DISSERTATION

Search for cosmological gravitational-wave
background at high frequencies

Graduate School of Human and Environmental Studies,
Kyoto University
NISHIZAWA, Atsushi

28 November, 2008: ver.1.0(22 July, 2009: updated ver.1.1)

Abstract
We describe the general framework for a cosmological gravitational-wave background(GWB) search with a laser-interferometric gravitational-wave (GW) detector. What
we studied in this thesis can be divided into two topics: (i) the search for non-tensorial
polarization modes (scalar- and vector-like polarizations) of a GWB with a large-scalelaser-interferometric detector pair on the ground, (ii) the GWB search at ultra-highfrequencies (∼ 100 MHz) with a pair of meter-sized laser-interferometric detectors.
The ﬁrst topic is involved in the theories with extra dimensions and the modiﬁed
gravity theories. In the general relativity, there are two polarization modes of a GW.
However, in the general theories of gravity, six polarization modes are allowed. If theextra polarization modes of GWB are detected, we can obtain some information about
new physics. The search has not been performed at all so far with an interferometer. So,
we extended the conventional formalism of a cross-correlation analysis to non-tensorial
polarization modes and calculated the sensitivity to the GWB. We also discussed thedetectability with real detector pairs.
The second topic is the search for a GWB at ultra high frequencies. It is also important because some models in cosmology and particle physics predict relatively large
GWB at ultra-high frequency ∼ 100 MHz. Upper limits on GWB in wide-frequency
ranges have been obtained from various observations. However, they are all indirectly
derived from the observations. As far as we know, little direct experiment has been
done above 100 kHz except for a few experiments, though the direct constraint is much
weaker than the constraints at other frequencies. Thus, a much tighter bound above
100 kHz is needed to test various theoretical models.
First, we investigated the laser-interferometric detector designs that can eﬀectively
respond to GW at high frequencies, and found that the conﬁguration, a so-calledsynchronous-recycling interferometer (SRI) is the best at these sensitivities. Then,
we investigated the location and orientation dependence of two SRIs in detail, andderived the optimal location of the two detectors and the cross-correlation sensitivity
to a GWB. We also describe the experiment done by our group and the results. Thesestudies are not limited at the search at ∼ 100 MHz, but can also be applied to thedetectors in which the wavelength of a GW is comparable with the detector size.
As a developed version of the SRI, we proposed a new detector design, a so-called
resonant speed meter (RSM). The remarkable feature of this interferometer is that, atcertain frequencies, gravitational-wave signals are ampliﬁed, while displacement noises
are not. We also studied the quantum noise in a RSM, and its ultimate sensitivity.

iContents
1 Introduction
1.1 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Gravitational-wave sources . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Gravitational-wave detectors . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Outline and notation of the thesis . . . . . . . . . . . . . . . . . . . . .

1
1
1
4
8

2 Stochastic gravitational-wave background
2.1 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Linearized Einstein equation . . . . . . . . . . . . . . . . . . . .
2.1.2 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Quadrupole nature of gravitational waves . . . . . . . . . . . . .
2.2 Stochastic gravitational-wave backgrounds . . . . . . . . . . . . . . . .
2.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Energy density . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Characteristic amplitude . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Number density of gravitons . . . . . . . . . . . . . . . . . . . .
2.2.5 Decoupling of gravitons . . . . . . . . . . . . . . . . . . . . . .
2.2.6 Characteristic frequency . . . . . . . . . . . . . . . . . . . . . .

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3 Creation of cosmological GWB
3.1 Inﬂation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 GWB creation in de-Sitter inﬂation . . . . . . . . . . . . . . . . . . . .
3.3 GWB creation in slow-roll inﬂation . . . . . . . . . . . . . . . . . . . .
3.4 GWB creation in quintessential inﬂation . . . . . . . . . . . . . . . . .
3.4.1 Quintessential inﬂation . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 GWB spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Pre-big-bang model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Other production mechanisms . . . . . . . . . . . . . . . . . . . . . . .

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39

4 Observational constraints on GWB
4.1 Big-bang nucleosynthesis limit . . . . . . . . . . . . . . . . . . . . . . .
4.2 CMB limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Pulsar-timing limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Limit from spinning pulsars . . . . . . . . . . . . . . . . . . . .
4.3.2 Limit from binary pulsars . . . . . . . . . . . . . . . . . . . . .

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45
46
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49

iii

CONTENTS

iv

5 Direct search for GWB
5.1 Correlation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Overlap reduction functions . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Tensorial expansion . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Optimal conﬁguration . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Overlap reduction functions of realistic detector pairs . . . . . .
5.3 Observational constraints on GWB by LIGO . . . . . . . . . . . . . . .
5.4 Searching for non-tensorial polarizations of gravitational waves . . . . .
5.4.1 Non-tensorial polarization mode . . . . . . . . . . . . . . . . . .
5.4.2 Angular response of a single detector . . . . . . . . . . . . . . .
5.4.3 Overlap reduction function . . . . . . . . . . . . . . . . . . . . .
5.4.4 Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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72

6 Direct GWB search at ultra-high frequencies
6.1 Gravitational-wave sources at ultra-high frequencies . . . . . . . . . . .
6.2 Optimal detector design . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Interferometric-detector designs . . . . . . . . . . . . . . . . . .
6.2.2 GW response functions . . . . . . . . . . . . . . . . . . . . . . .
6.2.3 Detector comparison . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Correlation of two detectors . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Identiﬁcation of GW response and noise in a SRI . . . . . . . .
6.3.2 Dependence of sensitivity on the relative locations between two
detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.3 Sensitivity to GWB . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.4 Cross-correlated noise . . . . . . . . . . . . . . . . . . . . . . .
6.4 Experimental search for GWB . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Resonant speed meter101
7.1 Detector design and sensitivity . . . . . . . . . . . . . . . . . . . . . . . 102
7.1.1 Detector response . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.1.2 Noise curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.2 Quantum noise in a RSM . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2.1 Input-output relation . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2.2 Spectral density . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.3 Sensitivity to GWB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8 Conclusions
117
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.2 Discussions and future prospects . . . . . . . . . . . . . . . . . . . . . . 118
A Bogolubov transformation and particle creation

121

B Formulae of spherical Bessel functions

123

v

CONTENTSC The calculation of the overlap reduction function for non-tensorial
modes
125
D GW polarizations in higher-dimensional spacetime
127
D.1 Pure 5-dimensional Minkowski spacetime . . . . . . . . . . . . . . . . . 127
D.2 Pure 6-dimensional Minkowski spacetime . . . . . . . . . . . . . . . . . 129
E GW response functions of interferometers: general expressions
131
E.1 Synchronous-recycling interferometer . . . . . . . . . . . . . . . . . . . 131
E.2 Fabry-Perot Michelson interferometer . . . . . . . . . . . . . . . . . . . 133
E.3 L-shaped cavity Michelson interferometer . . . . . . . . . . . . . . . . . 135
F Quantum theory in a laser interferometer139
F.1 Quantum formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
F.2 Conventional Fabry-Perot Michelson interferometer . . . . . . . . . . . 140
F.3 Optional quantum conﬁgurations . . . . . . . . . . . . . . . . . . . . . 143

Chapter 1
Introduction
1.1

Gravitational waves

A gravitational wave (GW) is a ripple of spacetime, which propagates as a wave withthe speed of light. The GW is predicted according to the theory of general relativity,
published in 1916 by Einstein [1]. Nowadays, the many predictions of the Generalrelativity have been conﬁrmed in many observations and experiments [2]: gravitational
lensing, Shapiro delay in the solar system, the perihelion advance of the planet Mercury,dragging of an inertial frame, etc.. In addition, the binary pulsars supply us with thesplendid opportunities to test the general relativity, owing to the considerable stability
and the strong gravity. From the observation of the change in the revolution periodof B1931+16 (Hulse-Taylar binary pulsar), general relativity has been tested at a levelof 1%, and the indirect evidence of the existence of GWs has been obtained [3, 4].
Recently, Valtonen et al. have claimed that a new indirect evidence is found by theobservation of a binary system of two candidate black holes in the quasar OJ 287 [5].
These evidences lead us to strongly believe the existence of GWs. Nevertheless,
GWs have not directly detected yet. Aiming for the ﬁrst direct detection of GWs,
many research groups have constructed large detectors and done observations with thesensitivity improved by degrees.

1.2

Gravitational-wave sources

GWs are radiated by objects whose motion involves acceleration, provided that the
motion is not spherically symmetric nor axisymmetric [6]. A number of GW sources
have been theoretically predicted, as shown in Fig. 1.1 together with the sensitivities
of GW detectors.

Gravitational waves from astrophysical sources
The promising GW sources with astronomical origins are violent events involving compact objects, such as supernovae, gamma-ray bursts, the binaries of neutron stars,
black holes, and white dwarfs and their mergers, and spinning-down pulsars. Recent
11. Introduction

2

study and observational data suggested a part of supernovae and gamma-ray bursts are
related to the mergers of neutron stars, black holes, and white dwarfs. The GWs fromthese systems are important in that not only do they provide the test in the strong
regime of gravity, but also they will bring new information that one cannot obtain by
the observations with electromagnetic waves. This is owing to the strong transparency
of GWs. Electromagnetic waves are scattered at the outer layer of the stars and theplasma around it, and prevent us from directly seeing the core of the stars. In contrast, GWs can escape from the dense regions of the stars and directly propagate to
the Earth. Thus, we can investigate the equation of state in extremely high-density
region such as an inner core of a neutron star, the strong regime of gravity and its
environment around a black hole, the explosion mechanism of a supernova, the central
engine of a gamma-ray burst, the inner region veiled by dust such as the accretion disk
around a compact object, etc..

Figure 1.1: Schematic view of GW sources and detector sensitivities. In the ﬁgure,the noise curves of LIGO, LCGT, LISA, and DECIGO are representationally shown.
As for the magnitude of the noise curve, VIRGO is almost the same as LIGO, BBO
is almost the same as DECIGO, and the second-generation interferometers (advanced
LIGO, advanced VIRGO, and AIGO) are almost the same as LCGT, though TAMA
and GEO are slightly worse than LIGO and VIRGO.

Cosmological gravitational-wave background
There is another type of GWs, called gravitational wave background (GWB), which isthe superposition of a huge number of GWs with random phases (for the review, [7]).
This is an analogy of cosmic microwave background (CMB). The CMB was discovered3

1.2 Gravitational-wave sources

by Penzias and Wilson in the mid 1960s as an isotropic radiation over the sky [8],and the COsmic Background Explorer (COBE) was ﬁrst detected the anisotropy of
the CMB in 1992 [9]. Then, the Wilkinson Microwave Anisotropy Probe (WMAP)
satellite was launched in 2001, precisely measured the CMB anisotropic temperatureﬂuctuations, and determined cosmological parameters with astonishing accuracy in2003 [10, 11, 12, 13, 14] (The WMAP has been yielded the observational data, andthe updated cosmological parameters with ﬁfth-year data have been published [15].).
The diﬀerence between CMB and GWB is the epoch of the decoupling. CMB photons
decouple when the energy scale of the universe is ∼ 0.3 eV, corresponding to the redshiftz ∼ 1100. On the other hand, gravitons decouple at the Planck energy scale of ∼1019 GeV, because of the small coupling rate of the gravitational interaction. Therefore,
the GWB created in the early universe survives without loss of the information of theuniverse when it created, and would be a powerful tool for a probe of the early universe
beyond the last scattering surface for the CMB. Such a GWB and the detection are
the main topic of this thesis.
The most popular mechanism generating cosmological GWB is inﬂation. The inﬂation is the most successful theoretical paradigm in the early universe. In the scenario, the universe undergoes a rapid accelerating expansion, and provides the seedsof inhomogeneity for large-scale structure today [16, 17, 18, 19]. During the inﬂation,
quantum ﬂuctuations in spacetime curvature are also extended beyond the horizon,
and may transit to classical ﬂuctuations. In particle-physics point of view, this process
is described as gravitational particle production. The relic gravitons would exist today
as a stochastic GWB. The basic predictions in the inﬂationary scenario have conﬁrmedby the analysis combining the data of the WMAP, the galaxy redshift surveys, the supernova observations [15, 20, 21, 22]. However, there are still many inﬂationary models
that are compatible with current observational constraints. Hence, complementary observations are necessarily required in order to shrink the parameter space. The GWB
is adequate for this purpose, since it is an only way to directly probe the inﬂation andthe involved high-energy physics.
Other processes in the early universe, such as preheating and phase transitions, andcosmic strings also produces GWB, whose spectrum strongly depends on the production
mechanism. Therefore, the detection or non-detection of the GWB bring us crucial
imformation of the early universe and high-energy physics, and can be used for thetests of theoretical models.

Astrophysical gravitational-wave background
GWBs also generated by astrophysical objects such as neutron stars, black holes, whitedwarfs, and massive stars, though this is not the topic of this thesis. If the event rate issmall, the superposition of GWs does not become a stochastic GWB, but a popcorn-like
GWB that individual sources can be resolved. However, if the event rate is large, such
GWBs cannot be distinguished from the inﬂationary GWB. So, the astrophysical GWB
would often be obstacle to detect the inﬂationary GWB. However, the astrophysical
GWB brings us crucial information in the early universe with the redshift from a few
to tens: the formation rate of the ﬁrst stars, the event rate of supernovae, branching1. Introduction

4

ratio between black hole and neutron star formation, mass and angular-momentum
distribution of black holes at the births, black-hole growth mechanisms. Hence, suchinformation plays very important role in the study of large-scale structures and theirevolutions.
Therefore, the GW is promising tool for cosmology and astronomy, since they brings
new information that one cannot obtain by the observations with electromagnetic
waves.

1.3

Gravitational-wave detectors

A gravitational-wave detector can be classiﬁed into two types: a laser-interferometrictype and a resonant-mass type. For the more detailed review of the up-to-date statusof the ground-based detectors, see [23].

Resonant-mass-type detectors
Surprisingly, the history of GW detector is long. In the 1960s, the research of theresonant mass was started by Weber [24]. The principle is simple, that the vibration ofan elastic (large metal cylinder) induced by GWs is electrically ampliﬁed and detectedby transducers. In 1969, Weber reported the coincident detection of the GW withresonant bars apart by 1000 km [25]. However, no GW was detected in subsequent
experiments by various people. Thus, in these days, it is not believed that the Weber’s
event was due to true GWs.
In the initial period of the history of GW search by the mid-1990s, the bar detectors had been a popular detector design, and a number of the bar detector in various
countries were evolved. However, the large community today has shrunk to four operating detectors [26, 27, 28]: EXPLORER (CERN, Geneva, Switzerland), NAUTILUS(INFN Frascati Laboratory, near Rome, Italy), AURIGA (INFN Legnaro Laboratory,
near Padua, Italy), ALLEGRO 1 (Louisiana State University, Louisiana, USA). Thesebar detectors is a ∼ 3 m-long aluminum cylinder with the mass of (2.2 − 2.3) × 103 kg,
and are operating at cryogenic temperatures to reduce thermal noise. The defect of
the bar detector is its narrow bandwidth in the sensitivity, which is ∼ ±20 Hz aroundthe resonant frequency of ∼ 900 Hz. Even at the resonant frequency, the sensitivity is
not signiﬁcantly high. Therefore, in these days, the resonant bars are superseded byinterferometers in the sensitivity. The bar detectors will continue to operate, but they
are gradually loosing scientiﬁc interests (Burst GWs of supernovae etc. in a kHz bandstill keep the interests of people.).

Ground-based interferometric detectors
The study of a laser-interferometric detector has been started in the 1970s, which was∼ 10 years later from a resonant-bar detector. In the 1990s, interferometers withthe arm length of tens of meter were constructed and had been developed toward a
1

It was announced in April 2007 that the ALLEGRO would soon cease operation.

5

1.3 Gravitational-wave detectors

Figure 1.2: Schematic view of laser-interferometric GW detector (Fabry-Perot Michelson interferometer). All optics are suspended with thin wires.

kilometer-sided laser-interferometric detector, which mainly aims for the detection of
GW sources from astrophysical compact objects with a few or several solar masses, asmentioned in the previous section. The sensitivity of the detector and the expected
GW targets are shown in Fig. 1.2. The ﬁrst large-scale interferometer in the world for
GWs is TAMA300 [29], which located at Mitaka, Tokyo, Japan, and has two arms withthe length of 300 m extended in perpendicular directions. Nowadays, other three largescale interferometer, LIGO [30], VIRGO [31], GEO600 [32] are also constructed andoperating. In Table 1.1, the characters of each detector are tabulated. The sensitivefrequency range is from 10−104 Hz. The goal sensitivity of the ﬁrst-generation detectors
is limited by seismic noise at lower frequencies, seismic noise at lower frequencies,
thermal noise at middle frequencies, shot noise at high frequencies.
There are two reasons why the interferometric GW detector is more popular thanresonant one today. For one reason, the advantage of the interferometers, comparedwith the resonant bars, is that the interferometers have broad-frequency-band sensitivity and can observe the waveform of GWs, though the resonant bars have narrowfrequency-band sensitivity. For another reason, the interferometric-type is easy to
improve the sensitivity by extending the baseline, since the longer baseline produces
the larger GW signal without amplifyng the displacement noise of mirrors. In fact, thebest sensitivity of the interferometric detector (LIGO and VIRGO) is higher than thatof resonant detectors at all frequencies in the observation band.
Among the ﬁrst-generation detectors (upper four interferometers in the Table 1.1),

1. Introduction
generation interferometercountry
1st
LIGO
USA
VIRGO
Italy and France
GEO600Germany and UK
TAMA300
Japan
2nd
Adv. LIGO
USA, UK, and Germany
Adv. VIRGO
Italy and France
AIGO
Australia
LCGT
Japan
3rd
ET
Europe6arm length starting date
4 km
operating
3 km
operating
600 m
operating
300 m
operating
4 km2014 3 km
2014 5 km
2017 3 km
201x 10 km
202x -

Table 1.1: Current and planned large-scale interferometric detectors on the ground.
The design parameters and the starting date of the second- and third-generation interferometers are not necessarily ﬁxed and could be changed in the future.
LIGO and VIRGO have almost accomplished the goal sensitivity at the time in 2008.
However, the event rate 2 is much smaller than unity per a year. Therefore, the second
generation of the interferometric detectors, whose sensitivity is roughly ten times better
than the ﬁrst generation, is needed to ﬁrmly detect GWs and to pioneer the GW
astronomy. Such detectors, which is currently planned and developed, are advanced
LIGO [33], advanced VIRGO [34], AIGO [35], and LCGT [36]. The characters ofeach detector are listed at the lower half of Table 1.1, and the sensitivity is drawn in
Fig. 1.1, together with that of the ﬁrst-generation interferometer. The advanced LIGO
and VIRGO are an upgraded version of current LIGO and VIRGO, respectively, while
AIGO and LCGT will be constructed at new places. The GW event rate is expected to
be a few or several in a year. In these detectors, the sensitivity is limited by quantum
noise in almost all frequencies, and the applied technique of quantum optics will beessential. Thus, quantum optical techniques for the interferometric GW detectors havebeen vigorously investigated and developed experimentally.
The project for the third-generation interferometer, Einstein Telescope (ET) [37],has been formally started in Europe since 2008, after the budget for the design study
was approved. The investigation of the detector conﬁguration is under way, aiming forroughly 10 times better sensitivity than that of the second-generation interferometers.

Space-based interferometric detectors
In the frequency band of a ground-based detector, the rich GW sources are expected.
More massive object, such as intermediate (∼ 103 − 105 M⊙ ) and supermassive (∼105 − M⊙ ) black holes, and the early stage of the evolution of the solar-mass compact
objects will be observed in a lower frequency band below 10 Hz. However, seismic disturbances prevent the ground-based detectors from searching such objects. Therefore,
three space-based detectors are planned, aiming for low-frequency GW astronomy.
2

The event rate depends on the GW sources and theoretical models that we consider. So, there is
still uncertainty of a factor or even of an order of the magnitude in the event rate.

7

1.3 Gravitational-wave detectors
interferometerfrequency bandarm length

LISA

country
or institution
ESA and NASA

10−4 − 10−2 Hz

BBO

NASA

10−2 − 10 Hz

DECIGOJapan

10−2 − 10 Hz

5 × 106 km(transponder-type)
5 × 104 km(transponder-type)1 × 103 km(FP-type)

starting
date2018 202x 2024 -Table 1.2: Planned interferometric detectors in space. DECIGO will use Fabry-Perotcavity, not a transponder type of LISA and BBO. The ﬁnesse is 10 so that the eﬀective
arm length is 104 km. The design parameters and the starting date are not ﬁxed andcould be changed in the future.
The space-based interferometer, Laser Interferometer Space Antenna (LISA) [38],was proposed by ESA in 1993 and is a joint program of ESA and NASA, which targets
GWs at 10−4 − 10−2 Hz. LISA is located at the place 20◦ behind the Earth in the
Heliocentric orbit, and consists of three spacecrafts apart by the distance of 5 × 106 km
each other, forming an equilateral triangle 3 . The relative distance between spacecrafts
is measured by a laser interferometry with drag-free test masses in the spacerafts. The
target GW source for LISA is the binary and merger of super massive black holes atcosmic distance up to the redshift z ∼ 20, extreme mass ratio inspirals, GWBs fromcosmic strings and electroweak phase transition. In the frequency range, there aremany resolved and unresolved galactic binaries composed of white dwarfs. To subtract
them, much eﬀort is devoted. However, it would be an obstacle limiting the sensitivity.
The frequency range between LISA and the ground-based detectors, 10−2 − 10 Hz,
is also important range for science, because there are intermediate black holes andcompact binaries before moving into the observational band of the ground detectors. In
addition, no signiﬁcant astrophysical foreground would exist there so that 10−2 − 10 Hz
is a good window for the inﬂationary GWB. At the frequencies, two space detectors
are proposed: DECi-hertz Interferometer Gravitational-wave Observatory (DECIGO)[39] and Big-Bang Observer (BBO) [40]. In Table 1.2, the parameters are tabulated.

GW detectors at ultra-high frequencies
Important other frequency range especially for cosmology is an ultra-high frequency
band. The frequency range is deﬁned and called in various ways in the literatures,
depending on the authors. We call the frequency range 10 kHz − 10 MHz very highfrequency (VHF), 10 MHz − 10 GHz ultra-high frequency (UHF), 10 GHz− super highfrequency (SHF). The generic term, high frequency (HF), describes the frequency range
above ∼ 10 Hz 4 .
3Exactly speaking, one percent of the arm length changes during the annual orbiting.
These deﬁnitions are not popular one and has not obtained the consensus, because it depends onthe frequencies, in which one is interested. Say, low frequencies is below ∼ 1−100 Hz and high frequen4

1. Introduction

8

There is no astronomical GW source at VHF, UHF and SHF, since the object mass
has to be much less than ∼ 1M⊙ to produce GWs at the frequencies. The reason
why GWs at UHF is important is because GWs at ∼ 100 MHz corresponds to theenergy scale ∼ 1015 GeV in the early universe, during which inﬂation may occur, andbecause, in some theoretical scenarios, the GW energy produced in the early universe
is much larger than that in the standard scenario. Furthermore, compact object withcosmological motivation, such as primordial black holes produced by inhomogeneous
ﬂuctuation in the early stage of universe and black strings in the braneworld scenario,
would emit UHF GWs. If the objects exist around the Earth, they can be detected
through GWs. Therefore, from GW detection at UHF, one can directly see traces of a
scenario in the early universe and obtain the implication in new physics.
Nevertheless, much eﬀort has not been devoted into GW search at this frequencies.
Consequently, the direct constraint due to GW detector is so loose, though the indirect
constraint due to the big-bang nucleosynthesis has been obtained. This is because theGW amplitude is much smaller at high frequencies than that at low frequencies, andit makes the detection diﬃcult. However, pioneering works are vital for future GW
astronomy and cosmology. At present, as far as I know, GW detectors are developedwith diﬀerent methods by four groups: at INFN, Genova, Italy [41, 42], Birmingham
University, UK [43, 44, 45], Chongqing University, China [46, 47], NAO, Tokyo, Japan[48, 49, 50].

1.4

Outline and notation of the thesis

This thesis is organized as follows:

Chapter 2
We brieﬂy introduce the property of GWs and the principle of GW detection with laserinterferometer. Then, as for stochastic GWB, various assumption and fundamental
quantities are provided.

Chapter 3
We overview the creation of GWB in inﬂation and several other models in the early
universe. Such backgrounds motivate us to search physics in the early universe.

Chapter 4This chapter is devoted to the current upper limit on GWB with indirect observations.
The limits are complementary to the direct search with interferometers.
cies is above ∼ 100 Hz − 10 kHz for people working on the ground-based large-scale interferometers.
We call the terms for convenience.

9

1.4 Outline and notation of the thesis

Chapter 5
In the former part of this chapter, we review cross-correlation analysis for GWB andcurrent tightest constraint from direct observation by LIGO. In the latter part, ouroriginal work that searches for non-tensorial polarization mode of GW is described.
This work will be published as
A. Nishizawa, A. Taruya, S. Kawamura, and M. Sakagami, ”Searching for nontensorial polarizations of a gravitational-wave background with laser interferometers”,
in preparation.

Chapter 6
We turn our attention to GW search at ultra-high frequencies. The experiment withinterferometers has not been studied well in the past. Thus, ﬁrst, we investigate theoptimal interferometer design and derive the cross-correlation sensitivity to GWB, as
described in
A. Nishizawa et al., ”Laser-interferometric detectors for gravitational wave backgrounds at 100 MHz: Detector design and sensitivity”, Phys. Rev. D 77, 022002(2008),
A. Nishizawa et al., ”Optimal location of two laser-interferometric detectors forgravitational wave backgrounds at 100 MHz”, Class. Quantum Grav. 25, 225011(2008).
Then, we give experimental results obtained by our group, based on
T. Akutsu et al., ”Search for a Stochastic Background of 100-MHz Gravitational
Waves with Laser Interferometers”, Phys. Rev. Lett. 101, 101101 (2008).

Chapter 7: Resonant speed meterAs the solution of displacement-ampliﬁed problem in a synchronous-recycling interferometer, we introduce a so-called resonant speed meter, which is a displacement noisecanceled conﬁguration based on a ring-shaped synchronous recycling interferometer,
proposed in
A. Nishizawa, S. Kawamura, and M. Sakagami, ”Resonant Speed Meter for
Gravitational-Wave Detection”, Phys. Rev. Lett. 101, 081101 (2008).
The remarkable feature of this interferometer is that, at certain frequencies,
gravitational-wave signals are ampliﬁed, while displacement noises are not. We also
show that the resonant speed meter works well for quantum noise. This work is pre-

1. Introduction

10

pared for a paper as
A. Nishizawa, S. Kawamura, and M. Sakagami, ”Quantum noise in a resonant speed
meter”, in preparation.

Chapter 8: Conclusions
This chapter is devoted to summarize this thesis and discuss the future work.

Notations• Natural units
In this thesis, we use c = ~ = kB = 1. However, they are written explicitly insome sections where is likely to be confused or at the place where it is easy tounderstand.
• Planck mass
In this thesis, the reduced Planck mass
MPl = (8πG)−1/2 ≈ 2.436 × 1018 GeV
is used, instead of mPl = G−1/2 ≈ 1.221 × 1019 GeV.
• Tensors indices
We follow the rules of super- and subscrpts for the relativistic theory in a Ddimensional spacetime:
greek alphabets for spacetime, µ, ν, α = 0, 1, 2, · · · , D,roman alphabets for space,i, j, k = 1, 2, · · · , D.
• Convention of metrics
We follow the convention of the metric sign for a Minkowski spacetime:ηµν = diag(−1, +1, +1, · · · , +1) .

Chapter 2
Stochastic gravitational-wave
background
2.1

Gravitational waves

2.1.1

Linearized Einstein equation

Einstein equation is given by1Gµν ≡ Rµν − gµν R = 8πG Tµν .
2

(2.1)

Gµν is the Einstein tensor, and Tµν is the stress-energy tensor. Christoﬀel symbol,
Riemann curvature tensor, Ricci tensor, and scalar curvature are deﬁned as
1 µα
g (gαν,λ + gαλ,ν − gνλ,α ) ,2µµR ναβ ≡ Γνβ,α − Γµνα,β + Γµγα Γγνβ − Γµγβ Γγνα ,Rµν ≡ Rαµαν ,R ≡ Rαα .
Γµνλ ≡

(2.2)
(2.3)

Let us consider gravitational waves propagating in a vacuum. The Einstein equation
can be written as
Gµν = 0 .

(2.4)

Suppose that a metric tensor is slightly deviated from the background spacetime,(0)+ hµν ,gµν = gµν

|hµν | ≪ 1 .

We use the superscript (0) for the zeroth order quantity and (1) for the ﬁrst order
quantity with respect to hµν . Taking the ﬁrst order of hµν , one can write Eqs. (2.1),11

2. Stochastic gravitational-wave background12(2.2) and (2.3) as
1 (0) (1) 1
(1)
G(1)
= Rµν
− gµν
R − hµν R(0) ,
µν
2
2
[
1
(0)
(0)
(1)
Rµν
=
hν α;αµ + hµ α;αν − hµν;α;α − h;µν + Rνβ hβµ + Rµβ hβν
2
]
(
)
(0)
(0)
βα
+ Rµβαν + Rνβαµ h,(0)R(1) = hµα;α;µ − h;α;α − hµν Rµν
,

(2.5)

(2.6)
(2.7)

where the symbol ; denotes a covariant derivative, and h ≡ hµµ . Substituting Eqs. (2.6)(0)
and (2.7) into Eq. (2.5), and using (2.4) and the fact Rµν = 0 and R(0) = 0 in thevacuum, one can obtain
(
)
(0)
(0)
α ;β
α
α
;α
(0)
;α
hν ;αµ + hµ ;αν − hµν ;α − h;µ;ν − gµν (hβ ;α − h ;α ) + Rµβαν + Rνβαµ hβα = 0 .
(2.8)
Now we deﬁne the traceless part of hµν ,1 (0)h.
h̄µν ≡ hµν − gµν
2

(2.9)

Using the new deﬁnition, we can rewrite Eq. (2.8) as(0)(0)
h̄β α;α;β + 2Rµανβ h̄αβ = 0 .
h̄µν;α;α − h̄ν α;αµ − h̄µ α;αν + gµν(2.10)In addition, to ﬁx the degrees of gauge freedom, we impose the following conditions(harmonic gauge conditions),h̄ν µ;ν = 0 .
(2.11)
Then, Eq. (2.10) gives(0)h̄µν;α;α + 2Rµανβ h̄αβ = 0 .

(2.12)

The contribution of each term in the left-hand side of the equation can be estimated.
For the GW wavelength λ and the background curvature scale L, the ﬁrst term isO(λ−2 ) and the second term is O(L−2 ). Therefore, if λ ≪ L, the second term can beignored and Eq. 2.12 results in the sourceless wave equation of GWs,¤h̄µν ≡ h̄µν;α;α = 0 .

2.1.2(2.13)

Gravitational waves

To investigate the components of h̄µν , we consider a plane GW propagating on the
background of a 4-dimensional spacetime with ﬂar curvature,αh̄µν = Aµν eikα x .

(2.14)

13

2.1 Gravitational waves

Substituting this into the harmonic gauge conditions, Eq. (2.11), and the wave equation,
Eq. (2.13), we can obtain two conditions,k ν Aµν = 0 ,Aµν kρ k ρ = 0 ,(2.15)in order to be a solution of the wave equation. Here we used the fact that h̄µν istraceless. The second condition is satisﬁed if kρ k ρ = 0. So, the GW has to propagate
along a null geodesic.
At Eq. (2.11), we have already imposed the gauge conditions, however, the gauge
freedom is not completely ﬁxed yet. To see it, let us consider an inﬁnitesimal gaugetransformation,x′µ = xµ + ξ µ .
(2.16)
Then, the metric tensor is transformed as∂xα ∂xβ∂x′µ ∂x′ν
∂xα ∂xβ= (ηαβ + hαβ ) ′µ ′ν ,
∂x ∂x

′
gµν
(xγ + ξ γ ) = gαβand, to the ﬁrst order in hµν and ξµ , it can be written as
′
(0)
gµν
(xγ ) = gµν+ (hµν − ξµ,ν − ξν,µ ) .

So, hµν is transformed ash′µν = hµν − ξµ,ν − ξν,µ .

(2.17)

Tracing Eq. (2.17) and using Eq. (2.9) giveh̄′µ ρ,ρ = h̄µρ ,ρ − ξµ,ρ,ρ .

(2.18)

Therefore, there remain the degrees of gauge freedom concerning ξµ . If the additional
conditions,¤ξµ = 0 ,is satisﬁed, Eq. (2.18) is consistent with the harmonic gauge conditions, and the gauge
is completely ﬁxed.
In conclusions, h̄µν or Aµν is traceless by deﬁnition of Eq. (2.9), and transverse by
the ﬁrst condition of Eq. (2.15). This gauge is a so-called transverse-traceless (TT)gauge. For GW propagating in the z direction, the complex amplitude of the GW canbe written as


0 0
0
0
 0 A+ A× 0 Aµν = (2.19)
 0 A× −A+ 0  ,0 0
00where the degrees of freedom A+ and A× , which does not vanish by the gauge transformation, are gravitational waves, and each polarization mode is called plus and crossmodes, respectively.

2. Stochastic gravitational-wave background

2.1.3

14

Quadrupole nature of gravitational waves

To see the eﬀect of GWs on test masses, let us consider the simple case that GW with+ polarization is propagating on the background ηµν = diag(−1, 1, 1, 1) in z direction,[][]
ds2 = −dt2 + 1 + h+ (t − z) dx2 + 1 − h+ (t − z) dy 2 + dz 2 ,h+ (t − z) = h+ cos[Ω(t − z)] ,(2.20)|h+ | ≪ 1 ,where Ω is the angular frequency of a GW.
Suppose that light pulse is emitted at the origin A and is received at the point
B on the surface of a sphere with the radius R0 . Taking a polar coordinate, (x =r sin θ cos φ, y = r sin θ sin φ, z = r cos θ), and setting ds2 = 0 for light, one can write
the metric, Eq. (2.20), as[]
12
dr = dt 1 − h+ cos[Ω(t − r cos θ)] sin θ cos 2φ .
2(2.21)The time delay due to GWs vanishes when θ = 0 and is maximum when θ = π/2. Thisis because GWs are transverse. Fixing θ = π/2 and integrating Eq. (2.21) from r = 0
to r = R0 give)h+=cos Ωt cos 2φ dt
1−
2
t0
[
]
h+
sin[Ω{t0 + ∆t(φ)}] − sin Ωt0 cos 2φ .
= ∆t(φ) −2Ω∫ t0 +∆t(φ) (R0(2.22)In the case when the wavelength of GWs is much larger than test mass separation (thedistance between the points A and B), ΩR0 ≪ 1, Eq. (2.22) can be written as][h+cos Ωt0 cos 2φ .
R(φ) ≡ c ∆t(φ) = R0 1 +2Thus, the strain induced by GWs with + polarization is proportional to cos 2φ. As for× polarization, the strain is proportional to sin 2φ. In Fig. 2.1, the eﬀects on test masses
are shown for GWs propagating vertical to the plane of a paper. Both polarization
modes are related by 45 degrees rotation. While, each polarization mode is symmetricwith respect to 180 degrees rotation. This indicates that gravitons have spin 2.

2.2

Stochastic gravitational-wave backgrounds

In this section, assumptions and physical quantities to describe stochastic GWB aredeﬁned, and common features involving GWB are discussed.

15

2.2 Stochastic gravitational-wave backgrounds

Figure 2.1: Inﬂuences of gravitational waves on test masses for each GW polarization
mode: + mode (left) and × mode (right).

2. Stochastic gravitational-wave background

2.2.1

16

Assumptions

A GWB is the superposition of a number of gravitational waves with random phases,⃗ the gravitational
which exists as a background in the universe. At generic point X,
metric perturbations in the TT gauge are given by
∫ ∞
∑∫
⃗
⃗ =
h(t, X)dΩ̂df h̃A (f, Ω̂) e2πif (t−Ω̂·X) eA (Ω̂) ,(2.23)A

S2−∞

where Ω̂ is a unit vector directed at GW propagation and h̃A (f, Ω̂) is the Fouriertransform of GW amplitude with polarizations A = +, ×. Note that h̃A (−f, Ω̂) =h̃∗A (f, Ω̂). Polarizarion tensors eA (Ω̂), are deﬁned ase+ (Ω̂) ≡ m̂ ⊗ m̂ − n̂ ⊗ n̂,e× (Ω̂) ≡ m̂ ⊗ n̂ + n̂ ⊗ m̂ ,(2.24)

where the unit vectors m̂, n̂ are orthogonal to Ω̂ and to each other. See Fig. 5.8. The
polarizarion tensors satisfy eA (Ω̂)eA′ (Ω̂) = 2δAA′ , A, A′ = +, ×.
The followings are usually assumed about stochastic GWB, (i) isotropic, (ii) unpolarized, (iii) stationary, and (iv) Gaussian [51]. (i) It is observationally known that
CMB is highly isotropic [52, 53]. However, this might not be true because a large
number of unresolved GW sources, for example, white dwarf binaries and neutron starbinaries, would exist as foreground in the direction of the galactic plane, or one canconsider non-standard cosmological scenarios in which GWB is not necessarily isotropic
as well as CMB. So, anisotropic GWB search in data analysis is important, but here
we assume GWB is isotropic for the simplest case. (ii) The assumption ”unpolarized”
GWB is natural since there is no strong evidence to believe that GWB has preferredpolarization, plus or cross (right- or left-handed), though it is possible that inﬂationary
background can be polarized with the gravitational Chern-Simons term that might bederived from string theory [54, 55]. (iii) The statistical property of GWB does notchange during the observation. This seems to be valid since the age of the universe is
far much longer than the observation time. (iv) The assumption that the amplitudeof GWB has Gaussian distribution is justiﬁed by the central limit theorem, because
GWB was created by a large number of independent gravitational wave sources. Theseassumptions (i) - (iv) are expressed by11〈h̃∗A (f, Ω̂)h̃A′ (f ′ , Ω̂′ )〉 = δ(f − f ′ ) δ 2 (Ω̂, Ω̂′ )δAA′ Sh (f ) ,(2.25)
4π
2where δ 2 (Ω̂, Ω̂′ ) ≡ δ(φ − φ′ )δ(cos θ − cos θ′ ) and 〈· · · 〉 denotes ensemble average. Sh (f )is the one-sided power spectral density deﬁned by Eq. (2.25), and satisﬁes Sh (−f ) =Sh (f ).

2.2.2

Energy density

The amount of GWB is deﬁned by the energy density per logarithmic frequency binnormalized by the critical energy density of the universe, that is,
1 dρgw,(2.26)Ωgw (f ) ≡ρc d ln f

17

2.2 Stochastic gravitational-wave backgrounds

2
where ρc = 3MPl
H02 and H0 is the Hubble constant. It is useful to represent the energy
density with the form h20 Ωgw (f ) and H0 = 100 h0 km s−1 Mpc−1 because H0 still has
observational uncertainty. Ωgw (f ) is related to Sh (f ) as follows. The energy density of
GW is given by [56, 57, 6]
gw
T00= ρgw =2MPl〈ḣab ḣab 〉λ ,4where 〈· · · 〉λ denotes the average over several GW wavelengths, in which the subscript
λ is ﬁxed so as not to be confused with the ensemble average. Substituting Eq. (2.23)
and using Eq. (2.25), we obtain∫ f =∞
2
ρgw = MPld(ln f )f (2πf )2 Sh (f ) ,
f =0

dρgw
d ln f2 3= 4π 2 MPl
f Sh (f ) .

Therefore, from Eq. (2.26), Ωgw is related to the power spectral density Sh (f ) as
( 2)4π
Ωgw (f ) =f 3 Sh (f ) .
(2.27)
2
3H0

2.2.3

Characteristic amplitude

A dimensionless characteristic GW amplitude can be deﬁned, using Eqs. (2.23) and(2.25), by∫ ∞ab〈hab (t)h (t)〉 = 2
df Sh (f )(2.28)−∞∫ f =∞= 4∫ f =∞d(ln f )h2c (f ) ,d(ln f )f Sh (f ) = 2
f =0

f =0

that is,h2c (f ) = 2f Sh (f ) .
Combining with Eq. (2.27), one can obtain the relation( 2)2π
f 2 h2c (f ) .
Ωgw (f ) =2
3H0(2.29)(2.30)

Dimensionless quantity, the tensor power spectrum
PT (k) ≡ 2h2c (k) ,(2.31)

is often used in cosmology, where the factor 2 results from two polarization modes. In
terms of PT (k), the energy density of GWB is given byΩgw (k) =k2
PT (k) .
12H02(2.32)

2. Stochastic gravitational-wave background

2.2.4

18

Number density of gravitons

The energy density Ωgw (f ) can also be expressed in terms of gravitons. Suppose thenumber of gravitons per polarization mode per unit volume in phase space, n(⃗x, ⃗k).
For a isotropic stochastic GWB, the number density only depends on frequency f andcan be written as n(⃗x, ⃗k) = nf . So,
∫
∫ ∞
d3 k
2
ρgw = 2
k nf = 16πd(ln f )f 4 nf
(2π)3
0
dρgw
= 16π 2 nf f 4d(ln f )By deﬁnition of Eq. (2.26), one can obtain(h20 Ωgw (f ) ≈ 2.3 × 10−6

2.2.5

nf
1042

)(

f
1 Hz

)4
.

(2.33)

Decoupling of gravitons

CMB photons are decoupled from other components of universe at the redshift z ∼ 1100
and carry information of the universe at the epoch to us. Similarly, gravitons aredecoupled far early in the history of the universe, because the gravitational interaction
is much weaker than the electromagnetic interaction. Let us estimate the time of thedecoupling of gravitons. In general, the rate of interaction is given by Γ = nσ|v|,
where n is the number density of particles, σ is cross section, and |v| is mean velocity−4 2of the particles. For gravitons, |v|=1 and σ ∼ MPl
T by dimensional analysis, then,−4 5
Γ ∼ MPl T . On the other hand, the rate of expansion of the universe is given by−2 4
the Friedmann equation, and is H 2 ∼ MPl
T for relativistic particles. Therefore,
the decoupling time of gravitons is at Γ ∼ H, that is, T ∼ MPl ∼ 1019 GeV. Thismeans that gravitons are in thermal equilibrium only at the epoch of Planck scaleand decoupled below Planck scale. This is the reason why GWB is a powerful tool
to directly prove the very early universe and a particle theory beyond the standard
model.
Is there thermal relic GWB on the analogy of CMB? One can infer the temperatureof GWB using entropy conservation. Entropy per unit volume is given by4
s = σSB T 3 ,3(2.34)

where σSB is Stefan-Boltzmann constant. Since the entropy is conserved in comoving
volume, it follows
gs (T )T 3 a3 = constant ,
( )3
( )3
∑
Ti
7 ∑
Ti
gs (T ) ≡
gi
+
,
gi
T
8 i=fermionT
i=bosonwhere a is a scale factor, and gs (T ) is the number of entropy degrees of freedom, whichis gs (T0 ) = 3.91 at present and gs (T∗ ) = 106.75 at energy scales above 300 GeV if we19

2.2 Stochastic gravitational-wave backgrounds

assume that the particle standard model holds. Thus, the temperature of thermal relicGWB at present is
(
)1/3
3.91
Tg =
Tγ ≈ 0.9 K .
(2.35)
106.75Here the temperature of CMB photon at present Tγ = 2.725 K is used. However, sucha thermal spectrum is unlikely to survive because (i) it is diluted if inﬂation occurs at
relatively low energy scales, (ii) it is unclear whether one can apply low energy physics
to Planck era or not, (iii) non-thermal gravitons are produced by various mechanisms
though the history of the universe and might hide the thermal spectrum.

2.2.6

Characteristic frequency

GWB observed today is produced at certain epoch of the universe and is red-shiftedby cosmic expansion. Thus, the frequency of the GWB and the energy scale of theuniverse can be related.
Suppose that GWB with the frequency f∗ is generated when the temperature of
the universe is T∗ by some mechanisms. Using entropy conservation, Eq. (2.34), one
can calculate the frequency today,(
f = f∗

a∗
a0

)

(
−14

≈ 8.00 × 10

f∗
1 Hz

)(

106.75gs∗)1/3 (

1 GeV
T∗

)
Hz .

(2.36)a∗ and a0 are the scale factors at T = T∗ and at present. The frequency f∗ depends onthe generation mechanism of GWB, however, at least, it should be generated within thehorizon of the universe. So, we will write the characteristic scale as λ∗ = ϵH∗−1 , (0 <ϵ < 1). The Hubble parameter is given by the Friedmann equation for relativistic
particles,
π 2 g∗ T∗4
,
2
90MPl
( )4
( )4
∑
Ti
Ti
7 ∑
≡
gi
+
gi
.
T
8 i=fermionT
i=bosonH∗2 =g∗

Substituting f∗ into Eq. (2.36), one can obtain()(
T∗g∗ )1/6−7 −1
f = 1.71 × 10 ϵ
Hz ,
1GeV
106.75or,(T∗ = 5.85 × 10 ϵ
6

f
1Hz

)(

(2.37)g∗ )−1/6
GeV .
106.75

Therefore, if the GWB is generated at the horizon scale (ϵ = 1), one can prove physics
in the early universe at T∗ > 1 TeV by investigating GWB with the frequencies f >0.1 mHz. This is the reason why GWB at high frequencies is so important for cosmology.

Chapter 3
Creation of cosmological GWB.

3.1

Inﬂation

Hot big-bang cosmology has several problems [58]: the horizon problem (Why is ouruniverse so homogeneous and isotropic over causally disconnected regions in the early
age of the universe?), the ﬂatness problem (Why is our universe considerably ﬂat?), andthe seed for cosmological structures (Small inhomogeneity must be needed to create
galaxies, stars, and clusters.). To overcome these problems, the theory of inﬂation, theidea that the universe underwent a brief period of rapid accelerating expansion, wasoriginally proposed by [16, 17, 18, 19].
Einstein equation with a perfect ﬂuid gives the Friedmann equation,
2

( )2
ȧ
1
H =
=
ρ,
2
a
3MPl

(3.1)

ä
1
=−
(ρ + 3p) ,
2
a
6MPl

(3.2)

and the acceleration equaiton,

where ”dot” denotes derivatives with respect to time t and MPl ≡ (8πG)−1/2 . From
Eq. (3.2), the expansion of the universe is accelerated if the condition ρ + 3p < 0 holds.
One of Such ﬂuids is a scalar ﬁeld. The stress-energy tensor of a scalar ﬁeld φ is givenbyTµν = φ,µ φ,ν − ηµν L ,1
L = − φ,α φ,α − V (φ) .
2

(3.3)

L is Lagrangian density of the scalar ﬁeld, and V is the potential. Here ﬂat spacetime,ηµν = diag(−1, +1, +1, +1), is assumed. In the homogeneous and isotropic universe,21

3. Creation of cosmological GWB

22

from Eq. (3.3), energy density ρ and pressure p can be simply written as
1 2
φ̇ + V (φ) ,
2
1 2
p =
φ̇ − V (φ) .
2ρ =The evolution of the ﬁeld φ can be derived using Tν µ;µ = 0 and the equation of motion
is given byφ̈ + 3H φ̇ + V ′ = 0 ,(3.4)
where the prime denotes the derivative with respect to φ.
If the scalar ﬁeld potential changes slowly, namely,¯ ′′ ¯( ′ )2
2
¯
¯
¯ ¯
MPl
V
2 ¯V ¯¯¯ϵV (φ) ≡≪1,
ηV (φ) ≡ MPl ¯ ¯ ≪ 1 ,2VVor, consistently,

(3.5)

¯
¯
¯ φ̈ ¯¯¯¯ H φ̇ ¯ ≪ 1 ,¯ 2¯¯ φ̇ ¯¯ ¯≪1,¯V ¯

the equation of state p ≈ −ρ is satisﬁed and hence the universe is accelerated. The
latter condition states that the scalar ﬁeld slowly rolls down the hill of a potential.
Therefore, the above condition and parameters ϵV and ηV are called slow-roll conditionand slow-roll parameters 1 , respectively [59]. In the slow-roll approximation, Eq. (3.4)
can be approximated to 3H φ̇ ≈ −V ′ . The Eq. (3.1) can also be approximated to√ȧ1≈V ≈ constant .
(3.6)
2
a
3MPl
Then, the solution is[√]Va(t) ∝ exp
t ,
2
3MPl

(3.7)

and the acceleration expansion can be obtained. Such a scalar ﬁeld is called inﬂaton.

3.2

GWB creation in de-Sitter inﬂation

Gravitational waves are an inevitable consequence of all inﬂationary models, being
created in the same manner as the density perturbation [60, 61, 62, 63]. In the context of quantum ﬁeld theory in a curved spacetime, which is more strict description,
1

Slow-roll parameters in Hamilton-Jacobi formulation are deﬁned as(ϵ ≡ 2MPl

H ′ (φ)H(φ))2,η ≡ 2MPl

H ′′ (φ).
H(φ)

To distinguish from them, the subscripts V are ﬁxed to the slow-roll parameters deﬁned by the inﬂaton
potential.

23

3.2 GWB creation in de-Sitter inﬂation

gravitational-wave production can be interpreted as the ampliﬁcation of vacuum ﬂuctuations by cosmic expansion (gravitational particle creation) [64, 65, 66].
We assume a spatially ﬂat universe, and that the background is homogeneous andisotropic, as well as the previous section. If there are gravitational waves, the spacetime
metric can be described by the Friedmann-Robertson-Walker (FRW) metric with tensor
perturbations,ds2 = a(τ )2 [−dτ 2 + (δij + hij )dxi dxj ] .
Here the conformal time τ is deﬁned by dτ = dt/a(t). The tensor perturbation can be
written in a form with a time-dependent function φAk (τ ) and a spatial plane-wave part,∑ ∑−1
ik·x Ae (Ω̂) ,h(τ, x) = MPl
φAk (τ )eA=+,×kwhere it is summed with respect to a GW polarization A and a comoving wave-number
vector k, and a unit vector Ω̂ ≡ k/|k| is deﬁned. From the linearized Einstein equation(2.13), the time evolution of the tensor perturbation can be given by
d2 φk
dφk
+ 2H
+ k 2 φk = 0 ,
2
dτ
dτ

(3.8)

where

1 da
.
a dτ
This equation is the same equation as for a massless scalar ﬁeld, and can be solved
if a(τ ) is given explicitly as a function of τ . Here we consider de-Sitter phase in theearly universe, followed by the radiation-dominated (RD) and matter-dominated (MD)phases. We do not consider the dark energy-dominated phase, since it would dominate
the universe only after the redshift z . 1 and hardly aﬀects a broad spectrum of GWB
2.
First we start with a qualitative look at the shape of a GWB spectrum. From
Eq. (3.8) by writing φk (τ ) as h(τ ),
H≡

d2 h
dh
+ 2H
+ k2h = 0 ,
2
dτ
dτ

(3.9)

is obtained. When k ≪ H, the solution is
h = constant .

for k ≪ H

On the other hand, when k ≫ H, substituting h(τ ) = A(τ ) exp[iB(τ )] into Eq. (3.9)
and keeping the leading terms, we can obtain a WKB solution,h∝21
exp[±ikτ ]afor k ≫ H .

However, the dark-energy domination indeed aﬀects the frequencies feq etc. through the cosmological parameters determined in the universe with the dark-energy dominated phase. Thus, inthis section, we take the dark-energy domination into account indirectly through the cosmologicalparameters.

3. Creation of cosmological GWB

24

Therefore, GW amplitude is frozen out after the horizon exit during inﬂation, and isdamped after the re-entry inside of the horizon during the RD or MD phases. GW
amplitude after the horizon re-entry can be written using the amplitude hini and thescale factor a(t∗ ) at the time of the horizon exit,|hobs | ≈ |hini |a(t∗ ),a(t0 )(3.10)

where t∗ and t0 are the time at the horizon exit and at present, respectively.
The frequency of GWs seen today is
2πf =ka(t∗ )=
H(t∗ ) .
a(t0 )a(t0 )(3.11)For the modes re-entering inside the horizon during the RD phase, using Eqs. (3.10)
and (3.11), and H 2 (t∗ ) ∝ a−4 (t∗ ), we obtain |hobs | ∝ f −1 , and for the modes reentering inside the horizon during the MD phase, using H 2 (t∗ ) ∝ a−3 (t∗ ), we obtain|hobs | ∝ f −2 . Substituting the these relations into Eq. (2.30) gives the frequency
dependence of Ωgw (f ),{2π 2 2constant (feq < f )2Ωgw (f ) =f |hobs | ∝,2
f −2(f < feq )3H0
where feq is the frequency corresponding to the matter-radiation equality.
Next, we treat GWB creation in a quantitative way, using a quantum ﬁeld theory.
In a particle point of view (graviton), gravitational waves can be written as∑ ∫d3 k [−1√h(τ, x) = MPlaA (k)φk (τ )eik·x + a†A (k)φ∗k (τ )e−ik·x ] eA (Ω̂) ,3(2π) 2kA=+,×where aA (k) and a†A (k) are the creation and annihilation operators. Vacuum states ineach phase of the universe are deﬁned using the annihilation operators by
aIA |0〉I = 0 ,aR
A |0〉R = 0 ,M
aA |0〉M = 0 .
The subscripts I, R, M represent the inﬂation, RD, MD, respectively. Transforming φk
to ψk = aφk in Eq. (3.8), the following equation can be obtained 3 ,
[
]
d2 ψk
1 d2 a2+ k −ψk = 0 .
(3.12)dτ 2
a dτ 2Hereafter we assume that the universe is isotropic and replace the wave number vector
k with k. Cosmic expansions in the de-Sitter phase, followed by the RD and MD
3

The variable a is used for both the scale factors and the annihilation operator. Be careful so as
not to be confused.

25

3.2 GWB creation in de-Sitter inﬂation

phases as a function of τ , are given by1,Hτ
1aR (τ ) =(τ − 2τ1 ) ,Hτ12(τ + τeq − 4τ1 )2
aM (τ ) =,4Hτ12 (τeq − 2τ1 )aI (τ ) = −(Inﬂation : −∞ < τ < τ1 ) ,(3.13)(RD : τ1 < τ < τeq ) ,(3.14)(MD : τeq < τ < τ0 ) .

(3.15)In the above expressions, the scale factor a(τ ) and its ﬁrst derivative are smoothly
continued at a phase transition of cosmic expansion. H is the Hubble parameter at thede-Sitter inﬂationary epoch, and has a constant value. Note that the sign of the conformal time is τ1 < 0 and τeq > 0 so that a(τ ) is positive and a monotonously growing
function. The equation (3.12) can be analytically solved by substituting Eqs. (3.13) (3.15) and gives solutions for positive frequency modes,
ψkI (kτ ) = −( π )1/2
2(2)(kτ )1/2 H3/2 (kτ ) ,(−∞ < τ < τ1 ) , (3.16)

ψkR (kτ ) = e−ikτ ,(τ1 < τ < τeq ) ,( π )1/2(2)
ψkM (kτ ) = −{k(τ + s)}1/2 H3/2 [k(τ + s)] , (τeq < τ < τ0 )2

(3.17)(3.18)(2)

where s ≡ τeq − 4τ1 , and H3/2 is the Hankel function of the second kind. Overall
coeﬃcients are chosen to give ψk → e−ikτ at the subhorizon limit, kτ → ∞. The above
expressions Eqs. (3.16) - (3.18) can be reduced, by using an explicit form of the Hankel(2)function H3/2 , to simpler forms,
[
]
i
= 1−
e−ikτ ,kτψkR (kτ ) = e−ikτ ,][iMe−ik(τ +s) .
ψk (kτ ) = 1 −k(τ + s)ψkI (kτ )(3.19)(3.20)(3.21)

These functions are the exact solutions for Eq. (3.12). However, no continuity between each expanding epoch is imposed. The continuity condition is given by Bogolubov transformation (Detailed explanation is given in Appendix A.) [67],ψkI = αkIR ψkR + βkIR (ψkR )∗ ,ψkR = αkRM ψkM + βkRM (ψkM )∗ .

(3.22)(3.23)

Substituting Eqs. (3.19) - (3.21) into Eqs. (3.22) and (3.23), we obtain the Bogolubov3. Creation of cosmological GWB

26

coeﬃcients,
αkIR = 1 −1i− 2 2 ,kτ1 2k τ1(3.24)

1 −2ikτ1e,
2k 2 τ12
[
]
i
1
= 1+
−
eiks ,
2k(τeq − 2τ1 ) 8k 2 (τeq − 2τ1 )2
1
e−ik(2τeq +s) .
= − 2
8k (τeq − 2τ1 )2

βkIR =(3.25)

αkRM(3.26)

βkRM(3.27)

What is needed to obtain a present GWB spectrum is the Bogolubov coeﬃcients fromthe inﬂationary phase to the MD phase. It is determined by the relation,
ψkI = αktotal ψkM + βktotal (ψkM )∗ .

(3.28)

Therefore, substituting Eqs. (3.22) and (3.23) into Eq. (3.28), we ﬁnd the total Bogolubov coeﬃcients,( total) ( IR) ( RM)αkβktotal
αk
βkIR
αk
βkRM
=
,
(3.29)
βktotal ∗ αktotal ∗βkIR ∗ αkIR ∗βkRM ∗ αkRM ∗together with Eqs. (3.24) - (3.27).
We assume the universe is in a vacuum state during the inﬂation phase. From
Eqs. (3.29), (A.4), and (A.6), the number of created gravitons is given by
nk = |βktotal |2=(|βkIR |2 + 2|βkRM |2

1|βkIR |2 +
2

)

[
]
IR IR ∗ RM RM+ 2 Re αk βk αk βk.

(3.30)

This equation clearly show that some gravitons are created from vacuum ﬂuctuations at
the inﬂation-RD transition, and that the created gravitons are subsequently ampliﬁed
at the RD-MD transition.
To evaluate Eq. (3.30), let us consider four cases shown in Fig. 3.1, with respect
to the epochs when the ﬂuctuations come into the cosmic horizon. As mentioned inAppendix A, particle creation does not occur at the mode within the horizon. Hence,
we do not consider the case 1. We do not consider case 4 either, as it is not observable
today. Consequently, what is important is case 2 (horizon re-entry during RD) andcase 3 (horizon re-entry during MD) 4
4
The number density of gravitons can be derived without the classiﬁcation of horizon re-entry. In
that case, a continuous solution isnk ≈is obtained.

( )4 [
(
)2 ]
1 f1
1 feq
1+
.
4 f
4 f

27

3.2 GWB creation in de-Sitter inﬂation

Figure 3.1: Schematic view of horizon exit and re-entry.

• Case 2: horizon re-entry during RD.
The wavelength of a mode is larger than the Hubble horizon at the inﬂation-RD.
The condition k ≪ H(τ1 ) is y1 ≡ k|τ1 | ≪ 1. Then, the wavelength of a modeis smaller than the Hubble horizon at the RD-MD transition. The conditionk ≫ H(τeq ) is y1 ≡ kτeq ≫ 1. The equation (3.30) being substituted Eqs. (3.24)- (3.27) and taken a leading-order term, gives
nk ≈

1
.
4y14

y1 can be related to a physical frequency at present f by
2πf a0
f
=
,
H a1
f1
( )1/2
H a1
H
t1
≡
=
,
2π a0
2π(1 + zeq ) teq

y1 = k|τ1 | =f1(3.31)

where Eq. (3.13) is used at the second equal sign of the upper equation, a0 = 1 isthe scale factor at present, and zeq is the redshift at the radiation-matter equality.
Using t1 ≈ 1/(2H) and the values 5 ,1 + zeq ≈ 2.41 × 104 Ωm h20 ,teq ≈ 3.23 × 1010 (Ωm h20 )−2 sec ,(3.32)

where Ωm is the energy-density parameter of matter in the universe today, we
obtain()1/2
H8
f1 ≈ 5.01 × 10
Hz .
(3.33)
10−4 MPl
5

Dark-energy domination is not considered here, and neutrinos are assumed to be relativistic.

3. Creation of cosmological GWB

28

Note that f1 does not depend on Ωm h20 . The frequency f1 is a cutoﬀ frequency
above which no GWB is produced. Therefore, the number density of gravitons
for the modes horizon re-entering during RD is
( )4
1 f1
nk ≈
.
4 f
• Case 3: horizon re-entry during MD.
The conditions for the wavelength of a mode are y1 ≪ 1 and yeq ≪ 1. The
equation (3.30) with leading-order terms is[]1
1
nk ≈ 4.
4y1 4(yeq + 2y1 )2(yeq + 2y1 ) can be written as
yeq + 2y1 = k(τeq − 2τ1 ) = kaeq Hτ12 = −
feq ≡

aeq f
f
aeq
kτ1 =
=
,
a1
a1 f1
feq

H(teq ) aeq
.
2π a0

√
Here we used Eqs. (3.13), (3.14), and (3.31). Using H(teq ) = 4( 2 − 1)/3teq [58]and Eq. (3.32) gives
feq ≈ 1.13 × 10−16 Ωm h20 Hz .
Therefore, the number density of gravitons for the modes horizon re-entering
during MD is( )4 ( )2
1 f1
feq
nk ≈
.
16 f
f
From (2.33), we ﬁnally obtain the GWB energy density

)2
(
H

−14

,
(feq < f < f1 ) , 3.6 × 10
−4 M
10
2
Pl
(
)2 ( )2h0 Ωgw (f ) ≈

H
feq

 3.6 × 10−14,(f0 < f < feq ) ,−4
10 MPlf(3.34)

where f0 = H0 /h0 = 3.24×10−18 Hz. The GWB spectrum is shown in Fig. 3.2 with H =2.7×10−5 MPl . The reason for this choice of the inﬂation energy scale is that a stringent
limit on the GWB spectrum of de-Sitter inﬂation comes from CMB observations, as
described in the next section.

3.3

GWB creation in slow-roll inﬂation

The Hubble parameter H is constant during de-Sitter inﬂation, however, is not constantbut slowly changes in slow-roll inﬂation, depending on the shape of the potential.

29

3.3 GWB creation in slow-roll inﬂation

Figure 3.2: GWB spectrum produced by de-Sitter inﬂation. The enegy scale of inﬂationis selected as H = 2.7 × 10−5 MPl .

Consequently, the spectrum of the produced gravitons is also modiﬁed. The evolutionof the Hubble parameter is governed by the equation of motion [68, 69],2φ̇ = −2MPl
H ′ (φ) ,[ ′ ]23
1H (φ)=
H 2 (φ) −V (φ) ,
2
4
2MPl
2MPl

(3.35)

which can be derived from Eqs. (3.1) and (3.4) by regarding the Hubble parameter asa function of φ and assuming that φ varies monotonically with time. The Eq. (3.35)
is called the Hamilton-Jacobi equation. The dynamics of the Hubble parameter isquantiﬁed by the generalized Hubble slow-roll parameters [70],[]2H ′ (φ),ϵ(φ) ≡H(φ)′ ℓ−1 (ℓ+1)
dHℓ2 ℓ (H ))λH ≡ (2MPl
,
ℓ
(ℓ+1)
H
dφ
2
2MPl

(ℓ ≥ 1) .

We can identify the ﬁrst and second slow-roll parameters 1 λH and 2 λH with η and ξ 2
in the literature [71] 6 .
The primordial power spectra of scalar and tensor ﬂuctuations at the end of inﬂation
6

Note that, in some papers, the squared index 2 is often omitted in the deﬁnition, though it is of
the second order.

3. Creation of cosmological GWB

30

are written by( )nS −1+(αS /2) ln(k/k0 )k
PS (k) = PS (k0 ),k0( )nT +(αT /2) ln(k/k0 )k
PT (k) = PT (k0 ),k0(3.36)

where k0 is a ﬁducial pivot scale observed at present in CMB, and is given by k0 =0.002 Mpc−1 , which corresponds to f0 , is conventionally used. The spectral indices nS
and nT up to the second order in the slow-roll parameters 7 are given by [72],¯d ln PS (k) ¯¯
nS − 1 ≡d ln k ¯αS →01
1≈ 2η − 4ϵ − 2(1 + C)ϵ2 − (3 − 5C)ϵη + (3 − C)ξ 2 ,22¯
d ln PT (k) ¯¯nT ≡d ln k ¯
αT →02≈ −2ϵ − (3 + C)ϵ + (1 + C)ϵη ,(3.37)and the derivatives [73, 74] areαS ≡d2 ln PS (k)≈ 16ϵη − 24ϵ2 − 2ξ 2 ,d ln k 2

αT ≡d2 ln PT (k)≈ 4ϵη − 4ϵ2 ,2d ln kwhere C ≡ 4(ln 2+γ)−5 ≈ 0.08145 and γ ≈ 0.57722 is the Euler-Mascheroni constant.
The tensor-to-scalar ratio is deﬁned by
¯
[
]
PT ¯¯
r≡
≈ 16ϵ 1 + 2C(ϵ − η) .
¯
PS k=k0
Note that nS , nT , and r are evaluated at the scale k0 = 0.002 Mpc−1 . From Eq. (3.37),
the tensor spectral index nT is negative, since ϵ is positively deﬁned in Eq. (3.5). Thismeans that the slow-roll inﬂation predicts smaller GWB spectrum than that of thede-Sitter inﬂation at high frequencies.
In Eq. (2.32), the energy density of the GWB is related to the tensor power spectrumat the present day PT (k) as
k2
PT (k) .
Ωgw (k) =12H02
7(3.38)The slow-roll parameters here are related to those in Hamilton-Jacobi formulation like ϵ = ϵV andη = ηV − ϵV , to the lowest order.

31

3.3 GWB creation in slow-roll inﬂation

The PT (k) can be related to the primordial tensor power spectrum PT (k) by a transfer
function T (k),PT (k) = T 2 (k)PT (k) .
(3.39)
By numerically integrating Eq. (3.8), the transfer function is found to be well approximated by the form [75]
√
( )
( )2
3j1 (kτ0 )kkT (k) =
1.0 + 1.34
+ 2.50.
(3.40)
kτ0
keq
keq
where keq = 0.073 Ωm h20 Mpc−1 is the wave number corresponding to the Hubble radiusat the time of the matter-radiation equality. Combining Eqs. (3.36), (3.38), (3.39), and(3.40) gives( )nT [( )2 ]( )3[j1 (kτ0 )]2kkk
PT (k0 )+ 2.50,Ωgw (k) =1.0 + 1.34
2 2
4H0 τ0k0
keq
keq
and, particularly for k ≫ keq ,
( )nT
15
k
Ωgw (k) ≈.
P (k0 )
2 2 4 T
16H0 keq τ0k0(3.41)

Comparing Eq. (3.41) with Eq. (3.34), one can obtain the GW energy spectrum()2 ( )nT
H
f

∗
−14

,
(feq < f < f1 ) , 3.6 × 10
−4
( 10 MPl )2 ( f0 )nT ( )2h20 Ωgw (f ) ≈

H∗
f
feq

 3.6 × 10−14,(f0 < f < feq ) ,10−4 MPlf0
f(3.42)
where H∗ is the Hubble parameter when the scale k0 crosses the horizon during theinﬂation. In Eq. (3.42), the extra factor (f /f0 )nT shows that the slow-roll inﬂation does
not produce larger GW spectrum than that of the de-Sitter inﬂation. Examples of thespectrum are shown in Fig. 3.3.
The GW spectrum is often written using the tensor-to-scalar ratio r [76]. The
amplitude of PT (k0 ) in Eq. (3.41) has not yet been observed in CMB, but the amplitudeof PS (k0 ) has been determined well [15],PS (k0 ) ≈ 2.457 × 10−9 .
Using the cosmological parameters determined by 5th year WMAP data, type Ia
supernovae and baryon acoustic oscillations [15], one ﬁnds keq = 0.010 Mpc−1 andτ0 = 1.44 × 104 Mpc. Therefore, Eq. (3.41) can be rewritten as
( )nT
f
2
−15
h0 Ωgw (f ) = 4.81 × 10 r,
f0
for f ≫ feq . A constraint on r has been obtained by the observations of 5th year
WMAP, type Ia supernovae and baryon acoustic oscillations, and the result is r < 0.54

3. Creation of cosmological GWB

32(95% C.L.) [15]. This is a conservative limit, in that the nonzero running nS is allowedin the analysis, where a prior on nS that has no running can impose a tighter limit onr, but in general inﬂation models, the running of nS would be nonzero. Consequently,
the upper limit of the GW energy spectrum due to slow-roll inﬂation ish20 Ωgw (f ) ≤ 2.60 × 10−15

for f > feq .

In terms of H∗ , this limit can be translated into H∗ ≤ 2.7 × 10−5 MPl .

Figure 3.3: GWB spectrum produced by slow-roll inﬂations. From the top on the left
side, the curves are plotted for the tensor-to-scalar ratios r = 0.5 (red), 10−2 (green),
10−4 (blue). For r = 0.5 (red), the tensor spectral index is selected as nT = −0.05(solid), −0.10 (dotted), −0.15 (dashed) with αT = 0. For r = 10−2 (green), therunning spectral index is selected as αT = 0 (solid), −10−3 (dotted), +10−3 (dashed).
The spectra of slow-roll inﬂations never exceed that of the de-Sitter inﬂation (black).

3.4

GWB creation in quintessential inﬂation

3.4.1

Quintessential inﬂation

Peebles and Vilenkin presented a model where the idea of a post-inﬂationary phase
stiﬀer than radiation is dynamically realized [77]. One of motivations of this model is
to explain two accelerations of the universe during the inﬂation and the current epoch,with a single scalar ﬁeld like the quintessence ﬁeld [78, 79]. The potential of the scalarﬁeld (inﬂaton) φ is given byλ(φ4 + M 4 ) ,(φ < 0) ,V (φ) =
λM 8


 4
,
(0 < φ)φ + M433

3.4 GWB creation in quintessential inﬂation

The λ is a normalization factor and can be determined by COBE normalization as
λ ≈ 1×10−14 . M is a characteristic energy parameter. If we want the energy density in
φ to be compatible with the present total energy (Ωφ ∼ 0.7) and moderately accelerate
the universe, it is required to take M ∼ 106 GeV. The potential is similar to that ofa chaotic inﬂation in the range φ < M , and is a slowly decaying function in φ > M .
Note that this scenario can be implemented with any other inﬂationary potential for
φ < 0, and the chaotic potential is one of illustrative examples.
The inﬂaton starts at φ < −MPl and rolls toward zero. The inﬂaton ends slowrolling at φ ∼ MPl , then the inﬂaton evolves toward a phase where the kinetic energyof the inﬂaton dominates. We will call the phase a kinetic energy-dominated (KD)phase 8 . In the phase, from Eqs. (3.1) and (3.4), the energy density of the inﬂaton ρφand the scale factor evolve as
( )6
a1
2
2
ρφ (τ ) = 3H1 MPl,awhere the Hubble parameter H1 at the end of the inﬂation is
H1 =

√
1
∼ λMPl .
a1 τ1Since ρ ∝ a−3(1+w) for a barotropic ﬂuid with the equation of state p = wρ, the kinetic
energy-dominated scalar ﬁeld behaves like the ﬂuid with w = 1, which is stiﬀer thanradiation.
At the end of the inﬂation, not only gravitons but also massless scalar ﬁelds areampliﬁed [67, 80]. Indeed there are various scalar, vector, and tensor degrees of freedom during the inﬂationary phase. However, free massless spinor and gauge ﬁelds areconformally invariant, so their contributions to the total energy density are suppressed.
Suppose that during the inﬂationary phase the number of massless scalar ﬁelds is Ns .
As well as gravitons, the ampliﬁed energy density has been calculated in the case ofnearly conformal coupling scalar ﬁelds [80], arbitrary coupling scalar ﬁelds [81], andarbitrary power-law expansion after inﬂation [82]. In all cases, the result is(ρr (τ ) = R H14

a1
a

)4
,

R=

Ns
∑

Ri ∼ 10−2 Ns .

(3.43)

i=1

Ri is the contribution of each massless scalar degrees of freedom to the energy densityof the ampliﬁed ﬂuctuations and it is of the order of 10−2 . As indicated in [77],the thermalization of the massless scalar ﬁelds occurs immediately after the particle−1/2
creation, when ath /a1 ∼ (102 − 103 ) Ns .
Since ρr decays more slowly than ρφ , a radiation-dominated phase follows the kinetic
energy-dominated phase. The onset of the radiation-dominated phase is computed fromρr (τr ) ∼ ρφ (τr ),
√
√
R H1
a1
∼
∼ λR .
ar
3 MPl
8

It is also called deﬂation or kinetion in other references.

3. Creation of cosmological GWB

34

By red-shifting the horizon sclale at τr up to now, we obtain( )( )3 ( )
Hr ar
H1 a1
ar
fr =
=
2π a0
2π ar
a0
(
)(
)1/3
λ
gs (τ0 )−3 3/4≈ 3.65 × 10 R
Hz ,
10−14
gs (τth )where gs (τth ) is the number of degrees of freedom after the thermalization of the massless scalar ﬁelds. It is also important to compute the frequency f1 corresponds to thehorizon scale at the end of inﬂation, which is given by( )( )( )
H1 a1
H1 a1
ar
f1 =
=
2π a0
2π ar
a0
(
)1/3
gs (τ0 )11 −1/4≈ 3.65 × 10 RHz .
gs (τth )The frequency corresponding to the matter-radiation equality is the same as that inde-Sitter inﬂation, and is given by feq ≈ 1.13 × 10−16 Ωm h20 Hz.

3.4.2

GWB spectrum

A GWB energy spectrum in the quintessential inﬂation has been calculated by themethod of particle creation in a curved spacetime, as described in Sec. 3.2 [82, 83]. In
quintessential inﬂation scenario, the smoothly continued solutions of cosmic expansion
are
1
,
Hτ√
3τ1 − 2τ1
aK (τ ) =,Hτ1
τ1
τ + τr − 3τ1√,aR (τ ) = −Hτ1 τ1 (3τ1 − 2τr )(τ + τeq + 2τr − 6τ1 )2√
aM (τ ) =,4Hτ1 (3τ1 − τr − τeq ) τ1 (3τ1 − 2τr )aI (τ ) = −(Inﬂation : −∞ < τ < τ1 ) ,(KD : τ1 < τ < τr ) ,(RD : τr < τ < τeq ) ,(MD : τeq < τ < τ0 ) .

Here we assume that the inﬂation is de-Sitter inﬂation. Given the cosmic expansions,
one can solve Eq. (3.12) and obtain solutions for positive frequency modes in eachphase,( π )1/2(2)
I(kτ )1/2 H3/2 (kτ ) ,(−∞ < τ < τ1 ) ,ψk (kτ ) =2( π )1/2(2)
ψkK (kτ ) ={k(τ + p)}1/2 H0 [k(τ + p)] ,(τ1 < τ < τr ) ,2(τr < τ < τeq ) ,ψkR (kτ ) = e−ikτ ,( π )1/2(2)
ψkM (kτ ) ={k(τ + q)}1/2 H3/2 [k(τ + q)] ,(τeq < τ < τ0 ) ,2

35

3.4 GWB creation in quintessential inﬂation

where p ≡ −3τ1 /2 and q ≡ τeq + 2τr − 6τ1 . The normalizaiton of the mode functions
is chosen to be ψk → e−ikτ when kτ → ∞.
Following procedures are the same as those in Sec. 3.2. The amount of createdgravitons is given by the Bogolubov coeﬃcients connecting from the inﬂationary phaseto the matter-dominated phase today. The Bogolubov transformations are deﬁned by
ψkI = αkIK ψkK + βkIK ψkK ∗ ,ψkK = αkKR ψkR + βkKR ψkR ∗ ,ψkR = αkRM ψkM + βkRM ψkM ∗ ,and the total Bogolubov coeﬃcients are given by)( totalβktotalαkβ total ∗ αtotal ∗)) ( RM(k IK k IK ) ( KR
βkKR
αk
βkRM
αk
αk
βk
.
=
βkRM ∗ αkRM ∗βkKR ∗ αkKR ∗βkIK ∗ αkIK ∗We assume the universe is in a vacuum state during the inﬂation phase. The numberof created gravitons is given by nk = |βktotal |2 , and the approximated GWB-energy
spectrum obtained in [83] is ( )[ ( )]2

f
f


K
ln
,
(fr < f < f1 ) ,


fr
f1






[ ( )]2frπΩgw (f ) =K ln,(feq < f < fr ) ,

4
f
1





( )2 [ ( )]2



π
feq
fr


ln
,
(f0 < f < feq ) ,
 K
4
f
f1
where

(
)1/3
(
)
81
gs (τ0 )λ−20 −2K≡
λ Ωγ ≈ 1.34 × 10 h0.
16π 3 gs (τth )10−14

Here we used gs (τ0 ) = 3.91 and gs (τth ) = 106.75 in the standard model of particle
physics. In supersymmetric extension of the standard model, the number of degrees
of freedom is doubled because of supersymmetric partners. However, the correction to
the GWB spectrum is small, and only a factor 21/3 ≈ 1.26 appears.
In quintessential inﬂationary models, the energy density of GWB has a peak at highfrequencies and becomes much larger than in the case of ordinary inﬂationary models.
The location of the peak, fpeak = e−2 f1 ∝ R−1/4 , weakly depends on R, but does notdepend on λ. The height of the peak is given byh20 Ωgw (fpeak ) ≈ 0.725 × 10−6 R−1 ,(3.44)and does not depend on λ either. Therefore, the peak cannot move one order ofmagnitude and is ﬁrmly localized around 100 GHz.

3. Creation of cosmological GWB

36

Figure 3.4: GWB spectrum produced by quintessential inﬂation for a ﬁxed value of
λ = 10−14 . Other parameters are selected as gs (τ0 ) = 3.91, gs (τth ) = 106.75, andh20 Ωm = 0.1369 [15]. Three curves corresponds to R = 0.15 (red, solid curve), R = 1(green, dotted curve), and R = 10 (blue, dashed curve).

The constraint on the spectrum comes from not CMB limit but big-bang nucleosynthesis limit. Using Eqs. (4.7) and (3.44), one can obtain 0.13 . R. From Eq. (3.43),
this means that the number of massless scalar degrees of freedom Ns has to exceed 13.
The GWB spectra with three values of R are plotted in Fig. 3.4.
Note that such a GWB spectrum with large peak at high frequencies is not a speciﬁcfeature appearing only in a quintessential inﬂation model, but is a common featurein scenarios with KD epoch of scalar ﬁelds. The gravitational reheating process weconsidered above is an ineﬃcient reheating process and is likely to result in a long KD
phase. Another reheating process, a so-called instant preheating has been proposed[84, 85, 86]. Instant reheating is much more eﬃcient than gravitational reheating, andcompletes the reheating process earlier. Consequently, the duration of the KD phase isshorter, and the peak on the GWB spectrum become smaller. The dependences of thespectrum on the reheating mechanisms have been investigated by Tashiro et al. [87].
Therefore, the GWB at high frequencies is important because it brings us information
regarding the reheating mechanism and thermal history of the universe after inﬂation.

3.5

Pre-big-bang modelPre-big-bang model is a cosmological model motivated by the string theory and was
originally proposed by Gasperini and Veneziano [88, 89]. In this model, the initial stateof the universe is assumed to be the string perturbative vacuum with the weak coupling
and low curvature. Then, the perturbative approach is well justiﬁed. The low-energy

37

3.5 Pre-big-bang model

string eﬀective action at the lowest order in the derivatives and in eφ is given by [88, 89]∫]√ [S ∼ d4 x −g e−φ (R + ∂µ φ ∂ µ φ) − V (φ) ,(3.45)
where φ is the dilaton ﬁeld. The dilaton potential V (φ) is due to nonperturbativeeﬀects and, thus, can be neglected in the low-energy perturbative regime. There is aphase, called a dilaton-dominated (DD) phase, that the kinetic energy of the dilatonﬁeld drives the universe through an superinﬂationary evolution (which is an accelerated
expansion in the string frame, or accelerated contraction in the Einstein frame). The
spacetime curvature increases in the DD phase , eventually reaching the string scale. In
this string phase, the low-energy string eﬀective action is not valid, and the perturbative
treatment breaks down. Although the transition between the pre-big-bang (DD) phaseand the post-big-bang phase is not well understood, some models, which can partiallydescribed it, have been proposed. The de-Sitter phase with linearly growing dilaton isconsidered to be the representative of a typical solution in the large curvature regime,
where non-perturbative corrections are expected to avoid the big-bang singularity andto stop the dilaton ﬁeld to grow. Then, it is assumed, at the end of the stringy phase,that the standard RD phase with the constant dilaton ﬁeld is restored. As we will see
later, MD phase does not much contribute to the GWB spectrum and is not important
for the computation. So, we do not consider MD phase here.
An exact GWB spectrum in the pre-big-bang model has derived in [90]. In ahomogeneous, isotropic, and spatially ﬂat background, for simplicity, the equation of
motion derived from the action in Eq. (3.45), gives the solutions of the evolving universein the string frame,
(
)−α
[
]
1
τ − (1 − α)τs
τ − (1 − α)τsa(τ ) = −,φ(τ ) = φs − γ ln
,
Hs τs
ατs
ατs
for the DD phase (−∞ < τ < τs ),1a(τ ) = −
,
Hs τ

( )
τ
φ(τ ) = φs − 2β ln,τsfor the stringy phase (τs < τ < τ1 ),a(τ ) =1(τ − 2τ1 ) ,Hs τ12φ = φ0 ,for the RD phase (τ1 < τ < τr ). The conformal time τs < 0, τ1 < 0, and τr > 0 are
deﬁned at the time of the DD-string, string-RD, RD-MD phase transition, respectively.
The solutions are connected so that the evolution is continuous acrosseach phase√√transition. The value α and γ are constants and given by α = 1/(1 + 3) and γ = 3,
in the absence of external matter [91]. In stringy phase, the Hubble parameter H
and the derivative of the dilaton ﬁeld φ̇ is set to be constants so that it produces thede-Sitter expansion.
The model has several parameters: α, β, γ, Hs , τs , and τ1 . Among these parameters,
α and γ concerns the solution in DD phase, but, they are ﬁxed, as mentioned above, if

3. Creation of cosmological GWB

38

we assume the absence of external matter. β and Hs are parameters that determines
the growth of the dilaton and the energy scale during the stringy phase, respectively.
However, the natural value of Hs is considered to be Hs ≈ 1/λs ≈ 0.03 MPl due tothe string length λs in the string theory [90]. The parameters τs and τ1 concerns theduration of the string era, and determines the phase when the ﬂuctuation with a certain
wave number crosses outside the horizon (the horizon crossing occurs at kτ ∼ 1). The
corresponding frequency today is given by redshifting the horizon scale at the time.
From Eq. (3.33), it follows(
f1 = 6.13 × 10

10

Hs0.03 MPl

)(

t1
λs

)1/2
Hz .

On the other hand, τs or fs are totally unknown, even as an order of magnitude. Since|τ1 | < |τs |, fs is in the range 0 < fs < f1 . Therefore, we ﬁx the model parameters
α and γ at the values above, and Hs and τ1 at the typical values Hs ≈ 0.03 MPl and
f1 = 6.13 × 1010 Hz. The unknown parameters are β and fs .
The equation for the Fourier modes of metric tensor perturbations is [92]]d2 ψk [ 2
+
k
−
V
(τ
)
ψk = 0 ,
dτ 2
)
1
d2 (
V (τ ) = eφ/2 2 a e−φ/2 .
adτ
The procedures to compute the created gravitons is the same as those for the de-Sitterand quintessential inﬂations in the previous sections. The exact GWB spectrum in the
pre-big-bang model is obtained in [90] as( )2µ+1 ( )5−2µ2π 3 fs4 f1fΩgw (f ) = b(µ) 2 2
H0 MPl fs
fs¯( ) ( )( ) ( )¯ (2) αf
f
αf
f
(2)
′
× ¯¯H0
Jµ
+ H1
Jµ
fs
fs
fs
fs
( ) ( )¯2
(1 − α) fs (2) αf
f ¯¯
−
H0
Jµ
,
2α f
fs
fs ¯

(3.46)

with

α 2µ
1
2 (2µ − 1)2 Γ2 (µ) ,µ ≡ |2β − 3| .
48
2In the spectrum, the new parameter µ explicitly appears instead of β. Thus, the shapeof the spectrum is completely determined by two parameters µ and fs .
For f ≪ fs ≪ f1 , the expression (3.46) can be expanded asb(µ) ≡(2µ − 1)2 2π 3 fs4Ωgw (f ) ≈2
192µ2 α H02 MPl
[
×

(

f1
fs

4
(2µα − 1 + α)2 + 2π)2µ+1 (

f
fs

)3

{
(
)
}2 ]
αf
(, 3.47)(2µα − 1 + α) ln+ γE − 22fs

39

3.6 Other production mechanisms

where γE ≈ 0.5772 is the Euler constant. On the other hand, for fs ≪ f ≪ f1 ,Eq. (3.46) can be approximated to4b(µ) 2π 3 fs4Ωgw (f ) ≈ 2
2
π α H02 MPl

(

f
f1

)3−2µ
.

(3.48)

It is important to stress that in the high frequency limit the unknown parameter fs cancels. The approximated expressions (3.47) and (3.48) give the frequency dependenceof the GWB spectrum, f 3 at low frequencies (the factor (ln f )2 is a small correction.),and f 3−2µ at high frequencies, which reproduce the frequency dependence of the approximated spectrum ﬁrst found in [93, 94, 95]. At the frequency f = f1 , there is a
sharp cutoﬀ in the spectrum because the ﬂuctuation never cross the horizon above thefrequencies, and its ampliﬁcation is suppressed.
In Fig. 7.8, the GWB spectra are plotted for three parameter sets (µ, fs ) = (1.5,
100 Hz), (1.3, 100 Hz), (1.4, 105 Hz). We select µ in the range 0 < µ < 3/2, because
the case with µ > 3/2 has large energy at lower frequency than f1 and easily violates
the observational bounds due to big-bang nucleosynthesis or GW detectors (LIGO)(see, Fig. 4.2 in Chap. 4). On the other hand, fs is allowed to have much lower values
than our choices of fs in Fig. 3.5, because the spectrum rapidly decreases as f 3 at
the frequencies f < fs , and the observational limits due to pulsar timing and CMB
can be easily avoided. So, the GWB spectrum can be extended toward much lower
frequencies, unless it violates the pulsar-timing limit. While, for this reason, MD phase
is unimportant for GWB spectrum in the pre-big-bang model.
The most stringent limit comes from those at high frequencies. Therefore, the prebig-bang model is signiﬁcantly interesting for GW experiments at high frequencies.
Advanced detectors will be able to search a part of a parameter space of the pre-bigbang model, and would be a test of the string theory. The observational data of 4th
science run of LIGO have already begun to bound the model parameters, though thelimit is still very weak [96].

3.6

Other production mechanisms

Preheating after inﬂation
At the end of inﬂation, the process by which the energy density driving inﬂation wasconverted into the matter and radiation we observe today, is called reheating. The
ﬁrst stage of the conversion, preheating, is known to be violent and accompanies rapid
growth of inhomogeneities by parametric resonance. Subsequently, the large concentrations of energy density in bubble structures collide, and generate a signiﬁcant fractionof energy in the form of gravitational waves. Then, the bubbles are dissipated throughturbulences, and are ﬁnally thermalized. The process also produces gravitational waves.
The computation of a GWB in the process is quite complicated, nonlinear problem,and needs a numerical simulation. Such a GWB in chaotic inﬂation has ﬁrst been
examined by Khlebnikov and Tkachev [97], and recently has been reanalyzed more accurately by Easther et al. [98, 99]. The GWB in hybrid inﬂation has been computed by

3. Creation of cosmological GWB

40

Figure 3.5: GWB spectrum produced in the minimal pre-bing-bang model with theﬁxed values of Hs ≈ 0.03 MPl and f1 = 6.13 × 1010 Hz.

Garcia-Bellido et al. [100, 101]. In both cases of inﬂations, the predicted GWB energy
density is h20 Ωgw ∼ 10−10 , localized at the frequencies 107 − 109 Hz on the spectrum
9, depending on the energy scale the preheating occurs. Thus, the GWB from thepreheating open a new window for searching for unknown physics in the early universe.

Phase transition
In the early universe, the restored symmetry is broken at a certain temperature. For
example, one of ﬁrst-order phase transitions from false vacuum to true vacuum might
be electroweak symmetry breaking, though it is not known yet whether the electroweak
phase transition is the ﬁrst-order or not. If the ﬁrst-order phase transitions occurred,
explosive thermal bubbles by the release of potential energy are generated, then collide,and form turbulent plasma. In the case, a GWB spectrum with characteristic peaks
is produced via the bubble collisions [102, 103] and the turbulence [104, 105], which issimilar to the preheating, and could produce larger spectrum than that due to inﬂation.
The detectability of the GWB has been discussed in the LISA band by Nicolis [106],and in the higher frequency band up to ∼ 100 Hz by Grojean and Servant [107]. Theenergy density is, at most, Ωgw ∼ 10−10 at the peak of the spectrum, being independentof the frequency band. If the ﬁrst-order phase transitions occurred at the temperature
T ∼ 100 GeV, the peak frequency of the GWB corresponds to the frequency range of
LISA, according to Eq. (2.37). If the transitions occurred at the higher temperature,say, T ∼ 107 GeV, advanced LIGO might detect the GWB.
The generation of the GWB at the frequencies, say, ∼ 1 − 100 Hz, is also possible, though theamount of the produced energy is relatively small.
941

3.6 Other production mechanisms

Cosmic strings
Cosmic strings were topological defects which may be formed during phase transitionsin the early universe [108]. Cosmic superstrings are produced in certain string theory.
A cosmological network of oscillating string loops generates a stochastic GWB [109,110, 111]. The spectrum is characterized by three parameters: the string tension (mass
per unit length) Gµ, the string reconnection probability p, and the parameter ϵ related
to the string loop size, where G is the gravitational constant. In the most cases, broad
GWB spectra are predicted in the frequency range from 10−12 Hz to 106 Hz, though
the bump of the GWB spectrum depends on ϵ. For relatively small ϵ, the maximal
amplitude could reach the big-bang nucleosynthesis bound, which is described in thenext chapter. The most-recent theoretical estimate of the upper limit on the parameters
has been discussed by Siemens et al. [112]. The large volume of the parameter spacecan be complementarily explored by the ground- and space-based GW detectors andthe pulsar timing.

Chapter 4
Observational constraints on GWB
As reviewed in Chap. 3, inﬂation creates a GWB spectrum broadly ranging in frequencies ∼ 10−18 - 109 Hz. The shape of the spectrum reﬂects inﬂation potentials and phasetransitions, predicted in extended theories of high-energy physics. Therefore, it is very
important to obtain constraints on the GWB spectrum not at a particular frequency
but several frequencies much diﬀerent one another. In this chapter, we will see howvarious observations impose the limits on the amount of GWB. At the end of thischapter, we summarize observational constraints on GWB and the predicted spectraof stochastic GWBs in Fig. 4.2.

4.1

Big-bang nucleosynthesis limit

Big-bang nucleosynthesis (BBN) admirably predicts the abundance of light-elements(H, D, 3 He, 4 He, 7 Li) in the universe currently observed. The abundance depends on
two parameters: a baryon-to-photon ratio τ and eﬀective degrees of freedom of particles
at the temperature ∼ 1 MeV, g∗ = g(T ∼ 1 MeV). These parameters are strongly
constrained by observations. Therefore, the energy density of GWs can not be muchlarger not so as to impair the predictions of the light-element abundances. Tightest
limit can be obtained by the abundance of 4 He, which is determined by the ratio of thenumber density of neutrons to protons available at the beginning of the nucleosynthesis,
(

nn
np

)

(

Q
≈ exp −Tf),

Q ≡ mn − mp ≈ 1.3 MeV .

The ratio signiﬁcantly depends on the freezing temperature Tf that neutrons decouple
from an weak interaction with protons.
The freezing temperature can be estimated from a balance between the rate ofcosmic expansion and of the weak interaction, which is the same argument as that inSec. 2.2.5. The rate of expansion of the universe is given by
H2 =

1
π2
ρ
=
g T4 ,
2
2 ∗
3MPl
90MPl
43

4. Observational constraints on GWB

44for relativistic particles, where g∗ is deﬁned as
g∗ ≡

(

∑

gi

i=boson

Ti
T

)4

( )4
7 ∑
Ti
+
gi
.
8 i=fermion
T

(4.1)

In the standard model of a particle theory, relativistic particles at T∗ ≈ 1 MeV are aphoton, an electron and positron, and three species of neutrinos (Nν = 3): γ, e± , ν, ν̄.
Then, g∗ = 43/3. If there would exist extra particles, g∗ increases. So, conventionally,
extra degrees of freedom are included into the species of neutrinos and are deﬁned by
the quantity Nνeﬀ , which is generally not an integer. Therefore, the degrees of freedom(4.1) is7
43 7 eﬀg∗ (Nνeﬀ ) = 2 + (4 + 2Nνeﬀ ) =+ (Nν − 3) .
(4.2)
8
4
8
On the other hand, the weak interaction can be approximated by the Fermi interaction if the energy scale considered is low enough compared to the mass of weakbosons, 80 − 90 GeV. Since the nucleosynthesis occurs at the energy scale around
1 MeV, the cross section is given by σ ∼ G2F T 2 , where the Fermi constant is GF =1.166 × 10−5 GeV−2 (This is really appropriate in the limit T ≫ Q. But we will use it
for simplicity.). Using particle velocities |v| ∼ 1 and the number density n ∼ T 3 , one
can obtain the rate of weak interaction Γ ∼ G2F T 5 .
Consequently, the condition Γ ∼ H gives
G2F Tf5 ≈( g )1/2 π T 2
∗

f

90

MPl

,

that is,Tf ∝ g∗1/6 .
If g∗ increases, the cosmic expansion at that time become rapid, and Tf rises. This
follows the increase of neutron abundance available at the time of freeze-out. Then,
more 4 He is produced. However, another parameter of the BBN, τ also aﬀects theamount of 4 He: lower τ leads to small amount of 4 He. Thus, we need to know both theupper limit of the 4 He abundance and the lower limit of τ in order to constrain g∗ orNνeﬀ . According to [113], although there are some astronomical uncertainty after theprimordial production of light elements due to the BBN, a conservative limit Nνeﬀ ≤ 4
is obtained.
From Eqs. (4.1) and (4.2), the energy density of gravitons has to be
(

ρgw
ργ

)

(
=
BBN

Tgw
T

)4

7
≤ (Nνeﬀ − 3) .
8
BBN

(4.3)

This quantity should be translated to the energy density today. Entropy conservation
in a comoving volume, as well as the discussion in Sec. 2.2.5, gives the relation
(

ρgw
ργ

)

(
=
0

ρgw
ργ

)

(
BBN

gs (T0 )gs (1 MeV)

)4/3
,

(4.4)

45

4.2 CMB limit

where gs (T0 ) = 3.91 and gs (1 MeV) = 10.75 1 . Combining Eqs. (4.3) and (4.4), we
obtain()ρgw≤ 0.227 (Nνeﬀ − 3) .
(4.5)
ργ 0
In the above equation, ρgw is an integrated energy density over frequencies. Using∫dρgw (f )ρgw = d(ln f ),d ln fthe observed value of ργ0 , and Eq. (2.26) gives∫ fmaxd(ln f ) h20 Ωgw (f ) ≤ 5.6 × 10−6 (Nνeﬀ − 3) .

(4.6)

fmin

The choice of the upper and lower cutoﬀ frequencies is discussed in [114]. The frequency fmin corresponds to the horizon scale at the time of the BBN, and fmin =fBBN ∼ 10−10 Hz, because GWs whose wavelength is larger than the horizon scale do
not contribute as radiation. As for fmax , it would be taken to be fmax = f1 ∼ 109 Hz
for an inﬂationary GWB and fmax = fPl ∼ 1043 Hz for a general situation.
Adopting Nνeﬀ ≤ 4 and assuming that the spectrum can be regarded as ﬂat in afrequency range ∆ ln f ∼ 1, we obtainh20 Ωgw (f ) ≤ 5.6 × 10−6 ,at 10−10 Hz . f . fmax .

(4.7)

Note that this limit is only valid for the GWB present at the time of the BBN andcannot be applied to GWB produced after that by astrophysical sources or some otherprocesses.

4.2

CMB limit

Gravitational waves contribute to temperature ﬂuctuations of CMB today. The amplitude of the ﬂuctuations cannot exceed that of current observations. Consequently, theamplitude of GWB has to be limited. As described in Sec. 3.3, 5th year WMAP data
combined with type Ia supernovae and baryon acoustic oscillation data tightly limits
the inﬂationary GWB energy spectrum in broad frequency ranges. However, this limit
can be derived, assuming that the tensor spectral index has a negative index of thetensor mode, nT < 0. In general, nT could be positive. Therefore, the CMB limit
derived in Sec. 3.3 is only valid at the pivot scale k0 = 0.002 Mpc−1 and gives
h20 Ωgw (k0 ) ≤ 5.93 × 10−14

at k0 = 0.002 Mpc−1 .

(4.8)

On the other hand, Seljak et al. [22] has analyzed cosmological parameters with
WMAP, SDSS galaxy clustering and SDSS Lyα forest data, taking the pivot scale at
kpivot = 0.05 Mpc−1 . They have obtained a constraint r < 0.45 (95% C.L.), allowing
1

We assumed that relativistic particles are γ, e± , ν, ν̄ at T = 1 MeV, and are γ, ν, ν̄ at T = T0 .

4. Observational constraints on GWB

46

a running spectral index of a scalar mode. Then, the constraint on GWB energy
spectrum is
h20 Ωgw (kpivot ) ≤ 3.41 × 10−15

at kpivot = 0.05 Mpc−1 .

(4.9)

Assuming that no steep peak exists on GWB energy spectrum between k0 and kpivot ,we can interpolate the CMB limits, Eqs. (4.8) and (4.9), and obtain the limit between
f0 = 3.24 × 10−18 Hz and fpivot = 8.10 × 10−17 Hz.
Another limit to GWB has been obtained by CMB and galaxy observations [115].
GWB with frequencies & 1015 Hz contributes to the radiation density of the universe at
the time of decoupling of CMB, and aﬀects on the expansion rate. It also aﬀects on thegrowth of perturbations at later times. As a result, CMB power spectrum and matter
spectrum today would be changed. This concept is similar to the constraint to GWB
due to BBN. However, the advantage is that the limit can be extended to the lower
frequencies than that of BBN. The frequency corresponds to wavelengths comparable
to the comoving horizon at CMB decoupling.
The excess energy of radiation at the time of CMB decoupling is constrained bycurrent observations in terms of eﬀective neutrino degrees of freedom at the time ofdecoupling of CMB Nνeﬀ (tdec ). Then, Nνeﬀ (tdec ) is translated into the limit to Ωgw . In
[115], the authors carried out likelihood analyses, using the constraints from CMB,
galaxy surveys, and the Lyα forest, and assuming that the number of neutrino degrees
of freedom (not including extra degrees of freedom) is Nν = 3.04 and that the neutrino
masses are free to vary. For GWB produced with homogeneous initial conditions (ininﬂation, pre-big-bang or ekpyrotic scenario) 2 , they have obtained
h20 Ωgw (f ) ≤ 8.4 × 10−6

at f & 10−15 Hz(95% C.L.) ,in a frequency range ∆ ln f ∼ 1. The lower end of frequencies that the limit can be applied is taken as 10−15 Hz. This is a conservative choice, because the gravitational-wave
wavelength must be within the horizon at the time of slightly before recombination.
The CMB limit is comparable to the big-bang nucleosynthesis limit.

4.3

Pulsar-timing limit

Pulsars are rapidly rotating compact objects, which are considered to be formed by
the supernovae of massive stars. The pulsars emit pulses with periods considerably
stable in time. Particularly, the periods of millisecond pulsars are extremely stable,
and has the time derivatives of the periods Ω̇ ∼ 10−19 sec sec−1 [116]. For instance,
the observations of the ﬁrst millisecond pulsar discovered, B1937+21, after 9 year ofdata, give a period of 1.557 806 468 819 794 5 ± 0.000 000 000 000 000 4 msec, which iscomparable to the accuracy of atomic clocks.
One can take advantage of the stability of pulsars to test general relativity. From theobservation of the change in the revolution period of B1931+16 (Hulse-Taylar binary
pulsar), general relativity has been tested at a level of 1%, and the indirect evidence
2

For adiabatic initial conditions, a limit at 95 % C.L. of h20 Ωgw (f ) ≤ 4.0 × 10−5 is obtained.

47

4.3 Pulsar-timing limitof the existence of GWs has been obtained [3, 4]. Such highly accurate clocks can also
be used for the direct detection of GWs.

4.3.1

Limit from spinning pulsars

Since the period of pulsars is extremely stable, one can detect the inﬂuence of gravitational waves on the pulses by monitoring the change of pulse arrival times. In otherwords, pulsar timing can be regarded as a gravitational-wave detector with a very longarm.
Suppose that pulses are emitted at a pulsar at the time t0 and received at the
Earth at the time t1 , and that a GW is propagating in the direction of z axis withthe polarization along x axis. The coordinates is shown in Fig. 4.1, where L is the
distance between the Earth and the pulsar, and θ is the angle between the direction of
the pulsar from the Earth and the direction the GW propagates. The change of arrival
times observed by the observer on the Earth can be written as [117, 118],∆ν(t1 )1
= − (1 − cos θ)[h(t1 ) − h(t0 − L cos θ)] ,ν(t0 )
2
in frequency shifts, or, equivalently,1∆t(t1 ) = − (1 − cos θ)2∫ L(1+cos θ)h(t0 − ξ)dξ ,0in time residuals.
The residual times in pulse arrival times are obtained by diﬀerentiating between
observed arrival times and the arrival times predicted with timing models (for thereview, see [116]). The timing model is written in a Taylor-expanded form,
1
φ(τ ) = φ0 + (τ − τ0 ) Ω0 + (τ − τ0 )2 Ω̇0 + . . . ,2(4.10)

where τ is the proper time of the pulsar, τ0 is the ﬁducial time, φ0 is the phase atτ0 , Ω0 and Ω̇0 are the pulse period and the derivative at τ0 . The proper time τ in
Eq. (4.10) is related to the time on the Earth by including various corrections on thelight path from the pulsar to the Earth: relativistic eﬀects in the solar system andaround the pulsar, revolution and peculiar motions of the Earth, dispersion measures,etc.. In addition, the timing model has plural parameters intrinsic to an individual
pulsar, such as pulsar’s pulse frequency, its ﬁrst derivative, astrometric, and binaryparameters. These parameters are ﬁtted so that the residual times have a zero-mean
Gaussian distribution after subtracting the predicted arrival times from the observed
one. If a systematic error is found in the residual, it will be the contributions from
GWs.
Denoting errors in the pulse arrival time ϵ and observation time T , one can estimate
the sensitivity to GWs asϵhc (f∗ ) ∼ ∼ ϵf∗ ,T

4. Observational constraints on GWB

48

Figure 4.1: Doppler tracking.

where f∗ is the frequency given by the observation time. If no GW signal would ﬁnd
in the residual data, the limit on hc is translated to the limit on Ωgw according to
Eq. (2.30). More detailed analysis of 7 years data of PSR B1855+09 by Kaspi et al.
[119] gives
h20 Ωgw (f∗ ) < 6 × 10−8 ,(95% C.L.) ,where f∗ = 1/7yr ≈ 4.5 × 10−9 Hz. Since hc ∝ T −1 ∝ f and h20 Ωgw ∝ f 2 h2c , the limiton h20 Ωgw for f > f∗ is(
h20 Ωgw (f ) < 6 × 10−8

f
f∗

)4
,

(95% C.L.) .

No constraint can be obtained for f < f∗ , because the systematic residuals are removedby the parameter ﬁtting.
Although the above limit results from the time residual of a single millisecond
pulsar, combining multiple pulsars provides us a powerful method that can remove
systematic errors, because the time residuals of the pulsars include identical noise
sources due to proper motion of the Earth, systematic errors of clocks, etc., and they
are correlated. Recent discovery of multiple millisecond pulsars allows such an analysis.
Consequently, the sensitivity to a GWB can also be improved.
More stringent limit will be placed soon by Parkes Pulsar Timing Array (PPTA)
project [120], which started in February 2004 as a collaborative eﬀort between the
ATNF, Swinburne University of Technology, the University of Brownsville, Texas. The
project aims to observe 20 millisecond pulsars with an RMS timing residual of 100 nsec

49

4.3 Pulsar-timing limit

over 5 years. According to Hobbs [120], the limit on a GWB of h20 Ωgw < 5 × 10−10
can be achieved after the 5 year observation. In the future, the Square-KilometreArray (SKA) [121] is planned by an international collaboration. SKA will be a radio
telescope with a collecting area that will exceed that of existing telescopes by a factor
of a hundred or so. Due to its outstanding sensitivity, sky and frequency coverage, SKA
will discover about 20,000 pulsars in our galaxy, including more than 1,000 millisecond
pulsars. Therefore, the sensitivity to a GWB of h20 Ωgw . 3 × 10−13 would be reached[121].
Recently, a technique to detect GWB by looking for correlations between pulsar
observations was developed by Jenet et al. [122], and they found a slightly diﬀerent
limit on a GWB with almost the same data set as the previous studies. Using the data
set with data spans ∼ 20 yr for PSR B1855+09 and ∼ 2 − 4 yr for seven other pulsars,
they obtained the limits on h20 Ωgw with ﬂat spectrum [123],h20 Ωgw (f ) < 2.0 × 10−8 ,at the frequencies 1/1 yr, 1/8 yr, and 1/20 yr. The threshold of the detection is deﬁned
so that detection rate is 95% and false alarm rate is 0.1%. If the full observation dataof PPTA (20 millisecond pulsars with an RMS timing residual of 100 nsec over 5 years)
will be available, the limit will be h20 Ωgw (f ) < 9.1 × 10−11 at the frequencies 1/1 yr,
1/8 yr, and 1/20 yr.

4.3.2

Limit from binary pulsars

Another important limit comes from pulsars in binary systems. The orbit of binary
pulsars is predicted by the general relativity and is well agreed with observations [3, 4].
So, it can be used as another clock. In most cases, the ﬁrst derivative of the period Ṗ
is negligibly small and no ﬁtting to unknown parameters is needed. This fact allows
us to constrain GWB at lower frequencies than 1/T by monitoring the gradual changeof the orbital period. Lower end of frequencies the limit can be obtained is f = 1/D,
where D is distance to the pulsar. The detailed analysis of PSR B1855+09 [124] gives
the limit
h20 Ωgw < 2.7 × 10−4

at 1.1 × 10−11 Hz < f < 4.5 × 10−9 Hz .

4. Observational constraints on GWB

50

Figure 4.2: Observational constraints on a GWB and the predicted spectra of stochastic
GWBs. The observational constraints denote already obtained limits with solid lines
and future limits with dotted lines. The GWB spectra are due to the de-Sitter inﬂation(H = 2.7 × 10−5 MPl ), the quintessential inﬂation (gs (τ0 ) = 3.91, gs (τth ) = 106.75,
λ = 10−14 , R = 0.15), and the pre-big-bang model (µ = 1.5, fs = 10 Hz). The
limits come from the big-bang nucleosynthesis (BBN), WMAP, CMB+galaxy+Lyα, thebinary pulsar (BPSR), the millisecond pulsar (MSP), the pulsar timing array (PPTA),
and the square kilometre array (SKA), which are all explained in this chapter. There isalso shown the limit by a spacecraft Doppler tracking (Cassini) [125], the ground-based
GW detector (LIGO and advanced LIGO), the space-based GW detector (LISA and
DECIGO) [126], and the ultra-high frequency GW detectors [50]. The LIGO boundand the sensitivity of advanced LIGO is described in Sec. 5.3.

Chapter 5
Direct search for GWB
5.1

Correlation analysis

Let us consider the outputs of a detector, s(t) = h(t) + n(t), where h(t) and n(t) are⃗ the
the GW signal and the noise of a detector. From Eq. (2.23), at generic point X,
gravitational metric perturbations in the transverse traceless gauge are given by
∫ ∞
∑∫
⃗
⃗ =
h(t, X)dΩ̂df h̃A (f, Ω̂) e2πif (t−Ω̂·X/c) eA (Ω̂) ,
(5.1)
A

S2

−∞

where Ω̂ is a unit vector directed at GW propagation and h̃A (f, Ω̂) is the Fouriertransform of GW amplitude with polarizations A = +, ×. Polarizarion tensors eA (Ω̂)
are deﬁned in Eq. (2.24).
GW signal h(t) from a detector is given by D(f, Ω̂) : h(f, Ω̂), where the symbol: denotes contraction between tensors, and D(f, Ω̂) is a so-called detector tensor,
which describes the total response of a detector and maps the gravitational metricperturbation to the GW signal from a detector. We deﬁne it including detector response
functions as]
1[
D(f, Ω̂) ≡(û ⊗ û)T (f, Ω̂ · û) − (v̂ ⊗ v̂)T (f, Ω̂ · v̂) .
(5.2)
2Here û and v̂ are unit vectors. We assume that they are orthogonal to each other andare directed to each detector arm. The function T is a detector response function that
describes the eﬀect of ﬁnite arm length on propagating light. In the detector whose
arm length is much smaller than the wavelength of GW, that is, in low frequency limit,
this function is approximated to unity, while in a detector whose size is comparable to
GW wavelength, the function signiﬁcantly aﬀects the response of the detector. Thisis true for a synchronous recycling interferometer at ultra-high frequencies, which wewill investigate in Chap. 6.
Using Eqs. (5.1) and (5.2), GW signal h(t) can be written as
∫ ∞
∑∫
⃗
⃗
h(t, X) =dΩ̂
df h̃A (f, Ω̂) e2πif (t−Ω̂·X/c) FA (f, Ω̂) ,
(5.3)
A

S2

−∞

51

5. Direct search for GWB

52

where an angular pattern function of a detector Fp (f, Ω̂) is deﬁned by
FA (f, Ω̂) ≡ D(f, Ω̂) : eA (Ω̂) .
Cross-correlation signal Y between two detectors is deﬁned as∫ T /2 ∫ T /2
Y ≡dt
dt′ s1 (t)s2 (t′ )Q(t − t′ ) ,
−T /2

(5.4)

(5.5)

−T /2

where s1 and s2 are an output from each detector, T is observation time. Q(t − t′ ) is anarbitrary real function, which is called an optimal ﬁlter. Its form is determined below
so that signal-to-noise ratio (SNR) is maximized. Fourier transforming s1 (t) and s2 (t),one can obtain∫ ∞ ∫ ∞Y =df
df ′ δT (f − f ′ )s̃∗1 (f )s̃2 (f ′ )Q̃(f ′ ),
(5.6)
−∞

−∞

where s̃1 (f ), s̃2 (f ) and Q̃(f ) are the Fourier transforms of s1 (t), s2 (t) and Q(t − t′ ),respectively. δT (f ) is the ﬁnite-time approximation to the Dirac delta function deﬁned
by∫ T /2sin(πf T ).
δT (f ) ≡dt e−2πif t =πf−T /2
In the above derivation, we took the limit of large T for one of the integrals. This isjustiﬁed by the fact that, in general, Q(t − t′ ) rapidly decreases for large |t − t′ |. The
correlation signal obtained above ideally has a contribution from only the GW signal
since we assume that noise has no correlation between two detectors. Thus, we takeensemble average of Eq. (5.6) and obtain the signal from GWB,
∫ ∞ ∫ ∞µ ≡ 〈Y 〉 =df
df ′ δT (f − f ′ )〈h̃∗1 (f )h̃2 (f ′ )〉Q̃(f ′ ) .
(5.7)
−∞

−∞

Substituting the Fourier transform of Eq. (5.3),∑∫⃗h̃(f ) =dΩ̂ h̃A (f, Ω̂)e−2πif Ω̂·X/c F A (f, Ω̂) ,
A

(5.8)

S2

into Eq. (5.7), and using Eqs. (2.25) and (2.27), one can obtain
∫ ∞
3H02
T
df |f |−3 Ωgw (|f |) γ(|f |) Q̃(f ).
µ=
20π 2
−∞
Here we deﬁned the overlap reduction function,∫1 ∑dΩ̂ 2πif Ω̂·∆X/c⃗γ(f ) ≡e
F1A ∗ (f, Ω̂)F2A (f, Ω̂) ,(2/5) A S 2 4π

(5.9)(5.10)⃗ ≡X⃗1−X⃗ 2 . The factor in a right-hand
where the separation of two detectors is ∆X
side of Eq. (5.10) is a normalization factor so that the overlap reduction function gives
unity in low frequency limit.

53

5.1 Correlation analysis

This deﬁnition is slightly diﬀerent from that in other papers [51, 7], because thedetector response function T is included in Eq. (5.2). In low frequency limit, T gives
unity, and the deﬁnition of Eq. (5.10) coincides with the equation in [51, 7]. The
overlap reduction function is deﬁned as meaning how GW signals in two detectors
are correlated, and equals unity for colocated and coaligned detectors. However, thediﬀerence is conspicuous at high frequencies. The overlap reduction function does notgive unity even for colocated and coaligned detector. Namely, the loss of GW signalsof detectors can be regarded as the reduction of overlap between two detectors. We
will return to this issue in Chap. 6 and investigate in detail. In this chapter, we willtreat detectors that GW wavelength is much larger than the size of the detectors andcan set the detector response function T = 1.
Next, we will calculate the variance of a correlation signal. Here we assume thatnoises in two detectors do not correlate at all and that the magnitude of GW signal ismuch smaller than that of noise. Consequently, the variance of correlation signal isσ 2 ≡ 〈Y 2 〉 − 〈Y 〉2 ≈ 〈Y 2 〉 .
Then, using Eq. (5.6), it follows∫ ∞ ∫ ∞
2σ ≈df
df ′ Q̃(f )Q̃∗ (f ′ ) 〈s̃∗1 (f )s̃1 (f ′ )〉 〈s̃2 (f )s̃∗2 (f ′ )〉
−∞
−∞
∫
T ∞
df P1 (|f |)P2 (|f |) |Q̃(f )|2 ,≈4 −∞(5.11)(5.12)

where the one-sided power spectrum density of noise is deﬁned by1〈ñ∗i (f )ñi (f ′ )〉 ≡ δ(f − f ′ )Pi (f ),i = 1, 2 .
2
Now we can determine the form of the optimal ﬁlter Q̃(f ). Equations (5.9) and(5.12) are expressed more simply, using an inner product∫ ∞(A, B) ≡df A∗ (f )B(f )P1 (|f |)P2 (|f |) ,
−∞

as

(
)
3H02
γ(|f |)Ωgw (|f |)T Q̃, 3,(5.13)µ =20π 2|f | P1 (|f |)P2 (|f |)
)
T (
2
σ ≈
Q̃, Q̃ .
(5.14)
4
From Eqs. (5.13) and (5.14), SNR for GWB is deﬁned as SNR ≡ µ/σ. Therefore, theoptimal ﬁlter function turns out to beQ̃(f ) = K

γ(f )Ωgw (|f |),|f |3 P1 (|f |)P2 (|f |)(5.15)

with an arbitrary normalization factor K. Applying this optimal ﬁlter to the above
equations, we obtain maximal SNR
[∫ ∞
]1/2
γ 2 (|f |)Ω2gw (|f |)3H02 √
T
df 6
SNR =
.
(5.16)
10π 2
f P1 (|f |)P2 (|f |)−∞5. Direct search for GWB

5.2

Overlap reduction functions

5.2.1

Tensorial expansion

54

The overlap reduction functions can be analytically investigated by expanding them intensorial bases and carrying out the angular integral with respect to the propagating
direction of GWs [127, 51].
For real detectors on the Earth such as LIGO etc., the wavelength of GWs is muchlarger than the size of detector. In this case, the angular pattern function deﬁned in
Eq. (5.4) does not depend on frequencies, and is simply given by
FA (Ω̂) = D(Ω̂) : eA (Ω̂) ,(5.17)
1
D(Ω̂) =[û ⊗ û − v̂ ⊗ v̂] .
(5.18)
2
Then, a non-normalized overlap reduction function γ̄ for a colocated and coaligned
detector pair (ideal case) is given by∑ ∫ dΩ̂
2
FA ∗ (Ω̂)FA (Ω̂) = ,(5.19)
γ̄ideal ≡5
S 2 4πA
which is exactly the same as the normalization factor appearing in Eq. (5.10). Deﬁning⃗ = X⃗1 − X⃗ 2 ≡ |∆X|⃗ dˆ ,∆X⃗
2πf |∆X|α(f ) ≡,
cand
Γijkℓ (α, dˆ) ≡∑∫1γ̄idealAdΩ̂ iαΩ̂·dˆ Aeẽij (Ω̂)ẽAkℓ (Ω̂) ,S 2 4π(5.20)

one can write the overlap reduction function asγ(f ) = Dij Dkℓ Γijkℓ (α, dˆ) .

(5.21)

Note that Γijkℓ satisﬁes the symmetric properties,
Γijkℓ = Γjikℓ ,Γijkℓ = Γijℓk ,Γijkℓ = Γkℓij .

Consequently, thank to the symmetries, Γijkℓ can be expanded in tensorial bases like
Γijkℓ (α, dˆ) = C1 (α)δij δkℓ + C2 (α)(δik δjℓ + δjk δiℓ ) + C3 (α)(δij dˆk dˆℓ + δkℓ dˆi dˆj )= C4 (α)(δik dˆj dˆℓ + δiℓ dˆj dˆk + δjk dˆi dˆℓ + δjℓ dˆi dˆk ) + C5 (α)dˆi dˆj dˆk dˆℓ .
(5.22)
Here we deﬁne the contracted quantities of Γijkℓ with the tensorial bases.
q1 ≡ Γijkℓ δ ij δ kℓ , q2 ≡ Γijkℓ (δ ik δ jℓ + δ jk δ iℓ ) , q3 ≡ Γijkℓ (δ ij dˆk dˆℓ + δ kℓ dˆi dˆj ) ,q4 ≡ Γijkℓ (δ ik dˆj dˆℓ + δ iℓ dˆj dˆk + δ jk dˆi dˆℓ + δ jℓ dˆi dˆk ) , q5 ≡ Γijkℓ dˆi dˆj dˆk dˆℓ ,(5.23)

55

5.2 Overlap reduction functions

Then, from Eqs. (5.22) and (5.23), the contractions q1 , · · · , q5 can be related to thecoeﬃcients C1 , · · · , C5 by


 


q1
9 6 6 4 1
C1
 q2   6 24 4 16 2   C2 

 


 q3  =  6 4 8 8 2   C3  ,

 


 q4   4 16 8 24 4   C4 
q5
1 2 2 4 1
C5or, conversely,




q1
3 −1 −3 1
1C1 −1 1
 C2 
1 −1 1   q2  C3  = 1  −3 1
5 −1 −5   q3 
 .

 8
 1 −1 −1 2 −5   q4  C4 
1
1 −5 −5 35
q5C5(5.24)On the other hand, q1 , · · · , q5 can be calculated explicitly from Eq. (5.20), by integratingwith respect to the propagation direction of GWs over the celestial sphere. Formulae of
spherical Bessel functions jn (x) needed for the calculation are provided in the Appendix
B. The results are
q1 = 0 ,q2 = 20j0 (α) ,q3 = 0 ,q4 = 40

j1 (α),
αq5 = 20

j2 (α),
α2and Eq. (5.24) can be reduced to
−28 803
C1 42 −60
 C2 
3 
j0




 C3  = 1  0 −120 −15   j2  .
 C4  42  0
90 −15  j4
C5
0
0
105


From Eqs. (5.21) and (5.22) together with the traceless property of Dij , the overlap
reduction function can be written asγ(f ) = ρ1 (α)Dij Dij + ρ2 (α)Dik Dkj di dj + ρ3 (α)Dij Dkℓ di dj dk dℓ ,(5.25)

with redeﬁnitions of the coeﬃcients, ρ1 (α) = 2C2 (α), ρ2 (α) = 4C4 (α), and ρ3 (α) =C5 (α). The new coeﬃcients are given by
28 −40 2
j0ρ1
 ρ2  = 1  0 120 −20   j2  .
14
j4
ρ3
0
0
35


5. Direct search for GWB

56

Figure 5.1: Coordinate on the Earth for a detector pair.

5.2.2

Optimal conﬁguration

To further investigate the dependence of the overlap reduction function on the detector
conﬁgurations, we introduce a coordinate, which simpliﬁes the expression in Eq. (5.25),
as shown in Fig. 5.1. The relative location and orientation of two detectors are characterized by the three parameters, (β, σ1 , σ2 ). The β is the separation angle between
two detectors measured from the center of the Earth. The angles σ1 and σ2 , are theorientations of the bisector of two arms of each detector, measured in counterclockwisemanner relative to the great circle connecting the two detectors. The distance between
two detectors is given by|∆X| = 2RE sin

β,2where the radius of the Earth is RE = 6371 km. Deﬁning new parameters,σ+ ≡σ1 + σ2,2σ− ≡σ1 − σ2,
2

one can characterize the overlap reduction function by the three parameters (β, σ+ , σ− ),which determine a detector conﬁguration. The overlap reduction function is given byγ T (α, β, σ+ , σ− ) = ΘT + (α, β) cos(4σ+ ) + ΘT − (α, β) cos(4σ− ) ,(5.26)

57

5.2 Overlap reduction functions

together with

(

)
3
45
169
ΘT + (α, β) ≡ −j0 − j2 +
j4
8
56
896
(
)
1
5
27
+
j0 − j2 −j4 cos β
2
7
224
(
)
5
3
1
−
j0 + j2 +j4 cos 2β ,
8
56
896
(
)
( )4
5
3
β
,
ΘT − (α, β) ≡j0 + j2 +j4 cos7112
2(5.27)(5.28)

Here the super- and sub-script ”T” are ﬁxed on the functions in order to distinguishthe tensor mode (+ and × modes) from scalar and vector mode, introduced in the later
section.
From Eq. (5.26), the candidates for the optimal conﬁguration of the detectors can
be classiﬁed into two types:
Type (i) :Type (ii) :cos(4σ+ ) = cos(4σ− ) = ±1 ,cos(4σ+ ) = − cos(4σ− ) = ±1 .

(5.29)In the type (i), the solutions are σ1 = 0 mod π, σ2 = 0 mod π for the plus sign, and
σ1 = π/2 mod π, σ2 = 0 mod π for the minus sign. This means that the great circle
connecting two detectors is parallel to the bisector of the two arms of the detector,
including the orientations when one of the detectors is rotated by multiples of a rightangle. In the type (ii), the solutions are σ1 = π/4 mod π, σ2 = −π/4 mod π for theplus sign, and σ1 = π/4 mod π, σ2 = π/4 mod π for the minus sign. This is the casewhen the great circle connecting two detectors is parallel to one of the arms of thedetectors, including the orientations when one of the detectors is rotated by multiples
of a right angle. Therefore, the optimal conﬁgurations is implemented when one of
the arms of two detectors is parallel or rotated by multiples of 45 degrees, relative tothe great circle connecting two detectors. Note that all of the conﬁgurations are notsimultaneously optimal one, since whether the conﬁguration is optimal or not depends
on the signs of the functions ΘT + and ΘT − .
In Fig. 5.2 and Fig. 5.3, the functions ΘT + and ΘT − are plotted as a function of
β for a ﬁxed frequency and as a function of frequency for a ﬁxed β (Tensor mode isshown with red, solid curve.). When two detectors are close each other (β → 0), or at
low frequencies (f → 0), the magnitude of overlap reduction function is maximized.

5.2.3

Overlap reduction functions of realistic detector pairs

The positions and orientations of the ongoing and planed kilometer-size interferometers
are listed in Table 5.1. We use a spherical coordinate system (θ, φ) with which thenorth pole is at θ = 0◦ , and φ represents longitude. The orientation angle ψ is the angle
between the local east direction and the bisecting line of two arms of each detector
measured counterclockwise. Hereafter we will consider only advanced (the secondgeneration) detectors: AIGO [35], LCGT [36], advanced LIGO (H1) and LIGO (L1)

5. Direct search for GWB

58

Figure 5.2: Plots of ΘM + and ΘM − , M = T, V, S, as a function of β for ﬁxed frequencies:
f = 10 Hz, 40 Hz, 80 Hz. Each curve shows tensor mode (red, solid), vector mode
(green, dotted), scalar mode (blue, dashed).

59

5.2 Overlap reduction functions

Figure 5.3: Plots of ΘM + and ΘM − , M = T, V, S, as a function of frequency for a ﬁxed
β: β = π/6, π/3, π/2, 2π/3. Each curve shows tensor mode (red, solid), vector mode
(green, dotted), scalar mode (blue, dashed).

5. Direct search for GWB

60[33], advanced VIRGO [34]. The reasons are because the number of pairs including
TAMA300 and GEO600 is too much to be presented here, and because the pair withinitial-generation interferometer is less sensitive to a GWB and has little opportunity todetect a GWB. From the positions and orientations in Table 5.1, the relative positions
and orientations (β, σ+ , σ− ) for each detector pair turn out to be those listed in Table
5.2. The combinations are also illustrated in Fig. 5.4.
Given a parameter set (β, σ+ , σ− ), the overlap reduction function for the detector
pair can be calculated. The overlap reduction functions are shown in Fig. 5.5 andFig. 5.6.
The functions start to oscillate and decay rapidly above the frequency whose wavelength corresponds to the separation between the detectors. The cutoﬀ frequency fc isgiven by fc ≡ c/(2|∆X|), and is listed in Table 5.3. Hence, largely separated detectors
are less sensitive to GWB at high frequencies. In addition, the functions approach
constant values at low frequencies. This value is completely determined by the relativeorientation of the detector pair.

5.3

Observational constraints on GWB by LIGO

LIGO has three power-recycled Michelson interferometers, with a Fabry-Perot cavity
in each orthogonal arm. They are located at two sites: Hanford, Washinton, and Livingston, Louisiana. There are two colocated interferometers at Hanford: H1 with 4km-long arm, and H2 with 2 km-long arm. At Livingston site, there is one interferometer: L1 with 4 km-long arm.
So far, ﬁve science runs (S1 - S5) have been done since 2002. S5 has been ﬁnishedin September, 2007. However, the cross-correlation analysis is proceeding now, and theresult has not been published yet. Thus, here we will refer the result of S4, which took
place between 2005 February 22 and March 23 [129].
Cross-correlation analysis of S4 data was performed, but no signal of GWB found.
Then, the upper limit on the energy density of GWB was obtained. During the S4 run,
three interferometers, H1, H2, and L1, were operated, but H1-H2 pair was contaminatedby instrumentally correlated noise. On the other hand, the instrumental noises of H1-L1
and H2-L1 pairs will, in general, be uncorrelated, though, in practice, negligible amountof correlated noise within statistically uncertainty exists. Therefore, the constraint on
the energy density of GWB comes from the cross-correlated data of H1-L1 and H2-L1
pairs. For a frequency independent spectrum, the upper limit ish20 Ωgw < 3.4 × 10−5 ,(90% C.L.) ,in the frequency range 51-150 Hz.
In the future, 1-year data of H1 and L1 with goal sensitivity will reach the sensitivity
to GWB, h20 Ωgw ≤ 1 × 10−6 in the 40-314-Hz band [130], which exceeds the current
BBN bound. Moreover, advanced LIGO, which is the updated version of H1 and L1
and whose sensitivity [131] is roughly 10 times better than that of initial LIGO in a
broad frequency range, will reach the sensitivity, h20 Ωgw ≤ 2 × 10−9 .

61

5.3 Observational constraints on GWB by LIGO
interferometer
θφψ
AIGO (A)
121.4 115.7 -45.0
LCGT (C)53.6 137.3 70.0
LIGO-H1 (H)
43.5 -119.4 171.8
LIGO-L1 (L)59.4 -90.8 243.0
VIRGO (V)
46.4
10.5 116.5
TAMA300 (T) 54.3 139.5 225.0
GEO600 (G)
47.7
9.8
68.8

Table 5.1: Positions and orientations of kilometer-sized interferometers on the Earth[128]. The ﬁgures are in an unit of degree.
detector pair
β
σ+
σ−
A-C
70.8 31.4 31.9
A-H
135.6 45.1 53.7
A-L
157.3 2.1 38.0
A-V
121.4 60.8 20.2
C-H
72.4 25.6 89.1
C-L
99.2 68.1 42.4
C-V
86.6 5.6 28.9
H-L
27.2 62.2 45.3
H-V
79.6 55.1 61.1
L-V
76.8 83.1 26.7
Table 5.2: Relative positions and orientations of a detector pair on the Earth. The
ﬁgures are in an unit of degree. Each detector is represented by initial letters indicatedin Table 5.1.

Figure 5.4: Relative positions and orientations of a realistic-detector pair. Leftpanel shows the combinations (cos 4σ− , cos 4σ+ ). Right panel shows the combinations(β, cos 4σ+ ). The condidates for the optimal conﬁguration, type (i) and (ii), are also
shown in the left panel.

5. Direct search for GWB

62

Figure 5.5: Overlap reduction functions for real-detector pairs on the Earth. Eachcurve shows tensor mode (red, solid), vector mode (green, dotted), scalar mode (blue,
dashed).

63

5.4 Searching for non-tensorial polarizations of gravitational waves
detector pair separation [km] fc [Hz]
A-C
7377
20
A-H
11796
13
A-L
12493
12
A-V
11113
13
C-H
7522
20
C-L
9707
15
C-V
8735
17
H-L
2996
51
H-V
8156
18
L-V
7912
19

Table 5.3: Separation between two detectors and the cutoﬀ frequency of the overlap
reduction function.

5.4

Searching for non-tensorial polarizations of gravitational waves

Recent cosmological observations surprisingly indicate that the expansion of the universe is accelerating [132, 133]. To explain the acceleration, various models are suggested. The models are divided into two classes: modiﬁed gravity theories beyond thegeneral relativity, and extra-dimensional models such as brane-world type and KaluzaKlein type. In these models, extra degrees of freedom of spacetime allows non-tensorial
polarization modes of a GW to exist. Since the extra dimensions must be compactiﬁed
so as not to contradict observations, we cannot directly see them. However, if the GW
propagates in the extra dimensions, the GWs projected onto our three-dimensional
space can be seen for us as non-tensorial polarization modes, other than plus and crossmodes. This is also true for a modiﬁed gravity theory. If the modiﬁed gravity theory
itself contains another degree of freedom such as a coupling with spacetime curvature,
there exist non-tensorial polarization modes. Such a GW can be used as a probe for
new physics concerning the extra dimensions and the modiﬁed gravity theory. In apulsar timing array, the detectability of the non-tensorial mode is discussed in [134].
However, in a laser-interferometric GW detector, little work can be found.

5.4.1

Non-tensorial polarization mode

In general, it is known that a metric gravity theory in the four dimensional spacetime
allows, at most, six polarization modes of a GW [135, 2]. For the GW propagating inz direction, the bases of the six polarizations are deﬁned by (the tensor components

5. Direct search for GWB

64

Figure 5.6: Overlap reduction functions for real-detector pairs on the Earth. Eachcurve shows tensor mode (red, solid), vector mode (green, dotted), scalar mode (blue,
dashed).

are x, y, z from the left or the top.),



1 0 0
0
+
×



0 −1 01ẽij =, ẽij =
0 0 0
0



0 0 1
0
y
x



0 0 0
0
ẽij =,ẽij =
1 0 0
0


1 0
0 0  ,0 0

0 0
0 1  ,
1 0




1 0 0
ẽbij =  0 1 0  ,
0 0 0


0 0 0
√
ẽℓij = 2  0 0 0  ,0 0 1(5.30)

where, in the ﬁrst line, +, ×, and b denote plus, cross, breathing mode, in the second
line, x, y, and ℓ denote vector-x, vector-y, longitudinal mode, respectively. The tilde
is ﬁxed at each polarization bases like ẽ in order to represent that they are projected
polarizations onto our three-dimensional spacetime in a general spacetime with extra
dimensions, and are deﬁned in a three-dimensional space. Each polarization mode is orij′
thogonal one another, and is normalized so that ẽAij ẽA′ = 2δAA′ , A, A = +, ×, b, ℓ, x, y.
Note that the breathing and longitudinal modes are not traceless. As we will see later,
the b and ℓ modes have a scalar-like property, and x and y have a vector-like property.
In Fig. 5.7, it is shown how each GW polarization aﬀects the test masses on a circle.

65

5.4 Searching for non-tensorial polarizations of gravitational waves

Figure 5.7: Six GW polarizations. Two ellipses⊙ show the eﬀect of the GW polarizationon the test masses on a circle. The symbol
and the arrow represent the propagating
direction of a GW.

Which modes appear in several alternative theories is arranged in Table 5.4. As
one can see in the Table, the extra scalar polarizations appear more easily than thevector polarizations.

5.4.2

Angular response of a single detector

The metric tensor with perturbations due to GWs is written as[]
ds2 = −dt2 + δij + hij (ωt − ⃗k · ⃗x) dxi dxj + (extra dim. terms) ,(5.31)hij (ωt − ⃗k · ⃗x) = h+ (ωt − ⃗k · ⃗x)ẽ+ + h× (ωt − ⃗k · ⃗x)ẽ× + hb (ωt − ⃗k · ⃗x)ẽb
ij

ij

ij

+hℓ (ωt − ⃗k · ⃗x)ẽℓij + hx (ωt − ⃗k · ⃗x)ẽxij + hy (ωt − ⃗k · ⃗x)ẽyij ,where hA , A = +, ×, b, ℓ, x, y are complex amplitudes of GWs for each mode. In
Eq. (5.31), ”(Extra dim. terms)” are added only in the theories with extra dimensions. The angular response function of a detector to GWs can be calculated from
Eqs. (5.17) and (5.18),
FA (Ω̂) = D(Ω̂) : ẽA (Ω̂) ,D(Ω̂) =

1[û ⊗ û − v̂ ⊗ v̂] ,2(5.32)

5. Direct search for GWB
theoretical modelẽ+ij
GR in pure 5D Minkowski spacetime⃝
GR in pure 6D Minkowski spacetime⃝
5D Kaluza-Klein compactiﬁcation⃝2
RS braneworld⃝
DGP braneworld (normal branch)⃝
DGP braneworld (self-accelerating branch) ⃝scalar-tensor theory⃝
Brans-Dicke theory⃝f (R) gravity⃝
bimetric theory
⃝

ẽ×
ij
⃝
⃝
⃝
⃝
⃝
⃝
⃝
⃝
⃝
⃝

66
ẽbij
⃝1
⃝
⃝2
—
⃝
⃝
⃝
⃝
⃝
⃝4

ẽℓij
⃝1
⃝
—
—
⃝
⃝
⃝
⃝
⃝
⃝4

ẽxij
⃝
⃝
△3
—
⃝
⃝
—
—
—
⃝4

ẽyij
⃝
⃝
△3
—
⃝
⃝
—
—
—
⃝4

Table 5.4: GW polarization modes in various theories. The polarization modes inpure Minkowski spacetimes are provided in Appendix D. Reference papers are [136]for the 5D Kaluza-Klein compactiﬁcation, [137] for the RS braneworld, [138] for the
DGP braneworld, [139, 140] for the scalar-tensor theory, [141, 139] for the Brans-Dicke
theory, [142, 143] for the f (R) gravity, and [144] for the bimetric theory. 1 In a generalﬁve-dimensional spacetime, ﬁve degrees of freedom for polarizations are allowed. In
general relativity (GR) in pure 5D Minkowski spacetime without the compactiﬁcation,
two scalar modes are correlated and behave as one degree of freedom. 2 The plus andbreathing modes are correlated. 3 No vector mode appears in the absence of anisotropy(a vector ﬁeld) at cosmological scales. 4 Extra polarizations can exist, but, would bemuch small at high frequencies.
in which it is implicitly assumed that the wavelength of GWs is much larger than the detector size. The absolute response of the detector, of course, depends on hA . However,given the polarization tensors, the angular response function FA can be discussed independently on the GW amplitude. We would not like to adhere to a model-dependent
search here, though it is important to investigate the prediction and parameter constraint on a theoretical model 1 . So, in this section, we describe only the detectabilityof the non-tensorial polarization of GWs with a laser-interferometric detector.
Suppose that an orthonormal coordinate in the three-dimensional space for a detector is û = (1, 0, 0)v̂ = (0, 1, 0) ,ŵ = (0, 0, 1)
and the coordinate rotated by angles (θ, φ) for a GW is ′ û = (cos θ cos φ, cos θ sin φ, − sin θ)v̂′ = (− sin φ, cos φ, 0).
 ′ŵ = (sin θ cos φ, sin θ sin φ, cos θ)
The most general choice is obtained with the rotation with respect to the angle ψ1

We will leave them as future works.

67

5.4 Searching for non-tensorial polarizations of gravitational waves

Figure 5.8: Coordinate systems are related by the rotation angles (φ, θ, ψ).

around the propagating axis of the GW,′′ m̂ = û cos ψ + v̂ sin ψn̂ = −v̂′ sin ψ + û′ cos ψ .
Ω̂ = ŵ′The coordinate systems are drawn in Fig. 5.8.
With the coordinate (m̂, n̂, Ω̂), the polarization tensor is
ẽ+
ẽ×
ẽb
ẽℓ
ẽx
ẽy

=
=
=
=
=
=

m̂ ⊗ m̂ − n̂ ⊗ n̂ ,m̂ ⊗ n̂ + n̂ ⊗ m̂ ,m̂ ⊗ m̂ + n̂ ⊗ n̂ ,√2 Ω̂ ⊗ Ω̂ ,m̂ ⊗ Ω̂ + Ω̂ ⊗ m̂ ,n̂ ⊗ Ω̂ + Ω̂ ⊗ n̂ .

From Eq. (5.32), the angular pattern functions for each polarization result in
1(1 + cos2 θ) cos 2φ cos 2ψ − cos θ sin 2φ sin 2ψ ,2
1
F× (θ, φ, ψ) = − (1 + cos2 θ) cos 2φ sin 2ψ − cos θ sin 2φ cos 2ψ ,2Fx (θ, φ, ψ) = sin θ (cos θ cos 2φ cos ψ − sin 2φ sin ψ) ,Fy (θ, φ, ψ) = − sin θ (cos θ cos 2φ sin ψ + sin 2φ cos ψ) ,F+ (θ, φ, ψ) =1
Fb (θ, φ) = − sin2 θ cos 2φ ,2
1Fℓ (θ, φ) = √ sin2 θ cos 2φ .
2(5.33)(5.34)(5.35)(5.36)

5. Direct search for GWB

68The ψ dependence indicates that the + and × modes have a tensor property (spin-2),
the x and y modes have a vector property (spin-1), the b and ℓ modes have a scalar
property (spin-0). Unfortunately, the angular pattern functions of the breathing andlongitudinal modes are completely degenerated, which prohibits to decompose themwith a laser-interferometric GW detector (It is also true for a laser-interferometric GW
detector with non-orthogonal arms.). In Fig. 5.9, the angular pattern functions for the
non-tensorial polarizations are shown. In Fig. 5.10, we also show the angular patternfunctions for the tensor, vector, scalar modes. These results are consistent with thoseobtained by Tobar, Suzuki, Kuroda [145].

5.4.3

Overlap reduction function

Next, we consider not GWs from a point-source but GWB. The reason is that, the
GWB has less parameters because of its stochastic nature. The non-Einstein mode ina GW from a point source is a interesting subject that should be studied. However,to decompose the polarizations, one has to identify the source location on the sky, in
addition to the six polarizations. Thus, more than eight detectors (for two sky coordinate parameters and six polarizations) are needed for the complete decomposition. On
the other hand, in the case of the GWB, there is no preferred direction. In addition,
in most of the cosmological scenarios, it is natural to assume that the + and × modes,
or x and y modes are not polarized. Since the number of parameters are less, threedetector is enough to decompose the polarization modes (tensor, vector and scalar).
Deﬁnitions
Let us deﬁne generalized overlap reduction functions for the non-tensorial polarizations.
From Eqs. (5.9), (5.10) and (5.19), the cross-correlation signal for the tensor mode canbe written as
∫ ∞
∑
3H02
−3 1
µ =
2ΩA
(5.37)
Tdf|f|
gw (|f |)γ̄A (|f |)Q̃(f ) ,
T
20π 2
γ̄
−∞
ideal A=+,×∫dΩ̂ 2πif Ω̂·∆X/c
⃗
γ̄A ≡
e
F1A F2A ,S 2 4πwhere we assume that + and × modes are not polarized (ΩA
gw = Ωgw /2), and γ̄A is a
non-normalized overlap reduction function for each polarization. We also assume thatx and y modes are not polarized. Then, the GWB energy density of tensor, vector,
scalar modes can be written as×ΩTgw ≡ Ω+
gw + Ωgw×(Ω+
gw = Ωgw ) ,(5.38)ΩVgw ≡ Ωxgw + Ωygw(Ωxgw = Ωygw ) ,(5.39)ΩSgw ≡ Ωbgw + Ωℓgw = Ωbgw (1 + κ) .

(5.40)

where the ratio of the energy density in the longitudinal mode to that in the breathingmode is κ ≡ Ωℓgw /Ωbgw . Using Eqs. (5.33) - (5.36), one can calculate the ideal values

69

5.4 Searching for non-tensorial polarizations of gravitational waves

Figure 5.9: Angular pattern functions of a detector for the non-tensorial polarizations.
(a) plus mode F+ , (b) cross mode F× , (c) x mode Fx , (d) y mode Fy , (e) longitudinalmode Fℓ . The angular pattern function of the breathing√ mode is the same as that of
the longitudinal mode except for an overall factor 1/ 2. At the center of the ﬁgure,the arms of an interferometer is shown.

5. Direct search for GWB

70

Figure 5.10: Angular
for the tensor, vector, scalar√ pattern functions√ 2 of2 a√detector2222modes. Each plot is F+ + F× , Fx + Fy , Fb + Fℓ , respectively.

of the non-normalized overlap reduction functions for the non-tensorial polarizations,which gives∫
Tγ̄ideal=∫Vγ̄ideal≡∫Sγ̄ideal≡dΩ̂ 2
2(F+ + F×2 ) = ,5
S 2 4πdΩ̂ 2
2(Fx + Fy2 ) = ,5
S 2 4π1 + 2κdΩ̂ 2(Fb + κFℓ2 ) =.
15
S 2 4πThe ﬁrst equation for the tensor mode is the same as Eq. (5.19). Then, the generalized
overlap reduction functions for the non-tensorial polarizations are deﬁned by∫dΩ̂ 2πif Ω̂·∆X/c⃗
e(FI+ FJ+ + FI× FJ× ) ,
4π
2
S
∫
1
dΩ̂ 2πif Ω̂·∆X/c⃗
V
γIJ(f ) ≡ Ve(FIx FJx + FIy FJy ) ,γ̄ideal S 2 4π∫
1dΩ̂ 2πif Ω̂·∆X/c⃗S
γIJ (f ) ≡ S
e(FIb FJb + κFIℓ FJℓ ) ,γ̄ideal S 2 4π
T
γIJ
(f )

=

1

T
γ̄ideal(5.41)(5.42)(5.43)The subscripts I, J represent I-th and J-th detectors, which are ﬁxed for a general casewith more than three detectors. The ﬁrst equation (5.41) for the tensor mode is thesame as Eq. (5.10). Here we assumed that gravitons propagate with the speed of light2.
2

Strictly speaking, the propagation speed is less than the speed of light if the gravitons are massive.
It is true in a spacetime with extra dimensions or in some modiﬁed gravity theories. However, the mass
is constrained by several observations of the galaxy [146], the solar system [147], and binary pulsars[148]. The limits from the galaxy and the solar system is obtained from the observation in static
gravitational ﬁelds, while the limit of binary pulsars comes from the change of the orbital period of

71

5.4 Searching for non-tensorial polarizations of gravitational waves

Therefore, for generalized case including the non-tensorial polarizations,Eq. (5.37) can be extended as
∫ ∞
∑
3H02
−3 1
µ =
T
df
|f
|
2 ΩA
gw γ̄A Q̃(f )
T
20π 2
γ̄
−∞
ideal A=+,×,x,y,b,ℓ
]
[
∫ ∞
3H02
S
V
S
T
V
−3
T
=
T
df |f | Ωgw γ + Ωgw γ + ξ Ωgw γ Q̃(f ) ,
20π 2
−∞
(
)
1 1 + 2κ
ξ ≡
,
3 1+κ
where, in the second line of µ, we used Eqs. (5.38) - (5.40) and (5.41) - (5.43). The
parameter ξ is in the range 1/3 ≤ ξ ≤ 2/3 and characterizes the ratio of the energy inthe longitudinal mode to the breathing mode.
Speciﬁc expressions for real detectors on the Earth
The overlap reduction functions for the vector and scalar modes can be analytically
expanded by tensor bases, as well as the tensor mode in Sec. 5.2.1. The calculation
is parallel with those for the tensor mode. So, here we give only the ﬁnal results (InAppendix C, the results in the course of the calculation is provided.),
ij J
kj
ij kℓ
M
M
i
M
γIJ
(f ) = ρM
1 (α)DI Dij + ρ2 (α)DI, k DJ di dj + ρ3 (α)DI DJ di dj dk dℓ ,(5.44)

with, for the vector mode,



 V 
j0
7 5
−2
ρ1
 ρV2  = 2  0 −15 20   j2  .
7
0 0 −35j4
ρV3
and, for the scalar mode,
 S 



ρ1
14 20
6
j0
 ρS2  = 1  0 −60 −60   j2  .
7
ρS3
0
0
105
j4

(5.45)The superscript M in Eq. (5.44) denotes the modes M = V (vector), S (scalar). Notethat the parameter κ vanishes in the overlap reduction function because Fb and Fℓhave the same response and γ is normalized. Equation (5.45) agrees with the previous
result by Maggiore and Nicolis [149].
the binary pulsars (PSR B1913+16 and PSR B1534+12), in dynamical gravitational ﬁelds. Finn andSutton considered energy loss from the binary system by emission of massive gravitons, and obtained
the limit on the mass of gravitons, (mg /ωorbit )2 < 0.003, where ωorbit is the orbital frequency of thebinary. This limit implies
√
)2
(
mg
vg
= 1−
& 0.998 ,cωorbit
The gravitons cannot change its speed by more than 0.2% from the speed of light. Thus, setting
vg = c does not aﬀect to cross-correlation analysis qualitatively.

5. Direct search for GWB

72

Next, we consider the overlap reduction functions of real detectors on the Earth.
The overlap reduction function (5.44) can be simply expressed by introducing thecoordinate system on the Earth deﬁned in Sec. 5.2.2, and is given byγ V (α, β, σ+ , σ− ) = ΘV + (α, β) cos(4σ+ ) + ΘV − (α, β) cos(4σ− ) ,γ S (α, β, σ+ , σ− ) = ΘS+ (α, β) cos(4σ+ ) + ΘS− (α, β) cos(4σ− ) ,with, for the vector mode,

(

)
3
45
169
ΘV + (α, β) ≡ −
j0 +
j2 −
j4
8
112
224
(
)
1
5
27
+
j0 + j2 + j4 cos β
2
14
56
)
(
5
3
1
j0 −
j2 −
j4 cos 2β ,
−
8
112
224
(
)
( )4
5
3
β
ΘV − (α, β) ≡j0 − j2 − j4 cos
,
14
28
2

and, for the scalar mode,(5.46)(5.47)

(

)
3
45
507
ΘS+ (α, β) ≡ −j0 + j2 +
j4
8
56
448
(
)
1
5
81
+
j0 + j2 −j4 cos β
2
7
112
(
)
1
5
9
j0 − j2 +j4 cos 2β ,
−
8
56
448
)
( )4
(
9
β
5
,
ΘS− (α, β) ≡j0 − j2 + j4 cos
756
2(5.48)(5.49)

ΘM + and ΘM − , M = V, S, are plotted as a function of β for ﬁxed frequencies inFig. 5.2 and as a function of frequency for a ﬁxed β in Fig. 5.3, together with thetensor mode (M = T ). The diﬀerence from the tensor mode appears at about the
cutoﬀ frequency of the overlap reduction function. Mathematically, this is because thecoeﬃcients of j0 in Eqs. (5.27) and (5.28) for the tensor mode, Eqs. (5.46) and (5.47)
for the vector mode, and Eqs. (5.48) and (5.49) for the scalar mode, are exactly same,and the diﬀerence comes from the terms of j2 and j4 . In other words, at low frequencies
where j0 → 1, j2 → 0, and j4 → 0, three modes are degenerated, while, at relatively
high frequencies, they behave diﬀerently. However, at much higher frequencies, theoverlap reduction function decays. Therefore, the most interesting frequency range
is around the cutoﬀ frequency, e.g. ∼ 10 − 100 Hz for detectors on the Earth. The
overlap reduction functions for a real ground-based detector pair is plotted in Fig. 5.5
and Fig. 5.6, together with the tensor mode.

5.4.4

Detectability

Considering the overlap reduction functions in the previous section, one can expect
to detect the non-Einstein modes as well as the tensor mode. However, the most73

5.4 Searching for non-tensorial polarizations of gravitational waves

important quantity in an observation is a SNR given by integrating with respect toa frequency. In this section, we calculate the SNR for each mode with two detectors,
assuming that only one mode exists.
The SNR can be calculated by using the formula deﬁned in Eq. (5.16),
[ ∫ ∞
]1/2
γ 2 (f )Ω2gw (f )3H02 √
SNR =
T 2
,
df 6
10π 2
f P1 (f )P2 (f )0
and replacing γ 2 Ω2gw with γT2 (ΩTgw )2 for the tensor mode, with γV2 (ΩVgw )2 for the vector
mode, and with ξ 2 γS2 (ΩSgw )2 for the scalar mode. As for the power spectra of thedetector noise Pi (f ), we assume that, for simplicity, all advanced detectors (A, C, H,
L, V) have the same noise as that of advanced LIGO. The analytical ﬁt of the noise
power spectrum of the advanced LIGO, based on [131], is given by [128],

(
)−4
(
)−1.7
f
f

−44
−47.25

10
+ 10
Hz−1



10
Hz
100
Hz


for 10 Hz ≤ f ≤ 240 Hz ,()3
P (f ) =

f

−46

10
Hz−1
for 240 Hz ≤ f ≤ 3000 Hz ,


1000
Hz


∞
otherwise .
In the calculation, we assume that the Ωgw is independent of frequency and the observation time is T = 3 yr. In Fig. 5.11, the SNRs of the optimal detector pairs,
type (i) and (ii), are shown as a function of β. As one can see, detector pairs
have {almost the same sensitivity
to the three modes. In Fig. 5.12, the optimal SNR,}
max SNR|type(i) , SNR|type(ii) , is shown as a function of β, together with SNRs of realdetector pairs for each mode. The SNRs of real-detector pairs, except for AIGO-LIGO(H1) and AIGO-LIGO (L1) pairs, are smaller than the optimal one due to the incomplete coincidence of the detector orientations. It is interesting to note that the scalarmode is enhanced in the SNR at the distance relatively close, and that the tensor modeis enhanced at the distance relatively far. This feature can be intuitively interpretedby the anglular responses of the detector shown in Fig. 5.10. At β ∼ π/2, the anglularresponses of the tensor mode between two detectors are less overlapped than those of
the scalar and vector modes. On the other hand, at β ∼ π, the anglular responses of
the tensor mode between two detectors are more overlapped. SNRs with a realistic detector pair are tabulated in Table 5.5. Therefore, a laser-interferometric GW detector
is sensitive to a GWB with the non-tensorial polarizations, having almost the same
SNR as the tensor mode.
In this section, we explored the detectability in the presence of an individual mode(tensor, vector, or scalar). However, in a general situation, three GWB modes are
mixed in the detector cross-correlation signal. So, the decomposition of the modes
is signiﬁcant problem, which should be addressed. It would be possible if more than
three detectors at diﬀerent sites are available, because of the diﬀerent dependence ofthe overlap reduction functions of the three modes on β. We will remain the problem
as future work.

5. Direct search for GWB

74

Figure 5.11: SNRs of the detector pair of type (i) (solid lines) and type (ii) (dashedlines) as a function of β. The tensor, vector, and scalar modes are represented byred, green, and blue curves, respectively. The energy density of a GWB is chosen as−8h20 ΩM
gw = 10 , M = T, V, S.

detector pair
A-C
A-H
A-L
A-V
C-H
C-L
C-V
H-L
H-V
L-V

h20 ΩTgw
8.6 × 10−9
3.6 × 10−9
3.4 × 10−9
8.7 × 10−9
1.2 × 10−8
4.5 × 10−8
5.7 × 10−9
1.6 × 10−9
7.1 × 10−9
6.7 × 10−9h20 ΩVgw
8.6 × 10−9
1.1 × 10−8
1.2 × 10−8
2.1 × 10−8
8.4 × 10−9
2.8 × 10−8
6.9 × 10−9
2.0 × 10−9
7.5 × 10−9
6.4 × 10−9ξh20 ΩSgw
4.5 × 10−9
7.3 × 10−9
8.8 × 10−9
1.4 × 10−8
8.4 × 10−9
2.5 × 10−8
4.7 × 10−9
1.7 × 10−9
4.5 × 10−9
4.3 × 10−9

Table 5.5: Detectable Ωgw for each mode with a real detector pair (SNR = 5). The
observation time is selected as T = 3 yr.

75

5.4 Searching for non-tensorial polarizations of gravitational waves

Figure 5.12: SNRs of the optimal and real detectors as a function of β. The en−8
ergy density of a GWB is chosen as h20 ΩM
gw = 10 , M = T, V, S. Each curve shows{}max SNR|type(i) , SNR|type(ii) for the tensor mode (red, solid), the vector mode (green,
dotted), and the scalar mode (blue, dashed). The ﬁlled circles are the SNR of real detector pairs for the tensor mode (red), the vector mode (green), and the scalar mode
(blue). At the arrow in the ﬁgure, four circles are overlapped: (detector pair, mode) =(2, tensor), (2, vector), (3, vector), and (3, scalar). The numbers in the ﬁgure represent the real detector pair: 1=HL, 2=AC, 3=CH, 4=LV, 5=HV, 6=CV, 7=CL, 8=AV,
9=AH, 10=AL.

Chapter 6
Direct GWB search at ultra-highfrequencies
At ultra-high frequencies, much eﬀort for the direct GW detection has not been devoted. Consequently, the direct constraint due to the GW detector is so loose, though
the indirect constraint due to the big-bang nucleosynthesis has been obtained. Thisis because the GW amplitude is much smaller at high frequencies than that at low
frequencies, and it makes the detection diﬃcult. However, pioneering works are vitalfor future GW astronomy and cosmology.
At present, as far as I know, GW detectors have been constructed so far by thegroups at INFN, Genova, Italy [41, 42], Birmingham University, UK [43, 44, 45], andis planned at Chongqing University, China [46, 47] 1 . The INFN detector is based on
the principle of the parametric conversion of power between the resonant modes of two
superconducting coupled microwave cavities. The Birmingham detector is based on
detecting the rotation of the polarization vector of an electromagnetic wave inducedby the interaction between a GW and the polarization vector of the electromagnetic
wave. The Chongqing detector uses the electromagnetic interaction of a Gaussian beam
propagating through a static magnetic ﬁeld. These detectors are based on the diﬀerent
principles from the phase measurement with the laser interferometry developed well in
the ground-based large-scale interferometers around 100 Hz.
We will ﬁrst use a laser interferometer for the GW search at ultra-high frequencies.
However, the application of the detection method to an ultra-high frequency band is
not as simple as one expects, because, at ultra-high frequencies, GW wavelength iscomparable to the size of a detector, which is, for example, the order of a few meters
to detect GWs at 100 MHz. In such a case, a long-wave approximation that the GW
wavelength is much larger than the detector size is not valid. The phase of GW changes
during the one-way trip of light between mirrors. Therefore, we have to use a detector
design that is able to integrate GW signals eﬃciently. Moreover, to detect GWB withsmaller amplitude than detector noise, one has to correlate signals from two detectors
in order to distinguish the GW signal. The analytical method has been well developed
by several authors [51, 150, 127]. However, it is not applicable to our situation at
1

The sensitive frequency ranges of the detectors is diﬀerent each other: the INFN detector is at
1 MHz, the Birmingham detector is at 100 MHz, and the Chongqing detector would be 10 GHz.

77

6. Direct GWB search at ultra-high frequencies

78

ultra-high frequencies, because in these references, they assume that GW wavelengthis much larger than the detector size. At ultra-high frequencies, the relative locationof the two detectors signiﬁcantly aﬀects the correlation sensitivity to GWB.
In the following sections after brieﬂy introducing GW sources at ultra-high frequencies, we investigate an optimal detector design at ultra-high frequencies by comparing
the GW response functions of detectors [48]. Next, we will use the extended formalismof cross-correlation analysis, derived in Sec. 5.1. The formalism properly includes theeﬀect of the response functions of the detectors. We investigate the dependence of thesensitivity on the relative location of two detectors [49]. For concreteness, we consider
GWs at 100 MHz and select detector conﬁguration to detect it. However, note that
our investigation is general, and is always applicable to GWs other than 100 MHz.

6.1

Gravitational-wave sources at ultra-high frequencies

GW searches at ultra-high frequencies are strongly motivated by theoretical predictions. As described in Chap. 3, some theoretical models, such as quintessential inﬂation and pre-big-bang model etc., produces much larger GWB spectra than that in thestandard slow-roll inﬂation, at ultra-high frequencies. There also exist GWs from apoint source at ultra-high frequencies. The GWs from ordinary astronomical objects
are radiated at the frequency range,(
fgw ∼ (Gρ)

1/2

∼

GM
R3

)1/2

(
. 10

M⊙
M

)
kHz .

(6.1)

Here we used the Schwarzschild radius R = 2GM for the most compact object. However, if there are compact objects with its mass smaller than 1 M⊙ , or, in the scenario
with extra dimensions, ultra-high frequency GWs are emitted.

Primordial black holes
Primordial black holes (PBHs) may have formed in the early universe either from initial
inhomogeneities or as a result of phase transition (for the recent review, see [151]).
Consequently, the mass of a PBH formed in the early universe roughly corresponds tothe total mass inside the Hubble horizon at that time, which is given by [152, 153]
)
(
t
15
g.
(6.2)
MH (t) ≈ 10
10−23 sec
By contrast, black holes forming at the present epoch could never be smaller thanabout 1 M⊙ . The origin of such low mass black holes is necessarily of primordial. From
Eq. (6.1), 100 − MHz GWs are radiated by the black holes of Jupiter mass, ∼ 10−4 M⊙ .
If such PBHs are produced in the early universe, they survive until the present
epoch, and emit GWs via binary evolution and coalescence [154, 155]. The quasinormal ringing frequency of the PBH with the mass ∼ 10−4 M⊙ is ∼ 100 MHz. On

79

6.2 Optimal detector design

the other hand, in many brane world models, PBHs with a lunar mass, ∼ 10−7 M⊙might have been produced when the temperature of the universe was at ∼ 1 TeV. If asigniﬁcant fraction of the dark matter of our galaxy consists of these lunar mass PBHs,
we have a chance to detect GWs from the PBHs at ultra-high frequencies, and could
ﬁnd some indications of extra dimensions in the phase evolution of the chirp waveformof GWs [156].

Black strings in a brane world
In braneworld scenario such as the Randall-Sundrum (RS) type [157, 158], one of blackhole solutions in the presence of the second brane is a so-called black string solution,
which is the Schwarzschild solution warped in the direction of an extra dimension. If
the black string is perturbed, it radiates massive gravitational waves, namely, KaluzaKlein (KK) modes, with a characteristic spectrum of an extra dimension [159, 160].
The waveform is quite diﬀerent from that in a 4-dimensional case, and shows lessdamping than the massless mode and late-time monochromatic oscillations insteadof a featureless power-law tail. The frequency is discrete, corresponding to the KK
tower of the gravitons, and is determined by the bulk curvature scale and the brane
separation, not depending on the black string and companion object masses. The
typical frequency and amplitude predicted in [160] are relevant to the GW detector
at ultra-high frequencies, ∼ 100 MHz or above. Therefore, it should be stressed thenecessity to develop the detectors capable of measuring such a high-frequency signatureof large extra dimensions.

6.2

Optimal detector design

6.2.1

Interferometric-detector designs

We will consider three detector designs, (i) synchronous recycling interferometer (SRI),(ii) Fabry-Perot Michelson interferometer (FPMI) and (iii) L-shaped cavity Michelsoninterferometer (LMI), and derive the response functions for GWs.
SRI (Fig.6.1) was ﬁrst proposed by R. W. P. Drever in [161] and detailed calculations
have been done in [162, 163]. Laser light is split at a beam splitter and sent into
an synchronous recycling (SR) cavity through a recycling mirror, which is mirror A
located at X1 in Fig.6.1. The beams circulating clockwise and counterclockwise in
the cavity experience gravitational waves and mirror displacements, leave the cavity,
and are recombined at the beam splitter. Then, the diﬀerential signal is detected ata photodetector. The advantage of SRI is that GW signals at certain frequencies areaccumulated and ampliﬁed because the light beams experience GWs with the same
phases during round trips in the folded cavity. Consider GW propagating normallyto the detector plane with an optimal polarization. In this case, the GW signal isampliﬁed at the frequencies f = (2n − 1) × c/4L, n = 1, 2, · · · , where c is the speed oflight and L is the arm length. More precisely, the arm length is the distance between
X1 and X2 (or X3 ) in Fig.6.1. On the other hand, the disadvantage of SRI is less

6. Direct GWB search at ultra-high frequencies

80

sensitivity for GWs at low frequencies, f < c/4L, because the GW signal is integrated
in the cavity and canceled out as the frequency is low.

Figure 6.1: Synchronous recycling interferometer (SRI).

The competitive design of detectors with SRI is an ordinary FPMI (Fig.6.2). FPMI
is the most popular design for current ground-based interferometers [30, 31, 32, 29] since
it has good sensitivity at low frequencies due to the ampliﬁcation of GW signals with
Fabry-Perot cavities. However, to amplify the GW signals at high frequencies, one
needs to use resonance due to the cavity. FPMI has the resonance of GW signals at
the frequencies, f = n × c/2L, n = 1, 2, · · · when the GW response is averaged overthe entire sky. To take advantage of the resonant response to GW at 100 MHz, the arm
length of FPMI should be 1.5 m. With this detector, one can achieve good sensitivity
with narrow bandwidth as well as SRI.
Another possible design of detectors is LMI (Fig.6.3), whose optical conﬁgurationis the same as the L-shaped FPMI. However, the GW response resembles SRI rather
than FPMI. Thus, this design can be regarded as being intermediate between SRI andFPMI.
To compare the detectors for GWB at 100 MHz, it is necessary to derive the GW
response functions for GW propagating in arbitrary directions and to compare those
averaged on the celestial sphere. For GW propagating normal to the detector planewith an optimal polarization, it is trivial that SRI and LMI with arm length 0.75 m
have maximal sensitivity, while FPMI with 1.5 m is not sensitive at all, at 100 MHz.
However, FPMI has nonzero sensitivity for GW not propagating orthogonally to thearms of the detector since the GW response of light going and coming diﬀers. Furthermore, the geometries of detectors also aﬀect the GW responses. Therefore, it isnontrivial which is the most sensitive detector.

81

6.2 Optimal detector design

Figure 6.2: Fabry-Perot Michelson interferometer (FPMI).

6.2.2

GW response functions

The general expression of the phase shift of light induced by GW propagating in anarbitrary direction has been derived in many references, for example, [164, 117]. We
will use the expression in [164]. When light travels between two test masses located at
XI and XJ , the phase shift of light created by GW in TT (transverse-traceless) gauge
is expressed as
δφIJ (t) = (nIJ ⊗ nIJ )∫ω LIJ /c ′ ∑ A
dt
e hA [t − (LIJ + ez · XI )/c + (1 − ez · nIJ )t′ ] ,
:
2 0
A
(6.3)
where t is the reception time of light at XJ , ω is the angular frequency of light, hA
is the amplitude of GW with plus or cross polarization and ez is a unit vector in the
direction of GW propagation. The arm length and unit vector in the direction of thearm are deﬁned as LIJ ≡ |XJ − XI | and nIJ ≡ (XJ − XI )/LIJ , respectively. The
symbol : means contraction between a tensor and vectors. The tensors eA , A = +, ×
are polarization tensors of GW and are deﬁned as well as Eq. (2.24) by
e+ = ex ⊗ ex − ey ⊗ ey ,e× = ex ⊗ ey + ey ⊗ ex ,(6.4)

where ex and ey are the unit vectors, which form the orthogonal coordinate with ez
2. For simplisity, we assume that there is no displacement noise, for example, thermal
2

In this section, we use ez instead of Ω̂ used in other sections, so as not to be confused with theangular frequency of a GW Ω. Note that the coordinate systems {ex , ey , ez } and {m̂, n̂, Ω̂} are
identical.

6. Direct GWB search at ultra-high frequencies

82

Figure 6.3: L-shaped cavity Michelson interferometer (LMI).

noise, seismic noise, radiation pressure noise, etc., at 100 MHz3 . In practice, such
undisirable noises deteriorate detector sensitivity in a real experiment. Treatments
to avoid the noises are described later in the section of correlated noises. Under theassumption of no displacement, the positions of mirrors in the absence of GW are not
perturbed and are just given by XI and XJ . The Fourier transform of Eq. (6.3) is givenby˜ (Ω) = nIJ ⊗ nIJ : ω
δφ
IJ
2

∑
A

eA h̃Ae−iΩ(ez ·XJ )/c − e−iΩ(LIJ +ez ·XI )/c,iΩ(1 − ez · nIJ )(6.5)

where h̃p is the Fourier component of the GW amplitude, and Ω is the angular frequency
of GW and is related to the GW frequency with Ω = 2πf .
In general, the response function of a detector is represented by the round-trip
signal in cavities multiplied by an ampliﬁcation factor in cavities. We denote the phase˜shift of the round-trip signal by δφ(Ω)
and the ampliﬁcation factor by α(Ω). Then,˜˜˜
the total output from the detector δΦ(Ω)
is written as δΦ(Ω)= α(Ω)δφ(Ω).
Detailed
calculations are described in the Appendix E. We show here only the results. Notethat we change the notation of the unit vectors directed in arms and the reﬂectivity of
mirrors in order to simplify the expression and make it easy to compare. The response
functions of each detector are
˜ all (Ω) = α(Ω, RF , RE )δφ˜ (Ω) ,δΦallα(Ω, RF , RE ) = −3

RE TF2,(RF − RE )(1 − RF RE e−4iΩτ )

(6.6)
(6.7)

In fact, rough estimates show that this assumption is at least valid as long as the detector sensitivity is above h ∼ 10−20 Hz−1/2 . In the experiment that reaches better sensitivity, thermal noise of
mirrors might limit the sensitivity, though other noises are far below.

83

6.2 Optimal detector design

and
• SRI
Replacing n12 → u, n13 → v and (RA , RC ) → (RF , RE ) in Eq. (E.7), the response
function is∑˜ (Ω) = (1 − e−2iΩτ ) ω e−iΩ(τ +ez ·X1 /c)δφeA h̃AallΩ
A: [(v ⊗ v) HSRI (Ω, vz ) − (u ⊗ u) HSRI (Ω, uz )] ,
(6.8)
with
HSRI (Ω, uz ) ≡]1 [−iΩτ uz
sin
Ωτ
−
iu
(e
−
cos
Ωτ
)
,
z
1 − u2z• FPMI
Doubling the arm length, τ → 2τ , so that the FPMI resonates at the same
frequencies as the SRI, and replacing n1 → u, n2 → v in Eq. (E.13), the response
function is∑˜ (Ω) = ω e−iΩ(2τ +ez ·XF /c)δφeA h̃AalliΩA: [(u ⊗ u) HFPM (2Ω, uz ) − (v ⊗ v) HFPM (2Ω, vz )] ,
(6.9)
with
HFPM (Ω, uz ) ≡ HSRI (Ω, uz ) ,• LMIReplacing n1 → u, n2 → v in Eq. (E.18), the response function is∑˜ (Ω) = ω e−iΩ(2τ +ez ·XF /c)δφeA h̃AallΩ
A: [(v ⊗ v) HLMI (Ω, vz ) − (u ⊗ u) HLMI (Ω, uz )] .
with
HLMI (Ω, uz ) ≡1[ sin Ωτ (e−ip1 + e−ip2 − 2 cos Ωτ )1 − u2z−iuz (e−i(p1 +p2 ) + cos 2Ωτ − (e−ip1 + e−ip2 ) cos Ωτ ) ] .

Here we deﬁned uz ≡ ez · u and vz ≡ ez · v. The phases p1 and p2 are deﬁned by
p1 ≡ Ωτ uz and p2 ≡ Ωτ vz , τ is deﬁned by τ ≡ L/c and RF and RE are the amplitude
reﬂectivities of front and end mirrors of cavities, respectively. TF is the amplitudetransmissivity of a front mirror of cavities. Note that, in the case of SRI, the front andend mirrors correspond to a recycling mirror and three other mirrors. Here we doubled
the arm length of FPMI so that the ﬁrst resonant frequency of GW signal coincides
with that of SRI and LMI. The formula for a FPMI is also calculated in [165, 166], andour result agrees with them.

6. Direct GWB search at ultra-high frequencies

6.2.3

84

Detector comparison

As mentioned in the previous section, in general, the GW response function has the
˜ all = α δφ
˜ . We will consider α, which is the common factor for all detectors,
form δΦall˜and δφ, which depends on the geometry of each detector, separately.
From Eq. (6.7), the magnitude of the optical ampliﬁcation factor in the cavities isdetermined only by the (composite) reﬂectivities of the front and end mirrors. The
frequencies of the peaks depend on the arm length of detectors. α is plotted in Fig. 6.4.
In the ﬁgure, we selected L = 0.75 m so that the ﬁrst resonant peak is located at
100 MHz. At higher frequencies, there are many resonant peaks. At lower frequencies,
optical ampliﬁcation is stronger as the frequency is lower, since the wavelength of lightis larger than the arm length of a detector. Here we selected the amplitude reﬂectivities
RF = 0.99 and RE = 1 for an illustrative purpose. However, in a real experiment, one
should select the reﬂectivities of the front mirrors much higher in order to achieve
better sensitivity, though the bandwidth becomes narrower.

Figure 6.4: The optical ampliﬁcation factor α(Ω). Parameters are selected L = 0.75m,
RF = 0.99 and RE = 1.
˜ , we introduce coordinates
To evaluate the round-trip phase shift due to GWs δφallhere. The detectors are located on the X-Y plane. Two unit vectors u and v are
written as u = (0, 1, 0) and v = (1, 0, 0), respectively. We denote the direction of
GW propagation by the unit vector ez , and the two unit vectors normal to ez andorthogonal to each other by ex and ey . These are written as
ex = ( cos θ cos φ, cos θ sin φ, − sin θ ) ,ey = ( − sin φ, cos φ, 0 ) ,ez = ( sin θ cos φ, sin θ sin φ, cos θ ) .
ex and ey deﬁne the GW polarization tensor in Eq. (6.4). Here we normalize and85

6.2 Optimal detector design

Figure 6.5: GW response function Trms (f ). Vertical lines are plotted at multiples of
100 MHz.

redeﬁne the GW response function as a dimensionless response function, namely,
T (Ω, φ, θ, ψ) ≡

˜ all
δφ
,
(ω h̃τ )(6.10)∑
where we assumed that GW has the formeA h̃A = h̃ (e+ cos 2ψ + e× sin 2ψ). ψ isthe polarization angle of GW. Integrating this function by φ, θ and ψ on the celestial
sphere and averaging lead to∫ 2π ∫ π
∫ 2π
1
dψ
2
Trms (Ω) ≡dφdθ sin θ|T (Ω, φ, θ, ψ)|2 .
4π 0
2π
0
0
The result of numerical calculation is shown in Fig. 6.5. All response functions
decrease above 100 MHz. This is the common feature of interferometers because theGW signal is destructively integrated in the cavity and is canceled out. Below 100 MHz,the response functions of SRI and LMI also decrease because the GW signal is partiallycanceled out during round trips of light beams in the folded cavities. On the other
hand, in the case of FPMI, the GW signal is more constructively integrated in thecavity and is more ampliﬁed, as the frequency is lower. At 100 MHz, SRI and LMI
have almost the same sensitivity, while FPMI has sensitivity worse by a factor ≈ 3.3.
This is because FPMI integrates GW signals less eﬃciently than other detectors. Thisdiﬀerence becomes signiﬁcant when one takes the correlation of two detectors into
account, which results in a factor of (3.3)2 ∼ 10 in sensitivity to GWB energy densityΩgw .
Let us consider the best sensitivity of SRI experimentally achievable with realistic
parameters, which is almost the same as that of LMI. We assume that the sensitivity is6. Direct GWB search at ultra-high frequencies

86

limited only by shot noise 4 . The magnitude of shot noise is determined by laser power,
the arm length and the reﬂectivities of mirrors, and is calculated by the condition thatthe phase shift due to GWs is equal to that of quantum noise [162],√
˜ all 〉rms = 2~ω ,|β| 〈δΦ(6.11)
ηI0
where ω is the angular frequency of laser, η is the quantum eﬃciency of a photodetector,
I0 is original laser power, andRF − REβ≡.
1 − RF RE
The reason why β is needed in the left-hand side of the Eq.(6.11) is that, in Eq.(E.4),
the phase shift due to GWs must be converted into the amplitude of a sideband ﬁeld.
Substituting Eqs.(6.6) and (6.10) leads to√12~
h(Ω) =,′
τ α (Ω)Trms (Ω) ηωI0
where we deﬁned¯¯¯¯RE TF2¯¯ .
α (Ω) ≡ |β α(Ω)| = ¯(1 − RF RE )(1 − RF RE e−4iΩτ ) ¯′(6.12)We select L = 0.75 m so that GW signal resonates at 100 MHz, ω = 1.77×1015 rad sec−1
and η = 1. The sensitivity achievable at 100 MHz in an ideal situation is)
1.6 × 104
Hz−1/2h ≈ 7.8 × 10
α′√
RF RE /(1 − RF RE ), which iswith bandwidth ∼ 108 /F Hz, where
ﬁnesseisF≡π√related to α′ by the relation F = π α′ RF /TF2 at 100 MHz. Note that α′ ≈ 1.6 × 104
2
is realized with reﬂectivities, say, RF2 = 0.99996 and RE= (0.99998)3 . When we selectthe arm length L = 0.75 m, laser power I0 = 1 W and |α′ | = 1.6 × 104 , the best
achievable sensitivity with SRI is h ≈ 7.8 × 10−21 Hz−1/2 at 100 MHz with bandwidth∼ 2 kHz. Note that our results can also be applied to detectors at other frequency
bands by tuning the arm length and shifting the peak of sensitivity.
In this section, we investigated the GW responses of interferometers at 100 MHz.
We considered three designs that took advantage of the ﬁrst optical resonance due tocavities and derived the GW response functions. In conclusion, SRI and LMI havealmost the same sensitivity at 100 MHz and FPMI has sensitivity worse by a factor of
3.3.
(

−21

4

1W
I0

)1/2 (

Rough estimates of the displacement noise in the L-shaped SRI have been done. For typicalparameters of the experiment, thermal and radiation-pressure noises have hn ≈ 6.8 × 10−21 Hz−1/2
and hn ≈ 1.6 × 10−24 Hz−1/2 , respectively. These noises are marginally below the goal sensitivityof our experiment. Therefore, no improvement of the sensitivity beyond would be expected. A newdetector design, which solves the issue of the displacement ampliﬁcation, is discussed in Chap. 7.

87

6.3 Correlation of two detectors

Above conclusion in the detector sensitivity is theoretical one. That is, we determined the best detector conﬁguration from the point of view of shot-noise limited
sensitivity to GWs. However, in real experiments, there are many advantages and disadvantages of detectors that we have not considered here. One of the advantages of SRI
is the simplicity of a control system. SRI has only one degree of freedom for locking
the interferometer because clockwise and counterclockwise lights share light paths in
the cavity and the Sagnac part, while FPMI and LMI have three degrees of freedom
in operation, which are for cavities in both arms and the Michelson part. Anotheradvantage of SRI is the symmetric optical conﬁguration of the cavity. This means thatclockwise and counterclockwise light in the cavity experience the same reﬂectivities ofmirrors. Thus, SRI is expected to have high tolerance to the imbalance of the reﬂectivities and relatively smaller laser frequency noise than other detectors. Even if we takethese facts into account, we can conclude that SRI is the best detector. These issues
should be investigated in more detail when one constructs real detectors.

6.3

Correlation of two detectors

6.3.1

Identiﬁcation of GW response and noise in a SRI

In the previous section, we derived the GW response of an SRI and found that it can be˜˜written in the form δΦ(f,Ω̂) = α(f ) δφ(f,Ω̂), using the Fourier component of phase˜shift due to GW during the round trip of light in a recycling cavity δφ(f,Ω̂) and an
optical ampliﬁcation factor of light in the cavity α(f ). From Eqs. (6.7) and (6.8), theconcrete expressions are written as
RE TF2,(RF − RE )(1 − RF RE e−8πif τ )∑ω −2πif (τ +Ω̂·X/c)
⃗
˜
δφ(f,
Ω̂) = (1 − e−4πif τ )eh̃A (f, Ω̂) eA (Ω̂)
2πf
A
[
]
: (v̂ ⊗ v̂)HSRI (f, Ω̂ · v̂) − (û ⊗ û)HSRI (f, Ω̂ · û) ,α(f ) = −(6.13)⃗ is a position vector
where Ω̂ can be identiﬁed with ez in the previous section, and X⃗ here,of the mirror M1 (n12 , n13 , X1 in Fig. 6.1 can be identiﬁed with u, v, and X
respectively.). We deﬁne an arm response function like− e−2πif τ
HSRI (f, Ω̂ · û) ,(6.14)
2πf τ
so that T gives unity in low frequency limit, as shown in Fig. 6.6. Using this response
function, we can rewrite Eq. (6.13) in a simple form,
T (f, Ω̂ · û) ≡

⃗
˜
δφ(f,
Ω̂) = 2 ωτ e−2πif Ω̂·X/c (1 − e−4πif τ )
[
]
∑
A 1
×
h̃A e :(û ⊗ û)T (f, Ω̂ · û) − (v̂ ⊗ v̂)T (f, Ω̂ · v̂)2A∑⃗= 2 ωτ e−2πif Ω̂·X/c (1 − e−4πif τ )h̃A (f, Ω̂) FA (f, Ω̂)(6.15)
A

6. Direct GWB search at ultra-high frequencies

88

Therefore, comparing Eq. (6.15) with Eq. (5.8), we obtain∫′˜˜δΦ (f ) ≡dΩ̂ δΦ(f,Ω̂)= κ(f ) α(f ) h̃(f ) .

(6.16)˜ ′ (f ), we incorporate the extra
where κ(f ) ≡ 2 ωτ (1 − e−4πif τ ). To identify h̃(f ) and δΦ
factor κ(f ) α(f ) into the noise spectrum5 , that is,√√
Pshot (f ) = |κ(f ) α′ (f )|−1 Pqnoise (f )√2~ω
= |κ(f ) α′ (f )|−1
(6.17)
ηI0
where Pqnoise is the noise due to vacuum ﬂuctuations. We assumed that shot noise isa dominant noise source around 100 MHz. In Eq. (6.17), α(f ) is replaced with α′ (f )deﬁned in Eq. (6.12), since phase shift due to GW has to be converted into photocurrent
at the photo detector by multiplying a constant factor.
We identiﬁed the speciﬁc forms of GW signal and noise in this subsection. The GW
˜ ′ (f ) completely corresponds to h̃(f ) in Eq. (5.8), and can be calculated using
signal δΦT in Eq. (6.14). The power spectrum of shot noise can be calculated with Eq. (6.17),
if experimental parameters are given.

6.3.2

Dependence of sensitivity on the relative locations between two detectors

The sensitivity to GWB is signiﬁcantly inﬂuenced by the geometrical conﬁgurationof two SRIs through the overlap reduction function γ when the wavelength of GW iscomparable to the size of a detector. In this section, we will perform detailed investigation of the dependence of the overlap reduction function on the relative location of
two detectors in an experiment for the detection of GWB at 100 MHz.
The overlap reduction function can be calculated numerically from Eq. (5.10) using⃗
the arm response function T given in Eq. (6.14), where the phase factor e2πif Ω̂·∆X/c
plays an important role. To see this, we consider the four conﬁgurations of detectors
and calculate γ(f ). The results are shown in Fig. 6.7 with the case of ”exact” and”long wavelength limit”. The former is calculated with the full arm response function
T . The latter is calculated with T = 1, which is just plotted for reference, though
the approximation is not valid around 100 MHz. Each conﬁguration of detectors is
⃗ =X
⃗1−X
⃗ 2 and the relative angle β. Notecharacterized by the relative position ∆X⃗
that XI is the position vector of M1 of I-th detector. In the case of (a) ideal, γ(f )rapidly decreases even though the detectors are completely colocated and coaligned,
because we deﬁned the γ(f ) including arm response functions. Case (b) T-shaped has
5Note that the derivation of shot noise is slightly diﬀerent from that in the previous section, where
we have incorporated the angular response to GW into shot noise. However, here an angular averaging
is contained in the GW signals.

89

6.3 Correlation of two detectors

Figure 6.6: Arm response functions of SRI as a function of f τ . Each plot is forarccos(Ω̂ · û) = π/2, π/3, π/6, respectively, as shown by plot labels in the ﬁgure.

a behavior similar to that of (a) for the same reason. The arm response function isneeded to take the eﬀect of the phase change of GW at high frequencies into account.
Cases (a) and (b) are similar, but have a subtle diﬀerence since the arms of detectors
are at diﬀerent locations and experience diﬀerent phases of GW. As a result, the overlapof case (b) is a little worse than case (a). In the cases of (c) crossed and (d) stacked,
γ(f ) also decrease more rapidly than in the long wavelength limit. This is because
the contribution of the GW phase change at high frequencies is added to that in thelong wavelength limit. Therefore, we cannot obtain γ(f ) = 1 at 100 MHz with thedetectors where detector size and GW wavelength are comparable. It follows that
SNR is worsened by a factor of (0.377)−1 ≈ 2.65 at 100 MHz in contrast to the casewhere long wave approximation is valid, even if the two detectors’ conﬁguration is
optimal. One may expect unit response to be obtained by constructing much smaller
˜ is worsened since the resonant
detectors. However, the total response of detectors δΦfrequency of GW signal also depends on the detector size and is shifted upward. Thus,
this loss of sensitivity due to the phase change of GW is inevitable.
Next, we will ﬁx the frequency at 100 MHz and consider γ(f ). In fact, SRI has anarrow frequency band and what we are most interested in is γ at 100 MHz. In Fig. 6.8,
the location of one detector is ﬁxed, while the other detector is located at the same site
⃗ = 0) and the directions of arms are rotated. In this case, the magnitude of γ(f )(∆Xoscillates, however, it has the maximum peak at β = 0 and the minimum peak at β = π.
It is intuitive that the GW signal is better correlated when the directions of arms arecoaligned. In Figs. 6.9 and 6.10, the angle of detectors is ﬁxed and the locations aretranslated. In an initially coaligned case (β = 0) in Fig. 6.9, as expected, γ(f ) has its
maximum at ∆X = 0 and keeps the moderate value in the range of ∆X = ±0.2 m. In

6. Direct GWB search at ultra-high frequencies

90

Figure 6.7: Overlap reduction function in the case of four detector conﬁgurations.
⃗ = 0, β = 0, (b) T-shaped, ∆X⃗ = 0, β = π/2, (c) crossed,
Each setup is (a) ideal, ∆X⃗ = (L/2, L/2, 0), β = π, (d) stacked, ∆X⃗ = (0, 0, L/2), β = 0, where the arm∆Xlength is selected as L = 0.75 m. The ”exact” means the calculation with arm response
function T and the ”long wavelength limit” T = 1. The latter is not valid around
100 MHz, but merely plotted for comparison. Note that the sign of γ(f ) in (b) isinversed for convenience of comparison.

91

6.3 Correlation of two detectors

Figure 6.8: Overlap reduction function when the detector with L = 0.75 m is initiallycolocated and coaligned and is rotated at the same location. β is the rotation angle.

an initially reversed case (β = π) in Fig. 6.10, an interesting
feature can be seen. When√the detector is translated to the direction (û + v̂)/ 2, the peak of γ(f ) is shifted. Thisis because the overlap of the two detectors is better when their arms are overlapped
geometrically like (c) in Fig. 6.7.
Here we have considered a SRI as a detector conﬁguration, and investigated thedegradation and location dependence of the overlap reduction function. However, theproperty also hold for a Fabry-Perot Michelson interferometer because the transfer
function is almost the same as that of a SRI, though the sensitivity is slightly worse,
see Eqs. (6.8) and (6.9). Thus, the results we obtained in this subsection are generalfeatures in an interferometric GW detector at high frequencies.

6.3.3

Sensitivity to GWB

We will describe the best location of detectors with respect to the sensitivity, andcalculate the sensitivity achievable with correlation analysis. From the results obtained
above, the best location is obviously colocated and coaligned, and gives γ(f )|100 MHz ≈0.377. As shown in Fig. 6.9, this value is hardly changed in the range of ∆X = ±0.2 m
for coaligned detectors. In an experiment, it is impossible to put the detectors incompletely colocated and coaligned location because of the restricted experimental
space of the optics. However, experimental detector conﬁguration does not signiﬁcantly
aﬀect γ(f ) if the detectors are nearly colocated and coaligned. Therefore, we ﬁx it toγopt = 0.377.
As for the power spectral density of noise in an SRI, one can calculate from

6. Direct GWB search at ultra-high frequencies

92

Figure 6.9: Overlap reduction function when the detector with L = 0.75 m is initially colocated and coaligned (β = 0) and is translated in certain directions. Each√curve means the direction of translation.
(+x,
+y,
0)
is
the
direction
of
(û
+
v̂)/
2,
√
(−x, +y, 0) is the direction of (û − v̂)/ 2, and (0, 0, z) is the direction perpendicular
to the û v̂ plane.

Figure 6.10: Overlap reduction function when the detector with L = 0.75 m is initiallycolocated and reversed (β = π) and is translated in certain directions. Each curvemeans translation in the same direction as shown in Fig. 6.9.

93

6.3 Correlation of two detectors

Eq. (6.17). We select the arm length of a detector as L = 0.75 m so that the GW
signal resonates at 100 MHz. With experimental parameters, ω = 1.77 × 1015 rad s−1
and τ = 1, the power spectral density of noise around 100 MHz is(−42

Pi (f ) ≈ 4.65 × 10

1.60 × 104
α′ (f )

)2 (

1W
I0

)
Hz−1 ,

i = 1, 2.

(6.18)

The factor α′ is called the optical ampliﬁcation factor in a cavity and gives α′ ≈ 1.6×104
with the reﬂectivity of the recycling mirror, RF2 = 0.99996, and the reﬂectivity of2the other three mirrors, RE= (0.99998)3 . The bandwidth is ∼ 2 kHz with thesereﬂectivities. Note that the noise power spectrum in Eq. (6.18) is that for optimally
incident GW and not angular-averaged, since the angular average is incorporated into
the overlap reduction function in the correlation analysis. To convert Eq. (6.18) toangular-averaged one, the factor (2πf τ / sin 2πf τ )2 has to be multiplied.
Substituting PI (f ) and γopt into Eq. (5.16), and assuming that observation time is
T = 1 yr and that Ωgw (f ) has a ﬂat spectrum around 100 MHz (which is suﬃcient forpractical purposes [167]), one can calculate the sensitivity of two SRIs to GWB andobtainh20 Ωgw ≈ 1.4 × 1014 at 100 MHz with L = 0.75 m.
(6.19)
This ﬁgure is obtained with practically feasible parameters in a laboratory and byassuming that the sensitivity is limited only by shot noise.
One might expect to improve the sensitivity with moderate technical developmentin the near future or with a longer armlength (lowering resonant frequency). However,it might be impossible to futher improved the sensitivity in this detector design, because
other noises would contribute to the sensitivity. The SRI ampliﬁes not only GW signal
but also displacement noise of mirrors. Cosequently, a rough estimate shows thatradiation-pressure or thermal noise prevent the SRI improving the sensitivity. This
problem can be avoided by changing the detector design and usign ring-shaped cavity,
a so-called resonant speed meter [168], which is described in a latter chapter.

6.3.4

Cross-correlated noise

In the real world, correlated environmental and instrumental noises would contributeto the cross-correlation signal of the detectors and mimic the presence of true GWB
signals. For the two detectors far separated, a number of correlated noise sources can beignored, while, for detectors close to each other like the case in this paper, there would
be many correlated noise sources. So, it is important to investigate the noise sourcesin advance. In this subsection, we describe the possible noise sources and suggest somepreventive treatments for them.
It is useful to derive the condition for the noise so as not to impair the crosscorrelation search. Correlated noise in two detectors can be described by the crossspectral function Pc (f ) deﬁned by1〈ñ∗1 (f )ñ2 (f ′ )〉 ≡ δ(f − f ′ )Pc (f ) .
2

6. Direct GWB search at ultra-high frequencies

94

If 〈S 2 〉 ≫ 〈S〉2 in (5.11), the correlated noise does not aﬀect the GWB search. From
Eqs. (5.7) and (5.12) by replacing h1,2 (f ) → n1,2 (f ), one can obtain the condition
[

T
2

∫ ∞
−∞

]2
df Pc (|f |)Q̃(f )

T
≪
4

∫ ∞
−∞

df P1 (|f |)P2 (|f |)|Q̃(f )|2 ,(6.20)

where P1 and P2 are uncorrelated noises of each detector. Narrow band approximation
allows replacement of the integral in Eq. (6.20) with characteristic bandwidth ∆f ≈f0 /|α′ |, f0 is a central frequency. Then, Eq. (6.20) isP1 (|f0 |)Pc (|f0 |) ≪ √,T ∆f(6.21)

where P1 (|f |) = P2 (|f |) is used. For the parameters we used in the previous section,
T = 1 yr and α′ = 1.6 × 104 , the condition is
Pc (|f0 |) ≪ 2.3 × 10−5 P1 (|f0 |) ≈ 1.0 × 10−47 Hz−1 .

(6.22)

Thus, the magnitude of noise with correlation should be much smaller than that of
uncorrelated noise. Note that this condition is not valid for transient correlated noise.
In the case, T in Eq. (6.21) is replaced with ∆T which is the duration of correlatedtransient disturbance, and the condition (6.22) is weakened.
In general, the properties (magnitude, angular pattern, time dependence, etc.) ofcorrelated noise strongly depend on the details of the instruments. So, we devote ourattention to qualitative aspects of correlated noise.
• Thermal and radiation pressure noises
Thermal noise includes those of all optics such as mirrors, suspensions, etc.,
and radiation noise results from vacuum ﬂuctuation. These noises would be theloudest noise if one goes beyond the sensitivity in Eq. (6.19). However, these haveno correlation between two detectors in nature.
• Seismic noise
Seismic noise is a dominant noise source at low frequencies below ∼ 10 Hz in large
ground-based interferometers. However, the noise can be signiﬁcantly suppressedby a seismic attenuation system at high frequencies. Furthermore, one can also
place the detector on a stiﬀ board to prevent the diﬀerential vibration mode
from coming into eﬀect. We can safely conclude that seismic noise at ultra-highfrequencies is low enough to be ignored.
• Acoustic noiseSound waves (supersonic waves at 100 MHz) around the two detectors would
produce a correlative change in the light path length by directly disturbing themirrors or perturbing the refraction index of gas. The noise can be reduced byputting the detector into a vacuum chamber. Another way to avoid the noise is95

6.4 Experimental search for GWB
to insert an acoustic absorbent between the detectors. It is recommended thatboth noise reduction methods are used together in case there is residual gas in
the vacuum chamber. In fact, the experiment in the air [50] shows no indicationof the acoustic noise, though the sensitivity is still far above than Eq. (6.19).
• Scattered light
Scattered light is also an awkward noise source when the detectors are closely situated. Preventive measures against scattered light should be taken. In addition,
it is eﬀective to use separate laser sources with frequencies signiﬁcantly shifted inrelation to one another, compared with the observation frequency, 100 MHz. A
simpler solution is to stack the detectors with a partition. According to Fig. 6.9,
the sensitivity is nearly optimized and hardly changed if the two detectors arelocated in a range ±0.2 m.
• External electromagnetic ﬁelds
There would be many radio-frequency electromagnetic ﬁelds around 100 MHz,such as those related to radio and television, which would depend on where
one lives. In experiment [50], such frequency bands were investigated and were
avoided before the experimental setup was constructed. Furthermore, an electromagnetic wave shield should be installed in case unanticipated electromagnetic
ﬁelds exist.
• Electric noise
One should be careful about electronic noises produced by measurement devices,
electric wires, and power supply. The most awkward thing among them is thepower supply because one usually uses an identical power supply. The problem
can be avoided if one uses separate batteries.

Therefore, in principle, all anticipated noise sources correlated between the two
detectors can be avoided. However, unexpected noise might exist in a real experiment.
It is useful for noise hunting to shift and rotate one of the detectors and compare thebehavior of the noise level, because most noises might strongly depend on the location
and angle of the detectors. Rotating one of the detectors by π/2, π, 3π/2 rad is a good
option for a detector conﬁguration, since it hardly worsens the sensitivity as shown in
Fig. 6.8. It is also useful for the conﬁrmation of the existence of real GWB to checkthe location dependence of the sensitivity with the results in the section above.

6.4

Experimental search for GWB

6.4.1

Experimental setup

In the laboratory of National Astronomical Observatory of Japan (NAOJ), we havedeveloped a GW detector that consists of two SRIs, which are constructed on a single

6. Direct GWB search at ultra-high frequencies

96

Figure 6.11: Optical design of the experiment with two SRIs, where optics are referred
to beam splitter (BS), recycling mirror (RM), transfer mirror (TM), end mirror (EM),symmetric port (SP), antisymmetric port (AP), polarization beam splitter (PBS), Faraday isolator (FI), Pockels cell (PC), electro-optic phase modulator (EOM), half-wave
plate (HWP), and quarter-wave plate (QWP). The suﬃxes ”1” or ”2” attached to eachcomponent indicate that the component belongs to IFO-1 or IFO-2, respectively. The
ﬁgure is provided by T. Akutsu [169].

97

6.4 Experimental search for GWB

optical table [50]. In Fig. 6.11, the optical design of the experiment is shown. Hereafter
we refer to the two interferometer as IFO-1 and IFO-2.
Each interferometer has a 75 cm-baseline cavity in order to be sensitive to GWs at
100 MHz. The size of Sagnac interferometer, which is the part in front of the recycling
cavity, is relatively small (∼ 12.5 cm) and its contribution to GW signal is insigniﬁcantcompared to that of the cavity. Two SRIs are put within a range ∼ 10 cm so that theoverlap reduction function has nearly ideal value. The laser source is a Nd:YAG laser
with the wavelength 1064 nm and the power 0.5 W. The laser frequency is stabilized to
the recycling cavity by the Pound-Drever-Hall technique [170], using a radio-frequency(RF) sideband (fRF = 85.4 MHz) modulated by EO1 in Fig. 6.12. The lasers of two
interferometers are independent and have no correlation between them. Pockels cells
are inserted in the recycling cavities and are used for calibration of GW signals. The
reﬂectivity of the recycling mirror is RF2 = 0.985 ± 0.005 and gives relatively low
optical ampliﬁcation, α(100 MHz) ∼ 100. This is because this experiment is the ﬁrst
step toward the direct detection of a GWB at 100 MHz.
GW signals are down-converted in frequency and are registered as output data,
as depicted in Fig. 6.12. Since the BS is not exactly balanced, small fraction of the
RF sideband leak into PD2. Hence, electrical signals at intermediate frequencies (IF),fIF ≡ fGW − fRF ∼ 15 MHz, can be detected at PD2, where fGW is the frequency
of GWs. However, IF frequency is still too high to be sampled with an inexpensive
data acquisition (DAQ) system. The IF signals are subsequently down-converted into
audio-frequency (AF) signals at fAF ≡ fIF − fLO ∼ 3 kHz with a local oscillator (LO),whose frequency is fLO = 14.696840 MHz.
The sensitivity to GW strain amplitude 6 is shown in Fig. 6.13. Red (solid) andgreen (dotted) curves are for IFO-1 and IFO-2, respectively. For IFO-1, the best
sensitivity is 6.4×10−17 Hz−1/2 at 100.0 MHz, for IFO-2, 8.5×10−17 Hz−1/2 at 100.1 MHz

6.4.2

Data analysis

We deﬁne a cross-correlation statistic:
1
Z≡
T∫ T /2∫ T /2
dt

−T /2

−T /2

dt′ s1 (t)s2 (t′ )Q(t − t′ ) ,which is the normalized (divided by T ) version of the cross-correlation signal, Eq. (5.5),
and is approximated to1Z≈T∫ fmaxdf s̃∗1 (f )s̃2 (f )Q̃(f ) .

(6.23)

fmin

The mean and variance of Z are calculated, as well as Eqs. (5.9) and (5.12) except
6

When the detector output is converted into detectable GW strain amplitude, it is assumed thata GW with optimal polarization are propagating vertical to the interferometer plane.

6. Direct GWB search at ultra-high frequencies

98

Figure 6.12: Schematic view of one of the interferometers [50]. DAQ: data aquisition
system, EO: electro-optic phase modulator, PD: photodetector, LO: local oscillator,
IF: intermediate frequency, and AF: audio frequency. Other acronyms are the same as
those in Fig. 6.11.

Figure 6.13: Noise curve of interferometers: IFO-1 (red, solid) and IFO-2 (green,
dotted) [50]. The peaks at 100.1 MHz are generated for calibration.

99

6.4 Experimental search for GWB

for the division by T and the ﬁnite integration range, and are given by∫ fmax3H02µZ = 〈Z〉 =2
df |f |−3 Ωgw (|f |)γ(|f |)Q̃(f ) ,
2
20π
fmin
∫ fmax1222σZ = 〈Z 〉 − 〈Z〉 =df P1 (|f |)P2 (|f |) |Q̃(f )|2 .
2T fmin(6.24)(6.25)

We will assume a frequency-independent spectrum of Ωgw . This assumption is wellmotivated by the narrow band sensitivity of the detector. For the choice of the optimalﬁlter in Eq. (5.15),
γ(f ),Q̃(f ) = K 3|f | P1 (|f |)P2 (|f |)Eqs. (6.24) and (6.25) becomeµZ
σZ2[ ∫ fmax]
3H02
γ 2 (f )=Ωgw K 2
,
df 6
20π 2
f P1 (f )P2 (f )fmin
[ ∫ fmax]K2
γ 2 (f )
=
2
.
df 6
4T
f P1 (f )P2 (f )fminChoosing the normalization constant 7 as
[ ∫ fmax
K= 2
df
fmin

γ 2 (f )f 6 P1 (f )P2 (f )]−1,

we obtain

3H02
K
2
Ω
,
σ
=
.
gw
Z
20π 2
4T
The SNR is deﬁned as |µZ |/σZ , which gives the same formula as Eq. (5.16). Notethat the above choice of the normalization constant does not aﬀect the SNR.
The observation has been done on 17 September 2007 (in Japanese standard time)at Mitaka, Tokyo, Japan. The measurement time is T = 1070.5 sec. The data record isdivided into N = 439 segments. The cross correlation and its uncertainty are calculated
for each segment, yielding a set {ZI , σZI } with I labeling the intervals. The ensemble
averages in Eqs. (6.24) and (6.25) are approximated by taking a weighted average over
all intervals:
NN∑
1
1 ∑ ZI−2=µZ = −2,σ.
(6.26)Z
σ2
σZ I=1 σZ2 I
I=1 ZIµZ =As for the integration range in Eq. (6.23), we choose the range from 2.08 to 4.19 kHz
for AF signal, corresponding to a 2-kHz bandwidth around 100.1 MHz for GW signals.
The cross-correlation estimates, µZ and σZ , are shown in Fig. 6.14. Estimation ofµZ is plotted as the red ﬁlled circles. The black curves enclose the two-sided 90% C.L.
interval of µZ : [µZ − 1.65 σZ , µZ + 1.65 σZ ]. We obtained µZ = 4.9 × 10−12 sec−2 with
σZ = 3.7 × 10−12 sec−2 after the integration of 1070.5 seconds, still including µZ = 0
7

The normalization constant has dimension of sec−3 , then µZ and σZ have dimensions of sec−2 .

6. Direct GWB search at ultra-high frequencies

100

Figure 6.14: Estimated µZ (red ﬁlled circle) and its uncertain range µZ ±1.65 σZ (blacksolid curve), which corresponds to 90% C.L. intervals [50].

within a 90% C.L. range, µZ ±1.65 σZ . Therefore, no signature of the existence of GWB
was found. We can also conclude that a random noise dominates the cross-correlation
signal, because the uncertainty σZ decreases proportional to T −1/2 .
No detection of GWB allows us to derive the upper limit on Ωgw . From Eq. (6.26),
we ﬁnds the upper limit [50], deﬁned as a one-sided 90% conﬁdence level corresponding
to µZ + 1.28 σZ ,h20 Ωgw |100 MHz ≤ 6.0 × 10+25 .
This limit is the ﬁrst step of the experiment, and is not stringent at all. However, we
improved the previous constraint on the GWB [45] by eight orders of magnitude.

Chapter 7
Resonant speed meterIn the previous chapter, it has been shown that a L-shaped SRI can amplify GW
signal at resonant frequencies. However, mirror displacement noise is also ampliﬁed.
When we increase the ﬁnesse of a cavity and improve the sensitivity beyond the goal
sensitivity in the previous section, the ampliﬁed displacement noise would limit thesensitivity. Thus, it is a crucial defect of the L-shaped SRI.
In this chapter, we propose a resonant speed meter [168], as a displacement noisecanceled conﬁguration 1 , based on a ring-shaped synchronous recycling interferometer.
In this method, the displacements of mirrors are sensed at diﬀerent times with theinterval ∆t. At the frequency (∆t)−1 , the phase of the mirror motion rotates by 2π
during one interval ∆t. This means that the beams experience the mirror displacementwith the same phase. Thus, the mirror displacements at diﬀerent times can be canceledby subtracting the signals of these beams. Note that this cancellation occurs at themultiple frequencies of (∆t)−1 , and that residual displacement noise remains at otherfrequencies, as is well known in a Sagnac interferometer [175]. On the other hand, thecavity has to be designed to amplify the GW signals. Such a design is possible if one
takes advantage of the quadrupole nature of GWs. At the noise cancellation frequency(∆t)−1 , if one beam propagates in one direction during the ﬁrst half period and in theright-angled direction during the other half period, the GW signal can survive. For the
beam propagating on the same path in the opposite direction, the GW signal with the
opposite sign can also survive. Then, subtracting the two beams gives an ampliﬁed GW
signal. The remarkable feature of this interferometer is that, at certain frequencies,
gravitational-wave signals are ampliﬁed, while displacement noises are not.

1

Displacement noise-free interferometry is originally proposed in Kawamura and Chen [171], andhas been developed by [172, 173, 174]. However, since all design uses no cavity, the shot-noise level isthe same as that of a Michelson interferometer. The resonant speed meter is the ﬁrst design, whichuses a cavity.

101

7. Resonant speed meter

102

Figure 7.1: Design of a resonant speed meter. A synchronous recycling cavity is composed of the four mirrors M1 − M4 . The electric ﬁelds of a CCW beam Aℓ , Bℓ , Cℓ , andDℓ are shown. The ﬁelds of CW beam are obtained by reversing along the y axis.

7.1

Detector design and sensitivity

7.1.1

Detector response

The detector conﬁguration of the resonant speed meter is shown in Fig. 7.1. Laserbeams divided at the balanced beam splitter Mb are reﬂected by completely reﬂective
mirrors Ma and Mc , and enter the (ring-shaped) synchronous recycling cavity, which
is formed by an input mirror M1 and three high-reﬂective mirrors M2 − M4 . In thecavity, each beam circulates clockwise (CW) and counterclockwise (CCW), then leaves
the cavity and is ﬁnally recombined at the beam splitter Mb 2 . First, we will derive
the detector responses to mirror displacements and GWs in the resonant speed meter,and show that, at certain frequencies, the displacement noises can be suppressed, while
GW signals are ampliﬁed. We also show that the detector sensitivity to GWs can be
improved proportional to the circulating number of light in the cavity.
M1 has an amplitude reﬂectivity and transmissivity (RF , TF ). As for the three othermirrors M2 , M3 , M4 , one can deal with the reﬂectivities introducing the composite
reﬂectivity RE . For simplicity, none of the mirrors have loss. Let us denote thedisplacement of M1 , M2 , M3 , and M4 , in the absence of GWs, as y1 (t), x2 (t), y3 (t), andx4 (t), respectively, where the coordinates x and y are deﬁned in Fig. 7.1. Electric ﬁelds
2This design is well known as a laser-ring gyroscope. It picks up the Earth rotation and shifts
the frequency of the counterpropagating beams, which is called the Sagnac eﬀect. The eﬀect can beignored when the size of the detector is relatively small. However, for a large detector, some treatment
must be done to compensate the phase shift.

103

7.1 Detector design and sensitivity

in the cavity are phase-shifted due to the GWs and the displacements of the mirrors.
The ﬁeld circulating CCW (denoted by ﬁxing the subscript ℓ) can be written asDℓ (t) = RE Cℓ (t − 4τ )
√
[ (g)
]
× exp iφℓ (t) + 2 iω/c {−x2 (t − τ ) + y3 (t − 2τ ) + x4 (t − 3τ )} ,(7.1)
where τ ≡ L/c, L is the side length of the cavity, and ω is the angular frequency of(g)laser light. φℓ (t) is the phase shift created by the GW and is expressed as the sum of
each-side contribution during the propagation in the cavity,(g)φℓ (t) ≡ φ21 (t) + φ32 (t − τ ) + φ43 (t − 2τ ) + φ14 (t − 3τ ) ,(7.2)

where, say, φ21 (t) is the phase shift due to the GW during the light trip from M2 to
M1 . The junction conditions at M1 are√Cℓ (t) = RF Dℓ (t) e− 2 i ω y1 (t)/c + TF Aℓ (t) ,(7.3)

Bℓ (t) = TF Dℓ (t) − RF Aℓ (t) e

(7.4)

√
2 i ω y1 (t)/c.

Equations (7.1), (7.3), and (7.4) are solved separately from the CW ﬁelds, and give√Bℓ (t) = −RF Aℓ (t) e 2 i ω y1 (t)/c +

∞
∑

k k−1
TF2 RERF Aℓ (t − 4kτ )k=1[√
× exp 2 i ω y1 (t − 4kτ )/c

}]
k {
∑
]
]
(g) [
(d) [
′
′
+i
φℓ t − 4(k − 1)τ + φℓ t − 4(k − 1)τ
,
k′ =1
(d)

where φℓ (t) is the phase shift due to the displacement of the mirrors and is expressed
by
√
]
2ω [
(d)
φℓ (t) ≡−x2 (t − τ ) + y3 (t − 2τ ) + x4 (t − 3τ ) − y1 (t − 4τ ) .
(7.5)
c
We assume that the carrier ﬁeld at M1 is Aℓ (t) = A0 eiωt and nonperturbed cavity length(g)satisﬁes the resonant condition 4ωτ = 2πn, n = 1, 2, · · · . In addition, |φℓ | ≪ 1,(d)
|φℓ | ≪ 1, and ω |y1 |/c ≪ 1 are assumed. Using these assumptions and deﬁning
Tℓ (t) ≡ Bℓ (t)/Aℓ (t), we obtain[√Tℓ (t) ≈ −RF 1 + 2i ω y1 (t)/c
−TF2
+i

∞
∑

k=1
k
∑(
k′ =1

k k−2
RF
RE

{
√
1 + 2i ω y1 (t − 4kτ )/c
)}]

]
]
(g) [
(d) [
φℓ t − 4(k ′ − 1) + φℓ t − 4(k ′ − 1)

.

(7.6)

7. Resonant speed meter

104

Fourier transformation of Eq. (7.6), it can be written in the form,√

2 i ωβ ′ (Ω)y˜1 /c + iα′ (Ω)(φ̃ℓ + φ̃ℓ ) ,−RF + RE e−4iΩτ
β ′ (Ω) ≡,1 − RF RE e−4iΩτT̃ℓ (Ω) =

(g)

(d)

(7.7)

and α′ (Ω) is deﬁned in (6.12). Here β ′ (Ω) and α′ (Ω) are optical ampliﬁcation factors
due to the cavity. According to the axisymmetry of the system along the y axis, thetransfer function for the CW beam is easily obtained,√(d)T̃r (Ω) = 2 i ωβ ′ (Ω)y˜1 /c + iα′ (Ω)(φ̃(g)(7.8)
r + φ̃r ).
The diﬀerential of the two beams is detected at the photodetector, and givesT̃ (Ω) ≡ T̃r (Ω) − T̃ℓ (Ω)
[
(g)
(d) ]
(d)
= −iα′ (φ̃(g)r − φ̃ℓ ) + (φ̃r − φ̃ℓ ) ,√2 2 i ω −2iΩτ(d)(d)φ̃r − φ̃ℓ=esin Ωτ (x̃2 + x̃4 ) .
c

(7.9)

At this stage, the displacements ỹ1 and ỹ3 in Eqs. (7.7) and (7.8) are automaticallycanceled out at any frequencies, because the CW and CCW beams simultaneously
experience the displacement of M1 and M3 3 . In addition, at the frequencies thatsatisfy Ωτ = nπ, n = 1, 2, · · · , Eq. (7.9) gives exactly zero, then all displacement
noises vanish. This is because the CW and CCW beams in the cavity experience
the displacement of M2 and M4 with the same phase, though the time of reﬂection isshifted by multiples of the period. Therefore, in our detector, displacement noises in
the cavity are not ampliﬁed around the cancellation frequencies, though the cavity is
on resonance.
Suppose that a GW propagates in the direction vertical to the detector plane withthe polarization along the direction M1 -M2 and M1 -M4 . In the transverse-tracelessgauge, it can be written as hT T (t, z) = h(t − z/c)[u ⊗ u − v ⊗ v] , where u and v areunit vectors directed from M1 toward M2 and M4 , respectively. Let the detector beon the x − y plane and set z = 0, for simplicity. The GW-induced phase shift of lightduring the trip from Mi to Mj is given by [176]φ̃ij (Ω) = ±h̃ω/Ω × e−iΩτ /2 sin(Ωτ /2).

(7.10)The positive and negative signs correspond to the vertical and horizontal propagationsin Fig. 7.1. h̃ is the Fourier component of GW amplitude. From Eq. (7.2) and thecounterpart for the CW beam, the Fourier component of the total GW response is
(g)

φ̃(g)
r − φ̃ℓ =38i h̃ω −2iΩτecos(Ωτ ) sin2 (Ωτ /2) .
Ω(7.11)

The arrival times of light at M1 , or M3 depend on the displacement of M2 and M4 . However, itis the order of O(x2 /L2 ) and can be ignored.

105

7.1 Detector design and sensitivity

So far, we have ignored the contribution of displacements at the mirrors in the
Sagnac part, Ma , Mb , and Mc , in Fig. 7.1. In fact, these displacements are not canceled,
and will contribute as residual noise. The contribution is√(s)
φ̃r(s) − φ̃ℓ = − 2ω/c (x̃a − x̃b + x̃c ),where x̃a , x̃b , and x̃c are Fourier components of the displacements of the mirrors, Ma ,Mb , and Mc .
Strictry speaking, in principle, the residual displacement noise can be canceled if
the Sagnac lengths are made comparable to the cavity length. However, the created
dip on a noise curve is too narrow to be practically useful, because beams sense themirror displacement at the diﬀerent moments with large time lag (before entering andafter leaving the cavity). Therefore, we will not consider such a option here.

(g)

(d)

Figure 7.2: GW response T̃rms (Ω) (upper ﬁgure), and displacement responses T̃rms (Ω)(s)
and T̃rms (Ω) (solid and dashed lines in the lower ﬁgure, respectively). Each quantity isnormalized to be dimensionless. The reﬂectivities of mirrors are RF = 0.99 and RE = 1
for the illustration, which corresponds to α′ ≈ 200.
Let us deﬁne each contribution of the total detector output asT̃ (g) (Ω) ≡ iα′ (φ̃(g)r − φ̃ℓ ),(g)T̃ (d) (Ω) ≡ iα′ (φ̃(d)r − φ̃ℓ ),
(d)

(s)

T̃ (s) (Ω) ≡ φ̃r(s) − φ̃ℓ .

(7.12)(7.13)

7. Resonant speed meter

106

Assuming that the magnitude√ of disturbances at each mirror is the same (the phase
is not), we deﬁne x̃rms = 〈|x̃i |2 〉, where i = 1, 2, 3, 4, a, b, c, and 〈· · · 〉 denotes theensemble average. Then, root mean squares of Eqs. (7.12) and (7.13) are4ω ′|α sin Ωτ | x̃rms ,
√c
6ω
(s)
T̃rms
(Ω) =
x̃rms .
c

(d)
T̃rms
(Ω) =

(7.14)(7.15)(d)(s)

The GW response T̃ (g) , and the displacement responses T̃rms and T̃rms , are shown in
Fig. 7.2 as a function of normalized frequency Ωτ /π. The GW signal resonates atΩres = 2π × (2m − 1)/2τ , while the cancellation of the displacement noise in the cavity
occurs at Ωcancel = 2π × n/2τ , where m, n = 1, 2, · · · . For m = n = 1, the GW signal isampliﬁed, while the cavity displacement noise is not, which can be explicitly seen from
Eqs. (7.9) and (7.11) (In a L-shaped synchronous recycling interferometer in Chap. 6,Ωres = 2π × (2m − 1)/4τ and Ωcancel = 2π × 2n/4τ . Then, no solution exists.). Note
that, in Fig. 7.2, the displacement response nulls at cancellation frequencies are so sharp
because of the cavity enhancement of the residual noise around the frequencies. The
essential is that the degeneracy between the ampliﬁcation frequencies for GW signal
and displacement noise is broken. Thus, there exist the frequencies where the GW
signal is ampliﬁed and the displacement noise is canceled.
The noise cancellation mechanism is the same as that of a speed meter [177, 178,
179, 180, 181]. This can be understood explicitly in a time domain. From Eq. (7.5)
and the CW counterpart, the round-trip displacement response in the cavity aroundthe cancellation frequency Ωres can be written as
√
2ω
(d)
(d)
φr (t) − φℓ (t) =[x2 (t − τ ) − x2 (t − 3τ ) + x4 (t − τ ) − x4 (t − 3τ )]√c()
2 2 ωτ δΩ[v2 (t − τ ) + v4 (t − τ )] ,∼c
Ωreswhere δΩ is detuned frequency from the resonance, and we used()δΩ
x2 (t − τ ) ≈ x2 (t − 3τ ) + 2τv2 (t − τ ) .
Ωres
Therefore, this detector senses not the mirror positions but the mirror velocities. Moreover, the GW signal is ampliﬁed by the resonant cavity. This is the reason why we callit a resonant speed meter.

7.1.2

Noise curves
(d)

(s)

The signal-to-noise ratio (SNR) can be calculated by SNR = T̃ (g) / (T̃rms + T̃rms + T̃ (q) ).
Here we added quantum noise T̃ (q) , which has a frequency-independent spectrum. In
Fig. 7.3, the noise curve of each component is shown. For the illustration of the noise
cancellation, we choose the displacement noise signiﬁcantly dominant, namely, x̃rms =10 x̃shot at Ωτ /π = 1, where x̃shot ≡ T̃ (q) c/ω (Note that whether the displacement

107

7.2 Quantum noise in a RSM

noise is dominant or not depends on the detection frequencies.). Around Ωτ /π = 1,
cavity displacement noise is canceled. The cancellation does not exactly occur at otherfrequencies and the residual noise creates a dip on the displacement noise spectrum atthe cancellation frequencies. The bandwidth of the dip is determined by the optical
path length between M2 and M4 , in other words, by the optical time lag that CW
or CCW beams experience the displacements of M2 and M4 . The time lag is 2τ andgives the factor sin Ωτ in Eq. (7.14). The total sensitivity at Ωτ /π = 1 is limited byresidual displacement noise in the Sagnac part (or shot noise if the displacement noiseis relatively small). However, both noises decrease proportional to α′ because of theampliﬁcation of the GW signal. On the other hand, the displacement noise of thecavity mirrors is independent of α′ , but is already canceled around Ωτ /π = 1. Thus,
within the narrow bandwidth, the total noise level diminishes as α increases.
In Fig. 7.4, the dependence of the relative total noise curve on α′ is shown. The
noise cancellation allows us to improve the sensitivity, being proportional to α′ without
being limited by displacement noises.

7.2

Quantum noise in a RSM

In the previous section, we derived the response of the RSM, and illustrated the sensitivity to GWs, where the displacement noise of a cavity is completely canceled at
the resonant frequencies. The analysis is done only in classical regime, where forceson the mirrors have classical origins. In quantum mechanics, however, vacuum ﬂuctuations unavoidably come into the system and contribute to noises (shot noise andradiation-pressure noise) [182, 183]. Particularly, radiation-pressure noise is caused by
the coupling between electromagnetic ﬁelds of vacuum ﬂuctuations and carriers, andhave quantum origins. Consequently, it should be checked whether the RSM works
well even for quantum noise or not. If the radiation-pressure noise is canceled, theRSM could be a quantum nondemolition (QND) interferometer [184, 185]. Hence, it
is also interesting to investigate how is the standard quantum limit (SQL) [186] (seealso, [184] for the review) of the RSM, and whether the sensitivity of the RSM could
overcome the SQL or not.
Recent works on quantum noise in a laser interferometer are based on a quantum
formalism by Kimble et al. [185]. According to the formalism, we will calculate quantum noise in the RSM. Brief review of the quantum noise in a Fabry-Perot Michelsoninterferometer (FPMI) is provided in Appendix F.

7.2.1

Input-output relation

An electromagnetic ﬁeld at the time t is written as
Ed (t) = [D + Ed1 (t)] cos ωt + Ed2 (t) sin ωt ,√∫dΩ
4π~ω ∞(d1,2 e−iΩt + d†1,2 eiΩt ).
Ed1,2 (t) =Ac 0
2π(7.16)

7. Resonant speed meter

108

Figure 7.3: Relative noise curves of a resonant speed meter. The curves, ”cavity”,”Sagnac”, and ”shot” are the displacement noise in the cavity, the displacement noisein the Sagnac part, and quantum (shot) noise, respectively. The curve ”total” is thesum of these three noises. The reﬂectivities are RF = 0.99 and RE = 1 (α′ ≈ 200 atΩτ /π = 1).

Figure 7.4: Dependence of relative total-noise curve on α. Solid, dashed, and dotted curves are when α′ ≈ 200, 2000, 20000 at Ωτ /π = 1, corresponding to RF =0.99, 0.999, 0.9999, respectively, and RE = 1.

109

7.2 Quantum noise in a RSM

The subscript of the ﬁeld E indicates at where the ﬁeld is deﬁned. D is the amplitudeof a carrier ﬁeld, ω is the angular frequency of carrier light, and A is the eﬀective
cross-section area of a beam. The ﬁelds in the RSM are deﬁned in Fig. 7.5. In thissection, we set the reﬂection of the mirrors RF = R and RE = 1 with no loss.

Figure 7.5: Electromagnetic ﬁelds in a RSM.

Carrier ﬁelds
With the input laser amplitude D, input laser power I0 is given by
Ac 2
D .
I0 =
8π
Since CW and CCW beams are symmetric in the case of carrier ﬁelds, we consider
only CW beam (ﬁxed the subscript r). From the relations√
Ur = D/ 2,
Ar = Br ,Vr = −R Ur + T Br ,Ar = R B r + T U r ,the carrier-ﬁeld amplitude Ar in the cavity is
T D
√ .
Ar =
1−R 2
Thus, the laser power inside the cavity is
(
)2
T
I0
.
Ic =
1−R
2

(7.17)

7. Resonant speed meter

110

Sideband ﬁelds
Each ﬁeld is related like1uℓ = √ (d + p),21
q = √ (vr − vℓ ) ,2

1
ur = √ (d − p) ,2
(7.18)

at the beam splitter, andvr = −R ur + T br ,ar = R b r + T u r ,vℓ = −R uℓ + T bℓ ,aℓ = R bℓ + T uℓ ,(7.19)

at the recycling mirror M1 .
In the recycling cavity, we will ﬁrst consider only CW beam (ﬁxed the subscriptr). The equations for the CCW beam is easily obtained from those for the CW beam.
The ﬁelds before and after one round trip in the cavity are related by
Ebr (t) = Ear (t − 4τ − ∆tr ) ,(7.20)

where the perturbation of the retarded time ∆t comes from that due to GWs andradiation pressure:∆tr = ∆tGW
+ ∆tRP.
rr
Here we assume no classical noise, perturbing mirror positions. Substituting Eq. (7.16)
into Eq. (7.20) and using the resonant condition 4ωτ = 2πn, n = 1, 2, · · · and thecondition ∆tr ≪ τ , the ﬁelds inside the cavity are approximately related as
br1 = e4iΩτ ar1
br2 = e4iΩτ ar2 + ω√(7.21)]
2Ic [ ˜ RP
˜ GW
∆tr + ∆t,r
~ω

(7.22)

where Eq. (7.17) is used. Note that Eqs. (7.21) and (7.22) are valid up to the ﬁrst order˜ r . As for the CCW beam, b̃ℓ1 and b̃ℓ2 can be obtainedof the sideband amplitude and ∆t
by changing the subscripts r ↔ ℓ and 2 ↔ 4 in Eqs. (7.21) and (7.22).
Radiation pressure
Radiation pressure is exerted on all mirrors in the cavity, and ﬂuctuates them. However, since CW and CCW beams senses the displacement of the mirrors M1 and M3
simultaneously, the radiation pressure noise at M1 and M3 cancels when two beams
are recombined at the beam splitter. Therefore, we do not take M1 and M3 into account, and the radiation-pressure noise comes from M2 and M4 . For the CW beam,
the retarded time due to the mirror displacements is√2RP[x2 (t − 3τ ) + x4 (t − τ )] .
(7.23)
∆tr =
c

111

7.2 Quantum noise in a RSM

The classical part of radiation pressure force exerted on M4 is given by√]A [ ¯22W
2FRP (t) == √Ear (t − 3τ ) + E¯aℓ(t − τ ) .
c
2 2π
W is the power of the electromagnetic ﬁeld, and E¯2 means E 2 is time-averaged over
the time scale ω −1 . The ﬂuctuating part of the radiation pressure isA√ Ar [Ear1 (t − 3τ ) + Eaℓ1 (t − τ )]
2 2π
√
∫ ∞
2 ~ωIcdΩ=(ar1 e3iΩτ + aℓ1 eiΩτ ) e−iΩt
.
c
2π
−∞

δFRP (t) =We assume that the mirrors are at the balanced position by classical radiation pressureand gravity. Consequently, from the equation of motion, ẍBA
4 (t) = δFRP (t)/m, theFourier component of x4 is obtained,√
2 ~ωIcx̃4 = −
(ar1 e3iΩτ + aℓ1 eiΩτ ) .
mΩ2 c
As for M2 , x̃2 can be obtained by changing the subscripts r ↔ ℓ and 2 ↔ 4,√
2 ~ωIc(aℓ1 e3iΩτ + ar1 eiΩτ ) .
x̃2 = −mΩ2 c
Therefore, the Fourier component of Eq. (7.23) is√]2 [ 3iΩτ
RP
˜r
∆t
=
x̃2 e
+ x̃4 eiΩτ ,c √]4 2~ωIc 4iΩτ [= −ear1 + aℓ1 cos 2Ωτ .
2
2
mΩ c

(7.24)

˜ RP
As for the CCW beam, ∆t
can be obtained by changing the subscripts r ↔ ℓ andℓ2 ↔ 4 in Eq. (7.24).
At ultra-high frequencies, there is an awkward issue that the mirror does not respond to the radiation pressure as a rigid body. In other words, the fractional pieceof the mirror can respond the radiation pressure. Consequently, one has to use aneﬀective mirror mass for m in the above equations. For an accurate estimate of theeﬀective mirror mass, one has to perform fully numerical calculations such as ﬁniteelement simulations. However, it can be roughly estimated with the beam cross-section
area and the sound speed in the material of the mirror asmeﬀ (f ) = ρ A

cs,fwhere ρ is the density of the mirror material, A is the beam spot size, cs is the soundspeed in the material of the mirror, and f0 is an observation frequency. In the case ofa fused silica, ρ = 2.22 g/cm3 , cs = 5.97 × 10−3 cm. For A ≈ 1 mm and f = 100 MHz,the eﬀective mass is meﬀ ≈ 1.4 × 10−4 g. This means that the radiation-pressure noiseis ∼ 7500 times larger than that in the rigid mirror at the same frequency.

7. Resonant speed meter112

GW signal
From Eq. (7.11), the retarded time due to GWs for the CW beam is
˜ GW = 4h̃ e2iΩτ sin2 (Ωτ /2) cos Ωτ .
∆triΩ

(7.25)

˜ GW can be obtained by changing the overall sign in the rightFor the CCW beam, ∆tℓhand side of Eq. (7.25).
Total input-output relation
Combining Eqs. (7.18), (7.19), (7.21), (7.22), (7.24), and (7.25) and expressing theoutput ﬁeld q with the input ﬁeld p, we can ﬁnally obtain the total input-output
relation:q1 = e2iΨ p12iΨ

q2 = e√[p2 − KRSM p1 ] + 2KRSM eiΨ

(

h̃

)

hSQL

where
T2
16ωIc
sin2 Ωτ ,mΩ2 c2 1 + R2 − 2R cos 4Ωτ
[
]
1+R
π
Ψ(Ω) ≡ arctantan 2Ωτ + ,
1−R
2
√
(
)
2~
tan ΩτhSQL (Ω) ≡.
mc2 sin2 Ωτ tan(Ωτ /2)

KRSM (Ω) ≡7.2.2(7.26)(7.27)

Spectral density

We deﬁne the output ﬁeld detected by the homodyne measurement with an arbitrary
phase ζ:qζ = q1 cos ζ + q2 sin ζ√= e2iΨ [p1 cos ζ + (p2 − KRSM p1 ) sin ζ] + 2KRSM eiΨ sin ζ

(

h̃
hSQL

)
.
(7.28)

We assume that the input state is in a vacuum state. The same procedure from
Eq. (F.7) to Eq. (F.8) in Appendix F yields the noise spectral density][
h2SQL 1 + (cot ζ − KRSM )2
Sh (Ω) =.
(7.29)
2
KRSM

113

7.2 Quantum noise in a RSM

Figure 7.6: KRSM when R = 0.99 and I0 = 100 W (solid curve), R = 0.999 andI0 = 100 W (dashed curve), R = 0.99 and I0 = 20 W (dotted curve), with parametersω = 1.77 × 1015 rad sec−1 , L = 1.5 m and m = 10−4 g (corresponding to the eﬀective
mirror mass at 100 MHz.). These parameters are just for an illustrative purpose, andthe corresponding radiation-pressure noise is negligibly small.

Since the ﬁelds in Eq. (7.28) are linear with respect to the amplitude, the spectral
density can be decomposed into three parts: shot noise, radiation-pressure noise, andcorrelation noise. Devoting our attention to the mirror mass, each noise contribution
in the spectral density is given by
Shshot
ShRP
Shcor

h2SQL
1
=
,
2 KRSM sin2 ζ
h2SQL=KRSM ,
2= −h2SQL cot ζ .

(7.30)(7.31)

Since the coeﬃcient KRSM is proportional to the laser power in the cavity, the shotnoise and the radiation-pressure noise are inversely proportional and proportional tothe power, respectively. This implies that, if the homodyne angle is selected as ζ = π/2
so that the quadrature amplitudes have no quantum correlation between them, there
is a minimum noise level at a certain frequency with ﬁxed power. The noise level gives
the SQL, which is()2tan Ωτ
2~
2.
Sh ≥ hSQL =mc2 sin2 Ωτ tan(Ωτ /2)
The equality is satisﬁed when KRSM = 1, and gives the deﬁnition in Eq. (7.27).
Can we overcome the SQL with quantum correlation in Eq. (7.31)? To minimize thenoise, it is obvious from Eq. (7.29) that the homodyne phase should be selected so that
ζFD (Ω) = arccot[KRSM (Ω)], which is frequency-dependent. However, since the RSM has

7. Resonant speed meter

114

Figure 7.7: hSQL with the mirror mass m = 10−4 g (corresponding to the eﬀective
mirror mass at 100 MHz.).

good sensitivity only in a narrow frequency band, the frequency-independent homodyne
phase is adequate for us to optimize the noise at a resonant frequency. Therefore, theoptimized homodyne phase at the ﬁrst resonance of the RSM, Ωres = π/τ , is ζres = π/2.
In this case, the quantum correlation vanishes, and one cannot circumvent the SQL.
This is a natural consequence, because, in the RSM, the radiation-pressure noise isalready canceled at the resonant frequency, as one can see from Eqs. (7.26) and (7.30).
The plots of KRSM and hSQL around the resonant frequency are shown in Fig. 7.6 andFig. 7.7.
In conclusion, in the RSM, the displacement noise including the radiation-pressure
noise is automatically canceled, in contrast to the L-shaped SRI, in which the displacement noise is ampliﬁed. However, the SQL is not circumvented in the RSM, because
the cancellation stems from a classical correlation. Thanks to the cancellation, the sensitivity at the resonant frequency is limited by shot noise, and is improved more andmore by increasing both the ﬁnesse of the cavity and the laser power. Furthermore, it
allows us to lower the resonant frequency by enlarging the detector size and to enhance
the sensitivity to GWs, not being hindered by the displacement noise.

7.3

Sensitivity to GWB

In this section, let us calculate how much sensitivity to GWB one can achieve with two
RSMs. According to the cancellation of the displacement noise of the cavity mirrors, thesensitivity of the RSM to GWB is limited only by shot noise 4 . Since the observational
bandwidth of the RSM is narrow, the integrand of the formula for the SNR can be
4

Residual noise in the Sagnac part cannot be canceled. However, it could be ignored because no
ampliﬁcation of the displacement noise occurs, while the GW signal is ampliﬁed.

115

7.3 Sensitivity to GWB

regarded as frequency-dependent. So, Eq. (5.16) can be approximated to[]1/2
γ 2 (f0 )Ω2gw (f0 )3H02 √SNR ≈T 2 ∆f (f0 ) 6,
10π 2
f0 P1 (f0 )P2 (f0 )where f0 is the resonant frequency, ∆f is the observational bandwidth. We assume
that the overlap reduction function is γopt = 0.377 at the resonant frequency, as wellas that for L-shaped SRI in Sec. 6.3.3 5 . For SNR = 1, the minimum reachable Ωgw is√
2 3
10π f0P1 (f0 )P2 (f0 )Ωmin.
gw (f0 ) ≈2
3H0 γopt
2 T ∆f (f0 )If we select the observation time T = 1 yr and conservative parameters I0 = 100 W
and R = 0.99999, Pi |100 MHz ≈ 1.86 × 10−47 and ∆f |100 MHz ≈ 100 MHz/F ≈ 160 Hz.
9Then, Ωmingw |100 MHz ≈ 1.5 × 10 . For advanced parameters I0 = 1000 W and R =0.999999, Pi |100 MHz ≈ 1.86 × 10−50 and ∆f |100 MHz ≈ 16 Hz. Then, Ωmingw |100 MHz ≈6
1.5×10 . At 100 MHz, the limit is still weak and is not so interesting for cosmologists. If
the detector size is larger, the resonant frequency f0 = c/(2L) is lower. It is interesting
to consider the sensitivity at relatively lower frequencies. The noise power and the
bandwidth also scale as Pi ∝ f02 and ∆f ∝ f0 , respectively. Then, Ωmingw (f0 ) scales
9/2
proportional to f0 . Consequently, the limit to Ωgw is drastically improved by enlarging
the detector size, though, of course, the detector size is practically restricted on the
Earth. In Fig. 7.8, the sensitivity achievable with two large RSM is plotted.
At 1 − 100 MHz, the sensitivity to GWB is not good and is less interest in a cosmological context. However, at lower frequencies corresponding to a relatively large-scaledetector, the sensitivity can overcome the critical energy density of the universe andeven the indirect bound due to the big-bang nucleosynthesis.

5

Strictly speaking, γ for RSMs and SRIs are slightly diﬀerent, but it is negligible.

7. Resonant speed meter

116

Figure 7.8: Sensitivity of the RSMs to GWB. The plots are the sensitivity to GWB
with the conservative parameters (red solid curve) and with the advanced parameters(red dashed curve), critical energy density of the universe (black dotted line), and thelimit due to big-bang nucleosynthesis (black dot-dashed line). The detector size L
corresponding to the resonant frequency is shown on the upper axis.

Chapter 8
Conclusions
8.1

Summary

In this thesis, we studied the general framework of the search for a cosmological GWB,
focused on a laser-interferometric GW detector. Such a search is well motivated by
the GWB created in the early universe. Particularly, the quintessential inﬂation andthe pre-big-bang model generate a large GWB at high frequencies. On the other hand,a large-scale laser-interferometric detector on the ground is well developed, and thesecond-generation detectors, which can reach the theoretically interesting range of themodel parameters and could directly detect the GWB, are underway.
First, we investigated the search for the non-tensorial polarization mode of a GWB
with large-scale advanced detectors on the Earth such as LIGO and VIRGO. In thetheories with extra dimensions and the modiﬁed gravity theories, the extra degrees of
freedom allows the GWB to have scalar and vector modes besides the tensor modein the general relativity. To search those extra polarization modes, we extended theconventional formalism of a cross-correlation analysis for the tensor mode to the nontensorial polarization modes. Then, we calculated the sensitivity to such GWBs. As
a result, we found that interferometric GW detectors is capable of detecting the nontensorial mode with almost the same SNR as the tensor mode. We also discussed thedetectability with the real detector pairs and derived the detectable energy density of
the GWB.
In the latter of the thesis, we studied the search for a GWB at ultra high frequencies.
Since the frequency band corresponds to the epoch of the extremely earth universe, it
is important not only for cosmology but also particle physics. In addition, some models
such as the quintessential inﬂation and the pre-big-bang model create the energy peakof the GWB at ∼ 100 MHz. So far only a few experiments have been done and no tightconstraint on the GWB with a direct search exists at ultra-high frequencies.
To construct GW detectors and impose a tighter limit, we ﬁrst investigated thelaser-interferometric detector designs that can eﬀectively respond to GW at high frequencies. Comparing the GW response functions of three detector design, we foundthat the synchronous recycling interferometer (SRI) is the best at these sensitivities,
which has better sensitivity than an ordinary Fabry-Perot Michelson interferometer(FPMI) by a factor of 3.3 at 100 MHz. Then, we considered the cross-correlation search1178. Conclusions

118

with two SRIs. However, the conventional analytical method of cross correlation is notapplicable to our situation around 100 MHz, because the GW wavelength is comparable
to the detector size, and the usual assumption of long wavelength limit breaks down.
Thus, we generalized the formalism so as to be applied in any frequencies, and investigated the location and orientation dependence of two SRIs. As a result, we found thatthe sensitivity is nearly optimized and hardly changed if two coaligned detectors arelocated in a range of ±0.2 m. For such a detector conﬁguration, we derived the optimal
location of the two detectors, and showed that two SRIs with conservative parameters
can reach the sensitivity of h20 Ωgw ≈ 1.4 × 1014 at 100 MHz. Finally, we described theexperiment done by our group with a pair of meter-sized laser-interferometric detectors, and obtained the results of h20 Ωgw < 6.0 × 1025 as the ﬁrst step of the experiment.
Though this is not stringent limit, we improved the previous constraint on the GWB[45] by eight orders of magnitude.
In the calculation of the sensitivity of the SRI, we assumed the sensitivity is shotnoise limited. But, the SRI has a fatal defect which ampliﬁes not only the GW signal
but also the displacement noise of mirrors, and the displacement noise eventually limits
the sensitivity. For the solution, we proposed a so-called resonant speed meter (RSM),
which is a new detector design based on a ring-shaped cavity. The remarkable featureof this interferometer is that, at certain frequencies, gravitational-wave signals areampliﬁed, while displacement noises are not. Therefore, the RSM has sharp sensitivity
to GWs at certain frequency. Next, we studied quantum noise of the RSM and foundthat the RSM cannot overcome the standard quantum limit, but the sensitivity is
shot-noise limited at the resonant frequency since radiation pressure noise is canceled.
Thanks to the cancellation of the displacement noise, we showed that a relatively largescale detector, corresponding to lower resonant frequency, can achieve the sensitivity
that overcome the critical energy density of the universe and even the indirect bounddue to the big-bang nucleosynthesis.

8.2

Discussions and future prospects

As we mentioned in the summary above, the non-tensorial polarization can be detectedwith interferometric GW detectors. However, we assumed that only one of the modes
is present. In general, the detector output one obtains is a mixture of the tensor,
vector, and scalar mode. So, we have to decompose the output signal by using atleast three independent signals, and calculate the SNR. It is expected to be possible
to decompose because there are several large-scale interferometric detectors on the
Earth. It is also interesting to decompose the tensor and scalar modes with a single
detector pair, under the assumption of the absence of the vector mode. If we assume
only two modes exist, our search method can also be applied to a space-based detector
pair. Further investigation on this issue is required. Furthermore, although we chieﬂyfocused on the experiment, it is also important to derive the prediction of a speciﬁctheoretical model and to restrict the model parameters.
We investigated the GWB search at ultra-high frequencies ∼ 100 MHz, however,
unfortunately, the detector sensitivity with current technologies is not enough to con-119

8.2 Discussions and future prospects

strain the theoretical model and obtain interesting scientiﬁc results, since, at 100 MHz,
we cannot surpass even the critical energy density of the universe.
Note that the application of our studies in the thesis is not limited at ultra-highfrequencies. It can also be applied in the search at low frequencies, if the wavelengthof a GW is comparable with the detector size, e.g. LISA. In addition, our results
are not speciﬁc feature of the SRI, because the detector response functions of a SRI
in Eqs. (6.8) and a FPMI (6.9) are identical except for the prefactor (1 − e−2iΩτ ).
Hence, it is possible to perform the same experiment with two FPMIs. One of theways to reach interesting ﬁelds of scientiﬁc results is to slightly lower the observation
frequency, as seen in Fig. 7.8. As another direction of research, a RSM can be used asa displacement noise-free interferometer [187]. Therefore, in the thesis, we provide thegeneral framework of a cosmological GWB search and the various detector topologies.

Appendix A
Bogolubov transformation andparticle creation
Particle creation in a dynamical gravitational ﬁeld is described by Bogolubov transformation, which is related to smooth connection of mode functions between two phases
before and after the phase transition. For the details, see [67].
Suppose that phase I and phase II are the phase before and after a phase transition,
respectively. In each phase, a quantum ﬁeld, for example, a quantized scalar ﬁeld φ(x)at a spacetime point can be expanded using complete sets of mode functions, {fk , fk∗ }in phase I and {Fk , Fk∗ } in phase II,∑[]ak fk + a†k fk∗ ,
k
∑
[
]
=
Ak Fk + A†k Fk∗ ,φ(x) =(A.1)

kwhere {ak , a†k } and {Ak , A†k } are creation and annihilation operators in phase I andphase II, respectively. Since {fk , fk∗ } and {Fk , Fk∗ } are the complete sets, we can relate
one in terns of the others. In general, Bogolubov transformation is deﬁned by∑
Fk =(αkk′ fk′ + βkk′ fk∗′ ) ,
k′
∑
(A.2)
fk =
(αk∗ ′ k Fk′ − βk′ k Fk∗′ ) .
k′

From Eqs. (A.1) and (A.2), the creation and annihilation operators between two phases
are related as∑ak =(αk′ k Ak′ + βk∗′ k A†k′ ) ,
k′
∑
†
∗
∗
Ak =
(αkk
′ ak′ − βkk′ ak′ ) ,k′

The vacuum states are deﬁned using the annihilation operators by
ak |0〉I = 0 ,Ak |0〉II = 0 .
121

A. Bogolubov transformation and particle creation

122

From the commutation relations of ak and Ak , the following relation between Bogolubov coeﬃcients can be found.
∑[]αk′ k αk∗ ′′ k − βk′ k βk∗′′ k = δk′ k′′ ,
(A.3)
k

∑[
]
αk′ k βk′′ k − βk′ k αk′′ k = 0 .
kIn the Friedmann-Robertson-Walker universe, according to isotropy and homogeneity, Bogolubov coeﬃcients do not depend on a direction of k and a spatial coordinate
x. Thus, Bogolubov coeﬃcients only depend on a frequency f , and can be written as
αkk′ = αf δkk′ ,βkk′ = βf δkk′ .
Consequantly, Eq. (A.3) gives a relation,|αk |2 − |βk |2 = 1 .

(A.4)

Next, particle creation at the phase transition is considered. Particle number operator is deﬁned, nf = a†f af in the phase I and Nf = A†f Af in the phase II. A quantum
state in the phase I is denoted |nf 〉. The number of particles created at the phasetransition signiﬁcantly depends on a time scale of the transition. If ∆T ≪ (2πf∗ )−1 ,where ∆T is the time scale of the transition, and f∗ is the frequency of the mode of theparticle in which we are interested, the transition is instantaneous and the quantum
state does not change. In that case, the expected number of the particle in the phase
II is
Nf = 〈nf |A†f Af |nf 〉()
1
2= nf + 2|βf | nf +
.
2

(A.5)

The second term in the right hand side of Eq. (A.5) show that pre-existing particles
before the phase transition are ampliﬁed and new particles are created. If the quantum
state is in a vacuum state in the phase I, Nf is simply
Nf = |βf |2 .

(A.6)

In a converse situation that the transition occurs very slowly, ∆T ≫ (2πf∗ )−1 , thenumber of the particle is exponentially suppressed because the quantum state canfollow a change due to the transition [67]. Therefore, no particle is created.
In a cosmological context, ∆T corresponds to the Hubble time H −1 . Hence, forthe superhorizon modes f∗ ≪ H −1 , particles are created, for the subhorizon modesf∗ ≪ H −1 , no particle is created. The boundary frequency, f ∼ H −1 gives a cutoﬀ
frequency of inﬂationary GWB spectrum.

Appendix B
Formulae of spherical Bessel
functions• Explicit expressions of a spherical Bessel function
sin α
,
α
sin α cos α
j1 (α) =
−
,
α2
α
sin α
cos α sin α
j2 (α) = 3 3 − 3 2 −.
ααα

j0 (α) =• Recurrence formulajn+1 (α) =n+1
jn (α) − jn−1 (α)α• Integrated forms
∫ 1
−1

∫ 1
−1

dx eiαx = 2j0 (α)
j1 (α)αj2 (α)= 16 2α[]
2=j0 (α) − 2j2 (α)
3
[
]
2
=
7j0 (α) − 20j2 (α) + 8j4 (α)
35dx eiαx (1 − x2 ) = 4∫ 1

dx eiαx (1 − x2 )2−1∫ 1
dx eiαx x2−1∫ 1
dx eiαx x4−1123

B. Formulae of spherical Bessel functions• Expressions of jn (α)/αn with jn (x)[]j1 (α)1=j0 (α) + j2 (α)
α
3
]
[
1
j2 (α)
7j0 (α) + 10j2 (α) + 3j4 (α)
=
α2
105

124

Appendix C
The calculation of the overlap
reduction function for non-tensorialmodes
We write down the equation obtained in the course of the calculation of the overlap
reduction functions for the vector and scalar modes. As for the procedure of thecalculation, see Sec. 5.2.1.
• Vector mode
q1 = 0 ,q2 = 20j0 (α) ,q3 = 0 ,][1
1
4
j0 (α) − j2 (α) − j4 (α) .
q5 = 20
15
21
35

[
]
20
q4 =
2j0 (α) − j2 (α) ,
3






C1
−28 −40 −12 C2 
 42
 j0
30
−12




 C3  = 1  0

 .
60
60 

 42 
 j2
 C4 
 0 −45 60  j4
C5
0
0 −420• Scalar mode
q1 =
q3 =
q4 =
q5 =

(
)
(
)
2+κ
1+κ
30
j0 (α) ,q2 = 60
j0 (α) ,1 + 2κ
1 + 2κ
[]20(2 + κ)j0 (α) + 2(1 − κ)j2 (α) ,1 + 2κ
[]40(1 + κ)j0 (α) + (1 − 2κ)j2 (α) ,1 + 2κ
[]2
7(4 + 3κ)j0 (α) + 20(2 − 3κ)j2 (α) + 12(1 + κ)j4 (α) .
7125

C. The calculation of the overlap reduction function for non-tensorial modes 126






C1
14(3 + κ) −20(3 − κ)
3(1 + 2κ)
 C2 
 7(1 + 2κ) 10(1 + 2κ)3(1 + 2κ) 
j0




1
 C3  = .
030(3 − κ) −15(1 + 2κ)  7(1 + 2κ)  j2
 C4 
0−15(1 + 2κ) −15(1 + 2κ)  j4
0
0
105(1 + 2κ)C5

Appendix D
GW polarizations inhigher-dimensional spacetime
In a 4-dimensional spacetime, there are two polarization modes of a GW: plus and crossmode. In a higher-dimensional spacetime, more polarization modes appear, because of
extra degrees of freedom.
The number of the polarization mode in a D-dimensional spacetime can easily be
counted in the same way in Sec. 2.1. The perturbed metric tensor hµν has D(D + 1)/2
independent components. However, by D harmonic gauge conditions, the components
are reduced by D the number of independent components. In addition, another D
conditions are imposed to completely ﬁx the gauge, which has residual degrees of
freedom under an inﬁnitesimal gauge transformation. Then, D(D − 3)/2 polarization
modes exist in the D-dimensional spacetime. For instance, there are 5 polarizations ina 5-dimensional spacetime, and 9 polarizations in a 6-dimensional spacetime.
In this Appendix below, for the simple examples, the GW polarization mode in ahigher-dimensional spacetime are provided.

D.1

Pure 5-dimensional Minkowski spacetime

We consider a 5-dimensional Minkowski spacetime without the compactiﬁcation ofan extra dimension. In other words, four spatial dimensions are equivalent. Such aspacetime is not realistic, but it is instructive to see how the polarization modes of a
GW appear.
To investigate the explicit components of the GW metric tensor, the requirement
for the coordinate system is to satisfy the harmonic gauge conditions and the gauge
conditions for an inﬁnitesimal gauge transformation. Suppose the inﬁnitesimal transformation in Eq. (2.16), asαξ µ = i C µ eikα x ,(D.1)
where C µ is the constant of complex number. Substituting Eqs. (2.14) and (D.1) into
Eq. (2.17), on can obtain
A′µν = Aµν + (Cµ kν + Cν kµ ) .
127

(D.2)

D. GW polarizations in higher-dimensional spacetime

128

We can choose the wave-number vector of a GW as kα = (−ω, 0, 0, ω cos ζ, ω sin ζ),
without the loss of generality. The angle ζ is the angle between the z axis and the
propagation direction of the GW in 5-dimensional spacetime. For the choice of thewave-number vector, the null condition kα k α = 0 holds. In addition, we choose thecoordinate so that A′0µ = 0. So, from Eq. (D.2), the coeﬃcients of the inﬁnitesimaltransformation are given by
A00
A01
A02
, C1 =, C2 =
,
2ω
ω
ω
A04
A03
+ C0 cos ζ,C4 =+ C0 sin ζ .
(D.3)
C3 =
ω
ω
On the other hand, to satisfy the harmonic gauge conditions, the ﬁrst condition of
Eq. (2.15),
k µ A′µν = 0 ,
(D.4)
C0 =

is needed. Hereafter we do not ﬁx the prime at the metric tensor, for simplicity. From
Eq. (D.4), one can obtain the relations
A13 = −A14 tan ζ ,A23 = −A24 tan ζ ,A33 = A44 tan2 ζ = −e34 . tan ζ .

(D.5)

With the help of the traceless condition and Eq. (D.5), A33 is related to A11 and A22
as
A33 = −(A11 + A22 ) sin2 ζ .
(D.6)
Therefore, the spatial coponents of Aµν can be written as


A11
A12
A13
−A13 cot ζ


A12
A22
A23
−A23 cot ζ .
Aij = 2

A13
A23−(A11 + A22 ) sin ζ (A11 + A22 ) sin ζ cos ζ −A13 cot ζ −A23 cot ζ (A11 + A22 ) sin ζ cos ζ −(A11 + A22 ) cos2 ζ(D.7)
The Aij has ﬁve independent components, which is consistent with the number of the
independent polarization modes, mentioned at the beginning of this chapter. When
the GW is propagating in our 3-space (ζ = 0), the extra-dimensional components of
the Aij diverges, because we express the tensor components with A13 and A23 . Notethat the Aij satisﬁes the transverse-traceless (TT) conditions, hµ µ = 0 and k ν Aµν = 0.
It is intuitive to express the Aij in terms of the projected components on our 3-space,which is deﬁned in Eq. (5.30). Redeﬁning
A11 = Ab + A+ ,A22 = Ab − A+ ,A12 = A× , A13 = Ax , A23 = Ay ,the components of Eq. (D.7) projected on our 3-space are
Ab + A+
A×
Ax
 .
A×
Ab − A+
Ay
Ãij = 2Ax
Ay−2Ab sin ζ

129

D.2 Pure 6-dimensional Minkowski spacetime

The spatial part of the metric tensor is given byαααikα xα ×hij (ωt − ⃗k · ⃗x) = A+ eikα x ẽ+ẽij + Ax eikα x ẽxij + Ay eikα x ẽyij
ij + A× e√αα+Ab eikα x ẽbij − 2 sin2 ζAb eikα x ẽℓij .

Note that, in 5-dimensional Minkowski spacetime, there is a correlation between b andℓ modes, and exist ﬁve polarization modes (two tensor, two vector, one scalar modes).

D.2

Pure 6-dimensional Minkowski spacetime

The calculation of the GW polarization mode in a 6-dimensional Minkowski spacetime without compactiﬁcation can be done in the same way as that in the case of
5-dimensional Minkowski spacetime, but it is complicated. Here we only show theresults.
The wave-number vector of a GW can be chosen askα = (−ω, 0, 0, ω cos ζ, ω sin ζ cos η, ω sin ζ sin η) ,where η is another angle of the propagation direction of a GW. As a result, one obtainthe spatial components of Aµν ,• independent components
A11 , A12 , A13 , A14 , A23 , A24 , A33 , A44 , A55 ,• dependent components
A22
A15
A25
A34
A35
A45

=
=
=
=
=
=

−A11 − A33 − A44 − A55 ,−A13 cotζ cscη − A14 cotη ,−A23 cotζ cscη − A24 cotη ,−[A33 cotζ secη + tan ζ(A44 cos η − A55 sin η tan η)) ]/2 ,−[A33 cotζ cscη − tan ζ (A44 cos η cotη − A55 sin η)) ]/2 ,[A33 cot2 ζ cscη secη − A44 cotη − A55 tan η ]/2 .

Redeﬁning
A11 = Ab + A+ ,A22 = Ab − A+ ,A12 = A× , A13 = Ax , A23 = Ay ,A33 = Aℓthe projected components on our 3-space are
Ab + A+
A×
Ax
A×
Ab − A+ Ay  .
Ãij = Ax
AyAℓ
The six components in our 3-space are all independent. Since there are 9 degrees of
freedom in a 6-dimensional Minkowski spacetime, other 3 degrees of freedom are hiddenin the extra dimensions.

Appendix E
GW response functions ofinterferometers: general expressions
E.1

Synchronous-recycling interferometer

The conﬁguration of SRI is shown in Fig. 6.1. We call the mirror A at X1 the recycling
mirror, the mirror B at X1 the transfer mirror and the mirrors at X2 and X3 the end
mirrors. The amplitude reﬂectivities and transmissivities of the transfer mirror andthe end mirrors at X2 and X3 are (RB , TB ), (R2 , T2 ), (R3 , T3 ), respectively. Those of
the recycling mirror are (+RA , +TA ) for the light incident from inside the cavity and(−RA , +TA ) for the light incident from outside the cavity. The angular frequency oflight is ω and the arm length is L. We deﬁne τ ≡ L/c. Electric ﬁelds at the recycling
mirror are deﬁned in Fig. E.1 and are related by the following equations,
Cℓ
Cr
Bℓ
Br

=
=
=
=

RA Dℓ + TA Aℓ ,RA Dr + TA Ar ,TA Dℓ − RA Aℓ ,TA Dr − RA Ar .

(E.1)

We assume that there is no displacement noise, for example, thermal noise, seismic
noise, radiation pressure noise etc., at 100 MHz. Then, D ﬁeld after circulating thecavity experiences the phase shift 4ωτ and the GW signal δφ(t),[]Dℓ (t) = Rc Cℓ (t − 4τ ) exp i{4ωτ + δφℓ (t)} ,[]Dr (t) = Rc Cr (t − 4τ ) exp i{4ωτ + δφr (t)} ,(E.2)
where we deﬁned the composite reﬂectivity of mirrors Rc ≡ RB R2 R3 andδφℓ (t) ≡ δφ21 (t) + δφ12 (t − τ ) + δφ31 (t − 2τ ) + δφ13 (t − 3τ ) ,δφr (t) ≡ δφ31 (t) + δφ13 (t − τ ) + δφ21 (t − 2τ ) + δφ12 (t − 3τ ) .

(E.3)

˜ denotes the phase shift due to GW when light travels from X2 to
For example, δφ
21
X1 in Fig. 6.1. This can be calculated using Eq. (6.3).
131

E. GW response functions of interferometers: general expressions

132

Figure E.1: Electric ﬁelds of SRI at the recycling mirror.

Equations (E.2) - (E.2) can be solved separately for right-handed and left-handedﬁelds. The input-output relation of the cavity for the left-handed ﬁelds becomes
]
[ k
∞
∑
∑
[
]
k−1
δφℓ t − 4(k ′ − 1)τ
.
TA2 Rck RA
Aℓ (t − 4kτ )e4iωτ exp i
Bℓ (t) = −RA Aℓ (t) +
k′ =1

k=1

We assume that the cavity is in resonance in the absence of GW, that is Aℓ (t) =Aℓ (t − 4τ ). Then,
]}
[ k
{
∞
∑
∑
[
]
k−2
δφℓ t − 4(k ′ − 1)τ
.
TA2 Rck RA
exp i
Bℓ (t) = −RA Aℓ (t) 1 −
k=1

k′ =1

Using the approximation that the GW signal is small (|δφ(t)| ≪ 1), we obtain
RA − RcAℓ (t)
1 − RA Rc
]
[
∞
k
[
]
1 − RA Rc ∑ 2 k k−1 ∑δφℓ t − 4(k ′ − 1)τ
.
T R R
exp −iRA − Rc k=1 A c A k′ =1Bℓ (t) ≈ −(E.4)

Therefore, the phase shift δΦℓ of left-handed light caused by GW isk∞[]
1 − RA Rc ∑ 2 k k−1 ∑δΦℓ (t) = −δφℓ t − 4(k ′ − 1)τ .
TA Rc RARA − Rc k=1
k′ =1

Fourier transforming Eq. (E.5) and using Eqs. (E.3) and (6.5) give˜ ℓ = α(Ω, Rc , RA )δφ
˜ ,
δΦ
ℓ
∑
ω
˜ =
δφ
e−iΩ(τ +ez ·X1 /c)eA h̃A
ℓ
Ω
A

(E.5)

(E.6)

n12 ⊗ n12
{ sin Ωτ − i(ez · n12 )(e−iΩτ ez ·n12 − cos Ωτ )}1 − (ez · n12 )2
n13 ⊗ n13e−2iΩτ { sin Ωτ − i(ez · n13 )(e−iΩτ ez ·n13 − cos Ωτ )} ] ,+1 − (ez · n13 )2

:[

133

E.2 Fabry-Perot Michelson interferometer
TA2 Rc.
α(Ω, RA , Rc ) ≡ −(RA − Rc )(1 − RA Rc e−4iΩτ )The GW signal for the right-handed light can be obtained by simply changing thesubscripts 2 ↔ 3 because of the symmetry of the system. Therefore, the output of thedetector is˜ all ≡ δΦ˜ r − δΦ˜ℓδΦ˜= α(Ω, RA , Rc )δφall∑−2iΩτ ω −iΩ(τ +ez ·X1 /c)
˜
δφ
=
(1
−
e
)
e
eA h̃AallΩ
A: [ (n13 ⊗ n13 ) HSRI (Ω, ez · n13 ) − (n12 ⊗ n12 ) HSRI (Ω, ez · n12 )} ] .
(E.7)
with
HSRI (Ω, ez · n12 ) ≡E.2

1
1 − (ez · n12 )2[]sin Ωτ − i(ez · n12 )(e−iΩτ ez ·n12 − cos Ωτ ) .

Fabry-Perot Michelson interferometer

The conﬁguration of FPMI is shown in Fig. 6.2. The amplitude reﬂectivities and transmissivities of end mirrors at XE1 and XE2 are (RE , TE ), and of front mirrors at XF 1
and XF 2 are (+RF , +TF ) for the light incident from inside the cavities and (−RF , +TF )for the light incident from outside the cavities. The arm length is L. Electric ﬁelds at
the front mirror are deﬁned in Fig. E.2. First, we will consider only one FP cavity andcalculate the input-output relation. At the end of our calculation, we will derive thefull output of FPMI.
The ﬁelds are related by the following equations,
Eout = −RF Ein + TF EB ,EA = RF EB + TF Ein .

(E.8)
(E.9)

Figure E.2: Electric ﬁelds of FPMI.

EA is reﬂected at the end mirror and returns to the front mirror after experiencing
the phase shift 2τ and the modulation due to GW. The relation between EA and EB

E. GW response functions of interferometers: general expressions

134

[
]
EB (t) = RE EA (t − 2τ ) exp i{2ωτ + δφcav (t)} ,δφcav (t) ≡ δφEF (t) + δφF E (t − τ ) ,(E.10)(E.11)

iswhere δφEF denotes the phase shift due to GW when light propagates from the end
mirror to the front mirror and the speciﬁc form is given by Eq. (6.3). Similarly, δφF E
is the one from the front mirror to the end mirror. From Eqs. (E.8) - (E.10), theinput-output relation for the FP cavity becomes
Eout (t) = −RF Ein (t)
[ k
]
∞
∑
∑
[
]
+RE TF2(RF RE )k−1 Ein (t) exp iδφcav t − 2(k ′ − 1)τ
k=1

k′ =1

RF − RE
Ein (t)1 − RF RE
]
[
k
∞
∑
[
]
RE TF2 (1 − RF RE ) ∑δφcav t − 2(k ′ − 1)τ
,× exp −i
(RF RE )k−1
RF − RE
k=1
k′ =1

≈ −

where we used the approximation |δφcav (t)| ≪ 1. Therefore, the phase shift δΦ caused
by GW isk∞∑[]
RE TF2 (1 − RF RE ) ∑k−1
δΦ(t) = −δφcav t − 2(k ′ − 1)τ .
(RF RE )RF − REk=1
k′ =1(E.12)

Fourier transforming Eq. (E.12) and using Eqs. (E.11) and (6.5) givesΦ̃(Ω) = α(Ω, RF , RE )δφ˜cav (Ω) ,
ω∑ A
n⊗n
˜
δφ
e h̃A e−iΩ(τ +ez ·XF /c)
cav =
Ω A
1 − (ez · n)2
× [ sin Ωτ − i(ez · n)(e−iΩτ ez ·n − cos Ωτ ) ] ,RE TF2α(Ω, RF , RE ) ≡ −,(RF − RE )(1 − RF RE e−2iΩτ )where n ≡ (XE − XF )/L. This formula is consistent with the previous result, Eq.(6)in [188], except for an overall constant factor.
Using the result obtained above, we can easily obtain the full-output signal for the
FPMI. We ﬁx the subscripts 1 and 2 to distinguish north and east arms in Fig. 6.2,
and deﬁne ni ≡ (XEi − XF i )/L, i = 1, 2. For simplicity, we assume that the two front
mirrors on both arms are located at the same place, that is, XF = XF 1 = XF 2 . This
assumption is valid because it hardly aﬀects the GW signal. Then, total output of

135

E.3 L-shaped cavity Michelson interferometer

FPMI is˜ all ≡ δΦ
˜ 1 − δΦ˜2δΦ˜= α(Ω, RF , RE )δφallω −iΩ(τ +ez ·XF /c) ∑ A
˜
δφ
e
e h̃A
all =
Ω
A
: [(n1 ⊗ n1 ) HFPM (Ω, ez · n1 ) − (n2 ⊗ n2 ) HFPM (Ω, ez · n2 )] .
(E.13)
with[]1−iΩτ ez ·n1
sin
Ωτ
−
i(e
·
n
)(e
−
cos
Ωτ
)
,
z
1
1 − (ez · n1 )2= HSRI (Ω, ez · n1 ) .

HFPM (Ω, ez · n1 ) ≡E.3

L-shaped cavity Michelson interferometer

The conﬁguration of LMI is shown in Fig. 6.3. The amplitude reﬂectivities and transmissivities of end mirrors at XE1 and XE2 are (RE , TE ), and of front mirrors at XF 1
and XF 2 are (+RF , +TF ) for the light incident from inside the cavities and (−RF , +TF )for the light incident from outside the cavities. The two mirrors at XC1 and XC2 arecompletely reﬂective. The arm length is L. Electric ﬁelds at the front mirror are deﬁned in Fig. E.3. First, we will consider only one L-shaped cavity and calculate theinput-output relation. At the end of our calculation, we will derive the full output of
LMI.
The relation between the ﬁelds is the same as Eqs. (E.8) and (E.9) for FPMI. The
relation between EB and EA is almost the same as FPMI. However, only diﬀerences
are that the round-trip time of LMI in the cavity is 4τ and that light is reﬂected by

Figure E.3: Electric ﬁelds of LMI.

E. GW response functions of interferometers: general expressions

136

the mirror at Xc during the trip. Thus, it is given by[]
EB (t) = RE EA (t − 4τ ) exp i{4ωτ + δφcav (t)} ,δφcav (t) ≡ δφCF (t) + δφEC (t − τ ) + δφCE (t − 2τ ) + δφF C (t − 3τ ) ,(E.14)(E.15)

where δφIJ , I, J = F, E, C denotes the phase shift due to GW when the light propagates from the I-th mirror to the J-th mirror. From Eqs. (E.8), (E.9) and (E.14), theinput-output relation for the L-shaped cavity becomes
Eout (t) = −RF Ein (t)
[ k
]
∞
∑
∑
[
]
+RE TF2(RF RE )k−1 Ein (t) exp iδφcav t − 4(k ′ − 1)τ
k=1

k′ =1

RF − RE
Ein (t)1 − RF RE
]
[
k
∞
∑
[
]
RE TF2 (1 − RF RE ) ∑δφcav t − 4(k ′ − 1)τ
,(RF RE )k−1
× exp −iRF − RE
k=1
k′ =1

≈ −

where we used the approximation |δφcav (t)| ≪ 1. Therefore, the phase shift δΦ caused
by GW isk∞∑[]
RE TF2 (1 − RF RE ) ∑k−1
δΦ(t) = −δφcav t − 4(k ′ − 1)τ ,(RF RE )RF − REk=1
k′ =1(E.16)

Fourier transforming Eq. (E.16) and using Eqs. (E.15) and (6.5) gives˜δΦ(Ω)= α(Ω, RF , RE )δφ˜cav (Ω) ,∑˜ cav = ωδφeA h̃A e−iΩ(2τ +ez ·XC /c)Ω A}
nCE ⊗ nCE {× [sin Ωτ − i(ez · nCE )(e−iΩτ ez ·nCE − cos Ωτ )2
1 − (ez · nCE )nCF ⊗ nCF−
{ sin Ωτ (1 − 2e−iΩτ ez ·nCF cos Ωτ )1 − (ez · nCF )2+i(ez · nCF )(e−iΩτ ez ·nCF cos 2Ωτ − cos Ωτ ) } ] ,α(Ω, RF , RE ) ≡ −RE TF2,(RF − RE )(1 − RF RE e−4iΩτ )where nCE ≡ (XE − XC )/L、nCF ≡ (XF − XC )/L.
Using the result obtained above, we can easily obtain the full-output signal for the
LMI. We ﬁx the subscripts 1 and 2 in order to distinguish the two arms in Fig. 6.3, anddeﬁne n1 ≡ nC2E = −nC1F and n2 ≡ −nC2F = nC1E . For simplicity, we assume thatthe two front mirrors and the two end mirrors on both arms are located at the sameplace, that is, XF = XF 1 = XF 2 and XE = XE1 = XE2 , respectively. This assumption

137

E.3 L-shaped cavity Michelson interferometeris valid because it hardly aﬀects the GW signal. Then, total output of LMI is
˜ all ≡ δΦ
˜ 1 − δΦ
˜2
δΦ
˜ all
= α(Ω, RF , RE )δφ(E.17)∑˜ all = ω e−iΩ(2τ +ez ·XF /c)eA h̃AδφΩA: [ (n2 ⊗ n2 ) HLMI (Ω, ez · n2 ) − (n1 ⊗ n1 ) HLMI (Ω, ez · n1 ) ] ,(E.18)
with
HLMI (Ω, ez · n1 ) ≡1[ sin Ωτ (e−ip1 + e−ip2 − 2 cos Ωτ )1 − (ez · n1 )2−i(ez · n1 )(e−i(p1 +p2 ) + cos 2Ωτ − (e−ip1 + e−ip2 ) cos Ωτ ) ] ,where we deﬁned p1 ≡ Ωτ (ez · n1 ) and p2 ≡ Ωτ (ez · n2 ).

Appendix F
Quantum theory in a laserinterferometer
Full-quantum treatment of quantum noise has been formulated by Kimble et al. (hereafter ”KLMTV”) [185]. Recent calculations of quantum noise in a laser interferometer
for GWs are based on the formalism. In this Appendix, we will review it brieﬂy.

F.1

Quantum formalism

In the Heisenberg picture, a quantized electromagnetic ﬁeld with positive frequencies
can be written as 1
∫ ∞√
dω
2π~ω
(+)
aω e−iωt
.
(F.1)
E =
Ac
2π
0
Here A is the eﬀective cross-section area of a beam, and aω is the annihilation operator
with a frequency ω, which holds the commutation relations:[aω , a†ω′ ] = 2πδ(ω − ω ′ ) .

[aω , aω′ ] = 0 ,(F.2)

We split the ﬁeld into side bands about the carrier frequency ω0 : ω = ω0 ± Ω, anddeﬁnea+ ≡ aω0 +Ω ,a− ≡ aω0 −Ω .
The commutation relation (F.2) implies[a+ , a†+′ ] = 2πδ(Ω − Ω′ ) ,[a− , a†−′ ] = 2πδ(Ω − Ω′ ) ,and Eq. (F.1) becomes
√
E

(+)

=

2π~ω0 −iω0 t
e
Ac

∫ ∞

(

0

1

a+ e−iΩt + a− e+iΩt

) dΩ
.
2π

We use ω for the angular frequency of an electromagnetic ﬁeld and ω0 for the angular frequencyof laser light, only in this chapter of Appendices.

139

F. Quantum theory in a laser interferometer

140

Here we used the approximation ω0 ± Ω ≈ ω0 in the square root, since, in general,ω0 ≈ 1.8 × 1015 rad sec−1 and Ω ≤ 109 rad sec−1 . We also formally extend the integrals
over Ω to inﬁnity, for ease of notation.
Next, we introduce two-photon mode formalism [189, 190], where the upper andlower sidebands a± are treated together. The two-photon mode formalism is adequate
for an interferometric GW detector, because mirror motions due to GWs and external
disturbances simultaneously create both sidebands. Deﬁning ﬁeld-amplitude operators
for quadrature mode 2
a1 ≡a+ + a†−
√
,
2

a2 ≡

a+ − a†−
√
,
2i

(F.3)

and using E = E (+) + E (−) together wtih E (−) = [E (+) ]† , one can write the electromagnetic ﬁeld into the following form,
√
[
∫ ∞(
) dΩ
4π~ω0
E =
cos(ω0 t)
a1 e−iΩt + a†1 eiΩtAc
2π
∫ 0∞ () dΩ ]+ sin(ω0 t)
a2 e−iΩt + a†2 eiΩt
.
(F.4)
2π
0
The new operators a1 and a2 satisfy the commutation relations:[a1 , a†2′ ] = −[a2 , a†1′ ] = 2πiδ(Ω − Ω′ ) ,[a1 , a1′ ] = [a1 , a†1′ ] = [a†1 , a†1′ ] = [a1 , a2′ ] = [a†1 , a†2′ ] = 0 .

F.2

Conventional Fabry-Perot Michelson interferometer

Hereafter we consider the case of Fabry-Perot Michelson interferometer (FPMI) [185].
However, the qualitative descriptions also hold for any other laser interferometers. In
the FPMI, quantum noise is caused by a vacuum ﬁeld a entering an interferometer fromthe dark port 3 [182, 183], as shown in Fig. F.1. The a ﬁeld is shot noise itself and also
produces radiation-pressure noise, coupled with the carrier light in the Fabry-Perot(FP) cavity. b is the output ﬁeld, which includes shot noise, radiation-pressure noise
and GW signal.
2

Strictly speaking, the ﬁeld amplitudes for quadrature modes are deﬁned by√√√√ω0 + Ω
ω0 − Ω †ω0 + Ω
ω0 − Ω †a+ +a− ,a2 ≡ −ia+ + i
a
a1 ≡
2ω0
2ω0
2ω0
2ω0 −

However, in our case, since ω0 ≫ Ω, they are approximated to Eq. (F.3)
3
In practice, losses of optics allow vacuum ﬂuctuation to come into the interferometer through not
only the dark port but also other paths, for example, end mirrors with ﬁnite transmissivity and loss,
or light scattering at mirrors and a beam splitter.

141

F.2 Conventional Fabry-Perot Michelson interferometer

Figure F.1: Input ﬁeld a and output ﬁeld b in a Fabry-Perot Michelson interferometer.

What we need to calculate the power spectrum of quantum noise is the relation
between the input and output ﬁelds. The input ﬁeld is given by Eq. (F.4) and theoutput ﬁeld is written as
√
[
∫ ∞(
) dΩ
4π~ω0
Eout =
cos(ω0 t)
b1 e−iΩt + b†1 eiΩtAc
2π
∫ 0∞ () dΩ ]+ sin(ω0 t)
b2 e−iΩt + b†2 eiΩt.
2π0Then, after long but straightforward calculation, one can obtain the input-output relation [185]:
b1 = a1 e2iβb2 = a2 e

2iβ− Ka1 e

2iβ

+

√

(
2K

h
hSQL

)
eiβ .

(F.5)

where various quantities are deﬁned as below.
γ =

T 2c
,
4L

β = arctan(Ω/γ) ,ISQL =

2(I0 /ISQL ),(Ω/γ)2 [1 + (Ω/γ)2 ]√8~=.
mΩ2 L2K =hSQL

mL2 γ 4
,
4ω0

(F.6)

F. Quantum theory in a laser interferometer142

In the calculation, it is assumed that all optics have no losses and that the end mirrors
are completely reﬂective. T is the amplitude transmissivity of FP cavity’s front mirror
4. γ is FP-cavities’ half bandwidth, which determines the characteristic frequency of
the FP cavity, and β is the eﬀective phase shift of a sideband ﬁeld in the FP cavity. K
is a coupling constant between a carrier ﬁeld and a sideband ﬁeld, which determines
the intensity of radiation-pressure. hSQL is the square root of the SQL spectral density
and ISQL is the laser power required to reach SQL at Ω = γ. In Eq. (F.5), the ﬁrst
term is shot noise, the second term is radiation-pressure noise, and the third term is
the GW signal.
Converting the noise signal to GW amplitude, hn is deﬁned as
hSQL
hn (Ω) = √ (a2 − Ka1 )eiβ .
2K

(F.7)

Then, spectral density is deﬁned as the variance of the reduced noise amplitude by12πδ(Ω − Ω′ )Sh (Ω) ≡ 〈in|hn (Ω)h†n (Ω′ )|in〉sym ,2
where subscript ”sym” means calculating by replacing hn (Ω)h†n (Ω′ ) with)1(hn (Ω)h†n (Ω′ ) + h†n (Ω′ )hn (Ω) ,2
and |in〉 is an input state. In our conﬁguration, input state at the dark port is in its
vacuum state, deﬁned using annihilation operators for each sideband by
a+ |0a 〉 = a− |0a 〉 = 0 .
Using the relation〈0a |aj (Ω)a†k (Ω′ )|0a 〉sym =12πδ(Ω − Ω′ )δjk ,2

we can obtain the spectral density of quantum noise in the FPMI,
h2SQL
Sh =
2

(

1
+K
K

)
.

(F.8)

This spectral density reaches the minimum noise level, the so-called standard quantum
limit (SQL), at Ω = γ when the laser power I0 is ISQL . This indicates that there exists
optimal laser power in order to minimize the noise at a certain frequency, since theshot noise and radiation pressure noise are proportional and inversely proportional to
the laser power, respectively. Therefore, in this conﬁguration of the FPMI, one cannot
overcome the SQL, because shot noise and radiation pressure noise have no dynamicalcorrelation.
4

This is diﬀerent from the deﬁnition in KLMTV’s paper.

143

F.3

F.3 Optional quantum conﬁgurations

Optional quantum conﬁgurations

Although one cannot overcome the SQL in the conventional conﬁguration of the FPMI,
but, quantum mechanically, the SQL can be circumvented. Several methods to overcome the SQL with quantum correlations have been proposed.

Squeezed input
If a squeezed vacuum ﬁeld is injected from the dark port, instead of the coherent
vacuum ﬁeld, shot noise level can be reduced. The original idea is proposed by Unruh[191] in the early 1980s. In the case of a coherent vacuum input, the quadrature modes
a1 and a2 in Eq. (F.5) have the same amplitude. If the vacuum state is squeezed, oneof the quadrature amplitude is reduced, while the other amplitude orthogonal to it
become worse. In other words, one can perform more accurate phase measurement ofa beam with shot noise less than that of the coherent vacuum, if the detection phaseis properly selected. This technique is completely based on quantum correlations.
The squeezing of light was ﬁrst conﬁrmed experimentally by Slusher et al. in 1985.
Soon after the ﬁrst detection, it was demonstrated that the shot-noise level is improved
below that of zero-point ﬂuctuations by using the squeezed vacuum incident into theinterferometer [192]. Then, the experimental demonstrations toward the installationof the squeezing into a large-scale interferometer has been reported by many authors[193, 194, 195, 196, 197, 198].

Variational output
Vyatchanin and Matsko [199, 200, 201] invented this design conceptually in the early
1990s. The detector input is the same as that in a conventional one, but the output
is modiﬁed. This method takes advantage of the fact that the output ﬁeld is ponderomotively squeezed due to the mirror motion induced by radiation pressure, which was
ﬁrst recognized by Braginsky and Manukin [202]. Thus, if the phase of the homodyne
detection is properly selected, the quantum noise can be reduced and circumvents the
SQL without modifying the input of the interferometer.

Signal recycling
This method uses one extra mirror, called a signal recycling (SR) mirror 5 . This additional mirror can reshape the noise curve and make two dips on it [203, 204]. Recently,
quantum noise has been calculated in the case of Advanced-LIGO, in which SR mirror
is put at the dark port of the interferometer [205, 206]. The SR mirror creates dynamical correlations between shot noise and radiation-pressure noise, and makes it possible
to circumvent the SQL. The signal recycling is planned to apply to the next-generationinterferometers such as Advanced LIGO [33], Advanced VIRGO [34], LCGT [36], and
5

The interferometer conﬁguration is called signal recycling, detuned signal recycling or resonant
sideband extraction, depending on the microscopic position of the extra mirror at the dark port. Here
we call all of them signal recycling.

F. Quantum theory in a laser interferometer

144

AIGO [35]. This method has been extended in another detector conﬁguration [207], aso-called diﬀerential-type FPMI, which detects signals for each arm independently andcombines (diﬀerentiates) them after the detection. In the conﬁguration with two SR
mirrors, three dips appear on the noise curve.

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Acknowledgement
I would like to thank all who have supported me, in Japanese.
本論文を書き上げるにあたっては、多くの人々にお世話になりました 6 。
京都大学の阪上雅昭 教授には修士課程入学当時から未熟な私にも辛抱強くご指導頂き、物理に限らず研究に対する心構えまで多岐に渡って教えて頂いた事は心から感謝致します。また、国立天文台の川村静児 準教授には博士課程より受託院生として快く受け入れて頂き、ご指導・助言を頂くと共に、自由に研究に専念出来る環境を提供して頂いた事に深く感謝致します。
この学位論文のテーマの１つである超高周波数帯における重力波探査に関して、次の方々にお世話になりました。国立天文台の新井宏二 助教、辰巳大輔 助教、山元一広
研究員には実験に関する質問に対応して頂き、また論文の内容に関して助言を頂きました。同研究室・大学院生の阿久津智忠 氏には本実験だけではなく、その他の様々な議論にも付き合って頂きました。名古屋大学の杉山直 教授および高橋龍一 研究員、日本大学の千葉剛 準教授には本研究の理論面において支えて頂きました。学位論文のもう１つのテーマである重力波の偏極モードに関する研究では、東京大学の樽家篤史 助
教にご多忙にも関わらず時間を割いて議論して頂きました。それ以外にも、カリフォ
ルニア工科大学の Yanbei Chen 助教および宗宮健太郎 研究員、東京大学の安東正樹
助教、国立天文台の瀬戸直樹 研究員、大阪市立大学の神田展行 教授は私の研究に興味を示して下さり、貴重なご意見を頂きました。
以下の方々には国立天文台で行われている実験に共同研究として快く参加させて頂きました。国立天文台の佐藤修一 研究員、同・大学院生の苔山圭以子 氏には変位雑音
フリー重力波検出器実験で、国立天文台の阪田紫帆里 研究員および Volker Leonhardt
研究員には量子非破壊測定実験で、同・宮川治 研究員および同・大学院生の川添史子
氏には RSE 実験でお世話になりました。また、同・事務支援員である近藤さん、西村さん、吉住さんには出張手続きを始めとして種々の事務作業に関してお世話になりました。京都大学での同研究室の丹羽佳人 氏および奥住聡 氏、国立天文台での同研究室の和泉究 氏とは研究の合間に研究に関する事やどうでも良い事をよく議論しました。
そのような日常会話も研究生活の一助となりました。
最後に、陰ながら暖かく見守り続けてくれた両親に感謝します。

6

役職名は主にお世話になった当時のものです。また、海外研究機関の場合の役職名は対応する日本
での役職名にしてあります。

155