed later. Of the remaining factors in the integrand we write (4.10) (xj + t) a(x− xj) 1≤j<k≤n (xk − xj) = (−1)a(n+1) l=0 l! l=0 cl (x+ t)−a × det k−1(xj) j=1,...,n k=1,...,N k−1(x) k=1,...,N Dj−1y L k−1(y)|y=−t j=1,...,a k=1,...,N where the confluent Vandermonde identity has been used and N = n+ a+ 1. Reassembling the integral with these two factors, then expanding the determi- nant in (4.9) we multiply each of n factors into the determinant of (4.10). Making use of the antisymmetry of the row ordering in the first n rows of the determinant we can perform the n integrals as long as we choose α = 2. Then (4.11) ∆n∆n+1pn(x; t) = (−1)a(n+1) l=0 l! l=0 cl l=0 cl (j + 2)! (x+ t)−a × det n+k−1(x) k=1,...,a+1 Dj−1y L n+k−1(y)|y=−t j=1,...,a k=1,...,a+1 Using the identities (4.12) L(α−1)n (x) = L n (x)− L n−1(x), along with elementary column operations, and then identity (4.4) we are lead to (4.6). The evaluation for the polynomial ratio (4.8) is a simple consequence of the first evaluation along with (4.13) L (−a+1+k) n+a+1−k (0) = n+ a+ 1− k 32 PETER J. FORRESTER AND NICHOLAS S. WITTE Proposition 4.3. When a ∈ Z≥0 the deformed Hankel determinant is given by (4.14) Dn(x, x) = cn+2,n+2+a(−1) (a+1)(a+2)(x+ t)−2a × det (j+k−1−a) n+a+3−j−k(−t) j=1,...,a k=1,...a+2 (j+k−1−a) n+a+3−j−k(x) j=1,2 k=1,...a+2 Proof. The quantity Dn(x, x) was essentially computed in [12] in Eq. (3.20), which can be recast as (4.15) Dn(x, x) = cn+2,n+2+a(−1) (a+1)(a+2)(x+ t)−2a × det Dj+k−2u L −(a−1) n+a+1 (u)|u=−t j=1,...,a k=1,...a+2 Dj+k−2u L −(a−1) n+a+1 (u)|u=x j=1,2 k=1,...a+2 Then (4.14) follows by application of the identity (4.4). � As a consequence the distribution of the first eigenvalue spacing is (4.16) An,a(y) = (−1) (a+1)(a+2)y2ey dt ta(t− y)−ae−(n+2)t × det (j+k−a−1) n+a+3−j−k(−t) j=1,...,a k=1,...a+2 (j+k−a−1) n+a+3−j−k(−y) j=1,2 k=1,...a+2 for a = 1, 2, 3, . . . and (4.17) An,0(y) = y2e−(n+1)y n+1(−y)L n−1(−y)− L(2)n (−y) for a = 0. 5. Hard Edge Scaling 5.1. General Case. We define new scaled spectral variables s, z by (5.1) t = , x = − in the triangular domain s > z > 0 and study the scaling of the finite distribu- tion (1.31) as the polynomial degree n → ∞. What is required here is not just the asymptotic scaling of the orthogonal polynomial coefficients but also of the polynomials themselves in the neighbourhood of an endpoint of the interval of or- thogonality. For the deformed Laguerre polynomials this would be a generalisation of the asymptotics of Hilb’s type for the Laguerre polynomials as found in Szegö (5.2) ex/2xµ/2L(µ)n (−x) = Γ(µ+ n+ 1) M)µ/2 Mx) + O(n DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 33 with M = 4n + 2µ + 2 as n → ∞ and Iµ(z) the standard modified Bessel func- tion. Despite a resurgence in activity around these questions, especially the use of Riemann-Hilbert techniques on these problems, there are no results available for our particular problem. A general review of the asymptotics of orthogonal polyno- mials can be found in [23], and an introduction to the Riemann-Hilbert approach to the asymptotics is the chapter in [22]. However to establish the nature of and the existence of limits for our variables we do not require such techniques. Lemma 5.1. Under the scaling of the (5.1) 4nθn(t) and κn(t) are bounded for all real positive t and n ≥ 1. Proof. From the result of Lemma 3.3 we see that (5.3) − s ≤ 4nθn(s/4n) ≤ 0, s/4n− 4nθn(s/4n) ≤ κn(s/4n) ≤ (n+ 1)s/4n, and the assertion follows. � Corollary 5.1. Under the above conditions (5.4) n+ κn(t)− t θn(t) t=s/4n = O(1), as n→ ∞. Proof. The above lemma states that (5.5) θn(s/4n) = O(1), as n→ ∞, and we find as a consequence that also (5.6) ε := n(n+ a+ 2)− s/4 + Γn(s/4n) = O(1), as n→ ∞. The discriminantD appearing in the workings of Proposition 3.3 can then be written (5.7) D2 := θ4n−2(2n+a+2−t)θ3n+[4(ε−t)+(a+2−t)2−4nt]θ2n+4t[ε+a+2−t]θn+4t2, and therefore (5.8) θn(t) t=s/4n = O(1), as n→ ∞. From the formula for κn in Proposition 3.3 we note that (5.9) n+ κn − t = −a+ 2 (θn + t)− and the result then follows. � Proposition 5.1. For bounded s ∈ R+ under the scaling (5.1) the variables θn(t) and Γn(t) converge to limits in the following manner 4nθn(t)|t=s/4n = µ(s),(5.10) n(n+ a+ 2) + Γn(t)|t=s/4n = ν(s).(5.11) 34 PETER J. FORRESTER AND NICHOLAS S. WITTE The variable κn(t) converges like (5.12) lim κn(t)|t=s/4n = − µ(s). Proof. Firstly we note that η := a2n(t)− n(n+ a+ 2) t=s/4n = O(1), as n→ ∞, as follows from (5.6). Starting with (3.8) we see that θn − θn−1 = 2θn + t+ (2n+ a+ 2)κn − [n2 + (n+ 1)(a+ 2)]t (2n+ a+ 2 + η)t+ 2(n+ a+ 2)θn κn − t ,(5.13) and thus nθn−(n−1)θn−1 = θn−1+ (2n+ a+ 2 + η)t+ 2(n+ a+ 2)θn κn − t Thus we have shown nθn(t)− (n− 1)θn−1(t)|t=s/4n = O(n −1), as n→ ∞. Now we can write the quantity of interest nθn(t)|t=s/4n − (n− 1)θn−1(t)|t=s/4(n−1) = [nθn(t)− (n− 1)θn−1(t)]|t=s/4n + (n− 1)θn−1(t)|t=s/4n − (n− 1)θn−1(t)|t=s/4(n−1) , so we require bounds on the difference of the last two terms on the right-hand side. Let t = s/4n and t> = s/4(n− 1). Because θn(t) is continuously differentiable and its derivative is given by (3.13) (n− 1) |θn−1(t)− θn−1(t>)| ≤ (n− 1)(t> − t) max u∈(t,t>) |θ̇n−1(u)|, u∈(t,t>) |θ̇n−1(u)|, u∈(t,t>) u−1|2κn−1(u) + θn−1(u)(2n+ a+ 1− u− θn−1(u))|, u∈(t,t>) u−1|θn−1(u)| n− 1 + κn−1(u)− u θn−1(u) + a+ 3− u− θn−1(u) + θn−1(u) ≤ t> max u∈(t,t>) n− 1 + κn−1(u)− u θn−1(u) + a+ 3 + u+ |θn−1(u)|+ |θn−1(u)| = O(n−1), as n→ ∞. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 35 Thus the limit shown in (5.10) exists. Turning our attention to (5.12) we can use (3.1) to compute that κn+1 − κn = −2κn − bnθn, = −2κn − (2n+ a+ 3− t− θn)θn, = −2θn κn − t − 2t− (a+ 3)θn + θn(θn + t), and therefore [κn+1(t)− κn(t)]|t=s/4n = O(n −1), as n→ ∞. Now in this case the quantity we require is κn+1(t)|t=s/4(n+1) − κn(t)|t=s/4n = κn+1(t)|t=s/4(n+1) − κn+1(t)|t=s/4n + [κn+1(t)− κn(t)]|t=s/4n , and therefore we need to bound the difference of first two terms on the right- hand side. Let us denote t< = s/4(n + 1). Again because κn(t) is continuously differentiable with derivative (3.22) we have |κn+1(t<)− κn+1(t)| ≤ (t− t<) max u∈(t<,t) |κ̇n+1(u)|, 4n(n+ 1) u∈(t<,t) |κ̇n+1(u)|, 4n(n+ 1) u∈(t<,t) u−1|κn+1(u)− a2n+1(u)(θn+1(u)− θn(u))|, u∈(t<,t) |κn+1(u)|+ (2n+ a+ 4 + |η(u)|)u + 2(n+ a+ 3)|θn+1(u)| n+ 1 + κn+1(u)− u θn+1(u) = O(n−1), as n→ ∞. In the last two steps we have used (5.13) and the subsequent estimates. Thus the limit in (5.12) follows. The fact that the limit of κn(t) under the hard edge scaling is related to the limit given in (5.10) follows from the relation (3.2). The limit (5.11) is a consequence of the limits in the primary variables. � In addition the following combinations of variables possess scaling limits which will subsequently be useful. 36 PETER J. FORRESTER AND NICHOLAS S. WITTE Corollary 5.2. For bounded s ∈ R+ the following limits as n→ ∞ exist 2κn(t) + θn(t)bn(t) θn(t) t=s/4n ,(5.14) κn(t)− t θn(t) t=s/4n C(s),(5.15) n+ a+ 2 + κn(t) + t θn−1(t) t=s/4n −C(s),(5.16) κn(t)− t θn(t) + n+ a+ 2 + κn(t) + t θn−1(t) t=s/4n ξ(s).(5.17) Proof. The limit in (5.15) is a consequence of (5.4) in a previous corollary. The scaling limit of (5.16) follows from that of (5.15) and the identity (3.11). The limit (5.17) can be derived from (3.12) and as a result one can deduce that κn − t + n+ a+ 2 + κn + t is of order O(n−1). � We note other relations amongst the scaling limit functions, namely (5.18) 2µC(s) = −[(a+ 2)µ+ 2s] [(a+ 2)µ+ 2s]2 + 4µ(µ+ s)ν − µ(µ+ s)2 (5.19) ξ(s) = − sC(C + a) (5.20) 2C + a+ 3 = s µ̇− 2 Proposition 5.2. The scaled variables µ(s), ν(s) are characterised by solutions to the PIII′ system with parameters v1 = a+ 2, v2 = a− 2. In particular (5.21) ν(s) = −σIII′(s) + s− a− 2, where σIII′(s) satisfies the Jimbo-Miwa-Okamoto σ-form for PIII′ with above param- eters. The boundary conditions to uniquely specify the solution ν(s) are (5.22) ν(s) = 8(a+ 3)(a+ 2)2(a+ 1) s2 +O(s3) 4a+3(a+ 2)(a+ 1)Γ(a+ 3)Γ(a+ 4) sa+3 (1 + O(s)) + O(s2a+6), assuming a /∈ Z≥0 and |arg(s)| < π. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 37 Proof. Introducing the scaling ansatzes (5.10) and (5.11) formally into the differen- tial equation (3.34) under the scaling (5.1) we find that the highest order nontrivial relation (5.23) µ(s) + s = 4sν̇(s), at order n−1. Proceeding in the same manner with the differential equation (3.35) we find the highest order relation is (5.24) sµ̇(s) = µ+ [(a+ 2)µ+ 2s]2 + 4µ(µ+ s)ν − µ(µ+ s)2 which occurs at order n−1 as well. Eliminating µ(s) using (5.23) we find that (5.24) yields (5.25) s2(ν̈)2 − (a+ 2)2(ν̇)2 + ν̇(4ν̇ − 1)(sν̇ − ν) + a(a+ 2)ν̇ − a2 = 0, which is almost the Jimbo-Miwa-Okamoto σ-form for PIII′ [28]. The boundary conditions follow from the application of the scaling limit (5.11) to the expansion about t = 0, Equation (3.29). � In addition we find the scaling behaviour of the Hankel determinants and poly- nomial evaluations to be given by the following propositions. Proposition 5.3. As n→ ∞ under the hard edge scaling the Hankel determinants scale as (5.26) ∆n(t)|t=s/4n ∼ 1! . . . (n− 1)!Γ(a+3) . . .Γ(n+ a+2) exp and the monic polynomials evaluated at x = 0 scale as (5.27) πn(0; t)|t=s/4n ∼ (−1)n(a+ 3)n exp Proof. The relation (5.26) arises from integrating (3.27) with respect to t and then employing the scaling form for the integrand as given by (5.11). The second relation is derived by integrating (3.39) and using (5.15) for the scaling of the resulting integrand. The factorial and Pochhammer prefactors arise from the normalisations at t = 0. � We find that the polynomial ratios have well defined scaling behaviour rather than the polynomial themselves. Proposition 5.4. The polynomial ratios Qn(x; t), Rn(x; t) scale as Qn(x; t)|x=−z/4n,t=s/4n = q(z; s),(5.28) nRn(x; t)|x=−z/4n,t=s/4n = p(z; s),(5.29) where q(z; s), p(z; s) are entire functions of z. 38 PETER J. FORRESTER AND NICHOLAS S. WITTE Proof. We start with the product form of the scaled polynomial (5.30) Qn(− 4nxj,n(s/4n) and seek bounds for the logarithm of the ratio of the n-th scaled polynomial to the (n− 1)st. When 0 < s, z < ∞ we can write the general bound as the sum of four contributions (5.31) Qn(−z/4n; s/4n) Qn−1(−z/4(n− 1); s/4(n− 1)) 4nxn,n(t) 4nxj,n(t) − log 4(n− 1)xj,n(t) 4(n− 1)xj,n(t) − log 4(n− 1)xj,n(t>) 4(n− 1)xj,n(t>) − log 4(n− 1)xj,n−1(t>) where t = s/4n and t> = s/4(n− 1). Using the inequality < A−B for A > B > 0 we can find a simpler bound (5.32) LHS of (5.31) ≤ z 4nxn,n(t) 4nxj,n(t) 4(n− 1)xj,n(t) 4(n− 1)xj,n(t) 4(n− 1)xj,n(t>) 4(n− 1)xj,n(t>) 4(n− 1)xj,n−1(t>) Considering the first sum of (5.32) we see that this is 4nxj,n(t) 4(n− 1)xj,n(t) 4n(n− 1) xj,n(t) 4n(n− 1) xj,n(t) 4n(n− 1) = zO(n−1), as n→ ∞, DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 39 for all bounded s. The second term of (5.32) is 4(n− 1)xj,n(t) 4(n− 1)xj,n(t>) 4(n− 1) xj,n(t) xj,n(t>) The summand appearing here can be bounded in the following way xj,n(t) xj,n(t>) |xj,n(t>)− xj,n(t)| xj,n(t>)xj,n(t) xj,n(t>)xj,n(t) (t> − t) max u∈(t,t>) |ẋj,n(u)|.(5.33) Now from (2.96) we note that |ẋj,n(u)| = u−1 θn(u) + u −θn(u) + xj,n(u) xj,n(u), and so furnishes a bound on (5.33) xj,n(t) xj,n(t>) xj,n(t>)xj,n(t) 4n(n− 1) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) xj,n(u), xj,n(t>)xj,n(t) xj,n(t) max u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) xj,n(t>) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) This means that xj,n(t) xj,n(t>) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) xj,n(t>) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) xj,n(t>) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) xj,n(t>) 2(n− 1) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) 2(n− 1) u∈(t,t>) θn(u) + u −θn(u) + xj,n(u) 2(n− 1) maxu∈(t,t>) κn(u)− u θn(u) where we have used (2.94) in the last step. The total contribution of the second term is therefore bounded by 8(n− 1)2 O(1) = zO(n−1). 40 PETER J. FORRESTER AND NICHOLAS S. WITTE The third sum in (5.32) is 4(n− 1) xj,n(t>) xj,n−1(t>) 4(n− 1) xj,n(t>) xj,n−1(t>) 4(n− 1) xj,n(t>) xj,n−1(t>) From the identity (3.48) and the scaling of the variables involved as given in (5.17) we conclude that the contribution of this sum is bounded by 4(n− 1) O(1) = zO(n−1). Finally we note that the isolated term in (5.32) is of order zO(n−2) as a leading estimate of the largest zero xn,n is of order O(n). This establishes that Qn under the hard edge scaling of the independent variables converges to a limit as n → ∞, for all real, positive and bounded z, s. � The spectral and deformation derivatives of the Qn, Rn system scale to the corresponding derivatives of the q, p system as in the following result. Proposition 5.5. Specify scaled quantities as in Proposition 5.1, Corollary 5.2 and Proposition 5.2. The spectral derivatives of the q, p system are (s− z)z∂zq = −zCq − (µ+ z)p,(5.34) (s− z)z∂zp = −z (z − s) q + [−2s+ z(C + a+ 2)]p,(5.35) and their deformation derivatives are (s− z)s∂sq = zCq + (µ+ s)p,(5.36) (s− z)s∂sp = zξq − [s(2C + a)− zC]p,(5.37) The boundary conditions satisfied by the solutions q(z; s) and p(z; s) of the above system on the domain s ≥ z ≥ 0 along z = 0 are q(0; s) = 1,(5.38) p(0; s) = 0,(5.39) for all s > 0. Proof. The first spectral derivative (5.34) follows from the scaling of (3.53) and employing (5.10), (5.12) and (5.15). The second member (5.35) is derived from the scaling of (3.54) and using (5.15) and (5.17). The first deformation derivative (5.36) follows from the scaling of (3.55) and utilising (5.10), (5.12) and (5.15). The DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 41 second deformation derivative (5.37) arises from the scaling of (3.56), employing (5.16) and (5.17) and noting that (κn + t) θn−1 + t − (κn − t) θn + t t=s/4n −s(2C + a) The boundary conditions (5.38) and (5.39) follow from the definitions (3.41) and (3.52) respectively and the scalings in Proposition 5.4. � Remark 5.1. Both the spectral derivative (3.43) and the deformation derivative (3.45) scale to (5.40) (µ+ s)z∂zq + (µ+ z)s∂sq + zC(s)q = 0, and this mixed derivative equation can be easily found from Proposition 5.5 by eliminating the variable p between (5.34) and (5.36), i.e. (µ+ s) times (5.34) plus (µ+ z) times (5.36). The three-term recurrence relation (3.42) scales to (5.41) z2∂2zq + 2sz∂z∂sq + s 2∂2sq + s µ̇− 2 [z∂zq + s∂sq]− zq = 0. This can also be recovered from the equations of Proposition 5.5. If we eliminate q between (5.34) and (5.36) by adding them we find that (5.42) p = z∂zq + s∂sq. Employing this we find that (5.41) is equivalent to (5.43) z∂zp+ s∂sp+ (−1 + s µ̇− 2 )p− 1 zq = 0, which can be found by adding (5.35) and (5.37) and noting the relation (5.20). Remark 5.2. In addition to the boundary conditions at z = 0 given by (5.38) and (5.39) there are also relations along s = z (5.44) q(s; s) = − C q(s; s), and further (5.45) p(s; s) q(s; s) C + a = − sC However these are a consequence of the spectral and deformation derivatives and so do not constitute independent boundary conditions. Remark 5.3. The compatibility of the two sets of derivatives, (5.34) and (5.35) on the one hand, and (5.36) and (5.37) on the other hand, affords a check on our results. We find that compatibility of (5.34) and (5.36) leads us to conclude that (5.46) ξ = sĊ + (µ+ s), 42 PETER J. FORRESTER AND NICHOLAS S. WITTE and we also recover (5.20). Similar considerations applied to (5.35) and (5.37) imply (5.47) sξ̇ = −(2C + a+ 2)ξ + 1 s(2C + a), and we get (5.46) again. Using (5.19) to eliminate µ we arrive at a coupled pair of first order ordinary differential equations sĊ = ξ + C(C + a),(5.48) sξ̇ = −(2C + a+ 2)ξ + 1 s(2C + a).(5.49) Using the latter equation to eliminate C we obtain a second order ordinary differ- ential equation for ξ, which by means of the transformation (5.50) ξ(s) = sy(s) y(s)− 1 is transformed into the standard equation for the fifth Painlevé transcendent. This is a degenerate case of PV which reduces to the third Painlevé transcendent be- cause the parameters are αV = 9/2, βV = −a2/2, γV = 1/2, δV = 0. Making an independent variable transformation s 7→ 2s so that γV = 1 we determine the PIII parameters to be αIII = 2(2 − a), βIII = 2(a + 3), γIII = 1, δIII = −1 which is consistent with those in Proposition 5.2. The matrix form of the spectral derivatives (5.34,5.35) and deformation deriva- tives (5.36,5.37) yield the Lax pairs ∂zΨ = z − s Ψ,(5.51) ∂sΨ = B − As z − s Ψ,(5.52) in the matrix variable (5.53) Ψ(z; s) = q(z; s) p(z; s) The system has two regular singularities at z = 0, s and an irregular one at z = ∞ with a Poincaré index of 1. This system is essentially equivalent to the isomon- odromic system of the fifth Painlevé equation but is the degenerate case. The DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 43 residue matrices are 0 −µ(s)s ,(5.54) µ(s) + s ξ(s) −C(s)− a ,(5.55) ,(5.56) −C(s) 0 −ξ(s) −C(s) .(5.57) Local convergent expansions about the regular singularities take the form for z = 0 q(z; s) = r0m(s)z χ0+m,(5.58) p(z; s) = u0m(s)z χ0+m,(5.59) for |z| < s and the initial relations amongst the coefficients are found to be (5.60) sχ0r 0 = −µu00, sχ0u00 = −2su00. This implies that χ0 = −2 or χ0 = 0 and u00 = 0. The latter case applies here as both q, p are analytic at z = 0, and in addition we also require p = 0 on z = 0. In addition r00 = 1. The recurrence relations for general m are s(m+ 2)u0m = (C + a+m+ 1)u m−1 + r0m−1 − r0m−2,(5.61) smr0m = −µu0m + (m− 1− C)r0m−1 − u0m−1.(5.62) For z = s we have the convergent expansion q(z; s) = rsm(s)(s− z)χs+m,(5.63) p(z; s) = usm(s)(s− z)χs+m,(5.64) for |z − s| < s and where in this case the initial relations are (5.65) s(−χs + C)rs0 = −(µ+ s)us0, (C + a+ χs)us0 = ξrs0. Combining these we get a relation which is identical to the second equality in (5.45) only if χs = 0,−a. The former case is the one we must choose as q, p are well-defined 44 PETER J. FORRESTER AND NICHOLAS S. WITTE and finite on z = s. For general m the recurrence relations are s(C −m)rsm + (µ+ s)usm = (C −m+ 1)rsm−1 + usm−1, (5.66) −ξrsm + s(C + a+m)usm = − rsm−1 + (C + a+m+ 1)u m−1 + rsm−2. (5.67) This system has a unique solution only if s2m(a + m) 6= 0 for s > 0 and m ≥ 1 which in turn means that a 6= −N. We also note that the two sets of coefficients are related by (5.68) rsm(s) = (−1)m snr0m+n(s), and an identical relation for usm(s). Proposition 5.6. The determinant Dn(x, x) scales as (5.69) Dn(x, x)|x=−z/4n,t=s/4n ∼ −4∆nπn(0)πn+1(0) [q∂zp− p∂zq] . Proof. We can employ the polynomials Qn, Rn in the evaluation (2.45) and find Dn(x, x) = ∆nπn(0)πn+1(0)[Qn+1R n+1 − Rn+1Q′n+1]. Applying the scaling of Proposition 5.4 to this expression we arrive at (5.69). � Proposition 5.7. The distribution An,a(y) scales to (5.70) An,a(y)|y=z/4n ∼ Aa(z). Proof. This is the only possible scaling consistent with the scaling of the indepen- dent variable to the hard edge, y = z/4n. � Proposition 5.8. The distribution of the first eigenvalue spacing at the hard edge is given by (5.71) Aa(z) = 42a+3Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3) ds sa(s− z)a exp [ν(v) + 2C(v)] [q∂zp− p∂zq] . Proof. We apply the scaling (5.70) to Eq. (1.31) and utilise the previous relation for the scaling of the integrand (5.69). For the first three factors on the right-hand side of (5.69) we can use the scaling results of (5.26) and (5.27), yielding the above integral representation. � We note that the factor of the integrand of (5.71) can be written as (5.72) q∂zp− p∂zq = q2 − 2 p2 − 1 (ξq − Cp)ξq − (C + a)p DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 45 The corresponding distribution A±(z) defined in (1.35) and (1.36) for the special cases a = ±1/2 is given by (5.73) A±(z) = 42a+2Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3) ds s2a+1(s− z)2a+1(2s− z)2 exp [ν(v) + 2C(v)] × [q∂zp− p∂zq]| s7→s2 z 7→z(2s−z) 5.2. Special Case a ∈ Z≥0. Proposition 5.9. In the special case a ∈ Z≥0 we have (5.74) ∆n(t)|t=s/4n ∼ cn+1,n+1+a (n+ a)! (−1)⌊a/2⌋(2n)as−a det Ij+2−k( j,k=1,...,a (5.75) ν(s) = s log s−a det Ij+2−k( j,k=1,...,a whilst for a = 0 we find ν(s) = 0. Proof. The first relation (5.74) follows from an application of the Hilb type asymp- totic formula (5.2) to (4.1) and the second follows by using this result in (3.27). � Another consequence of the Hilb formula is the scaling of the orthogonal poly- nomial ratio as given by (4.8). Proposition 5.10. The scaled orthogonal polynomial ratio q(z; s) is (5.76) q(z; s) = z−3/2 (z/s)k/2I3−k( k=1,...a+1 [Ij+2−k( s)] j=1,...,a k=1,...a+1 s, 1/2s, 1/s3/2, 0, . . . , 0 [Ij+2−k( s)] j=1,...,a k=1,...a+1 , a ≥ 1 and for a = 0 is (5.77) q(z; s) = Finally the eigenvalue spacing distribution (4.16) takes the following form in the scaling limit. 46 PETER J. FORRESTER AND NICHOLAS S. WITTE Proposition 5.11. The distribution of the first eigenvalue spacing at the hard edge for a ∈ Z>0 is (5.78) Aa(z) = 2 −4z1/2 e−s/4 × det [Ij+2−k( s)] j=1,...,a k=1,...a+2 (s/z)(2−k)/2Ij+2−k( j=1,2 k=1,...a+2 for a ≥ 1 and for a = 0 is (5.79) A0(z) = e−z/4 I22 ( z)− I1( z)I3( Proof. We apply the Hilb asymptotic formula (5.2) to (4.16). As 1 (a+1)(a+2)+ 1 + ⌊a/2⌋ is always even for a ∈ Z we have (5.78). � 6. Analytical Studies at the Hard Edge In this section of our study we intend to develop the analytical and non-formal theory of the solutions to the defining ordinary and partial differential equations described in the previous sections. This is because we wish to compute precision numerical data characterising the distribution function of the first eigenvalue spac- ing at the hard edge Aa(z) for arbitrary parameter a. For this purpose it is not sufficient to employ a single local expansion of the σ-function, about s = 0 say, because it has a finite convergence domain and one cannot use this to evaluate the s-integrals on the interval [0,∞). For this reason we construct a patchwork of over- lapping local expansions including Taylor series expansions about regular points s0 for positive and real values. A similar approach was undertaken by Prähofer and Spohn in their study [29] of the exact scaling functions for one-dimensional stationary KPZ growth. 6.1. The σ-function expansion about s = 0. The terms given in the expansion of the Painlevé III′ σ-function (5.22) are the minimum required to specify the full non-analytic Puiseux-type expansion of the particular solution for ν(s) about s = 0 in the sector −π ≤ arg(s) < π. This is the primary data specifying our particular solution and we need to use this in various ways in order to compute the distribution Aa(z). Proposition 6.1 ([18],[21]). The Painlevé III′ σ-function ν(s) has a Puiseux-type expansion about the fixed regular singular point s = 0 of the form (6.1) ν(s) = ck,js j+ka, a ∈ C, 0 ≤ Re(a) < 1, with |arg(s)| < π and is convergent in a finite domain s ∈ {z ∈ C : |z| < R, |za| < DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 47 The coefficients ck,j are determined by recurrences which follow from the sub- stitution of expansion (6.1) into the relation (5.25). We will assume that a is not rational for simplicity. Considering terms in the resulting equation with sp and p ∈ Z≥0 then the p = 0 case implies that if c0,0 = 0 then this choice fixes (6.2) c0,1 = 4(a+ 2) The p = 1 is automatically zero but for the p = 2 case one has the two options (6.3) c0,2 = − 8(a+ 3)(a+ 2)2(a+ 1) , or 0. The former case applies here by comparison with (5.22). For p ≥ 3 the recurrence (6.4) − a(p+ a+ 1)(p− a− 3) (a+ 3)(a+ 2)2(a+ 1) c0,p − j(p− j)c0,jc0,p+1−j + 8c0,2 (j − 1)(p− j)c0,j−1c0,p+1−j j(p+ 2− j)[(j − 1)(p+ 1− j)− (a+ 2)2]c0,jc0,p+2−j p+2−j jm(p+ 1− j −m)c0,jc0,mc0,p+2−j−m = 0, which allows for c0,p to be recursively found, valid for a /∈ Z. Terms with s−2+qa for q = 2k ≥ 2 imply that ck,0 = 0 given that c0,0 = 0 and a 6= 3/(k − 1),−1/(k + 1). Consequently all terms with s−1+qa vanish also. The generic recurrence relation following from examining the sp+qa term is (6.5) a(a+ 2)(p+ 1 + qa)cq,p+1 = (j + ka)[p− j + (q − k)a]ck,jcq−k,p+1−j (j+ka)[p+2−j+(q−k)a][(j−1+ka)[p+1−j+(q−k)a]−(a+2)2]ck,jcq−k,p+2−j p+2−j (j+ka)(m+la)[p+1−j−m+(q−k−l)a]ck,jcl,mcq−k−l,p+2−j−m, for q, p ≥ 0. The convention is taken that sums with upper limits less than their lower limits are zero, or equivalently coefficients with negative indices are zero. Contrary to appearances the highest coefficient in (6.5) will turn out to be cq,p as cq,p+2 occurs with c0,0 as a factor and cq,p+1 has a factor of (6.6) a(a+ 2) + c0,0 − 2(a+ 2)2c0,1 − 8c0,0c0,1, 48 PETER J. FORRESTER AND NICHOLAS S. WITTE which is zero as a consequence of c0,0 = 0 and (6.2). We find that the coefficient cq,p occurs as a linear term and has a factor of (6.7) (p−1+qa)c0,1(4c0,1−1)+4(p+qa)(p−1+qa−(a+2)2)c0,2−16(p+qa)c0,0c0,2. With the above evaluations (6.2) and (6.3) this is (6.8) − a(p+ 1 + (q + 1)a)(p− 3 + (q − 1)a) 2(a+ 3)(a+ 2)2(a+ 1) This vanishes when q = 1, p = 3 and we find that c1,3 is undetermined and therefore is a free parameter. With these choices a triangular subset of the coefficients vanish (6.9) ck,j = 0, j = 0, . . . , 3k − 1 for k = 1, . . . ,∞, so that the initial non-zero term in the j−sum is ck,3k. All other terms are fixed by c1,3 and a. For the variable µ(s) we can deduce the following Puiseux-type expansion from the above work (6.10) µ(s) ∼ (a+ 3)(a+ 2)2(a+ 1) 4a+2(a+ 2)(a+ 1)Γ2(a+ 3) sa+3. In regard to the quantity 2C(s) we require as much detail about this as for ν(s). This variable is a σ-function for an identical problem, where the fixed exponent 2 in the weight (1.32) is replaced by 3. If we define the expansion coefficients for this object (6.11) 2C(s) = ak,js j+ka, a ∈ C, 0 ≤ Re(a) < 1, then the defining recurrences for these using (5.20) are (6.12) a0,0 = 0, a0,1 = 4(a+ 2)(a+ 1)c0,2, and for j ≥ 2 (6.13) a0,j = 2(a+ 2) − (j + 1)(j − 2− a)c0,j+1 + la0,j+1−lc0,l For the case k ≥ 1 we have ak,j = 0 for j = 0, .., 3k− 1 and the remaining non-zero terms are given by (6.14) a1,j = 2(a+ 2) − (j + 1 + a)(j − 2)c1,j+1 (l + a)a0,j+1−lc1,l + la1,j+1−lc0,l DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 49 for k = 1 and the general case k ≥ 2 by (6.15) ak,j = 2(a+ 2) − (j + 1 + ka)(j − 2 + (k − 1)a)ck,j+1 (l + ka)a0,j+1−lck,l + lak,j+1−lc0,l (l +ma)ak−m,j+1−lcm,l The first few terms are (6.16) 2C(s) ∼ 2(a+ 3)(a+ 2) s− a(a 2 − 5a− 18) 8(a+ 4)(a+ 3)2(a+ 2)2(a+ 1) 4a+2(a+ 2)(a+ 1)Γ(a+ 3)Γ(a+ 4) sa+3. (6.17) ξ(s) ∼ 4(a+ 3) s− 3a 8(a+ 4)(a+ 3)2(a+ 2) 4a+7(a+ 3)(a+ 2)(a+ 1)Γ2(a+ 4) sa+4. 6.2. The q, p expansion about s = 0. In this subsection we seek local expansions about s = 0 for the coefficients functions r0m(s), u m(s), r m(s), u m(s) appearing in (5.58), (5.59), (5.63) and (5.64). In parallel with the transcendent quantities these will have Puisuex-type expansions of the form r0m(s) = k,j≥0 r0m,k,js j+ka,(6.18) rsm(s) = k,j≥0 rsm,k,js j+ka,(6.19) with analogous expansions for the remaining two coefficients. This is immediately clear because the recurrence relations for these imply that they are polynomial functions of the variables µ(s), C(s), ξ(s). These recurrence relations imply the following ones for the z = 0 coefficients r0m,k,j , u m,k,j (6.20) (a+m+ 2)u0m,k,0 = r0m−1,k,0, mr m,k,0 = u m,k,0, for m ≥ 1, k ≥ 0. For the general case j ≥ 1, k ≥ 0 we have (6.21) (m+ 2)u0m,k,j−1 = (a+m+ 1)u m−1,k,j + ak−q,j−pu m−1,q,p r0m−1,k,j−1− [j−p+(k−q)a] ck−q,j−p + ak−q,j−p r0m−1,q,p− r0m−2,k,j , 50 PETER J. FORRESTER AND NICHOLAS S. WITTE (6.22) mr0m,k,j−1 = (m− 1)r0m−1,k,j − ak−q,j−pr m−1,q,p + u0m,k,j−1 − 4 [j − p+ (k − q)a]ck−q,j−pu0m,q,p − u0m−1,k,j, These equations can be solved for successive values of m starting with the m = 0 values for all k, j (6.23) u00,k,j = 0, r 0,k,j = 0, except for k = j = 0 where r 0,0,0 = 1, which follow from r00(s) = 1, u 0(s) = 0. The next few coefficients can be read off (6.24) r01(s) ∼ 4(a+ 3) 16(a+ 4)(a+ 3)2(a+ 2) 128.4a(a+ 3)(a+ 2)(a+ 1)Γ(a+ 5)Γ(a+ 4) sa+3, (6.25) u01(s) ∼ 4(a+ 3) 8(a+ 4)(a+ 3)2(a+ 2) 128.4a(a+ 3)(a+ 2)(a+ 1)Γ2(a+ 4) sa+3, (6.26) r02(s) ∼ 32(a+ 4)(a+ 3) 64(a+ 5)(a+ 4)(a+ 3)2(a+ 2) 512.4a(a+ 3)(a+ 2)(a+ 1)Γ(a+ 6)Γ(a+ 4) sa+3, (6.27) u02(s) ∼ 16(a+ 4)(a+ 3) 64(a+ 5)(a+ 4)(a+ 3)2(a+ 2) 512.4a(a+ 3)(a+ 2)(a+ 1)Γ(a+ 5)Γ(a+ 4) sa+3. The analogous results for rsm,k,j , u m,k,j are (6.28) (a+m+ 2)usm,k,0 = − rsm−1,k,0, mr m,k,0 = u m,k,0, DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 51 for m ≥ 1, k ≥ 0. Again for the general case j ≥ 1, k ≥ 0 we have (6.29) (a+m)usm,k,j−1 + ak−q,j−1−pu m,q,p = (a+m+ 1)usm−1,k,j + ak−q,j−pu m−1,q,p [j − 1− p+ (k − q)a] ck−q,j−1−p + ak−q,j−1−p rsm,q,p rsm−1,k,j−1 − [j − p+ (k − q)a] ck−q,j−p + ak−q,j−p rsm−1,q,p r0m−2,k,j , (6.30) mrsm,k,j−1 − ak−q,j−1−pr m,q,p = (m− 1)rsm−1,k,j − ak−q,j−pr m−1,q,p − usm−1,k,j + 4 [j − p+ (k − q)a]ck−q,j−pusm,q,p, Again these can be solved for successive values of m starting with the initial values given by the relations (6.31) rs0,k,j = r0n,k,j−n, along with an identical formula for us0,k,j. 6.3. The σ-function expansion about s = ∞. The nature of the expansion of ν(s) about s = ∞ is rather different because this fixed singular point is irregular in the case of the third Painlevé transcendent. Proposition 6.2. The formal asymptotic expansion of ν(s) about s = ∞ has the (6.32) ν(s) ∼ −j/2, where d−1 = ± 12a. Proof. We start with the general ansatz of (6.33) ν(s) = dskα +O(s(k−1)α), 52 PETER J. FORRESTER AND NICHOLAS S. WITTE where k ∈ N and α ∈ C with 0 < Re(α) < 1. Using (5.25) we find that the only terms which can balance the O(1) term are the O(s2kα−1) and O(s3kα−2) terms. If kα < 1 then the former choice applies and we find kα = 1/2. If we assume kα > 1 then the latter case must be chosen but this leads to kα = 2/3 in contradiction to the hypothesis. With the correct choice of kα we find the above relation for leading coefficient, d−1. Considering the sub-leading terms we note that it is only possible for the terms of O(s−α) to balance those of O(s−1/2) so that in fact α = 1/2 and k = 1. This is entirely consistent with the expansion for the Painlevé III transcendent qIII(t) about t = ∞ as found in [15]. � The choice of the sign of the leading coefficient is positive in our application. Consequently we find the first few terms of the asymptotic expansions of the σ- function (6.34) ν(s) = as1/2 − 1 a(a+ 4) + as−1/2 + a2s−1 a(16a2 − 7)s−3/2 +O(s−2), and the auxiliary variables (6.35) µ(s) + s = as1/2 − 15 as−1/2 − 15 a2s−1 a(16a2 − 7)s−3/2 +O(s−2), (6.36) 2C(s) + a = as−1/2 + a2s−1 + a(8a2 − 21)s−3/2 +O(s−2), (6.37) ξ(s) = as1/2 − 35 as−1/2 − 35 a2s−1 +O(s−3/2). The large s-regime also implies a simplification of spectral derivative (5.51) which becomes (6.38) ∂zΨ = 2(z − s) 2(z − s) − I+O(s−1/2) Using the substitution Ψ 7→ exp(−a ln(s−z)I/2)Ψ this decouples and can be solved in terms of the modified Bessel functions. An application of the boundary condition (5.38) implies that q(z; s) ∼ z),(6.39) p(z; s) ∼ .(6.40) DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 53 6.4. The σ-function expansion about a regular point. In this subsection we seek Taylor series expansions for the sigma function and derived variables about regular points s0, taken to be positive and real without any loss of generality. Let us write (6.41) ν(s) = dj(s− s0)j , and using (5.25) we find the following recurrence relations for the coefficients dj (6.42) 4s20d a(a+2)d1+ d0d1 − [s0+(a+2)2]d21 +4(s0d1 − d0)d21 = 0, and for n ≥ 1, d2 6= 0 (6.43) 4s20(n+ 2)(n+ 1)d2dn+2 = − a(a+ 2)(n+ 1)dn+1 + [s0 + (a+ 2) (j + 1)(n− j + 1)dj+1dn−j+1 (j + 1)(n− j − 1)dj+1dn−j (j + 1)j(n− j + 1)(n− j)dj+1dn−j+1 − 2s0 (j + 2)(j + 1)(n− j + 1)(n− j)dj+2dn−j+1 − s20 (j + 2)(j + 1)(n− j + 2)(n− j + 1)dj+2dn−j+2 (i+ 1)(j + 1)(n− i− j − 1)di+1dj+1dn−i−j − 4s0 (i+ 1)(j + 1)(n− i− j + 1)di+1dj+1dn−i−j+1. These recurrences are solved subject to the initial values of ck,js 0 ,(6.44) (j + ka)ck,js j−1+ka 0 ,(6.45) 54 PETER J. FORRESTER AND NICHOLAS S. WITTE which in turn can be found from the solutions to the recurrences (6.4) and (6.5). In addition we define µ(s) = fj(s− s0)j ,(6.46) C(s) = gj(s− s0)j ,(6.47) ξ(s) = hj(s− s0)j .(6.48) The coefficients appearing here are computed using the recurrences (6.49) f0 = s0(4d1 − 1), f1 = 4d1 − 1 + 8s0d2, fj = 4[jdj + s0(j + 1)dj+1], j ≥ 2, and subject to f0 6= 0 2f0g0 = −2s0 − (a+ 3)f0 + s0f1,(6.50) 2f0g1 = −2− (a+ 2)f1 + 2s0f2 − 2f1g0,(6.51) 2f0gj = (j − a− 3)fj + (j + 1)s0fj+1 − 2 fj−kgk, j ≥ 2,(6.52) and provided s0 + f0 6= 0 (s0 + f0)h0 = −s0g0(a+ g0),(6.53) (s0 + f0)h1 = −as0g1 − 2s0g0g1 − (a+ g0)g0 − (1 + f1)h0,(6.54) (s0 + f0)hj = −as0gj − agj−1 − s0 gj−kgk − gj−kgk−1(6.55) − (1 + f1)hj−1 − fj−khk, j ≥ 2. 6.5. The q, p expansion about a regular point. We will seek a Taylor series ap- proximation for the scaled polynomial and associated function q(z; s), p(z; s) about the regular point (z0, s0) with 0 < z0 < s0. Let us write q(z; s) = j,k≥0 rj,k(z − z0)j(s− s0)k,(6.56) p(z; s) = j,k≥0 uj,k(z − z0)j(s− s0)k,(6.57) Using the first spectral derivative (5.34) we obtain the recurrence relation (6.58) z0(s0−z0)(j+1)rj+1,k+(s0−2z0)jrj,k−(j−1)rj−1,k+z0(j+1)rj+1,k−1+jrj,k−1 = −z0uj,k − uj−1,k − [fk−luj,l + gk−lrj−1,l + z0gk−lrj,l] , DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 55 for j, k ≥ 1. When k = 0 and j ≥ 1 we have the specialisation (6.59) z0(s0 − z0)(j + 1)rj+1,0 + [(s0 − 2z0)j + z0g0]rj,0 + [g0 − j + 1]rj−1,0 = −(f0 + z0)uj,0 − uj−1,0. The second spectral derivative (5.35) yields the recurrence relation (6.60) z0(s0 − z0)(j + 1)uj+1,k + [(s0 − 2z0)j + 2s0 − (a+ 2)z0]uj,k − (a+ j + 1)uj−1,k + z0(j + 1)uj+1,k−1 + (j + 2)uj,k−1 z0(s0 − z0)rj,k + (s0 − 2z0)rj−1,k − rj−2,k + z0rj,k−1 + rj−1,k−1 hk−l(rj−1,l + z0rj,l) + gk−l(uj−1,l + z0uj,l), for j, k ≥ 1. When k = 0 and j ≥ 1 we have the specialisation (6.61) z0(s0−z0)(j+1)uj+1,0+[(s0−2z0)j+2s0−(a+2)z0−z0g0]uj,0−[a+1+g0+j]uj−1,0 z0(s0 − z0)− h0z0]rj,0 + [ (s0 − 2z0)− h0]rj−1,0 − rj−2,0. Using the first deformation derivative (5.36) we obtain the recurrence relation (6.62) s0(s0−z0)(k+1)rj,k+1−s0(k+1)rj−1,k+1+(2s0−z0)krj,k−krj−1,k+(k−1)rj,k−1 = s0uj,k + uj,k−1 + [fk−luj,l + gk−lrj−1,l + z0gk−lrj,l] , for j, k ≥ 1. When j = 0 and k ≥ 1 we have the specialisation (6.63) s0(s0 − z0)(k + 1)r0,k+1 + (2s0 − z0)kr0,k + (k − 1)r0,k−1 = s0u0,k + u0,k−1 + [fk−lu0,l + z0gk−lr0,l]. The second deformation derivative (5.37) in turn gives us the recurrence relation (6.64) s0(s0 − z0)(k + 1)uj,k+1 − s0(k + 1)uj−1,k+1 + [(2s0 − z0)k + as0]uj,k − kuj−1,k + (a+ k − 1)uj,k−1 hk−l(rj−1,l+z0rj,l)−(2s0−z0) gk−luj,l+ gk−luj−1,l−2 gk−1−luj,l, 56 PETER J. FORRESTER AND NICHOLAS S. WITTE for j, k ≥ 1. When j = 0 and k ≥ 1 we have the specialisation (6.65) s0(s0 − z0)(k + 1)u0,k+1 + [(2s0 − z0)k + as0]u0,k + [a+ k − 1]u0,k−1 hk−lr0,l − (2s0 − z0) gk−lu0,l − 2 gk−1−lu0,l. These recurrences can be solved in the following way. First (6.59) and (6.61) are solved for rj,0, uj,0 for j ≥ 1 in terms of r0,0, u0,0. Then these solutions can be substituted into the boundary conditions rj,0(−z0)j = 1,(6.66) uj,0(−z0)j = 0,(6.67) and this allows for r0,0, u0,0 to be found. Next the sequence r0,k, u0,k can be found for k ≥ 1 using (6.63) and (6.65). Finally the two general systems (6.58),(6.60) and (6.62),(6.64) can be employed to compute rj,k, uj,k for j, k ≥ 1. DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 57 6.6. Numerical studies at the Hard Edge. Using the integral formula for the distribution Aa(z) as given by (5.71) it is possible to compute values of this and the examples of a = 0, 1, 2 are plotted in Figure 2. However we wish to characterise it in a precise quantitative way and evaluate the moments of this distribution (6.68) mk := dz zkAa(z), k ∈ Z≥0. These are easily seen to be (6.69) mk = 42a+3Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3) ds sk+2a+3e−s/4eF (s) du uk+2(1− u)aG(us; s), where (6.70) F (s) := [ν(v) + 2C(v)], (6.71) G(z; s) := q∂zp− p∂zq. We note that by employing the large s asymptotic form of q(z; s), p(z; s) as given in (6.39,6.40) we can deduce the asymptotic form of the spacing distribution is given (6.72) Aa(z) ∼ Ce−z/4+(a+2) zz−1/2−a where C is a constant which cannot be found from our methods. For small z we find that (6.73) Aa(z) ∼ 42a+3Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3) ds s2a − ξ(s) [ν(v) + 2C(v)] where we have used (5.58,5.59) and the fact that u01(s) = 1/12−ξ(s)/3s. Therefore we can conclude that the moments exist for Re(k) > −3, Re(a) > −1 and Re(k + 2a) > −4. An instance where exact evaluation of the moments can be made is the case a = 0 and the first four of these are m1 = 4e 2 [I0(2)− I1(2)] , m2 = 32e2I0(2),(6.74) m3 = 384e 2 [2I0(2) + I1(2)] , m4 = 2048e 2 [13I0(2) + 9I1(2)] .(6.75) We investigated the distributions Aa(z) and A ±(z) for the two special cases of a = ±1/2 in some detail because of the motivations provided by (1.6) and (1.7). 58 PETER J. FORRESTER AND NICHOLAS S. WITTE The analogue of (6.69) for A±(z) is (6.76) m±k = 24a+5Γ(a+ 1)Γ(a+ 2)Γ2(a+ 3) ds sk/2+2a+3e−s/4eF (s) du uk+2(1 − u)2a+1(2− u)2G(u(2− u)s; s), for a = ± 1 . The statistical data for Aa(z) for the cases a = −1/2, 0, 1/2, 1, 2 are given in Table 1 and the data for A±(z) is given in Table 2. Our strategy is that by employing local Puiseux-type and Taylor expansions for the two factors in the integrand, namely eF (s) and G(z; s), within a given finite member of the patchwork of local expansion domains the above integrals restricted to this domain can be exactly evaluated. This is essential as numerical quadrature algorithms implemented in either computer algebra software or compiled language packages (e.g. QUADPACK) have minimal attained error tolerances which cannot be reduced below a fixed bound. For the compiled language option with a floating point representation of 64 bits the best one could expect is a relative error of around 10−15 but often it is far worse and around 10−8 − 10−9. To illustrate this we have computed the statistical data for the a = 1, 2 cases using QUADPACK routines and the results are displayed in Table 1. In the case of the Puiseux-type expansions the integrals are (6.77) ds sk+3+l+n+(m+2)ae−s/4, (6.78) ∫ 1/2 du uk+2+n(1− u)a or du uk+2(1 − u)a+n, which for k, l,m, n ∈ Z≥0 can be evaluated in terms of radicals, the Gamma func- tion at integer arguments, a rational function of a and the Whittaker function M(α, β;S/4) or its specialisations depending on a. For example in the case a = 1/2 the s-integral reduces to the error and exponential functions. For the Taylor ex- pansion case we have the double integral (6.79) ds sk+3+2a(s− s0)me−s/4 du uk+2(1− u)a(us− z0)l, which for k, l,m ∈ Z≥0 and s0 ∈ (s1, s2) is evaluated in terms of a rational function of a, a polynomial function of s0, z0 and the Whittaker functions M(α, β; s1,2/4). Therefore the only sources of error are from the truncation of the expansions and the finite number of intervals, both of which can be adjusted to reduce the contributing errors. The computations were performed, in most cases, using the computer algebra system Maple with a sufficiently large number of decimal digits and found that 250 digits was more than adequate. In addition we found that we had to tailor DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 59 the numerical parameters for each case of a = ±1/2 differently as the errors varied quite strongly with a (this was especially pronounced as a approached −1). We discuss the case of a = 1/2 first. In regard to the truncations about the singular point s = 0, we found that 1890 terms in the expansion of the transcendent vari- able (6.1) with k ≤ 30, j + k ≤ 120 yielded an error for the second derivative of ν(s) at s = 2 which was estimated to be 4.6 × 10−98. At most 100 terms were re- tained in each of the expansions of the transcendent variables about regular points (6.41),(6.46),(6.47) and (6.48) because much fewer, of the order of j, k ≤ 20, were required in the corresponding expansions of the linear variables. The number of intervals in the s-direction was taken to be 19 with the sequence of s0 values being {0, 2, 5/2, 3, 4, 6, 9, 13, 19, 25, 30, 38, 54, 72, 90, 115, 150, 200, 300, 500}. The bound- aries of the s-interval with node s0 were taken to be located at the midpoints of s0 and its neighbouring nodes. This sequence of nodes was chosen to be close to an optimal situation yielding the largest separation of each node from its preceding node, yet close enough so to ensure that the error in ν′′(s) at the node was less than 9.8 × 10−37. For each s-interval with node s0 two expansion points in the z-direction were chosen because a single expansion point could never ensure that all of the integration region would fall within the domain of convergence about that point. The two points that together yielded the largest convergence domain were found to be located at z0 = 0 and z0 = s0. Subdividing the z-interval into three sub-intervals was found to contribute a variation of less than 3.2 × 10−19 to the normalisation. Another criteria that the sequence of s0 nodes had to sat- isfy was that each (s, z) integration region fell completely within the union of the convergence domains about (s0, 0) and (s0, s0). For the expansions of the linear variables about the lines z = 0, s and about the singular point s = 0, as defined in (6.18,6.19) along with (5.58,5.59,5.63,5.64), we chose the cut-off in the sum to be 20. The expansions of linear variables about the lines z = 0, s and about a regular point s = s0 > 0 as defined in (6.56,6.57) were cutoff at 25. An overall es- timate of the accuracy is provided by the normalisation, which was unity to within 1.6×10−18. The second case with a = −1/2 was more demanding computationally. We needed 5150 terms in the expansion of the transcendent variable (6.1) with k ≤ 50, j + k ≤ 200 and computed these with compiled code using the multiple- precision library MPFUN [3],[2]. The error for ν′′(s) at s = 2 was estimated to be around 7.9×10−59. Again only 100 terms were retained in each of the expansions of the transcendent variables about regular points. A larger number of intervals in the s-direction were employed, namely 24, and the sequence of s0 values was taken to be {0, 2, 5/2, 3, 7/2, 4, 5, 6, 7, 9, 11, 14, 18, 22, 28, 36, 46, 58, 72, 90, 114, 144, 180, 220, 300}. This time the sequence of nodes was chosen to ensure that the error in ν′′(s) at each node was less than 3.6×10−60. And again for each s-interval with node s0 two expansion points in the z-direction were chosen at z0 = 0, s0. In the expansions of 60 PETER J. FORRESTER AND NICHOLAS S. WITTE the linear variables about the lines z = 0, s and about the singular point s = 0 the cut-off in the sum was chosen to be 20 as before. The expansions of these variables about the lines z = 0, s and about a regular point s = s0 > 0 was terminated at the cutoff of 25 also. The estimate of the accuracy provided by the normalisation was 4.6× 10−20. Because the raw moments grow rapidly with order we have computed some standard statistical quantities instead using the definitions of the variance σ2, the skewness γ1 and the kurtosis excess γ2 (6.80) σ2 = µ2, γ1 = , γ2 = in terms of the central moments µ2 = m2 −m21,(6.81) µ3 = m3 − 3m1m2 + 2m31,(6.82) µ4 = m4 − 4m1m3 + 6m21m2 − 3m41.(6.83) DISTRIBUTION OF THE FIRST EIGENVALUE SPACING ... 61 Figure 2. The distribution of the first eigenvalue spacing at the hard edge of random hermitian matrices Aa(z) for integral values of a = 0, 1, 2. Acknowledgements This work was supported by the Australian Research Council. 62 PETER J. FORRESTER AND NICHOLAS S. WITTE Table 1. Low order statistics of the distribution Aa(z) for various values of the parameter a. References [1] A. I. Aptekarev. Asymptotics of orthogonal polynomials in a neighborhood of endpoints of the interval of orthogonality. Sb. Math., 76(1):35–50, 1993. [2] D.H. Bailey. A Portable High Performance Multiprecision Package. 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Colloquium Publications 23. American Mathematical Society, Providence, Rhode Island, third edition, 1967. [32] C. A. Tracy and H. Widom. Fredholm determinants, differential equations and matrix models. Comm. Math. Phys., 163(1):33–72, 1994. [33] C. A. Tracy and H. Widom. Level spacing distributions and the Bessel kernel. Comm. Math. Phys., 161(2):289–309, 1994. [34] A. M. Tulino and S. Verdú. Random matrix theory and wireless communications. Found. Trends Commun. Inf. Theory, 1(1):1–182, 2004. [35] V. B. Uvarov. Relation between polynomials orthogonal with different weights. Dokl. Akad. Nauk SSSR, 126:33–36, 1959. [36] J. J. M. Verbaarschot. The spectrum of the Dirac operator near zero virtuality for Nc = 2 and chiral random matrix theory. Nuclear Phys. B, 426(3):559–574, 1994. Department of Mathematics and Statistics, University of Melbourne,Victoria 3010, Australia E-mail address: p.forrester@ms.unimelb.edu.au E-mail address: n.witte@ms.unimelb.edu.au 1. Introduction 2. Orthogonal Polynomial System 2.1. Semi-classical Orthogonal Polynomials 2.2. Deformed Laguerre Orthogonal Polynomials 3. Difference and Differential Equations 3.1. Difference Equations 3.2. Reduction to Painlevé V 3.3. Inequalities and Bounds 4. Special Case a Z0 5. Hard Edge Scaling 5.1. General Case 5.2. Special Case a Z0 6. Analytical Studies at the Hard Edge 6.1. The -function expansion about s=0 6.2. The q, p expansion about s=0 6.3. The -function expansion about s= 6.4. The -function expansion about a regular point 6.5. The q, p expansion about a regular point 6.6. Numerical studies at the Hard Edge Acknowledgements References

Second-order perturbations of cosmological fluids: Relativistic effects of pressure, multi-component, curvature, and rotation Jai-chan Hwang∗ Department of Astronomy and Atmospheric Sciences, Kyungpook National University, Taegu, Korea Hyerim Noh† Korea Astronomy and Space Science Institute, Daejon, Korea (Dated: October 31, 2018) We present general relativistic correction terms appearing in Newton’s gravity to the second-order perturbations of cosmological fluids. In our previous work we have shown that to the second-order perturbations, the density and velocity perturbation equations of general relativistic zero-pressure, irrotational, single-component fluid in a flat background coincide exactly with the ones known in Newton’s theory. We also have shown the effect of gravitational waves to the second-order, and pure general relativistic correction terms appearing in the third-order perturbations. Here, we present results of second-order perturbations relaxing all the assumptions made in our previous works. We derive the general relativistic correction terms arising due to (i) pressure, (ii) multi-component, (iii) background curvature, and (iv) rotation. In case of multi-component zero-pressure, irrotational fluids under the flat background, we effectively do not have relativistic correction terms, thus the relativistic result again coincides with the Newtonian ones. In the other three cases we generally have pure general relativistic correction terms. In case of pressure, the relativistic corrections appear even in the level of background and linear perturbation equations. In the presence of background curvature, or rotation, pure relativistic correction terms directly appear in the Newtonian equations of motion of density and velocity perturbations to the second order. In the small-scale limit (far inside the horizon), relativistic equations including the rotation coincide with the ones in Newton’s gravity. All equations in this work include the cosmological constant in the background world model. We also present the case of multiple minimally coupled scalar fields, and properly derive the large-scale conservation properties of curvature perturbation variable in various temporal gauge conditions to the second order. PACS numbers: PACS numbers: 04.50.+h, 04.62.+v, 98.80.-k Contents I. Introduction 2 II. Newtonian nonlinear perturbations 3 III. Summary of previous work 5 IV. Relativistic fully nonlinear equations 6 A. Covariant equations 6 B. ADM equations 8 V. Second-order perturbations 8 A. Basic equations in the energy-frame 10 B. Decomposition 14 C. Comoving gauge and irrotational condition 16 VI. Effects of pressure 17 A. Irrotational case 17 ∗Electronic address: jchan@knu.ac.kr †Electronic address: hr@kasi.re.kr http://arxiv.org/abs/0704.1927v1 mailto:jchan@knu.ac.kr mailto:hr@kasi.re.kr B. Comoving gauge 17 C. Newtonian correspondence 19 D. Linear-order relativistic pressure corrections 20 VII. Effects of multi-component 20 A. Irrotational case 21 B. Linear perturbations 22 1. Exact solutions 24 C. Comoving gauge 25 D. Newtonian correspondence 26 VIII. Effects of curvature 28 A. Newtonian correspondence 28 IX. Effects of vector-type perturbation 29 A. Linear perturbations 29 B. Second-order perturbations 30 C. Zero-pressure case 31 D. Newtonian correspondence 31 E. Pure vector-type perturbations 33 X. Equations with fields 33 A. A minimally coupled scalar field 33 B. Minimally coupled scalar fields 35 C. Generalized gravity case 35 XI. Curvature perturbations and large-scale conservations 36 A. Linear-order equations 36 B. Second-order equations 38 C. Large-scale solutions 39 XII. Discussion 39 Acknowledgments 40 References 40 I. INTRODUCTION Large amount of cosmological data on the large-scale structures and motions of galaxies [1, 2], and the temperature and polarization anisotropies of cosmic microwave background radiation [3, 4] have been accumulating recently. In current standard cosmological scenario such structures are explained as small (linear) or large (nonlinear) deviations from spatially homogeneous and isotropic Friedmann background world model. In order to explain these data theo- retically, researchers rely on the linear perturbation theory based on relativistic gravity, and quasi-linear perturbation theories and nonlinear simulations based on Newton’s gravity. To the linear order in perturbation the general rel- ativistic result was first derived by Lifshitz in 1946 [5], and later shown to coincide with the Newtonian result in a zero-pressure medium [6]. The same is also known to be true for the background world model. That is, the general relativistic result was first derived by Friedmann in 1922 [7], and later shown to coincide with Newtonian result in a zero-pressure medium [8]. The observed large-scale distribution of galaxies shows that in the largest observed scale (say, larger than several hundred mega-parsec scale) the distribution may not be inconsistent with the linear assumption around the Friedmann background. However, as the scale becomes smaller the distribution apparently shows quasi-linear to fully nonlinear structures. The fully nonlinear processes occur in small scale where the relativistic effects characterized by GM/rc2 ∼ v2/c2 are quite small. If we could ignore such relativistic effects, Newton’s gravity would be sufficient to handle the relevant nonlinear processes. If we need to consider the weakly relativistic correction terms in fully nonlinear stage, instead of the relativistic perturbation approach which can handle the fully relativistic processes under weakly nonlinear assumption, we can use the post-Newtonian approximation developed in the context of cosmology in [9]. For structures in the quasi-linear evolution phase, previous researches were based on Newton’s gravity especially assuming the single component zero-pressure fluid without rotational perturbation [10]. In our previous works in [11, 12] we have shown that, in the single component zero-pressure fluid without rotational perturbation, cosmological scalar-type perturbation equations in a spatially flat background coincide exactly with the Newtonian ones up to the second order in perturbation. In [11, 12] we also have shown the contribution of gravitational wave perturbations to the hydrodynamic parts in the second-order perturbations. In Newton’s gravity the hydrodynamic equations of zero- pressure fluid contain only the quadratic order nonlinearity. In [13] we presented pure general relativistic correction terms appearing in the third-order perturbation, and showed that all third-order correction terms are 10−5 times smaller than the second-order relativistic/Newtonian terms, and independent of the horizon scale. In this work we will take into account of the pure general relativistic effects appearing in the second-order perturba- tions which were ignored in our previous work in [12]. We will consider general relativistic effects of (i) pressure, (ii) multi-component, (iii) background curvature, (iv) rotation in cosmological fluids to the second-order perturbations. As results we will show that, although in [12] we have shown the exact relativistic/Newtonian correspondence to the second-order perturbations by ignoring the above four conditions and the gravitational waves, as we take these four effects into account we often encounter pure general relativistic effects appearing in the corresponding Newtonian equa- tions even to the second order in perturbations. Our results will show that the relativistic/Newtonian correspondence continues even in the multi-component situation assuming zero-pressure irrotational fluid in a flat background, but in the presence of the cosmological constant. This is a practically useful result because the matter content of present universe is dominated by collisionless dark matter and baryon both of which practically have zero-pressure. In the other three cases, relaxing any of the assumptions about pressure, rotation, and background curvature generally leads to pure general relativistic correction terms to the second order. We will present such correction terms in the context of Newtonian hydrodynamics. One additional relativistic/Newtonian correspondence occurs in the case of rotation in small-scale (sub-horizon-scale) limit which is another practically important result. This correspondence allows us to use the Newtonian equations safely in such a small-scale limit even in the presence of rotational perturbation to the second order. In Sec. II we summarize Newtonian hydrodynamic perturbation equations valid to fully nonlinear order. In Sec. III we briefly summarize our previous result of relativistic/Newtonian correspondence to the second order, and pure general relativistic correction terms appearing in the third order. In Sec. IV we present parts of the covariant and the ADM (Arnowitt-Deser-Misner) equations which are valid in multi-component situation. In Sec. V we present the basic perturbation equations valid to second order. In [11] the basic set of equations was presented using fluid quantities based on the normal-frame four-vector. The fluid quantities in the present work are based on the energy- frame four-vector, and in this section we present the basic equations using such fluid quantities. In Secs. VI-IX we analyse the effects of the pressure, the multi-components, the background curvature, and the rotational perturbation, respectively. In Sec. X we present equations in the scalar fields and generalized gravity theories using the energy-frame fluid quantities. In Sec. XI we properly derive conservation properties of curvature perturbation in various temporal gauge (hypersurface) conditions to the second order in perturbations. Section XII is a discussion. In this work we follow notations used in [11, 12]. We set c ≡ 1, but when we compare with Newtonian case we often recover the speed of light c. II. NEWTONIAN NONLINEAR PERTURBATIONS In order to compare properly the relativistic results with the Newtonian ones, in this section we summarize the Newtonian cosmological perturbation theory in fully nonlinear context. We consider multi-component fluids in the presence of isotropic pressure. In case of n-fluids with the mass densities ̺i, the pressures pi, the velocities vi (i = 1, 2, . . . n), and the gravitational potential Φ, we have ˙̺i +∇ · (̺ivi) = 0, (1) v̇i + vi · ∇vi = − ∇pi −∇Φ, (2) ∇2Φ = 4πG ̺j . (3) Assuming the presence of spatially homogeneous and isotropic but temporally dynamic background, we introduce fully nonlinear perturbations as ̺i = ¯̺i + δ̺i, pi = p̄i + δpi, vi = Hr+ ui, Φ = Φ̄ + δΦ, (4) where H ≡ ȧ/a, and a(t) is a cosmic scale factor. We move to the comoving coordinate x where r ≡ a(t)x, (5) ∇ = ∇ −Hx · ∇ . (6) In the following we neglect the subindex x. To the background order we have ˙̺i + 3H̺i = 0, ̺j , H , (7) where E is an integration constant which can be interpreted as the specific total energy in Newton’s gravity; in Einstein’s gravity we have 2E = −Kc2 where K can be normalized to be the sign of spatial curvature. To the perturbed order we have [14] δ̇i + ∇ · ui = − ∇ · (δiui) , (8) u̇i +Hui + ui · ∇ui = − a ¯̺i 1 + δi ∇δΦ, (9) ∇2δΦ = 4πG ¯̺jδj . (10) By introducing the expansion θi and the rotation −→ω i of each component as ∇ · ui, −→ω i ≡ ∇× ui, (11) Eq. (9) gives θ̇i + 2Hθi + 4πG ¯̺jδj = − ∇ · (ui · ∇ui)− a2 ¯̺i 1 + δi , (12) −̇→ω i + 2H −→ω i = − ∇× (ui · ∇ui) + a2 ¯̺i (∇δi)×∇δpi (1 + δi)2 . (13) By introducing decomposition of perturbed velocity into the potential- and transverse parts as ui ≡ ∇ui + u i , ∇ · u i ≡ 0; θi = −→ω i = i , (14) instead of Eq. (13) we have i +Hu i = − ui · ∇ui + 1 + δi −∇∆−1∇ · ui · ∇ui + 1 + δi . (15) We note that the pure ui contributions in the right-hand-side of Eq. (13) or Eq. (15) vanish. Thus, under van- ishing pressure, pure irrotational perturbation cannot generate the rotational perturbation. Equation (13) shows that presence of pressure perturbation oblique (i.e., non-parallel) to the density perturbation can generate rotational perturbation. Combining Eqs. (8),(12) we can derive δ̈i + 2Hδ̇i − 4πG ¯̺jδj = − [a∇ · (δiui)] ∇ · (ui · ∇ui) + a2 ¯̺i 1 + δi . (16) Equations (8)-(16) are valid to fully nonlinear order. Notice that for vanishing pressure these equations have only quadratic order nonlinearity in perturbations. III. SUMMARY OF PREVIOUS WORK In [11, 12] we have derived second-order perturbation equations valid for the single component, irrotational, and zero-pressure medium in zero-curvature background. These are ∇ · u = − ∇ · (δu) , (17) ∇ · (u̇+Hu) + 4πG̺δ = − ∇ · (u · ∇u)− Ċ(t)αβ uα|β + Ċ , (18) δ̈ + 2Hδ̇ − 4πG̺δ = ∇ · (u · ∇u)− [a∇ · (δu)] + Ċ(t)αβ uα|β + Ċ . (19) Except for the presence of tensor-type perturbation, Eqs. (17)-(19) are exactly the same as the ones known in the Newtonian theory. We note that these equations are valid in the presence of Λ. To the linear order, these are valid in the presence of general K, see Sec. VIII. We have correctly identified the relativistic density and velocity perturbation variables which correspond to the Newtonian counterparts to the second order. In the relativistic context, our δ and u are the density perturbation and (related to) the perturbed expansion scalar, respectively, in the comoving gauge; the variables are equivalently gauge-invariant. However, we were not able to identify relativistic variable which corresponds to the Newtonian gravitational potential to the second order; this is understandable if we consider the factor of two difference between Einstein’s (post-Newtonian) and Newton’s gravity theories in predicting the light bending under the gravitational field. To the linear order the spatial curvature perturbation in the zero-shear gauge can be identified as the perturbed Newtonian potential [15, 16]. Equations (17)-(19) include effects of gravitational waves to the density and velocity perturbations. Equations of the gravitational waves can be found in [12]. To the third order, we have [13] ∇ · u = − ∇ · (δu) + 2ϕu−∇ · ∇δ, (20) + 4πGµδ = − ∇ · (u · ∇u) ϕu · ∇ (∇ · u) + u · ∇u− u∇ · u u · ∇ u · ∇X + X∇ · u,(21) δ̈ + 2 δ̇ − 4πGµδ = − [a∇ · (δu)] + ∇ · (u · ∇u) + 2ϕu−∇ ϕu · ∇ (∇ · u)− u · ∇u− u∇ · u u · ∇ u · ∇X − X∇ · u,(22) where X ≡ 2ϕ∇ · u− u · ∇ϕ+ ∆−1∇ · [u · ∇ (∇ϕ) + u∆ϕ] . (23) In these equations we ignored the role of tensor-type perturbation; for a complete set of equations, see [13]. The variable ϕ is a perturbed-order metric (spatial curvature) variable in the comoving gauge condition, see later. All the third-order correction terms in Eqs. (20)-(22) are simply of ϕ-order higher than the second-order rela- tivistic/Newtonian terms. Thus, the pure general relativistic effects are at least ϕ-order higher than the relativis- tic/Newtonian ones in the second order equations. Thus, we only need the behavior of ϕ to the linear order which is related to the other hydrodynamic variables as ϕ = −δΦ+ ȧ∆−1∇ · u. (24) It also satisfies [17] ϕ̇ = 0, (25) thus ϕ = C(x) with no decaying mode; this is true considering the presence of the cosmological constant, see [17]. IV. RELATIVISTIC FULLY NONLINEAR EQUATIONS In this section, for convenience, we present some additional covariant or ADM equations not available in [11]. In the multi-component situation, we have T̃ab ≡ T̃(j)ab. (26) The energy-momentum conservation gives T̃ b(i)a;b = Ĩ(i)a, Ĩ(j)a = 0. (27) Tildes indicate the covariant quantities. A. Covariant equations We introduce the fluid quantities as T̃(i)ab ≡ µ̃(i)ũ(i)aũ(i)b + p̃(i) g̃ab + ũ(i)aũ(i)b + q̃(i)aũ(i)b + q̃(i)bũ(i)a + π̃(i)ab, (28) where ũa(i)ũ(i)a ≡ −1, ũ (i)q̃(i)a ≡ 0 ≡ ũ (i)π̃(i)ab, π̃ (i)a ≡ 0. (29) The fluid quantities of each component are based on the fluid four-vector ũ(i)a as µ̃(i) ≡ T̃(i)abũ (i)ũ (i), p̃(i) ≡ T̃(i)abh̃ (i), q̃(i)a ≡ −T̃(i)cdũ (i)h̃ (i)a, π̃(i)ab ≡ T̃(i)cdh̃ (i)ah̃ (i)b − p̃(i)h̃(i)ab, (30) where h̃(i)ab ≡ g̃ab + ũ(i)aũ(i)b. Equation (27) gives µ(i) + µ̃(i) + p̃(i) θ̃(i) + q̃ (i);a + q̃ (i)ã(i)a + π̃ (i)σ̃(i)ab = −ũ (i)Ĩ(i)a, (31) µ̃(i) + p̃(i) ã(i)a + h̃ p̃(i),b + q(i)b + π̃ (i)b;c + q̃b(i) ω̃(i)ab + σ̃(i)ab + θ̃(i)h̃(i)ab = h̃ b(i)aĨ(i)b, (32) where the kinematic quantities are also based on the fluid four-vector ũ(i)a as h̃ c(i)ah̃ (i)b ũ(i)c;d = h̃ (i)[ah̃ (i)b]ũ(i)c;d + h̃ (i)(ah̃ (i)b)ũ(i)c;d ≡ ω̃(i)ab + θ̃(i)ab = ũ(i)a;b + ã(i)aũ(i)b, θ̃(i) ≡ u (i);a, σ̃(i)ab ≡ θ̃(i)ab − θ̃(i)h̃(i)ab, ã(i)a ≡ ũ(i)a;bũ (i) ≡ u(i)a, µ(i) ≡ µ̃(i),aũ (i). (33) In the multi-component situation, we can derive the corresponding equation of Raychaudhury equation for the indi- vidual component. From ũ(i)a;bc − ũ(i)a;cb ≡ ũ(i)dR̃ abc we can derive θ(i) + θ̃2(i) − ã (i);a + σ̃ (i)σ̃(i)ab − ω̃ (i)ω̃(i)ab = 4πG µ̃− 3p̃− 2T̃abũ (i)ũ + Λ. (34) In the energy-frame we take q̃(i)a ≡ 0 for each component of the fluids without losing any physical degree of freedom. In a single component situation, taking the energy-frame, the energy conservation equation, the momentum con- servation equation, and the Raychaudhury equation are [18] µ+ (µ̃+ p̃) θ̃ + π̃abσ̃ab = 0, (35) (µ̃+ p̃) ãa + h̃ p̃,b + π̃ = 0, (36) θ̃2 − ãa;a + σ̃ abσ̃ab − ω̃ abω̃ab + 4πG (µ̃+ 3p̃)− Λ = 0. (37) By combining Eqs. (35)-(37) we can derive µ+ π̃abσ̃ab µ̃+ p̃ µ+ π̃abσ̃ab µ̃+ p̃ = 4πG (µ̃+ 3p̃)− Λ + σ̃abσ̃ab − ω̃ abω̃ab + h̃ab(p̃,b + π̃ µ̃+ p̃ . (38) This equation was derived in Eq. (88) of [19], see also [20]. In the multi-component case, in the energy-frame, combining Eqs. (31)-(34) we can derive µ(i) + π̃ σ̃(i)ab + ũ Ĩ(i)a µ̃(i) + p̃(i) µ(i) + π̃ σ̃(i)ab + ũ Ĩ(i)a µ̃(i) + p̃(i) = −4πG µ̃− 3p̃− 2T̃abũ (i)ũ − Λ + σ̃ab(i)σ̃(i)ab − ω̃ (i)ω̃(i)ab + (p̃(i),b + π̃ (i)b;c − Ĩ(i)b) µ̃(i) + p̃(i) . (39) In [21] Langlois and Vernizzi derived a simple covariant relation which leads to one of the conserved variable in the large-scale limit. These authors introduced ζ̃a ≡ h̃ α̃,b + 3(µ̃+ p̃) α ≡ α̃,aũ θ̃. (40) Using only the energy conservation in Eq. (35) we can derive £ũζ̃a = − 3(µ̃+ p̃) p̃,a + (µ̃+ p̃)θ̃ ũaπ̃ bcσ̃bc 3(µ̃+ p̃) π̃bcσ̃bc 3(µ̃+ p̃) µ̃,aπ̃ bcσ̃bc 3(µ̃+ p̃)2 , (41) where £ũ is a Lie derivative along ũa with £ũζ̃a ≡ ζ̃a;bũ b + ζ̃bũb;a. Thus, for vanishing anisotropic pressure, we have the Langlois-Vernizzi relation [21] £ũζ̃a = − 3(µ̃+ p̃) p̃,b − . (42) These equations are valid in a single component fluid, or in multiple component fluids for the collective fluid variables. We can easily extend the relation to the individual fluid component as follows. We introduce ζ̃(i)a ≡ h̃ α̃(i),b + µ̃(i),b 3(µ̃(i) + p̃(i)) α(i) ≡ α̃(i),aũ (i) ≡ θ̃(i). (43) Using only Eq. (31) we can derive £ũ(i) ζ̃(i)a = − θ̃(i) 3(µ̃(i) + p̃(i)) p̃(i),a + (µ̃(i) + p̃(i))θ̃(i) µ̃(i),a ũ(i)a σ̃(i)bc + ũ Ĩ(i)b 3(µ̃(i) + p̃(i)) σ̃(i)bc + ũ Ĩ(i)b 3(µ̃(i) + p̃(i)) µ̃(i),a σ̃(i)bc + ũ Ĩ(i)b 3(µ̃(i) + p̃(i)) . (44) Thus, for vanishing anisotropic pressure and direct interactions among fluids, i.e., π̃(i)ab = 0 = Ĩ(i)a, we have £ũ(i) ζ̃(i)a = − θ̃(i) 3(µ̃+ p̃) h̃ b(i)a p̃(i),b − µ̃(i),b . (45) Application of these compact relations to large-scale conservation properties to the second order will be studied in Sec. XI. B. ADM equations The ADM formulation [22] is presented in Eqs. (2)-(13),(47),(48) of [11]. Interpretation of the ADM fluid quantities in Eqs. (45),(46) of [11] was based on the normal-frame fluid quantities; for relations to the energy-frame fluid quantities, see Eq. (57) below. The ADM fluid quantities of individual component are introduced as E(i) ≡ ñañbT̃ (i) = N 2T̃ 00(i) , J(i)α ≡ −ñbT̃ (i)α = NT̃ (i)α, S(i)αβ ≡ T̃(i)αβ , S(i) ≡ h αβS(i)αβ , S̄(i)αβ ≡ S(i)αβ − hαβS(i). (46) Equation (27) gives Eqs. (12),(13),(47),(48) in [11]. Equations (10),(12),(47) in [11] can be arranged as (∂0 −N α∂α)K (Kαα ) = 4πG (E + S)− Λ + K̄αβK̄αβ − N :αα, (47) Kαα = )−1 [ (∂0 −N α∂α)E + − S̄αβK̄αβ , (48) Kαα = E(i) + )−1 [ (∂0 −N α∂α)E(i) + N2Jα(i) K̄αβ + Ĩ(i)0 − Ĩ(i)αN . (49) Momentum conservation equations for the collective and individual components can be found in Eqs. (13),(48) of [11]. By combining Eqs. (47)-(49) we can derive the ADM counterpart of the density perturbation equations in Eqs. (38),(39). V. SECOND-ORDER PERTURBATIONS We use a metric convention in Eq. (49) of [11] g̃00 ≡ −a 2 (1 + 2A) , g̃0α ≡ −a 2Bα, g̃αβ ≡ a + 2Cαβ . (50) The subindex 0 indicates the conformal time η with adη ≡ cdt. To the second-order in perturbation we introduce the fluid four-vector of individual component as ũα(i) ≡ V α(i), ũ (i) = 1− A+ V α(i)V(i)α −B αV(i)α ũ(i)α = a V(i)α −Bα +ABα + 2V , ũ(i)0 = −a 1 +A− V α(i)V(i)α . (51) In this definition of fluid four-vector we follow the notation in Eq. (53) of [11]. If we introduce ũα ≡ V̄ α , we have V α(i) = (1−A)V̄ (i). The fluid quantities of individual component are introduced as µ̃(i) ≡ µ(i) + δµ(i), p̃(i) ≡ p(i) + δp(i), π̃(i)αβ ≡ a 2Π(i)αβ , π̃(i)0α = −a 2Π(i)αβV , π̃(i)00 = 0, (52) where from π̃ a ≡ 0 we have Π α(i)α − 2C αβΠ(i)αβ = 0. (53) The energy-momentum tensor of individual component in the energy-frame follows from Eq. (28) as T̃ 0(i)0 = −µ(i) − δµ(i) − µ(i) + p(i) V α(i) −B V(i)α, T̃ 0(i)α = µ(i) + p(i) V(i)α −Bα −AV(i)α + 2ABα + 2V δµ(i) + δp(i) V(i)α −Bα +Π(i)αβ T̃ α(i)β = p(i) + δp(i) δαβ + µ(i) + p(i) V α(i) V(i)β −Bβ +Π α(i)β − 2C αγΠ(i)βγ . (54) Using T̃ ab = , and the total fluid quantities in Eq. (82) of [11] we have µ(j), p = p(j), (55) for the background order fluid quantities, and δµ(j) + µ(j) + p(j) V α(j) −B V(j)α − Vα δp(j) + µ(j) + p(j) V α(j) −B V(j)α − Vα (µ+ p)Vα = µ(j) + p(j) V(j)α + δµ(j) + δp(j) V(j)α − Vα V(j)β − Vβ Παβ = Π α(j)β + µ(j) + p(j) vα(j) −B V(j)β − Vβ V(j)γ − Vγ , (56) for perturbed order fluid quantities to the second-order. From Eq. (46) we have the ADM fluid quantities based on the energy-frame fluid quantities E(i) = µ(i) + δµ(i) + µ(i) + p(i) V α(i) −B V(i)α −Bα J(i)α = a µ(i) + p(i) V(i)α −Bα +ABα + 2V δµ(i) + δp(i) V(i)α −Bα + Π(i)αβ S(i) = 3 p(i) + δp(i) µ(i) + p(i) V α(i) −B V(i)α −Bα S̄(i)αβ = a Π(i)αβ + µ(i) + p(i) V(i)α −Bα V(i)β −Bβ V(i)γ −Bγ . (57) We can compare Eq. (57) with the ADM fluid quantities based on the normal-frame fluid quantities in Eq. (76) of [11]; in a single component case we simply delete (i) subindices. In [11] the fluid quantities are based on the normal-frame vector. By taking ũ(i)α ≡ 0, ũ(i)a becomes the normal- frame vector ña, see Eq. (54) in [11]. Based on the normal-frame vector, to the second order, the fluid quantities have contributions due to the frame choice: for example, even in the zero-pressure fluid, the perturbed pressure based on the normal-frame does not necessarily vanish to the second-order, see Eq. (58) below. By comparing Eq. (57) with Eq. (76) of [11] we have δµN(i) = δµ(i) + µ(i) + p(i) V α(i) −B V(i)α −Bα δpN(i) = δp(i) + µ(i) + p(i) V α(i) −B V(i)α −Bα Q(i)α = µ(i) + p(i) V(i)α −Bα +ABα + 2V δµ(i) + δp(i) V(i)α −Bα +Π(i)αβ ΠN(i)αβ = Π(i)αβ + µ(i) + p(i) V(i)α −Bα V(i)β −Bβ V(i)γ −Bγ . (58) For the total fluid quantities the relations between the two frames are presented in Eq. (87) of [11]. Thus, by replacing all fluid quantities in Eqs. (99)-(107) of [11] using Eq. (58) and Eq. (87) of [11] we have the equations in the energy frame. Using Q(i)α in Eq. (58), Eq. (54) gives T̃ 0(i)α = (1−A)Q(i)α. (59) As the fluid four-velocity of i-th component we can use either Q(i)α or V(i)α −Bα related by Eq. (58). The kinematic quantities in the energy-frame are presented in Eqs. (63)-(66) of [11]. In Eq. (33) we introduced kinematic quantities for the individual component. To the second order we can show θ̃(i) = 3 Cα′α + V α(i)|α + A2 − 3 BαV(i)α − ACα′α + V α(i) A,α + C V α(i)V(i)α + V α(i) −B V(i)α −Bα CαβC′αβ , σ̃(i)αβ = a V(i)(α|β) + C αβ −AC V(i)(α −B(α V(i)β) −Bβ) + V(i)(αA,β) + V Cαβ|γ + 2V (i)|(α C(β)γ Cγ′γ + V (i)|γ (i)|γ + Cγ′γ −AC V(i)γ −Bγ A,γ + V Cδδ|γ − 2C γδC′γδ ω̃(i)αβ = a V(i)[α −B[α +AB[α + 2V V(i)[α −B[α V(i)β] −Bβ] +A,β] ã(i)α = A,α + V(i)α −Bα +ABα + 2V − 2AA,α −A V(i)α − Bα V(i)α −Bα +Bβ|α σ̃(i)α0 = −V σ̃(i)αβ , σ̃(i)00 = 0; ω̃(i)α0 = −V ω̃(i)αβ , ω̃(i)00 = 0; ã(i)0 = −V (i)ã(i)α, (60) where a prime indicates the time derivative based on η. The gauge transformation properties of the fluid quantities are presented in Eqs. (232)-(235) of [11] for the normal- frame, and Eqs. (238) of [11] for the energy-frame. A prescription to get the gauge transformation properties for individual fluid quantities is also presented below Eq. (235) of [11]. Under the gauge transformation we have δµ̂(i) = δµ(i) − µ′(i) + δµ ξ0 − δµ(i),αξ µ′′(i)ξ 0ξ0 + µ′(i) ξ0ξ0′ + ξαξ0,α δp̂(i) = δp(i) − p′(i) + δp ξ0 − δp(i),αξ p′′(i)ξ 0ξ0 + p′(i) ξ0ξ0′ + ξαξ0,α V̂(i)α − B̂α + ÂB̂α + 2V̂ Ĉαβ = V(i)α −Bα +ABα + 2V Cαβ + ξ V(i)α −Bα V(i)α −Bα V(i)β −Bβ ξβ,α − V(i)α −Bα A− ξ0′ − ξ0,α − ξ ,β − ξ 0ξ0′,α − ξ βξ0,αβ , Π̂(i)αβ = Π(i)αβ − Π′(i)αβ + 2 Π(i)αβ ξ0 −Π(i)αβ,γξ γ − 2Π(i)γ(αξ . (61) A. Basic equations in the energy-frame The basic set of equations with fluid quantities based on the normal-frame is presented in Eqs. (99)-(107) of [11]. By using Eq. (58) we can recover the equations with fluid quantities based on the energy frame. For convenience, in the following we present the complete set of equations with fluid quantities in the energy frame. These equations are written without taking any gauge conditions yet, thus in a sort of gauge-ready form. To the linear order this method was suggested by Bardeen [23, 24]. Definition of δK: K̄ + 3H + δK − 3HA+ Ċαα + HA− Ċαα − HBαBα + + 2Cαβ Ċαβ + ≡ n0. (62) Energy constraint equation: 16πGµ+ 2Λ− 6H2 − R(3) + 16πGδµ+ 4HδK − R(3)Cαα δK2 − 16πG (µ+ p) (V α −Bα) (Vα −Bα)− Ċαβ + B(α|β) Ċαβ + Ċαα + R(3)Cαγ C − Cα|βα + Cαβ|γ 3Cαβ|γ − 2Cαγ|β ≡ n1. (63) Momentum constraint equation: Ċβα + +B |βα Ċγγ + δK,α + 8πGa (µ+ p) Vα −Bα +ABα + 2V AδK,α − 8πGa (δµ+ δp) (Vα −Bα) + (µ+ p)A (Vα −Bα) + Παβ V β −Bβ Ċβα + + B |βα Ċγα + B |γα +B + 2Cβγ Ċαγ + B(α|γ) + Cγ|βα − C Ċβγ + Ċγγ + + 2Cγδ Ċγδ + ≡ n2α. (64) Trace of the ADM propagation equation: 3Ḣ + 3H2 + 4πG (µ+ 3p)− Λ + δK̇ + 2HδK − 4πG (δµ+ 3δp) + 3Ḣ + = AδK̇ − δK,αB δK2 + 8πG (µ+ p) (V α −Bα) (Vα −Bα) 3A2 −BαBα 2A∆A+A,αA,α − ∆ (BαBα) +A + 2CαβA,α|β Ċαβ + B(α|β) Ċαβ + Ċαα + ≡ n3. (65) Tracefree ADM propagation equation: Ċαβ + Bα|β +B Ċαβ + Bα|β +B Ċγγ + Ċγγ + A|γ γ R(3)Cαβ − R(3)Cγγ − 8πGΠαβ Ċαβ + Bα|β +B A+ 2Cαγ Ċβγ + B(β|γ) Ċαβ + Bα|β +B A+ 2Cαγ Ċβγ + B(β|γ) Ċαβ + Bα|β +B Ċαβ + Bα |β +B Bγ + δK Ċαβ + Bα|β +B −A2 +BγBγ − 2CαγA,β|γ − Ċγγ + A+ 2Cγδ Ċγδ + Ċγγ + A+ 2Cγδ Ċγδ + Ċγγ + Ċγγ + Bδ + δK Ċγγ + −AA|γ γ + −A2 +BδBδ − 2CγδA,γ|δ − − Cγ|δγ Ċαγ + Bα|γ +B Cαδ|βγ + C − Cαβ|δγ − C + 2Cαγ Cδγ|βδ + C β|γδ − C − Cδδ|γβ R(3)Cαγ C Cαδ |β + C − Cγδ|βC γδ|α + 2Cαγ|δ Cβδ|γ − Cβγ|δ Cǫγ|δǫ + C γ|ǫδ − C − Cǫǫ|γδ R(3)CδγC 2Cǫδ|ǫ − C − Cγ|δγ + Cγδ|ǫ 2Cγǫ|δ − 3Cγδ|ǫ −16πGCαγΠβγ + 8πG (µ+ p) (V α −Bα) (Vβ −Bβ)− δαβ (V γ −Bγ) (Vγ −Bγ) ≡ n α4β . (66) Energy conservation equation: [µ̇+ 3H (µ+ p)] + δµ̇+ 3H (δµ+ δp)− (µ+ p) (δK − 3HA) + (µ+ p) V α −Bα +ABα + 2V βCαβ δµ,αB α + (δµ+ δp) (δK − 3HA) + (µ+ p)AδK + H (µ+ p) A2 −BαBα a4 (µ+ p) (V α −Bα) (Vα −Bα) (δµ+ δp) (V α −Bα) + Παβ (Vβ −Bβ) (µ+ p) −A (V α −Bα)|α − 2A,α (V α −Bα) + 2 Cαβ (Vβ −Bβ) − Cαα|β V β −Bβ Ċαβ + ≡ n5. (67) Momentum conservation equation: a4 (µ+ p) Vα −Bα +ABα + 2V (µ+ p)A,α + δp,α +Π = (µ+ p) (δK − 3HA) (Vα −Bα)− (δµ+ δp) (Vα −Bα) + Π α (Vβ −Bβ) δp,α +Π A− (δµ+ δp)A,α − (µ+ p) −AA,α +Bβ|αV β + (Vα −Bα)|β V β + (Vα −Bα) V β −Bβ CβγΠαγ Πγα + C Πβγ −A,βΠ ≡ n6α. (68) In the multi-component situation we additionally have the energy and the momentum conservation of individual component. Using the energy-frame fluid quantities, Eqs. (106),(107) in [11] become µ̇(i) + 3H µ(i) + p(i) I(i)0 + δµ̇(i) + 3H δµ(i) + δp(i) µ(i) + p(i) (δK − 3HA) µ(i) + p(i) V α(i) −B α +ABα + 2V δI(i)0 δµ(i),αB δµ(i) + δp(i) (δK − 3HA) + µ(i) + p(i) AδK + µ(i) + p(i) A2 −BαBα µ(i) + p(i) V α(i) −B V(i)α −Bα δµ(i) + δp(i) V α(i) −B V(i)β −Bβ µ(i) + p(i) V α(i) −B − 2A,α V α(i) −B V(i)β −Bβ − Cα|βα V(i)β −Bβ Ċαβ + δI(i)αB α ≡ n(i)5, (69) µ(i) + p(i) V(i)α −Bα +ABα + 2V µ(i) + p(i) A,α + δp(i),α +Π (i)α|β − δI(i)α µ(i) + p(i) (δK − 3HA) V(i)α −Bα δµ(i) + δp(i) V(i)α −Bα V(i)β −Bβ δp(i),α +Π (i)α|β − δI(i)α δµ(i) + δp(i) µ(i) + p(i) −AA,α +Bβ|αV V(i)α −Bα V(i)α −Bα CβγΠ(i)αγ −A,βΠ ≡ n(i)6α. (70) By removing indices indicating the components in Eqs. (69),(70) we recover equations for the collective component which coincide with the equations in a single component situation in Eqs. (67),(68). To the background order, from Eqs. (69),(65) we have µ̇(i) + 3H µ(i) + p(i) I(i)0, (71) µ̇+ 3H (µ+ p) = 0, (72) 3Ḣ + 3H2 = − µ(j) + 3p(j) + Λc2, (73) µ(j) − , (74) where we recovered the speed of light c. Dimensions are [G̺] = T−2, [c] = LT−1, [η] = 1, [p] = [µ] = [̺c2], [a] = L, [K] = 1, [Λ] = L−2, [I(i)0] = [µ]. (75) Equation (72) follows from the sum of Eq. (71) over components. Equation (74) follows from integrating Eq. (73) where K-term can be regarded as an integration constant; in Einstein’s gravity K-term can be normalized as the sign of spatial curvature. Compared with the Newtonian background equations in Eq. (7), ignoring the direct inter- action terms in Eq. (71), the presence of pressure terms in Eqs. (71)-(74) is the pure general relativistic effect. The cosmological constant Λ can be introduced by hand even in the Newtonian case. B. Decomposition We decompose the metric to three perturbation types A ≡ α, Bα ≡ β,α +B α , Cαβ ≡ ϕg + γ,α|β + C (α|β) , (76) where superscripts (v) and (t) indicate the transverse vector-type, and transverse-tracefree tensor-type perturbations, respectively. We introduce χ ≡ a β + c−1aγ̇ , Ψ(v)α ≡ B α + c −1aĊ(v)α , (77) which are spatially gauge-invariant to the linear order. We set Kαα ≡ −3H + κ. (78) We will identify κ with Newtonian velocity variable which will be an important step in our analysis, see Eqs. (120),(194),(215). For the fluid quantities we decompose ũ(i)α = a V(i)α −Bα +ABα + 2V ≡ av(i)α ≡ a −v(i),α + v , (79) Π(i)αβ ≡ Π(i),α|β − ∆Π(i) (i)(α|β) (i)αβ , δI(i)α ≡ δI(i),α + δI . (80) The perturbed fluid velocity variables v(i) and v subtly differ from the ones introduced in [11]; see Sec. VC. For the collective fluid component or for a single component case, we simply delete (i) subindices. For isotropic pressure we introduce δp ≡ c2sδµ+ e, c . (81) The perturbation variable e is called an entropic perturbation. Defined in this way e is gauge-invariant only to the linear order. To the second order, from Eq. (61) we can derive the following gauge-invariant combination δpδµ ≡ e− . (82) In our notation, δpδµ is a gauge-invariant combination which is the same as δp in the δµ = 0 slicing (temporal gauge) condition to the second order; as the spatial gauge we take γ ≡ 0 to the second order; for the derivation, see the prescription below Eq. (266) of [11]. In the multi-component case, we similarly have δp(i)δµ(i) ≡ e(i) − δµ(i) µ̇(i) ė(i) + c2(i) δµ(i) , (83) where δp(i) ≡ c (i)δµ(i) + e(i), c (i) ≡ ṗ(i) µ̇(i) . (84) We have [25] e = erel + eint, erel ≡ c2(j) − c δµ(j), eint ≡ e(j). (85) Equation (56) gives δµ(j) + µ(j) + p(j) vα(j) v(j)α − vα δp(j) + µ(j) + p(j) vα(j) v(j)α − vα (µ+ p) vα = µ(j) + p(j) v(j)α + δµ(j) + δp(j) v(j)α − vα v(j)β − vβ Παβ = Π α(j)β + µ(j) + p(j) vα(j) v(j)β − vβ δαβ v v(j)γ − vγ , (86) By recovering c, dimensions of the variables are [g̃ab] = [g̃ ab] = [ũa] = 1, [T̃ab] = [µ], [g ] = 1, [∇] = [∆] = 1, [w] = [c2s] = 1, [A] = [Bα] = [Cαβ ] = 1, [α] = [ϕ] = [β] = [γ] = [B α ] = [C α ] = [Ψ α ] = [C ] = 1, [χ] = L, [κ] = T−1, [δµ] = [δp] = [e] = [Παβ ] = [Π ] = [µ], [δ] = [Vα] = 1, [v] = [v α ] = 1, [Π] = L 2[µ], [Π(v)α ] = L[µ]. (87) Scalar-type perturbation equations can be derived from Eqs. (62)-(70) κ− 3Hα+ 3ϕ̇+ χ = n0, (88) 4πGδµ+Hκ+ ∆+ 3K n1, (89) ∆+ 3K χ− 12πG (µ+ p) av = n2 ≡ ∆−1∇αn2α, (90) κ̇+ 2Hκ− 4πG (δµ+ 3δp) + 3Ḣ + α = n3, (91) χ̇+Hχ− ϕ− α− 8πGΠ = n4 ≡ a2 (∆ + 3K) ∆−1∇α∇βn 4α, (92) δµ̇+ 3H (δµ+ δp)− (µ+ p) κ− 3Hα+ = n5, (93) a4 (µ+ p) v a4 (µ+ p) a (µ+ p) ∆+ 3K = n6 ≡ − ∆−1∇αn6α, (94) δµ̇(i) + 3H δµ(i) + δp(i) µ(i) + p(i) κ− 3Hα+ δI(i)0 = n5(i), (95) µ(i) + p(i) µ(i) + p(i) µ(i) + p(i) δp(i) + ∆ + 3K Π(i) − δI(i) = n6(i) ≡ − µ(i) + p(i) ∆−1∇αn6α(i). (96) Equations for the vector-type perturbation follow from Eqs. (64),(66),(68),(70) ∆ + 2K Ψ(v)α + 8πG(µ+ p)v n2α −∇α∆ −1∇βn2β 2α , (97) Ψ̇(v)α + 2HΨ α − 8πGΠ α = 2a (∆ + 2K) 4α −∇α∆ −1∇γ∇βn 4α , (98) [a4(µ+ p)v a4(µ+ p) ∆+ 2K n6α −∇α∆ −1∇βn6β 6α , (99) [a4(µ(i) + p(i))v a4(µ(i) + p(i)) ∆ + 2K µ(i) + p(i) µ(i) + p(i) µ(i) + p(i) n6(i)α −∇α∆ −1∇βn6(i)β 6(i)α .(100) Equations for the tensor-type perturbation follow from Eq. (66) + 3HĊ ∆− 2K − 8πGΠ = n4αβ − ∇α∇β − (∆ + 3K) ∆−1∇γ∇δn −2∇(α (∆ + 2K) ∇γn4β)γ −∇β)∆ −1∇γ∇δn 4αβ . (101) In order to derive eqs. (92,98,101) it is convenient to show ∇α∇β − (χ̇+Hχ− ϕ− α− 8πGΠ) + (α|β) − 8πG (α|β) + 3HĊ ∆− 2K − 8πGΠ = n4αβ , (102) which follows from Eq. (66). Quadratic combinations of linear-order perturbation variables of all three types of perturbations contribute to all three types of perturbation to the second order. C. Comoving gauge and irrotational condition In Eq. (180) of [11] we introduced Q(i)α ≡ µ(i) + p(i) −v̄(i),α + v̄ , (103) where we put overbars to v̄(i) and v̄ in order to distinguish these from our new notations to be used in this paper; in [11] we didn’t have overbars. From Eq. (58) we have −v̄(i),α + v̄ ≡ V(i)α −Bα +ABα + 2V Cαβ + δµ(i) + δp(i) µ(i) + p(i) V(i)α −Bα Π(i)αβ µ(i) + p(i) . (104) It is more convenient to introduce the decomposition in Eq. (79). Thus, we have −v̄(i),α + v̄ ≡ −v(i),α + v δµ(i) + δp(i) µ(i) + p(i) −v(i),α + v Π(i)αβ µ(i) + p(i) . (105) The variable v̄ introduced in [11] cannot be regarded as a proper vector-type perturbation. However, if we also take the temporal comoving gauge in [11] which sets v̄ ≡ 0 together with v̄ = 0, we have v = 0 = v ; these are the same as taking the proper irrotational condition (v = 0) and the temporal comoving gauge (v = 0). Our analyses in [12, 13] are, in fact, based on taking these two conditions together. In the irrotational fluids, the temporal comoving gauge v ≡ 0 leads to ũα = 0, thus ũa coincides with the normal frame four-vector ña. From Eq. (60) we have ω̃(i)αβ = av (i)[α|β] µ(i) + p(i) −v(i),[α + v (i)[α δp(i) + ∆ + 3K Π(i) − δI(i) ∆+ 3K (i)β] (i)β] (106) where we used Eq. (70) to the linear order. For vanishing vector-type perturbation we set v ≡ 0, etc. In this case, we have ω̃(i)αβ = − µ(i) + p(i) v(i),[α δp(i) + ∆+ 3K Π(i) − δI(i) , (107) which vanishes for δp(i) = 0 = Π(i) and δI(i) = 0. VI. EFFECTS OF PRESSURE We consider a single component situation without rotational perturbations. Equations (65),(67),(68), and Eq. (64) to the linear order provide a complete set of equations we need in the following. A. Irrotational case As an irrotational fluid we ignore all vector-type perturbations, thus v α = B α = C α = Π α = 0. Quadratic combinations of linear-order perturbations of all three-types of perturbations contribute to each type of second-order perturbations. Thus, concerning the second-order scalar-type and vector-type perturbations, ignoring the pure vector- type perturbation corresponds to ignoring the vector-type perturbation only to the linear order; to the linear order the vector-type perturbation only has a decaying solution in the expanding phase. Assuming the background equations are valid Eqs. (67),(68),(65),(64) give δ̇ + 3H − (1 + w) v − 3Hα (κ− 3Hα) + (1 + w) H (1 + w) α2 − β,αβ,α χ,α|β + Ċ (1 + w) v,α|β + γ,α|β + C (1 + w) v,α v,α − δp,α +Π + ϕ,α − [(∆ + 4K)γ],α (δµ+ δp) v,α +Παβv,β , (108) a4 (µ+ p) v,α (µ+ p)α,α + δp,α +Π (µ+ p) (v − β) (v − β) +(µ+ p) v,α v − 3Hα (δµ+ δp) v,α +Π − (δµ+ δp)α,α + α (µ+ p)α,α − δp,α −Π − α,βΠ α + 2 CγβΠαγ −ΠγαC +ΠβγC , (109) κ̇+ 2Hκ− 4πG (δµ+ 3δp) + 3Ḣ + α = − ,α + ακ̇+ 3α2 − β,αβ,α + 8πG (µ+ p) v,αv,α 2∆α (α+ ϕ) + α,α [α− ϕ+ (∆ + 4K)γ] + 2α,α|β γ,α|β + C ∆ (β,αβ,α) χ,α|β + Ċ(t)αβ χ,α|β + Ċ , (110) κ = − ∆+ 3K χ+ 12πGa (µ+ p) v. (111) Equation (111) is valid to the linear order. B. Comoving gauge As the temporal and spatial gauge conditions we set v ≡ 0 ≡ γ, (112) thus, β = χ/a. Under these gauge conditions Eq. (109) gives α = − χ,αχ,α − 1 + w 1 + 3c2s 2(1 + w) + αΠ, (113) where the imperfect fluid contributions (stresses) are denoted by αΠ with ∆ + 3K 1 + w ∆ + 3K ∆−1∇α 1 + c2s 1 + w e,α +Π 1 + w δ,β + e,β +Π Πβα + 2 CβγΠαγ −ΠγαC +ΠβγC (114) Using this, Eqs. (108)-(111) give δ̇ − 3Hwδ − (1 + w) κ = − ,α + κδ + 1 + w δ2 + δΠ, (115) κ̇+ 2Hκ− 4πGµδ − 1 + w ∆+ 3K δ = − 2(1 + w) 4πGµ− 1 + 2c2s 1 + w ∆+ 3K 1 + w 2Hδκ+ ϕ,αδ,α − 2ϕ∆δ − 2δ ,α|βC χ,α|β + Ċ(t)αβ χ,α|β + Ċ +κΠ, (116) κ = − ∆+ 3K χ, (117) where Eq. (117) is valid only to the linear order, and δΠ ≡ 2H ∆+ 3K κ+ 3H 1 + w χ,α|β + Ċ ∆+ 3K 3Hδ − (1 + w) κ+ ∆ + 3K ∆−1∇α 1 + c2s 1 + w e,α +Π 1 + w δ,β + e,β +Π Πβα + 2 CβγΠαγ −ΠγαC +ΠβγC κΠ ≡ 12πGe 1 + w 3Ḣ + ∆ + 3K 1 + w ∆ + 3K ∆−1∇α 1 + c2s 1 + w e,α +Π 1 + w δ,β + e,β +Π CβγΠαγ −ΠγαC +ΠβγC −2Hκ+ 4πG 1 + 6c2s µδ + 12πGe− 1 + w ∆ + 3K ∆+ 3K ∆ + 3K ∆+ 3K 1 + w ∆ + 3K ϕ,α − 2 ∆+ 3K ),α|β . (118) Combining Eqs. (115),(116) we can derive 1 + w (µ+ p)a − c2s δ = (1 + w) χ,α|βχ,α|β − 1 + w a2κδ − δ,αχ 1 + w 4πGµ− 1 + 2c2s 1 + w ∆+ 3K 2a2Hδκ+ ϕ,αδ,α − 2ϕ∆δ − 2δ ,α|βC + (1 + w) Ċ(t)αβ χ,α|β + Ċ +(1 + w) κΠ + 1 + w 1 + w δΠ,αχ ,α. (119) Notice that the equations above are valid in the presence of general K and Λ. C. Newtonian correspondence Guided by our success in the zero-pressure case, we continue to identify κ ≡ − ∇ · u, (120) to the second order. Using Eq. (117), assuming K = 0, we can identify κ = −c χ = − ∇ · u, u ≡ ∇χ = −c∇vχ, (121) to the linear order. We have recovered the speed of light c. We have [u] = LT−1. Equation (113) becomes α = − 1 + w 1 + 3c2s 2(1 + w) + αΠ. (122) Equations (115), (116) give δ̇ − 3wHδ + (1 + w) ∇ · u = − ∇ · (δu) + 1 + w Hδ2 + δΠ, (123) ∇ · (u̇+Hu) + 1 + w δ = − ∇ · (u · ∇u)− Ċ(t)αβ uα|β + Ċ 1 + w 1 + 2c2s 1 + w δ2 + 2Hδ ∇ · u+ 2ϕ∆δ − (∇ϕ) · ∇δ + 2δ,α|βC − κΠ.(124) We have [αΠ] = 1, [δΠ] = T −1, and [κΠ] = T −2. Combining these equations or Eq. (119) gives 1 + w (µ+ p)a − c2sc 1 + w ∇ · (u · ∇u)− 1 + w 1 + w ∇ · (δu)− 1 + w + (1 + w) Ċ(t)αβ uα|β + Ċ 1 + 2c2s 1 + w δ2 − c2s 2aHδ∇ · u+ 2ϕc2∆δ − c2 (∇ϕ) · ∇δ + 2c2δ,α|βC +(1 + w) κΠ + 1 + w 1 + w u · ∇δΠ. (125) We note that, to the linear order, Eq. (125) is valid in the presence of general K and Λ, see Eq. (119). D. Linear-order relativistic pressure corrections To the linear order, ignoring the entropic perturbation e and the anisotropic stress Π, Eqs. (123)-(125) become δ̇ − 3wHδ + (1 + w) ∇ · u = 0, (126) ∇ · (u̇+Hu) + 4πG̺δ = − 1 + w , (127) 1 + w a(1 + w)̺ , (128) where ignoring the specific internal energy density ǫ we used µ = ̺c2, thus δ = δ̺/̺; in general we have µ = ̺ c2 + ǫ [18]. Equation (128) can be expanded to give 1 + w a(1 + w)̺ = δ̈ + 2− 6w + 3c2s Hδ̇ − 1 + 8w − 6c2s − 3w 4πG̺− 12 w − c2s ) Kc2 5w − 3c2s δ, (129) which is valid in the presence of K and Λ. Equation of density perturbation in the comoving gauge was first derived by Nariai in [26]. In a single component case the Newtonian equations in Eqs. (8),(12),(16) give ∇ · u = 0, (130) ∇ · (u̇+Hu) + 4πG̺δ = − , (131) δ̈ + 2Hδ̇ − 4πG̺δ = . (132) Comparing Eqs. (126),(127) with Eqs. (130),(131), we notice the presence of w ≡ p/(̺c2) term in three places in the relativistic equations. Even to the linear order the presence of these terms should be regarded as pure general relativistic effect of the isotropic pressure. The effects of pressures to the second order compared with the Newtonian equations can be found in Eqs. (123)-(125) which should be compared with Newtonian equations in Eqs. (8),(12),(16). Pressure has the genuine relativistic role in cosmology even in the background level. VII. EFFECTS OF MULTI-COMPONENT We assume zero-pressure medium, thus set p(i) ≡ 0, δp(i) ≡ 0 ≡ Π(i)αβ . (133) From Eq. (86) we notice that to the second order the collective fluid quantities differ from simple sum of individual fluid quantities. Even in the zero-pressure mediums, we have δµ(j) + µ(j)v v(j)α − vα µ(j)v v(j)α − vα µvα = µ(j)v(j)α + δµ(j) v(j)α − vα Παβ = vα(j) v(j)β − vβ δαβ v v(j)γ − vγ , (134) thus δp 6= 0 6= Παβ to the second order. Equations (69),(70),(65),(67),(68),(64) give δ̇(i) − δK + 3HA+ 1 + δ(i) vα(i) δ(i),αB α + δ(i) (δK − 3HA) +AδK + A2 −BαBα −v(i)α 2v̇α(i) +Hv Avα(i)|α − (i) + Cαβv(i)β Cα|βα v(i)β , (135) 1 + δ(i) v(i)α A,α = (δK − 3HA) v(i)α AA,α − δ(i)A,α −B βBβ|α − v(i)αv − v(i)α|βB β − v(i)βB , (136) δK̇ + 2HδK − 4πG (δµ+ 3δp) + 3Ḣ + A = AδK̇ − δK,αB δK2 + 3A2 −BαBα 2A∆A+A,αA,α − ∆ (BαBα) +A + 2CαβA,α|β Ċαβ + B(α|β) Ċαβ + Ċαα + + 8πGµvαvα, (137) δ̇ − δK + 3HA+ [(1 + δ) vα]|α = − α + δ (δK − 3HA) +AδK + A2 −BαBα −vα (2v̇ α +Hvα)− Avα|α − Cαβvβ Cα|βα vβ − 3H , (138) [a (1 + δ) vα] A,α = (δK − 3HA) vα AA,α − δ(i)A,α −B βBβ|α − − vα|βB β − vβB δp,α +Π , (139) Ċβα + +B |βα Ċγγ + δK,α = −8πGa µ(j)v(j)α. (140) where Eq. (140) is valid to the linear order. A. Irrotational case Assuming an irrotational condition we ignore all vector-type perturbations. As the spatial gauge condition we take γ ≡ 0, (141) thus, β ≡ χ/a. Equations (135)-(140) become δ̇(i) − κ+ 3Hα− 1 + δ(i) δ(i),αχ ,α + δ(i) (κ− 3Hα) + ακ+ χ,αχ,α +Hv(i),αv α∆v(i) + ϕ,αv(i),α − ϕ∆v(i) − C(t)αβv(i),α|β , (142) 1 + δ(i) = (κ− 3Hα) v(i),α + −αα,α + χ,βχ,β|α + δ(i)α,α + v(i),αv v(i),βχ , (143) κ̇+ 2Hκ− 4πG (δµ+ 3δp) + 3Ḣ + α = ακ̇− 3α2 − χ,αχ,α 2α∆α+ α,αα,α − (χ,αχ,α)− α ,αϕ,α + 2ϕ∆α+ 2C (t)αβα,α|β Ċ(t)αβ + χ,α|β χ,α|β + 8πGµv,αv,α, (144) δ̇ − κ+ 3Hα− [(1 + δ) v,α]|α = − ,α + δ (κ− 3Hα) + ακ+ χ,αχ,α +Hv,αv α∆v + ϕ,αv,α − ϕ∆v − C(t)αβv,α|β − 3H , (145) [a (1 + δ) v] α,α = (κ− 3Hα) v,α −αα,α + χ,βχ,β|α + δα,α + δp,α +Π , (146) ∆ + 3K χ+ κ = 12πGa µ(j)v(j). (147) Equation (147) is valid to the linear order. B. Linear perturbations To the linear order Eqs. (142)-(146) give δ̇(i) − κ+ 3Hα− c v(i) = 0, (148) v̇(i) +Hv(i) − α = 0, (149) κ̇+ 2Hκ− 4πG ̺(j)δ(j) + 3Ḣ + c2 α = 0, (150) ∆+ 3K χ+ κ = ̺(j)v(j), (151) δ̇ − κ+ 3Hα− c v = 0, (152) v̇ +Hv − α = 0, (153) where we have recovered the speed of light c. The following additional equations can be found in Eqs. (195),(196),(199) of [11] κ− 3Hα+ 3ϕ̇+ c χ = 0, (154) 4πG̺δ +Hκ+ c2 ∆+ 3K ϕ = 0, (155) (χ̇+Hχ)− ϕ− α = 0. (156) Equation (156) gives αχ = −ϕχ. (157) Equation (153) gives αv = 0. (158) Equations (151), (154) give ϕ̇v = χv. (159) Thus, for K = 0 we have ϕ̇v = 0, (160) which is valid even in the presence of multi-components and the cosmological constant. Equations (149), (153) give v̇χ +Hvχ + ϕχ = 0, (161) v̇(i)χ +Hv(i)χ + ϕχ = 0. (162) Equations (151), (155) give ∆+ 3K ϕχ = −4πG̺δv = −4πG ̺(j)δ(j)v = −4πG ̺(j)δ(j)v(j) . (163) We can derive density perturbation equation in many different temporal gauge (hypersurface) conditions all of which naturally correspond to gauge-invariant variables. In a single component case, density perturbation in the comoving gauge (v = 0) is known to give Newtonian result. In the multi-component situation we have many different comoving gauge conditions. Here, we consider two such gauges for δ(i) variable: one based on v = 0 gauge, and the other based on v(ℓ) = 0 gauge for a specific ℓ. (I) Equation (148) evaluated in the v = 0 gauge, and using Eq. (151) we can derive δ̇(i)v − c v(i)χ − c vχ = 0, (164) where we used χv ≡ χ − av ≡ −avχ, vv(i) ≡ v − v(i), and αv ≡ α − c −1 (av) = 0. Using Eqs. (161),(162),(163), we δ̈(i)v + 2Hδ̇(i)v − 4πG ̺(j)δ(j)v = 0. (165) This coincides exactly with the Newtonian result in Eq. (16) to the linear order, even in the presence of K. Thus, we may identify δ(i)v as the Newtonian density perturbation δi to the linear order even in the presence of K. For K = 0 we may also identify −c∇v(i)χ as the Newtonian velocity perturbation ui. However, in the presence of K, we cannot identify the relativistic variables which correspond to the Newtonian velocity perturbation of individual component. Therefore, to the linear order we have the following Newtonian correspondences δi = δ(i)v, ui = −c∇v(i)χ, (166) where the latter one is valid only for K = 0. (II) Evaluating Eq. (148) in the v(ℓ) = 0 gauge for a specific ℓ, and using Eq. (151) we can derive δ̇(i)v(ℓ) − c ∆+ 3K v(ℓ)χ = v(j) − v(ℓ) v(i) − v(ℓ) , (167) where we used χv(ℓ) ≡ χ − av(ℓ) ≡ −av(ℓ)χ, v(i)v(ℓ) ≡ v(i) − v(ℓ), and αv(ℓ) ≡ α − c av(ℓ) = 0. Using Eqs. (161),(162),(163), we have δ̈(i)v(ℓ) + 2Hδ̇(i)v(ℓ) − 4πG ̺(j)δ(j)v(j) = v(ℓ) − v(j) . (168) The terms in right-hand-sides of Eqs. (167),(168) look like relativistic correction terms present even to the linear order based on the variable δ(i)v(ℓ) . Since no such correction terms appear in Eq. (165) based on the variable δ(i)v, the relativistic correction terms in Eq. (168) can be regarded as being caused by a complicated hypersurface (gauge) choice. 1. Exact solutions Assuming K = 0, we can identify Newtonian perturbation variables as δ ≡ δv, κv ≡ − ∇ · u, u ≡ −c∇vχ, δΦ ≡ −c 2ϕχ, δi ≡ δ(i)v, ui ≡ −c∇v(i)χ. (169) Equations (152),(150),(148),(149),(163) become δ̇ = − ∇ · u, (170) u̇+Hu = − ∇δΦ, (171) δ̇i = − ∇ · ui, (172) u̇i +Hui = − ∇δΦ, (173) δΦ = 4πG̺δ = 4πG ̺jδj . (174) Under the identification in Eq. (169) these equations are valid in both Newton’s and Einstein’s gravity theories. Equations (170), (171), (174), and Eqs. (172),(173),(174), respectively, give δ̈ + 2Hδ̇ − 4πG̺δ = = 0, (175) δ̈i + 2Hδ̇i − 4πG ̺jδj = 0. (176) Equation (175) has an exact solution δ(x, t) = H cg(x) ∫ t dt + cd(x) , (177) where cg and cd are integration constants which indicate the relatively growing and decaying solutions in expanding phase; we do not consider the lower bound of integration which is absorbed to the cd mode. Equations (175), (176), and the solution in Eq. (177) are valid considering general K and Λ in the background world model. Equation (174) can be solved to give δΦ = −G̺a2 δ(x′, t) |x′ − x| d3x′. (178) From Eqs. (170),(171),(174) we can show [14] u = −a 4πG̺a2 D(x), ∇ ·D ≡ 0, (179) where the D term is the solution of the homogeneous part of Eq. (171); it decouples from the density inhomogeneity and corresponds to the peculiar velocity in the background world model. Since the D term is not connected to the density inhomogeneity and simply decays, we may ignore it to the linear order. Now, for the individual component, from Eqs. (172),(173) we have ui = −a 4πG̺a2 ∇di(x) + Di(x), ∇ ·Di ≡ 0, (180) δi = δ + ci(x)−∆di(x) ∫ t dt′ a2(t′) , (181) ̺jdj ≡ 0 ≡ ̺jcj . (182) The ci and di are the two isocurvature-type (δ = 0, thus δΦ = 0) solutions. It happens that the relatively decaying isocurvature-type solution, i.e., di-mode, temporally behaves the same as the peculiar velocity in the background, i.e., Di-mode. The relatively growing isocurvature-type solution ci does not contribute to the ui, see Eq. (172). From Eqs. (179),(180) we have ui − u = [∇di(x) +Di(x)−D(x)] , (183) which simply decays; the Di and D solutions are divergence-free and decoupled from the density perturbation, and are the peculiar velocity perturbation present in the background world model. C. Comoving gauge To the linear order, only the hypersurface condition v = 0 (the comoving temporal gauge) allows the density perturbation equation presented in the Newtonian form. Thus, we take v ≡ 0, (184) even to the second order. We take γ ≡ 0 in Eq. (141) as the spatial gauge condition. Equation (146) gives α = − χ,αχ,α − vα(j)v(j)α +∆ vα(j)v (j)|β . (185) Using this, Eqs. (142)-(145) give δ̇ − κ = − ,α + δκ+ µ(j)v (j)v(j)α + 3H ∆−1∇α vα(j)v (j)|β , (186) κ̇+ 2Hκ− 4πG̺δ = − Ċ(t)αβ + χ,α|β χ,α|β 3Ḣ + 8πG̺+ c2 µ(j)v (j)v(j)α + 3Ḣ + c2 ∆−1∇α vα(j)v (j)|β , (187) δ̇(i) − κ+ c 1 + δ(i) vα(i) δ(i),αχ ,α + δ(i)κ+Hv(i)αv ϕ,αv(i)α − 2ϕv (i)|α − 2v µ(j)v (j)v(j)α + 3H ∆−1∇α vα(j)v (j)|β , (188) 1 + δ(i) v(i)α v(i)βχ + κv(i)α − v(i)αv µ(j)∇α v(j)β (j)|β . (189) From Eqs. ((186),187), and Eqs. (187)-(189) we can derive, respectively, − 4πG̺δ (1 + δ) = − Ċ(t)αβ + χ,α|β χ,α|β 2Ḣ + 4πG̺+ µ(j)v (j)v(j)α + 6Ḣ + c2 ∆−1∇α vα(j)v (j)|β , (190) δ̇(i) + δ(i),αχ − 4πG̺δ 1 + δ(i) Ċ(t)αβ + χ,α|β χ,α|β (κ+ ϕ̇) v(i)α + 2 (κ− ϕ̇) v (i)|α v(i)αχ vα(i)v + Ḣv(i)αv (i) + 3Ḣ + 4πG̺ µ(j)v (j)v(j)α + 6Ḣ ∆−1∇α vα(j)v (j)|β . (191) Notice the O(vα v(j)α) correction terms are present in Eqs. (186),(187),(190) even in the single component situation. Except for these O(vα v(j)α) terms, the remaining parts of these equations coincide with the ones in the single component situation. D. Newtonian correspondence For K = 0, to the linear order, we have χv ≡ χ− av ≡ −avχ, κv = −c χv = c vχ, v(i)v ≡ v(i) − v = v(i)χ − vχ, ϕ̇v = 0. (192) To the linear order we identify u ≡ −c∇vχ, ui ≡ −c∇v(i)χ. (193) Now, to the second order, we attempt identifying the Newtonian perturbation variables δ, δi, u, and ui as κv ≡ − ∇ · u, χv ≡ u, u ≡ ∇u, v(i)v ≡ (ui − u) , δ ≡ δv, δi ≡ δ(i)v. (194) Using these identifications Eqs. (186)-(191) can be written as ∇ · u = − ∇ · (δu) +H |uj − u| 2 + 3∆−1∇ · [(uj − u)∇ · (uj − u)] , (195) ∇ · (u̇+Hu) + 4πG̺δ = − ∇ · (u · ∇u)− Ċ(t)αβ uα|β + Ċ 4πG̺− c2 |uj − u| 12πG̺− c2 ∆−1∇ · [(uj − u)∇ · (uj − u)] , (196) δ̇i + ∇ · ui = − ∇ · (δiui) + 2ϕ∇ · (ui − u)− (ui − u) · ∇ϕ+ 2 (u i − u |ui − u| 2 + 3H |uj − u| 2 +∆−1∇ · [(uj − u)∇ · (uj − u)] , (197) ∇ · (u̇i +Hui) + 4πG̺δ = − ∇ · (ui · ∇ui)− Ċ (t)αβ uα|β + Ċ |uj − u| 2 + 3∆−1∇ · [(uj − u)∇ · (uj − u)] , (198) − 4πG̺δ = − [a∇ · (δu)] ∇ · (u · ∇u) + Ċ(t)αβ uα|β + Ċ 4πG̺− |uj − u| 24πG̺− c2 ∆−1∇ · [(uj − u)∇ · (uj − u)] , (199) a2δ̇i − 4πG̺δ = − [a∇ · (δiui)] ∇ · (ui · ∇ui) + Ċ i + Ċ (t)αβ {∆ [u · (ui − u)]−∇ · [(ui − u) · ∇u+ u · ∇ (ui − u)]} |ui − u| 2 − 8πG |uj − u| 2 + 3∆−1∇ · [(uj − u)∇ · (uj − u)] . (200) In Sec. VII B 1 we have shown that, to the linear order, (ui − u) simply decays (∝ a −1) in an expanding phase. From Eq. (183) we have ui − u = [∇di(x) +Di(x)−D(x)] , ̺jdj ≡ 0, ∇ ·D ≡ 0 ≡ ∇ ·Di. (201) Thus, ui−u simply decays in an expanding background. If we ignore these contributions from the velocity differences, except for the presence of the tensor-type perturbation, Eq. (200) coincides exactly with the zero-pressure limit of Newtonian result in Eq. (16). Terms in the last two lines of Eq. (200) are relativistic correction terms which vanish for a single component case leading to Eq. (199); the second line in Eq. (199) also vanishes in the single component case. Ignoring quadratic combination of (ui − u) terms, we have ∇ · u = − ∇ · (δu) , (202) ∇ · (u̇+Hu) + 4πG̺δ = − ∇ · (u · ∇u)− Ċ(t)αβ uα|β + Ċ , (203) − 4πG̺δ = − [a∇ · (δu)] ∇ · (u · ∇u) + Ċ(t)αβ uα|β + Ċ , (204) δ̇i + ∇ · ui = − ∇ · (δiui) + 2ϕ∇ · (ui − u)− (ui − u) · ∇ϕ+ 2 (u i − u , (205) ∇ · (u̇i +Hui) + 4πG̺δ = − ∇ · (ui · ∇ui)− Ċ (t)αβ uα|β + Ċ , (206) a2δ̇i − 4πG̺δ = − [a∇ · (δiui)] ∇ · (ui · ∇ui) + Ċ i + Ċ (t)αβ {∆ [u · (ui − u)]−∇ · [(ui − u) · ∇u+ u · ∇ (ui − u)]} . (207) Equations (202)-(204) coincide with the density and velocity perturbation equations of a single component medium [12]; thus, except for the contribution from gravitational waves, these equations coincide with ones in the Newtonian context. If we further ignore (ui − u) terms appearing in the pure second-order combinations, Eqs. (205)-(207) become δ̇i + ∇ · ui = − ∇ · (δiui) , (208) ∇ · (u̇i +Hui) + 4πG̺δ = − ∇ · (ui · ∇ui)− Ċ (t)αβ uα|β + Ċ , (209) a2δ̇i − 4πG̺δ = − [a∇ · (δiui)] ∇ · (ui · ∇ui) + Ċ i + Ċ (t)αβ . (210) Notice that, by ignoring i-indices, Eqs. (208)-(210) coincide with Eqs. (202)-(204). In this context, except for the contribution from gravitational waves, the above equations coincide exactly with ones in the Newtonian context even in the multi-component case; compare with Eqs. (8),(9),(16) without pressure. In the single component situation such a relativistic/Newtonian correspondence to the second order was shown in [11, 12]. In the present case, the same equation valid in the single component is now valid in the multi-component case for the collective fluid variables. This justifies our identifications of Newtonian perturbation variables in Eq. (194). VIII. EFFECTS OF CURVATURE We consider a single zero-pressure, irrotational fluid. We take the temporal comoving gauge (v ≡ 0) and the spatial γ ≡ 0 gauge. In the presence of background curvature the basic equations are presented in Eqs. (115)-(117) for nonvanishing pressure, or Eqs. (188)-(191) for zero-pressure multiple component fluids. By setting pressures equal to zero in Eqs. (115)-(117), or from Eqs. (187)-(190), we have δ̇ − κ = − ,α + δκ, (211) κ̇+ 2Hκ− 4πG̺δ = − Ċ(t)αβ + χ,α|β χ,α|β , (212) δ̈ + 2Hδ̇ − 4πG̺δ = 4πG̺δ2 − (δ,αχ Ċ(t)αβ + χ,α|β χ,α|β , (213) ∆+ 3K χ+ κ = 0, (214) where Eq. (214) is valid to the linear order. Compared with the situation with vanishing curvature, the effects of curvature in the above perturbed set of equations appear only in the linear-order relation between κ and χ in Eq. (214). A. Newtonian correspondence Considering the successful Newtonian correspondence to the linear order even in the presence of the background curvature, we assume the identification in Eq. (120) is valid to the second order. Then, to the linear order, from Eq. (214) we have κ ≡ − ∇ · u ≡ − u = −c ∆+ 3K χ, (215) where u ≡ ∇u and χ = χv = −avχ. Thus, to the linear order, formally we have ∆+ 3K u. (216) In the presence of curvature, the scalar-type perturbation can be handled by solving Eqs. (211),(212),(214) together with the identifications made above. We can formally separate the effects of pure curvature contribution. Using Eqs. (215),(216), Eqs. (211),(212),(214) become ∇ · u = − ∇ · (δu) + (∇δ) · ∇ ∆+ 3K , (217) ∇ · (u̇+Hu) + 4πG̺δ = − ∇ · (u · ∇u)− Ċ(t)αβ u,α|β ∆+ 3K ∆+ 3K ∇ · u 2∆+ 3K ∆+ 3K ∇ · u ∆+ 3K ),α|β ∆+ 3K . (218) From Eq. (74), the K term can be written as (Ωt − 1) , Ωt ≡ Ω+ ΩΛ, Ω ≡ , ΩΛ ≡ . (219) IX. EFFECTS OF VECTOR-TYPE PERTURBATION The spatial C-gauge sets γ ≡ 0 ≡ C(v)α . (220) The remaining variables under this gauge condition are completely free of the spatial gauge modes and have unique spatially gauge-invariant counterparts. If we simultaneously take any temporal gauge which also removes the temporal gauge mode completely, all the remaining variables have corresponding unique gauge-invariant counterparts. The above statements are true to all orders in perturbations, see Sec. VI of [11]. From Eq. (77) we have χ, B(v)α ≡ Ψ α . (221) Thus, we have χ,α +Ψ α , Cαβ ≡ ϕg . (222) As the temporal comoving gauge we set v ≡ 0. (223) As we mentioned, the remaining variables under these gauge conditions are completely free of the gauge modes and have unique gauge-invariant counterparts to all orders in perturbation, see [11]. A. Linear perturbations To the linear order, the three types of perturbations decouple, and evolve independently. The rotational perturbation is described by Eqs. (206)-(209) in [11]: ∆ + 2K Ψ(v)α + (µ+ p) v(v)α = 0, (224) a4 (µ+ p) v a4 (µ+ p) ∆+ 2K , (225) Ψ̇(v)α + 2HΨ Π(v)α , (226) where the last equation follows from the first two. Compared with Bardeen’s notation in [16], we have B(v)α = B (1)Q(1)α , C α = − α , Ψ α = ΨQ B(1) − Q(1)α , v(v)α = vcQ v(1) −B(1) Q(1)α , v α = v v(1) − Q(1)α , (227) where Bardeen’s v c and v s , thus our v α and v α + Ψ α , are related to the vorticity and the shear, respectively. From Eq. (60), to the linear order, we have ω̃αβ = av [α|β] σ̃αβ = a ∇α∇β − . (228) Bardeen called Ψ α a ‘frame-dragging potential’. The difference between v c and v s is crucially important to show Mach’s principle including the linear order rotational perturbation in [27]. In the absence of anisotropic stress which can act as a sink or source of the angular momentum, we have Angular momentum ∝ a4 (µ+ p) v(v)α ∝ a 2Ψ(v)α ∝ constant in time. (229) Thus, for vanishing anisotropic stress, we have v(v)α ∝ a4(µ+ p) , Ψ(v)α ∝ . (230) In the zero-pressure limit we have v(v)α ∝ , Ψ(v)α ∝ . (231) B. Second-order perturbations We consider a single component situation with general pressure. We set K ≡ 0. To the linear-order we use χ ≡ χv ≡ −avχ, κ ≡ κv = c vχ. (232) The scalar-type perturbation is described by Eqs. (67),(65) which give δ̇ + 3H c2s − w δ + 3H − (1 + w) κ = −3H (1 + w) v ,αχ −Ψ vχ,α −Ψ v ,αχ − Ψ vχ − 3Hα + (1 + w)αc vχ + (1 + w) v 1− 3c2s Hv(v)α + Π(v)α (1 + w) − (2α+ ϕ) v(v)α + 2C(t)αβv Παα − vχ,α|β + (δµ+ δp) v(v)α + , (233) κ̇+ 2Hκ− (δµ+ 3δp) = − 3Ḣ + c2 v ,αχ −Ψ vχ,α −Ψ − 2Hαc (δµ+ 3δp)α+ Ḣα2 + (α+ 2ϕ) c2 α,α (α− ϕ) C(t)αβα,α|β + (µ+ p) v(v)αv(v)α v ,αχ −Ψ Ċ(t)αβ − v ,α|βχ + Ψ(v)(α|β) vχ,α|β + . (234) The vector-type perturbation is described by Eq. (68) which gives a4 (µ+ p) v(v)α (µ+ p)α,α + δp,α +Π −αδp+ (µ+ p) v ,βχ −Ψ vχ,β −Ψ α,αδµ +(µ+ p) vχ − 3Hα v(v)α − (µ+ p) −v ,βχ +Ψ v(v)β − v ,βχ +Ψ − ϕ,βΠ α + ϕ,αΠ + 2C(t)βγΠαβ|γ + C (δµ+ δp) v(v)α +Π ≡ µcAα. (235) We have [Aα] = L −1. From this we have (µ+ p)α+ δp+ Π = aµ∆−1∇ ·A, (236) a4 (µ+ p) v(v)α Π(v)α = µc ∆−1∇ ·A . (237) Equation for the tensor-type perturbation (gravitational waves) follows from Eqs. (66),(102). C. Zero-pressure case In the zero-pressure limit, Eqs. (236),(235) give α = a∆−1∇ ·A, (238) aAα ≡ − v ,βχ −Ψ vχ,β −Ψ v ,βχ −Ψ v ,βχ −Ψ v(v)β , (239) thus, α is purely second-order, and v ,βχ −Ψ vχ,β −Ψ = ∆−1∇α v ,βχ −Ψ v ,βχ −Ψ v(v)β .(240) Equations (233), (234), (237) give δ̇ − κ = v ,αχ −Ψ (v)α − v(v)α +Hv(v)α v (v)α + −ϕ,αv(v)α + 2C (t)αβv −3H∆−1∇α v ,βχ −Ψ v ,βχ −Ψ v(v)β , (241) κ̇+ 2Hκ− 4πG̺δ = v ,αχ −Ψ v ,βχ −Ψ + Ċ(t)αβ vχ,α|β −Ψ +8πG̺v(v)αv(v)α − 3Ḣ + c2 ∆−1∇α v ,βχ −Ψ v ,βχ −Ψ v(v)β ,(242) v̇(v)α +Hv v ,βχ −Ψ v ,βχ −Ψ v(v)β v ,βχ −Ψ v ,βχ −Ψ v(v)β . (243) D. Newtonian correspondence In order to compare with Newtonian equations we continue identifying κ as in Eq. (120) to the second order. To the linear order we identify u ≡ −c ∇vχ − v ≡ ∇u+ u(v), (244) u ≡ −cvχ, u (v) ≡ cv(v). (245) We introduce the following notations Uα ≡ uα + cΨ α , Ũα ≡ u,α + cΨ α . (246) As mentioned below Eq. (227), to the linear order, uα and Uα are related to the vorticity and the shear, respectively. Equations (241)-(243) become ∇ · u = − ∇ · (δU) + C(t)αβu (v) · ∇ϕ+ (v) · u(v) + 3∆−1∇α Uβ + u(v)βŨβ|α ,(247) ∇ · (u̇+Hu) + 4πG̺δ = − ∇ · (U · ∇U)− Ċ(t)αβ Ũα|β −u(v) · u(v) + ∆−1∇α Uβ + u(v)βŨβ|α , (248) (v) +Hu(v) = − U · ∇u−∇∆−1∇ · (U · ∇u) . (249) From Eq. (249) we notice that the tensor-type perturbation does not affect the vector-type perturbation to the second order. The pure scalar-type perturbation also cannot generate the vector-type perturbation to the second-order; the same is true in the Newtonian case, see below Eq. (15). From Eq. (224), to the linear order, we have v(v)α = Ψ(v)α , (250) where k is a comoving wavenumber with ∆ ≡ −k2, thus [k] = 1. Since (ck)/(aH) ∼ (visual-horizon)/(scale), we have far inside horizon : v(v)α ≫ Ψ α , U ≃ u = ∇u+ u (v), Ũ ≃ ∇u, far outside horizon : v(v)α ≪ Ψ α , U ≃ Ũ = u,α + cΨ α . (251) Apparently, contributions of vector-type perturbation to the second order depend on the visual-horizon scale. Far inside the horizon, we can ignore cΨ α compared with u α . In the matter dominated era we have δv, (252) where we used Eqs. (293),(329),(330) of [11]. Thus, the third term in the right-hand-side of Eq. (247) is (aH/kc)2-order smaller than the first term. The fourth (and last) term in the right-hand-side of Eq. (247) is (aH/kc)[u(v)/(cδ)]-order smaller than the first term. The third (and last) term in the right-hand-side of Eq. (248) is also (aH/kc)2-order smaller than the first term. Thus, Eqs. (247)-(249) give ∇ · u = − ∇ · (δu) + C(t)αβu , (253) ∇ · (u̇+Hu) + 4πG̺δ = − ∇ · (u · ∇u)− Ċ(t)αβ u,α|β + cΨ , (254) (v) +Hu(v) = − u · ∇u−∇∆−1∇ · (u · ∇u) . (255) Thus, if we could ignore the tensor-type combination in Eqs. (253),(254), Eqs. (253)-(255) coincide exactly with the Newtonian equations: see Eqs. (8),(12),(15) ignoring the pressure terms and the subindices i. The vector-tensor combinations in Eq. (253),(254) are new relativistic contributions of the vector-type perturbations; compare these two equations with Eqs. (17),(18) which are valid in the absence of the vector-type perturbations. Notice the form of last term u,α|β + cΨ in Eq. (254) which subtly differs from the expression uα|β in Eq. (18). Contributions from the vector-type perturbation become more complicated near and outside the horizon scale. The presence of vector-type metric perturbation Ψ α , the scalar-type curvature perturbation ϕ, and the tensor-type perturbation C coupled with the vector-type perturbation give additional effects. E. Pure vector-type perturbations In Sec. VII-E of [11] we have considered a situation with pure vector-type perturbation. As the analysis was made based on the fluid quantities in the normal frame, in the following we present the case based on the fluid quantities in the energy frame. Considering only the vector-type perturbation of a fluid, Eq. (68) gives a4(µ+ p) (µ+ p) v(v)α + (α|β) v(v)β c(∆ + 2K)Π 2a2(µ+ p) v(v)β +Ψ(v)|β v(v)βv . (256) Thus, for Π α = 0 we have a4(µ+ p) a4 (µ+ p) v(v)α v(v)β +Ψ(v)|β v(v)βv . (257) This differs from Eq. (365) of [11] which is due to the difference in the frame choice. The momentum constraint equation in Eq. (101) of [11] becomes ∆ + 2K Ψ(v)α + (µ+ p) v(v)α = − (α|β) v(v)β . (258) Under the gauge transformation, from Eq. (61) we have v̂(v)α = v α − v ξ(v)β,α − v ξ(v)β . (259) To the linear order, from Eq. (230) of [11] we have B̂ α = B α + ξ α . We consider a gauge transformation from the C-gauge (C α ≡ 0, without hat) to the B-gauge (B α ≡ 0, with hat). We have B̂ α ≡ 0, and B α |C−gauge = −ξ Thus, ξ(v)α = − B(v)α |C−gaugedη = −a 2Ψ(v)α ∫ η dη , (260) where we used B α |C−gauge = Ψ α ∝ a −2. Thus, Eq. (259) gives v(v)α |B−gauge = v(v)α − Ψ(v)β,α − v Ψ(v)β ∫ η dη C−gauge . (261) X. EQUATIONS WITH FIELDS A. A minimally coupled scalar field Equations in the case of a minimally coupled scalar field are presented in Eqs. (112)-(114) of [11]. The equation of motion in Eq. (112) and the full Einstein’s equations in Eqs. (99)-(105) expressed using the normal-frame fluid quantities together with the normal-frame fluid quantities for the scalar field in Eqs. (114) all in [11] provide a complete set of equations we need to the second-order. The fluid quantities in Eq. (114) of [11] are presented in the normal-frame four-vector and it is convenient to know the conventionally used fluid quantities which are based on the energy-frame four-vector. These latter quantities can be read from Eqs. (88),(114) of [11] and Eq. (79) as QN(φ)α = − φ̇δφ,α + δφ,α δφ̇− φ̇A µ(φ) + p(φ) V (φ)α −Bα +ABα + 2V (φ)βCαβ δµ(φ) + δp(φ) V (φ)α −Bα µ(φ) + p(φ) + δµ(φ) + δp(φ) −v(φ),α + v (φ,v) δµ(φ) = δµN(φ) − δφ,αδφ,α, δp (φ) = δpN(φ) − δφ,αδφ,α, δφ,αδφ,β − δφ,γδφ,γ . (262) Thus, using Eq. (114) of [11] we have v(φ) = ∆−1∇α δφ̇−Aφ̇ v(φ,v)α = δφ̇− φ̇A δφ,α −∇α∆ δφ̇− φ̇A δµ(φ) = φ̇δφ̇− φ̇2A+ V,φδφ+ δφ̇2 − δφ,αδφ,α + V,φφδφ 2 − 2φ̇δφ̇A+ φ̇δφ,αB α + 2φ̇2A2 − φ̇2BαBα, δp(φ) = φ̇δφ̇− φ̇2A− V,φδφ+ δφ̇2 − δφ,αδφ,α − V,φφδφ 2 − 2φ̇δφ̇A+ φ̇δφ,αB α + 2φ̇2A2 − φ̇2BαBα, = 0. (263) Notice that no anisotropic stress is caused by a minimally coupled scalar field even to the second order in perturbations. The uniform-field gauge takes δφ ≡ 0 as a temporal gauge (slicing) condition to the second-order in perturbation. The uniform-field gauge gives v(φ) = 0 which is the comoving gauge, and vice versa. Thus, δφ = 0 ↔ v(φ) = 0. (264) We also have δµ(φ) − δp(φ) = 2V,φδφ+ V,φφδφ 2, (265) and under the uniform-field gauge we have = −φ̇2 A (1− 2A) + . (266) Equation (263) apparently shows that the vector-type perturbation v (φ,v) α does not vanish to the second-order. However, the second-order quantities in right-hand-side depend on the temporal gauge condition for the scalar-type perturbations, and trivially vanish for the uniform-field gauge. We can also show that it vanishes for the uniform- density gauge or the uniform-pressure gauge where we have δφ̇ − φ̇A ∝ δφ to the linear order. In fact, we can show that the vector-type perturbation is not sourced by the scalar field to the second order by evaluating the rotational tensor ω̃αβ in Eq. (60) for the scalar field: i.e., for a minimally coupled scalar field we have = 0, (267) to the second order. Thus, we conclude that a minimally coupled scalar field does not contribute to the rotational perturbation to the second order in perturbations. In fact, we can show that a single scalar field do not support vector-type perturbations to all orders in perturbations. As we take the energy frame, thus q̃a ≡ 0, from Eq. (23) of [11] we can show ũa = −φ̃,cφ̃,c . (268) Using the definition of the vorticity tensor in Eq. (33) we can show that ω̃ab = 0. Using the fluid quantities in Eq. (263) we can handle the scalar field using our non-ideal fluid formulation. The fluid equations in the energy frame, like Eqs. (62)-(68), remain valid in the case of scalar field with the fluid quantities expressed as in Eq. (263). The anisotropic stress vanishes and the entropic perturbation e is given as e(φ) ≡ δp(φ) − c2(φ)δµ (φ), c2(φ) ≡ ṗ(φ) µ̇(φ) φ̈− V,φ φ̈+ V,φ . (269) Under the comoving gauge v ≡ 0, using Eq. (266) we have δp(φ)v = δµ v , e 1− c2(φ) δµ(φ)v , (270) which is a well known relation, now valid to the second order in perturbations. A fluid formulation of the scalar field to the linear order is presented in [24, 29]. Using Eq. (270) together with vanishing anisotropic stress, Eqs. (62)-(68) or (88)-(94) provide the fluid formulation for a minimally coupled scalar field to the second order in perturbations. The perturbed equation of motion of the scalar field is presented in Eq. (112) of [11]. B. Minimally coupled scalar fields In the case of multiple minimally coupled scalar fields, the equation of motions in Eq. (119) and the full Einstein’s equations in Eqs. (99)-(105) expressed using the normal-frame fluid quantities together with the normal-frame fluid quantities for the scalar field in Eqs. (121) all in [11] provide a complete set of equations we need to the second-order. The fluid quantities in Eq. (121) of [11] are presented in the normal-frame four-vector and it is convenient to know the conventionally used fluid quantities which are based on the energy-frame four-vector. These latter quantities can be read from Eqs. (121) of [11] and Eq. (58) as v(φ) = φ̇(k)δφ(k) +∆ δφ̇(k) − φ̇(k)A δφ(k) − 2φ̇(k) φ̇(l)δφ(l) v(φ,v)α = − δφ̇(k) − φ̇(k)A δφ(k) − 2φ̇(k) φ̇(l)δφ(l) δφ̇(k) − φ̇(k)A δφ(k) − 2φ̇(k) φ̇(l)δφ(l) δµ(φ) = δµN(φ) − φ̇(k)δφ φ̇(l)δφ(l),α δp(φ) = δpN(φ) − φ̇(k)δφ φ̇(l)δφ(l),α φ̇(k)δφ(k),α φ̇(l)δφ(l),β φ̇(k)δφ φ̇(l)δφ(l),γ (271) where the normal-frame fluid quantities are presented in Eq. (121) of [11]. Thus, by moving into the energy-frame fluid quantities from the normal-frame ones we have rather complicated terms which do not cancel out nicely any term in the normal-frame quantities. In single field case we had such cancelations, for example we have Π = 0 in Eq. (263). But, we do not have such a luxury in the multi-component situation. Apparently Π in the energy-frame does not vanish and looks more complicated. Thus, in the multi-field situation we had better use both the fluid quantities and Einstein’s equations all expressed in the normal frame: these are Eqs. (99)-(105),(121) in [11]. As the temporal gauge condition we can set any one field perturbation, say the specific ℓ-th one δφ(ℓ), equal to zero which might be called the uniform-φ(ℓ) gauge to the second order. This apparently differs from the comoving gauge which sets v(φ) ≡ 0. In the multi-component situation we cannot take a gauge condition which makes v (φ,v) α = 0. Thus, the multiple scalar fields source the vector-type perturbation to the second order in perturbations. The scalar-, vector-, and tensor-type decomposition of the anisotropic stress Π can be read by using decomposition formulae in Eq. (177) of [11]. In multiple-field situation, it is ad hoc and cumbersome (if not impossible) to introduce individual fluid quantity for each field variable even to the background and linear order perturbations, see [30]. C. Generalized gravity case In Sec. IV.D of [11]we presented the equation of motion and effective fluid quantities in a class of generalized gravity theories together with additional presence of fluids and fields to the second order. The equation of motion is in Eq. (128) of [11], and the full Einstein’s equations in Eqs. (99)-(105) expressed using the normal-frame fluid quantities together with the normal-frame effective fluid quantities in Eq. (130) of [11] provide a complete set of equations we need to the second-order. The effective fluid quantities in Eq. (130) of [11] are presented in the normal-frame four-vector and using Eq. (88) in [11] we can easily derive the effective fluid quantities based on the energy-frame four-vector. As in the multiple field case in a previous subsection, by moving into the energy-frame, the effective fluid quantities become more complicated compared with the ones in the normal-frame. Thus, in these class of generalized gravity theories we had better use both the fluid quantities and Einstein’s equations all expressed in the normal frame: these are Eqs. (99)-(105),(130) in [11]. XI. CURVATURE PERTURBATIONS AND LARGE-SCALE CONSERVATIONS In the large-scale limit the spatial curvature perturbation ϕ in several different gauge conditions is known to remain constant in expanding phase. Often the conservation properties are shown based on the first time derivative of the curvature perturbation. In order to show the conservation properties properly we have to construct the closed form second-order differential equations for the curvature perturbation. In the following we will derive such first-order and second-order differential equations for ϕ̇v, ϕ̇χ, ϕ̇κ, and ϕ̇δ. First we will derive equations for the linear perturbation including the background curvature and non-ideal fluid properties. Then we will derive equations for the second order perturbation assuming a flat background; including the background curvature is trivial, though. We consider a single-component fluid. A. Linear-order equations We introduce a combination Φ ≡ ϕv − 4πG (µ+ p) ϕχ. (272) This combination was first introduced by Field and Shepley in [28]. From Eqs. (88),(90),(92), Eqs. (88)-(90),(92),(94), Eqs. (88),(93), and Eqs. (88),(89),(91), respectively, we can derive 4πG (µ+ p) Φ− 8πGHΠ, (273) 4πG (µ+ p) , (274) ϕ̇δ = , (275) ϕ̇κ = − 3Ḣ +∆/a2 1 + 3c2s ) ∆+ 3K ϕκ − 12πGe χκ. (276) These equations were presented in [24, 31] We can derive closed form second-order differential equations for Φ, ϕχ, ϕδ, and ϕκ (µ+ p) a 8πGH2 = c2s ϕχ − 4πG , (277) H2c2s (µ+ p) a3 (µ+ p) a3 H2c2s = c2s , (278) 3Ḣ +∆/a2 (µ+ p) a3 (µ+ p) a3 3Ḣ +∆/a2 ϕ̇δ + 4πG (µ+ p) + c2s (∆ + 3K) /a 12πG (µ+ p)− (∆ + 3K) /a2 3Ḣ + , (279) ϕ̇κ + 3Ḣ +∆/a2 1 + 3c2s ) ∆+ 3K ϕκ − 12πGe 3Ḣ +∆/a2 1 + 3c2s ) ∆+ 3K ϕκ − 12πGe + 8πGΠ . (280) Equations (277),(278) follow by combining Eqs. (273),(274). Equations (279) and (280) follow from Eqs. (88),(89),(91),(92), and Eqs. (88),(89),(91),(92), respectively. From Eqs. (89),(90) we can derive ∆+ 3K 12πG (µ+ p) a2 ϕκ = ϕv − ∆+ 3K 12πG (µ+ p)a2 ϕχ = Φ− 12πG (µ+ p) a2 ϕχ. (281) These relations were presented in [17]. In the large-scale limit and in near flat background, thus ignoring ∆ and K terms, we have Φ ≃ ϕv ≃ ϕδ ≃ ϕκ, (282) to the leading order in the large-scale expansion. In the large-scale limit, ignoring ∆ terms, in near flat background and for an ideal fluid case, thus setting K = 0 and e = 0 = Π, Eqs. (277)-(280) give (µ+ p) a , (283) ϕ̇v ∝ H2c2s (µ+ p) a3 , (284) ϕ̇δ ∝ ϕ̇κ ∝ . (285) Notice that if we simply ignore the ∆ terms in Eqs. (274)-(276) we simply have ϕ̇v = ϕ̇δ = ϕ̇κ = 0. In such a way we cannot recover the terms in the right-hand-side of Eqs. (284),(285); these terms lead to decaying solutions (in an expanding era) in the large-scale limit and are higher order in the large-scale expansion compared with the decaying solution of ϕχ, see below. From Eqs. (283)-(285) we have general large-scale asymptotic solutions ϕχ = 4πGC(x) ∫ t a (µ+ p) dt+ d(x) , (286) ϕv = C(x) + ∫ t c2sH (µ+ p) a3 dt, (287) ϕδ = ϕκ = C(x) + ∫ t dt , (288) where C(x) and d(x) are integration constants which correspond to the relatively growing and decaying modes, respectively, in an expanding phase; in a collapsing phase the roles are reversed. The coefficients are fixed using the relations in Eqs. (273),(274),(281). Notice that for the C-mode the relation in Eq. (282) is satisfied, and simply remain constant. For the d-mode, ϕv, ϕδ, and ϕκ are ∆/(aH) 2-order higher compared with the d-mode of ϕχ. The ϕv is one of the well known conserved quantity in the large-scale even in the context of generalized gravity theories [24, 32]. In order to evaluate the solutions to the second order in the next section, we need complete sets of linear order solutions for different gauge conditions. For an ideal fluid, and for a minimally coupled scalar field such complete sets of solutions are presented in tabular forms in [29, 31]. In the following we summarize such sets of solutions in an ideal fluid case for four different gauge conditions. From Table 8 of [31] we have ϕχ = −αχ = C , δχ = − = −2C adt, vχ = −C ϕv = C, Hχv = C adt, δv = − 1 + w αv = − ϕδ = C, Hχδ = C = 3αδ = − C, vδ = ϕκ = C, Hχκ = C adt, δκ = −3 1 + w 1 + 3c2s ακ = − C, vκ = − adt. (289) For corresponding sets of solutions for a minimally coupled scalar field, see Table 1 of [29]. Compared with the notation used in [31] we have γΨ = −8πG(µ+p)av. The lower bounds of integration of solutions in Eq. (289) give behaviors of d-modes. For solutions without integration, the d-modes are ∆/(aH)2-order higher than the non-vanishing d-mode, for example, see the solutions in Table 2 of [31]. Thus, the d-modes are ϕχ = −αχ = d , δχ = − , vχ = d ∫ t c2sH (µ+ p) a3 dt, Hχv = − d, δv = − 1 + w αv = − ∫ t dt , Hχδ = − ∫ t dt , αδ = ∫ t dt , vδ = ∫ t dt , Hχκ = − d, δκ = −3 1 + w 1 + 3c2s ακ = − ∫ t dt , vκ = d. (290) B. Second-order equations We assume K = 0. From Eqs. (88),(90),(92), Eqs. (88)-(90),(94), Eqs. (88),(93), and Eqs. (88),(89),(91), respec- tively, we can derive (ϕ−Hχ) 4πG (µ+ p) (ϕ− aHv)− 8πGHΠ+ (n0 − n2)−Hn4, (291) (ϕ− aHv) 4πG (µ+ p) (ϕ−Hχ)− n1 +Hn2 (n0 − n2)− aHn6, (292) ϕ̇δ = , (293) ϕ̇κ = − 3Ḣ +∆/a2 1 + 3c2s ϕκ − 12πGe− 1 + 3c2s n1 − n3 n0. (294) The perturbed order variables in Eqs. (293),(294) are evaluated in the uniform-density gauge (δ ≡ 0), and the uniform-expansion gauge (κ ≡ 0), respectively. Equation (293) also follows from Eq. (41) evaluated to the second order. We can derive closed form second-order differential equations for ϕv, ϕχ, ϕδ, and ϕκ (ϕ−Hχ) + 8πGHΠ− (n0 − n2) +Hn4 = c2s (ϕ−Hχ)− 4πG + c2s n1 +Hn2 4πG (µ+ p) (n0 − n2)− aHn6 , (295) H2c2s 4πG (µ+ p) a3 4πG (µ+ p) a3 H2c2s (ϕ− aHv) (n0 − n2) + aHn6 n1 −Hn2 = c2s ϕ− aHv − Hc2s∆ 4πG (µ+ p) a2 (n0 − n2)−Hn4 , (296) 1 + ∆/(3a2Ḣ) 1 + ∆/(3a2Ḣ) ϕ̇δ + 3Ḣ +∆/a2 ṅ1 + n1 −Hn3 1− c2s∆/(a 3 + ∆/(a2Ḣ) 3Ḣ + 9Ḣa2 − n3 − 3Ḣ + , (297) ϕ̇κ + 3Ḣ +∆/a2 1 + 3c2s − 12πGe− n3 −ϕκ + 3Ḣ +∆/a2 1 + 3c2s − 12πGe− n3 − 8πGΠ− n4 . (298) Equations (295),(296) follow by combining Eqs. (291),(292). Equation (297) follows from Eqs. (88),(89),(91),(92). Equation (298) follows from Eqs. (88),(89),(91),(92). The perturbed order variables in Eqs. (297),(298) are evaluated in the uniform-density gauge (δ ≡ 0), and the uniform-expansion gauge (κ ≡ 0), respectively. C. Large-scale solutions Now, we assume an ideal fluid, thus set e = 0 = Π. In the large-scale limit, thus ignoring the ∆/(aH)2-order higher terms, Eqs. (296)-(298) give ϕ̇v − (n0 − n2) + aHn6 + 4πG (µ+ p) n1 +Hn2 H2c2s 4πG (µ+ p) a3 , (299) ϕ̇δ + ṅ1 + Hn1 −Hn3 , (300) ϕ̇κ − 1 + 3c2s n1 + n3 , (301) where the perturbed order variables in Eq. (299) are evaluated in the comoving gauge (v ≡ 0). We already used the behavior of linear order solutions in Eqs. (289),(290) in order to show that the right-hand-side of Eqs. (296)-(298) vanish. Using the solutions in Eqs. (289),(290) we can show that ϕv − ϕ +O(∆C2,∆2d2) ∝ H2c2s (µ+ p)a3 , (302) ϕδ − ϕ +O(∆C2,∆2d2) ∝ ϕκ − ϕ +O(∆C2,∆2d2) ∝ . (303) Thus, we have general large-scale asymptotic solutions ϕv − ϕ v = C(x) + ∫ t c2sH (µ+ p) a3 dt, (304) ϕδ − ϕ δ = ϕκ − ϕ κ = C(x) + ∫ t dt , (305) where C(x) and d(x) are integration constants now including the second-order contributions, i.e., C = C(1) + C(2), etc. Ignoring the transient solutions in an expanding phase we have ϕv = ϕδ = ϕκ = C(x), (306) even to the second order in perturbations in the large-scale limit. XII. DISCUSSION In this work we presented pure general relativistic effects of second-order perturbations in Friedmann cosmological world model. In our previous work we have shown that to the second-order perturbations, the density and velocity perturbation equations of general relativistic zero-pressure, irrotational, single-component fluid in a flat background coincide exactly with the ones known in Newton’s theory, [12]. We also have shown the effect of gravitational waves to the second-order, and pure general relativistic correction terms appearing in the third-order perturbations, [12, 13]. Here, we presented results of second-order perturbations relaxing all the assumptions made in our previous work in [12]. We derived the general relativistic correction terms arising due to (i) pressure, (ii) multi-component, (iii) background curvature, and (iv) rotation. We also presented a general proof of large-scale conserved behaviors of curvature perturbation variable in several gauge conditions, now to the second order. Effects of pressure can be found in Eqs. (123)-(125). As we emphasized, the effect of pressure is generically relativistic even in the background world model and the linear order perturbations. Still, our equations show the pure general relativistic effects of pressure (including stresses) appearing in the second-order perturbations. Effects of multi-component fluids can be found in Eqs. (195)-(200). Although these equations apparently show deviations from Newtonian situation, in Sec. VII D we showed that if we ignore purely decaying terms in an expanding phase the equations are effectively the same as in the Newtonian situation. Effects of background spatial curvature K can be read from Eqs. (217),(218) or Eqs. (211)-(214). Effects of vector-type perturbation can be read from Eqs. (247)-(249). In the small-scale limit we showed that, if we ignore the tensor-type perturbation, the equations coincide with the Newtonian ones. Our results may have important practical implications in cosmology and the large-scale structure formation. Our new result showing relativistic/Newtonian correspondence in the zero-pressure irrotational multi-component fluids is practically relevant in currently favored cosmology where baryon and dark matter are two important ingredients of the current matter content in addition to the cosmological constant. All equations in our work are valid in the presence of the cosmological constant. A related important result is the relativistic/Newtonian correspondence valid in the presence of rotational perturbation far inside horizon. Thus, inside the horizon scale, even in the presence of rotational perturbations we can still rely on the Newtonian equations to handle quasi-linear evolution of large-scale structures. As the spatial curvature in the present cosmological era is known to be small [33], the possible presence of small spatial curvature may not be important in the second-order perturbations. Still, while the Newtonian equation is exactly valid to the linear order even in the presence of the spatial curvature, we have nontrivial general relativistic correction terms present to the second order in perturbations. Our second-order perturbation equations in the presence of pressure may have an interesting role as we approach early stage of universe where the effect of radiation becomes important. The importance of pressure to the second-order perturbations, of course, depends on whether nonlinear effects are significant in the early evolution stage of the large-scale structure during the radiation era and in the early matter dominated era. Realistic estimations of the diverse pure general relativistic contributions using the complete set of equations presented in this work are left for future investigations. In an accompanying paper we will investigate the effects of third-order perturbations of zero-pressure irrotational multi-component fluids in a flat background. This is one obvious remaining issue in our series of investigation of nonlinear cosmological perturbations where nontrivial general relativistic effects are expected. In the case of a single fluid we presented the pure general relativistic effects appearing in the third order in [13]. Corresponding results in the case of multi-component will be presented in [34]. Acknowledgments H.N. was supported by grant No. C00022 from the Korea Research Foundation. [1] M. Colless, et al., 2003, Preprint astro-ph/0306581. [2] K. Abazajian, et al. in http://www.sdss.org/dr3/ (2004). [3] C.L. Bennett et al., Astrophys. J. 464, L1 (1996) Preprint astro-ph/9601067. [4] C.L. 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Hwang, Astrophys. J. 375, 443 (1991). [25] H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984). [26] H. Nariai, Prog. Theor. Phys. 41, 686 (1969). [27] C. Schmid, Phys. Rev. D 74, 044031 (2006), Preprint gr-qc/0508066. [28] G.B. Field and L.C. Shepley, Astrophys. Space. Sci. 1, 309 (1968). [29] J. Hwang, Astrophys. J. 427, 542 (1994). [30] J. Hwang and H. Noh, Phys. Lett. B, 495, 277 (2000), Preprint astro-ph/0009268; Class. Quant. Grav. 19, 527 (2002) Preprint astro-ph/0103244. [31] J. Hwang, Astrophys. J. 415, 486 (1993). [32] J. Hwang and H. Noh, Phys. Rev. D 71, 063536 (2005), Preprint gr-qc/0412126 . [33] P. de Bernardis, et al., Nature 404, 955 (2000), Preprint astro-ph/0004404. [34] J. Hwang and H. Noh, in progress. http://arxiv.org/abs/gr-qc/0508066 http://arxiv.org/abs/astro-ph/0009268 http://arxiv.org/abs/astro-ph/0103244 http://arxiv.org/abs/gr-qc/0412126 http://arxiv.org/abs/astro-ph/0004404 Contents Introduction Newtonian nonlinear perturbations Summary of previous work Relativistic fully nonlinear equations Covariant equations ADM equations Second-order perturbations Basic equations in the energy-frame Decomposition Comoving gauge and irrotational condition Effects of pressure Irrotational case Comoving gauge Newtonian correspondence Linear-order relativistic pressure corrections Effects of multi-component Irrotational case Linear perturbations Exact solutions Comoving gauge Newtonian correspondence Effects of curvature Newtonian correspondence Effects of vector-type perturbation Linear perturbations Second-order perturbations Zero-pressure case Newtonian correspondence Pure vector-type perturbations Equations with fields A minimally coupled scalar field Minimally coupled scalar fields Generalized gravity case Curvature perturbations and large-scale conservations Linear-order equations Second-order equations Large-scale solutions Discussion Acknowledgments References

We present general relativistic correction terms appearing in Newton's gravity to the second-order perturbations of cosmological fluids. In our previous work we have shown that to the second-order perturbations, the density and velocity perturbation equations of general relativistic zero-pressure, irrotational, single-component fluid in a flat background coincide exactly with the ones known in Newton's theory. Here, we present the general relativistic second-order correction terms arising due to (i) pressure, (ii) multi-component, (iii) background curvature, and (iv) rotation. In case of multi-component zero-pressure, irrotational fluids under the flat background, we effectively do not have relativistic correction terms, thus the relativistic result again coincides with the Newtonian ones. In the other three cases we generally have pure general relativistic correction terms. In case of pressure, the relativistic corrections appear even in the level of background and linear perturbation equations. In the presence of background curvature, or rotation, pure relativistic correction terms directly appear in the Newtonian equations of motion of density and velocity perturbations to the second order. In the small-scale limit (far inside the horizon), relativistic equations including the rotation coincide with the ones in Newton's gravity.

Introduction 2 II. Newtonian nonlinear perturbations 3 III. Summary of previous work 5 IV. Relativistic fully nonlinear equations 6 A. Covariant equations 6 B. ADM equations 8 V. Second-order perturbations 8 A. Basic equations in the energy-frame 10 B. Decomposition 14 C. Comoving gauge and irrotational condition 16 VI. Effects of pressure 17 A. Irrotational case 17 ∗Electronic address: jchan@knu.ac.kr †Electronic address: hr@kasi.re.kr http://arxiv.org/abs/0704.1927v1 mailto:jchan@knu.ac.kr mailto:hr@kasi.re.kr B. Comoving gauge 17 C. Newtonian correspondence 19 D. Linear-order relativistic pressure corrections 20 VII. Effects of multi-component 20 A. Irrotational case 21 B. Linear perturbations 22 1. Exact solutions 24 C. Comoving gauge 25 D. Newtonian correspondence 26 VIII. Effects of curvature 28 A. Newtonian correspondence 28 IX. Effects of vector-type perturbation 29 A. Linear perturbations 29 B. Second-order perturbations 30 C. Zero-pressure case 31 D. Newtonian correspondence 31 E. Pure vector-type perturbations 33 X. Equations with fields 33 A. A minimally coupled scalar field 33 B. Minimally coupled scalar fields 35 C. Generalized gravity case 35 XI. Curvature perturbations and large-scale conservations 36 A. Linear-order equations 36 B. Second-order equations 38 C. Large-scale solutions 39 XII. Discussion 39 Acknowledgments 40 References 40 I. INTRODUCTION Large amount of cosmological data on the large-scale structures and motions of galaxies [1, 2], and the temperature and polarization anisotropies of cosmic microwave background radiation [3, 4] have been accumulating recently. In current standard cosmological scenario such structures are explained as small (linear) or large (nonlinear) deviations from spatially homogeneous and isotropic Friedmann background world model. In order to explain these data theo- retically, researchers rely on the linear perturbation theory based on relativistic gravity, and quasi-linear perturbation theories and nonlinear simulations based on Newton’s gravity. To the linear order in perturbation the general rel- ativistic result was first derived by Lifshitz in 1946 [5], and later shown to coincide with the Newtonian result in a zero-pressure medium [6]. The same is also known to be true for the background world model. That is, the general relativistic result was first derived by Friedmann in 1922 [7], and later shown to coincide with Newtonian result in a zero-pressure medium [8]. The observed large-scale distribution of galaxies shows that in the largest observed scale (say, larger than several hundred mega-parsec scale) the distribution may not be inconsistent with the linear assumption around the Friedmann background. However, as the scale becomes smaller the distribution apparently shows quasi-linear to fully nonlinear structures. The fully nonlinear processes occur in small scale where the relativistic effects characterized by GM/rc2 ∼ v2/c2 are quite small. If we could ignore such relativistic effects, Newton’s gravity would be sufficient to handle the relevant nonlinear processes. If we need to consider the weakly relativistic correction terms in fully nonlinear stage, instead of the relativistic perturbation approach which can handle the fully relativistic processes under weakly nonlinear assumption, we can use the post-Newtonian approximation developed in the context of cosmology in [9]. For structures in the quasi-linear evolution phase, previous researches were based on Newton’s gravity especially assuming the single component zero-pressure fluid without rotational perturbation [10]. In our previous works in [11, 12] we have shown that, in the single component zero-pressure fluid without rotational perturbation, cosmological scalar-type perturbation equations in a spatially flat background coincide exactly with the Newtonian ones up to the second order in perturbation. In [11, 12] we also have shown the contribution of gravitational wave perturbations to the hydrodynamic parts in the second-order perturbations. In Newton’s gravity the hydrodynamic equations of zero- pressure fluid contain only the quadratic order nonlinearity. In [13] we presented pure general relativistic correction terms appearing in the third-order perturbation, and showed that all third-order correction terms are 10−5 times smaller than the second-order relativistic/Newtonian terms, and independent of the horizon scale. In this work we will take into account of the pure general relativistic effects appearing in the second-order perturba- tions which were ignored in our previous work in [12]. We will consider general relativistic effects of (i) pressure, (ii) multi-component, (iii) background curvature, (iv) rotation in cosmological fluids to the second-order perturbations. As results we will show that, although in [12] we have shown the exact relativistic/Newtonian correspondence to the second-order perturbations by ignoring the above four conditions and the gravitational waves, as we take these four effects into account we often encounter pure general relativistic effects appearing in the corresponding Newtonian equa- tions even to the second order in perturbations. Our results will show that the relativistic/Newtonian correspondence continues even in the multi-component situation assuming zero-pressure irrotational fluid in a flat background, but in the presence of the cosmological constant. This is a practically useful result because the matter content of present universe is dominated by collisionless dark matter and baryon both of which practically have zero-pressure. In the other three cases, relaxing any of the assumptions about pressure, rotation, and background curvature generally leads to pure general relativistic correction terms to the second order. We will present such correction terms in the context of Newtonian hydrodynamics. One additional relativistic/Newtonian correspondence occurs in the case of rotation in small-scale (sub-horizon-scale) limit which is another practically important result. This correspondence allows us to use the Newtonian equations safely in such a small-scale limit even in the presence of rotational perturbation to the second order. In Sec. II we summarize Newtonian hydrodynamic perturbation equations valid to fully nonlinear order. In Sec. III we briefly summarize our previous result of relativistic/Newtonian correspondence to the second order, and pure general relativistic correction terms appearing in the third order. In Sec. IV we present parts of the covariant and the ADM (Arnowitt-Deser-Misner) equations which are valid in multi-component situation. In Sec. V we present the basic perturbation equations valid to second order. In [11] the basic set of equations was presented using fluid quantities based on the normal-frame four-vector. The fluid quantities in the present work are based on the energy- frame four-vector, and in this section we present the basic equations using such fluid quantities. In Secs. VI-IX we analyse the effects of the pressure, the multi-components, the background curvature, and the rotational perturbation, respectively. In Sec. X we present equations in the scalar fields and generalized gravity theories using the energy-frame fluid quantities. In Sec. XI we properly derive conservation properties of curvature perturbation in various temporal gauge (hypersurface) conditions to the second order in perturbations. Section XII is a discussion. In this work we follow notations used in [11, 12]. We set c ≡ 1, but when we compare with Newtonian case we often recover the speed of light c. II. NEWTONIAN NONLINEAR PERTURBATIONS In order to compare properly the relativistic results with the Newtonian ones, in this section we summarize the Newtonian cosmological perturbation theory in fully nonlinear context. We consider multi-component fluids in the presence of isotropic pressure. In case of n-fluids with the mass densities ̺i, the pressures pi, the velocities vi (i = 1, 2, . . . n), and the gravitational potential Φ, we have ˙̺i +∇ · (̺ivi) = 0, (1) v̇i + vi · ∇vi = − ∇pi −∇Φ, (2) ∇2Φ = 4πG ̺j . (3) Assuming the presence of spatially homogeneous and isotropic but temporally dynamic background, we introduce fully nonlinear perturbations as ̺i = ¯̺i + δ̺i, pi = p̄i + δpi, vi = Hr+ ui, Φ = Φ̄ + δΦ, (4) where H ≡ ȧ/a, and a(t) is a cosmic scale factor. We move to the comoving coordinate x where r ≡ a(t)x, (5) ∇ = ∇ −Hx · ∇ . (6) In the following we neglect the subindex x. To the background order we have ˙̺i + 3H̺i = 0, ̺j , H , (7) where E is an integration constant which can be interpreted as the specific total energy in Newton’s gravity; in Einstein’s gravity we have 2E = −Kc2 where K can be normalized to be the sign of spatial curvature. To the perturbed order we have [14] δ̇i + ∇ · ui = − ∇ · (δiui) , (8) u̇i +Hui + ui · ∇ui = − a ¯̺i 1 + δi ∇δΦ, (9) ∇2δΦ = 4πG ¯̺jδj . (10) By introducing the expansion θi and the rotation −→ω i of each component as ∇ · ui, −→ω i ≡ ∇× ui, (11) Eq. (9) gives θ̇i + 2Hθi + 4πG ¯̺jδj = − ∇ · (ui · ∇ui)− a2 ¯̺i 1 + δi , (12) −̇→ω i + 2H −→ω i = − ∇× (ui · ∇ui) + a2 ¯̺i (∇δi)×∇δpi (1 + δi)2 . (13) By introducing decomposition of perturbed velocity into the potential- and transverse parts as ui ≡ ∇ui + u i , ∇ · u i ≡ 0; θi = −→ω i = i , (14) instead of Eq. (13) we have i +Hu i = − ui · ∇ui + 1 + δi −∇∆−1∇ · ui · ∇ui + 1 + δi . (15) We note that the pure ui contributions in the right-hand-side of Eq. (13) or Eq. (15) vanish. Thus, under van- ishing pressure, pure irrotational perturbation cannot generate the rotational perturbation. Equation (13) shows that presence of pressure perturbation oblique (i.e., non-parallel) to the density perturbation can generate rotational perturbation. Combining Eqs. (8),(12) we can derive δ̈i + 2Hδ̇i − 4πG ¯̺jδj = − [a∇ · (δiui)] ∇ · (ui · ∇ui) + a2 ¯̺i 1 + δi . (16) Equations (8)-(16) are valid to fully nonlinear order. Notice that for vanishing pressure these equations have only quadratic order nonlinearity in perturbations. III. SUMMARY OF PREVIOUS WORK In [11, 12] we have derived second-order perturbation equations valid for the single component, irrotational, and zero-pressure medium in zero-curvature background. These are ∇ · u = − ∇ · (δu) , (17) ∇ · (u̇+Hu) + 4πG̺δ = − ∇ · (u · ∇u)− Ċ(t)αβ uα|β + Ċ , (18) δ̈ + 2Hδ̇ − 4πG̺δ = ∇ · (u · ∇u)− [a∇ · (δu)] + Ċ(t)αβ uα|β + Ċ . (19) Except for the presence of tensor-type perturbation, Eqs. (17)-(19) are exactly the same as the ones known in the Newtonian theory. We note that these equations are valid in the presence of Λ. To the linear order, these are valid in the presence of general K, see Sec. VIII. We have correctly identified the relativistic density and velocity perturbation variables which correspond to the Newtonian counterparts to the second order. In the relativistic context, our δ and u are the density perturbation and (related to) the perturbed expansion scalar, respectively, in the comoving gauge; the variables are equivalently gauge-invariant. However, we were not able to identify relativistic variable which corresponds to the Newtonian gravitational potential to the second order; this is understandable if we consider the factor of two difference between Einstein’s (post-Newtonian) and Newton’s gravity theories in predicting the light bending under the gravitational field. To the linear order the spatial curvature perturbation in the zero-shear gauge can be identified as the perturbed Newtonian potential [15, 16]. Equations (17)-(19) include effects of gravitational waves to the density and velocity perturbations. Equations of the gravitational waves can be found in [12]. To the third order, we have [13] ∇ · u = − ∇ · (δu) + 2ϕu−∇ · ∇δ, (20) + 4πGµδ = − ∇ · (u · ∇u) ϕu · ∇ (∇ · u) + u · ∇u− u∇ · u u · ∇ u · ∇X + X∇ · u,(21) δ̈ + 2 δ̇ − 4πGµδ = − [a∇ · (δu)] + ∇ · (u · ∇u) + 2ϕu−∇ ϕu · ∇ (∇ · u)− u · ∇u− u∇ · u u · ∇ u · ∇X − X∇ · u,(22) where X ≡ 2ϕ∇ · u− u · ∇ϕ+ ∆−1∇ · [u · ∇ (∇ϕ) + u∆ϕ] . (23) In these equations we ignored the role of tensor-type perturbation; for a complete set of equations, see [13]. The variable ϕ is a perturbed-order metric (spatial curvature) variable in the comoving gauge condition, see later. All the third-order correction terms in Eqs. (20)-(22) are simply of ϕ-order higher than the second-order rela- tivistic/Newtonian terms. Thus, the pure general relativistic effects are at least ϕ-order higher than the relativis- tic/Newtonian ones in the second order equations. Thus, we only need the behavior of ϕ to the linear order which is related to the other hydrodynamic variables as ϕ = −δΦ+ ȧ∆−1∇ · u. (24) It also satisfies [17] ϕ̇ = 0, (25) thus ϕ = C(x) with no decaying mode; this is true considering the presence of the cosmological constant, see [17]. IV. RELATIVISTIC FULLY NONLINEAR EQUATIONS In this section, for convenience, we present some additional covariant or ADM equations not available in [11]. In the multi-component situation, we have T̃ab ≡ T̃(j)ab. (26) The energy-momentum conservation gives T̃ b(i)a;b = Ĩ(i)a, Ĩ(j)a = 0. (27) Tildes indicate the covariant quantities. A. Covariant equations We introduce the fluid quantities as T̃(i)ab ≡ µ̃(i)ũ(i)aũ(i)b + p̃(i) g̃ab + ũ(i)aũ(i)b + q̃(i)aũ(i)b + q̃(i)bũ(i)a + π̃(i)ab, (28) where ũa(i)ũ(i)a ≡ −1, ũ (i)q̃(i)a ≡ 0 ≡ ũ (i)π̃(i)ab, π̃ (i)a ≡ 0. (29) The fluid quantities of each component are based on the fluid four-vector ũ(i)a as µ̃(i) ≡ T̃(i)abũ (i)ũ (i), p̃(i) ≡ T̃(i)abh̃ (i), q̃(i)a ≡ −T̃(i)cdũ (i)h̃ (i)a, π̃(i)ab ≡ T̃(i)cdh̃ (i)ah̃ (i)b − p̃(i)h̃(i)ab, (30) where h̃(i)ab ≡ g̃ab + ũ(i)aũ(i)b. Equation (27) gives µ(i) + µ̃(i) + p̃(i) θ̃(i) + q̃ (i);a + q̃ (i)ã(i)a + π̃ (i)σ̃(i)ab = −ũ (i)Ĩ(i)a, (31) µ̃(i) + p̃(i) ã(i)a + h̃ p̃(i),b + q(i)b + π̃ (i)b;c + q̃b(i) ω̃(i)ab + σ̃(i)ab + θ̃(i)h̃(i)ab = h̃ b(i)aĨ(i)b, (32) where the kinematic quantities are also based on the fluid four-vector ũ(i)a as h̃ c(i)ah̃ (i)b ũ(i)c;d = h̃ (i)[ah̃ (i)b]ũ(i)c;d + h̃ (i)(ah̃ (i)b)ũ(i)c;d ≡ ω̃(i)ab + θ̃(i)ab = ũ(i)a;b + ã(i)aũ(i)b, θ̃(i) ≡ u (i);a, σ̃(i)ab ≡ θ̃(i)ab − θ̃(i)h̃(i)ab, ã(i)a ≡ ũ(i)a;bũ (i) ≡ u(i)a, µ(i) ≡ µ̃(i),aũ (i). (33) In the multi-component situation, we can derive the corresponding equation of Raychaudhury equation for the indi- vidual component. From ũ(i)a;bc − ũ(i)a;cb ≡ ũ(i)dR̃ abc we can derive θ(i) + θ̃2(i) − ã (i);a + σ̃ (i)σ̃(i)ab − ω̃ (i)ω̃(i)ab = 4πG µ̃− 3p̃− 2T̃abũ (i)ũ + Λ. (34) In the energy-frame we take q̃(i)a ≡ 0 for each component of the fluids without losing any physical degree of freedom. In a single component situation, taking the energy-frame, the energy conservation equation, the momentum con- servation equation, and the Raychaudhury equation are [18] µ+ (µ̃+ p̃) θ̃ + π̃abσ̃ab = 0, (35) (µ̃+ p̃) ãa + h̃ p̃,b + π̃ = 0, (36) θ̃2 − ãa;a + σ̃ abσ̃ab − ω̃ abω̃ab + 4πG (µ̃+ 3p̃)− Λ = 0. (37) By combining Eqs. (35)-(37) we can derive µ+ π̃abσ̃ab µ̃+ p̃ µ+ π̃abσ̃ab µ̃+ p̃ = 4πG (µ̃+ 3p̃)− Λ + σ̃abσ̃ab − ω̃ abω̃ab + h̃ab(p̃,b + π̃ µ̃+ p̃ . (38) This equation was derived in Eq. (88) of [19], see also [20]. In the multi-component case, in the energy-frame, combining Eqs. (31)-(34) we can derive µ(i) + π̃ σ̃(i)ab + ũ Ĩ(i)a µ̃(i) + p̃(i) µ(i) + π̃ σ̃(i)ab + ũ Ĩ(i)a µ̃(i) + p̃(i) = −4πG µ̃− 3p̃− 2T̃abũ (i)ũ − Λ + σ̃ab(i)σ̃(i)ab − ω̃ (i)ω̃(i)ab + (p̃(i),b + π̃ (i)b;c − Ĩ(i)b) µ̃(i) + p̃(i) . (39) In [21] Langlois and Vernizzi derived a simple covariant relation which leads to one of the conserved variable in the large-scale limit. These authors introduced ζ̃a ≡ h̃ α̃,b + 3(µ̃+ p̃) α ≡ α̃,aũ θ̃. (40) Using only the energy conservation in Eq. (35) we can derive £ũζ̃a = − 3(µ̃+ p̃) p̃,a + (µ̃+ p̃)θ̃ ũaπ̃ bcσ̃bc 3(µ̃+ p̃) π̃bcσ̃bc 3(µ̃+ p̃) µ̃,aπ̃ bcσ̃bc 3(µ̃+ p̃)2 , (41) where £ũ is a Lie derivative along ũa with £ũζ̃a ≡ ζ̃a;bũ b + ζ̃bũb;a. Thus, for vanishing anisotropic pressure, we have the Langlois-Vernizzi relation [21] £ũζ̃a = − 3(µ̃+ p̃) p̃,b − . (42) These equations are valid in a single component fluid, or in multiple component fluids for the collective fluid variables. We can easily extend the relation to the individual fluid component as follows. We introduce ζ̃(i)a ≡ h̃ α̃(i),b + µ̃(i),b 3(µ̃(i) + p̃(i)) α(i) ≡ α̃(i),aũ (i) ≡ θ̃(i). (43) Using only Eq. (31) we can derive £ũ(i) ζ̃(i)a = − θ̃(i) 3(µ̃(i) + p̃(i)) p̃(i),a + (µ̃(i) + p̃(i))θ̃(i) µ̃(i),a ũ(i)a σ̃(i)bc + ũ Ĩ(i)b 3(µ̃(i) + p̃(i)) σ̃(i)bc + ũ Ĩ(i)b 3(µ̃(i) + p̃(i)) µ̃(i),a σ̃(i)bc + ũ Ĩ(i)b 3(µ̃(i) + p̃(i)) . (44) Thus, for vanishing anisotropic pressure and direct interactions among fluids, i.e., π̃(i)ab = 0 = Ĩ(i)a, we have £ũ(i) ζ̃(i)a = − θ̃(i) 3(µ̃+ p̃) h̃ b(i)a p̃(i),b − µ̃(i),b . (45) Application of these compact relations to large-scale conservation properties to the second order will be studied in Sec. XI. B. ADM equations The ADM formulation [22] is presented in Eqs. (2)-(13),(47),(48) of [11]. Interpretation of the ADM fluid quantities in Eqs. (45),(46) of [11] was based on the normal-frame fluid quantities; for relations to the energy-frame fluid quantities, see Eq. (57) below. The ADM fluid quantities of individual component are introduced as E(i) ≡ ñañbT̃ (i) = N 2T̃ 00(i) , J(i)α ≡ −ñbT̃ (i)α = NT̃ (i)α, S(i)αβ ≡ T̃(i)αβ , S(i) ≡ h αβS(i)αβ , S̄(i)αβ ≡ S(i)αβ − hαβS(i). (46) Equation (27) gives Eqs. (12),(13),(47),(48) in [11]. Equations (10),(12),(47) in [11] can be arranged as (∂0 −N α∂α)K (Kαα ) = 4πG (E + S)− Λ + K̄αβK̄αβ − N :αα, (47) Kαα = )−1 [ (∂0 −N α∂α)E + − S̄αβK̄αβ , (48) Kαα = E(i) + )−1 [ (∂0 −N α∂α)E(i) + N2Jα(i) K̄αβ + Ĩ(i)0 − Ĩ(i)αN . (49) Momentum conservation equations for the collective and individual components can be found in Eqs. (13),(48) of [11]. By combining Eqs. (47)-(49) we can derive the ADM counterpart of the density perturbation equations in Eqs. (38),(39). V. SECOND-ORDER PERTURBATIONS We use a metric convention in Eq. (49) of [11] g̃00 ≡ −a 2 (1 + 2A) , g̃0α ≡ −a 2Bα, g̃αβ ≡ a + 2Cαβ . (50) The subindex 0 indicates the conformal time η with adη ≡ cdt. To the second-order in perturbation we introduce the fluid four-vector of individual component as ũα(i) ≡ V α(i), ũ (i) = 1− A+ V α(i)V(i)α −B αV(i)α ũ(i)α = a V(i)α −Bα +ABα + 2V , ũ(i)0 = −a 1 +A− V α(i)V(i)α . (51) In this definition of fluid four-vector we follow the notation in Eq. (53) of [11]. If we introduce ũα ≡ V̄ α , we have V α(i) = (1−A)V̄ (i). The fluid quantities of individual component are introduced as µ̃(i) ≡ µ(i) + δµ(i), p̃(i) ≡ p(i) + δp(i), π̃(i)αβ ≡ a 2Π(i)αβ , π̃(i)0α = −a 2Π(i)αβV , π̃(i)00 = 0, (52) where from π̃ a ≡ 0 we have Π α(i)α − 2C αβΠ(i)αβ = 0. (53) The energy-momentum tensor of individual component in the energy-frame follows from Eq. (28) as T̃ 0(i)0 = −µ(i) − δµ(i) − µ(i) + p(i) V α(i) −B V(i)α, T̃ 0(i)α = µ(i) + p(i) V(i)α −Bα −AV(i)α + 2ABα + 2V δµ(i) + δp(i) V(i)α −Bα +Π(i)αβ T̃ α(i)β = p(i) + δp(i) δαβ + µ(i) + p(i) V α(i) V(i)β −Bβ +Π α(i)β − 2C αγΠ(i)βγ . (54) Using T̃ ab = , and the total fluid quantities in Eq. (82) of [11] we have µ(j), p = p(j), (55) for the background order fluid quantities, and δµ(j) + µ(j) + p(j) V α(j) −B V(j)α − Vα δp(j) + µ(j) + p(j) V α(j) −B V(j)α − Vα (µ+ p)Vα = µ(j) + p(j) V(j)α + δµ(j) + δp(j) V(j)α − Vα V(j)β − Vβ Παβ = Π α(j)β + µ(j) + p(j) vα(j) −B V(j)β − Vβ V(j)γ − Vγ , (56) for perturbed order fluid quantities to the second-order. From Eq. (46) we have the ADM fluid quantities based on the energy-frame fluid quantities E(i) = µ(i) + δµ(i) + µ(i) + p(i) V α(i) −B V(i)α −Bα J(i)α = a µ(i) + p(i) V(i)α −Bα +ABα + 2V δµ(i) + δp(i) V(i)α −Bα + Π(i)αβ S(i) = 3 p(i) + δp(i) µ(i) + p(i) V α(i) −B V(i)α −Bα S̄(i)αβ = a Π(i)αβ + µ(i) + p(i) V(i)α −Bα V(i)β −Bβ V(i)γ −Bγ . (57) We can compare Eq. (57) with the ADM fluid quantities based on the normal-frame fluid quantities in Eq. (76) of [11]; in a single component case we simply delete (i) subindices. In [11] the fluid quantities are based on the normal-frame vector. By taking ũ(i)α ≡ 0, ũ(i)a becomes the normal- frame vector ña, see Eq. (54) in [11]. Based on the normal-frame vector, to the second order, the fluid quantities have contributions due to the frame choice: for example, even in the zero-pressure fluid, the perturbed pressure based on the normal-frame does not necessarily vanish to the second-order, see Eq. (58) below. By comparing Eq. (57) with Eq. (76) of [11] we have δµN(i) = δµ(i) + µ(i) + p(i) V α(i) −B V(i)α −Bα δpN(i) = δp(i) + µ(i) + p(i) V α(i) −B V(i)α −Bα Q(i)α = µ(i) + p(i) V(i)α −Bα +ABα + 2V δµ(i) + δp(i) V(i)α −Bα +Π(i)αβ ΠN(i)αβ = Π(i)αβ + µ(i) + p(i) V(i)α −Bα V(i)β −Bβ V(i)γ −Bγ . (58) For the total fluid quantities the relations between the two frames are presented in Eq. (87) of [11]. Thus, by replacing all fluid quantities in Eqs. (99)-(107) of [11] using Eq. (58) and Eq. (87) of [11] we have the equations in the energy frame. Using Q(i)α in Eq. (58), Eq. (54) gives T̃ 0(i)α = (1−A)Q(i)α. (59) As the fluid four-velocity of i-th component we can use either Q(i)α or V(i)α −Bα related by Eq. (58). The kinematic quantities in the energy-frame are presented in Eqs. (63)-(66) of [11]. In Eq. (33) we introduced kinematic quantities for the individual component. To the second order we can show θ̃(i) = 3 Cα′α + V α(i)|α + A2 − 3 BαV(i)α − ACα′α + V α(i) A,α + C V α(i)V(i)α + V α(i) −B V(i)α −Bα CαβC′αβ , σ̃(i)αβ = a V(i)(α|β) + C αβ −AC V(i)(α −B(α V(i)β) −Bβ) + V(i)(αA,β) + V Cαβ|γ + 2V (i)|(α C(β)γ Cγ′γ + V (i)|γ (i)|γ + Cγ′γ −AC V(i)γ −Bγ A,γ + V Cδδ|γ − 2C γδC′γδ ω̃(i)αβ = a V(i)[α −B[α +AB[α + 2V V(i)[α −B[α V(i)β] −Bβ] +A,β] ã(i)α = A,α + V(i)α −Bα +ABα + 2V − 2AA,α −A V(i)α − Bα V(i)α −Bα +Bβ|α σ̃(i)α0 = −V σ̃(i)αβ , σ̃(i)00 = 0; ω̃(i)α0 = −V ω̃(i)αβ , ω̃(i)00 = 0; ã(i)0 = −V (i)ã(i)α, (60) where a prime indicates the time derivative based on η. The gauge transformation properties of the fluid quantities are presented in Eqs. (232)-(235) of [11] for the normal- frame, and Eqs. (238) of [11] for the energy-frame. A prescription to get the gauge transformation properties for individual fluid quantities is also presented below Eq. (235) of [11]. Under the gauge transformation we have δµ̂(i) = δµ(i) − µ′(i) + δµ ξ0 − δµ(i),αξ µ′′(i)ξ 0ξ0 + µ′(i) ξ0ξ0′ + ξαξ0,α δp̂(i) = δp(i) − p′(i) + δp ξ0 − δp(i),αξ p′′(i)ξ 0ξ0 + p′(i) ξ0ξ0′ + ξαξ0,α V̂(i)α − B̂α + ÂB̂α + 2V̂ Ĉαβ = V(i)α −Bα +ABα + 2V Cαβ + ξ V(i)α −Bα V(i)α −Bα V(i)β −Bβ ξβ,α − V(i)α −Bα A− ξ0′ − ξ0,α − ξ ,β − ξ 0ξ0′,α − ξ βξ0,αβ , Π̂(i)αβ = Π(i)αβ − Π′(i)αβ + 2 Π(i)αβ ξ0 −Π(i)αβ,γξ γ − 2Π(i)γ(αξ . (61) A. Basic equations in the energy-frame The basic set of equations with fluid quantities based on the normal-frame is presented in Eqs. (99)-(107) of [11]. By using Eq. (58) we can recover the equations with fluid quantities based on the energy frame. For convenience, in the following we present the complete set of equations with fluid quantities in the energy frame. These equations are written without taking any gauge conditions yet, thus in a sort of gauge-ready form. To the linear order this method was suggested by Bardeen [23, 24]. Definition of δK: K̄ + 3H + δK − 3HA+ Ċαα + HA− Ċαα − HBαBα + + 2Cαβ Ċαβ + ≡ n0. (62) Energy constraint equation: 16πGµ+ 2Λ− 6H2 − R(3) + 16πGδµ+ 4HδK − R(3)Cαα δK2 − 16πG (µ+ p) (V α −Bα) (Vα −Bα)− Ċαβ + B(α|β) Ċαβ + Ċαα + R(3)Cαγ C − Cα|βα + Cαβ|γ 3Cαβ|γ − 2Cαγ|β ≡ n1. (63) Momentum constraint equation: Ċβα + +B |βα Ċγγ + δK,α + 8πGa (µ+ p) Vα −Bα +ABα + 2V AδK,α − 8πGa (δµ+ δp) (Vα −Bα) + (µ+ p)A (Vα −Bα) + Παβ V β −Bβ Ċβα + + B |βα Ċγα + B |γα +B + 2Cβγ Ċαγ + B(α|γ) + Cγ|βα − C Ċβγ + Ċγγ + + 2Cγδ Ċγδ + ≡ n2α. (64) Trace of the ADM propagation equation: 3Ḣ + 3H2 + 4πG (µ+ 3p)− Λ + δK̇ + 2HδK − 4πG (δµ+ 3δp) + 3Ḣ + = AδK̇ − δK,αB δK2 + 8πG (µ+ p) (V α −Bα) (Vα −Bα) 3A2 −BαBα 2A∆A+A,αA,α − ∆ (BαBα) +A + 2CαβA,α|β Ċαβ + B(α|β) Ċαβ + Ċαα + ≡ n3. (65) Tracefree ADM propagation equation: Ċαβ + Bα|β +B Ċαβ + Bα|β +B Ċγγ + Ċγγ + A|γ γ R(3)Cαβ − R(3)Cγγ − 8πGΠαβ Ċαβ + Bα|β +B A+ 2Cαγ Ċβγ + B(β|γ) Ċαβ + Bα|β +B A+ 2Cαγ Ċβγ + B(β|γ) Ċαβ + Bα|β +B Ċαβ + Bα |β +B Bγ + δK Ċαβ + Bα|β +B −A2 +BγBγ − 2CαγA,β|γ − Ċγγ + A+ 2Cγδ Ċγδ + Ċγγ + A+ 2Cγδ Ċγδ + Ċγγ + Ċγγ + Bδ + δK Ċγγ + −AA|γ γ + −A2 +BδBδ − 2CγδA,γ|δ − − Cγ|δγ Ċαγ + Bα|γ +B Cαδ|βγ + C − Cαβ|δγ − C + 2Cαγ Cδγ|βδ + C β|γδ − C − Cδδ|γβ R(3)Cαγ C Cαδ |β + C − Cγδ|βC γδ|α + 2Cαγ|δ Cβδ|γ − Cβγ|δ Cǫγ|δǫ + C γ|ǫδ − C − Cǫǫ|γδ R(3)CδγC 2Cǫδ|ǫ − C − Cγ|δγ + Cγδ|ǫ 2Cγǫ|δ − 3Cγδ|ǫ −16πGCαγΠβγ + 8πG (µ+ p) (V α −Bα) (Vβ −Bβ)− δαβ (V γ −Bγ) (Vγ −Bγ) ≡ n α4β . (66) Energy conservation equation: [µ̇+ 3H (µ+ p)] + δµ̇+ 3H (δµ+ δp)− (µ+ p) (δK − 3HA) + (µ+ p) V α −Bα +ABα + 2V βCαβ δµ,αB α + (δµ+ δp) (δK − 3HA) + (µ+ p)AδK + H (µ+ p) A2 −BαBα a4 (µ+ p) (V α −Bα) (Vα −Bα) (δµ+ δp) (V α −Bα) + Παβ (Vβ −Bβ) (µ+ p) −A (V α −Bα)|α − 2A,α (V α −Bα) + 2 Cαβ (Vβ −Bβ) − Cαα|β V β −Bβ Ċαβ + ≡ n5. (67) Momentum conservation equation: a4 (µ+ p) Vα −Bα +ABα + 2V (µ+ p)A,α + δp,α +Π = (µ+ p) (δK − 3HA) (Vα −Bα)− (δµ+ δp) (Vα −Bα) + Π α (Vβ −Bβ) δp,α +Π A− (δµ+ δp)A,α − (µ+ p) −AA,α +Bβ|αV β + (Vα −Bα)|β V β + (Vα −Bα) V β −Bβ CβγΠαγ Πγα + C Πβγ −A,βΠ ≡ n6α. (68) In the multi-component situation we additionally have the energy and the momentum conservation of individual component. Using the energy-frame fluid quantities, Eqs. (106),(107) in [11] become µ̇(i) + 3H µ(i) + p(i) I(i)0 + δµ̇(i) + 3H δµ(i) + δp(i) µ(i) + p(i) (δK − 3HA) µ(i) + p(i) V α(i) −B α +ABα + 2V δI(i)0 δµ(i),αB δµ(i) + δp(i) (δK − 3HA) + µ(i) + p(i) AδK + µ(i) + p(i) A2 −BαBα µ(i) + p(i) V α(i) −B V(i)α −Bα δµ(i) + δp(i) V α(i) −B V(i)β −Bβ µ(i) + p(i) V α(i) −B − 2A,α V α(i) −B V(i)β −Bβ − Cα|βα V(i)β −Bβ Ċαβ + δI(i)αB α ≡ n(i)5, (69) µ(i) + p(i) V(i)α −Bα +ABα + 2V µ(i) + p(i) A,α + δp(i),α +Π (i)α|β − δI(i)α µ(i) + p(i) (δK − 3HA) V(i)α −Bα δµ(i) + δp(i) V(i)α −Bα V(i)β −Bβ δp(i),α +Π (i)α|β − δI(i)α δµ(i) + δp(i) µ(i) + p(i) −AA,α +Bβ|αV V(i)α −Bα V(i)α −Bα CβγΠ(i)αγ −A,βΠ ≡ n(i)6α. (70) By removing indices indicating the components in Eqs. (69),(70) we recover equations for the collective component which coincide with the equations in a single component situation in Eqs. (67),(68). To the background order, from Eqs. (69),(65) we have µ̇(i) + 3H µ(i) + p(i) I(i)0, (71) µ̇+ 3H (µ+ p) = 0, (72) 3Ḣ + 3H2 = − µ(j) + 3p(j) + Λc2, (73) µ(j) − , (74) where we recovered the speed of light c. Dimensions are [G̺] = T−2, [c] = LT−1, [η] = 1, [p] = [µ] = [̺c2], [a] = L, [K] = 1, [Λ] = L−2, [I(i)0] = [µ]. (75) Equation (72) follows from the sum of Eq. (71) over components. Equation (74) follows from integrating Eq. (73) where K-term can be regarded as an integration constant; in Einstein’s gravity K-term can be normalized as the sign of spatial curvature. Compared with the Newtonian background equations in Eq. (7), ignoring the direct inter- action terms in Eq. (71), the presence of pressure terms in Eqs. (71)-(74) is the pure general relativistic effect. The cosmological constant Λ can be introduced by hand even in the Newtonian case. B. Decomposition We decompose the metric to three perturbation types A ≡ α, Bα ≡ β,α +B α , Cαβ ≡ ϕg + γ,α|β + C (α|β) , (76) where superscripts (v) and (t) indicate the transverse vector-type, and transverse-tracefree tensor-type perturbations, respectively. We introduce χ ≡ a β + c−1aγ̇ , Ψ(v)α ≡ B α + c −1aĊ(v)α , (77) which are spatially gauge-invariant to the linear order. We set Kαα ≡ −3H + κ. (78) We will identify κ with Newtonian velocity variable which will be an important step in our analysis, see Eqs. (120),(194),(215). For the fluid quantities we decompose ũ(i)α = a V(i)α −Bα +ABα + 2V ≡ av(i)α ≡ a −v(i),α + v , (79) Π(i)αβ ≡ Π(i),α|β − ∆Π(i) (i)(α|β) (i)αβ , δI(i)α ≡ δI(i),α + δI . (80) The perturbed fluid velocity variables v(i) and v subtly differ from the ones introduced in [11]; see Sec. VC. For the collective fluid component or for a single component case, we simply delete (i) subindices. For isotropic pressure we introduce δp ≡ c2sδµ+ e, c . (81) The perturbation variable e is called an entropic perturbation. Defined in this way e is gauge-invariant only to the linear order. To the second order, from Eq. (61) we can derive the following gauge-invariant combination δpδµ ≡ e− . (82) In our notation, δpδµ is a gauge-invariant combination which is the same as δp in the δµ = 0 slicing (temporal gauge) condition to the second order; as the spatial gauge we take γ ≡ 0 to the second order; for the derivation, see the prescription below Eq. (266) of [11]. In the multi-component case, we similarly have δp(i)δµ(i) ≡ e(i) − δµ(i) µ̇(i) ė(i) + c2(i) δµ(i) , (83) where δp(i) ≡ c (i)δµ(i) + e(i), c (i) ≡ ṗ(i) µ̇(i) . (84) We have [25] e = erel + eint, erel ≡ c2(j) − c δµ(j), eint ≡ e(j). (85) Equation (56) gives δµ(j) + µ(j) + p(j) vα(j) v(j)α − vα δp(j) + µ(j) + p(j) vα(j) v(j)α − vα (µ+ p) vα = µ(j) + p(j) v(j)α + δµ(j) + δp(j) v(j)α − vα v(j)β − vβ Παβ = Π α(j)β + µ(j) + p(j) vα(j) v(j)β − vβ δαβ v v(j)γ − vγ , (86) By recovering c, dimensions of the variables are [g̃ab] = [g̃ ab] = [ũa] = 1, [T̃ab] = [µ], [g ] = 1, [∇] = [∆] = 1, [w] = [c2s] = 1, [A] = [Bα] = [Cαβ ] = 1, [α] = [ϕ] = [β] = [γ] = [B α ] = [C α ] = [Ψ α ] = [C ] = 1, [χ] = L, [κ] = T−1, [δµ] = [δp] = [e] = [Παβ ] = [Π ] = [µ], [δ] = [Vα] = 1, [v] = [v α ] = 1, [Π] = L 2[µ], [Π(v)α ] = L[µ]. (87) Scalar-type perturbation equations can be derived from Eqs. (62)-(70) κ− 3Hα+ 3ϕ̇+ χ = n0, (88) 4πGδµ+Hκ+ ∆+ 3K n1, (89) ∆+ 3K χ− 12πG (µ+ p) av = n2 ≡ ∆−1∇αn2α, (90) κ̇+ 2Hκ− 4πG (δµ+ 3δp) + 3Ḣ + α = n3, (91) χ̇+Hχ− ϕ− α− 8πGΠ = n4 ≡ a2 (∆ + 3K) ∆−1∇α∇βn 4α, (92) δµ̇+ 3H (δµ+ δp)− (µ+ p) κ− 3Hα+ = n5, (93) a4 (µ+ p) v a4 (µ+ p) a (µ+ p) ∆+ 3K = n6 ≡ − ∆−1∇αn6α, (94) δµ̇(i) + 3H δµ(i) + δp(i) µ(i) + p(i) κ− 3Hα+ δI(i)0 = n5(i), (95) µ(i) + p(i) µ(i) + p(i) µ(i) + p(i) δp(i) + ∆ + 3K Π(i) − δI(i) = n6(i) ≡ − µ(i) + p(i) ∆−1∇αn6α(i). (96) Equations for the vector-type perturbation follow from Eqs. (64),(66),(68),(70) ∆ + 2K Ψ(v)α + 8πG(µ+ p)v n2α −∇α∆ −1∇βn2β 2α , (97) Ψ̇(v)α + 2HΨ α − 8πGΠ α = 2a (∆ + 2K) 4α −∇α∆ −1∇γ∇βn 4α , (98) [a4(µ+ p)v a4(µ+ p) ∆+ 2K n6α −∇α∆ −1∇βn6β 6α , (99) [a4(µ(i) + p(i))v a4(µ(i) + p(i)) ∆ + 2K µ(i) + p(i) µ(i) + p(i) µ(i) + p(i) n6(i)α −∇α∆ −1∇βn6(i)β 6(i)α .(100) Equations for the tensor-type perturbation follow from Eq. (66) + 3HĊ ∆− 2K − 8πGΠ = n4αβ − ∇α∇β − (∆ + 3K) ∆−1∇γ∇δn −2∇(α (∆ + 2K) ∇γn4β)γ −∇β)∆ −1∇γ∇δn 4αβ . (101) In order to derive eqs. (92,98,101) it is convenient to show ∇α∇β − (χ̇+Hχ− ϕ− α− 8πGΠ) + (α|β) − 8πG (α|β) + 3HĊ ∆− 2K − 8πGΠ = n4αβ , (102) which follows from Eq. (66). Quadratic combinations of linear-order perturbation variables of all three types of perturbations contribute to all three types of perturbation to the second order. C. Comoving gauge and irrotational condition In Eq. (180) of [11] we introduced Q(i)α ≡ µ(i) + p(i) −v̄(i),α + v̄ , (103) where we put overbars to v̄(i) and v̄ in order to distinguish these from our new notations to be used in this paper; in [11] we didn’t have overbars. From Eq. (58) we have −v̄(i),α + v̄ ≡ V(i)α −Bα +ABα + 2V Cαβ + δµ(i) + δp(i) µ(i) + p(i) V(i)α −Bα Π(i)αβ µ(i) + p(i) . (104) It is more convenient to introduce the decomposition in Eq. (79). Thus, we have −v̄(i),α + v̄ ≡ −v(i),α + v δµ(i) + δp(i) µ(i) + p(i) −v(i),α + v Π(i)αβ µ(i) + p(i) . (105) The variable v̄ introduced in [11] cannot be regarded as a proper vector-type perturbation. However, if we also take the temporal comoving gauge in [11] which sets v̄ ≡ 0 together with v̄ = 0, we have v = 0 = v ; these are the same as taking the proper irrotational condition (v = 0) and the temporal comoving gauge (v = 0). Our analyses in [12, 13] are, in fact, based on taking these two conditions together. In the irrotational fluids, the temporal comoving gauge v ≡ 0 leads to ũα = 0, thus ũa coincides with the normal frame four-vector ña. From Eq. (60) we have ω̃(i)αβ = av (i)[α|β] µ(i) + p(i) −v(i),[α + v (i)[α δp(i) + ∆ + 3K Π(i) − δI(i) ∆+ 3K (i)β] (i)β] (106) where we used Eq. (70) to the linear order. For vanishing vector-type perturbation we set v ≡ 0, etc. In this case, we have ω̃(i)αβ = − µ(i) + p(i) v(i),[α δp(i) + ∆+ 3K Π(i) − δI(i) , (107) which vanishes for δp(i) = 0 = Π(i) and δI(i) = 0. VI. EFFECTS OF PRESSURE We consider a single component situation without rotational perturbations. Equations (65),(67),(68), and Eq. (64) to the linear order provide a complete set of equations we need in the following. A. Irrotational case As an irrotational fluid we ignore all vector-type perturbations, thus v α = B α = C α = Π α = 0. Quadratic combinations of linear-order perturbations of all three-types of perturbations contribute to each type of second-order perturbations. Thus, concerning the second-order scalar-type and vector-type perturbations, ignoring the pure vector- type perturbation corresponds to ignoring the vector-type perturbation only to the linear order; to the linear order the vector-type perturbation only has a decaying solution in the expanding phase. Assuming the background equations are valid Eqs. (67),(68),(65),(64) give δ̇ + 3H − (1 + w) v − 3Hα (κ− 3Hα) + (1 + w) H (1 + w) α2 − β,αβ,α χ,α|β + Ċ (1 + w) v,α|β + γ,α|β + C (1 + w) v,α v,α − δp,α +Π + ϕ,α − [(∆ + 4K)γ],α (δµ+ δp) v,α +Παβv,β , (108) a4 (µ+ p) v,α (µ+ p)α,α + δp,α +Π (µ+ p) (v − β) (v − β) +(µ+ p) v,α v − 3Hα (δµ+ δp) v,α +Π − (δµ+ δp)α,α + α (µ+ p)α,α − δp,α −Π − α,βΠ α + 2 CγβΠαγ −ΠγαC +ΠβγC , (109) κ̇+ 2Hκ− 4πG (δµ+ 3δp) + 3Ḣ + α = − ,α + ακ̇+ 3α2 − β,αβ,α + 8πG (µ+ p) v,αv,α 2∆α (α+ ϕ) + α,α [α− ϕ+ (∆ + 4K)γ] + 2α,α|β γ,α|β + C ∆ (β,αβ,α) χ,α|β + Ċ(t)αβ χ,α|β + Ċ , (110) κ = − ∆+ 3K χ+ 12πGa (µ+ p) v. (111) Equation (111) is valid to the linear order. B. Comoving gauge As the temporal and spatial gauge conditions we set v ≡ 0 ≡ γ, (112) thus, β = χ/a. Under these gauge conditions Eq. (109) gives α = − χ,αχ,α − 1 + w 1 + 3c2s 2(1 + w) + αΠ, (113) where the imperfect fluid contributions (stresses) are denoted by αΠ with ∆ + 3K 1 + w ∆ + 3K ∆−1∇α 1 + c2s 1 + w e,α +Π 1 + w δ,β + e,β +Π Πβα + 2 CβγΠαγ −ΠγαC +ΠβγC (114) Using this, Eqs. (108)-(111) give δ̇ − 3Hwδ − (1 + w) κ = − ,α + κδ + 1 + w δ2 + δΠ, (115) κ̇+ 2Hκ− 4πGµδ − 1 + w ∆+ 3K δ = − 2(1 + w) 4πGµ− 1 + 2c2s 1 + w ∆+ 3K 1 + w 2Hδκ+ ϕ,αδ,α − 2ϕ∆δ − 2δ ,α|βC χ,α|β + Ċ(t)αβ χ,α|β + Ċ +κΠ, (116) κ = − ∆+ 3K χ, (117) where Eq. (117) is valid only to the linear order, and δΠ ≡ 2H ∆+ 3K κ+ 3H 1 + w χ,α|β + Ċ ∆+ 3K 3Hδ − (1 + w) κ+ ∆ + 3K ∆−1∇α 1 + c2s 1 + w e,α +Π 1 + w δ,β + e,β +Π Πβα + 2 CβγΠαγ −ΠγαC +ΠβγC κΠ ≡ 12πGe 1 + w 3Ḣ + ∆ + 3K 1 + w ∆ + 3K ∆−1∇α 1 + c2s 1 + w e,α +Π 1 + w δ,β + e,β +Π CβγΠαγ −ΠγαC +ΠβγC −2Hκ+ 4πG 1 + 6c2s µδ + 12πGe− 1 + w ∆ + 3K ∆+ 3K ∆ + 3K ∆+ 3K 1 + w ∆ + 3K ϕ,α − 2 ∆+ 3K ),α|β . (118) Combining Eqs. (115),(116) we can derive 1 + w (µ+ p)a − c2s δ = (1 + w) χ,α|βχ,α|β − 1 + w a2κδ − δ,αχ 1 + w 4πGµ− 1 + 2c2s 1 + w ∆+ 3K 2a2Hδκ+ ϕ,αδ,α − 2ϕ∆δ − 2δ ,α|βC + (1 + w) Ċ(t)αβ χ,α|β + Ċ +(1 + w) κΠ + 1 + w 1 + w δΠ,αχ ,α. (119) Notice that the equations above are valid in the presence of general K and Λ. C. Newtonian correspondence Guided by our success in the zero-pressure case, we continue to identify κ ≡ − ∇ · u, (120) to the second order. Using Eq. (117), assuming K = 0, we can identify κ = −c χ = − ∇ · u, u ≡ ∇χ = −c∇vχ, (121) to the linear order. We have recovered the speed of light c. We have [u] = LT−1. Equation (113) becomes α = − 1 + w 1 + 3c2s 2(1 + w) + αΠ. (122) Equations (115), (116) give δ̇ − 3wHδ + (1 + w) ∇ · u = − ∇ · (δu) + 1 + w Hδ2 + δΠ, (123) ∇ · (u̇+Hu) + 1 + w δ = − ∇ · (u · ∇u)− Ċ(t)αβ uα|β + Ċ 1 + w 1 + 2c2s 1 + w δ2 + 2Hδ ∇ · u+ 2ϕ∆δ − (∇ϕ) · ∇δ + 2δ,α|βC − κΠ.(124) We have [αΠ] = 1, [δΠ] = T −1, and [κΠ] = T −2. Combining these equations or Eq. (119) gives 1 + w (µ+ p)a − c2sc 1 + w ∇ · (u · ∇u)− 1 + w 1 + w ∇ · (δu)− 1 + w + (1 + w) Ċ(t)αβ uα|β + Ċ 1 + 2c2s 1 + w δ2 − c2s 2aHδ∇ · u+ 2ϕc2∆δ − c2 (∇ϕ) · ∇δ + 2c2δ,α|βC +(1 + w) κΠ + 1 + w 1 + w u · ∇δΠ. (125) We note that, to the linear order, Eq. (125) is valid in the presence of general K and Λ, see Eq. (119). D. Linear-order relativistic pressure corrections To the linear order, ignoring the entropic perturbation e and the anisotropic stress Π, Eqs. (123)-(125) become δ̇ − 3wHδ + (1 + w) ∇ · u = 0, (126) ∇ · (u̇+Hu) + 4πG̺δ = − 1 + w , (127) 1 + w a(1 + w)̺ , (128) where ignoring the specific internal energy density ǫ we used µ = ̺c2, thus δ = δ̺/̺; in general we have µ = ̺ c2 + ǫ [18]. Equation (128) can be expanded to give 1 + w a(1 + w)̺ = δ̈ + 2− 6w + 3c2s Hδ̇ − 1 + 8w − 6c2s − 3w 4πG̺− 12 w − c2s ) Kc2 5w − 3c2s δ, (129) which is valid in the presence of K and Λ. Equation of density perturbation in the comoving gauge was first derived by Nariai in [26]. In a single component case the Newtonian equations in Eqs. (8),(12),(16) give ∇ · u = 0, (130) ∇ · (u̇+Hu) + 4πG̺δ = − , (131) δ̈ + 2Hδ̇ − 4πG̺δ = . (132) Comparing Eqs. (126),(127) with Eqs. (130),(131), we notice the presence of w ≡ p/(̺c2) term in three places in the relativistic equations. Even to the linear order the presence of these terms should be regarded as pure general relativistic effect of the isotropic pressure. The effects of pressures to the second order compared with the Newtonian equations can be found in Eqs. (123)-(125) which should be compared with Newtonian equations in Eqs. (8),(12),(16). Pressure has the genuine relativistic role in cosmology even in the background level. VII. EFFECTS OF MULTI-COMPONENT We assume zero-pressure medium, thus set p(i) ≡ 0, δp(i) ≡ 0 ≡ Π(i)αβ . (133) From Eq. (86) we notice that to the second order the collective fluid quantities differ from simple sum of individual fluid quantities. Even in the zero-pressure mediums, we have δµ(j) + µ(j)v v(j)α − vα µ(j)v v(j)α − vα µvα = µ(j)v(j)α + δµ(j) v(j)α − vα Παβ = vα(j) v(j)β − vβ δαβ v v(j)γ − vγ , (134) thus δp 6= 0 6= Παβ to the second order. Equations (69),(70),(65),(67),(68),(64) give δ̇(i) − δK + 3HA+ 1 + δ(i) vα(i) δ(i),αB α + δ(i) (δK − 3HA) +AδK + A2 −BαBα −v(i)α 2v̇α(i) +Hv Avα(i)|α − (i) + Cαβv(i)β Cα|βα v(i)β , (135) 1 + δ(i) v(i)α A,α = (δK − 3HA) v(i)α AA,α − δ(i)A,α −B βBβ|α − v(i)αv − v(i)α|βB β − v(i)βB , (136) δK̇ + 2HδK − 4πG (δµ+ 3δp) + 3Ḣ + A = AδK̇ − δK,αB δK2 + 3A2 −BαBα 2A∆A+A,αA,α − ∆ (BαBα) +A + 2CαβA,α|β Ċαβ + B(α|β) Ċαβ + Ċαα + + 8πGµvαvα, (137) δ̇ − δK + 3HA+ [(1 + δ) vα]|α = − α + δ (δK − 3HA) +AδK + A2 −BαBα −vα (2v̇ α +Hvα)− Avα|α − Cαβvβ Cα|βα vβ − 3H , (138) [a (1 + δ) vα] A,α = (δK − 3HA) vα AA,α − δ(i)A,α −B βBβ|α − − vα|βB β − vβB δp,α +Π , (139) Ċβα + +B |βα Ċγγ + δK,α = −8πGa µ(j)v(j)α. (140) where Eq. (140) is valid to the linear order. A. Irrotational case Assuming an irrotational condition we ignore all vector-type perturbations. As the spatial gauge condition we take γ ≡ 0, (141) thus, β ≡ χ/a. Equations (135)-(140) become δ̇(i) − κ+ 3Hα− 1 + δ(i) δ(i),αχ ,α + δ(i) (κ− 3Hα) + ακ+ χ,αχ,α +Hv(i),αv α∆v(i) + ϕ,αv(i),α − ϕ∆v(i) − C(t)αβv(i),α|β , (142) 1 + δ(i) = (κ− 3Hα) v(i),α + −αα,α + χ,βχ,β|α + δ(i)α,α + v(i),αv v(i),βχ , (143) κ̇+ 2Hκ− 4πG (δµ+ 3δp) + 3Ḣ + α = ακ̇− 3α2 − χ,αχ,α 2α∆α+ α,αα,α − (χ,αχ,α)− α ,αϕ,α + 2ϕ∆α+ 2C (t)αβα,α|β Ċ(t)αβ + χ,α|β χ,α|β + 8πGµv,αv,α, (144) δ̇ − κ+ 3Hα− [(1 + δ) v,α]|α = − ,α + δ (κ− 3Hα) + ακ+ χ,αχ,α +Hv,αv α∆v + ϕ,αv,α − ϕ∆v − C(t)αβv,α|β − 3H , (145) [a (1 + δ) v] α,α = (κ− 3Hα) v,α −αα,α + χ,βχ,β|α + δα,α + δp,α +Π , (146) ∆ + 3K χ+ κ = 12πGa µ(j)v(j). (147) Equation (147) is valid to the linear order. B. Linear perturbations To the linear order Eqs. (142)-(146) give δ̇(i) − κ+ 3Hα− c v(i) = 0, (148) v̇(i) +Hv(i) − α = 0, (149) κ̇+ 2Hκ− 4πG ̺(j)δ(j) + 3Ḣ + c2 α = 0, (150) ∆+ 3K χ+ κ = ̺(j)v(j), (151) δ̇ − κ+ 3Hα− c v = 0, (152) v̇ +Hv − α = 0, (153) where we have recovered the speed of light c. The following additional equations can be found in Eqs. (195),(196),(199) of [11] κ− 3Hα+ 3ϕ̇+ c χ = 0, (154) 4πG̺δ +Hκ+ c2 ∆+ 3K ϕ = 0, (155) (χ̇+Hχ)− ϕ− α = 0. (156) Equation (156) gives αχ = −ϕχ. (157) Equation (153) gives αv = 0. (158) Equations (151), (154) give ϕ̇v = χv. (159) Thus, for K = 0 we have ϕ̇v = 0, (160) which is valid even in the presence of multi-components and the cosmological constant. Equations (149), (153) give v̇χ +Hvχ + ϕχ = 0, (161) v̇(i)χ +Hv(i)χ + ϕχ = 0. (162) Equations (151), (155) give ∆+ 3K ϕχ = −4πG̺δv = −4πG ̺(j)δ(j)v = −4πG ̺(j)δ(j)v(j) . (163) We can derive density perturbation equation in many different temporal gauge (hypersurface) conditions all of which naturally correspond to gauge-invariant variables. In a single component case, density perturbation in the comoving gauge (v = 0) is known to give Newtonian result. In the multi-component situation we have many different comoving gauge conditions. Here, we consider two such gauges for δ(i) variable: one based on v = 0 gauge, and the other based on v(ℓ) = 0 gauge for a specific ℓ. (I) Equation (148) evaluated in the v = 0 gauge, and using Eq. (151) we can derive δ̇(i)v − c v(i)χ − c vχ = 0, (164) where we used χv ≡ χ − av ≡ −avχ, vv(i) ≡ v − v(i), and αv ≡ α − c −1 (av) = 0. Using Eqs. (161),(162),(163), we δ̈(i)v + 2Hδ̇(i)v − 4πG ̺(j)δ(j)v = 0. (165) This coincides exactly with the Newtonian result in Eq. (16) to the linear order, even in the presence of K. Thus, we may identify δ(i)v as the Newtonian density perturbation δi to the linear order even in the presence of K. For K = 0 we may also identify −c∇v(i)χ as the Newtonian velocity perturbation ui. However, in the presence of K, we cannot identify the relativistic variables which correspond to the Newtonian velocity perturbation of individual component. Therefore, to the linear order we have the following Newtonian correspondences δi = δ(i)v, ui = −c∇v(i)χ, (166) where the latter one is valid only for K = 0. (II) Evaluating Eq. (148) in the v(ℓ) = 0 gauge for a specific ℓ, and using Eq. (151) we can derive δ̇(i)v(ℓ) − c ∆+ 3K v(ℓ)χ = v(j) − v(ℓ) v(i) − v(ℓ) , (167) where we used χv(ℓ) ≡ χ − av(ℓ) ≡ −av(ℓ)χ, v(i)v(ℓ) ≡ v(i) − v(ℓ), and αv(ℓ) ≡ α − c av(ℓ) = 0. Using Eqs. (161),(162),(163), we have δ̈(i)v(ℓ) + 2Hδ̇(i)v(ℓ) − 4πG ̺(j)δ(j)v(j) = v(ℓ) − v(j) . (168) The terms in right-hand-sides of Eqs. (167),(168) look like relativistic correction terms present even to the linear order based on the variable δ(i)v(ℓ) . Since no such correction terms appear in Eq. (165) based on the variable δ(i)v, the relativistic correction terms in Eq. (168) can be regarded as being caused by a complicated hypersurface (gauge) choice. 1. Exact solutions Assuming K = 0, we can identify Newtonian perturbation variables as δ ≡ δv, κv ≡ − ∇ · u, u ≡ −c∇vχ, δΦ ≡ −c 2ϕχ, δi ≡ δ(i)v, ui ≡ −c∇v(i)χ. (169) Equations (152),(150),(148),(149),(163) become δ̇ = − ∇ · u, (170) u̇+Hu = − ∇δΦ, (171) δ̇i = − ∇ · ui, (172) u̇i +Hui = − ∇δΦ, (173) δΦ = 4πG̺δ = 4πG ̺jδj . (174) Under the identification in Eq. (169) these equations are valid in both Newton’s and Einstein’s gravity theories. Equations (170), (171), (174), and Eqs. (172),(173),(174), respectively, give δ̈ + 2Hδ̇ − 4πG̺δ = = 0, (175) δ̈i + 2Hδ̇i − 4πG ̺jδj = 0. (176) Equation (175) has an exact solution δ(x, t) = H cg(x) ∫ t dt + cd(x) , (177) where cg and cd are integration constants which indicate the relatively growing and decaying solutions in expanding phase; we do not consider the lower bound of integration which is absorbed to the cd mode. Equations (175), (176), and the solution in Eq. (177) are valid considering general K and Λ in the background world model. Equation (174) can be solved to give δΦ = −G̺a2 δ(x′, t) |x′ − x| d3x′. (178) From Eqs. (170),(171),(174) we can show [14] u = −a 4πG̺a2 D(x), ∇ ·D ≡ 0, (179) where the D term is the solution of the homogeneous part of Eq. (171); it decouples from the density inhomogeneity and corresponds to the peculiar velocity in the background world model. Since the D term is not connected to the density inhomogeneity and simply decays, we may ignore it to the linear order. Now, for the individual component, from Eqs. (172),(173) we have ui = −a 4πG̺a2 ∇di(x) + Di(x), ∇ ·Di ≡ 0, (180) δi = δ + ci(x)−∆di(x) ∫ t dt′ a2(t′) , (181) ̺jdj ≡ 0 ≡ ̺jcj . (182) The ci and di are the two isocurvature-type (δ = 0, thus δΦ = 0) solutions. It happens that the relatively decaying isocurvature-type solution, i.e., di-mode, temporally behaves the same as the peculiar velocity in the background, i.e., Di-mode. The relatively growing isocurvature-type solution ci does not contribute to the ui, see Eq. (172). From Eqs. (179),(180) we have ui − u = [∇di(x) +Di(x)−D(x)] , (183) which simply decays; the Di and D solutions are divergence-free and decoupled from the density perturbation, and are the peculiar velocity perturbation present in the background world model. C. Comoving gauge To the linear order, only the hypersurface condition v = 0 (the comoving temporal gauge) allows the density perturbation equation presented in the Newtonian form. Thus, we take v ≡ 0, (184) even to the second order. We take γ ≡ 0 in Eq. (141) as the spatial gauge condition. Equation (146) gives α = − χ,αχ,α − vα(j)v(j)α +∆ vα(j)v (j)|β . (185) Using this, Eqs. (142)-(145) give δ̇ − κ = − ,α + δκ+ µ(j)v (j)v(j)α + 3H ∆−1∇α vα(j)v (j)|β , (186) κ̇+ 2Hκ− 4πG̺δ = − Ċ(t)αβ + χ,α|β χ,α|β 3Ḣ + 8πG̺+ c2 µ(j)v (j)v(j)α + 3Ḣ + c2 ∆−1∇α vα(j)v (j)|β , (187) δ̇(i) − κ+ c 1 + δ(i) vα(i) δ(i),αχ ,α + δ(i)κ+Hv(i)αv ϕ,αv(i)α − 2ϕv (i)|α − 2v µ(j)v (j)v(j)α + 3H ∆−1∇α vα(j)v (j)|β , (188) 1 + δ(i) v(i)α v(i)βχ + κv(i)α − v(i)αv µ(j)∇α v(j)β (j)|β . (189) From Eqs. ((186),187), and Eqs. (187)-(189) we can derive, respectively, − 4πG̺δ (1 + δ) = − Ċ(t)αβ + χ,α|β χ,α|β 2Ḣ + 4πG̺+ µ(j)v (j)v(j)α + 6Ḣ + c2 ∆−1∇α vα(j)v (j)|β , (190) δ̇(i) + δ(i),αχ − 4πG̺δ 1 + δ(i) Ċ(t)αβ + χ,α|β χ,α|β (κ+ ϕ̇) v(i)α + 2 (κ− ϕ̇) v (i)|α v(i)αχ vα(i)v + Ḣv(i)αv (i) + 3Ḣ + 4πG̺ µ(j)v (j)v(j)α + 6Ḣ ∆−1∇α vα(j)v (j)|β . (191) Notice the O(vα v(j)α) correction terms are present in Eqs. (186),(187),(190) even in the single component situation. Except for these O(vα v(j)α) terms, the remaining parts of these equations coincide with the ones in the single component situation. D. Newtonian correspondence For K = 0, to the linear order, we have χv ≡ χ− av ≡ −avχ, κv = −c χv = c vχ, v(i)v ≡ v(i) − v = v(i)χ − vχ, ϕ̇v = 0. (192) To the linear order we identify u ≡ −c∇vχ, ui ≡ −c∇v(i)χ. (193) Now, to the second order, we attempt identifying the Newtonian perturbation variables δ, δi, u, and ui as κv ≡ − ∇ · u, χv ≡ u, u ≡ ∇u, v(i)v ≡ (ui − u) , δ ≡ δv, δi ≡ δ(i)v. (194) Using these identifications Eqs. (186)-(191) can be written as ∇ · u = − ∇ · (δu) +H |uj − u| 2 + 3∆−1∇ · [(uj − u)∇ · (uj − u)] , (195) ∇ · (u̇+Hu) + 4πG̺δ = − ∇ · (u · ∇u)− Ċ(t)αβ uα|β + Ċ 4πG̺− c2 |uj − u| 12πG̺− c2 ∆−1∇ · [(uj − u)∇ · (uj − u)] , (196) δ̇i + ∇ · ui = − ∇ · (δiui) + 2ϕ∇ · (ui − u)− (ui − u) · ∇ϕ+ 2 (u i − u |ui − u| 2 + 3H |uj − u| 2 +∆−1∇ · [(uj − u)∇ · (uj − u)] , (197) ∇ · (u̇i +Hui) + 4πG̺δ = − ∇ · (ui · ∇ui)− Ċ (t)αβ uα|β + Ċ |uj − u| 2 + 3∆−1∇ · [(uj − u)∇ · (uj − u)] , (198) − 4πG̺δ = − [a∇ · (δu)] ∇ · (u · ∇u) + Ċ(t)αβ uα|β + Ċ 4πG̺− |uj − u| 24πG̺− c2 ∆−1∇ · [(uj − u)∇ · (uj − u)] , (199) a2δ̇i − 4πG̺δ = − [a∇ · (δiui)] ∇ · (ui · ∇ui) + Ċ i + Ċ (t)αβ {∆ [u · (ui − u)]−∇ · [(ui − u) · ∇u+ u · ∇ (ui − u)]} |ui − u| 2 − 8πG |uj − u| 2 + 3∆−1∇ · [(uj − u)∇ · (uj − u)] . (200) In Sec. VII B 1 we have shown that, to the linear order, (ui − u) simply decays (∝ a −1) in an expanding phase. From Eq. (183) we have ui − u = [∇di(x) +Di(x)−D(x)] , ̺jdj ≡ 0, ∇ ·D ≡ 0 ≡ ∇ ·Di. (201) Thus, ui−u simply decays in an expanding background. If we ignore these contributions from the velocity differences, except for the presence of the tensor-type perturbation, Eq. (200) coincides exactly with the zero-pressure limit of Newtonian result in Eq. (16). Terms in the last two lines of Eq. (200) are relativistic correction terms which vanish for a single component case leading to Eq. (199); the second line in Eq. (199) also vanishes in the single component case. Ignoring quadratic combination of (ui − u) terms, we have ∇ · u = − ∇ · (δu) , (202) ∇ · (u̇+Hu) + 4πG̺δ = − ∇ · (u · ∇u)− Ċ(t)αβ uα|β + Ċ , (203) − 4πG̺δ = − [a∇ · (δu)] ∇ · (u · ∇u) + Ċ(t)αβ uα|β + Ċ , (204) δ̇i + ∇ · ui = − ∇ · (δiui) + 2ϕ∇ · (ui − u)− (ui − u) · ∇ϕ+ 2 (u i − u , (205) ∇ · (u̇i +Hui) + 4πG̺δ = − ∇ · (ui · ∇ui)− Ċ (t)αβ uα|β + Ċ , (206) a2δ̇i − 4πG̺δ = − [a∇ · (δiui)] ∇ · (ui · ∇ui) + Ċ i + Ċ (t)αβ {∆ [u · (ui − u)]−∇ · [(ui − u) · ∇u+ u · ∇ (ui − u)]} . (207) Equations (202)-(204) coincide with the density and velocity perturbation equations of a single component medium [12]; thus, except for the contribution from gravitational waves, these equations coincide with ones in the Newtonian context. If we further ignore (ui − u) terms appearing in the pure second-order combinations, Eqs. (205)-(207) become δ̇i + ∇ · ui = − ∇ · (δiui) , (208) ∇ · (u̇i +Hui) + 4πG̺δ = − ∇ · (ui · ∇ui)− Ċ (t)αβ uα|β + Ċ , (209) a2δ̇i − 4πG̺δ = − [a∇ · (δiui)] ∇ · (ui · ∇ui) + Ċ i + Ċ (t)αβ . (210) Notice that, by ignoring i-indices, Eqs. (208)-(210) coincide with Eqs. (202)-(204). In this context, except for the contribution from gravitational waves, the above equations coincide exactly with ones in the Newtonian context even in the multi-component case; compare with Eqs. (8),(9),(16) without pressure. In the single component situation such a relativistic/Newtonian correspondence to the second order was shown in [11, 12]. In the present case, the same equation valid in the single component is now valid in the multi-component case for the collective fluid variables. This justifies our identifications of Newtonian perturbation variables in Eq. (194). VIII. EFFECTS OF CURVATURE We consider a single zero-pressure, irrotational fluid. We take the temporal comoving gauge (v ≡ 0) and the spatial γ ≡ 0 gauge. In the presence of background curvature the basic equations are presented in Eqs. (115)-(117) for nonvanishing pressure, or Eqs. (188)-(191) for zero-pressure multiple component fluids. By setting pressures equal to zero in Eqs. (115)-(117), or from Eqs. (187)-(190), we have δ̇ − κ = − ,α + δκ, (211) κ̇+ 2Hκ− 4πG̺δ = − Ċ(t)αβ + χ,α|β χ,α|β , (212) δ̈ + 2Hδ̇ − 4πG̺δ = 4πG̺δ2 − (δ,αχ Ċ(t)αβ + χ,α|β χ,α|β , (213) ∆+ 3K χ+ κ = 0, (214) where Eq. (214) is valid to the linear order. Compared with the situation with vanishing curvature, the effects of curvature in the above perturbed set of equations appear only in the linear-order relation between κ and χ in Eq. (214). A. Newtonian correspondence Considering the successful Newtonian correspondence to the linear order even in the presence of the background curvature, we assume the identification in Eq. (120) is valid to the second order. Then, to the linear order, from Eq. (214) we have κ ≡ − ∇ · u ≡ − u = −c ∆+ 3K χ, (215) where u ≡ ∇u and χ = χv = −avχ. Thus, to the linear order, formally we have ∆+ 3K u. (216) In the presence of curvature, the scalar-type perturbation can be handled by solving Eqs. (211),(212),(214) together with the identifications made above. We can formally separate the effects of pure curvature contribution. Using Eqs. (215),(216), Eqs. (211),(212),(214) become ∇ · u = − ∇ · (δu) + (∇δ) · ∇ ∆+ 3K , (217) ∇ · (u̇+Hu) + 4πG̺δ = − ∇ · (u · ∇u)− Ċ(t)αβ u,α|β ∆+ 3K ∆+ 3K ∇ · u 2∆+ 3K ∆+ 3K ∇ · u ∆+ 3K ),α|β ∆+ 3K . (218) From Eq. (74), the K term can be written as (Ωt − 1) , Ωt ≡ Ω+ ΩΛ, Ω ≡ , ΩΛ ≡ . (219) IX. EFFECTS OF VECTOR-TYPE PERTURBATION The spatial C-gauge sets γ ≡ 0 ≡ C(v)α . (220) The remaining variables under this gauge condition are completely free of the spatial gauge modes and have unique spatially gauge-invariant counterparts. If we simultaneously take any temporal gauge which also removes the temporal gauge mode completely, all the remaining variables have corresponding unique gauge-invariant counterparts. The above statements are true to all orders in perturbations, see Sec. VI of [11]. From Eq. (77) we have χ, B(v)α ≡ Ψ α . (221) Thus, we have χ,α +Ψ α , Cαβ ≡ ϕg . (222) As the temporal comoving gauge we set v ≡ 0. (223) As we mentioned, the remaining variables under these gauge conditions are completely free of the gauge modes and have unique gauge-invariant counterparts to all orders in perturbation, see [11]. A. Linear perturbations To the linear order, the three types of perturbations decouple, and evolve independently. The rotational perturbation is described by Eqs. (206)-(209) in [11]: ∆ + 2K Ψ(v)α + (µ+ p) v(v)α = 0, (224) a4 (µ+ p) v a4 (µ+ p) ∆+ 2K , (225) Ψ̇(v)α + 2HΨ Π(v)α , (226) where the last equation follows from the first two. Compared with Bardeen’s notation in [16], we have B(v)α = B (1)Q(1)α , C α = − α , Ψ α = ΨQ B(1) − Q(1)α , v(v)α = vcQ v(1) −B(1) Q(1)α , v α = v v(1) − Q(1)α , (227) where Bardeen’s v c and v s , thus our v α and v α + Ψ α , are related to the vorticity and the shear, respectively. From Eq. (60), to the linear order, we have ω̃αβ = av [α|β] σ̃αβ = a ∇α∇β − . (228) Bardeen called Ψ α a ‘frame-dragging potential’. The difference between v c and v s is crucially important to show Mach’s principle including the linear order rotational perturbation in [27]. In the absence of anisotropic stress which can act as a sink or source of the angular momentum, we have Angular momentum ∝ a4 (µ+ p) v(v)α ∝ a 2Ψ(v)α ∝ constant in time. (229) Thus, for vanishing anisotropic stress, we have v(v)α ∝ a4(µ+ p) , Ψ(v)α ∝ . (230) In the zero-pressure limit we have v(v)α ∝ , Ψ(v)α ∝ . (231) B. Second-order perturbations We consider a single component situation with general pressure. We set K ≡ 0. To the linear-order we use χ ≡ χv ≡ −avχ, κ ≡ κv = c vχ. (232) The scalar-type perturbation is described by Eqs. (67),(65) which give δ̇ + 3H c2s − w δ + 3H − (1 + w) κ = −3H (1 + w) v ,αχ −Ψ vχ,α −Ψ v ,αχ − Ψ vχ − 3Hα + (1 + w)αc vχ + (1 + w) v 1− 3c2s Hv(v)α + Π(v)α (1 + w) − (2α+ ϕ) v(v)α + 2C(t)αβv Παα − vχ,α|β + (δµ+ δp) v(v)α + , (233) κ̇+ 2Hκ− (δµ+ 3δp) = − 3Ḣ + c2 v ,αχ −Ψ vχ,α −Ψ − 2Hαc (δµ+ 3δp)α+ Ḣα2 + (α+ 2ϕ) c2 α,α (α− ϕ) C(t)αβα,α|β + (µ+ p) v(v)αv(v)α v ,αχ −Ψ Ċ(t)αβ − v ,α|βχ + Ψ(v)(α|β) vχ,α|β + . (234) The vector-type perturbation is described by Eq. (68) which gives a4 (µ+ p) v(v)α (µ+ p)α,α + δp,α +Π −αδp+ (µ+ p) v ,βχ −Ψ vχ,β −Ψ α,αδµ +(µ+ p) vχ − 3Hα v(v)α − (µ+ p) −v ,βχ +Ψ v(v)β − v ,βχ +Ψ − ϕ,βΠ α + ϕ,αΠ + 2C(t)βγΠαβ|γ + C (δµ+ δp) v(v)α +Π ≡ µcAα. (235) We have [Aα] = L −1. From this we have (µ+ p)α+ δp+ Π = aµ∆−1∇ ·A, (236) a4 (µ+ p) v(v)α Π(v)α = µc ∆−1∇ ·A . (237) Equation for the tensor-type perturbation (gravitational waves) follows from Eqs. (66),(102). C. Zero-pressure case In the zero-pressure limit, Eqs. (236),(235) give α = a∆−1∇ ·A, (238) aAα ≡ − v ,βχ −Ψ vχ,β −Ψ v ,βχ −Ψ v ,βχ −Ψ v(v)β , (239) thus, α is purely second-order, and v ,βχ −Ψ vχ,β −Ψ = ∆−1∇α v ,βχ −Ψ v ,βχ −Ψ v(v)β .(240) Equations (233), (234), (237) give δ̇ − κ = v ,αχ −Ψ (v)α − v(v)α +Hv(v)α v (v)α + −ϕ,αv(v)α + 2C (t)αβv −3H∆−1∇α v ,βχ −Ψ v ,βχ −Ψ v(v)β , (241) κ̇+ 2Hκ− 4πG̺δ = v ,αχ −Ψ v ,βχ −Ψ + Ċ(t)αβ vχ,α|β −Ψ +8πG̺v(v)αv(v)α − 3Ḣ + c2 ∆−1∇α v ,βχ −Ψ v ,βχ −Ψ v(v)β ,(242) v̇(v)α +Hv v ,βχ −Ψ v ,βχ −Ψ v(v)β v ,βχ −Ψ v ,βχ −Ψ v(v)β . (243) D. Newtonian correspondence In order to compare with Newtonian equations we continue identifying κ as in Eq. (120) to the second order. To the linear order we identify u ≡ −c ∇vχ − v ≡ ∇u+ u(v), (244) u ≡ −cvχ, u (v) ≡ cv(v). (245) We introduce the following notations Uα ≡ uα + cΨ α , Ũα ≡ u,α + cΨ α . (246) As mentioned below Eq. (227), to the linear order, uα and Uα are related to the vorticity and the shear, respectively. Equations (241)-(243) become ∇ · u = − ∇ · (δU) + C(t)αβu (v) · ∇ϕ+ (v) · u(v) + 3∆−1∇α Uβ + u(v)βŨβ|α ,(247) ∇ · (u̇+Hu) + 4πG̺δ = − ∇ · (U · ∇U)− Ċ(t)αβ Ũα|β −u(v) · u(v) + ∆−1∇α Uβ + u(v)βŨβ|α , (248) (v) +Hu(v) = − U · ∇u−∇∆−1∇ · (U · ∇u) . (249) From Eq. (249) we notice that the tensor-type perturbation does not affect the vector-type perturbation to the second order. The pure scalar-type perturbation also cannot generate the vector-type perturbation to the second-order; the same is true in the Newtonian case, see below Eq. (15). From Eq. (224), to the linear order, we have v(v)α = Ψ(v)α , (250) where k is a comoving wavenumber with ∆ ≡ −k2, thus [k] = 1. Since (ck)/(aH) ∼ (visual-horizon)/(scale), we have far inside horizon : v(v)α ≫ Ψ α , U ≃ u = ∇u+ u (v), Ũ ≃ ∇u, far outside horizon : v(v)α ≪ Ψ α , U ≃ Ũ = u,α + cΨ α . (251) Apparently, contributions of vector-type perturbation to the second order depend on the visual-horizon scale. Far inside the horizon, we can ignore cΨ α compared with u α . In the matter dominated era we have δv, (252) where we used Eqs. (293),(329),(330) of [11]. Thus, the third term in the right-hand-side of Eq. (247) is (aH/kc)2-order smaller than the first term. The fourth (and last) term in the right-hand-side of Eq. (247) is (aH/kc)[u(v)/(cδ)]-order smaller than the first term. The third (and last) term in the right-hand-side of Eq. (248) is also (aH/kc)2-order smaller than the first term. Thus, Eqs. (247)-(249) give ∇ · u = − ∇ · (δu) + C(t)αβu , (253) ∇ · (u̇+Hu) + 4πG̺δ = − ∇ · (u · ∇u)− Ċ(t)αβ u,α|β + cΨ , (254) (v) +Hu(v) = − u · ∇u−∇∆−1∇ · (u · ∇u) . (255) Thus, if we could ignore the tensor-type combination in Eqs. (253),(254), Eqs. (253)-(255) coincide exactly with the Newtonian equations: see Eqs. (8),(12),(15) ignoring the pressure terms and the subindices i. The vector-tensor combinations in Eq. (253),(254) are new relativistic contributions of the vector-type perturbations; compare these two equations with Eqs. (17),(18) which are valid in the absence of the vector-type perturbations. Notice the form of last term u,α|β + cΨ in Eq. (254) which subtly differs from the expression uα|β in Eq. (18). Contributions from the vector-type perturbation become more complicated near and outside the horizon scale. The presence of vector-type metric perturbation Ψ α , the scalar-type curvature perturbation ϕ, and the tensor-type perturbation C coupled with the vector-type perturbation give additional effects. E. Pure vector-type perturbations In Sec. VII-E of [11] we have considered a situation with pure vector-type perturbation. As the analysis was made based on the fluid quantities in the normal frame, in the following we present the case based on the fluid quantities in the energy frame. Considering only the vector-type perturbation of a fluid, Eq. (68) gives a4(µ+ p) (µ+ p) v(v)α + (α|β) v(v)β c(∆ + 2K)Π 2a2(µ+ p) v(v)β +Ψ(v)|β v(v)βv . (256) Thus, for Π α = 0 we have a4(µ+ p) a4 (µ+ p) v(v)α v(v)β +Ψ(v)|β v(v)βv . (257) This differs from Eq. (365) of [11] which is due to the difference in the frame choice. The momentum constraint equation in Eq. (101) of [11] becomes ∆ + 2K Ψ(v)α + (µ+ p) v(v)α = − (α|β) v(v)β . (258) Under the gauge transformation, from Eq. (61) we have v̂(v)α = v α − v ξ(v)β,α − v ξ(v)β . (259) To the linear order, from Eq. (230) of [11] we have B̂ α = B α + ξ α . We consider a gauge transformation from the C-gauge (C α ≡ 0, without hat) to the B-gauge (B α ≡ 0, with hat). We have B̂ α ≡ 0, and B α |C−gauge = −ξ Thus, ξ(v)α = − B(v)α |C−gaugedη = −a 2Ψ(v)α ∫ η dη , (260) where we used B α |C−gauge = Ψ α ∝ a −2. Thus, Eq. (259) gives v(v)α |B−gauge = v(v)α − Ψ(v)β,α − v Ψ(v)β ∫ η dη C−gauge . (261) X. EQUATIONS WITH FIELDS A. A minimally coupled scalar field Equations in the case of a minimally coupled scalar field are presented in Eqs. (112)-(114) of [11]. The equation of motion in Eq. (112) and the full Einstein’s equations in Eqs. (99)-(105) expressed using the normal-frame fluid quantities together with the normal-frame fluid quantities for the scalar field in Eqs. (114) all in [11] provide a complete set of equations we need to the second-order. The fluid quantities in Eq. (114) of [11] are presented in the normal-frame four-vector and it is convenient to know the conventionally used fluid quantities which are based on the energy-frame four-vector. These latter quantities can be read from Eqs. (88),(114) of [11] and Eq. (79) as QN(φ)α = − φ̇δφ,α + δφ,α δφ̇− φ̇A µ(φ) + p(φ) V (φ)α −Bα +ABα + 2V (φ)βCαβ δµ(φ) + δp(φ) V (φ)α −Bα µ(φ) + p(φ) + δµ(φ) + δp(φ) −v(φ),α + v (φ,v) δµ(φ) = δµN(φ) − δφ,αδφ,α, δp (φ) = δpN(φ) − δφ,αδφ,α, δφ,αδφ,β − δφ,γδφ,γ . (262) Thus, using Eq. (114) of [11] we have v(φ) = ∆−1∇α δφ̇−Aφ̇ v(φ,v)α = δφ̇− φ̇A δφ,α −∇α∆ δφ̇− φ̇A δµ(φ) = φ̇δφ̇− φ̇2A+ V,φδφ+ δφ̇2 − δφ,αδφ,α + V,φφδφ 2 − 2φ̇δφ̇A+ φ̇δφ,αB α + 2φ̇2A2 − φ̇2BαBα, δp(φ) = φ̇δφ̇− φ̇2A− V,φδφ+ δφ̇2 − δφ,αδφ,α − V,φφδφ 2 − 2φ̇δφ̇A+ φ̇δφ,αB α + 2φ̇2A2 − φ̇2BαBα, = 0. (263) Notice that no anisotropic stress is caused by a minimally coupled scalar field even to the second order in perturbations. The uniform-field gauge takes δφ ≡ 0 as a temporal gauge (slicing) condition to the second-order in perturbation. The uniform-field gauge gives v(φ) = 0 which is the comoving gauge, and vice versa. Thus, δφ = 0 ↔ v(φ) = 0. (264) We also have δµ(φ) − δp(φ) = 2V,φδφ+ V,φφδφ 2, (265) and under the uniform-field gauge we have = −φ̇2 A (1− 2A) + . (266) Equation (263) apparently shows that the vector-type perturbation v (φ,v) α does not vanish to the second-order. However, the second-order quantities in right-hand-side depend on the temporal gauge condition for the scalar-type perturbations, and trivially vanish for the uniform-field gauge. We can also show that it vanishes for the uniform- density gauge or the uniform-pressure gauge where we have δφ̇ − φ̇A ∝ δφ to the linear order. In fact, we can show that the vector-type perturbation is not sourced by the scalar field to the second order by evaluating the rotational tensor ω̃αβ in Eq. (60) for the scalar field: i.e., for a minimally coupled scalar field we have = 0, (267) to the second order. Thus, we conclude that a minimally coupled scalar field does not contribute to the rotational perturbation to the second order in perturbations. In fact, we can show that a single scalar field do not support vector-type perturbations to all orders in perturbations. As we take the energy frame, thus q̃a ≡ 0, from Eq. (23) of [11] we can show ũa = −φ̃,cφ̃,c . (268) Using the definition of the vorticity tensor in Eq. (33) we can show that ω̃ab = 0. Using the fluid quantities in Eq. (263) we can handle the scalar field using our non-ideal fluid formulation. The fluid equations in the energy frame, like Eqs. (62)-(68), remain valid in the case of scalar field with the fluid quantities expressed as in Eq. (263). The anisotropic stress vanishes and the entropic perturbation e is given as e(φ) ≡ δp(φ) − c2(φ)δµ (φ), c2(φ) ≡ ṗ(φ) µ̇(φ) φ̈− V,φ φ̈+ V,φ . (269) Under the comoving gauge v ≡ 0, using Eq. (266) we have δp(φ)v = δµ v , e 1− c2(φ) δµ(φ)v , (270) which is a well known relation, now valid to the second order in perturbations. A fluid formulation of the scalar field to the linear order is presented in [24, 29]. Using Eq. (270) together with vanishing anisotropic stress, Eqs. (62)-(68) or (88)-(94) provide the fluid formulation for a minimally coupled scalar field to the second order in perturbations. The perturbed equation of motion of the scalar field is presented in Eq. (112) of [11]. B. Minimally coupled scalar fields In the case of multiple minimally coupled scalar fields, the equation of motions in Eq. (119) and the full Einstein’s equations in Eqs. (99)-(105) expressed using the normal-frame fluid quantities together with the normal-frame fluid quantities for the scalar field in Eqs. (121) all in [11] provide a complete set of equations we need to the second-order. The fluid quantities in Eq. (121) of [11] are presented in the normal-frame four-vector and it is convenient to know the conventionally used fluid quantities which are based on the energy-frame four-vector. These latter quantities can be read from Eqs. (121) of [11] and Eq. (58) as v(φ) = φ̇(k)δφ(k) +∆ δφ̇(k) − φ̇(k)A δφ(k) − 2φ̇(k) φ̇(l)δφ(l) v(φ,v)α = − δφ̇(k) − φ̇(k)A δφ(k) − 2φ̇(k) φ̇(l)δφ(l) δφ̇(k) − φ̇(k)A δφ(k) − 2φ̇(k) φ̇(l)δφ(l) δµ(φ) = δµN(φ) − φ̇(k)δφ φ̇(l)δφ(l),α δp(φ) = δpN(φ) − φ̇(k)δφ φ̇(l)δφ(l),α φ̇(k)δφ(k),α φ̇(l)δφ(l),β φ̇(k)δφ φ̇(l)δφ(l),γ (271) where the normal-frame fluid quantities are presented in Eq. (121) of [11]. Thus, by moving into the energy-frame fluid quantities from the normal-frame ones we have rather complicated terms which do not cancel out nicely any term in the normal-frame quantities. In single field case we had such cancelations, for example we have Π = 0 in Eq. (263). But, we do not have such a luxury in the multi-component situation. Apparently Π in the energy-frame does not vanish and looks more complicated. Thus, in the multi-field situation we had better use both the fluid quantities and Einstein’s equations all expressed in the normal frame: these are Eqs. (99)-(105),(121) in [11]. As the temporal gauge condition we can set any one field perturbation, say the specific ℓ-th one δφ(ℓ), equal to zero which might be called the uniform-φ(ℓ) gauge to the second order. This apparently differs from the comoving gauge which sets v(φ) ≡ 0. In the multi-component situation we cannot take a gauge condition which makes v (φ,v) α = 0. Thus, the multiple scalar fields source the vector-type perturbation to the second order in perturbations. The scalar-, vector-, and tensor-type decomposition of the anisotropic stress Π can be read by using decomposition formulae in Eq. (177) of [11]. In multiple-field situation, it is ad hoc and cumbersome (if not impossible) to introduce individual fluid quantity for each field variable even to the background and linear order perturbations, see [30]. C. Generalized gravity case In Sec. IV.D of [11]we presented the equation of motion and effective fluid quantities in a class of generalized gravity theories together with additional presence of fluids and fields to the second order. The equation of motion is in Eq. (128) of [11], and the full Einstein’s equations in Eqs. (99)-(105) expressed using the normal-frame fluid quantities together with the normal-frame effective fluid quantities in Eq. (130) of [11] provide a complete set of equations we need to the second-order. The effective fluid quantities in Eq. (130) of [11] are presented in the normal-frame four-vector and using Eq. (88) in [11] we can easily derive the effective fluid quantities based on the energy-frame four-vector. As in the multiple field case in a previous subsection, by moving into the energy-frame, the effective fluid quantities become more complicated compared with the ones in the normal-frame. Thus, in these class of generalized gravity theories we had better use both the fluid quantities and Einstein’s equations all expressed in the normal frame: these are Eqs. (99)-(105),(130) in [11]. XI. CURVATURE PERTURBATIONS AND LARGE-SCALE CONSERVATIONS In the large-scale limit the spatial curvature perturbation ϕ in several different gauge conditions is known to remain constant in expanding phase. Often the conservation properties are shown based on the first time derivative of the curvature perturbation. In order to show the conservation properties properly we have to construct the closed form second-order differential equations for the curvature perturbation. In the following we will derive such first-order and second-order differential equations for ϕ̇v, ϕ̇χ, ϕ̇κ, and ϕ̇δ. First we will derive equations for the linear perturbation including the background curvature and non-ideal fluid properties. Then we will derive equations for the second order perturbation assuming a flat background; including the background curvature is trivial, though. We consider a single-component fluid. A. Linear-order equations We introduce a combination Φ ≡ ϕv − 4πG (µ+ p) ϕχ. (272) This combination was first introduced by Field and Shepley in [28]. From Eqs. (88),(90),(92), Eqs. (88)-(90),(92),(94), Eqs. (88),(93), and Eqs. (88),(89),(91), respectively, we can derive 4πG (µ+ p) Φ− 8πGHΠ, (273) 4πG (µ+ p) , (274) ϕ̇δ = , (275) ϕ̇κ = − 3Ḣ +∆/a2 1 + 3c2s ) ∆+ 3K ϕκ − 12πGe χκ. (276) These equations were presented in [24, 31] We can derive closed form second-order differential equations for Φ, ϕχ, ϕδ, and ϕκ (µ+ p) a 8πGH2 = c2s ϕχ − 4πG , (277) H2c2s (µ+ p) a3 (µ+ p) a3 H2c2s = c2s , (278) 3Ḣ +∆/a2 (µ+ p) a3 (µ+ p) a3 3Ḣ +∆/a2 ϕ̇δ + 4πG (µ+ p) + c2s (∆ + 3K) /a 12πG (µ+ p)− (∆ + 3K) /a2 3Ḣ + , (279) ϕ̇κ + 3Ḣ +∆/a2 1 + 3c2s ) ∆+ 3K ϕκ − 12πGe 3Ḣ +∆/a2 1 + 3c2s ) ∆+ 3K ϕκ − 12πGe + 8πGΠ . (280) Equations (277),(278) follow by combining Eqs. (273),(274). Equations (279) and (280) follow from Eqs. (88),(89),(91),(92), and Eqs. (88),(89),(91),(92), respectively. From Eqs. (89),(90) we can derive ∆+ 3K 12πG (µ+ p) a2 ϕκ = ϕv − ∆+ 3K 12πG (µ+ p)a2 ϕχ = Φ− 12πG (µ+ p) a2 ϕχ. (281) These relations were presented in [17]. In the large-scale limit and in near flat background, thus ignoring ∆ and K terms, we have Φ ≃ ϕv ≃ ϕδ ≃ ϕκ, (282) to the leading order in the large-scale expansion. In the large-scale limit, ignoring ∆ terms, in near flat background and for an ideal fluid case, thus setting K = 0 and e = 0 = Π, Eqs. (277)-(280) give (µ+ p) a , (283) ϕ̇v ∝ H2c2s (µ+ p) a3 , (284) ϕ̇δ ∝ ϕ̇κ ∝ . (285) Notice that if we simply ignore the ∆ terms in Eqs. (274)-(276) we simply have ϕ̇v = ϕ̇δ = ϕ̇κ = 0. In such a way we cannot recover the terms in the right-hand-side of Eqs. (284),(285); these terms lead to decaying solutions (in an expanding era) in the large-scale limit and are higher order in the large-scale expansion compared with the decaying solution of ϕχ, see below. From Eqs. (283)-(285) we have general large-scale asymptotic solutions ϕχ = 4πGC(x) ∫ t a (µ+ p) dt+ d(x) , (286) ϕv = C(x) + ∫ t c2sH (µ+ p) a3 dt, (287) ϕδ = ϕκ = C(x) + ∫ t dt , (288) where C(x) and d(x) are integration constants which correspond to the relatively growing and decaying modes, respectively, in an expanding phase; in a collapsing phase the roles are reversed. The coefficients are fixed using the relations in Eqs. (273),(274),(281). Notice that for the C-mode the relation in Eq. (282) is satisfied, and simply remain constant. For the d-mode, ϕv, ϕδ, and ϕκ are ∆/(aH) 2-order higher compared with the d-mode of ϕχ. The ϕv is one of the well known conserved quantity in the large-scale even in the context of generalized gravity theories [24, 32]. In order to evaluate the solutions to the second order in the next section, we need complete sets of linear order solutions for different gauge conditions. For an ideal fluid, and for a minimally coupled scalar field such complete sets of solutions are presented in tabular forms in [29, 31]. In the following we summarize such sets of solutions in an ideal fluid case for four different gauge conditions. From Table 8 of [31] we have ϕχ = −αχ = C , δχ = − = −2C adt, vχ = −C ϕv = C, Hχv = C adt, δv = − 1 + w αv = − ϕδ = C, Hχδ = C = 3αδ = − C, vδ = ϕκ = C, Hχκ = C adt, δκ = −3 1 + w 1 + 3c2s ακ = − C, vκ = − adt. (289) For corresponding sets of solutions for a minimally coupled scalar field, see Table 1 of [29]. Compared with the notation used in [31] we have γΨ = −8πG(µ+p)av. The lower bounds of integration of solutions in Eq. (289) give behaviors of d-modes. For solutions without integration, the d-modes are ∆/(aH)2-order higher than the non-vanishing d-mode, for example, see the solutions in Table 2 of [31]. Thus, the d-modes are ϕχ = −αχ = d , δχ = − , vχ = d ∫ t c2sH (µ+ p) a3 dt, Hχv = − d, δv = − 1 + w αv = − ∫ t dt , Hχδ = − ∫ t dt , αδ = ∫ t dt , vδ = ∫ t dt , Hχκ = − d, δκ = −3 1 + w 1 + 3c2s ακ = − ∫ t dt , vκ = d. (290) B. Second-order equations We assume K = 0. From Eqs. (88),(90),(92), Eqs. (88)-(90),(94), Eqs. (88),(93), and Eqs. (88),(89),(91), respec- tively, we can derive (ϕ−Hχ) 4πG (µ+ p) (ϕ− aHv)− 8πGHΠ+ (n0 − n2)−Hn4, (291) (ϕ− aHv) 4πG (µ+ p) (ϕ−Hχ)− n1 +Hn2 (n0 − n2)− aHn6, (292) ϕ̇δ = , (293) ϕ̇κ = − 3Ḣ +∆/a2 1 + 3c2s ϕκ − 12πGe− 1 + 3c2s n1 − n3 n0. (294) The perturbed order variables in Eqs. (293),(294) are evaluated in the uniform-density gauge (δ ≡ 0), and the uniform-expansion gauge (κ ≡ 0), respectively. Equation (293) also follows from Eq. (41) evaluated to the second order. We can derive closed form second-order differential equations for ϕv, ϕχ, ϕδ, and ϕκ (ϕ−Hχ) + 8πGHΠ− (n0 − n2) +Hn4 = c2s (ϕ−Hχ)− 4πG + c2s n1 +Hn2 4πG (µ+ p) (n0 − n2)− aHn6 , (295) H2c2s 4πG (µ+ p) a3 4πG (µ+ p) a3 H2c2s (ϕ− aHv) (n0 − n2) + aHn6 n1 −Hn2 = c2s ϕ− aHv − Hc2s∆ 4πG (µ+ p) a2 (n0 − n2)−Hn4 , (296) 1 + ∆/(3a2Ḣ) 1 + ∆/(3a2Ḣ) ϕ̇δ + 3Ḣ +∆/a2 ṅ1 + n1 −Hn3 1− c2s∆/(a 3 + ∆/(a2Ḣ) 3Ḣ + 9Ḣa2 − n3 − 3Ḣ + , (297) ϕ̇κ + 3Ḣ +∆/a2 1 + 3c2s − 12πGe− n3 −ϕκ + 3Ḣ +∆/a2 1 + 3c2s − 12πGe− n3 − 8πGΠ− n4 . (298) Equations (295),(296) follow by combining Eqs. (291),(292). Equation (297) follows from Eqs. (88),(89),(91),(92). Equation (298) follows from Eqs. (88),(89),(91),(92). The perturbed order variables in Eqs. (297),(298) are evaluated in the uniform-density gauge (δ ≡ 0), and the uniform-expansion gauge (κ ≡ 0), respectively. C. Large-scale solutions Now, we assume an ideal fluid, thus set e = 0 = Π. In the large-scale limit, thus ignoring the ∆/(aH)2-order higher terms, Eqs. (296)-(298) give ϕ̇v − (n0 − n2) + aHn6 + 4πG (µ+ p) n1 +Hn2 H2c2s 4πG (µ+ p) a3 , (299) ϕ̇δ + ṅ1 + Hn1 −Hn3 , (300) ϕ̇κ − 1 + 3c2s n1 + n3 , (301) where the perturbed order variables in Eq. (299) are evaluated in the comoving gauge (v ≡ 0). We already used the behavior of linear order solutions in Eqs. (289),(290) in order to show that the right-hand-side of Eqs. (296)-(298) vanish. Using the solutions in Eqs. (289),(290) we can show that ϕv − ϕ +O(∆C2,∆2d2) ∝ H2c2s (µ+ p)a3 , (302) ϕδ − ϕ +O(∆C2,∆2d2) ∝ ϕκ − ϕ +O(∆C2,∆2d2) ∝ . (303) Thus, we have general large-scale asymptotic solutions ϕv − ϕ v = C(x) + ∫ t c2sH (µ+ p) a3 dt, (304) ϕδ − ϕ δ = ϕκ − ϕ κ = C(x) + ∫ t dt , (305) where C(x) and d(x) are integration constants now including the second-order contributions, i.e., C = C(1) + C(2), etc. Ignoring the transient solutions in an expanding phase we have ϕv = ϕδ = ϕκ = C(x), (306) even to the second order in perturbations in the large-scale limit. XII. DISCUSSION In this work we presented pure general relativistic effects of second-order perturbations in Friedmann cosmological world model. In our previous work we have shown that to the second-order perturbations, the density and velocity perturbation equations of general relativistic zero-pressure, irrotational, single-component fluid in a flat background coincide exactly with the ones known in Newton’s theory, [12]. We also have shown the effect of gravitational waves to the second-order, and pure general relativistic correction terms appearing in the third-order perturbations, [12, 13]. Here, we presented results of second-order perturbations relaxing all the assumptions made in our previous work in [12]. We derived the general relativistic correction terms arising due to (i) pressure, (ii) multi-component, (iii) background curvature, and (iv) rotation. We also presented a general proof of large-scale conserved behaviors of curvature perturbation variable in several gauge conditions, now to the second order. Effects of pressure can be found in Eqs. (123)-(125). As we emphasized, the effect of pressure is generically relativistic even in the background world model and the linear order perturbations. Still, our equations show the pure general relativistic effects of pressure (including stresses) appearing in the second-order perturbations. Effects of multi-component fluids can be found in Eqs. (195)-(200). Although these equations apparently show deviations from Newtonian situation, in Sec. VII D we showed that if we ignore purely decaying terms in an expanding phase the equations are effectively the same as in the Newtonian situation. Effects of background spatial curvature K can be read from Eqs. (217),(218) or Eqs. (211)-(214). Effects of vector-type perturbation can be read from Eqs. (247)-(249). In the small-scale limit we showed that, if we ignore the tensor-type perturbation, the equations coincide with the Newtonian ones. Our results may have important practical implications in cosmology and the large-scale structure formation. Our new result showing relativistic/Newtonian correspondence in the zero-pressure irrotational multi-component fluids is practically relevant in currently favored cosmology where baryon and dark matter are two important ingredients of the current matter content in addition to the cosmological constant. All equations in our work are valid in the presence of the cosmological constant. A related important result is the relativistic/Newtonian correspondence valid in the presence of rotational perturbation far inside horizon. Thus, inside the horizon scale, even in the presence of rotational perturbations we can still rely on the Newtonian equations to handle quasi-linear evolution of large-scale structures. As the spatial curvature in the present cosmological era is known to be small [33], the possible presence of small spatial curvature may not be important in the second-order perturbations. Still, while the Newtonian equation is exactly valid to the linear order even in the presence of the spatial curvature, we have nontrivial general relativistic correction terms present to the second order in perturbations. Our second-order perturbation equations in the presence of pressure may have an interesting role as we approach early stage of universe where the effect of radiation becomes important. The importance of pressure to the second-order perturbations, of course, depends on whether nonlinear effects are significant in the early evolution stage of the large-scale structure during the radiation era and in the early matter dominated era. Realistic estimations of the diverse pure general relativistic contributions using the complete set of equations presented in this work are left for future investigations. In an accompanying paper we will investigate the effects of third-order perturbations of zero-pressure irrotational multi-component fluids in a flat background. This is one obvious remaining issue in our series of investigation of nonlinear cosmological perturbations where nontrivial general relativistic effects are expected. In the case of a single fluid we presented the pure general relativistic effects appearing in the third order in [13]. Corresponding results in the case of multi-component will be presented in [34]. Acknowledgments H.N. was supported by grant No. C00022 from the Korea Research Foundation. [1] M. Colless, et al., 2003, Preprint astro-ph/0306581. [2] K. Abazajian, et al. in http://www.sdss.org/dr3/ (2004). [3] C.L. Bennett et al., Astrophys. J. 464, L1 (1996) Preprint astro-ph/9601067. [4] C.L. 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Hwang, Astrophys. J. 375, 443 (1991). [25] H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984). [26] H. Nariai, Prog. Theor. Phys. 41, 686 (1969). [27] C. Schmid, Phys. Rev. D 74, 044031 (2006), Preprint gr-qc/0508066. [28] G.B. Field and L.C. Shepley, Astrophys. Space. Sci. 1, 309 (1968). [29] J. Hwang, Astrophys. J. 427, 542 (1994). [30] J. Hwang and H. Noh, Phys. Lett. B, 495, 277 (2000), Preprint astro-ph/0009268; Class. Quant. Grav. 19, 527 (2002) Preprint astro-ph/0103244. [31] J. Hwang, Astrophys. J. 415, 486 (1993). [32] J. Hwang and H. Noh, Phys. Rev. D 71, 063536 (2005), Preprint gr-qc/0412126 . [33] P. de Bernardis, et al., Nature 404, 955 (2000), Preprint astro-ph/0004404. [34] J. Hwang and H. Noh, in progress. http://arxiv.org/abs/gr-qc/0508066 http://arxiv.org/abs/astro-ph/0009268 http://arxiv.org/abs/astro-ph/0103244 http://arxiv.org/abs/gr-qc/0412126 http://arxiv.org/abs/astro-ph/0004404 Contents Introduction Newtonian nonlinear perturbations Summary of previous work Relativistic fully nonlinear equations Covariant equations ADM equations Second-order perturbations Basic equations in the energy-frame Decomposition Comoving gauge and irrotational condition Effects of pressure Irrotational case Comoving gauge Newtonian correspondence Linear-order relativistic pressure corrections Effects of multi-component Irrotational case Linear perturbations Exact solutions Comoving gauge Newtonian correspondence Effects of curvature Newtonian correspondence Effects of vector-type perturbation Linear perturbations Second-order perturbations Zero-pressure case Newtonian correspondence Pure vector-type perturbations Equations with fields A minimally coupled scalar field Minimally coupled scalar fields Generalized gravity case Curvature perturbations and large-scale conservations Linear-order equations Second-order equations Large-scale solutions Discussion Acknowledgments References

Highly turbulent solutions of LANS−α and their LES potential Jonathan Pietarila Graham,1, 2 Darryl D. Holm,3, 4 Pablo D. Mininni,1, 5 and Annick Pouquet1 1National Center for Atmospheric Research, ∗ P.O. Box 3000, Boulder, Colorado 80307, USA 2currently at Max-Planck-Institut für Sonnensystemforschung, 37191 Katlenburg-Lindau, Germany 3Department of Mathematics, Imperial College London, London SW7 2AZ, UK 4Computer and Computational Science Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 5Departamento de Fı́sica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina (Dated: March 2, 2022) ∗ The National Center for Atmospheric Research is sponsored by the National Science Foundation http://arxiv.org/abs/0704.1928v3 Abstract We compute solutions of the Lagrangian-Averaged Navier-Stokes α−model (LANS−α) for significantly higher Reynolds numbers (up to Re ≈ 8300) than have previously been accomplished. This allows suf- ficient separation of scales to observe a Navier-Stokes inertial range followed by a second inertial range specific to LANS−α. Both fully helical and non-helical flows are examined, up to Reynolds numbers of ∼ 1300. The analysis of the third-order structure function scaling supports the predicted l3 scaling; it cor- responds to a k−1 scaling of the energy spectrum for scales smaller than α. The energy spectrum itself shows a different scaling which goes as k1. This latter spectrum is consistent with the absence of stretching in the sub-filter scales due to the Taylor frozen-in hypothesis employed as a closure in the derivation of LANS−α. These two scalings are conjectured to coexist in different spatial portions of the flow. The l3 (E(k) ∼ k−1) scaling is subdominant to k1 in the energy spectrum, but the l3 scaling is responsible for the direct energy cascade, as no cascade can result from motions with no internal degrees of freedom. We demonstrate verification of the prediction for the size of the LANS−α attractor resulting from this scaling. From this, we give a methodology either for arriving at grid-independent solutions for LANS−α, or for obtaining a formulation of a Large Eddy Simulation (LES) optimal in the context of the alpha models. The fully-converged grid-independent LANS−α may not be the best approximation to a direct numerical sim- ulation of the Navier-Stokes equations since the minimum error is a balance between truncation errors and the approximation error due to using LANS−α instead of the primitive equations. Furthermore, the small- scale behavior of LANS−α contributes to a reduction of flux at constant energy, leading to a shallower energy spectrum for large α. These small-scale features, however, do not preclude LANS−α to reproduce correctly the intermittency properties of the high Reynolds number flow. PACS numbers: 47.27.ep; 47.27.E-; 47.27.Jv; 47.50.-d I. INTRODUCTION Since the degrees of freedom for high Reynolds number (Re) turbulence, as can be encountered in geophysical and astrophysical flows, can be very large, the implementation of their numerical modeling can easily exceed technological limits for computations. Furthermore, since truncation of the omitted scales removes important physics, e.g., of multi-scale interactions, the only ap- proach to a numerical study of such flows is to employ subgrid modeling of those scales. This is frequently accomplished with Large Eddy Simulations (LES–see [1, 2, 3] for recent reviews). This is of importance for geophysical, astrophysical and engineering applications and can have consequences for meteorological [4] and climate prediction simulations [5], for instance. While realistic Reynolds numbers will remain out of reach for the foreseeable future, subgrid modeling can be an extremely useful tool in the computation of simulations for such applications. The incompressible Lagrangian-averaged Navier-Stokes equations (LANS−α, α−model, or also the viscous Camassa-Holm equation) [6, 7, 8, 9, 10, 11] is one possible subgrid model. It can be derived, for instance, by temporal averaging applied to Hamilton’s principle (where Taylor’s frozen-in turbulence hypothesis is applied as the closure, and also as the only approximation of the derivation) [12, 13, 14]. For this reason, the momentum-conservation structure of the equations are retained. For scales smaller than the filter width, LANS−α reduces the steepness of steep gradients of the Lagrangian mean velocity and limits how thin vortex tubes become as they are transported (the effect on larger length scales is negligible) [9]. The α−model may also be derived from smoothing the transport velocity of a material loop in Kelvin’s circulation theorem [11]. Con- sequently, there is no attenuation of resolved circulation, which is important for many engineering and geophysical flows where accurate prediction of circulation is highly desirable. LANS−α has previously been compared to direct numerical simulations (DNS) of the Navier-Stokes equa- tions at modest Taylor Reynolds numbers (Rλ ≈ 72 [15], Rλ ≈ 130 [9], and Rλ ≈ 300 [16]). LANS−α was compared to a dynamic eddy-viscosity LES in 3D isotropic turbulence under two different forcing functions (Rλ ≈ 80 and 115) and for decaying turbulence with initial conditions peaked at a low wavenumber (Rλ ≈ 70) and at a moderate wavenumber (Rλ ≈ 220) [17]. In these comparisons, LANS−α was preferable in that it demonstrated correct alignment between eigenvectors of the subgrid stress tensor and the eigenvectors of the resolved stress tensor and vorticity vector. LANS−α and a related regularization, the Leray model, were contrasted with a dynamic mixed (similarity plus eddy-viscosity) model in a turbulent mixing shear layer (Re ≈ 50) [18, 19]. LANS−α, with relatively high subfilter resolutions, was the most accurate of these three LES tested at this moderate Re, but it was found that the effects of numerical contamination can be strong enough to lose most of this potential. This could pose some limitations on its practical use. Quantifying those limitations is one of the goals of this present work. We will also find in this study that, even with sufficient subfilter resolution, LANS−α fails to represent all the neglected physics in a more turbulent regime (higher Re). The α−model also describes an incompressible second-grade non-Newtonian fluid (under a modified dissipation) [11]. In this interpretation, α is a material parameter which measures the elastic response of the fluid. Either from this standpoint, from its status as a regularization of the Navier-Stokes equations, or, independently of any physically motivation, as a set of partial differential equations with proven unique regular solutions, we may analyze LANS−α without any LES considerations. Analyzing inertial-range scaling for LANS−α for moderate and large α, as well as identifying different scalings at scales larger and smaller than α is another of the goals of this work. In this context we also study the numerical resolution requirements to obtain well- resolved solutions of LANS−α (i.e., grid-independent solutions) which leads to a verification of the predictions of the size of the attractor in LANS−α [11, 20]. Section II presents the LANS−α model, our numerical experiments and technique. In Section III we analyze inertial-range scaling for LANS−α. In Section IV we determine the numerical resolution requirements to obtain well- resolved solutions of LANS−α. In Section V we address the LES potential of LANS−α by comparing α−model simulations to a 2563 DNS (Re ≈ 500, Rλ ≈ 300), a 5123 DNS (Re ≈ 670, Rλ ≈ 350), a 5123 DNS (Re ≈ 1300, Rλ ≈ 490), a 10243 DNS (Re ≈ 3300, Rλ ≈ 790), and a 20483 DNS (Re ≈ 8300, Rλ ≈ 1300). (The Re ≈ 3300 simulation has been previously described in a study of the imprint of large-scale flows on local energy transfer [21, 22].) In Section VI, we compare and contrast in more detail LANS−α solutions with DNS at Re ≈ 3300. Finally, in Section VII we summarize our results, present our conclusion, and propose future directions of investigation. II. TECHNIQUE We consider the incompressible Navier-Stokes equations for a fluid with constant density, ∂tvi + vj∂jvi = −∂ip+ ν∂jjvi + Fi ∂ivi = 0, (1) where vi denotes the component of the velocity field in the xi direction, p the pressure divided by the density, ν the kinematic viscosity, and Fi an external force that drives the turbulence (in all results, the time, t, is expressed in units of the eddy-turnover time). The LANS−α equations [6, 7, 8, 9, 10, 11] are given by ∂tvi + uj∂jvi + vj∂iuj = −∂iπ + ν∂jjvi + Fi ∂ivi = ∂iui = 0, (2) where ui denotes the filtered component of the velocity field and π the modified pressure. Filtering is accomplished by the application of a normalized convolution filter L : f 7→ f̄ where f is any scalar or vector field. By convention, we define ui ≡ v̄i. We choose as our filter the inverse of a Helmholtz operator, L = H−1 = (1 − α2∂kk)−1. Therefore, u = gα ⊗ v where gα is the Green’s function for the Helmholtz operator, gα(r) = exp(−r/α)/(4πα2r) (i.e., the well-known Yukawa potential), or in Fourier space, û(k) = v̂(k)/(1 + α2k2). We solve Eqs. (1) and (2) using a parallel pseudospectral code [23, 24] in a three-dimensional (3D) cube with periodic boundary conditions. In most of the runs, we employ a Taylor-Green forcing [25], sin k0x cos k0y cos k0z − cos k0x sin k0y cos k0z (generally, with k0 = 2), and employ dynamic control [26] to maintain a nearly constant energy with time. This expression Eq. (3) is not a solution of the Euler’s equations, and as a result small scales are generated fast when the fluid is stirred with this forcing. The resulting flow models the fluid between counter-rotating cylinders [27] and has been widely used to study turbulence, including studies in the context of the generation of magnetic fields through dynamo instability [28]. We also consider some runs with random and ABC [22] forcing. We define the Taylor mi- croscale as λ = 2π 〈v2〉/〈ω2〉, and the mean velocity fluctuation as vrms = E(k)dk The Taylor microscale Reynolds number is defined by Rλ = vrmsλ/ν and the Reynolds number based on a unit length is Re = vrms × 1/ν. III. INERTIAL RANGE SCALING OF LANS−α A. l3 scaling of third-order structure function derived from the Kármán-Howarth theorem for LANS−α For LANS−α, the H1α(u) norm is the quadratic invariant to be identified with the energy, = −2νΩα, (4) where (u− α2∇2u) · ud3x = 1 v · ud3x, (5) ω · ω̄d3x. (6) As usual, we define the (omni-directional) spectral energy density, Eα(k), from the relation Eα(k)dσdk = Eα(k)dk (7) where dσ represents integration over the surface of a sphere. The α−model possesses a theorem corresponding to the Kármán-Howarth theorem [29] for the Navier-Stokes equations and, as in the Navier-Stokes case, scaling of the inertial range energy spectra may be derived from it [30]. We summarize here the dimensional analysis argument for the LANS−α inertial range scaling that follows from this theorem, beginning from Equation (3.8) in Ref. [30]. We use the short notation vi ≡ vi(x), u i ≡ u , t) and r ≡ x′ − x. In the statistically isotropic and homogeneous case, without external forces and with ν = 0, taking the dot product of Eq. (2) with u j we can obtain the equation ∂tQij = T mij − α2Smij . (8) The trace of this equation is the Fourier transform of the detailed energy balance for LANS−α. Qij = j + vju is the second-order correlation tensor while T mij = j + vju i + v iuj + v jui)u , (10) Smij = (∂mul∂iul)u j + (∂mul∂jul)u i + (gα ⊗ τ ′ j )vi + (gα ⊗ τ ′ i )vj , (11) are the third-order correlation tensors for LANS−α and τ ji is the sub-filter scale stress tensor. For α = 0 this reduces to the well-known relation derived by Kármán and Howarth. The energy dissipation rate for LANS−α , εα, satisfies εα ∝ ∂tQij . By dimensional analysis in Eq. (8) we arrive at (vu2 + u3). (12) For large scales such that l ≫ α, the second right hand term is ignored, u ≈ v, εα ≈ ε, and we arrive at the scaling of the four-fifths law, < (δv‖(l)) 3 >∼ εl [31]. Here, δv‖(l) ≡ [v(x+ l) − v(x)] · l/l is the longitudinal increment of v. The four-fifths law expresses that the third-order longitudinal structure function of v, Sv3 ≡ 〈(δv‖)3〉, is given in the inertial range in terms of the mean energy dissipation per unit mass ε by Sv3 = − εl, (13) or, equivalently, that the flux of energy across scales in the inertial range is constant. We also obtain the Kolmogorov 1941 [32, 33, 34] (hereafter, K41) energy spectrum, E(k)k ∼ v2 ∼ ε2/3l2/3, or, equivalently, E(k) ∼ ε2/3k−5/3. (14) For small scales such that l ≪ α, however, v ∼ α2l−2u and both right hand terms are equivalent in Eq. (12), and our scaling law becomes Su3 ≡< (δu‖(l))3 >∼ εαα−2l3. (15) Note that this scaling differs in a substantial way from the Kolmogorov scaling (∼ l). For our small scale energy spectrum we then have Eα(k)k ∼ uv ∼ ε2/3α α2/3, (16) where we used u ∼ α−2l2v. The energy spectrum for scales smaller than α is then Eα(k) ∼ ε2/3α α2/3k−1. (17) This spectrum can also be derived from phenomenological arguments originally introduced by Kraichnan [35], and it differs from the Navier-Stokes spectrum due to the fact that the fluid is advected by the smoothed velocity u which does not directly correspond to the conserved energy Eα [11]. FIG. 1: Third-order longitudinal structure function of the smoothed velocity field u, Su3 , versus l for large α LANS−α (α = 2π/3 indicated by the vertical dotted line). The scales identified with an inertial range are marked by vertical dashed lines and the scaling predicted by Eq. (15), l3, is indicated by a solid line. The fitted scaling exponent ζu3 (S 3 (l) ∼ lζ 3 ) is found to be ζu3 = 2.39 ± .04. This is more consistent with the scaling given by Eq. (15) than K41 scaling, l1 Eq. (13), or other proposed LANS−α scalings (indicated by dotted lines, see text). We test this prediction for LANS−α scaling at a resolution of 2563 (ν = 1.2×10−4) by moving both the forcing (k0 = 1) and α (kα ≡ 2π/α = 3) to large scales in order to increase the number of resolved scales for which kα > 1. In so doing, we are assuming that the scaling for large α is the same as for small α and large k (for evidence to this effect, see [36]). Confirmation as given by Eq. (15) is presented in Fig. 1 where we plot Su3 as a function of l (by convention, we plot Su3 =< |δu‖(l)|3 > to reduce cancellation in the statistics). The scales identified with an inertial range k ∈ [6, 10] are marked by vertical dashed lines and the predicted scaling, l3, is indicated by a solid line. We fit a scaling exponent (Su3 (l) ∼ lζ 3 ) and find ζu3 = 2.39± .04. This is significantly steeper than the classical Kolmogorov scaling given by Eq. (13); it can thus be viewed as more consistent with the scaling given by Eq. (15). It is also more consistent with l3 than with other possible LANS−α scalings: under the assumption that the turnover time scale of eddies of size ∼ l is determined by the unsmoothed velocity v, we find Su3 (l) ∼ l5, and if it is determined by v · u, we find Su3 (l) ∼ l4 (see, e.g., Refs. [16, 36, 37, 38]). The observed scaling corresponds to none of these cases, and is actually closer to an evaluation of the turnover time tl at the scale l given by tl ∼ l/ul (with Su3 (l) ∼ l3). Note that for 2D LANS−α, however, it is the case that the scaling is determined by the unsmoothed velocity v [36]. We note that this is one of many differences between the 2D and 3D cases (e.g., ideal invariants and cascades). Another difference, which we shall show in Section VI, is that in 2D vorticity structures decrease in scale as α increases while in 3D there is a change in aspect ratio with structures getting both shorter and fatter. This may, in fact, be related to the shallower LANS−α energy spectrum for kα > 1 which we show in Section VI. While differences are observed between the scaling shown in Fig. 1 and Eq. (15), the error bars deny a K41 scaling (as well as the l4 and l5 scalings) at scales smaller than α. We believe the discrepancy between the observed and predicted scaling can be due to lack of resolution to resolve properly the inertial range at sub-filter scales. We have less than a decade of inertial range and only 2563 points for the statistics. As more computational resources become available, this scaling should be re-examined. B. Subdominance of the k−1 energy spectrum and rigid-body motions As a consequence of LANS−α’s Taylor’s frozen-in hypothesis closure, scales smaller than α can phase-lock into coherent structures and be swept along by the larger scales (see, e.g., [30]). If we assume, formally, that this “frozen-in turbulence” takes the form of “rigid bodies” in the smoothed velocity field (no stretching), we arrive at a much different spectrum than k−1, Eq. (17). All scales smaller than α are subject to the frozen-in hypothesis and we expect to find such rigid bodies at these scales. We note that collections of “rigid” portions of the flow (rotating or non-rotating) reduce the total degrees of freedom (dof) and make physical sense with LANS−α’s relation to second-grade fluids: these rigid bodies can be envisioned as polymerized portions of FIG. 2: Spectral energy density, E(k), versus wavenumber, k, for large−α LANS−α solution. Here forcing (k0 = 1) and α (kα ≡ 2π/α = 3, vertical dotted line) are set at the largest scales to increase the number of scales for which kα > 1. Spectra are plotted for three norms: H1α(u) norm (solid line), L 2(u) norm (dotted line), and the L2(v) norm (dashed line). As these last two norms are not quadratic invariants of LANS−α, we employ the H1α norm for all following results. All three spectra correspond to that derived from the assumption of rigid bodies in the smoothed velocity u, Eq. (19). The vertical dashed lines are at the same scales as those in Fig. 1. the fluid. As a matter of fact, in such structures all internal dof are frozen. These “rigid bodies” follow as well from the consideration of LANS−α as an initial value problem in Fourier space, for which we have û(k) = v̂(k)/(1+α2k2). In the limit as α approaches infinity, all wavenumber (and spatial) dependence for v̄ is eliminated and the entire flow is advected by a uniform velocity field (advection without internal degrees of freedom). For a rigid body there can be no stretching and, therefore, all the longitudinal velocity incre- ments, δu‖, must be identically zero (δu(l) = Ω × l from basic mechanics with Ω the rotation vector and, hence, δu‖(l) = δu(l) · l/l = 0). Note that in LANS−α Eq. (2) the vj∂iuj term contributes only a rotation and not a stretching of u. Such polymerization would have two con- sequences. Firstly, since there is no stretching, these rigid bodies would not contribute to the turbulent energy cascade, < (δu‖(l)) 3 >= 0. (18) Secondly, the energy spectrum from dimensional analysis (u2 ∼ const, for large α/l: u = (1 + α2/l2)−1v ∼ l2v, and Eα(k)k ∼ uv ∼ k2) is Eα(k) ∼ k. (19) This is, in fact, the observed LANS−α spectrum for kα ≫ 1 as is shown in Fig. 2. We verified that the spectrum is not the result of under-resolved runs, as is the case, e.g., in the k2 spectrum observed in truncated Euler systems [39] or in extremely under-resolved spectral simulations of the Navier-Stokes equations. Indeed, equipartition of the energy among all modes in a truncated Euler−α system should also lead to a k2 spectrum. Along with several experiments with different viscosities and also with statistically homogeneous and isotropic forcing (not shown here), these are assurances that the observed spectrum is not a result of inadequate numerical resolution. It should be noted that this is the same computation for which the third-order structure function is shown in Fig. 1. The third-order structure function is consistent with a l3 scaling (corresponding to a k−1 energy spectrum) while the spectrum itself is k1. (Also shown in Fig. 2 are the L2(u) ≡ 〈u2〉/2 and the L2(v) ≡ 〈v2〉/2 norms which (through u ∼ α2v/k2 for kα ≫ 1) correspond to k−1 and k3 spectra, respectively. Since the analytical properties of the LANS−α solution are based on the energy balance, dEα/dt = −2νΩα, in the H1α(u) norm, we employ this norm for all following results.) These two different scalings, l3 and k1, are consistent with a picture where a fluid has both rigid-body portions at scales smaller than α (wherein there is no turbulent cascade) and spatial regions between these where the cascade does take place. For the structure functions, a non-cascading rigid body does not contribute to the scaling and consequently the cascading contribution, Eq. (15), dominates. The energy spectrum, however, for the limit of k very large, is dominated by the k+1 term, and hence the k−1 component is subdominant. We further explore the validity of this picture by examining the spatial variation of the cubed longitudinal increment, (δv‖(l)) 3 in DNS, and (δu‖(l)) 3 in LANS−α for α/l ≫ 1, which in each case is proportional to the energy flux across a fixed scale l. (The presence of the hypothesized “rigid bodies” should be evident as significant portions of the flow where there is no energy flux.) In Fig. 3 we show visualizations of these quantities corresponding to l = 2π/10 (k = 10) for both the large-α LANS−α simulation and a highly turbulent DNS (k0 = 2, ν = 3 × 10−4). The scale (k = 10) is chosen as it is in the inertial ranges of both flows. We note that for LANS−α, a significant portion of the flow is not contributing to the flux of energy to smaller scales (the filling factor for (δu‖(2π/10)) 3 < 10−2 is 0.67 as compared to 0.26 for the Navier-Stokes case). These regions can be identified as “polymerized” or “rigid bodies” in u and their locations are found to be robust when the l used for (δu‖(l)) 3 is varied over a factor of 2. Moreover, this is highlighted in the probability distribution functions (pdfs), see Fig. 4, where we see the LANS−α pdf is more strongly concentrated around zero than the DNS. This is consistent with the idea that FIG. 3: Two-dimensional slice of the cubed longitudinal increment (δu‖(2π/10)) 3for LANS−α and (δv‖(2π/10)) 3 for DNS. For all black pixels, the cubed longitudinal increment is less than 10−2 (approxi- mately consistent with rigid bodies). On the top is the large-α simulation (k0 = 1, kα = 3, ν = 1.2×10−4) where the filling factor (computed over the entire 3D domain) is 0.67. On the bottom is a DNS of Navier- Stokes (k0 = 2, ν = 3 × 10−4) where the filling factor is 0.26. Thus, a much greater portion of the flow is consistent with collections of rigid bodies for the large−α simulation. FIG. 4: Pdfs of (δv‖(2π/10)) 3 for DNS (N = 1024, solid line), and of (δu‖(2π/10)) 3 for LANS−α (N = 256, dashed line), and of the DNS downgraded to lower resolution (N = 256, dotted line). See Fig. 3 for simulation parameters. Note that both pdfs have a slight positive asymmetry consistent with a positive dissipation rate ε(α). The LANS−α pdf is more strongly concentrated around zero consistent with the idea that portions of the flow (at scales smaller than α) are acting as rigid bodies. the internal dof of large portions of the flow (at scales smaller than α) are frozen. We point out that this comparison is not a LES validation, but, rather, a comparison between the dynamics of two different fluids at similar Reynolds numbers. One flow is a well-resolved numerical solution of the Navier-Stokes equations, and the other is a well-resolved solution of the LANS−α equations with large α. For this reason a reduced resolution (N = 256) representation for the DNS (for which N = 1024) is not depicted in Fig. 3. We have performed such a down-sampling, however, and find the filling factor is reduced even more, to 0.14, and the tails of the pdf increase over the full- resolution analysis (dotted line in Fig. 4). No inverse Helmholtz filtering, H−1 is applied to the DNS data. Note that this would amount to computing (δu‖(l)) 3 in the DNS, which has no meaning in the dynamics of the Navier-Stokes equations (the energy flux is proportional to (δv‖(l)) We end this section with further evidence of coexistent energy spectra, k−1 and k1, in separate spatial portions of the flow. We mask out all portions of the flow that we identify with rigid bodies ((δu‖(2π/10)) 3 < 10−2, a 2D slice of which is shown in Fig. 3). The energy spectrum of the remaining portion of the flow is shown in Fig. 5 as a dashed line to be compared with the spectrum of the entire flow shown as a solid line. The operation of spatially filtering the flow before computing the spectrum serves to “smear out” the energy spectrum by convolving it with the spectrum of the filter. Deconvolution in 3D with N = 256 is intractable and we are, therefore, unable to remove this “smearing” of the energy spectrum of the cascading portions of the flow. FIG. 5: Spectral energy density, E(k), versus wavenumber, k, for large−α LANS−α solution. The solid line indicates the spectrum as given in Fig. 2 but for a single snapshot (the same as selected for Fig. 3). The dashed line indicates the spectrum wherein all portions of the flow associated with “rigid bodies” (a 2D slice of which is shown in Fig. 3) are removed. This provides further evidence that the flow spatially in between the “rigid bodies” possesses a negative power law energy spectrum (the predicted k−1 power law is shown as a solid line). Nonetheless, after conducting what tests we could with the filtering process (not shown here), we conclude that the power law of the energy spectrum of these portions is negative and, thus, distinctly different from that of the rigid bodies. IV. RESOLUTION REQUIREMENTS FOR GRID-INDEPENDENT LANS−α SOLUTIONS: SIZE OF ATTRACTOR It is useful to make a distinction between the quality of a subgrid model and effects arising from nonlinear interactions with discretization errors at marginal spatial resolutions (which are more characteristic of the discretization employed than of the subgrid model) [19, 40, 41]. Before doing this, we require an estimate for the total degrees of freedom for the LANS−α attractor which as we show, unlike for the 2D case (see [36]), for the 3D case is reduced compared to Navier-Stokes. The subdominant l3 scaling is associated with the flux of energy to small scales and thus must be used to estimate the degrees of freedom of the LANS−α attractor, dofα. For dissipation the large wavenumbers dominate and, therefore, combining the LANS−α energy balance, Eq. (4), with its sub-filter scale energy spectrum, Eq. (17), allows us to implicitly specify its dissipation wavenumber, kαη , by ∫ kαη k2Eα(k)dk ∼ ∫ kαη 2/3α2/3k−1dk ∼ εα2/3α2/3(kαη )2. (20) Then we have, kαη ∼ ν1/2α1/3 . (21) Using that the linear numerical resolution, N , must be proportional to the dissipation wavenumber (N ≥ 3kαη ) and that Re ∼ ν−1, we arrive at N = C0kα 1/3Re1/2, (22) or, equivalently, dofα = Re3/2, (23) where C0 is an unknown constant (for further details see [11]). We verify this prediction and determine the constant C0 through the use of a database stemming from studies in which both the free parameter, α (or, equivalently, kα) and the linear resolution, N , for a set of DNS flows with Re ≈ 500, 670, 1300, and 3300 are varied. In so doing, we establish the necessary numerical resolution for convergence to a grid-independent solution. Convergence to the grid-independent solution is determined by comparison of the energy spec- trum, Eα(k), between runs with a constant filter and varying resolution. In Fig. 6(a), we make such a comparison for Re ≈ 500 (N = 256 for DNS) and kα = 14 (N = 84, 96, 108, 128, and 192 for LANS−α). We plot energy spectra compensated by k5/3 so that a K41 k−5/3 spectrum would be flat. We see, based on comparing the energy spectra at wavenumbers smaller than kα to the 1923 LANS−α spectrum, that simulations at resolutions of 963 and less are not converged while the one at 1283 is. That is, except for the very small scales at the end of the dissipative range, there is very little difference between the spectra at 1283 and at 1923 (i.e., the solution is “grid- independent”). Meanwhile, for resolutions of 963 and less the spectra vary greatly with resolution (i.e., they are “unresolved”). In Fig. 6(b), we collect all the results of similar studies (Re ≈ 500) in a plot of resolution, N , versus inverse filter width, kα. (We change N for a given α, then change α and iterate.) Pluses correspond to grid-independent solutions, X’s to under-resolved solutions, and squares to “undecided” runs (i.e., that are neither clearly resolved nor clearly under-resolved). The FIG. 6: (Color online.) Plots for Re ≈ 500 simulations demonstrating convergence to the grid-independent LANS−α solution. (a) Average energy spectra (t ∈ [20, 33], t is time in units of eddy turn-over time) compensated by K41 for LANS−α simulations, kα = 14: 1923 (black solid), 843 (red dotted), 963 (green dashed), 1083 (blue dash-dotted), and 1283 (pink dash-triple-dot). The vertical dashed line denotes kα. Inset is a blow-up near kα where convergence can be clearly seen. LANS−α at a linear resolution of 1283 is approximately converged to the grid-independent solution while resolutions of 963 and less are clearly not. (b) The linear resolution of α−model simulations, N , is plotted versus kα. Simulations with inadequate resolution are plotted as X’s, those with approximately grid-independent solutions as +’s, and experiments that are neither clearly resolved nor clearly unresolved as boxes. The dashed lines represent N = Ck α indicating that a constant in the range 43.2 < C < 50.2 agrees with our data. This partially confirms the prediction of Eq. (22) and provides a reliable method to determine the needed resolution for a grid-independent LANS−α solution at a fixed Re. dashed lines represent Eq. (22) with the minimal and maximal choice of C (where C0 = CRe 1/2), that agrees with our results (i.e., 43.2 < C < 50.2). In Fig. 7 we conduct similar studies for Re ≈ 670. We find 49.5 < C < 51.4 and again validate the predictive power of Eq. (22) for the necessary numerical resolution for grid-independent solutions. FIG. 7: As Fig. 6(b) but for Re ≈ 670 simulations. The dashed lines represent N = Ck1/3α indicating that a constant in the range 49.5 < C < 51.4 agrees with our data. Note also that any power law, N ∝ kβα, with 0.30 < β < 0.46 also agrees with the data. FIG. 8: Acceptable choices of C = C0Re 1/2, versus Reynolds number, Re, for grid-independent LANS−α. Error bars are not confidence levels, but depict the range of values consistent with our database (N = Ck at the four Reynolds numbers we tested. The dashed line depicts the least-squares fit with slope 0.54±0.14. This completes the validation of Eq. (22) which predicts 0.5. The greatest utility of the prediction, however, is due to the single constant C0 which is inde- pendent of Reynolds number. A determination of this constant can cheaply be achieved repeating this process for several runs for low and moderate Re, and determines the resolution requirement for the highest Re attainable. The ranges of acceptable constants, C = C0Re 1/2, for the four Reynolds number flows studied are plotted versus Re in Fig. 8. A power law C = C0Re γ fits our data with γ = 0.54± 0.14 demonstrating the final validation of the prediction, γ = 0.5, Eq. (22). The value of the constant is found to be C0 = 2.0 ± 0.2. We made one study for the maximally- helical ABC forcing at Re ≈ 1600 and α = 2π/25. It is consistent with a value of C0 = 1.8±0.1. We therefore conclude that the constant C0 is not a strong function of the forcing employed or of the scale at which the system is forced. As a result, and unlike in 2D LANS−α [36], we verify that the size of the attractor in 3D LANS−α is smaller than that in Navier-Stokes, which is a promising result if the LANS−α equation is going to be used as an LES. However, before doing this, an assessment of the truncation errors introduced in discretized systems (as used to solve the equations numerically) and a study of the optimal choice for α to capture the properties of a DNS is needed. We consider these problems in the following section. V. CAN LANS−α BE CONSIDERED AS A LARGE EDDY SIMULATION? In this section, we consider the LANS−α equations as a means to an end, and consider the solutions to their discretized equations as approximations to the Navier-Stokes solutions. We seek numerical approximations of LANS−α that minimize the difference to a fully resolved or direct numerical solution (DNS) of Navier-Stokes (i.e., we analyze the behavior of LANS−α solutions in the LES framework, and call here the model a “LANS−α LES”, or in short “α-LES”). In the LES framework, LANS−α’s turbulent stress tensor, τ̄αij , is given by (see, e.g., [42]) τ̄αij = H−1α2(∂kui∂kuj + ∂kui∂juk − ∂iuk∂juk). (24) Previous studies have not made the distinction between grid-independent LANS−α and LANS−α LES, though one did study convergence to grid-independent solutions at moderate Re [19]. We find, however, a definite difference between the two approaches. We show in this section that, in fact, LANS−α combined with truncation error yields a better fit to DNS than grid-independent LANS−α. The resolution that yields an optimal α-LES (a terminology to be defined below) is also found to follow Eq. (22). In the Section V A, we then address the quality and usability of the predictions of the LANS−α model viewed as an LES. A remark about nomenclature may be in order at this point. Traditionally, and for good reasons, LES attempt at capturing the large-scale properties of a flow with a huge Reynolds number, as found, e.g., in the atmosphere. In that case, the wavenumber at which the DNS is truncated is, at best, in the inertial range and it might even be in the energy-containing range, as for the atmospheric boundary layer with a Taylor Reynolds number Rλ ∼ 104. Of a different nature are the modeling methods sometimes called quasi-DNS. Here, the idea is to model a flow at a given, moderate Reynolds number but with an expense in computing resources lesser than if performing a DNS. Under-resolved DNS fall in that category; in that case, the large-scales are presumably well reproduced but the small scales are noisy. It is in that spirit that we now examine the properties of the LANS−α model. We thus qualify a model as optimal in the sense of being optimal for the class of LANS−α models examined herein; in order to avoid repetition, we also use the terminology of alpha-optimal. FIG. 9: (Color online.) Plot of Re ≈ 670 simulations. Average compensated energy spectra: DNS (solid black line) and LANS−α simulations, kα = 41: N = 162 (red dotted), N = 192 (green dashed), and N = 216 (blue dash-dotted). LANS−α at a linear resolution of 192 is approximately converged to the grid- independent solution while a resolution of 162 is not. N = 162 does correspond, however, more closely to the DNS spectrum. We observe, in general, that a combination of LANS−α and truncation error yields the optimal α-LES. In Fig. 9 with kα = 41, we plot the Re ≈ 670 DNS spectrum (solid black line) and LANS−α spectra at three different resolutions. We observe that, while the N = 162 solution (dotted line, red online) is not converged, it is a better approximation to the DNS than the grid-independent LANS−α solution. For all simulations we studied, the grid-independent LANS−α solution is not the best approximation to the DNS. Another example is given in Fig. 10 where we plot the mean square spectral error normalized to make fair comparisons between large and small kα results, Esq = (Eα(k)−E(k))2 E2(k) , (25) where kF is the wavenumber for the forcing scale, E(k) is the DNS spectrum (in the L 2(v) norm), Eα(k) is the LANS−α spectrum (in the H1α(u) norm), and n is the number of terms in the sum. These errors are calculated for spectra averaged over turbulent steady-state solutions: t ∈ [16, 19] for Re ≈ 670. We see that for a given filter or a given simulation resolution, there is a local minimum in the error. This minimum is a balance between truncation errors and the approximation error due to using LANS−α instead of the full Navier-Stokes equations. Due to these errors being, in some sense, in opposition, the optimal α-LES solution is found at a lower resolution than the grid-independent solution. Indeed, we see by examining Fig. 10 (a) that for a given filter the combination of truncation error and the LANS−α solution is a better approximation to the DNS. For fixed resolution, Fig. 10 (b), the optimal value for α is not zero but has some finite value. This local minimum error shown in the figure keeps α from going to zero (kα → ∞) in dynamical models [15]. We note, also, that the error is low for a finite range of N and kα near the minimum, indicating that an α-LES solution may perform well for a range of parameters near the optimal ones. We find the resolution for an optimal α-LES is also predicted by Eq. (22) (with C ≈ 47 for Re ≈ 670, or C0 ≈ 1.8). That is, optimal α-LES resolution is just below that for grid-independent LANS−α solutions. Having demonstrated the predictability of the resolution for grid-independent LANS−α and of LANS−α LES given a Reynolds number and a filter, in the following section we seek to determine sufficient conditions on the free parameter α for LANS−α to be a successful A. Free parameter α and quality of the α-LES In this section, we make an analysis of the LES potential of LANS−α by considering only the grid-independent LANS−α solutions identified using Eq. (22). Note that from the results discussed in the previous section, we expect LANS−α optimal grid-dependent α−LES approx- imations to have better performance. In the limit of α going to zero, LANS−α Eq. (2) re- covers the Navier-Stokes equations, Eqs. (1), but the question we address now is how small must α be for LANS−α solutions to be good approximations to Navier-Stokes solutions. There are several length scales that α could be related to: the forcing scale lF , the integral scale L = 2π E(k)k−1dk/ E(k)dk, the Taylor microscale λ, or the Kolmogorov dissipation scale ηK . Plots of the mean square spectral errors to DNS (see Eq. (25)) versus these scales are shown in Fig. 11. While the general trend of errors decreasing with α is apparent in all cases, in Fig. 11(a) we see a large difference between errors at varying Reynolds numbers and FIG. 10: Plots for Re ≈ 670 simulations. (a) Error (see Eq. (25)) versus simulation resolution for kα = 20. The optimal (grid-dependent) LES is for a resolution of N ≈ 128 and has a much smaller error compared to the DNS than the grid-independent LANS−α solution at higher resolution. (b) Error versus kα for N = 128. At a given resolution the optimal value for α is not zero but occurs at a local minimal error. Any kα ∈ [15, 25] has an error near the minimum indicating that an LES solution may perform well for a range of parameters near the optimal ones. A constant of C = C0Re 1/2 ≈ 47 in Eq. (22) is found to correspond with optimal α-LES approximations. similar ratios of α to the forcing scale, lF . For a linear least-squares fit, the goodness-of-fit, (Eactualsq − Efitsq )2, was found to be χ2 = 6.2× 10−2. The errors for Re ≈ 3300 are much larger than for the same ratio lF/α as results at both Re ≈ 500 and Re ≈ 670. This is also the case for the integral scale. However, the quality of the α-LES appears to be closely tied to the ratio of α to the Kolmogorov dissipation scale. In Fig. 11(b) the errors are plotted versus the ratio of the dissipation scale, ηK , to α. We see a very strong dependence (χ 2 = 2.5 × 10−2) between errors for several runs with four different Reynolds numbers indicating that the quality of the LANS−α LES approximation is a function of the ratio of α to the dissipative scale. Finally, in Fig. 11(c) FIG. 11: Plot of errors, Eq. (25), of grid-independent solutions compared to DNS. Asterisks are for Re ≈ 8300, squares for Re ≈ 3300, triangles for Re ≈ 670, and diamonds for Re ≈ 500. The single right- most triangle in all plots corresponds to a value of α in the dissipative range (kα = 60). The norm we employ to measure the error, Eq. (25), is no longer a good norm when dissipative scales are considered. (a) Errors versus lF /α. No clear correlation between LES quality and the ratio of the forcing scale to α holds independently of Reynolds numbers. (b) Errors versus ratio of dissipative scale, ηK , to α. The quality of the LES appears to be closely tied to this ratio. (c) Errors versus ratio of Taylor wavenumber, λ, to α. The Re ≈ 8300 experiment (asterisk) indicates that the quality of the α-LES is not tied to the Taylor scale. the errors are plotted versus the ratio of the Taylor Scale, λ, to α. We find χ2 = 3.1 × 10−2 for a linear least-squares fit. We note that a single experiment conducted at Re ≈ 8300 (the asterisks) confirms that the maximal value of α is tied to the dissipation scale and not the Taylor scale. This is more clearly demonstrated in Fig. 12 where we plot compensated energy spectra for a nearly constant ratio λ/α at three Reynolds numbers. We see that the maximum deviation from the DNS spectrum increases with Re. As λ/α is the same in all cases, the optimal α is not dependent on the Taylor scale. These findings were not accessible at lower Reynolds numbers due to inadequate separation of scales. For example, we give in Fig. 13(a) spectral flux for DNS at Re ≈ 500, 670, and 3300 respectively. We define the kinetic energy transfer function, T (k), in Fourier space as T (k) = − v̂k · ̂(ω × v)dV , where (̂·) represents the Fourier transform. For LANS−α we have Tα(k) = − ũk · ̂(ω × u)dV where ω = ∇ × v. The flux is defined as usual from the transfer function as Π(α)(k) = T(α)(k . (26) Only Re ≈ 3300 (and Re ≈ 8300 not pictured here) demonstrates a range of nearly constant flux (a well-defined inertial range) before the dissipation scales. Following the scaling arguments in Ref. [11], one effect of the α−model is to increase the time scale for the cascade of energy to small scales. This reduces the flux as α increases (kα decreases) as do the hypothesized “rigid bodies;” this can be seen in Fig. 13(b). (Note that in DNS at high resolution, 80% of the flux is from local interactions which is strongly suppressed at scales smaller than α [21].) As dissipation dominates the flux for low and moderate Reynolds number, the reduced flux of the α−model has little conse- quence for these simulations. With a substantial inertial range, however, this reduced flux results in a pile-up of energy for scales larger than the dissipative scale and the spectrum approaches the k1 spectrum discussed in Section III. As a consequence of the integral conservation of energy (Eα = u · v) there is a corresponding decrease of energy at large scales. Consequently, as the inertial range increases, α must be moved to smaller and smaller scales in order for LANS−α not to alter scales larger than α. In summary, the α−model’s reduced flux of energy to small scales is more crucial when the dissipation scale is farther away from α. FIG. 12: Compensated averaged grid-independent energy spectra for DNS (solid) and LANS−α (dotted) holding the ratio of Taylor scale λ to α nearly constant. Vertical dotted lines indicate kα. (a) Re ≈ 670 and kα = 35 (λ/α = 18). (b) Re ≈ 3300 and kα = 70 (λ/α = 17). (c) Re ≈ 8300 and kα = 110 (λ/α = 17). We see that the maximum deviation from the DNS increases with Re. This is due to the greater distance between α and the dissipative scale ηK . (Note that scales larger than k = 3 are affected by numerical truncation issues.) FIG. 13: (Color online.) (a) Energy flux, Eq. (26), for three DNS with Re ≈ 3300 (black, solid), Re ≈ 670 (red, dotted), and Re ≈ 500 (green, dashed). No inertial range is discernible on the flux functions except for the highest Reynolds number case. The initial plateau followed by a bump and another plateau (for the case at the highest Reynolds number) is a result of the forcing employed. (b) Energy flux at Re ≈ 3300 for both DNS and α−model runs; DNS is the black, solid line. See inset for LANS−α parameters. LANS−α gives a reduced flux which is linked to the significant pile-up of energy at high wavenumber as visible in the energy spectrum (see Fig 14). Plots of εα versus t (not shown) also show that flux decreases (on average, at long times) with increasing α. B. Numerical savings from employing LANS−α If α must be directly proportional to the Kolmogorov dissipation scale, we can estimate the LES computational savings of the LANS−α model. For the Navier-Stokes equations we have dofNS ∝ Re9/4 and, as we verified in Section IV, for LANS−α we have dofα = C30kαRe3/2/27. If kα is directly proportional to the Navier-Stokes dissipation wavenumber, kη, we arrive at kη ∝ Re3/4, (27) and, consequently, dofLESα ∝ Re9/4. (28) Note that for free α, dofα (dof of LANS−α) is much smaller than dofNS . But, to obtain an optimal LES, α is tied to kη; then the resolution requirements (dof α ) are different and the decrease in necessary computational resolution from employing LANS−α is fixed. In fact, for the forcing and boundary conditions employed, we find dofLESα ≈ dofNS. (29) We note that Eq. (28) is consistent with theoretical predictions given in Ref. [20]. Other LES such as the similarity model [43] and the nonlinear (or gradient) model [44, 45] have also exhib- ited the characteristic that they achieve only moderate reductions in resolution and are, therefore, frequently used in mixed models with a Smagorinsky term (see, e.g., [3]). That such additional terms will be required for LANS−α to reproduce the energy spectrum of high Re flows, may not be a significant factor in its usability. Note that the usual addition of extra dissipative subgrid- stress terms (as in the Smagorinsky model) also introduces a stronger dependence of the system of equations with the spatial resolution, since the filter width in such models is often associated to the maximum wavenumber in the box, kmax. In that case, it can make more sense to use grid- dependent solutions of LANS−α (discussed at the beginning of Section V) which give an optimal LANS−α LES, and can as a result give an extra gain in the computational costs. We also conclude that, with the scale α being tied to the dissipation scale ηK , the model LANS−α behaves more like a quasi-DNS by opposition to a traditional LES. Note however that a factor of ≈ 2.3 in resolution gain translates into a factor 27 in CPU and a factor 12 in memory savings, still a substantial gain. FIG. 14: (Color online.) Compensated energy spectra averaged over t ∈ [8, 9], Re ≈ 3300. DNS is the solid black line and grid-independent LANS−α solutions are shown as (red online) dotted (kα = 70), (green) dashed (kα = 40), and (blue) dash-dotted (kα = 13) lines, respectively. A single LANS−α LES is shown as a (pink) dash-triple-dotted line (kα = 40, N = 384). The LES is seen to better approximate the DNS spectrum than the grid-independent solution for the same value of α (2π/40). As α is increased the energy spectrum approaches the k1 spectrum discussed in Section III B. VI. LANS−α AT VERY HIGH REYNOLDS NUMBER In this section, we compare and contrast LANS−α and Navier-Stokes solutions at high Reynolds number. Using results of previous sections for optimal resolution and the necessary value of α to approximate DNS, we now evaluate both grid-independent LANS−α solutions and a single LANS−α LES for a highly turbulent flow (Re ≈ 3300, Rλ ≈ 790). We calculate grid- independent solutions for kα = 70 (N = 512), for kα = 40 (N = 512), and for kα = 13 (N = 384). A LANS−α LES solution is computed for kα = 40 (N = 384). Averaged com- pensated energy spectra are shown in Fig. 14. We see that the optimal LANS−α LES is a better approximation of the DNS spectra than the grid-independent LANS−α for the same value of α (2π/40). We also see that if α is increased further, the energy spectrum approaches the k1 spectrum discussed in Section III B. Fig 15 is a perspective volume rendering of the enstrophy density ω2 (ω · ω̄ for LANS−α) for the DNS, kα = 70 LANS−α, and kα = 13 LANS−α. Due to the late time depicted here (t = 9, longer than a Lyapunov time) there can be no point-by-point comparison between the simulations. However, we note that the helical structure of the vortex tubes is preserved by the α−model but that the tubes themselves are shorter and somewhat thicker for large values of α. As was noted for moderate Reynolds numbers, this is due to LANS−α suppressing vortex stretching dynamics FIG. 15: (Color online.) Rendering of enstrophy density ω2 (ω · ω̄ for LANS−α). Due to the late time depicted here (t = 9, longer than a Lyapunov time) there can be no point-by-point comparison between the simulations. Instead, regions with approximately the same dimensions are selected around vortex tubes. Velocity v field lines are also shown illustrating the helical nature of the tubes which is seen to be captured by LANS−α. (a) DNS. The thick bars represent, from top to bottom, the Taylor scale λ and the dissipative scale ηK , respectively. For LANS−α results the scale α is depicted between these two. (b) kα = 70, N = 512. (c) kα = 13, N = 384. We see that, for large values of α, the vortex tubes become shorter and somewhat thicker. without changing its qualitative features [9]. This is in contrast to 2D LANS−α where the vorticity structures are seen to get thinner as α increases [36]. This could also be related to the scaling differences between 2D and 3D LANS−α. It has been claimed that the development of helical structures in turbulent flows can lead to the depletion of nonlinearity and the quenching of local interactions [46, 47]. The depletion of energy transfer due to local interactions at some cutoff in wavenumber is also believed to bring about the bottleneck effect [22, 48, 49, 50]. Consistent with these results, in 2D LANS−α (where the vorticity structures are more fine than Navier-Stokes) the spectrum is steeper and in 3D LANS−α (where the vorticity structures are shorter but fatter than Navier-Stokes) the spectrum is shallower. FIG. 16: (Color online.) Compensated 3rd-order structure function versus length l (a horizontal line scales with l). Structure functions corresponding to the Kármán-Howarth theorem are depicted (S3 for DNS, 3 ≡ 〈(δu)2δv〉 for LANS−α). Labels are as in Fig. 14. The dotted vertical lines indicate the various α’s. A small inertial range for the DNS near l = 1 is reproduced by LANS−α. The largest α (2π/13) exhibits a second inertial range at scales just smaller than α (〈(δu)2δv〉 ∼ l is consistent with Eq. (15)). Figure 16 shows the third-order (mixed) structure functions corresponding to the Kármán- Howarth theorems versus length l. For the DNS, we show S3 ≡ 〈δv3〉 and Sα3 ≡ 〈(δu)2δv〉 for LANS−α. The dotted vertical lines indicate the various α’s. A small inertial range for the DNS near l = 1 is reproduced by all LANS−α results. The largest α (2π/13) exhibits a second inertial range at scales just smaller than α (〈(δu)2δv〉 ∼ l is consistent with Eq. (15)). We note this is the first demonstration of third-order structure functions in LANS−α consistent with a K41 inertial range followed by an α inertial range and finally a dissipative range. Next, we observe the scaling of the longitudinal structure functions, Sp(l) ≡ 〈|δv‖|p〉, (30) where we again replace the H1α norm for the L 2 norm in the case of LANS−α, Sαp (l) ≡ 〈|δu‖δv‖|p/2〉. (31) We utilize the extended self-similarity (ESS) hypothesis [51, 52, 53] which proposes the scaling Sp(l) ∝ S3(l)ξp (32) or, for LANS−α, Sαp (l) ∝ 〈(δu)2δv〉ξp. (33) We display our results in Fig. 17. We note that for LANS−α, the third-order exponent is not equal to unity, contrary to the Navier-Stokes case. The Kármán-Howarth theorem implies 〈(δu)2δv〉 ∼ l, not Sα3 (l) ∼ l. We measured the deviation from linearity for each experiment (not depicted here) and found that LANS−α becomes more intermittent as α increases (kα = 13 is slightly more intermittent than the DNS). As artificially dropping local small-scale interactions gives enhanced intermittency [54, 55], this increased intermittency is the expected result of LANS−α reducing in- teractions at scales smaller than α. We note, however, that even with such a large filter, LANS−α is a good approximation to the intermittency properties of the DNS. This is surprising given its en- ergy spectrum and reduced flux in the inertial range. It is probably linked to the fact that LANS−α preserves global properties (in an H1 sense) of the Navier-Stokes equations and that these proper- ties are important to the dynamics of small scales as measured by high-order structure functions. VII. CONCLUSIONS We computed solutions of the Lagrangian-Averaged Navier-Stokes α−model (LANS−α) in three dimensions for significantly higher Reynolds numbers (up to Re ≈ 8300) than have previ- ously been accomplished and performed numerous forced turbulence simulations of LANS−α to study their equilibrium states. The results were compared to DNS for Re ≈ 500, 670, 3300, and 8300 , the last performed on a grid of 20483 points. We note that there are two ways to view the LANS−α simulations: as converged or “grid-independent” solutions of the LANS−α equations or as large-eddy simulations (α−LES) which include grid effects. We found a definite difference between the two approaches in that the fully-converged grid-independent LANS−α is not the best FIG. 17: (Color online.) Structure function scaling exponent ξp versus order p. Black X’s are shown for the DNS. Grid-independent LANS−α are shown as (red online) boxes (kα = 70), as (green) triangles (kα = 40), as (blue) diamonds (kα = 13). LANS−α LES (kα = 40, N = 384) is shown as (pink) asterisks. The dashed line indicates K41 scaling and the solid line the She-Lévêque (SL) formula [56]. approximation to a DNS of Navier-Stokes. Instead, the minimum error is a balance between trun- cation errors and the approximation error due to using LANS−α instead of the full Navier-Stokes equations. Due to these errors being, in some sense, in opposition, the optimal α-LES solution was found at a lower resolution than the grid-independent solution (the error was low for a finite range of N and α near the minimum, indicating that a LANS−α viewed as an LES solution may perform well for a range of parameters). Unlike the 2D case [36], 3D LANS−α has been shown to be a subgrid model (i.e., it reduces the resolution requirements of a given computation). This difference between 2D and 3D LANS−α indicates that other α−models (as the LAMHD−α Eqs. [57, 58] or the BV−α Eqs. [42]) may behave differently and studies of these systems at high resolution may be required. We confirm the presence of the theoretically predicted l3 scaling of the third-order structure function (corresponding to a k−1 scaling of the energy spectrum) [11, 16, 37] through its bound on the number of degrees of freedom for LANS−α [11], in the structure functions of the smoothed velocity in simulations with large α, and in the spectrum of specific spatial portions of the flow. In so doing, we have validated the predictive power of the bound dofα < Cα−1Re3/2, for the nu- merical resolution for grid-independent LANS−α solutions and for optimal LANS−α LES (with a separate constant of proportionality). The great utility of the prediction is that the single con- stant can cheaply be determined at low and moderate Reynolds number and predicts the resolution requirement for the highest Reynolds numbers attainable. We further found no great change in this single constant when employing the non-helical Taylor-Green or the maximally-helical ABC forcings. However, the small scale (kα ≫ 1) LANS−α spectrum was observed to be k+1. We attribute this to the frozen-in-turbulence closure employed in deriving the α−model. For scales smaller than α, portions of the smoothed flow u are locked into “rigid bodies.” By “rigid bodies,” we mean the internal degrees of freedom are frozen and these portions give no contribution to the energy cascade. This is consistent both with the observed k+1 spectrum and with field increments δu‖ being observed to be approximately zero over a large portion (compared to Navier-Stokes) of the flow. The turbulent energy cascade occurs in the space between these “rigid” portions. While the k−1 portions are subdominant to the k+1 portions in the energy spectrum, they prevail in the cascade and hence both the structure functions and the degrees of freedom of the LANS−α attractor. We find that both of these scalings (k+1 and k−1) contribute to a reduction of flux at constant energy (i.e., the dissipation is reduced as has previously been observed in 2D calculations [59]). This leads to a shallower (or even growing) energy spectrum as α increases. Thus, for LANS−α viewed as an LES to reproduce the Navier-Stokes energy spectrum it is necessary that α be not much larger than the dissipation scale (α / 4ηK independent of Reynolds number); in that sense, it can be considered as a quasi-DNS as opposed to a traditional LES, substantially larger Reynolds numbers being modeled in the latter case, leading to substantially larger gain in resolution. As a consequence, the computational savings of LANS−α is fixed and not a function of Reynolds number. (However, and unlike the 2D case, the 3D α−model does give a computational saving when used as a LES.) This result was not accessible at lower Reynolds numbers due to inadequate separation of scales. However, in one previous study for decaying turbulence with energy initially mostly at low wavenumbers (k = 3), it was evident that as time evolved and energy moved to smaller scales, the resolution requirements of LANS−α increased [17]. Other LES such as the similarity model [43] and the nonlinear (or gradient) model [44, 45] have also exhibited the char- acteristic that resolution may be decreased only modestly and are, therefore, frequently used in mixed models with a Smagorinsky term (see e.g., [3]). That such additional terms will be required for LANS−α to reproduce the energy spectrum of high Re flows, may not be a significant factor in its usability. We compared and contrasted LANS−α to a DNS at Re ≈ 3300 considering both structures and high-order statistics such as the longitudinal structure functions which are related with inter- mittency. With an appropriate choice of α we were able to observe a Navier-Stokes inertial range followed by LANS−α inertial range at scales smaller than α. For this second inertial range we again observed a k+1 energy spectrum. As α increased, we noted a change in the aspect ratio of vortex tubes (they became shorter and fatter). This can be related to quenching of local small- scale interactions at scales smaller than α and, thus, to the shallower spectrum for 3D LANS−α [22, 46, 47, 48, 49, 50]. Therefore, in 2D LANS−α (where the vorticity structures are more fine than Navier-Stokes) the spectrum is steeper [36] and in 3D LANS−α (where the vorticity struc- tures are shorter but fatter than Navier-Stokes) the spectrum is shallower. Finally, an examination of the longitudinal structure functions indicate that intermittency is increased as the parameter α is increased consistent with the suppression of local small-scale interactions at scales smaller than α [54, 55]. The elimination of the faster and faster interactions among smaller and smaller scales through the modified nonlinearity in LANS−α (together with the discrepancy between its solutions and Navier-Stokes solutions) highlights the importance of these interactions down to scales only slightly larger than the dissipative scale. That is, by removing these interactions anywhere in the inertial range (e.g., α ' 4ηK), the resulting energy spectrum was found to differ from the DNS at scales larger than α. The intermittency properties of the DNS, however, were well reproduced even with large filters. Noting this, if LANS−α’s k1 energy spectrum is not important for a given application, much greater reductions in resolution can be achieved. Future work should address whether this may be remedied in a LANS−α LES by the inclusion of another (dissipative) model for these interactions, or (in the case of magneto-hydrodynamics [57, 58] whether this problem is less significant because of the presence of greater spectral nonlocality [60, 61, 62]. The effect of LANS−α on the detailed scale-by-scale energy transfer should also be investigated as our results indicate that a model for local small-scale interactions would improve the α−model. Another direction of future research is to explore other reduced LANS−α models, Clark−α and Leray-α, which break the frozen-in-turbulence closure and, also, the conservation of circulation. Finally, note that because of its greater mathematical tractability, LANS−α possibly allows for a better un- derstanding of multi-scale interactions in turbulent flows thus modeled; therefore, detailed studies such as the one presented here may, in fine, allow for a better understanding of turbulence itself. 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Introduction Technique Inertial range scaling of LANS- l3 scaling of third-order structure function derived from the Kármán-Howarth theorem for LANS- Subdominance of the k-1 energy spectrum and rigid-body motions Resolution requirements for grid-independent LANS- solutions: Size of attractor Can LANS- be considered as a Large Eddy Simulation? Free parameter and quality of the -LES Numerical savings from employing LANS- LANS- at very high Reynolds number Conclusions Acknowledgments References

We compute solutions of the Lagrangian-Averaged Navier-Stokes alpha-model (LANS) for significantly higher Reynolds numbers (up to Re 8300) than have previously been accomplished. This allows sufficient separation of scales to observe a Navier-Stokes (NS) inertial range followed by a 2nd LANS inertial range. The analysis of the third-order structure function scaling supports the predicted l^3 scaling; it corresponds to a k^(-1) scaling of the energy spectrum. The energy spectrum itself shows a different scaling which goes as k^1. This latter spectrum is consistent with the absence of stretching in the sub-filter scales due to the Taylor frozen-in hypothesis employed as a closure in the derivation of LANS. These two scalings are conjectured to coexist in different spatial portions of the flow. The l^3 (E(k) k^(-1)) scaling is subdominant to k^1 in the energy spectrum, but the l^3 scaling is responsible for the direct energy cascade, as no cascade can result from motions with no internal degrees of freedom. We verify the prediction for the size of the LANS attractor resulting from this scaling. From this, we give a methodology either for arriving at grid-independent solutions for LANS, or for obtaining a formulation of a LES optimal in the context of the alpha models. The fully converged grid-independent LANS may not be the best approximation to a direct numerical simulation of the NS equations since the minimum error is a balance between truncation errors and the approximation error due to using LANS instead of the primitive equations. Furthermore, the small-scale behavior of LANS contributes to a reduction of flux at constant energy, leading to a shallower energy spectrum for large alpha. These small-scale features, do not preclude LANS to reproduce correctly the intermittency properties of high Re flow.

Introduction to Turbulence (Kluwer Academic Publishers, Dordrecht, 2001). [48] J. R. Herring, D. Schertzer, M. Lesieur, G. R. Newman, J. P. Chollet, and M. Larcheveque, Journal of Fluid Mechanics 124, 411 (1982). [49] D. Lohse and A. Müller-Groeling, Physical Review Letters 74, 1747 (1995), 1994chao.dyn..5002L. [50] D. O. Martı́nez, S. Chen, G. D. Doolen, R. H. Kraichnan, L.-P. Wang, and Y. Zhou, Journal of Plasma Physics 57, 195 (1997). [51] R. Benzi, S. Ciliberto, C. Baudet, G. Ruiz Chavarria, and R. Tripiccione, Europhysics Letters 24, 275 (1993). [52] R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli, and S. Succi, Phys. Rev. E 48, R29 (1993). [53] R. Benzi, L. Biferale, S. Ciliberto, M. V. Struglia, and R. Tripiccione, Phys. Rev. E 53, R3025 (1996), 1995chao.dyn..9013B. [54] J.-P. Laval, B. Dubrulle, and S. Nazarenko, Physics of Fluids 13, 1995 (2001), physics/0101036. [55] B. Dubrulle, J.-P. Laval, S. Nazarenko, and O. Zaboronski, Journal of Fluid Mechanics 520, 1 (2004), physics/0304035. [56] Z. She and E. Lévêque, Physical Review Letters 72, 336 (1994). [57] J. Pietarila Graham, P. D. Mininni, and A. Pouquet, Phys. Rev. E 72, 045301(R) (2005). [58] J. Pietarila Graham, D. D. Holm, P. Mininni, and A. Pouquet, Physics of Fluids 18, 045106 (2006). [59] D. Biskamp and E. Schwarz, Physics of Plasmas 8, 3282 (2001). [60] A. Alexakis, P. D. Mininni, and A. Pouquet, Phys. Rev. E 72, 046301 (2005). [61] P. Mininni, A. Alexakis, and A. Pouquet, Phys. Rev. E 72, 046302 (2005). [62] A. Alexakis, P. D. Mininni, and A. Pouquet, Astrophys. J. 640, 335 (2006), physics/0509069. Introduction Technique Inertial range scaling of LANS- l3 scaling of third-order structure function derived from the Kármán-Howarth theorem for LANS- Subdominance of the k-1 energy spectrum and rigid-body motions Resolution requirements for grid-independent LANS- solutions: Size of attractor Can LANS- be considered as a Large Eddy Simulation? Free parameter and quality of the -LES Numerical savings from employing LANS- LANS- at very high Reynolds number Conclusions Acknowledgments References

Reentrant Orbital Order and the True Ground State of LaSr2Mn2O7 Qing’An Li,1, 2 K.E. Gray,2, ∗ H. Zheng,2 H. Claus,2 S. Rosenkranz,2 S. Nyborg Ancona,2 R. Osborn,2 J.F. Mitchell,2 Y. Chen,3, 4 and J.W. Lynn3 1Chinese Academy of Sciences, Beijing, CHINA 2Materials Sciences Division, Argonne National Laboratory, Argonne, IL 60439, USA 3NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 4 Department of Materials Science and Engineering, University of Maryland, College Park, MD 20742, USA (Dated: October 23, 2021) Contrary to conventional wisdom, our purified La2−2xSr1+2xMn2O7 crystals exhibit CE-type or- bital and charge order as the low-temperature ground state for a hole doping level h = 0.5. For small deviations from h = 0.5, the high temperature CE phase is replaced at low temperatures by an A-type antiferromagnet without coexistence. Larger deviations result in a lack of CE order at any temperature. Thus, small inhomogeneities in cation or oxygen composition could explain why others commonly see this reentrance with coexistence. The quest to better understand strongly corre- lated electrons is at the heart of condensed mat- ter inquiry. Colossal magnetoresistive manganites ex- hibit a particularly vigorous competition among or- bital, charge and spin order [1, 2]. The phase di- agrams, e.g., La1−xSrxMnO3 or the bilayer version La2−2xSr1+2xMn2O7, display interesting features near half doping (x ∼ 0.5) where, e.g., long-range orbital and “checkerboard” charge ordering ( CE type) is pre- dicted [3]. In bilayer manganites, it has been com- monly accepted that CE order at x = 0.5 is reentrant [4, 5, 6, 7, 8, 9, 10]: it forms below ∼210 K, but then is replaced by an A-type antiferromagnet (AAFM) be- low ∼100 K. The lack of a low-temperature CE ordered ground state is surprising as it is found at x = 0.5 in many perovskite manganites [11]. In bilayer manganites, coexistence of CE order and AAFM between ∼100 K and ∼200 K is also universally reported [4, 5, 6, 7, 9]. All reports of reentrance in LaSr2Mn2O7 are based on superlattice peaks in neutron [4, 5, 6, 7], x-ray [6, 7, 8, 9, 10] and electron [12, 13, 14] diffraction and/or a peak in the resistivity [4, 6, 8, 12]. However, consistent with our own experience, these reported properties for x = 0.5 doping are variable. This implies an exquisite sensitivity to the exact value of hole doping, h = x − δ in La2−2xSr1+2xMn2O7−δ. This dictates a vital burden to obtain sample uniformity. Thus, in the present study we have purified large single crystals, which often exhibit compositional gradients, by cleaving them into very small crystals (∼1 mg) and discarding those that do not pass our test for uniformity (see below). Combining conduc- tivity, magnetization, and neutron and high-energy x-ray diffraction data on such highly homogeneous crystals, we here show that the CE order predicted by Goodenough [3] is the low-temperature ground state, presumably at h = 0.5, and that coexistence of CE and AAFM order is absent. We argue that re-entrance only occurs for small deviations from 0.5 and propose a revised phase diagram for bilayer manganites near h = 0.5. Crystals were melt-grown in an optical image furnace [15]. The c axis is perpendicular to the platelike crystals (∼2 × 0.5 × 0.1 mm3) and four gold pads are deposited along the top and bottom surfaces for transport measure- ments. In order to obtain a direct measure of the different order parameters, we performed high-energy synchrotron X-ray scattering experiments at the 1-ID-C station of the Advanced Photon Source at Argonne National Lab- oratory and neutron scattering experiments at the BT-7 triple-axis instrument at the NIST Center for Neutron Research. We utilize the (9/4,1/4,0) reflection obtained using 80 keV X-rays and the neutron intensities of the (1/4,1/4,3) and (1,1,3) reflections as measures of the CE orbital order, the CE antiferromagnetic order (CEAFM), and the A-type antiferromagnetic order, respectively [4]. Since four-terminal methods are unreliable for bulk crystals, we use six terminals to determine each prin- cipal component of conductivity, i.e., along the c-axis, σc, and in the ab-plane, σab [16]. In transport method A, current is injected through the outermost contacts on one surface [Fig. 1(a)]. Laplace’s equation is solved and inverted to get σab and σc from voltages measured across the innermost contacts of each surface. In method B, Laplace’s equation is solved for current injection through the top and bottom contacts at one end of the crystal, while voltages are measured between pairs of contacts on the top and bottom of the crystal [Fig. 1(b)]. FIG. 1: Schematic of six-terminal configurations for method A (a) and method B (b). To test for homogeneity, we evaluate σab and σc in four configurations. Two use method A with current applied to the outer contacts of either the top or bottom surfaces and two use method B with current applied to the end contacts on either the right- or left-hand sides. We re- quire both σab(T ) and σc(T ) to be qualitatively the same for all four configurations and within a factor of 2-3 in magnitude. Crystals used for Fig. 2 and 3 all pass our criteria, as described in Ref. [17]. Homogeneous crystals with a nominal x = 0.5 fall into three batches: batch-A are reentrant, batch-B exhibits CE order as the predom- inant low-temperature ground state, and batch-C never exhibit CE order. Examples of these are indicated in the highly schematic phase diagram of Fig. 4. The fact that we see three, and only three, unique states using four FIG. 2: Temperature dependence, shown for reentrant crys- tals (a)−(e) and nonreentrant crystals (f)−(j), of magnetiza- tion (a),(f), conductivity in ab-plane (b),(g) and along c-axis (c),(h) and neutron diffraction for CEAFM (d),(i) and A-type AFM (e),(j). Open symbols refer to cooling and filled symbols to warming. different bulk probes is unmistakable evidence of suffi- cient compositional uniformity. Any broad distribution of h = x − δ would result in the coexistence of these states in some of the 12 crystals studied. The relevance of δ is seen in one reentrant crystal that transformed into a non-reentrant crystal after annealing in pure oxygen for 60 hrs at 600 ◦C. Our scattering probes show that our samples consist of up to two or three slightly misaligned crystallites but are free from any impurity phases within our detection limits. Most of our crystals of nominal x = 0.5 composition display a transition from a paramagnetic insulator (PMI) above ∼200 K into a CE-type orbital and charge ordered state. This is often (batch-A) followed by a hysteretic transition into a state with higher magnetization [Fig. 2(a)] and conductivity [Figs. 2(b) and 2(c)] below ∼100 K that does not exhibit the CE-type superlattice reflec- tions [Figs. 2(d) and 3]. Our neutron diffraction data [Fig. 2(e)] confirm that the low-temperature state is the previously identified AAFM [4, 5, 6, 7]. However, in sev- eral crystals (batch-B) no more than a few percent [Figs. 2(f)-2(j)] transformed into the AAFM. A natural expla- nation is that CE order is the stable ground state only over a very narrow range of h and there are crystal-to- FIG. 3: Temperature dependencies of (a) σc and (b) x-ray diffraction intensity of the superlattice reflection for CE order (9/4,1/4,0), measured on the same crystal. Reentrance and hysteresis show a perfect correlation upon cooling (open sym- bols) and warming (filled symbols). Also shown in (a) are the conductivity data (black diamonds) for a batch-C crystal. FIG. 4: Schematic, qualitative phase diagram near h = 0.5. Symbols are measured transition temperatures (open: cool- ing; filled: warming). Except for h = 0.46 and 0.54, the h val- ues are arranged to connect smoothly with other data since we cannot determine them with sufficient precision. Reen- trant crystal data are plotted symmetrically both above and below 0.5. The metastable CEAFM is found on cooling but not on warming. The width of the boxes labeled coexistent, A, B, and C represent suggested range of h-values (i.e., 〈h〉±∆h) for coexistent crystals and batches A, B and C. crystal variations of the average hole doping 〈h〉 and a fi- nite width of the distribution ∆h. Thus, when 〈h〉 is most favorable for CE order, presumably at 0.5 as predicted by Goodenough [3], a larger fraction of the crystal would exhibit the CE ground state at low temperatures. Then for sufficiently small ∆h, the CE ground state can be the majority phase. For crystals exhibiting re-entrance we believe 〈h〉 differs somewhat from 0.5. We also find crystals (batch-C) without a transition to the CE-state, implying that 〈h〉 is yet further from 0.5. The conductiv- ity data for batch-C crystals are very similar to that of Ref. [18] and are shown in Fig. 3(a) for a nominal dop- ing x = 0.48. Note that the sharp decrease in σc at ∼ 200 K, found in the reentrant and nonreentrant crystals, is entirely absent, and the conductivity changes directly from the PMI to the AAFM behavior. Data on batch-A crystals display a striking tempera- ture dependence and hysteresis in σab and σc and mag- netization, as shown in Figs. 2(a)-2(c) and 3(a). Similar hysteresis is also found in the diffraction data of Figs. 2(d), 2(f) and 3(b) and in the data of others [4, 7]. Such hysteresis was also reported in resonant x-ray studies [9], but we are unaware of reports of such dramatic hystere- sis in the conductivity or magnetization of LaSr2Mn2O7. The drops in our conductivity and magnetization data exhibit a similar temperature range and hysteresis as all the published diffraction data [4, 7, 9]. A striking con- clusion of our neutron diffraction data for this batch-A crystal is the lack of coexistence of CE and AAFM order, which is universally found by others [4, 6, 7]. Neutrons probe AAFM and CEAFM magnetic order [Figs. 2(d) and 2(e)], the latter of which only occurs below ∼130 K (above 130 K the CE state is paramagnetic). In warm- ing and cooling cycles, one, and only one, state is ever found. This conclusion is possible because of the small ∆h in our crystals. Near the lower-temperature transi- tion of batch-A crystals, we found evidence for sluggish kinetics that was assisted by magnetic fields of 7 T. This may imply a small energy difference between these states since the change in magnetization at the transition is only ∼2 emu/g. Further, the broad thermal hysteresis associated with reentrance implies there is only a slight difference in the temperature dependencies of their free energies. Crystals from batch-B exhibit a similar high- temperature transition, but only a trace of the lower tem- perature transition to AAFM states. This is clearly seen in magnetization [Fig. 2(f)] and conductivity [Figs. 2(g) and 2(h)]. Apparently the free energy for CE order, that is the majority phase of batch-B crystals, is sufficiently low so that CE order remains stable down to low temper- atures. Since CE order has a sharp minimum in its free energy for h exactly 0.5, batch-B crystals should have 〈h〉 very close to that value. We confirm the low-temperature CE state through the CEAFM reflection [Fig. 2(i)] by neutron diffraction in a slightly larger nonreentrant crys- tal. A minor part of batch-B crystals may transform into the AAFM below ∼100 K [Fig. 2(j)]. Others [7, 10] have identified a weak CE superlattice peak at low tempera- tures which decays upon warming to above ∼50 K, but then reappears as a strong peak at ∼120 K. This has the appearance of a nonequilibrium “quenched-in” CE state, that is then annealed out upon warming to ∼50 K. To dispel this possibility and address thermodynamic stabil- ity, we monitor these reflections during slow cooling (<∼ 1 K/min) that should minimize quenched-in CE-order due to sluggish kinetics. We find that the CEAFM reflection is reversible upon slow heating [Fig. 2(i)], and conclude that the CE phase is the low temperature ground state in a majority of each batch-B crystal. For batch-B crystals, the succession of states with de- creasing temperature (shown schematically in Fig. 4) can be cast in terms of entropy. Transforming from the PMI to a charge-ordered paramagnet (CEPM) gains orbital and charge order while the onset of AF in the CE state (CEAFM) at ∼130 K additionally gains magnetic or- der. Transitions between the AAFM and CEAFM states are more complex: the internal energy for a CEAFM increases as h deviates from 0.5 while the broad hys- teresis implies similar temperature dependenies of their free energies. Therefore the phase boundary is almost vertical versus h. Batch-A crystals also transform from PMI to CEPM at ∼200 K but CE order appears to be metastable at lower temperatures (dashed lines in Fig. 4) until it transforms at ∼100 K to the AAFM ground state. Within this metastable region, the CE state de- velops CEAFM magnetic order below ∼130 K due to a gain in magnetic entropy. Upon warming, CEAFM order is absent and AAFM order persists until CEPM order is thermodynamically stable (T > 130 K). Thus it appears that the magnetic entropy gain in the CEPM is necessary to overcome the barrier between AAFM and CE states upon warming. Batch-C crystals show no evidence in conductivity or high-energy x-ray diffraction for CE order although they were made with a nominal composition of x = 0.5. The CE superlattice reflections seen in batch-A and batch-B crystals were missing at all temperatures down to 100 K. Curiously, all previously published conductivity data [4, 6, 8, 11] and in particular [18] known to us for nominal x = 0.5 look more like our batch-C crystals, although some show a small hysteresis [4, 6, 8]. A resistivity comparison [19] among these indicates a remarkable consistency in temperature dependence and thus establishes our batch- C crystals as a commonly seen variant of the nominal x = 0.5 layered manganite. The data of Refs. [6, 8] do show a somewhat larger resistivity peak at ∼180 K that could be the signature of some CE-order in their crystals and each of these report the CE reflection. However, the resistivity peak associated with CE order in our batch-A crystals is at least 10 times larger than the maximum value reported by others [6, 8]. This may imply CE order occurs in a larger fraction of our purified batch-A crystals. The electronic nature of AAFM states near h = 0.5 is not so easily determined. Conductivity for x = 0.58 crystals indicate ab-plane metals [20] , whereas insulating behavior is seen for x = 0.46 [21]. Conductivity data for the AAFM states of batch-A and batch-C crystals fall between these two extremes. A possible scenario is a continuous ab-plane-metal to insulator transition as h decreases from ∼0.58 to ∼0.46. In this picture the CE order replaces the AAFM over a limited h-range centered at 0.5, but the h-dependence of the AAFM states are otherwise unaffected. In summary, contrary to published data and ac- cepted wisdom for LaSr2Mn2O7, we show that the zero- temperature ground state is the CE type predicted by Goodenough [3]. We also find no evidence of CE and AAFM coexistence. That we do not know the exact h- values is not critical to these two new observations, but compositional purity was crucial to these discoveries. It was accomplished by stringent testing of small (∼1 mg) crystals and verified by our observations with bulk probes of only three, unique states in the 12 crystals tested. The lack of sufficient purity could explain why others consis- tently see coexistence of CE and AAFM order. The authors thank Peter Lee for assistance with synchrotron x-ray diffraction at the Advanced Photon Source. This research was supported by the U.S. De- partment of Energy, Basic Energy Sciences-Materials Sci- ences, under contract No. DE-AC02-06CH11357. ∗ Electronic address: KenGray@anl.gov [1] S. Jin, T. H. Tiefel, M. McCormack, R. A. Fastnacht, R. Ramesh, and L. H. Chen, Science 264, 413 (1994). [2] Y. Moritomo, A. Asamitsu, H. Kuwahara, and Y. Tokura, Nature (London) 380, 141 (1996). [3] J.B. Goodenough, Phys. Rev. 100, 564 (1955). [4] M. Kubota, H. Yoshizawa, Y. Moritomo, H. Fujioka, K. Hirota, and Y. Endoh, J. Phys. Soc. Japan 68, 2202 (1999). [5] C.D. Ling, J.E. Millburn, J.F. Mitchell, D.N. Argyriou, J. Linton, and H. Bordallo, Phys. Rev. B. 62, 15096 (2000). [6] D.N. Argyriou, H.N. Bordallo, B.J. Campbell, A.K. Cheetham, D.E. Cox, J.S. Gardner, K. Hanif, A. dos San- tos, and G.F. Strouse, Phys. Rev. B 61, 15269 (2000). [7] T. Chatterji, G.J. McIntyre, W. Caliebe, R. Surya- narayanan, G. Dhaleene, and A. Revcolevschi, Phys. Rev. B 61, 570 (2000). [8] T. Kimura, R. Kumai, Y. Tokura , J.Q. Li, and Y. Mat- sui, Phys. Rev. B 58, 11081 (1998). [9] Y. Wakabayashi, Y. Murakami, I. Koyama, T. Kimura, Y. Tokura, Y. Moritomo, Y. Endoh, K. Hirota, J. Phys. Soc. Japan 72, 618 (2003). [10] S.B. Wilkins, P.D. Spencer, T.A.W. Beale, P.D. Hatton, M. v. Zimmermann, S.D. Brown, D. Prabhakaran, and A.T. Boothroyd, Phys. Rev. B 67, 205110 (2003). [11] Y. Tomioka and Y. Tokura, Colossal magneto-resistive oxides (Gordon Breach, Amsterdam, 2000), Chap. 8. [12] J.Q. Li, Y. Matsui, T. Kimura, and Y. Tokura, Phys. Rev. B 57, R3205 (1998). [13] J.Q. Li, C. Dong, L.H. Liu, and Y.M. Ni, Phys. Rev. B 64, 174413 (2001). [14] Z.P. Luo, D.J. Miller, and J.F. Mitchell, Phys. Rev. B 71, 014418 (2005). [15] J.F. Mitchell, D.N. Argyriou, J.D. Jorgensen, D.G. Hinks, C.D. Potter, and S.D. Bader, Phys. Rev. B 55, 63 (1997). [16] Qing’An Li, K.E. Gray, and J.F. Mitchell, Phys. Rev. B 59, 9357 (1999). [17] Qing’An Li, K.E. Gray, S. Nyborg Ancona. H. Zheng, S. Rosenkranz, R. Osborn, and J.F. Mitchell, Phys. Rev. Lett. 96, 087201 (2006). [18] X.J. Chen, C.L. Zhang, J.S. Gardner, J.L. Sarrao, and C.C. Almasan, Phys. Rev. B 68, 064405 (2003). [19] To compare the literature results on an equivalent basis, we use our own four-terminal data, along a flat c-axis face of the crystal that is the “top”voltage in Fig. 3(a) and Ref. [16]. We assume this configuration for cases in the literature where the lead configuration is unspecified. [20] E. Badica, K.E. Gray, J.F. Mitchell, and H. Zheng, Phys. Rev. B 70, 174435 (2004). [21] Qing’An Li, K.E. Gray, and J.F. Mitchell, Phys. Rev. B 67, 184426 (2003). N.B. The crystal used in this study was originally designated as x = 0.48, but extensive re- cent studies using boules made nominally at x = 0.48 and 0.46 lead us to believe it was actually closer to 0.46. mailto:KenGray@anl.gov Acknowledgments References

Contrary to conventional wisdom, our purified La2-2xSr1+2xMn2O7 crystals exhibit CE-type orbital and charge order as the low-temperature ground state for a hole doping level h = 0.5. For small deviations from h = 0.5, the high temperature CE phase is replaced at low temperatures by an A-type antiferromagnet without coexistence. Larger deviations result in a lack of CE order at any temperature. Thus, small inhomogeneities in cation or oxygen composition could explain why others commonly see this reentrance with coexistence.

Reentrant Orbital Order and the True Ground State of LaSr2Mn2O7 Qing’An Li,1, 2 K.E. Gray,2, ∗ H. Zheng,2 H. Claus,2 S. Rosenkranz,2 S. Nyborg Ancona,2 R. Osborn,2 J.F. Mitchell,2 Y. Chen,3, 4 and J.W. Lynn3 1Chinese Academy of Sciences, Beijing, CHINA 2Materials Sciences Division, Argonne National Laboratory, Argonne, IL 60439, USA 3NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 4 Department of Materials Science and Engineering, University of Maryland, College Park, MD 20742, USA (Dated: October 23, 2021) Contrary to conventional wisdom, our purified La2−2xSr1+2xMn2O7 crystals exhibit CE-type or- bital and charge order as the low-temperature ground state for a hole doping level h = 0.5. For small deviations from h = 0.5, the high temperature CE phase is replaced at low temperatures by an A-type antiferromagnet without coexistence. Larger deviations result in a lack of CE order at any temperature. Thus, small inhomogeneities in cation or oxygen composition could explain why others commonly see this reentrance with coexistence. The quest to better understand strongly corre- lated electrons is at the heart of condensed mat- ter inquiry. Colossal magnetoresistive manganites ex- hibit a particularly vigorous competition among or- bital, charge and spin order [1, 2]. The phase di- agrams, e.g., La1−xSrxMnO3 or the bilayer version La2−2xSr1+2xMn2O7, display interesting features near half doping (x ∼ 0.5) where, e.g., long-range orbital and “checkerboard” charge ordering ( CE type) is pre- dicted [3]. In bilayer manganites, it has been com- monly accepted that CE order at x = 0.5 is reentrant [4, 5, 6, 7, 8, 9, 10]: it forms below ∼210 K, but then is replaced by an A-type antiferromagnet (AAFM) be- low ∼100 K. The lack of a low-temperature CE ordered ground state is surprising as it is found at x = 0.5 in many perovskite manganites [11]. In bilayer manganites, coexistence of CE order and AAFM between ∼100 K and ∼200 K is also universally reported [4, 5, 6, 7, 9]. All reports of reentrance in LaSr2Mn2O7 are based on superlattice peaks in neutron [4, 5, 6, 7], x-ray [6, 7, 8, 9, 10] and electron [12, 13, 14] diffraction and/or a peak in the resistivity [4, 6, 8, 12]. However, consistent with our own experience, these reported properties for x = 0.5 doping are variable. This implies an exquisite sensitivity to the exact value of hole doping, h = x − δ in La2−2xSr1+2xMn2O7−δ. This dictates a vital burden to obtain sample uniformity. Thus, in the present study we have purified large single crystals, which often exhibit compositional gradients, by cleaving them into very small crystals (∼1 mg) and discarding those that do not pass our test for uniformity (see below). Combining conduc- tivity, magnetization, and neutron and high-energy x-ray diffraction data on such highly homogeneous crystals, we here show that the CE order predicted by Goodenough [3] is the low-temperature ground state, presumably at h = 0.5, and that coexistence of CE and AAFM order is absent. We argue that re-entrance only occurs for small deviations from 0.5 and propose a revised phase diagram for bilayer manganites near h = 0.5. Crystals were melt-grown in an optical image furnace [15]. The c axis is perpendicular to the platelike crystals (∼2 × 0.5 × 0.1 mm3) and four gold pads are deposited along the top and bottom surfaces for transport measure- ments. In order to obtain a direct measure of the different order parameters, we performed high-energy synchrotron X-ray scattering experiments at the 1-ID-C station of the Advanced Photon Source at Argonne National Lab- oratory and neutron scattering experiments at the BT-7 triple-axis instrument at the NIST Center for Neutron Research. We utilize the (9/4,1/4,0) reflection obtained using 80 keV X-rays and the neutron intensities of the (1/4,1/4,3) and (1,1,3) reflections as measures of the CE orbital order, the CE antiferromagnetic order (CEAFM), and the A-type antiferromagnetic order, respectively [4]. Since four-terminal methods are unreliable for bulk crystals, we use six terminals to determine each prin- cipal component of conductivity, i.e., along the c-axis, σc, and in the ab-plane, σab [16]. In transport method A, current is injected through the outermost contacts on one surface [Fig. 1(a)]. Laplace’s equation is solved and inverted to get σab and σc from voltages measured across the innermost contacts of each surface. In method B, Laplace’s equation is solved for current injection through the top and bottom contacts at one end of the crystal, while voltages are measured between pairs of contacts on the top and bottom of the crystal [Fig. 1(b)]. FIG. 1: Schematic of six-terminal configurations for method A (a) and method B (b). To test for homogeneity, we evaluate σab and σc in four configurations. Two use method A with current applied to the outer contacts of either the top or bottom surfaces and two use method B with current applied to the end contacts on either the right- or left-hand sides. We re- quire both σab(T ) and σc(T ) to be qualitatively the same for all four configurations and within a factor of 2-3 in magnitude. Crystals used for Fig. 2 and 3 all pass our criteria, as described in Ref. [17]. Homogeneous crystals with a nominal x = 0.5 fall into three batches: batch-A are reentrant, batch-B exhibits CE order as the predom- inant low-temperature ground state, and batch-C never exhibit CE order. Examples of these are indicated in the highly schematic phase diagram of Fig. 4. The fact that we see three, and only three, unique states using four FIG. 2: Temperature dependence, shown for reentrant crys- tals (a)−(e) and nonreentrant crystals (f)−(j), of magnetiza- tion (a),(f), conductivity in ab-plane (b),(g) and along c-axis (c),(h) and neutron diffraction for CEAFM (d),(i) and A-type AFM (e),(j). Open symbols refer to cooling and filled symbols to warming. different bulk probes is unmistakable evidence of suffi- cient compositional uniformity. Any broad distribution of h = x − δ would result in the coexistence of these states in some of the 12 crystals studied. The relevance of δ is seen in one reentrant crystal that transformed into a non-reentrant crystal after annealing in pure oxygen for 60 hrs at 600 ◦C. Our scattering probes show that our samples consist of up to two or three slightly misaligned crystallites but are free from any impurity phases within our detection limits. Most of our crystals of nominal x = 0.5 composition display a transition from a paramagnetic insulator (PMI) above ∼200 K into a CE-type orbital and charge ordered state. This is often (batch-A) followed by a hysteretic transition into a state with higher magnetization [Fig. 2(a)] and conductivity [Figs. 2(b) and 2(c)] below ∼100 K that does not exhibit the CE-type superlattice reflec- tions [Figs. 2(d) and 3]. Our neutron diffraction data [Fig. 2(e)] confirm that the low-temperature state is the previously identified AAFM [4, 5, 6, 7]. However, in sev- eral crystals (batch-B) no more than a few percent [Figs. 2(f)-2(j)] transformed into the AAFM. A natural expla- nation is that CE order is the stable ground state only over a very narrow range of h and there are crystal-to- FIG. 3: Temperature dependencies of (a) σc and (b) x-ray diffraction intensity of the superlattice reflection for CE order (9/4,1/4,0), measured on the same crystal. Reentrance and hysteresis show a perfect correlation upon cooling (open sym- bols) and warming (filled symbols). Also shown in (a) are the conductivity data (black diamonds) for a batch-C crystal. FIG. 4: Schematic, qualitative phase diagram near h = 0.5. Symbols are measured transition temperatures (open: cool- ing; filled: warming). Except for h = 0.46 and 0.54, the h val- ues are arranged to connect smoothly with other data since we cannot determine them with sufficient precision. Reen- trant crystal data are plotted symmetrically both above and below 0.5. The metastable CEAFM is found on cooling but not on warming. The width of the boxes labeled coexistent, A, B, and C represent suggested range of h-values (i.e., 〈h〉±∆h) for coexistent crystals and batches A, B and C. crystal variations of the average hole doping 〈h〉 and a fi- nite width of the distribution ∆h. Thus, when 〈h〉 is most favorable for CE order, presumably at 0.5 as predicted by Goodenough [3], a larger fraction of the crystal would exhibit the CE ground state at low temperatures. Then for sufficiently small ∆h, the CE ground state can be the majority phase. For crystals exhibiting re-entrance we believe 〈h〉 differs somewhat from 0.5. We also find crystals (batch-C) without a transition to the CE-state, implying that 〈h〉 is yet further from 0.5. The conductiv- ity data for batch-C crystals are very similar to that of Ref. [18] and are shown in Fig. 3(a) for a nominal dop- ing x = 0.48. Note that the sharp decrease in σc at ∼ 200 K, found in the reentrant and nonreentrant crystals, is entirely absent, and the conductivity changes directly from the PMI to the AAFM behavior. Data on batch-A crystals display a striking tempera- ture dependence and hysteresis in σab and σc and mag- netization, as shown in Figs. 2(a)-2(c) and 3(a). Similar hysteresis is also found in the diffraction data of Figs. 2(d), 2(f) and 3(b) and in the data of others [4, 7]. Such hysteresis was also reported in resonant x-ray studies [9], but we are unaware of reports of such dramatic hystere- sis in the conductivity or magnetization of LaSr2Mn2O7. The drops in our conductivity and magnetization data exhibit a similar temperature range and hysteresis as all the published diffraction data [4, 7, 9]. A striking con- clusion of our neutron diffraction data for this batch-A crystal is the lack of coexistence of CE and AAFM order, which is universally found by others [4, 6, 7]. Neutrons probe AAFM and CEAFM magnetic order [Figs. 2(d) and 2(e)], the latter of which only occurs below ∼130 K (above 130 K the CE state is paramagnetic). In warm- ing and cooling cycles, one, and only one, state is ever found. This conclusion is possible because of the small ∆h in our crystals. Near the lower-temperature transi- tion of batch-A crystals, we found evidence for sluggish kinetics that was assisted by magnetic fields of 7 T. This may imply a small energy difference between these states since the change in magnetization at the transition is only ∼2 emu/g. Further, the broad thermal hysteresis associated with reentrance implies there is only a slight difference in the temperature dependencies of their free energies. Crystals from batch-B exhibit a similar high- temperature transition, but only a trace of the lower tem- perature transition to AAFM states. This is clearly seen in magnetization [Fig. 2(f)] and conductivity [Figs. 2(g) and 2(h)]. Apparently the free energy for CE order, that is the majority phase of batch-B crystals, is sufficiently low so that CE order remains stable down to low temper- atures. Since CE order has a sharp minimum in its free energy for h exactly 0.5, batch-B crystals should have 〈h〉 very close to that value. We confirm the low-temperature CE state through the CEAFM reflection [Fig. 2(i)] by neutron diffraction in a slightly larger nonreentrant crys- tal. A minor part of batch-B crystals may transform into the AAFM below ∼100 K [Fig. 2(j)]. Others [7, 10] have identified a weak CE superlattice peak at low tempera- tures which decays upon warming to above ∼50 K, but then reappears as a strong peak at ∼120 K. This has the appearance of a nonequilibrium “quenched-in” CE state, that is then annealed out upon warming to ∼50 K. To dispel this possibility and address thermodynamic stabil- ity, we monitor these reflections during slow cooling (<∼ 1 K/min) that should minimize quenched-in CE-order due to sluggish kinetics. We find that the CEAFM reflection is reversible upon slow heating [Fig. 2(i)], and conclude that the CE phase is the low temperature ground state in a majority of each batch-B crystal. For batch-B crystals, the succession of states with de- creasing temperature (shown schematically in Fig. 4) can be cast in terms of entropy. Transforming from the PMI to a charge-ordered paramagnet (CEPM) gains orbital and charge order while the onset of AF in the CE state (CEAFM) at ∼130 K additionally gains magnetic or- der. Transitions between the AAFM and CEAFM states are more complex: the internal energy for a CEAFM increases as h deviates from 0.5 while the broad hys- teresis implies similar temperature dependenies of their free energies. Therefore the phase boundary is almost vertical versus h. Batch-A crystals also transform from PMI to CEPM at ∼200 K but CE order appears to be metastable at lower temperatures (dashed lines in Fig. 4) until it transforms at ∼100 K to the AAFM ground state. Within this metastable region, the CE state de- velops CEAFM magnetic order below ∼130 K due to a gain in magnetic entropy. Upon warming, CEAFM order is absent and AAFM order persists until CEPM order is thermodynamically stable (T > 130 K). Thus it appears that the magnetic entropy gain in the CEPM is necessary to overcome the barrier between AAFM and CE states upon warming. Batch-C crystals show no evidence in conductivity or high-energy x-ray diffraction for CE order although they were made with a nominal composition of x = 0.5. The CE superlattice reflections seen in batch-A and batch-B crystals were missing at all temperatures down to 100 K. Curiously, all previously published conductivity data [4, 6, 8, 11] and in particular [18] known to us for nominal x = 0.5 look more like our batch-C crystals, although some show a small hysteresis [4, 6, 8]. A resistivity comparison [19] among these indicates a remarkable consistency in temperature dependence and thus establishes our batch- C crystals as a commonly seen variant of the nominal x = 0.5 layered manganite. The data of Refs. [6, 8] do show a somewhat larger resistivity peak at ∼180 K that could be the signature of some CE-order in their crystals and each of these report the CE reflection. However, the resistivity peak associated with CE order in our batch-A crystals is at least 10 times larger than the maximum value reported by others [6, 8]. This may imply CE order occurs in a larger fraction of our purified batch-A crystals. The electronic nature of AAFM states near h = 0.5 is not so easily determined. Conductivity for x = 0.58 crystals indicate ab-plane metals [20] , whereas insulating behavior is seen for x = 0.46 [21]. Conductivity data for the AAFM states of batch-A and batch-C crystals fall between these two extremes. A possible scenario is a continuous ab-plane-metal to insulator transition as h decreases from ∼0.58 to ∼0.46. In this picture the CE order replaces the AAFM over a limited h-range centered at 0.5, but the h-dependence of the AAFM states are otherwise unaffected. In summary, contrary to published data and ac- cepted wisdom for LaSr2Mn2O7, we show that the zero- temperature ground state is the CE type predicted by Goodenough [3]. We also find no evidence of CE and AAFM coexistence. That we do not know the exact h- values is not critical to these two new observations, but compositional purity was crucial to these discoveries. It was accomplished by stringent testing of small (∼1 mg) crystals and verified by our observations with bulk probes of only three, unique states in the 12 crystals tested. The lack of sufficient purity could explain why others consis- tently see coexistence of CE and AAFM order. The authors thank Peter Lee for assistance with synchrotron x-ray diffraction at the Advanced Photon Source. This research was supported by the U.S. De- partment of Energy, Basic Energy Sciences-Materials Sci- ences, under contract No. DE-AC02-06CH11357. ∗ Electronic address: KenGray@anl.gov [1] S. Jin, T. H. Tiefel, M. McCormack, R. A. Fastnacht, R. Ramesh, and L. H. Chen, Science 264, 413 (1994). [2] Y. Moritomo, A. Asamitsu, H. Kuwahara, and Y. Tokura, Nature (London) 380, 141 (1996). [3] J.B. Goodenough, Phys. Rev. 100, 564 (1955). [4] M. Kubota, H. Yoshizawa, Y. Moritomo, H. Fujioka, K. Hirota, and Y. Endoh, J. Phys. Soc. Japan 68, 2202 (1999). [5] C.D. Ling, J.E. Millburn, J.F. Mitchell, D.N. Argyriou, J. Linton, and H. Bordallo, Phys. Rev. B. 62, 15096 (2000). [6] D.N. Argyriou, H.N. Bordallo, B.J. Campbell, A.K. Cheetham, D.E. Cox, J.S. Gardner, K. Hanif, A. dos San- tos, and G.F. Strouse, Phys. Rev. B 61, 15269 (2000). [7] T. Chatterji, G.J. McIntyre, W. Caliebe, R. Surya- narayanan, G. Dhaleene, and A. Revcolevschi, Phys. Rev. B 61, 570 (2000). [8] T. Kimura, R. Kumai, Y. Tokura , J.Q. Li, and Y. Mat- sui, Phys. Rev. B 58, 11081 (1998). [9] Y. Wakabayashi, Y. Murakami, I. Koyama, T. Kimura, Y. Tokura, Y. Moritomo, Y. Endoh, K. Hirota, J. Phys. Soc. Japan 72, 618 (2003). [10] S.B. Wilkins, P.D. Spencer, T.A.W. Beale, P.D. Hatton, M. v. Zimmermann, S.D. Brown, D. Prabhakaran, and A.T. Boothroyd, Phys. Rev. B 67, 205110 (2003). [11] Y. Tomioka and Y. Tokura, Colossal magneto-resistive oxides (Gordon Breach, Amsterdam, 2000), Chap. 8. [12] J.Q. Li, Y. Matsui, T. Kimura, and Y. Tokura, Phys. Rev. B 57, R3205 (1998). [13] J.Q. Li, C. Dong, L.H. Liu, and Y.M. Ni, Phys. Rev. B 64, 174413 (2001). [14] Z.P. Luo, D.J. Miller, and J.F. Mitchell, Phys. Rev. B 71, 014418 (2005). [15] J.F. Mitchell, D.N. Argyriou, J.D. Jorgensen, D.G. Hinks, C.D. Potter, and S.D. Bader, Phys. Rev. B 55, 63 (1997). [16] Qing’An Li, K.E. Gray, and J.F. Mitchell, Phys. Rev. B 59, 9357 (1999). [17] Qing’An Li, K.E. Gray, S. Nyborg Ancona. H. Zheng, S. Rosenkranz, R. Osborn, and J.F. Mitchell, Phys. Rev. Lett. 96, 087201 (2006). [18] X.J. Chen, C.L. Zhang, J.S. Gardner, J.L. Sarrao, and C.C. Almasan, Phys. Rev. B 68, 064405 (2003). [19] To compare the literature results on an equivalent basis, we use our own four-terminal data, along a flat c-axis face of the crystal that is the “top”voltage in Fig. 3(a) and Ref. [16]. We assume this configuration for cases in the literature where the lead configuration is unspecified. [20] E. Badica, K.E. Gray, J.F. Mitchell, and H. Zheng, Phys. Rev. B 70, 174435 (2004). [21] Qing’An Li, K.E. Gray, and J.F. Mitchell, Phys. Rev. B 67, 184426 (2003). N.B. The crystal used in this study was originally designated as x = 0.48, but extensive re- cent studies using boules made nominally at x = 0.48 and 0.46 lead us to believe it was actually closer to 0.46. mailto:KenGray@anl.gov Acknowledgments References

Addendum: A Classification of Plane Symmetric Kinematic Self-similar Solutions M. Sharif ∗and Sehar Aziz † Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan. In our recent paper, we classified plane symmetric kinematic self-similar perfect fluid and dust solutions of the second, zeroth and infinite kinds. How- ever, we have missed some solutions during the process. In this short com- munication, we add up those missing solutions. We have found a total of seven solutions, out of which five turn out to be independent and cannot be found in the earlier paper. Keywords: Plane symmetry, Self-similar variable Recently, we presented a classification of kinematic self-similar plane sym- metric spacetimes [1]. We have discussed the plane symmetric solutions that admit kinematic self-similar vectors of the second, zeroth, and infinite kinds when the perfect fluid is tilted to the fluid flow, parallel or orthogonal. How- ever, we missed some cases that could provide more solutions. In this ad- dendum, we present those missing solutions, which turn out to be five in number. Further, for the the self-similarity of the first kind (tilted), the two-fluid formalism does not work as the self-similar variable is ξ = x shall investigate a different approach to obtain the solution in this case. The tilted perfect fluid yields four more solutions (one first-kind solution, two 2nd-kind solutions and one zeroth-kind solution), the parallel perfect fluid gives one infinite kind solution, and the orthogonal perfect fluid provides two ∗msharif@math.pu.edu.pk †sehar aziz@yahoo.com http://arxiv.org/abs/0704.1930v1 solutions (one first-kind solution and one 2nd-kind solution). Thus, we ob- tain total seven solutions out of which five solutions are independent. The plane symmetric metric considered in the paper [1] is the following: ds2 = e2ν(t,x)dt2 − dx2 − e2λ(t,x)(dy2 + dz2). (1) We are skipping the details as the procedure can be seen elsewhere [1]. The tilted perfect fluid of the first kind implies that the energy density ρ and pressure p must take the following forms: ρ(ξ), (2) p(ξ), (3) where the self-similar variable is ξ = x/t. When the Einstein field equations (EFEs) and the equations of motion for the matter field are satisfied, a set of ordinary differential equations (ODEs) is obtained, hence, the EFEs and equations of motion [1] reduce to ρ̇ = −2λ̇(ρ+ p), (4) 2p− ṗ = ν̇(ρ+ p), (5) ρ = −4λ̇− 3λ̇2 − 2λ̈− 1, (6) 0 = λ̇2, (7) 0 = λ̈+ λ̇2 + λ̇− λ̇ν̇, (8) p = 1 + 2λ̇+ λ̇2 + 2ν̇ + 2λ̇ν̇, (9) 0 = 2λ̇ν̇ − 2λ̈− 3λ̇2 − 2λ̇, (10) p = λ̈+ λ̇2 + λ̇+ λ̇ν̇ + ν̈ + ν̇2, (11) 0 = −λ̈− λ̇2 − λ̇+ λ̇ν̇. (12) Only the EOS(3) is compatible with this kind. Equations (2) and (3) yield p = kρ while Eqs. (7) and (4) imply that λ and ρ, respectively, are arbitrary constants. Also, Eq. (5) gives ν̇ = 2k . Using this value in the remaining equations, we obtain the following solution: ν = ln (c0ξ 2)), λ = c2, ρ = constant, k = −3 ± 2. (13) The corresponding metric is ds2 = ( )(2∓2 2)dt2 − dx2 − x2(dy2 + dz2). (14) For the self-similarity of the second kind, we obtain solutions only with the EOS(3), and these solutions are missing in Ref. 1. The energy density ρ and pressure p can be written as [ρ1(ξ) + ρ2(ξ)], (15) [p1(ξ) + p2(ξ)], (16) where the self-similar variable is ξ = x/(αt) α . A set of ODEs yield ρ̇1 = −2λ̇(ρ1 + p1), (17) ρ̇2 + 2αρ2 = −2λ̇(ρ2 + p2), (18) −ṗ1 + 2p1 = ν̇(ρ1 + p1), (19) −ṗ2 = ν̇(ρ2 + p2), (20) ρ1 = −4λ̇− 3λ̇2 − 2λ̈− 1, (21) α2e2νρ2 = λ̇ 2, (22) 0 = λ̈+ λ̇2 + λ̇− λ̇ν̇, (23) p1 = 1 + 2λ̇+ λ̇ 2 + 2ν̇ + 2λ̇ν̇, (24) α2e2νp2 = −2λ̈− 3λ̇2 − 2αλ̇+ 2λ̇ν̇, (25) p1 = λ̈+ λ̇ 2 + λ̇+ λ̇ν̇ + ν̈ + ν̇2, (26) α2e2νp2 = −λ̈− λ̇2 − αλ̇+ λ̇ν̇. (27) Proceeding along the same lines with the EOS(3), as given in Ref. 1, for k 6= −1 we assume that ρ1 = 0 and that ρ2 is arbitrary. Thus, Eqs. (21), (23), and (24) show that ν̇ = 0 and λ̇ = −1, and Eq. (18) implies that α = k + 1. Equations (25) and (27) give α = 2, and we obtain the following spacetime: ν = c1, λ = − ln ξ + c2, ρ1 = 0 = p1, ρ2 = constant = p2, α = 2, k = 1. (28) The resulting plane symmetric metric becomes ds2 = dt2 − dx2 − 2t(dy2 + dz2). (29) When k 6= −1, we take ρ2 = 0, and ρ1 is arbitrary; hence, Eq. (22) implies that λ̇ = 0. For the first possibility, it follows that k + 1 ln ξ + c1, λ = c2, p1 = kρ1, ρ1 = constant, p2 = 0 = ρ2, k = −3± 2 2; (30) hence, the plane symmetric spacetime will take the following form: ds2 = ( (αt)1/α k+1dt2 − dx2 − x2(dy2 + dz2). (31) For the self-similarity of the zeroth kind, the EFEs show that the quan- tities ρ and p must be of the form [ρ1(ξ) + x 2ρ2(ξ)], (32) [p1(ξ) + x 2p2(ξ)], (33) where the self-similar variable is ξ = x . A set of ODEs follows such that ρ̇1 = −2λ̇(ρ1 + p1), (34) ρ̇2 = −2λ̇(ρ2 + p2), (35) −ṗ1 + 2p1 = ν̇(ρ1 + p1), (36) −ṗ2 = ν̇(ρ2 + p2), (37) ρ1 = −4λ̇− 3λ̇2 − 2λ̈− 1, (38) e2νρ2 = λ̇ 2, (39) 0 = λ̈+ λ̇2 + λ̇− λ̇ν̇, (40) p1 = 1 + 2λ̇+ λ̇ 2 + 2ν̇ + 2λ̇ν̇, (41) e2νp2 = 2λ̇ν̇ − 2λ̈− 3λ̇2, (42) p1 = λ̈+ λ̇ 2 + λ̇+ λ̇ν̇ + ν̈ + ν̇2, (43) e2νp2 = −λ̈− λ̇2 + λ̇ν̇. (44) For the EOS(3) when k 6= −1, ρ2 = 0, and ρ1 is arbitrary, Eq. (39) yields λ̇ = 0 while Eqs. (34) and (36) show that ν̇ = 2k . Finally, we obtain the same solution as in the case of the second kind with the EOS(3) given by Eq. (30) with α = 0. The corresponding metric is ds2 = (xe−t) k+1dt2 − dx2 − e2t(dy2 + dz2). (45) For the self-similarity of the first kind in the orthogonal perfect fluid case, the EFEs and the equations of motion give e2ν(1 + ρ) = λ′ , (46) e2ν(3− p) = 3λ′2 + 2λ′′ − 2λ′ν ′, (47) e2ν(1− p) = λ′′ + λ′2 − λ′ν ′, (48) 2λ′(ρ+ p) = −ρ′, (49) ρ = p. (50) Clearly, Eq. (50) shows that this is a system with a stiff fluid. If these equations are solved simultaneously, Eq. (49) provides the value of λ′, and Eq. (46) gives the value of ν in terms of p. Equations (47) and (48) impose a constraint on p, p′ p− 2(1+ p)(p′′p− p′2) = 0, and we arrive at the following solution: ν = ln ( (1 + p) ), λ = − ln (p) + ln (c1), ρ = p. (51) For the self-similarity of the second kind in the orthogonal perfect fluid case, the quantities ρ and p must be of the forms κρ = x−2ρ1(ξ) + x −2αρ2(ξ), (52) κp = x−2p1(ξ) + x −2αp2(ξ). (53) A set of ODEs gives ρ′1 = −2λ ′(ρ1 + p1), (54) ρ′2 = −2λ ′(ρ2 + p2), (55) 2p1 = α(ρ1 + p1), (56) ρ2 = p2 (57) ρ1 = −1, (58) e2νρ2 = λ ′2, (59) 0 = (1− α)λ′, (60) p1 = 1 + 2α, (61) e2νp2 = −2λ′′ + 2λ′ν ′ − 3λ′ , (62) p1 = α 2, (63) e2νp2 = −λ′′ − λ′ + λ′ν ′. (64) Equation (57) represents a stiff fluid, and Eq. (60) gives λ′ = 0; hence, we obtain the following solution: ν = arbitrary, λ = c4, p2 = 0 = ρ2, ρ1 = −1, p1 = 3± 2 2, α = 1± 2. (65) The corresponding metric is ds2 = x2(1± 2)dt2 − dx2 − x2(dy2 + dz2). (66) For the self-similarity of the infinite kind in the parallel perfect fluid, a set of ODEs is given as − ρ = 3λ′2 + 2λ′′, (67) p = λ′ + 2λ′ν ′, (68) p = λ′′ + λ′ + λ′ν ′ + ν ′′ + ν ′ , (69) −p′ = ν ′(ρ+ p). (70) Solving Eqs. (67)-(70) with the assumption that p is constant, we find that λ is a linear function of ξ. Finally, we arrive at the following solution: ν = c1, λ = c2ξ + c3, ρ = −3p = constant. (71) The metric is given by ds2 = dt2 − dx2 − e2x(dy2 + dz2). (72) We notice that the solutions given by Eqs.(14), (31) and (66) turn out to be dependent and the solutions given by Eqs. (29), (45), (51), and (72) are independent. Thus, we have a total of five independent solutions. It is worth mentioning here that the self-similar solutions in Eqs. (14), (31), and (66) correspond to the already classified solutions [3] under particular coordinate transformations. The metrics given by Eqs. (14), (31), and (66) correspond to the class of metrics ds2 = e2ν(x)dt2 − dx2 − e2λ(x)(dy2 + dz2), (73) which has four Killing vectors admitting G3 ⊗ℜ with a timelike ℜ. We also note that the density is either zero or positive in all the solutions, except for the solution given by Eq. (66). The physical properties of all these solutions can be seen in the Ref. 4. We would like to mention here that the paper in Ref. 1 focussed on a clas- sification of plane symmetric kinematic self-similar solutions under certain assumptions. A complete classification for the most general plane symmetric kinematic self-similar solutions appears elsewhere [2]. ACKNOWLEDGMENT One of us (SA) would like to acknowledge Higher Education Commission (HEC) for the financial support. References [1] M. Sharif and S. Aziz: J. Korean Physical Society 49, 21 (2006). [2] M. Sharif and S. Aziz: Class. Quantum Grav. 24, 605 (2007). [3] T. Feroze, A. Qadir and M. Ziad: J. Math. Phys. 42, 4947 (2001). [4] M. Sharif and S. Aziz: J. Korean Physical Society 47, 757 (2005).

In our recent paper, we classified plane symmetric kinematic self-similar perfect fluid and dust solutions of the second, zeroth and infinite kinds. However, we have missed some solutions during the process. In this short communication, we add up those missing solutions. We have found a total of seven solutions, out of which five turn out to be independent and cannot be found in the earlier paper

Addendum: A Classification of Plane Symmetric Kinematic Self-similar Solutions M. Sharif ∗and Sehar Aziz † Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan. In our recent paper, we classified plane symmetric kinematic self-similar perfect fluid and dust solutions of the second, zeroth and infinite kinds. How- ever, we have missed some solutions during the process. In this short com- munication, we add up those missing solutions. We have found a total of seven solutions, out of which five turn out to be independent and cannot be found in the earlier paper. Keywords: Plane symmetry, Self-similar variable Recently, we presented a classification of kinematic self-similar plane sym- metric spacetimes [1]. We have discussed the plane symmetric solutions that admit kinematic self-similar vectors of the second, zeroth, and infinite kinds when the perfect fluid is tilted to the fluid flow, parallel or orthogonal. How- ever, we missed some cases that could provide more solutions. In this ad- dendum, we present those missing solutions, which turn out to be five in number. Further, for the the self-similarity of the first kind (tilted), the two-fluid formalism does not work as the self-similar variable is ξ = x shall investigate a different approach to obtain the solution in this case. The tilted perfect fluid yields four more solutions (one first-kind solution, two 2nd-kind solutions and one zeroth-kind solution), the parallel perfect fluid gives one infinite kind solution, and the orthogonal perfect fluid provides two ∗msharif@math.pu.edu.pk †sehar aziz@yahoo.com http://arxiv.org/abs/0704.1930v1 solutions (one first-kind solution and one 2nd-kind solution). Thus, we ob- tain total seven solutions out of which five solutions are independent. The plane symmetric metric considered in the paper [1] is the following: ds2 = e2ν(t,x)dt2 − dx2 − e2λ(t,x)(dy2 + dz2). (1) We are skipping the details as the procedure can be seen elsewhere [1]. The tilted perfect fluid of the first kind implies that the energy density ρ and pressure p must take the following forms: ρ(ξ), (2) p(ξ), (3) where the self-similar variable is ξ = x/t. When the Einstein field equations (EFEs) and the equations of motion for the matter field are satisfied, a set of ordinary differential equations (ODEs) is obtained, hence, the EFEs and equations of motion [1] reduce to ρ̇ = −2λ̇(ρ+ p), (4) 2p− ṗ = ν̇(ρ+ p), (5) ρ = −4λ̇− 3λ̇2 − 2λ̈− 1, (6) 0 = λ̇2, (7) 0 = λ̈+ λ̇2 + λ̇− λ̇ν̇, (8) p = 1 + 2λ̇+ λ̇2 + 2ν̇ + 2λ̇ν̇, (9) 0 = 2λ̇ν̇ − 2λ̈− 3λ̇2 − 2λ̇, (10) p = λ̈+ λ̇2 + λ̇+ λ̇ν̇ + ν̈ + ν̇2, (11) 0 = −λ̈− λ̇2 − λ̇+ λ̇ν̇. (12) Only the EOS(3) is compatible with this kind. Equations (2) and (3) yield p = kρ while Eqs. (7) and (4) imply that λ and ρ, respectively, are arbitrary constants. Also, Eq. (5) gives ν̇ = 2k . Using this value in the remaining equations, we obtain the following solution: ν = ln (c0ξ 2)), λ = c2, ρ = constant, k = −3 ± 2. (13) The corresponding metric is ds2 = ( )(2∓2 2)dt2 − dx2 − x2(dy2 + dz2). (14) For the self-similarity of the second kind, we obtain solutions only with the EOS(3), and these solutions are missing in Ref. 1. The energy density ρ and pressure p can be written as [ρ1(ξ) + ρ2(ξ)], (15) [p1(ξ) + p2(ξ)], (16) where the self-similar variable is ξ = x/(αt) α . A set of ODEs yield ρ̇1 = −2λ̇(ρ1 + p1), (17) ρ̇2 + 2αρ2 = −2λ̇(ρ2 + p2), (18) −ṗ1 + 2p1 = ν̇(ρ1 + p1), (19) −ṗ2 = ν̇(ρ2 + p2), (20) ρ1 = −4λ̇− 3λ̇2 − 2λ̈− 1, (21) α2e2νρ2 = λ̇ 2, (22) 0 = λ̈+ λ̇2 + λ̇− λ̇ν̇, (23) p1 = 1 + 2λ̇+ λ̇ 2 + 2ν̇ + 2λ̇ν̇, (24) α2e2νp2 = −2λ̈− 3λ̇2 − 2αλ̇+ 2λ̇ν̇, (25) p1 = λ̈+ λ̇ 2 + λ̇+ λ̇ν̇ + ν̈ + ν̇2, (26) α2e2νp2 = −λ̈− λ̇2 − αλ̇+ λ̇ν̇. (27) Proceeding along the same lines with the EOS(3), as given in Ref. 1, for k 6= −1 we assume that ρ1 = 0 and that ρ2 is arbitrary. Thus, Eqs. (21), (23), and (24) show that ν̇ = 0 and λ̇ = −1, and Eq. (18) implies that α = k + 1. Equations (25) and (27) give α = 2, and we obtain the following spacetime: ν = c1, λ = − ln ξ + c2, ρ1 = 0 = p1, ρ2 = constant = p2, α = 2, k = 1. (28) The resulting plane symmetric metric becomes ds2 = dt2 − dx2 − 2t(dy2 + dz2). (29) When k 6= −1, we take ρ2 = 0, and ρ1 is arbitrary; hence, Eq. (22) implies that λ̇ = 0. For the first possibility, it follows that k + 1 ln ξ + c1, λ = c2, p1 = kρ1, ρ1 = constant, p2 = 0 = ρ2, k = −3± 2 2; (30) hence, the plane symmetric spacetime will take the following form: ds2 = ( (αt)1/α k+1dt2 − dx2 − x2(dy2 + dz2). (31) For the self-similarity of the zeroth kind, the EFEs show that the quan- tities ρ and p must be of the form [ρ1(ξ) + x 2ρ2(ξ)], (32) [p1(ξ) + x 2p2(ξ)], (33) where the self-similar variable is ξ = x . A set of ODEs follows such that ρ̇1 = −2λ̇(ρ1 + p1), (34) ρ̇2 = −2λ̇(ρ2 + p2), (35) −ṗ1 + 2p1 = ν̇(ρ1 + p1), (36) −ṗ2 = ν̇(ρ2 + p2), (37) ρ1 = −4λ̇− 3λ̇2 − 2λ̈− 1, (38) e2νρ2 = λ̇ 2, (39) 0 = λ̈+ λ̇2 + λ̇− λ̇ν̇, (40) p1 = 1 + 2λ̇+ λ̇ 2 + 2ν̇ + 2λ̇ν̇, (41) e2νp2 = 2λ̇ν̇ − 2λ̈− 3λ̇2, (42) p1 = λ̈+ λ̇ 2 + λ̇+ λ̇ν̇ + ν̈ + ν̇2, (43) e2νp2 = −λ̈− λ̇2 + λ̇ν̇. (44) For the EOS(3) when k 6= −1, ρ2 = 0, and ρ1 is arbitrary, Eq. (39) yields λ̇ = 0 while Eqs. (34) and (36) show that ν̇ = 2k . Finally, we obtain the same solution as in the case of the second kind with the EOS(3) given by Eq. (30) with α = 0. The corresponding metric is ds2 = (xe−t) k+1dt2 − dx2 − e2t(dy2 + dz2). (45) For the self-similarity of the first kind in the orthogonal perfect fluid case, the EFEs and the equations of motion give e2ν(1 + ρ) = λ′ , (46) e2ν(3− p) = 3λ′2 + 2λ′′ − 2λ′ν ′, (47) e2ν(1− p) = λ′′ + λ′2 − λ′ν ′, (48) 2λ′(ρ+ p) = −ρ′, (49) ρ = p. (50) Clearly, Eq. (50) shows that this is a system with a stiff fluid. If these equations are solved simultaneously, Eq. (49) provides the value of λ′, and Eq. (46) gives the value of ν in terms of p. Equations (47) and (48) impose a constraint on p, p′ p− 2(1+ p)(p′′p− p′2) = 0, and we arrive at the following solution: ν = ln ( (1 + p) ), λ = − ln (p) + ln (c1), ρ = p. (51) For the self-similarity of the second kind in the orthogonal perfect fluid case, the quantities ρ and p must be of the forms κρ = x−2ρ1(ξ) + x −2αρ2(ξ), (52) κp = x−2p1(ξ) + x −2αp2(ξ). (53) A set of ODEs gives ρ′1 = −2λ ′(ρ1 + p1), (54) ρ′2 = −2λ ′(ρ2 + p2), (55) 2p1 = α(ρ1 + p1), (56) ρ2 = p2 (57) ρ1 = −1, (58) e2νρ2 = λ ′2, (59) 0 = (1− α)λ′, (60) p1 = 1 + 2α, (61) e2νp2 = −2λ′′ + 2λ′ν ′ − 3λ′ , (62) p1 = α 2, (63) e2νp2 = −λ′′ − λ′ + λ′ν ′. (64) Equation (57) represents a stiff fluid, and Eq. (60) gives λ′ = 0; hence, we obtain the following solution: ν = arbitrary, λ = c4, p2 = 0 = ρ2, ρ1 = −1, p1 = 3± 2 2, α = 1± 2. (65) The corresponding metric is ds2 = x2(1± 2)dt2 − dx2 − x2(dy2 + dz2). (66) For the self-similarity of the infinite kind in the parallel perfect fluid, a set of ODEs is given as − ρ = 3λ′2 + 2λ′′, (67) p = λ′ + 2λ′ν ′, (68) p = λ′′ + λ′ + λ′ν ′ + ν ′′ + ν ′ , (69) −p′ = ν ′(ρ+ p). (70) Solving Eqs. (67)-(70) with the assumption that p is constant, we find that λ is a linear function of ξ. Finally, we arrive at the following solution: ν = c1, λ = c2ξ + c3, ρ = −3p = constant. (71) The metric is given by ds2 = dt2 − dx2 − e2x(dy2 + dz2). (72) We notice that the solutions given by Eqs.(14), (31) and (66) turn out to be dependent and the solutions given by Eqs. (29), (45), (51), and (72) are independent. Thus, we have a total of five independent solutions. It is worth mentioning here that the self-similar solutions in Eqs. (14), (31), and (66) correspond to the already classified solutions [3] under particular coordinate transformations. The metrics given by Eqs. (14), (31), and (66) correspond to the class of metrics ds2 = e2ν(x)dt2 − dx2 − e2λ(x)(dy2 + dz2), (73) which has four Killing vectors admitting G3 ⊗ℜ with a timelike ℜ. We also note that the density is either zero or positive in all the solutions, except for the solution given by Eq. (66). The physical properties of all these solutions can be seen in the Ref. 4. We would like to mention here that the paper in Ref. 1 focussed on a clas- sification of plane symmetric kinematic self-similar solutions under certain assumptions. A complete classification for the most general plane symmetric kinematic self-similar solutions appears elsewhere [2]. ACKNOWLEDGMENT One of us (SA) would like to acknowledge Higher Education Commission (HEC) for the financial support. References [1] M. Sharif and S. Aziz: J. Korean Physical Society 49, 21 (2006). [2] M. Sharif and S. Aziz: Class. Quantum Grav. 24, 605 (2007). [3] T. Feroze, A. Qadir and M. Ziad: J. Math. Phys. 42, 4947 (2001). [4] M. Sharif and S. Aziz: J. Korean Physical Society 47, 757 (2005).

Weak non-linearities and cluster states Sebastien G.R. Louis,1, 2, ∗ Kae Nemoto,1 W. J. Munro,3, 1 and T. P. Spiller3 1National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan 2Department of Informatics, School of Multidisciplinary Sciences, The Graduate University for Advanced Studies, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430 Japan 3Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom (Dated: February 6, 2020) We propose a scalable approach to building cluster states of matter qubits using coherent states of light. Recent work on the subject relies on the use of single photonic qubits in the measurement process. These schemes have a low initial success probability and low detector efficiencies cause a serious blowup in resources. In contrast, our approach uses continuous variables and highly efficient measurements. We present a two-qubit scheme, with a simple homodyne measurement system yielding an entangling operation with success probability 1/2. Then we extend this to a three-qubit interaction, increasing this probability to 3/4. We discuss the important issues of the overhead cost and the time scaling, showing how these can be vastly improved with access to this new probability range. PACS numbers: 03.67.Lx, 03.67.Mn, 42.50.Dv, 32.80.-t I. INTRODUCTION The intriguing idea of one-way or cluster state quan- tum computing was initially developed by Briegel and Raussendorf [1]. They showed that a two-dimensional ar- ray of qubits, entangled in a particular way (through Condi- tional Phase gates), combined with single qubit operations, feed forward and measurements are sufficient for universal quantum computation. All the required interactions are al- ready contained inside the system, and the computation pro- ceeds through a series of local measurements (with classical feed forward), efficiently simulating quantum circuits. In ef- fect, the logical gates are prepared off-line and imprinted onto the qubits as they are transmitted through the cluster. This approach was quickly applied [2, 3, 4] to linear op- tics quantum computing [5, 6] (and references therein), both having been experimentally demonstrated [7, 8]. It pushes the problem with the probabilistic nature of 2-qubit gates into the off-line preparation of the cluster [2, 3]. In this context it was shown that simple parity gates are sufficient for building the required states. These schemes are then bounded by the single beam-splitter success probability of 1/2 and in fact this initial probability is far reduced when the single photon detection in- efficiencies are taken into account. Supplementing the linear optical approaches with weak nonlinearities[9, 10, 11, 12, 13] allows for the construction of parity gates with significantly higher success probabilities (near unity in some cases). A core issue however with photonic qubits is their ‘flying’ nature and the storage requirements this mandates. A natural way around this issue is to move to solid state or condensed matter qubits and use single photons for com- munication between them. Many proposals make use of sin- gle photons to effectively mediate interactions between mat- ter qubits [14, 15, 16, 17, 18, 19]. Having interacted with them, the photons then interact with each other in a linear op- ∗Electronic address: seblouis@nii.ac.jp tical setup before being measured, thus projecting the matter qubits into the required state without them having interacted directly. It has been shown that entanglement and logical op- erations can be generated in this way. The next step was to use these probabilistic entangling schemes to prepare cluster states of matter qubits [20, 21] using techniques like double- heralding or repeat-until-success. However the schemes are generally limited by the detection of the single photons (more than one in some cases). This can severely limit the probabil- ity of realizing the entangling operation and hence the creation of the cluster state. An alternative is available and this is what we will describe in this letter. Instead of using single pho- tons, we can use coherent states of light (similar to the weak nonlinearity approach). Homodyne measurements on coher- ent light fields can be made much more efficient than single photon detection and so we can achieve entangling operations with a probability greater than 1/2. In this paper we will show how this and other factors make continuous variables a very powerful tool for growing matter qubit based cluster states. II. GATES There are quite a number of well studied systems where one has a natural interaction between the matter qubit and the electromagnetic field. These include atoms (real and artifical) in CQED (both at the optical and telecom wavelengths) [22], NV-centers in diamond [23], quantum dots with a single ex- cess electron [24], trapped ions [25] and SQUIDs [26] to name only a few. All these systems are likely to be suitable candi- dates for what we describe below but to illustrate the details a little more clearly let us consider a lambda based CQED sys- tem. One could use cesium atoms or an NV-diamond center embedded in the cavity. Both of these systems operate in the optical frequency range and so are well matched to efficient homodyne measurements. The interaction between the coher- ent field mode and our matter qubit can generally be described by the Jaynes-Cummings interaction h̄g(σ−a† + σ+a) and in the dispersive limit (large detunings) one obtains an effective http://arxiv.org/abs/0704.1931v1 mailto:seblouis@nii.ac.jp interaction Hamiltonian of the form [27, 28]: Hint = h̄χσza †a. (1) where a (a†) refers to the annihilation (creation) operator of an electromagnetic field mode in a cavity and the matter qubit is described using the conventional Pauli operators, with the computational basis being given by the eigenstates of σz ≡ |0〉〈0| − |1〉〈1|, with |0〉 ≡ | ↑z〉 and |1〉 ≡ | ↓z〉. The atom- light coupling strength is determined via the parameter χ = g2/∆, where 2g is the vacuum Rabi splitting for the dipole transition and ∆ is the detuning between the dipole transition and the cavity field. The interaction Hint applied for a time t generates a conditional phase-rotation ±θ (with θ = χt) on the field mode dependent upon the state of the matter qubit. We call this a conditional rotation and it is very similar to the cross-Kerr interaction between photons. This time dependent interaction implicitly requires a pulsed probe. Now the interaction in (1) forms the basis for an entangling operation. A two-qubit gate has been proposed [13] based on controlled bus rotations and a subsequent measurement. The probe field coherent state |α〉 interacts with both qubits, so an initial state of the system |Ψi〉 = 12 (|00〉 + |01〉 + |10〉 + |11〉)|α〉 evolves to |Ψf 〉 = (|00〉|αe2iθ〉+ |11〉|αe−2iθ〉) (|01〉+ |10〉)|α〉. (2) Here we quickly observe that the probe field has evolved into three potentially distinct states and appropriate measure- ments can project our two qubits into a number of interest- ing states. At this stage we can choose from different types of measurements on the probe beam. The first and simplest option we have is to perform a homodyne measurement of some field quadrature X(φ) = (a†eiφ + ae−iφ) which for a sufficiently strong local oscillator (compared to the signal strength) implements a projective measurement |x(φ)〉〈x(φ)| on the probe state [29]. The key advantages with homodyne measurement, at least in the optical regime are that it is highly efficient (99% plus [30]) and is a standard tool of continuous variable experimentalists. The simplest homodyne measure- ment to perform is the momentum (P = X(π/2)) quadra- ture. In this case the measurement probability distribution has three peaks with the overlap error between them given by Perr = erfc(α sin θ/ 2). As long as αθ ∼ π this over- lap error is small (< 10−3) and the peaks are well separated. If our P quadrature measurement projects us onto the cen- tral peak |α〉, our two matter qubits are conditioned into the entangled state (|01〉 + |10〉)/ 2. This occurs with a proba- bility of 1/2. Detecting either of the other two side peaks will project the qubits to the known product states |00〉 or |11〉. The probability of entangling the two qubits is interesting in that we have already reached the limits of conventional linear optical implementations. When realistic detector efficiencies (η ∼ 70%) are taken into account, their optimal success prob- ability of 1/2 decreases dramatically (proportional to η or η2 depending on the implementation) and so the probability of the operation succeeding is now significantly less than 1/2. In contrast homodyne measurements are highly efficient and so our success probability will remain very close to 1/2. This limit may be fundamental to the linear optical schemes but in our case we can exceed it by changing the nature of our mea- surement. In principle we could achieve a near deterministic gate if we measured the the position quadrature (X = X(0)), however the requirements to ensure the distinguishability of the probe beam states are much more severe. We could also in principle use a photon number measurement after displacing the probe beam [13], but we would fall back into the issues affecting the linear optical schemes. By restricting ourselves to P = X(π/2) quadrature measurements and single interac- tions between the qubits and the probe, we are opting for the most robust weak-nonlinear approach so far proposed. Within the same framework of conditional rotations and P measurements, one can envisage three qubits interact- ing with a single probe beam. GHZ states are for in- stance one particularly useful state [3]. One way of pro- jecting the qubits onto GHZ-type states is to vary the strength of the interactions between the qubits and the probe beam [12]. Let us represent a rotation of the coherent probe beam by R(θσz) = exp(iθa †aσz). The sequence R(θσz1)R(θσz2)R(−2θσz3)|α〉 which we depict in Fig (1) will give the optimal paths and end points in phase space. The peak centered on the origin will then correspond to the Qubit 1 Qubit 2 Qubit 3 Probe (b)(a) P 011+101 000+111 010+100 FIG. 1: Schematic diagram (a) of a three qubit entangling operation. In (b) the possible probe trajectories caused by the three conditional rotations. There are five different end-states. Upon measurement, three of these will project the qubits to entangled states of interest. GHZ state (|000〉 + |111〉) 2 (after being detected). This will happen with a probability of 1/4 (all qubits started in per- fect superpositions). Next the two peaks having been rotated through ±2θ will correspond to the qubit states (|01〉1,2 + |10〉1,2)|1〉3/ 2 and (|01〉1,2 + |10〉1,2)|0〉3/ 2 respectively. Now in both of these possible outcomes we obtain the same Bell state on qubits 1 and 2, disentangled from qubit 3. So overall we obtain a GHZ state with probability of 1/4 and a Bell state with probability of 1/2 (on two qubits of our choice), heralded by the probe beam P quadrature measurement out- come. The other two outcomes project the qubits to known product states. Consequently, if all we want to do is entangle a pair of qubits, we can now do this with a probability of 3/4. This method can be extended to larger numbers of qubits, but the 3-qubit case minimizes the ratio of operation time over success probability. We shall use this result in the remainder of the paper, observing how current work on the generation of cluster states is simply inadequate for probabilities exceeding 1/2. Until now strategies have been said to be scalable if the resources don’t scale exponentially with the size of the cluster (in general they will scale sub-exponentially). This is a purely theoretical notion which bares little relation to the practical scalability we obtain in our approach. We stress that although the 3-qubit operation is a proba- bilistic entangling operation with different outcomes, these outcomes are heralded by the measurement of the bus and so the operation is a very useful entangling primitive for the con- struction of cluster states. For example, applying it to join two sections of cluster with a third ancillary qubit works with probability 3/4, giving (heralded) outcomes of joined clusters with a new dangling bond (probability 1/2), or joined clusters and two new dangling bonds (probability 1/4). Applying the operation to join three sections of cluster gives (heralded) out- comes of two sections joined and a new dangling bond (prob- ability 1/2), or all three sections joined and two new dangling bonds (probability 1/4). All these outcomes contribute to clus- ter state construction. III. SCALING Now let’s turn to the issue of building up linear cluster states (chains). In order to efficiently grow a chain with probabilis- tic gates, one needs to first inefficiently build small chains ex- ceeding a critical length Lc = 1+2(1−p)/p and then try join- ing them to the main one. This critical length varies between different entangling operations. If an actual conditional phase gate can be immediately implemented, then Lc = 2(1− p)/p for example. Or if this logical gate requires the qubits from the cluster to interact directly (non-distributive approach) then Lc = 4(1− p)/p [31]. Starting from this, and adopting a ‘di- vide and conquer’ approach to building these minimal chains, scaling relations are obtained for the average number of entan- gling operations required and the average time taken, to build a chain of length L. Using our 2-qubit gate (Lc = 3) and these scaling relations we obtain N [L] = 12L− 38. This is already the limiting scenario for simple single photon applications. In the repeat-until-success method [21], for a failure probability of 0.6 (and equal success and insurance probabilities, on all results), the scaling is N [L] = 185L− 1115 and for a failure probability of 0.4 it becomes N [L] ≃ 16.6L − 47.7. Now if we switch to our 3-qubit gate, then Lc < 2 and our mini- mal chain is now simply a 2-qubit cluster (locally equivalent to a Bell state) yielding N [L] = 8L − 44/3. This is a vast improvement over previous proposals. For the two-qubit entangling gate, we essentially stand at the same point as the photonic cluster state approaches. Opti- mizing the resources boils down to finding the optimal strate- gies in combining elements of cluster states. This is a very complex task, which Gross et al. [32] analyzed in great detail. For higher probabilities however, this critical length insur- ing average growth is no longer existent. All previously de- rived strategies become trivial within this probability range. Additional scalable approaches such as sequential adding are at hand and we shall go over the obvious ones. From pre- vious works on generating cluster states [20, 31], we know that the simplest way to grow short chains with probabilistic gates is through a ‘divide and conquer’ approach. It also turns out to be much quicker than a sequential adding, as we allow for many gates to operate in parallel. This technique links up chains of equal length on each round, and discards the chains which failed to do so. In the context of higher success probabilities this approach can be extended to growing large chains in the aim of saving time. The corresponding average number of entangling oper- ations becomes: Ndc[L] = (2/p)log2(L−1) − 1 . (3) From the initial strategy we reach a value linear in L: N [L] = (2/p) L− 1− 2(1− p)/p 1− 2(1− p)/p − 1/p , (4) and a sequential adding yields: Nseq[L] = (L − 1)/(2p− 1) . (5) Obviously the latter represents a considerable saving, as can be verified in Fig. (2). Though the divide and conquer method doesn’t scale linearly, up till lengths of 250 qubits, it requires less entangling operations than the initial scheme (which in fact is a full recycling approach). This is due to the fact that the probabilities we are dealing with are significantly higher than in previous proposals, which were undertaken in two steps, the building of minimal elements and then their merg- ing, in order to be scalable. If we look at the qubit resources however, the less recycling we do, the more qubits we waste in the process. But as the success probability of the gate increases, the recycling strategies all converge with the no- recycling strategy (in terms of qubit resources), this being par- ticularly noticeable for success probabilities higher than 1/2. We can also compare the time scaling of these various strategies, in units of time t corresponding to a single mea- surement. For the complete divide and conquer scheme we simply have: Tdc[L] = t (1 + log2(L− 1)) . (6) and for the initial scheme: T [L] = (t/p) 1 + log2 L− Lc L0 − Lc . (7) For the sequential adding, the cumulative time obeys TL+1 = TL + t/p, and the general form for T becomes: Tseq[L] = t(L− 1)/p . (8) The first two approaches have a logarithmic dependence on the length L, however Tdc is significantly lower as might have sequential Pf=0.6 Pf=0.4 T[L](a) (b) 100 200 300 400 302010 sequential D&C one EO per round initial scheme initial scheme FIG. 2: a) Comparison of entangling operation requirements for chain production. We achieve much lower scalings in comparison with those obtained through the repeat-until-success (RUS) scheme (Pf being the failure probability). b) Time scaling for the different strategies. The three-qubit gate is used in both plots. been expected (see Fig. (2)). Overall we see that there is a clear advantage to divide the task up and to run parallel en- tangling operations. The linear time scaling for the sequential method is due to the fact that operations cannot be undertaken in parallel during the growth. If we didn’t have access to simultaneous entangling operations, the time scaling for the divide and conquer methods would be equivalent to Ndc[L] which is sub-exponential. One needs to keep in mind that by adopting a sequential method, the whole procedure is simpli- fied considerably and would be more accessible to physical implementations. IV. DISCUSSION The cluster state comprises of active regions in which it is being built or measured in the computation (both can be un- dertaken simultaneously) and regions in which the qubits are simply waiting. Now this waiting can be minimized in the building itself, through the appropriate protocols, and in the measurement process. That is, the cluster can be built only a few layers in advance, so that the qubits have less waiting to do, between the building and the actual measurement. In any case, there will be some waiting. Therefore the lowest decoherence realization would be preferred, but it may not be the easiest to manipulate. Thus we may envisage hav- ing two different physical realizations constituting the cluster state. For example, we could use single electron spins ini- tially in building the cluster. Once the links are made between one site and its nearest neighbors, the qubit could be switched into a nuclear spin state which has a significantly longer co- herence time, via a swap operation or some other coherent write and read actions. Most of the waiting would be done in the long-lived state, before being swapped again for the readout [33, 34]. This follows the same scenario as using a second physical system to mediate the interaction and make the measurements, in distributed quantum computing. In the present proposal, we use a continuous variable bus and homo- dyne measurements to generate the links. This physical sys- tem shows itself to be very efficient in this application. Then, for example, electron spins or superconducting charge qubits could be the matter realization interacting with the bus and serving for the final readout. These systems provide the use- ful static aspect required, they interact well with the mediating bus and ensure good single qubit measurements. Finally a low decoherence realization such as nuclear spin could be envis- aged, mainly as a storage medium. The swapping or write and read procedure should have a high fidelity for this storage to be beneficial. On the whole, we see that optimization will depend directly on the physical realization(s) we have cho- sen to work with. For systems with long dephasing times we would give priority to sequential adding approaches, as we have some freedom in the time scaling and thus we can make significant savings in resources. But for realizations with short dephasing times, we would probably want to divide the task up and run operations in parallel, in order to accelerate the fab- rication of the cluster state, at the expense of extra resources. V. CONCLUSION In summary we have shown how the concept of the quan- tum bus can be adapted to efficiently generating cluster states of matter qubits. We can straightforwardly gain access to entangling probabilities higher than 1/2, removing the need to break up the building process into inefficient and efficient parts. This opens up a new class of strategies, for which the resource consumption and the time scaling are consequently vastly improved. Clearly, within this class, detailed strategies can be envisaged and they will depend on the chosen physical realization and the levels of decoherence present. VI. ACKNOWLEDGEMENTS We would like to thank R. van Meter and J. Eisert for help- ful discussions. This work was supported in part by MEXT in Japan and the EU project QAP. [1] R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86, 5188 (2001). [2] M.A. Nielsen, Phys. Rev. Lett. 93, 040503 (2004). [3] D.E. Browne and T. Rudolph, Phys. Rev. Lett. 95, 010501 (2005). [4] G. Gilbert, M. Hamrick, Y.S. Weinstein, Phys. Rev. A 73, 064303 (2006). [5] E. Knill, R. Laflamme and G.J. Milburn, Nature 409, 46 (2001). [6] P. Kok, W.J. Munro , K. Nemoto, T.C. Ralph , J.P. Dowling and G.J. Milburn , Rev. Mod. Phys. 79, 135 (2007). [7] P. Walther et al., Nature 434, 169 (2005). [8] N. Kiesel et al., Phys. Rev. Lett. 95, 210502 (2005). [9] W.J. Munro et al., J. Opt. B: Quantum. Semiclass. Opt. 7, S135 (2005). [10] W.J. Munro et al., New J. Phys. 7, 137 (2005). [11] Kae Nemoto and W. J. Munro, Phys. Lett A 344, 104 (2005). [12] F. Yamaguchi, Kae Nemoto and W. J. Munro, Phys. Rev. A 73, 060302(R) (2006). [13] T. P. Spiller et al., New Journal of Physics 8, 30(2006). [14] S. Bose, P.L. Knight, M.B. Plenio and V. Vedral, Phys. Rev. Lett.83, 5158 (1999). [15] C. Cabrillo, J.I. Cirac, P. Garcia-Fernandez and P. Zoller, Phys. Rev. A 59, 1025 (1999). [16] X.-L. Feng, Z.M. Zhang, X.D. Li, S.Q. Gong and Z.Z. Xu, Phys. Rev. Lett. 90, 217902 (2003). [17] L.-M. Duan and H. J. Kimble, Phys. Rev. Lett. 90, 253601 (2003). [18] D.E. Browne, M. B. Plenio and S.F. Huelga, Phys. Rev. Lett. 91, 067901 (2003). [19] C. Simon and W. T. M. Irvine, Phys. Rev. Lett. 91, 110405(2003). [20] S. D. Barrett and P. Kok, Phys. Rev. A 71, 060310(R) (2005). [21] Y.L. Lim, S.D. Barrett, A. Beige, P. Kok and L.C. Kwek, Phys. Rev. A 73, 012304 (2006). [22] J. M. Raimond, M. Brune and S, Haroche, Rev. Mod. Phys. 73, 565 (2001). [23] F. Jelezko, T. Gaebel, I. Popa, A. Gruber and J. Wrachtrup, Phys. Rev. Lett. 92, 076401 (2004). [24] E. Pazy et al., Europhys. Lett. 62, 175 (2003). [25] J.I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). [26] A. Shnirman, G. Schon and Z. Hermon, Phys. Rev. Lett. 79, 2371 (1997). [27] C. C Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press (28 Oct 2004). [28] A. Blais, R.S. Huang, A. Wallraff, S.M. Girvin and R.J. Schoelkopf, Phys. Rev. A 69, 062320 (2004). [29] Tomas Tyc and Barry C. Sanders, J. Phys. A 37, 7341 (2004). [30] E.S. Polzik, J. Carri, and H.J. Kimble, Phys. Rev. Lett. 68, 3020 (1992). [31] L.-M. Duan and R. Raussendorf, Phys. Rev. Lett. 95, 080503 (2005). [32] D. Gross, K. Kieling and J. Eisert, Phys. Rev. A 74, 042343 (2006). [33] B. E. Kane, Nature 393, 133 (1998). [34] P. van Loock et al., Phys. Rev. Lett. 96, 240501 (2006) .

We propose a scalable approach to building cluster states of matter qubits using coherent states of light. Recent work on the subject relies on the use of single photonic qubits in the measurement process. These schemes have a low initial success probability and low detector efficiencies cause a serious blowup in resources. In contrast, our approach uses continuous variables and highly efficient measurements. We present a two-qubit scheme, with a simple homodyne measurement system yielding an entangling operation with success probability 1/2. Then we extend this to a three-qubit interaction, increasing this probability to 3/4. We discuss the important issues of the overhead cost and the time scaling, showing how these can be vastly improved with access to this new probability range.

Weak non-linearities and cluster states Sebastien G.R. Louis,1, 2, ∗ Kae Nemoto,1 W. J. Munro,3, 1 and T. P. Spiller3 1National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan 2Department of Informatics, School of Multidisciplinary Sciences, The Graduate University for Advanced Studies, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430 Japan 3Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom (Dated: February 6, 2020) We propose a scalable approach to building cluster states of matter qubits using coherent states of light. Recent work on the subject relies on the use of single photonic qubits in the measurement process. These schemes have a low initial success probability and low detector efficiencies cause a serious blowup in resources. In contrast, our approach uses continuous variables and highly efficient measurements. We present a two-qubit scheme, with a simple homodyne measurement system yielding an entangling operation with success probability 1/2. Then we extend this to a three-qubit interaction, increasing this probability to 3/4. We discuss the important issues of the overhead cost and the time scaling, showing how these can be vastly improved with access to this new probability range. PACS numbers: 03.67.Lx, 03.67.Mn, 42.50.Dv, 32.80.-t I. INTRODUCTION The intriguing idea of one-way or cluster state quan- tum computing was initially developed by Briegel and Raussendorf [1]. They showed that a two-dimensional ar- ray of qubits, entangled in a particular way (through Condi- tional Phase gates), combined with single qubit operations, feed forward and measurements are sufficient for universal quantum computation. All the required interactions are al- ready contained inside the system, and the computation pro- ceeds through a series of local measurements (with classical feed forward), efficiently simulating quantum circuits. In ef- fect, the logical gates are prepared off-line and imprinted onto the qubits as they are transmitted through the cluster. This approach was quickly applied [2, 3, 4] to linear op- tics quantum computing [5, 6] (and references therein), both having been experimentally demonstrated [7, 8]. It pushes the problem with the probabilistic nature of 2-qubit gates into the off-line preparation of the cluster [2, 3]. In this context it was shown that simple parity gates are sufficient for building the required states. These schemes are then bounded by the single beam-splitter success probability of 1/2 and in fact this initial probability is far reduced when the single photon detection in- efficiencies are taken into account. Supplementing the linear optical approaches with weak nonlinearities[9, 10, 11, 12, 13] allows for the construction of parity gates with significantly higher success probabilities (near unity in some cases). A core issue however with photonic qubits is their ‘flying’ nature and the storage requirements this mandates. A natural way around this issue is to move to solid state or condensed matter qubits and use single photons for com- munication between them. Many proposals make use of sin- gle photons to effectively mediate interactions between mat- ter qubits [14, 15, 16, 17, 18, 19]. Having interacted with them, the photons then interact with each other in a linear op- ∗Electronic address: seblouis@nii.ac.jp tical setup before being measured, thus projecting the matter qubits into the required state without them having interacted directly. It has been shown that entanglement and logical op- erations can be generated in this way. The next step was to use these probabilistic entangling schemes to prepare cluster states of matter qubits [20, 21] using techniques like double- heralding or repeat-until-success. However the schemes are generally limited by the detection of the single photons (more than one in some cases). This can severely limit the probabil- ity of realizing the entangling operation and hence the creation of the cluster state. An alternative is available and this is what we will describe in this letter. Instead of using single pho- tons, we can use coherent states of light (similar to the weak nonlinearity approach). Homodyne measurements on coher- ent light fields can be made much more efficient than single photon detection and so we can achieve entangling operations with a probability greater than 1/2. In this paper we will show how this and other factors make continuous variables a very powerful tool for growing matter qubit based cluster states. II. GATES There are quite a number of well studied systems where one has a natural interaction between the matter qubit and the electromagnetic field. These include atoms (real and artifical) in CQED (both at the optical and telecom wavelengths) [22], NV-centers in diamond [23], quantum dots with a single ex- cess electron [24], trapped ions [25] and SQUIDs [26] to name only a few. All these systems are likely to be suitable candi- dates for what we describe below but to illustrate the details a little more clearly let us consider a lambda based CQED sys- tem. One could use cesium atoms or an NV-diamond center embedded in the cavity. Both of these systems operate in the optical frequency range and so are well matched to efficient homodyne measurements. The interaction between the coher- ent field mode and our matter qubit can generally be described by the Jaynes-Cummings interaction h̄g(σ−a† + σ+a) and in the dispersive limit (large detunings) one obtains an effective http://arxiv.org/abs/0704.1931v1 mailto:seblouis@nii.ac.jp interaction Hamiltonian of the form [27, 28]: Hint = h̄χσza †a. (1) where a (a†) refers to the annihilation (creation) operator of an electromagnetic field mode in a cavity and the matter qubit is described using the conventional Pauli operators, with the computational basis being given by the eigenstates of σz ≡ |0〉〈0| − |1〉〈1|, with |0〉 ≡ | ↑z〉 and |1〉 ≡ | ↓z〉. The atom- light coupling strength is determined via the parameter χ = g2/∆, where 2g is the vacuum Rabi splitting for the dipole transition and ∆ is the detuning between the dipole transition and the cavity field. The interaction Hint applied for a time t generates a conditional phase-rotation ±θ (with θ = χt) on the field mode dependent upon the state of the matter qubit. We call this a conditional rotation and it is very similar to the cross-Kerr interaction between photons. This time dependent interaction implicitly requires a pulsed probe. Now the interaction in (1) forms the basis for an entangling operation. A two-qubit gate has been proposed [13] based on controlled bus rotations and a subsequent measurement. The probe field coherent state |α〉 interacts with both qubits, so an initial state of the system |Ψi〉 = 12 (|00〉 + |01〉 + |10〉 + |11〉)|α〉 evolves to |Ψf 〉 = (|00〉|αe2iθ〉+ |11〉|αe−2iθ〉) (|01〉+ |10〉)|α〉. (2) Here we quickly observe that the probe field has evolved into three potentially distinct states and appropriate measure- ments can project our two qubits into a number of interest- ing states. At this stage we can choose from different types of measurements on the probe beam. The first and simplest option we have is to perform a homodyne measurement of some field quadrature X(φ) = (a†eiφ + ae−iφ) which for a sufficiently strong local oscillator (compared to the signal strength) implements a projective measurement |x(φ)〉〈x(φ)| on the probe state [29]. The key advantages with homodyne measurement, at least in the optical regime are that it is highly efficient (99% plus [30]) and is a standard tool of continuous variable experimentalists. The simplest homodyne measure- ment to perform is the momentum (P = X(π/2)) quadra- ture. In this case the measurement probability distribution has three peaks with the overlap error between them given by Perr = erfc(α sin θ/ 2). As long as αθ ∼ π this over- lap error is small (< 10−3) and the peaks are well separated. If our P quadrature measurement projects us onto the cen- tral peak |α〉, our two matter qubits are conditioned into the entangled state (|01〉 + |10〉)/ 2. This occurs with a proba- bility of 1/2. Detecting either of the other two side peaks will project the qubits to the known product states |00〉 or |11〉. The probability of entangling the two qubits is interesting in that we have already reached the limits of conventional linear optical implementations. When realistic detector efficiencies (η ∼ 70%) are taken into account, their optimal success prob- ability of 1/2 decreases dramatically (proportional to η or η2 depending on the implementation) and so the probability of the operation succeeding is now significantly less than 1/2. In contrast homodyne measurements are highly efficient and so our success probability will remain very close to 1/2. This limit may be fundamental to the linear optical schemes but in our case we can exceed it by changing the nature of our mea- surement. In principle we could achieve a near deterministic gate if we measured the the position quadrature (X = X(0)), however the requirements to ensure the distinguishability of the probe beam states are much more severe. We could also in principle use a photon number measurement after displacing the probe beam [13], but we would fall back into the issues affecting the linear optical schemes. By restricting ourselves to P = X(π/2) quadrature measurements and single interac- tions between the qubits and the probe, we are opting for the most robust weak-nonlinear approach so far proposed. Within the same framework of conditional rotations and P measurements, one can envisage three qubits interact- ing with a single probe beam. GHZ states are for in- stance one particularly useful state [3]. One way of pro- jecting the qubits onto GHZ-type states is to vary the strength of the interactions between the qubits and the probe beam [12]. Let us represent a rotation of the coherent probe beam by R(θσz) = exp(iθa †aσz). The sequence R(θσz1)R(θσz2)R(−2θσz3)|α〉 which we depict in Fig (1) will give the optimal paths and end points in phase space. The peak centered on the origin will then correspond to the Qubit 1 Qubit 2 Qubit 3 Probe (b)(a) P 011+101 000+111 010+100 FIG. 1: Schematic diagram (a) of a three qubit entangling operation. In (b) the possible probe trajectories caused by the three conditional rotations. There are five different end-states. Upon measurement, three of these will project the qubits to entangled states of interest. GHZ state (|000〉 + |111〉) 2 (after being detected). This will happen with a probability of 1/4 (all qubits started in per- fect superpositions). Next the two peaks having been rotated through ±2θ will correspond to the qubit states (|01〉1,2 + |10〉1,2)|1〉3/ 2 and (|01〉1,2 + |10〉1,2)|0〉3/ 2 respectively. Now in both of these possible outcomes we obtain the same Bell state on qubits 1 and 2, disentangled from qubit 3. So overall we obtain a GHZ state with probability of 1/4 and a Bell state with probability of 1/2 (on two qubits of our choice), heralded by the probe beam P quadrature measurement out- come. The other two outcomes project the qubits to known product states. Consequently, if all we want to do is entangle a pair of qubits, we can now do this with a probability of 3/4. This method can be extended to larger numbers of qubits, but the 3-qubit case minimizes the ratio of operation time over success probability. We shall use this result in the remainder of the paper, observing how current work on the generation of cluster states is simply inadequate for probabilities exceeding 1/2. Until now strategies have been said to be scalable if the resources don’t scale exponentially with the size of the cluster (in general they will scale sub-exponentially). This is a purely theoretical notion which bares little relation to the practical scalability we obtain in our approach. We stress that although the 3-qubit operation is a proba- bilistic entangling operation with different outcomes, these outcomes are heralded by the measurement of the bus and so the operation is a very useful entangling primitive for the con- struction of cluster states. For example, applying it to join two sections of cluster with a third ancillary qubit works with probability 3/4, giving (heralded) outcomes of joined clusters with a new dangling bond (probability 1/2), or joined clusters and two new dangling bonds (probability 1/4). Applying the operation to join three sections of cluster gives (heralded) out- comes of two sections joined and a new dangling bond (prob- ability 1/2), or all three sections joined and two new dangling bonds (probability 1/4). All these outcomes contribute to clus- ter state construction. III. SCALING Now let’s turn to the issue of building up linear cluster states (chains). In order to efficiently grow a chain with probabilis- tic gates, one needs to first inefficiently build small chains ex- ceeding a critical length Lc = 1+2(1−p)/p and then try join- ing them to the main one. This critical length varies between different entangling operations. If an actual conditional phase gate can be immediately implemented, then Lc = 2(1− p)/p for example. Or if this logical gate requires the qubits from the cluster to interact directly (non-distributive approach) then Lc = 4(1− p)/p [31]. Starting from this, and adopting a ‘di- vide and conquer’ approach to building these minimal chains, scaling relations are obtained for the average number of entan- gling operations required and the average time taken, to build a chain of length L. Using our 2-qubit gate (Lc = 3) and these scaling relations we obtain N [L] = 12L− 38. This is already the limiting scenario for simple single photon applications. In the repeat-until-success method [21], for a failure probability of 0.6 (and equal success and insurance probabilities, on all results), the scaling is N [L] = 185L− 1115 and for a failure probability of 0.4 it becomes N [L] ≃ 16.6L − 47.7. Now if we switch to our 3-qubit gate, then Lc < 2 and our mini- mal chain is now simply a 2-qubit cluster (locally equivalent to a Bell state) yielding N [L] = 8L − 44/3. This is a vast improvement over previous proposals. For the two-qubit entangling gate, we essentially stand at the same point as the photonic cluster state approaches. Opti- mizing the resources boils down to finding the optimal strate- gies in combining elements of cluster states. This is a very complex task, which Gross et al. [32] analyzed in great detail. For higher probabilities however, this critical length insur- ing average growth is no longer existent. All previously de- rived strategies become trivial within this probability range. Additional scalable approaches such as sequential adding are at hand and we shall go over the obvious ones. From pre- vious works on generating cluster states [20, 31], we know that the simplest way to grow short chains with probabilistic gates is through a ‘divide and conquer’ approach. It also turns out to be much quicker than a sequential adding, as we allow for many gates to operate in parallel. This technique links up chains of equal length on each round, and discards the chains which failed to do so. In the context of higher success probabilities this approach can be extended to growing large chains in the aim of saving time. The corresponding average number of entangling oper- ations becomes: Ndc[L] = (2/p)log2(L−1) − 1 . (3) From the initial strategy we reach a value linear in L: N [L] = (2/p) L− 1− 2(1− p)/p 1− 2(1− p)/p − 1/p , (4) and a sequential adding yields: Nseq[L] = (L − 1)/(2p− 1) . (5) Obviously the latter represents a considerable saving, as can be verified in Fig. (2). Though the divide and conquer method doesn’t scale linearly, up till lengths of 250 qubits, it requires less entangling operations than the initial scheme (which in fact is a full recycling approach). This is due to the fact that the probabilities we are dealing with are significantly higher than in previous proposals, which were undertaken in two steps, the building of minimal elements and then their merg- ing, in order to be scalable. If we look at the qubit resources however, the less recycling we do, the more qubits we waste in the process. But as the success probability of the gate increases, the recycling strategies all converge with the no- recycling strategy (in terms of qubit resources), this being par- ticularly noticeable for success probabilities higher than 1/2. We can also compare the time scaling of these various strategies, in units of time t corresponding to a single mea- surement. For the complete divide and conquer scheme we simply have: Tdc[L] = t (1 + log2(L− 1)) . (6) and for the initial scheme: T [L] = (t/p) 1 + log2 L− Lc L0 − Lc . (7) For the sequential adding, the cumulative time obeys TL+1 = TL + t/p, and the general form for T becomes: Tseq[L] = t(L− 1)/p . (8) The first two approaches have a logarithmic dependence on the length L, however Tdc is significantly lower as might have sequential Pf=0.6 Pf=0.4 T[L](a) (b) 100 200 300 400 302010 sequential D&C one EO per round initial scheme initial scheme FIG. 2: a) Comparison of entangling operation requirements for chain production. We achieve much lower scalings in comparison with those obtained through the repeat-until-success (RUS) scheme (Pf being the failure probability). b) Time scaling for the different strategies. The three-qubit gate is used in both plots. been expected (see Fig. (2)). Overall we see that there is a clear advantage to divide the task up and to run parallel en- tangling operations. The linear time scaling for the sequential method is due to the fact that operations cannot be undertaken in parallel during the growth. If we didn’t have access to simultaneous entangling operations, the time scaling for the divide and conquer methods would be equivalent to Ndc[L] which is sub-exponential. One needs to keep in mind that by adopting a sequential method, the whole procedure is simpli- fied considerably and would be more accessible to physical implementations. IV. DISCUSSION The cluster state comprises of active regions in which it is being built or measured in the computation (both can be un- dertaken simultaneously) and regions in which the qubits are simply waiting. Now this waiting can be minimized in the building itself, through the appropriate protocols, and in the measurement process. That is, the cluster can be built only a few layers in advance, so that the qubits have less waiting to do, between the building and the actual measurement. In any case, there will be some waiting. Therefore the lowest decoherence realization would be preferred, but it may not be the easiest to manipulate. Thus we may envisage hav- ing two different physical realizations constituting the cluster state. For example, we could use single electron spins ini- tially in building the cluster. Once the links are made between one site and its nearest neighbors, the qubit could be switched into a nuclear spin state which has a significantly longer co- herence time, via a swap operation or some other coherent write and read actions. Most of the waiting would be done in the long-lived state, before being swapped again for the readout [33, 34]. This follows the same scenario as using a second physical system to mediate the interaction and make the measurements, in distributed quantum computing. In the present proposal, we use a continuous variable bus and homo- dyne measurements to generate the links. This physical sys- tem shows itself to be very efficient in this application. Then, for example, electron spins or superconducting charge qubits could be the matter realization interacting with the bus and serving for the final readout. These systems provide the use- ful static aspect required, they interact well with the mediating bus and ensure good single qubit measurements. Finally a low decoherence realization such as nuclear spin could be envis- aged, mainly as a storage medium. The swapping or write and read procedure should have a high fidelity for this storage to be beneficial. On the whole, we see that optimization will depend directly on the physical realization(s) we have cho- sen to work with. For systems with long dephasing times we would give priority to sequential adding approaches, as we have some freedom in the time scaling and thus we can make significant savings in resources. But for realizations with short dephasing times, we would probably want to divide the task up and run operations in parallel, in order to accelerate the fab- rication of the cluster state, at the expense of extra resources. V. CONCLUSION In summary we have shown how the concept of the quan- tum bus can be adapted to efficiently generating cluster states of matter qubits. We can straightforwardly gain access to entangling probabilities higher than 1/2, removing the need to break up the building process into inefficient and efficient parts. This opens up a new class of strategies, for which the resource consumption and the time scaling are consequently vastly improved. Clearly, within this class, detailed strategies can be envisaged and they will depend on the chosen physical realization and the levels of decoherence present. VI. ACKNOWLEDGEMENTS We would like to thank R. van Meter and J. Eisert for help- ful discussions. This work was supported in part by MEXT in Japan and the EU project QAP. [1] R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86, 5188 (2001). [2] M.A. Nielsen, Phys. Rev. Lett. 93, 040503 (2004). [3] D.E. Browne and T. Rudolph, Phys. Rev. Lett. 95, 010501 (2005). [4] G. Gilbert, M. Hamrick, Y.S. Weinstein, Phys. Rev. A 73, 064303 (2006). [5] E. Knill, R. Laflamme and G.J. Milburn, Nature 409, 46 (2001). [6] P. Kok, W.J. Munro , K. Nemoto, T.C. Ralph , J.P. Dowling and G.J. Milburn , Rev. Mod. Phys. 79, 135 (2007). [7] P. Walther et al., Nature 434, 169 (2005). [8] N. Kiesel et al., Phys. Rev. Lett. 95, 210502 (2005). [9] W.J. Munro et al., J. Opt. B: Quantum. Semiclass. Opt. 7, S135 (2005). [10] W.J. Munro et al., New J. Phys. 7, 137 (2005). [11] Kae Nemoto and W. J. Munro, Phys. Lett A 344, 104 (2005). [12] F. Yamaguchi, Kae Nemoto and W. J. Munro, Phys. Rev. A 73, 060302(R) (2006). [13] T. P. Spiller et al., New Journal of Physics 8, 30(2006). [14] S. Bose, P.L. Knight, M.B. Plenio and V. Vedral, Phys. Rev. Lett.83, 5158 (1999). [15] C. Cabrillo, J.I. Cirac, P. Garcia-Fernandez and P. Zoller, Phys. Rev. A 59, 1025 (1999). [16] X.-L. Feng, Z.M. Zhang, X.D. Li, S.Q. Gong and Z.Z. Xu, Phys. Rev. Lett. 90, 217902 (2003). [17] L.-M. Duan and H. J. Kimble, Phys. Rev. Lett. 90, 253601 (2003). [18] D.E. Browne, M. B. Plenio and S.F. Huelga, Phys. Rev. Lett. 91, 067901 (2003). [19] C. Simon and W. T. M. Irvine, Phys. Rev. Lett. 91, 110405(2003). [20] S. D. Barrett and P. Kok, Phys. Rev. A 71, 060310(R) (2005). [21] Y.L. Lim, S.D. Barrett, A. Beige, P. Kok and L.C. Kwek, Phys. Rev. A 73, 012304 (2006). [22] J. M. Raimond, M. Brune and S, Haroche, Rev. Mod. Phys. 73, 565 (2001). [23] F. Jelezko, T. Gaebel, I. Popa, A. Gruber and J. Wrachtrup, Phys. Rev. Lett. 92, 076401 (2004). [24] E. Pazy et al., Europhys. Lett. 62, 175 (2003). [25] J.I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). [26] A. Shnirman, G. Schon and Z. Hermon, Phys. Rev. Lett. 79, 2371 (1997). [27] C. C Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press (28 Oct 2004). [28] A. Blais, R.S. Huang, A. Wallraff, S.M. Girvin and R.J. Schoelkopf, Phys. Rev. A 69, 062320 (2004). [29] Tomas Tyc and Barry C. Sanders, J. Phys. A 37, 7341 (2004). [30] E.S. Polzik, J. Carri, and H.J. Kimble, Phys. Rev. Lett. 68, 3020 (1992). [31] L.-M. Duan and R. Raussendorf, Phys. Rev. Lett. 95, 080503 (2005). [32] D. Gross, K. Kieling and J. Eisert, Phys. Rev. A 74, 042343 (2006). [33] B. E. Kane, Nature 393, 133 (1998). [34] P. van Loock et al., Phys. Rev. Lett. 96, 240501 (2006) .

A discriminating probe of gravity at cosmological scales Pengjie Zhang,1, 2 Michele Liguori,3 Rachel Bean,4 and Scott Dodelson5, 6 1Shanghai Astronomical Observatory, Chinese Academy of Science, 80 Nandan Road, Shanghai, China, 200030 2Joint Institute for Galaxy and Cosmology (JOINGC) of SHAO and USTC 3Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom 4Department of Astronomy, Cornell University, Ithaca, NY 14853 5Center for Particle Astrophysics, Fermi National Accelerator Laboratory, Batavia, IL 60510-0500 6Department of Astronomy & Astrophysics, The University of Chicago, Chicago, IL 60637-1433∗ The standard cosmological model is based on general relativity and includes dark matter and dark energy. An important prediction of this model is a fixed relationship between the gravitational potentials responsible for gravitational lensing and the matter overdensity. Alternative theories of gravity often make different predictions for this relationship. We propose a set of measurements which can test the lensing/matter relationship, thereby distinguishing between dark energy/matter models and models in which gravity differs from general relativity. Planned optical, infrared and radio galaxy and lensing surveys will be able to measure EG, an observational quantity whose expectation value is equal to the ratio of the Laplacian of the Newtonian potentials to the peculiar velocity divergence, to percent accuracy. We show that this will easily separate alternatives such as ΛCDM, DGP, TeVeS and f(R) gravity. PACS numbers: Introduction.— Predictions based on general relativ- ity plus the Standard Model of particle physics are at odds with a variety of independent astronomical obser- vations on galactic and cosmological scales. This failure has led to modifications in particle physics. By introduc- ing dark matter and dark energy, cosmologists have been able to account for a wide range of observations, from the overall expansion of the universe to the large scale structure of the early and late universe [1]. Alternatively, attempts have been made to modify general relativity at galactic [2] or cosmological scales [3, 4]. A fundamental question then arises: Can the two sets of modifications be distinguished from one another? The answer is “No” if only the zero order expansion of the universe is considered. By allowing the dark energy equation of state wDE to be a free function, the expansion history H(z) produced by any modified gravity can be mimicked exactly. Fortunately, struc- ture formation in modified gravities in general differs [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] from that in general relativity. The difference we focus on here is the relationship between gravitational potentials respon- sible for gravitational lensing and the matter overden- sity. Lensing is sensitive to ∇2(φ − ψ) along the line of sight where φ and ψ are the two potentials in the perturbed Friedman-Robertson-Walker metric: ds2 = (1 + 2ψ)dt2 − a2(1 + 2φ)dx2 and a is the scale factor. In standard general relativity (GR), in the absence of anisotropic stresses, φ = −ψ, so lensing is sensitive to ∗Electronic address: pjzhang@shao.ac.cn ∇2φ. The Poisson equation algebraically relates ∇2φ to the fractional overdensity δ, so lensing is essentially de- termined by δ along the line of sight. This is a prediction of the standard, GR-based theory that is generally not obeyed by alternate theories of gravity. Testing this prediction is non-trivial. Astronomers of- ten use the galaxy overdensity as a probe of the underly- ing matter overdensity, but the two are not exactly equal. Here we propose a test of this prediction which is rela- tively insensitive to the problem of galaxy bias. The basic idea is simple: • Extract the matter overdensity at a given redshift by measuring the velocity field. Matter conser- vation relates velocities to the overdensities. The measurement of the velocity field can be accom- plished by studying the anisotropy of the galaxy power spectrum in redshift space. • Extract the lensing signal at this redshift by cross- correlating these galaxies and lensing maps recon- structed from background galaxies. More quantitatively, the galaxy-velocity cross power spectrum Pgθ ≡ −〈δg(k)θ(−k)〉 can be inferred from redshift distortions in a galaxy distribution. Here, θ ≡ ∇ · v/H(z) and v is the comoving peculiar velocity. In the linear regime, matter conservation relates θ to δ by θ = −δ̇/H = −βδ, where β ≡ d lnD/d lna and D is the linear density growth factor. So, Pgθ = βPgδ , sat- isfying the first goal above. Cross correlating the same galaxies with lensing maps constructed from galaxies at higher redshifts, P∇2(φ−ψ)g can be measured. The ra- tio of these two cross-spectra therefore is a direct probe http://arxiv.org/abs/0704.1932v3 mailto:pjzhang@shao.ac.cn of ∇2(φ − ψ)/(βδ). It does not depend on galaxy bias or on the initial matter fluctuations, at least in the lin- ear regime. Modifications in gravity will in general leave signatures in either β and/or the Poisson equation. Galaxy-Velocity Cross-correlation.— A galaxy’s peculiar motion shifts its apparent radial position from xz to x z = xz+vz/H(z) in redshift space, where vz is the comoving radial peculiar velocity. The coherent velocity component changes the galaxy number overdensity from δg to δ g ≃ δg−∇zvz/H(z). Galaxy random motions mix different scales and damps the power spectrum on small scales. The redshift space galaxy power spectrum there- fore has the general form ([17] and references therein) P sg (k) = Pg(k) + 2u 2Pgθ(k) + u 4Pθ(k) k2u2σ2v H2(z) where u = k‖/k is the cosine of the angle of the k vector with respect to radial direction; Pg, Pgθ , Pθ are the real space galaxy power spectra of galaxies, galaxy-θ and θ, respectively; σv is the 1D velocity dispersion; and F (x) is a smoothing function, normalized to unity at x = 0, determined by the velocity probability distribution. This simple formula has passed tests in simulations on scales where δ <∼ 1 [17]. The derivation of Eq. (1) is quite general, so it should be applicable even when gravity is modified. The distinctive dependence of P sg on u allows for si- multaneous determination of Pg, Pgθ and Pθ [18]. The parameters we want to determine are the band powers of Pgθ(k) 1 defined such that P (k) = Pα if kα ≤ k < kα+1, where k1 < k2 < · · · < kα < · · · . We denote P as the band power of Pgθ. For a ki in each k bin, we have a measurement of P sg , which we denote as Pi. The unbiased minimum variance estimator of P 1 Distance D and H are required to translate the observed galaxy angular and redshift separation to k. In general, errors in D and H measurements cause both horizontal and vertical shifts in the EG plot. Both D and H will be measured by methods like type Ia supernovae and baryon acoustic oscillations with 1% accuracy, much smaller than the k bin size adopted, so the horoziontal shift is negligible. Errors in D show up in both Pgθ and the Cκg → P∇2(ψ−φ)g inversion through l = kD and thus largely cancel in evaluating EG. Errors in H(z) only show up in Pgθ measurement and thus cause a net shift in the value of EG. For 1% error in H, the fractional error in EG is ≤ (neff + 3)1% ≤ 3%. Here, neff is the effective power index of the corresponding power spectra. For the fiducial ΛCDM cosmology, it is negative in relevant k range. Thus errors induced by uncertainties in D(z) and H(z) measurement will be sub-dominant, except for SKA, which requires better control over systematic errors in D and H measurement. For simplicity, we neglect this potential error source. Measuring Pgθ also requires to marginalize over σv. However, in the linear regime k <∼ 0.2h/Mpc, k 2σ2v/H 2 ≪ 1 and F (k2u2σ2v/H 2) ≃ 1, for typical value σv ∼ 300 km/s. Thus the exact value of σv is not required for our analysis. Without loss of generality, we adopt σv = 300 km/s. WiPi, where Wi = (λ1 + λ2u i + λ3u i ). Here, Fi ≡ F (kuiσv/H), σ i is the variance of Pi and the three Lagrange multipliers λα (α = 1, 2, 3) is determined by λ = (0, , 0) ·A−1 ; Amn = 2(m+n−2) . (2) Galaxy-galaxy lensing.— Weak lensing is sensitive to the convergence κ, the projected gravitational poten- tial along the line of sight: ∇2(ψ − φ)W (χ, χs)dχ . (3) Here, W is the lensing kernel. For a flat universe, χ, χs are the comoving angular diameter distance to the lens and source, respectively. Eq. 3 is a pure geometric result and can be applied to any modified gravity models where photons follow null geodesics. A standard method to recover the lens redshift infor- mation is by the lensing-galaxy cross correlation. For galaxies in the redshift range [z1, z2], the resulting cross correlation power spectrum under the Limber’s approxi- mation is Cκg(l) = ng(χ)dχ W (χ, χs)ng(χ)P∇2(ψ−φ)g( , z)χ−2dχ W (χ̄, χs) ∫ l/χ1 P∇2(ψ−φ)g(k, z̄)dk fα(l)P Here, χ1,2 are the comoving angular diameter distance to redshift z1,2 and χ̄ is the mean distance. The band power α of P∇2(ψ−φ)g is defined at the same k range as P In practice, we measure the band power Cκg(l,∆l), cen- tered at l with band width ∆l. The weighting fα(l,∆l) is defined correspondingly. For each l, only a fraction of α having fα(l,∆l) 6= 0 contribute. A discriminating probe of gravity.— With the above measurements, one can construct an estimator ÊG = Cκg(l,∆l) 3H20a α fα(l,∆l)P , (5) whose expectation value is 〈ÊG〉 = ∇2(ψ − φ) −3H20a ∇2(ψ − φ) 3H20a ≡ EG . The fractional error on ÊG is 〈∆E2G〉 α〈(δP α fαP , (7) 0.4<z<0.6 LAMOST/AS2+LSST 0.85<z<1.15SKA 0.01 0.1 ADEPT+LSST 1.3<z<1.7 0.01 0.1 1.8<z<2.2SKA FIG. 1: EG as a smoking gun of gravity. Error estimation is based on ΛCDM and error bars are centered on the ΛCDM prediction (black solid straight line). We only show those k modes well in the linear regime. For clarity, we shift the error bars of LAMOST/AS2+LSST and ADEPT+LSST slightly rightward. Irregularities in the error-bars are caused by irreg- ularities in the available discrete k modes of redshift distor- tion. Dotted lines are the results of a flat DGP model with Ω0 = 0.2. Dashed lines are for f(R) = −λ1H 0 exp(−R/λ2H with λ2 = 100. Differences in expansion histories of these models are of percent level at z < 2 and are not the main cause of differences in EG. Solid lines with wiggles are for TeVeS with KB = 0.08, 0.09, 0.1, where the lines with most significant wiggles have KB = 0.1. where ∆C2 = [C2κg + (Cκ + C κ )(Cg + C g )]/(2l∆lfsky). Here, Cκ, C κ , Cg, C g are the power spectra of weak lensing convergence, weak lensing shot noise, galaxy and galaxy shot noise, respectively, and fsky is the fractional sky coverage. Errors on EG at any two adjacent bins are correlated, since they always share some same k modes. However, by requiring lα/χ1 = lα+1/χ2, where l1 < l2 < · · · < lα < · · · and kα = lα/χ2, EG measurement at each l bin only involves two k bins and thus only errors in adjacent bins are correlated. We choose ongoing/proposed spectroscopic surveys LAMOST, AS2, ADEPT and SKA as targets of redshift distortion measurements, and LSST and SKA as targets of lensing map reconstruction. SKA lensing maps can be constructed through cosmic magnification utilizing its unique flux dependence, with S/N comparable to that of LSST through cosmic shear [19]. Survey specifications are summarized in TABLE I. The fiducial cosmology adopted is the ΛCDM cosmology, with the WMAP best TABLE I: Summary of target surveys. redshift deg2 Ngal band operation LAMOSTa z< 0.8 10,000 ∼ 106 optical 2008 AS2b z< 0.8 10,000 ∼ 106 optical ≥ 2009 ADEPTc 1 < z < 2 28,600 ∼ 108 infrared ≥ 2009 SKAd z <∼ 5 22, 000 ∼ 10 9 radio 2020 LSSTe z <∼ 3.5 10,000 ∼ 10 9 optical 2012 ahttp://www.lamost.org/en/ bPrivate communication with Daniel Eisenstein chttp://www7.nationalacademies.org/ssb/BE Nov 2006 bennett.pdf dhttp://www.skatelescope.org/ ehttp://www.lsst.org fit parameters Ω0 = 0.26,ΩΛ = 1 − Ω0, h = 0.72, σ8 = 0.77 and ns = 1. The result is shown in figure 1. In gen- eral, at k < 0.1h/Mpc, cosmic variance in Cκg and Pgθ measurements dominates the EG error budget, result- ing in decreasing error-bars toward larger k. This makes fsky and the lensing source redshifts the two most rele- vant survey parameters for EG error estimation. Since systematic errors in LSST photometric redshifts can be controlled to better than 1%, errors in EG measurements of LAMOST/AS2+LSST and ADEPT+LSST caused by source redshift uncertainties are sub-dominant. We restrict our discussion to sub-horizon scale pertur- bations and express equations hereafter in the Fourier form. Four independent linear equations are required to solve for four perturbation variables δ, θ, ψ and φ. The mass-energy conservation provides two: δ̇ +Hθ = 0 and Ḣθ + 2H2θ − k2ψ/a2 = 0. For at least ΛCDM, quintessence-CDM, DGP and f(R) gravity, the other two takes the general form φ = −η(k, a)ψ , k2(φ− ψ) = 3H20Ω0a −1δG̃eff(k, a) . (8) Here Ω0 is the cosmological matter density in unit of the critical density ρc ≡ 3H 0/8πG. Refer to [14, 15, 16] for other ways of parameterizations. MOND has extra scalar and vector perturbations and does not follow the general form of Eq. 8 [6, 7]. (1) ΛCDM: η = 1, G̃eff = 1 and EG = Ω0/β. Dynam- ical dark energy will have large-scale fluctuations [20]. Furthermore, it may also have non-negligible anisotropic stress and is thus able to mimic modifications in gravity [21]. But, for models with large sound speed and neg- ligible anisotropic stress, such as quintessence, these are negligible at sub-horizon scales and Eq. 8 still holds. (2) Flat DGP: η = [1 − 1/3βDGP]/[1 + 1/3βDGP], G̃eff = 1 [9] and EG = Ω0/β, where βDGP = 1−2rcH(1+ Ḣ/3H2) < 0 and rc = H0/(1−Ω0). Ω0 differs from that of ΛCDM, in order to mimic H(z) of ΛCDM. (3) f(R) gravity: in the sub-horizon limit, G̃eff = (1+fR) −1 [11] and η = 1 [12], with fR ≡ df/dR|B where B denotes the FRW background. This falls naturally out of a conformal transformation of the expression for EG in the Einstein frame into the Jordan frame, noting that Einstein frame scalar field fluctuations are negligible on sub-horizon scales [12]. We numerically solve the full per- turbation equations in the Einstein frame since it is com- putationally simpler [12] and then conformally transform to the Jordan frame, which we choose as the physical frame, evaluating β such that EG = Ω0/(1+fR)β. In the limit that fR → 0, e.g. for f(R) ∼ λ1H 0 exp(−R/λ2H [11] with λ1 ≪ λ2, the evolution is observationally equiv- alent to ΛCDM. For modes that entered the horizon prior to matter-radiation equality, as we consider here, β, and therefore EG, is scale invariant for IR modifications to gravity, with fR > 0. 2 The scale independence of EG holds in ΛCDM, Quintessence-CDM and DGP. An ob- served scale-independent deviation in EG from ΛCDM could signify a special class of modified gravity, as shown in Fig. 1. (4) TeVeS/MOND. Besides the gravitational met- ric, TeVeS [2] contains a scalar and a vector field. These new fields act as sources for the gravitational potential φ in the modified Poisson equation and can change the evolution of cosmological perturbations with respect to standard gravity [6, 7]. We considered a TeVeS model with Ωb = 0.05, Ων = 0.17, ΩΛ = 0.78 and we adopted a choice of the TeVeS parameters that produces a signifi- cant enhancement of the growth factor. The TeVeS EG is significantly different from the standard EG (Fig. 1). It exhibits scale dependence with accompanying baryon acoustic wiggles. Both features are due to the vector field fluctuations, which play a significant role in struc- ture formation [7]. These fluctuations decrease toward small scales and cause the scale dependency of EG. We also checked that they affect the final shape of the acous- tic oscillations of the other components significantly. As a result, oscillations in φ, ψ and δ do not cancel out per- fectly in TeVeS when we take the ratio, thus producing the wiggles in EG. For the four gravity models investigated, differences in EG are much larger than observational statistical un- certainties. Planned surveys are promising to detect per- cent level deviation from GR and should distinguish these modified gravity models unambiguously. 2 Scales larger than the horizon at matter-radiation equality are suppressed [12] and, if measurable, would have a scale dependent increase in the value of EG in comparison to the small scale value. 3 To simplify the numerical treatment of the TeVeS perturbations equations while retaining a good qualitative description of all the significant physical effects at the same time, we introduced sev- eral approximations. Namely we assumed instantaneous recom- bination and employed the tight coupling approximation between baryons and photons at all scales before decoupling; moreover we evolved perturbations in the massive neutrino component in a simplified way by switching off neutrinos perturbations when they were below the free steaming scale and treating them as a fluid above the free streaming scale. At large scales, gravity is the only force determining the acceleration of galaxies and dark matter particles. So we assumed no galaxy velocity bias. As statistical errors in EG measurements reach the 1% level (Fig. 1), sev- eral systematics, besides the one discussed in footnote 1, may become non-negligible. One is the accuracy of the redshift distortion formula (Eq. 1), which may be prob- lematic for those modes with large u, even at very linear scales [17]. A remedy is to exclude them when extracting Pgθ, at the expense of statistical accuracy. As discussed before, accuracy of EG measurement is dominated by accuracy of P∇2(ψ−φ)g measurements and is thus less af- fected. A less severe one is the nonlinear evolution, which becomes non-negligible where the matter power spectrum variance ∆2m ∼ 0.1. In general relativity, nonlinear cor- rections to density and velocity differ (Fig. 12, [22]). A direct consequence is that EG develops a dependence on the matter power spectrum. Similar effects in modified gravity models are expected. This can be corrected by high order perturbation calculations, which should work well where ∆2m ∼ 0.2. We thank R. Caldwell, D. Eisenstein, B. Jain, M. Kunz, J. Ostriker and J.P. Uzan for useful discussions and the anonymous referees for useful suggestions. PJZ is supported by the National Science Foundation of China grant 10533030 and CAS grant KJCX3-SYW-N2. RB’s work is supported by the National Science Foundation grants AST-0607018 and PHY-0555216. SD is supported by the US Department of Energy. 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The standard cosmological model is based on general relativity and includes dark matter and dark energy. An important prediction of this model is a fixed relationship between the gravitational potentials responsible for gravitational lensing and the matter overdensity. Alternative theories of gravity often make different predictions for this relationship. We propose a set of measurements which can test the lensing/matter relationship, thereby distinguishing between dark energy/matter models and models in which gravity differs from general relativity. Planned optical, infrared and radio galaxy and lensing surveys will be able to measure $E_G$, an observational quantity whose expectation value is equal to the ratio of the Laplacian of the Newtonian potentials to the peculiar velocity divergence, to percent accuracy. We show that this will easily separate alternatives such as $\Lambda$CDM, DGP, TeVeS and $f(R)$ gravity.

Introduction.— Predictions based on general relativ- ity plus the Standard Model of particle physics are at odds with a variety of independent astronomical obser- vations on galactic and cosmological scales. This failure has led to modifications in particle physics. By introduc- ing dark matter and dark energy, cosmologists have been able to account for a wide range of observations, from the overall expansion of the universe to the large scale structure of the early and late universe [1]. Alternatively, attempts have been made to modify general relativity at galactic [2] or cosmological scales [3, 4]. A fundamental question then arises: Can the two sets of modifications be distinguished from one another? The answer is “No” if only the zero order expansion of the universe is considered. By allowing the dark energy equation of state wDE to be a free function, the expansion history H(z) produced by any modified gravity can be mimicked exactly. Fortunately, struc- ture formation in modified gravities in general differs [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] from that in general relativity. The difference we focus on here is the relationship between gravitational potentials respon- sible for gravitational lensing and the matter overden- sity. Lensing is sensitive to ∇2(φ − ψ) along the line of sight where φ and ψ are the two potentials in the perturbed Friedman-Robertson-Walker metric: ds2 = (1 + 2ψ)dt2 − a2(1 + 2φ)dx2 and a is the scale factor. In standard general relativity (GR), in the absence of anisotropic stresses, φ = −ψ, so lensing is sensitive to ∗Electronic address: pjzhang@shao.ac.cn ∇2φ. The Poisson equation algebraically relates ∇2φ to the fractional overdensity δ, so lensing is essentially de- termined by δ along the line of sight. This is a prediction of the standard, GR-based theory that is generally not obeyed by alternate theories of gravity. Testing this prediction is non-trivial. Astronomers of- ten use the galaxy overdensity as a probe of the underly- ing matter overdensity, but the two are not exactly equal. Here we propose a test of this prediction which is rela- tively insensitive to the problem of galaxy bias. The basic idea is simple: • Extract the matter overdensity at a given redshift by measuring the velocity field. Matter conser- vation relates velocities to the overdensities. The measurement of the velocity field can be accom- plished by studying the anisotropy of the galaxy power spectrum in redshift space. • Extract the lensing signal at this redshift by cross- correlating these galaxies and lensing maps recon- structed from background galaxies. More quantitatively, the galaxy-velocity cross power spectrum Pgθ ≡ −〈δg(k)θ(−k)〉 can be inferred from redshift distortions in a galaxy distribution. Here, θ ≡ ∇ · v/H(z) and v is the comoving peculiar velocity. In the linear regime, matter conservation relates θ to δ by θ = −δ̇/H = −βδ, where β ≡ d lnD/d lna and D is the linear density growth factor. So, Pgθ = βPgδ , sat- isfying the first goal above. Cross correlating the same galaxies with lensing maps constructed from galaxies at higher redshifts, P∇2(φ−ψ)g can be measured. The ra- tio of these two cross-spectra therefore is a direct probe http://arxiv.org/abs/0704.1932v3 mailto:pjzhang@shao.ac.cn of ∇2(φ − ψ)/(βδ). It does not depend on galaxy bias or on the initial matter fluctuations, at least in the lin- ear regime. Modifications in gravity will in general leave signatures in either β and/or the Poisson equation. Galaxy-Velocity Cross-correlation.— A galaxy’s peculiar motion shifts its apparent radial position from xz to x z = xz+vz/H(z) in redshift space, where vz is the comoving radial peculiar velocity. The coherent velocity component changes the galaxy number overdensity from δg to δ g ≃ δg−∇zvz/H(z). Galaxy random motions mix different scales and damps the power spectrum on small scales. The redshift space galaxy power spectrum there- fore has the general form ([17] and references therein) P sg (k) = Pg(k) + 2u 2Pgθ(k) + u 4Pθ(k) k2u2σ2v H2(z) where u = k‖/k is the cosine of the angle of the k vector with respect to radial direction; Pg, Pgθ , Pθ are the real space galaxy power spectra of galaxies, galaxy-θ and θ, respectively; σv is the 1D velocity dispersion; and F (x) is a smoothing function, normalized to unity at x = 0, determined by the velocity probability distribution. This simple formula has passed tests in simulations on scales where δ <∼ 1 [17]. The derivation of Eq. (1) is quite general, so it should be applicable even when gravity is modified. The distinctive dependence of P sg on u allows for si- multaneous determination of Pg, Pgθ and Pθ [18]. The parameters we want to determine are the band powers of Pgθ(k) 1 defined such that P (k) = Pα if kα ≤ k < kα+1, where k1 < k2 < · · · < kα < · · · . We denote P as the band power of Pgθ. For a ki in each k bin, we have a measurement of P sg , which we denote as Pi. The unbiased minimum variance estimator of P 1 Distance D and H are required to translate the observed galaxy angular and redshift separation to k. In general, errors in D and H measurements cause both horizontal and vertical shifts in the EG plot. Both D and H will be measured by methods like type Ia supernovae and baryon acoustic oscillations with 1% accuracy, much smaller than the k bin size adopted, so the horoziontal shift is negligible. Errors in D show up in both Pgθ and the Cκg → P∇2(ψ−φ)g inversion through l = kD and thus largely cancel in evaluating EG. Errors in H(z) only show up in Pgθ measurement and thus cause a net shift in the value of EG. For 1% error in H, the fractional error in EG is ≤ (neff + 3)1% ≤ 3%. Here, neff is the effective power index of the corresponding power spectra. For the fiducial ΛCDM cosmology, it is negative in relevant k range. Thus errors induced by uncertainties in D(z) and H(z) measurement will be sub-dominant, except for SKA, which requires better control over systematic errors in D and H measurement. For simplicity, we neglect this potential error source. Measuring Pgθ also requires to marginalize over σv. However, in the linear regime k <∼ 0.2h/Mpc, k 2σ2v/H 2 ≪ 1 and F (k2u2σ2v/H 2) ≃ 1, for typical value σv ∼ 300 km/s. Thus the exact value of σv is not required for our analysis. Without loss of generality, we adopt σv = 300 km/s. WiPi, where Wi = (λ1 + λ2u i + λ3u i ). Here, Fi ≡ F (kuiσv/H), σ i is the variance of Pi and the three Lagrange multipliers λα (α = 1, 2, 3) is determined by λ = (0, , 0) ·A−1 ; Amn = 2(m+n−2) . (2) Galaxy-galaxy lensing.— Weak lensing is sensitive to the convergence κ, the projected gravitational poten- tial along the line of sight: ∇2(ψ − φ)W (χ, χs)dχ . (3) Here, W is the lensing kernel. For a flat universe, χ, χs are the comoving angular diameter distance to the lens and source, respectively. Eq. 3 is a pure geometric result and can be applied to any modified gravity models where photons follow null geodesics. A standard method to recover the lens redshift infor- mation is by the lensing-galaxy cross correlation. For galaxies in the redshift range [z1, z2], the resulting cross correlation power spectrum under the Limber’s approxi- mation is Cκg(l) = ng(χ)dχ W (χ, χs)ng(χ)P∇2(ψ−φ)g( , z)χ−2dχ W (χ̄, χs) ∫ l/χ1 P∇2(ψ−φ)g(k, z̄)dk fα(l)P Here, χ1,2 are the comoving angular diameter distance to redshift z1,2 and χ̄ is the mean distance. The band power α of P∇2(ψ−φ)g is defined at the same k range as P In practice, we measure the band power Cκg(l,∆l), cen- tered at l with band width ∆l. The weighting fα(l,∆l) is defined correspondingly. For each l, only a fraction of α having fα(l,∆l) 6= 0 contribute. A discriminating probe of gravity.— With the above measurements, one can construct an estimator ÊG = Cκg(l,∆l) 3H20a α fα(l,∆l)P , (5) whose expectation value is 〈ÊG〉 = ∇2(ψ − φ) −3H20a ∇2(ψ − φ) 3H20a ≡ EG . The fractional error on ÊG is 〈∆E2G〉 α〈(δP α fαP , (7) 0.4<z<0.6 LAMOST/AS2+LSST 0.85<z<1.15SKA 0.01 0.1 ADEPT+LSST 1.3<z<1.7 0.01 0.1 1.8<z<2.2SKA FIG. 1: EG as a smoking gun of gravity. Error estimation is based on ΛCDM and error bars are centered on the ΛCDM prediction (black solid straight line). We only show those k modes well in the linear regime. For clarity, we shift the error bars of LAMOST/AS2+LSST and ADEPT+LSST slightly rightward. Irregularities in the error-bars are caused by irreg- ularities in the available discrete k modes of redshift distor- tion. Dotted lines are the results of a flat DGP model with Ω0 = 0.2. Dashed lines are for f(R) = −λ1H 0 exp(−R/λ2H with λ2 = 100. Differences in expansion histories of these models are of percent level at z < 2 and are not the main cause of differences in EG. Solid lines with wiggles are for TeVeS with KB = 0.08, 0.09, 0.1, where the lines with most significant wiggles have KB = 0.1. where ∆C2 = [C2κg + (Cκ + C κ )(Cg + C g )]/(2l∆lfsky). Here, Cκ, C κ , Cg, C g are the power spectra of weak lensing convergence, weak lensing shot noise, galaxy and galaxy shot noise, respectively, and fsky is the fractional sky coverage. Errors on EG at any two adjacent bins are correlated, since they always share some same k modes. However, by requiring lα/χ1 = lα+1/χ2, where l1 < l2 < · · · < lα < · · · and kα = lα/χ2, EG measurement at each l bin only involves two k bins and thus only errors in adjacent bins are correlated. We choose ongoing/proposed spectroscopic surveys LAMOST, AS2, ADEPT and SKA as targets of redshift distortion measurements, and LSST and SKA as targets of lensing map reconstruction. SKA lensing maps can be constructed through cosmic magnification utilizing its unique flux dependence, with S/N comparable to that of LSST through cosmic shear [19]. Survey specifications are summarized in TABLE I. The fiducial cosmology adopted is the ΛCDM cosmology, with the WMAP best TABLE I: Summary of target surveys. redshift deg2 Ngal band operation LAMOSTa z< 0.8 10,000 ∼ 106 optical 2008 AS2b z< 0.8 10,000 ∼ 106 optical ≥ 2009 ADEPTc 1 < z < 2 28,600 ∼ 108 infrared ≥ 2009 SKAd z <∼ 5 22, 000 ∼ 10 9 radio 2020 LSSTe z <∼ 3.5 10,000 ∼ 10 9 optical 2012 ahttp://www.lamost.org/en/ bPrivate communication with Daniel Eisenstein chttp://www7.nationalacademies.org/ssb/BE Nov 2006 bennett.pdf dhttp://www.skatelescope.org/ ehttp://www.lsst.org fit parameters Ω0 = 0.26,ΩΛ = 1 − Ω0, h = 0.72, σ8 = 0.77 and ns = 1. The result is shown in figure 1. In gen- eral, at k < 0.1h/Mpc, cosmic variance in Cκg and Pgθ measurements dominates the EG error budget, result- ing in decreasing error-bars toward larger k. This makes fsky and the lensing source redshifts the two most rele- vant survey parameters for EG error estimation. Since systematic errors in LSST photometric redshifts can be controlled to better than 1%, errors in EG measurements of LAMOST/AS2+LSST and ADEPT+LSST caused by source redshift uncertainties are sub-dominant. We restrict our discussion to sub-horizon scale pertur- bations and express equations hereafter in the Fourier form. Four independent linear equations are required to solve for four perturbation variables δ, θ, ψ and φ. The mass-energy conservation provides two: δ̇ +Hθ = 0 and Ḣθ + 2H2θ − k2ψ/a2 = 0. For at least ΛCDM, quintessence-CDM, DGP and f(R) gravity, the other two takes the general form φ = −η(k, a)ψ , k2(φ− ψ) = 3H20Ω0a −1δG̃eff(k, a) . (8) Here Ω0 is the cosmological matter density in unit of the critical density ρc ≡ 3H 0/8πG. Refer to [14, 15, 16] for other ways of parameterizations. MOND has extra scalar and vector perturbations and does not follow the general form of Eq. 8 [6, 7]. (1) ΛCDM: η = 1, G̃eff = 1 and EG = Ω0/β. Dynam- ical dark energy will have large-scale fluctuations [20]. Furthermore, it may also have non-negligible anisotropic stress and is thus able to mimic modifications in gravity [21]. But, for models with large sound speed and neg- ligible anisotropic stress, such as quintessence, these are negligible at sub-horizon scales and Eq. 8 still holds. (2) Flat DGP: η = [1 − 1/3βDGP]/[1 + 1/3βDGP], G̃eff = 1 [9] and EG = Ω0/β, where βDGP = 1−2rcH(1+ Ḣ/3H2) < 0 and rc = H0/(1−Ω0). Ω0 differs from that of ΛCDM, in order to mimic H(z) of ΛCDM. (3) f(R) gravity: in the sub-horizon limit, G̃eff = (1+fR) −1 [11] and η = 1 [12], with fR ≡ df/dR|B where B denotes the FRW background. This falls naturally out of a conformal transformation of the expression for EG in the Einstein frame into the Jordan frame, noting that Einstein frame scalar field fluctuations are negligible on sub-horizon scales [12]. We numerically solve the full per- turbation equations in the Einstein frame since it is com- putationally simpler [12] and then conformally transform to the Jordan frame, which we choose as the physical frame, evaluating β such that EG = Ω0/(1+fR)β. In the limit that fR → 0, e.g. for f(R) ∼ λ1H 0 exp(−R/λ2H [11] with λ1 ≪ λ2, the evolution is observationally equiv- alent to ΛCDM. For modes that entered the horizon prior to matter-radiation equality, as we consider here, β, and therefore EG, is scale invariant for IR modifications to gravity, with fR > 0. 2 The scale independence of EG holds in ΛCDM, Quintessence-CDM and DGP. An ob- served scale-independent deviation in EG from ΛCDM could signify a special class of modified gravity, as shown in Fig. 1. (4) TeVeS/MOND. Besides the gravitational met- ric, TeVeS [2] contains a scalar and a vector field. These new fields act as sources for the gravitational potential φ in the modified Poisson equation and can change the evolution of cosmological perturbations with respect to standard gravity [6, 7]. We considered a TeVeS model with Ωb = 0.05, Ων = 0.17, ΩΛ = 0.78 and we adopted a choice of the TeVeS parameters that produces a signifi- cant enhancement of the growth factor. The TeVeS EG is significantly different from the standard EG (Fig. 1). It exhibits scale dependence with accompanying baryon acoustic wiggles. Both features are due to the vector field fluctuations, which play a significant role in struc- ture formation [7]. These fluctuations decrease toward small scales and cause the scale dependency of EG. We also checked that they affect the final shape of the acous- tic oscillations of the other components significantly. As a result, oscillations in φ, ψ and δ do not cancel out per- fectly in TeVeS when we take the ratio, thus producing the wiggles in EG. For the four gravity models investigated, differences in EG are much larger than observational statistical un- certainties. Planned surveys are promising to detect per- cent level deviation from GR and should distinguish these modified gravity models unambiguously. 2 Scales larger than the horizon at matter-radiation equality are suppressed [12] and, if measurable, would have a scale dependent increase in the value of EG in comparison to the small scale value. 3 To simplify the numerical treatment of the TeVeS perturbations equations while retaining a good qualitative description of all the significant physical effects at the same time, we introduced sev- eral approximations. Namely we assumed instantaneous recom- bination and employed the tight coupling approximation between baryons and photons at all scales before decoupling; moreover we evolved perturbations in the massive neutrino component in a simplified way by switching off neutrinos perturbations when they were below the free steaming scale and treating them as a fluid above the free streaming scale. At large scales, gravity is the only force determining the acceleration of galaxies and dark matter particles. So we assumed no galaxy velocity bias. As statistical errors in EG measurements reach the 1% level (Fig. 1), sev- eral systematics, besides the one discussed in footnote 1, may become non-negligible. One is the accuracy of the redshift distortion formula (Eq. 1), which may be prob- lematic for those modes with large u, even at very linear scales [17]. A remedy is to exclude them when extracting Pgθ, at the expense of statistical accuracy. As discussed before, accuracy of EG measurement is dominated by accuracy of P∇2(ψ−φ)g measurements and is thus less af- fected. A less severe one is the nonlinear evolution, which becomes non-negligible where the matter power spectrum variance ∆2m ∼ 0.1. In general relativity, nonlinear cor- rections to density and velocity differ (Fig. 12, [22]). A direct consequence is that EG develops a dependence on the matter power spectrum. Similar effects in modified gravity models are expected. This can be corrected by high order perturbation calculations, which should work well where ∆2m ∼ 0.2. We thank R. Caldwell, D. Eisenstein, B. Jain, M. Kunz, J. Ostriker and J.P. Uzan for useful discussions and the anonymous referees for useful suggestions. PJZ is supported by the National Science Foundation of China grant 10533030 and CAS grant KJCX3-SYW-N2. RB’s work is supported by the National Science Foundation grants AST-0607018 and PHY-0555216. SD is supported by the US Department of Energy. 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THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS OF QUADRATIC DIFFERENTIALS JONATHAN TSAI Abstract. We obtain a first order differential equation for the driving func- tion of the chordal Loewner differential equation in the case where the domain is slit by a curve which is a trajectory arc of certain quadratic differentials. In particular this includes the case when the curve is a path on the square, trian- gle or hexagonal lattice in the upper half-plane or, indeed, in any domain with boundary on the lattice. We also demonstrate how we use this to calculate the driving function numerically. Equivalent results for other variants of the Loewner differential equation are also obtained: Multiple slits in the chordal Loewner differential equation and the radial Loewner differential equation. The proof of our theorem uses a generalization of Schwarz-Christoffel map- ping to domains bounded by trajectory arcs of rotations of a given quadratic differential. Introduction Suppose that H = {z ∈ C : Im(z) > 0} is the upper half-plane and γ : [0, T ) 7→ H is a simple Jordan curve with γ(0) ∈ R and γ(0, T ) = {γ(t) : t ∈ (0, T )} ⊂ H. Then for each t ∈ (0, T ), Ht = H \ γ(0, t] is a simply-connected domain and hence by the Riemann mapping theorem, we can find a conformal map ft of H onto Ht. Moreover, we can require that ft has series expansion ft(z) = z − as z → ∞. Normalized in this way ft is unique and is said to be hydrodynamically normalized. The function C(t) is positive, continuous and strictly increasing: it is called the half- plane capacity of γ(0, t]. Thus we can reparameterize γ such that C(t) = 2t for all t, we will call this parameterization by half-plane capacity. With this normalization and parameterization, the function ft satisfies the differential equation (where f denotes differentiation with respect to z and ḟt denotes differentiation with respect to t): (0.1) ḟt(z) = − 2f ′t(z) z − ξ(t) , where ξ(t) = f−1t (γ(t)) is a continuous real-valued function. This is the chordal Loewner differential equation; ξ(t) is called the driving function of the slit γ. The converse is also true: given a measurable function ξ, the differential equation (0.1) with initial condition f0(z) ≡ z has solution ft which is a conformal map from H 2000 Mathematics Subject Classification. Primary 30C20; Secondary 30C30, 60K35 . http://arxiv.org/abs/0704.1933v3 2 J. TSAI −4 −2 0 2 4 6 8 0 100 200 300 400 500 600 Figure 1. A path on the hexagonal lattice on the upper half-plane (left) and a plot of its driving function on the y-axis against time on the x-axis (right). −2 0 2 4 6 8 10 12 0 50 100 150 200 250 Figure 2. A path on the square lattice on the upper half-plane (left) and a plot of its driving function on the y-axis against time on the x-axis (right). into itself (although f(H) is not necessarily a slit domain). Chapter 3 and 4 of [7] gives full details of this construction. Since Schramm’s discovery of stochastic Loewner evolution in 1999 (see [15]), there has been huge interest in the chordal Loewner differential equation and its variants. But the relationship between the slit inH and its resulting driving function is not well understood. There are a few papers that relate the behaviour of the slit with the behaviour of the driving function e.g. [10],[8]; also, the paper [2] calculates the slit arising from a few driving functions. In this paper, we will obtain a first order differential equation for ξ (which we can then solve numerically) that allows us to calculate the driving function ξ in the case where the curve γ is a trajectory arc of a certain type of quadratic differential. We will show that this includes, for example, the case when γ is a path on the square/triangle/hexagonal lattice in the upper half-plane or indeed, in any domain whose boundary lies on such a lattice. So for example, Figure 1 plots the driving function of a path on the hexagonal lattice in the upper half-plane and Figure 2 plots the driving function of a path on the square lattice in the upper half-plane. THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 3 We also note that we can obtain equivalent results for other variants of the Loewner differential equation for example, in the radial version or with multiple slits. We will discuss this in the paper as well. The proof of our formulae uses a generalization of Schwarz-Christoffel mapping to domains bounded by trajectory arcs of rotations of a given quadratic differential. We also mention that, currently, the common method used to find the driving function of a given slit is to use the Zipper algorithm discovered independently by D. E. Marshall and R. Kühnau to approximate the function ft which can then be used to determine the driving function. The Zipper algorithm can be viewed as a discrete version of the Loewner differential equation and hence is well suited to studying growth processes. It also has the advantage of being very fast. See [11] and [3]. 1. Main results To state our main results, we have to provide some background in the theory of quadratic differentials. Note that not all the terms used here are standard in the literature. See Chapter 8 of [13] and [16] for more details. A quadratic differential on a domain D ⊂ Ĉ = C ∪ {∞} is the formal expression Q(z)dz2, where Q(z) is a meromorphic function on D. Then for ω ∈ D with ω 6= ∞, Q(z) has Laurent series expansion about ω, Q(z) = ak(z − ω)k for some n > −∞ with an 6= 0. Then we define the degree of ω with respect to Q(z)dz2, degQ(ω), to be equal to n. If ∞ ∈ D, then near ∞, Q has Laurent series expansion given by Q(z) = then we define the degree of ∞ with respect to Q(z)dz2, degQ(∞) to be equal to m − 4. The “4” in the definition ensures that the degree is conformally invariant in a way which we will make precise later. Then ω ∈ D is: • a zero of Q(z)dz2 if degQ(ω) > 0. • a pole of Q(z)dz2 if degQ(ω) < 0. • an ordinary point of Q(z)dz2 if degQ(ω) = 0. A trajectory arc of Q(z)dz2 is a curve γ : (a, b) 7→ D that does not meet any zeroes and poles of Q(z)dz2 and satisfies Q(γ(t))γ̇(t)2 > 0 for all t ∈ (a, b). For θ ∈ [0, π), a θ-trajectory arc of Q(z)dz2 is a curve γ : (a, b) 7→ D that satisfies arg[Q(γ(t))γ̇(t)2] = 2θ for all t ∈ (a, b). Then γ is a θ-trajectory arc of Q(z)dz2 if and only if it is a trajectory arc of e−2iθQ(z)dz2. Hence, a 0-trajectory arc is simply a trajectory arc and we call a π/2-trajectory arc an orthogonal trajectory arc. It is clear that these definitions are invariant under reparameterization of γ so we will often call the point set of γ 4 J. TSAI a trajectory arc or θ-trajectory arc. We call a maximal trajectory arc a trajectory and similarly, a maximal θ-trajectory arc is called a θ-trajectory. For example, if we consider the quadratic differential 1dz2 in C, then the θ-trajectories are the straight lines with gradient exp(2θ). We now consider a special type of quadratic differential: Let D be a domain with piecewise analytic boundary. A Kühnau quadratic differential is a quadratic differential, Q(z)dz2, on D satisfying the following two properties: Definition: Let D be a domain with piecewise analytic boundary. A Kühnau qua- dratic differential is a quadratic differential, Q(z)dz2, on D satisfying the following two properties: (1) We can write such that each Γj is an open analytic arc with Γk ∩ Γj = ∅ for k 6= j and moreover, Q(z) extends continuously to each Γj and arg[Q(z)dz 2] is constant on each Γj i.e. each Γj is a θj-trajectory arc for some θj ∈ [0, π). (2) At z ∈ Γk ∩ Γj for all j 6= k, there are either only finitely many direction from which trajectories approach the point z or if there are infinitely many directions from which trajectories approach the point z, then for each such direction, there is only one trajectory that approaches z at this direction. These quadratic differentials are studied by Kühnau in [6] where he applies them to the study of certain Grötzsch-style extremal problems Property (i) above, also implies that D is locally connected. Thus each prime end of D corresponds to a unique point in ∂D (see [14, p. 27]). If, in addition, a point on ∂D corresponds to a unique prime end, then we make no distinction between the two. Let z be a prime end of D. Then we have 2 cases: Either z ∈ Γj for some j = 1, . . . , n; or there exist exactly 2 of the (Γj), that end at the prime end z. In the latter case, we will denote z by zk and assume that Γk, a θk-trajectory arc, and Γk−1, a θk−1-trajectory arc, are the only 2 arcs that end at zk. Then we can define the degree of zk in D with respect to Q(z)dz 2, degD,Q(zk), as follows: degD,Q(zk) = 2[|θk − θk−1|/π + Jk − 1] if θk 6= θk−1, 2Jk if θk = θk−1, where Jk is the number of trajectories of Q(z)dz 2 inside D that end at the prime end zk. If Jk is infinite, then the degree is not defined. Then for prime ends z such that z ∈ Γj for some j = 1, . . . , n, we define degD,Q(z) = 0. Although the motivation for this definition currently seems unclear, we will see that this indeed generalizes the concept of degree to points on the boundary. In particular, we will show that for x ∈ ∂H, if deg H,Q(x) ∈ Z, then Q can be extended to a meromorphic function in a neighbourhood of x with H,Q(x) = degQ(x). We then have the following theorem on Kühnau quadratic differentials in H: THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 5 Theorem 1.1. Suppose that Q(z)dz2 is a Kühnau quadratic differential on H. Then we have Q(z) = R (z − ζj)λj for some constant R 6= 0, ζj ∈ C, λj ∈ R for j = 1, . . . , n. This theorem can be viewed as a generalization of the Schwarz-Christoffel for- mula to domains bounded by θk-trajectory arcs of a given quadratic differential. We then have the following theorem on the Loewner driving function of a φ- trajectory arc of a Kühnau quadratic differential Q(z)dz2 that starts at a point ξ0 ∈ R with degH,Q(ξ0) = N ∈ {0, 1, . . .}. Theorem 1.2. Suppose that Q(z)dz2 is a Kühnau quadratic differential on H such that there is a point ξ0 ∈ R with degH,Q(ξ0) = N ∈ {0, 1, . . .}; then we have Q(w) = (w − ξ0)N (w − aj)αj where aj ∈ C and αj ∈ R. Let γ : [0, T ) 7→ ∂H be a simple curve such that γ(0) = ξ0, γ(0, T ) ⊂ H and γ(0, T ) is a φ-trajectory arc of Q(z)dz2 (φ ∈ [0, π)) that is parameterized by half-plane capacity. Suppose that the functions ft maps H conformally onto H \ γ(0, t] and are hydrodynamically normalized. Then for t ∈ (0, T ) (1.1) 2ξ(t) = −µ−C−(t)− µ+C+(t)− αjAj(t) +Σ0, (1.2) ξ̇(t) = − µ C−(t)− ξ(t) − C+(t)− ξ(t) − Aj(t)− ξ(t) with initial condition ξ(0) = ξ0. Where the functions Aj(t) are defined by Aj(t) = f t (aj) for j = 1, . . . , n, and C+(t) > C−(t) are the two preimages of ξ0 under ft; µ± = deg H\γ(0,t],Q(ft(C ±(t))), Nξ0 + We can then use Theorem 1.2 to find the driving function in the case when the slit γ consists of consecutive θk-trajectory arcs of given quadratic differentials. We will explain how to do this in further detail later. One difficulty with using Theorem 1.2 is that the parameterization is inherently given in terms of half-plane capacity. This makes it difficult to calculate the driving function ξ if we do not know anything about the half-plane capacity of the trajectory arc (which, in general, is the case). The next theorem will allow us to compare the parametrization with the length of the slit: 6 J. TSAI Theorem 1.3. Suppose that Q(z)dz2, γ and ft are as defined in Theorem 1.2. Let Φt(z) = Q(ft(z))f (z − ξ(t))2 . Then γ satisfies (1.3) γ̇(t) = −2 Φt(ξ(t)) Q(γ(t)) The rest of this paper is organized as follows: In the Section 2, we will state some basic results from the theory of quadratic differentials and use them to prove Theorem 1.1. Then we will use Theorem 1.1 to prove Theorems 1.2 and 1.3 in Sec- tion 3. In Section 4 we will discuss how to obtain the driving function numerically using Theorems 1.2 and 1.3. Finally in Section 5, we will discuss extensions of The- orem 1.2 to the case with multiple slits as well as to the radial Loewner differential equation. 2. Kühnau quadratic differentials and generalized Schwarz-Christoffel mapping The aim of this section is to prove Theorem 1.1. We will first look at some of the basic results in the theory quadratic differentials that we will need. Transformation Law. Suppose that f is a conformal map from a domain D2 onto a domain D1 and suppose that Q1(w)dw 2 is a quadratic differential on D1. If we define (2.1) Q2(z) ≡ Q1(f(z))f ′(z)2 then Q2(z)dz 2 is a quadratic differential on D2. Then, it is clear that θ-trajectory arcs are preserved by this transformation law i.e. γ is a θ-trajectory arc of Q2(z)dz 2 ⇔ f ◦ γ is a θ-trajectory arc of Q2(w)dw2 , and also, for z ∈ D2 degQ2(z) = degQ1(f(z)). Hence trajectories and degQ are conformally invariant in the above sense. The following lemma tells us that the behaviour of a quadratic differential at a neighbourhood of a point is determined by the degree of that point. Lemma 2.1 (Local behaviour of quadratic differentials). Let Q(z)dz2 be a qua- dratic differential on a domain D. Then for every ω ∈ D there is a conformal mapping w = φ(z) of some neighbourhood of ω such that Q(z)dz2 =   dw2 if degQ(ω) = 0, wndw2 if degQ(ω) = n ≥ 1, w−ndw2 if degQ(ω) = −n ≤ −1 with n odd, c2w−2dw2 if degQ(ω) = −2, (w−n + cw−1)2dw2 if degQ(ω) = −n ≤ −4 with n even. Here, c is the residue of a branch of Q(z) at ω. Proof. See Theorem 8.1 of [13] or Section 6 of [16]. � THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 7 So since trajectories are conformally invariant this lemma tells us that the local structure of trajectories around a point ω ∈ D is completely determined by degQ(ω) and the converse is true as well. Lemma 2.2. Suppose that ω ∈ D and degQ(ω) = n. Then (1) For n ≥ −1, there are exactly n + 2 trajectories of Q(z)dz2 that end at ω and form equal angles with each other. (2) For n ≤ −3, there are infinitely many trajectories ending at ω and more- over, there are |n| − 2 directions at ω forming equal angles such that the trajectories approach ω in these directions. (3) For n = −2, the behaviour depends on the value of c (as defined in Lemma 2.1). (a) If c is real, then the trajectories are the images of all radial lines under the map φ defined in Lemma 2.1. (b) If c is purely imaginary, then the trajectories are the images of all concentric circles under the map φ defined in Lemma 2.1. (c) If Re[c], Im[c] 6= 0 then the trajectories are the images of logarithmic spirals under the map φ defined in Lemma 2.1. Proof. See Section 7 of [16]. � This lemma shows that it makes sense for us to define degD,Q(x), the degree of a point on the boundary, in terms of the trajectories ending at x. If D = H and Q extends to a meromorphic function on a neighbourhood of some x ∈ R∪ {∞} with H,Q(x) finite. Then by studying the trajectory structure at x, we can see that H,Q(x) = degQ(x). This is the motivation for defining degD,Q in the way we have. The next lemma shows that the degD,Q is also conformally invariant: Lemma 2.3. Suppose that Q(z)dz2 is a Kühnau quadratic differential on a domain D and f is a conformal map of the upper half-plane H onto D. Then the quadratic differential Qf (w)dw 2 on H, defined by Qf(w) = Q(f(w))f ′(w)2, is also a Kühnau quadratic differential. Moreover, suppose that z ∈ ∂D is a prime end of D. Then degD,Q(z) = degH,Qf (f −1(z)). Proof. By Carathéodory’s theorem, f extends continuously to ∂H and by Schwarz’s reflection, f extends analytically across f−1(Γk) for all k = 1, . . . , n. Since θ- trajectory arcs are conformally invariant, this implies that Qf defined by (2.1) is a Kühnau quadratic differential on H. Moreover, for all k = 1, . . . , n, f(Γk) is a θk-trajectory arc of Qf (z)dz 2. Also each point on ∂H corresponds bijectively to a prime end of H. Hence there is a bijective correspondence between the points of ∂H and prime ends of D. Then degD,Q(z) = degH,Qf (f −1(z)) follows from the conformal invariance of trajectories. � Reflection across trajectories. Suppose that D is a domain such that Γ ⊂ ∂D is an open interval in R. Let Q(z)dz2 be a quadratic differential such that Γ is a trajectory arc or an orthogonal 8 J. TSAI trajectory arc of Q(z)dz2. Then let D− = {z : z ∈ D} be the reflection of D along Γ. Define Q−(z) = Q(z) for z ∈ D−. Then since Γ is a trajectory, we have Q(z) = Q−(z) ∈ R for z ∈ Γ. Thus by defining (2.2) Q∗(z) = Q(z) for z ∈ D, Q−(z) for z ∈ D−, Q(z) = Q−(z) for z ∈ Γ. it is easy to see that Q∗ is meromorphic in D ∪ D− and hence Q∗(z)dz2 is a quadratic differential on D ∪ D−. Thus by the transformation law (and using Schwarz reflection), this shows that we can extend quadratic differentials across trajectory arcs or orthogonal trajectory arcs. We will use reflection to prove the following lemma: Lemma 2.4. Suppose Q(z)dz2 is a Kühnau quadratic differential on H. Then for any z ∈ ∂H, deg H,Q(z) ∈ Z implies that Q(z)dz2 extends to a quadratic differential on a neighbourhood of z and hence H,Q(z) = degQ(z). Proof. Firstly, if z ∈ Γj for some j = 1, . . . , n. Then by definition degH,Q(z) = 0 and Q(z)dz2 can be extended to a neighbourhood of z by reflection. By definition, every z ∈ Γj is an ordinary point of Q(z)dz2 and hence degQ(z) = degH,Q(z) = 0. Otherwise we write z = zk and suppose that a θk−1-trajectory arc, Γk−1, and a θk-trajectory arc, Γk, end at zk. Then, by definition, degH,Q(zk) ∈ Z implies that θk − θk−1 is a multiple of π/2. Thus Γk−1 and Γk are trajectory arcs or orthogonal trajectory arcs of e−2iθk−1Q(z)dz2. Thus by reflection, e−2iθk−1Q(z)dz2 extends to a neighbourhood of zk. Hence, Q(z)dz 2 also extends to a neighbourhood of zk. � We can now prove Theorem 1.1; but first, we explain briefly why we can view Theorem 1.1 as a generalized form of Schwarz-Christoffelmapping: Schwarz-Christoffel mapping is a method of computing the conformal map between the upper half-plane and a domain bounded by a polygon. See [12] for more details. If we have a confor- mal map f fromH to some domainD such that the sides ofD consist of θ-trajectory arcs of the quadratic differential Q(w)dw2. Then Q(w)dw2 is a Kühnau quadratic differential on D and hence by Lemma 2.3, Q(f(z))f ′(z)dz2 is a Kühnau quadratic differential on H. Theorem 1.1 then implies that Q(f(z))f ′(z)2 = R (z − ζj)λj This is precisely the Schwarz-Christoffel formula when Q(z) ≡ 1. Also, we comment that the case when Q(w)dw2 is either negative or positive on R (i.e. the boundary of R consists only of trajectory arcs and orthogonal trajectory arcs) is easy to prove: we can use reflection to extend Q(z)dz2 to a quadratic differential on the Riemann sphere Ĉ. Hence Q(z) must be rational since property (ii) in the definition of Kühnau quadratic differentials guarantees that Q(z) does not THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 9 have any essential singularities and so Q(z) is rational (since the only meromorphic functions on Ĉ are rational). This proves Theorem 1.1 for this case. Proof of Theorem 1.1. Since Q(z)dz2 is a Kühnau quadratic differential, we can z1 < . . . < zm, (zk−1, zk) for k = 1, . . . ,m, (zm,∞) for k = m+ 1, (−∞, z0) for k = 0, such that each Γk is a θk-trajectory arc of Q(z)dz 2 for some θk ∈ [0, π). Let T = {Γ1, . . . ,Γm+1} . Then take any Γ ∈ T . Since Γ is a θ-trajectory for some θ, Γ is a trajectory arc of e−2iθQ(z)dz2; hence by reflection, we can reflect the quadratic differential e−2iθQ(z)dz2 across Γ to get a quadratic differential on H− = {Im(z) < 0} which we call Q̃(z)dz2. Similarly, by rotating Q̃(z)dz2, we can reflect it across another Υ ∈ T to get another quadratic differential Q∗(z)dz2 on H. Since Q∗ is obtained from Q by rotating twice, we have Q∗(z) = eiσQ(z) for some σ ∈ [0, 2π). This shows that Ψ(z) = Q′(z) (Q∗)′(z) Q∗(z) can be extended to a meromorphic function in C\{z1, . . . , zm}. Then part (ii) of the definition of Kühnau quadratic differentials implies that all the finite singularities of Ψ(z) are simple poles otherwise Q(z) would have an essential singularity which, by the great Picard theorem, contradicts part (ii) of the definition of Kühnau quadratic differentials. Thus we can write: Ψ(z) = h(z)− z − ζj where ζj ∈ C, and λj ∈ R, and h(z) is an entire function in C that does not does not vanish in C. This implies that Q(z) = exp h(ζ)dζ (z − ζj)λj (z − zk)νk Moreover, the singularity at ∞ of h(ζ)dζ cannot be essential by part (ii) of the definition of Kühnau quadratic differentials (otherwise we would get a contradiction with the great Picard theorem as above). This implies that h(ζ)dζ 10 J. TSAI is constant (since it has no zeroes or poles). Hence Q(z) = R (z − ζj)λj If ζj ∈ C \ R, then by definition, we must have λj = degQ(ζj). Moreover, if ζj ∈ R and degH,Q(ζj) <∞ we also have νk = degH,Q(zk). We will not prove this fact here but in the following corollary we will consider a special case. The general proof follows readily from it. We will prove the following corollary which is simply an application of Theorem 1.1 to domains slit by φ- trajectory arcs: Corollary 2.5. Suppose that Q(z)dz2 is a Kühnau quadratic differential on H such that there is a point ξ0 ∈ R with degH,Q(ξ0) = N ∈ {0, 1, . . .}; then we can write (2.3) Q(w) = R(w − ξ0)N (w − aj)αj where aj ∈ C, αj ∈ R, and also R is some non-zero constant. Let γ : [0, T ] 7→ H be a simple curve such that γ(0) = ξ0 and γ(0, T ) is a φ-trajectory arc of Q(w)dw 2 in H (φ ∈ [0, π)) and ζ = γ(T ) ∈ H is an ordinary point of Q(w)dw2 (i.e. degQ(ζ) = 0). Suppose that f maps H conformally onto H \ γ(0, T ]. Then f satisfies (2.4) Q(f(z))f ′(z)2 = R′(z − ξ)2(z − c−)µ (z − c+)µ (z −Aj)αj , where R′ is some constant; c−, c+ are the two preimages of ξ0 under f satisfying c− < c+; Aj is the preimage of aj under f ; and ξ is the preimage of ζ; and µ± = deg H\γ(0,s],Q(f(c Proof. Theorem 1.1 and Lemma 2.4 imply that Q(w) can be written as (2.3). Then Lemma 2.3 implies that Q̂(z) = Q(f(z))f ′(z)2 is a Kühnau quadratic differential. So by Theorem 1.1, we only need to look at the singularities of Q̂(z)dz2. Now, by Schwarz reflection, f extends to a conformal map on C \ {ξ, c−, c+}. Thus, by the conformal invariance of trajectories, this implies that Qf (z) = R ′(z − ξ)M (z − c−)µ (z − c+)µ (z −Aj)αj . Then by Lemma 2.2, there are exactly two φ-trajectory arcs of Q(z)dz2 ending at ζ = γ(T ) of which γ(0, T ) is one of them. So by the conformal invariance of trajectories, there is one φ-trajectory arcs of Q̂(z)dz2 ending at ξ that is contained in H. Hence, by definition, deg bQ,H(ξ) = 2. Using Lemma 2.4, this implies that degQ(ξ) = 2 i.e. M = 2. Thus we only need to determine µ − and µ+. THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 11 Note that since ξ0 has degree N with respect to Q(z)dz 2, we can determine, using Lemma 2.2, that the angle between γ(0, T ) and f((ξ, c−)) at ξ0 is πψ− = π degH,Q(f(c −)) + 2 N + 2 and similarly, the angle between γ(0, T ) and f((ξ, c−)) at ξ0 is πψ+ = π degH,Q(f(c +)) + 2 N + 2 Hence, by Schwarz reflection, the function F (z) = (f(z)− ξ0)1/ψ extends to a conformal mapping on a neighbourhood of c−. Thus in a neighbour- hood of z = c−, we can write (2.5) f(z) = ξ0 + (z − c−)ψ h(z)ψ where h is analytic in a neighbourhood of z = c− with h(c−) 6= 0. Now Q′f (z) Qf (z) Q′(f(z))f ′(z)2 Q(f(z)) f ′′(z) f ′(z) The residue at z = c− of the left-hand side of the equation is µ−, and we can use (2.5) to determine the residue at z = c− of the right-hand side. Thus we get µ− = degH,Q(f(c We apply the same method to c+ to get µ+. � 3. Domains slit by θ-trajectory arcs Let Q(w)dw2 be a Kühnau quadratic differential on H with deg H,Q(ξ0) = N ∈ {0, 1, . . .} for some ξ0 ∈ R. Then by Theorem 1.1 and Lemma 2.4, Q(w) = (w − ξ0)N (w − aj)αj where αj ∈ R and aj ∈ C. Now suppose that γ : [0, T ) 7→ H is a simple curve such that γ(0) = ξ0 and γ(0, T ) is a φ-trajectory of Q(z)dz 2 in H (φ = [0, π)) that is parameterized by half-plane capacity. As mentioned in the introduction, there exists conformal maps ft : H 7→ Ht = H \ γ(0, t] satisfying the hydrodynamic normalization. Then by restricting Q(w)dw2 to a quadratic differential on Ht we can induce via ft and (2.1), a quadratic differential on H: (3.1) Qt(z)dz 2 = Q(ft(z))f 2dz2. We now use Corollary 2.5 and (3.1) to prove Theorem 1.2. Proof of Theorem 1.2. Note that by Schwarz reflection, each ft can be extended to a conformal map on Ĉ \ {C−(t), C+(t), ξ(t)}. Then since ft(z) satisfies the hydrodynamic normalization, this implies that f ′t(z) 2 = 1 +O as z → ∞. 12 J. TSAI So by (3.1), (3.2) Qt(z) Q(ft(z)) = 1 +O as z → ∞. If we let ζ = 1/z, then we get (3.3) Qt(1/ζ) Q(ft(1/ζ)) = 1 +O as ζ → 0. Since ft is analytic in a neighbourhood of infinity, (3.3) is a Taylor series expansion and hence we can look at the Taylor series coefficients, in particular: C(0,ǫ) f ′(1/ζ)2 C(0,ǫ) Qt(1/ζ) ζ2Q(ft(1/ζ)) dζ = 0 for small enough ǫ > 0 where C(0, ǫ) is the anticlockwise contour about the circle with centre at zero and radius ǫ > 0. Then by Theorem 1.1 and Corollary 2.5, we can write Q(w) = R(w − ξ0)N (w − aj)αj Qt(z) = R ′(z − ξ)2(z − C−(t))µ (z − C+(t))µ (z −Aj(t))αj . Hence by the residue theorem (since ft(1/ζ) = 1/ζ + · · · as ζ → 0), this implies 2ξ(t) + µ (t) + µ (t) + αkAk(t) Nξ0 + This implies (1.1). To get (1.2), note that ft satisfies the chordal Loewner differen- tial equation (0.1) and hence if we let Gt = f t ◦ fs for some s ∈ (0, T ) fixed and t > s, then the chain rule implies that Gt satisfies the differential equation (z) = Gt(z)− ξ(t) Then for some s sufficiently close to t, we can write each Aj(t) = Gt(wj) wj ∈ C for all j = 1, . . . , n. Thus Ȧj(t) = Aj(t)− ξ(t) Similarly, we get Ċ±(t) = C±(t)− ξ(t) . Hence we get (1.2) from differentiating (1.1). � THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 13 An extension: We can extend Theorem 1.2 to the case when γ is made up of different θk-trajectory arcs of some quadratic differential Q(z)dz2: Let γ : (0, T ] 7→ H be a curve with γ(0) ∈ R such that there is a partition {0 = t0 < t1 < · · · < tr = T } such that γ(tk−1, tk) is a θk-trajectory arc of Q(z)dz 2 and γ(tk) is an ordinary point of Q(z)dz2 for k = 1, . . . , r. Then we can find the driving function ξ(t) of γ by applying Theorem 1.2 to the θ1-trajectory arc γ(0, t1) to get a driving function ξ1(t), and applying Theorem 1.2 inductively to each f (γ(tk, tk+1)) (which is a θk+1-trajectory arc of the quadratic differential Qtk(z)dz 2 = Q(ftk(z))f (z)2dz2) to get ξk(t). Then ξ(t) = ξk(t) for t ∈ [tk−1, tk). We also have the following corollary: Corollary 3.1. Suppose that Q(w)dw2 and γ are as defined in Theorem 1.2. Then the driving function ξ and Aj , C −, C+ as defined in Theorem 1.2 are in C∞(0, T ). Moreover, we can write any derivative of ξ, C−, C+, Aj explicitly in terms of ξ, C−, C+, Aj and the exponents µ −, µ+, αj. Proof. Recall that, in the proof of Theorem 1.2, we had the formulae Ȧj(t) = Aj(t)− ξ(t) , Ċ±(t) = C±(t)− ξ(t) . This implies that each term in (1.2) is differentiable so we can write the second derivative of ξ in terms of ξ(t), Aj(t), C ±(t). This in turn implies that we can write the third derivative of ξ in terms of ξ(t), Aj(t), C ±(t) and the exponents. Continuing inductively, we have showed that every derivative of ξ exists and can be expressed in terms of ξ(t), Aj(t), C ±(t) and the exponents. Note that each derivative of ξ is finite for t ∈ (0, T ) since |Aj(t)− ξ(t)|, |C±(t)− ξ(t)| > 0. Then ξ is smooth implies that Aj(t), C ±(t) are also smooth. � Theorem 1.3 then follows from Corollary 2.5 and Theorem 1.2: Proof of Theorem 1.3. First note that, by the definition of θ-trajectory arcs, γ̇ always exists and is never 0. Also by Corollary 2.5, Φt(ξ(t)) 6= 0,∞; thus the right hand side of (1.3) always exists since, by definition, γ avoids poles and zeroes of Q(w)dw2. Recall that ft(ξ(t)) = γ(t), this implies that γ̇(t) = ḟt(ξ(t)) + f t(ξ(t))ξ̇(t). Then combining the Loewner differential equation (0.1) with (3.1) we have ḟt(z) = − z − ξ(t) Qt(z) Q(ft(z)) Φt(z) Q(ft(z)) ⇒ ḟt(ξ(t)) = −2 Φt(ξ(t)) Q(γ(t)) Note that Φt(ξ(t)) 6= 0,∞ since, by Corollary 2.5, Qt(z) has a double zero at ξ(t). 14 J. TSAI Thus we have γ̇(t) = −2 Φt(ξ(t)) Q(γ(t)) Qt(ξ(t)) Q(γ(t)) ξ̇(t) = −2 Φt(ξ(t)) Q(γ(t)) since Theorem 1.2 implies that ξ̇ is finite for all t ∈ (0, T ) and Corollary 2.5 implies that Qt(ξ(t)) = 0. � 4. Applying Theorem 1.2 In practice, understanding ξ(t) via (1.1) is not possible: it is difficult to calculate the positions of the zeroes and poles of Qt because the information we have on them is all relative to ξ(t) (which we are trying to find). On the other hand, (1.2) is more useful in applications. In this section, we will demonstrate how we can use (1.2) to calculate numerically the driving function of a given slit that consists of θk- trajectory arcs of a given quadratic differential. The method is basically a modified version of Euler’s method. Firstly, for any smooth function h on (0,T), Taylor’s theorem implies that for all M = 1, 2, . . ., (4.1)∣∣∣∣∣h h(t) + )∣∣∣∣∣ ≤ M !KM s∈(t,t+ 1K ) for t, t + 1/K ∈ (0, T ). We will apply (4.1) to the functions ξ, Ak and C± (as defined in Theorem 1.2) noting that, by Corollary 3.1, they are smooth and all of their derivatives can be expressed in terms of ξ(t), Ak(t) and C ±(t). Thus if we know ξ(s), Ak(s) and C ±(s) we can use (4.1) to obtain an approximate formula for ξ(s+K−1), Ak(s+K −1) and C±(s+K−1) (choosing K to be small and/or M to be large so that the right-hand-side of (4.1) is small); then we can apply (4.1) to ξ(s+K−1), Ak(s+K −1) and C±(s+K−1) to find ξ(s+2K−1), Ak(s+2K −1) and C±(s+2K−1). Continuing like this, we obtain an approximation of ξ at the points {s+ nK−1}. So clearly what we need to do now is find the starting values ξ(s), Ak(s) and C±(s) so we can apply the above method. But because ξ is not differentiable at 0, we cannot use the formula (4.1) with t = 0. The way around this is to note that if deg H,Q(ξ0) = N ∈ {0, 1, 2, . . .} then since we know degH,Qt(C +(t)), we can calculate the angle that the trajectory makes with the line [ξ0,∞) (as in the proof of Corollary 2.5). Then we find that the angle is πψ where: πψ = π 2 deg (C+(t)) + 2 N + 2 So if we choose s small enough, we have fs ≈ Fψ,ξ0s , where Fψ,ξ0s is the conformal map that maps H conformally onto H s that is hydro- dynamically normalized where Hψs is the upper half-plane slit by the straight line starting at ξ0 making an angle πψ with [ξ0,∞), with half-plane capacity 2s. Then we also have Ak(s) ≈ (Fψ −1(ak), THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 15 and also, C−(s), C+(s) are approximately the two preimages of ξ0 under F ψ+,ξ0 Then we can use (1.1) to calculate ξ(s) approximately. We can then plug this information into (4.1) as described above. Note that F t can be found using the fact that (4.2) F λt (z) = z − (1− 2p) z − (1− 2p) t+ (1− p) for some λ. Then we reparameterize this formula to remove the λ and translate the point 0 to ξ0. Unfortunately, inverting this function cannot be done explicitly but it can be done numerically very efficiently using Newton’s method. Alternatively, by selecting a small s, we can assume that Ak(s) ≈ ak for all k. Then we note that the 2 preimages of ξ0 under F s can be determined explicitly (see [11]). This obviates the need to numerically invert F Another difficulty is that, in general, given a slit, we cannot parameterize it by half-plane capacity so it would be difficult, for example, to know at which t one should stop. Most formulae for calculating half-plane capacity of some compact set K rely on knowing the conformal map fK of H onto H\K (normalized hydrodynam- ically). One possibility would be to use the probabilistic definitions of half-plane capacity given in [7]. We will use the fact that Theorem 1.3 and Corollary 3.1 imply that we can give all derivatives of γ(t) in terms of ξ(t), Ak(t), C −(t), C+(t) and the exponents µ−, µ+, αk so if we know these, we can also use (4.1) to approximate γ. This in turn allows us to calculate the length of the slit γ. Thus if we know beforehand length of our slit, we can calculate at what value of t we stop. We now have everything we need in order to use (4.1) to calculate the driving function numerically of any slit that is made up of θk trajectory arcs of a qua- dratic differential Q(w)dw2. We will demonstrate how this is done in the following example: An example. Suppose that γ : (0, T ) → H is a piecewise linear arc parameterized by half-plane capacity that satisfies: • γ(0) = 0. • From t = 0 to t = t1, γ is the straight line arc from 0 to i; call this Γ1. • From t = t1 to t = t2, γ is the straight line arc from i to 2 + i; call this Γ2. • From t = t2 to t = t3 = T , γ is the straight line arc from 2 + i to 2 + 2i; call this Γ3. First note that γ is made up of alternating (π/2)- and 0-trajectory arcs of the qua- dratic differential 1dw2 in H and hence we can use Theorem 1.2 (or more specifically the extension of Theorem 1.2 detailed in Section 3) to calculate ξ̇. As mentioned previously, there is no easy way to know beforehand what t1, . . . , t3 are. For sim- plicity, we will only use M = 1 in (4.1) i.e. ≈ f(t) + ḟ(t) and fix a large K. Obviously Γ1 forms a right angle with real line; so we can use (4.2) to determine the function ft1 = F 1/2,0 (z) = z2 − 4t1. 16 J. TSAI −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 Figure 3. The example path in the upper half-plane (left) and a plot of its driving function on the y-axis against time on the x-axis (right). It is easy to see that in this case, t1 = 1/4 and ξ is constantly 0 for t ∈ (0, t1]. This induces the quadratic differential using (3.1): Qt1(z)dz z2dz2 (z + 1)(z − 1) . Hence, we let A1(t1) = −1, A2(t1) = 1. Also f−1t1 (γ2) is a 0-trajectory arc of Q1(z)dz 2 starting from ξ(t1) = 0 on R (by the conformal invariance of trajectories). Now note that f−1t1 (γ2) makes an angle of π/4 with the positive real axis. and hence ft1+K−1 ≈ F 1/4,0 sinceK is large. We can then use Newton’s method to find the preimages under the above approximation of ft1+K−1 of the points A1(t1), A2(t1) and the 2 preimages of zero to get the points A1(t1+K −1), A2(t1+K −1), C−(t1+K −1), C+(t1+K −1) and hence, using (1.1), we can find ξ(t1+K −1). Then inserting this into (4.1), as detailed above we can also find ξ(t1 + nK −1) and A1(t1 + nK −1), A2(t1 + nK −1), C−(t1 + nK−1), C+(t1 + nK −1); also, by Theorem 1.3, we can find |γ̇(t1 + nK−1)| if we let t2(K) = inf |γ̇(t1 + nK−1)| > (length of Γ2) = 2 then t2(K) ≈ t2 for K large. So we just assume that t2 = t2(K). Let A3(t2) = C−(t2) and A4(t2) = C +(t2). Hence by (3.1), Qt2(z)dz (z − ξ(t2))2(z −A3(t2))dz2 (z −A1(t2))(z −A2(t2))(z −A4(t2)) Then, by the conformal invariance of trajectories, f−1t2 (Γ3) is a π/2-trajectory of Qt2(z)dz 2 and also, f−1t2 (Γ3), makes an angle 3π/4 with (ξ(t2),∞) and so ft2+K−1 ≈ F 3/4,0 Then, as before, we can use Newton’s method to find the preimages under the above approximation of ft2+K−1 of the points A1(t2), . . . , A4(t2) and the 2 preimages of THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 17 ξ(t2) to get the points A1(t2+K −1), . . . , A4(t2+K −1), C−(t2+K −1), C+(t2+K and hence use (1.1) to get ξ(t2+K −1). We insert these into the formula iteratively to get ξ(t2+nK −1) and A1(t2+nK −1), . . . , A4(t2+nK −1), C−(t2+nK −1), C+(t2+ nK−1) until t2+nK −1 ≈ T . Thus the end result is that we found the driving func- tion of the first 3 steps of the slit given in Figure 3. Of course, our calculation of ξ will be more accurate by taking larger K. For example, we can use the above method to calculate the driving function of any path on the square/triangle/hexagonal lattice on H starting from some point in R. In fact we can calculate the driving function of a path on the square/triangle/hexagonal lattice in any polygon D by mapping the half-plane conformally onto D and pulling back the quadratic differential 1dw2 on D to Q(z)dz2 on H using the transforma- tion law. Also note that, in general, any curve γ can be approximated by a curve γδ which lies on the square lattice δZ 2. Then it can be shown that ξδ → ξ uniformly as δ ց 0, where ξδ is the driving function of γδ and ξ is the driving function of γ hence, we can use the above method to calculate ξδ then take the limit as δ ց 0 to obtain ξ. Another point to note is that using the above method, we do not need to know before hand what the trajectory arc of the given quadratic differential looks like; so for arbitrary Kühnau quadratic differentials, we can use this method to plot the trajectories starting at the boundary. We end this section by looking at what happens when the slit approaches the boundary: Proposition 4.1. Suppose that γ : [0, T ) 7→ H is a simple curve such that γ(0) ∈ R and γ(0, T ) is a θ-trajectory arc of some quadratic differential Q(z)dz2. Then let ξ be the driving function of γ. If γ(t) ∈ R ∪ γ(0, T ), i.e. γ makes a loop at time T . Then ∣∣∣∣→ ∞ as tր T for all n = 0, 1, . . .. Proof. For t ∈ (0, T ), we define Γ(t) = {γ(s) : s ∈ (t, T )}. Then Γt is a θ-trajectory arc in Ht = H\γ(0, t] of Q(w)dw2 and it is also a crosscut in Ht (see [14]). Then by the conformal invariance of θ-trajectories, f t (Γt) ⊂ H is a θ-trajectory arc of Qt(z)dz 2. Moreover, f−1t (Γt) is a crosscut of H with one end point at ξ(t) and the other end point in R such that either C+(t) or C−(t) is contained in the closure of the bounded component of H \ f−1t (Γt). Without loss of generality, assume it is C+(t). Then since diam(f−1t (Γt)) → 0 as t ր T , we must have ξ(t) = C+(T ) and hence by (1.2), ξ̇(t) → ∞ as t ր T . Similarly, we differentiate (1.2) as mentioned in Corollary 3.1 to obtain the result for higher order derivatives. � 18 J. TSAI This means that as γ gets closer and closer to making a loop, the approximation by (4.1) stops working no matter what M we choose. This phenomenon can be observed in Figure 3, as we turn the last corner in γ, we can see that ξ decreases faster even though the slit is not yet that close to the boundary. 5. Generalizing Theorem 1.2 5.1. Multiple slits. Suppose that γk : [0, T ) → H for k = 1, . . . , N are disjoint simple curves such that γk(0) ∈ R and γk(0, T ) ⊂ H. By the Riemann mapping theorem, there exists unique ft that map H conformally onto Ht = H\ k=1 γk(0, t] that satisfies the hydrodynamic normalization. We can reparameterize such that γk(0, T ] has half-plane capacity 2t. Then ft satisfies (5.1) ḟt(z) = −2f ′t(z) bk(t) z − ξk(t) where bk(t) = 1, and ξk(t) = f t (γk(t)). See [2] for more details. Theorem 5.1. Suppose that Q(w)dw2 is a Kühnau quadratic differential on H such that the points ξk(0) ∈ R satisfy H,Q(ξk(0)) = βk ∈ {0, 1, 2, . . .} for all k. Then we can write Q(w) = (w − ξk(0))βk (w − aj)αj with aj ∈ C and αj ∈ R. Then suppose that γk : [0, T ) → H for k = 1, . . . , N are disjoint simple curves such that γk(0) ∈ R and γk(0, T ) ⊂ H and are parameterized as above. Then (5.2) 2 ξk = − (µ−k C k (t) + µ k (t)) αjAj(t) +Σ0, (5.3) ξ̇l(t) = k=1,k 6=l bk(t) ξl(t)− ξk(t) bl(t) (t)− ξl(t) bl(t) (t)− ξl(t) αjbl(t) Aj(t)− ξl(t) for all l ∈ {1, . . . , N}. Where C−k (t) and C k (t) are the two preimages of ξk(0) under ft satisfying C k (t) < C k (t); µ±k = degHt,Q(ft(C k (t))); THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 19 Aj(t) = f t (aj); and βkξk(0) Proof. By Theorem 1.1 and Lemma 2.4, we can write Q(w) = (w − ξk(0))βk (w − aj)αj Then either by modifying the proof of Corollary 2.5 or iterating N slit functions and applying Corollary 2.5 N times, it is not too difficult to see that if we define Qt(z) by (3.1), then (5.4) Qt(z) = (z − ξk(t))2(z − C−k (t)) k (z − C+k (t)) (z −Aj(t))αj Then the proof of (5.2) is exactly the same as the proof of (1.1) in Theorem 1.2. The proof of (5.3) is more complicated. First let Pt(z) = −2 bk(t) z − ξk(t) Then (5.1) becomes ḟt(z) = f t(z)Pt(z). Now take the logarithmic derivative of Qt(z) with respect to z and t separately using the definition of Qt(z) given by (3.1) to get Q′t(z) Qt(z) Q′(ft(z))f Q(ft(z)) f ′′t (z) f ′t(z) Q̇t(z) Qt(z) Q′(ft(z))ḟt(z) Q(ft(z)) ḟ ′t(z) f ′t(z) Q′(ft(z))f t(z)Pt(z) Q(ft(z)) f ′′t (z)Pt(z) + f t(z)P t (z) f ′t(z) where we substitute (5.1) in for ḟt to get from the first to the second line. Thus we (5.5) Q̇t(z) Qt(z) Q′t(z) Qt(z) Pt(z) + 2P t(z). So then by (5.4), we note that Q′t(z) Qt(z) z − ξk(t) z − C−k (t) z − C+k (t) z −Aj(t) Q̇t(z) Qt(z) 2ξ̇k(t) z − ξk(t) µ−k Ċ k (t) z − C−k (t) µ+k Ċ k (t) z − C+k (t) αjȦj(t) z −Aj(t) 20 J. TSAI 0 2 4 6 8 10 Figure 4. A plot of the two driving functions ξ1 (top) and ξ2 (bottom) on the y-axis against time on the x-axis. Thus substituting this into (5.5) and comparing the coefficient of z − ξl(t) (i.e. the residue at z = ξl(t) of both sides of (5.5)), we find that this is exactly (5.3). � Similarly, we can prove a version of Theorem 1.3 and Corollary 3.1 for multiple slits. This means that we can use the method detailed in Section 4 with (5.3) to calculate the driving function for multiple θk-trajectory arc slits. For example Figure 4 plots the graph of the two driving functions ξ1 and ξ2 in the case when γ1 and γ2 are 2 vertical slits starting from -1 and 1 (i.e. orthogonal trajectories of 1dz2) and growing at the same speed. Compare this with Figure 7 in [2]. 5.2. Radial Loewner evolution. The chordal Loewner differential equation was introduced because the upper half-plane was an easier domain to work with for many applications but the original setting of the Loewner differential equation is in the unit disc D = {z ∈ C : |z| < 1}: Suppose that γ : [0, T ) 7→ D is a simple curve such that γ(0) ∈ T = {z : |z| = 1} and γ(0, T ) ⊂ D \ {0}. Then Dt = D \ γ(0, t] is simply-connected and 0 ∈ Dt for all t ∈ (0, T ). Hence the Riemann mapping theorem implies that there is unique conformal map ft mapping D conformally onto Dt such that ft(0) = 0 and f t(0) > 0. Then Schwarz’s lemma and the Carathéodory kernel theorem implies that f ′t(0). is strictly decreasing and continuous so we can reparameterize such that f ′t(0) = e −t. f ′t(0) is sometimes called the conformal radius of Dt; hence in this case we are parameterizing by conformal radius. Then the functions ft satisfy the radial Loewner differential equation: ḟt(z) = −zf ′t(z) z + eiξ(t) z − eiξ(t) See [9] for more details. THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 21 Theorem 5.2. Suppose that Q(w)dw2 is a Kühnau quadratic differential on D such that degQ(0) = K ∈ Z and eiξ0 ∈ T = {|z| = 1} satisfies = N ∈ {0, 1, 2, . . .} then we have Q(w) = wK(w − eiξ0)N (w − aj)αj where aj ∈ C and αj ∈ R. Then if γ : [0, T ) 7→ D is a simple curve such that γ(0) = eiξ0 and γ(0, T ) ⊂ D \ {0} is a φ-trajectory arc of Q(w)dw2 in D that does not meet 0 and is parameterized as above. Then we have (5.6) e2iξ(t) = e−2tΠ0C −(t)−µ C+(t)−µ Aj(t) −αj , (5.7) ξ̇(t) = − 1 C−(t) + eiξ(t) C−(t)− eiξ(t) C+(t) + eiξ(t) C+(t)− eiξ(t) Aj(t) + e iξ(t) Aj(t)− eiξ(t) where, as usual, the functions Aj(t) are defined by Aj(t) = f t (aj) for j = 1, . . . , n, µ± = degDt,Q(ft(C ±(t))), C+(t) > C−(t) are the two preimages of eiξ0 under ft; and also, Π0 = e Proof. The formula for Q(w) can be obtained from Theorem 1.1 by the transforma- tion law. We then define Qt by (3.1). Since the point 0 is fixed by ft, this implies that the degQt(0) = degQ(0) = K. Thus we can apply Corollary 2.5 (again, using the transformation law) to get Qt(z) = z K(z − eiξ(t))2(z − C−(t))µ (z − C+(t))µ (z −Aj(t))αj Then since by definition, Qt(z) = Q(ft(z))f This immediately implies (5.6) by substituting z = 0. Then we get (5.7) in the same way as we get (1.2) from (1.1) in the proof of Theorem 1.2. � As in the case of multiple slits, a version of Theorem 1.3 and Corollary 3.1 holds for this case. 22 J. TSAI 5.3. Other versions of the Loewner differential equation. There are several other versions of the Loewner differential equation for simply-connected domains in the literature; the methods in this paper should work in those cases as well and the proofs should be similar to the proofs of Theorem 1.2 etc. Also, [4], [5] generalizes the Loewner differential equation to multiply-connected domains and again, some of the methods should work in these cases possibly using methods in [1] to extend Theorem 1.1 to multiply-connected domains. Finally, even if we consider general 2-dimensional growth processes given by the Loewner-Kufarev differential equation (see Chapter 6 of [13]), some of the methods in this paper should still be applicable. Acknowledgement. The author would like to express his gratitude to his super- visor Dr. T. K. Carne for his constant guidance and helpful discussion when writing this paper. References 1. D. Crowdy. The Schwarz-Christoffel mapping to bounded multiply connected polygonal do- mains. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461(2061):2653–2678, 2005. 2. W. Kager, B. Nienhuis, and L. P. Kadanoff. Exact solutions for Loewner evolutions. J. Statist. Phys., 115(3-4):805–822, 2004. 3. T. Kennedy. Computing the Loewner driving process of random curves in the half plane. arXiv:math/0702071v1 4. Y. Komatu. Untersuchungen über konforme Abbildung von zweifach zusammenhängenden Gebieten. Proc. Phys.-Math. Soc. Japan (3), 25:1–42, 1943. 5. Y. Komatu. On conformal slit mapping of multiply-connected domains. Proc. Japan Acad., 26(7):26–31, 1950. 6. R. Kühnau. Über die analytische Darstellung von Abbildungsfunktionen insbesondere von Extremalfunktionen der Theorie der konformen Abbildung. J. Reine Angew. Math., 228:93– 132, 1967. 7. G. F. Lawler. Conformally invariant processes in the plane. American Mathematical Society, USA, 2005. 8. J. R. Lind. A sharp condition for the Loewner equation to generate slits. Ann. Acad. Sci. Fenn. Math., 30(1):143–158, 2005. 9. K. Löwner. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I. Math. Ann., 89, 1923. 10. D. E. Marshall and S. Rohde. The Loewner differential equation and slit mappings. J. Amer. Math. Soc., 18(4):763–778 (electronic), 2005. 11. D. E. Marshall and S. Rohde. Convergence of the zipper algorithm for conformal mapping. preprint, 2006. 12. Z. Nehari. Conformal mapping. Dover Publications, New York, 1982. 13. C. Pommerenke. Univalent functions. Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, Band XXV. 14. C. Pommerenke. Boundary behaviour of conformal maps. Springer-Verlag, Berlin, 1992. 15. O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118:221–288, 2000. 16. K. Strebel. Quadratic differentials. Springer-Verlag, Berlin, 1984. Department of Mathematics, Chinese University of Hong Kong, Shatin, New Terri- tories, Hong Kong E-mail address: jtsai@math.cuhk.edu.hk http://arxiv.org/abs/math/0702071 Introduction 1. Main results Definition: 2. Kühnau quadratic differentials and generalized Schwarz-Christoffel mapping Transformation Law Reflection across trajectories 3. Domains slit by -trajectory arcs An extension: 4. Applying Theorem ?? An example. 5. Generalizing Theorem ?? 5.1. Multiple slits 5.2. Radial Loewner evolution 5.3. Other versions of the Loewner differential equation Acknowledgement. References

We obtain a first order differential equation for the driving function of the chordal Loewner differential equation in the case where the domain is slit by a curve which is a trajectory arc of certain quadratic differentials. In particular this includes the case when the curve is a path on the square, triangle or hexagonal lattice in the upper halfplane or, indeed, in any domain with boundary on the lattice. We also demonstrate how we use this to calculate the driving function numerically. Equivalent results for other variants of the Loewner differential equation are also obtained: Multiple slits in the chordal Loewner differential equation and the radial Loewner differential equation. The method also works for other versions of the Loewner differential equation. The proof of our formula uses a generalization of Schwarz-Christoffel mapping to domains bounded by trajectory arcs of rotations of a given quadratic differential that is of interest in its own right.

Introduction Suppose that H = {z ∈ C : Im(z) > 0} is the upper half-plane and γ : [0, T ) 7→ H is a simple Jordan curve with γ(0) ∈ R and γ(0, T ) = {γ(t) : t ∈ (0, T )} ⊂ H. Then for each t ∈ (0, T ), Ht = H \ γ(0, t] is a simply-connected domain and hence by the Riemann mapping theorem, we can find a conformal map ft of H onto Ht. Moreover, we can require that ft has series expansion ft(z) = z − as z → ∞. Normalized in this way ft is unique and is said to be hydrodynamically normalized. The function C(t) is positive, continuous and strictly increasing: it is called the half- plane capacity of γ(0, t]. Thus we can reparameterize γ such that C(t) = 2t for all t, we will call this parameterization by half-plane capacity. With this normalization and parameterization, the function ft satisfies the differential equation (where f denotes differentiation with respect to z and ḟt denotes differentiation with respect to t): (0.1) ḟt(z) = − 2f ′t(z) z − ξ(t) , where ξ(t) = f−1t (γ(t)) is a continuous real-valued function. This is the chordal Loewner differential equation; ξ(t) is called the driving function of the slit γ. The converse is also true: given a measurable function ξ, the differential equation (0.1) with initial condition f0(z) ≡ z has solution ft which is a conformal map from H 2000 Mathematics Subject Classification. Primary 30C20; Secondary 30C30, 60K35 . http://arxiv.org/abs/0704.1933v3 2 J. TSAI −4 −2 0 2 4 6 8 0 100 200 300 400 500 600 Figure 1. A path on the hexagonal lattice on the upper half-plane (left) and a plot of its driving function on the y-axis against time on the x-axis (right). −2 0 2 4 6 8 10 12 0 50 100 150 200 250 Figure 2. A path on the square lattice on the upper half-plane (left) and a plot of its driving function on the y-axis against time on the x-axis (right). into itself (although f(H) is not necessarily a slit domain). Chapter 3 and 4 of [7] gives full details of this construction. Since Schramm’s discovery of stochastic Loewner evolution in 1999 (see [15]), there has been huge interest in the chordal Loewner differential equation and its variants. But the relationship between the slit inH and its resulting driving function is not well understood. There are a few papers that relate the behaviour of the slit with the behaviour of the driving function e.g. [10],[8]; also, the paper [2] calculates the slit arising from a few driving functions. In this paper, we will obtain a first order differential equation for ξ (which we can then solve numerically) that allows us to calculate the driving function ξ in the case where the curve γ is a trajectory arc of a certain type of quadratic differential. We will show that this includes, for example, the case when γ is a path on the square/triangle/hexagonal lattice in the upper half-plane or indeed, in any domain whose boundary lies on such a lattice. So for example, Figure 1 plots the driving function of a path on the hexagonal lattice in the upper half-plane and Figure 2 plots the driving function of a path on the square lattice in the upper half-plane. THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 3 We also note that we can obtain equivalent results for other variants of the Loewner differential equation for example, in the radial version or with multiple slits. We will discuss this in the paper as well. The proof of our formulae uses a generalization of Schwarz-Christoffel mapping to domains bounded by trajectory arcs of rotations of a given quadratic differential. We also mention that, currently, the common method used to find the driving function of a given slit is to use the Zipper algorithm discovered independently by D. E. Marshall and R. Kühnau to approximate the function ft which can then be used to determine the driving function. The Zipper algorithm can be viewed as a discrete version of the Loewner differential equation and hence is well suited to studying growth processes. It also has the advantage of being very fast. See [11] and [3]. 1. Main results To state our main results, we have to provide some background in the theory of quadratic differentials. Note that not all the terms used here are standard in the literature. See Chapter 8 of [13] and [16] for more details. A quadratic differential on a domain D ⊂ Ĉ = C ∪ {∞} is the formal expression Q(z)dz2, where Q(z) is a meromorphic function on D. Then for ω ∈ D with ω 6= ∞, Q(z) has Laurent series expansion about ω, Q(z) = ak(z − ω)k for some n > −∞ with an 6= 0. Then we define the degree of ω with respect to Q(z)dz2, degQ(ω), to be equal to n. If ∞ ∈ D, then near ∞, Q has Laurent series expansion given by Q(z) = then we define the degree of ∞ with respect to Q(z)dz2, degQ(∞) to be equal to m − 4. The “4” in the definition ensures that the degree is conformally invariant in a way which we will make precise later. Then ω ∈ D is: • a zero of Q(z)dz2 if degQ(ω) > 0. • a pole of Q(z)dz2 if degQ(ω) < 0. • an ordinary point of Q(z)dz2 if degQ(ω) = 0. A trajectory arc of Q(z)dz2 is a curve γ : (a, b) 7→ D that does not meet any zeroes and poles of Q(z)dz2 and satisfies Q(γ(t))γ̇(t)2 > 0 for all t ∈ (a, b). For θ ∈ [0, π), a θ-trajectory arc of Q(z)dz2 is a curve γ : (a, b) 7→ D that satisfies arg[Q(γ(t))γ̇(t)2] = 2θ for all t ∈ (a, b). Then γ is a θ-trajectory arc of Q(z)dz2 if and only if it is a trajectory arc of e−2iθQ(z)dz2. Hence, a 0-trajectory arc is simply a trajectory arc and we call a π/2-trajectory arc an orthogonal trajectory arc. It is clear that these definitions are invariant under reparameterization of γ so we will often call the point set of γ 4 J. TSAI a trajectory arc or θ-trajectory arc. We call a maximal trajectory arc a trajectory and similarly, a maximal θ-trajectory arc is called a θ-trajectory. For example, if we consider the quadratic differential 1dz2 in C, then the θ-trajectories are the straight lines with gradient exp(2θ). We now consider a special type of quadratic differential: Let D be a domain with piecewise analytic boundary. A Kühnau quadratic differential is a quadratic differential, Q(z)dz2, on D satisfying the following two properties: Definition: Let D be a domain with piecewise analytic boundary. A Kühnau qua- dratic differential is a quadratic differential, Q(z)dz2, on D satisfying the following two properties: (1) We can write such that each Γj is an open analytic arc with Γk ∩ Γj = ∅ for k 6= j and moreover, Q(z) extends continuously to each Γj and arg[Q(z)dz 2] is constant on each Γj i.e. each Γj is a θj-trajectory arc for some θj ∈ [0, π). (2) At z ∈ Γk ∩ Γj for all j 6= k, there are either only finitely many direction from which trajectories approach the point z or if there are infinitely many directions from which trajectories approach the point z, then for each such direction, there is only one trajectory that approaches z at this direction. These quadratic differentials are studied by Kühnau in [6] where he applies them to the study of certain Grötzsch-style extremal problems Property (i) above, also implies that D is locally connected. Thus each prime end of D corresponds to a unique point in ∂D (see [14, p. 27]). If, in addition, a point on ∂D corresponds to a unique prime end, then we make no distinction between the two. Let z be a prime end of D. Then we have 2 cases: Either z ∈ Γj for some j = 1, . . . , n; or there exist exactly 2 of the (Γj), that end at the prime end z. In the latter case, we will denote z by zk and assume that Γk, a θk-trajectory arc, and Γk−1, a θk−1-trajectory arc, are the only 2 arcs that end at zk. Then we can define the degree of zk in D with respect to Q(z)dz 2, degD,Q(zk), as follows: degD,Q(zk) = 2[|θk − θk−1|/π + Jk − 1] if θk 6= θk−1, 2Jk if θk = θk−1, where Jk is the number of trajectories of Q(z)dz 2 inside D that end at the prime end zk. If Jk is infinite, then the degree is not defined. Then for prime ends z such that z ∈ Γj for some j = 1, . . . , n, we define degD,Q(z) = 0. Although the motivation for this definition currently seems unclear, we will see that this indeed generalizes the concept of degree to points on the boundary. In particular, we will show that for x ∈ ∂H, if deg H,Q(x) ∈ Z, then Q can be extended to a meromorphic function in a neighbourhood of x with H,Q(x) = degQ(x). We then have the following theorem on Kühnau quadratic differentials in H: THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 5 Theorem 1.1. Suppose that Q(z)dz2 is a Kühnau quadratic differential on H. Then we have Q(z) = R (z − ζj)λj for some constant R 6= 0, ζj ∈ C, λj ∈ R for j = 1, . . . , n. This theorem can be viewed as a generalization of the Schwarz-Christoffel for- mula to domains bounded by θk-trajectory arcs of a given quadratic differential. We then have the following theorem on the Loewner driving function of a φ- trajectory arc of a Kühnau quadratic differential Q(z)dz2 that starts at a point ξ0 ∈ R with degH,Q(ξ0) = N ∈ {0, 1, . . .}. Theorem 1.2. Suppose that Q(z)dz2 is a Kühnau quadratic differential on H such that there is a point ξ0 ∈ R with degH,Q(ξ0) = N ∈ {0, 1, . . .}; then we have Q(w) = (w − ξ0)N (w − aj)αj where aj ∈ C and αj ∈ R. Let γ : [0, T ) 7→ ∂H be a simple curve such that γ(0) = ξ0, γ(0, T ) ⊂ H and γ(0, T ) is a φ-trajectory arc of Q(z)dz2 (φ ∈ [0, π)) that is parameterized by half-plane capacity. Suppose that the functions ft maps H conformally onto H \ γ(0, t] and are hydrodynamically normalized. Then for t ∈ (0, T ) (1.1) 2ξ(t) = −µ−C−(t)− µ+C+(t)− αjAj(t) +Σ0, (1.2) ξ̇(t) = − µ C−(t)− ξ(t) − C+(t)− ξ(t) − Aj(t)− ξ(t) with initial condition ξ(0) = ξ0. Where the functions Aj(t) are defined by Aj(t) = f t (aj) for j = 1, . . . , n, and C+(t) > C−(t) are the two preimages of ξ0 under ft; µ± = deg H\γ(0,t],Q(ft(C ±(t))), Nξ0 + We can then use Theorem 1.2 to find the driving function in the case when the slit γ consists of consecutive θk-trajectory arcs of given quadratic differentials. We will explain how to do this in further detail later. One difficulty with using Theorem 1.2 is that the parameterization is inherently given in terms of half-plane capacity. This makes it difficult to calculate the driving function ξ if we do not know anything about the half-plane capacity of the trajectory arc (which, in general, is the case). The next theorem will allow us to compare the parametrization with the length of the slit: 6 J. TSAI Theorem 1.3. Suppose that Q(z)dz2, γ and ft are as defined in Theorem 1.2. Let Φt(z) = Q(ft(z))f (z − ξ(t))2 . Then γ satisfies (1.3) γ̇(t) = −2 Φt(ξ(t)) Q(γ(t)) The rest of this paper is organized as follows: In the Section 2, we will state some basic results from the theory of quadratic differentials and use them to prove Theorem 1.1. Then we will use Theorem 1.1 to prove Theorems 1.2 and 1.3 in Sec- tion 3. In Section 4 we will discuss how to obtain the driving function numerically using Theorems 1.2 and 1.3. Finally in Section 5, we will discuss extensions of The- orem 1.2 to the case with multiple slits as well as to the radial Loewner differential equation. 2. Kühnau quadratic differentials and generalized Schwarz-Christoffel mapping The aim of this section is to prove Theorem 1.1. We will first look at some of the basic results in the theory quadratic differentials that we will need. Transformation Law. Suppose that f is a conformal map from a domain D2 onto a domain D1 and suppose that Q1(w)dw 2 is a quadratic differential on D1. If we define (2.1) Q2(z) ≡ Q1(f(z))f ′(z)2 then Q2(z)dz 2 is a quadratic differential on D2. Then, it is clear that θ-trajectory arcs are preserved by this transformation law i.e. γ is a θ-trajectory arc of Q2(z)dz 2 ⇔ f ◦ γ is a θ-trajectory arc of Q2(w)dw2 , and also, for z ∈ D2 degQ2(z) = degQ1(f(z)). Hence trajectories and degQ are conformally invariant in the above sense. The following lemma tells us that the behaviour of a quadratic differential at a neighbourhood of a point is determined by the degree of that point. Lemma 2.1 (Local behaviour of quadratic differentials). Let Q(z)dz2 be a qua- dratic differential on a domain D. Then for every ω ∈ D there is a conformal mapping w = φ(z) of some neighbourhood of ω such that Q(z)dz2 =   dw2 if degQ(ω) = 0, wndw2 if degQ(ω) = n ≥ 1, w−ndw2 if degQ(ω) = −n ≤ −1 with n odd, c2w−2dw2 if degQ(ω) = −2, (w−n + cw−1)2dw2 if degQ(ω) = −n ≤ −4 with n even. Here, c is the residue of a branch of Q(z) at ω. Proof. See Theorem 8.1 of [13] or Section 6 of [16]. � THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 7 So since trajectories are conformally invariant this lemma tells us that the local structure of trajectories around a point ω ∈ D is completely determined by degQ(ω) and the converse is true as well. Lemma 2.2. Suppose that ω ∈ D and degQ(ω) = n. Then (1) For n ≥ −1, there are exactly n + 2 trajectories of Q(z)dz2 that end at ω and form equal angles with each other. (2) For n ≤ −3, there are infinitely many trajectories ending at ω and more- over, there are |n| − 2 directions at ω forming equal angles such that the trajectories approach ω in these directions. (3) For n = −2, the behaviour depends on the value of c (as defined in Lemma 2.1). (a) If c is real, then the trajectories are the images of all radial lines under the map φ defined in Lemma 2.1. (b) If c is purely imaginary, then the trajectories are the images of all concentric circles under the map φ defined in Lemma 2.1. (c) If Re[c], Im[c] 6= 0 then the trajectories are the images of logarithmic spirals under the map φ defined in Lemma 2.1. Proof. See Section 7 of [16]. � This lemma shows that it makes sense for us to define degD,Q(x), the degree of a point on the boundary, in terms of the trajectories ending at x. If D = H and Q extends to a meromorphic function on a neighbourhood of some x ∈ R∪ {∞} with H,Q(x) finite. Then by studying the trajectory structure at x, we can see that H,Q(x) = degQ(x). This is the motivation for defining degD,Q in the way we have. The next lemma shows that the degD,Q is also conformally invariant: Lemma 2.3. Suppose that Q(z)dz2 is a Kühnau quadratic differential on a domain D and f is a conformal map of the upper half-plane H onto D. Then the quadratic differential Qf (w)dw 2 on H, defined by Qf(w) = Q(f(w))f ′(w)2, is also a Kühnau quadratic differential. Moreover, suppose that z ∈ ∂D is a prime end of D. Then degD,Q(z) = degH,Qf (f −1(z)). Proof. By Carathéodory’s theorem, f extends continuously to ∂H and by Schwarz’s reflection, f extends analytically across f−1(Γk) for all k = 1, . . . , n. Since θ- trajectory arcs are conformally invariant, this implies that Qf defined by (2.1) is a Kühnau quadratic differential on H. Moreover, for all k = 1, . . . , n, f(Γk) is a θk-trajectory arc of Qf (z)dz 2. Also each point on ∂H corresponds bijectively to a prime end of H. Hence there is a bijective correspondence between the points of ∂H and prime ends of D. Then degD,Q(z) = degH,Qf (f −1(z)) follows from the conformal invariance of trajectories. � Reflection across trajectories. Suppose that D is a domain such that Γ ⊂ ∂D is an open interval in R. Let Q(z)dz2 be a quadratic differential such that Γ is a trajectory arc or an orthogonal 8 J. TSAI trajectory arc of Q(z)dz2. Then let D− = {z : z ∈ D} be the reflection of D along Γ. Define Q−(z) = Q(z) for z ∈ D−. Then since Γ is a trajectory, we have Q(z) = Q−(z) ∈ R for z ∈ Γ. Thus by defining (2.2) Q∗(z) = Q(z) for z ∈ D, Q−(z) for z ∈ D−, Q(z) = Q−(z) for z ∈ Γ. it is easy to see that Q∗ is meromorphic in D ∪ D− and hence Q∗(z)dz2 is a quadratic differential on D ∪ D−. Thus by the transformation law (and using Schwarz reflection), this shows that we can extend quadratic differentials across trajectory arcs or orthogonal trajectory arcs. We will use reflection to prove the following lemma: Lemma 2.4. Suppose Q(z)dz2 is a Kühnau quadratic differential on H. Then for any z ∈ ∂H, deg H,Q(z) ∈ Z implies that Q(z)dz2 extends to a quadratic differential on a neighbourhood of z and hence H,Q(z) = degQ(z). Proof. Firstly, if z ∈ Γj for some j = 1, . . . , n. Then by definition degH,Q(z) = 0 and Q(z)dz2 can be extended to a neighbourhood of z by reflection. By definition, every z ∈ Γj is an ordinary point of Q(z)dz2 and hence degQ(z) = degH,Q(z) = 0. Otherwise we write z = zk and suppose that a θk−1-trajectory arc, Γk−1, and a θk-trajectory arc, Γk, end at zk. Then, by definition, degH,Q(zk) ∈ Z implies that θk − θk−1 is a multiple of π/2. Thus Γk−1 and Γk are trajectory arcs or orthogonal trajectory arcs of e−2iθk−1Q(z)dz2. Thus by reflection, e−2iθk−1Q(z)dz2 extends to a neighbourhood of zk. Hence, Q(z)dz 2 also extends to a neighbourhood of zk. � We can now prove Theorem 1.1; but first, we explain briefly why we can view Theorem 1.1 as a generalized form of Schwarz-Christoffelmapping: Schwarz-Christoffel mapping is a method of computing the conformal map between the upper half-plane and a domain bounded by a polygon. See [12] for more details. If we have a confor- mal map f fromH to some domainD such that the sides ofD consist of θ-trajectory arcs of the quadratic differential Q(w)dw2. Then Q(w)dw2 is a Kühnau quadratic differential on D and hence by Lemma 2.3, Q(f(z))f ′(z)dz2 is a Kühnau quadratic differential on H. Theorem 1.1 then implies that Q(f(z))f ′(z)2 = R (z − ζj)λj This is precisely the Schwarz-Christoffel formula when Q(z) ≡ 1. Also, we comment that the case when Q(w)dw2 is either negative or positive on R (i.e. the boundary of R consists only of trajectory arcs and orthogonal trajectory arcs) is easy to prove: we can use reflection to extend Q(z)dz2 to a quadratic differential on the Riemann sphere Ĉ. Hence Q(z) must be rational since property (ii) in the definition of Kühnau quadratic differentials guarantees that Q(z) does not THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 9 have any essential singularities and so Q(z) is rational (since the only meromorphic functions on Ĉ are rational). This proves Theorem 1.1 for this case. Proof of Theorem 1.1. Since Q(z)dz2 is a Kühnau quadratic differential, we can z1 < . . . < zm, (zk−1, zk) for k = 1, . . . ,m, (zm,∞) for k = m+ 1, (−∞, z0) for k = 0, such that each Γk is a θk-trajectory arc of Q(z)dz 2 for some θk ∈ [0, π). Let T = {Γ1, . . . ,Γm+1} . Then take any Γ ∈ T . Since Γ is a θ-trajectory for some θ, Γ is a trajectory arc of e−2iθQ(z)dz2; hence by reflection, we can reflect the quadratic differential e−2iθQ(z)dz2 across Γ to get a quadratic differential on H− = {Im(z) < 0} which we call Q̃(z)dz2. Similarly, by rotating Q̃(z)dz2, we can reflect it across another Υ ∈ T to get another quadratic differential Q∗(z)dz2 on H. Since Q∗ is obtained from Q by rotating twice, we have Q∗(z) = eiσQ(z) for some σ ∈ [0, 2π). This shows that Ψ(z) = Q′(z) (Q∗)′(z) Q∗(z) can be extended to a meromorphic function in C\{z1, . . . , zm}. Then part (ii) of the definition of Kühnau quadratic differentials implies that all the finite singularities of Ψ(z) are simple poles otherwise Q(z) would have an essential singularity which, by the great Picard theorem, contradicts part (ii) of the definition of Kühnau quadratic differentials. Thus we can write: Ψ(z) = h(z)− z − ζj where ζj ∈ C, and λj ∈ R, and h(z) is an entire function in C that does not does not vanish in C. This implies that Q(z) = exp h(ζ)dζ (z − ζj)λj (z − zk)νk Moreover, the singularity at ∞ of h(ζ)dζ cannot be essential by part (ii) of the definition of Kühnau quadratic differentials (otherwise we would get a contradiction with the great Picard theorem as above). This implies that h(ζ)dζ 10 J. TSAI is constant (since it has no zeroes or poles). Hence Q(z) = R (z − ζj)λj If ζj ∈ C \ R, then by definition, we must have λj = degQ(ζj). Moreover, if ζj ∈ R and degH,Q(ζj) <∞ we also have νk = degH,Q(zk). We will not prove this fact here but in the following corollary we will consider a special case. The general proof follows readily from it. We will prove the following corollary which is simply an application of Theorem 1.1 to domains slit by φ- trajectory arcs: Corollary 2.5. Suppose that Q(z)dz2 is a Kühnau quadratic differential on H such that there is a point ξ0 ∈ R with degH,Q(ξ0) = N ∈ {0, 1, . . .}; then we can write (2.3) Q(w) = R(w − ξ0)N (w − aj)αj where aj ∈ C, αj ∈ R, and also R is some non-zero constant. Let γ : [0, T ] 7→ H be a simple curve such that γ(0) = ξ0 and γ(0, T ) is a φ-trajectory arc of Q(w)dw 2 in H (φ ∈ [0, π)) and ζ = γ(T ) ∈ H is an ordinary point of Q(w)dw2 (i.e. degQ(ζ) = 0). Suppose that f maps H conformally onto H \ γ(0, T ]. Then f satisfies (2.4) Q(f(z))f ′(z)2 = R′(z − ξ)2(z − c−)µ (z − c+)µ (z −Aj)αj , where R′ is some constant; c−, c+ are the two preimages of ξ0 under f satisfying c− < c+; Aj is the preimage of aj under f ; and ξ is the preimage of ζ; and µ± = deg H\γ(0,s],Q(f(c Proof. Theorem 1.1 and Lemma 2.4 imply that Q(w) can be written as (2.3). Then Lemma 2.3 implies that Q̂(z) = Q(f(z))f ′(z)2 is a Kühnau quadratic differential. So by Theorem 1.1, we only need to look at the singularities of Q̂(z)dz2. Now, by Schwarz reflection, f extends to a conformal map on C \ {ξ, c−, c+}. Thus, by the conformal invariance of trajectories, this implies that Qf (z) = R ′(z − ξ)M (z − c−)µ (z − c+)µ (z −Aj)αj . Then by Lemma 2.2, there are exactly two φ-trajectory arcs of Q(z)dz2 ending at ζ = γ(T ) of which γ(0, T ) is one of them. So by the conformal invariance of trajectories, there is one φ-trajectory arcs of Q̂(z)dz2 ending at ξ that is contained in H. Hence, by definition, deg bQ,H(ξ) = 2. Using Lemma 2.4, this implies that degQ(ξ) = 2 i.e. M = 2. Thus we only need to determine µ − and µ+. THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 11 Note that since ξ0 has degree N with respect to Q(z)dz 2, we can determine, using Lemma 2.2, that the angle between γ(0, T ) and f((ξ, c−)) at ξ0 is πψ− = π degH,Q(f(c −)) + 2 N + 2 and similarly, the angle between γ(0, T ) and f((ξ, c−)) at ξ0 is πψ+ = π degH,Q(f(c +)) + 2 N + 2 Hence, by Schwarz reflection, the function F (z) = (f(z)− ξ0)1/ψ extends to a conformal mapping on a neighbourhood of c−. Thus in a neighbour- hood of z = c−, we can write (2.5) f(z) = ξ0 + (z − c−)ψ h(z)ψ where h is analytic in a neighbourhood of z = c− with h(c−) 6= 0. Now Q′f (z) Qf (z) Q′(f(z))f ′(z)2 Q(f(z)) f ′′(z) f ′(z) The residue at z = c− of the left-hand side of the equation is µ−, and we can use (2.5) to determine the residue at z = c− of the right-hand side. Thus we get µ− = degH,Q(f(c We apply the same method to c+ to get µ+. � 3. Domains slit by θ-trajectory arcs Let Q(w)dw2 be a Kühnau quadratic differential on H with deg H,Q(ξ0) = N ∈ {0, 1, . . .} for some ξ0 ∈ R. Then by Theorem 1.1 and Lemma 2.4, Q(w) = (w − ξ0)N (w − aj)αj where αj ∈ R and aj ∈ C. Now suppose that γ : [0, T ) 7→ H is a simple curve such that γ(0) = ξ0 and γ(0, T ) is a φ-trajectory of Q(z)dz 2 in H (φ = [0, π)) that is parameterized by half-plane capacity. As mentioned in the introduction, there exists conformal maps ft : H 7→ Ht = H \ γ(0, t] satisfying the hydrodynamic normalization. Then by restricting Q(w)dw2 to a quadratic differential on Ht we can induce via ft and (2.1), a quadratic differential on H: (3.1) Qt(z)dz 2 = Q(ft(z))f 2dz2. We now use Corollary 2.5 and (3.1) to prove Theorem 1.2. Proof of Theorem 1.2. Note that by Schwarz reflection, each ft can be extended to a conformal map on Ĉ \ {C−(t), C+(t), ξ(t)}. Then since ft(z) satisfies the hydrodynamic normalization, this implies that f ′t(z) 2 = 1 +O as z → ∞. 12 J. TSAI So by (3.1), (3.2) Qt(z) Q(ft(z)) = 1 +O as z → ∞. If we let ζ = 1/z, then we get (3.3) Qt(1/ζ) Q(ft(1/ζ)) = 1 +O as ζ → 0. Since ft is analytic in a neighbourhood of infinity, (3.3) is a Taylor series expansion and hence we can look at the Taylor series coefficients, in particular: C(0,ǫ) f ′(1/ζ)2 C(0,ǫ) Qt(1/ζ) ζ2Q(ft(1/ζ)) dζ = 0 for small enough ǫ > 0 where C(0, ǫ) is the anticlockwise contour about the circle with centre at zero and radius ǫ > 0. Then by Theorem 1.1 and Corollary 2.5, we can write Q(w) = R(w − ξ0)N (w − aj)αj Qt(z) = R ′(z − ξ)2(z − C−(t))µ (z − C+(t))µ (z −Aj(t))αj . Hence by the residue theorem (since ft(1/ζ) = 1/ζ + · · · as ζ → 0), this implies 2ξ(t) + µ (t) + µ (t) + αkAk(t) Nξ0 + This implies (1.1). To get (1.2), note that ft satisfies the chordal Loewner differen- tial equation (0.1) and hence if we let Gt = f t ◦ fs for some s ∈ (0, T ) fixed and t > s, then the chain rule implies that Gt satisfies the differential equation (z) = Gt(z)− ξ(t) Then for some s sufficiently close to t, we can write each Aj(t) = Gt(wj) wj ∈ C for all j = 1, . . . , n. Thus Ȧj(t) = Aj(t)− ξ(t) Similarly, we get Ċ±(t) = C±(t)− ξ(t) . Hence we get (1.2) from differentiating (1.1). � THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 13 An extension: We can extend Theorem 1.2 to the case when γ is made up of different θk-trajectory arcs of some quadratic differential Q(z)dz2: Let γ : (0, T ] 7→ H be a curve with γ(0) ∈ R such that there is a partition {0 = t0 < t1 < · · · < tr = T } such that γ(tk−1, tk) is a θk-trajectory arc of Q(z)dz 2 and γ(tk) is an ordinary point of Q(z)dz2 for k = 1, . . . , r. Then we can find the driving function ξ(t) of γ by applying Theorem 1.2 to the θ1-trajectory arc γ(0, t1) to get a driving function ξ1(t), and applying Theorem 1.2 inductively to each f (γ(tk, tk+1)) (which is a θk+1-trajectory arc of the quadratic differential Qtk(z)dz 2 = Q(ftk(z))f (z)2dz2) to get ξk(t). Then ξ(t) = ξk(t) for t ∈ [tk−1, tk). We also have the following corollary: Corollary 3.1. Suppose that Q(w)dw2 and γ are as defined in Theorem 1.2. Then the driving function ξ and Aj , C −, C+ as defined in Theorem 1.2 are in C∞(0, T ). Moreover, we can write any derivative of ξ, C−, C+, Aj explicitly in terms of ξ, C−, C+, Aj and the exponents µ −, µ+, αj. Proof. Recall that, in the proof of Theorem 1.2, we had the formulae Ȧj(t) = Aj(t)− ξ(t) , Ċ±(t) = C±(t)− ξ(t) . This implies that each term in (1.2) is differentiable so we can write the second derivative of ξ in terms of ξ(t), Aj(t), C ±(t). This in turn implies that we can write the third derivative of ξ in terms of ξ(t), Aj(t), C ±(t) and the exponents. Continuing inductively, we have showed that every derivative of ξ exists and can be expressed in terms of ξ(t), Aj(t), C ±(t) and the exponents. Note that each derivative of ξ is finite for t ∈ (0, T ) since |Aj(t)− ξ(t)|, |C±(t)− ξ(t)| > 0. Then ξ is smooth implies that Aj(t), C ±(t) are also smooth. � Theorem 1.3 then follows from Corollary 2.5 and Theorem 1.2: Proof of Theorem 1.3. First note that, by the definition of θ-trajectory arcs, γ̇ always exists and is never 0. Also by Corollary 2.5, Φt(ξ(t)) 6= 0,∞; thus the right hand side of (1.3) always exists since, by definition, γ avoids poles and zeroes of Q(w)dw2. Recall that ft(ξ(t)) = γ(t), this implies that γ̇(t) = ḟt(ξ(t)) + f t(ξ(t))ξ̇(t). Then combining the Loewner differential equation (0.1) with (3.1) we have ḟt(z) = − z − ξ(t) Qt(z) Q(ft(z)) Φt(z) Q(ft(z)) ⇒ ḟt(ξ(t)) = −2 Φt(ξ(t)) Q(γ(t)) Note that Φt(ξ(t)) 6= 0,∞ since, by Corollary 2.5, Qt(z) has a double zero at ξ(t). 14 J. TSAI Thus we have γ̇(t) = −2 Φt(ξ(t)) Q(γ(t)) Qt(ξ(t)) Q(γ(t)) ξ̇(t) = −2 Φt(ξ(t)) Q(γ(t)) since Theorem 1.2 implies that ξ̇ is finite for all t ∈ (0, T ) and Corollary 2.5 implies that Qt(ξ(t)) = 0. � 4. Applying Theorem 1.2 In practice, understanding ξ(t) via (1.1) is not possible: it is difficult to calculate the positions of the zeroes and poles of Qt because the information we have on them is all relative to ξ(t) (which we are trying to find). On the other hand, (1.2) is more useful in applications. In this section, we will demonstrate how we can use (1.2) to calculate numerically the driving function of a given slit that consists of θk- trajectory arcs of a given quadratic differential. The method is basically a modified version of Euler’s method. Firstly, for any smooth function h on (0,T), Taylor’s theorem implies that for all M = 1, 2, . . ., (4.1)∣∣∣∣∣h h(t) + )∣∣∣∣∣ ≤ M !KM s∈(t,t+ 1K ) for t, t + 1/K ∈ (0, T ). We will apply (4.1) to the functions ξ, Ak and C± (as defined in Theorem 1.2) noting that, by Corollary 3.1, they are smooth and all of their derivatives can be expressed in terms of ξ(t), Ak(t) and C ±(t). Thus if we know ξ(s), Ak(s) and C ±(s) we can use (4.1) to obtain an approximate formula for ξ(s+K−1), Ak(s+K −1) and C±(s+K−1) (choosing K to be small and/or M to be large so that the right-hand-side of (4.1) is small); then we can apply (4.1) to ξ(s+K−1), Ak(s+K −1) and C±(s+K−1) to find ξ(s+2K−1), Ak(s+2K −1) and C±(s+2K−1). Continuing like this, we obtain an approximation of ξ at the points {s+ nK−1}. So clearly what we need to do now is find the starting values ξ(s), Ak(s) and C±(s) so we can apply the above method. But because ξ is not differentiable at 0, we cannot use the formula (4.1) with t = 0. The way around this is to note that if deg H,Q(ξ0) = N ∈ {0, 1, 2, . . .} then since we know degH,Qt(C +(t)), we can calculate the angle that the trajectory makes with the line [ξ0,∞) (as in the proof of Corollary 2.5). Then we find that the angle is πψ where: πψ = π 2 deg (C+(t)) + 2 N + 2 So if we choose s small enough, we have fs ≈ Fψ,ξ0s , where Fψ,ξ0s is the conformal map that maps H conformally onto H s that is hydro- dynamically normalized where Hψs is the upper half-plane slit by the straight line starting at ξ0 making an angle πψ with [ξ0,∞), with half-plane capacity 2s. Then we also have Ak(s) ≈ (Fψ −1(ak), THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 15 and also, C−(s), C+(s) are approximately the two preimages of ξ0 under F ψ+,ξ0 Then we can use (1.1) to calculate ξ(s) approximately. We can then plug this information into (4.1) as described above. Note that F t can be found using the fact that (4.2) F λt (z) = z − (1− 2p) z − (1− 2p) t+ (1− p) for some λ. Then we reparameterize this formula to remove the λ and translate the point 0 to ξ0. Unfortunately, inverting this function cannot be done explicitly but it can be done numerically very efficiently using Newton’s method. Alternatively, by selecting a small s, we can assume that Ak(s) ≈ ak for all k. Then we note that the 2 preimages of ξ0 under F s can be determined explicitly (see [11]). This obviates the need to numerically invert F Another difficulty is that, in general, given a slit, we cannot parameterize it by half-plane capacity so it would be difficult, for example, to know at which t one should stop. Most formulae for calculating half-plane capacity of some compact set K rely on knowing the conformal map fK of H onto H\K (normalized hydrodynam- ically). One possibility would be to use the probabilistic definitions of half-plane capacity given in [7]. We will use the fact that Theorem 1.3 and Corollary 3.1 imply that we can give all derivatives of γ(t) in terms of ξ(t), Ak(t), C −(t), C+(t) and the exponents µ−, µ+, αk so if we know these, we can also use (4.1) to approximate γ. This in turn allows us to calculate the length of the slit γ. Thus if we know beforehand length of our slit, we can calculate at what value of t we stop. We now have everything we need in order to use (4.1) to calculate the driving function numerically of any slit that is made up of θk trajectory arcs of a qua- dratic differential Q(w)dw2. We will demonstrate how this is done in the following example: An example. Suppose that γ : (0, T ) → H is a piecewise linear arc parameterized by half-plane capacity that satisfies: • γ(0) = 0. • From t = 0 to t = t1, γ is the straight line arc from 0 to i; call this Γ1. • From t = t1 to t = t2, γ is the straight line arc from i to 2 + i; call this Γ2. • From t = t2 to t = t3 = T , γ is the straight line arc from 2 + i to 2 + 2i; call this Γ3. First note that γ is made up of alternating (π/2)- and 0-trajectory arcs of the qua- dratic differential 1dw2 in H and hence we can use Theorem 1.2 (or more specifically the extension of Theorem 1.2 detailed in Section 3) to calculate ξ̇. As mentioned previously, there is no easy way to know beforehand what t1, . . . , t3 are. For sim- plicity, we will only use M = 1 in (4.1) i.e. ≈ f(t) + ḟ(t) and fix a large K. Obviously Γ1 forms a right angle with real line; so we can use (4.2) to determine the function ft1 = F 1/2,0 (z) = z2 − 4t1. 16 J. TSAI −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 Figure 3. The example path in the upper half-plane (left) and a plot of its driving function on the y-axis against time on the x-axis (right). It is easy to see that in this case, t1 = 1/4 and ξ is constantly 0 for t ∈ (0, t1]. This induces the quadratic differential using (3.1): Qt1(z)dz z2dz2 (z + 1)(z − 1) . Hence, we let A1(t1) = −1, A2(t1) = 1. Also f−1t1 (γ2) is a 0-trajectory arc of Q1(z)dz 2 starting from ξ(t1) = 0 on R (by the conformal invariance of trajectories). Now note that f−1t1 (γ2) makes an angle of π/4 with the positive real axis. and hence ft1+K−1 ≈ F 1/4,0 sinceK is large. We can then use Newton’s method to find the preimages under the above approximation of ft1+K−1 of the points A1(t1), A2(t1) and the 2 preimages of zero to get the points A1(t1+K −1), A2(t1+K −1), C−(t1+K −1), C+(t1+K −1) and hence, using (1.1), we can find ξ(t1+K −1). Then inserting this into (4.1), as detailed above we can also find ξ(t1 + nK −1) and A1(t1 + nK −1), A2(t1 + nK −1), C−(t1 + nK−1), C+(t1 + nK −1); also, by Theorem 1.3, we can find |γ̇(t1 + nK−1)| if we let t2(K) = inf |γ̇(t1 + nK−1)| > (length of Γ2) = 2 then t2(K) ≈ t2 for K large. So we just assume that t2 = t2(K). Let A3(t2) = C−(t2) and A4(t2) = C +(t2). Hence by (3.1), Qt2(z)dz (z − ξ(t2))2(z −A3(t2))dz2 (z −A1(t2))(z −A2(t2))(z −A4(t2)) Then, by the conformal invariance of trajectories, f−1t2 (Γ3) is a π/2-trajectory of Qt2(z)dz 2 and also, f−1t2 (Γ3), makes an angle 3π/4 with (ξ(t2),∞) and so ft2+K−1 ≈ F 3/4,0 Then, as before, we can use Newton’s method to find the preimages under the above approximation of ft2+K−1 of the points A1(t2), . . . , A4(t2) and the 2 preimages of THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 17 ξ(t2) to get the points A1(t2+K −1), . . . , A4(t2+K −1), C−(t2+K −1), C+(t2+K and hence use (1.1) to get ξ(t2+K −1). We insert these into the formula iteratively to get ξ(t2+nK −1) and A1(t2+nK −1), . . . , A4(t2+nK −1), C−(t2+nK −1), C+(t2+ nK−1) until t2+nK −1 ≈ T . Thus the end result is that we found the driving func- tion of the first 3 steps of the slit given in Figure 3. Of course, our calculation of ξ will be more accurate by taking larger K. For example, we can use the above method to calculate the driving function of any path on the square/triangle/hexagonal lattice on H starting from some point in R. In fact we can calculate the driving function of a path on the square/triangle/hexagonal lattice in any polygon D by mapping the half-plane conformally onto D and pulling back the quadratic differential 1dw2 on D to Q(z)dz2 on H using the transforma- tion law. Also note that, in general, any curve γ can be approximated by a curve γδ which lies on the square lattice δZ 2. Then it can be shown that ξδ → ξ uniformly as δ ց 0, where ξδ is the driving function of γδ and ξ is the driving function of γ hence, we can use the above method to calculate ξδ then take the limit as δ ց 0 to obtain ξ. Another point to note is that using the above method, we do not need to know before hand what the trajectory arc of the given quadratic differential looks like; so for arbitrary Kühnau quadratic differentials, we can use this method to plot the trajectories starting at the boundary. We end this section by looking at what happens when the slit approaches the boundary: Proposition 4.1. Suppose that γ : [0, T ) 7→ H is a simple curve such that γ(0) ∈ R and γ(0, T ) is a θ-trajectory arc of some quadratic differential Q(z)dz2. Then let ξ be the driving function of γ. If γ(t) ∈ R ∪ γ(0, T ), i.e. γ makes a loop at time T . Then ∣∣∣∣→ ∞ as tր T for all n = 0, 1, . . .. Proof. For t ∈ (0, T ), we define Γ(t) = {γ(s) : s ∈ (t, T )}. Then Γt is a θ-trajectory arc in Ht = H\γ(0, t] of Q(w)dw2 and it is also a crosscut in Ht (see [14]). Then by the conformal invariance of θ-trajectories, f t (Γt) ⊂ H is a θ-trajectory arc of Qt(z)dz 2. Moreover, f−1t (Γt) is a crosscut of H with one end point at ξ(t) and the other end point in R such that either C+(t) or C−(t) is contained in the closure of the bounded component of H \ f−1t (Γt). Without loss of generality, assume it is C+(t). Then since diam(f−1t (Γt)) → 0 as t ր T , we must have ξ(t) = C+(T ) and hence by (1.2), ξ̇(t) → ∞ as t ր T . Similarly, we differentiate (1.2) as mentioned in Corollary 3.1 to obtain the result for higher order derivatives. � 18 J. TSAI This means that as γ gets closer and closer to making a loop, the approximation by (4.1) stops working no matter what M we choose. This phenomenon can be observed in Figure 3, as we turn the last corner in γ, we can see that ξ decreases faster even though the slit is not yet that close to the boundary. 5. Generalizing Theorem 1.2 5.1. Multiple slits. Suppose that γk : [0, T ) → H for k = 1, . . . , N are disjoint simple curves such that γk(0) ∈ R and γk(0, T ) ⊂ H. By the Riemann mapping theorem, there exists unique ft that map H conformally onto Ht = H\ k=1 γk(0, t] that satisfies the hydrodynamic normalization. We can reparameterize such that γk(0, T ] has half-plane capacity 2t. Then ft satisfies (5.1) ḟt(z) = −2f ′t(z) bk(t) z − ξk(t) where bk(t) = 1, and ξk(t) = f t (γk(t)). See [2] for more details. Theorem 5.1. Suppose that Q(w)dw2 is a Kühnau quadratic differential on H such that the points ξk(0) ∈ R satisfy H,Q(ξk(0)) = βk ∈ {0, 1, 2, . . .} for all k. Then we can write Q(w) = (w − ξk(0))βk (w − aj)αj with aj ∈ C and αj ∈ R. Then suppose that γk : [0, T ) → H for k = 1, . . . , N are disjoint simple curves such that γk(0) ∈ R and γk(0, T ) ⊂ H and are parameterized as above. Then (5.2) 2 ξk = − (µ−k C k (t) + µ k (t)) αjAj(t) +Σ0, (5.3) ξ̇l(t) = k=1,k 6=l bk(t) ξl(t)− ξk(t) bl(t) (t)− ξl(t) bl(t) (t)− ξl(t) αjbl(t) Aj(t)− ξl(t) for all l ∈ {1, . . . , N}. Where C−k (t) and C k (t) are the two preimages of ξk(0) under ft satisfying C k (t) < C k (t); µ±k = degHt,Q(ft(C k (t))); THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 19 Aj(t) = f t (aj); and βkξk(0) Proof. By Theorem 1.1 and Lemma 2.4, we can write Q(w) = (w − ξk(0))βk (w − aj)αj Then either by modifying the proof of Corollary 2.5 or iterating N slit functions and applying Corollary 2.5 N times, it is not too difficult to see that if we define Qt(z) by (3.1), then (5.4) Qt(z) = (z − ξk(t))2(z − C−k (t)) k (z − C+k (t)) (z −Aj(t))αj Then the proof of (5.2) is exactly the same as the proof of (1.1) in Theorem 1.2. The proof of (5.3) is more complicated. First let Pt(z) = −2 bk(t) z − ξk(t) Then (5.1) becomes ḟt(z) = f t(z)Pt(z). Now take the logarithmic derivative of Qt(z) with respect to z and t separately using the definition of Qt(z) given by (3.1) to get Q′t(z) Qt(z) Q′(ft(z))f Q(ft(z)) f ′′t (z) f ′t(z) Q̇t(z) Qt(z) Q′(ft(z))ḟt(z) Q(ft(z)) ḟ ′t(z) f ′t(z) Q′(ft(z))f t(z)Pt(z) Q(ft(z)) f ′′t (z)Pt(z) + f t(z)P t (z) f ′t(z) where we substitute (5.1) in for ḟt to get from the first to the second line. Thus we (5.5) Q̇t(z) Qt(z) Q′t(z) Qt(z) Pt(z) + 2P t(z). So then by (5.4), we note that Q′t(z) Qt(z) z − ξk(t) z − C−k (t) z − C+k (t) z −Aj(t) Q̇t(z) Qt(z) 2ξ̇k(t) z − ξk(t) µ−k Ċ k (t) z − C−k (t) µ+k Ċ k (t) z − C+k (t) αjȦj(t) z −Aj(t) 20 J. TSAI 0 2 4 6 8 10 Figure 4. A plot of the two driving functions ξ1 (top) and ξ2 (bottom) on the y-axis against time on the x-axis. Thus substituting this into (5.5) and comparing the coefficient of z − ξl(t) (i.e. the residue at z = ξl(t) of both sides of (5.5)), we find that this is exactly (5.3). � Similarly, we can prove a version of Theorem 1.3 and Corollary 3.1 for multiple slits. This means that we can use the method detailed in Section 4 with (5.3) to calculate the driving function for multiple θk-trajectory arc slits. For example Figure 4 plots the graph of the two driving functions ξ1 and ξ2 in the case when γ1 and γ2 are 2 vertical slits starting from -1 and 1 (i.e. orthogonal trajectories of 1dz2) and growing at the same speed. Compare this with Figure 7 in [2]. 5.2. Radial Loewner evolution. The chordal Loewner differential equation was introduced because the upper half-plane was an easier domain to work with for many applications but the original setting of the Loewner differential equation is in the unit disc D = {z ∈ C : |z| < 1}: Suppose that γ : [0, T ) 7→ D is a simple curve such that γ(0) ∈ T = {z : |z| = 1} and γ(0, T ) ⊂ D \ {0}. Then Dt = D \ γ(0, t] is simply-connected and 0 ∈ Dt for all t ∈ (0, T ). Hence the Riemann mapping theorem implies that there is unique conformal map ft mapping D conformally onto Dt such that ft(0) = 0 and f t(0) > 0. Then Schwarz’s lemma and the Carathéodory kernel theorem implies that f ′t(0). is strictly decreasing and continuous so we can reparameterize such that f ′t(0) = e −t. f ′t(0) is sometimes called the conformal radius of Dt; hence in this case we are parameterizing by conformal radius. Then the functions ft satisfy the radial Loewner differential equation: ḟt(z) = −zf ′t(z) z + eiξ(t) z − eiξ(t) See [9] for more details. THE LOEWNER DRIVING FUNCTION OF TRAJECTORY ARCS 21 Theorem 5.2. Suppose that Q(w)dw2 is a Kühnau quadratic differential on D such that degQ(0) = K ∈ Z and eiξ0 ∈ T = {|z| = 1} satisfies = N ∈ {0, 1, 2, . . .} then we have Q(w) = wK(w − eiξ0)N (w − aj)αj where aj ∈ C and αj ∈ R. Then if γ : [0, T ) 7→ D is a simple curve such that γ(0) = eiξ0 and γ(0, T ) ⊂ D \ {0} is a φ-trajectory arc of Q(w)dw2 in D that does not meet 0 and is parameterized as above. Then we have (5.6) e2iξ(t) = e−2tΠ0C −(t)−µ C+(t)−µ Aj(t) −αj , (5.7) ξ̇(t) = − 1 C−(t) + eiξ(t) C−(t)− eiξ(t) C+(t) + eiξ(t) C+(t)− eiξ(t) Aj(t) + e iξ(t) Aj(t)− eiξ(t) where, as usual, the functions Aj(t) are defined by Aj(t) = f t (aj) for j = 1, . . . , n, µ± = degDt,Q(ft(C ±(t))), C+(t) > C−(t) are the two preimages of eiξ0 under ft; and also, Π0 = e Proof. The formula for Q(w) can be obtained from Theorem 1.1 by the transforma- tion law. We then define Qt by (3.1). Since the point 0 is fixed by ft, this implies that the degQt(0) = degQ(0) = K. Thus we can apply Corollary 2.5 (again, using the transformation law) to get Qt(z) = z K(z − eiξ(t))2(z − C−(t))µ (z − C+(t))µ (z −Aj(t))αj Then since by definition, Qt(z) = Q(ft(z))f This immediately implies (5.6) by substituting z = 0. Then we get (5.7) in the same way as we get (1.2) from (1.1) in the proof of Theorem 1.2. � As in the case of multiple slits, a version of Theorem 1.3 and Corollary 3.1 holds for this case. 22 J. TSAI 5.3. Other versions of the Loewner differential equation. There are several other versions of the Loewner differential equation for simply-connected domains in the literature; the methods in this paper should work in those cases as well and the proofs should be similar to the proofs of Theorem 1.2 etc. Also, [4], [5] generalizes the Loewner differential equation to multiply-connected domains and again, some of the methods should work in these cases possibly using methods in [1] to extend Theorem 1.1 to multiply-connected domains. Finally, even if we consider general 2-dimensional growth processes given by the Loewner-Kufarev differential equation (see Chapter 6 of [13]), some of the methods in this paper should still be applicable. Acknowledgement. The author would like to express his gratitude to his super- visor Dr. T. K. Carne for his constant guidance and helpful discussion when writing this paper. References 1. D. Crowdy. The Schwarz-Christoffel mapping to bounded multiply connected polygonal do- mains. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461(2061):2653–2678, 2005. 2. W. Kager, B. Nienhuis, and L. P. Kadanoff. Exact solutions for Loewner evolutions. J. Statist. Phys., 115(3-4):805–822, 2004. 3. T. Kennedy. Computing the Loewner driving process of random curves in the half plane. arXiv:math/0702071v1 4. Y. Komatu. Untersuchungen über konforme Abbildung von zweifach zusammenhängenden Gebieten. Proc. Phys.-Math. Soc. Japan (3), 25:1–42, 1943. 5. Y. Komatu. On conformal slit mapping of multiply-connected domains. Proc. Japan Acad., 26(7):26–31, 1950. 6. R. Kühnau. Über die analytische Darstellung von Abbildungsfunktionen insbesondere von Extremalfunktionen der Theorie der konformen Abbildung. J. Reine Angew. Math., 228:93– 132, 1967. 7. G. F. Lawler. Conformally invariant processes in the plane. American Mathematical Society, USA, 2005. 8. J. R. Lind. A sharp condition for the Loewner equation to generate slits. Ann. Acad. Sci. Fenn. Math., 30(1):143–158, 2005. 9. K. Löwner. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I. Math. Ann., 89, 1923. 10. D. E. Marshall and S. Rohde. The Loewner differential equation and slit mappings. J. Amer. Math. Soc., 18(4):763–778 (electronic), 2005. 11. D. E. Marshall and S. Rohde. Convergence of the zipper algorithm for conformal mapping. preprint, 2006. 12. Z. Nehari. Conformal mapping. Dover Publications, New York, 1982. 13. C. Pommerenke. Univalent functions. Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, Band XXV. 14. C. Pommerenke. Boundary behaviour of conformal maps. Springer-Verlag, Berlin, 1992. 15. O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118:221–288, 2000. 16. K. Strebel. Quadratic differentials. Springer-Verlag, Berlin, 1984. Department of Mathematics, Chinese University of Hong Kong, Shatin, New Terri- tories, Hong Kong E-mail address: jtsai@math.cuhk.edu.hk http://arxiv.org/abs/math/0702071 Introduction 1. Main results Definition: 2. Kühnau quadratic differentials and generalized Schwarz-Christoffel mapping Transformation Law Reflection across trajectories 3. Domains slit by -trajectory arcs An extension: 4. Applying Theorem ?? An example. 5. Generalizing Theorem ?? 5.1. Multiple slits 5.2. Radial Loewner evolution 5.3. Other versions of the Loewner differential equation Acknowledgement. References

On the measurement problem for a two-level quantum system Alexey A. Kryukov ∗ November 12, 2018 A geometric approach to quantum mechanics with unitary evolution and non-unitary collapse processes is developed. In this approach the Schrödinger evolution of a quantum system is a geodesic motion on the space of states of the system furnished with an appropriate Riemannian metric. The measuring device is modeled by a perturbation of the metric. The process of measurement is identified with a geodesic motion of state of the system in the perturbed metric. Under the assumption of random fluctuations of the perturbed metric, the Born rule for probabilities of collapse is derived. The approach is applied to a two-level quantum system to obtain a simple geometric interpretation of quantum commutators, the uncertainty principle and Planck’s constant. In light of this, a lucid analysis of the double-slit experiment with collapse and an experiment on a pair of entangled particles is presented. KEY WORDS: measurement problem - Born rule - Berry’s phase - EPR-paradox 1 GEOMETRY AND QUANTUM MECHANICS Geometric ideas have played a well recognized role in modern physics, especially in general relativity (GR) and gauge theories (GT). They also found a well established position in quantum mechanics (QM) in considerations related to Berry’s phase Ref. [1]. However, whereas in GR and GT geometry (i.e., the metric or connection) defines the dynamics of the theory, the geometric methods pertaining to Berry’s phase do not enjoy such a sweeping significance. The reason for this difference is quite obvious. Indeed, the geometry underlying GR and GT is directly related to ∗Department of Mathematics, University of Wisconsin Colleges E-mail: alexey.kryukov@uwc.edu, aakrioukov@facstaff.wisc.edu http://arxiv.org/abs/0704.1934v1 2 Alexey A. Kryukov the physical fields (gravitational or gauge) in the theory. At the same time, the Fubini-Study metric in the geometric interpretation of Berry’s phase (Refs. [2],[3] amongst many others) depends only on the geometry of the Hilbert space of states of quantum system. The latter geometry (i.e. Hilbert metric) is insensitive to changes in the Hamiltonian of the system and, consequently, is not dynamical. At the same time, it turns out to be easy to make the metric on Hilbert space of states of a closed quantum system dynamical Refs. [4],[5]. For this, notice first of all that the Schrödinger equation = − i ĥϕt (1.1) is the equation for integral curves of the vector field hϕ : H −→ TH, hϕ = − ih̄ ĥϕ associated with the Hamiltonian ĥ of the system. Here H is the Hilbert space of states of the system and TH is the tangent bundle over H. Assume that the Hilbert space H is a space of functions that are square-integrable with respect to an appropriate measure. Because the evolution governed by Eq. (1.1) is unitary, the integral curve through initial point ϕ0 on the unit sphere S H in H will stay on the sphere. Since this holds true for any initial point (modulo the domain issues), one concludes that the restriction of the vector field hϕ to the sphere S H is a vector field on the sphere. In the ordinary QM spaces TϕH tangent to H at ϕ ∈ H are identified with the space H itself. Similarly, spaces TϕS H tangent to the sphere SH at ϕ ∈ SH are identified with (real) affine subspaces of H. In particular, the metric on SH , whenever used, is assumed to be induced by the embedding of SH into H. However, the sphere SH is a manifold and thus, can be defined independently of the ambient space H. As such, SH is a Banach manifold which means that it can be obtained by “gluing together” open sets in a Banach space. The Hilbert metric Gϕ : TϕH × TϕH −→ C on tangent spaces TϕH can be also defined independently of the metric on H as an Hermitian tensor field on H. Such a tensor field gives rise to a Riemannian metric GRϕ : TϕS H ×TϕSH −→ R on SH , defined at each ϕ ∈ SH GRϕ(X,Y ) = 2ReGϕ(ξ, η). (1.2) Here X = (ξ, ξ), Y = (η, η) ∈ TϕSH are considered as vectors in the realization of the tangent space TϕH. The manifold S H , furnished with the (2, 0)-tensor field GRϕ, is then a Riemannian manifold. In the following, the manifold S H with the metric GRϕ will be denoted by S The final step in making the metric Gϕ on the sphere of states S G dynamical is to ensure that the integral curves of hϕ (i.e. the solutions to Schrödinger equation Eq. (1.1)) are geodesics on SG. For this it turns out to be sufficient to define Gϕ : TϕH × TϕH −→ C by Gϕ(ξ, η) = h̄ (ĥĥ∗)−1ξ, η . (1.3) On the measurement problem 3 Here ĥ∗ is the adjoint of ĥ (normally equal to ĥ) and the Hamiltonian ĥ is assumed to be invertible. Incidentally, even if the Hamiltonian ĥ is not bounded on H, it becomes bounded as an operator mapping points ϕ ∈ H into tangent spaces TϕH with the metric Eq. (1.3) (see Ref. [5]). Further general results concerning QM on Hilbert manifolds can be found in Ref. [4]. These results demonstrate that QM can be formulated in terms of geometry of the space of states. The goal of the Letter is to provide such a geometric formulation in case of a simple two-level system and to establish its advantages. Namely, the point of view that the space of states represents a new arena for physical processes and the evolution of state is a motion along geodesic is shown to be effective in addressing the major conceptual difficulties of quantum mechanics. Although the discussion deals primarily with a simple model, the most important results can be shown to be quite general. Some of these generalizations are described in the Letter while others are left for the upcoming publications. 2 ELECTRON IN A HOMOGENEOUS MAGNETIC FIELD Consider a free non-relativistic electron propagating in the direction of the X-axis in a homogeneous magnetic field B. The evolution equation (the Pauli equation) for the electron is Ψ− µσ̂ ·BΨ, (2.1) where Ψ = Ψ(s, x, t), s = 1, 2 is a two-component state function of the electron, µ is the electron’s magnetic moment and σ̂ = (σ̂x, σ̂y, σ̂z) is the vector made of Pauli matrices. The substitution Ψ(s, x, t) = ψt(x)ϕt(s) separates variables and produces two independent evolution equations. The first describes the evolution governed by the free Hamiltonian ψt. (2.2) The second equation describes the evolution in the space C2 of spinors ϕ: = −µσ̂ ·Bϕt. (2.3) It follows that in the case of the product states Ψ(s, x, t) = ψt(x)ϕt(s), one can analyze the evolution of spin state ϕt in the space of states C 2 without needing to involve the infinite-dimensional Hilbert space of states Ψ. 2.1 Quantum Mechanics on the Space of States S3 Let us proceed to reformulation of quantum mechanics of the system in geometrical terms. In this, the fact that the sphere S3 of unit normalized spin states can be 4 Alexey A. Kryukov furnished with the group structure of the group SU(2) will be helpful. The group structure will allow us to exploit simple results from differential geometry of Lie groups which will make the resulting picture more transparent and complete. First of all, the Hamiltonian ĥ = −µσ̂ ·B defines the vector field hϕ = ih̄µσ̂ ·Bϕ on the sphere S3 in the space C2 with the metric (ξ, η)C2 = k ξkηk. The integral curve of hϕ (i.e. the solution of Eq. (2.3)) through ϕ0 ∈ S3 is given by ϕt = e µσ̂·Btϕ0. (2.4) Since ϕt is a path in C 2, it is natural to call the vector the velocity of evolution of the system. The speed of evolution in the C2 is the norm of dϕt in C2 metric. Using (σ̂ ·A)(σ̂ ·B) = A ·B+ iσ̂ ·A×B, (2.5) one has (σ̂ ·B)2 = B2. (2.6) Therefore, by Hermicity of the matrix σ̂ ·B, one obtains µσ̂ ·Bϕt, µσ̂ ·Bϕt , (2.7) where B is the norm of B. In particular, the speed of evolution of the system depends only on the magnitude of the field. To make the evolution of the system a motion along a geodesic, the metric Gϕ on S3 will be defined by Eq. (1.3). Since ĥ is self-adjoint, one obtains ĥĥ∗ = ĥ2 = µ2(σ̂ ·B)2 = µ2B2I, where I is the identity operator on C2 and Eq. (2.6) has been used at the last step. Therefore, up to the constant factor (h̄/µB)2, the metric Gϕ coincides with the one induced by the embedding of S 3 into C2. That means that the carriers of the geodesics on S3 are the intersections of S3 with the planes through the origin. The fact that the found Riemannian metric is so simple is due to an especially simple form of the Hamiltonian in the model. If S3 is identified with the group manifold SU(2), the obtained metric is the Killing metric on SU(2). To see this, let us identify in the standard way the space C2 of complex vectors ϕ = with the space Mat of 2× 2 matrices z1 z2 −z2 z1 . (2.8) The map ω : ϕ −→ ϕ̂ is an isomorphism of (real) vector spaces C2 and Mat. The sphere S3 of unit states in C2 is identified via ω with the subset of matrices with unit determinant. The latter subset is the group SU(2) under matrix multiplication. On the measurement problem 5 The Killing metric on the Lie algebra su(2) can be defined by (X̂, Ŷ )K = cTr(X̂Ŷ +), (2.9) where c 6= 0 is an arbitrary constant and Tr stands for the trace. The Killing metric on the group SU(2) is defined for all ϕ̂ ∈ SU(2) and all left-invariant vector fields (ϕ̂) = ϕ̂X̂, L (ϕ̂) = ϕ̂Ŷ , X̂, Ŷ ∈ su(2) by (ϕ̂), L (ϕ̂))K = (LX̂(e), LŶ (e))K = (X̂, Ŷ )K , (2.10) where e is the identity element in SU(2). By direct substitution one verifies that (L (e), L (e))K = 2cRe(X,Y )C2 when- ever X̂ = dω(X) and Ŷ = dω(Y ) with dω being the differential of the map ω. In other words, the Killing metric is proportional to the metric induced by the em- bedding of S3 into the Euclidean space C2 = R4. This verifies that the metric Gϕ = (h̄/µB) 2I obtained earlier, is the Killing metric. For any two left invariant vector fields L on SU(2) the connection ∇ on SU(2) can be defined by [X̂,Ŷ ] . (2.11) Notice that the left invariant vector fields form a basis on the tangent space T SU(2) for all ϕ̂ ∈ SU(2). In particular, Eq. (2.11) is sufficient to define a connection on SU(2). This connection is symmetric, as the torsion tensor vanishes: ) = ∇L [X̂,Ŷ ] [Ŷ ,X̂] [X̂,Ŷ ] = 0. (2.12) The connection Eq. (2.11) is also compatible with the Killing metric, that is, for any vector fields ξ, η, ζ on SU(2) the following is true: ∇ξ(η, ζ)K = (∇ξη, ζ)K + (η,∇ξζ)K . (2.13) Indeed, assuming that ξ = L , η = L , ζ = L are left invariant, one has (ϕ̂), L (ϕ̂)K = (LŶ (e), LẐ(e))K = const (2.14) and therefore the left hand side of Eq. (2.13) vanishes. For the right hand side, by definition Eq. (2.11) one obtains: )K + (LŶ ,∇LX̂LẐ)K = ([X̂, Ŷ ], Ẑ)K + (Ŷ , [X̂, Ẑ])K . (2.15) From the anti-Hermicity of elements of su(2) one also has: ([X̂, Ŷ ], Ẑ)K = −cTr(X̂Ŷ Ẑ) + cTr(Ŷ X̂Ẑ) (2.16) 6 Alexey A. Kryukov ([Ŷ , X̂ ], Ẑ)K = −cTr(Ŷ X̂Ẑ) + cTr(X̂Ŷ Ẑ). (2.17) As a result, the sum on the right hand side of Eq. (2.13) is also zero which verifies that the connection Eq. (2.11) is compatible with the metric. In other words, the connection ∇ is the Levi-Civita connection of the Killing metric. For any magnetic fieldB in the model the one parameter subgroup ϕ̂t = e µσ̂·Bt is a geodesic through the identity e ∈ SU(2). Indeed, since dϕ̂t = −ϕ̂t ih̄µσ̂ ·B, the path ϕ̂t is the integral curve of the left invariant vector field Lĥϕ̂ = −ϕ̂ µσ̂ · B. Using the definition Eq. (2.11) one then has: [̂h,̂h] = 0. (2.18) Geodesics through an arbitrary point ϕ̂0 ∈ SU(2) can be then written in the form ϕ̂t = ϕ̂0e µσ̂·Bt. Considered as paths with values in C2, these geodesics take the form ϕt = e µσ̂·Btϕ0. The curvature tensor of ∇ can be obtained directly from the definition ]LẐ . (2.19) In particular, [[X̂,Ŷ ],Ẑ] . (2.20) and ( [X̂, Ŷ ], [Ẑ, Ŵ ] . (2.21) The sectional curvature in the plane through L is defined by . (2.22) With the help of Eqs. (2.21) and (2.10) this becomes [X̂, Ŷ ], [X̂, Ŷ ] X̂, X̂ Ŷ , Ŷ X̂, Ŷ . (2.23) Suppose for example that X̂ , Ŷ , Ẑ and Ŵ correspond to the spin observables. Recall that in the Planck system of units the operator of spin ŝ has eigenvalues ±1/2 and can be expressed in terms of the Pauli matrices σ̂1, σ̂2, σ̂3 as σ̂, (2.24) On the measurement problem 7 where σ̂ = (σ̂1, σ̂2, σ̂3). The corresponding anti-Hermitian generators êk = k = 1, 2, 3, form a basis of the Lie algebra su(2) and satisfy the commutator relations [êk, êl] = ǫklmêm, (2.25) where ǫklm denotes the completely antisymmetric tensor of rank three. In the basis {êk} the curvature tensor Eqs. (2.20), (2.21) takes the form Rik,lm = (δilδkm − δimδkl), (2.26) Rik,lm = (δilδkm − δimδkl) (2.27) where δik is the Kronecker delta. The symmetry property Rik,lm +R l,mk +R m,kl = 0 (2.28) of the curvature tensor coincides in the model with the Jacobi identity [[X̂, Ŷ ], Ẑ] + [[Ŷ , Ẑ], X̂] + [[Ẑ, X̂ ], Ŷ ] = 0 (2.29) for the Lie algebra elements X̂, Ŷ , Ẑ. From the isomorphism given by Eq. (2.8) it follows that any vector x = (x1, x2, x3) in the Euclidean space R3 can be identified with the element kiσ̂k =∑ k êk of the Lie algebra su(2). Under such an identification the Euclidean norm ‖x‖R3 of x is equal to detx and rotations in R3 are represented by transformations x −→ ÛxÛ+ with Û ∈ SU(2). One can make this identification into an isometry by assuming the equality of Euclidean and Killing norms ∥∥∥∥∥ 2xkêk ∥∥∥∥∥ = ‖x‖R3 . (2.30) This will fix the constant factor in front of the Killing metric. Note that the Euclidean space R3 can be identified with the space of all possible classical angular momenta of a particle. The electron’s possible angular momenta form a sphere S2 in R3. By equating the norms according to Eq. (2.30), the spaces tangent to S2 are identified with affine subspaces of spaces tangent to the sphere of states S3 with the induced metric. In particular, the sphere S2 can be identified with a submanifold of the space of states S3 with the induced metric. Let us remark that this identification is analogous to the identification of the classical space with the submanifold of point supported states in an infinite-dimensional Hilbert space of states, considered in Refs. [4], [6]. In the Killing metric Eq. (2.10) on S3 one has kêk, k,m x kxmδkm. To satisfy Eq. (2.30) the constant c must be 1/2, that is, the needed metric in Planck units has the form (X̂, Ŷ )K = Tr(X̂Ŷ +). (2.31) 8 Alexey A. Kryukov Using the formula Eq. (2.23), one obtains the following expression for the sectional curvature R(p) in the plane p through orthogonal vectors L ([ê1, ê2], [ê1, ê2])K (ê1, ê1)K (ê2, ê2)K = 4 (ê3, ê3)K = 1. (2.32) So the sectional curvature of S3 in Planck units is equal to 1. Note that in an arbitrary system of units the Killing metric would be multiplied by h̄2 and the sectional curvature would be equal to 1/h̄2. The dimension of sectional curvature is consistent with the fact that the tangent space su(2) is spanned by the spin operators having the dimension of angular momentum. In this approach the Planck’s constant and the commutators of spin observables acquire a transparent geometric interpretation. According to Eq. (2.23) the commu- tator of two observables is directly related to the sectional curvature of the sphere S3. Indeed, assume for simplicity that the vector fields L are orthogonal and unit normalized in the Killing metric. Then from Eqs. (2.22), (2.23) one has the following expression for the norm of the commutator of X̂ and Ŷ : ∥∥∥[X̂, Ŷ ] = 4R(p). (2.33) Here R(p) is the sectional curvature of S3 in the plane p = L(L ) which for the considered Riemannian metric was found to be a constant equal to 1/h̄2. Note that for L which are orthogonal but not unit, the equation Eq. (2.23) takes the form ∥∥∥[X̂, Ŷ ] = 4R(p) ∥∥∥X̂ ∥∥∥Ŷ . In particular, if the norms of X̂ and Ŷ are of order h̄ (e.g., X̂, Ŷ are the spin observables), then the norm of the commutator [X̂, Ŷ ] is of order h̄ as well. Note that despite the fact that the commutator [X̂, Ŷ ] is small in these units, it is of the order of the radius of the sphere of states, making quantum effects on the sphere quite transparent. The results obtained so far in this section were model specific. It is then im- portant to know whether they can be generalized to the case of higher dimensional spaces of states and of arbitrary observables. Also, what if the Hamiltonian of the system is time-dependent? Here is a sketch of what can be done in these cases. For any n the sphere of states S2n−1 in the space Cn is a homogeneous space U(n)/U(n − 1), where U(n) denotes the unitary group on Cn. The Killing metric on U(n) can be used to induce a Riemannian metric on S2n−1 via the embedding. Namely, the 2n − 1 linearly independent generators in the Lie algebra u(n), which belong to the orthogonal complement of a (fixed) subalgebra u(n − 1), form a sub- space V ⊂ u(n). The one-parameter subgroups eX̂τ with X̂ in V sweep a sphere S2n−1 and yield geodesics in the induced metric. The curvature of S2n−1 can be then computed via equations Eqs. (2.20), (2.21) with generators in V . Furthermore, the commutators of generators in V are related to the sectional curvature of S2n−1 by the same formula Eq. (2.33). On the measurement problem 9 Note that there exist anti-Hermitian observables that are not in V . For example, if n = 2 so that S3 = U(2)/U(1) = SU(2), then V is the Lie subalgebra su(2) ⊂ u(2) and so V consists of the traceless elements of u(2). If X̂ ∈ u(2) is not traceless, the one-parameter subgroup eX̂τ is still a geodesic in U(2). However, this geodesic does not “stay” on the subgroup SU(2) = S3. Of course, one could still consider the curves on S3 given by ϕt = e X̂τϕ0, for some point ϕ0 in S 3 ⊂ C2. For any given X̂ ∈ u(2) and all initial points ϕ0 these curves are still geodesics in the appropriate Riemannian metric (see section 1 and Ref. [4]). However, the algebraic features of the model change and the formulas connecting the commutators with the curvature are different. For the time-dependent Hamiltonians ĥ in the Hilbert space Cn the approach can be generalized as follows. The sphere of states S2n−1 is replaced with the manifold M = S2n−1 ×R, where R is the time line. Then, there exists a Riemannian metric on M in which the paths (ϕt, t) with ϕt = e −îhtϕ0 are geodesics for all ϕ0 ∈ S2n−1. Finally, as already mentioned, in the infinite-dimensional case there still exists a Riemannian metric for which all solutions to the Schrödinger equation with an invertible (time-independent) Hamiltonian are geodesics. However, the algebraic properties of the model require further investigation in this case. 2.2 Quantum Mechanics on the Projective Space of States CP 1 In physical experiments one can only determine the state of a system up to a com- plex non-zero factor. That means that the space of physical states is the complex projective space CPH of complex lines in the space of states. In the considered example it is the one dimensional complex projective space CP 1. By definition, CP 1 is the quotient C2∗/C∗, were ∗ means “take away zero”. In other words, CP 1 is the base manifold of the fibre bundle π : C2∗ −→ CP 1 with the natural projection along the fibres C∗. By considering unit normalized states only, one obtains CP as a quotient S3/S1. It is then the base of the fibre bundle π : S3 −→ CP 1, which is a sub-bundle of the previous fibre bundle. If ϕt is a path of the electron’s state on S 3 and π : S3 −→ CP 1 is the bundle projector, then π(ϕt) is a path on the base CP 1. Since this latter path represents what can be measured in experiments, it is important to obtain an explicit formula for π. For this consider a point ϕ = on S3 and let {ϕ} be the complex line formed by vectors λϕ, λ ∈ C. Provided ϕ1 6= 0, there is a unique point of intersection of the line with the affine plane in C2 formed by vectors , ξ ∈ C. Namely, by setting , (2.34) 10 Alexey A. Kryukov one obtains . (2.35) The map ρ = ϕ −→ ξ provides a coordinate chart on CP 1 which identifies CP 1 without a point (complex line through ) with the set C of complex numbers. The affine plane of vectors form a subspace in the Lie algebra su(2). The algebra su(2) itself has been identified earlier with the Euclidean space R3 of vectors kiσ̂k. The stereographic projection then identifies the unit sphere S the origin of R3 with the above plane C plus a point, i.e., with CP 1 itself. The relationship of the coordinate ξ in the plane C with coordinates (x1 = x, x2 = y, x3 = z) of the corresponding point on the sphere S2 is given by x+ iy . (2.36) Solving this for x, y and z and using Eq. (2.35), one obtains: x = ϕ1ϕ2 + ϕ1ϕ2, (2.37) y = i(ϕ1ϕ2 − ϕ1ϕ2), (2.38) z = ϕ2ϕ2 − ϕ1ϕ1. (2.39) The resulting map π : S3 −→ S2 given by (ϕ1, ϕ2) −→ (x, y, z) is the needed projection on the space of physical states. The equation for the integral curve Eq. (2.4) can be simplified by choosing the coordinate axes properly. In particular, one can always assume that the Z-axis is parallel to the magnetic field B. In this case σ̂ ·B = σ̂3B and Eq. (2.4) simplifies to eiωtϕ0+ e−iωtϕ0− , (2.40) where ω = µB and the initial state ϕ0 is equal to . Recall that the speed of evolution of the electron along S3 was given by Eq. (2.7). Let us find the speed of the projection of this evolution on the space of physical states S2 = CP 1. For this recall that the Killing metric on su(2) coincides with the Euclidean metric. The embedding of S2 into su(2) = R3 induces the familiar metric on S2. Such a metric also coincides with the famous Fubini-Study metric on CP 1 identified with S2. Since this metric is just a restriction of the Euclidean metric on R3, one has( . Using Eqs. (2.37)-(2.40), one obtains then = 4ω|ϕ0+||ϕ0−| = |ϕ0+||ϕ0−|. (2.41) On the measurement problem 11 If θ is the angle between the Z-axis and the vector (x, y, z), then sin2 θ = 1− z2 and one finds with the help of Eq. (2.39): sin θ. (2.42) In particular, if the initial state is an eigenstate of σz so that θ = 0 or π, then the speed ds vanishes and the evolution of the electron is projectively trivial. In other words, the eigenstates of the Hamiltonian ĥ = −µσ̂ ·B are zeros of the “push- forward” vector field dπ(hϕ) on CP 1. Here dπ is the differential of the map π and µσ̂ ·Bϕ as before. Recall that the integral curves ϕt given by Eq. (2.4) are geodesics on S 3. At the same time, as one can see from Eqs. (2.37)-(2.40), the projected curves {ϕt} are not in general geodesics in the induced metric. Moreover, there does not exist a Riemannian metric on CP 1 = S2 in which the projection by π of an arbitrary geodesic on S3 yields a geodesic on S2. Indeed, the great circles of S2 (parametrized by arc length) are geodesics in the Fubini-Study metric. At the same time, they are projections of geodesics on S3. The latter fact is obvious for the equator of S2 if one chooses |ϕ0+| = |ϕ0−| in Eqs. (2.37)-(2.40). For any other great circle this fact is verified by a change in direction of the Z-axis. But the condition that the great circles are geodesics fixes the metric on S2 up to a constant factor. This, together with the above observation that the projection {ϕt} of a geodesic may not be a geodesic itself, proves the claim. Suppose, as it is advocated here, that evolution by geodesics is a valid principle of quantum dynamics. Then the obtained result suggests that the space of states S3 may be “more physical” than the projective space CP 1 of “physical states”. In general terms, the actual, “easy to describe” physics may be happening on the space of states of a quantum system. However, our experiments can so far access only the projection of physical processes into the space of physical states. The projection of a process may look much less “natural”, then the original process on the space of states. It was verified earlier that the commutators of spin observables are related to geometry of the space of states S3. Later on in the paper the geometry of a measure- ment process will be discussed. It is then important to know that the probability of transition from state ϕ to state ψ in a measurement also has ties to geometry. Namely, this probability depends only on the distance between {ϕ} and {ψ} in the Fubini-Study metric on the projective space of states (see Ref. [2]). This result is immediate in the model under consideration. In fact, assume that the state ψ is the spin-up eigenstate of σ̂z (one can always assure this by a proper choice of coordinate axes). Then, according to Eq. (2.37), one has z = |ϕ1|2 − |ϕ2|2. (2.43) 12 Alexey A. Kryukov Using the normalization condition |ϕ1|2 + |ϕ2|2 = 1, one obtains |ϕ1|2 = . (2.44) If θ ∈ [0, π] is the angle between the vectors representing {ϕ} and {ψ} on S2, then z = cos θ and the equation Eq. (2.44) takes the form P (θ) = cos2 . (2.45) Here P (θ) = |(ψ,ϕ)|2 = |ϕ1|2 denotes the probability of transition. Notice that θ is the length of the arc of a great circle through {ϕ} and {ψ}. In other words, it is a geodesic distance between the points {ϕ} and {ψ} on S2 in the Fubini-Study metric. In particular, the probability P (θ) of transition between two states depends only on the distance between them. Furthermore, the uncertainty principle for the spin observables also has a simple geometrical interpretation. Since uncertainties of observables do not depend on the overall phase of a state, the geometry underlying the uncertainty principle is, once again, projective. The uncertainty principle for non-commuting observables X̂, Ŷ can be written in the form ∆X∆Y ≥ ∣∣∣(ϕ, [X̂, Ŷ ]ϕ) ∣∣∣ , (2.46) where ∆X2 = (ϕ, X̂2ϕ) − (ϕ, X̂ϕ)2 and similarly for ∆Y 2. Assuming X̂, Ŷ are the components ŝx, ŝy of the spin observable ŝ = σ̂, one has ∆sx∆sy ≥ |(ϕ, ŝzϕ)| . (2.47) Let us connect this inequality with geometry of the projective space S2 = CP 1. In light of the geometric interpretation of the commutators of observables Eq. (2.33) and the equation Eq. (2.46), the existence of such a connection is not surprising. Indeed, from the geometry of the sphere S2 furnished with the Fubini-Study (i.e., the usual!) metric, for any point (x, y, z) on the sphere one has: (y2 + z2)(x2 + z2) ≥ z2. (2.48) The inequality simply says that the product of distances from a point (x, y, z) on the sphere S2 ⊂ R3 to the X and Y -axes is at least |z|. This equation is the geometric form of the uncertainty principle. Indeed, from Eqs. (2.37)-(2.39) it follows that (ϕ, σ̂xϕ) = x, (ϕ, σ̂yϕ) = y, (ϕ, σ̂zϕ) = z, (2.49) where x, y, z are coordinates of the point of S2 representing the state ϕ. In addittion, σ̂2x = σ̂ y = σ̂ z = I and so ∆σ2x = 1−x2 = y2+z2, ∆σ2y = 1−y2 = x2+z2, ∆σ2z = 1−z2 = x2+y2. (2.50) On the measurement problem 13 With the use of Eqs. (2.49), (2.50), the inequality Eq. (2.48) is now equivalent to the uncertainty principle Eq. (2.47). Note that using the angles θx, θy, θz between the coordinate axes and the vector x = (x, y, z) in R3, one can write the equations Eqs. (2.50) in the form ∆σx = sin θx ∆σy = sin θy, ∆σz = sin θz. (2.51) In particular, the uncertainty ∆E in energy for the electron in the state ϕ in the model takes the form ∆E = µB sin θ, (2.52) where θ is the angle between the vectors B and x. Recall that θ is a geodesic distance between the points {ϕ} and {ψ} on S2. Therefore, the uncertainty ∆E is the largest, when {ϕ} is furthest away from the eigenstates, i.e. when θ = π the other hand, ∆E vanishes at the eigenstates of ĥ. The latter statement can be generalized as follows. Let H = Cn and let  : H −→ H be an observable with a simple spectrum λ1 < λ2 < ... < λn and eigenfunctions ϕ1, ϕ2, ..., ϕn. Let {ϕt} be a geodesic through the states {ϕk}, {ϕl} in the Fubini-Study metric on the projective space of states CPH . Then for any point {ϕ} on the geodesic the variance ∆A2 = (ϕ, Â2ϕ)− (ϕ, Âϕ)2 is an increasing function of the distance from {ϕ} to the pair of eigenstates {ϕk}, {ϕl} in CPH . The latter distance is simply the shortest of the distances from {ϕ} to the states {ϕk}, {ϕl}. In fact, because the eigenstates are orthogonal, the probability of transition from a state {ϕ} on the geodesic to a state {ϕi} with i 6= k, l is equal to zero. In particular, {ϕ} = {ckϕk + clϕl} for some coefficients ck, cl. Therefore, ∆A2 = |ck|2λ2k + 1− |ck|2 λ2l − |ck|2λk + (1− |ck|2)λl . (2.53) If {ϕ} coincides with {ϕk} so that |ck| = 1, then ∆A vanishes. On the other hand, as |ck| decreases, ∆A increases until |ck| becomes equal to 1√2 . By Eq. (2.45) this means that ∆A increases with the distance from {ϕ} to {ϕk} until {ϕ} becomes equally distant from {ϕk} and {ϕl}. It is important to note that the “geometric” probability of transition formula Eq. (2.45) is valid in an arbitrary space of states H = Cn. Also, as already discussed, the relationship Eq. (2.33) between the commutators and the curvature holds true on the sphere of states S2n−1 for observables in the subspace V ⊂ u(n) (see the end of section 2.1). Because of that, the uncertainty principle Eq. (2.46) can be still interpreted geometrically. The infinite-dimensional case requires a different approach and needs further analysis. 14 Alexey A. Kryukov 3 TENSOR PROPERTIES OF EQUATIONS IN THE MODEL In the developed geometrical formulation of the model, the space of states is a manifold furnished with a Riemannian metric which defines quantum dynamics on the space. It is well known that differential geometry of manifolds admits two equivalent formulations: local coordinate and coordinate free. Let us discuss the role played by both formulations in the model. Consider a pair X̂, Ŷ of elements of the Lie algebra su(2) and the corresponding pair Xϕ = X̂ϕ, Yϕ = Ŷ ϕ of the associated vector fields. By direct computation one sees that [Xϕ, Yϕ] = −[X̂, Ŷ ]ϕ, (3.1) where [Xϕ, Yϕ] is the Lie bracket of the vector fields. Recall that the integral curves of non-commuting vector fields cannot form a coordinate grid on the manifold. In particular, the integral curves of vector fields sxϕ, syϕ, szϕ associated with the spin observables ŝx, ŝy, ŝz do not form a coordinate grid on S 3. By the above, these integral curves are geodesics in the Killing metric on S3. The fact that they do not form a coordinate grid is then a direct consequence of the curvature of S3. Instead, the fields sxϕ, syϕ, szϕ form a local (non-coordinate) basis at every point of S There are, of course, many ways of choosing coordinates on S3. One natural choice is the normal coordinate system given on a neighborhood of any point by the exponential map. If {ek} is a basis on the tangent space TϕS3 at ϕ ∈ S3 and kek is a tangent vector, then the equation of geodesic ϕt through ϕ in the direction of A in the normal coordinates ϕ1, ϕ2, ϕ3 is linear: ϕkt = a kt, k = 1, 2, 3. (3.2) The evolution of spin state ϕ in time could be thought of as a motion along the manifold S3×R with R being the time axis. The direct product of the Killing metric on SU(2) and the usual Euclidean metric on R makes S3 × R into a Riemannian manifold. If ϕt is a geodesic on S 3, then (ϕt, t) will be a geodesic on S 3 × R. Notice that for any evolution ϕt of the state along S 3 ×R one can find a co-moving coordinate system on S3×R in which (ϕt, t) = (ϕ0, t), i.e., the state is at rest. Such a co-moving system is directly related to the well known Heisenberg representation. The coordinates typically used on the projective space of physical states CP 1 are homogeneous (ϕ1, ϕ2) and inhomogeneous ξ = , η = coordinates. For example, the Fubini-Study metric is usually expressed in terms of these coordinates. Other coordinate systems may be useful in applications. In particular, according to Eq. (2.49), the expectation values of Pauli matrices for a system in a state ϕ coincide with the x, y and z coordinates of the point {ϕ} in S2. Therefore, these expectation values can be identified with local coordinates on CP 1. This fact was used in section 2 to describe the motion of state along CP 1. On the measurement problem 15 The group SU(2) acting on the space C2 is the symmetry group of the theory. In particular, the group SU(2) “extends” to act on tensor algebra over C2 and the Schrödinger equation is a tensor equation. For any fixed time t the Schrödinger evolution operator Û(t, 0) given by Eq. (2.4) is an active realization of SU(2)- transformations on C2. The corresponding passive realization consists in a unitary change of basis on C2. One could consider instead the sphere S3 ⊂ C2 as a base manifold for tensor bundles and make the group SU(2) act locally on tensor products of spaces tangent and cotangent to S3. For each tensor type this gives a subbundle of the correspond- ing tensor bundle over C2. The Schrödinger equation is then a tensor equation with terms which are vector fields on S3. In fact, it is the equation for integral curves of the vector field hϕ = − ih̄ ĥϕ on S 3. Alternatively, it is the equation of geodesics in the Killing metric on S3. The notion of symmetry in QM is usually understood as an invariance of the Hamiltonian of the system under a symmetry transformation. In this case the Hamiltonian commutes with the transformation and the generator of transformation becomes a constant of motion. Although this is certainly true for rotations about the field direction in the model, such a restricted understanding of symmetry is not suitable for this Letter. Mathematically, the Schrödinger equation hϕ = − ih̄ ĥϕ in the model is written in a specific basis on the space of states C2. Under a unitary transformation Û ∈ SU(2) of the basis the Schrödinger equation behaves as a vector equation. In particular, the right hand side of the equation becomes equal to − i Û−1ĥϕ. Since coordinates of points ϕ also change to become ψ = Û−1ϕ, the right hand side of the equation takes the form Û−1ĥÛψ. (3.3) Note that Û−1 and Û in Eq. (3.3) act on different spaces! Namely, Û−1 acts on the tangent space TϕC 2, while Û acts on C2 itself. It is common, however, to identify vector spaces with the spaces tangent to them. By following this practice, one sees that under a change of basis the Hamiltonian is transformed by ĥ −→ Û−1ĥÛ . In other words, it transforms as a tensor of rank (1, 1). If in addition ĥ and Û commute, then (and only then!) the Hamiltonian is invariant and the usual conserved quantities exist in accordance with the Noether’s theorem. A particular choice of an orthonormal basis on the space of states C2 (alterna- tively, on S3 or CP 1) has an affect on results of observations expressed in the basis. The reason for that is clear: which state is a spin-up state, for example, depends on the basis in C2. This is quite analogous to dependency of the state of rest in classical mechanics on a choice of reference system. Alternatively, under the identification Eq. (2.8) of space C2 with a space of 2×2- matrices the choice of an orthonormal basis in C2 dictates the choice of basis in the Lie algebra su(2), i.e., the choice of sigma-matrices. A unitary transformation of 16 Alexey A. Kryukov sigma-matrices changes their eigenvectors, thus affecting the results of observations expressed in terms of these eigenvectors. The above can be also rephrased in terms of the geometry of the projective space CP 1 = S2 ⊂ R3. Namely, a unitary transformation of basis in C2 induces an orthogonal transformation of basis on R3. But the choice of a basis in R3 determines the result of measurement of the Z-component of electron’s spin. One can, therefore, conclude that a choice of coordinates on manifolds C2, S3 and CP 1 has a physical meaning similar to the choice of a reference system on the classical space and should not be neglected. On the other hand, physical laws must be, of course, coordinate independent. In the considered case this is assured by the geometric nature of the model. In particular, the Schrödinger equation Eq. (1.1) can be written in a form independent of an orthonormal basis on the space of spin states C2, i.e., in a vector form. In this case a state is just a vector Φ of C2 rather than a column of components in a particular orthonormal basis on C2. The Hamiltonian is an operator ĥ rather than a matrix. The equation Eq. (1.1) then becomes a vector equation on the manifold S3 written in a coordinate-free form: = − i ĥΦt. (3.4) Note that the eigenvalue problems for observables in the model are also tensor equations. In particular, the eigenvalue problem for the Hamiltonian can be written in a coordinate free form: ĥΦ = λΦ. (3.5) The tensor character of equations of quantum theory in the model signifies that the principle of relativity holds true on the space of states. This means, first of all, that both, the active transformations on the space of states and the passive transformations of coordinates on the space are available. In particular, there exist various physically distinguishable reference systems on the space of states (say, dif- ferent bases on C2). Most importantly, the validity of principle of relativity in the theory means that the equations of the theory are the same when written in any such reference system, that is, they are tensor equations. One may doubt the significance of such a principle in the example. After all, the SU(2) symmetry in the model has been known for years. Why would such a “relativistic” view of this symmetry be useful? Note however, that the quantum dynamics in the advocated approach takes place on the space of states. In particular, the evolution of a quantum system is the motion along a geodesic on the (curved) space of states. Because of that the notions of a reference system, of passive and active transformations, of tensor equations, as well as other differential-geometric notions on the space of states, become physically meaningful. This meaning is very On the measurement problem 17 much in line with the meaning of similar notions in special and general relativity or in the theory of gauge fields. For instance, a change in direction of the magnetic field B induces an active transformation on the space of states. This transformation has physically measur- able consequences: it moves geodesics on the space of states and hence changes the evolution of electron’s state. At the same time, the change in B can be compensated by a passive transformation. Indeed, one can choose an orthonormal basis on C2 so that the components of B in the corresponding basis in R3 remain the same. The equation of the new electron’s path in this basis coincides then with the original Most certainly, the above principle is different from the principle of relativity in space-time. In fact, it deals with tensor properties of equations on a Hilbert space of states, rather than on space-time. At the same time, it has the same kind of underlying mathematics and the same spirit as the ordinary principle of relativity. The principle is in fact a particular instance of the principle of functional relativity introduced in Ref. [4]. As discussed, the Hamiltonian in the model is not in general invariant under unitary transformations. In particular, an active unitary transformation associated with a change in direction of the magnetic field B produces a new Hamiltonian. In other words, the Hamiltonian is not a scalar in the tensor approach to the model. On the other hand, due to geometric nature of the model, one can easily identify the most important scalars (or invariants), forming the “bone structure” of the theory. One such invariant is the distance between any two points on spaces of states C2, S3 and CP 1 furnished with the above discussed metrics. Another one is the speed of quantum evolution in S3 in a given magnetic field given by Eq. (2.7). Yet another one is the scalar curvature of S3 or CP 1. This curvature can be expressed in terms of the sectional curvature of S3 which was found to be 1/h̄2. Let us remark that tensor character of the theory allows one to extend the origi- nal symmetry group SU(2) to the group GL(2, C) of general linear transformations acting on fibers of the tangent bundle over S3. Moreover, Schrödinger dynamics on the space of states S3 can be formulated in a way invariant under general coordi- nate transformations on S3. This follows at once from the fact that the Schrödinger equation in the model is the equation of geodesics on the Riemannian manifold S3. As such, this equation is meaningful in arbitrary coordinates on the manifold. Finally, let us comment on a possible argument against the advocated geometric approach. In the model considered here the Riemannian metric on the sphere S3 has turned the sphere into a manifold of constant sectional curvature. This allowed us to relate the curvature of the metric with the Planck’s constant. However, in general the sectional curvature of the Riemannian metric defined by Eq. (1.3) will not be a constant. Even when it is, there is no reason for this constant to be the same as in the model under discussion. It seems therefore that by making the Riemannian metric depend on the Hamiltonian of the system, one shall in general loose the 18 Alexey A. Kryukov relationship between the curvature and the Planck’s constant! To reply to this argument recall that the model considered here is non-relativistic (in the usual sense). In particular, the equation Eq. (2.1) is a special case of the Pauli equation. The latter equation is well known to be the non-relativistic limit of the Dirac equation for electron in electromagnetic field. The Dirac equation can be written in the form = cσ̂ · p̂− e , (3.6) where ϕ̃, χ̃ are two-component spinors, (φ,A) is the 4-potential of the field, p̂ is the momentum operator and e is the electron’s charge (see, for example, Ref. [8]). The largest term in Eq. (3.6) is the one containing the mc2 factor. By substituting (3.7) into Eq. (3.6) one recovers in the standard way the Pauli equation for the spin state ϕ with values in C2. Assume that the 4-potential in Eq. (3.6) describes a weak homogeneous magnetic field B and let ĥD be the corresponding Hamiltonian. Consider the metric GD given by Eq. (1.3) with the Hamiltonian ĥD. Then solutions to the Dirac equation for electron in the field B are geodesics in this metric. Note that the Hamiltonian ĥD has in the non-relativistic limit the form mc2 + ĥ, where ĥ is the non-relativistic Hamiltonian used in Eq. (2.1). Accordingly, the metric GD can be written in the form (h̄/mc2)2(I + ǫ), where ǫ is a small correction due to the Hamiltonian ĥ and I is the identity. It follows that the sectional curvature of GD consists of the main term of the order (mc2/h̄)2 and a small correction due to ĥ. Since the main term is constant (i.e., it does not depend on the fields), the advocated geometric interpretation of the curvature remains possible. However, a more careful analysis of the situation requires a “functional relativistic” formulation of the problem and will be discussed elsewhere. 4 THE PROCESS OF MEASUREMENT In the model under consideration the Schrödinger equation is the equation of geodesics on the space of states S3 furnished with the Killing metric. That means that the dynamics in the theory takes place on the Hilbert space of states rather than on the classical space. In this and the following sections it will be argued that the space of states is also the most appropriate background for tackling problems related to quantum measurement. In particular, the process of collapse of a state can be re- garded as a geodesic motion in the space of states with the metric “skewed” by the measuring device. On the measurement problem 19 Consider a pair of spin-1/2 particles. In QM the most general spin state of such a pair has a form i,j=± cijϕiψj , (4.1) where ϕ+, ψ+ are spin-up, and ϕ−, ψ− spin-down states of the particles. The Hilbert space of states having such a form is the tensor product C2⊗C2. The unit normalized states form a sphere S7 in this four-dimensional complex space. Whenever Ψ is not a product of states of the particles, the state of the pair is called entangled. It is well known that, when the particles are microscopic (i.e., suf- ficiently small in mass and size), the entangled states do indeed exist. By assuming the universal validity of QM, one concludes that the entangled states can be also prepared when one of the particles is replaced with a macroscopic measuring device, designed to measure spin of the second particle. In this case the total state of the system has the form Ψ = aϕ+ψ+ + bϕ−ψ− where ψ± represent states of the device, corresponding to spin-up and spin-down outcomes of measurement. However, unlike the case of microscopic objects, the entangled states with macroscopic objects have never been observed in experiments. The phenomenon of decoherence does not help resolve this problem because the mixtures of states of macroscopic objects have not been observed in experiments either. Recall that in the classical physics the motion of a pair of interacting particles on a manifold can be thought of as a motion of a point in a higher dimensional configuration space. Suppose in particular that particles of massesm andM interact gravitationally and move in the space R3 in accordance with the Newton’s Second Law. Then the motion of the pair is represented by a trajectory in configuration space R6. However, if M ≫ m, the motion simplifies and can be thought of as a motion of the particle of mass m in the field created by the particle of mass M . In this case the configuration space R6 of the pair is “effectively reduced” to the space R3 and the field on R3 created by the heavier particle. An analogous “reduction” of configuration space is implicitly present in the unitary QM whenever the influence of a “macroscopic surrounding” of a quantum system is accounted for by an appropriate choice of potential in the Schrödinger equation. Let us explore the idea that a similar approach can be applied to the process of measurement in the model. Namely, assume that the motion of the total state function of the electron and the measuring device during their interaction can be effectively replaced with the motion of electron’s state function in S3 under the influence of a physical field on S3 created by the measuring device. This means that, in some sense, the state function of the device does not change much as a result of interaction. One immediate objection is that the observed states of the device are orthogonal and so the state cannot change “just a little”. Without addressing this problem in detail, let us point out that the metric on the space of states of the device may be 20 Alexey A. Kryukov “skewed”. As a result, two different position eigenstates of a pointer may become very close in this metric. For instance, let H be the Hilbert space obtained by completion of a space of ordinary functions on R3 with respect to the inner product (ϕ,ψ)H = e−(x−y) ϕ(x)ψ(y)d3xd3y. (4.2) Such a space contains in general the eigenstates of position operator, i.e., the delta- functions δ(x − a). Moreover, two different position eigenstates δ(x − a), δ(x − b) with ‖a− b‖R3 ≪ 1 are close in the metric Eq. (4.2) (see Ref. [4]). If, in particular, δ(x − a), δ(x − b) are the eigenstates of a pointer, then the fact that theses states are close in H can imply that the state of the pointer does not change much in the process of interaction with a measured microscopic system. This argument does not prove, of course, that the proposed “reduction formal- ism” can be consistently implemented into the theory. To validate the formalism one must demonstrate that all imaginable measurements in QM can be modeled (at least in terms of their outcomes) by a field on the Hilbert space of states of the mea- sured system. In what follows such a demonstration will be presented in the case of a finite dimensional space of states and a time-independent observable. Namely, in this case a specific working model of measurement based on a perturbation of the metric on the space of states of the system will be constructed. Moreover, the effec- tiveness and the scope of the proposed method suggest that it can be successfully applied in general. For guidance in modeling the process of measurement let us return to the ex- ample under consideration. Suppose that the device in the example measures the component of electron’s spin in the direction of magnetic field. Equivalently, since the Hamiltonian is given by ĥ = −µσ̂ · B, the device can be designed to measure the electron’s energy. Without loss of generality one may assume that the field is directed along the Z-axis. Then the two eigenstates {ψ1}, {ψ2} of ĥ in CP 1 are positioned on the Z-axis at the poles of the sphere S2 = CP 1. Recall that these eigenstates are zeros of the vector field hϕ = − ih̄ ĥϕ projected on CP 1. In particu- lar, the Schrödinger evolution of ψ1, ψ2 is projectively trivial. So, as a result of the interaction between the electron and the device, the original circular motion of the electron’s state along a parallel on S2 = CP 1 is changed to the state of rest at one of the poles on the sphere. The following hypothesis, which will be clarified and exemplified later on, seems to be in order: (H1) The measuring device creates a physical field on the sphere of states with sources at the eigenstates of the measured observable. This field is capable of driving the electron’s state toward one of the eigenstates. Note that in the position measuring experiments a measuring device is a system of counters distributed in space at the eigenstates of the position observable. The counters are indeed sources of interaction between the particle and the apparatus. On the measurement problem 21 The points in the classical space where the measured particle can be found, can be identified with the position eigenstates of the particle (see Refs. [4], [6]). Since the counters are also located at these points, one can identify them with sources in the space of states of the particle. Finally, because the sources are capable of catching the particles, the (loosely stated) hypothesis (H1) is satisfied. This gives one the hope that a specific form of the hypothesis can be, in fact, realized. What could be the nature of the field postulated in the hypothesis? Recall that the Riemannian metric in the model is dynamical, i.e., it drives the evolution of the electron’s state. Suppose that the field is nothing but a perturbation of the metric on the sphere of states S3, induced by the presence of the measuring device. The evolution of the electron’s state during the measurement is then the motion along a geodesic in the perturbed metric. Note that the metric on the total space S3 and not only on the base space CP 1 = S2 must be perturbed. In fact, as already discussed, the projection {ϕt} of a Schrödinger evolution ϕt is not in general a geodesic. The above approach is certainly attractive, in particular, it does not require any ad hoc features in the theory. Furthermore, the approach can be easily implemented by an appropriate “denting” of the sphere of states of the system. Namely, as shown below, by perturbing the metric on the sphere S3 one can “redirect” the evolution of electron’s state so that the state would become stationary. For a greater generality, assume that the Hilbert space H of states of the system is n-dimensional and let (ϕ1, ..., ϕn) be the usual coordinates on H = Cn. Let the Riemannian metric on H to be of the form gik = η 2δik, (4.3) where η = η(ϕ) is a function on H. The equation of geodesics in this metric can be obtained in the usual way by variation of the length functional on paths ϕt. Let s be the arc length parameter and let τ be a parameter defined by dτ = ds . Then the equation of geodesics in the metric Eq. (4.3) on H can be written in the form ∇η2, (4.4) where ϕτ is identified with ϕt(τ). A similar equation in R 3 is well known in geomet- rical optics where it describes propagation of rays in a media with refractive index η. The equation Eq. (4.4) is also similar to the Newton equation of motion for a unit mass in the field U = −η2/2. The form of equation Eq. (4.4) makes it easy to see that for any sufficiently smooth path ϕτ there exists a function η such that the equation is satisfied, at least on a neighborhood of ϕ0 = ϕτ |τ=0. That is, ϕτ is a geodesic through ϕ0 in the Riemannian metric Eq. (4.3) on H, at least for small values of τ > 0. That also means that an arbitrary sufficiently smooth path ϕτ with values on the sphere SH ⊂ H is a geodesic through ϕ0 = ϕτ |τ=0 in an appropriate Riemannian metric on SH , at least for small τ > 0. As a side remark, various global results of this kind 22 Alexey A. Kryukov can be obtained by applying the methods of geometrical optics to Eq. (4.4) (see Ref. [7] for a review of geometrical optics). Consider now an arbitrary non-stationary Schrödinger evolution ϕt on S H driven by an invertible time-independent Hamiltonian ĥ. Let SH be furnished with a Riemannian metric GRϕ in which the evolution is a motion along a geodesic. It is known that such a Riemannian metric on SH exists. Pick a moment of time t = a and an eigenstate ψ ∈ SH of the Hamiltonian ĥ. Consider the geodesic χt connecting ϕa and ψ. By Eq. (4.4), there exists a perturbation of the metric GRϕ on a small neighborhood of ϕa which transforms the geodesic ϕt into the geodesic χt. Note that one could similarly perturb the metric on a small neighborhood of ψ to transform χt into the stationary geodesics through ψ. The provided perturbation of the metric takes place on the sphere of state SH = S2n−1. What could one say about the metric and the motion on the projective space CPn−1 = S2n−1/S1? To answer, consider the Riemannian metric on Cn∗ (i.e., C without the origin), defined for all ϕ ∈ Cn∗ and all pairs of vectors ξ, η ∈ TϕCn∗ by GRϕ(ξ, η) = ĥ−2ξ, η ||ϕ||2Cn . (4.5) This metric, being restricted to the sphere S2n−1, is a particular case of the metric Eq. (1.3). Solutions to the Schrödinger equation are geodesics in the metric Eq. (4.5) on Cn∗ (and in the induced metric on the sphere, see Ref. [4]). Notice that the multiplication map λ : ϕ −→ λϕ with λ ∈ C∗ is an isometry of the metric Eq. (4.5), that is, GRλϕ(dλξ, dλη) = GRϕ(ξ, η). This is clear because dλ = λ by linearity of the map and Re(λξ, λη)Cn/||λϕ||2Cn = Re(ξ, η)Cn/||ϕ||2Cn . Because of that, the metric Eq. (4.5) “projects down” to CPn, giving a metric on the projective space. More precisely, the metric Eq. (4.5) is induced by the projection of Cn∗ onto CP n−1, furnished with a Riemannian metric. Provided ĥ2 is proportional to the identity operator, the latter metric coincides with the Fubini- Studi metric on CPn−1 (see Ref. [4]). Note however that the multiplication by a complex number may not remain an isometry of the perturbed metric on Cn∗ . That is, the metric on C ∗ , needed to “redirect” the Schrödinger evolution to account for the process of measurement, may not originate in a Riemannian metric on CPn−1. In mathematical terms, the projection of Cn∗ with a perturbed metric onto CP n−1 is not in general a Riemannian submersion. In particular, such is the case for the discussed local perturbation of the metric on Cn∗ . This suggests once again that the metric on the sphere S the approach under investigation has a greater significance than the metric on the projective space CPH . This analysis demonstrates that by an appropriate “denting” of the sphere of states SH , dimH <∞, one can locally affect geodesics on the sphere in a desirable fashion. In particular, by perturbing the Riemannian metric GRϕ on S H one can alter the Schrödinger evolution of the state and drive the state toward one of the On the measurement problem 23 eigenstates of the measured observable. It follows that the physical field in the hypothesis (H1) can be indeed identified with a perturbation of the metric GRϕ. Note that the resulting metric is time-independent. The electron’s state in the construction propagates along a geodesic on the sphere of states, runs into a region with perturbed metric and collapses. It is important to remark that the above demonstration is only an existence proof; it does not provide a realistic model of interaction between the system and the device. Moreover, a particular nature of the field in the hypothesis (H1) will not be essential in the following. The thorough analysis of this nature requires the equations of the field and is left for the upcoming publications. The mere existence of the field satisfying the needed properties will be sufficient for the purpose of this Letter. The postulated physical field may be able to drive the state to one of the eigen- states, i.e., it may be responsible for the collapse itself. However, in such a scenario collapse seems to be a deterministic process and the probabilistic nature of collapse to a particular eigenstate is not explained. Recall now that in accordance with Eq. (2.45), the probability of collapse of a given state ϕ to an eigenstate ψk of an observable depends only on the distance θ between the states in the Fubini-Study metric on the projective space of states. This crucial property allows one to resolve the remaining difficulty in creating a working probabilistic model of collapse. Indeed, suppose that the field sources in the hypothesis (H1) are not fixed at the eigenstates ψk but fluctuate randomly about the eigenstates. In particular, the projections of the sources fluctuate randomly about the points {ψk} on CPH . Suppose further that fluctuations with projections of a small (in the Fubini-Study metric) amplitude are more likely to occur. Suppose finally that if a source reaches a small neighborhood of the state ϕ, it alters the evolution of the state and diverts it to the corresponding eigenstate (say, by perturbing the metric on the neighborhood). Then, the closer the state {ϕ} is to a particular eigenstate {ψk} (consequently, the larger the modulus of the coefficient ck in the decomposition ϕ = i ciψi is), the more likely it becomes for the source fluctuating about ψk to reach (and collapse) the state. At the same time, the further {ϕ} and {ψm} are (and hence, the smaller the modulus of cm is), the less likely it becomes for the source fluctuating about ψm to reach the state. In such a way, the competition between the sources can lead to the standard Born rule for the probability of collapse. To prove the latter claim, let us first of all make the above assumptions precise. To keep the analysis simple, the assumptions will refer to the fibre bundle π : S3 −→ S2 corresponding to the model under discussion. However, it will be clear that the measuring process on any fibre bundle π : S2n−1 −→ CPn−1 and for any time- independent observable on the corresponding space of states can be treated in the same way. Let θ ∈ (−π, π] and α ∈ (−π/2, π/2] be the angular coordinates on the projective space CP 1 = S2. Here the coordinate curves α = α0 yield great circles (pairs of 24 Alexey A. Kryukov meridians) through the poles θ = 0, θ = π of S2 and the curves θ = θ0 yield half the parallels on S2. As before, the eigenstates {ψ1}, {ψ2} are located at the poles of S2. Let β ∈ (−π, π] be the phase of a state on the sphere S3. Then the triple (θ, α, β) form a coordinate system on S3. In terms of these coordinates the following hypothesis is now accepted: (H2a) Fluctuations of each source along the sphere of states S3 can be described by a three-dimensional stochastic process (θt, αt, βt). For instance, consider the source associated with the eigenstate at θ = 0. For any t = t0 the random variables θt0 , αt0 , βt0 describing the source are independent. For any t = t0 the probability density of the random variable θt0 ∈ (−π, π] is equal to 1πcos The random variables αt0 , βt0 are uniformly distributed. The mean function of each process is zero. For any two times t1, t2, t1 6= t2, the random variables θt1 , θt2 are practically statistically independent, so that the stochastic process θt is uncorrelated in time. In other words, θt is ideally a white noise process. The same is true about the processes αt, βt. The stochastic processes describing different sources are independent. (H2b) A source at a point ϕ ∈ S3 with coordinates (θ, α, β) may be identified with a perturbation of the metric on a small neighborhood U ⊂ S3 of the point. If at some time t the U -neighborhood of a particular source contains the electron’s state, the perturbation of the metric alters the evolution of the state and collapses it to the corresponding eigenstate. Is there a realization of the hypothesis? It was already verified that (H2b) can be realized for any Schrödinger evolution ϕt by a “lensing” effect, i.e., by redirecting ϕt toward the eigenstate. Also, the white noise process postulated in (H2a) certainly exists as a mathematical idealization. Moreover, the processes of this kind are common in physics. Probably the most appropriate example is the thermal noise, i.e., the random process describing the electric current created by the thermal motion of electrons inside a conductor. Could the random fluctuations of the sources in (H2a) be of a similar origin? During a measurement the measuring device interacts with the measured system. At the same time, the molecules (atoms, particles) of the device experience a random thermal motion. In a general (non-stationary) case fluctuations of molecules result in fluctuations of their states on the space of states. So the main new assumption made is that the interaction between the measured system and the device also takes place on the space of states of the system rather than on the classical space alone. Fluctuations of states of the molecules are then associated with fluctuations of the field sources along the space of states leading to the postulated stochastic process. What is the probability dP1 of collapse of a state ϕt to a particular eigenstate ψ1 at some specific time t = t0 in the hypothesis? Such a probability is equal to the probability for the U -neighborhood of the corresponding source to contain the state On the measurement problem 25 at this time. Let (θ0, α0, β0) be coordinates of ϕt0 in the chosen coordinate system on S3. If U is sufficiently small, the change in the probability density of θt0 across U can be neglected, and, therefore, dP1 = dV. (4.6) Here dV is the volume of U which in this simplest case is identified with dθdαdβ for some fixed values of the differentials. 1 According to the hypothesis, the random variables describing the positions of different sources at t = t0 are independent. Therefore, the probability dP2 of collapse of the state ϕt0 to another eigenstate ψ2 can be computed in the same way. Finally, since the stochastic processes describing the sources are uncorrelated in time, the probability for the U -neighborhood of a source to contain any particular point ϕ0 is not affected by the previous history of the source. In particular, the probability rule Eq. (4.6) is universally valid. On the other hand, according to Eq. (2.45), the expression |c1|2 = cos2 (4.7) represents the standard probability of transition from the state {ϕt0} to the state {ψ1}, provided θ0 is the distance between the states in the Fubini-Study metric and c1 is the coefficient of ψ1 in the decomposition of ϕt0 . Clearly, the distance θ0 in the formula Eq. (4.7) can be replaced with the the angle θ0 between the states, explaining the chosen notation. Of course, a similar formula holds true for the coefficient c2 of ψ2. It follows that the ratio dP1/dP2 coincides with |c1|2/|c2|2. The conclusion is that the postulated hypothesis yields the Born rule for the probability of collapse as was claimed. The “single-push” process of collapse of the state ϕt to an eigenstate can be replaced with a more elaborate stochastic process. Each encounter with a source in this process results in a decrease in the distance θ ∈ [0, π] between the state {ϕt} and the corresponding eigenstate {ψk} by a certain value δ. Between the encounters the state undergoes the ordinary Schrödinger evolution. Assume for simplicity that the frequency of encounters is sufficiently high. In this case one can neglect the Schrödinger evolution of the state during the measurement. The stochastic process of collapse can be then defined as a finite, time-homogeneous Markov chain with absorbing boundaries θ = 0 and θ = π and with the number of states equal to π/δ + 1. The transition matrix for the process can be found via simple formulas 1The volume element for the sphere S3 with the usual metric in the chosen coordinates is dSV = sin θ cos θ dθdαdβ. It would be more appropriate to associate dSV with the volume of U . Moreover, the expression sin θ cos θ is the derivative of cos2 θ . This leads one to interesting models in which at any t = t0 the random variable θt0 is uniformly distributed on (−π, π] and the coefficient cos2 θ appears in the (complementary) cumulative distribution function due to the factor sin θ cos θ in the volume element dSV . However, the element dV will be sufficient for the purpose of this Letter. 26 Alexey A. Kryukov from the condition that the steady-state transition matrix has the right transition probabilities from any state θi to the absorbing states {ψ1}, {ψ2} (namely, cos2 θi2 and cos2 π−θi = sin2 θi ). The resulting process is a generalization of the (biased) random walk with absorbing boundaries (also known as the gambler’s ruin), in which the transition probabilities vary with the state. Namely, the transition probabilities for a step toward an absorbing state {ψi} increase as the electron’s state moves closer to {ψi}. Various stochastic processes have been extensively used in modeling collapse (see reviews Refs. [9] and [10]). In general words, the existing models are based on adding an external random noise term and a term containing the measured observable to the Schrödinger equation. The term with the observable provides the “choices” for observations, while the random noise term is a “chooser” (see Ref. [9]). The probability density for a particular noise in the models is given by yet another equation. This equation makes it more probable for the noise to fluctuate around values associated with the eigenstates of the observable and in such a way that the probability of the noise also depends on the initial state ϕ0 of the system. When applied to the process of measurement, the models of this kind explain the probabilistic results of observations by relating them to the random noise, selected by the mentioned probability rule. At the same time, the physical reason for a particular random noise remains unexplained (see Ref. [9]). Even without analyzing the existing stochastic models of collapse in detail, one can pinpoint the essential difference of the model considered here. Namely, the noise in the advocated approach is a process on the space of states which does not depend on a particular state ϕ of the measured system. In particular, the noise in the model does not change when the state ϕ changes. This independence of the noise from the state of the measured system opens a way for associating it with the measuring device itself. For example, as already discussed, the noise may originate in the thermal motion of molecules of the device, considered as a process on the space of states. Another important observation is that the process of collapse in the model is a deterministic process on the space of states. In fact, by associating the random noise with a physical process, one should be able, in principle, to provide a specific functional form of the noise. In this case it becomes possible to predict the time and the outcome of collapse for an arbitrary evolution ϕt of the system. Note that the proposed mechanism of collapse, although particularly simple, is not the only one satisfying the hypotheses (H1), (H2). Furthermore, as already mentioned, the proposed mechanism is far from being realistic at this stage. The ultimate choice of a physically valid scenario of collapse depends crucially on the field equations on the space of states and cannot be provided at this time. Instead, let us demonstrate that under the above assumptions even such a simple mechanism sheds new light on the quantum measurement problem. On the measurement problem 27 5 THE MEASUREMENT PROBLEM The observations made so far, combined with the results of Ref. [4] suggest the following statements about objects and interactions in QM: (S1) Physical objects in QM are most adequately represented by points of a Hilbert manifold of states. In this sense, they have a functional nature. (S2) Physical interactions involving microscopic objects (in particular, the process of measurement) are most adequately described as processes on the manifold of states, rather than on the classical space alone. In other words, the manifold of states represents a new arena for description of physical processes. (S3) The interactions can be described in terms of the Riemannian metric on the manifold of states. In particular, the states of microscopic particles move along geodesics on the sphere of states furnished with a Riemannian metric. In this sense, the interactions may have a geometric origin. Let us investigate how these statements together with the hypotheses (H1),(H2) in the previous section help provide an understanding of the measurement process in QM. First of all, a particular measuring device can be modeled by a metric field with sources at the eigenstates of the measured observable. That is, the kind of measurement performed on the system determines a specific field created by the device on the manifold of states of the system. Provided the model based on the hypotheses can be developed into a consistent physical theory, the latter result would resolve the so-called preferred basis problem in QM. The problem can be formulated as follows: (P1) How could the electron’s state ϕ “know”, which basis {ek} to use to associate the right probabilities to the coefficients ck in decomposition ϕ = k ckek? The constructed model suggests the following answer: (R1) The coefficients of state of the system in the basis of eigenvectors of the ob- servable describe position of the state relative to the sources of the field created by the device. As already discussed, this position determines the probability for the state to be “pushed” by sources to a particular eigenstate point on the projective space of states. In other words, by creating a surrounding field in the space of states, the device itself defines the “preferred” basis. Next, the process of collapse in the model is an objective process driving the state of the system to an eigenstate of the measured observable. The stochastic nature of the process is due in the model to random fluctuations of sources associated with measuring “parts” of the device. These fluctuations could be directly related to the usual chaotic oscillations of the “parts” extended to the space of states of the system. 28 Alexey A. Kryukov The “classical world” in the approach is represented by eigenstates of observ- ables. The set of all eigenstates of an observable  for a quantum system will be called the set of Â-classical states (or points) for the system. So the set of Â-classical states is a subset in the Hilbert space of states (or the corresponding space of phys- ical states) of the system. Let us point out that there is nothing special about the classical states as what is “classical” with respect to one observable is “quantum” with respect to another one. In the model under investigation, the integral curves of vector fields associated with observables are geodesics in the Killing metric on the sphere of states. More generally, the integral curves of vector fields associated with any reasonable set of physical observables of a quantum system can be shown to be geodesics in an appropriate Riemannian metric on the sphere of states of the system. The non- commutativity of observables is then tied to the curvature of the metric. Let us investigate in this light the “mother of quantum mechanics”, i.e., the double-slit experiment. There are two main paradoxes associated with the experi- ment: (P2) How could the electron pass trough both slits at once? (P3) How could a measuring device inserted after the screen with the slits instanta- neously change the way in which the electron has passed through the screen? The Hilbert space of states in the double-slit experiment is infinite-dimensional. It would be helpful to consider at the same time a version of the experiment with a finite dimensional space of states. For this let us return to the motion of electron in a homogeneous magnetic field. Recall that the spin state of the electron evolves in accordance with equation Eq. (2.3). If the field is directed along the Y -axis and the initial state of the electron is , the solution of Eq. (2.3) is given by cos µB0 sin µB0 . (5.1) If t changes, say, between 0 and π , then the process of passing through the field results in a “splitting” of the original spin-up eigenstate of the operator σ̂z into a superposition of spin-up and spin-down states. In this respect the experiment is a finite dimensional version of the double-slit experiment in which a localized electron wave packet gets transformed by the screen with the slits into a superposition of two wave packets. With this in hand, let us address the above mentioned paradoxes (P2) and (P3) of the double-slit experiment. Let us call the original double-slit experiment and the experiment with an electron in a homogeneous magnetic field the E1 and E2 experiments respectively. The electron in the experiment E2 evolves from the On the measurement problem 29 original “classical” state , into superposition of two eigenstates of σ̂z. During this evolution the Z-component of the electron’s spin is unknown. The reason for that is clear: for 0 < t ≤ π , the trajectory of electron’s state on S2 = CP 1 does not pass through the classical states, i.e., through the eigenstates of σ̂z. Classically speaking, one has a paradox here: the electron’s intrinsic angular momentum is not defined. Instead, the electron is in a superposition of states of two different angular momenta. In a way, the electron’s spin is up and down at the same time. Note however that the state function ϕt is defined for t = 0 as well as for t > 0. In other words, it describes the classical and the non-classical states equally well. Moreover, any (physical) state of the electron is just a point on S2. The evolution of the electron’s state is just a path {ϕt} with values in S2. The classical way of thinking tells us that the electron somehow splits into two parts that evolve along different paths. However, the actual evolution of the electron is most adequately described by a single path ϕt, thereby confirming the statements (S1), (S2). The situation in experiment E1 is almost identical, although the paradox here is more dramatic as our classical intuition of position is very strong. Again, the intuition tells us that the electron splits into two parts which are passing through different slits. However, the electron’s evolution is most adequately described by a path in the space of states. Of course, such a path does not “split” and it describes the evolution of electron before and after the screen with the slits equally well. The resolution of the paradox (P2) is then as follows: (R2) The electron in the experiment E2 is not in the spin-up and spin-down states at once. Rather, it is in the state that is neither a spin-up, nor a spin-down state. Similarly, the electron in the experiment E1 does not pass through two slits at once. Rather, it does not pass through the slits at all! Indeed, for a state to be a spin-up state, for example, it must be at the north pole of the sphere S2 of states, which is not the case for the electron’s state in the experiment E2 for t > 0. Similarly, to pass through a slit is to have a state localized at that slit. But the state of the electron after its interaction with the screen in the experiment E1 is not localized. In other words, the electron (i.e., the electron’s state) is located at a point on the space of states that is different from the point at which an electron passing through the slit would have been. To put it figuratively, the electron passes over rather than through the slits. One can see that the paradox (P2) is resolved by considering the motion of electron in the experiments E1, E2 as happening in the functional space of states rather than on the classical space or on the space of angular momenta. Vaguely speaking, the “functional” (i.e., consistent with (S1) and (S2)) way of thinking makes the paradox disappear. In light of this, the resolution of the paradox (P3) 30 Alexey A. Kryukov is now immediate: (R3) How could a measuring device inserted after the screen change the way the electron has passed through the screen? The answer is: it does not! If a counter is inserted behind the screen (and sufficiently far from the screen), the process of “passing through” the screen is not affected by it. In particular, the counter can be placed after the electron has already passed “through” the screen and this will not change the history. Indeed, the evolution of electron is described by a path ϕt. If only one slit is open, this path passes through a point in the space of states which is represented by a state function localized at the slit. If, however, both slits are open, the path does not pass through such a point. This is true independently of any measurement done behind the screen. What the counter does is to change the path ϕt for larger values of the parameter t so as to produce a state localized at one of the slits in a way discussed in the previous section. As a result, the final state is as if the electron had passed through only one of the slits. However, no reality should be attached in this case to the event of passing though the slits. Once again, the electron in the experiment E1 does not pass through the slits. Likewise, the state function ϕt does not describe the probability of passing through one of the slits (but only the probability to be found by one of the slits). Rather, ϕt itself represents a new “functional” reality of the world which is more adequate in QM than the familiar classical reality. To summarize, the paradox (P3) is resolved by accepting the statements (S1) and (S2), i.e., by recognizing the evolution of electron in the space of states as physical (i.e., real) and by allowing a “deformation” of such an evolution in the presence of a measuring device. Let us finally analyze a measurement performed on a pair of spin-1/2 particles. This will give a hint as to how to proceed in more general cases. As already discussed in section 4, the total quantum mechanical state of the pair is a point in the tensor product of Hilbert spaces of each particle. In particular, the spin state of a pair of electrons is an element of C2 ⊗ C2. A unit normalized state is a point on the unit sphere S7 in this four dimensional complex space. Physical spin states of the pair are then points in the complex projective space CP 3. Note that there may be points in CP 3 that do not represent a physical state of the pair. In particular, if the total angular momentum of the pair vanishes, the state of the pair can only be of the form aϕ+ ⊗ ψ− + bϕ− ⊗ ψ+ with a, b ∈ C. Moreover, if the particles are identical, one must have a = −b. The σ̂z-classical points on CP 3 are the points where both particles have a specific value of the Z-component of spin. These points are represented by the products of ϕ± and ψ±. In the case when the total angular momentum of the pair vanishes, the points are represented by ϕ+ ⊗ ψ− and ϕ− ⊗ ψ+. The evolution of the pair is now a path with values in S7. This path projects down to a path with values on the underlying space CP 3. On the measurement problem 31 With these standard ingredients in place, one can analyze now a version of the famous Einstein-Podolsky-Rosen (EPR) paradox in QM: (P4) Given a pair of spin-1/2 particles in entangled state atϕ+ ⊗ψ− + btϕ− ⊗ψ+, how could it be, that by measuring the Z-component of spin of one of them one fixes the Z-component of spin of the other one, even if the particles are far apart? Note that this paradox is similar to the paradox (P3), taking place on the space of states of the pair. Indeed, the resolution of the paradox is almost identical: (R4) Physical reality is described by the path ϕt with values in the space S 7 (or CP 3) of states of the pair. Unless one of the coefficients at, bt in atϕ+ ⊗ ψ− + btϕ− ⊗ ψ+ is zero for some t, the path ϕt does not pass through the σ̂z-classical points ϕ+ ⊗ ψ− or ϕ− ⊗ ψ+. That is, the particles do not have any Z-component of spin. To measure the Z-component of spin of a particle is to make the path ϕt pass through one of the σ̂z-classical points ϕ+ ⊗ψ− or ϕ− ⊗ ψ+. In this case (and only in this case!) the Z-components of spin of both particles are defined and take opposite values. Furthermore, as with a single particle, the interaction with the measuring device is assumed to cause a “deformation” of the path ϕt. The resulting path ends up then at one of the classical points via a stochastic process on the space of states. The full version of the experiment involving a spatial separation of the particles is even more dramatic. How could the second particle at a point y “know” about measurement of the Z-component of spin performed on the first particle at a distant point x? Again, physical reality of the pair is most adequately described by a path ϕt = atϕx+ ⊗ ψy− + btϕx− ⊗ ψy+ in the space H of states of the pair. Here ϕx+ is the spin-up state of the first particle located at x and similarly for the other state functions in ϕt. The classical points in H have the form ϕx+⊗ψy− and ϕx−⊗ψy+. If ϕt does not pass through these points the spin of individual particles is not defined. Intuitively, we think that if a particle is “here” (at a point x), then it ought to have all attributes of a “real” particle, including spin. But before the measurement is performed, the particle in the experiment is not really here! Indeed, it is somewhere else on the sphere of states SH in H (or on the corresponding projective space CPH). So if reality is associated with the state function of the pair, the paradox is resolved. But what about this “spooky action at a distance”? Notice that the new “func- tional” reality does not use it! Indeed, the equation of geodesics is “local” in the space of states S3, because it is a differential equation of geodesics on S3. Of course, this locality in the space of states does not preclude a non-locality in the classical 32 Alexey A. Kryukov space. Indeed, what is a point in the space of states may represent a pair of well separated particles in the classical space. Furthermore, what is close in the metric on the space of functions does not have to be close in the metric on the classical space (see Ref. [4]). A detailed analysis of this will be, however, a subject for a different paper. REFERENCES 1. M. V. Berry, Proc. Roy. Soc. London A 392, 45 (1984) 2. J. Anandan and Y. Aharonov, Phys. Rev. Lett. 65, 1697 (1990) 3. B. Simon, Phys. Rev. Lett. 51, 2167 (1983) 4. A. Kryukov, Found. Phys. 36, 175 (2006) 5. A. Kryukov, Int. J. Math. & Math. Sci. 14, 2241 (2005) 6. A. Kryukov, Found. Phys. 34, 1225 (2004) 7. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990) 8. J. Bjorken and S. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1964) 9. P. Pearle, “Collapse models,” in Open Systems and Measurement in Relativis- tic Quantum Theory, F. Petruccione and H.P. Breuer, eds. (Springer, 1999) 10. A. Bassi & G. Ghirardi, Physics Reports 379, 257 (2003) GEOMETRY AND QUANTUM MECHANICS ELECTRON IN A HOMOGENEOUS MAGNETIC FIELD Quantum Mechanics on the Space of States S3 Quantum Mechanics on the Projective Space of States CP1 TENSOR PROPERTIES OF EQUATIONS IN THE MODEL THE PROCESS OF MEASUREMENT THE MEASUREMENT PROBLEM

A geometric approach to quantum mechanics with unitary evolution and non-unitary collapse processes is developed. In this approach the Schrodinger evolution of a quantum system is a geodesic motion on the space of states of the system furnished with an appropriate Riemannian metric. The measuring device is modeled by a perturbation of the metric. The process of measurement is identified with a geodesic motion of state of the system in the perturbed metric. Under the assumption of random fluctuations of the perturbed metric, the Born rule for probabilities of collapse is derived. The approach is applied to a two-level quantum system to obtain a simple geometric interpretation of quantum commutators, the uncertainty principle and Planck's constant. In light of this, a lucid analysis of the double-slit experiment with collapse and an experiment on a pair of entangled particles is presented.

On the measurement problem for a two-level quantum system Alexey A. Kryukov ∗ November 12, 2018 A geometric approach to quantum mechanics with unitary evolution and non-unitary collapse processes is developed. In this approach the Schrödinger evolution of a quantum system is a geodesic motion on the space of states of the system furnished with an appropriate Riemannian metric. The measuring device is modeled by a perturbation of the metric. The process of measurement is identified with a geodesic motion of state of the system in the perturbed metric. Under the assumption of random fluctuations of the perturbed metric, the Born rule for probabilities of collapse is derived. The approach is applied to a two-level quantum system to obtain a simple geometric interpretation of quantum commutators, the uncertainty principle and Planck’s constant. In light of this, a lucid analysis of the double-slit experiment with collapse and an experiment on a pair of entangled particles is presented. KEY WORDS: measurement problem - Born rule - Berry’s phase - EPR-paradox 1 GEOMETRY AND QUANTUM MECHANICS Geometric ideas have played a well recognized role in modern physics, especially in general relativity (GR) and gauge theories (GT). They also found a well established position in quantum mechanics (QM) in considerations related to Berry’s phase Ref. [1]. However, whereas in GR and GT geometry (i.e., the metric or connection) defines the dynamics of the theory, the geometric methods pertaining to Berry’s phase do not enjoy such a sweeping significance. The reason for this difference is quite obvious. Indeed, the geometry underlying GR and GT is directly related to ∗Department of Mathematics, University of Wisconsin Colleges E-mail: alexey.kryukov@uwc.edu, aakrioukov@facstaff.wisc.edu http://arxiv.org/abs/0704.1934v1 2 Alexey A. Kryukov the physical fields (gravitational or gauge) in the theory. At the same time, the Fubini-Study metric in the geometric interpretation of Berry’s phase (Refs. [2],[3] amongst many others) depends only on the geometry of the Hilbert space of states of quantum system. The latter geometry (i.e. Hilbert metric) is insensitive to changes in the Hamiltonian of the system and, consequently, is not dynamical. At the same time, it turns out to be easy to make the metric on Hilbert space of states of a closed quantum system dynamical Refs. [4],[5]. For this, notice first of all that the Schrödinger equation = − i ĥϕt (1.1) is the equation for integral curves of the vector field hϕ : H −→ TH, hϕ = − ih̄ ĥϕ associated with the Hamiltonian ĥ of the system. Here H is the Hilbert space of states of the system and TH is the tangent bundle over H. Assume that the Hilbert space H is a space of functions that are square-integrable with respect to an appropriate measure. Because the evolution governed by Eq. (1.1) is unitary, the integral curve through initial point ϕ0 on the unit sphere S H in H will stay on the sphere. Since this holds true for any initial point (modulo the domain issues), one concludes that the restriction of the vector field hϕ to the sphere S H is a vector field on the sphere. In the ordinary QM spaces TϕH tangent to H at ϕ ∈ H are identified with the space H itself. Similarly, spaces TϕS H tangent to the sphere SH at ϕ ∈ SH are identified with (real) affine subspaces of H. In particular, the metric on SH , whenever used, is assumed to be induced by the embedding of SH into H. However, the sphere SH is a manifold and thus, can be defined independently of the ambient space H. As such, SH is a Banach manifold which means that it can be obtained by “gluing together” open sets in a Banach space. The Hilbert metric Gϕ : TϕH × TϕH −→ C on tangent spaces TϕH can be also defined independently of the metric on H as an Hermitian tensor field on H. Such a tensor field gives rise to a Riemannian metric GRϕ : TϕS H ×TϕSH −→ R on SH , defined at each ϕ ∈ SH GRϕ(X,Y ) = 2ReGϕ(ξ, η). (1.2) Here X = (ξ, ξ), Y = (η, η) ∈ TϕSH are considered as vectors in the realization of the tangent space TϕH. The manifold S H , furnished with the (2, 0)-tensor field GRϕ, is then a Riemannian manifold. In the following, the manifold S H with the metric GRϕ will be denoted by S The final step in making the metric Gϕ on the sphere of states S G dynamical is to ensure that the integral curves of hϕ (i.e. the solutions to Schrödinger equation Eq. (1.1)) are geodesics on SG. For this it turns out to be sufficient to define Gϕ : TϕH × TϕH −→ C by Gϕ(ξ, η) = h̄ (ĥĥ∗)−1ξ, η . (1.3) On the measurement problem 3 Here ĥ∗ is the adjoint of ĥ (normally equal to ĥ) and the Hamiltonian ĥ is assumed to be invertible. Incidentally, even if the Hamiltonian ĥ is not bounded on H, it becomes bounded as an operator mapping points ϕ ∈ H into tangent spaces TϕH with the metric Eq. (1.3) (see Ref. [5]). Further general results concerning QM on Hilbert manifolds can be found in Ref. [4]. These results demonstrate that QM can be formulated in terms of geometry of the space of states. The goal of the Letter is to provide such a geometric formulation in case of a simple two-level system and to establish its advantages. Namely, the point of view that the space of states represents a new arena for physical processes and the evolution of state is a motion along geodesic is shown to be effective in addressing the major conceptual difficulties of quantum mechanics. Although the discussion deals primarily with a simple model, the most important results can be shown to be quite general. Some of these generalizations are described in the Letter while others are left for the upcoming publications. 2 ELECTRON IN A HOMOGENEOUS MAGNETIC FIELD Consider a free non-relativistic electron propagating in the direction of the X-axis in a homogeneous magnetic field B. The evolution equation (the Pauli equation) for the electron is Ψ− µσ̂ ·BΨ, (2.1) where Ψ = Ψ(s, x, t), s = 1, 2 is a two-component state function of the electron, µ is the electron’s magnetic moment and σ̂ = (σ̂x, σ̂y, σ̂z) is the vector made of Pauli matrices. The substitution Ψ(s, x, t) = ψt(x)ϕt(s) separates variables and produces two independent evolution equations. The first describes the evolution governed by the free Hamiltonian ψt. (2.2) The second equation describes the evolution in the space C2 of spinors ϕ: = −µσ̂ ·Bϕt. (2.3) It follows that in the case of the product states Ψ(s, x, t) = ψt(x)ϕt(s), one can analyze the evolution of spin state ϕt in the space of states C 2 without needing to involve the infinite-dimensional Hilbert space of states Ψ. 2.1 Quantum Mechanics on the Space of States S3 Let us proceed to reformulation of quantum mechanics of the system in geometrical terms. In this, the fact that the sphere S3 of unit normalized spin states can be 4 Alexey A. Kryukov furnished with the group structure of the group SU(2) will be helpful. The group structure will allow us to exploit simple results from differential geometry of Lie groups which will make the resulting picture more transparent and complete. First of all, the Hamiltonian ĥ = −µσ̂ ·B defines the vector field hϕ = ih̄µσ̂ ·Bϕ on the sphere S3 in the space C2 with the metric (ξ, η)C2 = k ξkηk. The integral curve of hϕ (i.e. the solution of Eq. (2.3)) through ϕ0 ∈ S3 is given by ϕt = e µσ̂·Btϕ0. (2.4) Since ϕt is a path in C 2, it is natural to call the vector the velocity of evolution of the system. The speed of evolution in the C2 is the norm of dϕt in C2 metric. Using (σ̂ ·A)(σ̂ ·B) = A ·B+ iσ̂ ·A×B, (2.5) one has (σ̂ ·B)2 = B2. (2.6) Therefore, by Hermicity of the matrix σ̂ ·B, one obtains µσ̂ ·Bϕt, µσ̂ ·Bϕt , (2.7) where B is the norm of B. In particular, the speed of evolution of the system depends only on the magnitude of the field. To make the evolution of the system a motion along a geodesic, the metric Gϕ on S3 will be defined by Eq. (1.3). Since ĥ is self-adjoint, one obtains ĥĥ∗ = ĥ2 = µ2(σ̂ ·B)2 = µ2B2I, where I is the identity operator on C2 and Eq. (2.6) has been used at the last step. Therefore, up to the constant factor (h̄/µB)2, the metric Gϕ coincides with the one induced by the embedding of S 3 into C2. That means that the carriers of the geodesics on S3 are the intersections of S3 with the planes through the origin. The fact that the found Riemannian metric is so simple is due to an especially simple form of the Hamiltonian in the model. If S3 is identified with the group manifold SU(2), the obtained metric is the Killing metric on SU(2). To see this, let us identify in the standard way the space C2 of complex vectors ϕ = with the space Mat of 2× 2 matrices z1 z2 −z2 z1 . (2.8) The map ω : ϕ −→ ϕ̂ is an isomorphism of (real) vector spaces C2 and Mat. The sphere S3 of unit states in C2 is identified via ω with the subset of matrices with unit determinant. The latter subset is the group SU(2) under matrix multiplication. On the measurement problem 5 The Killing metric on the Lie algebra su(2) can be defined by (X̂, Ŷ )K = cTr(X̂Ŷ +), (2.9) where c 6= 0 is an arbitrary constant and Tr stands for the trace. The Killing metric on the group SU(2) is defined for all ϕ̂ ∈ SU(2) and all left-invariant vector fields (ϕ̂) = ϕ̂X̂, L (ϕ̂) = ϕ̂Ŷ , X̂, Ŷ ∈ su(2) by (ϕ̂), L (ϕ̂))K = (LX̂(e), LŶ (e))K = (X̂, Ŷ )K , (2.10) where e is the identity element in SU(2). By direct substitution one verifies that (L (e), L (e))K = 2cRe(X,Y )C2 when- ever X̂ = dω(X) and Ŷ = dω(Y ) with dω being the differential of the map ω. In other words, the Killing metric is proportional to the metric induced by the em- bedding of S3 into the Euclidean space C2 = R4. This verifies that the metric Gϕ = (h̄/µB) 2I obtained earlier, is the Killing metric. For any two left invariant vector fields L on SU(2) the connection ∇ on SU(2) can be defined by [X̂,Ŷ ] . (2.11) Notice that the left invariant vector fields form a basis on the tangent space T SU(2) for all ϕ̂ ∈ SU(2). In particular, Eq. (2.11) is sufficient to define a connection on SU(2). This connection is symmetric, as the torsion tensor vanishes: ) = ∇L [X̂,Ŷ ] [Ŷ ,X̂] [X̂,Ŷ ] = 0. (2.12) The connection Eq. (2.11) is also compatible with the Killing metric, that is, for any vector fields ξ, η, ζ on SU(2) the following is true: ∇ξ(η, ζ)K = (∇ξη, ζ)K + (η,∇ξζ)K . (2.13) Indeed, assuming that ξ = L , η = L , ζ = L are left invariant, one has (ϕ̂), L (ϕ̂)K = (LŶ (e), LẐ(e))K = const (2.14) and therefore the left hand side of Eq. (2.13) vanishes. For the right hand side, by definition Eq. (2.11) one obtains: )K + (LŶ ,∇LX̂LẐ)K = ([X̂, Ŷ ], Ẑ)K + (Ŷ , [X̂, Ẑ])K . (2.15) From the anti-Hermicity of elements of su(2) one also has: ([X̂, Ŷ ], Ẑ)K = −cTr(X̂Ŷ Ẑ) + cTr(Ŷ X̂Ẑ) (2.16) 6 Alexey A. Kryukov ([Ŷ , X̂ ], Ẑ)K = −cTr(Ŷ X̂Ẑ) + cTr(X̂Ŷ Ẑ). (2.17) As a result, the sum on the right hand side of Eq. (2.13) is also zero which verifies that the connection Eq. (2.11) is compatible with the metric. In other words, the connection ∇ is the Levi-Civita connection of the Killing metric. For any magnetic fieldB in the model the one parameter subgroup ϕ̂t = e µσ̂·Bt is a geodesic through the identity e ∈ SU(2). Indeed, since dϕ̂t = −ϕ̂t ih̄µσ̂ ·B, the path ϕ̂t is the integral curve of the left invariant vector field Lĥϕ̂ = −ϕ̂ µσ̂ · B. Using the definition Eq. (2.11) one then has: [̂h,̂h] = 0. (2.18) Geodesics through an arbitrary point ϕ̂0 ∈ SU(2) can be then written in the form ϕ̂t = ϕ̂0e µσ̂·Bt. Considered as paths with values in C2, these geodesics take the form ϕt = e µσ̂·Btϕ0. The curvature tensor of ∇ can be obtained directly from the definition ]LẐ . (2.19) In particular, [[X̂,Ŷ ],Ẑ] . (2.20) and ( [X̂, Ŷ ], [Ẑ, Ŵ ] . (2.21) The sectional curvature in the plane through L is defined by . (2.22) With the help of Eqs. (2.21) and (2.10) this becomes [X̂, Ŷ ], [X̂, Ŷ ] X̂, X̂ Ŷ , Ŷ X̂, Ŷ . (2.23) Suppose for example that X̂ , Ŷ , Ẑ and Ŵ correspond to the spin observables. Recall that in the Planck system of units the operator of spin ŝ has eigenvalues ±1/2 and can be expressed in terms of the Pauli matrices σ̂1, σ̂2, σ̂3 as σ̂, (2.24) On the measurement problem 7 where σ̂ = (σ̂1, σ̂2, σ̂3). The corresponding anti-Hermitian generators êk = k = 1, 2, 3, form a basis of the Lie algebra su(2) and satisfy the commutator relations [êk, êl] = ǫklmêm, (2.25) where ǫklm denotes the completely antisymmetric tensor of rank three. In the basis {êk} the curvature tensor Eqs. (2.20), (2.21) takes the form Rik,lm = (δilδkm − δimδkl), (2.26) Rik,lm = (δilδkm − δimδkl) (2.27) where δik is the Kronecker delta. The symmetry property Rik,lm +R l,mk +R m,kl = 0 (2.28) of the curvature tensor coincides in the model with the Jacobi identity [[X̂, Ŷ ], Ẑ] + [[Ŷ , Ẑ], X̂] + [[Ẑ, X̂ ], Ŷ ] = 0 (2.29) for the Lie algebra elements X̂, Ŷ , Ẑ. From the isomorphism given by Eq. (2.8) it follows that any vector x = (x1, x2, x3) in the Euclidean space R3 can be identified with the element kiσ̂k =∑ k êk of the Lie algebra su(2). Under such an identification the Euclidean norm ‖x‖R3 of x is equal to detx and rotations in R3 are represented by transformations x −→ ÛxÛ+ with Û ∈ SU(2). One can make this identification into an isometry by assuming the equality of Euclidean and Killing norms ∥∥∥∥∥ 2xkêk ∥∥∥∥∥ = ‖x‖R3 . (2.30) This will fix the constant factor in front of the Killing metric. Note that the Euclidean space R3 can be identified with the space of all possible classical angular momenta of a particle. The electron’s possible angular momenta form a sphere S2 in R3. By equating the norms according to Eq. (2.30), the spaces tangent to S2 are identified with affine subspaces of spaces tangent to the sphere of states S3 with the induced metric. In particular, the sphere S2 can be identified with a submanifold of the space of states S3 with the induced metric. Let us remark that this identification is analogous to the identification of the classical space with the submanifold of point supported states in an infinite-dimensional Hilbert space of states, considered in Refs. [4], [6]. In the Killing metric Eq. (2.10) on S3 one has kêk, k,m x kxmδkm. To satisfy Eq. (2.30) the constant c must be 1/2, that is, the needed metric in Planck units has the form (X̂, Ŷ )K = Tr(X̂Ŷ +). (2.31) 8 Alexey A. Kryukov Using the formula Eq. (2.23), one obtains the following expression for the sectional curvature R(p) in the plane p through orthogonal vectors L ([ê1, ê2], [ê1, ê2])K (ê1, ê1)K (ê2, ê2)K = 4 (ê3, ê3)K = 1. (2.32) So the sectional curvature of S3 in Planck units is equal to 1. Note that in an arbitrary system of units the Killing metric would be multiplied by h̄2 and the sectional curvature would be equal to 1/h̄2. The dimension of sectional curvature is consistent with the fact that the tangent space su(2) is spanned by the spin operators having the dimension of angular momentum. In this approach the Planck’s constant and the commutators of spin observables acquire a transparent geometric interpretation. According to Eq. (2.23) the commu- tator of two observables is directly related to the sectional curvature of the sphere S3. Indeed, assume for simplicity that the vector fields L are orthogonal and unit normalized in the Killing metric. Then from Eqs. (2.22), (2.23) one has the following expression for the norm of the commutator of X̂ and Ŷ : ∥∥∥[X̂, Ŷ ] = 4R(p). (2.33) Here R(p) is the sectional curvature of S3 in the plane p = L(L ) which for the considered Riemannian metric was found to be a constant equal to 1/h̄2. Note that for L which are orthogonal but not unit, the equation Eq. (2.23) takes the form ∥∥∥[X̂, Ŷ ] = 4R(p) ∥∥∥X̂ ∥∥∥Ŷ . In particular, if the norms of X̂ and Ŷ are of order h̄ (e.g., X̂, Ŷ are the spin observables), then the norm of the commutator [X̂, Ŷ ] is of order h̄ as well. Note that despite the fact that the commutator [X̂, Ŷ ] is small in these units, it is of the order of the radius of the sphere of states, making quantum effects on the sphere quite transparent. The results obtained so far in this section were model specific. It is then im- portant to know whether they can be generalized to the case of higher dimensional spaces of states and of arbitrary observables. Also, what if the Hamiltonian of the system is time-dependent? Here is a sketch of what can be done in these cases. For any n the sphere of states S2n−1 in the space Cn is a homogeneous space U(n)/U(n − 1), where U(n) denotes the unitary group on Cn. The Killing metric on U(n) can be used to induce a Riemannian metric on S2n−1 via the embedding. Namely, the 2n − 1 linearly independent generators in the Lie algebra u(n), which belong to the orthogonal complement of a (fixed) subalgebra u(n − 1), form a sub- space V ⊂ u(n). The one-parameter subgroups eX̂τ with X̂ in V sweep a sphere S2n−1 and yield geodesics in the induced metric. The curvature of S2n−1 can be then computed via equations Eqs. (2.20), (2.21) with generators in V . Furthermore, the commutators of generators in V are related to the sectional curvature of S2n−1 by the same formula Eq. (2.33). On the measurement problem 9 Note that there exist anti-Hermitian observables that are not in V . For example, if n = 2 so that S3 = U(2)/U(1) = SU(2), then V is the Lie subalgebra su(2) ⊂ u(2) and so V consists of the traceless elements of u(2). If X̂ ∈ u(2) is not traceless, the one-parameter subgroup eX̂τ is still a geodesic in U(2). However, this geodesic does not “stay” on the subgroup SU(2) = S3. Of course, one could still consider the curves on S3 given by ϕt = e X̂τϕ0, for some point ϕ0 in S 3 ⊂ C2. For any given X̂ ∈ u(2) and all initial points ϕ0 these curves are still geodesics in the appropriate Riemannian metric (see section 1 and Ref. [4]). However, the algebraic features of the model change and the formulas connecting the commutators with the curvature are different. For the time-dependent Hamiltonians ĥ in the Hilbert space Cn the approach can be generalized as follows. The sphere of states S2n−1 is replaced with the manifold M = S2n−1 ×R, where R is the time line. Then, there exists a Riemannian metric on M in which the paths (ϕt, t) with ϕt = e −îhtϕ0 are geodesics for all ϕ0 ∈ S2n−1. Finally, as already mentioned, in the infinite-dimensional case there still exists a Riemannian metric for which all solutions to the Schrödinger equation with an invertible (time-independent) Hamiltonian are geodesics. However, the algebraic properties of the model require further investigation in this case. 2.2 Quantum Mechanics on the Projective Space of States CP 1 In physical experiments one can only determine the state of a system up to a com- plex non-zero factor. That means that the space of physical states is the complex projective space CPH of complex lines in the space of states. In the considered example it is the one dimensional complex projective space CP 1. By definition, CP 1 is the quotient C2∗/C∗, were ∗ means “take away zero”. In other words, CP 1 is the base manifold of the fibre bundle π : C2∗ −→ CP 1 with the natural projection along the fibres C∗. By considering unit normalized states only, one obtains CP as a quotient S3/S1. It is then the base of the fibre bundle π : S3 −→ CP 1, which is a sub-bundle of the previous fibre bundle. If ϕt is a path of the electron’s state on S 3 and π : S3 −→ CP 1 is the bundle projector, then π(ϕt) is a path on the base CP 1. Since this latter path represents what can be measured in experiments, it is important to obtain an explicit formula for π. For this consider a point ϕ = on S3 and let {ϕ} be the complex line formed by vectors λϕ, λ ∈ C. Provided ϕ1 6= 0, there is a unique point of intersection of the line with the affine plane in C2 formed by vectors , ξ ∈ C. Namely, by setting , (2.34) 10 Alexey A. Kryukov one obtains . (2.35) The map ρ = ϕ −→ ξ provides a coordinate chart on CP 1 which identifies CP 1 without a point (complex line through ) with the set C of complex numbers. The affine plane of vectors form a subspace in the Lie algebra su(2). The algebra su(2) itself has been identified earlier with the Euclidean space R3 of vectors kiσ̂k. The stereographic projection then identifies the unit sphere S the origin of R3 with the above plane C plus a point, i.e., with CP 1 itself. The relationship of the coordinate ξ in the plane C with coordinates (x1 = x, x2 = y, x3 = z) of the corresponding point on the sphere S2 is given by x+ iy . (2.36) Solving this for x, y and z and using Eq. (2.35), one obtains: x = ϕ1ϕ2 + ϕ1ϕ2, (2.37) y = i(ϕ1ϕ2 − ϕ1ϕ2), (2.38) z = ϕ2ϕ2 − ϕ1ϕ1. (2.39) The resulting map π : S3 −→ S2 given by (ϕ1, ϕ2) −→ (x, y, z) is the needed projection on the space of physical states. The equation for the integral curve Eq. (2.4) can be simplified by choosing the coordinate axes properly. In particular, one can always assume that the Z-axis is parallel to the magnetic field B. In this case σ̂ ·B = σ̂3B and Eq. (2.4) simplifies to eiωtϕ0+ e−iωtϕ0− , (2.40) where ω = µB and the initial state ϕ0 is equal to . Recall that the speed of evolution of the electron along S3 was given by Eq. (2.7). Let us find the speed of the projection of this evolution on the space of physical states S2 = CP 1. For this recall that the Killing metric on su(2) coincides with the Euclidean metric. The embedding of S2 into su(2) = R3 induces the familiar metric on S2. Such a metric also coincides with the famous Fubini-Study metric on CP 1 identified with S2. Since this metric is just a restriction of the Euclidean metric on R3, one has( . Using Eqs. (2.37)-(2.40), one obtains then = 4ω|ϕ0+||ϕ0−| = |ϕ0+||ϕ0−|. (2.41) On the measurement problem 11 If θ is the angle between the Z-axis and the vector (x, y, z), then sin2 θ = 1− z2 and one finds with the help of Eq. (2.39): sin θ. (2.42) In particular, if the initial state is an eigenstate of σz so that θ = 0 or π, then the speed ds vanishes and the evolution of the electron is projectively trivial. In other words, the eigenstates of the Hamiltonian ĥ = −µσ̂ ·B are zeros of the “push- forward” vector field dπ(hϕ) on CP 1. Here dπ is the differential of the map π and µσ̂ ·Bϕ as before. Recall that the integral curves ϕt given by Eq. (2.4) are geodesics on S 3. At the same time, as one can see from Eqs. (2.37)-(2.40), the projected curves {ϕt} are not in general geodesics in the induced metric. Moreover, there does not exist a Riemannian metric on CP 1 = S2 in which the projection by π of an arbitrary geodesic on S3 yields a geodesic on S2. Indeed, the great circles of S2 (parametrized by arc length) are geodesics in the Fubini-Study metric. At the same time, they are projections of geodesics on S3. The latter fact is obvious for the equator of S2 if one chooses |ϕ0+| = |ϕ0−| in Eqs. (2.37)-(2.40). For any other great circle this fact is verified by a change in direction of the Z-axis. But the condition that the great circles are geodesics fixes the metric on S2 up to a constant factor. This, together with the above observation that the projection {ϕt} of a geodesic may not be a geodesic itself, proves the claim. Suppose, as it is advocated here, that evolution by geodesics is a valid principle of quantum dynamics. Then the obtained result suggests that the space of states S3 may be “more physical” than the projective space CP 1 of “physical states”. In general terms, the actual, “easy to describe” physics may be happening on the space of states of a quantum system. However, our experiments can so far access only the projection of physical processes into the space of physical states. The projection of a process may look much less “natural”, then the original process on the space of states. It was verified earlier that the commutators of spin observables are related to geometry of the space of states S3. Later on in the paper the geometry of a measure- ment process will be discussed. It is then important to know that the probability of transition from state ϕ to state ψ in a measurement also has ties to geometry. Namely, this probability depends only on the distance between {ϕ} and {ψ} in the Fubini-Study metric on the projective space of states (see Ref. [2]). This result is immediate in the model under consideration. In fact, assume that the state ψ is the spin-up eigenstate of σ̂z (one can always assure this by a proper choice of coordinate axes). Then, according to Eq. (2.37), one has z = |ϕ1|2 − |ϕ2|2. (2.43) 12 Alexey A. Kryukov Using the normalization condition |ϕ1|2 + |ϕ2|2 = 1, one obtains |ϕ1|2 = . (2.44) If θ ∈ [0, π] is the angle between the vectors representing {ϕ} and {ψ} on S2, then z = cos θ and the equation Eq. (2.44) takes the form P (θ) = cos2 . (2.45) Here P (θ) = |(ψ,ϕ)|2 = |ϕ1|2 denotes the probability of transition. Notice that θ is the length of the arc of a great circle through {ϕ} and {ψ}. In other words, it is a geodesic distance between the points {ϕ} and {ψ} on S2 in the Fubini-Study metric. In particular, the probability P (θ) of transition between two states depends only on the distance between them. Furthermore, the uncertainty principle for the spin observables also has a simple geometrical interpretation. Since uncertainties of observables do not depend on the overall phase of a state, the geometry underlying the uncertainty principle is, once again, projective. The uncertainty principle for non-commuting observables X̂, Ŷ can be written in the form ∆X∆Y ≥ ∣∣∣(ϕ, [X̂, Ŷ ]ϕ) ∣∣∣ , (2.46) where ∆X2 = (ϕ, X̂2ϕ) − (ϕ, X̂ϕ)2 and similarly for ∆Y 2. Assuming X̂, Ŷ are the components ŝx, ŝy of the spin observable ŝ = σ̂, one has ∆sx∆sy ≥ |(ϕ, ŝzϕ)| . (2.47) Let us connect this inequality with geometry of the projective space S2 = CP 1. In light of the geometric interpretation of the commutators of observables Eq. (2.33) and the equation Eq. (2.46), the existence of such a connection is not surprising. Indeed, from the geometry of the sphere S2 furnished with the Fubini-Study (i.e., the usual!) metric, for any point (x, y, z) on the sphere one has: (y2 + z2)(x2 + z2) ≥ z2. (2.48) The inequality simply says that the product of distances from a point (x, y, z) on the sphere S2 ⊂ R3 to the X and Y -axes is at least |z|. This equation is the geometric form of the uncertainty principle. Indeed, from Eqs. (2.37)-(2.39) it follows that (ϕ, σ̂xϕ) = x, (ϕ, σ̂yϕ) = y, (ϕ, σ̂zϕ) = z, (2.49) where x, y, z are coordinates of the point of S2 representing the state ϕ. In addittion, σ̂2x = σ̂ y = σ̂ z = I and so ∆σ2x = 1−x2 = y2+z2, ∆σ2y = 1−y2 = x2+z2, ∆σ2z = 1−z2 = x2+y2. (2.50) On the measurement problem 13 With the use of Eqs. (2.49), (2.50), the inequality Eq. (2.48) is now equivalent to the uncertainty principle Eq. (2.47). Note that using the angles θx, θy, θz between the coordinate axes and the vector x = (x, y, z) in R3, one can write the equations Eqs. (2.50) in the form ∆σx = sin θx ∆σy = sin θy, ∆σz = sin θz. (2.51) In particular, the uncertainty ∆E in energy for the electron in the state ϕ in the model takes the form ∆E = µB sin θ, (2.52) where θ is the angle between the vectors B and x. Recall that θ is a geodesic distance between the points {ϕ} and {ψ} on S2. Therefore, the uncertainty ∆E is the largest, when {ϕ} is furthest away from the eigenstates, i.e. when θ = π the other hand, ∆E vanishes at the eigenstates of ĥ. The latter statement can be generalized as follows. Let H = Cn and let  : H −→ H be an observable with a simple spectrum λ1 < λ2 < ... < λn and eigenfunctions ϕ1, ϕ2, ..., ϕn. Let {ϕt} be a geodesic through the states {ϕk}, {ϕl} in the Fubini-Study metric on the projective space of states CPH . Then for any point {ϕ} on the geodesic the variance ∆A2 = (ϕ, Â2ϕ)− (ϕ, Âϕ)2 is an increasing function of the distance from {ϕ} to the pair of eigenstates {ϕk}, {ϕl} in CPH . The latter distance is simply the shortest of the distances from {ϕ} to the states {ϕk}, {ϕl}. In fact, because the eigenstates are orthogonal, the probability of transition from a state {ϕ} on the geodesic to a state {ϕi} with i 6= k, l is equal to zero. In particular, {ϕ} = {ckϕk + clϕl} for some coefficients ck, cl. Therefore, ∆A2 = |ck|2λ2k + 1− |ck|2 λ2l − |ck|2λk + (1− |ck|2)λl . (2.53) If {ϕ} coincides with {ϕk} so that |ck| = 1, then ∆A vanishes. On the other hand, as |ck| decreases, ∆A increases until |ck| becomes equal to 1√2 . By Eq. (2.45) this means that ∆A increases with the distance from {ϕ} to {ϕk} until {ϕ} becomes equally distant from {ϕk} and {ϕl}. It is important to note that the “geometric” probability of transition formula Eq. (2.45) is valid in an arbitrary space of states H = Cn. Also, as already discussed, the relationship Eq. (2.33) between the commutators and the curvature holds true on the sphere of states S2n−1 for observables in the subspace V ⊂ u(n) (see the end of section 2.1). Because of that, the uncertainty principle Eq. (2.46) can be still interpreted geometrically. The infinite-dimensional case requires a different approach and needs further analysis. 14 Alexey A. Kryukov 3 TENSOR PROPERTIES OF EQUATIONS IN THE MODEL In the developed geometrical formulation of the model, the space of states is a manifold furnished with a Riemannian metric which defines quantum dynamics on the space. It is well known that differential geometry of manifolds admits two equivalent formulations: local coordinate and coordinate free. Let us discuss the role played by both formulations in the model. Consider a pair X̂, Ŷ of elements of the Lie algebra su(2) and the corresponding pair Xϕ = X̂ϕ, Yϕ = Ŷ ϕ of the associated vector fields. By direct computation one sees that [Xϕ, Yϕ] = −[X̂, Ŷ ]ϕ, (3.1) where [Xϕ, Yϕ] is the Lie bracket of the vector fields. Recall that the integral curves of non-commuting vector fields cannot form a coordinate grid on the manifold. In particular, the integral curves of vector fields sxϕ, syϕ, szϕ associated with the spin observables ŝx, ŝy, ŝz do not form a coordinate grid on S 3. By the above, these integral curves are geodesics in the Killing metric on S3. The fact that they do not form a coordinate grid is then a direct consequence of the curvature of S3. Instead, the fields sxϕ, syϕ, szϕ form a local (non-coordinate) basis at every point of S There are, of course, many ways of choosing coordinates on S3. One natural choice is the normal coordinate system given on a neighborhood of any point by the exponential map. If {ek} is a basis on the tangent space TϕS3 at ϕ ∈ S3 and kek is a tangent vector, then the equation of geodesic ϕt through ϕ in the direction of A in the normal coordinates ϕ1, ϕ2, ϕ3 is linear: ϕkt = a kt, k = 1, 2, 3. (3.2) The evolution of spin state ϕ in time could be thought of as a motion along the manifold S3×R with R being the time axis. The direct product of the Killing metric on SU(2) and the usual Euclidean metric on R makes S3 × R into a Riemannian manifold. If ϕt is a geodesic on S 3, then (ϕt, t) will be a geodesic on S 3 × R. Notice that for any evolution ϕt of the state along S 3 ×R one can find a co-moving coordinate system on S3×R in which (ϕt, t) = (ϕ0, t), i.e., the state is at rest. Such a co-moving system is directly related to the well known Heisenberg representation. The coordinates typically used on the projective space of physical states CP 1 are homogeneous (ϕ1, ϕ2) and inhomogeneous ξ = , η = coordinates. For example, the Fubini-Study metric is usually expressed in terms of these coordinates. Other coordinate systems may be useful in applications. In particular, according to Eq. (2.49), the expectation values of Pauli matrices for a system in a state ϕ coincide with the x, y and z coordinates of the point {ϕ} in S2. Therefore, these expectation values can be identified with local coordinates on CP 1. This fact was used in section 2 to describe the motion of state along CP 1. On the measurement problem 15 The group SU(2) acting on the space C2 is the symmetry group of the theory. In particular, the group SU(2) “extends” to act on tensor algebra over C2 and the Schrödinger equation is a tensor equation. For any fixed time t the Schrödinger evolution operator Û(t, 0) given by Eq. (2.4) is an active realization of SU(2)- transformations on C2. The corresponding passive realization consists in a unitary change of basis on C2. One could consider instead the sphere S3 ⊂ C2 as a base manifold for tensor bundles and make the group SU(2) act locally on tensor products of spaces tangent and cotangent to S3. For each tensor type this gives a subbundle of the correspond- ing tensor bundle over C2. The Schrödinger equation is then a tensor equation with terms which are vector fields on S3. In fact, it is the equation for integral curves of the vector field hϕ = − ih̄ ĥϕ on S 3. Alternatively, it is the equation of geodesics in the Killing metric on S3. The notion of symmetry in QM is usually understood as an invariance of the Hamiltonian of the system under a symmetry transformation. In this case the Hamiltonian commutes with the transformation and the generator of transformation becomes a constant of motion. Although this is certainly true for rotations about the field direction in the model, such a restricted understanding of symmetry is not suitable for this Letter. Mathematically, the Schrödinger equation hϕ = − ih̄ ĥϕ in the model is written in a specific basis on the space of states C2. Under a unitary transformation Û ∈ SU(2) of the basis the Schrödinger equation behaves as a vector equation. In particular, the right hand side of the equation becomes equal to − i Û−1ĥϕ. Since coordinates of points ϕ also change to become ψ = Û−1ϕ, the right hand side of the equation takes the form Û−1ĥÛψ. (3.3) Note that Û−1 and Û in Eq. (3.3) act on different spaces! Namely, Û−1 acts on the tangent space TϕC 2, while Û acts on C2 itself. It is common, however, to identify vector spaces with the spaces tangent to them. By following this practice, one sees that under a change of basis the Hamiltonian is transformed by ĥ −→ Û−1ĥÛ . In other words, it transforms as a tensor of rank (1, 1). If in addition ĥ and Û commute, then (and only then!) the Hamiltonian is invariant and the usual conserved quantities exist in accordance with the Noether’s theorem. A particular choice of an orthonormal basis on the space of states C2 (alterna- tively, on S3 or CP 1) has an affect on results of observations expressed in the basis. The reason for that is clear: which state is a spin-up state, for example, depends on the basis in C2. This is quite analogous to dependency of the state of rest in classical mechanics on a choice of reference system. Alternatively, under the identification Eq. (2.8) of space C2 with a space of 2×2- matrices the choice of an orthonormal basis in C2 dictates the choice of basis in the Lie algebra su(2), i.e., the choice of sigma-matrices. A unitary transformation of 16 Alexey A. Kryukov sigma-matrices changes their eigenvectors, thus affecting the results of observations expressed in terms of these eigenvectors. The above can be also rephrased in terms of the geometry of the projective space CP 1 = S2 ⊂ R3. Namely, a unitary transformation of basis in C2 induces an orthogonal transformation of basis on R3. But the choice of a basis in R3 determines the result of measurement of the Z-component of electron’s spin. One can, therefore, conclude that a choice of coordinates on manifolds C2, S3 and CP 1 has a physical meaning similar to the choice of a reference system on the classical space and should not be neglected. On the other hand, physical laws must be, of course, coordinate independent. In the considered case this is assured by the geometric nature of the model. In particular, the Schrödinger equation Eq. (1.1) can be written in a form independent of an orthonormal basis on the space of spin states C2, i.e., in a vector form. In this case a state is just a vector Φ of C2 rather than a column of components in a particular orthonormal basis on C2. The Hamiltonian is an operator ĥ rather than a matrix. The equation Eq. (1.1) then becomes a vector equation on the manifold S3 written in a coordinate-free form: = − i ĥΦt. (3.4) Note that the eigenvalue problems for observables in the model are also tensor equations. In particular, the eigenvalue problem for the Hamiltonian can be written in a coordinate free form: ĥΦ = λΦ. (3.5) The tensor character of equations of quantum theory in the model signifies that the principle of relativity holds true on the space of states. This means, first of all, that both, the active transformations on the space of states and the passive transformations of coordinates on the space are available. In particular, there exist various physically distinguishable reference systems on the space of states (say, dif- ferent bases on C2). Most importantly, the validity of principle of relativity in the theory means that the equations of the theory are the same when written in any such reference system, that is, they are tensor equations. One may doubt the significance of such a principle in the example. After all, the SU(2) symmetry in the model has been known for years. Why would such a “relativistic” view of this symmetry be useful? Note however, that the quantum dynamics in the advocated approach takes place on the space of states. In particular, the evolution of a quantum system is the motion along a geodesic on the (curved) space of states. Because of that the notions of a reference system, of passive and active transformations, of tensor equations, as well as other differential-geometric notions on the space of states, become physically meaningful. This meaning is very On the measurement problem 17 much in line with the meaning of similar notions in special and general relativity or in the theory of gauge fields. For instance, a change in direction of the magnetic field B induces an active transformation on the space of states. This transformation has physically measur- able consequences: it moves geodesics on the space of states and hence changes the evolution of electron’s state. At the same time, the change in B can be compensated by a passive transformation. Indeed, one can choose an orthonormal basis on C2 so that the components of B in the corresponding basis in R3 remain the same. The equation of the new electron’s path in this basis coincides then with the original Most certainly, the above principle is different from the principle of relativity in space-time. In fact, it deals with tensor properties of equations on a Hilbert space of states, rather than on space-time. At the same time, it has the same kind of underlying mathematics and the same spirit as the ordinary principle of relativity. The principle is in fact a particular instance of the principle of functional relativity introduced in Ref. [4]. As discussed, the Hamiltonian in the model is not in general invariant under unitary transformations. In particular, an active unitary transformation associated with a change in direction of the magnetic field B produces a new Hamiltonian. In other words, the Hamiltonian is not a scalar in the tensor approach to the model. On the other hand, due to geometric nature of the model, one can easily identify the most important scalars (or invariants), forming the “bone structure” of the theory. One such invariant is the distance between any two points on spaces of states C2, S3 and CP 1 furnished with the above discussed metrics. Another one is the speed of quantum evolution in S3 in a given magnetic field given by Eq. (2.7). Yet another one is the scalar curvature of S3 or CP 1. This curvature can be expressed in terms of the sectional curvature of S3 which was found to be 1/h̄2. Let us remark that tensor character of the theory allows one to extend the origi- nal symmetry group SU(2) to the group GL(2, C) of general linear transformations acting on fibers of the tangent bundle over S3. Moreover, Schrödinger dynamics on the space of states S3 can be formulated in a way invariant under general coordi- nate transformations on S3. This follows at once from the fact that the Schrödinger equation in the model is the equation of geodesics on the Riemannian manifold S3. As such, this equation is meaningful in arbitrary coordinates on the manifold. Finally, let us comment on a possible argument against the advocated geometric approach. In the model considered here the Riemannian metric on the sphere S3 has turned the sphere into a manifold of constant sectional curvature. This allowed us to relate the curvature of the metric with the Planck’s constant. However, in general the sectional curvature of the Riemannian metric defined by Eq. (1.3) will not be a constant. Even when it is, there is no reason for this constant to be the same as in the model under discussion. It seems therefore that by making the Riemannian metric depend on the Hamiltonian of the system, one shall in general loose the 18 Alexey A. Kryukov relationship between the curvature and the Planck’s constant! To reply to this argument recall that the model considered here is non-relativistic (in the usual sense). In particular, the equation Eq. (2.1) is a special case of the Pauli equation. The latter equation is well known to be the non-relativistic limit of the Dirac equation for electron in electromagnetic field. The Dirac equation can be written in the form = cσ̂ · p̂− e , (3.6) where ϕ̃, χ̃ are two-component spinors, (φ,A) is the 4-potential of the field, p̂ is the momentum operator and e is the electron’s charge (see, for example, Ref. [8]). The largest term in Eq. (3.6) is the one containing the mc2 factor. By substituting (3.7) into Eq. (3.6) one recovers in the standard way the Pauli equation for the spin state ϕ with values in C2. Assume that the 4-potential in Eq. (3.6) describes a weak homogeneous magnetic field B and let ĥD be the corresponding Hamiltonian. Consider the metric GD given by Eq. (1.3) with the Hamiltonian ĥD. Then solutions to the Dirac equation for electron in the field B are geodesics in this metric. Note that the Hamiltonian ĥD has in the non-relativistic limit the form mc2 + ĥ, where ĥ is the non-relativistic Hamiltonian used in Eq. (2.1). Accordingly, the metric GD can be written in the form (h̄/mc2)2(I + ǫ), where ǫ is a small correction due to the Hamiltonian ĥ and I is the identity. It follows that the sectional curvature of GD consists of the main term of the order (mc2/h̄)2 and a small correction due to ĥ. Since the main term is constant (i.e., it does not depend on the fields), the advocated geometric interpretation of the curvature remains possible. However, a more careful analysis of the situation requires a “functional relativistic” formulation of the problem and will be discussed elsewhere. 4 THE PROCESS OF MEASUREMENT In the model under consideration the Schrödinger equation is the equation of geodesics on the space of states S3 furnished with the Killing metric. That means that the dynamics in the theory takes place on the Hilbert space of states rather than on the classical space. In this and the following sections it will be argued that the space of states is also the most appropriate background for tackling problems related to quantum measurement. In particular, the process of collapse of a state can be re- garded as a geodesic motion in the space of states with the metric “skewed” by the measuring device. On the measurement problem 19 Consider a pair of spin-1/2 particles. In QM the most general spin state of such a pair has a form i,j=± cijϕiψj , (4.1) where ϕ+, ψ+ are spin-up, and ϕ−, ψ− spin-down states of the particles. The Hilbert space of states having such a form is the tensor product C2⊗C2. The unit normalized states form a sphere S7 in this four-dimensional complex space. Whenever Ψ is not a product of states of the particles, the state of the pair is called entangled. It is well known that, when the particles are microscopic (i.e., suf- ficiently small in mass and size), the entangled states do indeed exist. By assuming the universal validity of QM, one concludes that the entangled states can be also prepared when one of the particles is replaced with a macroscopic measuring device, designed to measure spin of the second particle. In this case the total state of the system has the form Ψ = aϕ+ψ+ + bϕ−ψ− where ψ± represent states of the device, corresponding to spin-up and spin-down outcomes of measurement. However, unlike the case of microscopic objects, the entangled states with macroscopic objects have never been observed in experiments. The phenomenon of decoherence does not help resolve this problem because the mixtures of states of macroscopic objects have not been observed in experiments either. Recall that in the classical physics the motion of a pair of interacting particles on a manifold can be thought of as a motion of a point in a higher dimensional configuration space. Suppose in particular that particles of massesm andM interact gravitationally and move in the space R3 in accordance with the Newton’s Second Law. Then the motion of the pair is represented by a trajectory in configuration space R6. However, if M ≫ m, the motion simplifies and can be thought of as a motion of the particle of mass m in the field created by the particle of mass M . In this case the configuration space R6 of the pair is “effectively reduced” to the space R3 and the field on R3 created by the heavier particle. An analogous “reduction” of configuration space is implicitly present in the unitary QM whenever the influence of a “macroscopic surrounding” of a quantum system is accounted for by an appropriate choice of potential in the Schrödinger equation. Let us explore the idea that a similar approach can be applied to the process of measurement in the model. Namely, assume that the motion of the total state function of the electron and the measuring device during their interaction can be effectively replaced with the motion of electron’s state function in S3 under the influence of a physical field on S3 created by the measuring device. This means that, in some sense, the state function of the device does not change much as a result of interaction. One immediate objection is that the observed states of the device are orthogonal and so the state cannot change “just a little”. Without addressing this problem in detail, let us point out that the metric on the space of states of the device may be 20 Alexey A. Kryukov “skewed”. As a result, two different position eigenstates of a pointer may become very close in this metric. For instance, let H be the Hilbert space obtained by completion of a space of ordinary functions on R3 with respect to the inner product (ϕ,ψ)H = e−(x−y) ϕ(x)ψ(y)d3xd3y. (4.2) Such a space contains in general the eigenstates of position operator, i.e., the delta- functions δ(x − a). Moreover, two different position eigenstates δ(x − a), δ(x − b) with ‖a− b‖R3 ≪ 1 are close in the metric Eq. (4.2) (see Ref. [4]). If, in particular, δ(x − a), δ(x − b) are the eigenstates of a pointer, then the fact that theses states are close in H can imply that the state of the pointer does not change much in the process of interaction with a measured microscopic system. This argument does not prove, of course, that the proposed “reduction formal- ism” can be consistently implemented into the theory. To validate the formalism one must demonstrate that all imaginable measurements in QM can be modeled (at least in terms of their outcomes) by a field on the Hilbert space of states of the mea- sured system. In what follows such a demonstration will be presented in the case of a finite dimensional space of states and a time-independent observable. Namely, in this case a specific working model of measurement based on a perturbation of the metric on the space of states of the system will be constructed. Moreover, the effec- tiveness and the scope of the proposed method suggest that it can be successfully applied in general. For guidance in modeling the process of measurement let us return to the ex- ample under consideration. Suppose that the device in the example measures the component of electron’s spin in the direction of magnetic field. Equivalently, since the Hamiltonian is given by ĥ = −µσ̂ · B, the device can be designed to measure the electron’s energy. Without loss of generality one may assume that the field is directed along the Z-axis. Then the two eigenstates {ψ1}, {ψ2} of ĥ in CP 1 are positioned on the Z-axis at the poles of the sphere S2 = CP 1. Recall that these eigenstates are zeros of the vector field hϕ = − ih̄ ĥϕ projected on CP 1. In particu- lar, the Schrödinger evolution of ψ1, ψ2 is projectively trivial. So, as a result of the interaction between the electron and the device, the original circular motion of the electron’s state along a parallel on S2 = CP 1 is changed to the state of rest at one of the poles on the sphere. The following hypothesis, which will be clarified and exemplified later on, seems to be in order: (H1) The measuring device creates a physical field on the sphere of states with sources at the eigenstates of the measured observable. This field is capable of driving the electron’s state toward one of the eigenstates. Note that in the position measuring experiments a measuring device is a system of counters distributed in space at the eigenstates of the position observable. The counters are indeed sources of interaction between the particle and the apparatus. On the measurement problem 21 The points in the classical space where the measured particle can be found, can be identified with the position eigenstates of the particle (see Refs. [4], [6]). Since the counters are also located at these points, one can identify them with sources in the space of states of the particle. Finally, because the sources are capable of catching the particles, the (loosely stated) hypothesis (H1) is satisfied. This gives one the hope that a specific form of the hypothesis can be, in fact, realized. What could be the nature of the field postulated in the hypothesis? Recall that the Riemannian metric in the model is dynamical, i.e., it drives the evolution of the electron’s state. Suppose that the field is nothing but a perturbation of the metric on the sphere of states S3, induced by the presence of the measuring device. The evolution of the electron’s state during the measurement is then the motion along a geodesic in the perturbed metric. Note that the metric on the total space S3 and not only on the base space CP 1 = S2 must be perturbed. In fact, as already discussed, the projection {ϕt} of a Schrödinger evolution ϕt is not in general a geodesic. The above approach is certainly attractive, in particular, it does not require any ad hoc features in the theory. Furthermore, the approach can be easily implemented by an appropriate “denting” of the sphere of states of the system. Namely, as shown below, by perturbing the metric on the sphere S3 one can “redirect” the evolution of electron’s state so that the state would become stationary. For a greater generality, assume that the Hilbert space H of states of the system is n-dimensional and let (ϕ1, ..., ϕn) be the usual coordinates on H = Cn. Let the Riemannian metric on H to be of the form gik = η 2δik, (4.3) where η = η(ϕ) is a function on H. The equation of geodesics in this metric can be obtained in the usual way by variation of the length functional on paths ϕt. Let s be the arc length parameter and let τ be a parameter defined by dτ = ds . Then the equation of geodesics in the metric Eq. (4.3) on H can be written in the form ∇η2, (4.4) where ϕτ is identified with ϕt(τ). A similar equation in R 3 is well known in geomet- rical optics where it describes propagation of rays in a media with refractive index η. The equation Eq. (4.4) is also similar to the Newton equation of motion for a unit mass in the field U = −η2/2. The form of equation Eq. (4.4) makes it easy to see that for any sufficiently smooth path ϕτ there exists a function η such that the equation is satisfied, at least on a neighborhood of ϕ0 = ϕτ |τ=0. That is, ϕτ is a geodesic through ϕ0 in the Riemannian metric Eq. (4.3) on H, at least for small values of τ > 0. That also means that an arbitrary sufficiently smooth path ϕτ with values on the sphere SH ⊂ H is a geodesic through ϕ0 = ϕτ |τ=0 in an appropriate Riemannian metric on SH , at least for small τ > 0. As a side remark, various global results of this kind 22 Alexey A. Kryukov can be obtained by applying the methods of geometrical optics to Eq. (4.4) (see Ref. [7] for a review of geometrical optics). Consider now an arbitrary non-stationary Schrödinger evolution ϕt on S H driven by an invertible time-independent Hamiltonian ĥ. Let SH be furnished with a Riemannian metric GRϕ in which the evolution is a motion along a geodesic. It is known that such a Riemannian metric on SH exists. Pick a moment of time t = a and an eigenstate ψ ∈ SH of the Hamiltonian ĥ. Consider the geodesic χt connecting ϕa and ψ. By Eq. (4.4), there exists a perturbation of the metric GRϕ on a small neighborhood of ϕa which transforms the geodesic ϕt into the geodesic χt. Note that one could similarly perturb the metric on a small neighborhood of ψ to transform χt into the stationary geodesics through ψ. The provided perturbation of the metric takes place on the sphere of state SH = S2n−1. What could one say about the metric and the motion on the projective space CPn−1 = S2n−1/S1? To answer, consider the Riemannian metric on Cn∗ (i.e., C without the origin), defined for all ϕ ∈ Cn∗ and all pairs of vectors ξ, η ∈ TϕCn∗ by GRϕ(ξ, η) = ĥ−2ξ, η ||ϕ||2Cn . (4.5) This metric, being restricted to the sphere S2n−1, is a particular case of the metric Eq. (1.3). Solutions to the Schrödinger equation are geodesics in the metric Eq. (4.5) on Cn∗ (and in the induced metric on the sphere, see Ref. [4]). Notice that the multiplication map λ : ϕ −→ λϕ with λ ∈ C∗ is an isometry of the metric Eq. (4.5), that is, GRλϕ(dλξ, dλη) = GRϕ(ξ, η). This is clear because dλ = λ by linearity of the map and Re(λξ, λη)Cn/||λϕ||2Cn = Re(ξ, η)Cn/||ϕ||2Cn . Because of that, the metric Eq. (4.5) “projects down” to CPn, giving a metric on the projective space. More precisely, the metric Eq. (4.5) is induced by the projection of Cn∗ onto CP n−1, furnished with a Riemannian metric. Provided ĥ2 is proportional to the identity operator, the latter metric coincides with the Fubini- Studi metric on CPn−1 (see Ref. [4]). Note however that the multiplication by a complex number may not remain an isometry of the perturbed metric on Cn∗ . That is, the metric on C ∗ , needed to “redirect” the Schrödinger evolution to account for the process of measurement, may not originate in a Riemannian metric on CPn−1. In mathematical terms, the projection of Cn∗ with a perturbed metric onto CP n−1 is not in general a Riemannian submersion. In particular, such is the case for the discussed local perturbation of the metric on Cn∗ . This suggests once again that the metric on the sphere S the approach under investigation has a greater significance than the metric on the projective space CPH . This analysis demonstrates that by an appropriate “denting” of the sphere of states SH , dimH <∞, one can locally affect geodesics on the sphere in a desirable fashion. In particular, by perturbing the Riemannian metric GRϕ on S H one can alter the Schrödinger evolution of the state and drive the state toward one of the On the measurement problem 23 eigenstates of the measured observable. It follows that the physical field in the hypothesis (H1) can be indeed identified with a perturbation of the metric GRϕ. Note that the resulting metric is time-independent. The electron’s state in the construction propagates along a geodesic on the sphere of states, runs into a region with perturbed metric and collapses. It is important to remark that the above demonstration is only an existence proof; it does not provide a realistic model of interaction between the system and the device. Moreover, a particular nature of the field in the hypothesis (H1) will not be essential in the following. The thorough analysis of this nature requires the equations of the field and is left for the upcoming publications. The mere existence of the field satisfying the needed properties will be sufficient for the purpose of this Letter. The postulated physical field may be able to drive the state to one of the eigen- states, i.e., it may be responsible for the collapse itself. However, in such a scenario collapse seems to be a deterministic process and the probabilistic nature of collapse to a particular eigenstate is not explained. Recall now that in accordance with Eq. (2.45), the probability of collapse of a given state ϕ to an eigenstate ψk of an observable depends only on the distance θ between the states in the Fubini-Study metric on the projective space of states. This crucial property allows one to resolve the remaining difficulty in creating a working probabilistic model of collapse. Indeed, suppose that the field sources in the hypothesis (H1) are not fixed at the eigenstates ψk but fluctuate randomly about the eigenstates. In particular, the projections of the sources fluctuate randomly about the points {ψk} on CPH . Suppose further that fluctuations with projections of a small (in the Fubini-Study metric) amplitude are more likely to occur. Suppose finally that if a source reaches a small neighborhood of the state ϕ, it alters the evolution of the state and diverts it to the corresponding eigenstate (say, by perturbing the metric on the neighborhood). Then, the closer the state {ϕ} is to a particular eigenstate {ψk} (consequently, the larger the modulus of the coefficient ck in the decomposition ϕ = i ciψi is), the more likely it becomes for the source fluctuating about ψk to reach (and collapse) the state. At the same time, the further {ϕ} and {ψm} are (and hence, the smaller the modulus of cm is), the less likely it becomes for the source fluctuating about ψm to reach the state. In such a way, the competition between the sources can lead to the standard Born rule for the probability of collapse. To prove the latter claim, let us first of all make the above assumptions precise. To keep the analysis simple, the assumptions will refer to the fibre bundle π : S3 −→ S2 corresponding to the model under discussion. However, it will be clear that the measuring process on any fibre bundle π : S2n−1 −→ CPn−1 and for any time- independent observable on the corresponding space of states can be treated in the same way. Let θ ∈ (−π, π] and α ∈ (−π/2, π/2] be the angular coordinates on the projective space CP 1 = S2. Here the coordinate curves α = α0 yield great circles (pairs of 24 Alexey A. Kryukov meridians) through the poles θ = 0, θ = π of S2 and the curves θ = θ0 yield half the parallels on S2. As before, the eigenstates {ψ1}, {ψ2} are located at the poles of S2. Let β ∈ (−π, π] be the phase of a state on the sphere S3. Then the triple (θ, α, β) form a coordinate system on S3. In terms of these coordinates the following hypothesis is now accepted: (H2a) Fluctuations of each source along the sphere of states S3 can be described by a three-dimensional stochastic process (θt, αt, βt). For instance, consider the source associated with the eigenstate at θ = 0. For any t = t0 the random variables θt0 , αt0 , βt0 describing the source are independent. For any t = t0 the probability density of the random variable θt0 ∈ (−π, π] is equal to 1πcos The random variables αt0 , βt0 are uniformly distributed. The mean function of each process is zero. For any two times t1, t2, t1 6= t2, the random variables θt1 , θt2 are practically statistically independent, so that the stochastic process θt is uncorrelated in time. In other words, θt is ideally a white noise process. The same is true about the processes αt, βt. The stochastic processes describing different sources are independent. (H2b) A source at a point ϕ ∈ S3 with coordinates (θ, α, β) may be identified with a perturbation of the metric on a small neighborhood U ⊂ S3 of the point. If at some time t the U -neighborhood of a particular source contains the electron’s state, the perturbation of the metric alters the evolution of the state and collapses it to the corresponding eigenstate. Is there a realization of the hypothesis? It was already verified that (H2b) can be realized for any Schrödinger evolution ϕt by a “lensing” effect, i.e., by redirecting ϕt toward the eigenstate. Also, the white noise process postulated in (H2a) certainly exists as a mathematical idealization. Moreover, the processes of this kind are common in physics. Probably the most appropriate example is the thermal noise, i.e., the random process describing the electric current created by the thermal motion of electrons inside a conductor. Could the random fluctuations of the sources in (H2a) be of a similar origin? During a measurement the measuring device interacts with the measured system. At the same time, the molecules (atoms, particles) of the device experience a random thermal motion. In a general (non-stationary) case fluctuations of molecules result in fluctuations of their states on the space of states. So the main new assumption made is that the interaction between the measured system and the device also takes place on the space of states of the system rather than on the classical space alone. Fluctuations of states of the molecules are then associated with fluctuations of the field sources along the space of states leading to the postulated stochastic process. What is the probability dP1 of collapse of a state ϕt to a particular eigenstate ψ1 at some specific time t = t0 in the hypothesis? Such a probability is equal to the probability for the U -neighborhood of the corresponding source to contain the state On the measurement problem 25 at this time. Let (θ0, α0, β0) be coordinates of ϕt0 in the chosen coordinate system on S3. If U is sufficiently small, the change in the probability density of θt0 across U can be neglected, and, therefore, dP1 = dV. (4.6) Here dV is the volume of U which in this simplest case is identified with dθdαdβ for some fixed values of the differentials. 1 According to the hypothesis, the random variables describing the positions of different sources at t = t0 are independent. Therefore, the probability dP2 of collapse of the state ϕt0 to another eigenstate ψ2 can be computed in the same way. Finally, since the stochastic processes describing the sources are uncorrelated in time, the probability for the U -neighborhood of a source to contain any particular point ϕ0 is not affected by the previous history of the source. In particular, the probability rule Eq. (4.6) is universally valid. On the other hand, according to Eq. (2.45), the expression |c1|2 = cos2 (4.7) represents the standard probability of transition from the state {ϕt0} to the state {ψ1}, provided θ0 is the distance between the states in the Fubini-Study metric and c1 is the coefficient of ψ1 in the decomposition of ϕt0 . Clearly, the distance θ0 in the formula Eq. (4.7) can be replaced with the the angle θ0 between the states, explaining the chosen notation. Of course, a similar formula holds true for the coefficient c2 of ψ2. It follows that the ratio dP1/dP2 coincides with |c1|2/|c2|2. The conclusion is that the postulated hypothesis yields the Born rule for the probability of collapse as was claimed. The “single-push” process of collapse of the state ϕt to an eigenstate can be replaced with a more elaborate stochastic process. Each encounter with a source in this process results in a decrease in the distance θ ∈ [0, π] between the state {ϕt} and the corresponding eigenstate {ψk} by a certain value δ. Between the encounters the state undergoes the ordinary Schrödinger evolution. Assume for simplicity that the frequency of encounters is sufficiently high. In this case one can neglect the Schrödinger evolution of the state during the measurement. The stochastic process of collapse can be then defined as a finite, time-homogeneous Markov chain with absorbing boundaries θ = 0 and θ = π and with the number of states equal to π/δ + 1. The transition matrix for the process can be found via simple formulas 1The volume element for the sphere S3 with the usual metric in the chosen coordinates is dSV = sin θ cos θ dθdαdβ. It would be more appropriate to associate dSV with the volume of U . Moreover, the expression sin θ cos θ is the derivative of cos2 θ . This leads one to interesting models in which at any t = t0 the random variable θt0 is uniformly distributed on (−π, π] and the coefficient cos2 θ appears in the (complementary) cumulative distribution function due to the factor sin θ cos θ in the volume element dSV . However, the element dV will be sufficient for the purpose of this Letter. 26 Alexey A. Kryukov from the condition that the steady-state transition matrix has the right transition probabilities from any state θi to the absorbing states {ψ1}, {ψ2} (namely, cos2 θi2 and cos2 π−θi = sin2 θi ). The resulting process is a generalization of the (biased) random walk with absorbing boundaries (also known as the gambler’s ruin), in which the transition probabilities vary with the state. Namely, the transition probabilities for a step toward an absorbing state {ψi} increase as the electron’s state moves closer to {ψi}. Various stochastic processes have been extensively used in modeling collapse (see reviews Refs. [9] and [10]). In general words, the existing models are based on adding an external random noise term and a term containing the measured observable to the Schrödinger equation. The term with the observable provides the “choices” for observations, while the random noise term is a “chooser” (see Ref. [9]). The probability density for a particular noise in the models is given by yet another equation. This equation makes it more probable for the noise to fluctuate around values associated with the eigenstates of the observable and in such a way that the probability of the noise also depends on the initial state ϕ0 of the system. When applied to the process of measurement, the models of this kind explain the probabilistic results of observations by relating them to the random noise, selected by the mentioned probability rule. At the same time, the physical reason for a particular random noise remains unexplained (see Ref. [9]). Even without analyzing the existing stochastic models of collapse in detail, one can pinpoint the essential difference of the model considered here. Namely, the noise in the advocated approach is a process on the space of states which does not depend on a particular state ϕ of the measured system. In particular, the noise in the model does not change when the state ϕ changes. This independence of the noise from the state of the measured system opens a way for associating it with the measuring device itself. For example, as already discussed, the noise may originate in the thermal motion of molecules of the device, considered as a process on the space of states. Another important observation is that the process of collapse in the model is a deterministic process on the space of states. In fact, by associating the random noise with a physical process, one should be able, in principle, to provide a specific functional form of the noise. In this case it becomes possible to predict the time and the outcome of collapse for an arbitrary evolution ϕt of the system. Note that the proposed mechanism of collapse, although particularly simple, is not the only one satisfying the hypotheses (H1), (H2). Furthermore, as already mentioned, the proposed mechanism is far from being realistic at this stage. The ultimate choice of a physically valid scenario of collapse depends crucially on the field equations on the space of states and cannot be provided at this time. Instead, let us demonstrate that under the above assumptions even such a simple mechanism sheds new light on the quantum measurement problem. On the measurement problem 27 5 THE MEASUREMENT PROBLEM The observations made so far, combined with the results of Ref. [4] suggest the following statements about objects and interactions in QM: (S1) Physical objects in QM are most adequately represented by points of a Hilbert manifold of states. In this sense, they have a functional nature. (S2) Physical interactions involving microscopic objects (in particular, the process of measurement) are most adequately described as processes on the manifold of states, rather than on the classical space alone. In other words, the manifold of states represents a new arena for description of physical processes. (S3) The interactions can be described in terms of the Riemannian metric on the manifold of states. In particular, the states of microscopic particles move along geodesics on the sphere of states furnished with a Riemannian metric. In this sense, the interactions may have a geometric origin. Let us investigate how these statements together with the hypotheses (H1),(H2) in the previous section help provide an understanding of the measurement process in QM. First of all, a particular measuring device can be modeled by a metric field with sources at the eigenstates of the measured observable. That is, the kind of measurement performed on the system determines a specific field created by the device on the manifold of states of the system. Provided the model based on the hypotheses can be developed into a consistent physical theory, the latter result would resolve the so-called preferred basis problem in QM. The problem can be formulated as follows: (P1) How could the electron’s state ϕ “know”, which basis {ek} to use to associate the right probabilities to the coefficients ck in decomposition ϕ = k ckek? The constructed model suggests the following answer: (R1) The coefficients of state of the system in the basis of eigenvectors of the ob- servable describe position of the state relative to the sources of the field created by the device. As already discussed, this position determines the probability for the state to be “pushed” by sources to a particular eigenstate point on the projective space of states. In other words, by creating a surrounding field in the space of states, the device itself defines the “preferred” basis. Next, the process of collapse in the model is an objective process driving the state of the system to an eigenstate of the measured observable. The stochastic nature of the process is due in the model to random fluctuations of sources associated with measuring “parts” of the device. These fluctuations could be directly related to the usual chaotic oscillations of the “parts” extended to the space of states of the system. 28 Alexey A. Kryukov The “classical world” in the approach is represented by eigenstates of observ- ables. The set of all eigenstates of an observable  for a quantum system will be called the set of Â-classical states (or points) for the system. So the set of Â-classical states is a subset in the Hilbert space of states (or the corresponding space of phys- ical states) of the system. Let us point out that there is nothing special about the classical states as what is “classical” with respect to one observable is “quantum” with respect to another one. In the model under investigation, the integral curves of vector fields associated with observables are geodesics in the Killing metric on the sphere of states. More generally, the integral curves of vector fields associated with any reasonable set of physical observables of a quantum system can be shown to be geodesics in an appropriate Riemannian metric on the sphere of states of the system. The non- commutativity of observables is then tied to the curvature of the metric. Let us investigate in this light the “mother of quantum mechanics”, i.e., the double-slit experiment. There are two main paradoxes associated with the experi- ment: (P2) How could the electron pass trough both slits at once? (P3) How could a measuring device inserted after the screen with the slits instanta- neously change the way in which the electron has passed through the screen? The Hilbert space of states in the double-slit experiment is infinite-dimensional. It would be helpful to consider at the same time a version of the experiment with a finite dimensional space of states. For this let us return to the motion of electron in a homogeneous magnetic field. Recall that the spin state of the electron evolves in accordance with equation Eq. (2.3). If the field is directed along the Y -axis and the initial state of the electron is , the solution of Eq. (2.3) is given by cos µB0 sin µB0 . (5.1) If t changes, say, between 0 and π , then the process of passing through the field results in a “splitting” of the original spin-up eigenstate of the operator σ̂z into a superposition of spin-up and spin-down states. In this respect the experiment is a finite dimensional version of the double-slit experiment in which a localized electron wave packet gets transformed by the screen with the slits into a superposition of two wave packets. With this in hand, let us address the above mentioned paradoxes (P2) and (P3) of the double-slit experiment. Let us call the original double-slit experiment and the experiment with an electron in a homogeneous magnetic field the E1 and E2 experiments respectively. The electron in the experiment E2 evolves from the On the measurement problem 29 original “classical” state , into superposition of two eigenstates of σ̂z. During this evolution the Z-component of the electron’s spin is unknown. The reason for that is clear: for 0 < t ≤ π , the trajectory of electron’s state on S2 = CP 1 does not pass through the classical states, i.e., through the eigenstates of σ̂z. Classically speaking, one has a paradox here: the electron’s intrinsic angular momentum is not defined. Instead, the electron is in a superposition of states of two different angular momenta. In a way, the electron’s spin is up and down at the same time. Note however that the state function ϕt is defined for t = 0 as well as for t > 0. In other words, it describes the classical and the non-classical states equally well. Moreover, any (physical) state of the electron is just a point on S2. The evolution of the electron’s state is just a path {ϕt} with values in S2. The classical way of thinking tells us that the electron somehow splits into two parts that evolve along different paths. However, the actual evolution of the electron is most adequately described by a single path ϕt, thereby confirming the statements (S1), (S2). The situation in experiment E1 is almost identical, although the paradox here is more dramatic as our classical intuition of position is very strong. Again, the intuition tells us that the electron splits into two parts which are passing through different slits. However, the electron’s evolution is most adequately described by a path in the space of states. Of course, such a path does not “split” and it describes the evolution of electron before and after the screen with the slits equally well. The resolution of the paradox (P2) is then as follows: (R2) The electron in the experiment E2 is not in the spin-up and spin-down states at once. Rather, it is in the state that is neither a spin-up, nor a spin-down state. Similarly, the electron in the experiment E1 does not pass through two slits at once. Rather, it does not pass through the slits at all! Indeed, for a state to be a spin-up state, for example, it must be at the north pole of the sphere S2 of states, which is not the case for the electron’s state in the experiment E2 for t > 0. Similarly, to pass through a slit is to have a state localized at that slit. But the state of the electron after its interaction with the screen in the experiment E1 is not localized. In other words, the electron (i.e., the electron’s state) is located at a point on the space of states that is different from the point at which an electron passing through the slit would have been. To put it figuratively, the electron passes over rather than through the slits. One can see that the paradox (P2) is resolved by considering the motion of electron in the experiments E1, E2 as happening in the functional space of states rather than on the classical space or on the space of angular momenta. Vaguely speaking, the “functional” (i.e., consistent with (S1) and (S2)) way of thinking makes the paradox disappear. In light of this, the resolution of the paradox (P3) 30 Alexey A. Kryukov is now immediate: (R3) How could a measuring device inserted after the screen change the way the electron has passed through the screen? The answer is: it does not! If a counter is inserted behind the screen (and sufficiently far from the screen), the process of “passing through” the screen is not affected by it. In particular, the counter can be placed after the electron has already passed “through” the screen and this will not change the history. Indeed, the evolution of electron is described by a path ϕt. If only one slit is open, this path passes through a point in the space of states which is represented by a state function localized at the slit. If, however, both slits are open, the path does not pass through such a point. This is true independently of any measurement done behind the screen. What the counter does is to change the path ϕt for larger values of the parameter t so as to produce a state localized at one of the slits in a way discussed in the previous section. As a result, the final state is as if the electron had passed through only one of the slits. However, no reality should be attached in this case to the event of passing though the slits. Once again, the electron in the experiment E1 does not pass through the slits. Likewise, the state function ϕt does not describe the probability of passing through one of the slits (but only the probability to be found by one of the slits). Rather, ϕt itself represents a new “functional” reality of the world which is more adequate in QM than the familiar classical reality. To summarize, the paradox (P3) is resolved by accepting the statements (S1) and (S2), i.e., by recognizing the evolution of electron in the space of states as physical (i.e., real) and by allowing a “deformation” of such an evolution in the presence of a measuring device. Let us finally analyze a measurement performed on a pair of spin-1/2 particles. This will give a hint as to how to proceed in more general cases. As already discussed in section 4, the total quantum mechanical state of the pair is a point in the tensor product of Hilbert spaces of each particle. In particular, the spin state of a pair of electrons is an element of C2 ⊗ C2. A unit normalized state is a point on the unit sphere S7 in this four dimensional complex space. Physical spin states of the pair are then points in the complex projective space CP 3. Note that there may be points in CP 3 that do not represent a physical state of the pair. In particular, if the total angular momentum of the pair vanishes, the state of the pair can only be of the form aϕ+ ⊗ ψ− + bϕ− ⊗ ψ+ with a, b ∈ C. Moreover, if the particles are identical, one must have a = −b. The σ̂z-classical points on CP 3 are the points where both particles have a specific value of the Z-component of spin. These points are represented by the products of ϕ± and ψ±. In the case when the total angular momentum of the pair vanishes, the points are represented by ϕ+ ⊗ ψ− and ϕ− ⊗ ψ+. The evolution of the pair is now a path with values in S7. This path projects down to a path with values on the underlying space CP 3. On the measurement problem 31 With these standard ingredients in place, one can analyze now a version of the famous Einstein-Podolsky-Rosen (EPR) paradox in QM: (P4) Given a pair of spin-1/2 particles in entangled state atϕ+ ⊗ψ− + btϕ− ⊗ψ+, how could it be, that by measuring the Z-component of spin of one of them one fixes the Z-component of spin of the other one, even if the particles are far apart? Note that this paradox is similar to the paradox (P3), taking place on the space of states of the pair. Indeed, the resolution of the paradox is almost identical: (R4) Physical reality is described by the path ϕt with values in the space S 7 (or CP 3) of states of the pair. Unless one of the coefficients at, bt in atϕ+ ⊗ ψ− + btϕ− ⊗ ψ+ is zero for some t, the path ϕt does not pass through the σ̂z-classical points ϕ+ ⊗ ψ− or ϕ− ⊗ ψ+. That is, the particles do not have any Z-component of spin. To measure the Z-component of spin of a particle is to make the path ϕt pass through one of the σ̂z-classical points ϕ+ ⊗ψ− or ϕ− ⊗ ψ+. In this case (and only in this case!) the Z-components of spin of both particles are defined and take opposite values. Furthermore, as with a single particle, the interaction with the measuring device is assumed to cause a “deformation” of the path ϕt. The resulting path ends up then at one of the classical points via a stochastic process on the space of states. The full version of the experiment involving a spatial separation of the particles is even more dramatic. How could the second particle at a point y “know” about measurement of the Z-component of spin performed on the first particle at a distant point x? Again, physical reality of the pair is most adequately described by a path ϕt = atϕx+ ⊗ ψy− + btϕx− ⊗ ψy+ in the space H of states of the pair. Here ϕx+ is the spin-up state of the first particle located at x and similarly for the other state functions in ϕt. The classical points in H have the form ϕx+⊗ψy− and ϕx−⊗ψy+. If ϕt does not pass through these points the spin of individual particles is not defined. Intuitively, we think that if a particle is “here” (at a point x), then it ought to have all attributes of a “real” particle, including spin. But before the measurement is performed, the particle in the experiment is not really here! Indeed, it is somewhere else on the sphere of states SH in H (or on the corresponding projective space CPH). So if reality is associated with the state function of the pair, the paradox is resolved. But what about this “spooky action at a distance”? Notice that the new “func- tional” reality does not use it! Indeed, the equation of geodesics is “local” in the space of states S3, because it is a differential equation of geodesics on S3. Of course, this locality in the space of states does not preclude a non-locality in the classical 32 Alexey A. Kryukov space. Indeed, what is a point in the space of states may represent a pair of well separated particles in the classical space. Furthermore, what is close in the metric on the space of functions does not have to be close in the metric on the classical space (see Ref. [4]). A detailed analysis of this will be, however, a subject for a different paper. REFERENCES 1. M. V. Berry, Proc. Roy. Soc. London A 392, 45 (1984) 2. J. Anandan and Y. Aharonov, Phys. Rev. Lett. 65, 1697 (1990) 3. B. Simon, Phys. Rev. Lett. 51, 2167 (1983) 4. A. Kryukov, Found. Phys. 36, 175 (2006) 5. A. Kryukov, Int. J. Math. & Math. Sci. 14, 2241 (2005) 6. A. Kryukov, Found. Phys. 34, 1225 (2004) 7. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990) 8. J. Bjorken and S. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1964) 9. P. Pearle, “Collapse models,” in Open Systems and Measurement in Relativis- tic Quantum Theory, F. Petruccione and H.P. Breuer, eds. (Springer, 1999) 10. A. Bassi & G. Ghirardi, Physics Reports 379, 257 (2003) GEOMETRY AND QUANTUM MECHANICS ELECTRON IN A HOMOGENEOUS MAGNETIC FIELD Quantum Mechanics on the Space of States S3 Quantum Mechanics on the Projective Space of States CP1 TENSOR PROPERTIES OF EQUATIONS IN THE MODEL THE PROCESS OF MEASUREMENT THE MEASUREMENT PROBLEM

Global-String and Vortex Superfluids in a Supersymmetric Scenario C.N. Ferreira 1,∗ J. A. Helayël-Neto 2,† and W.G. Ney 1‡ Núcleo de Estudos em F́ısica, Centro Federal de Educação Tecnológica de Campos Rua Dr. Siqueira, 273, Campos dos Goytacazes, Rio de Janeiro, Brazil, CEP 28030-130 Centro Brasileiro de Pesquisas F́ısicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro, Brazil, CEP 22290-180 (Dated: November 1, 2018) The main goal of this work is to investigate the possibility of finding the supersymmetric version of the U(1)-global string model which behaves as a vortex-superfluid. To describe the superfluid phase, we introduce a Lorentz-symmetry breaking background that, in an approach based on su- persymmetry, leads to a discussion on the relation between the violation of Lorentz symmetry and explicit soft supersymmetry breakings. We also study the relation between the string configuration and the vortex-superfluid phase. In the framework we settle down in terms of superspace and su- perfields, we actually establish a duality between the vortex degrees of freedom and the component fields of the Kalb-Ramond superfield. We make also considerations about the fermionic excitations that may appear in connection with the vortex formation. PACS numbers: 12.60.Jv,11.27.+d I. INTRODUCTION Stable vortex states may appear as an interesting man- ifestation of superfluidity. As one of the motivations, these vortices have been observed in bosonic or fermionic diluted gases[1, 2]. In the case of the bosonic vortices, the detection has been confirmed by analyzing the den- sity variations in an expanding Bose-Einstein Condensate (BEC)[2, 3]. In Fermi systems, we do not expect signif- icant density variations[4] but, under certain conditions, the density variations may be induced by the presence of one or more vortices that can be present in nuclear matter[5]. The interior of a neutron star, that is the only known system close to nuclear and neutron matter, con- stitutes the appropriate scenario for the vortex formation induced by the rotational state of the star. There are important observations of astrophysical rele- vance that might be influenced by the presence of vortices in the interior of neutron stars; for instance, the pulsar glitches. The glitching events represent a direct manifes- tation of the presence of superfluid vortices in the inte- rior of the star, the triggering event being an unbalance between the hydrodynamical forces acting on the vor- tex and the force of interaction of the vortex with nuclei presence in the crust, pinning force [6, 7]; but, there are doubts about the value of the pinning force. One is related to the value of the energy gap in uni- form neutron matter whereas the second problem is due to the very outlined way of treating vortex states in neu- ∗Electronic address: crisnfer@pq.cnpq.br †Electronic address: helayel@pq.cnpq.br ‡Electronic address: wander@cefetcampos.br tron matter. Global strings, which behave as a vortex superfluidity states, appear when a discrete symmetry is broken. These strings, as the local strings [9, 10, 11], were most likely produced during phase transitions [12], and appear in some Grand-Unified Gauge Theories. They carry a large energy density [10]. Both global and lo- cal strings were mainly studied as a possible mechanism for the seed density perturbation which has become a structure of large scale of the Universe we observe today [13, 14]. Nowadays, the approach considering cosmic string configurations has been revisited in connection with string theory[12, 15, 16] and the Wilkinson Microwave Anisotropy Probe (WMAP)[17]. The importance of this context is related to a possible measurement at the level of string theory, which has supersymmetry (SUSY) as one of its main characteristics. SUSY is also related to cosmic strings in other contexts, where one contem- plates the possibility that the boson-fermion symmetry was manifest in the early Universe, but it was broken approximately at the same time when these topological defects were formed. Many recent works investigate local strings by adopt- ing a supersymmetric framework [18, 19, 20, 21, 22, 23]. In the context of the star formation [24], SUSY appears as one of the most interesting mechanisms to describe cold dark matter[25, 26]. Both, local and global strings, are also important for their contribution to the gravita- tional radiation background[27]; in the case of the global symmetry, instead of radiating gravitationally, the dom- inant radiation mechanism for these strings is the emis- sion of massless Nambu-Goldstone bosons [28]. Global strings which behave as a vortex superfluidity states are connected with a Kalb-Ramond field. In some publica- tions, it has been shown that, in the low-energy regime, http://arxiv.org/abs/0704.1935v3 mailto:crisnfer@pq.cnpq.br mailto:helayel@pq.cnpq.br mailto:wander@cefetcampos.br the effective action that presents the Kalb-Ramond fields, that also appear in string theory, provides an accurate description of the dynamics of global strings[29]. The Kalb-Ramond field[30, 31] is an antisymmetric tensor. This tensor, whenever interacting with a mas- sive Higgs field, gives us a source. The system may have applications to superfluid helium and axion cosmology. A global vortex behaves as a superfluid if the Kalb-Ramond field breaks Lorentz symmetry in the background. The Kalb-Ramond fields in context of the topological defects can be studied in [32], with SUSY framework [20, 23] and associated with Lorentz-symmetry breaking can be studied in [21, 22]. For these implications, in this work, we analyze the equivalence of the vortex-superfluids to global strings in a supersymmetric context. This analogy is important to propose alternative models to be consid- ered; the vortex stability and the fermionic and bosonic behaviors of the matter can in the future enlighten us how to understand the vortex states in fermionic matter. The outline of this paper is as follows: in Section 2, we present some considerations about vortex superfluid models. In Section II, we devote our attention to showing how the Lorentz-symmetry violation by the Kalb-Ramond background induces explicit SUSY breaking terms. In Section III, we focus on the general properties of the su- persymmetric model for the vortex and treats some spe- cific properties of the supersymmetric superfluid phase in the model we study. In this Section, we also carry out the superfield identifications at zero temperature and start a discussion on the fermionic excitations. Section IV Fermionic excitations are the main issue presented. In Section V, we propose a discussion on the non-zero temperature treatment of the model. Finally, in Section VI, we draw our General Conclusions. II. THE IMPLICATIONS OF THE LORENTZ-VIOLATING BACKGROUND FOR SOFT SUSY BREAKING In this Section, we discuss the implications of the pres- ence of the Lorentz-violating background for a soft SUSY breaking. The idea here is to understand how the explicit soft SUSY breaking works to yield mass to some of the field of the model. We consider the possibility to get the scalar masses in SUSY theories by working with a Lorentz-symmetry violating background. This framework is important for a better understanding of the relation between explicit SUSY breaking and Lorentz-symmetry violation. It is important to stress that the problem of the SUSY break- ing is a very important matter, related with the hiearchy problem and the mass constraints on the supersymmet- ric particles. In this section, let us start off with the following supersymmetric Lagrangian: LK = Φ †e4gGΦ|θθθ̄θ̄. (1) where the ingredient superfields of the model are: a chiral scalar supermultiplet, Φ(φ, χ, F ), that contains a com- plex scalar field, φ, a spinor, χa, and an auxiliary com- plex scalar field, F . The chiral scalar supermultiplet Φ can be θ-expanded according to the following expression Φ = e−iθσ µθ̄∂µ [φ(x) + 2θaχa(x) + θ 2F (x)], (2) G is the Kalb-Ramond field-strength superfield defined in terms of the chiral spinor superfield as (DaΣa − D̄ȧΣ̄ȧ) (3) where Σa = ψa(x) + θ bΩba(x) + θ ξa(x) + iσ aȧ∂µψ̄ ȧ(x) −iθσµθ̄∂µψ̄a(x)− iθσµθ̄θȧ∂µΩȧa(x) θ2θ̄2✷ψa(x) The chirality condition for this field is D̄ȧΣa = 0. The Kalb-Ramond field accommodated in Ωȧb(x) is given by Ωab = −ǫabρ(x) + (σµν )abBµν(x). (5) with ρ(x) and Bµν(x) being complex fields, ρ(x) = P (x) + iM(x), Bµν(x) = 14 Bµν − iB̃µν(x) B̃µν(x) = ǫµναβB αβ(x) (7) The components P and ψa are compensating fields and are not present in the θ-expanssion of G, as it shall be explicitly given below. The superfield G(M, ξ, G̃µ), which plays a central role in connection with local vortices ([20]), accomodates the real scalar, M, the fermion ξ and the dual of the Kalb- Ramond field strength G̃µ. It can be θ-expanded accord- ing to the following expression: G = − ξ̄ȧ + aȧθ̄ aȧθ̄ 2∂µξ̄ ȧ − 1 aȧθ̄ ȧ∂µξ θ2θ̄2✷M ; (8) Now, we have all the elements to illustrate how the Lorentz-symmetry violation, signaled by the background of the Kalb-Ramond field, is intimately connected to the appearance of explicit (soft) SUSY breaking terms. We can notice that the superfield G carries only some degrees of freedom of Σa, the fermionic field ψa does not appear, ρ appears only through M , and as G̃µ is related to the 2-form, the Kalb-Ramond field, Bµν G̃µ = ǫµναβG ναβ . (9) Gµνκ = ∂µBνκ + ∂νBκµ + ∂κBµν (10) ∂µχ+ g 2|φ|2G̃µG̃µ +gG̃µ χ̄γµχ− φ̄∂µφ+ φ∂µφ̄ where in this discussion we consider Φ → ΦegM . The Lagrangian L is the interaction Lagrangian, and its explicit form is not important to show the relation be- tween the Lorentz and SUSY breakings. Let us consider the split Gµνλ as Gµνλ = G (self) (ext) . (12) The external Lorentz-symmetry breaking background is given by (ext) ρǫ0ijk = ρǫijk. (13) The crucial point here is the justification of why the background value of Gµνκ in (13) yields an explicit soft breaking of SUSY. The whole idea here is that the back- ground for Gµνλ given in (13) lies on a θ-component of G, actually, the θσµθ̄G̃µ in (8), which necessarily signals an explicit SUSY breaking. If the first component ( the θ -independent one) set up a non-trivial background then SUSY may not background, then SUSY may not be bro- ken; however, whenever the background value sits on a non-trivial θ-component, SUSY is necessarily explicitly broken down, and this is the case here. The relevant bosonic part of the (11) important to ana- lyzed the Lorentz-Breaking relation with SUSY breaking in the background is L = ∂µφ ∗∂µφ+ g2|φ|2G̃µG̃µ − φ∗∂µφ− φ∂µφ∗ By splitting the Kalb-Ramond fields as (12) and by adopting the ansatz of a Lorentz-breaking background, (13) there emerges a mass term for the bosons, LL−SUSY−B = g ρ|φ|2, (15) Terms like that may appear as a result of sponta- neous breaking of SUSY[34]. Soft explicit SUSY breaking terms are very important in connection with the physics derived from the Minimally Supersymmetric Standard Model (MSSM). In view of that, we try to stress here on the connection between a Lorentz-symmetry violating background and the appearance of explicit SUSY break- ing terms. III. THE SUPERSYMMETRIC VERSION FOR A GLOBAL VORTEX AND THE SUPERFLUID BEHAVIOR In the present section, we study the supersymmetric framework setting up the general formalism that gives us the terms to construct the global vortex and study the su- perfluid behavior. The action studied in the previous Sec- tion (1) helps us in understanding the consequences of the Lorentz-breaking background in connection with SUSY. Actually, we have to see how the presence of a back- ground yielding Lorentz symmetry violation also leads to an explicit SUSY breaking. In the present Section, let us adopt the action to study the vortex configuration as in the sequel: LK = Φ †e4gGΦ|θ2θ̄2 + S†S|θ2θ̄2 +W |θ2 + W̄ |θ̄2 . (16) These superfields satisfy a chirality constraint, given by the condition D̄ȧΦ = 0 and D̄ȧS = 0. The superfield Φ is defined in (2) and G in (8); the superfield S has the same properties as Φ and can be θ- expanded according S = e−iθσ µ θ̄∂µ [S(x) + 2θaζa(x) + θ 2H(x)]. (17) In the expression (16), W is the superpotential whose general form is W = aiΦi + bijΦiΦj + cijkΦiΦjΦk. (18) The scalar-field potential is given by ĀiAi = |2 (19) where Ai is the auxiliary component of the φi-superfield. Let us study the possibility to obtain the supersym- metric version of the global vortex potential according to the model discussed in [33]. In the case of a global gauge symmetry, the chiral superfield Φi transforms as a phase under the U(1)- symmetry: Φ′i = e −iqiΛΦi (20) where qi are U(1) global charges and Λ is the rigid U(1) rotation angle. The qi and Λ are real constants. It is possible to build up a potential with spontaneously broken gauge symmetry using three chiral superfields. For cosmic strings with a local gauge symmetry, one usu- ally needs two charged fields φ±, with respective U(1) charges q±, and a neutral field, Φ0. This mechanism re- mains the same if we have a global transformation (20); in this case, the superpotential takes the form: W (Φi) = µΦ0 Φ+Φ− − η2 In this approach, the neutral field Φ0 is important to give us the term responsible for the mass of the scalar field of the theory, but, in a global vortex superfluid con- figuration, we have a Lorentz breaking background, as discussed in[33], and we have proven, in the previous Sec- tion, that the Lorentz breaking introduces masses for the scalars; then, we can adopt a simpler potential, with one superfield Φ, with charge qΦ, and another superfield, S, with charge qS , satisfying the constraint qS = −2qΦ, so W = hSΦ2 (22) This form of the superpotential, in connection with the Lorentz-symmetry breaking (15) of the previous Section, leads to the Mexican hat configuration, responsible for the global vortex behavior that characterises superfluid- The SUSY transformations read as below: ǭȧξ̄ ȧ − i ǫaξa, (23) δξa = 2σ aȧǭ ∂µM − iG̃µ , (24) δG̃µ = ǫb(σµν)ab∂νξa + ǭḃ(σ̄ µν)ḃȧ∂ν ξ̄ ȧ (25) It is important to point out here that the soft super- symmetry breaking terms do not invalidate the super- symmetric transformations; actually, Lorentz symmetry and SUSY are broken down by the background, but they are both symmetries of the action. So, SUSY transfor- mations as translations in superspace are not lost. Global strings appear whenever a U(1) global symme- try is spontaneously broken. After the breaking of the U(1) symmetry, a massless Goldstone boson emerges that yields a long-range force. The bosonic Lagrangian result- ing from this supersymmetric model and that is relevant for the superfluid can be written as LB = ∂µφ φ+∂µS |φ|2GµνρGµνρ−V ′+G̃µJµ. for simplicity, we also adopt the redefinition Φ′ → ΦegM . The current jµ is given by φ∗∂µφ− φ∂µφ∗ The bosonic potential V ′ , that comes from the eq.(22), is given by ′ = g2|φ|4 + 4g2|S|2|φ|2. (28) We perform the background splitting (12) of (49), so that the full potential is V = h2|φ|4 − (g2ρ− 4h2|S|2)|φ|2. (29) This potential shows us that the U(1)-breaking gives mass to the moduli field |φ| while the phase of the scalar field remains massless. In the low-energy limit, the com- plex scalar field of the Goldstone model can be repre- sented as below: φ = ϕ(r)eiα, (30) where r is the radial coordinate. The boundary condi- tions are given by ϕ(r) = 0 to r = δ ϕ(r) = η to r → ∞. (31) The configuration (30)-(31) is the same as the one of the local vortices, but the long-range interactions of global strings, happen due to their coupling to a massless Goldstone field, cause their dynamics to be substantially different from those of the local strings. Spontaneous symmetry breaking requires that ϕ(r) have mass and α be a massless Goldstone boson. This breaking triggered by the soft SUSY breaking term we introduce, takes place whenever (g2ρ− 4h2|S|2) > 0. (32) This relationship is crucial to ensure the stability of the potential, for it guarantees the vortex is formed around the right ground state. This justifies our claim, stated in the previous Section, on the importance of the term that breaks SUSY explicitly (and softly) for the stability of the global vortex. A solution to the vortex configuration exists if g2 ≥ 4h2|S|2. In this configuration, we consider the boundary conditions to ϕ, given by (31) and, for the S-field, we consider the ansatz S = s(r)eiΛ. Outside the string, we consider the field 〈S〉 = 0. The global vortex presents a minimum roll and a central maximum characterizes the Mexican hat potential. By analyzing the potential minimum outside the string, with 〈φ〉 = η and 〈S〉 = 0, we have η = gρ . The ansatz in the core of the string allows us to analyze it in comparison with the fermionic Yukawa potential that are the subject of the section IV. The effective Lagrangian LB = ∂µϕ∂ ϕ+ ϕ2∂µα∂ α+ ∂µs∂ s+ s2∂µΛ∂ ϕ2G(self)µνρ G (self) ρεijkϕ2G (self) µν − V. (33) Now, let us write, the current Jµν it in terms of the Kalb-Ramond field, according to the functional relation below: µd4x = ǫµαβγ∂ αBβγJµd4x ǫαµβγB µν (34) where Jµν = ǫµναβ∂ φ− φ∂βφ∗ The configuration in the core of the string, where the commutator is not zero, [∂µ, ∂ν ]α 6= 0, in the presence of a vortex. We can see this clearly by considering a straight vortex along the z- axis, the azimuthal angle, and inte- grating over a two-surface orthogonal to the string yields, [∂x, ∂y]αdxdy = 2π, or [∂x, ∂y]α = δ(x)δ(y) , then, in the presence of the vortex the α is a multi-valued function of the coordinates and Jµν 6= 0 on the vortex core. Outside the string core, φ can be represented as φ ∼ η exp(iα(x)) and 〈S〉 = 0; the effective Lagrangian for the Goldstone mode (in the presence of the global strings at large distances of the core, which are non-massive exci- tations) can be written as L = η2∂µα∂ (self) µνρ G (self) εijkG (self) ijk + µν . (36) We use the fact that a real massless scalar field in four- dimensional Minkowski space is equivalent to a rank-2 anti-symmetric tensor , Bµν [30, 31]; the nature of this equivalence in the case of the global strings can be found in [28]. In SUSY, this duality property can be understand by superfield identification Φ†Φ ∼ G. (37) In fact, the left part that contains the vortex superfield gives us the term ϕ2∂µα and the right side gives us a term related with the dual field, G̃µ. The identification (37) gives us other contributions, related to the scalar field ϕ2 = 2ηM (38) The fermionic part is ∗ = −iηξa (39) 2χ̄ȧϕ = iηξ̄ȧ (40) and the vortex identification part |ϕ2|σµ∂µα+ χ̄χ = ηǫµνλρσ µ∂νBλρ. (41) We can notice that the fermionic part modifies the usual vortex duality relation [33]. If we neglect the fermionic contribution, eq.(41) can be written as ϕ2∂µα = ηǫµνλρ∂ νBλρ. (42) The identification of the bosonic part given by (42) has the same form as the [33] for the global vortex configu- ration, but, with the supersymmetric invariance, we can always have a fermionic part. The only remaining dynamical degree of freedom is the scalar (Goldstone boson) field, α. In this approach, we have the action for the (global) static string: (Self) µνβ G (Self) ρεijkG (self) ijk + µνd4x (43) where β = 1 + g2η2. At this point, it is advisable to remind that an ex- plicit Lorentz-symmetry breaking, as stated above, may be rephrased in terms of a softly explicit SUSY breaking term as the one we consider here [8]. Now, let us study the solution at long distances compared to the string core, when the interaction of the vortex with the clas- sical Goldstone-boson field is described by an effective Lagrangian. The stress tensor in the background consid- ers a string at rest point in û direction we have T 00 = T ii = ρ = p (44) T 0i = β ρG0jk ǫijk. (45) The equation of the motion for the Kalb-Ramond field µαβ = Jαβ (46) We obtain the solution ûirj − ûjri It yields the stress tensor interaction part given by T 0i = 2 (û× r)i A single straight global string has a logarithmically di- vergent energy per unit of length. We can think that these strings could be ignored because they appear to be unphysical. However, following cosmological phase tran- sitions, global strings may form loops with finite total energy or open strings with finite energy per horizon. An interesting application that some authors have been envisaging is the possibility that radiative decay of closed loops be connected with density fluctuation in the process of structure formation. This approach, considering the data basis, has been ruled out alone, but together with in- flationary models and considering the noise of the exper- imental data, we can still consider them [17]. In the ap- proach of [33], that is considered here, the vortex configu- ration is stable in the presence of the special background that breaks the Lorentz invariance[33]. The fact that the superfluid vortex is immersed in a Lorentz-noninvariant fluid suggests that the correct model for a superfluid vor- tex involves the choice of a special background. The rela- tivistic force law for the response of a vortex to the local field Gµνρ is analogous to the Lorentz force law in Elec- trodynamics. The external force due the background field interaction is given as F i = JjkG jki = ρǫjkiJjk . The bosonic part of the solution has the same form as in the non-supersymmetric model, but, in our construction, the solution presents the explicit dependence on the param- eters h and g and on the effects of the fermions. In the supersymmetric version, the introduction of a Lorentz- symmetry violating background gives us important impli- cations on the fermionic background that we shall discuss in the next section, when we study the supersymmetric superfluid. IV. THE ANALYSIS OF THE FERMIONS AND THE SUPERFLUIDITY BEHAVIOR In a supersymmetric framework, besides the bosonic degrees of freedom, there are fermionic partners in the theory. In this section, let us analyse the behavior of the fermions that accompany the bosonic fields. The fermionic action can be written as: χ̄σµ∂µχ+ BµνJ µν + LFKR + Lint, (49) where Jµν is the fermionic current of the vorticity. The latter can also be expressed as follows: Jµν = ǫµναβ∂ αχ̄σβχ, (50) where χ̄σµχ. (51) The Lagrangian LFKR contains the fermionic Kalb- Ramond couplings and reads as below: LFKR = χ̄σµG̃µχ. (52) This Lagrangian (52) amounts to a mass contribution given by the Lorentz-breaking parameter present in (13), that is, LmassFKR = χ̄χ. (53) Lint is the interacting Lagrangian, where we include the Yukawa terms which induce masses to the fermions that couple to the vortex. These Yukawa couplings are collected in: Y = g 2φζaχa + 2φ ζ̄a + Sχ χa + S . (54) From (54) and (53), there follows an interesting pos- sibility. If we choose 〈S〉 = 0 to be zero in the core of the string, the mass (53) does not appear in the core, the fermionic interaction term vanishes in the core and the fermions and χ and ξ become massless. In this case, where the fermions is not have mass, the fermionic zero- modes propagate with the speed of light in the z-direction and particles can be ejected from the vortex. Outside the vortex, these particles have masses induced by the φ-interaction Yukawa term and by (53), as induced by the Lorentz-symmetry breaking [35]. V. SECOND-ORDER PHASE TRANSITIONS AND THE RELATION OF THE S-FIELD WITH THE TEMPERATURE In this Section, let us study a physical interpretation of the field S. Up to now, we know that the field S is important for the zero-modes. Now, let us study another interpretation, possibly related with second-order phase transitions. The potential (29) can represent a high- temperature effective potential[28], that can be written V (φ, T ) = m2(T )|φ|2 + h2|φ|4 (55) where we identify S with the temperature, T . We actu- ally consider |S|2 = T 2, then m2(T ) = h2(4T 2 − η2). (56) The term m(T ) is the mass for the φ-field, whenever the state is symmetric, 〈|φ|〉 = 0. This mass vanishes when T = Tc, and . (57) Another important case occurs for T > Tc; the effec- tive mass m2(T ) is positive and the minimum of V is at φ = 0. The physical interpretation of this result is that the expectation value of φ vanishes. This means that the symmetry is restored at high temperature. The symmet- ric vacuum becomes unstable and φ develops a non-zero expectation value. Minimizing V , as in (55), we obtain, for T < Tc, |φ| = T 2c − T 2 An important realization of the second phase transition is the fact that |φ| grows continuously from zero, as the temperature decreases from the critical temperature, Tc. The cosmological point of view, when the supersym- metric Universe cools through the critical temperature, is that the field φ develops an expectation value of mag- nitude (58). The evolution of the phases α of φ and Λ of S with the temperature is not determined only by local physics; their values outside depends on random fluctuations and α and Λ take different values in differ- ent regions of space during the evolution. But, since the free energy is minimized, these phases after the Universe expansion can become precedent sections with the tem- perature T = 0. We can define the correlation length, Π(t), to be the length scale above which the values of α and Λ are uncorrelated. The evolution of Π(t) depends on details of the relaxation processes. Indeed, Π(t) has to satisfy the causality bound. The correlation length cannot establish scales greater than the causal horizons related with the distance travelled by the light during the life-time of the Universe. For T < Tc, the scalar field de- velops an expectation value corresponding to some point in the manifold M of the minima of the effective poten- tial V . We can see in (58) that the term that in our model is given by the soft SUSY breaking, was presented in the high temperature state, but, in the case T ≫ Tc, SUSY breaking can be neglected, and we can consider the Uni- verse as being in a supersymmetric phase. To understand this fact, we need String Theory arguments, not contem- plated in this work. This analysis only gives us knowledge about the vortex formation, and it is not able to provide us with information on the Lorentz breaking. But, if we consider that, when the temperature becomes low vortex formation may take place, then Lorentz symmetry may be violated and there occurs a vortex-superfluid forma- tion, as we have analysed throughout this paper. VI. GENERAL CONCLUSIONS In this work, we have shown that it is possible to build up a string vortex by modelling the vortex superfluid in a supersymmetric context. We have analyzed the potential that gives us the correct string vortex configuration, in zero temperature, it presents a soft SUSY-breaking in- duced by the hidden sector. We have also analyzed the bosonic aspects of the duality representation of the vor- tex. We also analysed the physical interpretation of the extra field S related to the presence of the zero-mode and we show that we can relate it to the temperature. It is advisable to comment here that the violation of Lorentz symmetry introduced is independent of the soft SUSY explicit breaking terms. The latter has been considered to be correct taking into account stability aspects of the potential. It would be very interesting to eventually un- derstand if the violation of Lorentz symmetry and the explicit SUSY breaking could be related to one another. This would render our proposal more interesting, in that we would be dealing with less arbitrary parameters. Also, it would clarify the interplay between Lorentz-symmetry violation (in the sense of particle transformations) and SUSY explicit breaking that describes mass splittings among bosons and fermions that belong to the same su- permultiplet. This is an issue that could be investigated better in a future work. The interesting phenomenolog- ical aspect of this discussion would be checking whether properties like the masses of the SUSY particles, such as the photino and the higgsino, would necessarily signal to some type of Lorentz-symmetry breaking. In section V, we have proposed a discussion in a cosmological evolution context, but, as already pointed out, these vortices also appear in star cores. In the case of the neutron stars, as presented in the Introduction, the force that dictates the vortex stability is induced by the nuclear matter; but, we do not eliminate the possibility of the presence of a Lorentz-symmetry breaking to have an important role in- side the star. Nothing guarantees that the matter inside the star has a Lorentz-invariant behavior, because the high energy envolves, in analogy with high energy γ-rays from extragalatic sources[36]. Another point is if we con- sider that dark matter is mostly composed by supersym- metric particles, the relation between the Lorentz and SUSY breaking may become important to understand the parameters of the model. Then, the possibility of the Lorentz-symmetry breaking in supersymmetric mat- ter becomes relevant for the dark matter stability around the stars [25] and particles can then be ejected out of these astrophysical structures. In this work, we do not have an application for these objects, but we understand that our model can be an alternative possibility to under- stand some phenomena involving high energies. The next step is studying the fermionic implication of the Lorentz- symmetry breaking Kalb-Ramond background and how we could find out a mechanism to justify its appearance, the relation between Lorentz and SUSY breaking and the origin of the hidden sector represented by soft breaking of global SUSY. Acknowledgments: W. Bietenholz is acknowledged for a critical read- ing and for many helpful suggestions on an original manuscript. The authors would also like to thank (CNPq-Brasil) for the invaluable financial support. [1] K. W. Madison and F. Chevy, J. Mod. Opt. 47, 2715 (2000); F. Chevy, K. W. Madison and J. Dalibard, Phys. Rev. Lett. 85, 2223 (2000). [2] J. R. Abo-shaeer, et al.,Science 292.476 (2001). [3] E. A. Cornell and C. E. Wieman, Rev. Mod. Phys. 74, 875, (2002); W. Katterle, Rev. Mod. Phys. 74,1131, (2002). [4] N. Nygaard, G. M. Bruun, C. W. Clark and D. L. Feder, Phys. Rev. Lett. 90, 210402, (2003); O. Elgaroy and F. V. De Blasio, Astron.& Astrophys, 370, 939, (2001). [5] Y. Yu and A. Bulgac, Phys. Rev. Lett. 90, 161101, (2003). [6] P.M. Pizzochero, L. Viverit, R.A. Broglia, Phys.Rev.Lett.79 3347, 1997. [7] P. W. Anderson and N. Itoh, Nature, 256, 25 (1975); M. A. Alpar, Ap J. 213, 527 (1977): R. Epstein and G. Baym, Ap J. 328, 680 (1988). [8] C. E. Campos Lima, C. N. Ferreira and J. A. Helayel- Neto, work in progress. [9] A.Vilenkin, Phys. Rev. D 23,852 , (1981); W.A.Hiscock, Phys. Rev. D 31, 3288, (1985); J.R.GottIII, Astrophys. Journal, 288, 422, (1985); D. Garfinkle, Phys. Rev. D 32 1323, (1985). [10] M.B. Hindmarsh and T.W.B. Kibble, Rept. Prog. 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[24] E. Moulin, F. Mayet and D. Santos, astro-ph/0503436, 2005. [25] G. Jungman et al., Phys. Rept. 267, 195, (1996), (Ref- erences about the detection of Dark Matter to see D. Spergel et al., Astrophys. J. Suppl. 148, 175, (2003); A. Ben̂ıt et al., Astron. & Astrophys. 399, L25, (2003); S. Perlmutter, M. S. Turner and M. White, Phys. Rev Lett. 83, 670, (1999); M. Tagmark et al., Astrophys. J. 606, 702, (2004); S. Eidelman et al., Phys. Lett. B 592, 1, (2004)). [26] A. Masiero, S. Profumo and P. Ullio, Nucl. Phys. B 712, 86, 2005; M. Drees, M. M. Nojir, D. P. Roy and Y. Ya- mada, Erratum IBID D6, 039901, 2001. [27] D. P. Bennett and F. R. Bouchet, Phys. Rev. Lett. 48, 2733 (1991). [28] A.Vilenkin and E.P.S.Shellard, Cosmic Strings and other Topological Defects (Cambridge University Press, 1994). [29] R. A. Battye and E. P. S. Shellard, Phys. Rev D 53, 1811, (1996). [30] M.Kalb and P.Ramond, Phys.Rev. D 9, 2273, (1974). [31] F. Lund and T. Regge, Phys. Rev. D14, 1524, (1976). [32] N. 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The main goal of this work is to investigate the possibility of finding the supersymmetric version of the U(1)-global string model which behaves as a vortex-superfluid. To describe the superfluid phase, we introduce a Lorentz-symmetry breaking background that, in an approach based on supersymmetry, leads to a discussion on the relation between the violation of Lorentz symmetry and explicit soft supersymmetry breakings. We also study the relation between the string configuration and the vortex-superfluid phase. In the framework we settle down in terms of superspace and superfields, we actually establish a duality between the vortex degrees of freedom and the component fields of the Kalb-Ramond superfield. We make also considerations about the fermionic excitations that may appear in connection with the vortex formation.

Introduction, the force that dictates the vortex stability is induced by the nuclear matter; but, we do not eliminate the possibility of the presence of a Lorentz-symmetry breaking to have an important role in- side the star. Nothing guarantees that the matter inside the star has a Lorentz-invariant behavior, because the high energy envolves, in analogy with high energy γ-rays from extragalatic sources[36]. Another point is if we con- sider that dark matter is mostly composed by supersym- metric particles, the relation between the Lorentz and SUSY breaking may become important to understand the parameters of the model. Then, the possibility of the Lorentz-symmetry breaking in supersymmetric mat- ter becomes relevant for the dark matter stability around the stars [25] and particles can then be ejected out of these astrophysical structures. In this work, we do not have an application for these objects, but we understand that our model can be an alternative possibility to under- stand some phenomena involving high energies. The next step is studying the fermionic implication of the Lorentz- symmetry breaking Kalb-Ramond background and how we could find out a mechanism to justify its appearance, the relation between Lorentz and SUSY breaking and the origin of the hidden sector represented by soft breaking of global SUSY. Acknowledgments: W. Bietenholz is acknowledged for a critical read- ing and for many helpful suggestions on an original manuscript. The authors would also like to thank (CNPq-Brasil) for the invaluable financial support. [1] K. W. Madison and F. Chevy, J. Mod. Opt. 47, 2715 (2000); F. Chevy, K. W. Madison and J. Dalibard, Phys. Rev. Lett. 85, 2223 (2000). [2] J. R. Abo-shaeer, et al.,Science 292.476 (2001). [3] E. A. Cornell and C. E. Wieman, Rev. Mod. Phys. 74, 875, (2002); W. Katterle, Rev. Mod. Phys. 74,1131, (2002). [4] N. Nygaard, G. M. Bruun, C. W. Clark and D. L. Feder, Phys. Rev. Lett. 90, 210402, (2003); O. Elgaroy and F. V. De Blasio, Astron.& Astrophys, 370, 939, (2001). [5] Y. Yu and A. Bulgac, Phys. Rev. Lett. 90, 161101, (2003). [6] P.M. Pizzochero, L. Viverit, R.A. Broglia, Phys.Rev.Lett.79 3347, 1997. [7] P. W. Anderson and N. Itoh, Nature, 256, 25 (1975); M. A. Alpar, Ap J. 213, 527 (1977): R. Epstein and G. Baym, Ap J. 328, 680 (1988). [8] C. E. Campos Lima, C. N. Ferreira and J. A. Helayel- Neto, work in progress. [9] A.Vilenkin, Phys. Rev. D 23,852 , (1981); W.A.Hiscock, Phys. Rev. D 31, 3288, (1985); J.R.GottIII, Astrophys. Journal, 288, 422, (1985); D. Garfinkle, Phys. Rev. D 32 1323, (1985). [10] M.B. Hindmarsh and T.W.B. Kibble, Rept. Prog. Phys 58, 477, (1995). [11] T.W.B. Kibble, Phys. Rep 67, 183, (1980). [12] T.W. Kibble, J. of Phys. A9, 1387 (1976). [13] A.Stebbins, Ap. J. (Lett), 303, L21 (1986). [14] H. Sato, Prog. Theor. Phys. 75, 1342 (1986). [15] M. Majumdar, hep-th/0512062; C. Lin and J. MCDonald,hep-ph/0604245; [16] J. Polchinski, Int. J. Mod. Phys. A20, 3413, 2005; J. Polchinski and J. V. Rocha, Phys. Rev D74, 083504, 2006. [17] E. Jeong and G. F. Smoot, ”Vality of Cosmic String Pattern Search with Cosmic Microwave Background”, artro-ph/0612706. [18] J. R. Morris, Phys.Rev. D 53, 2078, (1996). [19] S.C.Davis, A.C.Davis and M.Trodden, Phys. Lett. B 405, 257 (1997). [20] C.N. Ferreira, M.B.D.S.M. Porto, J.A. Helayel-Neto, Nucl. Phys. B 620: 181, (2002). [21] C. N. Ferreira, H. Chavez and J. A . Helayel-Neto , Proc. Sci. WC2004: 036, (2004). [22] V.B. Bezerra, C.N. Ferreira and J.A. Helayel-Neto, Phys. Rev. D 71: 044018, (2005) [23] C. N. Ferreira and J. A . Helayel-Neto , Proc. Sci. WC2006: 036, (2006). [24] E. Moulin, F. Mayet and D. Santos, astro-ph/0503436, 2005. [25] G. Jungman et al., Phys. Rept. 267, 195, (1996), (Ref- erences about the detection of Dark Matter to see D. Spergel et al., Astrophys. J. Suppl. 148, 175, (2003); A. Ben̂ıt et al., Astron. & Astrophys. 399, L25, (2003); S. Perlmutter, M. S. Turner and M. White, Phys. Rev Lett. 83, 670, (1999); M. Tagmark et al., Astrophys. J. 606, 702, (2004); S. Eidelman et al., Phys. Lett. B 592, 1, (2004)). [26] A. Masiero, S. Profumo and P. Ullio, Nucl. Phys. B 712, 86, 2005; M. Drees, M. M. Nojir, D. P. Roy and Y. Ya- mada, Erratum IBID D6, 039901, 2001. [27] D. P. Bennett and F. R. Bouchet, Phys. Rev. Lett. 48, 2733 (1991). [28] A.Vilenkin and E.P.S.Shellard, Cosmic Strings and other Topological Defects (Cambridge University Press, 1994). [29] R. A. Battye and E. P. S. Shellard, Phys. Rev D 53, 1811, (1996). [30] M.Kalb and P.Ramond, Phys.Rev. D 9, 2273, (1974). [31] F. Lund and T. Regge, Phys. Rev. D14, 1524, (1976). [32] N. R. F. Braga and C. N. Ferreira, JHEP 0503, 039, (2005). [33] R. L. Davis and E.P.S. Shellard, Phys. Rev. Lett 63, 2021, (1989). [34] H. P. Nilles. Phys. Rep. 110, 1 (1984). [35] C. N. Ferreira, J. A. 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Identities for number series and their reciprocals: Dirac delta function approach S. M. Abrarov 1, R. M. Abrarov 2 April 20, 2007 Abstract Dirac delta function (delta-distribution) approach can be used as efficient method to derive identities for number series and their reciprocals. Applying this method, a simple proof for identity relating prime counting function (π-function) and logarithmic integral (Li-function) can be obtained. Keywords: delta function, delta-distribution, prime counting function, distribution of primes, Mertens’ formula, harmonic number I. Introduction Since Paul Dirac introduced the delta function [1], it remains a versatile mathematical tool in many applications of the modern sciences and information technologies. Being a physicist P. Dirac intuitively believed in the existence of the delta function as a true mathematical object and successfully applied it developing the fundamentals of the relativistic quantum mechanics in his classic works [1, 2]. Nowadays Dirac delta function is widely used in the pure and applied mathematics and engineering physics [3-8]. Particularly the sampling methods, based on Dirac delta function, proved their reliability and are commonly implemented in the systems of telecommunications and signal processing [8]. In this work we introduce some useful applications of delta-distribution in the theory of numbers. Compared to Stieltjes integration methods [9], Dirac delta function approach is more natural and obvious for understanding and can be successfully used in the derivations of the identities for the number series and their reciprocals. II. Some properties of δ-function We will apply two main properties of the δ-function. The first one is the sampling property ( ) ( ) ( )0 0f x f x x xδ = −∫ dx , (1) while the second one relates itself with Heaviside step function [10] ( ) 00 − = ⎨ via integration ( ) ( )0 0x x x xχ δ − = −∫ dx . (2) It should be noted that integral of the delta function can be determined by infinitely narrow area in the vicinity of x0. Therefore (1) and (2) can be rewritten as ( ) ( ) ( ) f x f x x x = −∫ dx ( ) ( ) x x x x − = −∫ dx , respectively, where ε is infinitely small positive value. III. Step function series 3.1 Number of pulses Consider a set consisting of numbers qN 1, q2, q3, … , qN. Define the series as a sum of reciprocals qi as follows ( ) min iq x i (3) where is the smallest number in this set. According to the sampling property (1) of the delta- distribution, the series (3) can be rewritten as ( ) ( ) h x y q dy = −∑∫ , which, in turn, can be expressed in the form ( ) ( ) ixh x x qδ ′ = −∑ . (4) In compliance with (2), integrating of (4) yields ( ) ( ) min i yh y dy x q ′ = −∑∫ i . (5) RHS of (5) counts the number of stepwise pulses, therefore: 1 1 1 ... N timesq yh y dy x ′ = + + + = ⎢ ⎥⎣ ⎦∫ , where symbol ⎣ ⎦ denotes the floor operator. Equation (5) contains derivative, which may be inconvenient for computing. In order to avoid it, we apply integration by part. This leads to ( ) ( ) N xh x h y dy= − ∫ , (6) where is the number of the stepwise pulses within interval from up to N minq x . 3.2 Sum of numbers Multiplying x to both parts of (4) and integrating the result, we can find that ( ) ( ) i i i y h y q x q ′ = ∑∫ − . (7) RHS of (7) represents the sum of all numbers up to x. Therefore this equation we can be rewritten as q x q q y h y ′=∑ ∫ . (8) The derivative in (8) can be excluded again. This provides the following relation ( ) ( ) q x q q x h x yh y dy = −∑ ∫ . 3.3 Power series More generally, (4) can be extended for any power of x as given by ( ) ( )1 x h x x x qδ+ ′ = −∑ , where k is any number. This leads to the relation of the power series q x q q y h y dy′=∑ ∫ . (9) Excluding the derivative in (9), we have ( ) ( ) ( ) k k k q x q q x h x k y h y d+ = − +∑ ∫ y . (10) 3.4 Sum of reciprocals Define the following function as q x q By analogy with (4), the superposition of the delta functions is given by ( ) ( )ix qx = −∑ . (11) Dividing both parts of (11) by kx and integrating the result yields ( ) ( ) i ik k q x q xi iq dy x q ′ ⎛ ⎞ = − = ⎜ ⎟ (12) Finally, integrating by part (12) results to ( ) ( ) ( ) q x i q g x g y q x y+≤ = + +⎜ ⎟ ∑ ∫ 2k dy+ . (13) IV. Natural and prime numbers 4.1 Natural numbers We will use the relations derived above for natural and prime numbers. Consider the harmonic number (further index will be omitted) i ( ) 1 H x n ⎢ ⎥⎣ ⎦ ∑ (14) where n = 1, 2, 3 … . Substituting (14) into (10), we can find that ( ) ( ) ( )1 1 2 3 4 ... 1 k k k k k k k n x H x k ⎢ ⎥⎣ ⎦ = + + + + = − +∑ ∫ y H y dy There are two interesting cases. For k = 0¸ we obtain an equation counting the number of pulses, similar to that of (6). However, as the gap between two closest integers is always unity, the number of pulses must be equal to the largest integer maxn ∈ , i.e.: ( ) ( )max n nn xH x H y d= − ∫ y H y dy For k = 1, we have the sum of the arithmetic progression of the natural numbers ( ) ( )2 1 2 3 4 ... 2 n x H x y ⎢ ⎥⎣ ⎦ = + + + + = −∑ ∫ . Therefore using the formula , we can write ( )2max max max/ 2 1 2 3 ...n n n+ = + + + + ( ) ( ) 2max max x H x yH y dy = − ∫ . To derive a relation for harmonic number, we use (13) and substitute . This provides the following identity x x y y H x dy + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦= + ∫ . (15) It should be noted that the algorithm, built on the basis of (15), significantly accelerates the computation of the harmonic number, especially for the large values of x . This is possible to achieve since RHS of (15) includes the set of integers instead of reciprocals. The computation performed directly through (14) involves rapidly decreasing terms as x increases. As a result of the multiple process of division, the calculation becomes problematic due to the decrease of the significant digits after floating point. The use of integers effectively resolves this problem. 4.2 Prime numbers Define a sum of the reciprocal primes in the following form , (16) where p = 2, 3, 5, 7 … , i.e. all primes smaller than x. Substituting (16) into (10), we get ( ) ( ) ( )1 2 3 5 7 ... 1 k k k k k k k p x H x k y H y dy+ = + + + + = − +∑ ∫ . Consider again two interesting cases. For k = 0, we have an equation showing a quantity of all primes up to x. In other words, primes counting function can be expressed through the following identity ( ) ( ) ( ) p px xH x H y dyπ = − ∫ . (17) For k =1, we have an equation showing the sum of all primes up to x, i.e.: ( ) ( )2 2 3 5 7 ... 2 p x H x yH y dy = + + + + = −∑ ∫ . The identity for , expressed through set of primes, can be readily found using (13) and substituting : ( )pH x ( ) 2 3 p x p x H x p p dy x y≤ ≤ = +∑ ∑∫ . (18) By analogy to (15), RHS of (18) does not include the set of reciprocals. As a result, the algorithm, built on the basis of identity (18), essentially decreases the computation time, especially for the large arguments x . The computation of can also be performed though ( )pH x ( )xπ . Rewrite (17) in the following form ( ) ( ) px yH y dy ′= ∫ . (19) Rearranging (19) and applying integration by part, we get ( ) ( ) ( )2 H x dy = + ∫ . V. Proof of π-function identity Dirac delta function approach can also be useful for any kinds of broken-line functions. As an example, we represent a simple proof for the identity relating prime counting function and Li-function [11] ( ) ( ) ( ) ( ) x yR y x Li x dy Li = + −∫ + , (20) where ( )R x is defined as ( ) log log = −∑ x . (21) Using the sampling property (1) of Dirac delta function, R(x) can be expressed as log log log log p x p x x y p dy x − = − −∑ ∑∫ . (22) The derivative of (22) yields ( ) ( )log 1 R x x p ′ = −∑ − , ( ) (1 log log p x + = −∑ . (23) Integrating of (23) results to ( ) ( ) ( ) Li x dy const x p + + =∑∫ − . The sum of Heaviside step functions ( ) −∑ is actually π-function counting the number of all primes up to x . A constant in LHS can be determined from the initial condition at 2x = when π-function is unity. This provides exact formula for prime counting function and completes the proof of (20). The proof can be obtained even simpler. Consider Mertens’ formula providing a relation for sum consisting of the reciprocal primes [12] ( ) ( ) ( ) log log 1 log log 2 loglog R y R H x x dy = + − + +∫ . (24) Taking derivative of (24) yields ( ) ( )1 log logp x x x ′ = + . (25) Ultimately, substituting (25) into (19) leads to (20). It is proven again. VI. Conclusion We applied Dirac delta function approach to derive identities for the sums of the numbers and their reciprocals. δ-Function is also useful to obtain a simple proof of the identity relating prime counting function and Li-function. Acknowledgements Authors express gratitude to Prof. T. D. Radjabov and Dr. E. N. Tsoy for their constructive remarks and suggestions. References [1] P. A. M. Dirac, Proc. R. Soc., Series A, 113 (1927) 621 [2] P. A. M. Dirac, Principles of Quantum Mechanics, Oxford University Press, 1930 [3] R. P. Kanwal, Generalized Functions: Theory and Applications, Birkhäuser, 2004 [4] P. Antosik, J. Mikusinski, R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier Scientific Publishing Company, 1973 [5] A. I. Khuri , Advanced Calculus with Applications in Statistics, Wiley, 2003 [6] A. I. Khuri, Int. J. Math. Educ. Sci. Technol., 35 (2004) 185 [7] C. Guilpin, J. Gacougnolle and Y. Simon, Appl. Num. Math., 48 (2004) 27 [8] M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation, The Morgan Kaufmann Series in Interactive 3D Technology, pp. 279-367, 2004 http://links.jstor.org/sici?sici=0950-1207%2819270101%29113%3A765%3C621%3ATPIOTQ%3E2.0.CO%3B2-8 http://dx.doi.org/10.1080/00207390310001638313 http://dx.doi.org/10.1016/S0168-9274(03)00104-1 [9] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, pp. 152-155, 1988 [10] We consider an alternative form of Heaviside function when ( )0χ 1≡ . See, e.g.: http://en.wikipedia.org/wiki/Heaviside_step_function [11] R. M. Abrarov, S. M. Abrarov (to be published) [12] G. Tenenbaum and M. M. France, The Prime Numbers and Their Distribution, AMS, p. 22, 2001 ________________________________________________________________________________ 1 York University, Toronto, Canada abrarov@yorku.ca Dongguk University, Seoul, South Korea abrarov@dongguk.edu 2 University of Toronto, Canada rabrarov@physics.utoronto.ca http://en.wikipedia.org/wiki/Heaviside_step_function mailto:abrarov@yorku.ca mailto:abrarov@dongguk.edu mailto:rabrarov@physics.utoronto.ca

Dirac delta function (delta-distribution) approach can be used as efficient method to derive identities for number series and their reciprocals. Applying this method, a simple proof for identity relating prime counting function (pi-function) and logarithmic integral (Li-function) can be obtained.

Introduction Since Paul Dirac introduced the delta function [1], it remains a versatile mathematical tool in many applications of the modern sciences and information technologies. Being a physicist P. Dirac intuitively believed in the existence of the delta function as a true mathematical object and successfully applied it developing the fundamentals of the relativistic quantum mechanics in his classic works [1, 2]. Nowadays Dirac delta function is widely used in the pure and applied mathematics and engineering physics [3-8]. Particularly the sampling methods, based on Dirac delta function, proved their reliability and are commonly implemented in the systems of telecommunications and signal processing [8]. In this work we introduce some useful applications of delta-distribution in the theory of numbers. Compared to Stieltjes integration methods [9], Dirac delta function approach is more natural and obvious for understanding and can be successfully used in the derivations of the identities for the number series and their reciprocals. II. Some properties of δ-function We will apply two main properties of the δ-function. The first one is the sampling property ( ) ( ) ( )0 0f x f x x xδ = −∫ dx , (1) while the second one relates itself with Heaviside step function [10] ( ) 00 − = ⎨ via integration ( ) ( )0 0x x x xχ δ − = −∫ dx . (2) It should be noted that integral of the delta function can be determined by infinitely narrow area in the vicinity of x0. Therefore (1) and (2) can be rewritten as ( ) ( ) ( ) f x f x x x = −∫ dx ( ) ( ) x x x x − = −∫ dx , respectively, where ε is infinitely small positive value. III. Step function series 3.1 Number of pulses Consider a set consisting of numbers qN 1, q2, q3, … , qN. Define the series as a sum of reciprocals qi as follows ( ) min iq x i (3) where is the smallest number in this set. According to the sampling property (1) of the delta- distribution, the series (3) can be rewritten as ( ) ( ) h x y q dy = −∑∫ , which, in turn, can be expressed in the form ( ) ( ) ixh x x qδ ′ = −∑ . (4) In compliance with (2), integrating of (4) yields ( ) ( ) min i yh y dy x q ′ = −∑∫ i . (5) RHS of (5) counts the number of stepwise pulses, therefore: 1 1 1 ... N timesq yh y dy x ′ = + + + = ⎢ ⎥⎣ ⎦∫ , where symbol ⎣ ⎦ denotes the floor operator. Equation (5) contains derivative, which may be inconvenient for computing. In order to avoid it, we apply integration by part. This leads to ( ) ( ) N xh x h y dy= − ∫ , (6) where is the number of the stepwise pulses within interval from up to N minq x . 3.2 Sum of numbers Multiplying x to both parts of (4) and integrating the result, we can find that ( ) ( ) i i i y h y q x q ′ = ∑∫ − . (7) RHS of (7) represents the sum of all numbers up to x. Therefore this equation we can be rewritten as q x q q y h y ′=∑ ∫ . (8) The derivative in (8) can be excluded again. This provides the following relation ( ) ( ) q x q q x h x yh y dy = −∑ ∫ . 3.3 Power series More generally, (4) can be extended for any power of x as given by ( ) ( )1 x h x x x qδ+ ′ = −∑ , where k is any number. This leads to the relation of the power series q x q q y h y dy′=∑ ∫ . (9) Excluding the derivative in (9), we have ( ) ( ) ( ) k k k q x q q x h x k y h y d+ = − +∑ ∫ y . (10) 3.4 Sum of reciprocals Define the following function as q x q By analogy with (4), the superposition of the delta functions is given by ( ) ( )ix qx = −∑ . (11) Dividing both parts of (11) by kx and integrating the result yields ( ) ( ) i ik k q x q xi iq dy x q ′ ⎛ ⎞ = − = ⎜ ⎟ (12) Finally, integrating by part (12) results to ( ) ( ) ( ) q x i q g x g y q x y+≤ = + +⎜ ⎟ ∑ ∫ 2k dy+ . (13) IV. Natural and prime numbers 4.1 Natural numbers We will use the relations derived above for natural and prime numbers. Consider the harmonic number (further index will be omitted) i ( ) 1 H x n ⎢ ⎥⎣ ⎦ ∑ (14) where n = 1, 2, 3 … . Substituting (14) into (10), we can find that ( ) ( ) ( )1 1 2 3 4 ... 1 k k k k k k k n x H x k ⎢ ⎥⎣ ⎦ = + + + + = − +∑ ∫ y H y dy There are two interesting cases. For k = 0¸ we obtain an equation counting the number of pulses, similar to that of (6). However, as the gap between two closest integers is always unity, the number of pulses must be equal to the largest integer maxn ∈ , i.e.: ( ) ( )max n nn xH x H y d= − ∫ y H y dy For k = 1, we have the sum of the arithmetic progression of the natural numbers ( ) ( )2 1 2 3 4 ... 2 n x H x y ⎢ ⎥⎣ ⎦ = + + + + = −∑ ∫ . Therefore using the formula , we can write ( )2max max max/ 2 1 2 3 ...n n n+ = + + + + ( ) ( ) 2max max x H x yH y dy = − ∫ . To derive a relation for harmonic number, we use (13) and substitute . This provides the following identity x x y y H x dy + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦= + ∫ . (15) It should be noted that the algorithm, built on the basis of (15), significantly accelerates the computation of the harmonic number, especially for the large values of x . This is possible to achieve since RHS of (15) includes the set of integers instead of reciprocals. The computation performed directly through (14) involves rapidly decreasing terms as x increases. As a result of the multiple process of division, the calculation becomes problematic due to the decrease of the significant digits after floating point. The use of integers effectively resolves this problem. 4.2 Prime numbers Define a sum of the reciprocal primes in the following form , (16) where p = 2, 3, 5, 7 … , i.e. all primes smaller than x. Substituting (16) into (10), we get ( ) ( ) ( )1 2 3 5 7 ... 1 k k k k k k k p x H x k y H y dy+ = + + + + = − +∑ ∫ . Consider again two interesting cases. For k = 0, we have an equation showing a quantity of all primes up to x. In other words, primes counting function can be expressed through the following identity ( ) ( ) ( ) p px xH x H y dyπ = − ∫ . (17) For k =1, we have an equation showing the sum of all primes up to x, i.e.: ( ) ( )2 2 3 5 7 ... 2 p x H x yH y dy = + + + + = −∑ ∫ . The identity for , expressed through set of primes, can be readily found using (13) and substituting : ( )pH x ( ) 2 3 p x p x H x p p dy x y≤ ≤ = +∑ ∑∫ . (18) By analogy to (15), RHS of (18) does not include the set of reciprocals. As a result, the algorithm, built on the basis of identity (18), essentially decreases the computation time, especially for the large arguments x . The computation of can also be performed though ( )pH x ( )xπ . Rewrite (17) in the following form ( ) ( ) px yH y dy ′= ∫ . (19) Rearranging (19) and applying integration by part, we get ( ) ( ) ( )2 H x dy = + ∫ . V. Proof of π-function identity Dirac delta function approach can also be useful for any kinds of broken-line functions. As an example, we represent a simple proof for the identity relating prime counting function and Li-function [11] ( ) ( ) ( ) ( ) x yR y x Li x dy Li = + −∫ + , (20) where ( )R x is defined as ( ) log log = −∑ x . (21) Using the sampling property (1) of Dirac delta function, R(x) can be expressed as log log log log p x p x x y p dy x − = − −∑ ∑∫ . (22) The derivative of (22) yields ( ) ( )log 1 R x x p ′ = −∑ − , ( ) (1 log log p x + = −∑ . (23) Integrating of (23) results to ( ) ( ) ( ) Li x dy const x p + + =∑∫ − . The sum of Heaviside step functions ( ) −∑ is actually π-function counting the number of all primes up to x . A constant in LHS can be determined from the initial condition at 2x = when π-function is unity. This provides exact formula for prime counting function and completes the proof of (20). The proof can be obtained even simpler. Consider Mertens’ formula providing a relation for sum consisting of the reciprocal primes [12] ( ) ( ) ( ) log log 1 log log 2 loglog R y R H x x dy = + − + +∫ . (24) Taking derivative of (24) yields ( ) ( )1 log logp x x x ′ = + . (25) Ultimately, substituting (25) into (19) leads to (20). It is proven again. VI. Conclusion We applied Dirac delta function approach to derive identities for the sums of the numbers and their reciprocals. δ-Function is also useful to obtain a simple proof of the identity relating prime counting function and Li-function. Acknowledgements Authors express gratitude to Prof. T. D. Radjabov and Dr. E. N. Tsoy for their constructive remarks and suggestions. References [1] P. A. M. Dirac, Proc. R. Soc., Series A, 113 (1927) 621 [2] P. A. M. Dirac, Principles of Quantum Mechanics, Oxford University Press, 1930 [3] R. P. Kanwal, Generalized Functions: Theory and Applications, Birkhäuser, 2004 [4] P. Antosik, J. Mikusinski, R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier Scientific Publishing Company, 1973 [5] A. I. Khuri , Advanced Calculus with Applications in Statistics, Wiley, 2003 [6] A. I. Khuri, Int. J. Math. Educ. Sci. Technol., 35 (2004) 185 [7] C. Guilpin, J. Gacougnolle and Y. Simon, Appl. Num. Math., 48 (2004) 27 [8] M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation, The Morgan Kaufmann Series in Interactive 3D Technology, pp. 279-367, 2004 http://links.jstor.org/sici?sici=0950-1207%2819270101%29113%3A765%3C621%3ATPIOTQ%3E2.0.CO%3B2-8 http://dx.doi.org/10.1080/00207390310001638313 http://dx.doi.org/10.1016/S0168-9274(03)00104-1 [9] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, pp. 152-155, 1988 [10] We consider an alternative form of Heaviside function when ( )0χ 1≡ . See, e.g.: http://en.wikipedia.org/wiki/Heaviside_step_function [11] R. M. Abrarov, S. M. Abrarov (to be published) [12] G. Tenenbaum and M. M. France, The Prime Numbers and Their Distribution, AMS, p. 22, 2001 ________________________________________________________________________________ 1 York University, Toronto, Canada abrarov@yorku.ca Dongguk University, Seoul, South Korea abrarov@dongguk.edu 2 University of Toronto, Canada rabrarov@physics.utoronto.ca http://en.wikipedia.org/wiki/Heaviside_step_function mailto:abrarov@yorku.ca mailto:abrarov@dongguk.edu mailto:rabrarov@physics.utoronto.ca

UTCCS-P-31, UTHEP-542, KEK-CP-193, HUPD-0702, RBRC-666 Light quark masses from unquenched lattice QCD T. Ishikawa,1, 2 S. Aoki,3, 2 M. Fukugita,4 S. Hashimoto,5, 6 K-I. Ishikawa,7 N. Ishizuka,1, 3 Y. Iwasaki,3 K. Kanaya,3 T. Kaneko,5, 6 Y. Kuramashi,1, 3 M. Okawa,7 Y. Taniguchi,1, 3 N. Tsutsui,5 A. Ukawa,1, 3 N. Yamada,5, 6 and T. Yoshié1, 3 (CP-PACS and JLQCD Collaborations) 1 Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan 2 RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA 3 Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan 4 Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan 5 High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan 6 School of High Energy Accelerator Science, The Graduate University for Advanced Studies (Sokendai), Tsukuba 305-0801, Japan 7 Department of Physics, Hiroshima University, Higashi-Hiroshima 739-8526, Japan (Dated: October 31, 2018) We calculate the light meson spectrum and the light quark masses by lattice QCD simulation, treating all light quarks dynamically and employing the Iwasaki gluon action and the nonperturba- tively O(a)-improved Wilson quark action. The calculations are made at the squared lattice spacings at an equal distance a2 ≃ 0.005, 0.01 and 0.015 fm2, and the continuum limit is taken assuming an O(a2) discretization error. The light meson spectrum is consistent with experiment. The up, down and strange quark masses in the MS scheme at 2 GeV are m = (mu +md)/2 = 3.55 +0.65 −0.28 MeV and ms = 90.1 +17.2 −6.1 MeV where the error includes statistical and all systematic errors added in quadra- ture. These values contain the previous estimates obtained with the dynamical u and d quarks within the error. The masses of light quarks are fundamental parame- ters of QCD. They cannot be measured experimentally since quarks are confined in hadrons. Lattice QCD en- ables calculations of hadron masses as functions of quark masses, and hence allows a determination of the quark masses from the experimental hadron masses. This ap- proach has been successfully applied, first in quenched QCD [1] and then in Nf = 2 QCD where degenerate up (u) and down (d) quarks are treated dynamically [2]. These studies have revealed that the light quark mass val- ues are significantly reduced by dynamical u and d quark effects. In this article, we present our attempt to deter- mine the quark masses in Nf = 2 + 1 QCD where the heavier strange (s) quark is also treated dynamically. We wish to examine to what extent the dynamical s quark affects the light quark masses. We determine the quark masses in the continuum limit and estimate all possible systematic errors. We also calculate the prerequisite light meson spectrum. A similar attempt has been made by the MILC Collaboration [3]. We adopt the Iwasaki RG gauge action [4] and the clover quark action with the improvement coefficient cSW determined nonperturbatively for the RG action [5]. The choice of the gauge action is made to avoid a first-order phase transition (lattice artifact) observed for the plaque- tte gauge action [6]. We employed the Wilson quark for- malism because we prefer an unambiguous quark-flavor interpretation over the computational ease of the stag- gered formalism adopted by the MILC collaboration [7]. Configurations are generated at three values of the cou- pling β ≡ 6/g2 = 2.05, 1.90 and 1.83 corresponding to the squared lattice spacing a2 ≃ 0.005, 0.01 and 0.015 fm2, with the physical volume fixed to about (2.0fm)3. At each β, we perform simulations for 10 quark mass com- binations using a combined algorithm [8] of the Hybrid Monte Carlo (HMC) for the degenerate u and d quarks and the polynomial Hybrid Monte Calro (PHMC) for the s quark. Table I summarizes the simulation parameters. The meson and quark masses at the simulation points are determined from single exponential correlated χ2 fits to the correlators 〈P (t)P (0)〉, 〈V (t)V (0)〉 and 〈A4(t)P (0)〉, where P , V and Aµ denote pseudoscalar, vector and nonperturbatively O(a)-improved [9] axial- vector current operators, respectively. We use an ex- ponentially smeared source and a point sink, and mea- surements are made at every 10 HMC trajectories in the Coulomb gauge. For the pseudoscalar sector, 〈P (t)P (0)〉 and 〈A4(t)P (0)〉 are fitted simultaneously ignoring cor- relations among them. Errors are estimated by the jack- knife method with a bin size of 100 HMC trajectories; errors do not increase for larger bin sizes. Chiral fits are made to the light-light (LL), light- strange (LS) and strange-strange (SS) meson masses simultaneously ignoring their correlations, using a quadratic polynomial function of the sea quark masses (mu,md,ms) and valence quark masses (mval1,mval2) in mesons; f(Ms,Mv) (1) = A+BStrMs +BV trMv +DSV trMstrMv +CS1trM s + CS2(trMs) 2 + CV 1trM v + CV 2(trMv) where f = m2PS or vector meson mass mV , MS = http://arxiv.org/abs/0704.1937v2 TABLE I: Simulation parameters; L3×T is the lattice size, (κud, κs) is the hopping parameter combination, 1/δτ is the number of molecular dynamics steps in one trajectory, Npoly is the PHMC polynomial order, and traj. is analyzed trajectory length. Pseudoscalar vector mass ratios are also listed for light-light (LL) and strange-strange (SS) mass combinations. β = 1.83, L3 × T = 163 × 32, cSW = 1.761 κud κs δτ Npoly traj. (SS) κud κs δτ Npoly traj. 0.13655 0.13710 1/80 80 7000 0.7772(13) 0.7522(15) 0.13655 0.13760 1/90 110 7000 0.7769(14) 0.7235(19) 0.13710 1/85 80 7000 0.7524(21) 0.7524(21) 0.13710 1/100 110 8600 0.7448(14) 0.7128(16) 0.13760 1/100 100 7000 0.7076(18) 0.7414(17) 0.13760 1/110 120 8000 0.7033(18) 0.7033(18) 0.13800 1/120 110 8000 0.6629(22) 0.7365(16) 0.13800 1/120 130 8100 0.6525(23) 0.6941(20) 0.13825 1/140 120 8000 0.6213(24) 0.7343(15) 0.13825 1/150 150 8100 0.6083(32) 0.6884(21) β = 1.90, L3 × T = 203 × 40, cSW = 1.715 κud κs δτ Npoly traj. (SS) κud κs δτ Npoly traj. 0.13580 0.13580 1/125 110 5000 0.7673(15) 0.7673(15) 0.13580 0.13640 1/125 140 5200 0.7667(16) 0.7211(21) 0.13610 1/125 110 6000 0.7435(18) 0.7647(17) 0.13610 1/125 140 8000 0.7444(15) 0.7182(17) 0.13640 1/140 110 7600 0.7204(19) 0.7687(15) 0.13640 1/140 140 9000 0.7145(16) 0.7145(16) 0.13680 1/160 110 8000 0.6701(27) 0.7673(17) 0.13680 1/160 140 9200 0.6630(21) 0.7127(17) 0.13700 1/180 110 7900 0.6390(22) 0.7691(15) 0.13700 1/180 140 7900 0.6243(28) 0.7102(20) β = 2.05, L3 × T = 283 × 56, cSW = 1.628 κud κs δτ Npoly traj. (SS) κud κs δτ Npoly traj. 0.13470 0.13510 1/175 200 6000 0.7757(26) 0.7273(29) 0.13470 0.13540 1/175 250 6000 0.7790(23) 0.6821(32) 0.13510 1/195 200 6000 0.7316(24) 0.7316(24) 0.13510 1/195 250 6000 0.7341(29) 0.6820(39) 0.13540 1/225 200 6000 0.6874(30) 0.7395(23) 0.13540 1/225 250 6000 0.6899(34) 0.6899(34) 0.13550 1/235 200 6500 0.6611(34) 0.7361(25) 0.13550 1/235 250 6500 0.6679(45) 0.6899(43) 0.13560 1/250 200 6500 0.6337(38) 0.7377(28) 0.13560 1/250 250 6500 0.6361(47) 0.6852(46) 0 0.02 0.04 0.06 0.08 Ks=0.13580 Ks=0.13640 β=1.90, L xT=20 x40, polynomial 0 0.02 0.04 0.06 0.08 Ks=0.13580 Ks=0.13640 β=1.90, L xT=20 x40, polynomial FIG. 1: Chiral fits of meson masses with mAWIq at β = 1.90. diag(mu,md,ms), MV = diag(mval1,mval2), and “tr” means the trace of matrices. In the fits, we use the axial-vector Ward identity quark mass mq = limt→∞〈∂4A4(t)P (0)〉/(2〈P (t)P (0)〉) and set A = BS = CS1 = CS2 = 0 for m PS . These fits reproduce mea- sured data well, as illustrated in Fig.1, with reasonable χ2/d.o.f. of at most 1.36. The physical quark mass point and the lattice spac- ing are determined from the experimental values of π0, ρ0 and K (K-input) or π0, ρ0 and φ (φ-input) meson masses. Taking the ρ0 mass as input may cause a large systematic error, because the ρ → ππ decay mode is not open for our mass range (the lightest pion mass in this simulation ∼ 620 MeV) and hence chiral extrapo- TABLE II: Lattice spacings in fm units. β K-input φ-input [π,K, φ]-input 1.83 0.1174(23) 0.1184(26) 0.1095(25) 1.90 0.0970(26) 0.0971(25) 0.0936(33) 2.05 0.0701(29) 0.0702(28) 0.0684(41) lation of mV for lighter quarks may be quite different from our fits. In order to estimate this uncertainty, we also check another combination [π0,K, φ]. We assume the ideal mixing for the vector isosinglets. Since our sim- ulation is made with degenerate u and d quarks, we con- sider the isospin averages m = {(m2 )/2}1/2 and m = (mK∗± + mK∗0)/2 and predict the aver- age light quark mass m = (mu + md)/2. The electro- magnetic (EM) effects, not included in our simulations, are removed from the mK± above using Dashen’s the- orem [10] (m2 − m2K0)EM = (m − m2π0)EXP. The isospin breaking effects and the EM effects for other mesons we consider are expected to be small and thus are not considered. The experimental values we use are taken from the PDG booklet [11]; mπ0 = 0.1350GeV, mπ± = 0.1396GeV, mK0 = 0.4976GeV, mK± = 0.4937GeV, mρ0 = 0.7755GeV, mK∗0 = 0.8960GeV, mK∗± = 0.8917GeV and mφ = 1.0195GeV. Lattice spacings (Ta- ble II) for theK- and φ- inputs are consistent, while those for the [π,K, φ]-input are slightly smaller by at most 7%. An agreement of the meson spectrum with experiment is a necessary condition for a reliable estimate of the quark masses. To confirm this, we extrapolate the me- TABLE III: Meson masses in the continuum limit (in MeV units), compared to experiment. The EM effect is subtracted using Dashen’s theorem. K-input φ-input [π,K, φ]-input EXP. K̂ - 491(19) - 495.0 ρ0 - - 761(32) 775.5 K̂∗ 900.5(9.9) 898.0(1.4) 891(16) 893.9 φ 1025(19) - - 1019.5 1.000 1.050 0.860 0.880 0.900 0.880 0.890 0.900 0 0.01 0.02 0.03 0.04 0.05 0.450 0.500 0.550 Nf=0 (IW+ PT clover) Nf=2 (IW+ PT clover) Nf=2+1 (IW+NPT clover) experiment φ (K-input) K* (K-input) K* (φ-input) K (φ-input) FIG. 2: Continuum extrapolation of meson masses for Nf = 2 + 1 QCD (circles), compared to experiment (stars) and re- sults in Nf = 2 (squares) and Nf = 0 (triangles) QCD [2]. son masses linearly in a2, because our action is O(a) improved and data are well fitted, as shown in Fig.2, with small χ2/d.o.f ≤ 1.4. The masses in the continuum limit, summarized in Table III, are consistent with exper- iment with at most 2.9σ deviation. The K̂∗ mass turns out to be slightly heavier than experiment, though the supplemental [π,K, φ]-input gives consistent results with experiment with large statistical error. Possible origin of the deviation is due to uncertainty of chiral fits. In fact, an alternative fit based on chiral perturbation the- ory (χPT) we discuss later yields m = 894(12) MeV (K-input). In Fig.2 we overlay the previous results of meson masses [2] in the Nf = 2 and quenched (Nf = 0) QCD with tadpole improved one-loop cSW . The dynam- ical u and d quarks significantly reduce the O(10%) de- viation of the quenched spectrum from experiment. We find no further dynamical s quark effect beyond statisti- cal errors. The quark masses are evaluated for the MS scheme at the scale µ = 2GeV using the tadpole improved one- loop matching [12] at µ = a−1 with an improved cou- pling determined from plaquette and rectangular loop and four-loop renormalization group equation. In the continuum extrapolation of the quark masses, we assume 0 0.01 0.02 0.03 0.04 0.05 Nf=0 (K, IW+ PT clover) Nf=2 (K, IW+ PT clover) Nf=2+1 (K, IW+NPT clover) 0 0.01 0.02 0.03 0.04 0.05 Nf=0 (K, IW+ PT clover) Nf=0 (φ, IW+ PT clover) Nf=2 (K, IW+ PT clover) Nf=2 (φ, IW+ PT clover) Nf=2+1(K, IW+NPT clover) Nf=2+1(φ, IW+NPT clover) FIG. 3: Continuum extrapolations of the up, down and strange quark masses. For comparison, results for Nf = 0 and Nf = 2 QCD [2] are overlaid. the O(g4ma) contributions are small and neglect it. As Fig. 3 shows, the quark masses are well described by a linear function in a2, and the values determined for either the K- or the φ-inputs, while different at finite lattice spacings, extrapolate to a common value in the continuum limit. Therefore the continuum limit is esti- mated from a combined linear fit with the K- and the φ-inputs. We obtain mMS(µ = 2GeV) = 3.55(19) MeV and mMSs (µ = 2GeV) = 90.1(4.3) MeV with a suffi- ciently small χ2/d.o.f. < 0.42. Note that the supplemen- tal [π,K, φ]-input gives larger statistical error and hence is not used to estimate central values. We now turn to estimates of possible systematic errors. Finite size effect (FSE) — The meson masses at the infinite volume are estimated at β = 1.90 using data on a V ∼ (2.0fm)3 lattice and those from our exploratory study on a V ∼ (1.6fm)3 lattice [13], and assuming a strong volume dependence of (mhad,V − mhad,V=∞)/mhad,V=∞ ∝ 1/V [14]. The chiral fits to the infinite volume values lead to less than a 4% change for the meson masses at the physical point. For the quark masses, however, we find a larger shift of 12.2% from a V ∼ (2.0fm)3 lattice to V = ∞ for m with φ-input and 8.1% for ms with K-input (differences are smaller for the other cases). Assuming that FSE is independent of lat- tice spacing, we take the differences as estimates of FSE for the quark masses in the continuum limit. Chiral extrapolation — In addition to the polynomial chiral fits, we fit the meson masses using χPT formu- laes modified for the Wilson quark action (WχPT) [15]. Namely, we fit mπ, mK̂ , mρ and mK̂∗ using the NLO Nf = 2+1 QCD WχPT formulae for the O(a) improved theory [16]. Since the formula in Ref. [16] is not appli- cable for the φ meson, we estimate the effect only for K-input. In the fits we obtain m to be 3.1% smaller and ms to be 1.2% larger than those of the polynomial fit. We note that our WχPT fits to data do not exhibit a clear chiral logarithm, probably because u and d quark masses in our simulation are not sufficiently small. Further pos- sible systematic error from a long chiral extrapolation for ρ0, mentioned above, is estimated by the supplemental [π,K, φ]-input, which gives 3.0% larger for m and 3.4% larger value for ms than the central one. For an estimate of systematic errors from chiral fits, differences of the two alternative fits from the central value are added linearly. Renormalization factor — Uncertainty of the one- loop calculation of the renormalization factor is esti- mated by shifting the matching scale from µ = 1/a to µ = π/a and also using an alternative tadpole improved coupling [2]. Continuum extrapolation — Possible O(a3) effects are investigated by performing the continuum extrapola- tion adding an O(a3) term to the fit function. Electromagnetic (EM) effects — Systematic error due to uncertainty of the EM effects is estimated fol- lowing extensive arguments [7, 17, 18] to Dashen’s theorem [10]. Namely, we estimate the effects by a further mass shift of our input m using a relation )EM = (1 + ∆E)(m )EXP assum- ing the EM effects for other mesons are negligeble. We vary the ∆E in range [−1,+1] as our estimate of the EM effects, and we find a quite small change in ms and no change in m. Isospin breaking effects — Isospin breaking effects are estimated by chiral fits with Eq. (1) for mu 6= md and taking mπ0 , mρ0 , mK± and mK0 as inputs. We find that mu/md = 0.577(25), and that m and ms have no change from the K̂ input result. We note that mu/md strongly depends on an estimate of the EM ef- fects; mu/md =0.663–0.498 for ∆E = [−1,+1], though m and ms almost do not. Finally we obtain mMS(µ = 2GeV) = 3.55(19)(+43 −0 )( −11)( −17)( −0 )( −0), (2) mMSs (µ = 2GeV) = 90.1(4.3)(+7.3 −0 )( −0 )( −4.3)( +12.8 −0 )( −0.2)( −0), (3) in MeV units, where the errors are statistical, systematic due to FSE, chiral extrapolation, renormalization factor, continuum extrapolation, EM effect and isospin break- ing effect, respectively. Adding the errors in quadrature yields the values quoted in the abstract. These values agree well with the latest report from the MILC Collab- oration [3] m = 3.3 ± 0.3 MeV and ms = 90 ± 6 MeV where we added the quoted errors in quadrature. They also include the Nf = 2 values [2] within the error. Scaling violation in the quark masses is unexpectedly large, while that for the meson masses are reasonably bounded at a percent level at a ≈ 0.1 fm. To gain a better control over systematic uncertainties, a significant reduction in the simulated light quark masses on a correspondingly larger lattice is needed. An attempt is underway to meet these challenges [19]. This work is supported by the Epoch Making Simula- tion Projects of Earth Simulator Center, the Large Scale Simulation Program No.132 (FY2005) of High Energy Accelerator Research Organization (KEK), the Large Scale Simulation Projects of Academic Computing and Communications Center of University of Tsukuba, Inter University Services of Super Computers of Information Technology Center of University of Tokyo, Super Sinet Projects of National Institute of Informatics, and also by the Grant-in-Aid of the Ministry of Education (Nos. 13135204, 13135216, 15540251, 16540228, 16470147, 17340066, 17540259, 18104005, 18540250, 18740130). [1] S. Aoki et al. [CP-PACS Collaboration], Phys. Rev. Lett. 84, 238 (2000); Phys. Rev. D 67, 034503 (2003). [2] A. Ali Khan et al. [CP-PACS Collaboration], Phys. Rev. D 65, 054505 (2002) [Erratum-ibid. D 67, 059901 (2003)]; Phys. Rev. Lett. 85, 4674 (2000) [Erratum-ibid. 90, 029902 (2003)]; S. Aoki et al. [JLQCD Collabora- tion], Phys. Rev. D 68, 054502 (2003). [3] C. Bernard et al. [MILC Collaboration], PoS LAT2006, 163 (2006). [4] Y. Iwasaki, Nucl. Phys. B 258, 141 (1985); Univ. of Tsukuba Report No. UTHEP-118, 1983. [5] S. Aoki et al. [CP-PACS/JLQCD Collaboration], Phys. Rev. D 73, 034501 (2006). [6] S. Aoki et al. [JLQCD Collaboration], Phys. Rev. D 72, 054510 (2005). [7] C. Aubin et al. [HPQCD Collaboration, MILC Collab- oration and UKQCD Collaboration], Phys. Rev. D 70, 031504(R) (2004); Phys. Rev. D 70, 114501 (2004). [8] S. Aoki et al. [JLQCD Collaboration], Phys. Rev. D 65, 094507 (2002). [9] T. Kaneko et al. [CP-PACS/JLQCD and ALPHA Col- laborations], arXiv:hep-lat/0703006. [10] R. F. Dashen, Phys. Rev. 183, 1245 (1969). [11] W. M. Yao et al. [Particle Data Group], J. Phys. G 33, 1 (2006). [12] S. Aoki et al., Phys. Rev. D 58, 074505 (1998). [13] T. Kaneko et al. [CP-PACS/JLQCD Collaboration], Nucl. Phys. Proc. Suppl. 129, 188 (2004). [14] M. Fukugita et al., Phys. Lett. B 294, 380 (1992). [15] S. R. Sharpe and R. L. . Singleton, Phys. Rev. D 58, 074501 (1998); S. Aoki, Phys. Rev. D 68, 054508 (2003). [16] S. Aoki et al., Phys. Rev. D 73, 014511 (2006); S. Aoki et al., Phys. Rev. D 73, 094501 (2006); S. Takeda, Doctral thesis at Univ. of Tsukuba (2005). [17] J. Bijnens and J. Prades, Nucl. Phys. B 490, 239 (1997); J. Bijnens and P. Gosdzinsky, Phys. Lett. B 388, 203 (1996). [18] D. R. Nelson, arXiv:hep-lat/0212009. [19] Y. Kuramashi et al. [PACS-CS Collaboration], PoS LAT2006, 029 (2006)); A. Ukawa et al. [PACS-CS Col- laboration], PoS LAT2006, 039 (2006). http://arxiv.org/abs/hep-lat/0703006 http://arxiv.org/abs/hep-lat/0212009

We calculate the light meson spectrum and the light quark masses by lattice QCD simulation, treating all light quarks dynamically and employing the Iwasaki gluon action and the nonperturbatively O(a)-improved Wilson quark action. The calculations are made at the squared lattice spacings at an equal distance a^2~0.005, 0.01 and 0.015 fm^2, and the continuum limit is taken assuming an O(a^2) discretization error. The light meson spectrum is consistent with experiment. The up, down and strange quark masses in the \bar{MS} scheme at 2 GeV are \bar{m}=(m_{u}+m_{d})/2=3.55^{+0.65}_{-0.28} MeV and m_s=90.1^{+17.2}_{-6.1} MeV where the error includes statistical and all systematic errors added in quadrature. These values contain the previous estimates obtained with the dynamical u and d quarks within the error.

UTCCS-P-31, UTHEP-542, KEK-CP-193, HUPD-0702, RBRC-666 Light quark masses from unquenched lattice QCD T. Ishikawa,1, 2 S. Aoki,3, 2 M. Fukugita,4 S. Hashimoto,5, 6 K-I. Ishikawa,7 N. Ishizuka,1, 3 Y. Iwasaki,3 K. Kanaya,3 T. Kaneko,5, 6 Y. Kuramashi,1, 3 M. Okawa,7 Y. Taniguchi,1, 3 N. Tsutsui,5 A. Ukawa,1, 3 N. Yamada,5, 6 and T. Yoshié1, 3 (CP-PACS and JLQCD Collaborations) 1 Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan 2 RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA 3 Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan 4 Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan 5 High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan 6 School of High Energy Accelerator Science, The Graduate University for Advanced Studies (Sokendai), Tsukuba 305-0801, Japan 7 Department of Physics, Hiroshima University, Higashi-Hiroshima 739-8526, Japan (Dated: October 31, 2018) We calculate the light meson spectrum and the light quark masses by lattice QCD simulation, treating all light quarks dynamically and employing the Iwasaki gluon action and the nonperturba- tively O(a)-improved Wilson quark action. The calculations are made at the squared lattice spacings at an equal distance a2 ≃ 0.005, 0.01 and 0.015 fm2, and the continuum limit is taken assuming an O(a2) discretization error. The light meson spectrum is consistent with experiment. The up, down and strange quark masses in the MS scheme at 2 GeV are m = (mu +md)/2 = 3.55 +0.65 −0.28 MeV and ms = 90.1 +17.2 −6.1 MeV where the error includes statistical and all systematic errors added in quadra- ture. These values contain the previous estimates obtained with the dynamical u and d quarks within the error. The masses of light quarks are fundamental parame- ters of QCD. They cannot be measured experimentally since quarks are confined in hadrons. Lattice QCD en- ables calculations of hadron masses as functions of quark masses, and hence allows a determination of the quark masses from the experimental hadron masses. This ap- proach has been successfully applied, first in quenched QCD [1] and then in Nf = 2 QCD where degenerate up (u) and down (d) quarks are treated dynamically [2]. These studies have revealed that the light quark mass val- ues are significantly reduced by dynamical u and d quark effects. In this article, we present our attempt to deter- mine the quark masses in Nf = 2 + 1 QCD where the heavier strange (s) quark is also treated dynamically. We wish to examine to what extent the dynamical s quark affects the light quark masses. We determine the quark masses in the continuum limit and estimate all possible systematic errors. We also calculate the prerequisite light meson spectrum. A similar attempt has been made by the MILC Collaboration [3]. We adopt the Iwasaki RG gauge action [4] and the clover quark action with the improvement coefficient cSW determined nonperturbatively for the RG action [5]. The choice of the gauge action is made to avoid a first-order phase transition (lattice artifact) observed for the plaque- tte gauge action [6]. We employed the Wilson quark for- malism because we prefer an unambiguous quark-flavor interpretation over the computational ease of the stag- gered formalism adopted by the MILC collaboration [7]. Configurations are generated at three values of the cou- pling β ≡ 6/g2 = 2.05, 1.90 and 1.83 corresponding to the squared lattice spacing a2 ≃ 0.005, 0.01 and 0.015 fm2, with the physical volume fixed to about (2.0fm)3. At each β, we perform simulations for 10 quark mass com- binations using a combined algorithm [8] of the Hybrid Monte Carlo (HMC) for the degenerate u and d quarks and the polynomial Hybrid Monte Calro (PHMC) for the s quark. Table I summarizes the simulation parameters. The meson and quark masses at the simulation points are determined from single exponential correlated χ2 fits to the correlators 〈P (t)P (0)〉, 〈V (t)V (0)〉 and 〈A4(t)P (0)〉, where P , V and Aµ denote pseudoscalar, vector and nonperturbatively O(a)-improved [9] axial- vector current operators, respectively. We use an ex- ponentially smeared source and a point sink, and mea- surements are made at every 10 HMC trajectories in the Coulomb gauge. For the pseudoscalar sector, 〈P (t)P (0)〉 and 〈A4(t)P (0)〉 are fitted simultaneously ignoring cor- relations among them. Errors are estimated by the jack- knife method with a bin size of 100 HMC trajectories; errors do not increase for larger bin sizes. Chiral fits are made to the light-light (LL), light- strange (LS) and strange-strange (SS) meson masses simultaneously ignoring their correlations, using a quadratic polynomial function of the sea quark masses (mu,md,ms) and valence quark masses (mval1,mval2) in mesons; f(Ms,Mv) (1) = A+BStrMs +BV trMv +DSV trMstrMv +CS1trM s + CS2(trMs) 2 + CV 1trM v + CV 2(trMv) where f = m2PS or vector meson mass mV , MS = http://arxiv.org/abs/0704.1937v2 TABLE I: Simulation parameters; L3×T is the lattice size, (κud, κs) is the hopping parameter combination, 1/δτ is the number of molecular dynamics steps in one trajectory, Npoly is the PHMC polynomial order, and traj. is analyzed trajectory length. Pseudoscalar vector mass ratios are also listed for light-light (LL) and strange-strange (SS) mass combinations. β = 1.83, L3 × T = 163 × 32, cSW = 1.761 κud κs δτ Npoly traj. (SS) κud κs δτ Npoly traj. 0.13655 0.13710 1/80 80 7000 0.7772(13) 0.7522(15) 0.13655 0.13760 1/90 110 7000 0.7769(14) 0.7235(19) 0.13710 1/85 80 7000 0.7524(21) 0.7524(21) 0.13710 1/100 110 8600 0.7448(14) 0.7128(16) 0.13760 1/100 100 7000 0.7076(18) 0.7414(17) 0.13760 1/110 120 8000 0.7033(18) 0.7033(18) 0.13800 1/120 110 8000 0.6629(22) 0.7365(16) 0.13800 1/120 130 8100 0.6525(23) 0.6941(20) 0.13825 1/140 120 8000 0.6213(24) 0.7343(15) 0.13825 1/150 150 8100 0.6083(32) 0.6884(21) β = 1.90, L3 × T = 203 × 40, cSW = 1.715 κud κs δτ Npoly traj. (SS) κud κs δτ Npoly traj. 0.13580 0.13580 1/125 110 5000 0.7673(15) 0.7673(15) 0.13580 0.13640 1/125 140 5200 0.7667(16) 0.7211(21) 0.13610 1/125 110 6000 0.7435(18) 0.7647(17) 0.13610 1/125 140 8000 0.7444(15) 0.7182(17) 0.13640 1/140 110 7600 0.7204(19) 0.7687(15) 0.13640 1/140 140 9000 0.7145(16) 0.7145(16) 0.13680 1/160 110 8000 0.6701(27) 0.7673(17) 0.13680 1/160 140 9200 0.6630(21) 0.7127(17) 0.13700 1/180 110 7900 0.6390(22) 0.7691(15) 0.13700 1/180 140 7900 0.6243(28) 0.7102(20) β = 2.05, L3 × T = 283 × 56, cSW = 1.628 κud κs δτ Npoly traj. (SS) κud κs δτ Npoly traj. 0.13470 0.13510 1/175 200 6000 0.7757(26) 0.7273(29) 0.13470 0.13540 1/175 250 6000 0.7790(23) 0.6821(32) 0.13510 1/195 200 6000 0.7316(24) 0.7316(24) 0.13510 1/195 250 6000 0.7341(29) 0.6820(39) 0.13540 1/225 200 6000 0.6874(30) 0.7395(23) 0.13540 1/225 250 6000 0.6899(34) 0.6899(34) 0.13550 1/235 200 6500 0.6611(34) 0.7361(25) 0.13550 1/235 250 6500 0.6679(45) 0.6899(43) 0.13560 1/250 200 6500 0.6337(38) 0.7377(28) 0.13560 1/250 250 6500 0.6361(47) 0.6852(46) 0 0.02 0.04 0.06 0.08 Ks=0.13580 Ks=0.13640 β=1.90, L xT=20 x40, polynomial 0 0.02 0.04 0.06 0.08 Ks=0.13580 Ks=0.13640 β=1.90, L xT=20 x40, polynomial FIG. 1: Chiral fits of meson masses with mAWIq at β = 1.90. diag(mu,md,ms), MV = diag(mval1,mval2), and “tr” means the trace of matrices. In the fits, we use the axial-vector Ward identity quark mass mq = limt→∞〈∂4A4(t)P (0)〉/(2〈P (t)P (0)〉) and set A = BS = CS1 = CS2 = 0 for m PS . These fits reproduce mea- sured data well, as illustrated in Fig.1, with reasonable χ2/d.o.f. of at most 1.36. The physical quark mass point and the lattice spac- ing are determined from the experimental values of π0, ρ0 and K (K-input) or π0, ρ0 and φ (φ-input) meson masses. Taking the ρ0 mass as input may cause a large systematic error, because the ρ → ππ decay mode is not open for our mass range (the lightest pion mass in this simulation ∼ 620 MeV) and hence chiral extrapo- TABLE II: Lattice spacings in fm units. β K-input φ-input [π,K, φ]-input 1.83 0.1174(23) 0.1184(26) 0.1095(25) 1.90 0.0970(26) 0.0971(25) 0.0936(33) 2.05 0.0701(29) 0.0702(28) 0.0684(41) lation of mV for lighter quarks may be quite different from our fits. In order to estimate this uncertainty, we also check another combination [π0,K, φ]. We assume the ideal mixing for the vector isosinglets. Since our sim- ulation is made with degenerate u and d quarks, we con- sider the isospin averages m = {(m2 )/2}1/2 and m = (mK∗± + mK∗0)/2 and predict the aver- age light quark mass m = (mu + md)/2. The electro- magnetic (EM) effects, not included in our simulations, are removed from the mK± above using Dashen’s the- orem [10] (m2 − m2K0)EM = (m − m2π0)EXP. The isospin breaking effects and the EM effects for other mesons we consider are expected to be small and thus are not considered. The experimental values we use are taken from the PDG booklet [11]; mπ0 = 0.1350GeV, mπ± = 0.1396GeV, mK0 = 0.4976GeV, mK± = 0.4937GeV, mρ0 = 0.7755GeV, mK∗0 = 0.8960GeV, mK∗± = 0.8917GeV and mφ = 1.0195GeV. Lattice spacings (Ta- ble II) for theK- and φ- inputs are consistent, while those for the [π,K, φ]-input are slightly smaller by at most 7%. An agreement of the meson spectrum with experiment is a necessary condition for a reliable estimate of the quark masses. To confirm this, we extrapolate the me- TABLE III: Meson masses in the continuum limit (in MeV units), compared to experiment. The EM effect is subtracted using Dashen’s theorem. K-input φ-input [π,K, φ]-input EXP. K̂ - 491(19) - 495.0 ρ0 - - 761(32) 775.5 K̂∗ 900.5(9.9) 898.0(1.4) 891(16) 893.9 φ 1025(19) - - 1019.5 1.000 1.050 0.860 0.880 0.900 0.880 0.890 0.900 0 0.01 0.02 0.03 0.04 0.05 0.450 0.500 0.550 Nf=0 (IW+ PT clover) Nf=2 (IW+ PT clover) Nf=2+1 (IW+NPT clover) experiment φ (K-input) K* (K-input) K* (φ-input) K (φ-input) FIG. 2: Continuum extrapolation of meson masses for Nf = 2 + 1 QCD (circles), compared to experiment (stars) and re- sults in Nf = 2 (squares) and Nf = 0 (triangles) QCD [2]. son masses linearly in a2, because our action is O(a) improved and data are well fitted, as shown in Fig.2, with small χ2/d.o.f ≤ 1.4. The masses in the continuum limit, summarized in Table III, are consistent with exper- iment with at most 2.9σ deviation. The K̂∗ mass turns out to be slightly heavier than experiment, though the supplemental [π,K, φ]-input gives consistent results with experiment with large statistical error. Possible origin of the deviation is due to uncertainty of chiral fits. In fact, an alternative fit based on chiral perturbation the- ory (χPT) we discuss later yields m = 894(12) MeV (K-input). In Fig.2 we overlay the previous results of meson masses [2] in the Nf = 2 and quenched (Nf = 0) QCD with tadpole improved one-loop cSW . The dynam- ical u and d quarks significantly reduce the O(10%) de- viation of the quenched spectrum from experiment. We find no further dynamical s quark effect beyond statisti- cal errors. The quark masses are evaluated for the MS scheme at the scale µ = 2GeV using the tadpole improved one- loop matching [12] at µ = a−1 with an improved cou- pling determined from plaquette and rectangular loop and four-loop renormalization group equation. In the continuum extrapolation of the quark masses, we assume 0 0.01 0.02 0.03 0.04 0.05 Nf=0 (K, IW+ PT clover) Nf=2 (K, IW+ PT clover) Nf=2+1 (K, IW+NPT clover) 0 0.01 0.02 0.03 0.04 0.05 Nf=0 (K, IW+ PT clover) Nf=0 (φ, IW+ PT clover) Nf=2 (K, IW+ PT clover) Nf=2 (φ, IW+ PT clover) Nf=2+1(K, IW+NPT clover) Nf=2+1(φ, IW+NPT clover) FIG. 3: Continuum extrapolations of the up, down and strange quark masses. For comparison, results for Nf = 0 and Nf = 2 QCD [2] are overlaid. the O(g4ma) contributions are small and neglect it. As Fig. 3 shows, the quark masses are well described by a linear function in a2, and the values determined for either the K- or the φ-inputs, while different at finite lattice spacings, extrapolate to a common value in the continuum limit. Therefore the continuum limit is esti- mated from a combined linear fit with the K- and the φ-inputs. We obtain mMS(µ = 2GeV) = 3.55(19) MeV and mMSs (µ = 2GeV) = 90.1(4.3) MeV with a suffi- ciently small χ2/d.o.f. < 0.42. Note that the supplemen- tal [π,K, φ]-input gives larger statistical error and hence is not used to estimate central values. We now turn to estimates of possible systematic errors. Finite size effect (FSE) — The meson masses at the infinite volume are estimated at β = 1.90 using data on a V ∼ (2.0fm)3 lattice and those from our exploratory study on a V ∼ (1.6fm)3 lattice [13], and assuming a strong volume dependence of (mhad,V − mhad,V=∞)/mhad,V=∞ ∝ 1/V [14]. The chiral fits to the infinite volume values lead to less than a 4% change for the meson masses at the physical point. For the quark masses, however, we find a larger shift of 12.2% from a V ∼ (2.0fm)3 lattice to V = ∞ for m with φ-input and 8.1% for ms with K-input (differences are smaller for the other cases). Assuming that FSE is independent of lat- tice spacing, we take the differences as estimates of FSE for the quark masses in the continuum limit. Chiral extrapolation — In addition to the polynomial chiral fits, we fit the meson masses using χPT formu- laes modified for the Wilson quark action (WχPT) [15]. Namely, we fit mπ, mK̂ , mρ and mK̂∗ using the NLO Nf = 2+1 QCD WχPT formulae for the O(a) improved theory [16]. Since the formula in Ref. [16] is not appli- cable for the φ meson, we estimate the effect only for K-input. In the fits we obtain m to be 3.1% smaller and ms to be 1.2% larger than those of the polynomial fit. We note that our WχPT fits to data do not exhibit a clear chiral logarithm, probably because u and d quark masses in our simulation are not sufficiently small. Further pos- sible systematic error from a long chiral extrapolation for ρ0, mentioned above, is estimated by the supplemental [π,K, φ]-input, which gives 3.0% larger for m and 3.4% larger value for ms than the central one. For an estimate of systematic errors from chiral fits, differences of the two alternative fits from the central value are added linearly. Renormalization factor — Uncertainty of the one- loop calculation of the renormalization factor is esti- mated by shifting the matching scale from µ = 1/a to µ = π/a and also using an alternative tadpole improved coupling [2]. Continuum extrapolation — Possible O(a3) effects are investigated by performing the continuum extrapola- tion adding an O(a3) term to the fit function. Electromagnetic (EM) effects — Systematic error due to uncertainty of the EM effects is estimated fol- lowing extensive arguments [7, 17, 18] to Dashen’s theorem [10]. Namely, we estimate the effects by a further mass shift of our input m using a relation )EM = (1 + ∆E)(m )EXP assum- ing the EM effects for other mesons are negligeble. We vary the ∆E in range [−1,+1] as our estimate of the EM effects, and we find a quite small change in ms and no change in m. Isospin breaking effects — Isospin breaking effects are estimated by chiral fits with Eq. (1) for mu 6= md and taking mπ0 , mρ0 , mK± and mK0 as inputs. We find that mu/md = 0.577(25), and that m and ms have no change from the K̂ input result. We note that mu/md strongly depends on an estimate of the EM ef- fects; mu/md =0.663–0.498 for ∆E = [−1,+1], though m and ms almost do not. Finally we obtain mMS(µ = 2GeV) = 3.55(19)(+43 −0 )( −11)( −17)( −0 )( −0), (2) mMSs (µ = 2GeV) = 90.1(4.3)(+7.3 −0 )( −0 )( −4.3)( +12.8 −0 )( −0.2)( −0), (3) in MeV units, where the errors are statistical, systematic due to FSE, chiral extrapolation, renormalization factor, continuum extrapolation, EM effect and isospin break- ing effect, respectively. Adding the errors in quadrature yields the values quoted in the abstract. These values agree well with the latest report from the MILC Collab- oration [3] m = 3.3 ± 0.3 MeV and ms = 90 ± 6 MeV where we added the quoted errors in quadrature. They also include the Nf = 2 values [2] within the error. Scaling violation in the quark masses is unexpectedly large, while that for the meson masses are reasonably bounded at a percent level at a ≈ 0.1 fm. To gain a better control over systematic uncertainties, a significant reduction in the simulated light quark masses on a correspondingly larger lattice is needed. An attempt is underway to meet these challenges [19]. This work is supported by the Epoch Making Simula- tion Projects of Earth Simulator Center, the Large Scale Simulation Program No.132 (FY2005) of High Energy Accelerator Research Organization (KEK), the Large Scale Simulation Projects of Academic Computing and Communications Center of University of Tsukuba, Inter University Services of Super Computers of Information Technology Center of University of Tokyo, Super Sinet Projects of National Institute of Informatics, and also by the Grant-in-Aid of the Ministry of Education (Nos. 13135204, 13135216, 15540251, 16540228, 16470147, 17340066, 17540259, 18104005, 18540250, 18740130). [1] S. Aoki et al. [CP-PACS Collaboration], Phys. Rev. Lett. 84, 238 (2000); Phys. Rev. D 67, 034503 (2003). [2] A. Ali Khan et al. [CP-PACS Collaboration], Phys. Rev. D 65, 054505 (2002) [Erratum-ibid. D 67, 059901 (2003)]; Phys. Rev. Lett. 85, 4674 (2000) [Erratum-ibid. 90, 029902 (2003)]; S. Aoki et al. [JLQCD Collabora- tion], Phys. Rev. D 68, 054502 (2003). 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Singleton, Phys. Rev. D 58, 074501 (1998); S. Aoki, Phys. Rev. D 68, 054508 (2003). [16] S. Aoki et al., Phys. Rev. D 73, 014511 (2006); S. Aoki et al., Phys. Rev. D 73, 094501 (2006); S. Takeda, Doctral thesis at Univ. of Tsukuba (2005). [17] J. Bijnens and J. Prades, Nucl. Phys. B 490, 239 (1997); J. Bijnens and P. Gosdzinsky, Phys. Lett. B 388, 203 (1996). [18] D. R. Nelson, arXiv:hep-lat/0212009. [19] Y. Kuramashi et al. [PACS-CS Collaboration], PoS LAT2006, 029 (2006)); A. Ukawa et al. [PACS-CS Col- laboration], PoS LAT2006, 039 (2006). http://arxiv.org/abs/hep-lat/0703006 http://arxiv.org/abs/hep-lat/0212009

THE CHOW RING OF THE MODULI SPACE AND ITS RELATED HOMOGENEOUS SPACE OF BUNDLES ON P2 WITH CHARGE 1 YASUHIKO KAMIYAMA AND MICHISHIGE TEZUKA Abstract. For an algebraically closed field K with ch(K) 6= 2, let OM(1, SO(n,K)) denote the moduli space of holomorphic bundles on P2 with the structure group SO(n,K) and half the first Pon- tryagin index being equal to 1, each of which is trivial on a fixed line l and has a fixed holomorphic trivialization there. In this paper we determine the Chow ring of OM(1, SO(n,K)). 1. Introduction Let G be one of the classical groups SU(n), SO(n) or Sp(n), and let k ≥ 0 be half the first Pontryagin index of a G-bundle P over S4 = R4 ∪ {∞}. Denote by M(k,G) the framed moduli space whose points represent isomorphism classes of pairs: (self-dual G-connections on P , isomorphism P∞ ≃ G). Let OM(k,GC) denote the moduli space of holomorphic bundles on 2 for the associated complex group, trivial on a fixed line l∞ and with a fixed holomorphic trivialization there. Then Donaldson ([7]) showed a diffeomorphism M(k,G) ≃ OM(k,GC). In [12] the topology ofM(1, SO(n)) ≃ OM(1, SO(n,C)) was studied in detail. The result was used in [11] to prove the fact that the natural homomorphism J : H·(M(1, SO(n)),Z/2) → H·(Ω30Spin(n),Z/2) is injective. Moreover, the image of J was determined. To prove this, the following description of OM(1, SO(n,C)) by a homogeneous space was used: We set Wn = SO(n)/(SO(n− 4)× SU(2)). Then there is a diffeomorphism (1.1) OM(1, SO(n,C)) ≃ R5 ×Wn. 2000 Mathematics Subject Classification. 14M17 (14N10). Key words and phrases. Moduli space, homogeneous space, Chow ring, cycle http://arxiv.org/abs/0704.1938v1 The purpose of this paper is to generalize the definition ofOM(1, SO(n,C)) for any algebraically closed field K with ch(K) 6= 2 and to determine the Chow ring of this. The Chow ring of a classifying space was studied by Totaro [19]. A loop space is considered to be a dual situation of a classifying space in a certain sense. Our result and the result of [11] are the first step for a loop space. Definition 1.1. Let K be an algebraically closed field with ch(K) 6= 2. Let OM(1, SO(n,K)) denote the moduli space of holomorphic bundles on P2 with the structure group SO(n,K) and half the first Pontryagin index being equal to 1, each of which is trivial on a fixed line l∞ and has a fixed holomorphic trivialization there. The moduli space OM(1, SO(n,K)) is a quasi-projective variety and defines the Chow ring. More explicitly, the diffeomorphism (1.1) is generalized (in the sense of a biregular map) as follows: We set Xn = SO(n,K)/(SO(n− 4, K)× SL(2, K)) · Pu, where Pu denotes the unipotent radical. (Recall that for a parabolic subgroup P of an algebraic group G, P is a semidirect product of a reductive group and its unipotent radical Pu.) Then there is a biregular (1.2) OM(1, SO(n,K)) ≃ A2 ×Xn. (For the proof of (1.2), see Proposition 2.1.) A formula of Grothendieck [3] shows that CH ·(OM(1, SO(n,K))) ≃ CH ·(Xn). The purpose of this paper is to determine the Chow rings of Xn and its related algebraic variety Yn explicitly. The Schubert cell approach of the Chow ring of Yn by using a Young diagram is done in [15], [16]. However it needs further work to deter- mine the Chow ring of Xn from this. Hence we first calculate the Chow ring of Yn more explicitly by a different method. Then we calculate the Chow ring from the results. Our results for the Chow ring of Xn are This paper is organized as follows. In Sect. 2 we first prove (1.2). Then we recall basic facts on the Chow ring. In Sect. 3 we determine an integral basis and the ring structure of CH ·(Yn), where Yn is an algebraic variety which is related to Xn. (See Theorems 3.7 and 3.9.) The ring structure of CH ·(Yn) proved in Theorem 3.9 is one of our main results. Since the results are long, we give them in tables in Sect. 5. (See 5.2-5.5.) Using the results of Sect. 3, we determine CH ·(Xn) in Sect. 4. (See Theorem 4.1.) We thank N. Yagita for turning our interest to the Chow ring and explaining the paper [17]. 2. Preliminaries We first prove (1.2): Proposition 2.1. For an algebraically closed field K with ch(K) 6= 2, there is a biregular map OM(1, SO(n,K)) ≃ A2 ×Xn. Proof . Recall that a monad description of OM(k, SO(n,C)) was in- dicated in [7] and given explicitly in [14] and [18]. It is easy to see that the description remains valid for any algebraically closed field K. In particular, OM(1, SO(n,K)) is given as follows: Lemma 2.2. Let Cn be the space of n× 2 matrices z1 w1 z2 w2 zn wn with coefficients in K satisfying: a) cT c = O, that is: z2i = 0, w2i = 0 and ziwi = 0, b) The rank of c over K is 2. The group SL(2, K) acts on Cn from the right by the multiplication of matrices. Then there is a biregular map OM(1, SO(n,K)) ≃ A2 × (Cn/SL(2, K)). From the lemma, it suffices to prove Xn ≃ Cn/SL(2, K). We prove this for the case n = 2m. (The case n = 2m + 1 can be proved similarly.) Recall that in [2], SO(n,K) was defined as follows: Let q(x) be a quadratic form on An defined by q(x) = i=1 xixm+i, and let B(x, y) be the associated bilinear form. Then SO(n,K) is defined SO(n,K) = {σ ∈ Aut(An) : B(σ(x), σ(y)) = B(x, y) for x, y ∈ An}. We set xj = zj+ −1zj , xm+j = zj− −1zj , yj = wj+ −1wj and ym+j = wj− −1wj, where 1 ≤ j ≤ m. Then the defining equations of Cn are given by q(x) = q(y) = 0 and B(x, y) = 0. Clearly SO(n,K) acts on Cn. It is easy to prove the following lemma. (See [2, V 23.4].) Lemma 2.3. SO(n,K)/SO(n− 4, K) · Pu ≃ Cn, where Pu is the unipotent radical of a parabolic subgroup with a Levi factor SO(n− 4, K)×GL(2, K). Now Proposition 2.1 follows from Lemma 2.3. This completes the proof of Proposition 2.1. � Next we recall basic facts on the Chow ring. We suppose that an algebraic variety V is defined over K. Let CH ·(V ) denote the Chow ring and CH i(V ) the subgroup of CH ·(V ) generated by the cycles of codimension i. Theorem 2.4 ([3]). (i) Let V be a nonsingular variety, X a nonsin- gular closed subvariety of V , and U = X − V . Then there exists an exact sequence CH ·(X) i∗→ CH ·(V ) j → CH ·(U) → 0, where i : X → V (resp. j : U → V ) is a closed immersion (resp. an open immersion). For the definitions of i∗ and j ∗, see also [10]. (ii) Let π : E → V be a fiber bundle with an affine space An as a fiber. Then the induced map π∗ : CH ·(V ) → CH ·(E) is an isomorphism. The Chow ring of the following projective variety is well-known. Theorem 2.5 ([1], [6]). Let G be a reductive algebraic group and P a maximal parabolic subgroup. Then (i) a quotient G/P is a nonsingular projective variety. (ii) CH ·(G/P ) is generated by the Schubert varieties. (iii) CH ·(G/P ) is independent of ch(K). Moreover, CH ·(G/P ) ≃ H ·(G/P,Z) for K = C. 3. The ring structure of CH ·(Yn) Before describing the results, we need some notations and results. We set Yn = SO(n,K)/(SO(n− 4, K)×GL(2, K)) · Pu. Then we have a principal bundle (3.1) Gm → Xn π→ Yn. In this section we determine an integral basis and the ring structure of CH ·(Yn). By Theorem 2.5 (ii), (iii), we obtain the following theorem: Theorem 3.1 ([12]). We have an isomorphism as modules: (1) For n = 2m, CH ·(Yn)⊗ Z/2 ≃ Z/2[c1, c2]/(bm−1, c2bm−2)⊗∆(v2m−4, v2m−2). (2) For n = 2m+ 1, CH ·(Yn)⊗ Z/2 ≃ Z/2[c1, c2]/(bm−1, c2bm−2)⊗∆(v2m−2, v2m), where |c1| = 2, |c2| = 4, |bi| = 2i and |vi| = i. Theorem 3.2 ([12]). Let p be an odd prime. Then we have a ring isomorphism: (1) For n = 2m, CH ·(Yn)⊗ Z/p ≃ Z/p[c1, c2, χ2m−4]/(c2χ2m−4, χ22m−4 − dm−2, dm−1), where χ2m−4 ∈ H2m−4(BSO2m−4,Z/p) is the Euler class. (2) For n = 2m+ 1, CH ·(Yn)⊗ Z/p ≃ Z/p[c1, c2]/(dm−1, c22dm−2). We recall the definitions of bi, di and vi. In a polynomial ring Z[α, β], we set c1 = α+ β and c2 = αβ. Then bk and dk are defined by bk = (−1)k αiβk−i dk = (−1)k α2iβ2k−2i. The element v2r ∈ CH2r(Yn) is defined by (1) For n = 2m, (3.2) 2v2m−4 = χ2m−4 − bm−2 2v2m−2 = bm−1. (2) For n = 2m+ 1, (3.3) 2v2m−2 = bm−1 2v2m = c2bm−2. The following formulas are easily proved. Lemma 3.3. We have bk = (−1)k (−1)µ k − µ dk = (−1)k (−1)µ 2k − µ+ 1 2k−2µ The following lemmas are also easily shown. Lemma 3.4. (−1)µch−µ2 b2µ = dh. Lemma 3.5. We set fn(x) = (1 + x) n − (1 + xn) and write fn(x) as fn(x) = [n2 ] µ(1 + x)n−2µ. Then we have aµ = (−1)µ+1 n− 1− µ Especially, the last term is given by (−1)s+12xs for n = 2s (−1)s+1(2s+ 1)xs(1 + x) for n = 2s+ 1. For n = 2m or 2m+ 1, we define a subgroup An of CH ·(Yn) by (3.4) An = Z[c1]/(c m−1−i 1 ){ci2} where we set ∆Z(v2m−4, v2m−2) n = 2m ∆Z(v2m−2, v2m) n = 2m+ 1. The generators v2i is specified in (3.2) and (3.3). The following lemma is proved in the same way as in [12, Lemma 3.8]. Lemma 3.6. For a prime p, we abbreviate CH ·(Yn)⊗Z(p) as CH ·(Yn)(p). If p is odd, we have the following isomorphism of modules: (i) For n = 2m, CH ·(Yn)(p) ≃ Z(p)[c1]/(c2(m−1)1 ){1, χ2m−4}⊕ Z(p)[c1]/(c 2(m−1−i) 1 ){c2i−12 , c2i2 }. (ii) For n = 2m+ 1, CH ·(Yn)(p) ≃ Z(p)[c1]/(c 2(m−1−i) 1 ){c2i2 , c2i+12 }. Theorem 3.7. An integral basis of CH ·(Yn) is constructed from the monomial basis of An in (3.4). The results are summed up in Sect. 5, The proof is based on a rather complicated calculation. Its outline is as follows: We construct a set of suitable generators starting from the basis of An. It is easily verified that it is a basis of CH ·(Yn)⊗ Z/p by using the presentation of Lemma 3.6. Then it is a Z-basis of CH ·(Yn). We only prove the case n = 2m + 1 and m is even. To simplify the proof, we set: C = CH ·(Yn) (3.5) bm−1 = (−1)m−1bm−1 = i+j=m−1 αiβj, em = c2bm−2 dm−1 = (−1)m−1dm−1 Z[c1]/(c 1 )⊕ · · · ⊕ Z[c1]/(cm−1−i1 ){ci2} ⊕ · · · ⊕ {cm−22 } ⊗ Z{1, bm−1, em, bm−1em}. Let An the set of a basis {ci1c m : i+ j ≤ m− 2, ǫk = 0 or 1 (k = 1, 2)}. Then we have C ⊂ Q[c1, c2]/(dm−1, c22dm−2), A ⊂ Q[c1, c2]/(dm−1, c22dm−2) and c2i2 dm−1−i ∈ (dm−1, c22dm−2), i ≥ 0. Our concern is a homogeneous polynomial algebra SZ(V ) of V = {c1, c2}. It is identified with an inhomogeneous ring Z[x] by putting x = β . Then we have c1 = 1 + x, c2 = x, bm−1 = xm − 1 , and dm−1 = x2m − 1 x2 − 1 Using the identification, the next formulas are directly checked: In Z[c1, c2], − c2i+1+j2 bm−2−2i = c 2 b2i−1bm−1 − c 2b2iem − c2i+2+j2 bm−3−2i = c 2 b2ibm−1 − c 2b2i+1em. We set c2i−11 c 2 bm−1 2 b2i−1bm−1 − c 2b2iem = −c 2i+j+1 2 bm−2−2i(3.6) c2i1 c 2 bm−1 2 b2ibm−1 − c 2b2i+1em = −c 2i+j+2 2 bm−3−2i. Noting c22dm−2 = 0, we have c22bm−2bm−1−c22c1dm−2 = (x− 1)2 (xm−2−1)(xm−1−1) = c32bm−3bm−2. Using c2i2 dm−1−i = 0, 1 ≤ i ≤ m, we repeat the argument and get (3.7) ci2bm−1em = c 2 bm−2−ibm−1−i. In Yn, we see that dm−1 = 0. Recall that we set f2i+3(x) = (1 + x)2i+3 − (1 + x2i+3) (see Lemma 3.5). Then we have c2i+11 c 2bm−1em − c 2 didm−1 xj+1(xm − 1) (x2 − 1)2 (1 + x)2i+3(xm−1 − 1)− (x2i+2 − 1)(xm + 1) xj+1(xm − 1) (x2 − 1)2 (1 + x2i+3)(xm−1 − 1)− (x2i+2 − 1)(xm + 1) + f2i+3(x)(xm−1 − 1) x2i+j+3 (x− 1)(x2 − 1) (xm − 1)(xm−2i−3 − 1) + x (x2 − 1)2 (xm−1 − 1)(xm − 1)f2i+3(x). We set x2i+j+3 (x− 1)(x2 − 1) (xm − 1)(xm−2i−3 − 1) (x2 − 1)2 (xm−1 − 1)(xm − 1)f2i+3(x). Since x2λdm−1−λ = 0 for 0 ≤ λ ≤ m−1, we see that x2i+j+3+λdm−1−(i+2)−λ = 0 for 0 ≤ λ ≤ j − 1. Hence x2i+j+3+λdm−1−(i+2)−λ = x2i+2j+3 (x− 1)(x2 − 1) (xm−j−1)(xm−2i−j−3−1). From Lemma 3.5, 2i+1−2µ 1 bm−1em + (−1)ikxi+j+2 (xm−1 − 1)(xm − 1) (x− 1)(x2 − 1) where k = 2i+ 3. We set J = xi+j+2 (xm−1−1)(xm−1) (x−1)(x2−1) . Using xi+j+2+λdm−2−λ = 0 for 1 ≤ λ ≤ i+ j, we see that xi+j+2+λdm−2−λ = x2i+2j+3 (x− 1)(x2 − 1) (xm−2−i−j−1)(xm−1−i−j−1). Afterwards, we introduce the notation (3.8) c2i+11 c 2bm−1em := c2i+11 c 2bm−1em − 2i+1−2µ 1 bm−1em, where f2i+3(x) = µ=1 aµx µ(1 + x)2i+3−2µ (see Lemma 3.5). We sum up these arguments: (3.9) c2i+11 c 2bm−1em = I1 + (−1)ikJ, where x2i+2j+3 (x− 1)(x2 − 1) (xm−j − 1)(xm−2i−j−3 − 1), x2i+2j+3 (x− 1)(x2 − 1) (xm−2−i−j − 1)(xm−1−i−j − 1), and k = 2i+ 3. A direct calculation shows J − I1 = x (x−1)(x2−1) (xi+1 − 1)(xi+2 − 1). Hence we get a key formula (3.10) c2i+11 c 2bm−1em (−1)i(2i+ 3) + 1 I1+(−1)i(2i+3) (x− 1)(x2 − 1) (xi+1−1)(xi+2−1). We note that (3.9) holds for m even or odd. From now on, we as- sume that m is even. When we put j = 1 in (3.9), we see that I1 = c −i−3em. We set 〈c2i+11 c2bm−1em〉 := c2i+11 c2bm−1em (−1)i(2i+ 3) + 1 c2i+42 dm2 −i−3em. Then we have (3.11) 〈c2i+11 c2bm−1em〉 2i+ 3 = (−1)icm+12 bibi+1 We call a generator cm−3−2i1 c 2 bm−1em to be the head of the pre- sentation An in (3.4). For the head generator, it is directly shown that I1 = 0 by (3.9). Hence the formula (3.9) implies (3.12) cm−3−2i1 c 2 bm−1em m− 1− 2i = (−1)i −i−2bm Then the formulas (3.10) and (3.11) imply the following relations: We 〈c2i+11 bm−1em〉 := c2i+11 bm−1em (−1)i(2i+ 3) + 1 c2i+22 dm2 −i−2em 〈c2α+1−2β1 c 2 bm−1em〉 := 2α+1−2β 2 bm−1em (−1)α−β(2α− 2β + 3) + 1 c2α+42 dm2 −α−3em. Then we have 〈c2i+11 bm−1em〉 − (−1)i(2i+ 3) + 1 (−1)i−1(2i+ 1) 〈c2i−11 c2bm−1em〉 = − bibi+1 (3.13) 〈c2α+1−2β1 c 2 bm−1em〉 − (−1)β(2α− 2β + 3) + (−1)α 2α + 3 〈c2α+11 c2bm−1em〉 (3.14) m+β+2 bα−β−1bα−β m−3−2α−2β 2α+β+1 (−1)β(m− 1− 2α+ 2β) + (−1)m2 −α m− 1− 2α cm−3−2α1 c 2 bm−1em (3.15) m+2α+β+1 bγbγ+1, where γ = − 2− α− β. Last we consider a generator c2i+21 c 2bm−1em. We set 〈c2i+21 c 2bm−1em〉 :=c2i+21 c 2bm−1em − 2i+2−2µ 1 bm−1em (−1)i(2i+ 3) + 1 i+j+1 2 bm−1em, where f2i+3(x) = µ=1 aµx µ(1 + x)2i+3−2µ (see Lemma 3.5). Then the formula (3.9) implies that (3.16) 〈c2i+11 c 2bm−1em〉 = −c 2 bibi+1. We consider a set {(3.6), (3.10), (3.11), (3.12), (3.13), (3.14)} ∪ {x ∈ An : x = ci1c m, x = c 1bm−1, i+ j ≤ m− 2, ǫk = 0 or 1}. When we reduce it to the mod p reduction for an odd prime p, the identities from (3.6) to (3.14) show that they are linearly independent from Lemma 3.6. Hence, replacing bm−1 and em by v2m−2 and v2m respectively, we obtain an integral basis of CH ·(Y2m+1) for even m. An integral basis of CH ·(Yn) for other cases is written down in a table of Sect. 5. For a group G and a subgroup H , let [G : H ] denote the index, i.e. the cardinality of G/H . As a corollary of the above theorem, we have Corollary 3.8. (i) For n = 2m, [CH ·(Yn) : An] = 12 · 32 · . . . · (m− 3)2 · (m− 1) m: even 12 · 32 · . . . · (m− 2)2 m: odd. (ii) For n = 2m+ 1, [CH ·(Yn) : An] = 12 · 32 · . . . · (m− 3)2 · (m− 1) m: even 12 · 32 · . . . · (m− 2)2 ·m m: odd. The following theorem is proved by using the integral basis of CH ·(Yn) given in Theorem 3.7. Theorem 3.9. The ring structure of CH ·(Yn) is determined. The results are summed up in tables in 5.3 and 5.4 in Sect. 5. (See also 5.5.) Proof . We only show the formula (iii) in Sect. 5, 5.4. We use the notations of the proof of Theorem 3.7. Using x2idm−1−i = 0, we have cm−2i−11 c 2 bm−1−x2idm−1−i = x2 − 1 (x2i−1)+ x x2 − 1 (xm−1)fm−2i(x). We set I1 = (x2i − 1) and I2 = x (xm − 1)fm−2i(x). Then we m−2i−1−2µ 2 bm−1 + (−1) −i+12x m − 1 x2 − 1 We set J = x +i xm−1 . From (3.6), we have 2 em− (−1)µ +i−1−µ 2 bm−1  = (−1) x2i − 1 x2 − 1 Hence cm−2i−11 c 2 bm−1− m−2i−1−2µ 2 bm−1 (3.17) + (−1) 2 em + (−1)µ +i−1−µ 2 bm−1  = −xmx 2i − 1 x2 − 1 On the other hand, we see from (3.13) that (3.18) − xmx 2i − 1 x2 − 1 = 〈c2i−11 bm−1em〉+ 2i+ 1 2i− 1 〈c2i−31 c2bm−1em〉. We calculate S := (−1) 2 em + (−1)µ +i−1−µ 2 bm−1 By (3.6), S = (−1) −i2(S1 + S2), where (−1)µc +i+1−µ 2 b2µ+1bm−1 and S2 = c 2 em− (−1)µc +i−2−µ 2 b2µ+2em. We have from Lemma 3.3 that −1−µ−i m− 1− 2i− µ m−2i−1−2µ 2 bm−1. Using µ=0(−1)µc 2 b2µ = (−1)hdh (see Lemma 3.4), S2 = c (−1)µc −i−1−µ 2 b2µ  bm−1 = (−1) −1−ic2i2 dm2 −1−ibm−1. The formula (3.17) is cm−2i−11 c 2 bm−1 + (−1)µ m− 2i− 1− µ cm−2i−11 c 2 bm−1 − 2c2i2 dm2 −1−iem = −xm x x2 − 1 Comparing with (3.18), we obtain cm−2i−11 c 2 bm−1 = (−1)µ+1 m− 2i− 1− µ cm−2i−11 c 2 bm−1 2i− 1 c2i2 dm2 −1−iem + c2i−11 bm−1em 2i+ 1 2i− 1 c2i−31 c2bm−1em This completes the proof of the formula. � 4. The Chow ring of Xn Theorem 4.1. Let T (Xn) and F (Xn) be the torsion part and the free part of CH ·(Xn), respectively. Then we have (i) For n = 4t, F (Xn) ≃ Z[c2]/(ct2){1, v4t−4} T (Xn) ≃ Z/2[c2]/(ct2){v4t−2, v4t−4v4t−2}. (ii) For n = 4t+ 1, F (Xn) ≃ Z[c2]/(ct2)⊕ Z[c2]/(ct−12 ){v4t} T (Xn) ≃ Z/2[c2]/(ct2){v4t−2, v4t−2v4t} ⊕ Z/(2t){ct−12 v4t}. (iii) For n = 4t+ 2, F (Xn) ≃ Z[c2]/(ct2){1, v4t} ⊕ Z{v4t−2} T (Xn) ≃ Z/2[c2]/(ct−12 ){c2v4t−2, c2v4t−2v4t} ⊕ Z/4{v4t−2v4t}. (iv) For n = 4t+ 3, F (Xn) ≃ Z[c2]/(ct2){1, v4t} T (Xn) ≃ Z/2[c2]/(ct2){v4t+2, v4tv4t+2} ⊕ Z/(2t+ 1){ct2v4t}. Proof . Let X̃n = Xn ×Gm A1 be the associated bundle of (3.1) and s : Yn → X̃n the 0-section. Since s∗ : CH ·(X̃n) ∼→ CH ·(Yn) by Theorem 2.4 (ii), the first assertion of the same theorem for V = X̃n and X = s(Yn) gives an exact sequence CH ·(Yn) ·c1→ CH ·(Yn) π∗→ CH ·(Xn) → 0. Theorem 4.1 follows from this and the ring structure of CH ·(Yn) in Theorems 3.7 and 3.9. � Next we consider the cycle map. The cohomology groups mean an etale cohomology [9], [13]. All varieties are defined over K ′, which is a subfield of an algebraically closed field K. Let l be a prime with (l, ch(K)) = 1. We denote a locally constant sheaf µ⊗i by Z/l(i). Corollary 4.2. The homomorphism cl : CH i(Xn) → H2i(Xn,Z/l(i)) is injective. Proof . Since (X̃n, Yn) is a smooth pair, we have the Gysin sequence as in [5, Appendice 1.3.3] and [13, VI Remark 5.4]. Since the cycle map and the Gysin map are commutative, we have the following commuta- tive diagram, where each row is exact: CH i(Yn) ·c1−−−→ CH i+1(Yn) π∗−−−→ CH i+1(Xn) −−−→ 0 H2i(Yn,Z/l ·c1−−−→ H2(i+1)(Yn,Z/l(i+1)) π∗−−−→ H2(i+1)(Xn,Z/l(i+1)) Corollary 4.2 follows from Theorem 4.1 and this diagram. � Remark 4.3. Assume that we have aK ′-isomorphism Yn ≃ SO(n,K)/(SO(n− 4, K) × GL(2, K)), where SO(n,K) and SO(n − 4, K) are split over K ′, and that π : Yn → Xn is a K ′-map. Then the Galois actions G = Gal(K /K ′) on H ·(Yn,Z/l (i)) and H ·(Xn,Z/l (i)) are described by the character group X(T ) of a K ′-split maximal torus T of SO(n,K). It follows from a result of [5, 8-2]. 5. Tables of the ring structure of CH ·(Yn) 5.1. Notations. (i) For k ∈ N ∪ {0}, we define bk and dk ∈ Z[c1, c2] as follows: bk = (−1)k [ k2 ] (−1)µ k − µ dk = (−1)k (−1)µ 2k − µ+ 1 2k−2µ (ii) For g ∈ N and µ ∈ N ∪ {0,−1}, we define ag,µ ∈ Z by ag,µ = (−1)1+µ g g−1−µ µ ≥ 1 −1 µ = 0 0 µ = −1. Then the integers ag,µ are characterized by (1 + x)g = 1 + xg + [ g2 ] ag,µx µ(1 + x)g−2µ. (iii) The generators v2i are given by (3.2) and (3.3). 5.2. An integral basis of CH ·(Yn). In the following (I) and (II), we give an integral basis of CH ·(Yn). The notations are explained as follows: Let Sn be the set of the monomial basis of An in (3.4). Let T be a subset of Sn. Then for an element ξ ∈ T , 〈ξ〉 (resp. 〈ξ〉′) is defined to be the right-hand side of an equation (1)-(8) below. We consider a : ξ ∈ T ∪ {η : η ∈ Sn − T} , where lξ ∈ N. Following this procedure, we obtain an integral basis of CH ·(Yn). We abbreviate this basis as : ξ ∈ T (I) The case n = 2m. (i) For even m, 〈c2i+11 c2v2m−4v2m−2〉 2i+ 3 〈cm−2j−31 c 2 v2m−4v2m−2〉 m− 2j − 1 : 0 ≤ i ≤ − 2, 1 ≤ j ≤ (ii) For odd m, 〈c2i+11 c2v2m−4v2m−2〉′ 2i+ 3 〈cm−2j−21 c 2 v2m−4v2m−2〉′ m− 2j : 0 ≤ i ≤ m− 5 , 1 ≤ j ≤ m− 3 (II) The case n = 2m+ 1. (iii) For even m, 〈c2i+11 c2v2m−2v2m〉 2i+ 3 〈cm−2j−31 c 2 v2m−2v2m〉 m− 2j − 1 : 0 ≤ i ≤ m − 2, 1 ≤ j ≤ m (iv) For odd m, 〈c2i+11 v2m−2v2m〉 2i+ 3 〈cm−2j−21 c 2 v2m−2v2m〉 m− 2j : 0 ≤ i ≤ , 1 ≤ j ≤ Here 〈 〉 and 〈 〉′ are defined as follows: 〈c2i+11 c2v2m−4v2m−2〉 = c2i+11 c2v2m−4v2m−2 + (−1) m+2i+2 (−1)i(2i+ 3) + 1 c2i+42 dm−2i−6 v2m−4 a2i+3,µc 2i+1−2µ 2 v2m−4v2m−2. 〈cm−2j−31 c 2 v2m−4v2m−2〉 = c m−2j−3 2 v2m−4v2m−2 m−2j−4 am−2j−1,µc m−2j−3−2µ 2j+1+µ 2 v2m−4v2m−2. 〈c2i+11 c2v2m−4v2m−2〉′ = c2i+11 c2v2m−4v2m−2 + (−1) m+2i+1 (−1)i(2i+ 3) + 1 c2i+32 dm−2i−5 v2m−2 a2i+3,µc 2i+1−2µ 2 v2m−4v2m−2. 〈cm−2j−21 c 2 v2m−4v2m−2〉′ = c m−2j−2 2 v2m−4v2m−2 m−2j−3 am−2j,µc m−2j−2−2µ 2 v2m−4v2m−2. 〈c2i+11 c2v2m−2v2m〉 = c2i+11 c2v2m−2v2m + (−1) m+2i+2 (−1)i(2i+ 3) + 1 c2i+42 dm−2i−6 a2i+3,µc 2i+1−2µ 2 v2m−2v2m. 〈cm−2j−31 c 2 v2m−2v2m〉 = c m−2j−3 2 v2m−2v2m m−2j−4 am−2j−1,µc m−2j−3−2µ 2j+1+µ 2 v2m−2v2m. 〈c2i+11 v2m−2v2m〉 = c2i+11 v2m−2v2m + (−1) m+2i+3 (−1)i(2i+ 3) + 1 c2i+32 dm−2i−5 v2m−2 a2i+3,µc 2i+1−2µ 2v2m−2v2m. 〈cm−2j−21 c 2 v2m−2v2m〉 = c m−2j−2 2 v2m−2v2m m−2j−3 am−2j,µc m−2j−2−2µ 2 v2m−2v2m. 5.3. The ring structure of CH ·(Yn)(2) for n = 2m. even m odd m cm−11 (1) cm−k−11 c 2 (k ≥ 1) (2) cm−11 v2m−4 (3) cm−2i−11 c 2 v2m−4 (i ≥ 1) (4) cm−2i−21 c 2 v2m−4 (i ≥ 0) (5) (6) cm−2i−11 c 2 v2m−2 (i ≥ 0) (7) (8) cm−2i−21 c 2 v2m−2 (i ≥ 0) (9) cm−2i−11 c 2 v2m−4v2m−2 (i ≥ 0) (10) (11) cm−2i−21 c 2 v2m−4v2m−2 (i ≥ 0) (12) (13) v22m−4 (14) (15) v22m−2 (16) (17) (1) = [m−12 ] (−1)1+µ m− 1− µ m−1−2µ + (−1)m+12v2m−2. (2) = [m−k−12 ] (−1)1+µ m− k − 1− µ m−k−1−2µ (−1)m+k2c2bk−1 v2m−4 + (−1)m+k2c2bk−2 v2m−2. (3) = [m−12 ] (−1)1+µ m− 1− µ m−1−2µ v2m−4 + (−1)m+12v2m−4v2m−2. (4) = [m−2i−12 ] (−1)1+µ m− 2i− 1− µ m−2i−1−2µ v2m−4 (−1)m2 a2i−1,µc 2i−2−2µ v2m−4v2m−2. (5) = m+2i+2 2i+ 1 c2i+22 dm−2i−4 m−2i−2 am−2i−1,µc m−2i−2−2µ 2i+1+µ v2m−4 a2i−1,µ + 2i− 1 2i+ 1 a2i+1,1+µ 2i−3−2µ v2m−4v2m−2. (6) = m−2i−3 (−1)1+µ m− 2i− 2− µ m−2i−2−2µ 2i+1+µ v2m−4 m+2i+1 2i+ 1 c2i+12 dm−2i−3 v2m−2 a2i−1,µ + 2i− 1 2i+ 1 a2i+1,1+µ 2i−3−2µ v2m−4v2m−2. (7) = 2i+ 1 c2i+22 dm−2i−4 v2m−4 m−2i−2 (−1)1+µ m− 2i− 1− µ m−2i−1−2µ v2m−2 a2i−1,µ + 2i− 1 2i+ 1 a2i+1,1+µ 2i−3−2µ v2m−4v2m−2. (8) = m+2i+3 2i+ 1 c2i+12 dm−2i−3 m−2i−1 am−2i,µc m−2i−1−2µ v2m−2 a2i−1,µ + 2i− 1 2i+ 1 a2i+1,1+µ 2i−3−2µ v2m−4v2m−2. (9) = [m−2i−22 ] (−1)1+µ m− 2i− 2− µ m−2i−2−2µ 2i+1+µ v2m−2 (−1)m2 a2i+1,µc 2i−2µ v2m−4v2m−2. (10) = m−2i−4 m− 2i+ 1 m− 2i− 1 am−2i−1,µ + am−2i+1,1+µ m−2i−3−2µ 2i+1+µ v2m−4v2m−2. (11) = m−2i−1 am−2i,µc m−2i−1−2µ v2m−4v2m−2. (12) = m−2i−2 am−2i−1,µc m−2i−2−2µ 2i+1+µ v2m−4v2m−2. (13) = m−2i−5 m− 2i m− 2i− 2 am−2i−2,µ + am−2i,1+µ m−2i−4−2µ 2i+2+µ v2m−4v2m−2. (14) = (−1) 2 dm−2 v2m−4. (15) = − bm−2v2m−4 + (−1) 2 dm−3 v2m−2. (16) = (−1) 2 c22dm−4 v2m−4. (17) = (−1) 2 c2dm−3 v2m−2. 5.4. The ring structure of CH ·(Yn)(2) for n = 2m+ 1. even m odd m cm−11 (i) cm−k−11 c 2 (k ≥ 1) (ii) cm−2i−11 c 2 v2m−2 (i ≥ 0) (iii) (iv) cm−2i−21 c 2 v2m−2 (i ≥ 0) (v) cm−11 v2m (vi) cm−2i−11 c 2 v2m (i ≥ 1) (vii) cm−2i−21 c 2 v2m (i ≥ 0) (viii) (ix) cm−2i−11 c 2 v2m−2v2m (i ≥ 0) (x) (xi) cm−2i−21 c 2 v2m−2v2m (i ≥ 0) (xii) (xiii) v22m−2 (xiv) (xv) v22m (xvi) (xvii) (i) = [m−12 ] (−1)1+µ m− 1− µ m−1−2µ + (−1)m+12v2m−2. (ii) = [m−k−12 ] (−1)1+µ m− k − 1− µ m−k−1−2µ (−1)m+k2c2bk−2 v2m−2 + (−1)m+k+12bk−1 (iii) = m−2i−2 (−1)1+µ m− 2i− 1− µ m−2i−1−2µ v2m−2 m+2i+2 2i− 1 c2i2 dm−2i−2 2i+ 1 2i− 1 a2i−1,−1+µ + a2i+1,µ 2i−1−2µ v2m−2v2m. (iv) = m+2i+3 2i+ 1 c2i+12 dm−2i−3 m−2i−1 am−2i,µc m−2i−1−2µ v2m−2 a2i−1,µ + 2i− 1 2i+ 1 a2i+1,1+µ 2i−3−2µ v2m−2v2m. (v) = [m−2i−22 ] (−1)1+µ m− 2i− 2− µ m−2i−2−2µ 2i+1+µ v2m−2 (−1)m+12 a2i+1,µc 2i−2µ v2m−2v2m. (vi) = [m−12 ] (−1)1+µ m− 1− µ m−1−2µ v2m + (−1)m+12v2m−2v2m. (vii) = [m−2i−12 ] (−1)1+µ m− 2i− 1− µ m−2i−1−2µ (−1)m2 a2i−1,µc 2i−2−2µ v2m−2v2m. (viii) = m+2i+2 2i+ 1 c2i+22 dm−2i−4 m−2i−2 am−2i−1,µc m−2i−2−2µ 2i+1+µ a2i−1,µ + 2i− 1 2i+ 1 a2i+1,1+µ 2i−3−2µ v2m−2v2m. (ix) = m+2i+3 2i+ 3 c2i+32 dm−2i−5 v2m−2 m−2i−3 (−1)1+µ m− 2i− 2− µ m−2i−2−2µ 2i+1+µ a2i+1,µ − a2i+1,1+µ + 2i+ 1 2i+ 3 a2i+3,1+µ 2i−1−2µ v2m−2v2m. (x) = m−2i−4 m− 2i+ 1 m− 2i− 1 am−2i−1,µ + am−2i+1,1+µ m−2i−3−2µ 2i+1+µ v2m−2v2m. (xi) = m−2i−1 am−2i,µc m−2i−1−2µ v2m−2v2m. (xii) = m−2i−2 am−2i−1,µc m−2i−2−2µ 2i+1+µ v2m−2v2m. (xiii) = m−2i−5 m− 2i m− 2i− 2 am−2i−2,µ + am−2i,1+µ m−2i−4−2µ 2i+2+µ v2m−2v2m. (xiv) = (−1) 2 dm−2 (xv) = (−1) 2 c2dm−3 v2m−2. (xvi) = (−1) 2 c22dm−4 (xvii) = (−1) c32dm−5 v2m−2 − c1v2m−2v2m. 5.5. A remark on the ring structure of CH ·(Yn). We have given the ring structure of CH ·(Yn)(2) in 5.3 and 5.4. But actually, it is easy to determine the ring structure of CH ·(Yn) from 5.2, 5.3 and 5.4. For example, by the basis in 5.2, the formula 5.3 (5) is rewritten as follows: (5)′ cm−2i−21 c 2 v2m−4 = m+2i+2 (−1)i(2i− 1) + 1 c2i+22 dm−2i−4 m−2i−2 am−2i−1,µc m−2i−2−2µ 2i+1+µ v2m−4 a2i−1,µc 2i−3−2µ v2m−4v2m−2 − 4i− 2 2i+ 1 〈c2i−11 c2v2m−4v2m−2〉. The other cases can be calculated similarly. References [1] I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Schubert cells and cohomology of the spaces G/P , Russian Math. Surveys 28 (1973), 1–26. [2] A. Borel, Linear Algebraic Groups, Graduate Texts in Mathematics 126, Springer-Verlag, New York, 1991. [3] C. Chevalley, Anneaux de Chow et Applications, Secrétariat Mathématique, Paris, 1958. [4] P. Deligne, Théorie de Hodge III, Publ. Math. IHES 44 (1974), 5–77. [5] P. Deligne, Cohomologie Étale (SGA 4 1 ), Lecture Notes in Math. 569, Springer-Verlag, New York, 1977. [6] M. Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301. [7] S. K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), 453–460. [8] E. M. Friedlander, Étale Homotopy of Simplicial Schemes, Ann. of Math. Stud- ies 104, Princeton Univ. Press, Princeton, 1982. [9] A. Grothendieck, Théorie des Topos et Cohomologie Et́ale des Schémas (SGA 4). Tomes I–III, Lecture Notes in Math. 269, 270, 305, Springer-Verlag, New York, 1972–1973. [10] R. Hartshorne, Algebraic Geometry, Lecture Notes in Math. 52, Springer- Verlag, New York, 1977. [11] Y. Kamiyama, Generating varieties for the triple loop space of classical Lie groups, Fund. Math. 177 (2003), 269–283. [12] Y. Kamiyama, A. Kono and M. Tezuka, Cohomology of the moduli space of SO(n)-instantons with instanton number 1, Topology Appl. 146 (2005), 471– [13] J. S. Milne, Étale Cohomology, Princeton Univ. Press, Princeton, 1980. [14] P. Norbury and M. Sanders, Real instantons, Dirac operators and quaternionic classifying spaces, Proc. Amer. Math. Soc. 124 (1996), 2193–2201. [15] P. Pragacz and J. Ratajski, A pieri-type theorem for Lagrangian and odd or- thogonal Grassmannians, J. Reine Angew. Math. 476 (1996), 143–189. [16] P. Pragacz and J. Ratajski, A Pieri-type formula for even orthogonal Grass- mannians, Fund. Math. 178 (2003), 49–96. [17] B. Schuster and N. Yagita, Transfers of Chern classes in BP-cohomology and Chow rings, Trans. Amer. Math. Soc. 353 (2001), 1039–1054. [18] Y. Tian, The Atiyah-Jones conjecture for classical groups and Bott periodicity, J. Differential Geom. 44 (1996), 178–199. [19] B. Totaro, Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc. 10 (1997), 467–493. Department of Mathematics, University of the Ryukyus, Nishihara- Cho, Okinawa 903-0213, Japan Department of Mathematics, University of the Ryukyus, Nishihara- Cho, Okinawa 903-0213, Japan E-mail address : tez@sci.u-ryukyu.ac.jp 1. Introduction 2. Preliminaries 3. The ring structure of CH(Yn) 4. The Chow ring of Xn 5. Tables of the ring structure of CH(Yn) References

For an algebraically closed field K with ch K \not = 2, we determine the Chow ring of the moduli space of holomorphic bundles on a projective plane with the structure group SO(n,K) and half the first Pontryagin index being equal to 1, each of which is trivial on a fixed line and has a fixed holomorphic trivialization there.

Introduction Let G be one of the classical groups SU(n), SO(n) or Sp(n), and let k ≥ 0 be half the first Pontryagin index of a G-bundle P over S4 = R4 ∪ {∞}. Denote by M(k,G) the framed moduli space whose points represent isomorphism classes of pairs: (self-dual G-connections on P , isomorphism P∞ ≃ G). Let OM(k,GC) denote the moduli space of holomorphic bundles on 2 for the associated complex group, trivial on a fixed line l∞ and with a fixed holomorphic trivialization there. Then Donaldson ([7]) showed a diffeomorphism M(k,G) ≃ OM(k,GC). In [12] the topology ofM(1, SO(n)) ≃ OM(1, SO(n,C)) was studied in detail. The result was used in [11] to prove the fact that the natural homomorphism J : H·(M(1, SO(n)),Z/2) → H·(Ω30Spin(n),Z/2) is injective. Moreover, the image of J was determined. To prove this, the following description of OM(1, SO(n,C)) by a homogeneous space was used: We set Wn = SO(n)/(SO(n− 4)× SU(2)). Then there is a diffeomorphism (1.1) OM(1, SO(n,C)) ≃ R5 ×Wn. 2000 Mathematics Subject Classification. 14M17 (14N10). Key words and phrases. Moduli space, homogeneous space, Chow ring, cycle http://arxiv.org/abs/0704.1938v1 The purpose of this paper is to generalize the definition ofOM(1, SO(n,C)) for any algebraically closed field K with ch(K) 6= 2 and to determine the Chow ring of this. The Chow ring of a classifying space was studied by Totaro [19]. A loop space is considered to be a dual situation of a classifying space in a certain sense. Our result and the result of [11] are the first step for a loop space. Definition 1.1. Let K be an algebraically closed field with ch(K) 6= 2. Let OM(1, SO(n,K)) denote the moduli space of holomorphic bundles on P2 with the structure group SO(n,K) and half the first Pontryagin index being equal to 1, each of which is trivial on a fixed line l∞ and has a fixed holomorphic trivialization there. The moduli space OM(1, SO(n,K)) is a quasi-projective variety and defines the Chow ring. More explicitly, the diffeomorphism (1.1) is generalized (in the sense of a biregular map) as follows: We set Xn = SO(n,K)/(SO(n− 4, K)× SL(2, K)) · Pu, where Pu denotes the unipotent radical. (Recall that for a parabolic subgroup P of an algebraic group G, P is a semidirect product of a reductive group and its unipotent radical Pu.) Then there is a biregular (1.2) OM(1, SO(n,K)) ≃ A2 ×Xn. (For the proof of (1.2), see Proposition 2.1.) A formula of Grothendieck [3] shows that CH ·(OM(1, SO(n,K))) ≃ CH ·(Xn). The purpose of this paper is to determine the Chow rings of Xn and its related algebraic variety Yn explicitly. The Schubert cell approach of the Chow ring of Yn by using a Young diagram is done in [15], [16]. However it needs further work to deter- mine the Chow ring of Xn from this. Hence we first calculate the Chow ring of Yn more explicitly by a different method. Then we calculate the Chow ring from the results. Our results for the Chow ring of Xn are This paper is organized as follows. In Sect. 2 we first prove (1.2). Then we recall basic facts on the Chow ring. In Sect. 3 we determine an integral basis and the ring structure of CH ·(Yn), where Yn is an algebraic variety which is related to Xn. (See Theorems 3.7 and 3.9.) The ring structure of CH ·(Yn) proved in Theorem 3.9 is one of our main results. Since the results are long, we give them in tables in Sect. 5. (See 5.2-5.5.) Using the results of Sect. 3, we determine CH ·(Xn) in Sect. 4. (See Theorem 4.1.) We thank N. Yagita for turning our interest to the Chow ring and explaining the paper [17]. 2. Preliminaries We first prove (1.2): Proposition 2.1. For an algebraically closed field K with ch(K) 6= 2, there is a biregular map OM(1, SO(n,K)) ≃ A2 ×Xn. Proof . Recall that a monad description of OM(k, SO(n,C)) was in- dicated in [7] and given explicitly in [14] and [18]. It is easy to see that the description remains valid for any algebraically closed field K. In particular, OM(1, SO(n,K)) is given as follows: Lemma 2.2. Let Cn be the space of n× 2 matrices z1 w1 z2 w2 zn wn with coefficients in K satisfying: a) cT c = O, that is: z2i = 0, w2i = 0 and ziwi = 0, b) The rank of c over K is 2. The group SL(2, K) acts on Cn from the right by the multiplication of matrices. Then there is a biregular map OM(1, SO(n,K)) ≃ A2 × (Cn/SL(2, K)). From the lemma, it suffices to prove Xn ≃ Cn/SL(2, K). We prove this for the case n = 2m. (The case n = 2m + 1 can be proved similarly.) Recall that in [2], SO(n,K) was defined as follows: Let q(x) be a quadratic form on An defined by q(x) = i=1 xixm+i, and let B(x, y) be the associated bilinear form. Then SO(n,K) is defined SO(n,K) = {σ ∈ Aut(An) : B(σ(x), σ(y)) = B(x, y) for x, y ∈ An}. We set xj = zj+ −1zj , xm+j = zj− −1zj , yj = wj+ −1wj and ym+j = wj− −1wj, where 1 ≤ j ≤ m. Then the defining equations of Cn are given by q(x) = q(y) = 0 and B(x, y) = 0. Clearly SO(n,K) acts on Cn. It is easy to prove the following lemma. (See [2, V 23.4].) Lemma 2.3. SO(n,K)/SO(n− 4, K) · Pu ≃ Cn, where Pu is the unipotent radical of a parabolic subgroup with a Levi factor SO(n− 4, K)×GL(2, K). Now Proposition 2.1 follows from Lemma 2.3. This completes the proof of Proposition 2.1. � Next we recall basic facts on the Chow ring. We suppose that an algebraic variety V is defined over K. Let CH ·(V ) denote the Chow ring and CH i(V ) the subgroup of CH ·(V ) generated by the cycles of codimension i. Theorem 2.4 ([3]). (i) Let V be a nonsingular variety, X a nonsin- gular closed subvariety of V , and U = X − V . Then there exists an exact sequence CH ·(X) i∗→ CH ·(V ) j → CH ·(U) → 0, where i : X → V (resp. j : U → V ) is a closed immersion (resp. an open immersion). For the definitions of i∗ and j ∗, see also [10]. (ii) Let π : E → V be a fiber bundle with an affine space An as a fiber. Then the induced map π∗ : CH ·(V ) → CH ·(E) is an isomorphism. The Chow ring of the following projective variety is well-known. Theorem 2.5 ([1], [6]). Let G be a reductive algebraic group and P a maximal parabolic subgroup. Then (i) a quotient G/P is a nonsingular projective variety. (ii) CH ·(G/P ) is generated by the Schubert varieties. (iii) CH ·(G/P ) is independent of ch(K). Moreover, CH ·(G/P ) ≃ H ·(G/P,Z) for K = C. 3. The ring structure of CH ·(Yn) Before describing the results, we need some notations and results. We set Yn = SO(n,K)/(SO(n− 4, K)×GL(2, K)) · Pu. Then we have a principal bundle (3.1) Gm → Xn π→ Yn. In this section we determine an integral basis and the ring structure of CH ·(Yn). By Theorem 2.5 (ii), (iii), we obtain the following theorem: Theorem 3.1 ([12]). We have an isomorphism as modules: (1) For n = 2m, CH ·(Yn)⊗ Z/2 ≃ Z/2[c1, c2]/(bm−1, c2bm−2)⊗∆(v2m−4, v2m−2). (2) For n = 2m+ 1, CH ·(Yn)⊗ Z/2 ≃ Z/2[c1, c2]/(bm−1, c2bm−2)⊗∆(v2m−2, v2m), where |c1| = 2, |c2| = 4, |bi| = 2i and |vi| = i. Theorem 3.2 ([12]). Let p be an odd prime. Then we have a ring isomorphism: (1) For n = 2m, CH ·(Yn)⊗ Z/p ≃ Z/p[c1, c2, χ2m−4]/(c2χ2m−4, χ22m−4 − dm−2, dm−1), where χ2m−4 ∈ H2m−4(BSO2m−4,Z/p) is the Euler class. (2) For n = 2m+ 1, CH ·(Yn)⊗ Z/p ≃ Z/p[c1, c2]/(dm−1, c22dm−2). We recall the definitions of bi, di and vi. In a polynomial ring Z[α, β], we set c1 = α+ β and c2 = αβ. Then bk and dk are defined by bk = (−1)k αiβk−i dk = (−1)k α2iβ2k−2i. The element v2r ∈ CH2r(Yn) is defined by (1) For n = 2m, (3.2) 2v2m−4 = χ2m−4 − bm−2 2v2m−2 = bm−1. (2) For n = 2m+ 1, (3.3) 2v2m−2 = bm−1 2v2m = c2bm−2. The following formulas are easily proved. Lemma 3.3. We have bk = (−1)k (−1)µ k − µ dk = (−1)k (−1)µ 2k − µ+ 1 2k−2µ The following lemmas are also easily shown. Lemma 3.4. (−1)µch−µ2 b2µ = dh. Lemma 3.5. We set fn(x) = (1 + x) n − (1 + xn) and write fn(x) as fn(x) = [n2 ] µ(1 + x)n−2µ. Then we have aµ = (−1)µ+1 n− 1− µ Especially, the last term is given by (−1)s+12xs for n = 2s (−1)s+1(2s+ 1)xs(1 + x) for n = 2s+ 1. For n = 2m or 2m+ 1, we define a subgroup An of CH ·(Yn) by (3.4) An = Z[c1]/(c m−1−i 1 ){ci2} where we set ∆Z(v2m−4, v2m−2) n = 2m ∆Z(v2m−2, v2m) n = 2m+ 1. The generators v2i is specified in (3.2) and (3.3). The following lemma is proved in the same way as in [12, Lemma 3.8]. Lemma 3.6. For a prime p, we abbreviate CH ·(Yn)⊗Z(p) as CH ·(Yn)(p). If p is odd, we have the following isomorphism of modules: (i) For n = 2m, CH ·(Yn)(p) ≃ Z(p)[c1]/(c2(m−1)1 ){1, χ2m−4}⊕ Z(p)[c1]/(c 2(m−1−i) 1 ){c2i−12 , c2i2 }. (ii) For n = 2m+ 1, CH ·(Yn)(p) ≃ Z(p)[c1]/(c 2(m−1−i) 1 ){c2i2 , c2i+12 }. Theorem 3.7. An integral basis of CH ·(Yn) is constructed from the monomial basis of An in (3.4). The results are summed up in Sect. 5, The proof is based on a rather complicated calculation. Its outline is as follows: We construct a set of suitable generators starting from the basis of An. It is easily verified that it is a basis of CH ·(Yn)⊗ Z/p by using the presentation of Lemma 3.6. Then it is a Z-basis of CH ·(Yn). We only prove the case n = 2m + 1 and m is even. To simplify the proof, we set: C = CH ·(Yn) (3.5) bm−1 = (−1)m−1bm−1 = i+j=m−1 αiβj, em = c2bm−2 dm−1 = (−1)m−1dm−1 Z[c1]/(c 1 )⊕ · · · ⊕ Z[c1]/(cm−1−i1 ){ci2} ⊕ · · · ⊕ {cm−22 } ⊗ Z{1, bm−1, em, bm−1em}. Let An the set of a basis {ci1c m : i+ j ≤ m− 2, ǫk = 0 or 1 (k = 1, 2)}. Then we have C ⊂ Q[c1, c2]/(dm−1, c22dm−2), A ⊂ Q[c1, c2]/(dm−1, c22dm−2) and c2i2 dm−1−i ∈ (dm−1, c22dm−2), i ≥ 0. Our concern is a homogeneous polynomial algebra SZ(V ) of V = {c1, c2}. It is identified with an inhomogeneous ring Z[x] by putting x = β . Then we have c1 = 1 + x, c2 = x, bm−1 = xm − 1 , and dm−1 = x2m − 1 x2 − 1 Using the identification, the next formulas are directly checked: In Z[c1, c2], − c2i+1+j2 bm−2−2i = c 2 b2i−1bm−1 − c 2b2iem − c2i+2+j2 bm−3−2i = c 2 b2ibm−1 − c 2b2i+1em. We set c2i−11 c 2 bm−1 2 b2i−1bm−1 − c 2b2iem = −c 2i+j+1 2 bm−2−2i(3.6) c2i1 c 2 bm−1 2 b2ibm−1 − c 2b2i+1em = −c 2i+j+2 2 bm−3−2i. Noting c22dm−2 = 0, we have c22bm−2bm−1−c22c1dm−2 = (x− 1)2 (xm−2−1)(xm−1−1) = c32bm−3bm−2. Using c2i2 dm−1−i = 0, 1 ≤ i ≤ m, we repeat the argument and get (3.7) ci2bm−1em = c 2 bm−2−ibm−1−i. In Yn, we see that dm−1 = 0. Recall that we set f2i+3(x) = (1 + x)2i+3 − (1 + x2i+3) (see Lemma 3.5). Then we have c2i+11 c 2bm−1em − c 2 didm−1 xj+1(xm − 1) (x2 − 1)2 (1 + x)2i+3(xm−1 − 1)− (x2i+2 − 1)(xm + 1) xj+1(xm − 1) (x2 − 1)2 (1 + x2i+3)(xm−1 − 1)− (x2i+2 − 1)(xm + 1) + f2i+3(x)(xm−1 − 1) x2i+j+3 (x− 1)(x2 − 1) (xm − 1)(xm−2i−3 − 1) + x (x2 − 1)2 (xm−1 − 1)(xm − 1)f2i+3(x). We set x2i+j+3 (x− 1)(x2 − 1) (xm − 1)(xm−2i−3 − 1) (x2 − 1)2 (xm−1 − 1)(xm − 1)f2i+3(x). Since x2λdm−1−λ = 0 for 0 ≤ λ ≤ m−1, we see that x2i+j+3+λdm−1−(i+2)−λ = 0 for 0 ≤ λ ≤ j − 1. Hence x2i+j+3+λdm−1−(i+2)−λ = x2i+2j+3 (x− 1)(x2 − 1) (xm−j−1)(xm−2i−j−3−1). From Lemma 3.5, 2i+1−2µ 1 bm−1em + (−1)ikxi+j+2 (xm−1 − 1)(xm − 1) (x− 1)(x2 − 1) where k = 2i+ 3. We set J = xi+j+2 (xm−1−1)(xm−1) (x−1)(x2−1) . Using xi+j+2+λdm−2−λ = 0 for 1 ≤ λ ≤ i+ j, we see that xi+j+2+λdm−2−λ = x2i+2j+3 (x− 1)(x2 − 1) (xm−2−i−j−1)(xm−1−i−j−1). Afterwards, we introduce the notation (3.8) c2i+11 c 2bm−1em := c2i+11 c 2bm−1em − 2i+1−2µ 1 bm−1em, where f2i+3(x) = µ=1 aµx µ(1 + x)2i+3−2µ (see Lemma 3.5). We sum up these arguments: (3.9) c2i+11 c 2bm−1em = I1 + (−1)ikJ, where x2i+2j+3 (x− 1)(x2 − 1) (xm−j − 1)(xm−2i−j−3 − 1), x2i+2j+3 (x− 1)(x2 − 1) (xm−2−i−j − 1)(xm−1−i−j − 1), and k = 2i+ 3. A direct calculation shows J − I1 = x (x−1)(x2−1) (xi+1 − 1)(xi+2 − 1). Hence we get a key formula (3.10) c2i+11 c 2bm−1em (−1)i(2i+ 3) + 1 I1+(−1)i(2i+3) (x− 1)(x2 − 1) (xi+1−1)(xi+2−1). We note that (3.9) holds for m even or odd. From now on, we as- sume that m is even. When we put j = 1 in (3.9), we see that I1 = c −i−3em. We set 〈c2i+11 c2bm−1em〉 := c2i+11 c2bm−1em (−1)i(2i+ 3) + 1 c2i+42 dm2 −i−3em. Then we have (3.11) 〈c2i+11 c2bm−1em〉 2i+ 3 = (−1)icm+12 bibi+1 We call a generator cm−3−2i1 c 2 bm−1em to be the head of the pre- sentation An in (3.4). For the head generator, it is directly shown that I1 = 0 by (3.9). Hence the formula (3.9) implies (3.12) cm−3−2i1 c 2 bm−1em m− 1− 2i = (−1)i −i−2bm Then the formulas (3.10) and (3.11) imply the following relations: We 〈c2i+11 bm−1em〉 := c2i+11 bm−1em (−1)i(2i+ 3) + 1 c2i+22 dm2 −i−2em 〈c2α+1−2β1 c 2 bm−1em〉 := 2α+1−2β 2 bm−1em (−1)α−β(2α− 2β + 3) + 1 c2α+42 dm2 −α−3em. Then we have 〈c2i+11 bm−1em〉 − (−1)i(2i+ 3) + 1 (−1)i−1(2i+ 1) 〈c2i−11 c2bm−1em〉 = − bibi+1 (3.13) 〈c2α+1−2β1 c 2 bm−1em〉 − (−1)β(2α− 2β + 3) + (−1)α 2α + 3 〈c2α+11 c2bm−1em〉 (3.14) m+β+2 bα−β−1bα−β m−3−2α−2β 2α+β+1 (−1)β(m− 1− 2α+ 2β) + (−1)m2 −α m− 1− 2α cm−3−2α1 c 2 bm−1em (3.15) m+2α+β+1 bγbγ+1, where γ = − 2− α− β. Last we consider a generator c2i+21 c 2bm−1em. We set 〈c2i+21 c 2bm−1em〉 :=c2i+21 c 2bm−1em − 2i+2−2µ 1 bm−1em (−1)i(2i+ 3) + 1 i+j+1 2 bm−1em, where f2i+3(x) = µ=1 aµx µ(1 + x)2i+3−2µ (see Lemma 3.5). Then the formula (3.9) implies that (3.16) 〈c2i+11 c 2bm−1em〉 = −c 2 bibi+1. We consider a set {(3.6), (3.10), (3.11), (3.12), (3.13), (3.14)} ∪ {x ∈ An : x = ci1c m, x = c 1bm−1, i+ j ≤ m− 2, ǫk = 0 or 1}. When we reduce it to the mod p reduction for an odd prime p, the identities from (3.6) to (3.14) show that they are linearly independent from Lemma 3.6. Hence, replacing bm−1 and em by v2m−2 and v2m respectively, we obtain an integral basis of CH ·(Y2m+1) for even m. An integral basis of CH ·(Yn) for other cases is written down in a table of Sect. 5. For a group G and a subgroup H , let [G : H ] denote the index, i.e. the cardinality of G/H . As a corollary of the above theorem, we have Corollary 3.8. (i) For n = 2m, [CH ·(Yn) : An] = 12 · 32 · . . . · (m− 3)2 · (m− 1) m: even 12 · 32 · . . . · (m− 2)2 m: odd. (ii) For n = 2m+ 1, [CH ·(Yn) : An] = 12 · 32 · . . . · (m− 3)2 · (m− 1) m: even 12 · 32 · . . . · (m− 2)2 ·m m: odd. The following theorem is proved by using the integral basis of CH ·(Yn) given in Theorem 3.7. Theorem 3.9. The ring structure of CH ·(Yn) is determined. The results are summed up in tables in 5.3 and 5.4 in Sect. 5. (See also 5.5.) Proof . We only show the formula (iii) in Sect. 5, 5.4. We use the notations of the proof of Theorem 3.7. Using x2idm−1−i = 0, we have cm−2i−11 c 2 bm−1−x2idm−1−i = x2 − 1 (x2i−1)+ x x2 − 1 (xm−1)fm−2i(x). We set I1 = (x2i − 1) and I2 = x (xm − 1)fm−2i(x). Then we m−2i−1−2µ 2 bm−1 + (−1) −i+12x m − 1 x2 − 1 We set J = x +i xm−1 . From (3.6), we have 2 em− (−1)µ +i−1−µ 2 bm−1  = (−1) x2i − 1 x2 − 1 Hence cm−2i−11 c 2 bm−1− m−2i−1−2µ 2 bm−1 (3.17) + (−1) 2 em + (−1)µ +i−1−µ 2 bm−1  = −xmx 2i − 1 x2 − 1 On the other hand, we see from (3.13) that (3.18) − xmx 2i − 1 x2 − 1 = 〈c2i−11 bm−1em〉+ 2i+ 1 2i− 1 〈c2i−31 c2bm−1em〉. We calculate S := (−1) 2 em + (−1)µ +i−1−µ 2 bm−1 By (3.6), S = (−1) −i2(S1 + S2), where (−1)µc +i+1−µ 2 b2µ+1bm−1 and S2 = c 2 em− (−1)µc +i−2−µ 2 b2µ+2em. We have from Lemma 3.3 that −1−µ−i m− 1− 2i− µ m−2i−1−2µ 2 bm−1. Using µ=0(−1)µc 2 b2µ = (−1)hdh (see Lemma 3.4), S2 = c (−1)µc −i−1−µ 2 b2µ  bm−1 = (−1) −1−ic2i2 dm2 −1−ibm−1. The formula (3.17) is cm−2i−11 c 2 bm−1 + (−1)µ m− 2i− 1− µ cm−2i−11 c 2 bm−1 − 2c2i2 dm2 −1−iem = −xm x x2 − 1 Comparing with (3.18), we obtain cm−2i−11 c 2 bm−1 = (−1)µ+1 m− 2i− 1− µ cm−2i−11 c 2 bm−1 2i− 1 c2i2 dm2 −1−iem + c2i−11 bm−1em 2i+ 1 2i− 1 c2i−31 c2bm−1em This completes the proof of the formula. � 4. The Chow ring of Xn Theorem 4.1. Let T (Xn) and F (Xn) be the torsion part and the free part of CH ·(Xn), respectively. Then we have (i) For n = 4t, F (Xn) ≃ Z[c2]/(ct2){1, v4t−4} T (Xn) ≃ Z/2[c2]/(ct2){v4t−2, v4t−4v4t−2}. (ii) For n = 4t+ 1, F (Xn) ≃ Z[c2]/(ct2)⊕ Z[c2]/(ct−12 ){v4t} T (Xn) ≃ Z/2[c2]/(ct2){v4t−2, v4t−2v4t} ⊕ Z/(2t){ct−12 v4t}. (iii) For n = 4t+ 2, F (Xn) ≃ Z[c2]/(ct2){1, v4t} ⊕ Z{v4t−2} T (Xn) ≃ Z/2[c2]/(ct−12 ){c2v4t−2, c2v4t−2v4t} ⊕ Z/4{v4t−2v4t}. (iv) For n = 4t+ 3, F (Xn) ≃ Z[c2]/(ct2){1, v4t} T (Xn) ≃ Z/2[c2]/(ct2){v4t+2, v4tv4t+2} ⊕ Z/(2t+ 1){ct2v4t}. Proof . Let X̃n = Xn ×Gm A1 be the associated bundle of (3.1) and s : Yn → X̃n the 0-section. Since s∗ : CH ·(X̃n) ∼→ CH ·(Yn) by Theorem 2.4 (ii), the first assertion of the same theorem for V = X̃n and X = s(Yn) gives an exact sequence CH ·(Yn) ·c1→ CH ·(Yn) π∗→ CH ·(Xn) → 0. Theorem 4.1 follows from this and the ring structure of CH ·(Yn) in Theorems 3.7 and 3.9. � Next we consider the cycle map. The cohomology groups mean an etale cohomology [9], [13]. All varieties are defined over K ′, which is a subfield of an algebraically closed field K. Let l be a prime with (l, ch(K)) = 1. We denote a locally constant sheaf µ⊗i by Z/l(i). Corollary 4.2. The homomorphism cl : CH i(Xn) → H2i(Xn,Z/l(i)) is injective. Proof . Since (X̃n, Yn) is a smooth pair, we have the Gysin sequence as in [5, Appendice 1.3.3] and [13, VI Remark 5.4]. Since the cycle map and the Gysin map are commutative, we have the following commuta- tive diagram, where each row is exact: CH i(Yn) ·c1−−−→ CH i+1(Yn) π∗−−−→ CH i+1(Xn) −−−→ 0 H2i(Yn,Z/l ·c1−−−→ H2(i+1)(Yn,Z/l(i+1)) π∗−−−→ H2(i+1)(Xn,Z/l(i+1)) Corollary 4.2 follows from Theorem 4.1 and this diagram. � Remark 4.3. Assume that we have aK ′-isomorphism Yn ≃ SO(n,K)/(SO(n− 4, K) × GL(2, K)), where SO(n,K) and SO(n − 4, K) are split over K ′, and that π : Yn → Xn is a K ′-map. Then the Galois actions G = Gal(K /K ′) on H ·(Yn,Z/l (i)) and H ·(Xn,Z/l (i)) are described by the character group X(T ) of a K ′-split maximal torus T of SO(n,K). It follows from a result of [5, 8-2]. 5. Tables of the ring structure of CH ·(Yn) 5.1. Notations. (i) For k ∈ N ∪ {0}, we define bk and dk ∈ Z[c1, c2] as follows: bk = (−1)k [ k2 ] (−1)µ k − µ dk = (−1)k (−1)µ 2k − µ+ 1 2k−2µ (ii) For g ∈ N and µ ∈ N ∪ {0,−1}, we define ag,µ ∈ Z by ag,µ = (−1)1+µ g g−1−µ µ ≥ 1 −1 µ = 0 0 µ = −1. Then the integers ag,µ are characterized by (1 + x)g = 1 + xg + [ g2 ] ag,µx µ(1 + x)g−2µ. (iii) The generators v2i are given by (3.2) and (3.3). 5.2. An integral basis of CH ·(Yn). In the following (I) and (II), we give an integral basis of CH ·(Yn). The notations are explained as follows: Let Sn be the set of the monomial basis of An in (3.4). Let T be a subset of Sn. Then for an element ξ ∈ T , 〈ξ〉 (resp. 〈ξ〉′) is defined to be the right-hand side of an equation (1)-(8) below. We consider a : ξ ∈ T ∪ {η : η ∈ Sn − T} , where lξ ∈ N. Following this procedure, we obtain an integral basis of CH ·(Yn). We abbreviate this basis as : ξ ∈ T (I) The case n = 2m. (i) For even m, 〈c2i+11 c2v2m−4v2m−2〉 2i+ 3 〈cm−2j−31 c 2 v2m−4v2m−2〉 m− 2j − 1 : 0 ≤ i ≤ − 2, 1 ≤ j ≤ (ii) For odd m, 〈c2i+11 c2v2m−4v2m−2〉′ 2i+ 3 〈cm−2j−21 c 2 v2m−4v2m−2〉′ m− 2j : 0 ≤ i ≤ m− 5 , 1 ≤ j ≤ m− 3 (II) The case n = 2m+ 1. (iii) For even m, 〈c2i+11 c2v2m−2v2m〉 2i+ 3 〈cm−2j−31 c 2 v2m−2v2m〉 m− 2j − 1 : 0 ≤ i ≤ m − 2, 1 ≤ j ≤ m (iv) For odd m, 〈c2i+11 v2m−2v2m〉 2i+ 3 〈cm−2j−21 c 2 v2m−2v2m〉 m− 2j : 0 ≤ i ≤ , 1 ≤ j ≤ Here 〈 〉 and 〈 〉′ are defined as follows: 〈c2i+11 c2v2m−4v2m−2〉 = c2i+11 c2v2m−4v2m−2 + (−1) m+2i+2 (−1)i(2i+ 3) + 1 c2i+42 dm−2i−6 v2m−4 a2i+3,µc 2i+1−2µ 2 v2m−4v2m−2. 〈cm−2j−31 c 2 v2m−4v2m−2〉 = c m−2j−3 2 v2m−4v2m−2 m−2j−4 am−2j−1,µc m−2j−3−2µ 2j+1+µ 2 v2m−4v2m−2. 〈c2i+11 c2v2m−4v2m−2〉′ = c2i+11 c2v2m−4v2m−2 + (−1) m+2i+1 (−1)i(2i+ 3) + 1 c2i+32 dm−2i−5 v2m−2 a2i+3,µc 2i+1−2µ 2 v2m−4v2m−2. 〈cm−2j−21 c 2 v2m−4v2m−2〉′ = c m−2j−2 2 v2m−4v2m−2 m−2j−3 am−2j,µc m−2j−2−2µ 2 v2m−4v2m−2. 〈c2i+11 c2v2m−2v2m〉 = c2i+11 c2v2m−2v2m + (−1) m+2i+2 (−1)i(2i+ 3) + 1 c2i+42 dm−2i−6 a2i+3,µc 2i+1−2µ 2 v2m−2v2m. 〈cm−2j−31 c 2 v2m−2v2m〉 = c m−2j−3 2 v2m−2v2m m−2j−4 am−2j−1,µc m−2j−3−2µ 2j+1+µ 2 v2m−2v2m. 〈c2i+11 v2m−2v2m〉 = c2i+11 v2m−2v2m + (−1) m+2i+3 (−1)i(2i+ 3) + 1 c2i+32 dm−2i−5 v2m−2 a2i+3,µc 2i+1−2µ 2v2m−2v2m. 〈cm−2j−21 c 2 v2m−2v2m〉 = c m−2j−2 2 v2m−2v2m m−2j−3 am−2j,µc m−2j−2−2µ 2 v2m−2v2m. 5.3. The ring structure of CH ·(Yn)(2) for n = 2m. even m odd m cm−11 (1) cm−k−11 c 2 (k ≥ 1) (2) cm−11 v2m−4 (3) cm−2i−11 c 2 v2m−4 (i ≥ 1) (4) cm−2i−21 c 2 v2m−4 (i ≥ 0) (5) (6) cm−2i−11 c 2 v2m−2 (i ≥ 0) (7) (8) cm−2i−21 c 2 v2m−2 (i ≥ 0) (9) cm−2i−11 c 2 v2m−4v2m−2 (i ≥ 0) (10) (11) cm−2i−21 c 2 v2m−4v2m−2 (i ≥ 0) (12) (13) v22m−4 (14) (15) v22m−2 (16) (17) (1) = [m−12 ] (−1)1+µ m− 1− µ m−1−2µ + (−1)m+12v2m−2. (2) = [m−k−12 ] (−1)1+µ m− k − 1− µ m−k−1−2µ (−1)m+k2c2bk−1 v2m−4 + (−1)m+k2c2bk−2 v2m−2. (3) = [m−12 ] (−1)1+µ m− 1− µ m−1−2µ v2m−4 + (−1)m+12v2m−4v2m−2. (4) = [m−2i−12 ] (−1)1+µ m− 2i− 1− µ m−2i−1−2µ v2m−4 (−1)m2 a2i−1,µc 2i−2−2µ v2m−4v2m−2. (5) = m+2i+2 2i+ 1 c2i+22 dm−2i−4 m−2i−2 am−2i−1,µc m−2i−2−2µ 2i+1+µ v2m−4 a2i−1,µ + 2i− 1 2i+ 1 a2i+1,1+µ 2i−3−2µ v2m−4v2m−2. (6) = m−2i−3 (−1)1+µ m− 2i− 2− µ m−2i−2−2µ 2i+1+µ v2m−4 m+2i+1 2i+ 1 c2i+12 dm−2i−3 v2m−2 a2i−1,µ + 2i− 1 2i+ 1 a2i+1,1+µ 2i−3−2µ v2m−4v2m−2. (7) = 2i+ 1 c2i+22 dm−2i−4 v2m−4 m−2i−2 (−1)1+µ m− 2i− 1− µ m−2i−1−2µ v2m−2 a2i−1,µ + 2i− 1 2i+ 1 a2i+1,1+µ 2i−3−2µ v2m−4v2m−2. (8) = m+2i+3 2i+ 1 c2i+12 dm−2i−3 m−2i−1 am−2i,µc m−2i−1−2µ v2m−2 a2i−1,µ + 2i− 1 2i+ 1 a2i+1,1+µ 2i−3−2µ v2m−4v2m−2. (9) = [m−2i−22 ] (−1)1+µ m− 2i− 2− µ m−2i−2−2µ 2i+1+µ v2m−2 (−1)m2 a2i+1,µc 2i−2µ v2m−4v2m−2. (10) = m−2i−4 m− 2i+ 1 m− 2i− 1 am−2i−1,µ + am−2i+1,1+µ m−2i−3−2µ 2i+1+µ v2m−4v2m−2. (11) = m−2i−1 am−2i,µc m−2i−1−2µ v2m−4v2m−2. (12) = m−2i−2 am−2i−1,µc m−2i−2−2µ 2i+1+µ v2m−4v2m−2. (13) = m−2i−5 m− 2i m− 2i− 2 am−2i−2,µ + am−2i,1+µ m−2i−4−2µ 2i+2+µ v2m−4v2m−2. (14) = (−1) 2 dm−2 v2m−4. (15) = − bm−2v2m−4 + (−1) 2 dm−3 v2m−2. (16) = (−1) 2 c22dm−4 v2m−4. (17) = (−1) 2 c2dm−3 v2m−2. 5.4. The ring structure of CH ·(Yn)(2) for n = 2m+ 1. even m odd m cm−11 (i) cm−k−11 c 2 (k ≥ 1) (ii) cm−2i−11 c 2 v2m−2 (i ≥ 0) (iii) (iv) cm−2i−21 c 2 v2m−2 (i ≥ 0) (v) cm−11 v2m (vi) cm−2i−11 c 2 v2m (i ≥ 1) (vii) cm−2i−21 c 2 v2m (i ≥ 0) (viii) (ix) cm−2i−11 c 2 v2m−2v2m (i ≥ 0) (x) (xi) cm−2i−21 c 2 v2m−2v2m (i ≥ 0) (xii) (xiii) v22m−2 (xiv) (xv) v22m (xvi) (xvii) (i) = [m−12 ] (−1)1+µ m− 1− µ m−1−2µ + (−1)m+12v2m−2. (ii) = [m−k−12 ] (−1)1+µ m− k − 1− µ m−k−1−2µ (−1)m+k2c2bk−2 v2m−2 + (−1)m+k+12bk−1 (iii) = m−2i−2 (−1)1+µ m− 2i− 1− µ m−2i−1−2µ v2m−2 m+2i+2 2i− 1 c2i2 dm−2i−2 2i+ 1 2i− 1 a2i−1,−1+µ + a2i+1,µ 2i−1−2µ v2m−2v2m. (iv) = m+2i+3 2i+ 1 c2i+12 dm−2i−3 m−2i−1 am−2i,µc m−2i−1−2µ v2m−2 a2i−1,µ + 2i− 1 2i+ 1 a2i+1,1+µ 2i−3−2µ v2m−2v2m. (v) = [m−2i−22 ] (−1)1+µ m− 2i− 2− µ m−2i−2−2µ 2i+1+µ v2m−2 (−1)m+12 a2i+1,µc 2i−2µ v2m−2v2m. (vi) = [m−12 ] (−1)1+µ m− 1− µ m−1−2µ v2m + (−1)m+12v2m−2v2m. (vii) = [m−2i−12 ] (−1)1+µ m− 2i− 1− µ m−2i−1−2µ (−1)m2 a2i−1,µc 2i−2−2µ v2m−2v2m. (viii) = m+2i+2 2i+ 1 c2i+22 dm−2i−4 m−2i−2 am−2i−1,µc m−2i−2−2µ 2i+1+µ a2i−1,µ + 2i− 1 2i+ 1 a2i+1,1+µ 2i−3−2µ v2m−2v2m. (ix) = m+2i+3 2i+ 3 c2i+32 dm−2i−5 v2m−2 m−2i−3 (−1)1+µ m− 2i− 2− µ m−2i−2−2µ 2i+1+µ a2i+1,µ − a2i+1,1+µ + 2i+ 1 2i+ 3 a2i+3,1+µ 2i−1−2µ v2m−2v2m. (x) = m−2i−4 m− 2i+ 1 m− 2i− 1 am−2i−1,µ + am−2i+1,1+µ m−2i−3−2µ 2i+1+µ v2m−2v2m. (xi) = m−2i−1 am−2i,µc m−2i−1−2µ v2m−2v2m. (xii) = m−2i−2 am−2i−1,µc m−2i−2−2µ 2i+1+µ v2m−2v2m. (xiii) = m−2i−5 m− 2i m− 2i− 2 am−2i−2,µ + am−2i,1+µ m−2i−4−2µ 2i+2+µ v2m−2v2m. (xiv) = (−1) 2 dm−2 (xv) = (−1) 2 c2dm−3 v2m−2. (xvi) = (−1) 2 c22dm−4 (xvii) = (−1) c32dm−5 v2m−2 − c1v2m−2v2m. 5.5. A remark on the ring structure of CH ·(Yn). We have given the ring structure of CH ·(Yn)(2) in 5.3 and 5.4. But actually, it is easy to determine the ring structure of CH ·(Yn) from 5.2, 5.3 and 5.4. For example, by the basis in 5.2, the formula 5.3 (5) is rewritten as follows: (5)′ cm−2i−21 c 2 v2m−4 = m+2i+2 (−1)i(2i− 1) + 1 c2i+22 dm−2i−4 m−2i−2 am−2i−1,µc m−2i−2−2µ 2i+1+µ v2m−4 a2i−1,µc 2i−3−2µ v2m−4v2m−2 − 4i− 2 2i+ 1 〈c2i−11 c2v2m−4v2m−2〉. The other cases can be calculated similarly. References [1] I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Schubert cells and cohomology of the spaces G/P , Russian Math. Surveys 28 (1973), 1–26. [2] A. Borel, Linear Algebraic Groups, Graduate Texts in Mathematics 126, Springer-Verlag, New York, 1991. [3] C. Chevalley, Anneaux de Chow et Applications, Secrétariat Mathématique, Paris, 1958. [4] P. Deligne, Théorie de Hodge III, Publ. Math. IHES 44 (1974), 5–77. [5] P. Deligne, Cohomologie Étale (SGA 4 1 ), Lecture Notes in Math. 569, Springer-Verlag, New York, 1977. [6] M. Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301. [7] S. K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), 453–460. [8] E. M. Friedlander, Étale Homotopy of Simplicial Schemes, Ann. of Math. Stud- ies 104, Princeton Univ. Press, Princeton, 1982. [9] A. Grothendieck, Théorie des Topos et Cohomologie Et́ale des Schémas (SGA 4). Tomes I–III, Lecture Notes in Math. 269, 270, 305, Springer-Verlag, New York, 1972–1973. [10] R. Hartshorne, Algebraic Geometry, Lecture Notes in Math. 52, Springer- Verlag, New York, 1977. [11] Y. Kamiyama, Generating varieties for the triple loop space of classical Lie groups, Fund. Math. 177 (2003), 269–283. [12] Y. Kamiyama, A. Kono and M. Tezuka, Cohomology of the moduli space of SO(n)-instantons with instanton number 1, Topology Appl. 146 (2005), 471– [13] J. S. Milne, Étale Cohomology, Princeton Univ. Press, Princeton, 1980. [14] P. Norbury and M. Sanders, Real instantons, Dirac operators and quaternionic classifying spaces, Proc. Amer. Math. Soc. 124 (1996), 2193–2201. [15] P. Pragacz and J. Ratajski, A pieri-type theorem for Lagrangian and odd or- thogonal Grassmannians, J. Reine Angew. Math. 476 (1996), 143–189. [16] P. Pragacz and J. Ratajski, A Pieri-type formula for even orthogonal Grass- mannians, Fund. Math. 178 (2003), 49–96. [17] B. Schuster and N. Yagita, Transfers of Chern classes in BP-cohomology and Chow rings, Trans. Amer. Math. Soc. 353 (2001), 1039–1054. [18] Y. Tian, The Atiyah-Jones conjecture for classical groups and Bott periodicity, J. Differential Geom. 44 (1996), 178–199. [19] B. Totaro, Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc. 10 (1997), 467–493. Department of Mathematics, University of the Ryukyus, Nishihara- Cho, Okinawa 903-0213, Japan Department of Mathematics, University of the Ryukyus, Nishihara- Cho, Okinawa 903-0213, Japan E-mail address : tez@sci.u-ryukyu.ac.jp 1. Introduction 2. Preliminaries 3. The ring structure of CH(Yn) 4. The Chow ring of Xn 5. Tables of the ring structure of CH(Yn) References

Entanglement condition via su(2) and su(1,1) algebra using Schrödinger-Robertson uncertainty relation Hyunchul Nha∗ ARC Center of Excellence for Quantum Computer Technology, University of Queensland, Australia School of Computational Sciences, Korea Institute for Advanced Study, Korea (Dated: August 25, 2021) The Schrödinger-Robertson inequality generally provides a stronger bound on the product of uncertainties for two noncommuting observables than the Heisenberg uncertainty relation, and as such, it can yield a stricter separability condition in conjunction with partial transposition. In this paper, using the Schrödinger-Robertson uncertainty relation, the separability condition previously derived from the su(2) and the su(1,1) algebra is made stricter and refined to a form invariant with respect to local phase shifts. Furthermore, a linear optical scheme is proposed to test this invariant separability condition. PACS numbers: 03.67.Mn, 03.65.Ud, 42.50.Dv I. INTRODUCTION When a quantum system is subject to measurements corresponding to two noncommuting observables {A, B}, the product of uncertainties in measurement outcomes, 〈(∆A)2〉〈(∆B)2〉, has a certain lower bound. The Heisen- berg uncertainty relation (HUR)[1], which is most widely used, provides the bound as 〈(∆A)2〉〈(∆B)2〉 ≥ |〈[A,B]〉|2. (1) On the other hand, the Schrödinger-Robertson rela- tion(SRR) [2, 3] in general provides a stronger bound 〈(∆A)2〉〈(∆B)2〉 ≥ |〈[A,B]〉|2 + 〈∆A∆B〉2S , (2) where the cross correlation 〈∆A∆B〉S is defined in a sym- metric form as 〈∆A∆B〉S ≡ 〈∆A∆B +∆B∆A〉. (3) The SRR can be derived from the Cauchy-Schwartz in- equality, 〈f |f〉〈g|g〉 ≥ |〈f |g〉|2, where |f〉 = ∆A|Ψ〉 and |g〉 = ∆B|Ψ〉 for a generic quantum state |Ψ〉[4]. The HUR describes a special case of the SRR under the con- dition 〈∆A∆B〉S = 0, which is of course not always met. Recently, one of the important issues in quantum in- formatics has been to obtain conditions by which one can distinguish entangled states from separable ones. Some of such entanglement criteria derived so far have relied on the bounds set by various forms of uncertainty relations[5, 6, 7, 8], and remarkably for certain cases, in explicit conjunction with partial transposition(PT)[9, 10, 11]. More precisely, separable states can represent a certain physical state even under PT[12] and all uncer- tainty relations must therefore be satisfied by separable states under PT. The uncertainty relations in combina- tion with PT can thereby provide necessary conditions for separability. For continuous variables (CVs), earlier works were focused on Gaussian entangled states[13, 14, 15], but considerable attention has also been directed to non- Gaussian entangled states[16]. Most of all, the separa- bility conditions applicable to non-Gaussian entangled states have recently emerged[8, 9, 10, 11], and in partic- ular, Refs. [8, 10, 11] employed the su(2) and the su(1,1) algebra to derive such entanglement criteria. Using the HUR along with those two algebras, Nha and Kim have particularly derived the optimal separability condition among a certain class of inequalities[11]. This condi- tion has also been proposed to detect multipartite en- tanglement of photonic W states and shown to be robust against the detector inefficiency[17]. In this paper, it is our aim to refine the separability condition in Refs. [10, 11] by employing the SRR instead of the HUR. By doing this, we obtain a stricter separa- bility condition given by a form invariant with respect to local phase shifts. This invariance is a very adequate attribute as entanglement condition, for entanglement property must be invariant under any local unitary oper- ations. Furthermore, we propose how to experimentally test this invariant condition using linear optics and also discuss the practical connection of the previous condition in [10, 11] to the present one. II. SEPARABILITY CONDITION First, we briefly introduce how to derive the separabil- ity condition via the uncertainty relations in the su(2) and the su(1,1) algebra[11]. The su(2) algebra deals with the angular momentum operators Jx, Jy and Jz, which obey the commutation relations [Ji, Jj ] = iǫijkJk (i, j, k = x, y, z). This algebra can be represented by two http://arxiv.org/abs/0704.1939v2 bosonic operators a and b, as a†b+ ab† a†b− ab† a†a− b†b . (4) On the other hand, the operators Kx,Ky and Kz in the su(1,1) algebra can be represented by a†b† + ab a†b† − ab a†a+ b†b+ 1 , (5) which results in the commutation relations, [Kx,Ky] = −iKz, [Ky,Kz] = iKx, and [Kz,Kx] = iKy, different in sign from those of the su(2) algebra. Specifically, the commutator [Kx,Ky] = −iKz in the su(1,1) algebra gives the uncertainty relation via the HUR as 〈(∆Kx) 2〉〈(∆Ky) |〈Kz〉| 2, (6) which must be satisfied by any quantum states. Most importantly, the inequality (6) must be satisfied under PT by every separable state, since it can still describe a certain physical state[12]. That is, one obtains the separability condition as 〈(∆Kx) 2〉PT〈(∆Ky) 2〉PT ≥ |〈Kz〉| PT, (7) where the subscript PT means that the quantum average is calculated after taking partial transposition. Using a general relation 〈a†manb†pbq〉ρPT = 〈a †manb†qbp〉ρ (8) between the quantum average for the partially trans- posed density operator ρPT and that for the original den- sity operator ρ[11], the inequality (7) can be recast to give the separability condition expressed as + 〈(∆Jx) + 〈(∆Jy) [1 + 〈N+〉] , (9) where N+ = a †a + b†b is the total excitation number. Note that the inequality (9) is the optimal condition de- rived in [11], where the HUR was employed in a sum form to obtain a class of separability conditions[18]. III. STRICTER SEPARABILITY CONDITION In this section, let us now start from the SRR for the commutator [Kx,Ky] = −iKz, i.e., 〈(∆Kx) 2〉〈(∆Ky) |〈Kz〉| 2 + 〈∆Kx∆Ky〉 S , (10) instead of the HUR, then follow the same steps as below Eq. (6). Using the relation 〈∆Kx∆Ky〉S,PT = 〈∆Jx∆Jy〉S (11) via Eq. (8), we obtain a separability condition stricter than the one in (9) as + 〈(∆Jx) + 〈(∆Jy) [1 + 〈N+〉] + 〈∆Jx∆Jy〉 S .(12) Compared with the inequality (9), the new inequal- ity (12) prodvides a stronger condition for separabil- ity as long as the off-diagonal covariance 〈∆Jx∆Jy〉S is nonzero. As an example, consider the two-photon en- tangled states of the type |Ψ〉 = cos θ|2, 0〉+ i sin θ|0, 2〉. All these states satisfy the inequality (9), but violate the stricter one in (12), regardless of the parameter θ. There- fore, only the inequality (12) can detect entanglement for those two-photon states. We next show that the inequality (12) is invariant with respect to local phase shifts. Let us consider a 2 × 2 covariance matrix C of which elements are defined as Cij ≡ 〈∆Ji∆Jj +∆Jj∆Ji〉, (13) where {i, j} = {x, y}. The inequality (12) is then ex- pressed as Det{C}+ Tr{C} ≥ 2 + 2〈N+〉 , (14) where Det{} and Tr{} denote the determinant and the trace of a matrix. If one takes a local phase shift for mode b as b′ = be−iφ, the su(2) operators Jx and Jy are transformed into cosφ sinφ − sinφ cosφ . (15) The determinant and the trace of a matrix are unchanged under rotation, and the total photon number 〈N+〉 is also preserved through passive optical elements. The in- equality (14) is therefore invariant with respect to local phase shifts. This is an attribute very adequate as entan- glement condition, for entanglement should be invariant under local unitary operations. Note that a phase shift is the only local unitary operation that preserves the total photon number. IV. MEASUREMENT SCHEME We now discuss how the separability condition (12) can be tested in experiment. In Ref.[11], a linear optical scheme was proposed to measure the observables Jx, Jy and 〈N+〉 for the inequality (9), as depicted in Fig. 1. The mode b first undergoes a phase shift by φ and the FIG. 1: Experimental scheme for measuring the quantities necessary to test the inequality (12). All the quantum av- erages in (12) can be measured by detecting the photon number difference at the output, N{−,φ} ≡ c c − d −iφ + ab†eiφ, with four different phase shifts, φ = 0, π and −π . (See the main text.) BS: 50:50 beam-splitter, PS: phase-shifter, and PD: photo detector. two modes a and b are then injected to a 50:50 beam splitter. The modes c and d at the output are given by c = 1√ (a+ be−iφ) and d = 1√ (−a+ be−iφ), respectively. One needs to measure the photon number difference at the output, i.e., N{−,φ} ≡ c †c− d†d = a†be−iφ + ab†eiφ, (16) which becomes 2Jx (2Jy) for φ = 0 (φ = ). (See Eq. (4).) The total photon number 〈N+〉 is simply given by the sum, c†c+ d†d, at the output. In the present inequality (12), in addition to Jx, Jy and 〈N+〉, one also needs to measure the off-diagonal covari- ance 〈∆Jx∆Jy〉S . Note that 〈∆Jx∆Jy〉S = 〈JxJy + JyJx〉 − 〈Jx〉〈Jy〉, where JxJy + JyJx = a†2b2 − a2b†2 {−,φ=π {−,φ=−π . (17) Thus, by choosing two different phase shifts φ = π φ = −π in Fig. 1, the quantum average 〈JxJy + JyJx〉 can be measured in two pieces as shown in Eq. (17). In summary, the single experimental setup in Fig. 1 can be used to measure all the quantities necessary to test the inequality (12). Finally, we discuss how the inequality (9) can be re- garded as ”equivalent” to the stricter inequality (12). Us- ing the relation in Eq. (15) implemented by a local phase shift, one has the covariance in the rotated frame as 〈∆J ′x∆J y〉S = sin 2φ 〈(∆Jy) 2〉 − 〈(∆Jx) +cos 2φ〈∆Jx∆Jy〉S . (18) Thus, by choosing the phase shift as tan 2φ = 2〈∆Jx∆Jy〉S 〈(∆Jx)2〉 − 〈(∆Jy)2〉 , (19) the covariance in the rotated frame can be made vanish. In this situation, the inequality (12) is reduced to the inequality (9). In other words, as long as one is allowed to perform a local phase shift, which does not alter the entanglement property at all, the two inequalities can be interpreted as equivalently useful. However, this relies on the capability of measuring all the covariances and of performing a phase shift very accurately required by Eq. (19). It is then of no practical advantage to adhere to the inequality (9): One can simply test the inequal- ity (12) if one is able to measure the off-diagonal covari- ance 〈∆Jx∆Jy〉S in addition. V. SUMMARY In this paper, we have derived a stricter separability condition via the su(2) and the su(1,1) algebra using the Schrödinger-Robertson inequality instead of the Heisen- berg uncertainty relation. It has been shown that this refined condition is expressed in a form invariant with respect to local phase shifts. A linear optical setup has been proposed to test the invariant separability condition and the practical connection of the previously obtained condition to the present one was also discussed. Note added in proof. Recently, the author has learned that a similar linear optical method was proposed to mea- sure the same quantities as the ones in this paper, but in a different context[19]. VI. ACKNOWLEDGMENT This work was supported by the University of Queens- land. *email:phylove00@gmail.com [1] W. Heisenberg, Z. Phys. 43, 122 (1927). [2] E. Schrödinger, Sitzunsber. Preuss. Akad. Wiss., Phys. Math. Kl. 19, 296 (1930). [3] H. R. Robertson, Phys. Rev. 46 794 (1934). [4] V. V. 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Lett. 88, 120401 (2002). [16] H. Nha and H. J. Carmichael, Phys. Rev. Lett. 93, 020401 (2004); R. Garcia-Patron, J. Fiurasek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett. 93, 130409 (2004); S. Olivares, M. G. A. Paris, Phys. Rev. A 70, 032112 (2004); R. Garcia-Patron, J. Fi- urasek, and N. J. Cerf, Phys. Rev. A 71, 022105 (2005). [17] H. Nha and J. Kim, Phys. Rev. A 75, 012326 (2007). [18] In this paper, we directly use the product form of un- certainty relations, like Eqs. (1) and (2), which broadly provides a stronger separability condition than the sum form. See also P. Hyllus and J. Eisert , New J. Phys. 8, 51 (2006). [19] R. A. Campos and C. C. Gerry, Phys. Rev. A 60, 1572 (1999). http://arxiv.org/abs/quant-ph/9711028

The Schr{\"o}dinger-Robertson inequality generally provides a stronger bound on the product of uncertainties for two noncommuting observables than the Heisenberg uncertainty relation, and as such, it can yield a stricter separability condition in conjunction with partial transposition. In this paper, using the Schr{\"o}dinger-Robertson uncertainty relation, the separability condition previously derived from the su(2) and the su(1,1) algebra is made stricter and refined to a form invariant with respect to local phase shifts. Furthermore, a linear optical scheme is proposed to test this invariant separability condition.

Entanglement condition via su(2) and su(1,1) algebra using Schrödinger-Robertson uncertainty relation Hyunchul Nha∗ ARC Center of Excellence for Quantum Computer Technology, University of Queensland, Australia School of Computational Sciences, Korea Institute for Advanced Study, Korea (Dated: August 25, 2021) The Schrödinger-Robertson inequality generally provides a stronger bound on the product of uncertainties for two noncommuting observables than the Heisenberg uncertainty relation, and as such, it can yield a stricter separability condition in conjunction with partial transposition. In this paper, using the Schrödinger-Robertson uncertainty relation, the separability condition previously derived from the su(2) and the su(1,1) algebra is made stricter and refined to a form invariant with respect to local phase shifts. Furthermore, a linear optical scheme is proposed to test this invariant separability condition. PACS numbers: 03.67.Mn, 03.65.Ud, 42.50.Dv I. INTRODUCTION When a quantum system is subject to measurements corresponding to two noncommuting observables {A, B}, the product of uncertainties in measurement outcomes, 〈(∆A)2〉〈(∆B)2〉, has a certain lower bound. The Heisen- berg uncertainty relation (HUR)[1], which is most widely used, provides the bound as 〈(∆A)2〉〈(∆B)2〉 ≥ |〈[A,B]〉|2. (1) On the other hand, the Schrödinger-Robertson rela- tion(SRR) [2, 3] in general provides a stronger bound 〈(∆A)2〉〈(∆B)2〉 ≥ |〈[A,B]〉|2 + 〈∆A∆B〉2S , (2) where the cross correlation 〈∆A∆B〉S is defined in a sym- metric form as 〈∆A∆B〉S ≡ 〈∆A∆B +∆B∆A〉. (3) The SRR can be derived from the Cauchy-Schwartz in- equality, 〈f |f〉〈g|g〉 ≥ |〈f |g〉|2, where |f〉 = ∆A|Ψ〉 and |g〉 = ∆B|Ψ〉 for a generic quantum state |Ψ〉[4]. The HUR describes a special case of the SRR under the con- dition 〈∆A∆B〉S = 0, which is of course not always met. Recently, one of the important issues in quantum in- formatics has been to obtain conditions by which one can distinguish entangled states from separable ones. Some of such entanglement criteria derived so far have relied on the bounds set by various forms of uncertainty relations[5, 6, 7, 8], and remarkably for certain cases, in explicit conjunction with partial transposition(PT)[9, 10, 11]. More precisely, separable states can represent a certain physical state even under PT[12] and all uncer- tainty relations must therefore be satisfied by separable states under PT. The uncertainty relations in combina- tion with PT can thereby provide necessary conditions for separability. For continuous variables (CVs), earlier works were focused on Gaussian entangled states[13, 14, 15], but considerable attention has also been directed to non- Gaussian entangled states[16]. Most of all, the separa- bility conditions applicable to non-Gaussian entangled states have recently emerged[8, 9, 10, 11], and in partic- ular, Refs. [8, 10, 11] employed the su(2) and the su(1,1) algebra to derive such entanglement criteria. Using the HUR along with those two algebras, Nha and Kim have particularly derived the optimal separability condition among a certain class of inequalities[11]. This condi- tion has also been proposed to detect multipartite en- tanglement of photonic W states and shown to be robust against the detector inefficiency[17]. In this paper, it is our aim to refine the separability condition in Refs. [10, 11] by employing the SRR instead of the HUR. By doing this, we obtain a stricter separa- bility condition given by a form invariant with respect to local phase shifts. This invariance is a very adequate attribute as entanglement condition, for entanglement property must be invariant under any local unitary oper- ations. Furthermore, we propose how to experimentally test this invariant condition using linear optics and also discuss the practical connection of the previous condition in [10, 11] to the present one. II. SEPARABILITY CONDITION First, we briefly introduce how to derive the separabil- ity condition via the uncertainty relations in the su(2) and the su(1,1) algebra[11]. The su(2) algebra deals with the angular momentum operators Jx, Jy and Jz, which obey the commutation relations [Ji, Jj ] = iǫijkJk (i, j, k = x, y, z). This algebra can be represented by two http://arxiv.org/abs/0704.1939v2 bosonic operators a and b, as a†b+ ab† a†b− ab† a†a− b†b . (4) On the other hand, the operators Kx,Ky and Kz in the su(1,1) algebra can be represented by a†b† + ab a†b† − ab a†a+ b†b+ 1 , (5) which results in the commutation relations, [Kx,Ky] = −iKz, [Ky,Kz] = iKx, and [Kz,Kx] = iKy, different in sign from those of the su(2) algebra. Specifically, the commutator [Kx,Ky] = −iKz in the su(1,1) algebra gives the uncertainty relation via the HUR as 〈(∆Kx) 2〉〈(∆Ky) |〈Kz〉| 2, (6) which must be satisfied by any quantum states. Most importantly, the inequality (6) must be satisfied under PT by every separable state, since it can still describe a certain physical state[12]. That is, one obtains the separability condition as 〈(∆Kx) 2〉PT〈(∆Ky) 2〉PT ≥ |〈Kz〉| PT, (7) where the subscript PT means that the quantum average is calculated after taking partial transposition. Using a general relation 〈a†manb†pbq〉ρPT = 〈a †manb†qbp〉ρ (8) between the quantum average for the partially trans- posed density operator ρPT and that for the original den- sity operator ρ[11], the inequality (7) can be recast to give the separability condition expressed as + 〈(∆Jx) + 〈(∆Jy) [1 + 〈N+〉] , (9) where N+ = a †a + b†b is the total excitation number. Note that the inequality (9) is the optimal condition de- rived in [11], where the HUR was employed in a sum form to obtain a class of separability conditions[18]. III. STRICTER SEPARABILITY CONDITION In this section, let us now start from the SRR for the commutator [Kx,Ky] = −iKz, i.e., 〈(∆Kx) 2〉〈(∆Ky) |〈Kz〉| 2 + 〈∆Kx∆Ky〉 S , (10) instead of the HUR, then follow the same steps as below Eq. (6). Using the relation 〈∆Kx∆Ky〉S,PT = 〈∆Jx∆Jy〉S (11) via Eq. (8), we obtain a separability condition stricter than the one in (9) as + 〈(∆Jx) + 〈(∆Jy) [1 + 〈N+〉] + 〈∆Jx∆Jy〉 S .(12) Compared with the inequality (9), the new inequal- ity (12) prodvides a stronger condition for separabil- ity as long as the off-diagonal covariance 〈∆Jx∆Jy〉S is nonzero. As an example, consider the two-photon en- tangled states of the type |Ψ〉 = cos θ|2, 0〉+ i sin θ|0, 2〉. All these states satisfy the inequality (9), but violate the stricter one in (12), regardless of the parameter θ. There- fore, only the inequality (12) can detect entanglement for those two-photon states. We next show that the inequality (12) is invariant with respect to local phase shifts. Let us consider a 2 × 2 covariance matrix C of which elements are defined as Cij ≡ 〈∆Ji∆Jj +∆Jj∆Ji〉, (13) where {i, j} = {x, y}. The inequality (12) is then ex- pressed as Det{C}+ Tr{C} ≥ 2 + 2〈N+〉 , (14) where Det{} and Tr{} denote the determinant and the trace of a matrix. If one takes a local phase shift for mode b as b′ = be−iφ, the su(2) operators Jx and Jy are transformed into cosφ sinφ − sinφ cosφ . (15) The determinant and the trace of a matrix are unchanged under rotation, and the total photon number 〈N+〉 is also preserved through passive optical elements. The in- equality (14) is therefore invariant with respect to local phase shifts. This is an attribute very adequate as entan- glement condition, for entanglement should be invariant under local unitary operations. Note that a phase shift is the only local unitary operation that preserves the total photon number. IV. MEASUREMENT SCHEME We now discuss how the separability condition (12) can be tested in experiment. In Ref.[11], a linear optical scheme was proposed to measure the observables Jx, Jy and 〈N+〉 for the inequality (9), as depicted in Fig. 1. The mode b first undergoes a phase shift by φ and the FIG. 1: Experimental scheme for measuring the quantities necessary to test the inequality (12). All the quantum av- erages in (12) can be measured by detecting the photon number difference at the output, N{−,φ} ≡ c c − d −iφ + ab†eiφ, with four different phase shifts, φ = 0, π and −π . (See the main text.) BS: 50:50 beam-splitter, PS: phase-shifter, and PD: photo detector. two modes a and b are then injected to a 50:50 beam splitter. The modes c and d at the output are given by c = 1√ (a+ be−iφ) and d = 1√ (−a+ be−iφ), respectively. One needs to measure the photon number difference at the output, i.e., N{−,φ} ≡ c †c− d†d = a†be−iφ + ab†eiφ, (16) which becomes 2Jx (2Jy) for φ = 0 (φ = ). (See Eq. (4).) The total photon number 〈N+〉 is simply given by the sum, c†c+ d†d, at the output. In the present inequality (12), in addition to Jx, Jy and 〈N+〉, one also needs to measure the off-diagonal covari- ance 〈∆Jx∆Jy〉S . Note that 〈∆Jx∆Jy〉S = 〈JxJy + JyJx〉 − 〈Jx〉〈Jy〉, where JxJy + JyJx = a†2b2 − a2b†2 {−,φ=π {−,φ=−π . (17) Thus, by choosing two different phase shifts φ = π φ = −π in Fig. 1, the quantum average 〈JxJy + JyJx〉 can be measured in two pieces as shown in Eq. (17). In summary, the single experimental setup in Fig. 1 can be used to measure all the quantities necessary to test the inequality (12). Finally, we discuss how the inequality (9) can be re- garded as ”equivalent” to the stricter inequality (12). Us- ing the relation in Eq. (15) implemented by a local phase shift, one has the covariance in the rotated frame as 〈∆J ′x∆J y〉S = sin 2φ 〈(∆Jy) 2〉 − 〈(∆Jx) +cos 2φ〈∆Jx∆Jy〉S . (18) Thus, by choosing the phase shift as tan 2φ = 2〈∆Jx∆Jy〉S 〈(∆Jx)2〉 − 〈(∆Jy)2〉 , (19) the covariance in the rotated frame can be made vanish. In this situation, the inequality (12) is reduced to the inequality (9). In other words, as long as one is allowed to perform a local phase shift, which does not alter the entanglement property at all, the two inequalities can be interpreted as equivalently useful. However, this relies on the capability of measuring all the covariances and of performing a phase shift very accurately required by Eq. (19). It is then of no practical advantage to adhere to the inequality (9): One can simply test the inequal- ity (12) if one is able to measure the off-diagonal covari- ance 〈∆Jx∆Jy〉S in addition. V. SUMMARY In this paper, we have derived a stricter separability condition via the su(2) and the su(1,1) algebra using the Schrödinger-Robertson inequality instead of the Heisen- berg uncertainty relation. It has been shown that this refined condition is expressed in a form invariant with respect to local phase shifts. A linear optical setup has been proposed to test the invariant separability condition and the practical connection of the previously obtained condition to the present one was also discussed. Note added in proof. Recently, the author has learned that a similar linear optical method was proposed to mea- sure the same quantities as the ones in this paper, but in a different context[19]. VI. ACKNOWLEDGMENT This work was supported by the University of Queens- land. *email:phylove00@gmail.com [1] W. Heisenberg, Z. Phys. 43, 122 (1927). [2] E. Schrödinger, Sitzunsber. Preuss. Akad. Wiss., Phys. Math. Kl. 19, 296 (1930). [3] H. R. Robertson, Phys. Rev. 46 794 (1934). [4] V. V. Dodonov, E. V. Kurmyshev, and V. I. Man’ko, Phys. Lett. 79A, 150 (1980);B. Nagel, eprint arXiv:quant-ph/9711028. [5] H. F. Hofmann and S. Takeuchi, Phys. Rev. A 68, 032103 (2003); H. F. Hofmann, Phys. Rev. A 68, 034307 (2003). [6] O. Gühne,Phys. Rev. Lett. 92, 117903 (2004); G. Toth and O. Gühne,Phys. Rev. A 72, 022340 (2005). [7] M. G. Raymer, A. C. Funk, B. C. Sanders, and H. de Guise, Phys. Rev. A 67, 052104 (2003). [8] M. Hillery and M. Zubairy, Phys. Rev. Lett. 96, 050503 (2006); M. Hillery and M. Zubairy, Phys. Rev. A 74, 032333 (2006). [9] E. Shchukin and W. Vogel, Phys. Rev. Lett. 95, 230502 (2005). [10] G. S. Agarwal and A. Biswas, New J. Phys. 7, 211 (2005). [11] H. Nha and J. Kim, Phys. Rev. A 74, 012317 (2006). [12] A. Peres, Phys. Rev. Lett. 77, 1413 (1996). [13] L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722 (2000). [14] R. Simon, Phys. Rev. Lett. 84, 2726 (2000). [15] S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, Phys. Rev. Lett. 88, 120401 (2002). [16] H. Nha and H. J. Carmichael, Phys. Rev. Lett. 93, 020401 (2004); R. Garcia-Patron, J. Fiurasek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett. 93, 130409 (2004); S. Olivares, M. G. A. Paris, Phys. Rev. A 70, 032112 (2004); R. Garcia-Patron, J. Fi- urasek, and N. J. Cerf, Phys. Rev. A 71, 022105 (2005). [17] H. Nha and J. Kim, Phys. Rev. A 75, 012326 (2007). [18] In this paper, we directly use the product form of un- certainty relations, like Eqs. (1) and (2), which broadly provides a stronger separability condition than the sum form. See also P. Hyllus and J. Eisert , New J. Phys. 8, 51 (2006). [19] R. A. Campos and C. C. Gerry, Phys. Rev. A 60, 1572 (1999). http://arxiv.org/abs/quant-ph/9711028

Additional Explanatory Notes on the Analytic Proof of the Finite Generation of the Canonical Ring Yum-Tong Siu 1 Introduction. This set of notes is put together to provide some addi- tional explanatory material on the analytic proof of the finite generation of the canonical ring for use by the participants in the Workshop on Minimal and Canonical Models in Algebraic Geometry in April 16–20, 2007 at the Mathematical Sciences Research Institute at Berkeley, California. In late March 2007 before the Workshop Mihai Paun of Strasbourg came to Harvard for two weeks to pose to me some questions which he and other people have about the analytic proof of the finite generation of the canoni- cal ring which I posted in October 2006 on the arXiv.org server [Siu 2006]. Besides orally answering his questions I also wrote up some notes for him to give him more precise details. This set of notes is compiled by putting together the notes which I wrote up to answer his questions. This set of notes consists of two parts. Part I is about how to apply the general nonvanishing theorem to prove the precise achievement of stable vanishing order in codimension one. Part II gives the argument for the precise achievement of the stable vanishing order for higher codimension. For both Part I and Part II the new powerful tool in the analytic proof is the use of the second case of the dichotomy of the modified restriction of the curvature current of the metric of minimum singularity for the canonical line bundle. Let V = Vk be an irreducible subvariety of codimension k in the compact complex algebraic manifold X of finite type whose canonical ring is to be proved to be finitely generated such that V is an embedded branch of the stable base point set. A modified restriction ΘV to V of the curvature current of the canonical line bundle is given as follows. For some ην ≥ 0 and Vν irreducible of codimension 1 in Vν−1 (k < ν < 0) with V0 = X and η0 = 0, let Θ0 be the curvature current ΘKX on X of of the metric of minimum singularity for the canonical line bundle KX and inductively Θν = (Θν−1 − ην−1Vν−1) |Vν . 1Partially supported by a grant from the National Science Foundation. http://arxiv.org/abs/0704.1940v1 Then Θk is the modified restriction ΘV to V of the curvature current of KX for the sequence Vk−1 ⊂ · · · ⊂ V1 ⊂ X of nested subvarieties. For the canonical decomposition γj [Yj] +R of ΘV on V there are two cases. The first case is either J = ∞ or R 6= 0 and the second case is J < ∞ and R = 0, where Yj is a subvariety of codimension one in V and R is a closed positive current on V whose Lelong number is zero outside a countable union of subvarieties of codimension at least two in V . For Part I on how to apply the general nonvanishing theorem to prove the precise achievement of stable vanishing order in codimension one, the result of the use of this new powerful tool is that the second case of the dichotomy must always eventually occur at some positive dimensional V , because if the first case occurs all the time down to dimension zero then there is some improvement in the stable vanishing at some point of V in the multi-directions defined by the sequence of nested subvarieties. The second case of the dichotomy gives a explicitly constructed section on V (unique up to a nonzero constant factor) which belongs to the multiplier ideal sheaf corresponding to ΘV . In Part I of these additional explanatory notes, in order to make the arguments of the general nonvanishing theorem more transparent, I treat separately the two cases of the dichotomy. We use the process of using the techniques for the Fuijta conjedture by constructing singular metrics successively to cut down on the dimension of (the projection of) the zero-set of the multiplier ideal sheaf until we end up with the inevitable second case of the dichotomy and in the second case of the dichotomy we use the sections explicitly constructed from the special form of the canonical decomposition of the modified restriction of the curvature current. In extending the explicitly constructed sections all the way back to the ambient manifoldX we introduce the technique of constrained minimum center of log canonical singularity. In Part II of these additional explanatory notes about the argument for the precise achievement of the stable vanishing order for higher codimension, we give its illustration in the low dimensional cases of complex surfaces and complex threefolds. The illustration in the low dimensional cases avoids the encumberment of complicated notations in the case of general dimension and at the same time contains the essence of the argument of the general case. For the case of a threefoldX in Part II, the key case of higher codimension is that of an irreducible curve C which is an embedded curve in the stable base point set. The important ingredient of using the second case of the dichotomy is the following. For simplicity assume that there is no codimension-one base point set. Suppose s1, · · · , sk are pluricanonical sections on X such that C is cut out by the family Sσ of surfaces defined by pluricanonical sections sσ = j=1 σjsj parametrized by σ = (σ1, · · · , σk) ∈ Ck − {0}. Suppose s is a pluricanonical section whose restriction to some Sσ achieves the restriction of the stable vanishing order ασ on Sσ at some point Pσ of C. Then, after taking away the restriction of the stable vanishing order ασ of s|Sσ across C, the resulting section (sC,σ) ασ on Sσ (where sC,σ is the canonical section of C in Sσ), when restricted to C, is the section (up to a nonzero constant factor) explicitly constructed from the second case of the dichotomy for C in Sσ. This means that at every point of C outside the finite zero-set Zσ of the resulting section (sC,σ) ασ on C, the section (sC,σ) ασ achieves the restriction of the stable vanishing order ασ on Sσ. Suppose for every parameter σ there is a parameter τ and some point Pσ,τ in C such that the restriction of the pluricanonical section sτ to Sσ achieves the restriction of the stable vanishing order on Sσ at the point Pσ,τ of C. Let Z be the finite subset of C such that as a point P varies in C outside Z the Artinian subscheme defined by s1, · · · , sk in the normal direction of C at P does not change. In other words, every point Q ∈ C − Z admits an open neighborhood U in C − Z and a holomorphic family of local nonsingular complex surfaces MP in X parametrized by P ∈ U such that each MP intersecting C normally at P and dimC OMP OMP (sj|MP ) is independent of P ∈ U . From the above discussion about s being the explicitly constructed section and about the achievement of the restriction to Sσ of stable vanishing order at points of C outside Zσ it follows that the stable vanishing order is achieved at points of C − Z. This use of the sections, explicitly constructed from the curvature current in the second case of the dichotomy, in order to come up with a finite sub- set Z of C can be regarded as the higher-codimensional way of constructing sections of vector bundles explicitly from the second case of the dichotomy of the modified restriction of the curvature current. Note that the codimension one case for a surface is invoked for us to get the rationality of the restric- tion of the stable vanishing order to a surface and its achievement on the surface at some point of C. This part of the hard work already done for the codimension one case is used here. This vector bundle section on C explicitly constructed from the second case of the dichotomy of the modified restric- tion of the curvature current is obtained, so to speak, by piecing together the line bundle sections on C explicitly constructed by using the surface slices S from the second case of the dichotomy of the modified restriction to the sur- face slice S of the curvature current. This is the reason why the usual worry about interminable blowups to transform into hypersurfaces newly appearing higher-codimensional base point sets is not a problem here. Before presenting Part I and Part II, we would like to make two remarks. The first remark is about the finite generation of the canonical ring without the assumption of finite type. In analysis, when KX is not big, adding a line bundle E to KX to have a big bundle KX + E and working with KX + E instead of KX would only mean a standard modification of the argument for the finite generation of the canonical ring for the case of general type. (This is actually used in the form of KX − γY , with γ being the stable vanishing order across the hypersurface Y , in the proof of the finite generation of the canonical ring for the case of general type.) However, getting rid of E afterwards needs the justification of a rather involved limiting process. Such a limiting process is of the same nature as the limiting process which is used to extend the proof of the deformational invariance of the plurigenera for the case of general type [Siu 1998] to the general algebraic case [Siu 2002]. For the finite generation of the canonical ring, when KX is big, we intro- duce the curvature ΘKX of the metric of minimum singularity constructed from pluricanonical sections. When KX is not big, we use an ample line bundle A on X and the limit curvature current Θ̃KX := lim ΘKX+ 1mA to replace ΘKX . All the diophantine approximation arguments deal with the modified restriction of the canonical decomposition of Θ̃KX . This makes it possible to go ahead with the limiting process. In the case of the deformational invariance of the plurigenera the limiting process is handled by abandoning the metric of minimum singularity and replacing it by another metric with maximum allowable singularity which is still good enough for the finiteness of the L2 norm of the section to be extended, because there is a need to bound the dimension of the space of L2 holomorphic m-canonical sections with the bound independent of m or at least growing more slowly than m. It is not yet clear what the analogous procedure is for the problem of the finite generation of the canonical ring. The second remark is to answer the question why one does not simply take a pluricanonical section and get the finite generation of the canonical ring of its divisor and then use extension of pluricanonical sections from that divisor to the entire ambient space. This procedure is explained in §8 of the posted notes of my analytic proof [Siu 2006]. The extension technique for pluricanonical sections in the problem of deformational invariance of the plurigenera was developed with that particular application in mind [Siu 1998, Siu 2006]. Though the pluricanonical extension technique for deformational invariance of the plurigenera is for extension from the fiber at a point to the open unit disk, in analysis it does not matter whether one deals with a compact manifold or a Stein manifold if the Stein manifold is a Zariski open subset and one keeps track of the L2 bounds. However, the situation of the singularities in the pluricanonical divisor poses technical problems in analysis. In my article in the proceedings of a 2001 conference in Hanoi [Siu 2004] I addressed the problem of pluricanonical extension from a singular divisor. There are technical problems concerning finite L2 bounds. The difficulty in handling such technical problems concern- ing finite L2 bounds is the same as getting the comparability of the metric of minimum singularity for KX and that from an appropriate truncated fi- nite sum in the application of Skoda’s theorem of ideal generation [Skoda 1972]. Moreover, the Ohsawa-Takegoshi-type extension theorem used in the pluricanonical extension is closely related to Skoda’s theorem of ideal gen- eration (see, for example, Ohsawa’s article in the Festschrift for Grauert’s 70th birthday [Ohsawa 2002]) and the constant in Skoda’s theorem is more precise. That is why we use the approach by Skoda’s theorem instead of the essentially equivalent approach of pluricanonical extension. It seems that at this point any approach to the problem of finite gener- ation of the canonical ring has to rely, as starting point, on the technique of pluricanonical extension. Any difference in the different approaches are more a matter of technical handling of the singularities of the pluricanonical divisor. In the definition of a stable metric in [Siu 2006, (6.1)] there are some typographical errors and inaccuracies. The correct definition is as follows. A metric e−κ of a line bundle L on a compact Kähler manifold M is said to be stable if κ is locally plurisubharmonic and there exists some ε > 0 with the following property. If U is an open neighborhood of a point P ∈ M , and ϕ and ψ are plurisubharmonic functions on U such that the total mass, with respect to the Kähler form of M , of the sum of the two closed positive (1, 1)-currents Θϕ and Θψ is less than ǫ and if κ − ϕ is plurisubharmonic, then there exists an open neighborhood U ′ of P inM such that the multiplier ideal sheaf Iκ+ψ of the metric e−κ−ψ on U ′ is equal to the multiplier ideal sheaf Iκ−ϕ of the metric e−κ+ϕ on U ′. PART I How to Apply Nonvanishing Theorem to Precisely Achieve Stable Vanishing Order in Codimension One The proof of the precise achievement of stable vanishing order in codi- mension one has the following ingredients. (a) The techniques for the Fujita conjecture which consists of (i) constructing singular metrics with curvature current of strict pos- itive lower bound whose multiplier ideal sheaf has high vanishing order at a prescribed point, (ii) blowing up the zero-set of the multiplier ideal sheaf of the new singular metric, (iii) repeating the procedure so that the zero-set of the multiplier ideal sheaf of the new singular metric on a blow-up space is projected down to lower and lower dimension in the original manifold, until one gets to the case of a singular point in the original manifold, (iv) extending a section defined on the single point to over all of the original manifold by the vanishing theorem of Kawamata- Viehweg-Nadel. Because of the need of adding one canonical line bundle in the ap- plication of the vanishing theorem of Kawamata-Viehweg-Nadel, the singular metric is for a multiple of the line bundle in question minus the canonical line bundle. (b) Introducing a dichotomy depending on the canonical decomposition of some curvature current so that (i) in the first case of the dichotomy the technique of the Fujita con- jecture for the construction of singular metric can be carried out, (ii) in the second case of the dichotomy a section can be explicitly constructed which can be extended to all of the original manifold. Once we get to the second case of the dichotomy there is no need to use the technique of the Fujita conjecture to construct any more singular metric. The process is stopped and complete. For the precise achievement of stable vanishing order, the second case of the dichotomy must arise, otherwise the stable vanishing order can be improved, which contradicts the definition of a stable vanishing order. The explicitly constructed section is rather rigid in the sense that there is no choice and the section comes from the curvature current in a rather unique way. This uniqueness and rigidity of the explicitly constructed section will be a key ingredient in the proof of the precise achievement of the stable vanishing order in higher codimension. (c) The new technique of constrained minimum center of log canonical sin- gularity, whose motivation and precise application will be explained below. This new technique is needed, because of the undesirable addi- tional vanishing order in the process of constructing singular metrics, which will be explained in detail below. (d) Use of diophantine approximation to handle irrational coefficients. Since the use of diophantine approximation has already been described in detail in the posted notes of my proof of the finite generation of the canonical ring [Siu 2006], we will not discuss diophantine approxima- tion in these additional explanatory notes and we just assume that the relevant coefficients are known to be rational numbers. As the first step we will consider the first case of the dichotomy and describe how to construct an appropriate singular metric. Then we will give the motivation for constrained minimum center of log canonical singularity by reviewing the goal and the strategy of the standard technique for the Fujita conjecture. Finally we consider the second case of the dichotomy where a rather rigid section is explicitly constructed. Proposition(Construction of Metric with Multiplier Ideal Sheaf Vanishing to High Order at a Prescribed Point for the First Case of the Dichotomy of the Curvature Current). Let M be a compact complex projective algebraic manifold of complex dimension n. Let L be a holomorphic line bundle on M with a (possibly singular) metric e−ϕ along its fibers whose curvature current Θϕ is a closed positive (1, 1)-current. Let τj [Vj] +R be the canonical decomposition of Θϕ, where J ∈ N∪{0,∞} and the Lelong number of R is zero outside a countable union Z of subvarieties of codi- mension at least two in M and Vj is an irreducible hypersurface in M and τj > 0. Assume that either J = ∞ or R 6= 0 (that is, one is in the first case of the dichotomy for the curvature current Θϕ). Assume that for some positive integer p0 there is a (possibly singular) metric e −χ along the fibers of p0L−KM which is stable and whose curvature current Θχ is a closed positive (1, 1)-current which dominates some strictly positive smooth (1, 1)-form on M . Let P0 be a point of M such that the Lelong number of Θϕ is zero at P0. Let q ∈ N. Then for some sufficiently divisible m ∈ N the line bundle (m+ p0)L − KM admits a metric e−χ̃ whose curvature current dominates some strictly positive smooth (1, 1)-form on M such that its multiplier ideal sheaf Iχ̃ is contained in the maximum ideal of M at P0 raised to the q-th power. Moreover, if M is a hypersurface in some compact complex algebraic manifold X of general type so that L is the restriction of some line bundle L̃ on X and KM = KX +M and the metric e −χ is defined by a convergent infinite sum of multi-valued holomorphic sections of p0L̃−KX −M over X , then the metric e−χ̃ can be chosen to be defined also by a convergent infinite sum of multi-valued holomorphic sections of (p + p0) L̃−KX −M over X . Proof. The idea of the proof is to use the techniques for Fujita’s conjecture (see, for example, [Angehrn-Siu 1995]). Slicing by an Ample Divisor. Let A be a very ample line bundle over M such that A−KM is ample. Let hA be a smooth metric of A whose curvature form ωA is positive on M . We assume that A is chosen to be sufficiently ample so that for each point P ∈M the proper transform of A in the manifold obtained from M by blowing up P is still very ample. This technical assumption will enable us to choose a generic element of Γ (M, A) vanishing at P0 which is not a zero-divisor of a prescribed coherent ideal sheaf. Let p and k be positive integers and we will impose more conditions on p and k later. Let s1 be a generic element of Γ (M, A) vanishing at P0 so that the short exact sequence 0 → Ipϕ+χ ((p+ p0)L−KM + kA) θs1−→ Ipϕ+χ ((p+ p0)L−KM + (k + 1)A) → (Ipϕ+χ /s1Ipϕ+χ ) ((p+ p0)L−KM + (k + 1)A) → 0 is exact, where θs1 is defined by multiplication by s1. Let M1 be the zero-set of s1 and OM1 = (OM /s1OM ) |M1, which we can assume to be regular with ideal sheaf equal to s1OM because s1 is generic element of Γ (M,A) vanishing at P0. By choosing s1 generically we can also assume that I(pϕ+χ)|M1 = Ipϕ+χ /s1Ipϕ+χ and I(pϕ)|M1 = Ipϕ /s1Ipϕ and Iχ|M1 = Iχ /s1Iχ . We use χ (·, ·) to denote the arithmetic genus which means χ (·, ·) = (−1)ν dimCHν (·, ·) . From the long cohomology exact sequence of the above short exact sequence we obtain χ (M, Ipϕ+χ ((p+ p0)L−KM + (k + 1)A)) = χ (M, Ipϕ+χ ((p+ p0)L−KM + kA))+χ M1, I(pϕ+χ)|M1 ((p+ p0)L−KM + (k + 1)A) |M1 Since A − KM is ample and 2A − KM1 = A − KM is also ample, when we assume k ≥ 1, by the theorem of Kawamata-Viehweg-Nadel Hν (M, Ipϕ+χ ((p+ p0)L−KM + kA)) = 0 for ν ≥ 1, M1, I(pϕ+χ)|M1 (((p+ p0)L−KM + (k + 1)A) |M1) = 0 for ν ≥ 1 so that dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + (k + 1)A)) = dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + kA)) + dimC Γ M1, I(pϕ+χ)|M1 (((p+ p0)L−KM + (k + 1)A) |M1) ≥ dimC Γ M1, I(pϕ+χ)|M1 (((p+ p0)L−KM + (k + 1)A) |M1) Slicing by Ample Divisors Down to a Curve. Instead of one single element s ∈ Γ (M,A), we can choose generically s1, · · · , sn−1 ∈ Γ (M, A) all vanishing at P0 so that inductively for 1 ≤ ν ≤ n−1 the common zero-set Mν of s1, · · · , sν with the structure sheaf OMν := is regular and we end up with the inequality dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + (k + n− 1)A)) ≥ dimC Γ Mn−1, I(pϕ+χ)|Mn−1 ((p+ p0)L−KM + (k + n− 1)A) |Mn−1 Since Mn−1 is a curve, all coherent ideal sheaves on it are principal and are locally free and they come from holomorphic line bundles. We would like to remark also that this particular step of slicing by n− 1 ample divisors to get down to a curve roughly corresponds to the step in Shokurov’s proof of his non-vanishing theorem [Shokurov 1985] where he takes the product of his numerically effective divisor in his n-dimensional manifold with the (n−1)-th power of a numerically effective big line bundle. Application of the Theorem of Riemann-Roch to a Curve and Comparing Contributions from the Curvature Current and the Multiplier Ideal Sheaves. Let b be the Chern class of the line bundle onMn−1 defined by the multiplier ideal sheaf Iχ|Mn−1 of the restriction to Mn−1 of the metric e −χ. Let c be the nonnegative number R ∧ (ωA)n−1 . (†) dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + (k + n− 1)A)) ≥ dimC Γ Mn−1, I(pϕ+χ|Mn−1) ((p+ p0)L−KM + (k + n− 1)A) |Mn−1 ≥ 1− genus (Mn−1) + b+ (k + n− 1)An−1Mn−1 (pτj − ⌊pτj⌋)Vj · An−1 + p where the last identity is from the theorem of Riemann-Roch applied to the regular curve Mn−1 and the locally free sheaf I(pϕ+χ|Mn−1) ((p+ p0)L−KM + (k + n− 1)A) |Mn−1 on Mn−1. From the assumption that J = ∞ or R 6= 0, we conclude that the right-hand side of (†) goes to ∞ as p goes to ∞ through an appropriate sequence, where for the case of J = ∞ and R = 0 a diophantine argument has to be used whereas for the case R 6= 0 we simply need to use c > 0. Construction of Sections with Extra Vanishing Order from Dimension Count- ing and Construction of Metrics by Canceling Contributions from Ample Di- visors by Using the General Type Property. For any ℓ ∈ N the number of terms in a polynomial of degree ℓ in d variables is . Take a positive integer N and we will impose more condition on N later. By the behavior of the right-hand side of (†) as p→ ∞, there exists p ∈ Z such that dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + (k + n− 1)A)) ≥ 1+ n+N (k + n− 1) q and we can find some non identically zero element s of Γ (M, Ipϕ+χ ((p + p0)L−KM + (k + n− 1)A)) which vanishes to order at least N (k + n− 1) q at P0 so that s N(k+n−1) is a multi-valued holomorphic section of the Q-line-bundle p N(k+n−1) A over M which vanishes to order at least q at P0. We assume that N is chosen so large that the curvature current Θχ dominates ωA. Let p̂ to be the round-up of p N(k+n−1) and δp = p̂− pN(k+n−1) . We introduce the metric e−χ̃ := e−χ−δpϕ N |s| of (p+ p0)L − KM so that the multiplier ideal of Iχ̃ at P0 is contained in (mM,P0) . Q.E.D. Remark on Application of the Proposition on Construction of Metric for the First Case of the Dichotomy. The above application is applied in the following manner. Let X be the the compact complex algebraic manifold of finite type whose canonical ring is to be proved to be finitely generated. Let Y be a hypersurface in X across which the stable vanishing order is γ > 0. Let P0 be a generic point of Y . We start out withM = Y and L = KX−γY . After we get the new metric e−χ̃, we use an interpolation between e−χ̃ and e−pϕ−χ and a slight modification to get to a minimum center of log canonical singularity which, after blow-up, projects down to a proper subvariety of M containing P0. Then we replace X by its blow-up and replace M by the new minimum center of log canonical singularity and replace L by its pullback to the blowup of X . We continue doing this until we inevitably come to the second case of dichotomy eventually as explained in the Introduction. Remark on the Second Case of the Dichotomy. Suppose τj [Vj] with J < ∞. Then we can explicitly construct a section s0 of pL over M . The reason why a minimal center of log canonical singularity is used in the techniques for the Fujita conjecture is to make sure that when we take the subspace defined by the multiplier ideal sheaf, the subspace has a reduced structure. In our case we have to introduce the notion of constrained minimal center of log canonical singularity so that the center is not completely contained in the zero-set of s0. For that we have to pay the price that the subspace defined by the multiplier ideal sheaf may have an unreduced structure, but the set where nonzero nilpotent elements of its structure sheaf occurs is contained in the zero-set of s0. By raising s0 to a sufficiently high power, we can handle the unreduced structure and get the extension of a sufficiently high power of s0 to the ambient manifoldX by using the vanishing theorem from the metric of pL−KM . We are going to elaborate on this by reviewing the goal of the techniques for the Fujita conjecture and the use of minimal center of log canonical singularity and also how we are naturally and by necessity led to the concept of constrained minimal center of log canonical singularity. Main Idea of the Techniques for the Fujita Conjecture. For the discussion about the main idea of the technique for the Fujita conjecture, we forget the above meaning of X and L and for the time being use the symbols X and L in another context. We will so indicate when we later return to the above meaning of X and L. The goal of the technique for the Fujita conjecture is to find global sections to generate some positive power mL of a line bundle L over a compact complex algebraic manifold X . For a more general setting, the goal is to find global sections to globally generate J (mL) over X for some given coherent ideal sheaf J at points outside some given subvariety Z of X . The problem of proving the finite generation of the canonical ring by verifying the precise achievement of stable vanishing orders actually involves this more general setting. There the even more complicated situation of supremum norm is used instead of just the L2 norm. However, for the sake of simplicity in our discussion of the main idea of the techniques of the Fujita conjecture, we stick with the simpler goal of find global sections to generate some positive power mL of a line bundle L over a compact complex algebraic manifold X . The main idea of the technique is to focus on the subvariety where the global generation fails. We take a basis of s1, · · · , sk ∈ Γ (X,mL) and let Y be their common zero-set so that global generation precisely fails at points of Y . The main idea of the technique is simply to focus on Y if Y is nonempty. We seek elements of Γ (Y,mL|Y ) which are not identically zero and then extend them to elements of Γ (X,mL), which would then contradict the def- inition of Y . Hopefully the extension of elements of Γ (Y,mL|Y ) to elements of Γ (X,mL) could be done by the vanishing of some appropriate first co- homology group. Usually this first cohomology comes from the vanishing theorem of Kawamata-Viehweg-Nadel. We seek a metric e−ϕ of mL−KX so (i) the zero-set of its multiplier ideal sheaf Iϕ is Y , and (ii) the curvature current of e−ϕ dominates some strictly positive smooth (1, 1)-form on X . In such a case we have 1 (X, Iϕ (mL)) = 0 from the vanishing theorem of Kawamata-Viehweg-Nadel and the map Γ (X,mL) → Γ (Y, (OX /Iϕ ) (mL)) is surjective. The next step is to come up with some element of Γ (Y, (OX /Iϕ ) (mL)) which induces a non identically zero element of Γ (Y,mL|Y ). It is at this point that the question of a possibly unreduced complex structure OX /Iϕ arises. It means that the structure sheaf OX /Iϕ for Y may have nonzero nilpotent elements. This is the case when Iϕ is a proper subsheaf of the full ideal sheaf IY of Y and is not equal to IY . An element of IY which is not in Iϕ would yield a nonzero nilpotent element of the structure sheaf OX /Iϕ for Y . When we have an unreduced structure OX /Iϕ for Y , it is more difficult to produce some element of Γ (Y, (OX /Iϕ ) (mL)) which induces a non identically zero element of Γ (Y,mL|Y ). To handle the problem of unreduced structure sheaf, the technique is to use minimum cen- ters of log canonical singularity whose role we are going to explain. Minimum Center of Log Canonical Singularity. The idea is to seek a metric e−ψ for mL −K which is less singular than e−ϕ so that the multiplier ideal sheaf Iψ of e−ψ contains the multiplier ideal sheaf Iϕ of e−ϕ. This procedure usually involves the interpolation of two metrics and a slight modification of the result of the interpolation. We seek to make the metric of e−ψ for mL −K to be as least singular as possible, with just enough singularity to make the multiplier ideal sheaf Iψ of e−ψ not equal to OX . Let Y ′ be the support of OX /Iψ . This kind of least or minimum singularity for the choice of e−ψ gives us a reduced complex structure OX /Iψ for Y ′. The reduced complex subspace (Y ′,OX /Iψ ) is called a minimum center of log canonical singularity. (Usually for this technique of minimum center of log canonical singularity one requires, in addition, that the proper transform of Y ′ in some appropriate blow-up X̃ of X to be a nonsingular hypersurface in X̃ .) Now one replaces Y by Y ′ and uses the vanishing of H1 (X, Iψ (mL)) = 0 and the surjectivity of the map Γ (X,mL) → Γ (Y ′, (OX /Iψ ) (mL)) to reduce the problem to the construction of nonzero element of Γ (Y ′, (OX /Iψ ) (mL)) = Γ (Y ′, mL|Y ′) . Constrained Minimum Center of Log Canonical Singularity. For our case at hand for the finite generation of the canonical ring one modification has to be added in the application of the technique of minimum center of log canonical singularity. This modification necessitates the introduction of a new concept which we give the name constrained minimum center of log canonical singularity just to make it easier to refer to. Let us now describe the situation. In the second case of the dichotomy of the curvature current, there is some non identically zero element s0 of Γ (Y,mL|Y ) which is explicitly constructed from the canonical decomposition on Y of a modified restriction of the curvature current. The section s0 may have a nonempty zero-set W in Y . If we just use the technique of minimum center of log canonical singularity without any modification, we may end up with a minimum center of log canonical sin- gularity Y ′ which is completely contained inside the zero-set W of s0. In such a case the extension of s0|Y ′ is useless, because the restriction s0|Y ′ of s0 to Y ′ is identically zero on Y ′. So we need to introduce a modification to the technique of minimum center of log canonical singularity. In the proce- dure of using a metric e−ψ with least singularity to get the minimum center of log canonical singularity, we introduce the additional condition that the support of OX /Iψ is not contained entirely in the zero-set W of s0. With this additional condition we can no longer require that the structure sheaf OX /Iψ of Y ′ is reduced, but we can require that the set E of points where the structure sheaf OX /Iψ of Y ′ fails to be reduced is entirely contained in W . We call Y ′, which is obtained from this procedure of the additional condition, a constrained minimum center of log canonical singularity. The key point about the use of a constrained minimum center of log canonical singularity is the following. Though the restriction s|Y ′ to Y ′ is only holomorphic on the reduced structure of Y ′, yet since s0 vanishes on E we can take a sufficiently high power sN of s so that sN |Y ′ is holomorphic on the unreduced structure OX /Iψ of Y ′. We now extend sN |Y ′ to X . Of course, we have to replace m by Nm. Proposition (Global Generation of the Pluricanonical Bundle at Points of Zero Stable Vanishing Order). Let X be a compact complex algebraic manifold of complex dimension n. Let e−ϕ be the metric of KX of minimum singularity and let Θϕ be its curvature current. Then there exist a positive integer m0 such that the common zero-set W of a C-basis of Γ (X,m0KX) is precisely the set of points where the Lelong number of Θϕ is positive. Proof. We use the technique for the Fujita conjecture and constrained min- imum centers of log canonical singularity. We use W as the set of constraint for the constrained minimum center of log canonical singularity. We will not go into further details here, because a similar but harder situation will be handled in the proof of precisely achieving stable vanishing order γ > 0 for codimension one in the case of a hypersurface Y whose coefficient in Θϕ is γ. The only difference is that here the number γ is replaced by 0 and we do the argument in the ambient space X instead of in the hypersurface Y taking the place of X . We now finish the use of the temporary meaning of X and L and return to the earlier meaning of X and L. Proposition(Extension of Explicitly Constructed Section to Ambient Man- ifold by Constrained Minimum Center of Log Canonical Singularity). Let X be a compact complex algebraic manifold of complex dimension n. Let e−ϕ be the metric of KX of minimum singularity and let Θϕ be its curvature current. Let M be a nonsingular hypersurface in X such that the stable vanishing order η for M is a positive rational number. Let (Θϕ − ηM) |M = be the canonical decomposition of the closed positive (1, 1)-current (Θϕ − ηM) |M on M with J < ∞ and each γj being rational. Then the stable vanishing order η for M is precisely achieved. Proof. By the previous Proposition we find a positive integer m0 such that the common zero-set of a C-basis s1, · · · , sk of Γ (X,m0KX) is precisely the set of points where the Lelong number of Θϕ is positive. If any of the elements of Γ (X,m0KX) precisely achieves the stable vanishing order η forM , we are already done. By replacing m0 by another sufficiently large integer, we can make the vanishing order across M of j=1 |sj| m0 to be as close to η as prescribed (though still > η). On the other hand, by raising j=1 |sj| m0 to a positive integral power afterwards and using interpolation, we can adjust the stable vanishing order across M to η plus any positive prescribed number. Let L = KX − ηM . We can thus use s1, · · · , sk and interpolation of metrics and their slight modifications to construct a metric e−χ ofmL−KX = (m− 1)L− (η + 1)M of strictly positive curvature current so that (i) the zero-set of its multiplier ideal sheaf Iχ is contained in the set of points where the Lelong number of Θϕ is positive, and (ii) the generic vanishing order of the its multiplier ideal sheaf Iχ across M is precisely 1. We are able to fulfill Condition(i), because we can construct the metric e−χ by using the (m−1)-th power j=1 |sj| m0 and since the common zero-set of a C-basis s1, · · · , sk of Γ (X,m0KX) is precisely the set of points where the Lelong number of Θϕ is positive. We are able to fulfill Condition(ii), because we have the additional order η+1 acrossM to spare when we use the (m−1)- th power j=1 |sj| m0 and require only the higher generic vanishing order of mη + 1 = (m − 1)η + (η + 1) across M instead of the order (m − 1)η from the stable vanishing order η across M . The vanishing theorem of Kawamata-Viehweg-Nadel gives H1 (X, Iχ (mL)) = 0. To get elements of Γ (X,mL), we need to use elements of Γ (X, (OX /Iχ ) (mL)). At points of Y the additional vanishing order of the multiplier ideal sheaf Iχ of e−χ occurs only at points where the Lelong number of Θϕ is positive. When we construct a constrained minimum centerM ′ of log canonical singularity in M by interpolation (with the subvariety ∪Jj=1Yj as the set of constraint), the complex structure of M ′ is reduced outside ∪Jj=1Yj. Now we can explicitly construct the section of the line bundle NL|M on the reduced structure ofM for some appropriately chosen positive integerN , where sYj is the canonical section of Yj onM . Since s0 vanishes on sYj , by replacing N by a large positive integral multiple, we can assume that the restriction of s0 toM ′ can be extended to a holomorphic section over any unreduced structure of M ′ which is reduced outside ∪Jj=1Yj and, moreover, can be extended to an element of Γ (X, (OX /Iχ ) (NL)). Note that the support of OX /Iχ may be more than just Y , but its intersection with Y is contained in the zero-set ∪Jj=1Yj of s0 so that the last extension to an element of Γ (X, (OX /Iχ ) (NL)) is possible for a sufficiently large N . Since s0 is nonzero at some point P0 of M ′, the stable vanishing order across M is precisely achieved by s̃0 (sM) ∈ Γ (X,NqKX) at the point P0 of M . Q.E.D. PART II Illustration in Low Dimension of the Argument of Precise Achievement of Stable Vanishing Order for Higher Codimension We now illustrate the argument of the precise achievement of the stable vanishing order for higher codimension by using the low dimensional cases of complex surfaces and complex threefolds. First we consider the case of surfaces. For surfaces codimension two means isolated points. For isolated points for any dimension there is a simple direct argument, which is given in the following proposition. After we present the case of threefolds, we will remark on how this simple direct argument can be interpreted in the context of the argument for any dimension. Proposition (Precise Achievement of Stable Vanishing Order at a Finite Set). The stable vanishing order is automatically precisely achieved every- where when it is precisely achieved outside a finite set of a compact complex algebraic manifold X . Proof. Let X be a compact complex algebraic manifold of general type. Let e−ϕ be the metric of minimum singularity for the canonical line bundle KX of X . Suppose it has been proved that the stable vanishing order is precisely achieved outside a finite number of points P1, · · · , Pk of X by using σ1, · · · , σℓ ∈ Γ (X,m0KX) . We are going to show that this finite number of points must be the empty set, otherwise there is a contradiction. Let e−ψ = j=1 |σj | We take p ∈ N sufficiently large to magnify the discrepancy of the vanishing orders of e−pm0ϕ and e−pψ so that the support of the quotient Ipm0ϕ /Ipψ of the multiplier ideal sheaves Ipm0ϕ and Ipψ of e−pm0ϕ and e−pψ respectively is the finite set {P1, · · · , Pk}. We now apply a slight modification to Ipψ to get a metric with strictly positive curvature current. Let e−θ be a metric of KX with strictly positive curvature current. Since e −ϕ is the metric of minimum singularity for KX , it follows that when ε > 0 is sufficiently small (which we assume to be the case) the multiplier ideal sheaf I(p−ε)ψ+εm0θ of the metric e−((p−ε)ψ+εm0θ) agrees with the multiplier ideal sheaves Ipm0ϕ of e−pm0ϕ on X − {P1, · · · , Pk} and the support of the quotient Ipm0ϕ I(p−ε)ψ+εm0θ of the multiplier ideal sheaves Ipm0ϕ and Ipψ of e−pm0ϕ and e−((p−ε)ψ+εm0θ) respectively is the finite set {P1, · · · , Pk}. By the vanishing theorem of Kawamata-Viehweg-Nadel, we have H1 (X, Ipψ ((pm0 + 1)KX)) = 0. Then the map Γ (X, Ipm0ϕ ((pm0 + 1)KX)) → Γ Ipm0ϕ I(p−ε)ψ+εm0θ ((pm0 + 1)KX) is surjective. Note that for this we do not need the vanishing of the coho- mology group H1 (X, Ipm0ϕ ((pm0 + 1)KX)) . Since Ipm0ϕ I(p−ε)ψ+εm0θ ((pm0 + 1)KX) Ipm0ϕ I(p−ε)ψ+εm0θ it follows from Nakayama’s lemma and the surjectivity of Γ (X, Ipm0ϕ ((pm0 + 1)KX)) → Ipm0ϕ I(p−ε)ψ+εm0θ that the map Γ (X, Ipm0ϕ ((pm0 + 1)KX)) → (Ipm0ϕ)Pj is surjective. This actually gives a contradiction, because the stable vanishing order of (pm0 + 1)KX should be given by e −(pm0+1)ϕ instead of by e−pm0ϕ. Q.E.D. Remarks. (a) With this proposition, to get the finite generation of the canon- ical ring for the case of a compact complex algebraic surface of general type it suffices to show that the stable vanishing order is precisely achieved at codimension one. (b) For the analytic proof of the finite generation of the canonical ring, when we get down to the point of having already verified the precise achievement of the stable vanishing order outside a finite set of points, we do not need to blow up the points to reduce the argument to the case of a hypersurface in the new blown-up manifold. Therefore the difficulty does not exist, in the case of a surface, of blowing up a point P0 to get a curve C1 and then locating some bad point P1 (where the precise achievement of stable vanishing order fails) in the curve C1 and then blowing up P1 to get a curve C2 and locating some bad point P2 (where the precise achievement of stable vanishing order fails) in the curve C2 and then blowing up P2 and possibly finally ending up with an unending infinite sequence of bad points, each in a tower of successively blown-up surfaces. Higher Codimension Argument for Threefold Case. Let X be a complex complex algebraic threefold of general type. Let e−ϕ be the metric of mini- mum singularity for the canonical line bundle KX of X . Suppose it has been proved that the stable vanishing order is precisely achieved outside a curve Cj (where each Cj is irreducible) by s1, · · · , sk ∈ Γ (X,m0KX) . Note that here we have used the Proposition given above to rule out the possibility that, besides at the curve C, the stable vanishing order may fail to be precisely achieved at a finite set of points in X − C. On X we have the canonical decomposition of ΘKX = γjYj +R of the curvature current of the metric of minimum singularity for KX . We are skipping the diophantine argument which is explained in detail in [Siu 2006] and consider as verified the rationality of each γj. By replacing m0 by an appropriate integral multiple, we can assume without loss of generality that each m0γj is a positive integer. Let s )m0γj , where sYj is the canonical section of Yj. Let L = KX − j=1 γjYj. Since we can replace σ1, · · · , σℓ by , · · · , sk ∈ Γ (X,m0L) and the essence of the rest of the argument that is to follows does not change, for notational simplicity we are going to assume that all γj = 0 so that L = KX . Also, for notational simplicity we are going to assume that the curve C is irreducible. For σ = (σ1, · · · , σk) ∈ Ck − {0} let sσ = j=1 σjsj and let Sσ be the surface in X defined by sσ. We consider those σ for which Sσ is irreducible across which sσ vanishes to order 1. By considering the blow-up of X by the ideal generated by s1, · · · , sk and the precise achievement of stable vanishing order for codimension one, after replacing m0 by a positive integral multiple if necessary, we have the following situation. (a) For σ = (σ1, · · · , σk) ∈ Ck − {0} for which Sσ is irreducible where sσ vanishes to order 1, there exist (i) some τ = (τ1, · · · , τk) ∈ Ck − {0} and (ii) some finite subset Zτ,σ of C such that the section sτ is not identically zero on Sσ and the multi- valued section sτ |Sσ (sC,σ) on a neighborhood of C − Zτ,σ in Sσ is nonzero at points of C − Zτ,σ, where ησ is the stable vanishing order at C on the surface Sσ and sC,σ is the canonical section of C in Sσ. (b) One has the second case of the dichotomy for the curvature current (ΘKX |Sσ − ησ [C])|C on C. (c) The multi-valued section sτ |Sσ (sC,σ) on C is constructed from the second case of the dichotomy for the curvature current (ΘKX |Sσ − ησ [C])|C on C. (c) The stable vanishing order ησ for C on Sσ is achieved by sτ at points of C − Zτ,σ. Note that, because of Condition(b) and Condition(c) we can choose Zτ,σ in- dependent of τ so that Zτ,σ = Zσ for some finite subset Zσ of C depending only on σ. To finish the proof of the precise achievement of the stable vanish- ing order for the threefold X , it suffices to show that there is a finite subset Z of C such that every Zσ is contained in Z. At a regular point P0 of C, there is some open neighborhood U of P0 in X such that (i) the pair (U,C) is biholomorphic to the pair (∆3, {(0, 0)} ×∆), where ∆ is the open unit disk in C, and (ii) sj|U is represented by a holomorphic function fj (z1, z2, t) on ∆3 for 1 ≤ j ≤ k. The existence of the finite subset Z of C with the property that Zτ,σ ⊂ Z follows from Lemma 2 given below. Lemma 1. Let f0, · · · , fk be holomorphic function germs on C2 at the origin so that the origin is the common zero-set of any two of the holomorphic function germs f0, · · · , fk. Let C0 be the complex curve germ at the origin defined by f0 = 0, which is assumed to be irreducible and across which f0 vanishes to order 1. Then the following numbers are the same. (i) The multiplicity of the ideal j=0OC2,0fj at the origin. (ii) The dimension over C of OC2,0 OC2,0fj . (iii) The Lelong number λ of Θ̂ := ∂∂̄ log |fj |2 on C0 at the origin. Here the Lelong number λ means the Lelong number of the pullback of the closed positive (1, 1)-current Θ̂ to the normalization of C0. (iv) The number η such that j=1 |fj| j=1 |zj | is bounded between two positive numbers near the origin on C0. Lemma 2. Let k ≥ 2 and let fj(z1, z2, t), for 1 ≤ j ≤ k, be holomorphic functions on the tri-disk ∆3 with coordinates z1, z2, t. Assume that the com- mon zero-set of any two of f1, · · · , fk is {(0, 0)} × ∆ = {z1 = z2 = 0}. For any k-tuple (a1, · · · , ak) of complex numbers not all zero let Sa1,··· ,ak be the zero-set of j=1 ajfj . Then there exists a discrete subset Z of {(0, 0)} ×∆ with the following property. For any k-tuple (a1, · · · , ak) of complex numbers not all zero there exists some nonnegative number γa1,··· ,ak such that j=1 |fj | |zj |2 )γa1,··· ,ak is continuous nonzero on some neighborhood of (Sa1,··· ,ak ∩ {(0, 0)} ×∆)−Z in Sa1,··· ,ak if Sa1,··· ,ak is irreducible and j=1 ajfj vanishes to order 1 across Sa1,··· ,ak . Proof. Observe that for fixed t if a1 6= 0 then each of j=1 |fj | |a1f1 + · · ·+ akfk|2 + j=2 |fj | is bounded by a positive constant times the other on some neighborhood of the origin in C2. Use Lemma 1. Q.E.D. Remarks. (a) We would like to highlight the intuitive geometric reason for the existence of a discrete set Z in C such that the “bad set” Zσ in C for the restriction of the closed positive (1, 1)-current ΘKX restricted to Sσ is contained in Z. The key point is that the surface Sσ is sliced out by a C- linear combination of the pluricanonical sections s1, · · · , sk and that these pluricanonical sections s1, · · · , sk have the property that the “bad set” Zσ of C can be described by the extra vanishing of j=1 |sj| beyond their generic vanishing order at points of C. (b) We need to restrict ΘKX to Sσ, because we have to take away the vanish- ing order of ΘKX at C and it is only for codimension one we can take away the vanishing order. The vanishing order on different Sσ may be different. (c) The “bad set” Zσ in C describes the points where the relative position of the pair of two Artinian subschemes in the normal direction of C jumps. One Artinian subscheme comes from the restriction of s1, · · · , sk to a local surface T normal to C and the other one comes from s1, · · · , sk plus a C-linear combination σ1s1+ · · ·+ σksk after restricting them to T . The main point is that there is a finite subset Z of C such that Zσ ⊂ Z with Z independent of the choice of σ. References. [Angehrn-Siu 1995] U. Angehrn and Y.-T. Siu, Effective freeness and point separation for adjoint bundles. Invent. Math. 122 (1995), 291–308. [Ohsawa 2002] T. Ohsawa, A precise L2 division theorem. In: Complex Geometry (Selected papers dedicated to Hans Grauert from the International Conference on Analytic and Algebraic Methods in Complex Geometry held in Göttingen, April 3–8, 2000), ed. I. Bauer, F, Catanese, Y. Kawamata, Th. Peternell and Y.-T. Siu. Springer-Verlag, Berlin, 2002, pp.185–191. [Siu 1998] Y.-T. Siu, Invariance of plurigenera. Invent. Math. 134 (1998), no. 3, 661–673. [Siu 2002] Y.-T. Siu, Extension of twisted pluricanonical sections with plurisub- harmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. In: Complex Geometry (Selected papers dedicated to Hans Grauert from the International Conference on An- alytic and Algebraic Methods in Complex Geometry held in Göttingen, April 3–8, 2000), ed. I. Bauer, F, Catanese, Y. Kawamata, Th. Peternell and Y.-T. Siu. Springer-Verlag, Berlin, 2002, pp.223–277. [Siu 2004] Y.-T. Siu, Invariance of plurigenera and torsion-freeness of direct image sheaves of pluricanonical bundles. In: Finite or infinite dimensional complex analysis and applications, Adv. Complex Anal. Appl., 2, Kluwer Acad. Publ., Dordrecht, 2004, pp. 45–83. [Siu 2006] Y.-T. Siu, A general non-vanishing theorem and an analytic proof of the finite generation of the canonical ring, arXiv:math/0610740. [Shokurov 1985] V. V. Shokurov, A nonvanishing theorem. Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), 635–651. [Skoda 1972] H. Skoda, Application des techniques L2 à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids. Ann. Sci. École Norm. Sup. 5 (1972), 545-579. http://arxiv.org/abs/math/0610740

This set of notes provides some additional explanatory material on the analytic proof of the finite generation of the canonical ring for a compact complex algebraic manifold of general type.

Introduction. This set of notes is put together to provide some addi- tional explanatory material on the analytic proof of the finite generation of the canonical ring for use by the participants in the Workshop on Minimal and Canonical Models in Algebraic Geometry in April 16–20, 2007 at the Mathematical Sciences Research Institute at Berkeley, California. In late March 2007 before the Workshop Mihai Paun of Strasbourg came to Harvard for two weeks to pose to me some questions which he and other people have about the analytic proof of the finite generation of the canoni- cal ring which I posted in October 2006 on the arXiv.org server [Siu 2006]. Besides orally answering his questions I also wrote up some notes for him to give him more precise details. This set of notes is compiled by putting together the notes which I wrote up to answer his questions. This set of notes consists of two parts. Part I is about how to apply the general nonvanishing theorem to prove the precise achievement of stable vanishing order in codimension one. Part II gives the argument for the precise achievement of the stable vanishing order for higher codimension. For both Part I and Part II the new powerful tool in the analytic proof is the use of the second case of the dichotomy of the modified restriction of the curvature current of the metric of minimum singularity for the canonical line bundle. Let V = Vk be an irreducible subvariety of codimension k in the compact complex algebraic manifold X of finite type whose canonical ring is to be proved to be finitely generated such that V is an embedded branch of the stable base point set. A modified restriction ΘV to V of the curvature current of the canonical line bundle is given as follows. For some ην ≥ 0 and Vν irreducible of codimension 1 in Vν−1 (k < ν < 0) with V0 = X and η0 = 0, let Θ0 be the curvature current ΘKX on X of of the metric of minimum singularity for the canonical line bundle KX and inductively Θν = (Θν−1 − ην−1Vν−1) |Vν . 1Partially supported by a grant from the National Science Foundation. http://arxiv.org/abs/0704.1940v1 Then Θk is the modified restriction ΘV to V of the curvature current of KX for the sequence Vk−1 ⊂ · · · ⊂ V1 ⊂ X of nested subvarieties. For the canonical decomposition γj [Yj] +R of ΘV on V there are two cases. The first case is either J = ∞ or R 6= 0 and the second case is J < ∞ and R = 0, where Yj is a subvariety of codimension one in V and R is a closed positive current on V whose Lelong number is zero outside a countable union of subvarieties of codimension at least two in V . For Part I on how to apply the general nonvanishing theorem to prove the precise achievement of stable vanishing order in codimension one, the result of the use of this new powerful tool is that the second case of the dichotomy must always eventually occur at some positive dimensional V , because if the first case occurs all the time down to dimension zero then there is some improvement in the stable vanishing at some point of V in the multi-directions defined by the sequence of nested subvarieties. The second case of the dichotomy gives a explicitly constructed section on V (unique up to a nonzero constant factor) which belongs to the multiplier ideal sheaf corresponding to ΘV . In Part I of these additional explanatory notes, in order to make the arguments of the general nonvanishing theorem more transparent, I treat separately the two cases of the dichotomy. We use the process of using the techniques for the Fuijta conjedture by constructing singular metrics successively to cut down on the dimension of (the projection of) the zero-set of the multiplier ideal sheaf until we end up with the inevitable second case of the dichotomy and in the second case of the dichotomy we use the sections explicitly constructed from the special form of the canonical decomposition of the modified restriction of the curvature current. In extending the explicitly constructed sections all the way back to the ambient manifoldX we introduce the technique of constrained minimum center of log canonical singularity. In Part II of these additional explanatory notes about the argument for the precise achievement of the stable vanishing order for higher codimension, we give its illustration in the low dimensional cases of complex surfaces and complex threefolds. The illustration in the low dimensional cases avoids the encumberment of complicated notations in the case of general dimension and at the same time contains the essence of the argument of the general case. For the case of a threefoldX in Part II, the key case of higher codimension is that of an irreducible curve C which is an embedded curve in the stable base point set. The important ingredient of using the second case of the dichotomy is the following. For simplicity assume that there is no codimension-one base point set. Suppose s1, · · · , sk are pluricanonical sections on X such that C is cut out by the family Sσ of surfaces defined by pluricanonical sections sσ = j=1 σjsj parametrized by σ = (σ1, · · · , σk) ∈ Ck − {0}. Suppose s is a pluricanonical section whose restriction to some Sσ achieves the restriction of the stable vanishing order ασ on Sσ at some point Pσ of C. Then, after taking away the restriction of the stable vanishing order ασ of s|Sσ across C, the resulting section (sC,σ) ασ on Sσ (where sC,σ is the canonical section of C in Sσ), when restricted to C, is the section (up to a nonzero constant factor) explicitly constructed from the second case of the dichotomy for C in Sσ. This means that at every point of C outside the finite zero-set Zσ of the resulting section (sC,σ) ασ on C, the section (sC,σ) ασ achieves the restriction of the stable vanishing order ασ on Sσ. Suppose for every parameter σ there is a parameter τ and some point Pσ,τ in C such that the restriction of the pluricanonical section sτ to Sσ achieves the restriction of the stable vanishing order on Sσ at the point Pσ,τ of C. Let Z be the finite subset of C such that as a point P varies in C outside Z the Artinian subscheme defined by s1, · · · , sk in the normal direction of C at P does not change. In other words, every point Q ∈ C − Z admits an open neighborhood U in C − Z and a holomorphic family of local nonsingular complex surfaces MP in X parametrized by P ∈ U such that each MP intersecting C normally at P and dimC OMP OMP (sj|MP ) is independent of P ∈ U . From the above discussion about s being the explicitly constructed section and about the achievement of the restriction to Sσ of stable vanishing order at points of C outside Zσ it follows that the stable vanishing order is achieved at points of C − Z. This use of the sections, explicitly constructed from the curvature current in the second case of the dichotomy, in order to come up with a finite sub- set Z of C can be regarded as the higher-codimensional way of constructing sections of vector bundles explicitly from the second case of the dichotomy of the modified restriction of the curvature current. Note that the codimension one case for a surface is invoked for us to get the rationality of the restric- tion of the stable vanishing order to a surface and its achievement on the surface at some point of C. This part of the hard work already done for the codimension one case is used here. This vector bundle section on C explicitly constructed from the second case of the dichotomy of the modified restric- tion of the curvature current is obtained, so to speak, by piecing together the line bundle sections on C explicitly constructed by using the surface slices S from the second case of the dichotomy of the modified restriction to the sur- face slice S of the curvature current. This is the reason why the usual worry about interminable blowups to transform into hypersurfaces newly appearing higher-codimensional base point sets is not a problem here. Before presenting Part I and Part II, we would like to make two remarks. The first remark is about the finite generation of the canonical ring without the assumption of finite type. In analysis, when KX is not big, adding a line bundle E to KX to have a big bundle KX + E and working with KX + E instead of KX would only mean a standard modification of the argument for the finite generation of the canonical ring for the case of general type. (This is actually used in the form of KX − γY , with γ being the stable vanishing order across the hypersurface Y , in the proof of the finite generation of the canonical ring for the case of general type.) However, getting rid of E afterwards needs the justification of a rather involved limiting process. Such a limiting process is of the same nature as the limiting process which is used to extend the proof of the deformational invariance of the plurigenera for the case of general type [Siu 1998] to the general algebraic case [Siu 2002]. For the finite generation of the canonical ring, when KX is big, we intro- duce the curvature ΘKX of the metric of minimum singularity constructed from pluricanonical sections. When KX is not big, we use an ample line bundle A on X and the limit curvature current Θ̃KX := lim ΘKX+ 1mA to replace ΘKX . All the diophantine approximation arguments deal with the modified restriction of the canonical decomposition of Θ̃KX . This makes it possible to go ahead with the limiting process. In the case of the deformational invariance of the plurigenera the limiting process is handled by abandoning the metric of minimum singularity and replacing it by another metric with maximum allowable singularity which is still good enough for the finiteness of the L2 norm of the section to be extended, because there is a need to bound the dimension of the space of L2 holomorphic m-canonical sections with the bound independent of m or at least growing more slowly than m. It is not yet clear what the analogous procedure is for the problem of the finite generation of the canonical ring. The second remark is to answer the question why one does not simply take a pluricanonical section and get the finite generation of the canonical ring of its divisor and then use extension of pluricanonical sections from that divisor to the entire ambient space. This procedure is explained in §8 of the posted notes of my analytic proof [Siu 2006]. The extension technique for pluricanonical sections in the problem of deformational invariance of the plurigenera was developed with that particular application in mind [Siu 1998, Siu 2006]. Though the pluricanonical extension technique for deformational invariance of the plurigenera is for extension from the fiber at a point to the open unit disk, in analysis it does not matter whether one deals with a compact manifold or a Stein manifold if the Stein manifold is a Zariski open subset and one keeps track of the L2 bounds. However, the situation of the singularities in the pluricanonical divisor poses technical problems in analysis. In my article in the proceedings of a 2001 conference in Hanoi [Siu 2004] I addressed the problem of pluricanonical extension from a singular divisor. There are technical problems concerning finite L2 bounds. The difficulty in handling such technical problems concern- ing finite L2 bounds is the same as getting the comparability of the metric of minimum singularity for KX and that from an appropriate truncated fi- nite sum in the application of Skoda’s theorem of ideal generation [Skoda 1972]. Moreover, the Ohsawa-Takegoshi-type extension theorem used in the pluricanonical extension is closely related to Skoda’s theorem of ideal gen- eration (see, for example, Ohsawa’s article in the Festschrift for Grauert’s 70th birthday [Ohsawa 2002]) and the constant in Skoda’s theorem is more precise. That is why we use the approach by Skoda’s theorem instead of the essentially equivalent approach of pluricanonical extension. It seems that at this point any approach to the problem of finite gener- ation of the canonical ring has to rely, as starting point, on the technique of pluricanonical extension. Any difference in the different approaches are more a matter of technical handling of the singularities of the pluricanonical divisor. In the definition of a stable metric in [Siu 2006, (6.1)] there are some typographical errors and inaccuracies. The correct definition is as follows. A metric e−κ of a line bundle L on a compact Kähler manifold M is said to be stable if κ is locally plurisubharmonic and there exists some ε > 0 with the following property. If U is an open neighborhood of a point P ∈ M , and ϕ and ψ are plurisubharmonic functions on U such that the total mass, with respect to the Kähler form of M , of the sum of the two closed positive (1, 1)-currents Θϕ and Θψ is less than ǫ and if κ − ϕ is plurisubharmonic, then there exists an open neighborhood U ′ of P inM such that the multiplier ideal sheaf Iκ+ψ of the metric e−κ−ψ on U ′ is equal to the multiplier ideal sheaf Iκ−ϕ of the metric e−κ+ϕ on U ′. PART I How to Apply Nonvanishing Theorem to Precisely Achieve Stable Vanishing Order in Codimension One The proof of the precise achievement of stable vanishing order in codi- mension one has the following ingredients. (a) The techniques for the Fujita conjecture which consists of (i) constructing singular metrics with curvature current of strict pos- itive lower bound whose multiplier ideal sheaf has high vanishing order at a prescribed point, (ii) blowing up the zero-set of the multiplier ideal sheaf of the new singular metric, (iii) repeating the procedure so that the zero-set of the multiplier ideal sheaf of the new singular metric on a blow-up space is projected down to lower and lower dimension in the original manifold, until one gets to the case of a singular point in the original manifold, (iv) extending a section defined on the single point to over all of the original manifold by the vanishing theorem of Kawamata- Viehweg-Nadel. Because of the need of adding one canonical line bundle in the ap- plication of the vanishing theorem of Kawamata-Viehweg-Nadel, the singular metric is for a multiple of the line bundle in question minus the canonical line bundle. (b) Introducing a dichotomy depending on the canonical decomposition of some curvature current so that (i) in the first case of the dichotomy the technique of the Fujita con- jecture for the construction of singular metric can be carried out, (ii) in the second case of the dichotomy a section can be explicitly constructed which can be extended to all of the original manifold. Once we get to the second case of the dichotomy there is no need to use the technique of the Fujita conjecture to construct any more singular metric. The process is stopped and complete. For the precise achievement of stable vanishing order, the second case of the dichotomy must arise, otherwise the stable vanishing order can be improved, which contradicts the definition of a stable vanishing order. The explicitly constructed section is rather rigid in the sense that there is no choice and the section comes from the curvature current in a rather unique way. This uniqueness and rigidity of the explicitly constructed section will be a key ingredient in the proof of the precise achievement of the stable vanishing order in higher codimension. (c) The new technique of constrained minimum center of log canonical sin- gularity, whose motivation and precise application will be explained below. This new technique is needed, because of the undesirable addi- tional vanishing order in the process of constructing singular metrics, which will be explained in detail below. (d) Use of diophantine approximation to handle irrational coefficients. Since the use of diophantine approximation has already been described in detail in the posted notes of my proof of the finite generation of the canonical ring [Siu 2006], we will not discuss diophantine approxima- tion in these additional explanatory notes and we just assume that the relevant coefficients are known to be rational numbers. As the first step we will consider the first case of the dichotomy and describe how to construct an appropriate singular metric. Then we will give the motivation for constrained minimum center of log canonical singularity by reviewing the goal and the strategy of the standard technique for the Fujita conjecture. Finally we consider the second case of the dichotomy where a rather rigid section is explicitly constructed. Proposition(Construction of Metric with Multiplier Ideal Sheaf Vanishing to High Order at a Prescribed Point for the First Case of the Dichotomy of the Curvature Current). Let M be a compact complex projective algebraic manifold of complex dimension n. Let L be a holomorphic line bundle on M with a (possibly singular) metric e−ϕ along its fibers whose curvature current Θϕ is a closed positive (1, 1)-current. Let τj [Vj] +R be the canonical decomposition of Θϕ, where J ∈ N∪{0,∞} and the Lelong number of R is zero outside a countable union Z of subvarieties of codi- mension at least two in M and Vj is an irreducible hypersurface in M and τj > 0. Assume that either J = ∞ or R 6= 0 (that is, one is in the first case of the dichotomy for the curvature current Θϕ). Assume that for some positive integer p0 there is a (possibly singular) metric e −χ along the fibers of p0L−KM which is stable and whose curvature current Θχ is a closed positive (1, 1)-current which dominates some strictly positive smooth (1, 1)-form on M . Let P0 be a point of M such that the Lelong number of Θϕ is zero at P0. Let q ∈ N. Then for some sufficiently divisible m ∈ N the line bundle (m+ p0)L − KM admits a metric e−χ̃ whose curvature current dominates some strictly positive smooth (1, 1)-form on M such that its multiplier ideal sheaf Iχ̃ is contained in the maximum ideal of M at P0 raised to the q-th power. Moreover, if M is a hypersurface in some compact complex algebraic manifold X of general type so that L is the restriction of some line bundle L̃ on X and KM = KX +M and the metric e −χ is defined by a convergent infinite sum of multi-valued holomorphic sections of p0L̃−KX −M over X , then the metric e−χ̃ can be chosen to be defined also by a convergent infinite sum of multi-valued holomorphic sections of (p + p0) L̃−KX −M over X . Proof. The idea of the proof is to use the techniques for Fujita’s conjecture (see, for example, [Angehrn-Siu 1995]). Slicing by an Ample Divisor. Let A be a very ample line bundle over M such that A−KM is ample. Let hA be a smooth metric of A whose curvature form ωA is positive on M . We assume that A is chosen to be sufficiently ample so that for each point P ∈M the proper transform of A in the manifold obtained from M by blowing up P is still very ample. This technical assumption will enable us to choose a generic element of Γ (M, A) vanishing at P0 which is not a zero-divisor of a prescribed coherent ideal sheaf. Let p and k be positive integers and we will impose more conditions on p and k later. Let s1 be a generic element of Γ (M, A) vanishing at P0 so that the short exact sequence 0 → Ipϕ+χ ((p+ p0)L−KM + kA) θs1−→ Ipϕ+χ ((p+ p0)L−KM + (k + 1)A) → (Ipϕ+χ /s1Ipϕ+χ ) ((p+ p0)L−KM + (k + 1)A) → 0 is exact, where θs1 is defined by multiplication by s1. Let M1 be the zero-set of s1 and OM1 = (OM /s1OM ) |M1, which we can assume to be regular with ideal sheaf equal to s1OM because s1 is generic element of Γ (M,A) vanishing at P0. By choosing s1 generically we can also assume that I(pϕ+χ)|M1 = Ipϕ+χ /s1Ipϕ+χ and I(pϕ)|M1 = Ipϕ /s1Ipϕ and Iχ|M1 = Iχ /s1Iχ . We use χ (·, ·) to denote the arithmetic genus which means χ (·, ·) = (−1)ν dimCHν (·, ·) . From the long cohomology exact sequence of the above short exact sequence we obtain χ (M, Ipϕ+χ ((p+ p0)L−KM + (k + 1)A)) = χ (M, Ipϕ+χ ((p+ p0)L−KM + kA))+χ M1, I(pϕ+χ)|M1 ((p+ p0)L−KM + (k + 1)A) |M1 Since A − KM is ample and 2A − KM1 = A − KM is also ample, when we assume k ≥ 1, by the theorem of Kawamata-Viehweg-Nadel Hν (M, Ipϕ+χ ((p+ p0)L−KM + kA)) = 0 for ν ≥ 1, M1, I(pϕ+χ)|M1 (((p+ p0)L−KM + (k + 1)A) |M1) = 0 for ν ≥ 1 so that dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + (k + 1)A)) = dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + kA)) + dimC Γ M1, I(pϕ+χ)|M1 (((p+ p0)L−KM + (k + 1)A) |M1) ≥ dimC Γ M1, I(pϕ+χ)|M1 (((p+ p0)L−KM + (k + 1)A) |M1) Slicing by Ample Divisors Down to a Curve. Instead of one single element s ∈ Γ (M,A), we can choose generically s1, · · · , sn−1 ∈ Γ (M, A) all vanishing at P0 so that inductively for 1 ≤ ν ≤ n−1 the common zero-set Mν of s1, · · · , sν with the structure sheaf OMν := is regular and we end up with the inequality dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + (k + n− 1)A)) ≥ dimC Γ Mn−1, I(pϕ+χ)|Mn−1 ((p+ p0)L−KM + (k + n− 1)A) |Mn−1 Since Mn−1 is a curve, all coherent ideal sheaves on it are principal and are locally free and they come from holomorphic line bundles. We would like to remark also that this particular step of slicing by n− 1 ample divisors to get down to a curve roughly corresponds to the step in Shokurov’s proof of his non-vanishing theorem [Shokurov 1985] where he takes the product of his numerically effective divisor in his n-dimensional manifold with the (n−1)-th power of a numerically effective big line bundle. Application of the Theorem of Riemann-Roch to a Curve and Comparing Contributions from the Curvature Current and the Multiplier Ideal Sheaves. Let b be the Chern class of the line bundle onMn−1 defined by the multiplier ideal sheaf Iχ|Mn−1 of the restriction to Mn−1 of the metric e −χ. Let c be the nonnegative number R ∧ (ωA)n−1 . (†) dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + (k + n− 1)A)) ≥ dimC Γ Mn−1, I(pϕ+χ|Mn−1) ((p+ p0)L−KM + (k + n− 1)A) |Mn−1 ≥ 1− genus (Mn−1) + b+ (k + n− 1)An−1Mn−1 (pτj − ⌊pτj⌋)Vj · An−1 + p where the last identity is from the theorem of Riemann-Roch applied to the regular curve Mn−1 and the locally free sheaf I(pϕ+χ|Mn−1) ((p+ p0)L−KM + (k + n− 1)A) |Mn−1 on Mn−1. From the assumption that J = ∞ or R 6= 0, we conclude that the right-hand side of (†) goes to ∞ as p goes to ∞ through an appropriate sequence, where for the case of J = ∞ and R = 0 a diophantine argument has to be used whereas for the case R 6= 0 we simply need to use c > 0. Construction of Sections with Extra Vanishing Order from Dimension Count- ing and Construction of Metrics by Canceling Contributions from Ample Di- visors by Using the General Type Property. For any ℓ ∈ N the number of terms in a polynomial of degree ℓ in d variables is . Take a positive integer N and we will impose more condition on N later. By the behavior of the right-hand side of (†) as p→ ∞, there exists p ∈ Z such that dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + (k + n− 1)A)) ≥ 1+ n+N (k + n− 1) q and we can find some non identically zero element s of Γ (M, Ipϕ+χ ((p + p0)L−KM + (k + n− 1)A)) which vanishes to order at least N (k + n− 1) q at P0 so that s N(k+n−1) is a multi-valued holomorphic section of the Q-line-bundle p N(k+n−1) A over M which vanishes to order at least q at P0. We assume that N is chosen so large that the curvature current Θχ dominates ωA. Let p̂ to be the round-up of p N(k+n−1) and δp = p̂− pN(k+n−1) . We introduce the metric e−χ̃ := e−χ−δpϕ N |s| of (p+ p0)L − KM so that the multiplier ideal of Iχ̃ at P0 is contained in (mM,P0) . Q.E.D. Remark on Application of the Proposition on Construction of Metric for the First Case of the Dichotomy. The above application is applied in the following manner. Let X be the the compact complex algebraic manifold of finite type whose canonical ring is to be proved to be finitely generated. Let Y be a hypersurface in X across which the stable vanishing order is γ > 0. Let P0 be a generic point of Y . We start out withM = Y and L = KX−γY . After we get the new metric e−χ̃, we use an interpolation between e−χ̃ and e−pϕ−χ and a slight modification to get to a minimum center of log canonical singularity which, after blow-up, projects down to a proper subvariety of M containing P0. Then we replace X by its blow-up and replace M by the new minimum center of log canonical singularity and replace L by its pullback to the blowup of X . We continue doing this until we inevitably come to the second case of dichotomy eventually as explained in the Introduction. Remark on the Second Case of the Dichotomy. Suppose τj [Vj] with J < ∞. Then we can explicitly construct a section s0 of pL over M . The reason why a minimal center of log canonical singularity is used in the techniques for the Fujita conjecture is to make sure that when we take the subspace defined by the multiplier ideal sheaf, the subspace has a reduced structure. In our case we have to introduce the notion of constrained minimal center of log canonical singularity so that the center is not completely contained in the zero-set of s0. For that we have to pay the price that the subspace defined by the multiplier ideal sheaf may have an unreduced structure, but the set where nonzero nilpotent elements of its structure sheaf occurs is contained in the zero-set of s0. By raising s0 to a sufficiently high power, we can handle the unreduced structure and get the extension of a sufficiently high power of s0 to the ambient manifoldX by using the vanishing theorem from the metric of pL−KM . We are going to elaborate on this by reviewing the goal of the techniques for the Fujita conjecture and the use of minimal center of log canonical singularity and also how we are naturally and by necessity led to the concept of constrained minimal center of log canonical singularity. Main Idea of the Techniques for the Fujita Conjecture. For the discussion about the main idea of the technique for the Fujita conjecture, we forget the above meaning of X and L and for the time being use the symbols X and L in another context. We will so indicate when we later return to the above meaning of X and L. The goal of the technique for the Fujita conjecture is to find global sections to generate some positive power mL of a line bundle L over a compact complex algebraic manifold X . For a more general setting, the goal is to find global sections to globally generate J (mL) over X for some given coherent ideal sheaf J at points outside some given subvariety Z of X . The problem of proving the finite generation of the canonical ring by verifying the precise achievement of stable vanishing orders actually involves this more general setting. There the even more complicated situation of supremum norm is used instead of just the L2 norm. However, for the sake of simplicity in our discussion of the main idea of the techniques of the Fujita conjecture, we stick with the simpler goal of find global sections to generate some positive power mL of a line bundle L over a compact complex algebraic manifold X . The main idea of the technique is to focus on the subvariety where the global generation fails. We take a basis of s1, · · · , sk ∈ Γ (X,mL) and let Y be their common zero-set so that global generation precisely fails at points of Y . The main idea of the technique is simply to focus on Y if Y is nonempty. We seek elements of Γ (Y,mL|Y ) which are not identically zero and then extend them to elements of Γ (X,mL), which would then contradict the def- inition of Y . Hopefully the extension of elements of Γ (Y,mL|Y ) to elements of Γ (X,mL) could be done by the vanishing of some appropriate first co- homology group. Usually this first cohomology comes from the vanishing theorem of Kawamata-Viehweg-Nadel. We seek a metric e−ϕ of mL−KX so (i) the zero-set of its multiplier ideal sheaf Iϕ is Y , and (ii) the curvature current of e−ϕ dominates some strictly positive smooth (1, 1)-form on X . In such a case we have 1 (X, Iϕ (mL)) = 0 from the vanishing theorem of Kawamata-Viehweg-Nadel and the map Γ (X,mL) → Γ (Y, (OX /Iϕ ) (mL)) is surjective. The next step is to come up with some element of Γ (Y, (OX /Iϕ ) (mL)) which induces a non identically zero element of Γ (Y,mL|Y ). It is at this point that the question of a possibly unreduced complex structure OX /Iϕ arises. It means that the structure sheaf OX /Iϕ for Y may have nonzero nilpotent elements. This is the case when Iϕ is a proper subsheaf of the full ideal sheaf IY of Y and is not equal to IY . An element of IY which is not in Iϕ would yield a nonzero nilpotent element of the structure sheaf OX /Iϕ for Y . When we have an unreduced structure OX /Iϕ for Y , it is more difficult to produce some element of Γ (Y, (OX /Iϕ ) (mL)) which induces a non identically zero element of Γ (Y,mL|Y ). To handle the problem of unreduced structure sheaf, the technique is to use minimum cen- ters of log canonical singularity whose role we are going to explain. Minimum Center of Log Canonical Singularity. The idea is to seek a metric e−ψ for mL −K which is less singular than e−ϕ so that the multiplier ideal sheaf Iψ of e−ψ contains the multiplier ideal sheaf Iϕ of e−ϕ. This procedure usually involves the interpolation of two metrics and a slight modification of the result of the interpolation. We seek to make the metric of e−ψ for mL −K to be as least singular as possible, with just enough singularity to make the multiplier ideal sheaf Iψ of e−ψ not equal to OX . Let Y ′ be the support of OX /Iψ . This kind of least or minimum singularity for the choice of e−ψ gives us a reduced complex structure OX /Iψ for Y ′. The reduced complex subspace (Y ′,OX /Iψ ) is called a minimum center of log canonical singularity. (Usually for this technique of minimum center of log canonical singularity one requires, in addition, that the proper transform of Y ′ in some appropriate blow-up X̃ of X to be a nonsingular hypersurface in X̃ .) Now one replaces Y by Y ′ and uses the vanishing of H1 (X, Iψ (mL)) = 0 and the surjectivity of the map Γ (X,mL) → Γ (Y ′, (OX /Iψ ) (mL)) to reduce the problem to the construction of nonzero element of Γ (Y ′, (OX /Iψ ) (mL)) = Γ (Y ′, mL|Y ′) . Constrained Minimum Center of Log Canonical Singularity. For our case at hand for the finite generation of the canonical ring one modification has to be added in the application of the technique of minimum center of log canonical singularity. This modification necessitates the introduction of a new concept which we give the name constrained minimum center of log canonical singularity just to make it easier to refer to. Let us now describe the situation. In the second case of the dichotomy of the curvature current, there is some non identically zero element s0 of Γ (Y,mL|Y ) which is explicitly constructed from the canonical decomposition on Y of a modified restriction of the curvature current. The section s0 may have a nonempty zero-set W in Y . If we just use the technique of minimum center of log canonical singularity without any modification, we may end up with a minimum center of log canonical sin- gularity Y ′ which is completely contained inside the zero-set W of s0. In such a case the extension of s0|Y ′ is useless, because the restriction s0|Y ′ of s0 to Y ′ is identically zero on Y ′. So we need to introduce a modification to the technique of minimum center of log canonical singularity. In the proce- dure of using a metric e−ψ with least singularity to get the minimum center of log canonical singularity, we introduce the additional condition that the support of OX /Iψ is not contained entirely in the zero-set W of s0. With this additional condition we can no longer require that the structure sheaf OX /Iψ of Y ′ is reduced, but we can require that the set E of points where the structure sheaf OX /Iψ of Y ′ fails to be reduced is entirely contained in W . We call Y ′, which is obtained from this procedure of the additional condition, a constrained minimum center of log canonical singularity. The key point about the use of a constrained minimum center of log canonical singularity is the following. Though the restriction s|Y ′ to Y ′ is only holomorphic on the reduced structure of Y ′, yet since s0 vanishes on E we can take a sufficiently high power sN of s so that sN |Y ′ is holomorphic on the unreduced structure OX /Iψ of Y ′. We now extend sN |Y ′ to X . Of course, we have to replace m by Nm. Proposition (Global Generation of the Pluricanonical Bundle at Points of Zero Stable Vanishing Order). Let X be a compact complex algebraic manifold of complex dimension n. Let e−ϕ be the metric of KX of minimum singularity and let Θϕ be its curvature current. Then there exist a positive integer m0 such that the common zero-set W of a C-basis of Γ (X,m0KX) is precisely the set of points where the Lelong number of Θϕ is positive. Proof. We use the technique for the Fujita conjecture and constrained min- imum centers of log canonical singularity. We use W as the set of constraint for the constrained minimum center of log canonical singularity. We will not go into further details here, because a similar but harder situation will be handled in the proof of precisely achieving stable vanishing order γ > 0 for codimension one in the case of a hypersurface Y whose coefficient in Θϕ is γ. The only difference is that here the number γ is replaced by 0 and we do the argument in the ambient space X instead of in the hypersurface Y taking the place of X . We now finish the use of the temporary meaning of X and L and return to the earlier meaning of X and L. Proposition(Extension of Explicitly Constructed Section to Ambient Man- ifold by Constrained Minimum Center of Log Canonical Singularity). Let X be a compact complex algebraic manifold of complex dimension n. Let e−ϕ be the metric of KX of minimum singularity and let Θϕ be its curvature current. Let M be a nonsingular hypersurface in X such that the stable vanishing order η for M is a positive rational number. Let (Θϕ − ηM) |M = be the canonical decomposition of the closed positive (1, 1)-current (Θϕ − ηM) |M on M with J < ∞ and each γj being rational. Then the stable vanishing order η for M is precisely achieved. Proof. By the previous Proposition we find a positive integer m0 such that the common zero-set of a C-basis s1, · · · , sk of Γ (X,m0KX) is precisely the set of points where the Lelong number of Θϕ is positive. If any of the elements of Γ (X,m0KX) precisely achieves the stable vanishing order η forM , we are already done. By replacing m0 by another sufficiently large integer, we can make the vanishing order across M of j=1 |sj| m0 to be as close to η as prescribed (though still > η). On the other hand, by raising j=1 |sj| m0 to a positive integral power afterwards and using interpolation, we can adjust the stable vanishing order across M to η plus any positive prescribed number. Let L = KX − ηM . We can thus use s1, · · · , sk and interpolation of metrics and their slight modifications to construct a metric e−χ ofmL−KX = (m− 1)L− (η + 1)M of strictly positive curvature current so that (i) the zero-set of its multiplier ideal sheaf Iχ is contained in the set of points where the Lelong number of Θϕ is positive, and (ii) the generic vanishing order of the its multiplier ideal sheaf Iχ across M is precisely 1. We are able to fulfill Condition(i), because we can construct the metric e−χ by using the (m−1)-th power j=1 |sj| m0 and since the common zero-set of a C-basis s1, · · · , sk of Γ (X,m0KX) is precisely the set of points where the Lelong number of Θϕ is positive. We are able to fulfill Condition(ii), because we have the additional order η+1 acrossM to spare when we use the (m−1)- th power j=1 |sj| m0 and require only the higher generic vanishing order of mη + 1 = (m − 1)η + (η + 1) across M instead of the order (m − 1)η from the stable vanishing order η across M . The vanishing theorem of Kawamata-Viehweg-Nadel gives H1 (X, Iχ (mL)) = 0. To get elements of Γ (X,mL), we need to use elements of Γ (X, (OX /Iχ ) (mL)). At points of Y the additional vanishing order of the multiplier ideal sheaf Iχ of e−χ occurs only at points where the Lelong number of Θϕ is positive. When we construct a constrained minimum centerM ′ of log canonical singularity in M by interpolation (with the subvariety ∪Jj=1Yj as the set of constraint), the complex structure of M ′ is reduced outside ∪Jj=1Yj. Now we can explicitly construct the section of the line bundle NL|M on the reduced structure ofM for some appropriately chosen positive integerN , where sYj is the canonical section of Yj onM . Since s0 vanishes on sYj , by replacing N by a large positive integral multiple, we can assume that the restriction of s0 toM ′ can be extended to a holomorphic section over any unreduced structure of M ′ which is reduced outside ∪Jj=1Yj and, moreover, can be extended to an element of Γ (X, (OX /Iχ ) (NL)). Note that the support of OX /Iχ may be more than just Y , but its intersection with Y is contained in the zero-set ∪Jj=1Yj of s0 so that the last extension to an element of Γ (X, (OX /Iχ ) (NL)) is possible for a sufficiently large N . Since s0 is nonzero at some point P0 of M ′, the stable vanishing order across M is precisely achieved by s̃0 (sM) ∈ Γ (X,NqKX) at the point P0 of M . Q.E.D. PART II Illustration in Low Dimension of the Argument of Precise Achievement of Stable Vanishing Order for Higher Codimension We now illustrate the argument of the precise achievement of the stable vanishing order for higher codimension by using the low dimensional cases of complex surfaces and complex threefolds. First we consider the case of surfaces. For surfaces codimension two means isolated points. For isolated points for any dimension there is a simple direct argument, which is given in the following proposition. After we present the case of threefolds, we will remark on how this simple direct argument can be interpreted in the context of the argument for any dimension. Proposition (Precise Achievement of Stable Vanishing Order at a Finite Set). The stable vanishing order is automatically precisely achieved every- where when it is precisely achieved outside a finite set of a compact complex algebraic manifold X . Proof. Let X be a compact complex algebraic manifold of general type. Let e−ϕ be the metric of minimum singularity for the canonical line bundle KX of X . Suppose it has been proved that the stable vanishing order is precisely achieved outside a finite number of points P1, · · · , Pk of X by using σ1, · · · , σℓ ∈ Γ (X,m0KX) . We are going to show that this finite number of points must be the empty set, otherwise there is a contradiction. Let e−ψ = j=1 |σj | We take p ∈ N sufficiently large to magnify the discrepancy of the vanishing orders of e−pm0ϕ and e−pψ so that the support of the quotient Ipm0ϕ /Ipψ of the multiplier ideal sheaves Ipm0ϕ and Ipψ of e−pm0ϕ and e−pψ respectively is the finite set {P1, · · · , Pk}. We now apply a slight modification to Ipψ to get a metric with strictly positive curvature current. Let e−θ be a metric of KX with strictly positive curvature current. Since e −ϕ is the metric of minimum singularity for KX , it follows that when ε > 0 is sufficiently small (which we assume to be the case) the multiplier ideal sheaf I(p−ε)ψ+εm0θ of the metric e−((p−ε)ψ+εm0θ) agrees with the multiplier ideal sheaves Ipm0ϕ of e−pm0ϕ on X − {P1, · · · , Pk} and the support of the quotient Ipm0ϕ I(p−ε)ψ+εm0θ of the multiplier ideal sheaves Ipm0ϕ and Ipψ of e−pm0ϕ and e−((p−ε)ψ+εm0θ) respectively is the finite set {P1, · · · , Pk}. By the vanishing theorem of Kawamata-Viehweg-Nadel, we have H1 (X, Ipψ ((pm0 + 1)KX)) = 0. Then the map Γ (X, Ipm0ϕ ((pm0 + 1)KX)) → Γ Ipm0ϕ I(p−ε)ψ+εm0θ ((pm0 + 1)KX) is surjective. Note that for this we do not need the vanishing of the coho- mology group H1 (X, Ipm0ϕ ((pm0 + 1)KX)) . Since Ipm0ϕ I(p−ε)ψ+εm0θ ((pm0 + 1)KX) Ipm0ϕ I(p−ε)ψ+εm0θ it follows from Nakayama’s lemma and the surjectivity of Γ (X, Ipm0ϕ ((pm0 + 1)KX)) → Ipm0ϕ I(p−ε)ψ+εm0θ that the map Γ (X, Ipm0ϕ ((pm0 + 1)KX)) → (Ipm0ϕ)Pj is surjective. This actually gives a contradiction, because the stable vanishing order of (pm0 + 1)KX should be given by e −(pm0+1)ϕ instead of by e−pm0ϕ. Q.E.D. Remarks. (a) With this proposition, to get the finite generation of the canon- ical ring for the case of a compact complex algebraic surface of general type it suffices to show that the stable vanishing order is precisely achieved at codimension one. (b) For the analytic proof of the finite generation of the canonical ring, when we get down to the point of having already verified the precise achievement of the stable vanishing order outside a finite set of points, we do not need to blow up the points to reduce the argument to the case of a hypersurface in the new blown-up manifold. Therefore the difficulty does not exist, in the case of a surface, of blowing up a point P0 to get a curve C1 and then locating some bad point P1 (where the precise achievement of stable vanishing order fails) in the curve C1 and then blowing up P1 to get a curve C2 and locating some bad point P2 (where the precise achievement of stable vanishing order fails) in the curve C2 and then blowing up P2 and possibly finally ending up with an unending infinite sequence of bad points, each in a tower of successively blown-up surfaces. Higher Codimension Argument for Threefold Case. Let X be a complex complex algebraic threefold of general type. Let e−ϕ be the metric of mini- mum singularity for the canonical line bundle KX of X . Suppose it has been proved that the stable vanishing order is precisely achieved outside a curve Cj (where each Cj is irreducible) by s1, · · · , sk ∈ Γ (X,m0KX) . Note that here we have used the Proposition given above to rule out the possibility that, besides at the curve C, the stable vanishing order may fail to be precisely achieved at a finite set of points in X − C. On X we have the canonical decomposition of ΘKX = γjYj +R of the curvature current of the metric of minimum singularity for KX . We are skipping the diophantine argument which is explained in detail in [Siu 2006] and consider as verified the rationality of each γj. By replacing m0 by an appropriate integral multiple, we can assume without loss of generality that each m0γj is a positive integer. Let s )m0γj , where sYj is the canonical section of Yj. Let L = KX − j=1 γjYj. Since we can replace σ1, · · · , σℓ by , · · · , sk ∈ Γ (X,m0L) and the essence of the rest of the argument that is to follows does not change, for notational simplicity we are going to assume that all γj = 0 so that L = KX . Also, for notational simplicity we are going to assume that the curve C is irreducible. For σ = (σ1, · · · , σk) ∈ Ck − {0} let sσ = j=1 σjsj and let Sσ be the surface in X defined by sσ. We consider those σ for which Sσ is irreducible across which sσ vanishes to order 1. By considering the blow-up of X by the ideal generated by s1, · · · , sk and the precise achievement of stable vanishing order for codimension one, after replacing m0 by a positive integral multiple if necessary, we have the following situation. (a) For σ = (σ1, · · · , σk) ∈ Ck − {0} for which Sσ is irreducible where sσ vanishes to order 1, there exist (i) some τ = (τ1, · · · , τk) ∈ Ck − {0} and (ii) some finite subset Zτ,σ of C such that the section sτ is not identically zero on Sσ and the multi- valued section sτ |Sσ (sC,σ) on a neighborhood of C − Zτ,σ in Sσ is nonzero at points of C − Zτ,σ, where ησ is the stable vanishing order at C on the surface Sσ and sC,σ is the canonical section of C in Sσ. (b) One has the second case of the dichotomy for the curvature current (ΘKX |Sσ − ησ [C])|C on C. (c) The multi-valued section sτ |Sσ (sC,σ) on C is constructed from the second case of the dichotomy for the curvature current (ΘKX |Sσ − ησ [C])|C on C. (c) The stable vanishing order ησ for C on Sσ is achieved by sτ at points of C − Zτ,σ. Note that, because of Condition(b) and Condition(c) we can choose Zτ,σ in- dependent of τ so that Zτ,σ = Zσ for some finite subset Zσ of C depending only on σ. To finish the proof of the precise achievement of the stable vanish- ing order for the threefold X , it suffices to show that there is a finite subset Z of C such that every Zσ is contained in Z. At a regular point P0 of C, there is some open neighborhood U of P0 in X such that (i) the pair (U,C) is biholomorphic to the pair (∆3, {(0, 0)} ×∆), where ∆ is the open unit disk in C, and (ii) sj|U is represented by a holomorphic function fj (z1, z2, t) on ∆3 for 1 ≤ j ≤ k. The existence of the finite subset Z of C with the property that Zτ,σ ⊂ Z follows from Lemma 2 given below. Lemma 1. Let f0, · · · , fk be holomorphic function germs on C2 at the origin so that the origin is the common zero-set of any two of the holomorphic function germs f0, · · · , fk. Let C0 be the complex curve germ at the origin defined by f0 = 0, which is assumed to be irreducible and across which f0 vanishes to order 1. Then the following numbers are the same. (i) The multiplicity of the ideal j=0OC2,0fj at the origin. (ii) The dimension over C of OC2,0 OC2,0fj . (iii) The Lelong number λ of Θ̂ := ∂∂̄ log |fj |2 on C0 at the origin. Here the Lelong number λ means the Lelong number of the pullback of the closed positive (1, 1)-current Θ̂ to the normalization of C0. (iv) The number η such that j=1 |fj| j=1 |zj | is bounded between two positive numbers near the origin on C0. Lemma 2. Let k ≥ 2 and let fj(z1, z2, t), for 1 ≤ j ≤ k, be holomorphic functions on the tri-disk ∆3 with coordinates z1, z2, t. Assume that the com- mon zero-set of any two of f1, · · · , fk is {(0, 0)} × ∆ = {z1 = z2 = 0}. For any k-tuple (a1, · · · , ak) of complex numbers not all zero let Sa1,··· ,ak be the zero-set of j=1 ajfj . Then there exists a discrete subset Z of {(0, 0)} ×∆ with the following property. For any k-tuple (a1, · · · , ak) of complex numbers not all zero there exists some nonnegative number γa1,··· ,ak such that j=1 |fj | |zj |2 )γa1,··· ,ak is continuous nonzero on some neighborhood of (Sa1,··· ,ak ∩ {(0, 0)} ×∆)−Z in Sa1,··· ,ak if Sa1,··· ,ak is irreducible and j=1 ajfj vanishes to order 1 across Sa1,··· ,ak . Proof. Observe that for fixed t if a1 6= 0 then each of j=1 |fj | |a1f1 + · · ·+ akfk|2 + j=2 |fj | is bounded by a positive constant times the other on some neighborhood of the origin in C2. Use Lemma 1. Q.E.D. Remarks. (a) We would like to highlight the intuitive geometric reason for the existence of a discrete set Z in C such that the “bad set” Zσ in C for the restriction of the closed positive (1, 1)-current ΘKX restricted to Sσ is contained in Z. The key point is that the surface Sσ is sliced out by a C- linear combination of the pluricanonical sections s1, · · · , sk and that these pluricanonical sections s1, · · · , sk have the property that the “bad set” Zσ of C can be described by the extra vanishing of j=1 |sj| beyond their generic vanishing order at points of C. (b) We need to restrict ΘKX to Sσ, because we have to take away the vanish- ing order of ΘKX at C and it is only for codimension one we can take away the vanishing order. The vanishing order on different Sσ may be different. (c) The “bad set” Zσ in C describes the points where the relative position of the pair of two Artinian subschemes in the normal direction of C jumps. One Artinian subscheme comes from the restriction of s1, · · · , sk to a local surface T normal to C and the other one comes from s1, · · · , sk plus a C-linear combination σ1s1+ · · ·+ σksk after restricting them to T . The main point is that there is a finite subset Z of C such that Zσ ⊂ Z with Z independent of the choice of σ. References. [Angehrn-Siu 1995] U. Angehrn and Y.-T. Siu, Effective freeness and point separation for adjoint bundles. Invent. Math. 122 (1995), 291–308. [Ohsawa 2002] T. Ohsawa, A precise L2 division theorem. In: Complex Geometry (Selected papers dedicated to Hans Grauert from the International Conference on Analytic and Algebraic Methods in Complex Geometry held in Göttingen, April 3–8, 2000), ed. I. Bauer, F, Catanese, Y. Kawamata, Th. Peternell and Y.-T. Siu. Springer-Verlag, Berlin, 2002, pp.185–191. [Siu 1998] Y.-T. Siu, Invariance of plurigenera. Invent. Math. 134 (1998), no. 3, 661–673. [Siu 2002] Y.-T. Siu, Extension of twisted pluricanonical sections with plurisub- harmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. In: Complex Geometry (Selected papers dedicated to Hans Grauert from the International Conference on An- alytic and Algebraic Methods in Complex Geometry held in Göttingen, April 3–8, 2000), ed. I. Bauer, F, Catanese, Y. Kawamata, Th. Peternell and Y.-T. Siu. Springer-Verlag, Berlin, 2002, pp.223–277. [Siu 2004] Y.-T. Siu, Invariance of plurigenera and torsion-freeness of direct image sheaves of pluricanonical bundles. In: Finite or infinite dimensional complex analysis and applications, Adv. Complex Anal. Appl., 2, Kluwer Acad. Publ., Dordrecht, 2004, pp. 45–83. [Siu 2006] Y.-T. Siu, A general non-vanishing theorem and an analytic proof of the finite generation of the canonical ring, arXiv:math/0610740. [Shokurov 1985] V. V. Shokurov, A nonvanishing theorem. Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), 635–651. [Skoda 1972] H. Skoda, Application des techniques L2 à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids. Ann. Sci. École Norm. Sup. 5 (1972), 545-579. http://arxiv.org/abs/math/0610740

arXiv:0704.1941v1 [math.GT] 16 Apr 2007 TAIT’S CONJECTURES AND ODD CROSSING NUMBER AMPHICHEIRAL KNOTS A. Stoimenow∗ Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan e-mail: stoimeno@kurims.kyoto-u.ac.jp WWW: http://www.kurims.kyoto-u.ac.jp/˜stoimeno/ Abstract. We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his pioneering work in tabulating the simplest knots, and solved a century later using the Jones polynomial. We announce the solution, again based on a substantial study of the Jones polynomial, of one (possibly his last re- maining?) problem of Tait, with the construction of amphicheiral knots of almost all odd crossing numbers. An application to the non-triviality problem for the Jones polynomial is also outlined. Keywords: Jones polynomial, amphicheiral knot, crossing number AMS subject classification: 57M25 (primary), 01A55, 01A60 (secondary) 1. The first knot tables Knot theory took its origins in the late 19th century. At that time, W. Thomson (“Lord Kelvin”), P. G. Tait and J. Maxwell propagated the vortex-atom theory, in an attempt to explain the structure of the universe. They believed that a super-substance, ether, makes up all of matter, and atoms are knotted tubes of ether. Knotting is hereby understood as tying a piece of rope, and then identifying its both ends so that the tying cannot be any more undone. Thus, in the realm of constructing a periodic table of elements, Tait began the catalogisation of the “simplest” knots. He depicted knots (as we still do today) by diagrams, consisting of a (smooth) plane curve with transverse self-intersections, or crossings. At each crossing one of the two strands passes over the other. The above notion of simplicity refers to the number of crossings of the diagram. Tait’s list aimed at presenting, among (knots with) diagrams of few crossings, each knot by exactly one diagram. Alternatively we can define the crossing number of a knot as the minimal crossing number of all its diagrams, and say that we seek the list of knots with given (small) crossing number. We also like knots represented by different diagrams in the list to be inequivalent, in the sense that one cannot turn a (closed) piece of rope knotted the one way into one knotted the other way, without cutting the rope. The simplest knots are shown in figure 1. The leftmost one, of crossing number 0, is the trivial knot or unknot. It has some special importance, much like the unit element in a group. Tait completed the list up to 7 crossings. Little, Kirkman, later Conway [Co] and others took over and continued his work. In the modern computer age, tables have reached the knots of 17 crossings, with millions of entries, even though Tait’s vortex-atom theory has long been dismissed. An account on knot tabulation is given, with emphasis on its history, in [S], and from a more contemporary point of view in [HTW, H]. 2. Tait’s conjectures Tait’s accomplishment allows us to call him with some right the first knot theorist. Yet Tait worked mainly by intuition. He had at his time no rigorous way of showing knots inequivalent. Tait’s reasoning is not easy to interpret precisely nowadays, had it been formulated in a language quite different from (and far less developed than) our present. Nonetheless he evidently observed several phenomena, which, apart from knot tabulation, would become a legacy to his successors. This is a preprint. I would be grateful for any comments and corrections. Current version: November 2, 2018 First version: April 13, 2007 ∗Financial support by the 21st Century COE Program is acknowledged. http://arxiv.org/abs/0704.1941v1 2 2 Tait’s conjectures unknot left-hand trefoil right-hand trefoil figure-8 knot Figure 1 It seems to remain unclear whether Tait was convinced certain properties to hold for all, or just for alternating knots. A knot is alternating, if in some (alternating) diagram the curve passes crossings interchangingly over-under like , i.e. not containing or . The knots in figure 1 are such. In fact, this is true for all knots up to 7 crossings, catalogued by Tait, and at least for a large portion of the slightly more complicated ones he was shown by his successors in his lifetime (even though it is known now that alternation is a rare property for generic crossing numbers [Th3]). Thus, certainly Tait was guided by evidence from alternating diagrams. Their occurrence in the tables suggested to him Conjecture 2.1 (Tait’s conjecture I) A reduced, i.e. not of the form , alternating diagram has min- imal crossing number (for the knot it represents). For the next problem, we need to define the writhe. If one equips (the curve of) a knot diagram with an orientation, then each crossing looks, if observed from an appropriate angle, locally like (positive crossing) or (negative crossing). The writhe is the difference between the number of former crossings and the number of latter. (An easy observation shows that the writhe is the same for either orientation.) Conjecture 2.2 (Tait’s conjecture II) Minimal crossing number diagrams of the same (alternating?) knot have the same writhe. For alternating diagrams, he conjectured more precisely the following: Conjecture 2.3 (Tait’s conjecture III) Alternating diagrams of the same knot are related by a sequence of flypes: PQ ←→ One can easily observe that a flype preserves the writhe, and so conjecture III implies conjecture II (for alternating knots1). For Tait’s last problem, we consider an amphicheiral knot. Such a knot can be turned into its mirror image. From the knots in figure 1, the unknot is obviously amphicheiral. So is the figure-eight knot, as shows a simple exercise. In contrast, the trefoil is not amphicheiral. In other words, the left-hand trefoil and its mirror image, the right-hand trefoil, are two distinct knots (a fact that stubbed knot theorists for a while, and was first proved by Max Dehn). Tait was wondering what crossing numbers amphicheiral knots can have. The evidence he had in mind can probably be formulated so: 1up to some technical issues of primeness and whether all minimal crossing diagrams are alternating. These issues are settled, but we like to skip them here for simplicity. Conjecture 2.4 (Tait’s conjecture IV) Amphicheiral (alternating?) knots have even crossing number. Note that (for alternating knots, and with the remark in footnote 1) this is a consequence of conjecture II, for mirroring a diagram interchanges positive and negative crossings, and so negates the writhe. 3. Reidemeister moves and invariants A few decades after their genesis, Tait’s knot lists would be proved right. With the work of Alexander, Reidemeister and others knot theory began to be put on a mathematical fundament. Reidemeister showed that three types of local moves (i.e. moves altering only a fragment of the diagrams, as shown below) suffice to interrelate all diagrams of the same knot. ←→ ←→ ←→ ←→ ←→ ←→ To distinguish two knots thus translates into the question how to prove that two diagrams (of these knots) are not connected by a sequence of such moves. This is done with the help of an invariant, that is, a map { knot diagrams }→ “something” , whose value in “something” does not change (is invariant) when the argument (diagram) is changed by a Reide- meister move. The question what “something” should be is justified. The answer is that it would suffice to be any set of objects whose distinctness is easy to verify, yet which is large enough to allow the invariant to take many different values. For us it will be the ring Z[t, t−1] of Laurent polynomials in one variable with integer coefficients. Clearly one can compare coefficients easier than wondering about a sequence of Reidemeister moves, which may be arbitrarily long and pass over arbitrarily complicated intermediate diagrams. (There is a fundamental, but out of our focus here, issue how to, and that one can to some extent, control these sequences [HL].) An obvious other desirable feature of an invariant is that we could easily evaluate it from a diagram. Alexander’s merit was to construct precisely such an invariant. The Alexander polynomial [Al] remained (and still remains) a main theme in knot theory for decades to come. Let us observe, though, that we actually already came across one other knot invariant, the crossing number. It is an invariant directly by definition, since it was defined on the whole Reidemeister move equivalence class of diagrams. However, this definition makes it difficult to evaluate from a diagram – in contrast to Alexander’s polynomial, for which its creator, and later many others, gave several simple procedures. 4. The Jones polynomial 60 years after Alexander, a new chapter of knot theory was opened by V. Jones, with the discovery of a successor to Alexander’s polynomial. The developments the Jones polynomial V [J] has sparked in the 20 years since its appearance are impossible even to be vaguely sketched in completeness, and go far beyond both the competence and expository intention of the author here. Several more concepts, like links, braids, geometric, Vassiliev and quantum invariants etc., are left out for simplicity and length reasons, for which the author likes to apologise at this point. A good (though still partial, and now no longer very recent) account on these issues was given by Birman [Bi]. Let us recall, though, one of the first achievements the Jones polynomial became famous with – the solution of Tait’s conjectures. Conjectures I, II and IV, for alternating knots, were proved by Kauffman [Kf2], Murasugi [Mu, Mu2] and Thistlethwaite [Th2]. We will need a few more words on Kauffman’s proof, since it uses a calculation procedure for V called state model. This state model also gives a very elementary proof that the Jones polynomial is an invariant. (Kauffman had previously developed a similar model for the Alexander polynomial, too [Kf3].) Based on Kauffman’s state model, Lickorish and Thistlethwaite [LT] defined a (semi)adequate knot and diagram. An advantage of this concept is that alternating diagrams/knots are adequate. Many details would better be skipped here, but let us say that a diagram is adequate/semiadequate if it is +adequate and/or−adequate, and that taking the mirror image of a diagram transforms the property for + into the one for −. A knot with some of these properties 4 6 Semiadequacy invariants and the non-triviality problem is defined as one that has a diagram with the feature of the same name. Thistlethwaite extended the proof of the Tait’s three conjectures to the class of adequate knots [Th], applying a 2-variable generalisation of V , the Kauffman polynomial [Kf]. For a general (non-alternating) knot, in case of conjecture II, Tait’s intuition had been proved misleading. In the 1970’s, K. Perko [Pe] observed that there are two 10 crossing knots in the tables [Ro, appendix], which are equivalent, even though their 10 crossing diagrams have different writhe. This duplication had remained unnoticed for quite a while, possibly due to the belief in Tait’s conjecture. Remedying this error (and a few other long- remained ones) in the tables still causes some confusion in their use. Tait’s conjecture III was settled a few years after the others by Menasco and Thistlethwaite [MT], mostly using geometric techniques, though again with some (now subordinate) appearance of the Jones polynomial. 5. The crossing numbers of amphicheiral knots Kauffman, Murasugi and Thistlethwaite’s proof of Tait’s conjecture IV shows that an alternating knot K of odd crossing number and its mirror image !K always have distinct Jones polynomials. (Let us in contrast remark that K and !K have the same Alexander polynomial for every knot K.) In the opposite direction, their work allows to easily find a(n alternating) amphicheiral knot of every even crossing number at least 4. But their results could not decide what crossing number non-alternating amphicheiral knots can have. The main aim of this note is to announce the complete solution to Tait’s (last?) problem. Theorem 5.1 For each odd natural number n≥ 15, there exists an amphicheiral knot of crossing number n. Similarly to Perko’s knot, a particular instance disproving Tait’s conjecture IV for non-alternating knots was found accidentally: Hoste and Thistlethwaite, in the course of routine knot tabulation, discovered an amphicheiral 15 crossing knot. (Their compilational work had previously shown that there are no amphicheiral knots of odd crossing numbers up to 13.) Settling the other crossing numbers is a major problem, though, since exhaustive enumeration is no longer a feasible approach – we face the above noticed difficulty that we do not know (generally) how to determine the crossing number. A few other methods are known, but all they fail on such examples. Thus the way to our result is rather far, and below we will conclude by just giving a brief outline of the proof. The details will appear in a separate (long) paper. We also mention another application of our approach, which addresses the non-triviality problem for the Jones polynomial. 6. Semiadequacy invariants and the non-triviality problem The Alexander polynomial was, from its very beginning, connected to topological features of knots. The situation is rather different for its successor. The problem to give a topological meaning to the Jones polynomial has bothered many knot theorists ever since this invariant appeared. So far we still find ourselves in the embarrassing state where we “can quickly fill pages with the coefficients and exponents of V for not-too-complicated knots without having the slightest idea what they mean” ([Bi, end of §3]). Similarly unsolved, and intriguing, remains the problem, formulated by Jones, if his polynomial detects the unknot. Again, “our lack of knowledge about this problem is in striking contrast to the control mathematicians now have over the Alexander polynomial: understanding its topological meaning, we also know precisely how to construct knots with Alexander polynomial 1” (ibid., rem. (iii) after theorem 3, §8; see, though, also [EKT]). In an attempt to gain more insight into the appearance of the Jones polynomial, the author, and (up to minor interaction, independently) Dasbach and Lin [DL, DL2], initiated a detailed study of some coefficients of V in semiadequate diagrams. Let us remark here that, while adequacy is only a slight extension of alternation, semiad- equacy is a rather wide extension of adequacy. (For the experts: semiadequate knots contain completely positive, Montesinos and 3-braid knots.) Semiadequacy is still a fairly general condition, yet it helps settle many technical issues. For semiadequate knots the first coefficient of the Jones polynomial is ±1, almost by definition [LT]. The outcome of our work was that we gained an understanding of coefficients 2 and 3. Their invariance allows to derive 3 invariant quantities each from a +adequate, and similarly from a −adequate diagram, called below semiadequacy References 5 invariants. Their merit is that they reflect directly certain features of the diagram, and so we have a precise idea how a semiadequate diagram with given invariants must look like. The first of them allows to prove: Theorem 6.1 Semiadequate knots have non-trivial Jones polynomial. This implies (for experts) the result also for Montesinos and 3-braid knots, but it can be proved also for their Whitehead doubles, some strongly n-trivial knots and k-almost positive knots with k ≤ 3. Our three semiadequacy invariants become also, joined by a relative obtained from the Kauffman polynomial and Thistlethwaite’s results [Th], the main tool for the proof of theorem 5.1. For given odd n ≥ 15, we start with an amphicheiral knot K that has an n crossing diagram, which is semiadequate. Luckily, such examples can be obtained by leaning on Hoste-Thistlethwaite’s knot. The work in [Th] shows then that the crossing number of K is at least n− 1, and were it n− 1, a minimal crossing diagram D would be adequate. Then we have 4 invariants for both +adequacy and −adequacy each available. A detailed study of how an n− 1 crossing diagram with such invariants must look like is necessary to exclude most cases for D. Hereby, among the various generalisations of Thistlethwaite’s knot, one must choose carefully the one whose invariants make the exclusion argument most convenient (or better to say, feasible at all). Only a small fraction of possibilities for D remain, which are easy to check, and rule out, by computer. This allows us to conclude that in fact D cannot exist. References [Al] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), 275–306. [Bi] J. S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. 28(2) (1993), 253–287. [Co] J. H. Conway, On enumeration of knots and links, in “Computational Problems in abstract algebra” (J. Leech, ed.), Pergamon Press, 1969, 329–358. [DL] O. Dasbach and X.-S. Lin, A volume-ish theorem for the Jones polynomial of alternating knots, math.GT/0403448, to appear in Pacific J. Math. [DL2] ” and ” , On the Head and the Tail of the Colored Jones Polynomial, Compositio Math. 142(5) (2006), 1332–1342. [EKT] S. Eliahou, L. H. Kauffman and M. Thistlethwaite, Infinite families of links with trivial Jones polynomial, Topology 42(1) (2003), 155–169. [HL] J. Hass and J. C. Lagarias, The number of Reidemeister moves needed for unknotting, J. Am. Math. Soc. 14(2) (2001), 399–428. [H] J. Hoste, The enumeration and classification of knots and links, “Handbook of Knot Theory” (W. Menasco and M. Thistlethwaite, eds.), Elsevier (2005), 209–232. [HTW] ” , M. Thistlethwaite and J. Weeks, The first 1,701,936 knots, Math. Intell. 20 (4) (1998), 33–48. [J] V. F. R. Jones, A polynomial invariant of knots and links via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985), 103–111. [Kf] L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), 417–471. [Kf2] ” , State models and the Jones polynomial, Topology 26 (1987), 395–407. [Kf3] ” , Formal knot theory, Mathematical Notes 30, Princeton University Press, Princeton, NJ, 1983. [LT] W. B. R. Lickorish and M. B. Thistlethwaite, Some links with non-trivial polynomials and their crossing numbers, Comment. Math. Helv. 63 (1988), 527–539. [MT] W. W. Menasco and M. B. Thistlethwaite, The Tait flyping conjecture, Bull. Amer. Math. Soc. 25 (2) (1991), 403–412. [Mu] K. Murasugi, Jones polynomial and classical conjectures in knot theory, Topology 26 (1987), 187–194. [Mu2] ” , Jones polynomials and classical conjectures in knot theory II, Math. Proc. Cambridge Philos. Soc. 102(2) (1987), 317–318. [Pe] K. A. Perko Jr., On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262–266. [Ro] D. Rolfsen, Knots and links, Publish or Perish, 1976. [S] D. S. Silver, Knot Theory’s Odd Origins, American Scientist 94(2) (2006), 158–165. [Th] M. B. Thistlethwaite, On the Kauffman polynomial of an adequate link, Invent. Math. 93(2) (1988), 285–296. [Th2] ” , A spanning tree expansion of the Jones polynomial, Topology 26(3) (1987), 297–309. [Th3] ” , On the structure and scarcity of alternating links and tangles, J. Knot Theory Ramifications 7(7) (1998), 981–1004.

We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his pioneering work in tabulating the simplest knots, and solved a century later using the Jones polynomial. We announce the solution, again based on a substantial study of the Jones polynomial, of one (possibly his last remaining?) problem of Tait, with the construction of amphicheiral knots of almost all odd crossing numbers. An application to the non-triviality problem for the Jones polynomial is also outlined.

arXiv:0704.1941v1 [math.GT] 16 Apr 2007 TAIT’S CONJECTURES AND ODD CROSSING NUMBER AMPHICHEIRAL KNOTS A. Stoimenow∗ Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan e-mail: stoimeno@kurims.kyoto-u.ac.jp WWW: http://www.kurims.kyoto-u.ac.jp/˜stoimeno/ Abstract. We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his pioneering work in tabulating the simplest knots, and solved a century later using the Jones polynomial. We announce the solution, again based on a substantial study of the Jones polynomial, of one (possibly his last re- maining?) problem of Tait, with the construction of amphicheiral knots of almost all odd crossing numbers. An application to the non-triviality problem for the Jones polynomial is also outlined. Keywords: Jones polynomial, amphicheiral knot, crossing number AMS subject classification: 57M25 (primary), 01A55, 01A60 (secondary) 1. The first knot tables Knot theory took its origins in the late 19th century. At that time, W. Thomson (“Lord Kelvin”), P. G. Tait and J. Maxwell propagated the vortex-atom theory, in an attempt to explain the structure of the universe. They believed that a super-substance, ether, makes up all of matter, and atoms are knotted tubes of ether. Knotting is hereby understood as tying a piece of rope, and then identifying its both ends so that the tying cannot be any more undone. Thus, in the realm of constructing a periodic table of elements, Tait began the catalogisation of the “simplest” knots. He depicted knots (as we still do today) by diagrams, consisting of a (smooth) plane curve with transverse self-intersections, or crossings. At each crossing one of the two strands passes over the other. The above notion of simplicity refers to the number of crossings of the diagram. Tait’s list aimed at presenting, among (knots with) diagrams of few crossings, each knot by exactly one diagram. Alternatively we can define the crossing number of a knot as the minimal crossing number of all its diagrams, and say that we seek the list of knots with given (small) crossing number. We also like knots represented by different diagrams in the list to be inequivalent, in the sense that one cannot turn a (closed) piece of rope knotted the one way into one knotted the other way, without cutting the rope. The simplest knots are shown in figure 1. The leftmost one, of crossing number 0, is the trivial knot or unknot. It has some special importance, much like the unit element in a group. Tait completed the list up to 7 crossings. Little, Kirkman, later Conway [Co] and others took over and continued his work. In the modern computer age, tables have reached the knots of 17 crossings, with millions of entries, even though Tait’s vortex-atom theory has long been dismissed. An account on knot tabulation is given, with emphasis on its history, in [S], and from a more contemporary point of view in [HTW, H]. 2. Tait’s conjectures Tait’s accomplishment allows us to call him with some right the first knot theorist. Yet Tait worked mainly by intuition. He had at his time no rigorous way of showing knots inequivalent. Tait’s reasoning is not easy to interpret precisely nowadays, had it been formulated in a language quite different from (and far less developed than) our present. Nonetheless he evidently observed several phenomena, which, apart from knot tabulation, would become a legacy to his successors. This is a preprint. I would be grateful for any comments and corrections. Current version: November 2, 2018 First version: April 13, 2007 ∗Financial support by the 21st Century COE Program is acknowledged. http://arxiv.org/abs/0704.1941v1 2 2 Tait’s conjectures unknot left-hand trefoil right-hand trefoil figure-8 knot Figure 1 It seems to remain unclear whether Tait was convinced certain properties to hold for all, or just for alternating knots. A knot is alternating, if in some (alternating) diagram the curve passes crossings interchangingly over-under like , i.e. not containing or . The knots in figure 1 are such. In fact, this is true for all knots up to 7 crossings, catalogued by Tait, and at least for a large portion of the slightly more complicated ones he was shown by his successors in his lifetime (even though it is known now that alternation is a rare property for generic crossing numbers [Th3]). Thus, certainly Tait was guided by evidence from alternating diagrams. Their occurrence in the tables suggested to him Conjecture 2.1 (Tait’s conjecture I) A reduced, i.e. not of the form , alternating diagram has min- imal crossing number (for the knot it represents). For the next problem, we need to define the writhe. If one equips (the curve of) a knot diagram with an orientation, then each crossing looks, if observed from an appropriate angle, locally like (positive crossing) or (negative crossing). The writhe is the difference between the number of former crossings and the number of latter. (An easy observation shows that the writhe is the same for either orientation.) Conjecture 2.2 (Tait’s conjecture II) Minimal crossing number diagrams of the same (alternating?) knot have the same writhe. For alternating diagrams, he conjectured more precisely the following: Conjecture 2.3 (Tait’s conjecture III) Alternating diagrams of the same knot are related by a sequence of flypes: PQ ←→ One can easily observe that a flype preserves the writhe, and so conjecture III implies conjecture II (for alternating knots1). For Tait’s last problem, we consider an amphicheiral knot. Such a knot can be turned into its mirror image. From the knots in figure 1, the unknot is obviously amphicheiral. So is the figure-eight knot, as shows a simple exercise. In contrast, the trefoil is not amphicheiral. In other words, the left-hand trefoil and its mirror image, the right-hand trefoil, are two distinct knots (a fact that stubbed knot theorists for a while, and was first proved by Max Dehn). Tait was wondering what crossing numbers amphicheiral knots can have. The evidence he had in mind can probably be formulated so: 1up to some technical issues of primeness and whether all minimal crossing diagrams are alternating. These issues are settled, but we like to skip them here for simplicity. Conjecture 2.4 (Tait’s conjecture IV) Amphicheiral (alternating?) knots have even crossing number. Note that (for alternating knots, and with the remark in footnote 1) this is a consequence of conjecture II, for mirroring a diagram interchanges positive and negative crossings, and so negates the writhe. 3. Reidemeister moves and invariants A few decades after their genesis, Tait’s knot lists would be proved right. With the work of Alexander, Reidemeister and others knot theory began to be put on a mathematical fundament. Reidemeister showed that three types of local moves (i.e. moves altering only a fragment of the diagrams, as shown below) suffice to interrelate all diagrams of the same knot. ←→ ←→ ←→ ←→ ←→ ←→ To distinguish two knots thus translates into the question how to prove that two diagrams (of these knots) are not connected by a sequence of such moves. This is done with the help of an invariant, that is, a map { knot diagrams }→ “something” , whose value in “something” does not change (is invariant) when the argument (diagram) is changed by a Reide- meister move. The question what “something” should be is justified. The answer is that it would suffice to be any set of objects whose distinctness is easy to verify, yet which is large enough to allow the invariant to take many different values. For us it will be the ring Z[t, t−1] of Laurent polynomials in one variable with integer coefficients. Clearly one can compare coefficients easier than wondering about a sequence of Reidemeister moves, which may be arbitrarily long and pass over arbitrarily complicated intermediate diagrams. (There is a fundamental, but out of our focus here, issue how to, and that one can to some extent, control these sequences [HL].) An obvious other desirable feature of an invariant is that we could easily evaluate it from a diagram. Alexander’s merit was to construct precisely such an invariant. The Alexander polynomial [Al] remained (and still remains) a main theme in knot theory for decades to come. Let us observe, though, that we actually already came across one other knot invariant, the crossing number. It is an invariant directly by definition, since it was defined on the whole Reidemeister move equivalence class of diagrams. However, this definition makes it difficult to evaluate from a diagram – in contrast to Alexander’s polynomial, for which its creator, and later many others, gave several simple procedures. 4. The Jones polynomial 60 years after Alexander, a new chapter of knot theory was opened by V. Jones, with the discovery of a successor to Alexander’s polynomial. The developments the Jones polynomial V [J] has sparked in the 20 years since its appearance are impossible even to be vaguely sketched in completeness, and go far beyond both the competence and expository intention of the author here. Several more concepts, like links, braids, geometric, Vassiliev and quantum invariants etc., are left out for simplicity and length reasons, for which the author likes to apologise at this point. A good (though still partial, and now no longer very recent) account on these issues was given by Birman [Bi]. Let us recall, though, one of the first achievements the Jones polynomial became famous with – the solution of Tait’s conjectures. Conjectures I, II and IV, for alternating knots, were proved by Kauffman [Kf2], Murasugi [Mu, Mu2] and Thistlethwaite [Th2]. We will need a few more words on Kauffman’s proof, since it uses a calculation procedure for V called state model. This state model also gives a very elementary proof that the Jones polynomial is an invariant. (Kauffman had previously developed a similar model for the Alexander polynomial, too [Kf3].) Based on Kauffman’s state model, Lickorish and Thistlethwaite [LT] defined a (semi)adequate knot and diagram. An advantage of this concept is that alternating diagrams/knots are adequate. Many details would better be skipped here, but let us say that a diagram is adequate/semiadequate if it is +adequate and/or−adequate, and that taking the mirror image of a diagram transforms the property for + into the one for −. A knot with some of these properties 4 6 Semiadequacy invariants and the non-triviality problem is defined as one that has a diagram with the feature of the same name. Thistlethwaite extended the proof of the Tait’s three conjectures to the class of adequate knots [Th], applying a 2-variable generalisation of V , the Kauffman polynomial [Kf]. For a general (non-alternating) knot, in case of conjecture II, Tait’s intuition had been proved misleading. In the 1970’s, K. Perko [Pe] observed that there are two 10 crossing knots in the tables [Ro, appendix], which are equivalent, even though their 10 crossing diagrams have different writhe. This duplication had remained unnoticed for quite a while, possibly due to the belief in Tait’s conjecture. Remedying this error (and a few other long- remained ones) in the tables still causes some confusion in their use. Tait’s conjecture III was settled a few years after the others by Menasco and Thistlethwaite [MT], mostly using geometric techniques, though again with some (now subordinate) appearance of the Jones polynomial. 5. The crossing numbers of amphicheiral knots Kauffman, Murasugi and Thistlethwaite’s proof of Tait’s conjecture IV shows that an alternating knot K of odd crossing number and its mirror image !K always have distinct Jones polynomials. (Let us in contrast remark that K and !K have the same Alexander polynomial for every knot K.) In the opposite direction, their work allows to easily find a(n alternating) amphicheiral knot of every even crossing number at least 4. But their results could not decide what crossing number non-alternating amphicheiral knots can have. The main aim of this note is to announce the complete solution to Tait’s (last?) problem. Theorem 5.1 For each odd natural number n≥ 15, there exists an amphicheiral knot of crossing number n. Similarly to Perko’s knot, a particular instance disproving Tait’s conjecture IV for non-alternating knots was found accidentally: Hoste and Thistlethwaite, in the course of routine knot tabulation, discovered an amphicheiral 15 crossing knot. (Their compilational work had previously shown that there are no amphicheiral knots of odd crossing numbers up to 13.) Settling the other crossing numbers is a major problem, though, since exhaustive enumeration is no longer a feasible approach – we face the above noticed difficulty that we do not know (generally) how to determine the crossing number. A few other methods are known, but all they fail on such examples. Thus the way to our result is rather far, and below we will conclude by just giving a brief outline of the proof. The details will appear in a separate (long) paper. We also mention another application of our approach, which addresses the non-triviality problem for the Jones polynomial. 6. Semiadequacy invariants and the non-triviality problem The Alexander polynomial was, from its very beginning, connected to topological features of knots. The situation is rather different for its successor. The problem to give a topological meaning to the Jones polynomial has bothered many knot theorists ever since this invariant appeared. So far we still find ourselves in the embarrassing state where we “can quickly fill pages with the coefficients and exponents of V for not-too-complicated knots without having the slightest idea what they mean” ([Bi, end of §3]). Similarly unsolved, and intriguing, remains the problem, formulated by Jones, if his polynomial detects the unknot. Again, “our lack of knowledge about this problem is in striking contrast to the control mathematicians now have over the Alexander polynomial: understanding its topological meaning, we also know precisely how to construct knots with Alexander polynomial 1” (ibid., rem. (iii) after theorem 3, §8; see, though, also [EKT]). In an attempt to gain more insight into the appearance of the Jones polynomial, the author, and (up to minor interaction, independently) Dasbach and Lin [DL, DL2], initiated a detailed study of some coefficients of V in semiadequate diagrams. Let us remark here that, while adequacy is only a slight extension of alternation, semiad- equacy is a rather wide extension of adequacy. (For the experts: semiadequate knots contain completely positive, Montesinos and 3-braid knots.) Semiadequacy is still a fairly general condition, yet it helps settle many technical issues. For semiadequate knots the first coefficient of the Jones polynomial is ±1, almost by definition [LT]. The outcome of our work was that we gained an understanding of coefficients 2 and 3. Their invariance allows to derive 3 invariant quantities each from a +adequate, and similarly from a −adequate diagram, called below semiadequacy References 5 invariants. Their merit is that they reflect directly certain features of the diagram, and so we have a precise idea how a semiadequate diagram with given invariants must look like. The first of them allows to prove: Theorem 6.1 Semiadequate knots have non-trivial Jones polynomial. This implies (for experts) the result also for Montesinos and 3-braid knots, but it can be proved also for their Whitehead doubles, some strongly n-trivial knots and k-almost positive knots with k ≤ 3. Our three semiadequacy invariants become also, joined by a relative obtained from the Kauffman polynomial and Thistlethwaite’s results [Th], the main tool for the proof of theorem 5.1. For given odd n ≥ 15, we start with an amphicheiral knot K that has an n crossing diagram, which is semiadequate. Luckily, such examples can be obtained by leaning on Hoste-Thistlethwaite’s knot. The work in [Th] shows then that the crossing number of K is at least n− 1, and were it n− 1, a minimal crossing diagram D would be adequate. Then we have 4 invariants for both +adequacy and −adequacy each available. A detailed study of how an n− 1 crossing diagram with such invariants must look like is necessary to exclude most cases for D. Hereby, among the various generalisations of Thistlethwaite’s knot, one must choose carefully the one whose invariants make the exclusion argument most convenient (or better to say, feasible at all). Only a small fraction of possibilities for D remain, which are easy to check, and rule out, by computer. This allows us to conclude that in fact D cannot exist. References [Al] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), 275–306. [Bi] J. S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. 28(2) (1993), 253–287. [Co] J. H. Conway, On enumeration of knots and links, in “Computational Problems in abstract algebra” (J. Leech, ed.), Pergamon Press, 1969, 329–358. [DL] O. Dasbach and X.-S. Lin, A volume-ish theorem for the Jones polynomial of alternating knots, math.GT/0403448, to appear in Pacific J. Math. [DL2] ” and ” , On the Head and the Tail of the Colored Jones Polynomial, Compositio Math. 142(5) (2006), 1332–1342. [EKT] S. Eliahou, L. H. Kauffman and M. Thistlethwaite, Infinite families of links with trivial Jones polynomial, Topology 42(1) (2003), 155–169. [HL] J. Hass and J. C. Lagarias, The number of Reidemeister moves needed for unknotting, J. Am. Math. Soc. 14(2) (2001), 399–428. [H] J. Hoste, The enumeration and classification of knots and links, “Handbook of Knot Theory” (W. Menasco and M. Thistlethwaite, eds.), Elsevier (2005), 209–232. [HTW] ” , M. Thistlethwaite and J. Weeks, The first 1,701,936 knots, Math. Intell. 20 (4) (1998), 33–48. [J] V. F. R. Jones, A polynomial invariant of knots and links via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985), 103–111. [Kf] L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), 417–471. [Kf2] ” , State models and the Jones polynomial, Topology 26 (1987), 395–407. [Kf3] ” , Formal knot theory, Mathematical Notes 30, Princeton University Press, Princeton, NJ, 1983. [LT] W. B. R. Lickorish and M. B. Thistlethwaite, Some links with non-trivial polynomials and their crossing numbers, Comment. Math. Helv. 63 (1988), 527–539. [MT] W. W. Menasco and M. B. Thistlethwaite, The Tait flyping conjecture, Bull. Amer. Math. Soc. 25 (2) (1991), 403–412. [Mu] K. Murasugi, Jones polynomial and classical conjectures in knot theory, Topology 26 (1987), 187–194. [Mu2] ” , Jones polynomials and classical conjectures in knot theory II, Math. Proc. Cambridge Philos. Soc. 102(2) (1987), 317–318. [Pe] K. A. Perko Jr., On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262–266. [Ro] D. Rolfsen, Knots and links, Publish or Perish, 1976. [S] D. S. Silver, Knot Theory’s Odd Origins, American Scientist 94(2) (2006), 158–165. [Th] M. B. Thistlethwaite, On the Kauffman polynomial of an adequate link, Invent. Math. 93(2) (1988), 285–296. [Th2] ” , A spanning tree expansion of the Jones polynomial, Topology 26(3) (1987), 297–309. [Th3] ” , On the structure and scarcity of alternating links and tangles, J. Knot Theory Ramifications 7(7) (1998), 981–1004.

untitled A revisit of the papers on the theory of relativity: Reconsideration of the hypothesis of ether-dragging Masanori Sato Honda Electronics Co., Ltd., 20 Oyamazuka, Oiwa-cho, Toyohashi, Aichi 441-3193, Japan Abstract: This paper revisits previous papers related to the theory of relativity. Afterwards, a reconsideration of the hypothesis of ether-dragging is discussed. The ether is compatible with the theory of relativity and historical experiments; this paper explains the Michelson-Morley experiment using the ether-dragging hypothesis without the orthodox interpretation that the speed c is a fixed constant in terms of any system of inertial coordinates. Key words: The theory of relativity, ether-dragging, the global positioning system (GPS), the earth-centered locally inertial (ECI) coordinate system, aberration, Michelson-Morley experiment, Sagnac effect PACS numbers: 03.30.+p 1. Introduction The theory of special relativity, proposed by Einstein [1] in 1905, was simple and intuitive. The principle is the invariance of the speed of light: light in a vacuum propagates with the speed c (a fixed constant) regardless of the motion of the light source. However, the orthodox interpretation of the theory of special relativity was derived later from the results of the Michelson-Morley experiment [2], which suggested that “the speed c is a fixed constant in terms of any system of inertial coordinates”. This orthodox interpretation is rather difficult to illustrate in the theory of special relativity: the Michelson-Morley experiment in the gravitational field of the earth cannot be discussed in the inertial coordinate; that is, the orthodox interpretation cannot be applied to the Michelson-Morley experiment. This is the starting point of this paper. The idea that light in a vacuum propagates with the speed c (a fixed constant), regardless of the state of motion of the light source was commonly accepted by the end of the 19th and early 20th centuries; this idea was represented in Maxwell’s equations and the wave equation as, , (1) where E is the amplitude of the wave, and c is the phase velocity of the wave. In those days, Maxwell and other scientists considered equation (1) to be defined in stationary coordinates. They considered that the speed of light, c, is defined in stationary coordinates, which is in the stationary ether. First of all, I would like to make this point clear: the orthodox interpretation is correct on the condition of the uniform flow of the ether; this is in the limitation of the theory of special relativity. Equation (1) has a solution )(exp kxtiE −∝ ω , (ω: frequency, k: wave number), which has a constant phase velocity kc /ω= . This representation indicates that in the ether in uniform flow, the phase velocity is always constant. For example, the Doppler shift is detected as not only the frequency ω but also the wave number k that satisfy the constant phase velocity, c: this is because the wave number k is always proportional to the frequency ω in the inertial coordinate. Figure 1 (a) illustrates the idea that light in a vacuum propagates with the speed c regardless of the state of motion of the light source. This also illustrates an idea from Einstein's 1905 paper [1]. Stationary light source Moving observer 1 Moving observer 2 Fig. 1 (b) Moving observers 1 and 2 detect the speed of light c Fig. 1 (a) Light in vacuum propagates with the speed c regardless of the state of motion of the light source Moving light source Stationary observer Figure 1 (b) shows that moving observers 1 and 2 detect the constant phase velocity of light c regardless of their motion. The constant speed of light is satisfied on the condition that both the light source and the observer are in inertial motion. As far as wave equation (1) is discussed in the inertial coordinate system, the speed of light c is always constant. On the other hand, the interpretation of the theory of special relativity that the speed, c, is a fixed constant in terms of any system of inertial coordinates is rather ambiguous. This is due to the fact that when we discuss the ether-dragging, we have to assume the gravitational field. The ether-wind or ether-drift is not observed in the inertial coordinate system; the discussion should be carried out in the theory of general relativity. To make the discussion more clear, let us consider an acoustic wave in the atmosphere, which has an isotropic constancy: although the earth moves in the solar system, we never consider the speed of the acoustic wave to be a fixed constant in any system of inertial coordinates. This is interpreted as the acoustic wave traveling in the atmosphere, which is completely dragged by the gravity of the earth. Although the motion of the earth in the solar system does not affect the speed of the acoustic wave, we never consider that the speed of an acoustic wave is a fixed constant in terms of any system of inertial coordinates. This is because, we know the gravitational field of the earth is not in the inertial coordinate system. Again, this paper starts from the simple question of how to illustrate the orthodox interpretation of this theory. I must conclude that this orthodox interpretation can be applied only in the case of a uniform ether. The frame of the wave equation does not appear to adhere to stationary coordinates; however, it does adhere to the physical constants of the medium. In the case of an acoustic wave in the atmosphere, the inertial coordinates of equation (1) are assumed to be the atmosphere. The physical coefficients are the density and the coefficient of stiffness. The wave equation of the electromagnetic wave can be interpreted using this analogy; that is, the inertial coordinates are the permittivity and the permeability around the earth. As a counterargument for the orthodox interpretation, let us consider the earth-centered locally inertial (ECI) coordinate system in the global positioning system (GPS) experiment [3]. The reason why GPS works precisely in the ECI coordinate system is that the ECI coordinate system can be considered as the stationary state. It is difficult to calculate GPS in the solar system; only the ECI coordinate system works as the stationary frame. This experimental result is an analogy to an acoustic wave in the dragged-atmosphere that is, the electromagnetic wave in the dragged-ether by the gravity around the earth. Using the analogy of an acoustic wave in the atmosphere, I use a classic hypothesis that the ether is the permittivity of free space, ε0, and the permeability of free space, µ0. This classic hypothesis was derived from the proposal by Lorentz of luminiferous ether that the absolute stationary coordinate is defined in the stationary ether. Thus, to explain the Michelson-Morley experiment, he proposed the Lorentz contraction of length. I will also show that the complete ether-dragging hypothesis is compatible with the Michelson-Morley experiment. This hypothesis was derived from the proposal by Maxwell that the Maxwell equation and wave equation are satisfied in the stationary coordinate system, i.e., the stationary ether. Maxwell predicted an ether-wind; however, the GPS experiment showed that the ether-wind was not observed at least up to 20,000 km from the ground level. Figure 2 shows that the ether is not only dragged, but also modified by gravity. The modification of the permittivity and the permeability by gravity causes a decrease in the speed of light, =c . Diffraction around the gravitational potential of the sun, as observed by Eddington [4], can be explained using this proposal that light propagates toward regions of high refractive index, that is, toward the sun. In the hypothesis that the ether is the permittivity and the permeability, the modification (increase) in the permittivity and permeability is used, rather than the curvature of spacetime. In the early 20th century, there were many great scientists who held very rigid beliefs in their own thoughts. Michelson accepted Einstein's work; however, he believed in the ether. He was not satisfied with the results of the Mt. Wilson experiment [2], and he repeated the experiment [5]. He worried that "shimmers" of air between the mountains might have fouled his results. In 1930, Fig. 2 Gravitational field illustrated by the analogy of the atmosphere of the earth. The values of the permittivity of free space, ε0, and permeability of free space, µ0, vary depending on the height. That is, the values are changed in order to satisfy the effect of the gravitational field on time dilation. Michelson’s belief in the ether brought him to his last and most ambitious test [6], the measurement of the velocity of light in a partial vacuum. His daughter, Dorothy Michelson Livingston [7], wrote that Michelson never gave up his belief in the ether. In 1924, Michelson-Gale-Pearson experiment [8] was carried out to observe the effect of the earth's rotation on the velocity of light. They assumed a fixed ether and the theory of special relativity. A fixed ether means the ether fixed to the ECI coordinate system; that is, the earth rotates in the ether. The theory of special relativity means that light in a vacuum propagates with the speed c regardless of the motion of the light source. They constructed the experimental setups using long pipes of partial vacuum. The experimental results showed the angular velocity of the earth in accordance with the theory of special relativity and the fixed ether. In those days, Michelson tried to prove the fixed ether experimentally; however the hypothesis of ether gradually disappeared. In 1985, on the Sagnac experiment using GPS [9], there was no discussion of the ether. This is because Sagnac effect as well as Michelson-Gale-Pearson experiment can be reasonably explained without the hypothesis of ether. Miller [10] was also a great scientist; he carried out the Michelson-Morley experiment with incredible enthusiasm. He was also a great experimentalist, and never changed his belief in the ether. In 1933, he reported experimental data that showed a slight seasonal and sidereal periodic fringe shift in the Michelson-Morley experiment. However, in 1955, his experimental results were re-evaluated and found to be thermal artifacts [11]. As far as the complete ether-dragging hypothesis is concerned, the null results are inevitable; thus, I believe that Miller’s experimental results showed that the interferometer measurements are affected by the motion of the earth. The null results of the GPS experiments were obtained by direct one-way measurement, which has a very high sensitivity compared to interferometer measurements. In a one-way (from the GPS satellite to the GPS station on earth) direct measurement of the speed of light, the sensitivity to a velocity of 30 km/s is calculated as 4101/000,300/30 −×=÷ skmskm . The sensitivity of the Michelson interferometer is estimated as 1050.0 000,300 1 −×= vE . Thus, the sensitivity of the direct one-way measurement is 4102 × higher than that of the Michelson interferometer. The null results are confirmed by one-way direct measurement in the GPS experiments. At that time, it was hypothesized that ether-dragging occurs around the ground level. To check this hypothesis experimentally, the Michelson-Morley experiment was carried out using massive lead blocks (one path of the interferometer was set between two lead blocks); there was no fringe shift [12]. Michelson, Miller, and others discussed partial ether-dragging at the global magnetic field level; today, the GPS experiments show that if there is ether-dragging, it will be observed as an ether-wind more than 20,000 km from the ground level. In 1951, Dirac [13, 14] referred to the ether in the context of his new electromagnetic theory. He suggested describing the ether from the viewpoint of quantum mechanics, that is, quantization of the ether. However, his interest in and discussion of the ether gradually disappeared. He wrote a book entitled “General Theory of Relativity” in 1975 [15], in which the ether was not described at all. As discussed by Dirac in the early 1950’s, the ether may exhibit physical effects in quantum phenomena. In this paper, a hypothesis of complete ether-dragging that is based on the beliefs of the great scientists is described. Thereafter, I will show that the historical experimental results are compatible with the ether. Although the hypothesis of no ether is compatible with the historical experimental results; however, they do not rule out the ether hypothesis. 2. Interpretation of historic experiments 2.1 Aberration of light is compatible with ether-dragging The aberration of light was observed by Bradley in 1725. He explained the aberration using Newton’s particle property of photons, as shown in Fig. 3. The aberration was considered to be one of the experimental results that show there is no ether-dragging around earth. Fresnel explained the aberration by assuming that the ether is unaffected by the motion of the earth [16]. This aberration is difficult to explain using the wave nature of the photon; however, it is easily explained using the particle nature of the photon. Let us consider the hypothesis that the refractive index is dragged by the earth. The refractive index of air (n) is 1.000292, the speed of light (c) is 300,000 km/s, and the velocity of the earth in the solar system (v) is 30 km/s. Thus, using Fresnel’s equation (2), we obtain 1 (2) 912,2990175.0912,29930 000292.1 000292.1 000,3001 −+∴ v This calculation shows that the refractive index of the air modifies the speed of light. However, the contribution from the dragging of the refractive index by the earth is calculated as 9105839.0912,2990175.0 −×=÷ . For simplicity, let us assume a vacuum, that is, n=1. From Fresnel’s equation (2), it can be concluded that the speed of light c is not affected by the velocity v. Therefore, we conclude that the aberration of light is clearly observed in the ether-dragging scenario. However, according to Fresnel’s equation, the refractive index n depends on the frequency of the light. Of course, this dispersion is not observed, as the ether was considered to be frequency-independent. Thus, it was said that the aberration cannot be compatible with ether-dragging. This shows that it is difficult to interpret the aberration with the wave nature of the photon. The wave nature of light does not explain the compatibility between ether-dragging and the aberration. This is because the wave property shows that a photon is dragged by the ether. Bradley explained the aberration using Newton’s particle property of photons, which gives another simple illustration of a photon traveling in a straight line in the moving ether, without changing its direction, as shown in Fig. 3. Therefore, the aberration does not rule out ether-dragging. The particle property, in the particle-wave duality of the photon, makes the explanation simple. 2.2. Doppler shift In 1842, Doppler proposed the Doppler shift of light in his treatise "On the colored light of the binary stars and some other stars of the heaven". If the star is moving toward us, the speed of a Fig 3 Aberration of light is observed in the ether-dragging: Bradley explained the aberration using Newton’s particle property of photons; however, it is difficult to explain using the wave nature of the photon. The ether is not only dragged but also fixed to the ECI coordinate system. The earth rotates in the fixed ether. v=30 km/s Wave front Dragged ether by the earth: Fixed ether to the ECI coordinate system Photon Stationary ether The earth radiated photon is not changed by the motion of the light source; however, the photon has more energy and momentum, observed as the blue Doppler shift. It is interesting that the energy and momentum of a photon depend on the motion of the light source. It appears that the energy and momentum follow the Lorentz transformation, although the speed of the photon does not. The energy and momentum of a photon depend on the motion of the light source: if the star is moving toward us, the photon has increased energy and momentum, which is the blue Doppler shift. If the observer is moving towards the light source, he detects a higher energy and momentum for the photon. Although the speed of a photon does not follow the Galilean transformation, the energy and momentum appear to follow the Lorentz transformation. Let us discuss the Doppler shift of light. Equation (3) shows the longitudinal Doppler shift, 0νν . (3) Here, νD is the Doppler frequency, and ν0 is the frequency of the source in the stationary state. To make the discussion simple, both the observer and the source are in free space; furthermore, either the observer or the source is in a stationary state, as shown in Fig. 4. The relative velocity is u. Thus, equation (3) can be used. The observer detects a Doppler frequency νD. Stationary observer Photon νD Moving source (a) Stationary observer sees moving source c u Moving observer Photon ν0 Stationary source (b) Moving observer sees stationary source Fig. 4 The Doppler shift from the viewpoint of quantum mechanics. Both the observer and the source are in free space. Either the observer or the source is in a stationary state. The relative velocity is u. Thus, equation (3) can be used. u c In Fig. 4 (a), a moving source radiates a photon of energy hνD (h: Planck’s constant); in Fig. 4 (b), the moving observer detects a photon of energy hνD. Although the speed of light, c, is constant, the moving source radiates a photon with higher energy and momentum. The moving observer then detects a photon with higher energy and momentum. In quantum mechanics, the phase velocity is defined as c=ε/µ, (ε: energy, µ: momentum), thus, the phase velocity is always constant. Let us discuss the moving observer in Fig. 4 (b) who detects a photon with speed c. It is possible to say that the moving observer detects a modified frequency and wave number as the Doppler shift. Thus, the phase velocity becomes a constant, that is, the speed of light, c. This interpretation is compatible with constant light speed, regardless of the motion of the light source and the orthodox interpretation. The discussion of the Doppler shift assumes stationary coordinates. That is, when we use equation (3), it is assumed that either the observer or the source is in a stationary state. 2.3 Sagnac effect in GPS Ashby [3] summarized the Doppler shift of carrier and modulated waves in GPS. He described that the Doppler shift of modulated wave is proportional to that of carrier. In the GPS, pulse coded modulation is used for the measurement of the distance. The Doppler shift of modulated wave is observed as the frequency change of modulated wave (i.e., wave packet). The Doppler shift of modulated wave is equivalent to the Sagnac effect in GPS. As shown in Fig. 5, the station on earth detects the Sagnac effect as well as the Doppler shift of modulated wave (Appendix). Sagnac effect shows the distance change between the light source and the observer by the motion of the observer. If the observer moves on the flight time of light, the distance between the light Fig. 5 Sagnac effect and Doppler shift of modulated wave in GPS Station on earth GPS satellite Modulated wave source and the observer is changed. For example, let the distance between the GPS satellite (signal source) and the observer on earth set 30,000 km (the distance that light travels at 0.1 second). On the equator, the speed of the ground is around 0.47 km/s. Thus the Sagnac effect is 0.047 km at the measurement of 30,000 km for the observer on the equator, which means that at the flight time of light of 0.1 second, the observer moves 0.047 km. Sagnac experiment using the GPS showed the accuracy within 2 % [3, 9]. From the Sagnac effect, it is assumed that the ether has two properties: 1) the ether is dragged with the earth, 2) the ether does not rotate with the earth; the ether is fixed to the ECI coordinate system. Thus the earth rotates in the fixed ether as shown in Fig. 5. It looks like a comet. From these discussions, there remains two possible selections; one is there is no ether, and the other is fixed ether to the ECI coordinate system. The historical experiments do not rule out the ether hypothesis. 2.4 Summary of the historic experiments revisited In this section, the previous experimental results are reconsidered with the hypothesis that the ether is the permittivity, ε0, and the permeability, µ0 of free space (i.e., the refractive index n). Table 1 shows the historic experiments and their interpretation under the ether hypothesis. Table 1 Experiments from historical papers Experiments Comments 1 Aberration of light (1725) The particle property of the photon gives a simple illustration; the photon travels in a straight line in the moving ether without changing direction. The aberration is compatible with the assumption of ether-dragging by the earth. 2 Doppler shift (1842) The energy and momentum of the photon appear to satisfy the Lorentz transformation, although the speed of the photon does not. The moving light source causes a Doppler shift. The moving observer also detects the Doppler shift. The speed of light appears to be defined in the stationary frame; that is, the ether. 3 Michelson-Morley experiment [2] (1887) In the 1887 experiment, Michelson and Morley ruled out a stationary ether. The experimental results (i.e., null results) can be interpreted as evidence for complete ether-dragging by the gravitational field of the earth. The Michelson-Morley experimental results can be explained using ether-dragging. 4 Sagnac (1913) Sagnac effect is discussed in GPS-Sagnac effect. 5 Eddington [4] (1920) Diffraction around the gravitational potential of the sun, as observed by Eddington, can be explained with the hypothesis that light propagates toward regions of high refractive index, that is, toward the sun. 6 Michelson-Gale -Pearson experiment [16] (1924) Michelson-Gale-Pearson experiment was carried out to observe the effect of the earth's rotation on the velocity of light. They assumed a fixed ether and the theory of special relativity. The experimental results showed the angular velocity of the earth in accordance with the theory of special relativity and the fixed ether. 7 GPS-Sagnac [15] (1985) Sagnac effect shows the distance change between the light source and the observer by the motion of the observer on the flight time of photon. The ether is dragged with the earth as well as fixed to the ECI coordinate system. The earth rotates in the ether. In the GPS, only the relative velocity defined in the ECI coordinate system is correct, we cannot use the relative velocity defined in the solar system. This experimental evidence reasonably explains the hypothesis of the fixed ether to the earth. 8. GPS [3] GPS satellites orbit 20,000 km above ground level, at a velocity of 4 km/s. GPS uses an earth-centered locally inertial (ECI) coordinate system. This coordinate system can be assumed to be a stationary frame. In the GPS experiment, 20,000 km from ground level, the ether-wind and Fresnel’s ether-dragging are not observed. The Michelson-Morley experiment conducted with GPS, i.e., by direct one-way measurement rather than with the use of a Michelson interferometer, confirms the constancy of the speed of light. The GPS experiments can be considered to be a reconfirmation of complete frame-dragging by the gravitational field of the earth. As described above, the complete ether-dragging hypothesis is compatible with the historical experimental results. 3. Ether-dragging and the stationary state Figure 6 illustrates ether-dragging by the gravitational fields of the earth and the sun. The ECI coordinate system (the gravitational field of the earth) drags the ether in the solar system. The solar system simultaneously drags the ether in the galaxy, and the galaxy also drags the ether in the cosmic microwave background (CMB). Thus, there are many stationary states: the ECI coordinate system, the galaxy and the CMB. If we leave the gravitational field of the earth, we are in the gravitational field of the sun (solar system), that is, the sun-centered locally inertial coordinate system. When we travel at the speed of the earth (i.e., the relative velocity to the earth is 0) in the solar system, we observe a time dilation at a velocity of 30 km/s. If we reach the gravitational field of mars, we will be in another stationary state; that is, we will be in the mars-centered locally inertial coordinate system. When we leave the solar system, we will be in the galaxy; if we travel parallel to the solar system, we observe a time dilation caused by a speed of 230 km/s (the speed of the solar system in the galaxy). There are many stationary states in the solar system, for example, on the earth and mars. This is similar to the example of an acoustic wave, in which we have many stationary states, for example, in a moving train or in a flying airplane. It is possible that ether-dragging occurs on larger scales (that is, the gravitational field scale) than were predicted by Michelson and Miller. They believed in the ether, and it is possible that their belief could be confirmed in space physics. 4. Discussion 4.1 The wave-particle duality in the theory of relativity In the above discussions, I used the wave and particle properties. It is not easy to explain the experimental results using only one property. For example, the aberration is explained using the particle property, and I cannot explain the aberration with the wave property. On the other hand, the Doppler shift is difficult to explain only by the particle property; the wave property is needed. In the interpretation of the theory of relativity, particle-wave duality is required. Thus, the ether-dragging hypothesis is explained using the wave-particle duality; I cannot consistently explain this hypothesis using only one of the two properties. Therefore, the ether-dragging hypothesis should be checked experimentally with space physics. 4.2 Merits and drawbacks of the ether Let us discuss the merits and faults of the ether hypothesis. The faults of the ether hypothesis are that it is unnecessary as far as explaining observed phenomena, and that it has never been observed; however, it is harmless and eases the understanding of the theory of special relativity. The concept of the ether is very convenient; it defines a stationary reference frame. Thus, a relative velocity is also defined. Fig. 6 Illustration of the ether-dragging by the gravitational fields of the earth and the sun. Not only the earth but the sun also drags the ether in the galaxy, which moves in the cosmic microwave background (CMB). Thus, there are many stationary states; this is because the gravitational fields are the stationary states. For examples, the ECI coordinate system, the solar system, the galaxy and the CMB. ECI coordinate system v=230 km/s Cosmic microwave background Solar system v=30 km/s Galaxy v=700 km/s As described above, one of the candidates for the ether is the permittivity of free space, ε0, and the permeability of free space, µ0. These are very familiar from electromagnetic theory; however, the physical meanings of the permittivity and the permeability have not been studied thoroughly enough. The merit of the ether hypothesis is that a stationary reference frame can be defined. This stationary reference frame is compatible with the theory of special relativity; however, it is not compatible with the interpretation of special relativity. I only disagree with the interpretation, which is very strong and usually used in the discussion of the gravitational field of the earth. At first, I considered the ether to be an additional concept to aid in understanding the theory of special relativity; at this stage, however, I have come to believe in the physical reality of the ether. I believe that the ether is compatible with the theory of relativity; the Michelson-Morley experiment can be solved using the theory of general relativity without the orthodox interpretation. 5. Conclusion The consideration of the ether began because I could not illustrate the expression “the speed, c, is a fixed constant in any system of inertial coordinates.” In this study, I found that many great scientists believed in the ether. It was surprising for me to read their splendid papers. The ether is compatible with the theory of relativity; furthermore, I think the ether is useful for understanding the theory of relativity. Although the ether is said to be superfluous, it appears to have a physical reality. I believe that searching for the ether opens up possibilities for new physics. For example, the quantization of the ether, as pointed out by Dirac, is very attractive. There is a possibility that experiments in space physics may obtain new results regarding the ether. Today, we have new information from quantum mechanics and the GPS experiments. In the days of these highly valuable historic papers, the authors did not have such information. Therefore, it is important to revisit historic papers from the viewpoint of modern physics. References 1) A. Einstein, "On the Electrodynamics of Moving Bodies," Annalen der Physik, 17, 891, (1905), in English translation; http://www.fourmilab.ch/etexts/einstein/specrel/www/. 2) A. Michelson, and E. Morley, "On the Relative Motion of the Earth and the Luminiferous Ether", American Journal of Science, Third Series, 34, 333, (1887), http://www.aip.org/history/gap/PDF/michelson.pdf. 3) N. Ashby, “Relativity in the Global Positioning System,” www.livingreviews.org/Articles/Volume6/2003-1ashby, (2003). 4) A. Eddington and C. Davidson, "A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919", Phil. Trans. Roy. Soc. A 220, 291, (1920). 5) A. Michelson, F. Pease, and F. Pearson, "Repetition of the Michelson-Morley Experiment", Nature, 123, 88, (1929). 6) A. Michelson, F. Pease, and F. Pearson: "Measurement of the Velocity of Light in a Partial Vacuum", Astrophysical J., 82, 26, (1935). 7) Dorothy Michelson Livingston, "Master of Light: A Biography of Albert A. Michelson," Univ of Chicago Pr (Tx), (1979). 8) A. Michelson, H. Gale, "The Effect of the Earth's Rotation on the Velocity of Light", The Astrophysical Journal, 61, 140-145, (1924), http://adsabs.harvard.edu/abs/1925ApJ....61..140M. 9) D. Allan, M. Weiss, and N. Ashby, “Around-the-World Relativistic Sagnac Experiment”, Science, 228, 69–70, (1985). 10) D. Miller, "The Ether-Drift Experiment and the Determination of the Absolute Motion of the Earth", Reviews of Modern Physics, 5, 203, (1933). 11) R. Shankland, S. McCuskey, F. Leone, and G. Kuerti, "New Analysis of the Interferometer Observations of Dayton C. Miller," Rev. Mod. Phys., 27, 167, (1955). 12) G. Hammar, "The Velocity of Light within a Massive Enclosure," Physical Review 48, 462, (1935). 13) P. Dirac "Is there an ether?" Nature, 168, 906, (1951). 14) P. Dirac "Is there an ether?" Nature, 169, 702, (1952). 15) P. Dirac, "General Theory of Relativity", (Princeton Landmarks in Mathematics and Physics), (1975). 16) C. Misner, K. Thorne, J. Wheeler, “Gravitation”, Freeman, (1973). Appendix: Sagnac effect on the group velocity Let us discuss the group velocity change depending on the observer’s velocity. Sagnac effect is observed as the distance change between the light source and the observer by the motion of the observer. Let us consider two observers on the equator. The two observers 1 and 2 are combined by a rigid rod of length L as shown in Fig. A. Not only observer 1 but also observer 2 observes the Sagnac effect. That is, observer 2 moves during the flight time of light between observers 1 and 2. Thus, observer 2 also detects Sagnac effect of 61057.1 −× ; which indicates that a light reaches earlier to the observers than in the stationary state. Considering the Lorentz transformation, the group velocity is calculated as, . (A-1) Where, γ is the Lorentz factor, t1 is the time when a light reaches observer 1, t2 is the time when a light reaches observer 2, and 12 tttS −=∆ . Let the differential time of stationary observers to set 0t∆ , thus we obtain, The Lorentz factor appears both in length and time; therefore it is cancelled as shown in equation (2). Figure B shows the Sagnac effect between observers 1 and 2; after observer 1 detects the coded wave, observer 2 moves towards the GPS satellite. At the time t2, observer 2 moves vtS ×∆ . From Fig. B we obtain, vtLct SS ×∆−=×∆ . (A-2) =∆∴ . (A-3) G +=⎟ . (A-5) The Sagnac effect in GPS data shows group velocity change due to the motion of the observer. Fig. A Sagnac effect on the group velocity measurement using GPS: pulse coded signal is detected by observers 1 and 2. The detected times t1 and t2 suffer the Sagnac effect. Thus, the differential time 12 tttS −=∆ becomes smaller than 0t∆ in which observers 1 and 2 are in the stationary states. Observer 2 Observer 1 GPS satellite Coded wave: group velocity c t2 t1 L Observer’s velocity v vtS ×∆ ctS ×∆ Fig. B Derivation of the equation of Sagnac effect: after observer 1 detects the coded wave, observer 2 moves towards the GPS satellite. At the time t2, when the coded wave reaches at observer 2, which moves vtS ×∆ . Thus, we obtain vtLct SS ×∆−=×∆ . t2 t1 Observer 2 Observer 1

This paper revisits previous papers related to the theory of relativity. Afterwards, a reconsideration of the hypothesis of ether-dragging is discussed. The ether is compatible with the theory of relativity and historical experiments; this paper explains the Michelson-Morley experiment using the ether-dragging hypothesis without the orthodox interpretation that the speed c is a fixed constant in terms of any system of inertial coordinates.

Introduction The theory of special relativity, proposed by Einstein [1] in 1905, was simple and intuitive. The principle is the invariance of the speed of light: light in a vacuum propagates with the speed c (a fixed constant) regardless of the motion of the light source. However, the orthodox interpretation of the theory of special relativity was derived later from the results of the Michelson-Morley experiment [2], which suggested that “the speed c is a fixed constant in terms of any system of inertial coordinates”. This orthodox interpretation is rather difficult to illustrate in the theory of special relativity: the Michelson-Morley experiment in the gravitational field of the earth cannot be discussed in the inertial coordinate; that is, the orthodox interpretation cannot be applied to the Michelson-Morley experiment. This is the starting point of this paper. The idea that light in a vacuum propagates with the speed c (a fixed constant), regardless of the state of motion of the light source was commonly accepted by the end of the 19th and early 20th centuries; this idea was represented in Maxwell’s equations and the wave equation as, , (1) where E is the amplitude of the wave, and c is the phase velocity of the wave. In those days, Maxwell and other scientists considered equation (1) to be defined in stationary coordinates. They considered that the speed of light, c, is defined in stationary coordinates, which is in the stationary ether. First of all, I would like to make this point clear: the orthodox interpretation is correct on the condition of the uniform flow of the ether; this is in the limitation of the theory of special relativity. Equation (1) has a solution )(exp kxtiE −∝ ω , (ω: frequency, k: wave number), which has a constant phase velocity kc /ω= . This representation indicates that in the ether in uniform flow, the phase velocity is always constant. For example, the Doppler shift is detected as not only the frequency ω but also the wave number k that satisfy the constant phase velocity, c: this is because the wave number k is always proportional to the frequency ω in the inertial coordinate. Figure 1 (a) illustrates the idea that light in a vacuum propagates with the speed c regardless of the state of motion of the light source. This also illustrates an idea from Einstein's 1905 paper [1]. Stationary light source Moving observer 1 Moving observer 2 Fig. 1 (b) Moving observers 1 and 2 detect the speed of light c Fig. 1 (a) Light in vacuum propagates with the speed c regardless of the state of motion of the light source Moving light source Stationary observer Figure 1 (b) shows that moving observers 1 and 2 detect the constant phase velocity of light c regardless of their motion. The constant speed of light is satisfied on the condition that both the light source and the observer are in inertial motion. As far as wave equation (1) is discussed in the inertial coordinate system, the speed of light c is always constant. On the other hand, the interpretation of the theory of special relativity that the speed, c, is a fixed constant in terms of any system of inertial coordinates is rather ambiguous. This is due to the fact that when we discuss the ether-dragging, we have to assume the gravitational field. The ether-wind or ether-drift is not observed in the inertial coordinate system; the discussion should be carried out in the theory of general relativity. To make the discussion more clear, let us consider an acoustic wave in the atmosphere, which has an isotropic constancy: although the earth moves in the solar system, we never consider the speed of the acoustic wave to be a fixed constant in any system of inertial coordinates. This is interpreted as the acoustic wave traveling in the atmosphere, which is completely dragged by the gravity of the earth. Although the motion of the earth in the solar system does not affect the speed of the acoustic wave, we never consider that the speed of an acoustic wave is a fixed constant in terms of any system of inertial coordinates. This is because, we know the gravitational field of the earth is not in the inertial coordinate system. Again, this paper starts from the simple question of how to illustrate the orthodox interpretation of this theory. I must conclude that this orthodox interpretation can be applied only in the case of a uniform ether. The frame of the wave equation does not appear to adhere to stationary coordinates; however, it does adhere to the physical constants of the medium. In the case of an acoustic wave in the atmosphere, the inertial coordinates of equation (1) are assumed to be the atmosphere. The physical coefficients are the density and the coefficient of stiffness. The wave equation of the electromagnetic wave can be interpreted using this analogy; that is, the inertial coordinates are the permittivity and the permeability around the earth. As a counterargument for the orthodox interpretation, let us consider the earth-centered locally inertial (ECI) coordinate system in the global positioning system (GPS) experiment [3]. The reason why GPS works precisely in the ECI coordinate system is that the ECI coordinate system can be considered as the stationary state. It is difficult to calculate GPS in the solar system; only the ECI coordinate system works as the stationary frame. This experimental result is an analogy to an acoustic wave in the dragged-atmosphere that is, the electromagnetic wave in the dragged-ether by the gravity around the earth. Using the analogy of an acoustic wave in the atmosphere, I use a classic hypothesis that the ether is the permittivity of free space, ε0, and the permeability of free space, µ0. This classic hypothesis was derived from the proposal by Lorentz of luminiferous ether that the absolute stationary coordinate is defined in the stationary ether. Thus, to explain the Michelson-Morley experiment, he proposed the Lorentz contraction of length. I will also show that the complete ether-dragging hypothesis is compatible with the Michelson-Morley experiment. This hypothesis was derived from the proposal by Maxwell that the Maxwell equation and wave equation are satisfied in the stationary coordinate system, i.e., the stationary ether. Maxwell predicted an ether-wind; however, the GPS experiment showed that the ether-wind was not observed at least up to 20,000 km from the ground level. Figure 2 shows that the ether is not only dragged, but also modified by gravity. The modification of the permittivity and the permeability by gravity causes a decrease in the speed of light, =c . Diffraction around the gravitational potential of the sun, as observed by Eddington [4], can be explained using this proposal that light propagates toward regions of high refractive index, that is, toward the sun. In the hypothesis that the ether is the permittivity and the permeability, the modification (increase) in the permittivity and permeability is used, rather than the curvature of spacetime. In the early 20th century, there were many great scientists who held very rigid beliefs in their own thoughts. Michelson accepted Einstein's work; however, he believed in the ether. He was not satisfied with the results of the Mt. Wilson experiment [2], and he repeated the experiment [5]. He worried that "shimmers" of air between the mountains might have fouled his results. In 1930, Fig. 2 Gravitational field illustrated by the analogy of the atmosphere of the earth. The values of the permittivity of free space, ε0, and permeability of free space, µ0, vary depending on the height. That is, the values are changed in order to satisfy the effect of the gravitational field on time dilation. Michelson’s belief in the ether brought him to his last and most ambitious test [6], the measurement of the velocity of light in a partial vacuum. His daughter, Dorothy Michelson Livingston [7], wrote that Michelson never gave up his belief in the ether. In 1924, Michelson-Gale-Pearson experiment [8] was carried out to observe the effect of the earth's rotation on the velocity of light. They assumed a fixed ether and the theory of special relativity. A fixed ether means the ether fixed to the ECI coordinate system; that is, the earth rotates in the ether. The theory of special relativity means that light in a vacuum propagates with the speed c regardless of the motion of the light source. They constructed the experimental setups using long pipes of partial vacuum. The experimental results showed the angular velocity of the earth in accordance with the theory of special relativity and the fixed ether. In those days, Michelson tried to prove the fixed ether experimentally; however the hypothesis of ether gradually disappeared. In 1985, on the Sagnac experiment using GPS [9], there was no discussion of the ether. This is because Sagnac effect as well as Michelson-Gale-Pearson experiment can be reasonably explained without the hypothesis of ether. Miller [10] was also a great scientist; he carried out the Michelson-Morley experiment with incredible enthusiasm. He was also a great experimentalist, and never changed his belief in the ether. In 1933, he reported experimental data that showed a slight seasonal and sidereal periodic fringe shift in the Michelson-Morley experiment. However, in 1955, his experimental results were re-evaluated and found to be thermal artifacts [11]. As far as the complete ether-dragging hypothesis is concerned, the null results are inevitable; thus, I believe that Miller’s experimental results showed that the interferometer measurements are affected by the motion of the earth. The null results of the GPS experiments were obtained by direct one-way measurement, which has a very high sensitivity compared to interferometer measurements. In a one-way (from the GPS satellite to the GPS station on earth) direct measurement of the speed of light, the sensitivity to a velocity of 30 km/s is calculated as 4101/000,300/30 −×=÷ skmskm . The sensitivity of the Michelson interferometer is estimated as 1050.0 000,300 1 −×= vE . Thus, the sensitivity of the direct one-way measurement is 4102 × higher than that of the Michelson interferometer. The null results are confirmed by one-way direct measurement in the GPS experiments. At that time, it was hypothesized that ether-dragging occurs around the ground level. To check this hypothesis experimentally, the Michelson-Morley experiment was carried out using massive lead blocks (one path of the interferometer was set between two lead blocks); there was no fringe shift [12]. Michelson, Miller, and others discussed partial ether-dragging at the global magnetic field level; today, the GPS experiments show that if there is ether-dragging, it will be observed as an ether-wind more than 20,000 km from the ground level. In 1951, Dirac [13, 14] referred to the ether in the context of his new electromagnetic theory. He suggested describing the ether from the viewpoint of quantum mechanics, that is, quantization of the ether. However, his interest in and discussion of the ether gradually disappeared. He wrote a book entitled “General Theory of Relativity” in 1975 [15], in which the ether was not described at all. As discussed by Dirac in the early 1950’s, the ether may exhibit physical effects in quantum phenomena. In this paper, a hypothesis of complete ether-dragging that is based on the beliefs of the great scientists is described. Thereafter, I will show that the historical experimental results are compatible with the ether. Although the hypothesis of no ether is compatible with the historical experimental results; however, they do not rule out the ether hypothesis. 2. Interpretation of historic experiments 2.1 Aberration of light is compatible with ether-dragging The aberration of light was observed by Bradley in 1725. He explained the aberration using Newton’s particle property of photons, as shown in Fig. 3. The aberration was considered to be one of the experimental results that show there is no ether-dragging around earth. Fresnel explained the aberration by assuming that the ether is unaffected by the motion of the earth [16]. This aberration is difficult to explain using the wave nature of the photon; however, it is easily explained using the particle nature of the photon. Let us consider the hypothesis that the refractive index is dragged by the earth. The refractive index of air (n) is 1.000292, the speed of light (c) is 300,000 km/s, and the velocity of the earth in the solar system (v) is 30 km/s. Thus, using Fresnel’s equation (2), we obtain 1 (2) 912,2990175.0912,29930 000292.1 000292.1 000,3001 −+∴ v This calculation shows that the refractive index of the air modifies the speed of light. However, the contribution from the dragging of the refractive index by the earth is calculated as 9105839.0912,2990175.0 −×=÷ . For simplicity, let us assume a vacuum, that is, n=1. From Fresnel’s equation (2), it can be concluded that the speed of light c is not affected by the velocity v. Therefore, we conclude that the aberration of light is clearly observed in the ether-dragging scenario. However, according to Fresnel’s equation, the refractive index n depends on the frequency of the light. Of course, this dispersion is not observed, as the ether was considered to be frequency-independent. Thus, it was said that the aberration cannot be compatible with ether-dragging. This shows that it is difficult to interpret the aberration with the wave nature of the photon. The wave nature of light does not explain the compatibility between ether-dragging and the aberration. This is because the wave property shows that a photon is dragged by the ether. Bradley explained the aberration using Newton’s particle property of photons, which gives another simple illustration of a photon traveling in a straight line in the moving ether, without changing its direction, as shown in Fig. 3. Therefore, the aberration does not rule out ether-dragging. The particle property, in the particle-wave duality of the photon, makes the explanation simple. 2.2. Doppler shift In 1842, Doppler proposed the Doppler shift of light in his treatise "On the colored light of the binary stars and some other stars of the heaven". If the star is moving toward us, the speed of a Fig 3 Aberration of light is observed in the ether-dragging: Bradley explained the aberration using Newton’s particle property of photons; however, it is difficult to explain using the wave nature of the photon. The ether is not only dragged but also fixed to the ECI coordinate system. The earth rotates in the fixed ether. v=30 km/s Wave front Dragged ether by the earth: Fixed ether to the ECI coordinate system Photon Stationary ether The earth radiated photon is not changed by the motion of the light source; however, the photon has more energy and momentum, observed as the blue Doppler shift. It is interesting that the energy and momentum of a photon depend on the motion of the light source. It appears that the energy and momentum follow the Lorentz transformation, although the speed of the photon does not. The energy and momentum of a photon depend on the motion of the light source: if the star is moving toward us, the photon has increased energy and momentum, which is the blue Doppler shift. If the observer is moving towards the light source, he detects a higher energy and momentum for the photon. Although the speed of a photon does not follow the Galilean transformation, the energy and momentum appear to follow the Lorentz transformation. Let us discuss the Doppler shift of light. Equation (3) shows the longitudinal Doppler shift, 0νν . (3) Here, νD is the Doppler frequency, and ν0 is the frequency of the source in the stationary state. To make the discussion simple, both the observer and the source are in free space; furthermore, either the observer or the source is in a stationary state, as shown in Fig. 4. The relative velocity is u. Thus, equation (3) can be used. The observer detects a Doppler frequency νD. Stationary observer Photon νD Moving source (a) Stationary observer sees moving source c u Moving observer Photon ν0 Stationary source (b) Moving observer sees stationary source Fig. 4 The Doppler shift from the viewpoint of quantum mechanics. Both the observer and the source are in free space. Either the observer or the source is in a stationary state. The relative velocity is u. Thus, equation (3) can be used. u c In Fig. 4 (a), a moving source radiates a photon of energy hνD (h: Planck’s constant); in Fig. 4 (b), the moving observer detects a photon of energy hνD. Although the speed of light, c, is constant, the moving source radiates a photon with higher energy and momentum. The moving observer then detects a photon with higher energy and momentum. In quantum mechanics, the phase velocity is defined as c=ε/µ, (ε: energy, µ: momentum), thus, the phase velocity is always constant. Let us discuss the moving observer in Fig. 4 (b) who detects a photon with speed c. It is possible to say that the moving observer detects a modified frequency and wave number as the Doppler shift. Thus, the phase velocity becomes a constant, that is, the speed of light, c. This interpretation is compatible with constant light speed, regardless of the motion of the light source and the orthodox interpretation. The discussion of the Doppler shift assumes stationary coordinates. That is, when we use equation (3), it is assumed that either the observer or the source is in a stationary state. 2.3 Sagnac effect in GPS Ashby [3] summarized the Doppler shift of carrier and modulated waves in GPS. He described that the Doppler shift of modulated wave is proportional to that of carrier. In the GPS, pulse coded modulation is used for the measurement of the distance. The Doppler shift of modulated wave is observed as the frequency change of modulated wave (i.e., wave packet). The Doppler shift of modulated wave is equivalent to the Sagnac effect in GPS. As shown in Fig. 5, the station on earth detects the Sagnac effect as well as the Doppler shift of modulated wave (Appendix). Sagnac effect shows the distance change between the light source and the observer by the motion of the observer. If the observer moves on the flight time of light, the distance between the light Fig. 5 Sagnac effect and Doppler shift of modulated wave in GPS Station on earth GPS satellite Modulated wave source and the observer is changed. For example, let the distance between the GPS satellite (signal source) and the observer on earth set 30,000 km (the distance that light travels at 0.1 second). On the equator, the speed of the ground is around 0.47 km/s. Thus the Sagnac effect is 0.047 km at the measurement of 30,000 km for the observer on the equator, which means that at the flight time of light of 0.1 second, the observer moves 0.047 km. Sagnac experiment using the GPS showed the accuracy within 2 % [3, 9]. From the Sagnac effect, it is assumed that the ether has two properties: 1) the ether is dragged with the earth, 2) the ether does not rotate with the earth; the ether is fixed to the ECI coordinate system. Thus the earth rotates in the fixed ether as shown in Fig. 5. It looks like a comet. From these discussions, there remains two possible selections; one is there is no ether, and the other is fixed ether to the ECI coordinate system. The historical experiments do not rule out the ether hypothesis. 2.4 Summary of the historic experiments revisited In this section, the previous experimental results are reconsidered with the hypothesis that the ether is the permittivity, ε0, and the permeability, µ0 of free space (i.e., the refractive index n). Table 1 shows the historic experiments and their interpretation under the ether hypothesis. Table 1 Experiments from historical papers Experiments Comments 1 Aberration of light (1725) The particle property of the photon gives a simple illustration; the photon travels in a straight line in the moving ether without changing direction. The aberration is compatible with the assumption of ether-dragging by the earth. 2 Doppler shift (1842) The energy and momentum of the photon appear to satisfy the Lorentz transformation, although the speed of the photon does not. The moving light source causes a Doppler shift. The moving observer also detects the Doppler shift. The speed of light appears to be defined in the stationary frame; that is, the ether. 3 Michelson-Morley experiment [2] (1887) In the 1887 experiment, Michelson and Morley ruled out a stationary ether. The experimental results (i.e., null results) can be interpreted as evidence for complete ether-dragging by the gravitational field of the earth. The Michelson-Morley experimental results can be explained using ether-dragging. 4 Sagnac (1913) Sagnac effect is discussed in GPS-Sagnac effect. 5 Eddington [4] (1920) Diffraction around the gravitational potential of the sun, as observed by Eddington, can be explained with the hypothesis that light propagates toward regions of high refractive index, that is, toward the sun. 6 Michelson-Gale -Pearson experiment [16] (1924) Michelson-Gale-Pearson experiment was carried out to observe the effect of the earth's rotation on the velocity of light. They assumed a fixed ether and the theory of special relativity. The experimental results showed the angular velocity of the earth in accordance with the theory of special relativity and the fixed ether. 7 GPS-Sagnac [15] (1985) Sagnac effect shows the distance change between the light source and the observer by the motion of the observer on the flight time of photon. The ether is dragged with the earth as well as fixed to the ECI coordinate system. The earth rotates in the ether. In the GPS, only the relative velocity defined in the ECI coordinate system is correct, we cannot use the relative velocity defined in the solar system. This experimental evidence reasonably explains the hypothesis of the fixed ether to the earth. 8. GPS [3] GPS satellites orbit 20,000 km above ground level, at a velocity of 4 km/s. GPS uses an earth-centered locally inertial (ECI) coordinate system. This coordinate system can be assumed to be a stationary frame. In the GPS experiment, 20,000 km from ground level, the ether-wind and Fresnel’s ether-dragging are not observed. The Michelson-Morley experiment conducted with GPS, i.e., by direct one-way measurement rather than with the use of a Michelson interferometer, confirms the constancy of the speed of light. The GPS experiments can be considered to be a reconfirmation of complete frame-dragging by the gravitational field of the earth. As described above, the complete ether-dragging hypothesis is compatible with the historical experimental results. 3. Ether-dragging and the stationary state Figure 6 illustrates ether-dragging by the gravitational fields of the earth and the sun. The ECI coordinate system (the gravitational field of the earth) drags the ether in the solar system. The solar system simultaneously drags the ether in the galaxy, and the galaxy also drags the ether in the cosmic microwave background (CMB). Thus, there are many stationary states: the ECI coordinate system, the galaxy and the CMB. If we leave the gravitational field of the earth, we are in the gravitational field of the sun (solar system), that is, the sun-centered locally inertial coordinate system. When we travel at the speed of the earth (i.e., the relative velocity to the earth is 0) in the solar system, we observe a time dilation at a velocity of 30 km/s. If we reach the gravitational field of mars, we will be in another stationary state; that is, we will be in the mars-centered locally inertial coordinate system. When we leave the solar system, we will be in the galaxy; if we travel parallel to the solar system, we observe a time dilation caused by a speed of 230 km/s (the speed of the solar system in the galaxy). There are many stationary states in the solar system, for example, on the earth and mars. This is similar to the example of an acoustic wave, in which we have many stationary states, for example, in a moving train or in a flying airplane. It is possible that ether-dragging occurs on larger scales (that is, the gravitational field scale) than were predicted by Michelson and Miller. They believed in the ether, and it is possible that their belief could be confirmed in space physics. 4. Discussion 4.1 The wave-particle duality in the theory of relativity In the above discussions, I used the wave and particle properties. It is not easy to explain the experimental results using only one property. For example, the aberration is explained using the particle property, and I cannot explain the aberration with the wave property. On the other hand, the Doppler shift is difficult to explain only by the particle property; the wave property is needed. In the interpretation of the theory of relativity, particle-wave duality is required. Thus, the ether-dragging hypothesis is explained using the wave-particle duality; I cannot consistently explain this hypothesis using only one of the two properties. Therefore, the ether-dragging hypothesis should be checked experimentally with space physics. 4.2 Merits and drawbacks of the ether Let us discuss the merits and faults of the ether hypothesis. The faults of the ether hypothesis are that it is unnecessary as far as explaining observed phenomena, and that it has never been observed; however, it is harmless and eases the understanding of the theory of special relativity. The concept of the ether is very convenient; it defines a stationary reference frame. Thus, a relative velocity is also defined. Fig. 6 Illustration of the ether-dragging by the gravitational fields of the earth and the sun. Not only the earth but the sun also drags the ether in the galaxy, which moves in the cosmic microwave background (CMB). Thus, there are many stationary states; this is because the gravitational fields are the stationary states. For examples, the ECI coordinate system, the solar system, the galaxy and the CMB. ECI coordinate system v=230 km/s Cosmic microwave background Solar system v=30 km/s Galaxy v=700 km/s As described above, one of the candidates for the ether is the permittivity of free space, ε0, and the permeability of free space, µ0. These are very familiar from electromagnetic theory; however, the physical meanings of the permittivity and the permeability have not been studied thoroughly enough. The merit of the ether hypothesis is that a stationary reference frame can be defined. This stationary reference frame is compatible with the theory of special relativity; however, it is not compatible with the interpretation of special relativity. I only disagree with the interpretation, which is very strong and usually used in the discussion of the gravitational field of the earth. At first, I considered the ether to be an additional concept to aid in understanding the theory of special relativity; at this stage, however, I have come to believe in the physical reality of the ether. I believe that the ether is compatible with the theory of relativity; the Michelson-Morley experiment can be solved using the theory of general relativity without the orthodox interpretation. 5. Conclusion The consideration of the ether began because I could not illustrate the expression “the speed, c, is a fixed constant in any system of inertial coordinates.” In this study, I found that many great scientists believed in the ether. It was surprising for me to read their splendid papers. The ether is compatible with the theory of relativity; furthermore, I think the ether is useful for understanding the theory of relativity. Although the ether is said to be superfluous, it appears to have a physical reality. I believe that searching for the ether opens up possibilities for new physics. For example, the quantization of the ether, as pointed out by Dirac, is very attractive. There is a possibility that experiments in space physics may obtain new results regarding the ether. Today, we have new information from quantum mechanics and the GPS experiments. In the days of these highly valuable historic papers, the authors did not have such information. Therefore, it is important to revisit historic papers from the viewpoint of modern physics. References 1) A. Einstein, "On the Electrodynamics of Moving Bodies," Annalen der Physik, 17, 891, (1905), in English translation; http://www.fourmilab.ch/etexts/einstein/specrel/www/. 2) A. Michelson, and E. Morley, "On the Relative Motion of the Earth and the Luminiferous Ether", American Journal of Science, Third Series, 34, 333, (1887), http://www.aip.org/history/gap/PDF/michelson.pdf. 3) N. Ashby, “Relativity in the Global Positioning System,” www.livingreviews.org/Articles/Volume6/2003-1ashby, (2003). 4) A. Eddington and C. Davidson, "A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919", Phil. Trans. Roy. Soc. A 220, 291, (1920). 5) A. Michelson, F. Pease, and F. Pearson, "Repetition of the Michelson-Morley Experiment", Nature, 123, 88, (1929). 6) A. Michelson, F. Pease, and F. Pearson: "Measurement of the Velocity of Light in a Partial Vacuum", Astrophysical J., 82, 26, (1935). 7) Dorothy Michelson Livingston, "Master of Light: A Biography of Albert A. Michelson," Univ of Chicago Pr (Tx), (1979). 8) A. Michelson, H. Gale, "The Effect of the Earth's Rotation on the Velocity of Light", The Astrophysical Journal, 61, 140-145, (1924), http://adsabs.harvard.edu/abs/1925ApJ....61..140M. 9) D. Allan, M. Weiss, and N. Ashby, “Around-the-World Relativistic Sagnac Experiment”, Science, 228, 69–70, (1985). 10) D. Miller, "The Ether-Drift Experiment and the Determination of the Absolute Motion of the Earth", Reviews of Modern Physics, 5, 203, (1933). 11) R. Shankland, S. McCuskey, F. Leone, and G. Kuerti, "New Analysis of the Interferometer Observations of Dayton C. Miller," Rev. Mod. Phys., 27, 167, (1955). 12) G. Hammar, "The Velocity of Light within a Massive Enclosure," Physical Review 48, 462, (1935). 13) P. Dirac "Is there an ether?" Nature, 168, 906, (1951). 14) P. Dirac "Is there an ether?" Nature, 169, 702, (1952). 15) P. Dirac, "General Theory of Relativity", (Princeton Landmarks in Mathematics and Physics), (1975). 16) C. Misner, K. Thorne, J. Wheeler, “Gravitation”, Freeman, (1973). Appendix: Sagnac effect on the group velocity Let us discuss the group velocity change depending on the observer’s velocity. Sagnac effect is observed as the distance change between the light source and the observer by the motion of the observer. Let us consider two observers on the equator. The two observers 1 and 2 are combined by a rigid rod of length L as shown in Fig. A. Not only observer 1 but also observer 2 observes the Sagnac effect. That is, observer 2 moves during the flight time of light between observers 1 and 2. Thus, observer 2 also detects Sagnac effect of 61057.1 −× ; which indicates that a light reaches earlier to the observers than in the stationary state. Considering the Lorentz transformation, the group velocity is calculated as, . (A-1) Where, γ is the Lorentz factor, t1 is the time when a light reaches observer 1, t2 is the time when a light reaches observer 2, and 12 tttS −=∆ . Let the differential time of stationary observers to set 0t∆ , thus we obtain, The Lorentz factor appears both in length and time; therefore it is cancelled as shown in equation (2). Figure B shows the Sagnac effect between observers 1 and 2; after observer 1 detects the coded wave, observer 2 moves towards the GPS satellite. At the time t2, observer 2 moves vtS ×∆ . From Fig. B we obtain, vtLct SS ×∆−=×∆ . (A-2) =∆∴ . (A-3) G +=⎟ . (A-5) The Sagnac effect in GPS data shows group velocity change due to the motion of the observer. Fig. A Sagnac effect on the group velocity measurement using GPS: pulse coded signal is detected by observers 1 and 2. The detected times t1 and t2 suffer the Sagnac effect. Thus, the differential time 12 tttS −=∆ becomes smaller than 0t∆ in which observers 1 and 2 are in the stationary states. Observer 2 Observer 1 GPS satellite Coded wave: group velocity c t2 t1 L Observer’s velocity v vtS ×∆ ctS ×∆ Fig. B Derivation of the equation of Sagnac effect: after observer 1 detects the coded wave, observer 2 moves towards the GPS satellite. At the time t2, when the coded wave reaches at observer 2, which moves vtS ×∆ . Thus, we obtain vtLct SS ×∆−=×∆ . t2 t1 Observer 2 Observer 1